# Classical modular forms downloaded from the LMFDB on 28 July 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1280/ # Query "{'level': 1280}" returned 225 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. "1280.1.e.a" 2 0.638803216170243 "2.0.4.1" [-4, -20] [5] [0, 0, 0, 0] NULL "q-i q^{5}+q^{9}-q^{25}-2 i q^{29}+2 q^{41}+\\cdots" "1280.1.h.a" 2 0.638803216170243 "2.2.8.1" [-20] [] [0, 0, -2, 0] NULL "q-\\beta q^{3}-q^{5}-\\beta q^{7}+q^{9}+\\beta q^{15}+\\cdots" "1280.1.h.b" 2 0.638803216170243 "2.0.4.1" [-4, -40] [40] [0, 0, 0, 0] NULL "q-i q^{5}-q^{9}-2 i q^{13}-q^{25}-2 i q^{37}+\\cdots" "1280.1.h.c" 2 0.638803216170243 "2.2.8.1" [-20] [] [0, 0, 2, 0] NULL "q-\\beta q^{3}+q^{5}+\\beta q^{7}+q^{9}-\\beta q^{15}+\\cdots" "1280.1.m.a" 2 0.638803216170243 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q-q^{5}+i q^{9}+(i-1)q^{13}+(i-1)q^{17}+\\cdots" "1280.1.m.b" 2 0.638803216170243 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+q^{5}+i q^{9}+(-i+1)q^{13}+(i-1)q^{17}+\\cdots" "1280.1.p.a" 2 0.638803216170243 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q-q^{5}-i q^{9}+(i+1)q^{13}+(-i+1)q^{17}+\\cdots" "1280.1.p.b" 2 0.638803216170243 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+q^{5}-i q^{9}+(-i-1)q^{13}+(-i+1)q^{17}+\\cdots" "1280.2.a.a" 2 10.220851458723889 "2.2.12.1" [] [] [0, -2, -2, 2] 1 "q+(-1+\\beta )q^{3}-q^{5}+(1+\\beta )q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.2.a.b" 2 10.220851458723889 "2.2.12.1" [] [] [0, -2, -2, 6] -1 "q+(-1+\\beta )q^{3}-q^{5}+(3-\\beta )q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.2.a.c" 2 10.220851458723889 "2.2.12.1" [] [] [0, -2, 2, -6] 1 "q+(-1+\\beta )q^{3}+q^{5}+(-3+\\beta )q^{7}+\\cdots" "1280.2.a.d" 2 10.220851458723889 "2.2.12.1" [] [] [0, -2, 2, -2] 1 "q+(-1+\\beta )q^{3}+q^{5}+(-1-\\beta )q^{7}+\\cdots" "1280.2.a.e" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, -2, 0] 1 "q+\\beta q^{3}-q^{5}-\\beta q^{7}-q^{9}-2\\beta q^{11}+\\cdots" "1280.2.a.f" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, -2, 0] -1 "q+\\beta q^{3}-q^{5}-3\\beta q^{7}-q^{9}+4\\beta q^{11}+\\cdots" "1280.2.a.g" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, -2, 0] -1 "q+\\beta q^{3}-q^{5}+3\\beta q^{7}-q^{9}+2\\beta q^{11}+\\cdots" "1280.2.a.h" 2 10.220851458723889 "2.2.40.1" [] [] [0, 0, -2, 0] -1 "q+\\beta q^{3}-q^{5}+\\beta q^{7}+7q^{9}-6q^{13}+\\cdots" "1280.2.a.i" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, 2, 0] 1 "q+\\beta q^{3}+q^{5}-3\\beta q^{7}-q^{9}+2\\beta q^{11}+\\cdots" "1280.2.a.j" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, 2, 0] -1 "q+\\beta q^{3}+q^{5}+3\\beta q^{7}-q^{9}+4\\beta q^{11}+\\cdots" "1280.2.a.k" 2 10.220851458723889 "2.2.8.1" [] [] [0, 0, 2, 0] -1 "q+\\beta q^{3}+q^{5}+\\beta q^{7}-q^{9}-2\\beta q^{11}+\\cdots" "1280.2.a.l" 2 10.220851458723889 "2.2.40.1" [] [] [0, 0, 2, 0] -1 "q+\\beta q^{3}+q^{5}-\\beta q^{7}+7q^{9}+6q^{13}+\\cdots" "1280.2.a.m" 2 10.220851458723889 "2.2.12.1" [] [] [0, 2, -2, -6] 1 "q+(1+\\beta )q^{3}-q^{5}+(-3-\\beta )q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.2.a.n" 2 10.220851458723889 "2.2.12.1" [] [] [0, 2, -2, -2] -1 "q+(1+\\beta )q^{3}-q^{5}+(-1+\\beta )q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.2.a.o" 2 10.220851458723889 "2.2.12.1" [] [] [0, 2, 2, 2] -1 "q+(1+\\beta )q^{3}+q^{5}+(1-\\beta )q^{7}+(1+2\\beta )q^{9}+\\cdots" "1280.2.a.p" 2 10.220851458723889 "2.2.12.1" [] [] [0, 2, 2, 6] -1 "q+(1+\\beta )q^{3}+q^{5}+(3+\\beta )q^{7}+(1+2\\beta )q^{9}+\\cdots" "1280.2.c.a" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -4, 0] NULL "q+(i-2)q^{5}+3 q^{9}-6 i q^{13}+8 i q^{17}+\\cdots" "1280.2.c.b" 2 10.220851458723889 "2.0.20.1" [-40] [] [0, 0, 0, 0] NULL "q+\\beta q^{5}-2\\beta q^{7}+3q^{9}-2q^{11}-2\\beta q^{13}+\\cdots" "1280.2.c.c" 2 10.220851458723889 "2.0.20.1" [-40] [] [0, 0, 0, 0] NULL "q+\\beta q^{5}+2\\beta q^{7}+3q^{9}+2q^{11}-2\\beta q^{13}+\\cdots" "1280.2.c.d" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 4, 