# Classical modular forms downloaded from the LMFDB on 23 May 2024.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/?level=165&weight=4&char_order=1&atkin_lehner_string=++-
# Query "{'level': 165, 'weight': 4, 'char_order': 1, 'atkin_lehner_string': ' -'}" returned 0 forms, sorted by analytic conductor.
# Each entry in the following data list has the form:
# [Label, Dim, $A$, Field, CM, Traces, $q$-expansion]
# For more details, see the definitions at the bottom of the file.
# 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.
#$q$-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.