Newform orbit 861.2.bp.a

• Label: 861.2.bp.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.bp
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.bp (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(108\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 864 q - 4 q^{3} - 16 q^{6} - 16 q^{7} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
44.1	−2.64718	+	0.709309i	1.64993	+	0.526991i	4.77238	−	2.75533i	−0.303737	−	1.13356i	−4.74147	−	0.224726i	−1.57558	+	2.12545i	−6.80321	+	6.80321i	2.44456	+	1.73900i	1.60809	+	2.78530i
44.2	−2.63317	+	0.705555i	−1.61399	+	0.628523i	4.70371	−	2.71569i	−0.456996	−	1.70553i	3.80644	−	2.79376i	−2.39058	−	1.13364i	−6.61436	+	6.61436i	2.20992	−	2.02886i	2.40669	+	4.16851i
44.3	−2.59021	+	0.694046i	−1.13140	−	1.31146i	4.49546	−	2.59546i	−0.954798	−	3.56336i	3.84078	+	2.61173i	2.64189	+	0.142946i	−6.05050	+	6.05050i	−0.439871	+	2.96758i	4.94626	+	8.56718i
44.4	−2.58395	+	0.692367i	−1.63421	+	0.573902i	4.46537	−	2.57808i	0.775226	+	2.89318i	3.82536	−	2.61441i	2.09207	−	1.61964i	−5.97015	+	5.97015i	2.34127	−	1.87575i	−4.00629	−	6.93910i
44.5	−2.57312	+	0.689466i	0.317179	−	1.70276i	4.41355	−	2.54816i	0.0913994	+	0.341107i	0.357855	+	4.60010i	−1.33734	+	2.28288i	−5.83241	+	5.83241i	−2.79879	−	1.08016i	−0.470363	−	0.814693i
44.6	−2.46524	+	0.660558i	0.283183	+	1.70874i	3.90900	−	2.25686i	0.688446	+	2.56932i	−1.82684	−	4.02540i	1.34867	+	2.27620i	−4.53647	+	4.53647i	−2.83962	+	0.967773i	−3.39437	−	5.87921i
44.7	−2.46479	+	0.660440i	0.326149	+	1.70107i	3.90698	−	2.25570i	−0.383702	−	1.43199i	−1.92734	−	3.97738i	0.174843	−	2.63997i	−4.53144	+	4.53144i	−2.78725	+	1.10960i	1.89149	+	3.27616i
44.8	−2.42057	+	0.648591i	1.35412	−	1.07998i	3.70646	−	2.13992i	0.887879	+	3.31361i	−2.57727	+	3.49245i	−1.87878	−	1.86284i	−4.03985	+	4.03985i	0.667267	−	2.92485i	−4.29835	−	7.44496i
44.9	−2.40613	+	0.644720i	−0.177666	−	1.72291i	3.64173	−	2.10256i	0.696788	+	2.60045i	1.53828	+	4.03101i	2.64200	−	0.140894i	−3.88410	+	3.88410i	−2.93687	+	0.612207i	−3.35312	−	5.80778i
44.10	−2.36090	+	0.632601i	1.18311	−	1.26501i	3.44161	−	1.98701i	−0.515500	−	1.92387i	−1.99295	+	3.73500i	0.0819307	−	2.64448i	−3.41170	+	3.41170i	−0.200510	−	2.99329i	2.43408	+	4.21596i
44.11	−2.28246	+	0.611583i	−1.69021	−	0.378422i	3.10353	−	1.79182i	−0.0134746	−	0.0502881i	4.08926	−	0.169968i	0.161587	+	2.64081i	−2.64607	+	2.64607i	2.71359	+	1.27922i	0.0615106	+	0.106540i
