# Classical modular forms downloaded from the LMFDB on 01 March 2024.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/4840/
# Query "{'level': 4840}" returned 34 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 = [
["4840.2.a.a", 1, 38.6475945782997, "1.1.1.1", [], [0, -2, -1, 0], [[2, -1], [5, 1], [11, 1]], "q-2q^{3}-q^{5}+q^{9}-2q^{13}+2q^{15}+\\cdots"],
["4840.2.a.b", 1, 38.6475945782997, "1.1.1.1", [], [0, -2, -1, 0], [[2, 1], [5, 1], [11, 1]], "q-2q^{3}-q^{5}+q^{9}+2q^{13}+2q^{15}+\\cdots"],
["4840.2.a.c", 1, 38.6475945782997, "1.1.1.1", [], [0, 0, -1, 2], [[2, -1], [5, 1], [11, -1]], "q-q^{5}+2q^{7}-3q^{9}+8q^{19}-8q^{23}+\\cdots"],
["4840.2.a.d", 1, 38.6475945782997, "1.1.1.1", [], [0, 0, -1, 2], [[2, 1], [5, 1], [11, -1]], "q-q^{5}+2q^{7}-3q^{9}+4q^{13}+4q^{17}+\\cdots"],
["4840.2.a.e", 1, 38.6475945782997, "1.1.1.1", [], [0, 0, 1, -4], [[2, -1], [5, -1], [11, -1]], "q+q^{5}-4q^{7}-3q^{9}-6q^{13}+6q^{17}+\\cdots"],
["4840.2.a.f", 1, 38.6475945782997, "1.1.1.1", [], [0, 0, 1, 4], [[2, -1], [5, -1], [11, -1]], "q+q^{5}+4q^{7}-3q^{9}+2q^{13}-2q^{17}+\\cdots"],
["4840.2.a.g", 1, 38.6475945782997, "1.1.1.1", [], [0, 1, -1, -3], [[2, -1], [5, 1], [11, 1]], "q+q^{3}-q^{5}-3q^{7}-2q^{9}+4q^{13}+\\cdots"],
["4840.2.a.h", 1, 38.6475945782997, "1.1.1.1", [], [0, 1, -1, 3], [[2, 1], [5, 1], [11, 1]], "q+q^{3}-q^{5}+3q^{7}-2q^{9}-4q^{13}+\\cdots"],
["4840.2.a.i", 1, 38.6475945782997, "1.1.1.1", [], [0, 3, 1, -1], [[2, -1], [5, -1], [11, -1]], "q+3q^{3}+q^{5}-q^{7}+6q^{9}+6q^{13}+\\cdots"],
["4840.2.a.j", 2, 38.6475945782997, "2.2.17.1", [], [0, -1, 2, 1], [[2, 1], [5, -1], [11, -1]], "q-\\beta q^{3}+q^{5}+\\beta q^{7}+(1+\\beta )q^{9}-2q^{13}+\\cdots"],
["4840.2.a.k", 2, 38.6475945782997, "2.2.12.1", [], [0, 0, 2, -4], [[2, 1], [5, -1], [11, -1]], "q+\\beta q^{3}+q^{5}+(-2-\\beta )q^{7}+2\\beta q^{13}+\\cdots"],
["4840.2.a.l", 2, 38.6475945782997, "2.2.12.1", [], [0, 0, 2, 4], [[2, -1], [5, -1], [11, -1]], "q+\\beta q^{3}+q^{5}+(2+\\beta )q^{7}-2\\beta q^{13}+\\cdots"],
["4840.2.a.m", 2, 38.6475945782997, "2.2.17.1", [], [0, 1, -2, -5], [[2, 1], [5, 1], [11, -1]], "q+\\beta q^{3}-q^{5}+(-2-\\beta )q^{7}+(1+\\beta )q^{9}+\\cdots"],
["4840.2.a.n", 2, 38.6475945782997, "2.2.17.1", [], [0, 1, -2, -3], [[2, -1], [5, 1], [11, -1]], "q+\\beta q^{3}-q^{5}+(-2+\\beta )q^{7}+(1+\\beta )q^{9}+\\cdots"],
["4840.2.a.o", 2, 38.6475945782997, "2.2.8.1", [], [0, 2, 2, -2], [[2, 1], [5, -1], [11, -1]], "q+(1+\\beta )q^{3}+q^{5}+(-1-\\beta )q^{7}+2\\beta q^{9}+\\cdots"],
["4840.2.a.p", 2, 38.6475945782997, "2.2.8.1", [], [0, 2, 2, 2], [[2, -1], [5, -1], [11, -1]], "q+(1+\\beta )q^{3}+q^{5}+(1+\\beta )q^{7}+2\\beta q^{9}+\\cdots"],
["4840.2.a.q", 3, 38.6475945782997, "3.3.568.1", [], [0, -1, 3, -1], [[2, 1], [5, -1], [11, 1]], "q+\\beta _{2}q^{3}+q^{5}+\\beta _{2}q^{7}+(3-\\beta _{1})q^{9}+\\cdots"],
["4840.2.a.r", 3, 38.6475945782997, "3.3.568.1", [], [0, -1, 3, 1], [[2, -1], [5, -1], [11, 1]], "q+\\beta _{2}q^{3}+q^{5}-\\beta _{2}q^{7}+(3-\\beta _{1})q^{9}+\\cdots"],