0] NULL "q+(i+2)q^{5}+3 q^{9}-6 i q^{13}-8 i q^{17}+\\cdots" "1280.2.c.e" 4 10.220851458723889 "4.0.2304.2" [] [] [0, 0, -4, 0] NULL "q+\\beta _{2}q^{3}+(-1-\\beta _{1})q^{5}+\\beta _{2}q^{7}-3q^{9}+\\cdots" "1280.2.c.f" 4 10.220851458723889 "4.0.1600.1" [-20] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{3}+\\beta _{3}q^{5}+3\\beta _{1}q^{7}-7q^{9}+\\cdots" "1280.2.c.g" 4 10.220851458723889 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{3}q^{3}+(-\\beta _{1}+\\beta _{2})q^{5}-\\beta _{1}q^{7}+\\cdots" "1280.2.c.h" 4 10.220851458723889 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{3}q^{3}+(\\beta _{1}+\\beta _{2})q^{5}-\\beta _{1}q^{7}-3q^{9}+\\cdots" "1280.2.c.i" 4 10.220851458723889 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+(-\\beta _{1}-\\beta _{2})q^{5}+\\beta _{3}q^{7}+\\cdots" "1280.2.c.j" 4 10.220851458723889 "4.0.1600.1" [-20] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}-\\beta _{3}q^{5}-\\beta _{2}q^{7}+q^{9}+\\beta _{2}q^{15}+\\cdots" "1280.2.c.k" 4 10.220851458723889 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+(\\beta _{1}-\\beta _{2})q^{5}+\\beta _{3}q^{7}+q^{9}+\\cdots" "1280.2.c.l" 4 10.220851458723889 "4.0.6400.1" [-40] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{5}-\\beta _{1}q^{7}+3q^{9}-\\beta _{3}q^{11}+\\cdots" "1280.2.c.m" 4 10.220851458723889 "4.0.2304.2" [] [] [0, 0, 4, 0] NULL "q+\\beta _{2}q^{3}+(1+\\beta _{1})q^{5}-\\beta _{2}q^{7}-3q^{9}+\\cdots" "1280.2.d.a" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, -8] NULL "q+i q^{5}-4 q^{7}+3 q^{9}-4 i q^{11}+\\cdots" "1280.2.d.b" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, -4] NULL "q+2 i q^{3}+i q^{5}-2 q^{7}-q^{9}-4 i q^{11}+\\cdots" "1280.2.d.c" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, -4] NULL "q+2 i q^{3}-i q^{5}-2 q^{7}-q^{9}-2 i q^{13}+\\cdots" "1280.2.d.d" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, -4] NULL "q-i q^{5}-2 q^{7}+3 q^{9}-6 i q^{11}+\\cdots" "1280.2.d.e" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+2 i q^{3}+i q^{5}-q^{9}-2 i q^{11}+\\cdots" "1280.2.d.f" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+2 i q^{3}-i q^{5}-q^{9}-2 i q^{11}+\\cdots" "1280.2.d.g" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 4] NULL "q+2 i q^{3}+i q^{5}+2 q^{7}-q^{9}+2 i q^{13}+\\cdots" "1280.2.d.h" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 4] NULL "q+2 i q^{3}-i q^{5}+2 q^{7}-q^{9}-4 i q^{11}+\\cdots" "1280.2.d.i" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 4] NULL "q+i q^{5}+2 q^{7}+3 q^{9}-6 i q^{11}+\\cdots" "1280.2.d.j" 2 10.220851458723889 "2.0.4.1" [] [] [0, 0, 0, 8] NULL "q+i q^{5}+4 q^{7}+3 q^{9}+4 i q^{11}+\\cdots" "1280.2.d.k" 4 10.220851458723889 "4.0.400.1" [] [] [0, 0, 0, -4] NULL "q+(-\\beta _{1}+\\beta _{2})q^{3}-\\beta _{1}q^{5}+(-1-\\beta _{3})q^{7}+\\cdots" "1280.2.d.l" 4 10.220851458723889 "4.0.256.1" [] [] [0, 0, 0, 0] NULL "q+\\beta_{2} q^{3}+\\beta_1 q^{5}-\\beta_{3} q^{7}-5 q^{9}+\\cdots" "1280.2.d.m" 4 10.220851458723889 "4.0.400.1" [] [] [0, 0, 0, 4] NULL "q+(-\\beta _{1}+\\beta _{2})q^{3}+\\beta _{1}q^{5}+(1+\\beta _{3})q^{7}+\\cdots" "1280.2.f.a" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, -4, 0] NULL "q-2 q^{3}+(-i-2)q^{5}-2 i q^{7}+q^{9}+\\cdots" "1280.2.f.b" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, 4, 0] NULL "q-2 q^{3}+(-i+2)q^{5}-2 i q^{7}+q^{9}+\\cdots" "1280.2.f.c" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -4, 0] NULL "q+(i-2)q^{5}-3 q^{9}+4 q^{13}-8 i q^{17}+\\cdots" "1280.2.f.d" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 4, 0] NULL "q+(i+2)q^{5}-3 q^{9}-4 q^{13}+8 i q^{17}+\\cdots" "1280.2.f.e" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, -4, 0] NULL "q+2 q^{3}+(-i-2)q^{5}+2 i q^{7}+q^{9}+\\cdots" "1280.2.f.f" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, 4, 0] NULL "q+2 q^{3}+(-i+2)q^{5}+2 i q^{7}+q^{9}+\\cdots" "1280.2.f.g" 4 10.220851458723889 "4.0.400.1" [-20] [] [0, -4, 0, 0] NULL "q+(-1-\\beta _{3})q^{3}-\\beta _{2}q^{5}+(3\\beta _{1}-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.f.h" 4 10.220851458723889 "4.0.400.1" [-20] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{3})q^{3}-\\beta _{2}q^{5}+(-3\\beta _{1}+\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.f.i" 6 10.220851458723889 "6.0.350464.1" [] [] [0, -4, 0, 0] NULL "q+(-1-\\beta _{1})q^{3}+(\\beta _{1}+\\beta _{3})q^{5}+(\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.f.j" 6 10.220851458723889 "6.0.350464.1" [] [] [0, -4, 0, 0] NULL "q+(-1-\\beta _{1})q^{3}+(-\\beta _{1}+\\beta _{5})q^{5}+(\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.f.k" 6 10.220851458723889 "6.0.350464.1" [] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{1})q^{3}+(\\beta _{1}+\\beta _{3})q^{5}+(-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.f.l" 6 10.220851458723889 "6.0.350464.1" [] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{1})q^{3}+(-\\beta _{1}+\\beta _{5})q^{5}+(-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.j.a" 16 10.220851458723889 NULL [] [] [0, 0, 0, 0] NULL "q+(-\\beta _{9}-\\beta _{12})q^{3}+(\\beta _{5}-\\beta _{6})q^{5}+\\beta _{2}q^{7}+\\cdots" "1280.2.j.b" 16 10.220851458723889 NULL [] [] [0, 0, 0, 0] NULL "q+(-\\beta _{9}-\\beta _{12})q^{3}+(-\\beta _{5}+\\beta _{6})q^{5}+\\cdots" "1280.2.j.c" 32 10.220851458723889 NULL [] [] [0, 0, 0, 0] NULL NULL "1280.2.j.d" 32 10.220851458723889 NULL [] [] [0, 0, 0, 0] NULL NULL "1280.2.l.a" 8 10.220851458723889 "8.0.5308416.1" [] [] [0, -4, 0, 0] NULL "q+(\\beta_{6}+\\beta_{5})q^{3}-\\beta_1 q^{5}+(\\beta_{5}+\\beta_{4}+\\cdots-2\\beta_1)q^{7}+\\cdots" "1280.2.l.b" 8 10.220851458723889 "8.0.349241344.2" [] [] [0, -4, 0, 0] NULL "q+(-1+\\beta _{4})q^{3}-\\beta _{2}q^{5}+(-2\\beta _{6}-\\beta _{7})q^{7}+\\cdots" "1280.2.l.c" 8 10.220851458723889 "8.0.349241344.2" [] [] [0, -4, 0, 0] NULL "q+(-1+\\beta _{4})q^{3}+\\beta _{2}q^{5}+(2\\beta _{6}+\\beta _{7})q^{7}+\\cdots" "1280.2.l.d" 8 10.220851458723889 "8.0.5308416.1" [] [] [0, -4, 0, 0] NULL "q+(\\beta_{6}+\\beta_{5})q^{3}+\\beta_1 q^{5}+(-\\beta_{5}-\\beta_{4}+\\cdots+2\\beta_1)q^{7}+\\cdots" "1280.2.l.e" 8 10.220851458723889 "8.0.349241344.2" [] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{5}+\\beta _{6})q^{3}+\\beta _{3}q^{5}+(-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.2.l.f" 8 10.220851458723889 "8.0.5308416.1" [] [] [0, 4, 0, 0] NULL "q+(\\beta_{3}-\\beta_{2}+\\beta_1+1)q^{3}+\\beta_{5} q^{5}+\\cdots" "1280.2.l.g" 8 10.220851458723889 "8.0.5308416.1" [] [] [0, 4, 0, 0] NULL "q+(\\beta_{3}-\\beta_{2}+\\beta_1+1)q^{3}-\\beta_{5} q^{5}+\\cdots" "1280.2.l.h" 8 10.220851458723889 "8.0.349241344.2" [] [] [0, 4, 0, 0] NULL "q+(1+\\beta _{5}+\\beta _{6})q^{3}-\\beta _{3}q^{5}+(\\beta _{2}+\\beta _{3}+\\cdots)q^{7}+\\cdots" "1280.2.n.a" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, -2, -4] NULL "q+(-2 i-2)q^{3}+(-2 i-1)q^{5}+\\cdots" "1280.2.n.b" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, 2, 4] NULL "q+(-2 i-2)q^{3}+(2 i+1)q^{5}+(-2 i+2)q^{7}+\\cdots" "1280.2.n.c" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, -4, 2] NULL "q+(-i-1)q^{3}+(i-2)q^{5}+(-i+1)q^{7}+\\cdots" "1280.2.n.d" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, 4, -2] NULL "q+(-i-1)q^{3}+(-i+2)q^{5}+(i-1)q^{7}+\\cdots" "1280.2.n.e" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q+(-2 i-1)q^{5}-3 i q^{9}+(5 i-5)q^{13}+\\cdots" "1280.2.n.f" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q+(2 i-1)q^{5}-3 i q^{9}+(i-1)q^{13}+\\cdots" "1280.2.n.g" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+(-2 i+1)q^{5}-3 i q^{9}+(-i+1)q^{13}+\\cdots" "1280.2.n.h" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+(2 i+1)q^{5}-3 i q^{9}+(-5 i+5)q^{13}+\\cdots" "1280.2.n.i" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, -4, -2] NULL "q+(i+1)q^{3}+(i-2)q^{5}+(i-1)q^{7}+\\cdots" "1280.2.n.j" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, 4, 2] NULL "q+(i+1)q^{3}+(-i+2)q^{5}+(-i+1)q^{7}+\\cdots" "1280.2.n.k" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, -2, 4] NULL "q+(2 i+2)q^{3}+(-2 i-1)q^{5}+(-2 i+2)q^{7}+\\cdots" "1280.2.n.l" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, 2, -4] NULL "q+(2 i+2)q^{3}+(2 i+1)q^{5}+(2 i-2)q^{7}+\\cdots" "1280.2.n.m" 8 10.220851458723889 "8.0.4000000.1" [] [] [0, -4, 0, 0] NULL "q+(\\beta_{2}-1)q^{3}+(\\beta_{6}-\\beta_{4})q^{5}-\\beta_{6} q^{7}+\\cdots" "1280.2.n.n" 8 10.220851458723889 "8.0.49787136.1" [] [] [0, 0, 0, -4] NULL "q+(-\\beta _{2}-\\beta _{7})q^{3}+(\\beta _{2}+\\beta _{4}+\\beta _{7})q^{5}+\\cdots" "1280.2.n.o" 8 10.220851458723889 "8.0.40960000.