# Classical modular forms downloaded from the LMFDB on 25 May 2026.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/261/
# Query "{'level': 261}" returned 112 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.



"261.1.f.a"	4	0.13025596829721361	"4.0.256.1"	[]	[]	[0, 0, 0, -4]	NULL	"q-\\zeta_{8}q^{2}+(-\\zeta_{8}-\\zeta_{8}^{3})q^{5}-q^{7}+\\zeta_{8}^{3}q^{8}+\\cdots"
"261.2.a.a"	2	2.084095492755418	"2.2.5.1"	[]	[]	[-1, 0, -4, 0]	1	"q-\\beta q^{2}+(-1+\\beta )q^{4}-2q^{5}+(-1+\\cdots)q^{7}+\\cdots"
"261.2.a.b"	2	2.084095492755418	"2.2.5.1"	[]	[]	[-1, 0, -2, -4]	1	"q-\\beta q^{2}+(-1+\\beta )q^{4}+(-2+2\\beta )q^{5}+\\cdots"
"261.2.a.c"	2	2.084095492755418	"2.2.5.1"	[]	[]	[1, 0, 4, 0]	-1	"q+\\beta q^{2}+(-1+\\beta )q^{4}+2q^{5}+(-1+\\cdots)q^{7}+\\cdots"
"261.2.a.d"	2	2.084095492755418	"2.2.8.1"	[]	[]	[2, 0, 2, 0]	-1	"q+(1+\\beta )q^{2}+(1+2\\beta )q^{4}+q^{5}-2\\beta q^{7}+\\cdots"
"261.2.a.e"	3	2.084095492755418	"3.3.229.1"	[]	[]	[-2, 0, 0, 4]	-1	"q+(-1-\\beta _{2})q^{2}+(2+\\beta _{1})q^{4}+2\\beta _{1}q^{5}+\\cdots"
"261.2.c.a"	2	2.084095492755418	"2.0.20.1"	[]	[]	[0, 0, 6, 4]	NULL	"q+\\beta q^{2}-3q^{4}+3q^{5}+2q^{7}-\\beta q^{8}+\\cdots"
"261.2.c.b"	4	2.084095492755418	"4.0.400.1"	[]	[]	[0, 0, -4, -8]	NULL	"q+\\beta _{1}q^{2}+(1+\\beta _{2})q^{4}+2\\beta _{2}q^{5}+(-1+\\cdots)q^{7}+\\cdots"
"261.2.c.c"	6	2.084095492755418	"6.0.10241038656.1"	[-87]	[]	[0, 0, 0, 0]	NULL	"q+\\beta _{1}q^{2}+(-2+\\beta _{2})q^{4}+(\\beta _{2}-\\beta _{4}+\\cdots)q^{7}+\\cdots"
"261.2.e.a"	22	2.084095492755418	NULL	[]	[]	[-1, -2, 1, 7]	NULL	NULL
"261.2.e.b"	34	2.084095492755418	NULL	[]	[]	[-1, -2, 1, -9]	NULL	NULL
"261.2.g.a"	4	2.084095492755418	"4.0.256.1"	[]	[]	[0, 0, 0, 16]	NULL	"q+\\zeta_{8}q^{2}-\\zeta_{8}^{2}q^{4}+(-\\zeta_{8}+\\zeta_{8}^{3})q^{5}+\\cdots"
"261.2.g.b"	8	2.084095492755418	"8.0.40960000.1"	[]	[]	[0, 0, 0, -8]	NULL	"q-\\beta _{1}q^{2}+(2\\beta _{2}+\\beta _{6})q^{4}+(\\beta _{4}-\\beta _{7})q^{5}+\\cdots"
"261.2.g.c"	8	2.084095492755418	"8.0.1871773696.1"	[]	[]	[0, 0, 0, -8]	NULL	"q+\\beta _{1}q^{2}+(2\\beta _{3}+\\beta _{5})q^{4}+(\\beta _{1}+\\beta _{2}+\\cdots)q^{5}+\\cdots"
"261.2.i.a"	56	2.084095492755418	NULL	[]	[]	[0, 0, 2, -2]	NULL	NULL
"261.2.k.a"	6	2.084095492755418	"6.0.16807.1"	[]	[]	[2, 0, -1, 1]	NULL	"q+(1-\\zeta_{14}-\\zeta_{14}^{3}+\\zeta_{14}^{4}-\\zeta_{14}^{5})q^{2}+\\cdots"
