# Classical modular forms downloaded from the LMFDB on 26 July 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/96/ # Query "{'level': 96}" returned 207 forms, sorted by analytic conductor. # Each entry in the following data list has the form: # [Label, Dim, $A$, Field, CM, Traces, Fricke sign, $q$-expansion] # For more details, see the definitions at the bottom of the file. "96.2.a.a" 1 0.7665638594042917 "1.1.1.1" [] [0, -1, 2, 4] -1 "q-q^{3}+2q^{5}+4q^{7}+q^{9}-4q^{11}+\\cdots" "96.2.a.b" 1 0.7665638594042917 "1.1.1.1" [] [0, 1, 2, -4] -1 "q+q^{3}+2q^{5}-4q^{7}+q^{9}+4q^{11}+\\cdots" "96.2.c.a" 4 0.7665638594042917 "4.0.256.1" [] [0, 0, 0, 0] NULL "q-\\beta_1 q^{3}+\\beta_{3} q^{5}+(\\beta_{2}-\\beta_1)q^{7}+\\cdots" "96.2.d.a" 2 0.7665638594042917 "2.0.4.1" [] [0, 0, 0, 4] NULL "q+i q^{3}+2 i q^{5}+2 q^{7}-q^{9}-4 i q^{13}+\\cdots" "96.2.f.a" 2 0.7665638594042917 "2.0.8.1" [-8] [0, 2, 0, 0] NULL "q+(1+\\beta )q^{3}+(-1+2\\beta )q^{9}-2\\beta q^{11}+\\cdots" "96.2.n.a" 32 0.7665638594042917 NULL [] [0, 0, 0, 0] NULL NULL "96.2.o.a" 56 0.7665638594042917 NULL [] [0, -4, 0, -8] NULL NULL "96.3.b.a" 4 2.61581053785885 "4.0.4752.1" [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}-\\beta _{2}q^{5}+(-\\beta _{2}-\\beta _{3})q^{7}+\\cdots" "96.3.e.a" 4 2.61581053785885 "4.0.2304.1" [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}+(2\\beta _{1}-\\beta _{3})q^{7}+(-3+\\cdots)q^{9}+\\cdots" "96.3.e.b" 4 2.61581053785885 "4.0.12544.1" [] [0, 0, 0, 0] NULL "q+(\\beta _{1}+\\beta _{3})q^{3}-\\beta _{2}q^{5}+(-\\beta _{1}-2\\beta _{3})q^{7}+\\cdots" "96.3.g.a" 4 2.61581053785885 "4.0.144.1" [] [0, 0, -8, 0] NULL "q+\\beta_{2} q^{3}+(-\\beta_{3}-2)q^{5}+(4\\beta_{2}+\\beta_1)q^{7}+\\cdots" "96.3.h.a" 1 2.61581053785885 "1.1.1.1" [-24] [0, -3, 2, 10] NULL "q-3q^{3}+2q^{5}+10q^{7}+9q^{9}+10q^{11}+\\cdots" "96.3.h.b" 1 2.61581053785885 "1.1.1.1" [-24] [0, 3, -2, 10] NULL "q+3q^{3}-2q^{5}+10q^{7}+9q^{9}-10q^{11}+\\cdots" "96.3.h.c" 4 2.61581053785885 "4.0.3136.3" [] [0, 0, 0, -16] NULL "q+(-\\beta _{1}-\\beta _{2})q^{3}-2\\beta _{1}q^{5}-4q^{7}+\\cdots" "96.3.m.a" 64 2.61581053785885 NULL [] [0, 0, 0, 0] NULL NULL "96.3.p.a" 120 2.61581053785885 NULL [] [0, -4, 0, -8] NULL NULL "96.4.a.a" 1 5.664183360551099 "1.1.1.1" [] [0, -3, -14, 36] 1 "q-3q^{3}-14q^{5}+6^{2}q^{7}+9q^{9}+6^{2}q^{11}+\\cdots" "96.4.a.b" 1 5.664183360551099 "1.1.1.1" [] [0, -3, 2, -12] -1 "q-3q^{3}+2q^{5}-12q^{7}+9q^{9}-60q^{11}+\\cdots" "96.4.a.c" 1 5.664183360551099 "1.1.1.1" [] [0, -3, 10, -4] 1 "q-3q^{3}+10q^{5}-4q^{7}+9q^{9}+20q^{11}+\\cdots" "96.4.a.d" 1 5.664183360551099 "1.1.1.1" [] [0, 3, -14, -36] -1 "q+3q^{3}-14q^{5}-6^{2}q^{7}+9q^{9}-6^{2}q^{11}+\\cdots" "96.4.a.e" 1 5.664183360551099 "1.1.1.1" [] [0, 3, 2, 12] 1 "q+3q^{3}+2q^{5}+12q^{7}+9q^{9}+60q^{11}+\\cdots" "96.4.a.f" 1 5.664183360551099 "1.1.1.1" [] [0, 3, 10, 4] 1 "q+3q^{3}+10q^{5}+4q^{7}+9q^{9}-20q^{11}+\\cdots" "96.4.c.a" 12 5.664183360551099 "12.0.26525057735983104.1" [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{3}-\\beta _{8}q^{5}-\\beta _{1}q^{7}+(-2+\\beta _{5}+\\cdots)q^{9}+\\cdots" "96.4.d.a" 6 5.664183360551099 "6.0.8248384.1" [] [0, 0, 0, -28] NULL "q+\\beta _{2}q^{3}+(-\\beta _{2}+\\beta _{3})q^{5}+(-5+\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.4.f.a" 2 5.664183360551099 "2.0.8.1" [-8] [0, -10, 0, 0] NULL "q+(-5+\\beta )q^{3}+(23-10\\beta )q^{9}-50\\beta q^{11}+\\cdots" "96.4.f.b" 8 5.664183360551099 NULL [] [0, 12, 0, 0] NULL "q+(1+\\beta _{2})q^{3}+\\beta _{4}q^{5}-\\beta _{6}q^{7}+(-9+\\cdots)q^{9}+\\cdots" "96.4.n.a" 96 5.664183360551099 NULL [] [0, 0, 0, 0] NULL NULL "96.4.o.a" 184 5.664183360551099 NULL [] [0, -4, 0, -8] NULL NULL "96.5.b.a" 8 9.923516456049056 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+\\beta _{2}q^{5}+(-\\beta _{2}+\\beta _{4}-\\beta _{5}+\\cdots)q^{7}+\\cdots" "96.5.e.a" 8 9.923516456049056 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}-\\beta _{3}q^{7}+(-11-2\\beta _{2}+\\cdots)q^{9}+\\cdots" "96.5.e.b" 8 9.923516456049056 "8.0.3317760000.8" [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+\\beta _{2}q^{5}+(\\beta _{1}+\\beta _{4})q^{7}+(-3+\\cdots)q^{9}+\\cdots" "96.5.g.a" 4 9.923516456049056 "4.0.144.1" [] [0, 0, -40, 0] NULL "q+3\\beta_{2} q^{3}+(-\\beta_{3}-10)q^{5}+(-20\\beta_{2}+11\\beta_1)q^{7}+\\cdots" "96.5.g.b" 4 9.923516456049056 "4.0.144.1" [] [0, 0, 88, 0] NULL "q-\\beta_{2} q^{3}+(-\\beta_{3}+22)q^{5}+(-4\\beta_{2}-\\beta_1)q^{7}+\\cdots" "96.5.h.a" 1 9.923516456049056 "1.1.1.1" [-24] [0, -9, -46, -2] NULL "q-9q^{3}-46q^{5}-2q^{7}+3^{4}q^{9}+142q^{11}+\\cdots" "96.5.h.b" 1 9.923516456049056 "1.1.1.1" [-24] [0, 9, 46, -2] NULL "q+9q^{3}+46q^{5}-2q^{7}+3^{4}q^{9}-142q^{11}+\\cdots" "96.5.h.c" 12 9.923516456049056 NULL [] [0, 0, 0, 8] NULL "q-\\beta _{1}q^{3}+(\\beta _{1}-\\beta _{3})q^{5}+(1-\\beta _{4})q^{7}+\\cdots" "96.5.m.a" 128 9.923516456049056 NULL [] [0, 0, 0, 0] NULL NULL "96.5.p.a" 248 9.923516456049056 NULL [] [0, -4, 0, -8] NULL NULL "96.6.a.a" 1 15.39684670204614 "1.1.1.1" [] [0, -9, -86, -180] -1 "q-9q^{3}-86q^{5}-180q^{7}+3^{4}q^{9}+\\cdots" "96.6.a.b" 1 15.39684670204614 "1.1.1.1" [] [0, -9, -14, 100] 1 "q-9q^{3}-14q^{5}+10^{2}q^{7}+3^{4}q^{9}+\\cdots" "96.6.a.c" 1 15.39684670204614 "1.1.1.1" [] [0, -9, 26, -36] 1 "q-9q^{3}+26q^{5}-6^{2}q^{7}+3^{4}q^{9}-180q^{11}+\\cdots" "96.6.a.d" 1 15.39684670204614 "1.1.1.1" [] [0, 9, -86, 180] 1 "q+9q^{3}-86q^{5}+180q^{7}+3^{4}q^{9}+\\cdots" "96.6.a.e" 1 15.39684670204614 "1.1.1.1" [] [0, 9, -14, -100] 1 "q+9q^{3}-14q^{5}-10^{2}q^{7}+3^{4}q^{9}+\\cdots" "96.6.a.f" 1 15.39684670204614 "1.1.1.1" [] [0, 9, 26, 36] -1 "q+9q^{3}+26q^{5}+6^{2}q^{7}+3^{4}q^{9}+180q^{11}+\\cdots" "96.6.a.g" 2 15.39684670204614 "2.2.124.1" [] [0, -18, 36, -120] -1 "q-9q^{3}+(18+\\beta )q^{5}+(-60-\\beta )q^{7}+\\cdots" "96.6.a.h" 2 15.39684670204614 "2.2.124.1" [] [0, 18, 36, 120] -1 "q+9q^{3}+(18+\\beta )q^{5}+(60+\\beta )q^{7}+3^{4}q^{9}+\\cdots" "96.6.c.a" 20 15.39684670204614 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{3}+\\beta _{6}q^{5}+\\beta _{2}q^{7}+(-2-\\beta _{7}+\\cdots)q^{9}+\\cdots" "96.6.d.a" 10 15.39684670204614 NULL [] [0, 0, 0, 196] NULL "q-\\beta _{1}q^{3}+(-\\beta _{1}+\\beta _{3})q^{5}+(20+\\beta _{4}+\\cdots)q^{7}+\\cdots" "96.6.f.a" 2 15.39684670204614 "2.0.8.1" [-8] [0, 2, 0, 0] NULL "q+(1-11\\beta )q^{3}+(-241-22\\beta )q^{9}+\\cdots" "96.6.f.b" 16 15.39684670204614 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}-\\beta _{3}q^{5}+\\beta _{4}q^{7}+(30+\\beta _{7}+\\cdots)q^{9}+\\cdots" "96.6.n.a" 160 15.39684670204614 NULL [] [0, 0, 0, 0] NULL NULL "96.6.o.a" 312 15.39684670204614 NULL [] [0, -4, 0, -8] NULL NULL "96.7.b.a" 12 22.085192027505695 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}-\\beta _{2}q^{5}+(-\\beta _{2}+\\beta _{4})q^{7}+\\cdots" "96.7.e.a" 12 22.085192027505695 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+\\beta _{3}q^{5}+\\beta _{1}q^{7}+(-2^{6}+\\beta _{3}+\\cdots)q^{9}+\\cdots" "96.7.e.b" 12 22.085192027505695 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{3}+\\beta _{2}q^{5}+(-2\\beta _{3}+\\beta _{6})q^{7}+\\cdots" "96.7.g.a" 4 22.085192027505695 "4.0.144.1" [] [0, 0, -200, 0] NULL "q-3\\beta_{2} q^{3}+(-5\\beta_{3}-50)q^{5}+(20\\beta_{2}+73\\beta_1)q^{7}+\\cdots" "96.7.g.b" 8 22.085192027505695 "8.0.19752615936.5" [] [0, 0, 112, 0] NULL "q-\\beta _{1}q^{3}+(14-\\beta _{2})q^{5}+(7\\beta _{1}+\\beta _{4}+\\cdots)q^{7}+\\cdots" "96.7.h.a" 1 22.085192027505695 "1.1.1.1" [-24] [0, -27, -142, -470] NULL "q-3^{3}q^{3}-142q^{5}-470q^{7}+3^{6}q^{9}+\\cdots" "96.7.h.b" 1 22.085192027505695 "1.1.1.1" [-24] [0, 27, 142, -470] NULL "q+3^{3}q^{3}+142q^{5}-470q^{7}+3^{6}q^{9}+\\cdots" "96.7.h.c" 20 22.085192027505695 NULL [] [0, 0, 0, 944] NULL "q+\\beta _{1}q^{3}-\\beta _{3}q^{5}+(47-\\beta _{2})q^{7}+(-73+\\cdots)q^{9}+\\cdots" "96.7.m.a" 192 22.085192027505695 NULL [] [0, 0, 0, 0] NULL NULL "96.7.p.a" 376 22.085192027505695 NULL [] [0, -4, 0, -8] NULL NULL "96.8.a.a" 1 29.98896244647569 "1.1.1.1" [] [0, -27, -70, 92] -1 "q-3^{3}q^{3}-70q^{5}+92q^{7}+3^{6}q^{9}+\\cdots" "96.8.a.b" 1 29.98896244647569 "1.1.1.1" [] [0, 27, -70, -92] -1 "q+3^{3}q^{3}-70q^{5}-92q^{7}+3^{6}q^{9}+\\cdots" "96.8.a.c" 2 29.98896244647569 "2.2.24.1" [] [0, -54, -28, 936] 1 "q-3^{3}q^{3}+(-14+3\\beta )q^{5}+(468+7\\beta )q^{7}+\\cdots" "96.8.a.d" 2 29.98896244647569 "2.2.940.1" [] [0, -54, 180, -1032] 1 "q-3^{3}q^{3}+(90+\\beta )q^{5}+(-516+5\\beta )q^{7}+\\cdots" "96.8.a.e" 2 29.98896244647569 "2.2.184.1" [] [0, -54, 196, -504] -1 "q-3^{3}q^{3}+(98+\\beta )q^{5}+(-252-3\\beta )q^{7}+\\cdots" "96.8.a.f" 2 29.98896244647569 "2.2.24.1" [] [0, 54, -28, -936] -1 "q+3^{3}q^{3}+(-14+3\\beta )q^{5}+(-468+\\cdots)q^{7}+\\cdots" "96.8.a.g" 2 29.98896244647569 "2.2.940.1" [] [0, 54, 180, 1032] 1 "q+3^{3}q^{3}+(90+\\beta )q^{5}+(516-5\\beta )q^{7}+\\cdots" "96.8.a.h" 2 29.98896244647569 "2.2.184.1" [] [0, 54, 196, 504] 1 "q+3^{3}q^{3}+(98+\\beta )q^{5}+(252+3\\beta )q^{7}+\\cdots" "96.8.c.a" 28 29.98896244647569 NULL [] [0, 0, 0, 0] NULL NULL "96.8.d.a" 14 29.98896244647569 NULL [] [0, 0, 0, -1372] NULL "q+\\beta _{1}q^{3}+(-\\beta _{1}+\\beta _{3})q^{5}+(-98-\\beta _{4}+\\cdots)q^{7}+\\cdots" "96.8.f.a" 2 29.98896244647569 "2.0.8.1" [-8] [0, 86, 0, 0] NULL "q+(43-13\\beta )q^{3}+(1511-1118\\beta )q^{9}+\\cdots" "96.8.f.b" 4 29.98896244647569 "4.0.97344.2" [] [0, 36, 0, 0] NULL "q+(9+9\\beta _{1})q^{3}-5\\beta _{2}q^{5}+31\\beta _{3}q^{7}+\\cdots" "96.8.f.c" 20 29.98896244647569 NULL [] [0, -120, 0, 0] NULL "q+(-6+\\beta _{1})q^{3}-\\beta _{2}q^{5}+\\beta _{6}q^{7}+(254+\\cdots)q^{9}+\\cdots" "96.8.n.a" 224 29.98896244647569 NULL [] [0, 0, 0, 0] NULL NULL "96.8.o.a" 440 29.98896244647569 NULL [] [0, -4, 0, -8] NULL NULL "96.9.b.a" 16 39.10834656586821 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}-\\beta _{2}q^{5}+\\beta _{3}q^{7}+3^{7}q^{9}+\\cdots" "96.9.e.a" 16 39.10834656586821 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{3}+\\beta _{3}q^{5}+(\\beta _{1}-3\\beta _{2})q^{7}+(-435+\\cdots)q^{9}+\\cdots" "96.9.e.b" 16 39.10834656586821 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}-\\beta _{3}q^{5}+(-\\beta _{1}+\\beta _{2})q^{7}+\\cdots" "96.9.g.a" 8 39.10834656586821 NULL [] [0, 0, -1232, 0] NULL "q+\\beta _{1}q^{3}+(-154+\\beta _{6})q^{5}+(17\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.9.g.b" 8 39.10834656586821 "8.0.3468738816.6" [] [0, 0, 560, 0] NULL "q+\\beta _{2}q^{3}+(70+\\beta _{4})q^{5}+(-7\\beta _{1}-7\\beta _{3}+\\cdots)q^{7}+\\cdots" "96.9.h.a" 1 39.10834656586821 "1.1.1.1" [-24] [0, -81, 866, 4798] NULL "q-3^{4}q^{3}+866q^{5}+4798q^{7}+3^{8}q^{9}+\\cdots" "96.9.h.b" 1 39.10834656586821 "1.1.1.1" [-24] [0, 81, -866, 4798] NULL "q+3^{4}q^{3}-866q^{5}+4798q^{7}+3^{8}q^{9}+\\cdots" "96.9.h.c" 28 39.10834656586821 NULL [] [0, 0, 0, -9592] NULL NULL "96.9.m.a" 256 39.10834656586821 NULL [] [0, 0, 0, 0] NULL NULL "96.9.p.a" 504 39.10834656586821 NULL [] [0, -4, 0, -8] NULL NULL "96.10.a.a" 2 49.44344027173182 "2.2.8236.1" [] [0, -162, -1404, 3528] 1 "q-3^{4}q^{3}+(-702+\\beta )q^{5}+(42^{2}+7\\beta )q^{7}+\\cdots" "96.10.a.b" 2 49.44344027173182 "2.2.536.1" [] [0, -162, 244, -3816] 1 "q-3^{4}q^{3}+(122+\\beta )q^{5}+(-1908-\\beta )q^{7}+\\cdots" "96.10.a.c" 2 49.44344027173182 "2.2.184.1" [] [0, -162, 596, 5304] -1 "q-3^{4}q^{3}+(298+\\beta )q^{5}+(2652-17\\beta )q^{7}+\\cdots" "96.10.a.d" 2 49.44344027173182 "2.2.8236.1" [] [0, 162, -1404, -3528] 1 "q+3^{4}q^{3}+(-702+\\beta )q^{5}+(-42^{2}+\\cdots)q^{7}+\\cdots" "96.10.a.e" 2 49.44344027173182 "2.2.536.1" [] [0, 162, 244, 3816] -1 "q+3^{4}q^{3}+(122+\\beta )q^{5}+(1908+\\beta )q^{7}+\\cdots" "96.10.a.f" 2 49.44344027173182 "2.2.184.1" [] [0, 162, 596, -5304] 1 "q+3^{4}q^{3}+(298+\\beta )q^{5}+(-2652+17\\beta )q^{7}+\\cdots" "96.10.a.g" 3 49.44344027173182 "3.3.723301.1" [] [0, -243, -154, 7564] -1 "q-3^{4}q^{3}+(-51+\\beta _{2})q^{5}+(2521+\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.10.a.h" 3 49.44344027173182 "3.3.723301.1" [] [0, 243, -154, -7564] -1 "q+3^{4}q^{3}+(-51+\\beta _{2})q^{5}+(-2521+\\cdots)q^{7}+\\cdots" "96.10.c.a" 36 49.44344027173182 NULL [] [0, 0, 0, 0] NULL NULL "96.10.d.a" 18 49.44344027173182 NULL [] [0, 0, 0, 9604] NULL "q+\\beta _{9}q^{3}+(2\\beta _{9}+\\beta _{10})q^{5}+(534-\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.10.f.a" 2 49.44344027173182 "2.0.8.1" [-8] [0, -190, 0, 0] NULL "q+(-95+73\\beta )q^{3}+(-1633-13870\\beta )q^{9}+\\cdots" "96.10.f.b" 32 49.44344027173182 NULL [] [0, 192, 0, 0] NULL NULL "96.10.n.a" 288 49.44344027173182 NULL [] [0, 