44.12	−2.27845	+	0.610508i	−1.24637	−	1.20273i	3.08654	−	1.78202i	0.474142	+	1.76952i	3.57406	+	1.97945i	−1.32874	−	2.28789i	−2.60871	+	2.60871i	0.106861	+	2.99810i	−2.16061	−	3.74229i
44.13	−2.23674	+	0.599332i	−0.695004	+	1.58650i	2.91174	−	1.68109i	0.563627	+	2.10348i	0.603704	−	3.96511i	−2.63651	+	0.220953i	−2.23045	+	2.23045i	−2.03394	−	2.20524i	−2.52137	−	4.36714i
44.14	−2.06700	+	0.553850i	−0.887150	+	1.48760i	2.23368	−	1.28961i	−0.730063	−	2.72463i	1.00983	−	3.56622i	−1.25040	+	2.33163i	−0.876460	+	0.876460i	−1.42593	−	2.63946i	3.01808	+	5.22747i
44.15	−2.04994	+	0.549280i	1.11458	+	1.32579i	2.16850	−	1.25199i	−0.797684	−	2.97700i	−3.01305	−	2.10558i	2.16757	+	1.51712i	−0.756292	+	0.756292i	−0.515435	+	2.95539i	3.27041	+	5.66452i
44.16	−2.01791	+	0.540698i	1.67191	+	0.452440i	2.04757	−	1.18217i	0.248112	+	0.925968i	−3.61841	−	0.00898391i	2.38161	−	1.15236i	−0.538197	+	0.538197i	2.59060	+	1.51288i	−1.00134	−	1.73437i
44.17	−2.00135	+	0.536260i	1.73099	−	0.0606305i	1.98577	−	1.14648i	−0.705212	−	2.63189i	−3.43180	+	1.04960i	−2.63210	−	0.268405i	−0.429224	+	0.429224i	2.99265	−	0.209901i	2.82275	+	4.88914i
44.18	−1.99563	+	0.534726i	1.71424	−	0.247750i	1.96454	−	1.13423i	1.06405	+	3.97110i	−3.28850	+	1.41106i	−0.0984622	+	2.64392i	−0.392183	+	0.392183i	2.87724	−	0.849405i	−4.24690	−	7.35585i
44.19	−1.90624	+	0.510776i	−1.34012	+	1.09730i	1.64081	−	0.947324i	−0.860585	−	3.21175i	1.99412	−	2.77622i	2.16964	−	1.51416i	0.147015	−	0.147015i	0.591863	−	2.94104i	3.28097	+	5.68280i
44.20	−1.88845	+	0.506008i	−1.21451	+	1.23490i	1.57814	−	0.911137i	0.230970	+	0.861992i	1.66867	−	2.94659i	2.64162	−	0.147740i	0.245696	−	0.245696i	−0.0499376	−	2.99958i	−0.872349	−	1.51095i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner
41.e	odd	8	1	inner
123.i	even	8	1	inner
287.v	odd	24	1	inner
861.bp	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.bp.a	✓	864
3.b	odd	2	1	inner	861.2.bp.a	✓	864
7.c	even	3	1	inner	861.2.bp.a	✓	864
21.h	odd	6	1	inner	861.2.bp.a	✓	864
41.e	odd	8	1	inner	861.2.bp.a	✓	864
123.i	even	8	1	inner	861.2.bp.a	✓	864
287.v	odd	24	1	inner	861.2.bp.a	✓	864
861.bp	even	24	1	inner	861.2.bp.a	✓	864

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.bp.a	✓	864	1.a	even	1	1	trivial
861.2.bp.a	✓	864	3.b	odd	2	1	inner
861.2.bp.a	✓	864	7.c	even	3	1	inner
861.2.bp.a	✓	864	21.h	odd	6	1	inner
861.2.bp.a	✓	864	41.e	odd	8	1	inner
861.2.bp.a	✓	864	123.i	even	8	1	inner
861.2.bp.a	✓	864	287.v	odd	24	1	inner
861.2.bp.a	✓	864	861.bp	even	24	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).