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 4.72 Definition 1.81 Definition 1.94 Definition 1.125

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 4.51

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 4.61 Definition 4.69 Definition 1.110

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 4.36 Definition 4.51 Definition 4.83 Definition 2.61

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 4.36 Definition 4.51 Definition 4.83

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 4.37 Definition 4.83

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 4.46

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 4.68

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 4.31 Definition 4.37

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 4.1 Definition 4.10 Definition 4.37 Definition 4.51 Definition 2.56

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 4.14 Definition 4.31 Definition 4.83 Definition 1.81 Definition 1.94

Cuspidal modular form (mathlib). (LMFDB)

This is CuspForm \Gamma k in mathlib. In the LMFDB definition, k is a positive integer and \Gamma is a finite-index subgroup of \SL_2(\mathbb{Z}). Mathlib is more general: k is an integer and \Gamma is a subgroup of \GL_2(\mathbb{R}).

Cuspidal subspace (mathlib). (LMFDB)

This is cuspFormSubmodule \Gamma k in mathlib. It is the submodule of the ModularForm \Gamma k consisting of cusp forms.

Depends on: Definition 4.13 Definition 4.72

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 4.37 Definition 4.51

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 4.19 Definition 4.35 Definition 4.51 Definition 4.52

Dimension (mathlib). (LMFDB)

This is defined as 'finrank ℂ (ModularForm Γ k)' in mathlib, where ModularForm \Gamma k is the space of modular forms and finrank ℂ gives its module rank over \mathbb{C}.

Depends on: Definition 4.71 Definition 4.72

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 4.35 Definition 4.37 Definition 4.51 Definition 4.53

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 4.51 Definition 4.61 Definition 4.69 Definition 1.115 Definition 2.8