0, 0, 0] NULL NULL "96.10.o.a" 568 49.44344027173182 NULL [] [0, -4, 0, -8] NULL NULL "96.11.b.a" 20 60.994296256677515 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{3}+\\beta _{1}q^{5}+\\beta _{11}q^{7}+3^{9}q^{9}+\\cdots" "96.11.e.a" 20 60.994296256677515 NULL [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{3}+\\beta _{6}q^{5}+(10\\beta _{3}+\\beta _{4})q^{7}+\\cdots" "96.11.e.b" 20 60.994296256677515 NULL [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}+(15\\beta _{1}+\\beta _{3})q^{7}+\\cdots" "96.11.g.a" 8 60.994296256677515 NULL [] [0, 0, 2800, 0] NULL "q+3\\beta _{1}q^{3}+(350+\\beta _{5})q^{5}+(176\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.11.g.b" 12 60.994296256677515 NULL [] [0, 0, 3432, 0] NULL "q+\\beta _{2}q^{3}+(286+\\beta _{1})q^{5}+(-9\\beta _{2}+\\beta _{8}+\\cdots)q^{7}+\\cdots" "96.11.h.a" 1 60.994296256677515 "1.1.1.1" [-24] [0, -243, 5282, -24950] NULL "q-3^{5}q^{3}+5282q^{5}-24950q^{7}+3^{10}q^{9}+\\cdots" "96.11.h.b" 1 60.994296256677515 "1.1.1.1" [-24] [0, 243, -5282, -24950] NULL "q+3^{5}q^{3}-5282q^{5}-24950q^{7}+3^{10}q^{9}+\\cdots" "96.11.h.c" 36 60.994296256677515 NULL [] [0, 0, 0, 49904] NULL NULL "96.11.m.a" 320 60.994296256677515 NULL [] [0, 0, 0, 0] NULL NULL "96.11.p.a" 632 60.994296256677515 NULL [] [0, -4, 0, -8] NULL NULL "96.12.a.a" 1 73.76094533372053 "1.1.1.1" [] [0, -243, -6070, -16100] 1 "q-3^{5}q^{3}-6070q^{5}-16100q^{7}+3^{10}q^{9}+\\cdots" "96.12.a.b" 1 73.76094533372053 "1.1.1.1" [] [0, 243, -6070, 16100] 1 "q+3^{5}q^{3}-6070q^{5}+16100q^{7}+3^{10}q^{9}+\\cdots" "96.12.a.c" 2 73.76094533372053 "2.2.49564.1" [] [0, -486, -7020, -56520] -1 "q-3^{5}q^{3}+(-3510+5\\beta )q^{5}+(-28260+\\cdots)q^{7}+\\cdots" "96.12.a.d" 2 73.76094533372053 "2.2.1945.1" [] [0, -486, 5300, 38872] 1 "q-3^{5}q^{3}+(2650-3\\beta )q^{5}+(19436+\\cdots)q^{7}+\\cdots" "96.12.a.e" 2 73.76094533372053 "2.2.49564.1" [] [0, 486, -7020, 56520] -1 "q+3^{5}q^{3}+(-3510+5\\beta )q^{5}+(28260+\\cdots)q^{7}+\\cdots" "96.12.a.f" 2 73.76094533372053 "2.2.1945.1" [] [0, 486, 5300, -38872] 1 "q+3^{5}q^{3}+(2650-3\\beta )q^{5}+(-19436+\\cdots)q^{7}+\\cdots" "96.12.a.g" 3 73.76094533372053 "3.3.2158836.1" [] [0, -729, 726, -6516] 1 "q-3^{5}q^{3}+(242+\\beta _{1})q^{5}+(-2172+\\cdots)q^{7}+\\cdots" "96.12.a.h" 3 73.76094533372053 NULL [] [0, -729, 4422, 39996] -1 "q-3^{5}q^{3}+(1474+\\beta _{1})q^{5}+(13332+\\cdots)q^{7}+\\cdots" "96.12.a.i" 3 73.76094533372053 "3.3.2158836.1" [] [0, 729, 726, 6516] -1 "q+3^{5}q^{3}+(242+\\beta _{1})q^{5}+(2172-3\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.12.a.j" 3 73.76094533372053 NULL [] [0, 729, 4422, -39996] 1 "q+3^{5}q^{3}+(1474+\\beta _{1})q^{5}+(-13332+\\cdots)q^{7}+\\cdots" "96.12.c.a" 44 73.76094533372053 NULL [] [0, 0, 0, 0] NULL NULL "96.12.d.a" 22 73.76094533372053 NULL [] [0, 0, 0, -67228] NULL NULL "96.12.f.a" 2 73.76094533372053 "2.0.8.1" [-8] [0, -394, 0, 0] NULL "q+(-197+263\\beta )q^{3}+(-99529-103622\\beta )q^{9}+\\cdots" "96.12.f.b" 40 73.76094533372053 NULL [] [0, 396, 0, 0] NULL NULL "96.12.n.a" 352 73.76094533372053 NULL [] [0, 0, 0, 0] NULL NULL "96.12.o.a" 696 73.76094533372053 