# Classical modular forms downloaded from the LMFDB on 21 June 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1125/ # Query "{'level': 1125}" returned 47 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. "1125.1.g.a" 8 0.5614481392121277 "8.0.1024000000.2" [-15] [] [0, 0, 0, 0] NULL "q-\\beta _{5}q^{2}+(\\beta _{2}-\\beta _{6})q^{4}+(-\\beta _{3}+\\beta _{7})q^{8}+\\cdots" "1125.2.a.a" 2 8.983170227394043 "2.2.5.1" [] [] [-3, 0, 0, 5] -1 "q+(-1-\\beta )q^{2}+3\\beta q^{4}+(3-\\beta )q^{7}+\\cdots" "1125.2.a.b" 2 8.983170227394043 "2.2.5.1" [] [] [-1, 0, 0, 1] 1 "q-\\beta q^{2}+(-1+\\beta )q^{4}+\\beta q^{7}+(-1+\\cdots)q^{8}+\\cdots" "1125.2.a.c" 2 8.983170227394043 "2.2.5.1" [] [] [-1, 0, 0, 6] -1 "q-\\beta q^{2}+(-1+\\beta )q^{4}+3q^{7}+(-1+\\cdots)q^{8}+\\cdots" "1125.2.a.d" 2 8.983170227394043 "2.2.5.1" [] [] [1, 0, 0, -6] 1 "q+\\beta q^{2}+(-1+\\beta )q^{4}-3q^{7}+(1-2\\beta )q^{8}+\\cdots" "1125.2.a.e" 2 8.983170227394043 "2.2.5.1" [] [] [1, 0, 0, -1] 1 "q+\\beta q^{2}+(-1+\\beta )q^{4}-\\beta q^{7}+(1-2\\beta )q^{8}+\\cdots" "1125.2.a.f" 2 8.983170227394043 "2.2.5.1" [] [] [3, 0, 0, -5] -1 "q+(1+\\beta )q^{2}+3\\beta q^{4}+(-3+\\beta )q^{7}+\\cdots" "1125.2.a.g" 4 8.983170227394043 "4.4.1125.1" [-15] [] [-5, 0, 0, 0] 1 "q+(-1+\\beta _{2}+\\beta _{3})q^{2}+(1+\\beta _{1}-2\\beta _{2}+\\cdots)q^{4}+\\cdots" "1125.2.a.h" 4 8.983170227394043 "4.4.2525.1" [] [] [-3, 0, 0, -4] 1 "q+(-1-\\beta _{3})q^{2}+(1+\\beta _{1}-\\beta _{2}+\\beta _{3})q^{4}+\\cdots" "1125.2.a.i" 4 8.983170227394043 "4.4.2000.1" [] [] [0, 0, 0, 0] 1 "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}+(-\\beta _{1}-2\\beta _{3})q^{7}+\\cdots" "1125.2.a.j" 4 8.983170227394043 "4.4.2000.1" [] [] [0, 0, 0, 0] -1 "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}+(\\beta _{1}+2\\beta _{3})q^{7}+\\cdots" "1125.2.a.k" 4 8.983170227394043 "4.4.4400.1" [] [] [0, 0, 0, 0] -1 "q-\\beta _{3}q^{2}+(1-2\\beta _{2})q^{4}+(-\\beta _{1}-\\beta _{3})q^{7}+\\cdots" "1125.2.a.l" 4 8.983170227394043 "4.4.2525.1" [] [] [3, 0, 0, 4] -1 "q+(1+\\beta _{2}-\\beta _{3})q^{2}+(2-\\beta _{1}-\\beta _{3})q^{4}+\\cdots" "1125.2.a.m" 4 8.983170227394043 "4.4.1125.1" [-15] [] [5, 0, 0, 0] -1 "q+(1-\\beta _{2}-\\beta _{3})q^{2}+(1+\\beta _{1}-2\\beta _{2}+\\cdots)q^{4}+\\cdots" "1125.2.b.a" 4 8.983170227394043 "4.0.4400.