["4840.2.a.s", 3, 38.6475945782997, "3.3.788.1", [], [0, 1, -3, -3], [[2, -1], [5, 1], [11, -1]], "q+\\beta _{1}q^{3}-q^{5}+(-1-\\beta _{1}+\\beta _{2})q^{7}+\\cdots"],
["4840.2.a.t", 3, 38.6475945782997, "3.3.404.1", [], [0, 1, -3, -1], [[2, -1], [5, 1], [11, -1]], "q+\\beta _{2}q^{3}-q^{5}-\\beta _{1}q^{7}+(2+\\beta _{1}-\\beta _{2})q^{9}+\\cdots"],
["4840.2.a.u", 3, 38.6475945782997, "3.3.404.1", [], [0, 1, -3, 1], [[2, 1], [5, 1], [11, -1]], "q+\\beta _{2}q^{3}-q^{5}+\\beta _{1}q^{7}+(2+\\beta _{1}-\\beta _{2})q^{9}+\\cdots"],
["4840.2.a.v", 3, 38.6475945782997, "3.3.788.1", [], [0, 1, -3, 3], [[2, 1], [5, 1], [11, -1]], "q+\\beta _{1}q^{3}-q^{5}+(1+\\beta _{1}-\\beta _{2})q^{7}+(1+\\cdots)q^{9}+\\cdots"],
["4840.2.a.w", 4, 38.6475945782997, "4.4.4752.1", [], [0, -2, -4, -6], [[2, 1], [5, 1], [11, 1]], "q+(-1-\\beta _{2})q^{3}-q^{5}+(-2-\\beta _{2}+\\beta _{3})q^{7}+\\cdots"],
["4840.2.a.x", 4, 38.6475945782997, "4.4.4752.1", [], [0, -2, -4, 6], [[2, -1], [5, 1], [11, 1]], "q+(-1-\\beta _{2})q^{3}-q^{5}+(2+\\beta _{2}-\\beta _{3})q^{7}+\\cdots"],
["4840.2.a.y", 4, 38.6475945782997, "4.4.725.1", [], [0, -2, 4, -7], [[2, -1], [5, -1], [11, 1]], "q-\\beta _{2}q^{3}+q^{5}+(-2+\\beta _{1}+\\beta _{2}-\\beta _{3})q^{7}+\\cdots"],
["4840.2.a.z", 4, 38.6475945782997, "4.4.725.1", [], [0, -2, 4, 7], [[2, 1], [5, -1], [11, -1]], "q-\\beta _{2}q^{3}+q^{5}+(2-\\beta _{1}-\\beta _{2}+\\beta _{3})q^{7}+\\cdots"],
["4840.2.a.ba", 6, 38.6475945782997, "6.6.25903625.1", [], [0, -3, -6, -7], [[2, -1], [5, 1], [11, -1]], "q-\\beta _{1}q^{3}-q^{5}+(-1+\\beta _{2}-\\beta _{3}+\\beta _{5})q^{7}+\\cdots"],
["4840.2.a.bb", 6, 38.6475945782997, "6.6.25903625.1", [], [0, -3, -6, 7], [[2, 1], [5, 1], [11, 1]], "q-\\beta _{1}q^{3}-q^{5}+(1-\\beta _{2}+\\beta _{3}-\\beta _{5})q^{7}+\\cdots"],
["4840.2.a.bc", 6, 38.6475945782997, "6.6.22733568.1", [], [0, -2, 6, -4], [[2, -1], [5, -1], [11, 1]], "q+\\beta _{1}q^{3}+q^{5}+(-\\beta _{3}-\\beta _{4})q^{7}+(-\\beta _{1}+\\cdots)q^{9}+\\cdots"],
["4840.2.a.bd", 6, 38.6475945782997, "6.6.22733568.1", [], [0, -2, 6, 4], [[2, 1], [5, -1], [11, 1]], "q+\\beta _{1}q^{3}+q^{5}+(\\beta _{3}+\\beta _{4})q^{7}+(-\\beta _{1}+\\cdots)q^{9}+\\cdots"],
["4840.2.a.be", 6, 38.6475945782997, "6.6.45753625.1", [], [0, 2, -6, -6], [[2, 1], [5, 1], [11, -1]], "q-\\beta _{3}q^{3}-q^{5}+(-1+\\beta _{2}+\\beta _{5})q^{7}+\\cdots"],
["4840.2.a.bf", 6, 38.6475945782997, "6.6.45753625.1", [], [0, 2, -6, 6], [[2, -1], [5, 1], [11, 1]], "q-\\beta _{3}q^{3}-q^{5}+(1-\\beta _{2}-\\beta _{5})q^{7}+(-2\\beta _{2}+\\cdots)q^{9}+\\cdots"],
["4840.2.a.bg", 8, 38.6475945782997, None, [], [0, 1, 8, -6], [[2, 1], [5, -1], [11, 1]], "q+\\beta _{1}q^{3}+q^{5}+(-1-\\beta _{4})q^{7}+(3+\\beta _{4}+\\cdots)q^{9}+\\cdots"],
["4840.2.a.bh", 8, 38.6475945782997, None, [], [0, 1, 8, 6], [[2, -1], [5, -1], [11, -1]], "q+\\beta _{1}q^{3}+q^{5}+(1+\\beta _{4})q^{7}+(3+\\beta _{4}+\\cdots)q^{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.