# Classical modular forms downloaded from the LMFDB on 18 April 2026.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/370/
# Query "{'level': 370}" returned 153 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.



"370.2.a.a"	1	2.9544648747873743	"1.1.1.1"	[]	[-1, -2, -1, -1]	-1	"q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\\cdots"
"370.2.a.b"	1	2.9544648747873743	"1.1.1.1"	[]	[-1, 0, -1, 0]	1	"q-q^{2}+q^{4}-q^{5}-q^{8}-3q^{9}+q^{10}+\\cdots"
"370.2.a.c"	1	2.9544648747873743	"1.1.1.1"	[]	[-1, 2, 1, 1]	-1	"q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\\cdots"
"370.2.a.d"	1	2.9544648747873743	"1.1.1.1"	[]	[1, -2, 1, 2]	-1	"q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+2q^{7}+\\cdots"
"370.2.a.e"	2	2.9544648747873743	"2.2.12.1"	[]	[-2, -2, 2, -6]	1	"q-q^{2}+(-1+\\beta )q^{3}+q^{4}+q^{5}+(1+\\cdots)q^{6}+\\cdots"
"370.2.a.f"	2	2.9544648747873743	"2.2.33.1"	[]	[2, 4, 2, -3]	-1	"q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+(-1+\\cdots)q^{7}+\\cdots"
"370.2.a.g"	3	2.9544648747873743	"3.3.892.1"	[]	[3, 0, -3, -1]	-1	"q+q^{2}-\\beta _{2}q^{3}+q^{4}-q^{5}-\\beta _{2}q^{6}+\\cdots"
"370.2.b.a"	2	2.9544648747873743	"2.0.4.1"	[]	[0, 0, -4, 0]	NULL	"q+i q^{2}-q^{4}+(-i-2)q^{5}-2 i q^{7}+\\cdots"
"370.2.b.b"	2	2.9544648747873743	"2.0.4.1"	[]	[0, 0, 2, 0]	NULL	"q+i q^{2}-q^{4}+(2 i+1)q^{5}+i q^{7}+\\cdots"
"370.2.b.c"	4	2.9544648747873743	"4.0.2304.2"	[]	[0, 0, -8, 0]	NULL	"q+\\beta _{1}q^{2}+\\beta _{2}q^{3}-q^{4}+(-2+\\beta _{1}+\\cdots)q^{5}+\\cdots"
"370.2.b.d"	10	2.9544648747873743	"10.0.12837029094400.1"	[]	[0, 0, 6, 0]	NULL	"q-\\beta _{2}q^{2}+(\\beta _{4}-\\beta _{5})q^{3}-q^{4}+(1-\\beta _{1}+\\cdots)q^{5}+\\cdots"
"370.2.c.a"	10	2.9544648747873743	NULL	[]	[-10, 0, -3, 0]	NULL	"q-q^{2}+\\beta _{1}q^{3}+q^{4}-\\beta _{3}q^{5}-\\beta _{1}q^{6}+\\cdots"
"370.2.c.b"	10	2.9544648747873743	NULL	[]	[10, 0, 3, 0]	NULL	"q+q^{2}+\\beta _{1}q^{3}+q^{4}+\\beta _{3}q^{5}+\\beta _{1}q^{6}+\\cdots"
"370.2.d.a"	2	2.9544648747873743	"2.0.4.1"	[]	[0, 0, 0, -4]	NULL	"q+i q^{2}-q^{4}+i q^{5}-2 q^{7}-i q^{8}+\\cdots"
"370.2.d.b"	2	2.9544648747873743	"2.0.4.1"	[]	[0, 0, 0, 10]	NULL	"q+i q^{2}-q^{4}+i q^{5}+5 q^{7}-i q^{8}+\\cdots"
