# Classical modular forms downloaded from the LMFDB on 24 June 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/2325/ # Query "{'level': 2325}" returned 108 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. "2325.1.m.a" 4 1.1603261543717305 "4.0.256.1" [-15, -155] [93] [0, 0, 0, 0] NULL "q-\\zeta_{8}^{3}q^{3}+\\zeta_{8}^{2}q^{4}-\\zeta_{8}^{2}q^{9}+\\zeta_{8}q^{12}+\\cdots" "2325.1.m.b" 4 1.1603261543717305 "4.0.256.1" [-3, -155] [465] [0, 0, 0, 0] NULL "q-\\zeta_{8}q^{3}+\\zeta_{8}^{2}q^{4}+\\zeta_{8}^{2}q^{9}-\\zeta_{8}^{3}q^{12}+\\cdots" "2325.1.u.a" 2 1.1603261543717305 "2.0.3.1" [-3] [] [0, -1, 0, 1] NULL "q-\\zeta_{6}q^{3}+q^{4}+\\zeta_{6}q^{7}+\\zeta_{6}^{2}q^{9}-\\zeta_{6}q^{12}+\\cdots" "2325.1.u.b" 2 1.1603261543717305 "2.0.3.1" [-3] [] [0, 1, 0, -1] NULL "q+\\zeta_{6}q^{3}+q^{4}-\\zeta_{6}q^{7}+\\zeta_{6}^{2}q^{9}+\\zeta_{6}q^{12}+\\cdots" "2325.1.u.c" 4 1.1603261543717305 "4.0.576.1" [] [] [0, -2, 0, 2] NULL "q-\\beta _{3}q^{2}-\\beta _{2}q^{3}-q^{4}+(-\\beta _{1}+\\beta _{3})q^{6}+\\cdots" "2325.1.u.d" 4 1.1603261543717305 "4.0.144.1" [-15] [] [0, 0, 0, 0] NULL "q-\\zeta_{12}^{3}q^{2}+\\zeta_{12}q^{3}-\\zeta_{12}^{4}q^{6}-\\zeta_{12}^{3}q^{8}+\\cdots" "2325.1.u.e" 4 1.1603261543717305 "4.0.576.1" [] [] [0, 2, 0, -2] NULL "q-\\beta _{3}q^{2}+\\beta _{2}q^{3}-q^{4}+(\\beta _{1}-\\beta _{3})q^{6}+\\cdots" "2325.1.z.a" 4 1.1603261543717305 "4.0.144.1" [-3] [] [0, 0, 0, 0] NULL "q+\\zeta_{12}^{5}q^{3}-q^{4}+\\zeta_{12}^{5}q^{7}-\\zeta_{12}^{4}q^{9}+\\cdots" "2325.1.z.b" 8 1.1603261543717305 "8.0.5308416.1" [] [] [0, 0, 0, 0] NULL "q+(-\\zeta_{24}^{3}+\\zeta_{24}^{9})q^{2}-\\zeta_{24}^{10}q^{3}+\\cdots" "2325.1.bq.a" 8 1.1603261543717305 "8.0.4000000.1" [-3] [] [0, 0, 0, 0] NULL "q-\\zeta_{20}^{7}q^{3}+\\zeta_{20}^{6}q^{4}+(\\zeta_{20}^{3}-\\zeta_{20}^{9}+\\cdots)q^{7}+\\cdots" "2325.1.bv.a" 8 1.1603261543717305 "8.0.4000000.1" [] [] [0, 0, 0, 2] NULL "q+\\zeta_{20}^{5}q^{3}-\\zeta_{20}^{6}q^{4}+\\zeta_{20}q^{5}-\\zeta_{20}^{4}q^{7}+\\cdots" "2325.1.bz.a" 8 1.1603261543717305 "8.0.4000000.1" [] [] [0, 0, 0, 2] NULL "q-\\zeta_{20}^{3}q^{3}-\\zeta_{20}^{6}q^{4}-\\zeta_{20}^{7}q^{5}+\\cdots" "2325.1.ca.a" 4 1.1603261543717305 "4.0.125.1" [-3] [] [0, 1, 0, 2] NULL "q-\\zeta_{10}^{2}q^{3}-\\zeta_{10}q^{4}+(\\zeta_{10}^{3}-\\zeta_{10}^{4}+\\cdots)q^{7}+\\cdots" "2325.1.ca.b" 8 1.1603261543717305 "8.0.4000000.1" [-15] [] [0, 0, 0, 0] NULL "q+(\\zeta_{20}^{3}-\\zeta_{20}^{9})q^{2}+\\zeta_{20}^{9}q^{3}+(\\zeta_{20}^{2}+\\cdots)q^{4}+\\cdots" "2325.1.cq.a" 8 1.1603261543717305 "8.0.5308416.1" [-3] [] [0, 0, 0, 0] NULL "q-\\zeta_{24}^{11}q^{3}+\\zeta_{24}^{6}q^{4}+(\\zeta_{24}-\\zeta_{24}^{9}+\\cdots)q^{7}+\\cdots" "2325.1.cq.b" 8 1.1603261543717305 "8.0.5308416.1" [-15] [] [0, 0, 0, 0] NULL "q+(\\zeta_{24}^{7}+\\zeta_{24}^{11})q^{2}-\\zeta_{24}^{5}q^{3}+(-\\zeta_{24}^{2}+\\cdots)q^{4}+\\cdots" "2325.1.dj.a" 16 1.1603261543717305 "16.0.1048576000000000000.1" [-3] [] [0, 0, 0, 0] NULL "q-\\zeta_{40}^{3}q^{3}+\\zeta_{40}^{14}q^{4}+(\\zeta_{40}-\\zeta_{40}^{17}+\\cdots)q^{7}+\\cdots" "2325.1.dj.b" 16 1.1603261543717305 "16.0.1048576000000000000.1" [-15] [] [0, 0, 0, 0] NULL "q+(\\zeta_{40}^{3}-\\zeta_{40}^{11})q^{2}+\\zeta_{40}^{13}q^{3}+\\cdots" "2325.1.ee.a" 16 1.1603261543717305 "16.0.104976000000000000.1" [-3] [] [0, 0, 0, 0] NULL "q+\\zeta_{60}^{13}q^{3}-\\zeta_{60}^{24}q^{4}+(\\zeta_{60}^{11}+\\cdots)q^{7}+\\cdots" "2325.1.fd.a" 8 1.1603261543717305 "8.0.1265625.1" [-3] [] [0, -1, 0, 1] NULL "q+\\zeta_{30}^{13}q^{3}-\\zeta_{30}^{9}q^{4}+(\\zeta_{30}^{11}-\\zeta_{30}^{12}+\\cdots)q^{7}+\\cdots" "2325.1.fd.b" 8 1.1603261543717305 "8.0.1265625.1" [-3] [] [0, 1, 0, -1] NULL "q-\\zeta_{30}^{13}q^{3}-\\zeta_{30}^{9}q^{4}+(-\\zeta_{30}^{11}+\\cdots)q^{7}+\\cdots" "2325.1.fd.c" 16 1.1603261543717305 "16.0.104976000000000000.1" [-15] [] [0, 0, 0, 0] NULL "q+(-\\zeta_{60}^{11}+\\zeta_{60}^{13})q^{2}+\\zeta_{60}^{23}q^{3}+\\cdots" "2325.1.gg.a" 32 1.1603261543717305 "32.0.47330370277129322496000000000000000000000000.1" [-15] [] [0, 0, 0, 0] NULL "q+(\\zeta_{120}^{23}-\\zeta_{120}^{31})q^{2}-\\zeta_{120}^{13}q^{3}+\\cdots" "2325.1.gg.b" 32 1.1603261543717305 "32.0.47330370277129322496000000000000000000000000.1" [-3] [] [0, 0, 0, 0] NULL "q+\\zeta_{120}^{43}q^{3}+\\zeta_{120}^{54}q^{4}+(-\\zeta_{120}^{41}+\\cdots)q^{7}+\\cdots" "2325.2.a.a" 1 18.565218469947688 "1.1.1.1" [] [] [-2, -1, 0, 4] 1 "q-2q^{2}-q^{3}+2q^{4}+2q^{6}+4q^{7}+\\cdots" "2325.2.a.b" 1 18.565218469947688 "1.1.1.1" [] [] [-2, 1, 0, -4] 1 "q-2q^{2}+q^{3}+2q^{4}-2q^{6}-4q^{7}+\\cdots" "2325.2.a.c" 1 18.565218469947688 "1.1.1.1" [] [] [-1, -1, 0, -2] 1 "q-q^{2}-q^{3}-q^{4}+q^{6}-2q^{7}+3q^{8}+\\cdots" "2325.2.a.d" 1 18.565218469947688 "1.1.1.1" [] [] [-1, 1, 0, 2] 1 "q-q^{2}+q^{3}-q^{4}-q^{6}+2q^{7}+3q^{8}+\\cdots" "2325.2.a.e" 1 18.565218469947688 "1.1.1.1" [] [] [-1, 1, 0, 2] 1 "q-q^{2}+q^{3}-q^{4}-q^{6}+2q^{7}+3q^{8}+\\cdots" "2325.2.a.f" 1 18.565218469947688 "1.1.1.1" [] [] [0, -1, 0, 0] 1 "q-q^{3}-2q^{4}+q^{9}-3q^{11}+2q^{12}+\\cdots" "2325.2.a.g" 1 18.565218469947688 "1.1.1.1" [] [] [0, 1, 0, 0] 1 "q+q^{3}-2q^{4}+q^{9}-3q^{11}-2q^{12}+\\cdots" "2325.2.a.h" 1 18.565218469947688 "1.1.1.1" [] [] [1, -1, 0, -2] 1 "q+q^{2}-q^{3}-q^{4}-q^{6}-2q^{7}-3q^{8}+\\cdots" "2325.2.a.i" 1 18.565218469947688 "1.1.1.1" [] [] [1, -1, 0, 4] 1 "q+q^{2}-q^{3}-q^{4}-q^{6}+4q^{7}-3q^{8}+\\cdots" "2325.2.a.j" 1 18.565218469947688 "1.1.1.1" [] [] [1, 1, 0, 2] 1 "q+q^{2}+q^{3}-q^{4}+q^{6}+2q^{7}-3q^{8}+\\cdots" "2325.2.a.k" 1 18.565218469947688 "1.1.1.1" [] [] [2, -1, 0, 4] 1 "q+2q^{2}-q^{3}+2q^{4}-2q^{6}+4q^{7}+\\cdots" "2325.2.a.l" 1 18.565218469947688 "1.1.1.1" [] [] [2, 1, 0, -4] 1 "q+2q^{2}+q^{3}+2q^{4}+2q^{6}-4q^{7}+\\cdots" "2325.2.a.m" 2 18.565218469947688 "2.2.12.1" [] [] [0, 2, 0, 6] -1 "q+\\beta q^{2}+q^{3}+q^{4}+\\beta q^{6}+(3-\\beta )q^{7}+\\cdots" "2325.2.a.n" 2 18.565218469947688 "2.2.8.1" [] [] [2, -2, 0, 4] -1 "q+(1+\\beta )q^{2}-q^{3}+(1+2\\beta )q^{4}+(-1+\\cdots)q^{6}+\\cdots" "2325.2.a.o" 2 18.565218469947688 "2.2.5.1" [] [] [3, 2, 0, 4] -1 "q+(1+\\beta )q^{2}+q^{3}+3\\beta q^{4}+(1+\\beta )q^{6}+\\cdots" "2325.2.a.p" 3 18.565218469947688 "3.3.148.1" [] [] [-3, -3, 0, -2] 1 "q+(-1-\\beta _{2})q^{2}-q^{3}+(2-\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "2325.2.a.q" 3 18.565218469947688 "3.3.148.1" [] [] [-1, -3, 0, -2] -1 "q-\\beta _{1}q^{2}-q^{3}+(\\beta _{1}+\\beta _{2})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.a.r" 3 18.565218469947688 "3.3.564.1" [] [] [-1, 3, 0, -8] 1 "q-\\beta _{1}q^{2}+q^{3}+(2+\\beta _{2})q^{4}-\\beta _{1}q^{6}+\\cdots" "2325.2.a.s" 3 18.565218469947688 "3.3.229.1" [] [] [0, -3, 0, -4] 1 "q+\\beta _{1}q^{2}-q^{3}+(1+\\beta _{2})q^{4}-\\beta _{1}q^{6}+\\cdots" "2325.2.a.t" 3 18.565218469947688 "3.3.148.1" [] [] [0, -3, 0, 4] 1 "q-\\beta _{2}q^{2}-q^{3}+(1-\\beta _{1}-\\beta _{2})q^{4}+\\beta _{2}q^{6}+\\cdots" "2325.2.a.u" 3 18.565218469947688 "3.3.148.1" [] [] [0, 3, 0, -4] 1 "q+\\beta _{2}q^{2}+q^{3}+(1-\\beta _{1}-\\beta _{2})q^{4}+\\beta _{2}q^{6}+\\cdots" "2325.2.a.v" 4 18.565218469947688 "4.4.8468.1" [] [] [-2, 4, 0, -4] -1 "q+\\beta _{2}q^{2}+q^{3}+(2-\\beta _{1})q^{4}+\\beta _{2}q^{6}+\\cdots" "2325.2.a.w" 5 18.565218469947688 "5.5.126032.1" [] [] [-3, 5, 0, -8] 1 "q+(-1-\\beta _{3})q^{2}+q^{3}+(1-\\beta _{1}+\\beta _{3}+\\cdots)q^{4}+\\cdots" "2325.2.a.x" 5 18.565218469947688 "5.5.126032.1" [] [] [3, -5, 0, 8] -1 "q+(1+\\beta _{3})q^{2}-q^{3}+(1-\\beta _{1}+\\beta _{3})q^{4}+\\cdots" "2325.2.a.y" 6 18.565218469947688 "6.6.75968016.1" [] [] [-1, -6, 0, -2] -1 "q-\\beta _{1}q^{2}-q^{3}+(1+\\beta _{4}+\\beta _{5})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.a.z" 