# Classical modular forms downloaded from the LMFDB on 13 June 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/2312/ # Query "{'level': 2312}" returned 75 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. "2312.1.e.a" 2 1.1538383092075015 "2.0.8.1" [-8] [] [-2, 0, 0, 0] NULL "q-q^{2}-\\beta q^{3}+q^{4}+\\beta q^{6}-q^{8}-q^{9}+\\cdots" "2312.1.e.b" 4 1.1538383092075015 "4.0.2048.2" [-8] [] [4, 0, 0, 0] NULL "q+q^{2}-\\beta _{1}q^{3}+q^{4}-\\beta _{1}q^{6}+q^{8}+\\cdots" "2312.1.f.a" 1 1.1538383092075015 "1.1.1.1" [-8, -136] [17] [1, 0, 0, 0] NULL "q+q^{2}+q^{4}+q^{8}-q^{9}+q^{16}-q^{18}+\\cdots" "2312.1.f.b" 2 1.1538383092075015 "2.2.8.1" [-8] [] [2, 0, 0, 0] NULL "q+q^{2}-\\beta q^{3}+q^{4}-\\beta q^{6}+q^{8}+q^{9}+\\cdots" "2312.1.f.c" 4 1.1538383092075015 "4.4.2048.1" [-8] [] [-4, 0, 0, 0] NULL "q-q^{2}-\\beta _{1}q^{3}+q^{4}+\\beta _{1}q^{6}-q^{8}+\\cdots" "2312.1.j.a" 2 1.1538383092075015 "2.0.4.1" [-8, -136] [17] [0, 0, 0, 0] NULL "q+i q^{2}-q^{4}-i q^{8}+i q^{9}+q^{16}+\\cdots" "2312.1.j.b" 2 1.1538383092075015 "2.0.4.1" [-8] [] [0, 2, 0, 0] NULL "q+i q^{2}+(-i+1)q^{3}-q^{4}+(i+1)q^{6}+\\cdots" "2312.1.j.c" 8 1.1538383092075015 "8.0.16777216.1" [-8] [] [0, 0, 0, 0] NULL "q+\\zeta_{16}^{4}q^{2}+(\\zeta_{16}^{5}-\\zeta_{16}^{7})q^{3}-q^{4}+\\cdots" "2312.1.p.a" 4 1.1538383092075015 "4.0.256.1" [-8] [] [0, -4, 0, 0] NULL "q+\\zeta_{8}^{3}q^{2}+(-1-\\zeta_{8})q^{3}-\\zeta_{8}^{2}q^{4}+\\cdots" "2312.1.p.b" 4 1.1538383092075015 "4.0.256.1" [-8] [] [0, 0, 0, 0] NULL "q+\\zeta_{8}^{3}q^{2}+(-\\zeta_{8}^{2}+\\zeta_{8}^{3})q^{3}-\\zeta_{8}^{2}q^{4}+\\cdots" "2312.1.p.c" 4 1.1538383092075015 "4.0.256.1" [-8, -136] [17] [0, 0, 0, 0] NULL "q-\\zeta_{8}^{3}q^{2}-\\zeta_{8}^{2}q^{4}-\\zeta_{8}q^{8}-\\zeta_{8}q^{9}+\\cdots" "2312.1.p.d" 4 1.1538383092075015 "4.0.256.1" [-8] [] [0, 4, 0, 0] NULL "q+\\zeta_{8}^{3}q^{2}+(1+\\zeta_{8})q^{3}-\\zeta_{8}^{2}q^{4}+(-1+\\cdots)q^{6}+\\cdots" "2312.1.p.e" 8 1.1538383092075015 "8.0.16777216.1" [-8] [] [0, 0, 0, 0] NULL "q+\\zeta_{16}^{6}q^{2}+(-\\zeta_{16}^{3}-\\zeta_{16}^{7})q^{3}+\\cdots" "2312.1.q.a" 16 1.1538383092075015 "16.0.18446744073709551616.1" [] [136] [0, 0, 0, 0] NULL "q-\\zeta_{32}^{2}q^{2}+(-\\zeta_{32}^{3}-\\zeta_{32}^{11})q^{3}+\\cdots" "2312.1.x.a" 