# Classical modular forms downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1089/ # Query "{'level': 1089}" returned 157 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. "1089.1.c.a" 2 0.5434817987573396 "2.0.8.1" [-3] [] [0, 0, 0, 0] NULL "q+q^{4}-\\beta q^{7}+\\beta q^{13}+q^{16}-\\beta q^{19}+\\cdots" "1089.1.h.a" 4 0.5434817987573396 "4.0.576.1" [] [] [0, 2, 2, 0] NULL "q+(\\beta _{1}-\\beta _{3})q^{2}+(1-\\beta _{2})q^{3}+(1-\\beta _{2}+\\cdots)q^{4}+\\cdots" "1089.1.i.a" 2 0.5434817987573396 "2.0.3.1" [-11] [] [0, 1, 3, 0] NULL "q-\\zeta_{6}^{2}q^{3}+\\zeta_{6}^{2}q^{4}+(1+\\zeta_{6})q^{5}-\\zeta_{6}q^{9}+\\cdots" "1089.1.k.a" 8 0.5434817987573396 "8.0.64000000.1" [-3] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{4}-\\beta _{7}q^{7}-\\beta _{1}q^{13}+\\beta _{4}q^{16}+\\cdots" "1089.1.r.a" 8 0.5434817987573396 "8.0.1265625.1" [-11] [] [0, -1, -3, 0] NULL "q+\\zeta_{30}^{7}q^{3}-\\zeta_{30}^{13}q^{4}+(\\zeta_{30}^{6}+\\zeta_{30}^{11}+\\cdots)q^{5}+\\cdots" "1089.1.s.a" 8 0.5434817987573396 "8.0.1265625.1" [-11] [] [0, 1, -1, 0] NULL "q-\\zeta_{30}^{7}q^{3}-\\zeta_{30}^{13}q^{4}+\\zeta_{30}q^{5}+\\cdots" "1089.1.s.b" 16 0.5434817987573396 "16.0.26873856000000000000.1" [] [] [0, -2, -2, 0] NULL "q+\\beta _{1}q^{2}-\\beta _{7}q^{3}+\\beta _{2}q^{4}+(\\beta _{3}+\\beta _{15})q^{5}+\\cdots" "1089.2.a.a" 1 8.695708780117434 "1.1.1.1" [] [] [-2, 0, -4, -1] 1 "q-2q^{2}+2q^{4}-4q^{5}-q^{7}+8q^{10}+\\cdots" "1089.2.a.b" 1 8.695708780117434 "1.1.1.1" [] [] [-2, 0, -1, 2] 1 "q-2q^{2}+2q^{4}-q^{5}+2q^{7}+2q^{10}+\\cdots" "1089.2.a.c" 1 8.695708780117434 "1.1.1.1" [] [] [-1, 0, -1, -2] 1 "q-q^{2}-q^{4}-q^{5}-2q^{7}+3q^{8}+q^{10}+\\cdots" "1089.2.a.d" 1 8.695708780117434 "1.1.1.1" [] [] [-1, 0, 4, 2] -1 "q-q^{2}-q^{4}+4q^{5}+2q^{7}+3q^{8}-4q^{10}+\\cdots" "1089.2.a.e" 1 8.695708780117434 "1.1.1.1" [-3] [] [0, 0, 0, -5] -1 "q-2q^{4}-5q^{7}-2q^{13}+4q^{16}+7q^{19}+\\cdots" "1089.2.a.f" 1 8.695708780117434 "1.1.1.1" [-3] [] [0, 0, 0, 5] -1 "q-2q^{4}+5q^{7}+2q^{13}+4q^{16}-7q^{19}+\\cdots" "1089.2.a.g" 1 8.695708780117434 "1.1.1.1" [-11] [] [0, 0, 3, 0] -1 "q-2q^{4}+3q^{5}+4q^{16}-6q^{20}+9q^{23}+\\cdots" "1089.2.a.h" 1 8.695708780117434 "1.1.1.1" [] [] [1, 0, -4, 2] -1 "q+q^{2}-q^{4}-4q^{5}+2q^{7}-3q^{8}-4q^{10}+\\cdots" "1089.2.a.i" 1 8.695708780117434 "1.1.1.1" [] [] [1, 0, -1, 2] 1 "q+q^{2}-q^{4}-q^{5}+2q^{7}-3q^{8}-q^{10}+\\cdots" "1089.2.a.j" 1 8.695708780117434 "1.1.1.1" [] [] [1, 0, 2, -4] 1 "q+q^{2}-q^{4}+2q^{5}-4q^{7}-3q^{8}+2q^{10}+\\cdots" "1089.2.a.k" 1 8.695708780117434 "1.1.1.1" [] [] [2, 0, -4, 1] 1 "q+2q^{2}+2q^{4}-4q^{5}+q^{7}-8q^{10}+\\cdots" "1089.2.a.l" 2 8.695708780117434 "2.2.5.1" [] [] [-3, 0, -1, 2] 1 "q+(-1-\\beta )q^{2}+3\\beta q^{4}+(-1+\\beta )q^{5}+\\cdots" "1089.2.a.m" 2 8.695708780117434 "2.2.5.1" [] [] [-1, 0, 3, 6] -1 "q-\\beta