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{6}q^{3}-\\beta _{4}q^{5}-\\beta _{2}q^{7}-\\beta _{3}q^{9}+\\cdots" "1280.2.n.p" 8 10.220851458723889 "8.0.49787136.1" [] [] [0, 0, 0, 4] NULL "q+(-\\beta _{2}-\\beta _{7})q^{3}+(-\\beta _{2}-\\beta _{4}-\\beta _{7})q^{5}+\\cdots" "1280.2.n.q" 8 10.220851458723889 "8.0.4000000.1" [] [] [0, 4, 0, 0] NULL "q+(-\\beta_{2}+1)q^{3}+(-\\beta_{6}+\\beta_{4})q^{5}+\\cdots" "1280.2.n.r" 12 10.220851458723889 "12.0.125772815663104.1" [] [] [0, 0, -4, 0] NULL "q+\\beta _{2}q^{3}+\\beta _{7}q^{5}+\\beta _{6}q^{7}+(\\beta _{5}+\\beta _{7}+\\cdots)q^{9}+\\cdots" "1280.2.n.s" 12 10.220851458723889 "12.0.125772815663104.1" [] [] [0, 0, 4, 0] NULL "q+\\beta _{2}q^{3}-\\beta _{7}q^{5}-\\beta _{6}q^{7}+(\\beta _{5}+\\beta _{7}+\\cdots)q^{9}+\\cdots" "1280.2.o.a" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, -2, 4] NULL "q+(2 i-2)q^{3}+(-2 i-1)q^{5}+(-2 i+2)q^{7}+\\cdots" "1280.2.o.b" 2 10.220851458723889 "2.0.4.1" [] [] [0, -4, 2, -4] NULL "q+(2 i-2)q^{3}+(2 i+1)q^{5}+(2 i-2)q^{7}+\\cdots" "1280.2.o.c" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, -4, -6] NULL "q+(i-1)q^{3}+(i-2)q^{5}+(3 i-3)q^{7}+\\cdots" "1280.2.o.d" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, -4, -2] NULL "q+(i-1)q^{3}+(-i-2)q^{5}+(i-1)q^{7}+\\cdots" "1280.2.o.e" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, 4, 2] NULL "q+(i-1)q^{3}+(i+2)q^{5}+(-i+1)q^{7}+\\cdots" "1280.2.o.f" 2 10.220851458723889 "2.0.4.1" [] [] [0, -2, 4, 6] NULL "q+(i-1)q^{3}+(-i+2)q^{5}+(-3 i+3)q^{7}+\\cdots" "1280.2.o.g" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q+(-2 i-1)q^{5}+3 i q^{9}+(i+1)q^{13}+\\cdots" "1280.2.o.h" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, -2, 0] NULL "q+(2 i-1)q^{5}+3 i q^{9}+(5 i+5)q^{13}+\\cdots" "1280.2.o.i" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+(-2 i+1)q^{5}+3 i q^{9}+(-5 i-5)q^{13}+\\cdots" "1280.2.o.j" 2 10.220851458723889 "2.0.4.1" [-4] [] [0, 0, 2, 0] NULL "q+(2 i+1)q^{5}+3 i q^{9}+(-i-1)q^{13}+\\cdots" "1280.2.o.k" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, -4, 2] NULL "q+(-i+1)q^{3}+(-i-2)q^{5}+(-i+1)q^{7}+\\cdots" "1280.2.o.l" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, -4, 6] NULL "q+(-i+1)q^{3}+(i-2)q^{5}+(-3 i+3)q^{7}+\\cdots" "1280.2.o.m" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, 4, -6] NULL "q+(-i+1)q^{3}+(-i+2)q^{5}+(3 i-3)q^{7}+\\cdots" "1280.2.o.n" 2 10.220851458723889 "2.0.4.1" [] [] [0, 2, 4, -2] NULL "q+(-i+1)q^{3}+(i+2)q^{5}+(i-1)q^{7}+\\cdots" "1280.2.o.o" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, -2, -4] NULL "q+(-2 i+2)q^{3}+(-2 i-1)q^{5}+\\cdots" "1280.2.o.p" 2 10.220851458723889 "2.0.4.1" [] [] [0, 4, 2, 4] NULL "q+(-2 i+2)q^{3}+(2 i+1)q^{5}+(-2 i+2)q^{7}+\\cdots" "1280.2.o.q" 4 10.220851458723889 "4.0.144.1" [] [] [0, 0, -8, 0] NULL "q-\\beta_{2} q^{3}+(-\\beta_1-2)q^{5}+\\beta_{2} q^{7}+\\cdots" "1280.2.o.r" 4 10.220851458723889 "4.0.144.1" [] [] [0, 0, 8, 0] NULL "q-\\beta_{2} q^{3}+(\\beta_1+2)q^{5}-\\beta_{2} q^{7}+\\cdots" "1280.2.o.s" 12 10.220851458723889 "12.0.74906393903104.1" [] [] [0, 0, -4, -8] NULL "q-\\beta _{2}q^{3}-\\beta _{7}q^{5}+(-1-\\beta _{2}+\\beta _{4}+\\cdots)q^{7}+\\cdots" "1280.2.o.t" 12 10.220851458723889 "12.0.74906393903104.1" [] [] [0, 0, -4, 8] NULL "q+\\beta _{3}q^{3}-\\beta _{6}q^{5}+(1+\\beta _{3}+\\beta _{5}-\\beta _{7}+\\cdots)q^{7}+\\cdots" "1280.2.o.u" 12 