16 1.1538383092075015 "16.0.2862423051509815793.1" [-8] [] [-1, -2, 0, 0] NULL "q-\\zeta_{34}q^{2}+(-\\zeta_{34}^{5}-\\zeta_{34}^{7})q^{3}+\\zeta_{34}^{2}q^{4}+\\cdots" "2312.1.y.a" 16 1.1538383092075015 "16.0.2862423051509815793.1" [-8] [] [1, 0, 0, 0] NULL "q+\\zeta_{34}^{5}q^{2}+(\\zeta_{34}+\\zeta_{34}^{8})q^{3}+\\zeta_{34}^{10}q^{4}+\\cdots" "2312.1.be.a" 32 1.1538383092075015 "32.0.35190667333271321019306672876612934335729762304.1" [-8] [] [0, 2, 0, 0] NULL "q-\\zeta_{68}^{13}q^{2}+(\\zeta_{68}^{14}+\\zeta_{68}^{23})q^{3}+\\cdots" "2312.1.bg.a" 64 1.1538383092075015 NULL [-8] [] [0, 0, 0, 0] NULL "q+\\zeta_{136}^{35}q^{2}+(\\zeta_{136}^{7}-\\zeta_{136}^{22})q^{3}+\\cdots" "2312.2.a.a" 1 18.461412947320024 "1.1.1.1" [] [] [0, -2, 0, 0] -1 "q-2q^{3}+q^{9}-2q^{11}-6q^{13}+4q^{19}+\\cdots" "2312.2.a.b" 1 18.461412947320024 "1.1.1.1" [] [] [0, -2, 0, 2] 1 "q-2q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\\cdots" "2312.2.a.c" 1 18.461412947320024 "1.1.1.1" [] [] [0, 2, 0, -2] 1 "q+2q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\\cdots" "2312.2.a.d" 1 18.461412947320024 "1.1.1.1" [] [] [0, 2, 2, 2] -1 "q+2q^{3}+2q^{5}+2q^{7}+q^{9}+6q^{11}+\\cdots" "2312.2.a.e" 2 18.461412947320024 "2.2.13.1" [] [] [0, -3, 1, 1] 1 "q+(-1-\\beta )q^{3}+\\beta q^{5}+\\beta q^{7}+(1+3\\beta )q^{9}+\\cdots" "2312.2.a.f" 2 18.461412947320024 "2.2.5.1" [] [] [0, -1, -1, -5] 1 "q-\\beta q^{3}+(1-3\\beta )q^{5}+(-3+\\beta )q^{7}+\\cdots" "2312.2.a.g" 2 18.461412947320024 "2.2.8.1" [] [] [0, 0, 0, 0] -1 "q-\\beta q^{5}+\\beta q^{7}-3q^{9}-2\\beta q^{11}+2q^{13}+\\cdots" "2312.2.a.h" 2 18.461412947320024 "2.2.8.1" [] [] [0, 0, 0, 0] -1 "q+\\beta q^{3}-3\\beta q^{5}-3\\beta q^{7}-q^{9}-\\beta q^{11}+\\cdots" "2312.2.a.i" 2 18.461412947320024 "2.2.8.1" [] [] [0, 0, 0, 0] 1 "q+\\beta q^{3}-\\beta q^{5}+\\beta q^{7}-q^{9}-\\beta q^{11}+\\cdots" "2312.2.a.j" 2 18.461412947320024 "2.2.8.1" [] [] [0, 0, 0, 0] -1 "q+\\beta q^{3}+2\\beta q^{5}+2\\beta q^{7}-q^{9}-\\beta q^{11}+\\cdots" "2312.2.a.k" 2 18.461412947320024 "2.2.8.1" [] [] [0, 0, 0, 0] -1 "q+2\\beta q^{3}+\\beta q^{5}-2\\beta q^{7}+5q^{9}-2\\beta q^{11}+\\cdots" "2312.2.a.l" 2 18.461412947320024 "2.2.5.1" [] [] [0, 1, 1, 5] -1 "q+\\beta q^{3}+(-1+3\\beta )q^{5}+(3-\\beta )q^{7}+\\cdots" "2312.2.a.m" 2 18.461412947320024 "2.2.5.1" [] [] [0, 2, -4, -2] 1 "q+(1+\\beta )q^{3}-2q^{5}+(-1-\\beta )q^{7}+(3+\\cdots)q^{9}+\\cdots" "2312.2.a.n" 2 18.461412947320024 "2.2.13.1" [] [] [0, 3, -1, -1] -1 "q+(1+\\beta )q^{3}-\\beta q^{5}-\\beta q^{7}+(1+3\\beta )q^{9}+\\cdots" "2312.2.a.o" 3 18.461412947320024 "3.3.81.1" [] [] [0, -3, -6, -6] 1 "q+(-1-2\\beta _{1}+\\beta _{2})q^{3}+(-2-\\beta _{1}+\\cdots)q^{5}+\\cdots" "2312.2.a.p" 3 18.461412947320024 "3.3.81.1" [] [] [0, -3, 6, -3] 1 "q+(-1-\\beta _{1})q^{3}+(2+\\beta _{1}-\\beta _{2})q^{5}+\\cdots" "2312.2.a.q" 3 18.461412947320024 "3.3.81.1" [] [] [0, 3, -6, 3] -1 "q+(1+\\beta _{1})q^{3}+(-2-\\beta _{1}+\\beta _{2})q^{5}+\\cdots" "2312.2.a.r" 3 18.461412947320024 "3.3.81.1" [] [] [0, 3, 6, 6] -1 "q+(1+2\\beta _{1}-\\beta _{2})q^{3}+(2+\\beta _{1})q^{5}+(2+\\cdots)q^{7}+\\cdots" "2312.2.a.s" 4 18.461412947320024 "4.4.2048.1" [] [] [0, 0, 0, 0] 1 "q+\\beta _{1}q^{3}+\\beta _{3}q^{5}+(-2\\beta _{1}-\\beta _{3})q^{7}+\\cdots" "2312.2.a.t" 4 18.461412947320024 "4.4.2048.1" [] [] [0, 0, 0, 0] 1 "q+(\\beta _{1}+\\beta _{3})q^{3}+(-2\\beta _{1}+\\beta _{3})q^{5}+(\\beta _{1}+\\cdots)q^{7}+\\cdots" "2312.2.a.u" 6 18.461412947320024 "6.6.3418281.1" [] [] [0, 0, -6, -3] 1 "q-\\beta _{1}q^{3}+(-1-\\beta _{3})q^{5}+(-\\beta _{1}-2\\beta _{2}+\\cdots)q^{7}+\\cdots" "2312.2.a.v" 6 18.461412947320024 "6.6.3418281.1" [] [] [0, 0, 6, 3] -1 "q+\\beta _{1}q^{3}+(1+\\beta _{3})q^{5}+(\\beta _{1}+2\\beta _{2}+\\beta _{3}+\\cdots)q^{7}+\\cdots" "2312.2.a.w" 12 18.461412947320024 NULL [] [] [0, 0, 0, 0] -1 "q+\\beta _{1}q^{3}-\\beta _{10}q^{5}+\\beta _{6}q^{7}+(2+\\beta _{3}+\\cdots)q^{9}+\\cdots" "2312.2.b.a" 2 18.461412947320024 "2.0.8.1" [] [] [0, 0, 0, 0] NULL "q+2\\beta q^{3}+\\beta q^{5}+2\\beta q^{7}-5q^{9}+2\\beta q^{11}+\\cdots" "2312.2.b.b" 2 18.461412947320024 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+\\beta q^{3}-q^{9}-\\beta q^{11}-6 q^{13}-4 q^{19}+\\cdots" "2312.2.b.c" 2 18.461412947320024 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+\\beta q^{3}+\\beta q^{5}-\\beta q^{7}-q^{9}-3\\beta q^{11}+\\cdots" "2312.2.b.d" 2 18.461412947320024 "2.0.8.1" [] [] [0, 0, 0, 0] NULL "q+\\beta q^{3}+2\\beta q^{5}-2\\beta