"261.2.k.b"	18	2.084095492755418	NULL	[]	[]	[2, 0, 7, -4]	NULL	"q+\\beta _{3}q^{2}+(-\\beta _{10}-\\beta _{16})q^{4}+(-\\beta _{6}+\\cdots)q^{5}+\\cdots"
"261.2.k.c"	18	2.084095492755418	NULL	[]	[]	[4, 0, 1, -4]	NULL	"q+(\\beta _{3}-\\beta _{9})q^{2}+(2\\beta _{1}+\\beta _{5}-\\beta _{6}+2\\beta _{12}+\\cdots)q^{4}+\\cdots"
"261.2.k.d"	24	2.084095492755418	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.2.l.a"	112	2.084095492755418	NULL	[]	[]	[-6, -4, 0, -4]	NULL	NULL
"261.2.o.a"	12	2.084095492755418	"12.0.7877952219361.1"	[]	[]	[7, 0, 1, -11]	NULL	"q+(1-\\beta _{3}-\\beta _{7}-\\beta _{9}-\\beta _{10})q^{2}+(\\beta _{1}+\\cdots)q^{4}+\\cdots"
"261.2.o.b"	24	2.084095492755418	NULL	[]	[]	[0, 0, 4, 8]	NULL	NULL
"261.2.o.c"	36	2.084095492755418	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.2.q.a"	336	2.084095492755418	NULL	[]	[]	[-5, -10, -9, -5]	NULL	NULL
"261.2.r.a"	120	2.084095492755418	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.2.u.a"	336	2.084095492755418	NULL	[]	[]	[-7, -14, -9, -5]	NULL	NULL
"261.2.x.a"	672	2.084095492755418	NULL	[]	[]	[-36, -24, -42, -10]	NULL	NULL
"261.3.b.a"	20	7.111734899803749	NULL	[]	[]	[0, 0, 0, 16]	NULL	"q+\\beta _{1}q^{2}+(-2+\\beta _{2})q^{4}+\\beta _{12}q^{5}+\\cdots"
"261.3.d.a"	20	7.111734899803749	NULL	[]	[]	[0, 0, 0, -16]	NULL	"q+\\beta _{10}q^{2}+(2+\\beta _{1})q^{4}-\\beta _{13}q^{5}+(-1+\\cdots)q^{7}+\\cdots"
"261.3.f.a"	8	7.111734899803749	NULL	[]	[]	[-2, 0, 0, -4]	NULL	"q-\\beta _{5}q^{2}+(-\\beta _{3}-\\beta _{4}+\\beta _{5}+\\beta _{6})q^{4}+\\cdots"
"261.3.f.b"	8	7.111734899803749	NULL	[]	[]	[8, 0, 0, 12]	NULL	"q+(1+\\beta _{2}-\\beta _{3})q^{2}+(\\beta _{2}-2\\beta _{3}-2\\beta _{4}+\\cdots)q^{4}+\\cdots"
"261.3.f.c"	12	7.111734899803749	NULL	[]	[]	[0, 0, 0, -12]	NULL	"q+(\\beta _{1}-\\beta _{9})q^{2}+(-\\beta _{2}-\\beta _{6}+3\\beta _{7}+\\cdots)q^{4}+\\cdots"
"261.3.f.d"	20	7.111734899803749	NULL	[]	[]	[0, 0, 0, 0]	NULL	"q+\\beta _{1}q^{2}+(-2\\beta _{10}+\\beta _{13})q^{4}-\\beta _{16}q^{5}+\\cdots"
"261.3.h.a"	116	7.111734899803749	NULL	[]	[]	[0, 0, -6, -2]	NULL	NULL
"261.3.j.a"	112	7.111734899803749	NULL	[]	[]	[0, 2, -18, 2]	NULL	NULL
"261.3.m.a"	232	7.111734899803749	NULL	[]	[]	[-2, -4, 0, -4]	NULL	NULL