"370.2.d.c"	6	2.9544648747873743	"6.0.399424.1"	[]	[0, -4, 0, 10]	NULL	"q+\\beta _{2}q^{2}+(-1-\\beta _{1}-\\beta _{3})q^{3}-q^{4}+\\cdots"
"370.2.e.a"	2	2.9544648747873743	"2.0.3.1"	[]	[1, -2, -1, 0]	NULL	"q+(1-\\zeta_{6})q^{2}-2\\zeta_{6}q^{3}-\\zeta_{6}q^{4}-\\zeta_{6}q^{5}+\\cdots"
"370.2.e.b"	2	2.9544648747873743	"2.0.3.1"	[]	[1, -1, 1, -2]	NULL	"q+(1-\\zeta_{6})q^{2}-\\zeta_{6}q^{3}-\\zeta_{6}q^{4}+\\zeta_{6}q^{5}+\\cdots"
"370.2.e.c"	2	2.9544648747873743	"2.0.3.1"	[]	[1, 3, 1, 0]	NULL	"q+(1-\\zeta_{6})q^{2}+3\\zeta_{6}q^{3}-\\zeta_{6}q^{4}+\\zeta_{6}q^{5}+\\cdots"
"370.2.e.d"	4	2.9544648747873743	"4.0.144.1"	[]	[-2, 2, -2, 0]	NULL	"q+(\\beta_1-1)q^{2}+\\beta_1 q^{3}-\\beta_1 q^{4}+\\cdots"
"370.2.e.e"	4	2.9544648747873743	"4.0.14400.7"	[]	[2, 2, -2, -4]	NULL	"q-\\beta _{2}q^{2}+(1+\\beta _{2})q^{3}+(-1-\\beta _{2})q^{4}+\\cdots"
"370.2.e.f"	6	2.9544648747873743	"6.0.2696112.1"	[]	[-3, 0, 3, -2]	NULL	"q-\\beta _{4}q^{2}+\\beta _{5}q^{3}+(-1+\\beta _{4})q^{4}+(1+\\cdots)q^{5}+\\cdots"
"370.2.g.a"	2	2.9544648747873743	"2.0.4.1"	[]	[0, -2, 4, 2]	NULL	"q+i q^{2}+(-i-1)q^{3}-q^{4}+(i+2)q^{5}+\\cdots"
"370.2.g.b"	2	2.9544648747873743	"2.0.4.1"	[]	[0, 0, -4, -4]	NULL	"q+i q^{2}-q^{4}+(i-2)q^{5}+(-2 i-2)q^{7}+\\cdots"
"370.2.g.c"	4	2.9544648747873743	"4.0.256.1"	[]	[0, 0, 0, 8]	NULL	"q+\\zeta_{8}^{2}q^{2}+2\\zeta_{8}q^{3}-q^{4}+(2\\zeta_{8}-\\zeta_{8}^{3})q^{5}+\\cdots"
"370.2.g.d"	10	2.9544648747873743	NULL	[]	[0, -2, 2, -4]	NULL	"q-\\beta _{6}q^{2}+(-\\beta _{2}-\\beta _{3}-\\beta _{8})q^{3}-q^{4}+\\cdots"
"370.2.g.e"	20	2.9544648747873743	NULL	[]	[0, -4, 2, -2]	NULL	"q-\\beta _{10}q^{2}-\\beta _{1}q^{3}-q^{4}-\\beta _{8}q^{5}+\\beta _{4}q^{6}+\\cdots"
"370.2.h.a"	2	2.9544648747873743	"2.0.4.1"	[]	[2, 0, -2, -4]	NULL	"q+q^{2}+q^{4}+(2 i-1)q^{5}+(2 i-2)q^{7}+\\cdots"
"370.2.h.b"	2	2.9544648747873743	"2.0.4.1"	[]	[2, 2, -2, 2]	NULL	"q+q^{2}+(-i+1)q^{3}+q^{4}+(-2 i-1)q^{5}+\\cdots"
"370.2.h.c"	4	2.9544648747873743	"4.0.256.1"	[]	[4, 0, 0, 8]	NULL	"q+q^{2}+2\\zeta_{8}q^{3}+q^{4}+(-\\zeta_{8}-2\\zeta_{8}^{3})q^{5}+\\cdots"
"370.2.h.d"	10	2.9544648747873743	NULL	[]	[10, 2, 2, -4]	NULL	"q+q^{2}+(\\beta _{1}-\\beta _{4})q^{3}+q^{4}+(-\\beta _{1}+\\beta _{3}+\\cdots)q^{5}+\\cdots"
"370.2.h.e"	20	2.9544648747873743	NULL	[]	[-20, 4, -4, -2]	NULL	"q-q^{2}-\\beta _{4}q^{3}+q^{4}+\\beta _{13}q^{5}+\\beta _{4}q^{6}+\\cdots"
