# Classical modular forms downloaded from the LMFDB on 15 June 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/340/ # Query "{'level': 340}" returned 112 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. "340.1.ba.a" 4 0.16968210429522082 "4.0.256.1" [-4] [] [0, 0, 0, 0] NULL "q-\\zeta_{8}q^{2}+\\zeta_{8}^{2}q^{4}-\\zeta_{8}^{2}q^{5}-\\zeta_{8}^{3}q^{8}+\\cdots" "340.1.ba.b" 4 0.16968210429522082 "4.0.256.1" [-4] [] [0, 0, 0, 0] NULL "q+\\zeta_{8}q^{2}+\\zeta_{8}^{2}q^{4}-\\zeta_{8}^{3}q^{5}+\\zeta_{8}^{3}q^{8}+\\cdots" "340.1.bc.a" 8 0.16968210429522082 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+\\zeta_{16}^{5}q^{2}-\\zeta_{16}^{2}q^{4}-\\zeta_{16}^{6}q^{5}+\\cdots" "340.1.bj.a" 8 0.16968210429522082 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+\\zeta_{16}^{5}q^{2}-\\zeta_{16}^{2}q^{4}+\\zeta_{16}^{3}q^{5}+\\cdots" "340.2.a.a" 1 2.714913668723533 "1.1.1.1" [] [] [0, 0, -1, -4] 1 "q-q^{5}-4q^{7}-3q^{9}+2q^{11}-6q^{13}+\\cdots" "340.2.a.b" 3 2.714913668723533 "3.3.404.1" [] [] [0, 0, 3, 0] -1 "q-\\beta _{2}q^{3}+q^{5}-\\beta _{2}q^{7}+(2-\\beta _{1}+\\beta _{2})q^{9}+\\cdots" "340.2.c.a" 6 2.714913668723533 "6.0.37161216.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}-\\beta _{4})q^{3}+\\beta _{4}q^{5}+(\\beta _{1}+\\beta _{4})q^{7}+\\cdots" "340.2.e.a" 8 2.714913668723533 "8.0.2058981376.2" [] [] [0, 0, -2, 0] NULL "q+\\beta _{6}q^{3}+\\beta _{2}q^{5}+(\\beta _{5}-\\beta _{6}+\\beta _{7})q^{7}+\\cdots" "340.2.g.a" 8 2.714913668723533 "8.0.4441101041664.4" [] [] [0, 0, 0, 0] NULL "q+\\beta _{6}q^{3}-\\beta _{1}q^{5}+\\beta _{2}q^{7}-\\beta _{3}q^{9}+\\cdots" "340.2.i.a" 2 2.714913668723533 "2.0.4.1" [-4] [] [2, 0, 2, 0] NULL "q+(-i+1)q^{2}-2 i q^{4}+(2 i+1)q^{5}+\\cdots" "340.2.i.b" 2 2.714913668723533 "2.0.4.1" [-4] [] [2, 0, 4, 0] NULL "q+(-i+1)q^{2}-2 i q^{4}+(-i+2)q^{5}+\\cdots" "340.2.i.c" 96 2.714913668723533 NULL [] [] [-4, 0, -8, 0] NULL NULL "340.2.l.a" 8 2.714913668723533 "8.0.18939904.2" [] [] [4, 0, -16, 0] NULL "q+(1-\\beta _{2}+\\beta _{5}-\\beta _{7})q^{2}+(-1-2\\beta _{3}+\\cdots)q^{3}+\\cdots" "340.2.l.b" 88 2.714913668723533 NULL [] [] [-4, 0, 16, 0] NULL NULL "340.2.m.a" 2 2.714913668723533 "2.0.4.1" [] [] [0, -4, -4, 0] NULL "q+(2 i-2)q^{3}+(i-2)q^{5}-5 i q^{9}+\\cdots" "340.2.m.b" 2 2.714913668723533 "2.0.4.1" [] [] [0, 4, 2, 0] NULL "q+(-2 i+2)q^{3}+(-2 i+1)q^{5}+\\cdots" "340.2.m.c" 12 2.714913668723533 NULL [] [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{3}+(\\beta _{3}-\\beta _{9}+\\beta _{10})q^{5}+(-\\beta _{1}+\\cdots)q^{7}+\\cdots" "340.2.o.a" 12 2.714913668723533 NULL [] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{6}+\\beta _{7}+\\beta _{10})q^{3}+\\beta _{3}q^{5}+\\cdots" "340.2.r.a" 2 2.714913668723533 "2.0.4.1" [-4] [] [2, 0, -4, 0] NULL "q+(i+1)q^{2}+2 i q^{4}+(i-2)q^{5}+\\cdots" "340.2.r.b" 2 2.714913668723533 "2.0.4.1" [-4] [] [2, 0, 4, 0] NULL "q+(i+1)q^{2}+2 i q^{4}+(-i+2)q^{5}+\\cdots" "340.2.r.c" 96 2.714913668723533 NULL [] [] [-8, 0, 0, 0] NULL NULL "340.2.s.a" 2 2.714913668723533 "2.0.4.1" [-4] [] [-2, 0, -2, 0] NULL "q+(-i-1)q^{2}+2 i q^{4}+(2 i-1)q^{5}+\\cdots" "340.2.s.b" 2 2.714913668723533 "2.0.4.1" [-4] [] [-2, 0, 4, 0] NULL "q+(-i-1)q^{2}+2 i q^{4}+(i+2)q^{5}+\\cdots" "340.2.s.c" 96 2.714913668723533 NULL [] [] [4, 0, -8, 0] NULL NULL "340.2.u.a" 24 2.714913668723533 NULL [] [] [0, 0, 0, 0] NULL NULL "340.2.w.a" 4 2.714913668723533 "4.0.256.1" [-4] [] [0, 0, -4, 0] NULL "q+(\\zeta_{8}-\\zeta_{8}^{3})q^{2}+2q^{4}+(-1+2\\zeta_{8}^{2}+\\cdots)q^{5}+\\cdots" "340.2.w.b" 4 2.714913668723533 "4.0.256.1" [-4] [] [0, 0, 0, 0] NULL "q+(-\\zeta_{8}+\\zeta_{8}^{3})q^{2}+2q^{4}+(-\\zeta_{8}+2\\zeta_{8}^{3})q^{5}+\\cdots" "340.2.w.c" 8 2.714913668723533 "8.0.205520896.4" [] [] [0, 0, -8, 0] NULL "q+(-\\beta _{1}+\\beta _{3})q^{2}+\\beta _{7}q^{3}+2q^{4}+(-1+\\cdots)q^{5}+\\cdots" "340.2.w.d" 184 2.714913668723533 NULL [] [] [-8, 0, 0, 0] NULL NULL "340.2.z.a" 4 2.714913668723533 "4.0.256.1" [-4] [] [0, 0, 0, 0] NULL "q+(-\\zeta_{8}-\\zeta_{8}^{3})q^{2}-2q^{4}+(\\zeta_{8}+2\\zeta_{8}^{3})q^{5}+\\cdots" "340.2.z.b" 4 2.714913668723533 "4.0.256.1" [-4] [] [0, 0, 4, 0] NULL "q+(\\zeta_{8}+\\zeta_{8}^{3})q^{2}-2q^{4}+(1+2\\zeta_{8}^{2}+\\cdots)q^{5}+\\cdots" "340.2.z.c" 8 2.714913668723533 "8.0.205520896.4" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}+\\beta _{3})q^{2}+\\beta _{5}q^{3}-2q^{4}+(\\beta _{1}+\\cdots)q^{5}+\\cdots" "340.2.z.d" 184 2.714913668723533 NULL [] [] [0, 0, -8, 0] NULL NULL "340.2.bb.a" 40 2.714913668723533 NULL [] [] [0, 0, 4, 0] NULL NULL "340.2.bd.a" 72 2.714913668723533 NULL [] [] [0, 0, 0, 0] NULL NULL "340.2.bf.a" 288 2.714913668723533 NULL [] [] [0, 0, 0, 0] NULL NULL "340.2.bg.a" 8 2.714913668723533 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+(-\\zeta_{16}-\\zeta_{16}^{5})q^{2}+2\\zeta_{16}^{6}q^{4}+\\cdots" "340.2.bg.b" 8 2.714913668723533 