Let \(k\) be a positive integer and let \(\Gamma \) be a finite index subgroup of the modular group \(\textrm{SL}(2,\mathbb {Z})\).

A (classical) modular form \(f\) of weight \(k\) on \(\Gamma \), is a holomorphic function defined on the upper half plane \(\mathcal{H}\), which satisfies the transformation property

for all \(z\in \mathcal{H}\) and \(\gamma =\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \Gamma \) and is holomorphic at all the cusps of \(\Gamma \).

If \(\Gamma \) contains the principal congruence subgroup \(\Gamma (N)\) then \(f\) is said to be a modular form of level \(N\).

For each fixed choice of \(k\) and \(\Gamma \) the set of modular forms of weight \(k\) on \(G\) form a finite-dimensional \(\mathbb {C}\)-vector space denoted \(M_k(\Gamma )\).

For the congruence subgroup \(\Gamma _1(N)\) the space \(M_k(\Gamma _1(N))\) decomposes as a direct sum of subspaces \(M_k(N,\chi )\) over the group of Dirichlet characters \(\chi \) of modulus \(N\), where \(M_k(N,\chi )\) is the subspace of forms \(f\in M_k(N)\) that satisfy

for all \(\gamma =\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) in \(\Gamma _0(N)\).

Elements of \(M_k(N,\chi )\) are said to be modular forms of weight \(k\), level \(N\), and character \(\chi \).

For trivial character \(\chi \) of modulus \(N\) we have \(M_k(N,\chi )=M_k(\Gamma _0(N))\).

The analytic conductor of a newform \(f \in S_k^{\mathrm{new}}(N,\chi )\) is the positive real number

where \(\psi (x):=\Gamma '(x)/\Gamma (x)\) is the logarithmic derivative of the Gamma function.

The analytic rank of a cuspidal modular form \(f\) is the analytic rank of the L-function

where the \(a_n\) are the complex coefficients that appear in the \(q\)-expansion of the modular form: \(f(z)=\sum _{n\ge 1}a_nq^n\), where \(q=e^{2\pi i z}\).

The complex coefficients \(a_n\) depend on a choice of embedding of the coefficient field of \(f\) into the complex numbers. It is conjectured that the analytic rank does not depend on this choice, and this conjecture has been verified for all classical modular forms stored in the LMFDB.

In general, analytic ranks of L-functions listed in the LMFDB are upper bounds that are believed (but not proven) to be tight.

For modular forms, the analytic ranks listed in the LMFDB are provably correct whenever the listed analytic rank is 0, or the listed analytic rank is 1 and the modular form is self dual (in the self dual case the sign of the functional equation determines the parity of the analytic rank).

The Artin field of a weight one newform is the number field fixed by the kernel of its associated Galois representation \(\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{GL}_2(\mathbb {C})\).

This number field is typically identified as the Galois closure of a sibling subfield with minimal degree and absolute discriminant.

The Artin image of a weight one newform is the image of its associated Galois representation \(\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{GL}_2(\mathbb {C})\).

The Artin image is a finite subgroup of \(\textrm{GL}_2(\mathbb {C})\) whose cardinality is equal to the degree of the Artin field.

Let \(N\) be a positive integer, and let \(Q\) be a positive divisor of \(N\) satisfying \(\gcd (Q,N/Q)=1\). Then there exist \(x,y,z,t \in \mathbb {Z}\) for which the matrix

has determinant \(Q\). The matrix \(W_Q\) normalizes the group \(\Gamma _0(N)\), and for any weight \(k\) it induces a linear operator \(w_Q\) on the space of cusp forms \(S_k(\Gamma _0(N))\) that commutes with the Hecke operators \(T_p\) for all \(p \nmid Q\) and acts as its own inverse.

The linear operator \(w_Q\) does not depend on the choice of \(x,y,z,t\) and is called the Atkin-Lehner involution of \(S_k(\Gamma _0(N))\). Any cusp form \(f\) in \(S_k(\Gamma _0(N))\) which is an eigenform for all \(T_p\) with \(p \nmid N\) is also an eigenform for \(w_Q\), with eigenvalue \(\pm 1\).

The matrix \(W_Q\) induces an automorphism of the modular curve \(X_0(N)\) that is also denoted \(w_Q\).

In the case \(Q=N\), the Atkin-Lehner involution \(w_N\) is also called the Fricke involution.

A bad prime for a modular form \( f\) is a prime dividing the level of \(f\).

A good prime is a prime that is not a bad prime. In other words, a prime that does not divide the level.

The character of an elliptic modular form \(f\) of weight \(k\) for the group \(\Gamma \) is the Dirichlet character \(\chi \) that appears in its transformation under the action of the defining group \(\Gamma \). Namely,

for any \(z\in \mathcal{H}\) and \(\gamma = \begin{pmatrix} * & * \\ c & d \end{pmatrix}\in \Gamma \). Here \(\Gamma \) is a subgroup of \(\rm {SL}_2(\mathbb {Z})\) containing the principal congruence subgroup \(\Gamma (N)\), and \(\chi \) is a character mod \(N\).

A classical modular form is said to have complex multiplication if it admits a self twist by the Kronecker character of an imaginary quadratic field.

The coefficient field of a modular form is the subfield of \(\mathbb {C}\) generated by the coefficients \(a_n\) of its \(q\)-expansion \(\sum a_nq^n\). The space of cusp forms \(S_k^\mathrm {new}(N,\chi )\) has a basis of modular forms that are simultaneous eigenforms for all Hecke operators and with algebraic Fourier coefficients. For such eigenforms the coefficient field will be a number field, and Galois conjugate eigenforms will share the same coefficient field. Moreover, if \(m\) is the smallest positive integer such that the values of the character \(\chi \) are contained in the cyclotomic field \(\mathbb {Q}(\zeta _m)\), the coefficient field will contain \(\mathbb {Q}(\zeta _m)\) For eigenforms, the coefficient field is also known as the Hecke field.