q^{2}+(-1+\\beta )q^{4}+(2-\\beta )q^{5}+3q^{7}+\\cdots" "1089.2.a.n" 2 8.695708780117434 "2.2.12.1" [-3] [] [0, 0, 0, 0] 1 "q-2q^{4}-\\beta q^{7}+4\\beta q^{13}+4q^{16}-3\\beta q^{19}+\\cdots" "1089.2.a.o" 2 8.695708780117434 "2.2.12.1" [] [] [0, 0, 6, 0] -1 "q+\\beta q^{2}+q^{4}+3q^{5}+2\\beta q^{7}-\\beta q^{8}+\\cdots" "1089.2.a.p" 2 8.695708780117434 "2.2.5.1" [] [] [0, 0, -4, 0] -1 "q-\\beta q^{2}+3q^{4}-2q^{5}-2\\beta q^{7}-\\beta q^{8}+\\cdots" "1089.2.a.q" 2 8.695708780117434 "2.2.28.1" [] [] [0, 0, 0, -4] -1 "q+\\beta q^{2}+5q^{4}+\\beta q^{5}-2q^{7}+3\\beta q^{8}+\\cdots" "1089.2.a.r" 2 8.695708780117434 "2.2.28.1" [] [] [0, 0, 0, 4] -1 "q+\\beta q^{2}+5q^{4}-\\beta q^{5}+2q^{7}+3\\beta q^{8}+\\cdots" "1089.2.a.s" 2 8.695708780117434 "2.2.5.1" [] [] [1, 0, 3, -6] 1 "q+\\beta q^{2}+(-1+\\beta )q^{4}+(2-\\beta )q^{5}-3q^{7}+\\cdots" "1089.2.a.t" 2 8.695708780117434 "2.2.5.1" [] [] [3, 0, -1, -2] -1 "q+(1+\\beta )q^{2}+3\\beta q^{4}+(-1+\\beta )q^{5}+\\cdots" "1089.2.a.u" 4 8.695708780117434 "4.4.17424.1" [] [] [0, 0, -2, 0] -1 "q+\\beta _{1}q^{2}+(2+\\beta _{3})q^{4}+\\beta _{3}q^{5}+(-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1089.2.a.v" 4 8.695708780117434 "4.4.4400.1" [] [] [0, 0, 0, -8] 1 "q+\\beta _{1}q^{2}+(2+\\beta _{2})q^{4}-\\beta _{1}q^{5}+(-3+\\cdots)q^{7}+\\cdots" "1089.2.a.w" 4 8.695708780117434 "4.4.4400.1" [] [] [0, 0, 0, 8] -1 "q+\\beta _{1}q^{2}+(2+\\beta _{2})q^{4}+\\beta _{1}q^{5}+(3+2\\beta _{2}+\\cdots)q^{7}+\\cdots" "1089.2.d.a" 2 8.695708780117434 "2.0.8.1" [] [] [-2, 0, 0, 0] NULL "q-q^{2}-q^{4}-\\beta q^{5}-2\\beta q^{7}+3q^{8}+\\cdots" "1089.2.d.b" 2 8.695708780117434 "2.0.8.1" [] [] [2, 0, 0, 0] NULL "q+q^{2}-q^{4}+\\beta q^{5}-2\\beta q^{7}-3q^{8}+\\cdots" "1089.2.d.c" 4 8.695708780117434 "4.0.2304.1" [] [] [-4, 0, 0, 0] NULL "q+(-1-\\beta _{2})q^{2}+(2+2\\beta _{2})q^{4}-2\\beta _{1}q^{5}+\\cdots" "1089.2.d.d" 4 8.695708780117434 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{2}q^{2}+q^{4}+(-2\\beta _{1}+\\beta _{3})q^{5}+2\\beta _{3}q^{7}+\\cdots" "1089.2.d.e" 4 8.695708780117434 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{2}+q^{4}+(2\\beta _{1}-\\beta _{3})q^{5}+2\\beta _{3}q^{7}+\\cdots" "1089.2.d.f" 4 8.695708780117434 "4.0.2304.1" [] [] [4, 0, 0, 0] NULL "q+(1+\\beta _{2})q^{2}+(2+2\\beta _{2})q^{4}-2\\beta _{1}q^{5}+\\cdots" "1089.2.d.g" 16 8.695708780117434 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{15}q^{2}+(1+\\beta _{4})q^{4}+(-\\beta _{2}+\\beta _{5}+\\cdots)q^{5}+\\cdots" "1089.2.e.a" 2 8.695708780117434 "2.0.3.1" [] [] [-2, 0, 2, 4] NULL "q+(-2+2\\zeta_{6})q^{2}+(1-2\\zeta_{6})q^{3}-2\\zeta_{6}q^{4}+\\cdots" "1089.2.e.b" 2 8.695708780117434 "2.0.3.1" [] [] [0, -3, 3, -4] NULL "q+(-1-\\zeta_{6})q^{3}+2\\zeta_{6}q^{4}+3\\zeta_{6}q^{5}+\\cdots" "1089.2.e.c" 2 8.695708780117434 "2.0.3.1" [] [] [1, -3, -1, 