# Classical modular forms downloaded from the LMFDB on 09 June 2026.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/444/
# Query "{'level': 444}" returned 134 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.



"444.1.h.a"	2	0.22158486560905308	"2.0.3.1"	[]	[37]	[0, -1, 0, 2]	NULL	"q+\\zeta_{6}^{2}q^{3}+q^{7}-\\zeta_{6}q^{9}+(\\zeta_{6}+\\zeta_{6}^{2}+\\cdots)q^{11}+\\cdots"
"444.1.bc.a"	6	0.22158486560905308	"6.0.19683.1"	[-3]	[]	[0, 0, 0, 3]	NULL	"q-\\zeta_{18}^{7}q^{3}+(\\zeta_{18}^{5}-\\zeta_{18}^{6})q^{7}-\\zeta_{18}^{5}q^{9}+\\cdots"
"444.1.bf.a"	6	0.22158486560905308	"6.0.19683.1"	[-3]	[]	[0, 0, 0, -3]	NULL	"q+\\zeta_{18}^{8}q^{3}+(-\\zeta_{18}^{3}-\\zeta_{18}^{7})q^{7}+\\cdots"
"444.2.a.a"	1	3.5453578497448492	"1.1.1.1"	[]	[]	[0, -1, 0, 0]	-1	"q-q^{3}+q^{9}+4q^{11}-2q^{13}+6q^{19}+\\cdots"
"444.2.a.b"	1	3.5453578497448492	"1.1.1.1"	[]	[]	[0, 1, -2, -4]	1	"q+q^{3}-2q^{5}-4q^{7}+q^{9}-4q^{11}+\\cdots"
"444.2.a.c"	2	3.5453578497448492	"2.2.12.1"	[]	[]	[0, -2, -2, 0]	1	"q-q^{3}+(-1+\\beta )q^{5}-2\\beta q^{7}+q^{9}+\\cdots"
"444.2.a.d"	2	3.5453578497448492	"2.2.24.1"	[]	[]	[0, 2, 0, 4]	-1	"q+q^{3}+\\beta q^{5}+2q^{7}+q^{9}+(2-2\\beta )q^{13}+\\cdots"
"444.2.c.a"	8	3.5453578497448492	"8.0.386672896.3"	[]	[]	[-2, 2, 0, 0]	NULL	"q+\\beta _{7}q^{2}+(-\\beta _{2}-\\beta _{7})q^{3}+(\\beta _{3}+\\beta _{4}+\\cdots)q^{4}+\\cdots"
"444.2.c.b"	8	3.5453578497448492	"8.0.386672896.3"	[]	[]	[2, -2, 0, 0]	NULL	"q-\\beta _{7}q^{2}+(-1-\\beta _{4}-\\beta _{7})q^{3}+(\\beta _{3}+\\cdots)q^{4}+\\cdots"
"444.2.c.c"	20	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	"q+\\beta _{1}q^{2}-\\beta _{5}q^{3}+\\beta _{2}q^{4}+(-\\beta _{13}+\\cdots)q^{5}+\\cdots"
"444.2.c.d"	36	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.e.a"	4	3.5453578497448492	"4.0.1600.1"	[]	[]	[0, -4, 0, 4]	NULL	"q-q^{3}-\\beta _{1}q^{5}+(1-\\beta _{3})q^{7}+q^{9}+(-2+\\cdots)q^{11}+\\cdots"
"444.2.e.b"	4	3.5453578497448492	"4.0.32448.1"	[]	[]	[0, 4, 0, -4]	NULL	"q+q^{3}+\\beta _{1}q^{5}+(-1+\\beta _{2})q^{7}+q^{9}+\\cdots"
"444.2.g.a"	4	3.5453578497448492	"4.0.7056.2"	[]	[]	[-2, 0, -8, 0]	NULL	"q+(-1+\\beta _{2})q^{2}-\\beta _{3}q^{3}+(-1-\\beta _{2}+\\cdots)q^{4}+\\cdots"