The coefficient ring of a modular form is the subring \(\mathbb {Z}[a_1,a_2,a_3,\ldots ]\) of \(\mathbb {C}\) generated by the coefficients \(a_n\) of its \(q\)-expansion \(\sum a_nq^n\). In the case of a newform the coefficients \(a_n\) are algebraic integers and the coefficient ring is a finite index subring of the ring of integers of the coefficient field of the newform. It is also known as the Hecke ring, since the \(a_n\) are eigenvalues of Hecke operators.

A congruence subgroup \(\Gamma \) of \(\textrm{SL}_2(\mathbb {Z})\) is a subgroup that contains a principal congruence subgroup \(\Gamma (N) := \ker \left( \operatorname {SL}_2(\mathbb {Z}) \to \operatorname {SL}_2(\mathbb {Z}/N\mathbb {Z}) \right)\) for some \(N\ge 1\). The least such \(N\) is the level of \(\Gamma \).

Let \(k\) be a positive integer and let \(\Gamma \) be a finite index subgroup of the modular group \(\textrm{SL}(2,\mathbb {Z})\).

A cusp form of weight \(k\) on \(\Gamma \) is a modular form \(f\in M_k(\Gamma )\) that vanishes at all cusps of \(\Gamma \). In particular, the constant term in the Fourier expansion of \(f\) about any cusp is zero.

The cusp forms in \(M_k(\Gamma )\) form a subspace \(S_k(\Gamma )\). For each Dirichlet character \(\chi \) of modulus \(N\) the cusp forms in \(M_k(N,\chi )\) form a subspace \(S_k(N,\chi )\); these are the cusp forms of weight \(k\), level \(N\), and character \(\chi \).

The Hecke algebra acts on \(S_k^{\mathrm{new}}(N, \chi )\), breaking it up into irreducible pieces. Each piece is spanned by a set of conjugate eigenforms with Fourier coefficients in a number field of degree equal to the dimension of the subspace. We refer to an irreducible orbit as a newform.

The coefficient field of a modular form is a number field. A defining polynomial for this number field is explicitly recorded, because some of the data associated to the modular form will be expressed in terms of roots of this polynomial.

The dimension of a space of modular forms is its dimension as a complex vector space; for spaces of newforms \(S_k^{\rm new}(N,\chi )\) this is the same as the dimension of the \(\mathbb {Q}\)-vector space spanned by its eigenforms.

The dimension of a newform refers to the dimension of its newform subspace, equivalently, the cardinality of its newform orbit. This is equal to the degree of its coefficient field (as an extension of \(\mathbb {Q}\)).

The relative dimension of \(S_k^{\rm new}(N,\chi )\) is its dimension as a \(\mathbb {Q}(\chi )\)-vector space, where \(\mathbb {Q}(\chi )\) is the field generated by the values of \(\chi \), and similarly for newform subspaces.

For a newspace \(S^\mathrm {new}_k(N,\chi )\) we say that a set of Hecke operators \(\mathcal T:=\{ T_{p_1},\ldots ,T_{p_r}\} \) distinguishes the newforms in the space if the sets \(X_f(\mathcal T)\) of characteristic polynomials of the \(T_p\in \mathcal T\) acting on the subspace \(V_f\) spanned by the Galois orbit of \(f\) in \(S_k^\mathrm {new}(N,\chi )\) are distinct as \(f\) ranges over (non-conjugate) newforms in \(S_k^\mathrm {new}(N,\chi )\).

The set \(\mathcal T\) can be identified by a list of primes \(p\). For convenience we restrict to primes \(p\) that do not divide the level \(N\) and list the unique ordered sequence of primes \(p_1,\ldots ,p_n\) for which the sequence of integers \(c_1,\ldots ,c_n\) defined by

is strictly increasing. The length of the sequence \(p_1,\ldots p_n\) is always less then the number of newforms in \(S_k^\mathrm {new}(N,\chi )\) and we obtain the empty sequence when \(S_k^\mathrm {new}(N,\chi )\) contains just one newform.

The dual of a cuspidal modular form \(f\) is the form whose coefficients \(a_n\) in its \(q\)-expansion are the complex conjugates of those of \(f\). The L-function of the dual form is the dual of the L-function of \(f\).

The coefficient field of a non-self-dual newform is a CM field.

For an even integer \(k\geq 4\), we define the (normalized) holomorphic Eisenstein series of level 1

where \(\Gamma _z=\{ \gamma \in \Gamma : \gamma z=z\} \) is the isotropy group of the cusp \(z\).

The Eisenstein series \(E_k\) are modular forms of weight \(k\) and level \(1\) on the modular group.

They have the following \(q\)-expansion:

where the \(B_k\) are the Bernoulli numbers, \(\sigma _{k-1}(n)\) is a divisor function, and \(q=e^{2 \pi i z}\).

Let \(k\) be a positive integer and let \(\Gamma \) be a finite index subgroup of the modular group \(\textrm{SL}(2,\mathbb {Z})\).

the Eisenstein subspace \(E_k(\Gamma )\) is the orthogonal complement in \(M_k(\Gamma )\) to the subspace \(S_k(\Gamma )\) under the Petersson inner product.