"370.2.l.a"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, 0, 2]	NULL	"q+\\zeta_{12}q^{2}+(-\\zeta_{12}-\\zeta_{12}^{3})q^{3}+\\zeta_{12}^{2}q^{4}+\\cdots"
"370.2.l.b"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, 0, 4]	NULL	"q+\\zeta_{12}q^{2}+(\\zeta_{12}+\\zeta_{12}^{3})q^{3}+\\zeta_{12}^{2}q^{4}+\\cdots"
"370.2.l.c"	12	2.9544648747873743	"12.0.116304318664704.2"	[]	[0, 4, 0, 2]	NULL	"q-\\beta _{2}q^{2}+(1-\\beta _{6}+\\beta _{10})q^{3}+(1-\\beta _{6}+\\cdots)q^{4}+\\cdots"
"370.2.m.a"	4	2.9544648747873743	"4.0.1089.1"	[]	[-2, -3, -6, -12]	NULL	"q-\\beta _{2}q^{2}+(-1+\\beta _{1})q^{3}+(-1+\\beta _{2}+\\cdots)q^{4}+\\cdots"
"370.2.m.b"	4	2.9544648747873743	"4.0.1089.1"	[]	[2, 3, -3, 12]	NULL	"q+\\beta _{2}q^{2}+(1-\\beta _{1})q^{3}+(-1+\\beta _{2})q^{4}+\\cdots"
"370.2.m.c"	16	2.9544648747873743	NULL	[]	[-8, 3, 6, 12]	NULL	"q+(-1-\\beta _{1})q^{2}+\\beta _{10}q^{3}+\\beta _{1}q^{4}+\\cdots"
"370.2.m.d"	16	2.9544648747873743	NULL	[]	[8, -3, 0, -12]	NULL	"q+(1+\\beta _{1})q^{2}-\\beta _{10}q^{3}+\\beta _{1}q^{4}+\\beta _{8}q^{5}+\\cdots"
"370.2.n.a"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, -8, -6]	NULL	"q+\\zeta_{12}q^{2}+\\zeta_{12}^{2}q^{4}+(-2+\\zeta_{12}^{3})q^{5}+\\cdots"
"370.2.n.b"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, 4, 0]	NULL	"q+\\zeta_{12}q^{2}+\\zeta_{12}^{2}q^{4}+(-\\zeta_{12}+2\\zeta_{12}^{2}+\\cdots)q^{5}+\\cdots"
"370.2.n.c"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, 4, 6]	NULL	"q+\\zeta_{12}q^{2}+\\zeta_{12}^{2}q^{4}+(2-\\zeta_{12}-2\\zeta_{12}^{2}+\\cdots)q^{5}+\\cdots"
"370.2.n.d"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 0, 4, 0]	NULL	"q+\\zeta_{12}q^{2}+(3\\zeta_{12}-3\\zeta_{12}^{3})q^{3}+\\zeta_{12}^{2}q^{4}+\\cdots"
"370.2.n.e"	8	2.9544648747873743	"8.0.303595776.1"	[]	[0, 0, 0, 0]	NULL	"q-\\beta _{3}q^{2}-2\\beta _{5}q^{3}-\\beta _{4}q^{4}+(\\beta _{1}+\\beta _{5}+\\cdots)q^{5}+\\cdots"
"370.2.n.f"	12	2.9544648747873743	"12.0.89539436150784.1"	[]	[0, 0, 0, 0]	NULL	"q+\\beta _{3}q^{2}+(\\beta _{5}-\\beta _{6}-\\beta _{7}+\\beta _{9})q^{3}+\\cdots"
"370.2.o.a"	18	2.9544648747873743	NULL	[]	[0, 0, 0, -3]	NULL	"q-\\beta _{7}q^{2}+(-\\beta _{2}+\\beta _{8}-\\beta _{17})q^{3}+\\beta _{6}q^{4}+\\cdots"
