Newform orbit 503.2.c.a

• Label: 503.2.c.a
• Level: $503$
• Weight: $2$
• Character orbit: 503.c
• Analytic conductor: $4.016$
• Analytic rank: $0$
• Dimension: $10250$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	503.c (of order \(251\), degree \(250\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.01647522167\)
Analytic rank:	\(0\)
Dimension:	\(10250\)
Relative dimension:	\(41\) over \(\Q(\zeta_{251})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{251}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.73497	−	2.13612i	−2.22490	+	1.92567i	−1.14279	+	5.45470i	−0.109819	−	0.346829i	7.97361	+	1.41168i	0.0493067	−	0.143165i	8.74162	−	4.50242i	0.811649	−	5.59990i	−0.550336	+	0.836323i
2.2	−1.72191	−	2.12004i	1.69918	−	1.47066i	−1.11950	+	5.34354i	−0.0846785	−	0.267431i	−6.04369	−	1.07000i	−1.43085	+	4.15454i	8.40007	−	4.32651i	0.294060	−	2.02884i	−0.421156	+	0.640014i
2.3	−1.53474	−	1.88959i	0.404749	−	0.350314i	−0.805040	+	3.84256i	1.19914	+	3.78712i	−1.28313	−	0.227171i	1.44952	−	4.20876i	4.16812	−	2.14681i	−0.389221	+	2.68540i	5.31575	−	8.07812i
2.4	−1.49378	−	1.83917i	−0.567738	+	0.491382i	−0.741049	+	3.53712i	−0.276099	−	0.871972i	1.75181	+	0.310147i	0.530619	−	1.54068i	3.39955	−	1.75096i	−0.349454	+	2.41102i	−1.19127	+	1.81033i
2.5	−1.47995	−	1.82214i	1.66985	−	1.44527i	−0.719831	+	3.43585i	0.408432	+	1.28990i	−5.10478	−	0.903771i	0.254080	−	0.737735i	3.15213	−	1.62353i	0.269269	−	1.85780i	1.74593	−	2.65321i
2.6	−1.43081	−	1.76164i	−1.19346	+	1.03295i	−0.646042	+	3.08364i	0.511937	+	1.61679i	3.52731	+	0.624490i	−1.13380	+	3.29205i	2.32143	−	1.19567i	−0.0729611	+	0.503389i	2.11572	−	3.21517i
2.7	−1.35722	−	1.67103i	2.35620	−	2.03931i	−0.540193	+	2.57841i	−0.767603	−	2.42424i	−6.60563	−	1.16949i	0.954936	−	2.77271i	1.21411	−	0.625336i	0.962553	−	6.64105i	−3.00917	+	4.57291i
2.8	−1.31051	−	1.61352i	−0.0751196	+	0.0650167i	−0.475908	+	2.27157i	−1.09154	−	3.44730i	0.203351	+	0.0360021i	−1.53778	+	4.46504i	0.592995	−	0.305425i	−0.428907	+	2.95921i	−4.13181	+	6.27895i
2.9	−1.15193	−	1.41827i	−2.28946	+	1.98155i	−0.274449	+	1.30998i	1.06426	+	3.36113i	5.44767	+	0.964477i	0.658051	−	1.91069i	−1.07463	+	0.553494i	0.884773	−	6.10441i	3.54105	−	5.38118i
2.10	−0.994976	−	1.22503i	−1.98352	+	1.71676i	−0.100616	+	0.480254i	−1.27158	−	4.01588i	4.07664	+	0.721744i	1.44819	−	4.20490i	−2.11761	+	1.09069i	0.556781	−	3.84146i	−3.65440	+	5.55343i
2.11	−0.968162	−	1.19202i	0.996021	−	0.862065i	−0.0734591	+	0.350630i	−0.521680	−	1.64756i	−1.99191	−	0.352655i	0.222510	−	0.646070i	−2.24135	+	1.15442i	−0.181422	+	1.25171i	−1.45885	+	2.21696i
2.12	−0.948756	−	1.16812i	1.18005	−	1.02135i	−0.0542683	+	0.259030i	0.841155	+	2.65653i	−2.31264	−	0.409440i	−1.01850	+	2.95726i	−2.32163	+	1.19577i	−0.0809476	+	0.558491i	2.30510	−	3.50297i
2.13	−0.945803	−	1.16449i	−0.790504	+	0.684188i	−0.0513814	+	0.245250i	−0.107766	−	0.340345i	1.54439	+	0.273425i	0.296825	−	0.861848i	−2.33318	+	1.20172i	−0.273541	+	1.88727i	−0.294402	+	0.447391i
2.14	−0.766891	−	0.944209i	−2.44234	+	2.11387i	0.106699	−	0.509291i	−0.375743	−	1.18667i	3.86894	+	0.684973i	−1.33365	+	3.87232i	−2.72550	+	1.40379i	1.06626	−	7.35660i	−0.832307	+	1.26482i
2.15	−0.604817	−	0.744661i	1.49016	−	1.28975i	0.221392	−	1.05673i	−1.03109	−	3.25639i	−1.86170	−	0.329604i	−0.182067	+	0.528642i	−2.62653	+	1.35281i	0.126808	−	0.874904i	−1.80128	+	2.73733i
2.16	−0.604152	−	0.743843i	−0.138203	+	0.119616i	0.221807	−	1.05871i	0.634410	+	2.00359i	0.172471	+	0.0305350i	0.803660	−	2.33347i	−2.62536	+	1.35221i	−0.425531	+	2.93591i	1.10707	−	1.68237i
2.17	−0.448047	−	0.551642i	2.42081	−	2.09524i	0.306545	−	1.46318i	−0.150619	−	0.475684i	−2.24046	−	0.396660i	−0.834374	+	2.42265i	−2.20809	+	1.13729i	1.04000	−	7.17539i	−0.194923	+	0.296217i
2.18	−0.380517	−	0.468498i	−1.24266	+	1.07553i	0.335410	−	1.60096i	−0.464781	−	1.46787i	0.976735	+	0.172925i	−0.519527	+	1.50848i	−1.95081	+	1.00478i	−0.0428949	+	0.295950i	−0.510836	+	0.776297i
2.19	−0.225034	−	0.277066i	−0.186491	+	0.161409i	0.383983	−	1.83280i	0.927472	+	2.92913i	0.0866877	+	0.0153475i	−0.576054	+	1.67260i	−1.22886	+	0.632932i	−0.421597	+	2.90877i	0.602849	−	0.916125i
2.20	−0.102711	−	0.126460i	0.592852	−	0.513119i	0.404666	−	1.93152i	−0.990176	−	3.12716i	−0.125782	−	0.0222689i	1.03743	−	3.01224i	−0.575493	+	0.296411i	−0.342141	+	2.36057i	−0.293759	+	0.446413i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
503.c	even	251	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	503.2.c.a	✓	10250
503.c	even	251	1	inner	503.2.c.a	✓	10250

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
503.2.c.a	✓	10250	1.a	even	1	1	trivial
503.2.c.a	✓	10250	503.c	even	251	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(503, [\chi])\).