Classical modular form. (LMFDB)

Let k be a positive integer and let \Gamma be a finite index subgroup of the modular group \SL(2,\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

f(\gamma z) = (cz+d)^k f(z)

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

f(\gamma z) = \chi(d)(cz+d)^k f(z)

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)).

Depends on: Definition 1.45 Definition 1.81 Definition 1.94 Definition 1.124

Analytic conductor of a classical newform. (LMFDB)

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

N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2,

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

Depends on: Definition 4.47

Analytic rank. (LMFDB)

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

L(f,s) = \sum_{n\ge 1} a_nn^{-s}

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).

Depends on: Definition 4.10 Definition 4.13 Definition 4.56 Definition 4.63 Definition 1.109

Artin field. (LMFDB)

The Artin field of a weight one newform is the number field fixed by the kernel of its associated Galois representation \rho\colon \Gal(\overline{\Q}/\Q)\to \GL_2(\C).

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

Depends on: Definition 4.32 Definition 4.47 Definition 4.76 Definition 2.57

Artin image. (LMFDB)

The Artin image of a weight one newform is the image of its associated Galois representation \rho\colon \Gal(\overline{\Q}/\Q)\to \GL_2(\C).

The Artin image is a finite subgroup of \GL_2(\C) whose cardinality is equal to the degree of the Artin field.

Depends on: Definition 4.4 Definition 4.32 Definition 4.47 Definition 4.76

Atkin-Lehner involution w_Q. (LMFDB)

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 \Z for which the matrix

W_Q=\left( \begin{matrix} Qx & y \\ Nz & Qt\end{matrix} \right)

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.

Depends on: Definition 4.13 Definition 4.33 Definition 4.76

Bad prime. (LMFDB)

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.

Depends on: Definition 4.1 Definition 4.42

Character of a modular form. (LMFDB)

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,

f(\gamma z) = \chi(d)(cz+d)^k f(z)

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.

Depends on: Definition 1.45 Definition 4.1

CM form. (LMFDB)

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.

Depends on: Definition 4.1 Definition 4.62

Coefficient field for newforms. (LMFDB)

The coefficient field of a modular form is the subfield of \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 \Q(\zeta_m), the coefficient field will contain \Q(\zeta_m) For eigenforms, the coefficient field is also known as the Hecke field.

Depends on: Definition 4.13 Definition 4.28 Definition 4.33

Coefficient ring. (LMFDB)

The coefficient ring of a modular form is the subring \Z[a_1,a_2,a_3,\ldots] of \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.

Depends on: Definition 4.10 Definition 4.47

Congruence subgroup. (LMFDB)

A congruence subgroup \Gamma of \SL_2(\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.

Cuspidal modular form. (LMFDB)

Let k be a positive integer and let \Gamma be a finite index subgroup of the modular group \SL(2,\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.

Depends on: Definition 1.45 Definition 4.1 Definition 4.28 Definition 4.76 Definition 1.81 Definition 1.94

Decomposition into newforms. (LMFDB)

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.

Depends on: Definition 4.33 Definition 4.47

Defining polynomial. (LMFDB)

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.

Depends on: Definition 4.10 Definition 2.1

Dimension. (LMFDB)

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 \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 \Q).

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

Depends on: Definition 4.31 Definition 4.47 Definition 4.48

Distinguishing Hecke operators. (LMFDB)

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

c_m := \#\bigl\{X_f(\{T_{p_i}:i < m\}): \mathrm{newforms}\ f\in S_k^\mathrm{new}(N,\chi)\bigr\}

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.

Depends on: Definition 4.31 Definition 4.33 Definition 4.47 Definition 4.49

Dual cuspform. (LMFDB)

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.