An Eisenstein form of weight \(k\) on \(\Gamma \) is a modular form \(f\in E_k(\Gamma )\). For each Dirichlet character \(\chi \) of modulus \(N\) the Eisenstein forms in \(M_k(N,\chi )\) form a subspace \(E_k(N,\chi )\); these are the Eisenstein forms of weight \(k\), level \(N\), and character \(\chi \).

The space \(E_k(N, \chi )\) is spanned by the \(E_k^{\chi _1, \chi _2}(d \\ tau)\) where \(\chi _1 \chi _2 = \chi \) and \(d N_1 N_2 \mid N\), unless \(k = 2\) and \(\chi = 1\), in which case \(E_2^{1,1}(d \tau )\) is not holomorphic, and is replaced by \(E_2^{1,1}(\tau ) - d E_2^{1,1}(d \tau )\).

The label of an Eisenstein newform \(f\in E_k^{\rm new}(N,\chi )\) has the format \( N.k.E.a.x \), where

• \( N\) is the level;

• \(k\) is the weight;

• \(N.a\) is the label of the Galois orbit of the Dirichlet character \(\chi \);

• \(x\) is the label of the Galois orbit of the newform \(f\).

For each embedding of the coefficient field of \(f\) into the complex numbers, the corresponding modular form over \(\mathbb {C}\) has a label of the form \(N.k.E.a.x.n.i\), where

• \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi \);

• \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character \(\chi \).

An Eisenstein newform is an Eisenstein form \(f\in E_k^{\rm new}(N,\chi )\) in the Eisenstein new subspace that is also an eigenform of all Hecke operators, normalized so that the \(q\)-expansion \(f(z)=\sum a_n q^n\), where \(q=e^{2\pi i z}\), has coefficient \(a_1=1\). The Eisenstein newforms are a basis for the Eisenstein new subspace.

The space \(E_k(N,\chi )\) of Eisenstein modular forms of level \(N\), weight \(k\), and character \(\chi \) can be decomposed

into old and new subspaces, defined as follows.

If \(M\) is a proper divisor of \(N\) and \(\chi _M\) is a Dirichlet character of modulus \(M\) that induces \(\chi \), then for all \(d \mid (N/M)\), there is a map from \(E_k(M,\chi _M) \to E_k(N,\chi )\) via \(f(z) \mapsto f(dz)\). The span of the images of all of these maps is the old subspace \(E_k^{\rm old}(N,\chi ) \subseteq E_k(N,\chi )\).

The new subspace \(E_k^{\rm new}(N,\chi )\) is the subspace spanned by the newforms \(E_k^{\chi _1, \chi _2}(\\ tau)\) such that \(\chi _1 \chi _2 = \chi \) and \(N_1 N_2 = N\), unless \(k = 2\) and \(\chi = 1\), in which case \(E_2^{\rm new}(N) = 0\) when \(N\) is not a prime, and when \(N = p\) is prime it is spanned by \(E_2^{1,1}(\tau ) - p E_2^{1,1}(p \tau )\).

Let \(k, N_1, N_2\) be positive integers, and let \(\chi _1, \chi _2\) be primitive Dirichlet characters modulo \(N_1\) and \(N_2\) respectively.

The Eisenstein series of weight \(k\) associated to \(\chi _1\) and \(\chi _2\) is

where \(q = e^{2 \pi i \tau }\), \(L(s,\chi _i)\) is the Dirichlet \(L\)-function associated to \(\chi _i\), and

The coefficients in the \(q\)-expansion \(\sum a_nq^n\) of a newform \(f\) are algebraic integers that generate the coefficient field \(\mathbb {Q}(f)\) of \(f\).

Each embedding \(\iota \colon \mathbb {Q}(f)\to \mathbb {C}\) gives rise to a modular form \(\iota (f)\) with \(q\)-expansion \(\sum \iota (a_n)q^n\); the modular form \(\iota (f)\) is an embedding of the newform \(f\).

Distinct embeddings give rise to modular forms that lie in the same galois orbit but have distinct \(L\)-functions \(L(s):=\sum \iota (a_n)n^{-s}\).

If \(f\) is a newform of character \(\chi \), each embedding \(\mathbb {Q}(f)\to \mathbb {C}\) induces an embedding \(\mathbb {Q}(\chi )\to \mathbb {C}\) of the value field of \(\chi \). The embeddings of \(f\) may be grouped into blocks with the same Dirichlet character; distinct blocks correspond to modular forms with distinct (but Galois conjugate) Dirichlet characters.

The label complex embedded holomorphic cusp form \( f\) is \( N.k.a.x.c.j \) (sometimes shortened as \( a.j\) ), where

• \( N\) is the level,

• \(k\) is the weight,

• \(N.a\) is the label of the Galois orbit of the Dirichlet character,

• \(x\) is the Hecke Galois orbit label,

• \(N.c\) is the Conrey label for the character corresponding to the embedding, and

• \(j\) is the index for the embedding within those with the same Dirichlet character, these are ordered by the vector \(\iota (a_n)\), where we order the complex numbers first by their real part and then by their imaginary part.

An eta quotient is any function \(f\) of the form

where \(m_i\in \mathbb {N}\) and \(r_i\in \mathbb {Z}\) and \(\eta (z)\) is the Dedekind eta function.

An eta product is an eta quotient in which all the \(r_i\) are non-negative.