"261.3.n.a"	120	7.111734899803749	NULL	[]	[]	[0, 0, 0, 16]	NULL	NULL
"261.3.p.a"	120	7.111734899803749	NULL	[]	[]	[0, 0, 0, -16]	NULL	NULL
"261.3.s.a"	48	7.111734899803749	NULL	[]	[]	[16, 0, 14, -10]	NULL	NULL
"261.3.s.b"	120	7.111734899803749	NULL	[]	[]	[-8, 0, 0, 0]	NULL	NULL
"261.3.s.c"	120	7.111734899803749	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.3.t.a"	696	7.111734899803749	NULL	[]	[]	[-15, -10, -15, -5]	NULL	NULL
"261.3.v.a"	696	7.111734899803749	NULL	[]	[]	[-21, -14, -15, -5]	NULL	NULL
"261.3.w.a"	1392	7.111734899803749	NULL	[]	[]	[-12, -24, -14, -10]	NULL	NULL
"261.4.a.a"	2	15.399498511498301	"2.2.41.1"	[]	[]	[1, 0, 1, -24]	-1	"q+\\beta q^{2}+(2+\\beta )q^{4}+(2-3\\beta )q^{5}+(-13+\\cdots)q^{7}+\\cdots"
"261.4.a.b"	2	15.399498511498301	"2.2.8.1"	[]	[]	[2, 0, 10, -16]	-1	"q+(1+\\beta )q^{2}+(-5+2\\beta )q^{4}+(5+4\\beta )q^{5}+\\cdots"
"261.4.a.c"	2	15.399498511498301	"2.2.17.1"	[]	[]	[5, 0, 11, -24]	1	"q+(3-\\beta )q^{2}+(5-5\\beta )q^{4}+(7-3\\beta )q^{5}+\\cdots"
"261.4.a.d"	5	15.399498511498301	NULL	[]	[]	[-3, 0, -29, 4]	-1	"q+(-1+\\beta _{1})q^{2}+(6+\\beta _{2}+\\beta _{3})q^{4}+\\cdots"
"261.4.a.e"	5	15.399498511498301	NULL	[]	[]	[-3, 0, 1, 4]	1	"q+(-1+\\beta _{1})q^{2}+(6-2\\beta _{1}+\\beta _{2}+\\beta _{3}+\\cdots)q^{4}+\\cdots"
"261.4.a.f"	5	15.399498511498301	"5.5.13458092.1"	[]	[]	[0, 0, -10, 40]	1	"q+\\beta _{1}q^{2}+(6+2\\beta _{1}+2\\beta _{3}+\\beta _{4})q^{4}+\\cdots"
"261.4.a.g"	7	15.399498511498301	NULL	[]	[]	[-2, 0, -20, -12]	-1	"q-\\beta _{1}q^{2}+(5+\\beta _{1}+\\beta _{3})q^{4}+(-3+\\beta _{1}+\\cdots)q^{5}+\\cdots"
"261.4.a.h"	7	15.399498511498301	NULL	[]	[]	[2, 0, 20, -12]	1	"q+\\beta _{1}q^{2}+(5+\\beta _{1}+\\beta _{3})q^{4}+(3-\\beta _{1}+\\cdots)q^{5}+\\cdots"
"261.4.c.a"	2	15.399498511498301	"2.0.116.1"	[-87]	[]	[0, 0, 0, 64]	NULL	"q+\\beta q^{2}-21q^{4}+2^{5}q^{7}-13\\beta q^{8}+\\cdots"
"261.4.c.b"	4	15.399498511498301	"4.0.13456.1"	[-87]	[]	[0, 0, 0, -64]	NULL	"q-\\beta _{1}q^{2}+(-2+\\beta _{3})q^{4}+(-15-2\\beta _{3})q^{7}+\\cdots"
"261.4.c.c"	6	15.399498511498301	NULL	[]	[]	[0, 0, -22, -28]	NULL	"q+\\beta _{1}q^{2}+(-5+\\beta _{4}+\\beta _{5})q^{4}+(-4+\\cdots)q^{5}+\\cdots"
"261.4.c.d"	8	15.399498511498301	NULL	[]	[]	[0, 0, 0, -16]	NULL	"q+\\beta _{2}q^{2}+(-2-\\beta _{1})q^{4}-\\beta _{3}q^{5}+(-1+\\cdots)q^{7}+\\cdots"
