Newform orbit 572.2.bf.a

• Label: 572.2.bf.a
• Level: $572$
• Weight: $2$
• Character orbit: 572.bf
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.41338	+	0.0484642i	−2.30576	+	1.33123i	1.99530	−	0.136997i	0.670105	−	0.670105i	3.19441	−	1.99329i	1.27385	+	4.75407i	−2.81349	+	0.290330i	2.04436	−	3.54093i	−0.914639	+	0.979591i
67.2	−1.41251	−	0.0694338i	1.39356	−	0.804570i	1.99036	+	0.196152i	−1.40920	+	1.40920i	−2.02427	+	1.03970i	−0.135295	−	0.504929i	−2.79778	−	0.415264i	−0.205335	+	0.355651i	2.08835	−	1.89266i
67.3	−1.40492	−	0.161847i	−0.747603	+	0.431629i	1.94761	+	0.454764i	−1.77001	+	1.77001i	1.12018	−	0.485407i	−0.913100	−	3.40773i	−2.66264	−	0.954122i	−1.12739	+	1.95270i	2.77319	−	2.20025i
67.4	−1.39566	+	0.228338i	−2.91789	+	1.68465i	1.89572	−	0.637363i	1.61438	−	1.61438i	3.68771	−	3.01745i	−1.10646	−	4.12937i	−2.50025	+	1.32241i	4.17606	−	7.23315i	−1.88450	+	2.62174i
67.5	−1.39359	+	0.240656i	2.08344	−	1.20287i	1.88417	−	0.670751i	1.79541	−	1.79541i	−2.61397	+	2.17770i	−0.472870	−	1.76477i	−2.46433	+	1.38819i	1.39380	−	2.41414i	−2.06998	+	2.93413i
67.6	−1.39307	+	0.243637i	−1.43186	+	0.826687i	1.88128	−	0.678806i	−1.59868	+	1.59868i	1.79327	−	1.50049i	−0.175707	−	0.655746i	−2.45537	+	1.40397i	−0.133176	+	0.230667i	1.83757	−	2.61657i
67.7	−1.36347	+	0.375447i	−0.276166	+	0.159445i	1.71808	−	1.02382i	0.625185	−	0.625185i	0.316680	−	0.321083i	0.928719	+	3.46603i	−1.95815	+	2.04099i	−1.44915	+	2.51001i	−0.617695	+	1.08714i
67.8	−1.33591	+	0.464039i	2.67338	−	1.54348i	1.56934	−	1.23983i	−1.23437	+	1.23437i	−2.85518	+	3.30251i	0.844074	+	3.15013i	−1.52117	+	2.38454i	3.26465	−	5.65455i	1.07622	−	2.22181i
67.9	−1.29762	−	0.562298i	0.747603	−	0.431629i	1.36764	+	1.45930i	−1.77001	+	1.77001i	−1.21281	+	0.139715i	0.913100	+	3.40773i	−0.954122	−	2.66264i	−1.12739	+	1.95270i	3.29207	−	1.30153i
67.10	−1.28912	+	0.581515i	−0.101031	+	0.0583303i	1.32368	−	1.49929i	1.51781	−	1.51781i	0.0963216	−	0.133946i	−0.585492	−	2.18508i	−0.834531	+	2.70251i	−1.49320	+	2.58629i	−1.07402	+	2.83927i
67.11	−1.25798	−	0.646123i	−1.39356	+	0.804570i	1.16505	+	1.62562i	−1.40920	+	1.40920i	2.27292	−	0.111729i	0.135295	+	0.504929i	−0.415264	−	2.79778i	−0.205335	+	0.355651i	2.68326	−	0.862235i
67.12	−1.19979	−	0.748663i	2.30576	−	1.33123i	0.879008	+	1.79648i	0.670105	−	0.670105i	−3.76308	−	0.129034i	−1.27385	−	4.75407i	0.290330	−	2.81349i	2.04436	−	3.54093i	−1.30567	+	0.302305i
67.13	−1.16821	+	0.797043i	1.53999	−	0.889113i	0.729444	−	1.86223i	−2.51999	+	2.51999i	−1.09037	+	2.26611i	−1.06525	−	3.97557i	0.632133	+	2.75688i	0.0810425	−	0.140370i	0.935344	−	4.95242i
67.14	−1.09451	−	0.895575i	2.91789	−	1.68465i	0.395890	+	1.96043i	1.61438	−	1.61438i	−4.70238	−	0.769336i	1.10646	+	4.12937i	1.32241	−	2.50025i	4.17606	−	7.23315i	−3.21274	+	0.321150i
67.15	−1.08655	−	0.905208i	−2.08344	+	1.20287i	0.361197	+	1.96711i	1.79541	−	1.79541i	3.35261	+	0.578958i	0.472870	+	1.76477i	1.38819	−	2.46433i	1.39380	−	2.41414i	−3.57602	+	0.325589i
67.16	−1.08461	−	0.907530i	1.43186	−	0.826687i	0.352778	+	1.96864i	−1.59868	+	1.59868i	−2.30327	−	0.402823i	0.175707	+	0.655746i	1.40397	−	2.45537i	−0.133176	+	0.230667i	3.18480	−	0.283101i
67.17	−1.07247	+	0.921853i	0.790362	−	0.456316i	0.300374	−	1.97732i	0.301911	−	0.301911i	−0.426981	+	1.21798i	0.768929	+	2.86968i	1.50065	+	2.39751i	−1.08355	+	1.87677i	−0.0454722	+	0.602108i
67.18	−1.01057	+	0.989319i	−1.57035	+	0.906642i	0.0424954	−	1.99955i	2.19271	−	2.19271i	0.689987	−	2.46980i	−0.0595437	−	0.222220i	1.93525	+	2.06272i	0.143999	−	0.249414i	−0.0465927	+	4.38518i
67.19	−0.993073	−	1.00688i	0.276166	−	0.159445i	−0.0276135	+	1.99981i	0.625185	−	0.625185i	−0.434795	−	0.119726i	−0.928719	−	3.46603i	2.04099	−	1.95815i	−1.44915	+	2.51001i	−1.25034	−	0.00863198i
67.20	−0.924916	−	1.06983i	−2.67338	+	1.54348i	−0.289059	+	1.97900i	−1.23437	+	1.23437i	4.12391	+	1.43247i	−0.844074	−	3.15013i	2.38454	−	1.52117i	3.26465	−	5.65455i	2.46226	+	0.178873i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.f	odd	12	1	inner
52.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	572.2.bf.a	✓	280
4.b	odd	2	1	inner	572.2.bf.a	✓	280
13.f	odd	12	1	inner	572.2.bf.a	✓	280
52.l	even	12	1	inner	572.2.bf.a	✓	280

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
572.2.bf.a	✓	280	1.a	even	1	1	trivial
572.2.bf.a	✓	280	4.b	odd	2	1	inner
572.2.bf.a	✓	280	13.f	odd	12	1	inner
572.2.bf.a	✓	280	52.l	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).