Let \(f\) be a modular form on a finite index subgroup \(\Gamma \) of \(\textrm{SL}_2(\mathbb {Z})\), and suppose \(\Gamma \) contains the matrix \(T:=\left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)\). Then \(f\) is periodic with period 1, so it has a Fourier expansion of the form

where \(q=e^{2 \pi i z}\). That is the Fourier expansion of \(f\) around the cusp \(\infty \), with Fourier coefficients \(a_n\). If one says "the Fourier expansion of \(f\)", is it understood to refer to the expansion at \(\infty \).

For other cusps of \(\Gamma \), suppose \(w\) is the width of the cusp \(\gamma \infty \), for some cusp representative \(\gamma \). Then we can write \(f\) as \(f(z)=g_{\gamma }(e^{2\pi iz/w})\) for some holomorphic function \(g_{\gamma }\) on the punctured unit disk. We can expand \(g\) as a Laurent series:

We then define the Fourier expansion of \(f\) around the cusp \(\gamma \infty \) to be

where \(q=e^{2\pi iz}\).

The \(a_{\gamma }(n)\) are called the Fourier coefficients of \(f\) with respect to the cusp \(\gamma \infty \).

The Fricke involution is the Atkin-Lehner involution \(w_N\) on the space \(S_k(\Gamma _0(N))\) (induced by the corresponding involution on the modular curve \(X_0(N)\)).

For a newform \(f \in S_k^{\textup{new}}(\Gamma _0(N))\), the sign of the functional equation satisfied by the L-function attached to \(f\) is \(i^{-k}\) times the eigenvalue of \(\omega _N\) on \(f\). So, for example when \(k=2\), the signs swap, and the analytic rank of \(f\) is even when \(w_N f = -f\) and odd when \(w_N f = +f\).

Two newforms \(f=\sum a_nq^n\) and \(g=\sum b_nq^n\) are Galois conjugate if there is an automorphism \(\sigma \in \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\) such that \(b_n=\sigma (a_n)\) for all \(n\ge 1\), in which case we write \(g=\sigma (f)\).

The set \(\{ \sigma (f):\sigma \in \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\} \) of all Galois conjugates of \(f\) is the Galois orbit of \(f\); it has cardinality equal to the dimension of \(f\), equivalently, the degree of its coefficient field

The Galois orbit of a newform \(f\in S_k^{\rm new}(N,\chi )\) is the finite set

of its Galois conjugates, which forms a canonical \(\mathbb {Q}\)-basis for the corresponding newform subspace.

Galois orbits of newforms are also called newform orbits.

As shown by Deligne and Serre [ , every newform of weight one has an associated Galois representation \(\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{GL}_2(\mathbb {C})\).

This representation corresponds to an Artin representation of dimension two whose conductor is the level \(N\) of the modular form.

Conversely, every odd irreducible two-dimensional Artin representation of conductor \(N\) gives rise to a modular form of weight one and level \(N\).

Composing the representation \(\rho \) with the natural map \(\textrm{GL}_2(\mathbb {C})\to \textrm{PGL}_2(\mathbb {C})\) yields the projective Galois representation \(\bar\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{PGL}_2(\mathbb {C})\).

Let \(f\) be a modular form of weight \(k\), level \(N\), and character \(\chi \).

For each positive integer \(n\) the Hecke operator \(T_n\) is a linear operator on the vector space \(M_k(N,\chi )\) whose action on \(f\in M_k(N,\chi )\) can be defined as follows. If \(f(z)=\sum a_n (f)q^n\) is the \(q\)-expansion of \(f\in M_k(N,\chi )\), where \(q=e^{2\pi i z}\), then the \(q\)-expansion of \(T_nf\in M_k(N,\chi )\) has coefficients

The Hecke operators pairwise commute, and when restricted to the subspace \(S_k(N,\chi )\) of cusp forms, they commute with their adjoints with respect to the Petersson scalar product. This implies that \(S_k(N,\chi )\) has a canonical basis whose elements are eigenforms for all the Hecke operators. If we normalize such an eigenform \(f(z)=\sum a_n q^n\) so that \(a_1=1\), then for all \(n\ge 1\) we have

The newspace \(S_k^{\rm new}(N,\chi )\subseteq S_k(N,\chi )\) is invariant under the action of the Hecke operators, so the canonical basis of normalized eigenforms for \(S_k(N,\chi )\) includes a basis of newforms for \(S_k^{\rm new}(N,\chi )\).

The Hecke orbit of a cusp form \(f\) in \(S_k(N,\chi )\) is defined as the space generated by \(T_p(f)\) for all Hecke operators \(T_p\) for \(p\) coprime to the level.

The coefficient ring generator bound of a newform with \(q\)-expansion \(\sum a_nq^n\) is the least positive integer \(n\) such that \(\mathbb {Z}[a_1,\ldots ,a_n]\) is the entire coefficient ring \(\mathbb {Z}[a_1,a_2,a_3,\ldots ]\).

The Hecke characteristic polynomial of a newform \(f\) at a prime \(p\) is the characteristic polynomial of the Hecke operator \(T_p\) acting on the newform subspace \(V_f\).

Galois conjugate newforms \(f\) and \(g\) are inner twists if there is a Dirichlet character \(\chi \) such that

for all but finitely many primes \(p\). Without loss of generality, we may assume that \(\chi \) is a primitive Dirichlet character, and by a theorem of Ribet [ , the newform \(g\) is conjugate to \(f\) via a \(\mathbb {Q}\)-automorphism \(\sigma \) of the coefficient field of \(f\). The set of pairs \((\chi ,\sigma )\) form the group of inner twists of \(f\).