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{2}+(-1-2\\beta _{2})q^{4}+(\\beta _{1}-\\beta _{3})q^{7}+\\cdots" "1125.2.b.b" 4 8.983170227394043 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}-\\beta _{3})q^{2}+3\\beta _{2}q^{4}+(\\beta _{1}+3\\beta _{3})q^{7}+\\cdots" "1125.2.b.c" 4 8.983170227394043 "4.0.125.1" [] [] [0, 0, 0, 0] NULL "q+\\beta_{3} q^{2}+\\beta_{2} q^{4}+(-\\beta_{3}+2\\beta_1)q^{7}+\\cdots" "1125.2.b.d" 4 8.983170227394043 "4.0.125.1" [] [] [0, 0, 0, 0] NULL "q+\\beta_{3} q^{2}+\\beta_{2} q^{4}+(\\beta_{3}-2\\beta_1)q^{7}+\\cdots" "1125.2.b.e" 4 8.983170227394043 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}-\\beta _{1}q^{7}+(2\\beta _{1}+\\cdots)q^{8}+\\cdots" "1125.2.b.f" 4 8.983170227394043 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}+3\\beta _{3}q^{7}+(2\\beta _{1}+\\cdots)q^{8}+\\cdots" "1125.2.b.g" 8 8.983170227394043 "8.0.1632160000.5" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{4}+\\beta _{7})q^{2}+(-1+\\beta _{2}+\\beta _{5}+\\beta _{6}+\\cdots)q^{4}+\\cdots" "1125.2.b.h" 8 8.983170227394043 "8.0.324000000.1" [-15] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{2}+(-2+\\beta _{2})q^{4}+(2\\beta _{1}+\\beta _{7})q^{8}+\\cdots" "1125.2.f.a" 32 8.983170227394043 NULL [] [] [0, 0, 0, 0] NULL NULL "1125.2.f.b" 32 8.983170227394043 NULL [] [] [0, 0, 0, 0] NULL NULL "1125.3.c.a" 32 30.654029740533403 NULL [] [] [0, 0, 0, 0] NULL NULL "1125.3.c.b" 32 30.654029740533403 NULL [] [] [0, 0, 0, 0] NULL NULL "1125.3.d.a" 16 30.654029740533403 NULL [] [] [-10, 0, 0, 0] NULL "q+(-1+\\beta _{2})q^{2}+(3-\\beta _{2}-\\beta _{3})q^{4}+\\cdots" "1125.3.d.b" 16 30.654029740533403 NULL [] [] [10, 0, 0, 0] NULL "q+(1-\\beta _{2})q^{2}+(3-\\beta _{2}-\\beta _{3})q^{4}+(-\\beta _{6}+\\cdots)q^{7}+\\cdots" "1125.3.d.c" 32 30.654029740533403 NULL [] [] [0, 0, 0, 0] NULL NULL "1125.4.a.a" 4 66.3771487564582 "4.4.1125.1" [-15] [] [-5, 0, 0, 0] -1 "q+(-1-\\beta _{2})q^{2}+(13-6\\beta _{1}+2\\beta _{2}+\\cdots)q^{4}+\\cdots" "1125.4.a.b" 4 66.3771487564582 "4.4.11025.1" [] [] [-4, 0, 0, 3] -1 "q+(-1+\\beta _{1}+\\beta _{3})q^{2}+(-1-3\\beta _{1}+\\cdots)q^{4}+\\cdots" "1125.4.a.c" 4 66.3771487564582 "4.4.12400.1" [] [] [0, 0, 0, 0] -1 "q+\\beta _{1}q^{2}+(-1+2\\beta _{2})q^{4}+(2\\beta _{1}-9\\beta _{3})q^{7}+\\cdots" "1125.4.a.d" 4 66.3771487564582 "4.4.11025.1" [] [] [4, 0, 0, -3] 1 "q+(2-\\beta _{1}+\\beta _{3})q^{2}+(2-\\beta _{1}-2\\beta _{2}+\\cdots)q^{4}+\\cdots" "1125.4.a.e" 4 66.3771487564582 "4.4.1125.1" [-15] [] [5, 0, 0, 0] 1 "q+(1+\\beta _{3})q^{2}+(7+7\\beta _{1}-\\beta _{2}+2\\beta _{3})q^{4}+\\cdots" "1125.4.a.f" 6 66.3771487564582 "6.6.497918125.1" [] [] [-7, 0, 0, 67] 1 "q+(-1+\\beta _{2})q^{2}+(5-\\beta _{2}+\\beta _{3}+\\beta _{4}+\\cdots)q^{4}+\\cdots" "1125.4.a.g" 6 66.3771487564582 "6.6.920588125.1" [] [] [-5, 0, 0, 28] 1 "q+(-1+\\beta _{2})q^{2}+(4-\\beta _{1}-2\\beta _{2}+\\beta _{5})q^{4}+\\cdots" "1125.4.a.h" 6 66.3771487564582 "6.6.3126413125.1" [] [] [-3, 0, 0, 44] -1 "q+(-1+\\beta _{1})q^{2}+(5-\\beta _{1}+\\beta _{3})q^{4}+\\cdots" "1125.4.a.i" 6 66.3771487564582 "6.6.3126413125.1" [] [] [3, 0, 0, -44] -1 "q+(1-\\beta _{1})q^{2}+(5-\\beta _{1}+\\beta _{3})q^{4}+(-7+\\cdots)q^{7}+\\cdots" "1125.4.a.j" 6 66.3771487564582 "6.6.920588125.1" [] [] [5, 0, 0, -28] 1 "q+(1-\\beta _{2})q^{2}+(4-\\beta _{1}-2\\beta _{2}+\\beta _{5})q^{4}+\\cdots" "1125.4.a.k" 6 66.3771487564582 "6.6.497918125.1" [] [] [7, 0, 0, -67] -1 "q+(1-\\beta _{2})q^{2}+(5-\\beta _{2}+\\beta _{3}+\\beta _{4})q^{4}+\\cdots" "1125.4.a.l" 8 66.3771487564582 NULL [] [] [-7, 0, 0, 19] -1 "q+(-1+\\beta _{1})q^{2}+(5-\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "1125.4.a.m" 8 66.3771487564582 NULL [] [] [0, 0, 0, 0] 1 "q+(-\\beta _{1}+\\beta _{4})q^{2}+(\\beta _{1}-\\beta _{4}-2\\beta _{5}+\\cdots)q^{4}+\\cdots" "1125.4.a.n" 8 66.3771487564582 NULL [] [] [0, 0, 0, 0] -1 "q+(\\beta _{1}-\\beta _{4})q^{2}+(\\beta _{1}-\\beta _{4}-2\\beta _{5})q^{4}+\\cdots" "1125.4.a.o" 8 66.3771487564582 NULL [] [] [0, 0, 0, 0] 1 "q-\\beta _{5}q^{2}+(5-\\beta _{3})q^{4}+(-2\\beta _{4}-\\beta _{5}+\\cdots)q^{7}+\\cdots" "1125.4.a.p" 8 66.3771487564582 NULL [] [] [7, 0, 0, -19] 1 "q+(1-\\beta _{1})q^{2}+(5-\\beta _{1}+\\beta _{2})q^{4}+(-3+\\cdots)q^{7}+\\cdots" "1125.4.a.q" 12 66.3771487564582 NULL [] [] [0, 0, 0, 0] -1 "q+\\beta _{1}q^{2}+(5+\\beta _{2})q^{4}+(-2\\beta _{1}-\\beta _{10}+\\cdots)q^{7}+\\cdots" "1125.4.a.r" 12 66.3771487564582 NULL [] [] [0, 0, 0, 0] 1 "q+\\beta _{1}q^{2}+(5+\\beta _{2})q^{4}+(2\\beta _{1}+\\beta _{10}+\\cdots)q^{7}+\\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. #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.