Newform orbit 663.2.m.a

• Label: 663.2.m.a
• Level: $663$
• Weight: $2$
• Character orbit: 663.m
• Analytic conductor: $5.294$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	663.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29408165401\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
200.1	−1.97019	+	1.97019i	0.243461	+	1.71485i		−	5.76329i		−	0.424909i	−3.85825	−	2.89893i	1.45224			7.41440	+	7.41440i	−2.88145	+	0.835000i	0.837150	+	0.837150i
200.2	−1.94250	+	1.94250i	−1.46216	−	0.928482i		−	5.54663i		−	1.78814i	4.64383	−	1.03668i	0.329480			6.88934	+	6.88934i	1.27584	+	2.71519i	3.47347	+	3.47347i
200.3	−1.88525	+	1.88525i	1.44285	−	0.958221i		−	5.10835i		−	3.73637i	−0.913646	+	4.52662i	1.74614			5.86001	+	5.86001i	1.16363	−	2.76514i	7.04401	+	7.04401i
200.4	−1.86332	+	1.86332i	−0.0952885	−	1.72943i		−	4.94395i			3.22382i	3.40004	+	3.04493i	−2.10101			5.48553	+	5.48553i	−2.98184	+	0.329589i	−6.00701	−	6.00701i
200.5	−1.78759	+	1.78759i	1.71974	−	0.206162i		−	4.39097i			3.81701i	−2.70566	+	3.44272i	2.23922			4.27409	+	4.27409i	2.91499	−	0.709089i	−6.82326	−	6.82326i
200.6	−1.78331	+	1.78331i	−1.49921	+	0.867392i		−	4.36038i		−	0.627214i	1.12673	−	4.22038i	−3.96678			4.20930	+	4.20930i	1.49526	−	2.60081i	1.11852	+	1.11852i
200.7	−1.71102	+	1.71102i	1.40662	+	1.01065i		−	3.85520i		−	0.0207272i	−4.13601	+	0.677516i	1.42087			3.17430	+	3.17430i	0.957170	+	2.84321i	0.0354647	+	0.0354647i
200.8	−1.66061	+	1.66061i	1.13643	−	1.30711i		−	3.51528i		−	0.370198i	0.283426	+	4.05777i	−2.52362			2.51630	+	2.51630i	−0.417051	−	2.97087i	0.614756	+	0.614756i
200.9	−1.65824	+	1.65824i	−1.53131	+	0.809367i		−	3.49949i			1.69318i	1.19716	−	3.88140i	1.65666			2.48651	+	2.48651i	1.68985	−	2.47879i	−2.80769	−	2.80769i
200.10	−1.59537	+	1.59537i	−0.316862	−	1.70282i		−	3.09038i		−	0.771102i	3.22213	+	2.21111i	5.04617			1.73956	+	1.73956i	−2.79920	+	1.07912i	1.23019	+	1.23019i
200.11	−1.53345	+	1.53345i	1.14670	+	1.29811i		−	2.70291i		−	4.16533i	−3.74898	−	0.232178i	−2.64794			1.07787	+	1.07787i	−0.370166	+	2.97708i	6.38730	+	6.38730i
200.12	−1.51460	+	1.51460i	−1.66316	−	0.483637i		−	2.58801i			2.77750i	3.25153	−	1.78650i	2.49850			0.890605	+	0.890605i	2.53219	+	1.60873i	−4.20680	−	4.20680i
200.13	−1.47397	+	1.47397i	−0.238850	+	1.71550i		−	2.34519i			2.02949i	−2.17655	−	2.88066i	−3.58092			0.508804	+	0.508804i	−2.88590	−	0.819496i	−2.99141	−	2.99141i
200.14	−1.38931	+	1.38931i	−1.55104	+	0.770889i		−	1.86039i		−	3.62652i	1.08388	−	3.22589i	3.52499			−0.193960	−	0.193960i	1.81146	−	2.39136i	5.03837	+	5.03837i
200.15	−1.35457	+	1.35457i	−1.43470	−	0.970379i		−	1.66974i		−	3.37596i	3.25786	−	0.628958i	−1.50328			−0.447358	−	0.447358i	1.11673	+	2.78441i	4.57298	+	4.57298i
200.16	−1.33113	+	1.33113i	−0.114832	+	1.72824i		−	1.54382i		−	1.40836i	−2.14766	−	2.45337i	0.136228			−0.607232	−	0.607232i	−2.97363	−	0.396916i	1.87472	+	1.87472i
200.17	−1.29601	+	1.29601i	0.952245	−	1.44680i		−	1.35929i			1.75702i	0.640948	+	3.10919i	2.98090			−0.830369	−	0.830369i	−1.18646	−	2.75542i	−2.27711	−	2.27711i
200.18	−1.28128	+	1.28128i	−1.44048	−	0.961772i		−	1.28338i			2.95648i	3.07797	−	0.613366i	−4.29419			−0.918199	−	0.918199i	1.14999	+	2.77083i	−3.78809	−	3.78809i
200.19	−1.22930	+	1.22930i	1.71137	+	0.266837i		−	1.02238i		−	0.642053i	−2.43182	+	1.77577i	−3.73389			−1.20180	−	1.20180i	2.85760	+	0.913314i	0.789278	+	0.789278i
200.20	−1.14596	+	1.14596i	−0.579944	−	1.63207i		−	0.626468i		−	1.37360i	2.53489	+	1.20570i	−2.20444			−1.57402	−	1.57402i	−2.32733	+	1.89302i	1.57409	+	1.57409i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
221.i	odd	4	1	inner
663.m	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	663.2.m.a	✓	160
3.b	odd	2	1	inner	663.2.m.a	✓	160
13.d	odd	4	1		663.2.r.a	yes	160
17.c	even	4	1		663.2.r.a	yes	160
39.f	even	4	1		663.2.r.a	yes	160
51.f	odd	4	1		663.2.r.a	yes	160
221.i	odd	4	1	inner	663.2.m.a	✓	160
663.m	even	4	1	inner	663.2.m.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
663.2.m.a	✓	160	1.a	even	1	1	trivial
663.2.m.a	✓	160	3.b	odd	2	1	inner
663.2.m.a	✓	160	221.i	odd	4	1	inner
663.2.m.a	✓	160	663.m	even	4	1	inner
663.2.r.a	yes	160	13.d	odd	4	1
663.2.r.a	yes	160	17.c	even	4	1
663.2.r.a	yes	160	39.f	even	4	1
663.2.r.a	yes	160	51.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).