# Classical modular forms downloaded from the LMFDB on 26 July 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/8664/ # Query "{'level': 8664}" returned 45 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. "8664.2.a.a" 1 69.18238831123732 "1.1.1.1" [] [0, -1, -3, -3] [[2, -1], [3, 1], [19, -1]] "q-q^{3}-3q^{5}-3q^{7}+q^{9}-q^{11}+2q^{13}+\\cdots" "8664.2.a.b" 1 69.18238831123732 "1.1.1.1" [] [0, -1, -2, -5] [[2, 1], [3, 1], [19, 1]] "q-q^{3}-2q^{5}-5q^{7}+q^{9}-4q^{11}+\\cdots" "8664.2.a.c" 1 69.18238831123732 "1.1.1.1" [] [0, -1, -2, -4] [[2, 1], [3, 1], [19, 1]] "q-q^{3}-2q^{5}-4q^{7}+q^{9}+2q^{11}+\\cdots" "8664.2.a.d" 1 69.18238831123732 "1.1.1.1" [] [0, -1, 0, 3] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+3q^{7}+q^{9}+2q^{11}-q^{13}+\\cdots" "8664.2.a.e" 1 69.18238831123732 "1.1.1.1" [] [0, -1, 2, -3] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+2q^{5}-3q^{7}+q^{9}-5q^{13}+\\cdots" "8664.2.a.f" 1 69.18238831123732 "1.1.1.1" [] [0, -1, 2, 0] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+2q^{5}+q^{9}-2q^{13}-2q^{15}+\\cdots" "8664.2.a.g" 1 69.18238831123732 "1.1.1.1" [] [0, -1, 3, 1] [[2, 1], [3, 1], [19, 1]] "q-q^{3}+3q^{5}+q^{7}+q^{9}-3q^{11}-3q^{15}+\\cdots" "8664.2.a.h" 1 69.18238831123732 "1.1.1.1" [] [0, 1, -2, -5] [[2, -1], [3, -1], [19, -1]] "q+q^{3}-2q^{5}-5q^{7}+q^{9}-4q^{11}+\\cdots" "8664.2.a.i" 1 69.18238831123732 "1.1.1.1" [] [0, 1, -2, -4] [[2, -1], [3, -1], [19, 1]] "q+q^{3}-2q^{5}-4q^{7}+q^{9}+2q^{11}+\\cdots" "8664.2.a.j" 1 69.18238831123732 "1.1.1.1" [] [0, 1, -2, 0] [[2, 1], [3, -1], [19, -1]] "q+q^{3}-2q^{5}+q^{9}+4q^{11}+2q^{13}+\\cdots" "8664.2.a.k" 1 69.18238831123732 "1.1.1.1" [] [0, 1, 0, 3] [[2, 1], [3, -1], [19, 1]] "q+q^{3}+3q^{7}+q^{9}+2q^{11}+q^{13}+\\cdots" "8664.2.a.l" 1 69.18238831123732 "1.1.1.1" [] [0, 1, 1, -3] [[2, 1], [3, -1], [19, -1]] "q+q^{3}+q^{5}-3q^{7}+q^{9}-5q^{11}+2q^{13}+\\cdots" "8664.2.a.m" 1 69.18238831123732 "1.1.1.1" [] [0, 1, 2, -3] [[2, 1], [3, -1], [19, 1]] "q+q^{3}+2q^{5}-3q^{7}+q^{9}+5q^{13}+\\cdots" "8664.2.a.n" 1 69.18238831123732 "1.1.1.1" [] [0, 1, 3, 1] [[2, -1], [3, -1], [19, 1]] "q+q^{3}+3q^{5}+q^{7}+q^{9}-3q^{11}+3q^{15}+\\cdots" "8664.2.a.o" 1 69.18238831123732 "1.1.1.1" [] [0, 1, 4, 4] [[2, -1], [3, -1], [19, -1]] "q+q^{3}+4q^{5}+4q^{7}+q^{9}-4q^{11}+\\cdots" "8664.2.a.p" 2 69.18238831123732 "2.2.5.1" [] [0, -2, -3, -1] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+(-1-\\beta )q^{5}+(-1+\\beta )q^{7}+\\cdots" "8664.2.a.q" 2 69.18238831123732 "2.2.5.1" [] [0, -2, -2, 4] [[2, 1], [3, 1], [19, -1]] "q-q^{3}+(-1-\\beta )q^{5}+(2-\\beta )q^{7}+q^{9}+\\cdots" "8664.2.a.r" 2 69.18238831123732 "2.2.17.1" [] [0, -2, 1, 1] [[2, 1], [3, 1], [19, -1]] "q-q^{3}+\\beta q^{5}+\\beta q^{7}+q^{9}+(4-\\beta )q^{11}+\\cdots" "8664.2.a.s" 2 69.18238831123732 "2.2.5.1" [] [0, -2, 2, 0] [[2, 1], [3, 1], [19, 1]] "q-q^{3}+(1+\\beta )q^{5}-\\beta q^{7}+q^{9}+(3+\\beta )q^{11}+\\cdots" "8664.2.a.t" 2 69.18238831123732 "2.2.5.1" [] [0, 2, -3, -1] [[2, 1], [3, -1], [19, -1]] "q+q^{3}+(-1-\\beta )q^{5}+(-1+\\beta )q^{7}+\\cdots" "8664.2.a.u" 2 69.18238831123732 "2.2.5.1" [] [0, 2, -2, 4] [[2, -1], [3, -1], [19, 1]] "q+q^{3}+(-1-\\beta )q^{5}+(2-\\beta )q^{7}+q^{9}+\\cdots" "8664.2.a.v" 2 69.18238831123732 "2.2.41.1" [] [0, 2, -1, -3] [[2, -1], [3, -1], [19, -1]] "q+q^{3}-\\beta q^{5}+(-2+\\beta )q^{7}+q^{9}+(2+\\cdots)q^{11}+\\cdots" "8664.2.a.w" 2 69.18238831123732 "2.2.5.1" [] [0, 2, 2, 0] [[2, -1], [3, -1], [19, -1]] "q+q^{3}+(1+\\beta )q^{5}-\\beta q^{7}+q^{9}+(3+\\beta )q^{11}+\\cdots" "8664.2.a.x" 3 69.18238831123732 "3.3.81.1" [] [0, -3, 0, 3] [[2, -1], [3, 1], [19, 1]] "q-q^{3}-\\beta _{1}q^{5}+(1+\\beta _{1})q^{7}+q^{9}+(-2+\\cdots)q^{11}+\\cdots" "8664.2.a.y" 3 69.18238831123732 "3.3.316.1" [] [0, -3, 3, 1] [[2, -1], [3, 1], [19, 1]] "q-q^{3}+(1-\\beta _{1})q^{5}-\\beta _{2}q^{7}+q^{9}+(-2\\beta _{1}+\\cdots)q^{11}+\\cdots" "8664.2.a.z" 3 69.18238831123732 "3.3.81.1" [] [0, 3, 0, 3] [[2, 1], [3, -1], [19, -1]] "q+q^{3}-\\beta _{1}q^{5}+(1+\\beta _{1})q^{7}+q^{9}+(-2+\\cdots)q^{11}+\\cdots" "8664.2.a.ba" 3 69.18238831123732 "3.3.316.1" [] [0, 3, 3, 1] [[2, 1], [3, -1], [19, 1]] "q+q^{3}+(1-\\beta _{1})q^{5}-\\beta _{2}q^{7}+q^{9}+(-2\\beta _{1}+\\cdots)q^{11}+\\cdots" "8664.2.a.bb" 4 69.18238831123732 "4.4.34025.1" [] [0, -4, -2, -1] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+\\beta _{2}q^{5}+(\\beta _{2}-\\beta _{3})q^{7}+q^{9}+\\cdots" "8664.2.a.bc" 4 69.18238831123732 "4.4.5225.1" [] [0, -4, 2, 2] [[2, 1], [3, 1], [19, -1]] "q-q^{3}+\\beta _{3}q^{5}+\\beta _{3}q^{7}+q^{9}+(-2+\\cdots)q^{11}+\\cdots" "8664.2.a.bd" 4 69.18238831123732 "4.4.34025.1" [] [0, 4, -2, -1] [[2, 1], [3, -1], [19, -1]] "q+q^{3}+\\beta _{2}q^{5}+(\\beta _{2}-\\beta _{3})q^{7}+q^{9}+\\cdots" "8664.2.a.be" 4 69.18238831123732 "4.4.5225.1" [] [0, 4, 2, 2] [[2, -1], [3, -1], [19, -1]] "q+q^{3}+\\beta _{3}q^{5}+\\beta _{3}q^{7}+q^{9}+(-2+\\cdots)q^{11}+\\cdots" "8664.2.a.bf" 