"444.2.g.b"	4	3.5453578497448492	"4.0.7056.2"	[]	[]	[2, 0, 8, 0]	NULL	"q+(1-\\beta _{2})q^{2}+\\beta _{3}q^{3}+(-1-\\beta _{2})q^{4}+\\cdots"
"444.2.g.c"	16	3.5453578497448492	NULL	[-111]	[]	[0, 0, 0, 0]	NULL	"q-\\beta _{1}q^{2}-\\beta _{4}q^{3}-\\beta _{8}q^{4}-\\beta _{12}q^{5}+\\cdots"
"444.2.g.d"	48	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.i.a"	2	3.5453578497448492	"2.0.3.1"	[]	[]	[0, 1, -2, 3]	NULL	"q+\\zeta_{6}q^{3}-2\\zeta_{6}q^{5}+3\\zeta_{6}q^{7}+(-1+\\cdots)q^{9}+\\cdots"
"444.2.i.b"	4	3.5453578497448492	"4.0.12321.1"	[]	[]	[0, 2, 1, -6]	NULL	"q+\\beta _{2}q^{3}+\\beta _{1}q^{5}-3\\beta _{2}q^{7}+(-1+\\beta _{2}+\\cdots)q^{9}+\\cdots"
"444.2.i.c"	6	3.5453578497448492	"6.0.27379323.1"	[]	[]	[0, -3, -1, 3]	NULL	"q+(-1-\\beta _{4})q^{3}+(\\beta _{1}+\\beta _{2})q^{5}+(1+\\beta _{4}+\\cdots)q^{7}+\\cdots"
"444.2.k.a"	2	3.5453578497448492	"2.0.4.1"	[]	[]	[-2, 2, -4, 0]	NULL	"q+(i-1)q^{2}+q^{3}-2 i q^{4}+(2 i-2)q^{5}+\\cdots"
"444.2.k.b"	2	3.5453578497448492	"2.0.4.1"	[]	[]	[2, -2, -4, 0]	NULL	"q+(-i+1)q^{2}-q^{3}-2 i q^{4}+(2 i-2)q^{5}+\\cdots"
"444.2.k.c"	36	3.5453578497448492	NULL	[]	[]	[-4, -36, 2, 0]	NULL	NULL
"444.2.k.d"	36	3.5453578497448492	NULL	[]	[]	[0, 36, 2, 0]	NULL	NULL
"444.2.m.a"	4	3.5453578497448492	"4.0.1936.1"	[]	[]	[0, 0, 0, -4]	NULL	"q+\\beta _{1}q^{3}-q^{7}+(3+\\beta _{3})q^{9}+(2\\beta _{1}-\\beta _{2}+\\cdots)q^{11}+\\cdots"
"444.2.m.b"	4	3.5453578497448492	"4.0.144.1"	[-3]	[]	[0, 0, 0, 0]	NULL	"q-\\beta_{2} q^{3}+2\\beta_{3} q^{7}-3 q^{9}+(2\\beta_{3}-2\\beta_{2}-\\beta_1+1)q^{13}+\\cdots"
"444.2.m.c"	4	3.5453578497448492	"4.0.144.1"	[]	[]	[0, 0, 0, 12]	NULL	"q+(-\\zeta_{12}-\\zeta_{12}^{3})q^{3}+(-1-2\\zeta_{12}+\\cdots)q^{5}+\\cdots"
"444.2.m.d"	12	3.5453578497448492	NULL	[]	[]	[0, 0, 0, -8]	NULL	"q+(\\beta _{1}-\\beta _{7})q^{3}-\\beta _{11}q^{5}+(-1-\\beta _{2}+\\cdots)q^{7}+\\cdots"
"444.2.p.a"	4	3.5453578497448492	"4.0.576.2"	[]	[]	[0, -6, 0, -6]	NULL	"q+\\beta _{1}q^{2}+(-2-\\beta _{2})q^{3}+2\\beta _{2}q^{4}+\\cdots"
"444.2.p.b"	4	3.5453578497448492	"4.0.576.2"	[]	[]	[0, 6, 0, 6]	NULL	"q+\\beta _{1}q^{2}+(2+\\beta _{2})q^{3}+2\\beta _{2}q^{4}+(-2\\beta _{1}+\\cdots)q^{5}+\\cdots"