Each pair \((\chi ,\sigma )\) corresponding to an inner twist of \(f\) is uniquely determined by the the primitive character \(\chi \), and we say that \(f\) admits an inner twist by \(\chi \). When \(\sigma =1\) is is the trivial automorphism, we have \(g=f\) and say that \(f\) admits a self twist by \(\chi \); in this case \(\chi \) is either the trivial character or the Kronecker character of a quadratic field.

The number of inner twists of \(f\) is an invariant of its Galois orbit, as is the number of inner twists by characters in any particular Galois orbit of Dirichlet characters.

The home page of each newform in the LMFDB includes a list of inner twists, in which non-trivial self twists are distinguished by listing the associated quadratic field (the CM or RM field), while inner twists that are not self twists are simply marked as "inner".

The inner twist count of a newform \(f\) is the number of distinct inner twists of \(f\).

Associated to each inner twist is a pair \((\chi ,\sigma )\), where \(\chi \) is a primitive Dirichlet character and \(\sigma \) is a \(\mathbb {Q}\)-automorphism of the coefficient field of \(f\).

Pairs with \(\sigma =1\) are self twists \((\chi ,1)\), including the pair \((1,1)\) corresponding to the twist of \(f\) by the trivial character; self twists are included in the count of inner twists.

The set of pairs \((\chi ,\sigma )\) forms the group of inner twists; the inner twist count is the cardinality of this group.

Not all of the inner twists included in the inner twist count have necessarily been proved; those that have are explicitly identified in the table of inner twists on the newforms home page. In cases where not every inner twist has been proved the inner twist should be viewed as a rigorous upper bound that is believed to be tight.

Inner twist data is available only for newforms for which exact eigenvalue data has been computed; this includes all newforms of dimension up to \(20\) and all newforms of weight 1; when the inner twist count is specified in a search the results include only newforms for which inner twists have been computed.

It is possible for a newform \(f\) to admit an inner twist by more than one Dirichlet character \(\varphi \) in the same Galois orbit. Different embeddings of \(f\) into \(\mathbb {C}\) will yield different \(\varphi \), but the number of such \(\varphi \) is the same for every embedding; this number is the multiplicity.

The label of a newform \(f\in S_k^{\rm new}(N,\chi )\) has the format \( N.k.a.x \), where

• \( N\) is the level;

• \(k\) is the weight;

• \(N.a\) is the label of the Galois orbit of the Dirichlet character \(\chi \);

• \(x\) is the label of the Galois orbit of the newform \(f\).

For each embedding of the coefficient field of \(f\) into the complex numbers, the corresponding modular form over \(\mathbb {C}\) has a label of the form \(N.k.a.x.n.i\), where

• \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi \);

• \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character \(\chi \).

The label of a \(f\in M_k^{\\ rm new}(N,\chi )\) has the format \( N.k.A.a.x \), where

• \( N\) is the level;

• \(k\) is the weight;

• \(N.a\) is the label of the Galois orbit of the Dirichlet character \(\chi \);

• \( A\) is a character signifying the automorphic type of the modular form - \(E\) for Eisenstein or \(C\) for cuspidal.

• \(x\) is the label of the Galois orbit of the newform \(f\).

For each embedding of the coefficient field of \(f\) into the complex numbers, the corresponding modular form over \(\mathbb {C}\) has a label of the form \(N.k.A.a.x.n.i\), where

• \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi \);

• \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character \(\chi \).

If \(f \in S_k^{\rm new}(N, \chi )\) is cuspidal, the automorphic type may be omitted from both labels.

A level of a modular form \(f\) is a positive integer \(N\) such that \(f\) is a modular form on a subgroup \(\Gamma \) of \(\operatorname {SL}_2(\mathbb {Z})\) that contains the principal congruence subgroup \(\Gamma (N)\).

The level of a newform is the least such integer \(N\).

A newform is maximal if its Galois orbit spans the ambient subspace that contains it (its Atkin-Lehner subspace when the character is trivial, the entire newspace otherwise).

A newform is the largest newform in its ambient subspace if its dimension is strictly larger than that of any other newform in the same subspace (this includes newforms that are maximal).

A modular form is minimal if it is not a twist of a form of lower level.

The minimal twist of a newform \(f\) is the twist \(g\) of \(f\) whose label is lexicographically minimal among all twists of \(f\) that are both twist minimal and have minimal character \(\chi \).

A key feature of the minimal twist \(g\) (and more generally, of any twist minimal \(g\) of level \(N\) and minimal character \(\chi \)) is that for any character \(\psi \), the level \(M\) of the twist \(g\otimes \psi \) can be computed as \(M={\rm lcm}(N,{\rm cond}(\psi ){\rm cond}(\chi \psi ))\).

The minus subspace of \(S_k(\Gamma _0(N))\) is the eigenspace of the Fricke involution \(w_N\) with eigenvalue \(-1\).

A newform is a cusp form \(f\in S_k^{\rm new}(N,\chi )\) in the new subspace that is also an eigenform of all Hecke operators, normalized so that the \(q\)-expansion \(f(z)=\sum a_n q^n\), where \(q=e^{2\pi i z}\), begins with the coefficient \(a_1=1\). The newforms are a basis for the new subspace.

The newform subspace of a newform \(f\) in \(S_k^{\rm new}(N,\chi )\) is the subspace generated by \(T_p(f)\) for all Hecke operators \(T_p\) for \(p\) coprime to the level, equivalently, the subspace generated by the Galois conjugates of \(f\).

Every newspace has a canonical decomposition into newform subspaces.

The space \(S_k(N,\chi )\) of cuspidal modular forms of level \(N\), weight \(k\), and character \(\chi \) can be decomposed

into old and new subspaces, defined as follows.