"261.4.c.e"	16	15.399498511498301	NULL	[]	[]	[0, 0, 28, 40]	NULL	"q+\\beta _{1}q^{2}+(-3+\\beta _{2})q^{4}+(2+\\beta _{8})q^{5}+\\cdots"
"261.4.e.a"	78	15.399498511498301	NULL	[]	[]	[2, 1, 4, 62]	NULL	NULL
"261.4.e.b"	90	15.399498511498301	NULL	[]	[]	[2, 1, 4, -50]	NULL	NULL
"261.4.g.a"	60	15.399498511498301	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.5.b.a"	36	26.97956036488337	NULL	[]	[]	[0, 0, 0, -48]	NULL	NULL
"261.5.d.a"	40	26.97956036488337	NULL	[]	[]	[0, 0, 0, 160]	NULL	NULL
"261.5.f.a"	18	26.97956036488337	NULL	[]	[]	[8, 0, 0, -4]	NULL	"q+\\beta _{2}q^{2}+(-\\beta _{1}+\\beta _{2}-6\\beta _{4}-\\beta _{13}+\\cdots)q^{4}+\\cdots"
"261.5.f.b"	40	26.97956036488337	NULL	[]	[]	[-12, 0, 0, 0]	NULL	NULL
"261.5.f.c"	40	26.97956036488337	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.6.a.a"	4	41.86017697118795	"4.4.3257317.1"	[]	[]	[0, 0, 68, -208]	1	"q+\\beta _{2}q^{2}+(2-3\\beta _{1}+5\\beta _{2}-\\beta _{3})q^{4}+\\cdots"
"261.6.a.b"	4	41.86017697118795	"4.4.8167381.1"	[]	[]	[3, 0, 136, -28]	-1	"q+(1+\\beta _{1})q^{2}+(14+6\\beta _{1}+3\\beta _{2}+2\\beta _{3})q^{4}+\\cdots"
"261.6.a.c"	5	41.86017697118795	NULL	[]	[]	[9, 0, -14, -68]	1	"q+(2-\\beta _{1})q^{2}+(11-\\beta _{1}+\\beta _{3})q^{4}+(-3+\\cdots)q^{5}+\\cdots"
"261.6.a.d"	7	41.86017697118795	NULL	[]	[]	[-13, 0, -64, 168]	1	"q+(-2+\\beta _{1})q^{2}+(21-2\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.6.a.e"	7	41.86017697118795	NULL	[]	[]	[-4, 0, -32, 184]	-1	"q+(-1+\\beta _{1})q^{2}+(23-3\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.6.a.f"	8	41.86017697118795	NULL	[]	[]	[1, 0, -14, 128]	-1	"q+\\beta _{1}q^{2}+(19+\\beta _{1}+\\beta _{2})q^{4}+(-2+\\cdots)q^{5}+\\cdots"
"261.6.a.g"	12	41.86017697118795	NULL	[]	[]	[-4, 0, -100, -136]	1	"q-\\beta _{1}q^{2}+(17-\\beta _{1}+\\beta _{2})q^{4}+(-9+\\cdots)q^{5}+\\cdots"
"261.6.a.h"	12	41.86017697118795	NULL	[]	[]	[4, 0, 100, -136]	-1	"q+\\beta _{1}q^{2}+(17-\\beta _{1}+\\beta _{2})q^{4}+(9-\\beta _{1}+\\cdots)q^{5}+\\cdots"
"261.6.c.a"	6	41.86017697118795	"6.0.10241038656.1"	[-87]	[]	[0, 0, 0, 0]	NULL	"q+(3\\beta _{1}+\\beta _{5})q^{2}+(-2^{5}+10\\beta _{2}-13\\beta _{4}+\\cdots)q^{4}+\\cdots"