q^{7}+q^{9}+\\beta q^{11}+\\cdots" "2312.2.b.e" 2 18.461412947320024 "2.0.8.1" [] [] [0, 0, 0, 0] NULL "q+\\beta q^{3}-\\beta q^{5}-\\beta q^{7}+q^{9}+\\beta q^{11}+\\cdots" "2312.2.b.f" 2 18.461412947320024 "2.0.8.1" [] [] [0, 0, 0, 0] NULL "q+\\beta q^{3}-3\\beta q^{5}+3\\beta q^{7}+q^{9}+\\beta q^{11}+\\cdots" "2312.2.b.g" 4 18.461412947320024 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}+\\beta _{1}q^{7}+(-3+\\beta _{3})q^{9}+\\cdots" "2312.2.b.h" 4 18.461412947320024 "4.0.2704.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}-\\beta _{2})q^{3}-\\beta _{1}q^{5}+\\beta _{1}q^{7}+(-4+\\cdots)q^{9}+\\cdots" "2312.2.b.i" 4 18.461412947320024 "4.0.2048.2" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}+\\beta _{3})q^{3}+(\\beta _{1}-2\\beta _{3})q^{5}+(\\beta _{1}+\\cdots)q^{7}+\\cdots" "2312.2.b.j" 4 18.461412947320024 "4.0.2048.2" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}-\\beta _{3}q^{5}+(2\\beta _{1}-\\beta _{3})q^{7}+(1+\\cdots)q^{9}+\\cdots" "2312.2.b.k" 4 18.461412947320024 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+(3\\beta _{1}+\\beta _{3})q^{5}+(\\beta _{1}+3\\beta _{3})q^{7}+\\cdots" "2312.2.b.l" 6 18.461412947320024 "6.0.419904.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}-\\beta _{3}-\\beta _{5})q^{3}+(\\beta _{1}-2\\beta _{3})q^{5}+\\cdots" "2312.2.b.m" 6 18.461412947320024 "6.0.419904.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}+\\beta _{3}+\\beta _{5})q^{3}+(-\\beta _{1}-2\\beta _{3}+\\cdots)q^{5}+\\cdots" "2312.2.b.n" 12 18.461412947320024 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{10}q^{5}+\\beta _{6}q^{7}+(-2+\\beta _{3}+\\cdots)q^{9}+\\cdots" "2312.2.b.o" 12 18.461412947320024 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+(-\\beta _{1}-\\beta _{3}-\\beta _{4}-\\beta _{7}-\\beta _{10}+\\cdots)q^{5}+\\cdots" "2312.4.a.a" 1 136.4124159332723 "1.1.1.1" [] [] [0, 4, 2, -24] 1 "q+4q^{3}+2q^{5}-24q^{7}-11q^{9}+44q^{11}+\\cdots" "2312.4.a.b" 2 136.4124159332723 "2.2.12.1" [] [] [0, -4, 12, 36] 1 "q+(-2+\\beta )q^{3}+(6-2\\beta )q^{5}+(18-3\\beta )q^{7}+\\cdots" "2312.4.a.c" 3 136.4124159332723 "3.3.1556.1" [] [] [0, -8, -2, -12] -1 "q+(-3+\\beta _{2})q^{3}+(-1+\\beta _{1})q^{5}+(-4+\\cdots)q^{7}+\\cdots" "2312.4.a.d" 3 