# Classical modular forms downloaded from the LMFDB on 15 April 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/83/ # Query "{'level': 83}" returned 48 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. "83.1.b.a" 1 0.0414223960485392 "1.1.1.1" [-83] [] [0, -1, 0, -1] NULL "q-q^{3}+q^{4}-q^{7}-q^{11}-q^{12}+q^{16}+\\cdots" "83.2.a.a" 1 0.6627583367766272 "1.1.1.1" [] [] [-1, -1, -2, -3] 1 "q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}-3q^{7}+\\cdots" "83.2.a.b" 6 0.6627583367766272 "6.6.9059636.1" [] [] [1, 1, 2, 3] -1 "q+(-\\beta _{1}-\\beta _{2})q^{2}-\\beta _{3}q^{3}+(1-\\beta _{5})q^{4}+\\cdots" "83.2.c.a" 240 0.6627583367766272 NULL [] [] [-38, -37, -35, -33] NULL NULL "83.3.b.a" 3 2.2615861941904645 "3.3.2241.1" [-83] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}+\\beta _{2})q^{3}+4q^{4}+(-3\\beta _{1}-2\\beta _{2})q^{7}+\\cdots" "83.3.b.b" 10 2.2615861941904645 NULL [] [] [0, -6, 0, 12] NULL "q+\\beta _{1}q^{2}+(-1+\\beta _{2})q^{3}+(-3+\\beta _{7}+\\cdots)q^{4}+\\cdots" "83.3.d.a" 520 2.2615861941904645 NULL [] [] [-41, -35, -41, -53] NULL NULL "83.4.a.a" 7 4.897158530476471 NULL [] [] [-5, -11, -17, -46] -1 "q+(-1-\\beta _{4})q^{2}+(-1+\\beta _{1}+\\beta _{4})q^{3}+\\cdots" "83.4.a.b" 13 4.897158530476471 NULL [] [] [5, 7, 33, 52] 1 "q+\\beta _{1}q^{2}+(1+\\beta _{8})q^{3}+(6+\\beta _{2})q^{4}+\\cdots" "83.4.c.a" 800 4.897158530476471 NULL [] [] [-41, -37, -57, -47] NULL NULL "83.5.b.a" 3 8.579706935959079 "3.3.2241.1" [-83] [] [0, 0, 0, 0] NULL "q+(2\\beta _{1}-3\\beta _{2})q^{3}+2^{4}q^{4}+(21\\beta _{1}-11\\beta _{2})q^{7}+\\cdots" "83.5.b.b" 24 8.579706935959079 NULL [] [] [0, 6, 0, -112] NULL NULL "83.5.d.a" 1080 8.579706935959079 NULL [] [] [-41, -47, -41, 71] NULL NULL "83.6.a.a" 14 13.311857044477392 NULL [] [] [-13, -17, -162, -247] 1 "q+(-1+\\beta _{1})q^{2}+(-1+\\beta _{7})q^{3}+(10+\\cdots)q^{4}+\\cdots" "83.6.a.b" 20 13.311857044477392 NULL [] [] [7, 37, 88, 439] -1 "q+\\beta _{1}q^{2}+(2+\\beta _{5})q^{3}+(19+\\beta _{2})q^{4}+\\cdots" "83.6.c.a" 1360 13.311857044477392 NULL [] [] [-35, -61, 33, -233] NULL NULL "83.7.b.a" 1 19.094488940447633 "1.1.1.1" [-83] [] [0, -29, 0, -61] NULL "q-29q^{3}+2^{6}q^{4}-61q^{7}+112q^{9}+\\cdots" "83.7.b.b" 2 19.094488940447633 "2.2.249.1" [-83] [] [0, 29, 0, 61] NULL "q+(15+\\beta )q^{3}+2^{6}q^{4}+(23-15\\beta )q^{7}+\\cdots" "83.7.b.c" 38 19.094488940447633 NULL [] [] [0, 18, 0, 262] NULL NULL "83.7.d.a" 1640 19.094488940447633 NULL [] [] [-41, -59, -41, -303] NULL NULL "83.8.a.a" 21 25.927957115182107 NULL [] [] [-17, -95, -502, -3273] -1 NULL "83.8.a.b" 27 25.927957115182107 NULL [] [] [23, 67, 748, 1529] 1 NULL "83.8.c.a" 1920 25.927957115182107 NULL [] [] [-47, -13, -287, 1703] NULL NULL "83.9.b.a" 3 33.81242463507356 "3.3.2241.1" [-83] [] [0, 0, 0, 0] NULL "q+(-47\\beta _{1}-2\\beta _{2})q^{3}+2^{8}q^{4}+(-846\\beta _{1}+\\cdots)q^{7}+\\cdots" "83.9.b.b" 52 33.81242463507356 NULL [] [] [0, -114, 0, 2028] NULL NULL "83.9.d.a" 2200 33.81242463507356 NULL [] [] [-41, 73, -41, -2069] NULL NULL "83.10.a.a" 28 42.74797440160147 NULL [] [] [-49, -317, -3467, -13836] 1 NULL "83.10.a.b" 34 42.74797440160147 NULL [] [] [31, 169, 2783, 19778] -1 NULL "83.10.c.a" 2480 42.74797440160147 NULL [] [] [-23, 107, 643, -5983] NULL NULL "83.11.b.a" 3 52.7346519719191 "3.3.2241.1" [-83] [] [0, 0, 0, 0] NULL "q+(-139\\beta _{1}+6\\beta _{2})q^{3}+2^{10}q^{4}+(5922\\beta _{1}+\\cdots)q^{7}+\\cdots" "83.11.b.b" 66 52.7346519719191 NULL [] [] [0, 42, 0, -22288] NULL NULL "83.11.d.a" 2760 52.7346519719191 NULL [] [] [-41, -83, -41, 22247] NULL NULL "83.12.a.a" 34 63.77248398644588 NULL [] [] [-41, -719, -19862, -97477] -1 NULL "83.12.a.b" 40 63.77248398644588 NULL [] [] [119, 739, 11388, 137821] 1 NULL "83.12.c.a" 3040 63.77248398644588 NULL [] [] [-71, -565, -1227, -6897] NULL NULL "83.13.b.a" 1 75.8614868339317 "1.1.1.1" [-83] [] [0, -617, 0, -231577] NULL "q-617q^{3}+2^{12}q^{4}-231577q^{7}+\\cdots" "83.13.b.b" 2 75.8614868339317 "2.2.249.1" [-83] [] [0, 617, 0, 231577] NULL "q+(294-29\\beta )q^{3}+2^{12}q^{4}+(116246+\\cdots)q^{7}+\\cdots" "83.13.b.c" 80 75.8614868339317 NULL [] [] [0, 918, 0, 103918] NULL NULL "83.14.a.a" 42 89.00167103008754 NULL [] [] [-193, -2057, -78072, -907547] 1 NULL "83.14.a.b" 48 89.00167103008754 NULL [] [] [127, 2317, 78178, 739539] -1 NULL "83.15.b.a" 3 103.19304356608197 "3.3.2241.1" [-83] [] [0, 0, 0, 0] NULL "q+(-63\\beta _{1}+152\\beta _{2})q^{3}+2^{14}q^{4}+\\cdots" "83.15.b.b" 94 103.19304356608197 NULL [] [] [0, -2238, 0, -386388] NULL NULL "83.16.a.a" 48 118.43560923293394 NULL [] [] [-473, -5843, -442447, -7927854] -1 NULL "83.16.a.b" 54 118.43560923293394 NULL [] [] [167, 7279, 338803, 3601748] 1 NULL "83.18.a.a" 55 152.07433249598452 NULL [] [] [-241, -11321, -1009352, -50694303] 1 NULL "83.18.a.b" 61 152.07433249598452 NULL [] [] [1039, 28045, 2896898, 30012911] -1 NULL "83.20.a.a" 62 189.91785814179605 NULL [] [] [-1481, -94199, -6266132, -202919777] -1 NULL "83.20.a.b" 68 189.91785814179605 NULL [] [] [1079, 23899, 13265118, 362030721] 1 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.