6 18.565218469947688 "6.6.136751504.1" [] [] [-1, 6, 0, 6] -1 "q-\\beta _{1}q^{2}+q^{3}+(1+\\beta _{2})q^{4}-\\beta _{1}q^{6}+\\cdots" "2325.2.a.ba" 6 18.565218469947688 "6.6.136751504.1" [] [] [1, -6, 0, -6] -1 "q+\\beta _{1}q^{2}-q^{3}+(1+\\beta _{2})q^{4}-\\beta _{1}q^{6}+\\cdots" "2325.2.a.bb" 6 18.565218469947688 "6.6.75968016.1" [] [] [1, 6, 0, 2] -1 "q+\\beta _{1}q^{2}+q^{3}+(1+\\beta _{4}+\\beta _{5})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.a.bc" 11 18.565218469947688 NULL [] [] [-3, -11, 0, -8] 1 "q-\\beta _{1}q^{2}-q^{3}+(1+\\beta _{2})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.a.bd" 11 18.565218469947688 NULL [] [] [3, 11, 0, 8] -1 "q+\\beta _{1}q^{2}+q^{3}+(1+\\beta _{2})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.c.a" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+2 i q^{2}+i q^{3}-2 q^{4}-2 q^{6}+4 i q^{7}+\\cdots" "2325.2.c.b" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+2 i q^{2}-i q^{3}-2 q^{4}+2 q^{6}-4 i q^{7}+\\cdots" "2325.2.c.c" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+i q^{2}+i q^{3}+q^{4}-q^{6}+4 i q^{7}+\\cdots" "2325.2.c.d" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+i q^{2}+i q^{3}+q^{4}-q^{6}-2 i q^{7}+\\cdots" "2325.2.c.e" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+i q^{2}+i q^{3}+q^{4}-q^{6}-2 i q^{7}+\\cdots" "2325.2.c.f" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q+i q^{2}-i q^{3}+q^{4}+q^{6}+2 i q^{7}+\\cdots" "2325.2.c.g" 2 18.565218469947688 "2.0.4.1" [] [] [0, 0, 0, 0] NULL "q-i q^{3}+2 q^{4}-q^{9}-3 q^{11}-2 i q^{12}+\\cdots" "2325.2.c.h" 4 18.565218469947688 "4.0.400.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{1}-\\beta _{3})q^{2}+\\beta _{3}q^{3}+3\\beta _{2}q^{4}+(1+\\cdots)q^{6}+\\cdots" "2325.2.c.i" 4 18.565218469947688 "4.0.256.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta_{2}+\\beta_1)q^{2}+\\beta_1 q^{3}+(-2\\beta_{3}-1)q^{4}+\\cdots" "2325.2.c.j" 4 18.565218469947688 "4.0.144.1" [] [] [0, 0, 0, 0] NULL "q-\\beta_{2} q^{2}+\\beta_1 q^{3}-q^{4}+\\beta_{3} q^{6}+\\cdots" "2325.2.c.k" 6 18.565218469947688 "6.0.5089536.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{5}q^{2}-\\beta _{4}q^{3}+(-2+\\beta _{3})q^{4}+\\beta _{1}q^{6}+\\cdots" "2325.2.c.l" 6 18.565218469947688 "6.0.350464.1" [] [] [0, 0, 0, 0] NULL "q+(\\beta _{3}-\\beta _{5})q^{2}-\\beta _{3}q^{3}+(-2-\\beta _{1}+\\cdots)q^{4}+\\cdots" "2325.2.c.m" 6 18.565218469947688 "6.0.350464.