Depends on: Definition 4.10 Definition 4.13 Definition 4.47 Definition 4.56 Definition 4.63 Definition 1.114 Definition 2.8

Holomorphic Eisenstein series of level 1. (LMFDB)

For an even integer k\geq4, we define the (normalized) holomorphic Eisenstein series of level 1

E_k(z)=\frac{1}{2\zeta(k)}\sum_{(c,d)\ne(0,0)}(cz+d)^{-k}=\sum_{\left(\begin{matrix} a&b\\c&d \end{matrix}\right)\in\ \Gamma_{\infty}\setminus \SL(2,\Z) }(cz+d)^{-k},

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:

E_k(z)=1-\frac{2k}{B_k}\sum_{n\geq1}\sigma_{k-1}(n)q^n,

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

Depends on: Definition 1.2 Definition 1.3 Definition 4.1 Definition 4.42 Definition 4.56 Definition 4.76 Definition 1.94

Holomorphic Eisenstein modular form. (LMFDB)

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

Depends on: Definition 1.45 Definition 4.1 Definition 4.13 Definition 4.24 Definition 4.52 Definition 4.76 Definition 1.94

Label of a classical Eisenstein modular form. (LMFDB)

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 \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.

Depends on: Definition 1.45 Definition 1.47 Definition 1.49 Definition 4.10 Definition 4.22 Definition 4.25 Definition 4.31 Definition 4.42 Definition 4.57 Definition 4.76

Eisenstein newform. (LMFDB)

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.

Depends on: Definition 4.23 Definition 4.28 Definition 4.33

New Eisenstein subspace. (LMFDB)

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

E_k(N,\chi) = E_k^{\rm old}(N,\chi) \oplus E_k^{\rm new}(N,\chi)

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).

Depends on: Definition 1.45 Definition 1.50 Definition 1.52 Definition 4.1 Definition 4.8 Definition 4.20 Definition 4.24 Definition 4.42 Definition 4.76

Holomorphic Eisenstein series. (LMFDB)

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

E_{k}^{\chi_1, \chi_2}(\tau) = \frac{1}{2} \left ( \delta_{\chi_1=1} L(1-k, \chi_2) + \delta_{k=1} \delta_{\chi_2=1} L(0,\chi_1) \right) + \sum_{n=1}^{\infty} \sigma_{k-1}^{\chi_1, \chi_2}(n) q^n,

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

\sigma_{k-1}^{\chi_1, \chi_2}(n) = \sum_{m \mid n} \chi_1(n/m) \chi_2(m) m^{k-1}.

Depends on: Definition 1.45 Definition 1.54 Definition 4.76

Embedding of a modular form. (LMFDB)

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

Each embedding \iota\colon \Q(f)\to \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 \Q(f)\to \C induces an embedding \Q(\chi)\to \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.

Depends on: Definition 1.45 Definition 1.56 Definition 4.8 Definition 4.10 Definition 4.31 Definition 4.56 Definition 1.108 Definition 2.16

Complex embedding label. (LMFDB)

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.

Depends on: Definition 1.49 Definition 4.13 Definition 4.25 Definition 4.42 Definition 4.76

Eta quotient. (LMFDB)

An eta quotient is any function f of the form

f(z)=\prod_{1\leq i\leq s}\eta^{r_i}(m_iz),

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.

Depends on: Definition 1.123

Fourier coefficients of a modular form. (LMFDB)

Let f be a modular form on a finite index subgroup \Gamma of \SL_2(\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

f(z)=\sum_{n\ge 0}a_n q^n ,

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:

g_{\gamma}(q^{1/w})=\sum_{n\geq0}a_{\gamma}(n)q^{n/w}\quad\text{for}\quad0<|q|<1.

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

f(z)=\sum_{n \geq0}a_{\gamma}(n)q^{n/w},

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

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

Depends on: Definition 4.1 Definition 1.81 Definition 1.82

Fricke involution. (LMFDB)

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.