6 69.18238831123732 "6.6.1528713.1" [] [0, -6, -3, 0] [[2, 1], [3, 1], [19, -1]] "q-q^{3}-\\beta _{1}q^{5}+(-1+2\\beta _{1}+\\beta _{2})q^{7}+\\cdots" "8664.2.a.bg" 6 69.18238831123732 "6.6.142368125.1" [] [0, -6, 1, 2] [[2, 1], [3, 1], [19, -1]] "q-q^{3}+\\beta _{5}q^{5}+\\beta _{2}q^{7}+q^{9}+(\\beta _{1}+\\beta _{2}+\\cdots)q^{11}+\\cdots" "8664.2.a.bh" 6 69.18238831123732 "6.6.65669049.1" [] [0, -6, 3, 6] [[2, -1], [3, 1], [19, -1]] "q-q^{3}+(1-\\beta _{1})q^{5}+(1+\\beta _{4})q^{7}+q^{9}+\\cdots" "8664.2.a.bi" 6 69.18238831123732 "6.6.1528713.1" [] [0, 6, -3, 0] [[2, -1], [3, -1], [19, 1]] "q+q^{3}-\\beta _{1}q^{5}+(-1+2\\beta _{1}+\\beta _{2})q^{7}+\\cdots" "8664.2.a.bj" 6 69.18238831123732 "6.6.142368125.1" [] [0, 6, 1, 2] [[2, -1], [3, -1], [19, -1]] "q+q^{3}+\\beta _{5}q^{5}+\\beta _{2}q^{7}+q^{9}+(\\beta _{1}+\\beta _{2}+\\cdots)q^{11}+\\cdots" "8664.2.a.bk" 6 69.18238831123732 "6.6.65669049.1" [] [0, 6, 3, 6] [[2, 1], [3, -1], [19, 1]] "q+q^{3}+(1-\\beta _{1})q^{5}+(1+\\beta _{4})q^{7}+q^{9}+\\cdots" "8664.2.a.bl" 8 69.18238831123732 "8.8.38020000000.1" [] [0, -8, -6, -2] [[2, 1], [3, 1], [19, 1]] "q-q^{3}+(-1-\\beta _{2})q^{5}-\\beta _{1}q^{7}+q^{9}+\\cdots" "8664.2.a.bm" 8 69.18238831123732 "8.8.38020000000.1" [] [0, 8, -6, -2] [[2, -1], [3, -1], [19, 1]] "q+q^{3}+(-1-\\beta _{2})q^{5}-\\beta _{1}q^{7}+q^{9}+\\cdots" "8664.2.a.bn" 9 69.18238831123732 NULL [] [0, -9, -3, -6] [[2, -1], [3, 1], [19, 1]] "q-q^{3}+(-\\beta _{1}-\\beta _{3}+\\beta _{5})q^{5}+(-1+\\cdots)q^{7}+\\cdots" "8664.2.a.bo" 9 69.18238831123732 NULL [] [0, -9, 3, 0] [[2, 1], [3, 1], [19, 1]] "q-q^{3}+\\beta _{1}q^{5}+\\beta _{7}q^{7}+q^{9}+(1-\\beta _{1}+\\cdots)q^{11}+\\cdots" "8664.2.a.bp" 9 69.18238831123732 NULL [] [0, 9, -3, -6] [[2, 1], [3, -1], [19, -1]] "q+q^{3}+(-\\beta _{1}-\\beta _{3}+\\beta _{5})q^{5}+(-1+\\cdots)q^{7}+\\cdots" "8664.2.a.bq" 9 69.18238831123732 NULL [] [0, 9, 3, 0] [[2, -1], [3, -1], [19, -1]] "q+q^{3}+\\beta _{1}q^{5}+\\beta _{7}q^{7}+q^{9}+(1-\\beta _{1}+\\cdots)q^{11}+\\cdots" "8664.2.a.br" 12 69.18238831123732 NULL [] [0, -12, 2, 4] [[2, -1], [3, 1], [19, 1]] "q-q^{3}+\\beta _{1}q^{5}+\\beta _{10}q^{7}+q^{9}+(1+\\beta _{4}+\\cdots)q^{11}+\\cdots" "8664.2.a.bs" 12 69.18238831123732 NULL [] [0, 12, 2, 4] [[2, 1], [3, -1], [19, 1]] "q+q^{3}+\\beta _{1}q^{5}+\\beta _{10}q^{7}+q^{9}+(1+\\beta _{4}+\\cdots)q^{11}+\\cdots" # 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.