"370.2.o.b"	18	2.9544648747873743	NULL	[]	[0, 0, 0, -9]	NULL	"q+\\beta _{4}q^{2}+\\beta _{8}q^{3}+(-\\beta _{10}+\\beta _{11})q^{4}+\\cdots"
"370.2.o.c"	24	2.9544648747873743	NULL	[]	[0, 0, 0, -3]	NULL	NULL
"370.2.o.d"	24	2.9544648747873743	NULL	[]	[0, 0, 0, -9]	NULL	NULL
"370.2.q.a"	4	2.9544648747873743	"4.0.144.1"	[]	[-2, 4, -4, 4]	NULL	"q-\\zeta_{12}^{2}q^{2}+(1+\\zeta_{12}-\\zeta_{12}^{3})q^{3}+\\cdots"
"370.2.q.b"	4	2.9544648747873743	"4.0.144.1"	[]	[-2, 6, 2, -2]	NULL	"q-\\zeta_{12}^{2}q^{2}+(1+\\zeta_{12}+\\zeta_{12}^{2}-2\\zeta_{12}^{3})q^{3}+\\cdots"
"370.2.q.c"	8	2.9544648747873743	"8.0.5308416.1"	[]	[4, 4, 4, 12]	NULL	"q+\\zeta_{24}^{4}q^{2}+(\\zeta_{24}^{4}+\\zeta_{24}^{5}+\\zeta_{24}^{6}+\\cdots)q^{3}+\\cdots"
"370.2.q.d"	12	2.9544648747873743	NULL	[]	[-6, 0, 2, -4]	NULL	"q+(-1-\\beta _{7})q^{2}+(1-\\beta _{1}-\\beta _{3}+2\\beta _{4}+\\cdots)q^{3}+\\cdots"
"370.2.q.e"	16	2.9544648747873743	NULL	[]	[-8, -8, -4, 0]	NULL	"q+(-1-\\beta _{14})q^{2}+(-1+\\beta _{2}+\\beta _{13}+\\cdots)q^{3}+\\cdots"
"370.2.q.f"	32	2.9544648747873743	NULL	[]	[16, -2, 6, -10]	NULL	NULL
"370.2.r.a"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 2, 2, 4]	NULL	"q+(-\\zeta_{12}+\\zeta_{12}^{3})q^{2}+(\\zeta_{12}^{2}+\\zeta_{12}^{3})q^{3}+\\cdots"
"370.2.r.b"	4	2.9544648747873743	"4.0.144.1"	[]	[0, 6, 4, -2]	NULL	"q+(-\\zeta_{12}+\\zeta_{12}^{3})q^{2}+(1-\\zeta_{12}+\\zeta_{12}^{2}+\\cdots)q^{3}+\\cdots"
"370.2.r.c"	8	2.9544648747873743	"8.0.5308416.1"	[]	[0, 8, -8, 0]	NULL	"q+(-\\zeta_{24}^{2}+\\zeta_{24}^{6})q^{2}+(1+\\zeta_{24}^{2}+\\cdots)q^{3}+\\cdots"
"370.2.r.d"	12	2.9544648747873743	NULL	[]	[0, 6, -8, -4]	NULL	"q+\\beta _{8}q^{2}+(1-\\beta _{1}-\\beta _{3}+\\beta _{4}+\\beta _{8}+\\cdots)q^{3}+\\cdots"
"370.2.r.e"	16	2.9544648747873743	NULL	[]	[0, -16, 0, 0]	NULL	"q+(\\beta _{5}-\\beta _{13})q^{2}+(-1-\\beta _{5}+\\beta _{13}+\\cdots)q^{3}+\\cdots"
"370.2.r.f"	32	2.9544648747873743	NULL	[]	[0, -10, 6, 2]	NULL	NULL
"370.2.v.a"	60	2.9544648747873743	NULL	[]	[0, 0, -3, 0]	NULL	NULL
"370.2.v.b"	60	2.9544648747873743	NULL	[]	[0, 0, 6, 0]	NULL	NULL
"370.2.w.a"	36	2.9544648747873743	NULL	[]	[0, 0, 0, 12]	NULL	NULL
"370.2.w.b"	48	2.9544648747873743	NULL	[]	[0, 0, 0, 12]	NULL	NULL
