# Classical modular forms downloaded from the LMFDB on 08 May 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3936/ # Query "{'level': 3936}" returned 45 forms, sorted by analytic conductor. # Each entry in the following data list has the form: # [Label, Dim, $A$, Field, CM, RM, Traces, Fricke sign, $q$-expansion] # For more details, see the definitions at the bottom of the file. "3936.1.b.a" 16 1.9643198897234975 "16.0.18446744073709551616.1" [-164] [] [0, 0, 0, 0] NULL "q-\\zeta_{32}^{3}q^{3}+(\\zeta_{32}^{2}+\\zeta_{32}^{14})q^{5}+(-\\zeta_{32}^{7}+\\cdots)q^{7}+\\cdots" "3936.1.m.a" 1 1.9643198897234975 "1.1.1.1" [-984] [] [0, -1, -1, 0] NULL "q-q^{3}-q^{5}+q^{9}-q^{13}+q^{15}-q^{17}+\\cdots" "3936.1.m.b" 1 1.9643198897234975 "1.1.1.1" [-984] [] [0, -1, 1, 0] NULL "q-q^{3}+q^{5}+q^{9}-q^{13}-q^{15}+q^{17}+\\cdots" "3936.1.m.c" 1 1.9643198897234975 "1.1.1.1" [-984] [] [0, 1, -1, 0] NULL "q+q^{3}-q^{5}+q^{9}+q^{13}-q^{15}+q^{17}+\\cdots" "3936.1.m.d" 1 1.9643198897234975 "1.1.1.1" [-984] [] [0, 1, 1, 0] NULL "q+q^{3}+q^{5}+q^{9}+q^{13}+q^{15}-q^{17}+\\cdots" "3936.1.m.e" 4 1.9643198897234975 "4.0.256.1" [] [328] [0, 0, 0, 0] NULL "q+\\zeta_{8}^{3}q^{3}-\\zeta_{8}^{2}q^{9}+(\\zeta_{8}+\\zeta_{8}^{3})q^{11}+\\cdots" "3936.1.cl.a" 4 1.9643198897234975 "4.0.256.1" [-8] [] [0, -4, 0, 0] NULL "q-q^{3}+q^{9}+(\\zeta_{8}^{2}+\\zeta_{8}^{3})q^{11}+(-1+\\cdots)q^{17}+\\cdots" "3936.1.cl.b" 4 1.9643198897234975 "4.0.256.1" [-8] [] [0, 0, 0, 0] NULL "q+\\zeta_{8}q^{3}+\\zeta_{8}^{2}q^{9}+(-\\zeta_{8}^{2}-\\zeta_{8}^{3}+\\cdots)q^{11}+\\cdots" "3936.1.fx.a" 16 1.9643198897234975 "16.0.1048576000000000000.1" [-8] [] [0, 0, 0, 0] NULL "q-\\zeta_{40}^{17}q^{3}-\\zeta_{40}^{14}q^{9}+(-\\zeta_{40}^{15}+\\cdots)q^{11}+\\cdots" "3936.1.fx.b" 16 1.9643198897234975 "16.0.1048576000000000000.1" [-8] [] [0, 4, 0, 0] NULL "q-\\zeta_{40}^{8}q^{3}+\\zeta_{40}^{16}q^{9}+(\\zeta_{40}^{15}+\\zeta_{40}^{18}+\\cdots)q^{11}+\\cdots" "3936.2.a.a" 1 31.42911823557596 "1.1.1.1" [] [] [0, -1, -2, 4] 1 "q-q^{3}-2q^{5}+4q^{7}+q^{9}-4q^{11}+\\cdots" "3936.2.a.b" 1 31.42911823557596 "1.1.1.1" [] [] [0, -1, 1, -2] 1 "q-q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}-5q^{13}+\\cdots" "3936.2.a.c" 1 31.42911823557596 "1.1.1.1" [] [] [0, -1, 2, -2] -1 "q-q^{3}+2q^{5}-2q^{7}+q^{9}-4q^{11}+\\cdots" "3936.2.a.d" 1 31.42911823557596 "1.1.1.1" [] [] [0, 1, -2, -4] -1 "q+q^{3}-2q^{5}-4q^{7}+q^{9}+4q^{11}+\\cdots" "3936.2.a.e" 1 31.42911823557596 "1.1.1.1" [] [] [0, 1, 1, 2] -1 "q+q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-5q^{13}+\\cdots" "3936.2.a.f" 1 31.42911823557596 "1.1.1.1" [] [] [0, 1, 2, 2] -1 "q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{11}+\\cdots" "3936.2.a.g" 3 31.42911823557596 "3.3.148.1" [] [] [0, -3, -2, 0] 1 "q-q^{3}+(-1+\\beta _{1})q^{5}+\\beta _{2}q^{7}+q^{9}+\\cdots" "3936.2.a.h" 3 31.42911823557596 "3.3.148.1" [] [] [0, 3, -2, 0] 1 "q+q^{3}+(-1+\\beta _{1})q^{5}-\\beta _{2}q^{7}+q^{9}+\\cdots" "3936.2.a.i" 4 31.42911823557596 "4.4.15188.1" [] [] [0, -4, -4, 6] 1 "q-q^{3}+(-1-\\beta _{3})q^{5}+(1-\\beta _{2})q^{7}+\\cdots" "3936.2.a.j" 4 31.42911823557596 "4.4.202932.1" [] [] [0, -4, -2, -2] 1 "q-q^{3}+(-1-\\beta _{2})q^{5}+(-1+\\beta _{1})q^{7}+\\cdots" "3936.2.a.k" 4 31.42911823557596 "4.4.17428.1" [] [] [0, -4, -1, 4] -1 "q-q^{3}-\\beta _{1}q^{5}+(1-\\beta _{1}+\\beta _{2})q^{7}+q^{9}+\\cdots" "3936.2.a.l" 4 31.42911823557596 "4.4.8468.1" [] [] [0, -4, 1, 0] -1 "q-q^{3}-\\beta _{3}q^{5}+(\\beta _{2}-\\beta _{3})q^{7}+q^{9}+\\cdots" "3936.2.a.m" 4 31.42911823557596 "4.4.15188.1" [] [] [0, 4, -4, -6] 1 "q+q^{3}+(-1-\\beta _{3})q^{5}+(-1+\\beta _{2})q^{7}+\\cdots" "3936.2.a.n" 4 31.42911823557596 "4.4.202932.1" [] [] [0, 4, -2, 2] -1 "q+q^{3}+(-1-\\beta _{2})q^{5}+(1-\\beta _{1})q^{7}+\\cdots" "3936.2.a.o" 4 31.42911823557596 "4.4.17428.1" [] [] [0, 4, -1, -4] 1 "q+q^{3}-\\beta _{1}q^{5}+(-1+\\beta _{1}-\\beta _{2})q^{7}+\\cdots" "3936.2.a.p" 4 31.42911823557596 "4.4.8468.1" [] [] [0, 4, 1, 0] 1 "q+q^{3}-\\beta _{3}q^{5}+(-\\beta _{2}+\\beta _{3})q^{7}+q^{9}+\\cdots" "3936.2.a.q" 5 31.42911823557596 "5.5.6852676.1" [] [] [0, -5, -2, 0] -1 "q-q^{3}-\\beta _{1}q^{5}+\\beta _{4}q^{7}+q^{9}+(1-\\beta _{1}+\\cdots)q^{11}+\\cdots" "3936.2.a.r" 5 31.42911823557596 "5.5.6852676.1" [] [] [0, 5, -2, 0] -1 "q+q^{3}-\\beta _{1}q^{5}-\\beta _{4}q^{7}+q^{9}+(-1+\\cdots)q^{11}+\\cdots" "3936.2.a.s" 6 31.42911823557596 "6.6.121506628.1" [] [] [0, -6, 3, -8] 1 "q-q^{3}-\\beta _{4}q^{5}+(-1+\\beta _{3})q^{7}+q^{9}+\\cdots" "3936.2.a.t" 6 31.42911823557596 "6.6.121506628.1" [] [] [0, 6, 3, 8] -1 "q+q^{3}-\\beta _{4}q^{5}+(1-\\beta _{3})q^{7}+q^{9}+(-\\beta _{2}+\\cdots)q^{11}+\\cdots" "3936.2.a.u" 7 31.42911823557596 NULL [] [] [0, -7, 6, -8] -1 "q-q^{3}+(1+\\beta _{1})q^{5}+(-1-\\beta _{2})q^{7}+\\cdots" "3936.2.a.v" 7 31.42911823557596 NULL [] [] [0, 7, 6, 8] -1 "q+q^{3}+(1+\\beta _{1})q^{5}+(1+\\beta _{2})q^{7}+q^{9}+\\cdots" "3936.2.e.a" 42 31.42911823557596 NULL [] [] [0, -42, 0, 0] NULL NULL "3936.2.e.b" 42 31.42911823557596 NULL [] [] [0, 42, 0, 0] NULL NULL "3936.2.f.a" 40 31.42911823557596 NULL [] [] [0, 0, 0, -4] NULL NULL "3936.2.f.b" 40 31.42911823557596 NULL [] [] [0, 0, 0, -4] NULL NULL "3936.2.j.a" 2 31.42911823557596 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q-i q^{3}+4 i q^{7}-q^{9}-3 i q^{11}+\\cdots" "3936.2.j.b" 2 31.42911823557596 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+i q^{3}-4 i q^{7}-q^{9}+3 i q^{11}+\\cdots" "3936.2.j.c" 2 31.42911823557596 "2.0.4.1" [] [] [0, 0, 6, 0] NULL "q-i q^{3}+3 q^{5}-2 i q^{7}-q^{9}-5 i q^{13}+\\cdots" "3936.2.j.d" 2 31.42911823557596 "2.0.4.1" [] [] [0, 0, 6, 0] NULL "q-i q^{3}+3 q^{5}-2 i q^{7}-q^{9}+5 i q^{13}+\\cdots" "3936.2.j.e" 4 31.42911823557596 "4.0.400.1" [] [] [0, 0, 8, 0] NULL "q-\\beta _{1}q^{3}+2q^{5}+2\\beta _{1}q^{7}-q^{9}+3\\beta _{1}q^{11}+\\cdots" "3936.2.j.f" 8 31.42911823557596 "8.0.401550342400.7" [] [] [0, 0, -16, 0] NULL "q-\\beta _{2}q^{3}-2q^{5}-2\\beta _{2}q^{7}-q^{9}+(\\beta _{2}+\\cdots)q^{11}+\\cdots" "3936.2.j.g" 20 31.42911823557596 NULL [] [] [0, 0, -4, 0] NULL "q-\\beta _{3}q^{3}-\\beta _{7}q^{5}+\\beta _{16}q^{7}-q^{9}+\\beta _{9}q^{11}+\\cdots" "3936.2.j.h" 22 31.42911823557596 NULL [] [] [0, 0, -4, 0] NULL NULL "3936.2.j.i" 22 31.42911823557596 NULL [] [] [0, 0, -4, 0] NULL NULL # 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. #RM (rm_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. #Fricke sign (fricke_eigenval) -- # 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 $w_N f = +f$. #$q$-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.