"444.2.p.c"	136	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.r.a"	2	3.5453578497448492	"2.0.3.1"	[]	[]	[0, -1, -6, -1]	NULL	"q-\\zeta_{6}q^{3}+(-4+2\\zeta_{6})q^{5}-\\zeta_{6}q^{7}+\\cdots"
"444.2.r.b"	2	3.5453578497448492	"2.0.3.1"	[]	[]	[0, -1, -3, 2]	NULL	"q-\\zeta_{6}q^{3}+(-2+\\zeta_{6})q^{5}+2\\zeta_{6}q^{7}+\\cdots"
"444.2.r.c"	4	3.5453578497448492	"4.0.441.1"	[]	[]	[0, -2, 9, 0]	NULL	"q-\\beta _{2}q^{3}+(3-2\\beta _{2}+\\beta _{3})q^{5}+(-2+\\cdots)q^{7}+\\cdots"
"444.2.r.d"	8	3.5453578497448492	"8.0.147748753161.1"	[]	[]	[0, 4, 0, -1]	NULL	"q+(1+\\beta _{3})q^{3}+\\beta _{5}q^{5}+(-\\beta _{4}-\\beta _{7})q^{7}+\\cdots"
"444.2.t.a"	144	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.u.a"	12	3.5453578497448492	NULL	[]	[]	[0, 0, 3, 0]	NULL	"q+(\\beta _{3}-\\beta _{4})q^{3}+(1-\\beta _{9}+\\beta _{11})q^{5}+\\cdots"
"444.2.u.b"	18	3.5453578497448492	NULL	[]	[]	[0, 0, 3, 3]	NULL	"q-\\beta _{8}q^{3}-\\beta _{11}q^{5}+(-\\beta _{5}+\\beta _{6})q^{7}+\\cdots"
"444.2.w.a"	4	3.5453578497448492	"4.0.144.1"	[-3]	[]	[0, -6, 0, 0]	NULL	"q+(-1-\\zeta_{12}^{2})q^{3}+(-\\zeta_{12}+2\\zeta_{12}^{3})q^{7}+\\cdots"
"444.2.w.b"	4	3.5453578497448492	"4.0.144.1"	[-3]	[]	[0, 6, 0, 0]	NULL	"q+(1+\\zeta_{12}^{2})q^{3}+(-3\\zeta_{12}+6\\zeta_{12}^{3})q^{7}+\\cdots"
"444.2.w.c"	40	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.y.a"	8	3.5453578497448492	"8.0.3468738816.6"	[]	[]	[-4, -4, 0, 12]	NULL	"q+(-1+\\beta _{4}-\\beta _{6})q^{2}+\\beta _{6}q^{3}+(2\\beta _{2}+\\cdots)q^{4}+\\cdots"
"444.2.y.b"	8	3.5453578497448492	"8.0.3468738816.6"	[]	[]	[-4, 4, 0, -12]	NULL	"q+(-\\beta _{2}+\\beta _{4}+\\beta _{6})q^{2}+(1+\\beta _{6})q^{3}+\\cdots"
"444.2.y.c"	68	3.5453578497448492	NULL	[]	[]	[6, -34, -4, -12]	NULL	NULL
"444.2.y.d"	68	3.5453578497448492	NULL	[]	[]	[6, 34, -4, 12]	NULL	NULL
"444.2.z.a"	432	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.ba.a"	432	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.bb.a"	18	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	"q+(-\\beta _{1}-\\beta _{5})q^{3}-\\beta _{3}q^{5}+(-\\beta _{1}-\\beta _{6}+\\cdots)q^{7}+\\cdots"