Depends on: Definition 4.3 Definition 4.6 Definition 1.108 Definition 1.116

Galois conjugate newforms. (LMFDB)

Two newforms f=\sum a_nq^n and g=\sum b_nq^n are Galois conjugate if there is an automorphism \sigma\in \Gal(\overline{\Q}/\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\Gal(\overline{\Q}/\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

Depends on: Definition 4.47 Definition 2.12

Galois orbit of a newform. (LMFDB)

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

[f]:=\{\sigma(f):\sigma\in \mathrm{Gal}(\overline{\Q}/\Q)\}

of its Galois conjugates, which forms a canonical \Q-basis for the corresponding newform subspace.

Galois orbits of newforms are also called newform orbits.

Depends on: Definition 4.30 Definition 4.47 Definition 4.48

Galois representation. (LMFDB)

As shown by Deligne and Serre , every newform of weight one has an associated Galois representation \rho\colon \Gal(\overline{\Q}/\Q)\to \GL_2(\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 \GL_2(\C)\to \PGL_2(\C) yields the projective Galois representation \bar\rho\colon \Gal(\overline{\Q}/\Q)\to \PGL_2(\C).

Depends on: Definition 1.35 Definition 1.36 Definition 1.38 Definition 4.47 Definition 4.76

Hecke operator. (LMFDB)

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

a_m(T_nf) := \sum_{d|\gcd(m,n)}\chi(d)d^{k-1}a_{mn/d^2}(f).

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

T_n f = a_nf.

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).

Depends on: Definition 1.45 Definition 4.1 Definition 4.13 Definition 4.42 Definition 4.52 Definition 4.76

Hecke orbit. (LMFDB)

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.

Depends on: Definition 4.13 Definition 4.33 Definition 4.42

Coefficient ring generator bound. (LMFDB)

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

Depends on: Definition 4.11 Definition 4.47 Definition 4.56

Hecke characteristic polynomial. (LMFDB)

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.

Depends on: Definition 4.33 Definition 4.47 Definition 4.48

Inner twist. (LMFDB)

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

a_p(g) = \chi(p)a_p(f)

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 \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".

Depends on: Definition 1.45 Definition 1.47 Definition 1.54 Definition 4.9 Definition 4.10 Definition 4.30 Definition 4.31 Definition 4.47 Definition 4.58 Definition 4.62

Inner twist count. (LMFDB)

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 \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.

Depends on: Definition 1.45 Definition 4.10 Definition 4.16 Definition 4.37 Definition 4.47 Definition 4.62 Definition 4.76

Inner twist multiplicity. (LMFDB)

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 \C will yield different \varphi, but the number of such \varphi is the same for every embedding; this number is the multiplicity.

Depends on: Definition 1.45 Definition 1.47 Definition 4.25 Definition 4.37 Definition 4.47

Label of a classical modular form. (LMFDB)

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 \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.

Depends on: Definition 1.45 Definition 1.47 Definition 1.49 Definition 4.10 Definition 4.25 Definition 4.31 Definition 4.42 Definition 4.47 Definition 4.57 Definition 4.76

Label of a classical modular form. (LMFDB)

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 \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.

Depends on: Definition 1.45 Definition 1.47 Definition 1.49 Definition 4.10 Definition 4.25 Definition 4.31 Definition 4.42 Definition 4.57 Definition 4.76

Level of a modular form. (LMFDB)

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.

Depends on: Definition 4.1 Definition 1.94

Maximal newform. (LMFDB)

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).

Depends on: Definition 4.8 Definition 4.16 Definition 4.31 Definition 4.47

Minimal modular form. (LMFDB)

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

Minimal twist. (LMFDB)

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)).

Depends on: Definition 1.51 Definition 4.8 Definition 4.40 Definition 4.42 Definition 4.47 Definition 4.73 Definition 4.74

Minus space. (LMFDB)

The minus subspace of S_k(\Gamma_0(N)) is the eigenspace of the Fricke involution w_N with eigenvalue -1.

Depends on: Definition 4.29

Newform. (LMFDB)

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.

Depends on: Definition 4.28 Definition 4.33 Definition 4.49

Newform subspace. (LMFDB)

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.