10.220851458723889 "12.0.74906393903104.1" [] [] [0, 0, 4, -8] NULL "q+\\beta _{3}q^{3}+\\beta _{6}q^{5}+(-1-\\beta _{3}-\\beta _{5}+\\cdots)q^{7}+\\cdots" "1280.2.o.v" 12 10.220851458723889 "12.0.74906393903104.1" [] [] [0, 0, 4, 8] NULL "q-\\beta _{2}q^{3}+\\beta _{7}q^{5}+(1+\\beta _{2}-\\beta _{4}-\\beta _{6}+\\cdots)q^{7}+\\cdots" "1280.2.q.a" 16 10.220851458723889 "16.0.9349208943630483456.9" [] [] [0, 0, -8, 0] NULL "q-\\beta _{5}q^{3}+(-\\beta _{4}+\\beta _{9}-\\beta _{15})q^{5}+(\\beta _{1}+\\cdots)q^{7}+\\cdots" "1280.2.q.b" 16 10.220851458723889 "16.0.9349208943630483456.9" [] [] [0, 0, 8, 0] NULL "q-\\beta _{5}q^{3}+(\\beta _{4}-\\beta _{9}+\\beta _{15})q^{5}+(-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1280.2.q.c" 32 10.220851458723889 NULL [] [] [0, 0, -8, 0] NULL NULL "1280.2.q.d" 32 10.220851458723889 NULL [] [] [0, 0, 8, 0] NULL NULL "1280.2.s.a" 16 10.220851458723889 NULL [] [] [0, 0, -8, 0] NULL "q+(\\beta _{10}-\\beta _{11})q^{3}+(-1+\\beta _{4})q^{5}-\\beta _{2}q^{7}+\\cdots" "1280.2.s.b" 16 10.220851458723889 NULL [] [] [0, 0, 8, 0] NULL "q+(\\beta _{10}-\\beta _{11})q^{3}+(1-\\beta _{4})q^{5}+\\beta _{2}q^{7}+\\cdots" "1280.2.s.c" 32 10.220851458723889 NULL [] [] [0, 0, -8, 0] NULL NULL "1280.2.s.d" 32 10.220851458723889 NULL [] [] [0, 0, 8, 0] NULL NULL "1280.3.b.a" 4 34.877473838118 "4.0.46400.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}-\\beta _{3}q^{5}+(\\beta _{1}-\\beta _{2})q^{7}+(-5+\\cdots)q^{9}+\\cdots" "1280.3.b.b" 4 34.877473838118 "4.0.46400.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+\\beta _{3}q^{5}+(-\\beta _{1}+\\beta _{2})q^{7}+\\cdots" "1280.3.b.c" 4 34.877473838118 "4.0.1600.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}+\\beta _{3}q^{7}-q^{9}-2\\beta _{2}q^{13}+\\cdots" "1280.3.b.d" 4 34.877473838118 "4.0.1600.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{3}q^{5}+3\\beta _{2}q^{7}+7q^{9}+\\cdots" "1280.3.b.e" 8 34.877473838118 "8.0.2342560000.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{6}q^{3}-\\beta _{1}q^{5}+(-2\\beta _{5}+\\beta _{7})q^{7}+\\cdots" "1280.3.b.f" 8 34.877473838118 "8.0.2152960000.5" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}+(\\beta _{1}-\\beta _{5})q^{7}+(-5+\\cdots)q^{9}+\\cdots" "1280.3.b.g" 8 34.877473838118 "8.0.2152960000.5" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}-\\beta _{2}q^{5}+(-\\beta _{1}+\\beta _{5})q^{7}+\\cdots" "1280.3.b.h" 8 34.877473838118 "8.0.12960000.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{3}-\\beta _{1}q^{5}+(3\\beta _{3}-\\beta _{5})q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.3.b.i" 16 34.877473838118 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{4}q^{3}-\\beta _{1}q^{5}+\\beta _{12}q^{7}+(-3+\\beta _{2}+\\cdots)q^{9}+\\cdots" "1280.3.e.a" 2 34.877473838118 "2.0.4.1" [-4] [] [0, 0, -8, 0] NULL "q+(3 i-4)q^{5}+9 q^{9}-24 q^{13}+16 i q^{17}+\\cdots" "1280.3.e.b" 2 34.877473838118 "2.0.4.1" [-20] [] [0, 0, 0, -8] NULL "q+4 i q^{3}-5 i q^{5}-4 q^{7}-7 q^{9}+\\cdots" "1280.3.e.c" 2 34.877473838118 "2.0.4.1" [-20] [] [0, 0, 0, 8] NULL "q+4 i q^{3}+5 i q^{5}+4 q^{7}-7 q^{9}+\\cdots" "1280.3.e.d" 2 34.877473838118 "2.0.4.1" [-4] [] [0, 0, 8, 0] NULL "q+(3 i+4)q^{5}+9 q^{9}+24 q^{13}-16 i q^{17}+\\cdots" "1280.3.e.e" 4 34.877473838118 "4.0.400.1" [-20] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+5\\beta _{1}q^{5}+3\\beta _{3}q^{7}-11q^{9}+\\cdots" "1280.3.e.f" 6 34.877473838118 "6.0.1827904.1" [] [] [0, 0, -8, -12] NULL "q+\\beta _{1}q^{3}+(-2-\\beta _{2}+\\beta _{5})q^{5}+(-2+\\cdots)q^{7}+\\cdots" "1280.3.e.g" 6 34.877473838118 "6.0.1827904.1" [] [] [0, 0, -8, 12] NULL "q+\\beta _{1}q^{3}+(-1+\\beta _{4})q^{5}+(2+\\beta _{4}+\\beta _{5})q^{7}+\\cdots" "1280.3.e.h" 6 34.877473838118 "6.0.1827904.1" [] [] [0, 0, 8, -12] NULL "q+\\beta _{1}q^{3}+(1-\\beta _{4})q^{5}+(-2-\\beta _{4}-\\beta _{5})q^{7}+\\cdots" "1280.3.e.i" 6 34.877473838118 "6.0.1827904.1" [] [] [0, 0, 8, 12] NULL "q+\\beta _{1}q^{3}+(2+\\beta _{2}-\\beta _{5})q^{5}+(2+\\beta _{4}+\\cdots)q^{7}+\\cdots" "1280.3.e.j" 8 34.877473838118 "8.0.5308416.1" [] [] [0, 0, 0, 0] NULL "q+\\beta_{2} q^{3}+(-\\beta_{3}-\\beta_1)q^{5}-3\\beta_{4} q^{7}+\\cdots" "1280.3.e.k" 24 34.877473838118 NULL [] [] [0, 0, 0, 0] NULL NULL "1280.3.e.l" 24 34.877473838118 NULL [] [] [0, 0, 0, 0] NULL NULL "1280.3.g.a" 4 34.877473838118 "4.0.400.1" [] [] [0, -12, 0, 0] NULL "q+(-3-\\beta _{3})q^{3}-\\beta _{2}q^{5}+(\\beta _{1}-5\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.3.g.b" 4 34.877473838118 "4.0.400.1" [] [] [0, -4, 0, 0] NULL "q+(-1+\\beta _{3})q^{3}-\\beta _{2}q^{5}+(-\\beta _{1}+\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.3.g.c" 4 34.877473838118 "4.0.400.1" [] [] [0, 4, 0, 0] NULL "q+(1-\\beta _{3})q^{3}+\\beta _{2}q^{5}+(-\\beta _{1}+\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.3.g.d" 4 34.877473838118 "4.0.400.1" [] [] [0, 12, 0, 0] NULL "q+(3+\\beta _{3})q^{3}-\\beta _{2}q^{5}+(-\\beta _{1}+5\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.3.g.e" 8 34.877473838118 "8.0.4000000.1" [] [] [0, 0, 0, 0] NULL "q-\\beta_{4} q^{3}+\\beta_1 q^{5}+(-\\beta_{5}-\\beta_{3})q^{7}+\\cdots" "1280.3.g.f" 8 34.877473838118 "8.0.12960000.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{3}+\\beta _{4}q^{5}+\\beta _{6}q^{7}+(9+3\\beta _{1}+\\cdots)q^{9}+\\cdots" "1280.3.g.g" 16 34.877473838118 NULL [] [] [0, -16, 0, 0] NULL "q+(-1+\\beta _{4})q^{3}-\\beta _{7}q^{5}+(\\beta _{7}+\\beta _{9}+\\cdots)q^{7}+\\cdots" "1280.3.g.h" 16 34.877473838118 NULL [] [] [0, 16, 0, 0] NULL "q+(1-\\beta _{4})q^{3}-\\beta _{7}q^{5}+(-\\beta _{7}-\\beta _{9}+\\cdots)q^{7}+\\cdots" "1280.3.h.a" 2 34.877473838118 "2.0.4.1" [-4] [] [0, 0, -8, 0] NULL "q+(-3 i-4)q^{5}-9 q^{9}+10 i q^{13}+\\cdots" "1280.3.h.b" 2 34.877473838118 "2.0.4.1" [-40] [] [0, 0, 0, -12] NULL "q+5 i q^{5}-6 q^{7}-9 q^{9}-18 i q^{11}+\\cdots" "1280.3.h.c" 2 34.877473838118 "2.0.4.1" [-40] [] [0, 0, 0, 12] NULL "q+5 i q^{5}+6 q^{7}-9 q^{9}+18 i q^{11}+\\cdots" "1280.3.h.d" 2 34.877473838118 "2.0.4.1" [-4] [] [0, 0, 8, 0] NULL "q+(3 i+4)q^{5}-9 q^{9}-10 i q^{13}+\\cdots" "1280.3.h.e" 4 34.877473838118 "4.4.1600.1" [-20] [] [0, 0, -20, 0] NULL "q-\\beta _{1}q^{3}-5q^{5}+(\\beta _{1}+\\beta _{2})q^{7}+(9+\\beta _{3})q^{9}+\\cdots" "1280.3.h.f" 4 34.877473838118 "4.0.6400.3" [-40] [] [0, 0, 0, 0] NULL "q+5\\beta _{1}q^{5}-\\beta _{3}q^{7}-9q^{9}+\\beta _{2}q^{11}+\\cdots" "1280.3.h.g" 4 34.877473838118 "4.4.1600.1" [-20] [] [0, 0, 20, 0] NULL "q-\\beta _{1}q^{3}+5q^{5}+(-\\beta _{1}-\\beta _{2})q^{7}+(9+\\cdots)q^{9}+\\cdots" "1280.3.h.h" 8 34.877473838118 "8.0.493995311104.27" [] [] [0, 0, -12, 0] NULL "q+\\beta _{4}q^{3}+(-2+\\beta _{1}-\\beta _{2})q^{5}+(-\\beta _{4}+\\cdots)q^{7}+\\cdots" "1280.3.h.i" 8 34.877473838118 "8.0.691798081536.3" [] [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{3}+\\beta _{6}q^{5}-\\beta _{2}q^{7}-3q^{9}-\\beta _{7}q^{11}+\\cdots" "1280.3.h.j" 8 34.877473838118 "8.0.12745506816.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{3}+(\\beta _{1}+\\beta _{2})q^{5}-\\beta _{3}q^{7}+5q^{9}+\\cdots" "1280.3.h.k" 8 34.877473838118 "8.0.599695360000.8" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+(-\\beta _{1}-\\beta _{5})q^{5}-\\beta _{3}q^{7}+\\cdots" "1280.3.h.l" 8 34.877473838118 "8.0.493995311104.27" [] [] [0, 0, 12, 0] NULL "q-\\beta _{4}q^{3}+(1+\\beta _{1})q^{5}+(-\\beta _{4}+\\beta _{5}+\\cdots)q^{7}+\\cdots" "1280.3.h.m" 16 34.877473838118 "16.0.741637881856000000000000.5" [] [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{3}-\\beta _{10}q^{5}-\\beta _{9}q^{7}+(5-\\beta _{3}+\\cdots)q^{9}+\\cdots" "1280.3.h.n" 16 34.877473838118 NULL [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+\\beta _{4}q^{5}-\\beta _{13}q^{7}+(6+\\beta _{3}+\\cdots)q^{9}+\\cdots" "1280.4.a.a" 2 75.52244480734798 "2.2.76.1" [] [] [0, -6, -10, 10] 1 "q+(-3+\\beta )q^{3}-5q^{5}+(5-5\\beta )q^{7}+\\cdots" "1280.4.a.b" 2 75.52244480734798 "2.2.76.1" [] [] [0, -6, 10, -10] -1 "q+(-3+\\beta )q^{3}+5q^{5}+(-5+5\\beta )q^{7}+\\cdots" "1280.4.a.c" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, -10, 0] -1 "q+\\beta q^{3}-5q^{5}+15\\beta q^{7}-5^{2}q^{9}-22\\beta q^{11}+\\cdots" "1280.4.a.d" 2 75.52244480734798 "2.2.40.1" [] [] [0, 0, -10, 0] -1 "q+\\beta q^{3}-5q^{5}+5\\beta q^{7}-17q^{9}+8\\beta q^{11}+\\cdots" "1280.4.a.e" 2 75.52244480734798 "2.2.40.1" [] [] [0, 0, -10, 0] 1 "q+\\beta q^{3}-5q^{5}+\\beta q^{7}-17q^{9}+20\\beta q^{11}+\\cdots" "1280.4.a.f" 2 75.52244480734798 "2.2.136.1" [] [] [0, 0, -10, 0] -1 "q+\\beta q^{3}-5q^{5}+\\beta q^{7}+7q^{9}-4\\beta q^{11}+\\cdots" "1280.4.a.g" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, -10, 0] 1 "q+\\beta q^{3}-5q^{5}-3\\beta q^{7}+23q^{9}-8\\beta q^{11}+\\cdots" "1280.4.a.h" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, -10, 0] 1 "q+7\\beta q^{3}-5q^{5}+13\\beta q^{7}+71q^{9}+\\cdots" "1280.4.a.i" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, 10, 0] 1 "q+\\beta q^{3}+5q^{5}-15\\beta q^{7}-5^{2}q^{9}-22\\beta q^{11}+\\cdots" "1280.4.a.j" 2 75.52244480734798 "2.2.40.1" [] [] [0, 0, 10, 0] 1 "q+\\beta q^{3}+5q^{5}-\\beta q^{7}-17q^{9}+20\\beta q^{11}+\\cdots" "1280.4.a.k" 2 75.52244480734798 "2.2.40.1" [] [] [0, 0, 10, 0] -1 "q+\\beta q^{3}+5q^{5}-5\\beta q^{7}-17q^{9}+8\\beta q^{11}+\\cdots" "1280.4.a.l" 2 75.52244480734798 "2.2.136.1" [] [] [0, 0, 10, 0] -1 "q+\\beta q^{3}+5q^{5}-\\beta q^{7}+7q^{9}-4\\beta q^{11}+\\cdots" "1280.4.a.m" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, 10, 0] 1 "q+\\beta q^{3}+5q^{5}+3\\beta q^{7}+23q^{9}-8\\beta q^{11}+\\cdots" "1280.4.a.n" 2 75.52244480734798 "2.2.8.1" [] [] [0, 0, 10, 0] -1 "q+7\\beta q^{3}+5q^{5}-13\\beta q^{7}+71q^{9}+\\cdots" "1280.4.a.o" 2 75.52244480734798 "2.2.76.1" [] [] [0, 6, -10, -10] -1 "q+(3+\\beta )q^{3}-5q^{5}+(-5-5\\beta )q^{7}+\\cdots" "1280.4.a.p" 2 75.52244480734798 "2.2.76.1" [] [] [0, 6, 10, 10] 1 "q+(3+\\beta )q^{3}+5q^{5}+(5+5\\beta )q^{7}+(1+\\cdots)q^{9}+\\cdots" "1280.4.a.q" 4 75.52244480734798 "4.4.190224.1" [] [] [0, 0, -20, -32] -1 "q+\\beta _{1}q^{3}-5q^{5}+(-8+\\beta _{1}-3\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.4.a.r" 4 75.52244480734798 "4.4.1600.1" [] [] [0, 0, -20, 0] 1 "q+\\beta _{1}q^{3}-5q^{5}+(-5\\beta _{1}-\\beta _{2})q^{7}+\\cdots" "1280.4.a.s" 4 75.52244480734798 "4.4.13845312.2" [] [] [0, 0, -20, 0] 1 "q-\\beta _{2}q^{3}-5q^{5}+(-\\beta _{1}-\\beta _{2})q^{7}+(19+\\cdots)q^{9}+\\cdots" "1280.4.a.t" 4 75.52244480734798 "4.4.53824.1" [] [] [0, 0, -20, 0] -1 "q-\\beta _{2}q^{3}-5q^{5}+(-3\\beta _{1}+\\beta _{2})q^{7}+\\cdots" "1280.4.a.u" 4 75.52244480734798 "4.4.190224.1" [] [] [0, 0, -20, 32] 1 "q-\\beta _{1}q^{3}-5q^{5}+(8-\\beta _{1}+3\\beta _{2})q^{7}+\\cdots" "1280.4.a.v" 4 75.52244480734798 "4.4.190224.1" [] [] [0, 0, 20, -32] -1 "q-\\beta _{1}q^{3}+5q^{5}+(-8+\\beta _{1}-3\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.4.a.w" 4 75.52244480734798 "4.4.1600.1" [] [] [0, 0, 20, 0] -1 "q-\\beta _{1}q^{3}+5q^{5}+(-5\\beta _{1}-\\beta _{2})q^{7}+\\cdots" "1280.4.a.x" 4 75.52244480734798 "4.4.13845312.2" [] [] [0, 0, 20, 0] 1 "q-\\beta _{2}q^{3}+5q^{5}+(\\beta _{1}+\\beta _{2})q^{7}+(19+\\cdots)q^{9}+\\cdots" "1280.4.a.y" 4 75.52244480734798 "4.4.53824.1" [] [] [0, 0, 20, 0] 1 "q-\\beta _{2}q^{3}+5q^{5}+(3\\beta _{1}-\\beta _{2})q^{7}+31q^{9}+\\cdots" "1280.4.a.z" 4 75.52244480734798 "4.4.190224.1" [] [] [0, 0, 20, 32] 1 "q+\\beta _{1}q^{3}+5q^{5}+(8-\\beta _{1}+3\\beta _{2})q^{7}+\\cdots" "1280.4.a.ba" 6 75.52244480734798 NULL [] [] [0, -6, -30, 14] -1 "q+(-1-\\beta _{1})q^{3}-5q^{5}+(2+\\beta _{4})q^{7}+\\cdots" "1280.4.a.bb" 6 75.52244480734798 NULL [] [] [0, -6, 30, -14] -1 "q+(-1-\\beta _{1})q^{3}+5q^{5}+(-2-\\beta _{4}+\\cdots)q^{7}+\\cdots" "1280.4.a.bc" 6 75.52244480734798 NULL [] [] [0, 6, -30, -14] 1 "q+(1+\\beta _{1})q^{3}-5q^{5}+(-2-\\beta _{4})q^{7}+\\cdots" "1280.4.a.bd" 6 75.52244480734798 NULL [] [] [0, 6, 30, 14] 1 "q+(1+\\beta _{1})q^{3}+5q^{5}+(2+\\beta _{4})q^{7}+(9+\\cdots)q^{9}+\\cdots" "1280.4.d.a" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -68] NULL "q+6 i q^{3}+5 i q^{5}-34 q^{7}-9 q^{9}+\\cdots" "1280.4.d.b" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -36] NULL "q+10 i q^{3}-5 i q^{5}-18 q^{7}-73 q^{9}+\\cdots" "1280.4.d.c" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -32] NULL "q+4 i q^{3}+5 i q^{5}-16 q^{7}+11 q^{9}+\\cdots" "1280.4.d.d" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -32] NULL "q+4 i q^{3}-5 i q^{5}-16 q^{7}+11 q^{9}+\\cdots" "1280.4.d.e" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -12] NULL "q+2 i q^{3}+5 i q^{5}-6 q^{7}+23 q^{9}+\\cdots" "1280.4.d.f" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -12] NULL "q+2 i q^{3}-5 i q^{5}-6 q^{7}+23 q^{9}+\\cdots" "1280.4.d.g" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -8] NULL "q+8 i q^{3}-5 i q^{5}-4 q^{7}-37 q^{9}+\\cdots" "1280.4.d.h" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, -4] NULL "q+8 i q^{3}-5 i q^{5}-2 q^{7}-37 q^{9}+\\cdots" "1280.4.d.i" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 4] NULL "q+8 i q^{3}+5 i q^{5}+2 q^{7}-37 q^{9}+\\cdots" "1280.4.d.j" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 8] NULL "q+8 i q^{3}+5 i q^{5}+4 q^{7}-37 q^{9}+\\cdots" "1280.4.d.k" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 12] NULL "q+2 i q^{3}+5 i q^{5}+6 q^{7}+23 q^{9}+\\cdots" "1280.4.d.l" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 12] NULL "q+2 i q^{3}-5 i q^{5}+6 q^{7}+23 q^{9}+\\cdots" "1280.4.d.m" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 32] NULL "q+4 i q^{3}+5 i q^{5}+16 q^{7}+11 q^{9}+\\cdots" "1280.4.d.n" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 32] NULL "q+4 i q^{3}-5 i q^{5}+16 q^{7}+11 q^{9}+\\cdots" "1280.4.d.o" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 36] NULL "q+10 i q^{3}+5 i q^{5}+18 q^{7}-73 q^{9}+\\cdots" "1280.4.d.p" 2 75.52244480734798 "2.0.4.1" [] [] [0, 0, 0, 68] NULL "q+6 i q^{3}-5 i q^{5}+34 q^{7}-9 q^{9}+\\cdots" "1280.4.d.q" 4 75.52244480734798 "4.0.2304.2" [] [] [0, 0, 0, -16] NULL "q+(4\\beta _{1}-\\beta _{2})q^{3}-5\\beta _{1}q^{5}+(-4-5\\beta _{3})q^{7}+\\cdots" "1280.4.d.r" 4 75.52244480734798 "4.0.41616.2" [] [] [0, 0, 0, -16] NULL "q+2\\beta _{1}q^{3}-5\\beta _{1}q^{5}+(-4+\\beta _{2})q^{7}+\\cdots" "1280.4.d.s" 4 75.52244480734798 "4.0.7056.1" [] [] [0, 0, 0, -12] NULL "q+(-\\beta _{1}+\\beta _{2})q^{3}+5\\beta _{1}q^{5}+(-3-3\\beta _{3})q^{7}+\\cdots" "1280.4.d.t" 4 75.52244480734798 "4.0.2704.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+5\\beta _{1}q^{5}+\\beta _{3}q^{7}-5^{2}q^{9}+\\cdots" "1280.4.d.u" 4 75.52244480734798 "4.0.6400.3" [] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{3}-5\\beta _{1}q^{5}-3\\beta _{3}q^{7}-13q^{9}+\\cdots" "1280.4.d.v" 4 75.52244480734798 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{3}+5\\beta _{1}q^{5}-7\\beta _{3}q^{7}+7q^{9}+\\cdots" "1280.4.d.w" 4 75.52244480734798 "4.0.7056.1" [] [] [0, 0, 0, 12] NULL "q+(-\\beta _{1}+\\beta _{2})q^{3}-5\\beta _{1}q^{5}+(3+3\\beta _{3})q^{7}+\\cdots" "1280.4.d.x" 4 75.52244480734798 "4.0.2304.2" [] [] [0, 0, 0, 16] NULL "q+(4\\beta _{1}-\\beta _{2})q^{3}+5\\beta _{1}q^{5}+(4+5\\beta _{3})q^{7}+\\cdots" "1280.4.d.y" 4 75.52244480734798 "4.0.41616.2" [] [] [0, 0, 0, 16] NULL "q+2\\beta _{1}q^{3}+5\\beta _{1}q^{5}+(4-\\beta _{2})q^{7}+23q^{9}+\\cdots" "1280.4.d.z" 6 75.52244480734798 "6.0.12559936.1" [] [] [0, 0, 0, -40] NULL "q+(-\\beta _{2}+\\beta _{4})q^{3}+5\\beta _{2}q^{5}+(-7-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1280.4.d.ba" 6 75.52244480734798 "6.0.12559936.1" [] [] [0, 0, 0, 40] NULL "q+(-\\beta _{2}+\\beta _{4})q^{3}-5\\beta _{2}q^{5}+(7+\\beta _{1}+\\cdots)q^{7}+\\cdots" "1280.4.d.bb" 8 75.52244480734798 NULL [] [] [0, 0, 0, -40] NULL "q+(\\beta _{1}-\\beta _{5})q^{3}+5\\beta _{1}q^{5}+(-5+\\beta _{2}+\\cdots)q^{7}+\\cdots" "1280.4.d.bc" 8 75.52244480734798 NULL [] [] [0, 0, 0, 40] NULL "q+(\\beta _{1}-\\beta _{5})q^{3}-5\\beta _{1}q^{5}+(5-\\beta _{2})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.