"444.2.bb.b"	24	3.5453578497448492	NULL	[]	[]	[0, 0, 0, -3]	NULL	NULL
"444.2.bg.a"	12	3.5453578497448492	"12.0.1586874322944.1"	[-3]	[]	[0, 0, 0, 0]	NULL	"q+(-\\zeta_{36}^{4}+2\\zeta_{36}^{10})q^{3}+(-\\zeta_{36}^{3}+\\cdots)q^{7}+\\cdots"
"444.2.bg.b"	144	3.5453578497448492	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.2.bh.a"	228	3.5453578497448492	NULL	[]	[]	[0, 0, 6, 0]	NULL	NULL
"444.2.bh.b"	228	3.5453578497448492	NULL	[]	[]	[0, 0, 6, 0]	NULL	NULL
"444.3.b.a"	2	12.098123737597183	"2.0.3.1"	[]	[]	[-2, 0, 0, 0]	NULL	"q+(\\beta-1)q^{2}-\\beta q^{3}+(-2\\beta-2)q^{4}+\\cdots"
"444.3.b.b"	2	12.098123737597183	"2.0.3.1"	[]	[]	[2, 0, 0, 0]	NULL	"q+(\\beta+1)q^{2}+\\beta q^{3}+(2\\beta-2)q^{4}+\\cdots"
"444.3.b.c"	72	12.098123737597183	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.3.d.a"	24	12.098123737597183	NULL	[]	[]	[0, 4, 0, -8]	NULL	NULL
"444.3.f.a"	72	12.098123737597183	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.3.h.a"	2	12.098123737597183	"2.0.3.1"	[-3]	[]	[0, -6, 0, -4]	NULL	"q-3 q^{3}-2 q^{7}+9 q^{9}-2\\beta q^{13}+\\cdots"
"444.3.h.b"	24	12.098123737597183	NULL	[]	[]	[0, 8, 0, 8]	NULL	NULL
"444.3.j.a"	296	12.098123737597183	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.3.l.a"	24	12.098123737597183	NULL	[]	[]	[0, 0, -12, 0]	NULL	NULL
"444.3.n.a"	2	12.098123737597183	"2.0.3.1"	[-3]	[]	[0, 3, 0, -13]	NULL	"q+3\\zeta_{6}q^{3}-13\\zeta_{6}q^{7}+(-9+9\\zeta_{6})q^{9}+\\cdots"
"444.3.n.b"	2	12.098123737597183	"2.0.3.1"	[-3]	[]	[0, 3, 0, 11]	NULL	"q+3\\zeta_{6}q^{3}+11\\zeta_{6}q^{7}+(-9+9\\zeta_{6})q^{9}+\\cdots"
"444.3.n.c"	48	12.098123737597183	NULL	[]	[]	[0, -5, 0, 4]	NULL	NULL
"444.3.o.a"	76	12.098123737597183	NULL	[]	[]	[-1, -114, -2, -12]	NULL	NULL
"444.3.o.b"	76	12.098123737597183	NULL	[]	[]	[-1, 114, -2, 12]	NULL	NULL
"444.3.q.a"	2	12.098123737597183	"2.0.3.1"	[-3]	[]	[0, 3, 0, -11]	NULL	"q+(3-3\\zeta_{6})q^{3}+(-11+11\\zeta_{6})q^{7}+\\cdots"
"444.3.q.b"	2	12.098123737597183	"2.0.3.1"	[-3]	[]	[0, 3, 0, 13]	NULL	"q+(3-3\\zeta_{6})q^{3}+(13-13\\zeta_{6})q^{7}-9\\zeta_{6}q^{9}+\\cdots"
"444.3.q.c"	48	12.098123737597183	NULL	[]	[]	[0, -7, 0, -4]	NULL	NULL
"444.3.s.a"	76	12.098123737597183	NULL	[]	[]	[3, -114, -6, 12]	NULL	NULL