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+(\\zeta_{16}+\\zeta_{16}^{5})q^{2}+2\\zeta_{16}^{6}q^{4}+(-\\zeta_{16}^{2}+\\cdots)q^{5}+\\cdots" "340.2.bg.c" 384 2.714913668723533 NULL [] [] [0, 0, -16, 0] NULL NULL "340.2.bi.a" 72 2.714913668723533 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.b.a" 64 9.264328988250094 NULL [] [] [4, 0, 0, 0] NULL NULL "340.3.d.a" 1 9.264328988250094 "1.1.1.1" [-340] [] [-2, 0, -5, 0] NULL "q-2q^{2}+4q^{4}-5q^{5}-8q^{8}+9q^{9}+\\cdots" "340.3.d.b" 1 9.264328988250094 "1.1.1.1" [-340] [] [-2, 0, 5, 0] NULL "q-2q^{2}+4q^{4}+5q^{5}-8q^{8}+9q^{9}+\\cdots" "340.3.d.c" 1 9.264328988250094 "1.1.1.1" [-340] [] [2, 0, -5, 0] NULL "q+2q^{2}+4q^{4}-5q^{5}+8q^{8}+9q^{9}+\\cdots" "340.3.d.d" 1 9.264328988250094 "1.1.1.1" [-340] [] [2, 0, 5, 0] NULL "q+2q^{2}+4q^{4}+5q^{5}+8q^{8}+9q^{9}+\\cdots" "340.3.d.e" 2 9.264328988250094 "2.0.4.1" [-4] [] [0, 0, -6, 0] NULL "q+\\beta q^{2}-4 q^{4}+(-2\\beta-3)q^{5}-4\\beta q^{8}+\\cdots" "340.3.d.f" 2 9.264328988250094 "2.0.4.1" [-4] [] [0, 0, 6, 0] NULL "q+\\beta q^{2}-4 q^{4}+(2\\beta+3)q^{5}-4\\beta q^{8}+\\cdots" "340.3.d.g" 96 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.f.a" 96 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.h.a" 72 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.j.a" 36 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.k.a" 36 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.n.a" 2 9.264328988250094 "2.0.4.1" [-4] [] [-4, 0, -8, 0] NULL "q-2 q^{2}+4 q^{4}+(-3 i-4)q^{5}-8 q^{8}+\\cdots" "340.3.n.b" 2 9.264328988250094 "2.0.4.1" [-4] [] [-4, 0, -6, 0] NULL "q-2 q^{2}+4 q^{4}+(4 i-3)q^{5}-8 q^{8}+\\cdots" "340.3.n.c" 2 9.264328988250094 "2.0.4.1" [-4] [] [4, 0, -6, 0] NULL "q+2 q^{2}+4 q^{4}+(-4 i-3)q^{5}+8 q^{8}+\\cdots" "340.3.n.d" 2 9.264328988250094 "2.0.4.1" [-4] [] [4, 0, 8, 0] NULL "q+2 q^{2}+4 q^{4}+(-3 i+4)q^{5}+8 q^{8}+\\cdots" "340.3.n.e" 200 9.264328988250094 NULL [] [] [0, 0, 12, 0] NULL NULL "340.3.p.a" 144 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.q.a" 32 9.264328988250094 NULL [] [] [0, -4, 0, 16] NULL NULL "340.3.t.a" 36 9.264328988250094 NULL [] [] [0, 0, 4, 12] NULL NULL "340.3.v.a" 288 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.x.a" 72 9.264328988250094 NULL [] [] [0, 0, 8, 12] NULL NULL "340.3.y.a" 72 9.264328988250094 NULL [] [] [0, 0, 0, -12] NULL NULL "340.3.ba.a" 416 9.264328988250094 NULL [] [] [0, 0, -8, 0] NULL NULL "340.3.bc.a" 8 9.264328988250094 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+2\\zeta_{16}^{5}q^{2}-4\\zeta_{16}^{2}q^{4}+(4\\zeta_{16}^{3}+\\cdots)q^{5}+\\cdots" "340.3.bc.b" 