4] NULL "q+(1-\\zeta_{6})q^{2}+(-2+\\zeta_{6})q^{3}+\\zeta_{6}q^{4}+\\cdots" "1089.2.e.d" 4 8.695708780117434 "4.0.225.1" [] [] [-3, -6, -2, -2] NULL "q+(-1-\\beta _{1}-\\beta _{3})q^{2}+(-2-\\beta _{3})q^{3}+\\cdots" "1089.2.e.e" 4 8.695708780117434 "4.0.144.1" [] [] [0, -6, 6, 0] NULL "q-\\beta_{2} q^{2}+(-\\beta_1-1)q^{3}+(\\beta_1-1)q^{4}+\\cdots" "1089.2.e.f" 4 8.695708780117434 "4.0.1089.1" [-11] [] [0, 1, -3, 0] NULL "q-\\beta _{3}q^{3}+2\\beta _{2}q^{4}+(1-2\\beta _{1}-2\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.2.e.g" 4 8.695708780117434 "4.0.225.1" [] [] [3, -6, -2, 2] NULL "q+(1+\\beta _{1}+\\beta _{3})q^{2}+(-2-\\beta _{3})q^{3}+\\cdots" "1089.2.e.h" 6 8.695708780117434 "6.0.19683.1" [] [] [0, 0, -3, -3] NULL "q+(-\\beta_{5}+\\beta_{4}+\\cdots-\\beta_{2})q^{2}+\\cdots+(\\beta_{4}+\\beta_{2})q^{3}+\\cdots" "1089.2.e.i" 8 8.695708780117434 "8.0.508277025.1" [] [] [1, 5, -4, 1] NULL "q+(\\beta _{2}+\\beta _{7})q^{2}+(1+\\beta _{7})q^{3}+(-2+\\beta _{1}+\\cdots)q^{4}+\\cdots" "1089.2.e.j" 12 8.695708780117434 NULL [] [] [0, 4, 2, 0] NULL "q+\\beta _{7}q^{2}+(1+\\beta _{2}-\\beta _{6})q^{3}+(-1-2\\beta _{4}+\\cdots)q^{4}+\\cdots" "1089.2.e.k" 16 8.695708780117434 "16.0.1333317747165888577536.1" [] [] [0, -2, 4, 0] NULL "q+(-\\beta _{1}-\\beta _{2}-\\beta _{6})q^{2}-\\beta _{13}q^{3}+(2\\beta _{5}+\\cdots)q^{4}+\\cdots" "1089.2.e.l" 20 8.695708780117434 NULL [] [] [0, -1, -3, 1] NULL "q-\\beta _{3}q^{2}-\\beta _{16}q^{3}+(-\\beta _{12}+\\beta _{17}+\\cdots)q^{4}+\\cdots" "1089.2.e.m" 20 8.695708780117434 NULL [] [] [0, -1, -3, -1] NULL "q+\\beta _{3}q^{2}-\\beta _{16}q^{3}+(-\\beta _{12}+\\beta _{17}+\\cdots)q^{4}+\\cdots" "1089.2.e.n" 24 8.695708780117434 NULL [] [] [0, 0, 6, 0] NULL NULL "1089.2.e.o" 36 8.695708780117434 NULL [] [] [-2, 9, 1, 1] NULL NULL "1089.2.e.p" 36 8.695708780117434 NULL [] [] [2, 9, 1, -1] NULL NULL "1089.3.b.a" 2 29.673100788836333 "2.0.8.1" [] [] [0, 0, 0, -2] NULL "q+\\beta q^{2}+2q^{4}-4\\beta q^{5}-q^{7}+6\\beta q^{8}+\\cdots" "1089.3.b.b" 2 29.673100788836333 "2.0.8.1" [] [] [0, 0, 0, 2] NULL "q+\\beta q^{2}+2q^{4}+4\\beta q^{5}+q^{7}+6\\beta q^{8}+\\cdots" "1089.3.b.c" 4 29.673100788836333 "4.0.57600.2" [] [] [0, 0, 0, -20] NULL "q+\\beta _{1}q^{2}+(-4+\\beta _{3})q^{4}+(-\\beta _{1}+2\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.3.b.d" 4 29.673100788836333 "4.0.57600.2" [] [] [0, 0, 0, 20] NULL "q+\\beta _{1}q^{2}+(-4+\\beta _{3})q^{4}+(\\beta _{1}-2\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.3.b.e" 4 29.673100788836333 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{2}-2q^{4}+2\\beta _{3}q^{5}-7\\beta _{2}q^{7}+\\cdots" "1089.3.b.f" 8 29.673100788836333 "8.0.1105425137664.10" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}+(-4-\\beta _{2})q^{4}+(-\\beta _{6}-\\beta _{7})q^{5}+\\cdots" "1089.3.b.g" 8 29.673100788836333 "8.0.65306824704.6" [] [] [0, 0, 0, -16] NULL "q-\\beta _{2}q^{2}+(-2-\\beta _{3})q^{4}+(-\\beta _{1}+\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.3.b.h" 8 29.673100788836333 "8.0.3317760000.7" [] [] [0, 0, 0, 0] NULL "q-\\beta _{7}q^{2}+\\beta _{3}q^{4}-\\beta _{1}q^{5}+(\\beta _{2}+\\beta _{4}+\\cdots)q^{7}+\\cdots" "1089.3.b.i" 16 29.673100788836333 NULL [] [] [0, 0, 0, -8] NULL "q+\\beta _{1}q^{2}+(-2+\\beta _{2})q^{4}+\\beta _{13}q^{5}+\\cdots" "1089.3.b.j" 16 29.673100788836333 NULL [] [] [0, 0, 0, 8] NULL "q+\\beta _{1}q^{2}+(-2+\\beta _{2})q^{4}-\\beta _{13}q^{5}+\\cdots" "1089.3.c.a" 2 29.673100788836333 "2.0.8.1" [] [] [0, 0, 14, 0] NULL "q+\\beta q^{2}+2q^{4}+7q^{5}-5\\beta q^{7}+6\\beta q^{8}+\\cdots" "1089.3.c.b" 4 29.673100788836333 "4.0.7744.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{2}-7q^{4}-2\\beta _{1}q^{5}+7\\beta _{2}q^{7}+\\cdots" "1089.3.c.c" 4 29.673100788836333 "4.0.2304.1" [] [] [0, 0, 8, 0] NULL "q+2\\beta _{1}q^{2}+(-4+4\\beta _{2})q^{4}+(2-2\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.3.c.d" 4 29.673100788836333 "4.0.2304.1" [] [] [0, 0, 0, 0] NULL "q+(-2\\beta _{1}+\\beta _{3})q^{2}+(-2+3\\beta _{2})q^{4}+\\cdots" "1089.3.c.e" 4 29.673100788836333 "4.0.125.1" [] [] [0, 0, -16, 0] NULL "q+(-\\beta_{3}+\\beta_1)q^{2}+(4\\beta_{2}+1)q^{4}+\\cdots" "1089.3.c.f" 4 29.673100788836333 "4.0.576.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{2}q^{2}+q^{4}-2\\beta _{3}q^{5}-5\\beta _{1}q^{7}+\\cdots" "1089.3.c.g" 4 29.673100788836333 "4.0.2304.1" [] [] [0, 0, 8, 0] NULL "q+\\beta _{1}q^{2}+(2+\\beta _{2})q^{4}+(2+3\\beta _{2})q^{5}+\\cdots" "1089.3.c.h" 4 29.673100788836333 "4.0.2304.1" [-3] [] [0, 0, 0, 0] NULL "q+4q^{4}+(-3\\beta _{1}-\\beta _{2})q^{7}+(-5\\beta _{1}+\\cdots)q^{13}+\\cdots" "1089.3.c.i" 8 29.673100788836333 "8.0.79010463744.2" [] [] [0, 0, 0, 0] NULL "q+\\beta _{4}q^{2}+(-4+\\beta _{2})q^{4}+(-\\beta _{3}+\\beta _{7})q^{5}+\\cdots" "1089.3.c.j" 8 29.673100788836333 "8.0.41108373504.15" [] [] [0, 0, -16, 0] NULL "q+(\\beta _{1}+\\beta _{4})q^{2}+(-3-\\beta _{6})q^{4}+(-2+\\cdots)q^{5}+\\cdots" "1089.3.c.k" 8 29.673100788836333 "8.0.523388583936.1" [] [] [0, 0, 4, 0] NULL "q+\\beta _{1}q^{2}+(-3+\\beta _{4}+\\beta _{5})q^{4}+(\\beta _{3}+\\cdots)q^{5}+\\cdots" "1089.3.c.l" 16 29.673100788836333 NULL [] [] [0, 0, 0, 0] NULL "q-\\beta _{3}q^{2}+(-3+\\beta _{2})q^{4}+\\beta _{8}q^{5}+(-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1089.3.c.m" 16 29.673100788836333 NULL [] [] [0, 0, 4, 0] NULL "q+\\beta _{6}q^{2}+(-1+\\beta _{2}-\\beta _{3})q^{4}+(\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.4.a.a" 1 64.25307999625153 "1.1.1.1" [] [] [-5, 0, 14, 32] 1 "q-5q^{2}+17q^{4}+14q^{5}+2^{5}q^{7}-45q^{8}+\\cdots" "1089.4.a.b" 1 64.25307999625153 "1.1.1.1" [] [] [-4, 0, 13, 26] 1 "q-4q^{2}+8q^{4}+13q^{5}+26q^{7}-52q^{10}+\\cdots" "1089.4.a.c" 1 64.25307999625153 "1.1.1.1" [] [] [-3, 0, 12, 12] -1 "q-3q^{2}+q^{4}+12q^{5}+12q^{7}+21q^{8}+\\cdots" "1089.4.a.d" 1 64.25307999625153 "1.1.1.1" [] [] [-1, 0, -7, 4] 1 "q-q^{2}-7q^{4}-7q^{5}+4q^{7}+15q^{8}+\\cdots" "1089.4.a.e" 1 64.25307999625153 "1.1.1.1" [] [] [-1, 0, 4, 26] 1 "q-q^{2}-7q^{4}+4q^{5}+26q^{7}+15q^{8}+\\cdots" "1089.4.a.f" 1 64.25307999625153 "1.1.1.1" [-11] [] [0, 0, -18, 0] -1 "q-8q^{4}-18q^{5}+2^{6}q^{16}+12^{2}q^{20}+\\cdots" "1089.4.a.g" 1 64.25307999625153 "1.1.1.1" [-3] [] [0, 0, 0, -20] -1 "q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\\cdots" "1089.4.a.h" 1 64.25307999625153 "1.1.1.1" [] [] [1, 0, -7, -4] 1 "q+q^{2}-7q^{4}-7q^{5}-4q^{7}-15q^{8}+\\cdots" "1089.4.a.i" 1 64.25307999625153 "1.1.1.1" [] [] [3, 0, 12, -12] -1 "q+3q^{2}+q^{4}+12q^{5}-12q^{7}-21q^{8}+\\cdots" "1089.4.a.j" 1 64.25307999625153 "1.1.1.1" [] [] [4, 0, 13, -26] 1 "q+4q^{2}+8q^{4}+13q^{5}-26q^{7}+52q^{10}+\\cdots" "1089.4.a.k" 2 64.25307999625153 "2.2.12.1" [] [] [-2, 0, 10, 8] 1 "q+(-1+\\beta )q^{2}+(5-2\\beta )q^{4}+(5-\\beta )q^{5}+\\cdots" "1089.4.a.l" 2 64.25307999625153 "2.2.12.1" [] [] [-2, 0, 20, 16] -1 "q+(-1+\\beta )q^{2}+(5-2\\beta )q^{4}+(10+\\beta )q^{5}+\\cdots" "1089.4.a.m" 2 64.25307999625153 "2.2.12.1" [-3] [] [0, 0, 0, 0] 1 "q-8q^{4}-\\beta q^{7}-2\\beta q^{13}+2^{6}q^{16}+\\cdots" "1089.4.a.n" 2 64.25307999625153 "2.2.12.1" [] [] [0, 0, 6, 0] -1 "q+\\beta q^{2}-5q^{4}+3q^{5}-2\\beta q^{7}-13\\beta q^{8}+\\cdots" "1089.4.a.o" 2 64.25307999625153 "2.2.5.1" [] [] [0, 0, 4, 0] -1 "q-\\beta q^{2}-3q^{4}+2q^{5}-10\\beta q^{7}+11\\beta q^{8}+\\cdots" "1089.4.a.p" 2 64.25307999625153 "2.2.5.1" [] [] [0, 0, 13, -44] -1 "q+(2-4\\beta )q^{2}+12q^{4}+(1+11\\beta )q^{5}+\\cdots" "1089.4.a.q" 2 64.25307999625153 "2.2.5.1" [] [] [0, 0, 13, 44] 1 "q+(2-4\\beta )q^{2}+12q^{4}+(12-11\\beta )q^{5}+\\cdots" "1089.4.a.r" 2 64.25307999625153 "2.2.104.1" [] [] [0, 0, -10, 0] -1 "q+\\beta q^{2}+18q^{4}-5q^{5}-4\\beta q^{7}+10\\beta q^{8}+\\cdots" "1089.4.a.s" 2 64.25307999625153 "2.2.12.1" [] [] [0, 0, -18, 0] -1 "q+3\\beta q^{2}+19q^{4}-9q^{5}-14\\beta q^{7}+\\cdots" "1089.4.a.t" 2 64.25307999625153 "2.2.33.1" [] [] [1, 0, -16, -2] 1 "q+\\beta q^{2}+\\beta q^{4}+(-10+4\\beta )q^{5}+(-2+\\cdots)q^{7}+\\cdots" "1089.4.a.u" 2 64.25307999625153 "2.2.97.1" [] [] [1, 0, 14, -24] 1 "q+\\beta q^{2}+(2^{4}+\\beta )q^{4}+(6+2\\beta )q^{5}+(-14+\\cdots)q^{7}+\\cdots" "1089.4.a.v" 2 64.25307999625153 "2.2.12.1" [] [] [2, 0, -2, -20] 1 "q+(1+\\beta )q^{2}+(-4+2\\beta )q^{4}+(-1-8\\beta )q^{5}+\\cdots" "1089.4.a.w" 2 64.25307999625153 "2.2.12.1" [] [] [2, 0, -20, 16] -1 "q+(1+\\beta )q^{2}+(5+2\\beta )q^{4}+(-10+\\beta )q^{5}+\\cdots" "1089.4.a.x" 2 64.25307999625153 "2.2.12.1" [] [] [2, 0, 