If \(M\) is a proper divisor of \(N\) and \(\chi _M\) is a Dirichlet character of modulus \(M\) that induces \(\chi \), then for all \(d \mid (N/M)\), there is a map from \(S_k(M,\chi _M) \to S_k(N,\chi )\) via \(f(z) \mapsto f(dz)\). The span of the images of all of these maps is the old subspace \(S_k^{\rm old}(N,\chi ) \subseteq S_k(N,\chi )\).

The new subspace \(S_k^{\rm new}(N,\chi )\) is the orthogonal complement of \(S_k^{\rm old}(N,\chi )\) with respect to the Petersson inner product.

A basis for the new subspace is given by newforms.

An inner twist is nontrivial if it is not the self twist by the trivial character.

Each space of \(S_k(N,\chi )\) of cuspidal modular forms of weight \(k\), level \(N\), and character \(\chi \) contains an old subspace \(S_k^{\rm old}(N,\chi )\) that can be expressed as a direct sum of spaces of newforms \(S_k^{\rm new}(N_i,\chi _i)\), where each \(N_i\) is a proper divisor of \(N\) divisible by the conductor of \(\chi \), and each \(\chi _i\) is the unique character of modulus \(N_i\) induced by the primitive character that induces \(\chi \).

This decomposition arises from the injective maps

that exist for each divisor \(d\) of \(N/N_i\). The image of each \(\iota _d\) is isomorphic to \(S_k(N_i,\chi _i)\), and we have the decomposition

where \(m_i\) is the number of divisors of \(N/N_i\). Restricting the direct sum to proper divisors \(N_i\) of \(N\) yields a decomposition for \(S_k^{\rm old}(N,\chi )\).

Let \(f\) and \(g\) be two modular forms with respect to a finite index subgroup \(G\) of \(\Gamma \). When it exists, we define the Petersson scalar product of \(f\) and \(g\) with respect to the group \(G\) by

where \(\mathfrak {F}\) is a fundamental domain for \(G\) and \(d\mu =dxdy/y^2\) is the measure associated to the hyperbolic metric.

Note that the Petersson scalar product exists if at least one of \(f\), \(g\) is a cusp form.

The plus subspace of \(S_k(\Gamma _0(N)\) is the eigenspace of the Fricke involution \(\omega _N\) with eigenvalue \(1\).

The projective field of a weight one newform is the number field fixed by the kernel of its associated projective Galois representation \(\bar\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{PGL}_2(\mathbb {C})\).

This number field is typically identified as the Galois closure of a sibling subfield with minimal degree and absolute discriminant.

The projective image of a weight one newform is the image of its associated projective Galois representation \(\rho \colon \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \textrm{PGL}_2(\mathbb {C})\). It is a finite subgroup of \(\textrm{PGL}_2(\mathbb {C})\) that can be classified as one of four types: It is either isomorphic to a dihedral group \(D_n\) for some integer \(n\ge 2\) (where \(D_2:=C_2\times C_2\) is the Klein group), or to one of \(A_4, S_4, A_5\), where \(A_n\) and \(S_n\) respectively denote the alternating and symmetric groups on \(n\) letters.

The \(q\)-expansion of a modular form \(f(z)\) is its Fourier expansion at the cusp \(z=i\infty \), expressed as a power series \(\sum _{n=0}^{\infty } a_n q^n\) in the variable \(q=e^{2\pi iz}\).

For cusp forms, the constant coefficient \(a_0\) of the \(q\)-expansion is zero.

For newforms, we have \(a_1=1\) and the coefficients \(a_n\) are algebraic integers in a number field \(K \subseteq \mathbb {C}\).

Accordingly, we define the \(q\)-expansion of a newform orbit \([f]\) to be the \(q\)-expansion of any newform \(f\) in the orbit, but with coefficients \(a_n \in K\) (without an embedding into \(\mathbb {C}\)). Each embedding \(K \hookrightarrow \mathbb {C}\) then gives rise to an embedded newform whose \(q\)-expansion has \(a_n \in \mathbb {C}\), as above.

The relative dimension of a newform in a space of modular forms \(S_k^{\mathrm{new}}(\Gamma _0(N),\chi )\) is the dimension of its coefficient field as an extension of the character field \(\mathbb {Q}(\chi )\) (the number field generated by the values of \(\chi \)).

A modular form is said to have real multiplication if it admits a self twist by the Kronecker character of a real quadratic field.

Only modular forms of weight one can have real multiplication.

The Satake angles \(\theta _p = \arg \alpha _p \in [-\pi , \pi ]\) are the arguments of a complex embedding of the Satake parameters \(\alpha _p\).

Let \(f\) be newform of level \(N\), weight \(k\) and character \(\chi \). Let \(p\) be a good prime, i.e., \(p \nmid N\).

The Satake parameters \(\alpha _p\) are the reciprocal roots of \(L_p\left(p^{-(k-1)/2} t \right)\), where

\(T_p\) is Hecke operator, and \(a_p\) its trace.

The Sato-Tate group of a newform is a compact Lie group that one can attach to the Galois representation associated to the newform.

For newforms of weight \(k=1\), the Sato-Tate group is simply the image of the corresponding 2-dimensional Artin representation, a finite subgroup of \(\textrm{SL}_2(\mathbb {C})\).

For newforms of weight \(k{\gt}1\) the Sato-Tate group is a subgroup of \(\mathrm{U}(2)\) whose identity component is either \(\mathrm{SU(2)}\) (for newforms without CM) or \(\mathrm{U}(1)\) (for CM newforms) diagonally embedded in \(\mathrm{U}(2)\).