Depends on: Definition 4.33 Definition 4.42 Definition 4.47 Definition 4.49

New subspace. (LMFDB)

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

S_k(N,\chi) = S_k^{\rm old}(N,\chi) \oplus S_k^{\rm new}(N,\chi)

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.

Depends on: Definition 1.45 Definition 1.50 Definition 1.52 Definition 4.1 Definition 4.8 Definition 4.13 Definition 4.42 Definition 4.52 Definition 4.76

Nontrivial inner twist. (LMFDB)

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

Depends on: Definition 4.37 Definition 4.62

Old subspace of modular forms. (LMFDB)

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

\begin{aligned} \iota_d\colon S_k(N_i,\chi_i)&\to S_k(N,\chi)\\ f\ &\mapsto f(d\tau) \end{aligned}

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

S_k(N,\chi)\simeq\!\!\!\! \bigoplus_{\mathrm{cond}(\chi)|N_i|N}\!\!\!\! S_k^{\rm new}(N_i,\chi_i)^{\oplus m_i},

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).

Depends on: Definition 1.46 Definition 1.50 Definition 1.54 Definition 4.1 Definition 4.8 Definition 4.13 Definition 4.42 Definition 4.49 Definition 4.76

Petersson scalar product. (LMFDB)

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

\langle f,g\rangle_G=\frac{1}{[\Gamma:G]}\int_{\mathfrak{F}}f(z)\overline{ g(z)}y^kd\mu,

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.

Depends on: Definition 4.13 Definition 1.83 Definition 1.96

Plus space. (LMFDB)

The plus subspace of S_k(\Gamma_0(N) is the eigenspace of the Fricke involution \omega_N with eigenvalue 1.

Depends on: Definition 4.29

Projective field. (LMFDB)

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 \Gal(\overline{\Q}/\Q)\to \PGL_2(\C).

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

Depends on: Definition 4.32 Definition 4.47 Definition 4.76 Definition 2.1 Definition 2.57

Projective image. (LMFDB)

The projective image of a weight one newform is the image of its associated projective Galois representation \rho\colon \Gal(\overline{\Q}/\Q)\to \PGL_2(\C). It is a finite subgroup of \PGL_2(\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.

Depends on: Definition 4.32 Definition 4.47 Definition 4.76

q-expansion of a modular form. (LMFDB)

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 \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 \C). Each embedding K \hookrightarrow \C then gives rise to an embedded newform whose q-expansion has a_n \in \C, as above.

Depends on: Definition 4.13 Definition 4.28 Definition 4.31 Definition 4.47 Definition 1.81 Definition 2.1 Definition 2.16

Relative dimension. (LMFDB)

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 \Q(\chi) (the number field generated by the values of \chi).

Depends on: Definition 4.47

Real multiplication. (LMFDB)

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.

Depends on: Definition 4.1 Definition 4.62

Satake Angles. (LMFDB)

The Satake angles \theta_p = \arg \alpha_p \in [-\pi, \pi] are the arguments of a complex embedding of the Satake parameters \alpha_p.

Depends on: Definition 4.60

Satake parameters. (LMFDB)

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

L_p\left( t \right) = 1 -a_p t + \chi(p) p^{k-1} t^2 = \det(1 - t \, T_p) = (1 - \alpha_p p^{\frac{k-1}{2}}t )(1 - \alpha_p^{-1} \chi(p) p^{\frac{k-1}{2}} t),

T_p is Hecke operator, and a_p its trace.

Depends on: Definition 4.7 Definition 4.8 Definition 4.33 Definition 4.42 Definition 4.47 Definition 4.76

Sato-Tate group of a modular form. (LMFDB)

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 \SL_2(\C).

For newforms of weight k>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 .

Depends on: Definition 4.9 Definition 4.47 Definition 4.76

Self-twist. (LMFDB)

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

a_p(f) = \chi(p)a_p(f)

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.

Depends on: Definition 1.45 Definition 1.53 Definition 1.54 Definition 4.42 Definition 4.47 Definition 4.55 Definition 4.76 Definition 2.14