"444.3.s.b"	76	12.098123737597183	NULL	[]	[]	[3, 114, -6, -12]	NULL	NULL
"444.3.v.a"	24	12.098123737597183	NULL	[]	[]	[0, -36, 0, 0]	NULL	NULL
"444.3.v.b"	24	12.098123737597183	NULL	[]	[]	[0, 36, 0, 0]	NULL	NULL
"444.3.x.a"	592	12.098123737597183	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.3.bc.a"	6	12.098123737597183	"6.0.19683.1"	[-3]	[]	[0, 0, 0, -33]	NULL	"q-3\\zeta_{18}^{4}q^{3}+(-3+5\\zeta_{18}^{2}-5\\zeta_{18}^{3}+\\cdots)q^{7}+\\cdots"
"444.3.bc.b"	144	12.098123737597183	NULL	[]	[]	[0, -3, 0, 6]	NULL	NULL
"444.3.bd.a"	228	12.098123737597183	NULL	[]	[]	[-3, 0, 6, -36]	NULL	NULL
"444.3.bd.b"	228	12.098123737597183	NULL	[]	[]	[-3, 0, 6, 36]	NULL	NULL
"444.3.be.a"	228	12.098123737597183	NULL	[]	[]	[3, 0, 6, -36]	NULL	NULL
"444.3.be.b"	228	12.098123737597183	NULL	[]	[]	[3, 0, 6, 36]	NULL	NULL
"444.3.bf.a"	6	12.098123737597183	"6.0.19683.1"	[-3]	[]	[0, 0, 0, 33]	NULL	"q+(3\\zeta_{18}^{2}-3\\zeta_{18}^{5})q^{3}+(8-3\\zeta_{18}+\\cdots)q^{7}+\\cdots"
"444.3.bf.b"	144	12.098123737597183	NULL	[]	[]	[0, 3, 0, -6]	NULL	NULL
"444.3.bi.a"	1776	12.098123737597183	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.3.bj.a"	72	12.098123737597183	NULL	[]	[]	[0, 0, 6, 0]	NULL	NULL
"444.3.bj.b"	84	12.098123737597183	NULL	[]	[]	[0, 0, 6, 0]	NULL	NULL
"444.4.a.a"	1	26.196848042548833	"1.1.1.1"	[]	[]	[0, -3, -4, -25]	-1	"q-3q^{3}-4q^{5}-5^{2}q^{7}+9q^{9}+67q^{11}+\\cdots"
"444.4.a.b"	4	26.196848042548833	"4.4.835792.1"	[]	[]	[0, -12, -6, 20]	-1	"q-3q^{3}+(-1-\\beta _{1}+\\beta _{3})q^{5}+(4+\\beta _{1}+\\cdots)q^{7}+\\cdots"
"444.4.a.c"	4	26.196848042548833	NULL	[]	[]	[0, -12, 0, -5]	1	"q-3q^{3}-\\beta _{2}q^{5}+(-2\\beta _{1}-\\beta _{2}+\\beta _{3})q^{7}+\\cdots"
"444.4.a.d"	4	26.196848042548833	"4.4.2123604.1"	[]	[]	[0, 12, 8, -33]	-1	"q+3q^{3}+(2-\\beta _{1}+\\beta _{2})q^{5}+(-8+3\\beta _{1}+\\cdots)q^{7}+\\cdots"
"444.4.a.e"	5	26.196848042548833	NULL	[]	[]	[0, 15, 18, 23]	1	"q+3q^{3}+(4-\\beta _{3})q^{5}+(5-\\beta _{2}-\\beta _{3}+\\cdots)q^{7}+\\cdots"
"444.4.e.a"	8	26.196848042548833	NULL	[]	[]	[0, -24, 0, -18]	NULL	"q-3q^{3}-\\beta _{4}q^{5}+(-2-\\beta _{2})q^{7}+9q^{9}+\\cdots"