The Sato-Tate conjecture implies that as \(p\to \infty \) the limiting distribution of normalized Hecke eigenvalues \(a_p/p^{(k-1)/2}\) converges to the trace distribution induced by the Haar measure of the Sato-Tate group.

The Sato-Tate conjecture for classical modular forms has been proved [ .

A newform \(f\) admits a self-twist by a primitive Dirichlet character \(\chi \) if the equality

holds for all but finitely many primes \(p\).

For non-trivial \(\chi \) this can hold only when \(\chi \) has order \(2\) and \(a_p=0\) for all primes \(p\) not dividing the level of \(f\) for which \(\chi (p)=-1\). The character \(\chi \) is then the Kronecker character of a quadratic field \(K\) and may be identified by the discriminant \(D\) of \(K\).

If \(D\) is negative, the modular form \(f\) is said to have complex multiplication (CM) by \(K\), and if \(D\) is positive, \(f\) is said to have real multiplication (RM) by \(K\). The latter can occur only when \(f\) is a modular form of weight \(1\) whose projective image is dihedral.

It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when \(f\) is a modular form of weight one whose projective image is isomorphic to \(D_2:=C_2\times C_2\); in this case \(f\) admits three non-trivial self twists, two of which are CM and one of which is RM.

A cuspidal modular form \(f\) is said to be self dual if the coefficients \(a_n\) that appear in its \(q\)-expansion are real numbers; equivalently, the L-function of the modular form is self dual.

The coefficient field of a newform is either a totally real number field or a CM field, depending on whether the newform is self dual or not.

Let \(k\) be an odd integer, and let \(N\) a positive integer divisible by \(4\). Let \(\chi \) be a character modulo \(N\). Let \(t\) be a square-free integer. The Shimura correspondence is the linear map \(Sh_t:S_{k/2}(N, \chi )\to S_{k-1}(N/2, \chi ^2)\) defined by the equation

where

• \(\lambda =(k-1)/2\).

• \(\chi _t\) is the character given by \(\chi _t(m) = \chi (m) \left(\frac{-1}{m}\right) \left(\frac{t}{m}\right)\).

• \(g(z) = \sum _{n\geq 1} a_n q^n\) is the \(q\)-expansion of \(g\).

This map is Hecke linear. If \(k\geq 5\), it takes cusp forms to cusp forms.

The space of modular forms of level \(N\), weight \(k\), and character \(\chi \) is denoted \(M_k(N,\chi )\).

The space \(M_k(N,\chi )\) is a finite-dimensional complex vector space which further decmoposes into subspaces. In particular, we have a subspace of cusp forms \(S_k(N,\chi ) \subseteq M_k(N,\chi )\).

The trace form of a newspace \(S_k(N,\chi )\) is the modular form obtained by summing its canonical basis of newforms.

Stark’s conjecture applied to the associated Galois representation of a newform \(f(z)=\sum a_n q^n\) of weight one [ states the following. Let \(E=\mathbb {Q}((a_n)_{n \in \mathbb {N}})\), \(\Delta =\text{Gal}(E/\mathbb {Q})\) and \(f^\alpha (z)=\sum \alpha (a_n) q^n\) for \(\alpha \in \Delta \). Let \(L(s, f)\) be the L-function of \(f\). Then, for all \(b \in E^*\) there exists an integer \(m \geq 1\) and a unit \(\varepsilon \) in the Artin field of \(f\), called the Stark unit, such that

In the case where the coefficients of \(\text{Tr}(bf)\) are in \(\mathbb {Z}\), Chinburg further conjectured that there exists a Stark unit for \(m=1\) [ . Notice that if we choose \(b = 1\), the preceding condition always holds. Here, we compute the Stark unit of the newform for \(b=1\) and \(m=1\).

The Sturm bound is an upper bound on the least index where the coefficients of the Fourier expansions of distinct modular forms in the same space \(M_k(N,\chi )\) must differ.

More precisely, for any space \(M_k(N,\chi )\) of modular forms of weight \(k\), level \(N\), and character \(\chi \), the Sturm bound is the integer

where

If \(f=\sum _{n\ge 0}a_n q^n\) and \(g=\sum _{n\ge 0}b_n q^n\) are elements of \(M_k(N,\chi )\) with \(a_n=b_n\) for all \(n\le B(M_k(N,\chi ))\) then \(f=g\); see Corollary 9.20 in [ for \(k{\gt}1\) and Lemma 5 in [ for \(k=1\).

The Sturm bound applies, in particular, to newforms of the same level, weight, and character. Better bounds for newforms are known in certain cases (see Corollary 9.19 and Theorem 9.21 in [ , for example), but for consistency we always take the Sturm bound to be the integer \(B(M_k(N,\chi ))\) defined above.

Note that the Sturm bound for \(S_k^{\mathrm{new}}(N,\chi )\) does not apply (in general) to the space

associated to the Galois orbit \([\chi ]\); rather, it applies to each direct summand \(S_k^{\rm new}(N,\chi ')\).

The Sturm bound is an upper bound on the least index where the coefficients of the Fourier expansions of distinct modular forms in the same space must differ.

More precisely, for any space \(M_k(\Gamma _1(N))\) of modular forms of weight \(k\) and level \(N\), the Sturm bound is the integer

where

If \(f=\sum _{n\ge 0}a_n q^n\) and \(g=\sum _{n\ge 0}b_n q^n\) are elements of \(M_k(\Gamma _1(N))\) with \(a_n=b_n\) for all \(n\le B(M_k(\Gamma _1(N)))\) then \(f=g\); see Corollary 9.19 in [ for \(k{\gt}1\).

