# Classical modular forms downloaded from the LMFDB on 11 December 2023.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/
# Query "{'level': 768, 'weight': 4, 'char_order': 1, 'fricke_eigenval': -1}" returned 10 forms, sorted by analytic conductor.
# Each entry in the following data list has the form:
# [Label, Dim, $A$, Field, CM, Traces, A-L signs, $q$-expansion]
# For more details, see the definitions at the bottom of the file.
# To create a list of forms, type "forms = make_data()"
columns = ["label", "dim", "analytic_conductor", "nf_label", "cm_discs", "trace_display", "atkin_lehner_eigenvals", "qexp_display"]
data = [
["768.4.a.b", 1, 45.31346688440879, "1.1.1.1", [], [0, -3, 8, -12], [[2, -1], [3, 1]], "q-3q^{3}+8q^{5}-12q^{7}+9q^{9}+12q^{11}+\\cdots"],
["768.4.a.c", 1, 45.31346688440879, "1.1.1.1", [], [0, 3, -8, -12], [[2, 1], [3, -1]], "q+3q^{3}-8q^{5}-12q^{7}+9q^{9}-12q^{11}+\\cdots"],
["768.4.a.e", 2, 45.31346688440879, "2.2.13.1", [], [0, -6, -8, -16], [[2, -1], [3, 1]], "q-3q^{3}+(-4-\\beta )q^{5}+(-8-\\beta )q^{7}+\\cdots"],
["768.4.a.g", 2, 45.31346688440879, "2.2.12.1", [], [0, -6, 0, 0], [[2, -1], [3, 1]], "q-3q^{3}+3\\beta q^{5}+\\beta q^{7}+9q^{9}-2^{4}\\beta q^{13}+\\cdots"],
["768.4.a.m", 2, 45.31346688440879, "2.2.8.1", [], [0, 6, 0, 0], [[2, 1], [3, -1]], "q+3q^{3}+\\beta q^{5}-5\\beta q^{7}+9q^{9}-20q^{11}+\\cdots"],
["768.4.a.n", 2, 45.31346688440879, "2.2.12.1", [], [0, 6, 0, 0], [[2, 1], [3, -1]], "q+3q^{3}+3\\beta q^{5}-\\beta q^{7}+9q^{9}-2^{4}\\beta q^{13}+\\cdots"],
["768.4.a.p", 2, 45.31346688440879, "2.2.13.1", [], [0, 6, 8, -16], [[2, 1], [3, -1]], "q+3q^{3}+(4+\\beta )q^{5}+(-8-\\beta )q^{7}+9q^{9}+\\cdots"],
["768.4.a.q", 3, 45.31346688440879, "3.3.1436.1", [], [0, -9, -10, 14], [[2, -1], [3, 1]], "q-3q^{3}+(-3-\\beta _{2})q^{5}+(5-\\beta _{1})q^{7}+\\cdots"],
["768.4.a.s", 3, 45.31346688440879, "3.3.1436.1", [], [0, 9, -10, -14], [[2, 1], [3, -1]], "q+3q^{3}+(-3-\\beta _{2})q^{5}+(-5+\\beta _{1}+\\cdots)q^{7}+\\cdots"],
["768.4.a.u", 4, 45.31346688440879, "4.4.9792.1", [], [0, -12, 0, 0], [[2, -1], [3, 1]], "q-3q^{3}+\\beta _{1}q^{5}+(-\\beta _{1}+\\beta _{3})q^{7}+9q^{9}+\\cdots"]
]
def create_record(row):
out = {col: val for col, val in zip(columns, row)}
return out
def make_data():
return [create_record(row) for row in data]
# Label --
# 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 $\chi$.
# Dim --
# 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.
#$A$ (analytic_conductor) --
# 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.
#Field (nf_label) --
# 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**.
#CM (cm_discs) --
# 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.
#Traces (trace_display) --
# 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 $\mathrm{Aut}(\C)$ (equivalently, the sum under all embeddings of the coefficient field into $\C$). The trace form is a modular form $\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma_1(N))$ whose $q$-expansion has integral coefficients $a_n(\mathrm{Tr}(f)) \in \Z$.
# 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.
#A-L signs (atkin_lehner_eigenvals) --
# 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.
#$q$-expansion (qexp_display) --
# 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.