"370.2.x.a"	108	2.9544648747873743	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"370.2.ba.a"	108	2.9544648747873743	NULL	[]	[0, -6, 6, 0]	NULL	NULL
"370.2.ba.b"	120	2.9544648747873743	NULL	[]	[0, -6, -6, 0]	NULL	NULL
"370.2.bd.a"	108	2.9544648747873743	NULL	[]	[0, 6, 0, 0]	NULL	NULL
"370.2.bd.b"	120	2.9544648747873743	NULL	[]	[0, 6, 0, 0]	NULL	NULL
"370.3.f.a"	28	10.081769781330985	NULL	[]	[-28, 0, 0, 8]	NULL	NULL
"370.3.f.b"	28	10.081769781330985	NULL	[]	[28, 0, 0, -8]	NULL	NULL
"370.3.i.a"	36	10.081769781330985	NULL	[]	[-36, 4, -12, -16]	NULL	NULL
"370.3.i.b"	36	10.081769781330985	NULL	[]	[36, -4, 4, 8]	NULL	NULL
"370.3.j.a"	38	10.081769781330985	NULL	[]	[-38, 4, -8, 0]	NULL	NULL
"370.3.j.b"	38	10.081769781330985	NULL	[]	[38, 4, 8, 0]	NULL	NULL
"370.3.k.a"	2	10.081769781330985	"2.0.4.1"	[]	[-2, 2, -10, 0]	NULL	"q+(i-1)q^{2}+q^{3}-2 i q^{4}-5 q^{5}+\\cdots"
"370.3.k.b"	2	10.081769781330985	"2.0.4.1"	[]	[2, -2, 0, 0]	NULL	"q+(-i+1)q^{2}-q^{3}-2 i q^{4}+5 i q^{5}+\\cdots"
"370.3.k.c"	36	10.081769781330985	NULL	[]	[-36, -2, 14, 0]	NULL	NULL
"370.3.k.d"	36	10.081769781330985	NULL	[]	[36, 2, 10, 0]	NULL	NULL
"370.3.p.a"	76	10.081769781330985	NULL	[]	[-38, 0, -16, 0]	NULL	NULL
"370.3.p.b"	76	10.081769781330985	NULL	[]	[38, 0, -4, 0]	NULL	NULL
"370.3.s.a"	76	10.081769781330985	NULL	[]	[-38, 2, 10, 0]	NULL	NULL
"370.3.s.b"	76	10.081769781330985	NULL	[]	[38, 2, 2, 0]	NULL	NULL
"370.3.t.a"	72	10.081769781330985	NULL	[]	[-36, -2, 2, -8]	NULL	NULL
"370.3.t.b"	80	10.081769781330985	NULL	[]	[40, -2, -6, 8]	NULL	NULL
"370.3.u.a"	56	10.081769781330985	NULL	[]	[-28, 0, 0, 8]	NULL	NULL
"370.3.u.b"	56	10.081769781330985	NULL	[]	[28, 0, 0, -8]	NULL	NULL
"370.3.y.a"	228	10.081769781330985	NULL	[]	[0, -6, -18, 0]	NULL	NULL
"370.3.y.b"	228	10.081769781330985	NULL	[]	[0, -6, 6, 0]	NULL	NULL
"370.3.z.a"	216	10.081769781330985	NULL	[]	[0, 6, -6, 0]	NULL	NULL
"370.3.z.b"	240	10.081769781330985	NULL	[]	[0, 6, 18, 0]	NULL	NULL
"370.3.bb.a"	228	10.081769781330985	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"370.3.bb.b"	228	10.081769781330985	NULL	[]	[0, 0, 6, 0]	NULL	NULL