The Sturm bound applies, in particular, to newforms of the same level and weight. Better bounds for newforms are known in certain cases (see Corollary 9.19 and Theorem 9.21 in [ , for example), but for consistency we always take the Sturm bound to be the integer \(B(M_k(\Gamma _1(N)))\) defined above.

The space \(M_k(N,\chi )\) of modular forms of level \(N\), weight \(k\), and character \(\chi \) can be decomposed as

where \(E_k(N,\chi )\) is the Eisenstein subspace (the span of Eisenstein series) and \(S_k(N,\chi )\) the subspace of cusp forms.

These spaces further decompose into old and new subspaces as follows. If \(M\) is a proper divisor of \(N\) and \(\chi _M\) is a Dirichlet character of modulus \(M\) that induces \(\chi \), then for every divisor \(d \mid (N/M)\), there is a map from \(M_k(M,\chi _M) \to M_k(N,\chi )\) via \(f(z) \mapsto f(dz)\). The span of the images of all of these maps is the old subspace \(M_k^{\rm old}(N,\chi ) \subseteq M_k(N,\chi )\).

The cuspidal subspace decomposes as

where the new subspace \(S_k^{\rm new}(N,\chi )\) is the orthogonal complement of \(S_k^{\rm old}(N,\chi )\) with respect to the Petersson inner product.

The Eisenstein subspace similarly decomposes as

where \(E_k^{\rm new}(N,\chi )\) is the span of those Eisenstein series attached to a pair \((\chi _1,\chi _2)\) of (primitive) characters of conductor \(N\).

The trace bound for a space of newforms \(S_k^{new}(N, \chi )\) is the least positive integer \(m\) such that taking traces down to \(\mathbb {Q}\) of the coefficients \(a_n\) for \(n \le m\) suffices to distinguish all the Galois orbits of newforms in the space; here \(a_n\) denotes the \(n\)th coefficient of the \(q\)-expansion \(\sum a_n q^n\) of a newform.

If the newforms in the space all have distinct dimensions then the trace bound is 1, because the trace of \(a_1=1\) from the coefficient field of the newform down to \(\mathbb {Q}\) is equal to the dimension of its Galois orbit.

For a newform \(f \in S_k^{\rm new}(\Gamma _1(N))\), its trace form \(\mathrm{Tr}(f)\) is the sum of its distinct conjugates under \(\mathrm{Aut}(\mathbb {C})\) (equivalently, the sum under all embeddings of the coefficient field into \(\mathbb {C}\)). The trace form is a modular form \(\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma _1(N))\) whose \(q\)-expansion has integral coefficients \(a_n(\mathrm{Tr}(f)) \in \mathbb {Z}\).

The coefficient \(a_1\) is equal to the dimension of the newform.

For \(p\) prime, the coefficient \(a_p\) is the trace of Frobenius in the direct sum of the \(\ell \)-adic Galois representations attached to the conjugates of \(f\) (for any prime \(\ell \)). When \(f\) has weight \(k=2\), the coefficient \(a_p(f)\) is the trace of Frobenius acting on the modular abelian variety associated to \(f\).

For a newspace \(S_k^{\rm new}(N,\chi )\), its trace form is the sum of the trace forms \(\mathrm{Tr}(f)\) over all newforms \(f\in S_k^{\rm new}(N,k)\); it is also a modular form in \(S_k^{\rm new}(\Gamma _1(N))\).

The graphical plot displayed in the properties box on the home page of each newform or newspace is computed using the trace form.

Associated to each newform \(f\) and primitive Dirichlet character \(\psi \), there is a unique newform \(g:=f\otimes \psi \), the twist of \(f\) by \(\psi \), that satisfies

for all integers \(n\ge 1\) coprime to \(N\) and the conductor of \(\psi \). The newforms \(f\) and \(g\) are then twist equivalent. When \(g\) is a Galois conjugate of \(f\), it is said to be an inner twist.

The newform orbit \([g]\) is a twist of the newform orbit \([f]\) by the character orbit \([\psi ]\) if some \(g\in [g]\) is a twist of \(f\) by some \(\psi \) in \([\psi ]\). This may occur with multiplicity.

Twist equivalence is an equivalence relation. The twist class of a newform or newform orbit is its equivalence class under this relation.

In the LMFDB each twist class is identified by the label of its minimal twist.

A newform \(f\) is twist minimal if its level achieves the minimum within its twist class.

A twist minimal newform \(f\) need not have minimal character, but if this is not the case there will be a twist of \(f\) that is both twist minimal and has minimal character.

In the LMFDB, the designated representative of each twist class is the twist minimal newform \(g\) of minimal character whose label is lexicographically minimal among all such newforms. This newform \(g\) is called the minimal twist of the newforms in its twist equivalence class and is identified by a checkmark () in tables of twists.

These conventions also apply to newform orbits.

The multiplicity of a newform orbit \([g]\) as a twist of a newform orbit \([f]\) by a primitive character orbit \([\psi ]\) is the number of distinct \(\psi \in [\psi ]\) for which \(f\otimes \psi \in [g]\). This number is the same for every \(f\in [f]\) and depends only on the Galois orbits \([g]\), \([f]\), and \([\psi ]\).

When \(g\) is an inner twist of \(f\), this multiplicity is equal to the inner twist count of \(f\).

The weight of an elliptic modular form \(f\) is the integer or half-integer power of \((cz+d)\) that occurs in the modular transformation property of \(f\) under the action of \(\gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right)\) on the upper half plane. That is, the weight is the number \(k\) in the transformation law