10, -8] 1 "q+(1+\\beta )q^{2}+(5+2\\beta )q^{4}+(5+\\beta )q^{5}+\\cdots" "1089.4.a.y" 4 64.25307999625153 "4.4.34225.1" [] [] [-4, 0, 11, 25] -1 "q+(-1+\\beta _{1}+\\beta _{2})q^{2}+(3-3\\beta _{1}-3\\beta _{2}+\\cdots)q^{4}+\\cdots" "1089.4.a.z" 4 64.25307999625153 "4.4.5225.1" [] [] [-3, 0, -12, 11] 1 "q+(-1-\\beta _{1}-\\beta _{2})q^{2}+(-2+3\\beta _{1}+\\cdots)q^{4}+\\cdots" "1089.4.a.ba" 4 64.25307999625153 NULL [] [] [-1, 0, -14, -20] 1 "q-\\beta _{1}q^{2}+(6+\\beta _{1}+\\beta _{3})q^{4}+(-3+\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.4.a.bb" 4 64.25307999625153 "4.4.3600.1" [] [] [0, 0, -16, 0] -1 "q+(-\\beta _{1}+\\beta _{3})q^{2}+2\\beta _{2}q^{4}+(-4+4\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.4.a.bc" 4 64.25307999625153 "4.4.14221152.1" [] [] [0, 0, 0, -2] -1 "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}-\\beta _{3}q^{5}+\\beta _{2}q^{7}+\\cdots" "1089.4.a.bd" 4 64.25307999625153 "4.4.14221152.1" [] [] [0, 0, 0, 2] -1 "q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}+\\beta _{3}q^{5}-\\beta _{2}q^{7}+\\cdots" "1089.4.a.be" 4 64.25307999625153 "4.4.7225.1" [] [] [0, 0, 0, 0] 1 "q+\\beta _{1}q^{2}+9q^{4}-\\beta _{3}q^{5}+\\beta _{2}q^{7}+\\beta _{1}q^{8}+\\cdots" "1089.4.a.bf" 4 64.25307999625153 NULL [] [] [1, 0, -14, 20] 1 "q+\\beta _{1}q^{2}+(6+\\beta _{1}+\\beta _{3})q^{4}+(-3+\\beta _{2}+\\cdots)q^{5}+\\cdots" "1089.4.a.bg" 4 64.25307999625153 "4.4.5225.1" [] [] [3, 0, -12, -11] -1 "q+(1+\\beta _{1}+\\beta _{2})q^{2}+(-2+3\\beta _{1}+2\\beta _{2}+\\cdots)q^{4}+\\cdots" "1089.4.a.bh" 4 64.25307999625153 "4.4.34225.1" [] [] [4, 0, 11, -25] 1 "q+(1+\\beta _{1}+\\beta _{2})q^{2}+(3+\\beta _{1}+\\beta _{2}+2\\beta _{3})q^{4}+\\cdots" "1089.4.a.bi" 6 64.25307999625153 NULL [] [] [-5, 0, -9, 1] -1 "q+(-1+\\beta _{1})q^{2}+(3-\\beta _{1}+\\beta _{2}-\\beta _{3}+\\cdots)q^{4}+\\cdots" "1089.4.a.bj" 6 64.25307999625153 "6.6.22606886592.1" [] [] [0, 0, -14, 0] -1 "q+\\beta _{1}q^{2}+(3+\\beta _{3}+\\beta _{4})q^{4}+(-2-2\\beta _{3}+\\cdots)q^{5}+\\cdots" "1089.4.a.bk" 6 64.25307999625153 NULL [] [] [5, 0, -9, -1] 1 "q+(1-\\beta _{1})q^{2}+(3-\\beta _{1}+\\beta _{2}-\\beta _{3})q^{4}+\\cdots" "1089.4.a.bl" 12 64.25307999625153 NULL [] [] [0, 0, 0, -66] -1 "q+\\beta _{1}q^{2}+(5+\\beta _{2})q^{4}+(-\\beta _{1}-\\beta _{5}+\\cdots)q^{5}+\\cdots" "1089.4.a.bm" 12 64.25307999625153 NULL [] [] [0, 0, 0, 66] 1 "q+\\beta _{1}q^{2}+(5+\\beta _{2})q^{4}+(\\beta _{1}+\\beta _{5})q^{5}+\\cdots" "1089.4.a.bn" 12 64.25307999625153 NULL [] [] [0, 0, 0, 0] 1 "q+\\beta _{1}q^{2}+(6+\\beta _{7})q^{4}+\\beta _{4}q^{5}+(-2\\beta _{2}+\\cdots)q^{7}+\\cdots" "1089.6.a.a" 1 174.6579797763359 "1.1.1.1" [] [] [-9, 0, -24, -72] -1 "q-9q^{2}+7^{2}q^{4}-24q^{5}-72q^{7}-153q^{8}+\\cdots" "1089.6.a.b" 1 174.6579797763359 "1.1.1.1" [] [] [-6, 0, -6, 40] 1 "q-6q^{2}+4q^{4}-6q^{5}+40q^{7}+168q^{8}+\\cdots" "1089.6.a.c" 1 174.6579797763359 "1.1.1.1" [] [] [-4, 0, 19, -10] 1 "q-4q^{2}-2^{4}q^{4}+19q^{5}-10q^{7}+192q^{8}+\\cdots" "1089.6.a.d" 1 174.6579797763359 "1.1.1.1" [] [] [-2, 0, -46, -148] 1 "q-2q^{2}-28q^{4}-46q^{5}-148q^{7}+\\cdots" "1089.6.a.e" 1 174.6579797763359 "1.1.1.1" [-11] [] [0, 0, -57, 0] -1 "q-2^{5}q^{4}-57q^{5}+2^{10}q^{16}+1824q^{20}+\\cdots" "1089.6.a.f" 1 174.6579797763359 "1.1.1.1" [-3] [] [0, 0, 0, -25] -1 "q-2^{5}q^{4}-5^{2}q^{7}+1202q^{13}+2^{10}q^{16}+\\cdots" "1089.6.a.g" 1 174.6579797763359 "1.1.1.1" [-3] [] [0, 0, 0, 25] -1 "q-2^{5}q^{4}+5^{2}q^{7}-1202q^{13}+2^{10}q^{16}+\\cdots" "1089.6.a.h" 1 174.6579797763359 "1.1.1.1" [] [] [1, 0, 92, 26] 1 "q+q^{2}-31q^{4}+92q^{5}+26q^{7}-63q^{8}+\\cdots" "1089.6.a.i" 1 174.6579797763359 "1.1.1.1" [] [] [9, 0, -24, 72] -1 "q+9q^{2}+7^{2}q^{4}-24q^{5}+72q^{7}+153q^{8}+\\cdots" "1089.6.a.j" 2 174.6579797763359 "2.2.177.1" [] [] [-5, 0, -58, 286] 1 "q+(-2-\\beta )q^{2}+(2^{4}+5\\beta )q^{4}+(-24+\\cdots)q^{5}+\\cdots" "1089.6.a.k" 2 174.6579797763359 "2.2.12.1" [-3] [] [0, 0, 0, 0] 1 "q-2^{5}q^{4}-149\\beta q^{7}+116\\beta q^{13}+2^{10}q^{16}+\\cdots" "1089.6.a.l" 2 174.6579797763359 "2.2.5.1" [] [] [0, 0, 196, 0] -1 "q-\\beta q^{2}-12q^{4}+98q^{5}+53\\beta q^{7}+\\cdots" "1089.6.a.m" 2 174.6579797763359 "2.2.152.1" [] [] [0, 0, 38, 0] -1 "q+\\beta q^{2}+6q^{4}+19q^{5}+8\\beta q^{7}-26\\beta q^{8}+\\cdots" "1089.6.a.n" 2 174.6579797763359 "2.2.12.1" [] [] [0, 0, -48, 0] -1 "q+4\\beta q^{2}+2^{4}q^{4}-24q^{5}-\\beta q^{7}-2^{6}\\beta q^{8}+\\cdots" "1089.6.a.o" 2 174.6579797763359 "2.2.313.1" [] [] [1, 0, 38, 18] 1 "q+\\beta q^{2}+(46+\\beta )q^{4}+(24-10\\beta )q^{5}+\\cdots" "1089.6.a.p" 2 174.6579797763359 "2.2.33.1" [] [] [13, 0, -58, -146] 1 "q+(7-\\beta )q^{2}+(5^{2}-13\\beta )q^{4}+(-24+\\cdots)q^{5}+\\cdots" "1089.6.a.q" 3 174.6579797763359 "3.3.193425.1" [] [] [-1, 0, 58, 117] 1 "q-\\beta _{1}q^{2}+(6+\\beta _{1}+2\\beta _{2})q^{4}+(17+6\\beta _{1}+\\cdots)q^{5}+\\cdots" "1089.6.a.r" 3 174.6579797763359 "3.3.54492.1" [] [] [0, 0, -24, -84] 1 "q+\\beta _{2}q^{2}+(28-2\\beta _{1}-4\\beta _{2})q^{4}+(-8+\\cdots)q^{5}+\\cdots" "1089.6.a.s" 3 174.6579797763359 "3.3.193425.1" [] [] [1, 0, 58, -117] 1 "q+\\beta _{1}q^{2}+(6+\\beta _{1}+2\\beta _{2})q^{4}+(17+6\\beta _{1}+\\cdots)q^{5}+\\cdots" "1089.6.a.t" 4 174.6579797763359 NULL [] [] [-9, 0, -42, -14] 1 "q+(-2-\\beta _{1})q^{2}+(12+6\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "1089.6.a.u" 4 174.6579797763359 NULL [] [] [9, 0, -42, 14] 1 "q+(2+\\beta _{1})q^{2}+(12+6\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "1089.6.a.v" 5 174.6579797763359 NULL [] [] [-4, 0, -29, -102] 1 "q+(-1+\\beta _{1})q^{2}+(21-\\beta _{1}+\\beta _{3})q^{4}+\\cdots" "1089.6.a.w" 5 174.6579797763359 NULL [] [] [-4, 0, 100, 