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{5}q^{2}-\\beta _{3}q^{3}+(-1+\\beta _{1}-\\beta _{2}+\\cdots)q^{4}+\\cdots" "2325.2.c.n" 6 18.565218469947688 "6.0.3356224.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}-\\beta _{3}q^{3}+(-1+\\beta _{2})q^{4}-\\beta _{4}q^{6}+\\cdots" "2325.2.c.o" 6 18.565218469947688 "6.0.350464.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{4}q^{2}+\\beta _{3}q^{3}+(-\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "2325.2.c.p" 8 18.565218469947688 "8.0.4589249536.2" [] [] [0, 0, 0, 0] NULL "q+(-\\beta _{1}-\\beta _{5})q^{2}-\\beta _{5}q^{3}+(-2+\\beta _{2}+\\cdots)q^{4}+\\cdots" "2325.2.c.q" 12 18.565218469947688 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}+\\beta _{6}q^{3}+(-1+\\beta _{2})q^{4}+\\beta _{7}q^{6}+\\cdots" "2325.2.c.r" 12 18.565218469947688 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{2}+\\beta _{5}q^{3}+(-1-\\beta _{6}+\\beta _{7}+\\cdots)q^{4}+\\cdots" "2325.4.a.a" 1 137.17944076334692 "1.1.1.1" [] [] [-3, 3, 0, 34] 1 "q-3q^{2}+3q^{3}+q^{4}-9q^{6}+34q^{7}+\\cdots" "2325.4.a.b" 1 137.17944076334692 "1.1.1.1" [] [] [-1, -3, 0, 23] -1 "q-q^{2}-3q^{3}-7q^{4}+3q^{6}+23q^{7}+\\cdots" "2325.4.a.c" 1 137.17944076334692 "1.1.1.1" [] [] [0, -3, 0, 0] -1 "q-3q^{3}-8q^{4}+9q^{9}-2^{6}q^{11}+24q^{12}+\\cdots" "2325.4.a.d" 1 137.17944076334692 "1.1.1.1" [] [] [0, 3, 0, 0] -1 "q+3q^{3}-8q^{4}+9q^{9}-2^{6}q^{11}-24q^{12}+\\cdots" "2325.4.a.e" 1 137.17944076334692 "1.1.1.1" [] [] [1, -3, 0, -7] -1 "q+q^{2}-3q^{3}-7q^{4}-3q^{6}-7q^{7}+\\cdots" "2325.4.a.f" 1 137.17944076334692 "1.1.1.1" [] [] [1, 3, 0, -23] -1 "q+q^{2}+3q^{3}-7q^{4}+3q^{6}-23q^{7}+\\cdots" "2325.4.a.g" 1 137.17944076334692 "1.1.1.1" [] [] [3, 3, 0, 24] -1 "q+3q^{2}+3q^{3}+q^{4}+9q^{6}+24q^{7}+\\cdots" "2325.4.a.h" 1 137.17944076334692 "1.1.1.1" [] [] [4, -3, 0, -22] -1 "q+4q^{2}-3q^{3}+8q^{4}-12q^{6}-22q^{7}+\\cdots" "2325.4.a.i" 2 137.17944076334692 "2.2.41.1" [] [] [-2, -6, 0, 7] -1 "q-q^{2}-3q^{3}-7q^{4}+3q^{6}+(5+3\\beta )q^{7}+\\cdots" "2325.4.a.j" 2 137.17944076334692 "2.2.5.1" [] [] [-2, 6, 0, 14] -1 "q-q^{2}+3q^{3}-7q^{4}-3q^{6}+(7+4\\beta )q^{7}+\\cdots" "2325.4.a.k" 2 137.17944076334692 "2.2.41.1" [] [] [1, 6, 0, -29] 1 "q+\\beta q^{2}+3q^{3}+(2+\\beta )q^{4}+3\\beta q^{6}+\\cdots" "2325.4.a.l" 2 137.17944076334692 "2.2.5.1" [] [] [2, -6, 0, -14] -1 "q+q^{2}-3q^{3}-7q^{4}-3q^{6}+(-7-4\\beta )q^{7}+\\cdots" "2325.4.a.m" 2 137.17944076334692 "2.2.17.1" [] [] [3, -6, 0, 37] 1 "q+(1+\\beta )q^{2}-3q^{3}+(-3+3\\beta )q^{4}+\\cdots" "2325.4.a.n" 2 137.17944076334692 "2.2.29.1" [] [] [5, 6, 0, 4] 1 "q+(3-\\beta )q^{2}+3q^{3}+(8-5\\beta )q^{4}+(9+\\cdots)q^{6}+\\cdots" "2325.4.a.o" 3 137.17944076334692 "3.3.2089.1" [] [] [-3, 9, 0, -19] -1 "q+(-1-\\beta _{1})q^{2}+3q^{3}+(1+3\\beta _{1}+2\\beta _{2})q^{4}+\\cdots" "2325.4.a.p" 4 137.17944076334692 "4.4.2862868.2" [] [] [2, -12, 0, 64] -1 "q+\\beta _{1}q^{2}-3q^{3}+(3+2\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "2325.4.a.q" 6 137.17944076334692 NULL [] [] [-3, -18, 0, -47] -1 "q+(-1+\\beta _{1})q^{2}-3q^{3}+(6+\\beta _{2})q^{4}+\\cdots" "2325.4.a.r" 6 137.17944076334692 NULL [] [] [-3, -18, 0, 0] -1 "q+(-1+\\beta _{1})q^{2}-3q^{3}+(10-\\beta _{1}+\\beta _{4}+\\cdots)q^{4}+\\cdots" "2325.4.a.s" 6 137.17944076334692 NULL [] [] [3, 18, 0, 23] -1 "q+(1-\\beta _{1})q^{2}+3q^{3}+(3-\\beta _{1}+\\beta _{2}+\\cdots)q^{4}+\\cdots" "2325.4.a.t" 7 137.17944076334692 NULL [] [] [-4, 21, 0, -61] -1 "q+(-1+\\beta _{1})q^{2}+3q^{3}+(6-\\beta _{1}+\\beta _{2}+\\cdots)q^{4}+\\cdots" "2325.4.a.u" 7 137.17944076334692 NULL [] [] [1, 21, 0, 61] 1 "q+\\beta _{1}q^{2}+3q^{3}+(4+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.v" 7 137.17944076334692 NULL [] [] [5, -21, 0, 49] 1 "q+(1-\\beta _{1})q^{2}-3q^{3}+(4-\\beta _{1}+\\beta _{2}+\\cdots)q^{4}+\\cdots" "2325.4.a.w" 9 137.17944076334692 NULL [] [] [-3, -27, 0, -63] 1 "q-\\beta _{1}q^{2}-3q^{3}+(6+\\beta _{1}+\\beta _{2})q^{4}+\\cdots" "2325.4.a.x" 9 137.17944076334692 NULL [] [] [1, 27, 0, -19] 1 "q+\\beta _{1}q^{2}+3q^{3}+(6+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.y" 11 137.17944076334692 NULL [] [] [-2, -33, 0, 33] -1 "q-\\beta _{1}q^{2}-3q^{3}+(6+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.z" 11 137.17944076334692 NULL [] [] [2, 33, 0, -33] -1 "q+\\beta _{1}q^{2}+3q^{3}+(6+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.ba" 13 137.17944076334692 NULL [] [] [-1, 39, 0, -14] -1 "q-\\beta _{1}q^{2}+3q^{3}+(4+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bb" 13 137.17944076334692 NULL [] [] [1, -39, 0, 14] -1 "q+\\beta _{1}q^{2}-3q^{3}+(4+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bc" 16 137.17944076334692 NULL [] [] [-1, -48, 0, -14] 1 "q-\\beta _{1}q^{2}-3q^{3}+(4+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bd" 16 137.17944076334692 NULL [] [] [-1, 48, 0, 42] 1 "q-\\beta _{1}q^{2}+3q^{3}+(4+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.be" 16 137.17944076334692 NULL [] [] [1, -48, 0, -42] 1 "q+\\beta _{1}q^{2}-3q^{3}+(4+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bf" 16 137.17944076334692 NULL [] [] [1, 48, 0, 14] 1 "q+\\beta _{1}q^{2}+3q^{3}+(4+\\beta _{2})q^{4}+3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bg" 19 137.17944076334692 NULL [] [] [-6, 57, 0, -56] -1 "q-\\beta _{1}q^{2}+3q^{3}+(3+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bh" 19 137.17944076334692 NULL [] [] [6, -57, 0, 56] 1 "q+\\beta _{1}q^{2}-3q^{3}+(3+\\beta _{2})q^{4}-3\\beta _{1}q^{6}+\\cdots" "2325.4.a.bi" 25 137.17944076334692 NULL [] [] [-6, -75, 0, -56] -1 NULL "2325.4.a.bj" 25 137.17944076334692 NULL [] [] [6, 75, 0, 56] 1 NULL # Label -- # The **label** of a newform $f\in S_k^{\rm new}(N,\chi)$ has the format \( N.k.a.x \), where # - \( N\) is the level; # - \(k\) is the weight; # - \(N.a\) is the label of the Galois orbit of the Dirichlet character $\chi$; # - \(x\) is the label of the Galois orbit of the newform $f$. # For each embedding of the coefficient field of $f$ into the complex numbers, the corresponding modular form over $\C$ has a label of the form \(N.k.a.x.n.i\), where # - \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi\); # - \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character $\chi$. # Dim -- # The **dimension** of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms. # The **dimension** of a newform refers to the dimension of its newform subspace, equivalently, the cardinality of its newform orbit. This is equal to the degree of its coefficient field (as an extension of $\Q$). # The **relative dimension** of $S_k^{\rm new}(N,\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newform subspaces. #$A$ (analytic_conductor) -- # The **analytic conductor** of a newform $f \in S_k^{\mathrm{new}}(N,\chi)$ is the positive real number # \[ # N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2, # \] # where $\psi(x):=\Gamma'(x)/\Gamma(x)$ is the logarithmic derivative of the Gamma function. #Field (nf_label) -- # The **coefficient field** of a modular form is the subfield of $\C$ generated by the coefficients $a_n$ of its $q$-expansion $\sum a_nq^n$. The space of cusp forms $S_k^\mathrm{new}(N,\chi)$ has a basis of modular forms that are simultaneous eigenforms for all Hecke operators and with algebraic Fourier coefficients. For such eigenforms the coefficient field will be a number field, and Galois conjugate eigenforms will share the same coefficient field. Moreover, if $m$ is the smallest positive integer such that the values of the character $\chi$ are contained in the cyclotomic field $\Q(\zeta_m)$, the coefficient field will contain $\Q(\zeta_m)$ # For eigenforms, the coefficient field is also known as the **Hecke field**. #CM (cm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #RM (rm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #Traces (trace_display) -- # For a newform $f \in S_k^{\rm new}(\Gamma_1(N))$, its **trace form** $\mathrm{Tr}(f)$ is the sum of its distinct conjugates under $\mathrm{Aut}(\C)$ (equivalently, the sum under all embeddings of the coefficient field into $\C$). The trace form is a modular form $\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma_1(N))$ whose $q$-expansion has integral coefficients $a_n(\mathrm{Tr}(f)) \in \Z$. # The coefficient $a_1$ is equal to the dimension of the newform. # For $p$ prime, the coefficient $a_p$ is the trace of Frobenius in the direct sum of the $\ell$-adic Galois representations attached to the conjugates of $f$ (for any prime $\ell$). When $f$ has weight $k=2$, the coefficient $a_p(f)$ is the trace of Frobenius acting on the modular abelian variety associated to $f$. # For a newspace $S_k^{\rm new}(N,\chi)$, its trace form is the sum of the trace forms $\mathrm{Tr}(f)$ over all newforms $f\in S_k^{\rm new}(N,k)$; it is also a modular form in $S_k^{\rm new}(\Gamma_1(N))$. # The graphical plot displayed in the properties box on the home page of each newform or newspace is computed using the trace form. #Fricke sign (fricke_eigenval) -- # The **Fricke involution** is the Atkin-Lehner involution $w_N$ on the space $S_k(\Gamma_0(N))$ (induced by the corresponding involution on the modular curve $X_0(N)$). # For a newform $f \in S_k^{\textup{new}}(\Gamma_0(N))$, the sign of the functional equation satisfied by the L-function attached to $f$ is $i^{-k}$ times the eigenvalue of $\omega_N$ on $f$. So, for example when $k=2$, the signs swap, and the analytic rank of $f$ is even when $w_N f = -f$ and odd when $w_N f = +f$. #$q$-expansion (qexp_display) -- # The **$q$-expansion** of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$, expressed as a power series $\sum_{n=0}^{\infty} a_n q^n$ in the variable $q=e^{2\pi iz}$. # For cusp forms, the constant coefficient $a_0$ of the $q$-expansion is zero. # For newforms, we have $a_1=1$ and the coefficients $a_n$ are algebraic integers in a number field $K \subseteq \C$. # Accordingly, we define the **$q$-expansion** of a newform orbit $[f]$ to be the $q$-expansion of any newform $f$ in the orbit, but with coefficients $a_n \in K$ (without an embedding into $\C$). Each embedding $K \hookrightarrow \C$ then gives rise to an embedded newform whose $q$-expansion has $a_n \in \C$, as above.