"261.6.c.b"	12	41.86017697118795	NULL	[]	[]	[0, 0, -46, 20]	NULL	"q+\\beta _{1}q^{2}+(-14+\\beta _{2})q^{4}+(-4-\\beta _{6}+\\cdots)q^{5}+\\cdots"
"261.6.c.c"	20	41.86017697118795	NULL	[]	[]	[0, 0, 0, 272]	NULL	"q+\\beta _{11}q^{2}+(-11+\\beta _{1})q^{4}+\\beta _{12}q^{5}+\\cdots"
"261.6.c.d"	24	41.86017697118795	NULL	[]	[]	[0, 0, 196, -120]	NULL	NULL
"261.6.g.a"	100	41.86017697118795	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"261.7.b.a"	56	60.04411582478111	NULL	[]	[]	[0, 0, 0, -1280]	NULL	NULL
"261.7.d.a"	60	60.04411582478111	NULL	[]	[]	[0, 0, 0, -816]	NULL	NULL
"261.8.a.a"	6	81.53249165135578	NULL	[]	[]	[7, 0, 126, -628]	-1	"q+(1+\\beta _{1})q^{2}+(53+2\\beta _{1}+\\beta _{3})q^{4}+\\cdots"
"261.8.a.b"	7	81.53249165135578	NULL	[]	[]	[8, 0, 320, -1704]	-1	"q+(1+\\beta _{1})q^{2}+(50-\\beta _{1}+2\\beta _{2}+\\beta _{3}+\\cdots)q^{4}+\\cdots"
"261.8.a.c"	7	81.53249165135578	NULL	[]	[]	[17, 0, 376, -692]	1	"q+(2+\\beta _{1})q^{2}+(31+3\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.8.a.d"	9	81.53249165135578	NULL	[]	[]	[-9, 0, 126, 744]	1	"q+(-1-\\beta _{1})q^{2}+(78+3\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.8.a.e"	10	81.53249165135578	NULL	[]	[]	[-15, 0, -624, 680]	-1	"q+(-2+\\beta _{1})q^{2}+(62-\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.8.a.f"	10	81.53249165135578	NULL	[]	[]	[0, 0, -180, 1040]	1	"q+\\beta _{1}q^{2}+(92+2\\beta _{1}+\\beta _{2})q^{4}+(-18+\\cdots)q^{5}+\\cdots"
"261.8.a.g"	16	81.53249165135578	NULL	[]	[]	[-8, 0, -500, 840]	-1	"q-\\beta _{1}q^{2}+(57+2\\beta _{1}+\\beta _{2})q^{4}+(-30+\\cdots)q^{5}+\\cdots"
"261.8.a.h"	16	81.53249165135578	NULL	[]	[]	[8, 0, 500, 840]	1	"q+\\beta _{1}q^{2}+(57+2\\beta _{1}+\\beta _{2})q^{4}+(30+\\cdots)q^{5}+\\cdots"
"261.8.c.a"	6	81.53249165135578	"6.0.10241038656.1"	[-87]	[]	[0, 0, 0, 0]	NULL	"q+(7\\beta _{1}-2\\beta _{5})q^{2}+(-2^{7}+\\beta _{2}+52\\beta _{4}+\\cdots)q^{4}+\\cdots"
"261.8.c.b"	16	81.53249165135578	NULL	[]	[]	[0, 0, 198, 660]	NULL	"q+\\beta _{1}q^{2}+(-45+\\beta _{2})q^{4}+(12+\\beta _{4}+\\cdots)q^{5}+\\cdots"
"261.8.c.c"	28	81.53249165135578	NULL	[]	[]	[0, 0, 0, -2720]	NULL	NULL
"261.8.c.d"	36	81.53249165135578	NULL	[]	[]	[0, 0, -892, 24]	NULL	NULL
"261.9.b.a"	76	106.3258172259542	NULL	[]	[]	[0, 0, 0, -6928]	NULL	NULL
"261.9.d.a"	80	106.3258172259542	NULL	[]	[]	[0, 0, 0, 320]	NULL	NULL