"370.3.bc.a"	144	10.081769781330985	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"370.3.bc.b"	144	10.081769781330985	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"370.4.a.a"	1	21.830706702124026	"1.1.1.1"	[]	[2, -2, 5, 2]	-1	"q+2q^{2}-2q^{3}+4q^{4}+5q^{5}-4q^{6}+\\cdots"
"370.4.a.b"	1	21.830706702124026	"1.1.1.1"	[]	[2, 1, 5, -25]	-1	"q+2q^{2}+q^{3}+4q^{4}+5q^{5}+2q^{6}+\\cdots"
"370.4.a.c"	1	21.830706702124026	"1.1.1.1"	[]	[2, 6, 5, 3]	1	"q+2q^{2}+6q^{3}+4q^{4}+5q^{5}+12q^{6}+\\cdots"
"370.4.a.d"	3	21.830706702124026	"3.3.4860.1"	[]	[6, -3, -15, 3]	-1	"q+2q^{2}+(-1+\\beta _{2})q^{3}+4q^{4}-5q^{5}+\\cdots"
"370.4.a.e"	4	21.830706702124026	NULL	[]	[-8, 3, 20, 16]	1	"q-2q^{2}+(1-\\beta _{1})q^{3}+4q^{4}+5q^{5}+\\cdots"
"370.4.a.f"	5	21.830706702124026	NULL	[]	[-10, -9, 25, -33]	-1	"q-2q^{2}+(-2+\\beta _{1})q^{3}+4q^{4}+5q^{5}+\\cdots"
"370.4.a.g"	5	21.830706702124026	NULL	[]	[-10, -3, -25, 4]	1	"q-2q^{2}+(-\\beta _{1}+\\beta _{2})q^{3}+4q^{4}-5q^{5}+\\cdots"
"370.4.a.h"	5	21.830706702124026	NULL	[]	[-10, 3, -25, 11]	-1	"q-2q^{2}+(1-\\beta _{1})q^{3}+4q^{4}-5q^{5}+\\cdots"
"370.4.a.i"	5	21.830706702124026	NULL	[]	[10, 11, 25, 23]	1	"q+2q^{2}+(2+\\beta _{1})q^{3}+4q^{4}+5q^{5}+\\cdots"
"370.4.a.j"	6	21.830706702124026	NULL	[]	[12, -3, -30, -4]	1	"q+2q^{2}+(-1+\\beta _{1})q^{3}+4q^{4}-5q^{5}+\\cdots"
"370.4.b.a"	22	21.830706702124026	NULL	[]	[0, 0, 8, 0]	NULL	NULL
"370.4.b.b"	32	21.830706702124026	NULL	[]	[0, 0, 8, 0]	NULL	NULL
"370.4.c.a"	28	21.830706702124026	NULL	[]	[-56, 0, 1, 0]	NULL	NULL
"370.4.c.b"	28	21.830706702124026	NULL	[]	[56, 0, -1, 0]	NULL	NULL
"370.4.d.a"	18	21.830706702124026	NULL	[]	[0, 0, 0, -10]	NULL	"q-2\\beta _{10}q^{2}-\\beta _{4}q^{3}-4q^{4}+5\\beta _{10}q^{5}+\\cdots"
"370.4.d.b"	20	21.830706702124026	NULL	[]	[0, -12, 0, 18]	NULL	"q-2\\beta _{11}q^{2}+(-1-\\beta _{3})q^{3}-4q^{4}+\\cdots"
"370.4.e.a"	2	21.830706702124026	"2.0.3.1"	[]	[2, -1, -5, 20]	NULL	"q+(2-2\\zeta_{6})q^{2}-\\zeta_{6}q^{3}-4\\zeta_{6}q^{4}-5\\zeta_{6}q^{5}+\\cdots"
"370.4.e.b"	16	21.830706702124026	NULL	[]	[16, 7, -40, 0]	NULL	"q+2\\beta _{2}q^{2}+(1-\\beta _{1}-\\beta _{2})q^{3}+(-4+\\cdots)q^{4}+\\cdots"
"370.4.e.c"	18	21.830706702124026	NULL	[]	[-18, 6, 45, -8]	NULL	"q+(-2-2\\beta _{4})q^{2}+(\\beta _{1}-\\beta _{2}-\\beta _{4})q^{3}+\\cdots"