136.4124159332723 "3.3.8396.1" [] [] [0, 4, 8, 2] -1 "q+(1-\\beta _{2})q^{3}+(3-\\beta _{1}+2\\beta _{2})q^{5}+(2+\\cdots)q^{7}+\\cdots" "2312.4.a.e" 4 136.4124159332723 "4.4.550476.1" [] [] [0, 2, -8, 22] 1 "q+\\beta _{1}q^{3}+(-1+\\beta _{1}+\\beta _{2}+2\\beta _{3})q^{5}+\\cdots" "2312.4.a.f" 6 136.4124159332723 NULL [] [] [0, -7, -3, -7] -1 "q+(-1-\\beta _{1})q^{3}+(-1+\\beta _{2})q^{5}+(-1+\\cdots)q^{7}+\\cdots" "2312.4.a.g" 6 136.4124159332723 NULL [] [] [0, -1, -13, -1] -1 "q+\\beta _{3}q^{3}+(-2+\\beta _{2})q^{5}+\\beta _{1}q^{7}+(10+\\cdots)q^{9}+\\cdots" "2312.4.a.h" 6 136.4124159332723 NULL [] [] [0, 0, 0, 0] -1 "q+\\beta _{1}q^{3}+\\beta _{4}q^{5}+(-\\beta _{1}+\\beta _{2}+\\beta _{4}+\\cdots)q^{7}+\\cdots" "2312.4.a.i" 6 136.4124159332723 NULL [] [] [0, 1, 13, 1] 1 "q-\\beta _{3}q^{3}+(2-\\beta _{2})q^{5}-\\beta _{1}q^{7}+(10+\\cdots)q^{9}+\\cdots" "2312.4.a.j" 6 136.4124159332723 NULL [] [] [0, 7, 3, 7] 1 "q+(1+\\beta _{1})q^{3}+(1-\\beta _{2})q^{5}+(1+\\beta _{5})q^{7}+\\cdots" "2312.4.a.k" 8 136.4124159332723 NULL [] [] [0, 0, 0, 0] 1 "q-\\beta _{2}q^{3}+\\beta _{4}q^{5}+(-\\beta _{1}-\\beta _{2}-\\beta _{4}+\\cdots)q^{7}+\\cdots" "2312.4.a.l" 14 136.4124159332723 NULL [] [] [0, 0, 0, 0] 1 "q+\\beta _{3}q^{3}+\\beta _{8}q^{5}+(-\\beta _{1}+\\beta _{4})q^{7}+\\cdots" "2312.4.a.m" 14 136.4124159332723 NULL [] [] [0, 0, 0, 0] -1 "q+\\beta _{3}q^{3}+(-\\beta _{1}+\\beta _{9})q^{5}+(\\beta _{1}+\\beta _{5}+\\cdots)q^{7}+\\cdots" "2312.4.a.n" 18 136.4124159332723 NULL [] [] [0, -18, 0, -51] -1 "q+(-1+\\beta _{1})q^{3}-\\beta _{5}q^{5}+(-3-\\beta _{4}+\\cdots)q^{7}+\\cdots" "2312.4.a.o" 18 136.4124159332723 NULL [] [] [0, 0, -30, -33] -1 "q+\\beta _{1}q^{3}+(-2-\\beta _{11})q^{5}+(-2-\\beta _{1}+\\cdots)q^{7}+\\cdots" "2312.4.a.p" 18 136.4124159332723 NULL [] [] [0, 0, 30, 33] 1 "q-\\beta _{1}q^{3}+(2+\\beta _{11})q^{5}+(2+\\beta _{1}+\\beta _{12}+\\cdots)q^{7}+\\cdots" "2312.4.a.q" 18 136.4124159332723 NULL [] [] [0, 18, 0, 51] 1 "q+(1-\\beta _{1})q^{3}+\\beta _{5}q^{5}+(3+\\beta _{4}-\\beta _{10}+\\cdots)q^{7}+\\cdots" "2312.4.a.r" 24 136.4124159332723 NULL [] [] [0, 0, 0, 0] -1 NULL "2312.4.a.s" 28 136.4124159332723 NULL [] [] [0, 0, 0, 0] 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.