8 9.264328988250094 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q-2\\zeta_{16}^{5}q^{2}-4\\zeta_{16}^{2}q^{4}+(-4\\zeta_{16}^{3}+\\cdots)q^{5}+\\cdots" "340.3.bc.c" 816 9.264328988250094 NULL [] [] [-8, 0, -16, 0] NULL NULL "340.3.be.a" 96 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.bh.a" 144 9.264328988250094 NULL [] [] [0, 0, 0, 0] NULL NULL "340.3.bj.a" 8 9.264328988250094 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q+2\\zeta_{16}q^{2}+4\\zeta_{16}^{2}q^{4}+(-4\\zeta_{16}^{2}+\\cdots)q^{5}+\\cdots" "340.3.bj.b" 8 9.264328988250094 "8.0.16777216.1" [-4] [] [0, 0, 0, 0] NULL "q-2\\zeta_{16}q^{2}+4\\zeta_{16}^{2}q^{4}+(4\\zeta_{16}^{2}-3\\zeta_{16}^{6}+\\cdots)q^{5}+\\cdots" "340.3.bj.c" 816 9.264328988250094 NULL [] [] [-8, 0, -16, 0] NULL NULL "340.4.a.a" 1 20.06064940195181 "1.1.1.1" [] [] [0, -5, 5, 2] -1 "q-5q^{3}+5q^{5}+2q^{7}-2q^{9}+12q^{11}+\\cdots" "340.4.a.b" 1 20.06064940195181 "1.1.1.1" [] [] [0, 2, 5, 2] -1 "q+2q^{3}+5q^{5}+2q^{7}-23q^{9}-30q^{11}+\\cdots" "340.4.a.c" 4 20.06064940195181 "4.4.3507444.3" [] [] [0, -3, -20, -8] 1 "q+(-1-\\beta _{2})q^{3}-5q^{5}+(-2-\\beta _{2}+\\cdots)q^{7}+\\cdots" "340.4.a.d" 5 20.06064940195181 NULL [] [] [0, -3, -25, -2] -1 "q+(-1-\\beta _{3})q^{3}-5q^{5}+(\\beta _{3}+\\beta _{4})q^{7}+\\cdots" "340.4.a.e" 5 20.06064940195181 NULL [] [] [0, -3, 25, 38] 1 "q+(-1+\\beta _{1})q^{3}+5q^{5}+(8+\\beta _{4})q^{7}+\\cdots" "340.4.c.a" 18 20.06064940195181 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{10}q^{5}+\\beta _{12}q^{7}+(-6+\\cdots)q^{9}+\\cdots" "340.4.e.a" 24 20.06064940195181 NULL [] [] [0, 0, -8, 0] NULL NULL "340.4.g.a" 28 20.06064940195181 NULL [] [] [0, 0, 0, 0] NULL NULL "340.4.m.a" 56 20.06064940195181 NULL [] [] [0, 0, -4, 0] NULL NULL "340.4.o.a" 36 20.06064940195181 NULL [] [] [0, 4, 0, 0] NULL NULL "340.4.u.a" 72 20.06064940195181 NULL [] [] [0, 0, 0, 0] NULL NULL "340.5.j.a" 72 35.14578744850707 NULL [] [] [0, 0, 54, 0] NULL NULL "340.5.k.a" 72 35.14578744850707 NULL [] [] [0, 0, 0, 0] NULL NULL "340.5.q.a" 64 35.14578744850707 NULL [] [] [0, 20, 0, -160] NULL NULL "340.5.t.a" 72 35.14578744850707 NULL [] [] [0, 0, 30, -60] NULL NULL "340.6.a.a" 6 54.53049873641341 NULL [] [] [0, 0, -150, -108] -1 "q-\\beta _{1}q^{3}-5^{2}q^{5}+(-18+2\\beta _{1}+\\beta _{4}+\\cdots)q^{7}+\\cdots" "340.6.a.b" 6 54.53049873641341 NULL [] [] [0, 0, 150, -148] 1 "q-\\beta _{1}q^{3}+5^{2}q^{5}+(-5^{2}+2\\beta _{1}-\\beta _{4}+\\cdots)q^{7}+\\cdots" "340.6.a.c" 7 54.53049873641341 NULL [] [] [0, 0, -175, -168] 1 "q+\\beta _{1}q^{3}-5^{2}q^{5}+(-24-\\beta _{1}-\\beta _{5}+\\cdots)q^{7}+\\cdots" "340.6.a.d" 9 54.53049873641341 NULL [] [] [0, 0, 225, -12] -1 "q-\\beta _{1}q^{3}+5^{2}q^{5}+(-1-2\\beta _{1}+\\beta _{3}+\\cdots)q^{7}+\\cdots" "340.6.c.a" 30 54.53049873641341 NULL [] [] [0, 0, 0, 0] NULL NULL "340.6.e.a" 40 54.53049873641341 NULL [] [] [0, 0, 58, 0] NULL NULL "340.6.g.a" 44 54.53049873641341 NULL [] [] [0, 0, 0, 0] NULL NULL "340.6.m.a" 88 54.53049873641341 NULL [] [] [0, 0, -82, 0] NULL NULL "340.6.o.a" 60 54.53049873641341 NULL [] [] [0, -44, 0, 0] NULL NULL "340.7.q.a" 96 78.21838843074934 NULL [] [] [0, -64, 0, 816] NULL NULL "340.8.a.a" 7 106.21090866460139 NULL [] [] [0, 0, 875, -736] -1 "q-\\beta _{1}q^{3}+5^{3}q^{5}+(-105+2\\beta _{1}-\\beta _{2}+\\cdots)q^{7}+\\cdots" "340.8.a.b" 9 106.21090866460139 NULL [] [] [0, 0, -1125, 876] 1 "q-\\beta _{1}q^{3}-5^{3}q^{5}+(97+\\beta _{1}-\\beta _{4})q^{7}+\\cdots" "340.8.a.c" 10 106.21090866460139 NULL [] [] [0, 0, -1250, -904] -1 "q+\\beta _{1}q^{3}-5^{3}q^{5}+(-90-2\\beta _{1}-\\beta _{4}+\\cdots)q^{7}+\\cdots" "340.8.a.d" 10 106.21090866460139 NULL [] [] [0, 0, 1250, -1144] 1 "q+\\beta _{1}q^{3}+5^{3}q^{5}+(-114+3\\beta _{1}-\\beta _{3}+\\cdots)q^{7}+\\cdots" "340.8.c.a" 42 106.21090866460139 NULL [] [] [0, 0, 0, 0] NULL NULL "340.8.e.a" 56 106.21090866460139 NULL [] [] [0, 0, 142, 0] NULL NULL "340.8.g.a" 64 106.21090866460139 NULL [] [] [0, 0, 0, 0] NULL NULL "340.8.o.a" 84 106.21090866460139 NULL [] [] [0, 52, 0, 0] NULL NULL "340.10.a.a" 11 175.11218429571687 NULL [] [] [0, 81, -6875, 6346] -1 "q+(7+\\beta _{1})q^{3}-5^{4}q^{5}+(578-3\\beta _{1}+\\cdots)q^{7}+\\cdots" "340.10.a.b" 11 175.11218429571687 NULL [] [] [0, 81, 6875, -6854] 1 "q+(7+\\beta _{1})q^{3}+5^{4}q^{5}+(-623+\\beta _{4}+\\cdots)q^{7}+\\cdots" "340.10.a.c" 12 175.11218429571687 NULL [] [] [0, 81, -7500, 2964] 1 "q+(7-\\beta _{1})q^{3}-5^{4}q^{5}+(244+11\\beta _{1}+\\cdots)q^{7}+\\cdots" "340.10.a.d" 14 175.11218429571687 NULL [] [] [0, 81, 8750, -632] -1 "q+(6-\\beta _{1})q^{3}+5^{4}q^{5}+(-45-2\\beta _{1}+\\cdots)q^{7}+\\cdots" "340.10.c.a" 54 175.11218429571687 NULL [] [] [0, 0, 0, 0] NULL NULL "340.10.e.a" 72 175.11218429571687 NULL [] [] [0, 0, -1592, 0] NULL NULL "340.10.g.a" 80 175.11218429571687 NULL [] [] [0, 0, 0, 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.