"370.4.e.d"	20	21.830706702124026	NULL	[]	[-20, 0, -50, 8]	NULL	"q+(-2-2\\beta _{5})q^{2}+(\\beta _{1}-\\beta _{2})q^{3}+4\\beta _{5}q^{4}+\\cdots"
"370.4.e.e"	20	21.830706702124026	NULL	[]	[20, 0, 50, -24]	NULL	"q+2\\beta _{4}q^{2}+(-\\beta _{1}-\\beta _{3})q^{3}+(-4+4\\beta _{4}+\\cdots)q^{4}+\\cdots"
"370.4.l.a"	36	21.830706702124026	NULL	[]	[0, 0, 0, 16]	NULL	NULL
"370.4.l.b"	40	21.830706702124026	NULL	[]	[0, 12, 0, -12]	NULL	NULL
"370.5.f.a"	48	38.2468863410224	NULL	[]	[-96, 0, 0, -48]	NULL	NULL
"370.5.f.b"	48	38.2468863410224	NULL	[]	[96, 0, 0, 48]	NULL	NULL
"370.6.a.a"	6	59.342013330802835	NULL	[]	[24, -36, 150, -230]	1	"q+4q^{2}+(-6-\\beta _{3})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"370.6.a.b"	6	59.342013330802835	NULL	[]	[24, -6, -150, 100]	1	"q+4q^{2}+(-1+\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"370.6.a.c"	7	59.342013330802835	NULL	[]	[-28, -6, 175, 279]	-1	"q-4q^{2}+(-1+\\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"370.6.a.d"	7	59.342013330802835	NULL	[]	[-28, 8, -175, -149]	-1	"q-4q^{2}+(1+\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"370.6.a.e"	7	59.342013330802835	NULL	[]	[-28, 26, -175, -100]	1	"q-4q^{2}+(4-\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"370.6.a.f"	8	59.342013330802835	NULL	[]	[-32, -42, 200, -64]	1	"q-4q^{2}+(-5-\\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"370.6.a.g"	9	59.342013330802835	NULL	[]	[36, -6, -225, 51]	-1	"q+4q^{2}+(-1+\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"370.6.a.h"	10	59.342013330802835	NULL	[]	[40, 18, 250, 113]	-1	"q+4q^{2}+(2-\\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"370.6.b.a"	40	59.342013330802835	NULL	[]	[0, 0, -22, 0]	NULL	NULL
"370.6.b.b"	50	59.342013330802835	NULL	[]	[0, 0, -22, 0]	NULL	NULL
"370.6.c.a"	48	59.342013330802835	NULL	[]	[-192, 0, -37, 0]	NULL	NULL
"370.6.c.b"	48	59.342013330802835	NULL	[]	[192, 0, 37, 0]	NULL	NULL
"370.6.d.a"	32	59.342013330802835	NULL	[]	[0, 0, 0, 214]	NULL	NULL
"370.6.d.b"	34	59.342013330802835	NULL	[]	[0, -36, 0, 410]	NULL	NULL
"370.8.a.a"	8	115.58245942912505	NULL	[]	[64, -82, 1000, -1393]	-1	"q+8q^{2}+(-10-\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"370.8.a.b"	9	115.58245942912505	NULL	[]	[72, 0, -1125, 31]	-1	"q+8q^{2}-\\beta _{1}q^{3}+2^{6}q^{4}-5^{3}q^{5}-8\\beta _{1}q^{6}+\\cdots"