"444.4.e.b"	10	26.196848042548833	NULL	[]	[]	[0, 30, 0, 22]	NULL	"q+3q^{3}+\\beta _{1}q^{5}+(2-\\beta _{2})q^{7}+9q^{9}+\\cdots"
"444.4.i.a"	18	26.196848042548833	NULL	[]	[]	[0, 27, -5, 16]	NULL	"q+(3-3\\beta _{2})q^{3}+(-1+\\beta _{2}-\\beta _{3})q^{5}+\\cdots"
"444.4.i.b"	22	26.196848042548833	NULL	[]	[]	[0, -33, -5, -18]	NULL	NULL
"444.4.m.a"	76	26.196848042548833	NULL	[]	[]	[0, 0, 0, 0]	NULL	NULL
"444.4.r.a"	2	26.196848042548833	"2.0.3.1"	[]	[]	[0, -3, 18, 17]	NULL	"q-3\\zeta_{6}q^{3}+(12-6\\zeta_{6})q^{5}+17\\zeta_{6}q^{7}+\\cdots"
"444.4.r.b"	16	26.196848042548833	NULL	[]	[]	[0, 24, 18, 12]	NULL	"q+(3-3\\beta _{5})q^{3}+(1-\\beta _{1}+\\beta _{5})q^{5}+(2+\\cdots)q^{7}+\\cdots"
"444.4.r.c"	18	26.196848042548833	NULL	[]	[]	[0, -27, 0, -27]	NULL	"q+(-3+3\\beta _{6})q^{3}+\\beta _{2}q^{5}+(-3+3\\beta _{6}+\\cdots)q^{7}+\\cdots"
"444.5.d.a"	48	45.89626360922688	NULL	[]	[]	[0, -8, 0, 80]	NULL	NULL
"444.5.h.a"	2	45.89626360922688	"2.0.3.1"	[-3]	[]	[0, 18, 0, 188]	NULL	"q+9 q^{3}+94 q^{7}+81 q^{9}-22\\beta q^{13}+\\cdots"
"444.5.h.b"	48	45.89626360922688	NULL	[]	[]	[0, -28, 0, -80]	NULL	NULL
"444.5.l.a"	52	45.89626360922688	NULL	[]	[]	[0, 0, -24, 0]	NULL	NULL
"444.5.n.a"	2	45.89626360922688	"2.0.3.1"	[-3]	[]	[0, -9, 0, 23]	NULL	"q-9\\zeta_{6}q^{3}+23\\zeta_{6}q^{7}+(-3^{4}+3^{4}\\zeta_{6})q^{9}+\\cdots"
"444.5.n.b"	2	45.89626360922688	"2.0.3.1"	[-3]	[]	[0, -9, 0, 71]	NULL	"q-9\\zeta_{6}q^{3}+71\\zeta_{6}q^{7}+(-3^{4}+3^{4}\\zeta_{6})q^{9}+\\cdots"
"444.5.n.c"	96	45.89626360922688	NULL	[]	[]	[0, 13, 0, -40]	NULL	NULL
"444.5.q.a"	2	45.89626360922688	"2.0.3.1"	[-3]	[]	[0, -9, 0, -71]	NULL	"q+(-9+9\\zeta_{6})q^{3}+(-71+71\\zeta_{6})q^{7}+\\cdots"
"444.5.q.b"	2	45.89626360922688	"2.0.3.1"	[-3]	[]	[0, -9, 0, -23]	NULL	"q+(-9+9\\zeta_{6})q^{3}+(-23+23\\zeta_{6})q^{7}+\\cdots"
"444.5.q.c"	96	45.89626360922688	NULL	[]	[]	[0, 23, 0, 40]	NULL	NULL
"444.6.a.a"	7	71.2104159969634	NULL	[]	[]	[0, -63, 14, 40]	-1	"q-9q^{3}+(2+\\beta _{1})q^{5}+(6-\\beta _{1}-\\beta _{2}+\\cdots)q^{7}+\\cdots"
"444.6.a.b"	7	71.2104159969634	NULL	[]	[]	[0, 63, -36, -156]	1	"q+9q^{3}+(-5+\\beta _{2})q^{5}+(-23+\\beta _{1}+\\cdots)q^{7}+\\cdots"