NULL [] [0, -4, 0, -8] NULL NULL "96.13.b.a" 24 87.74340645852341 NULL [] [0, 0, 0, 0] NULL NULL "96.13.e.a" 24 87.74340645852341 NULL [] [0, 0, 0, 0] NULL NULL "96.13.e.b" 24 87.74340645852341 NULL [] [0, 0, 0, 0] NULL NULL "96.13.g.a" 12 87.74340645852341 NULL [] [0, 0, -37752, 0] NULL "q-\\beta _{1}q^{3}+(-3146-\\beta _{2})q^{5}+(63\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.13.g.b" 12 87.74340645852341 NULL [] [0, 0, 17160, 0] NULL "q-\\beta _{7}q^{3}+(1430-\\beta _{1})q^{5}+(18\\beta _{6}+8\\beta _{7}+\\cdots)q^{7}+\\cdots" "96.13.h.a" 1 87.74340645852341 "1.1.1.1" [-24] [0, -729, -11086, 14398] NULL "q-3^{6}q^{3}-11086q^{5}+14398q^{7}+\\cdots" "96.13.h.b" 1 87.74340645852341 "1.1.1.1" [-24] [0, 729, 11086, 14398] NULL "q+3^{6}q^{3}+11086q^{5}+14398q^{7}+\\cdots" "96.13.h.c" 44 87.74340645852341 NULL [] [0, 0, 0, -28792] NULL NULL "96.14.a.a" 3 102.94169179383618 NULL [] [0, -2187, -20226, 140868] -1 "q-3^{6}q^{3}+(-6742+\\beta _{1}-2\\beta _{2})q^{5}+\\cdots" "96.14.a.b" 3 102.94169179383618 NULL [] [0, -2187, -11594, -175316] 1 "q-3^{6}q^{3}+(-3865-\\beta _{1})q^{5}+(-58438+\\cdots)q^{7}+\\cdots" "96.14.a.c" 3 102.94169179383618 NULL [] [0, -2187, 45966, -684108] 1 "q-3^{6}q^{3}+(15322-\\beta _{1})q^{5}+(-228036+\\cdots)q^{7}+\\cdots" "96.14.a.d" 3 102.94169179383618 NULL [] [0, 2187, -20226, -140868] 1 "q+3^{6}q^{3}+(-6742+\\beta _{1}-2\\beta _{2})q^{5}+\\cdots" "96.14.a.e" 3 102.94169179383618 NULL [] [0, 2187, -11594, 175316] 1 "q+3^{6}q^{3}+(-3865-\\beta _{1})q^{5}+(58438+\\cdots)q^{7}+\\cdots" "96.14.a.f" 3 102.94169179383618 NULL [] [0, 2187, 45966, 684108] -1 "q+3^{6}q^{3}+(15322-\\beta _{1})q^{5}+(228036+\\cdots)q^{7}+\\cdots" "96.14.a.g" 4 102.94169179383618 NULL [] [0, -2916, 19656, 102672] -1 "q-3^{6}q^{3}+(4914+\\beta _{1})q^{5}+(25668+\\cdots)q^{7}+\\cdots" "96.14.a.h" 4 102.94169179383618 NULL [] [0, 2916, 19656, -102672] -1 "q+3^{6}q^{3}+(4914+\\beta _{1})q^{5}+(-25668+\\cdots)q^{7}+\\cdots" "96.14.c.a" 52 102.94169179383618 NULL [] [0, 0, 0, 0] NULL NULL "96.14.d.a" 26 102.94169179383618 NULL [] [0, 0, 0, 470596] NULL NULL "96.14.f.a" 2 102.94169179383618 "2.0.8.1" [-8] [0, 2498, 0, 0] NULL "q+(1249+131\\beta )q^{3}+(1525679+327238\\beta )q^{9}+\\cdots" "96.14.f.b" 48 102.94169179383618 NULL [] [0, -2496, 0, 0] NULL NULL "96.15.b.a" 28 119.35580942582975 NULL [] [0, 0, 0, 0] NULL NULL "96.15.e.a" 28 119.35580942582975 NULL [] [0, 0, 0, 0] NULL NULL "96.15.e.b" 28 119.35580942582975 NULL [] [0, 0, 0, 0] NULL NULL "96.15.g.a" 12 119.35580942582975 NULL [] [0, 0, 85800, 0] NULL "q-\\beta _{2}q^{3}+(7150+\\beta _{3})q^{5}+(-172\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.15.g.b" 16 119.35580942582975 NULL [] [0, 0, -118048, 0] NULL "q+\\beta _{1}q^{3}+(-7378+\\beta _{3})q^{5}+(-87\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.15.h.a" 1 119.35580942582975 "1.1.1.1" [-24] [0, -2187, -154222, 1078570] NULL "q-3^{7}q^{3}-154222q^{5}+1078570q^{7}+\\cdots" "96.15.h.b" 1 119.35580942582975 "1.1.1.1" [-24] [0, 2187, 154222, 1078570] NULL "q+3^{7}q^{3}+154222q^{5}+1078570q^{7}+\\cdots" "96.15.h.c" 52 119.35580942582975 NULL [] [0, 0, 