"261.10.a.a"	9	134.4243532387709	NULL	[]	[]	[-3, 0, -909, -360]	1	"q-\\beta _{1}q^{2}+(179-\\beta _{1}+\\beta _{2})q^{4}+(-108+\\cdots)q^{5}+\\cdots"
"261.10.a.b"	9	134.4243532387709	NULL	[]	[]	[0, 0, 738, -7128]	1	"q+\\beta _{1}q^{2}+(133+2\\beta _{1}+\\beta _{2})q^{4}+(83+\\cdots)q^{5}+\\cdots"
"261.10.a.c"	9	134.4243532387709	NULL	[]	[]	[51, 0, 2841, -13968]	-1	"q+(6-\\beta _{1})q^{2}+(288-6\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.10.a.d"	12	134.4243532387709	NULL	[]	[]	[-35, 0, -909, 9244]	-1	"q+(-3+\\beta _{1})q^{2}+(262-\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.10.a.e"	12	134.4243532387709	NULL	[]	[]	[-16, 0, -1762, 12080]	-1	"q+(-1-\\beta _{1})q^{2}+(291+3\\beta _{1}+\\beta _{2}+\\cdots)q^{4}+\\cdots"
"261.10.a.f"	12	134.4243532387709	NULL	[]	[]	[-13, 0, -2159, -4364]	1	"q+(-1-\\beta _{1})q^{2}+(7^{3}+3\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.10.a.g"	21	134.4243532387709	NULL	[]	[]	[-16, 0, -2500, 2900]	1	NULL
"261.10.a.h"	21	134.4243532387709	NULL	[]	[]	[16, 0, 2500, 2900]	-1	NULL
"261.11.b.a"	92	165.828242947842	NULL	[]	[]	[0, 0, 0, 52144]	NULL	NULL
"261.11.d.a"	100	165.828242947842	NULL	[]	[]	[0, 0, 0, 36784]	NULL	NULL
"261.12.a.a"	11	200.53757012605269	NULL	[]	[]	[32, 0, 2740, -49432]	-1	"q+(3-\\beta _{1})q^{2}+(832-6\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.12.a.b"	11	200.53757012605269	NULL	[]	[]	[55, 0, 5656, -14784]	-1	"q+(5-\\beta _{1})q^{2}+(704-8\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.12.a.c"	12	200.53757012605269	NULL	[]	[]	[41, 0, 11906, -42544]	1	"q+(3+\\beta _{1})q^{2}+(979+7\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.12.a.d"	14	200.53757012605269	NULL	[]	[]	[-9, 0, 5656, 52444]	1	"q+(-1+\\beta _{1})q^{2}+(992+\\beta _{1}+\\beta _{2})q^{4}+\\cdots"
"261.12.a.e"	14	200.53757012605269	NULL	[]	[]	[0, 0, -9760, 85024]	1	"q-\\beta _{1}q^{2}+(1312+3\\beta _{1}+\\beta _{2})q^{4}+(-698+\\cdots)q^{5}+\\cdots"
"261.12.a.f"	15	200.53757012605269	NULL	[]	[]	[-87, 0, -13094, 24684]	-1	"q+(-6+\\beta _{1})q^{2}+(1196-7\\beta _{1}+\\beta _{2}+\\cdots)q^{4}+\\cdots"
"261.12.a.g"	26	200.53757012605269	NULL	[]	[]	[-32, 0, -12500, -66832]	-1	NULL
"261.12.a.h"	26	200.53757012605269	NULL	[]	[]	[32, 0, 12500, -66832]	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.