18] -1 "q+(-1+\\beta _{1})q^{2}+(21-3\\beta _{1}+\\beta _{4})q^{4}+\\cdots" "1089.6.a.x" 5 174.6579797763359 NULL [] [] [4, 0, -100, 18] -1 "q+(1-\\beta _{1})q^{2}+(21-3\\beta _{1}+\\beta _{4})q^{4}+\\cdots" "1089.6.a.y" 5 174.6579797763359 NULL [] [] [4, 0, -29, 102] 1 "q+(1-\\beta _{1})q^{2}+(21-\\beta _{1}+\\beta _{3})q^{4}+(-6+\\cdots)q^{5}+\\cdots" "1089.6.a.z" 6 174.6579797763359 "6.6.54016037568.1" [] [] [0, 0, 50, 0] -1 "q+(-\\beta _{1}+3\\beta _{2})q^{2}+(14+\\beta _{3}-7\\beta _{4}+\\cdots)q^{4}+\\cdots" "1089.6.a.ba" 6 174.6579797763359 NULL [] [] [0, 0, 48, 0] -1 "q+\\beta _{1}q^{2}+(28+\\beta _{2})q^{4}+(8+\\beta _{2}+\\beta _{4}+\\cdots)q^{5}+\\cdots" "1089.6.a.bb" 8 174.6579797763359 NULL [] [] [-8, 0, -70, 292] 1 "q+(-1+\\beta _{1})q^{2}+(11-\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "1089.6.a.bc" 8 174.6579797763359 NULL [] [] [0, 0, 0, 0] 1 "q-\\beta _{4}q^{2}+(19-\\beta _{1})q^{4}-\\beta _{2}q^{5}+\\beta _{7}q^{7}+\\cdots" "1089.6.a.bd" 8 174.6579797763359 NULL [] [] [0, 0, 256, 0] -1 "q+\\beta _{1}q^{2}+(19-2\\beta _{2}+\\beta _{6})q^{4}+(33+\\cdots)q^{5}+\\cdots" "1089.6.a.be" 8 174.6579797763359 NULL [] [] [0, 0, 0, -106] -1 "q+\\beta _{1}q^{2}+(5^{2}+\\beta _{2})q^{4}+(-3\\beta _{1}-\\beta _{3}+\\cdots)q^{5}+\\cdots" "1089.6.a.bf" 8 174.6579797763359 NULL [] [] [0, 0, 0, 106] -1 "q+\\beta _{1}q^{2}+(5^{2}+\\beta _{2})q^{4}+(3\\beta _{1}+\\beta _{3}+\\cdots)q^{5}+\\cdots" "1089.6.a.bg" 8 174.6579797763359 NULL [] [] [8, 0, -70, -292] -1 "q+(1-\\beta _{1})q^{2}+(11-\\beta _{1}+\\beta _{2})q^{4}+(-9+\\cdots)q^{5}+\\cdots" "1089.6.a.bh" 10 174.6579797763359 NULL [] [] [-9, 0, -11, 470] -1 "q+(-1+\\beta _{1})q^{2}+(19-\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "1089.6.a.bi" 10 174.6579797763359 NULL [] [] [-7, 0, 33, -78] 1 "q+(-1+\\beta _{1})q^{2}+(15-\\beta _{1}+\\beta _{2}-\\beta _{3}+\\cdots)q^{4}+\\cdots" "1089.6.a.bj" 10 174.6579797763359 NULL [] [] [0, 0, -198, 0] -1 "q+\\beta _{6}q^{2}+(18+\\beta _{1})q^{4}+(-20-\\beta _{1}+\\cdots)q^{5}+\\cdots" "1089.6.a.bk" 10 174.6579797763359 NULL [] [] [7, 0, 33, 78] -1 "q+(1-\\beta _{1})q^{2}+(15-\\beta _{1}+\\beta _{2}-\\beta _{3}+\\cdots)q^{4}+\\cdots" "1089.6.a.bl" 10 174.6579797763359 NULL [] [] [9, 0, -11, -470] 1 "q+(1-\\beta _{1})q^{2}+(19-\\beta _{1}+\\beta _{2})q^{4}+(-1+\\cdots)q^{5}+\\cdots" "1089.6.a.bm" 12 174.6579797763359 NULL [] [] [0, 0, 0, 0] 1 "q+\\beta _{6}q^{2}+(14+\\beta _{3})q^{4}+(\\beta _{1}-\\beta _{2})q^{5}+\\cdots" "1089.6.a.bn" 20 174.6579797763359 NULL [] [] [0, 0, 0, -472] 1 "q+\\beta _{1}q^{2}+(15+\\beta _{2})q^{4}+(-\\beta _{1}-\\beta _{8}+\\cdots)q^{5}+\\cdots" "1089.6.a.bo" 20 174.6579797763359 NULL [] [] [0, 0, 0, 472] -1 "q+\\beta _{1}q^{2}+(15+\\beta _{2})q^{4}+(\\beta _{1}+\\beta _{8})q^{5}+\\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.