0, -2157136] NULL NULL "96.16.a.a" 3 136.98576489592358 NULL [] [0, -6561, -57970, 596372] -1 "q-3^{7}q^{3}+(-19323-\\beta _{1})q^{5}+(198793+\\cdots)q^{7}+\\cdots" "96.16.a.b" 3 136.98576489592358 NULL [] [0, 6561, -57970, -596372] -1 "q+3^{7}q^{3}+(-19323-\\beta _{1})q^{5}+(-198793+\\cdots)q^{7}+\\cdots" "96.16.a.c" 4 136.98576489592358 NULL [] [0, -8748, -233528, 973008] 1 "q-3^{7}q^{3}+(-58382+\\beta _{1})q^{5}+(243252+\\cdots)q^{7}+\\cdots" "96.16.a.d" 4 136.98576489592358 NULL [] [0, -8748, 56456, 1296528] -1 "q-3^{7}q^{3}+(14114+\\beta _{1})q^{5}+(324132+\\cdots)q^{7}+\\cdots" "96.16.a.e" 4 136.98576489592358 NULL [] [0, -8748, 98280, -1621008] 1 "q-3^{7}q^{3}+(24570-\\beta _{1})q^{5}+(-405252+\\cdots)q^{7}+\\cdots" "96.16.a.f" 4 136.98576489592358 NULL [] [0, 8748, -233528, -973008] -1 "q+3^{7}q^{3}+(-58382+\\beta _{1})q^{5}+(-243252+\\cdots)q^{7}+\\cdots" "96.16.a.g" 4 136.98576489592358 NULL [] [0, 8748, 56456, -1296528] 1 "q+3^{7}q^{3}+(14114+\\beta _{1})q^{5}+(-324132+\\cdots)q^{7}+\\cdots" "96.16.a.h" 4 136.98576489592358 NULL [] [0, 8748, 98280, 1621008] 1 "q+3^{7}q^{3}+(24570-\\beta _{1})q^{5}+(405252+\\cdots)q^{7}+\\cdots" "96.16.c.a" 60 136.98576489592358 NULL [] [0, 0, 0, 0] NULL NULL "96.16.d.a" 30 136.98576489592358 NULL [] [0, 0, 0, -3294172] NULL NULL "96.16.f.a" 2 136.98576489592358 "2.0.8.1" [-8] [0, -1450, 0, 0] NULL "q+(-725-2629\\beta )q^{3}+(-13297657+\\cdots)q^{9}+\\cdots" "96.16.f.b" 56 136.98576489592358 NULL [] [0, 1452, 0, 0] NULL NULL "96.17.b.a" 32 155.83156210236208 NULL [] [0, 0, 0, 0] NULL NULL "96.17.e.a" 32 155.83156210236208 NULL [] [0, 0, 0, 0] NULL NULL "96.17.e.b" 32 155.83156210236208 NULL [] [0, 0, 0, 0] NULL NULL "96.17.g.a" 16 155.83156210236208 NULL [] [0, 0, -590240, 0] NULL "q+\\beta _{8}q^{3}+(-36890+\\beta _{1})q^{5}+(20^{2}\\beta _{8}+\\cdots)q^{7}+\\cdots" "96.17.g.b" 16 155.83156210236208 NULL [] [0, 0, 1298528, 0] NULL "q-\\beta _{1}q^{3}+(81158-\\beta _{3})q^{5}+(53\\beta _{1}+\\cdots)q^{7}+\\cdots" "96.17.h.a" 1 155.83156210236208 "1.1.1.1" [-24] [0, -6561, -31294, -11491202] NULL "q-3^{8}q^{3}-31294q^{5}-11491202q^{7}+\\cdots" "96.17.h.b" 1 155.83156210236208 "1.1.1.1" [-24] [0, 6561, 31294, -11491202] NULL "q+3^{8}q^{3}+31294q^{5}-11491202q^{7}+\\cdots" "96.17.h.c" 60 155.83156210236208 NULL [] [0, 0, 0, 22982408] NULL NULL "96.18.a.a" 4 175.89320385077727 NULL [] [0, -26244, -530264, 2328048] -1 "q-3^{8}q^{3}+(-132566+\\beta _{2})q^{5}+(582012+\\cdots)q^{7}+\\cdots" "96.18.a.b" 4 175.89320385077727 NULL [] [0, -26244, -355096, -13392] 1 "q-3^{8}q^{3}+(-88774+\\beta _{1})q^{5}+(-3348+\\cdots)q^{7}+\\cdots" "96.18.a.c" 4 175.89320385077727 NULL [] [0, -26244, 39816, 26149776] 1 "q-3^{8}q^{3}+(9954-\\beta _{1})q^{5}+(6537444+\\cdots)q^{7}+\\cdots" "96.18.a.d" 4 175.89320385077727 NULL [] [0, 26244, -530264, -2328048] 1 "q+3^{8}q^{3}+(-132566+\\beta _{2})q^{5}+(-582012+\\cdots)q^{7}+\\cdots" "96.18.a.e" 4 175.89320385077727 NULL [] [0, 26244, -355096, 13392] -1 "q+3^{8}q^{3}+(-88774+\\beta _{1})q^{5}+(3348+\\cdots)q^{7}+\\cdots" "96.18.a.f" 4 175.89320385077727 NULL [] [0, 26244, 39816, -26149776] 1 "q+3^{8}q^{3}+(9954-\\beta _{1})q^{5}+(-6537444+\\cdots)q^{7}+\\cdots" "96.18.a.g" 5 175.89320385077727 NULL [] [0, -32805, 821066, -775916] -1 "q-3^{8}q^{3}+(164213+\\beta _{1})q^{5}+(-155183+\\cdots)q^{7}+\\cdots" "96.18.a.h" 5 175.89320385077727 NULL [] [0, 32805, 821066, 775916] -1 "q+3^{8}q^{3}+(164213+\\beta _{1})q^{5}+(155183+\\cdots)q^{7}+\\cdots" "96.18.c.a" 68 175.89320385077727 NULL [] [0, 0, 0, 0] NULL NULL "96.18.d.a" 34 175.89320385077727 NULL [] [0, 0, 0, 23059204] NULL NULL "96.18.f.a" 2 175.89320385077727 "2.0.8.1" [-8] [0, -19582, 0, 0] NULL "q+(-9791+4079\\beta )q^{3}+(62587199+\\cdots)q^{9}+\\cdots" "96.18.f.b" 64 175.89320385077727 NULL [] [0, 19584, 0, 0] NULL NULL "96.19.b.a" 36 197.1706922013731 NULL [] [0, 0, 0, 0] NULL NULL "96.19.e.a" 36 197.1706922013731 NULL [] [0, 0, 0, 0] NULL NULL "96.19.e.b" 36 197.1706922013731 NULL [] [0, 0, 0, 0] NULL NULL "96.19.g.a" 16 197.1706922013731 NULL [] [0, 0, -2951200, 0] NULL "q+\\beta _{2}q^{3}+(-184450+\\beta _{1})q^{5}+(-247\\beta _{2}+\\cdots)q^{7}+\\cdots" "96.19.g.b" 20 197.1706922013731 NULL [] [0, 0, -492328, 0] NULL "q-\\beta _{10}q^{3}+(-24616-\\beta _{1})q^{5}+(-394\\beta _{10}+\\cdots)q^{7}+\\cdots" "96.19.h.a" 1 197.1706922013731 "1.1.1.1" [-24] [0, -19683, 3792962, 62062090] NULL "q-3^{9}q^{3}+3792962q^{5}+62062090q^{7}+\\cdots" "96.19.h.b" 1 197.1706922013731 "1.1.1.1" [-24] [0, 19683, -3792962, 62062090] NULL "q+3^{9}q^{3}-3792962q^{5}+62062090q^{7}+\\cdots" "96.19.h.c" 68 197.1706922013731 NULL [] [0, 0, 0, -124124176] NULL NULL "96.20.a.a" 4 219.66402869412553 NULL [] [0, -78732, 199080, -42996624] -1 "q-3^{9}q^{3}+(49770-\\beta _{1})q^{5}+(-10749156+\\cdots)q^{7}+\\cdots" "96.20.a.b" 4 219.66402869412553 NULL [] [0, 78732, 199080, 42996624] -1 "q+3^{9}q^{3}+(49770-\\beta _{1})q^{5}+(10749156+\\cdots)q^{7}+\\cdots" "96.20.a.c" 5 219.66402869412553 NULL [] [0, -98415, -693878, -123314940] -1 "q-3^{9}q^{3}+(-138776+\\beta _{1})q^{5}+(-24662985+\\cdots)q^{7}+\\cdots" "96.20.a.d" 5 219.66402869412553 NULL [] [0, -98415, -44614, 145833588] 1 "q-3^{9}q^{3}+(-8923-\\beta _{1})q^{5}+(29166716+\\cdots)q^{7}+\\cdots" "96.20.a.e" 5 219.66402869412553 NULL [] [0, -98415, 4105330, -95899156] 1 "q-3^{9}q^{3}+(821066+\\beta _{1})q^{5}+(-19179831+\\cdots)q^{7}+\\cdots" "96.20.a.f" 5 219.66402869412553 NULL [] [0, 98415, -693878, 123314940] 1 "q+3^{9}q^{3}+(-138776+\\beta _{1})q^{5}+(24662985+\\cdots)q^{7}+\\cdots" "96.20.a.g" 5 219.66402869412553 NULL [] [0, 98415, -44614, -145833588] -1 "q+3^{9}q^{3}+(-8923-\\beta _{1})q^{5}+(-29166716+\\cdots)q^{7}+\\cdots" "96.20.a.h" 5 219.66402869412553 NULL [] [0, 98415, 4105330, 95899156] 1 "q+3^{9}q^{3}+(821066+\\beta _{1})q^{5}+(19179831+\\cdots)q^{7}+\\cdots" "96.20.c.a" 76 219.66402869412553 NULL [] [0, 0, 0, 0] NULL NULL "96.20.d.a" 38 219.66402869412553 NULL [] [0, 0, 0, -161414428] NULL NULL "96.20.f.a" 2 219.66402869412553 "2.0.8.1" [-8] [0, 52214, 0, 0] NULL "q+(26107+15503\\beta )q^{3}+(200889431+\\cdots)q^{9}+\\cdots" "96.20.f.b" 72 219.66402869412553 NULL [] [0, -52212, 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. #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.