"370.8.a.c"	10	115.58245942912505	NULL	[]	[-80, 54, 1250, 1050]	1	"q-8q^{2}+(5+\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"370.8.a.d"	11	115.58245942912505	NULL	[]	[-88, -54, 1375, -1351]	-1	"q-8q^{2}+(-5+\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"370.8.a.e"	11	115.58245942912505	NULL	[]	[-88, 0, -1375, 312]	1	"q-8q^{2}+\\beta _{1}q^{3}+2^{6}q^{4}-5^{3}q^{5}-8\\beta _{1}q^{6}+\\cdots"
"370.8.a.f"	11	115.58245942912505	NULL	[]	[-88, 54, -1375, 655]	-1	"q-8q^{2}+(5-\\beta _{1})q^{3}+2^{6}q^{4}-5^{3}q^{5}+\\cdots"
"370.8.a.g"	12	115.58245942912505	NULL	[]	[96, 0, -1500, -312]	1	"q+8q^{2}+\\beta _{1}q^{3}+2^{6}q^{4}-5^{3}q^{5}+8\\beta _{1}q^{6}+\\cdots"
"370.8.a.h"	12	115.58245942912505	NULL	[]	[96, 80, 1500, 1008]	1	"q+8q^{2}+(7-\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"370.8.d.a"	42	115.58245942912505	NULL	[]	[0, 108, 0, -790]	NULL	NULL
"370.8.d.b"	44	115.58245942912505	NULL	[]	[0, 0, 0, 582]	NULL	NULL
"370.10.a.a"	12	190.56325938063307	NULL	[]	[192, -243, 7500, -6192]	1	"q+2^{4}q^{2}+(-20-\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"370.10.a.b"	12	190.56325938063307	NULL	[]	[192, -93, -7500, -6792]	1	"q+2^{4}q^{2}+(-8+\\beta _{1})q^{3}+2^{8}q^{4}-5^{4}q^{5}+\\cdots"
"370.10.a.c"	13	190.56325938063307	NULL	[]	[-208, -127, -8125, 4391]	-1	"q-2^{4}q^{2}+(-10+\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"370.10.a.d"	13	190.56325938063307	NULL	[]	[-208, 35, -8125, 6792]	1	"q-2^{4}q^{2}+(3-\\beta _{1})q^{3}+2^{8}q^{4}-5^{4}q^{5}+\\cdots"
"370.10.a.e"	13	190.56325938063307	NULL	[]	[-208, 447, 8125, 8593]	-1	"q-2^{4}q^{2}+(34+\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}q^{5}+\\cdots"
"370.10.a.f"	14	190.56325938063307	NULL	[]	[-224, 123, 8750, -8214]	1	"q-2^{4}q^{2}+(9-\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}q^{5}+\\cdots"
"370.10.a.g"	15	190.56325938063307	NULL	[]	[240, -93, -9375, -9193]	-1	"q+2^{4}q^{2}+(-6-\\beta _{1})q^{3}+2^{8}q^{4}-5^{4}q^{5}+\\cdots"
"370.10.a.h"	16	190.56325938063307	NULL	[]	[256, 243, 10000, 10615]	-1	"q+2^{4}q^{2}+(15+\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}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.



#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.