"444.6.a.c"	8	71.2104159969634	NULL	[]	[]	[0, -72, -36, 40]	1	"q-9q^{3}+(-4-\\beta _{1})q^{5}+(5-\\beta _{4})q^{7}+\\cdots"
"444.6.a.d"	8	71.2104159969634	NULL	[]	[]	[0, 72, 14, 236]	-1	"q+9q^{3}+(2-\\beta _{1})q^{5}+(30-\\beta _{3})q^{7}+\\cdots"
"444.6.e.a"	16	71.2104159969634	NULL	[]	[]	[0, -144, 0, 116]	NULL	"q-9q^{3}+\\beta _{1}q^{5}+(7+\\beta _{2})q^{7}+3^{4}q^{9}+\\cdots"
"444.6.e.b"	16	71.2104159969634	NULL	[]	[]	[0, 144, 0, -276]	NULL	"q+9q^{3}+\\beta _{1}q^{5}+(-17-\\beta _{2})q^{7}+3^{4}q^{9}+\\cdots"
"444.6.i.a"	30	71.2104159969634	NULL	[]	[]	[0, -135, -11, 187]	NULL	NULL
"444.6.i.b"	30	71.2104159969634	NULL	[]	[]	[0, 135, -11, -107]	NULL	NULL
"444.6.r.a"	32	71.2104159969634	NULL	[]	[]	[0, -144, 90, 9]	NULL	NULL
"444.6.r.b"	32	71.2104159969634	NULL	[]	[]	[0, 144, 90, -89]	NULL	NULL
"444.7.d.a"	72	102.14401312721384	NULL	[]	[]	[0, -20, 0, -408]	NULL	NULL
"444.7.h.a"	76	102.14401312721384	NULL	[]	[]	[0, 32, 0, -560]	NULL	NULL
"444.7.l.a"	76	102.14401312721384	NULL	[]	[]	[0, 0, -132, 0]	NULL	NULL
"444.8.a.a"	10	138.69895131495005	NULL	[]	[]	[0, -270, 448, 140]	1	"q-3^{3}q^{3}+(45-\\beta _{1})q^{5}+(14-\\beta _{3})q^{7}+\\cdots"
"444.8.a.b"	10	138.69895131495005	NULL	[]	[]	[0, 270, -450, -1232]	-1	"q+3^{3}q^{3}+(-45-\\beta _{2})q^{5}+(-123+\\cdots)q^{7}+\\cdots"
"444.8.a.c"	11	138.69895131495005	NULL	[]	[]	[0, -297, 198, 140]	-1	"q-3^{3}q^{3}+(18-\\beta _{1})q^{5}+(13+\\beta _{2})q^{7}+\\cdots"
"444.8.a.d"	11	138.69895131495005	NULL	[]	[]	[0, 297, -200, 1512]	1	"q+3^{3}q^{3}+(-18-\\beta _{1})q^{5}+(137-\\beta _{2}+\\cdots)q^{7}+\\cdots"
"444.8.e.a"	22	138.69895131495005	NULL	[]	[]	[0, -594, 0, 12]	NULL	NULL
"444.8.e.b"	22	138.69895131495005	NULL	[]	[]	[0, 594, 0, -1132]	NULL	NULL
"444.8.i.a"	46	138.69895131495005	NULL	[]	[]	[0, -621, -55, 223]	NULL	NULL
"444.8.i.b"	46	138.69895131495005	NULL	[]	[]	[0, 621, -55, 337]	NULL	NULL
"444.8.r.a"	44	138.69895131495005	NULL	[]	[]	[0, -594, -252, -1595]	NULL	NULL
"444.8.r.b"	44	138.69895131495005	NULL	[]	[]	[0, 594, -252, 1035]	NULL	NULL
"444.9.d.a"	96	180.87610286714047	NULL	[]	[]	[0, 112, 0, 160]	NULL	NULL
"444.9.l.a"	100	180.87610286714047	NULL	[]	[]	[0, 0, 336, 0]	NULL	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.