Newform orbit 572.2.bm.a

• Label: 572.2.bm.a
• Level: $572$
• Weight: $2$
• Character orbit: 572.bm
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bm (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 5 q^{2} - 3 q^{4} - 24 q^{5} - 5 q^{6} - 20 q^{8} - 78 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.41408	−	0.0197050i	−2.48287	+	0.260960i	1.99922	+	0.0557288i	−0.329087	−	1.01283i	3.51611	−	0.320092i	−0.0624841	+	0.594497i	−2.82596	−	0.118200i	3.16210	−	0.672124i	0.445397	+	1.43870i
35.2	−1.41200	+	0.0790274i	−0.146087	+	0.0153543i	1.98751	−	0.223174i	1.02366	+	3.15050i	0.205061	−	0.0332252i	0.530846	−	5.05066i	−2.78873	+	0.472190i	−2.91334	+	0.619249i	−1.69439	−	4.36763i
35.3	−1.40161	−	0.188417i	2.98914	−	0.314171i	1.92900	+	0.528172i	1.02633	+	3.15871i	−4.24879	−	0.122859i	−0.494362	+	4.70354i	−2.60418	−	1.10374i	5.90180	−	1.25447i	−0.843352	−	4.62064i
35.4	−1.40108	−	0.192317i	1.35212	−	0.142114i	1.92603	+	0.538900i	−0.229666	−	0.706840i	−1.92176	−	0.0609233i	−0.0980876	+	0.933242i	−2.59487	−	1.12545i	−1.12640	+	0.239425i	0.185843	+	1.03451i
35.5	−1.39027	+	0.259143i	0.781568	−	0.0821461i	1.86569	−	0.720556i	0.119255	+	0.367030i	−1.06530	+	0.316743i	0.0805893	−	0.766756i	−2.40708	+	1.48525i	−2.33034	+	0.495330i	−0.260910	−	0.479366i
35.6	−1.36069	−	0.385373i	−3.20597	+	0.336961i	1.70298	+	1.04875i	0.704069	+	2.16690i	4.49220	+	0.776993i	−0.0455161	+	0.433057i	−1.91307	−	2.08331i	7.23025	−	1.53684i	−0.122957	−	3.21982i
35.7	−1.35772	−	0.395706i	−0.705374	+	0.0741378i	1.68683	+	1.07452i	−1.19366	−	3.67372i	0.987040	+	0.178462i	0.389527	−	3.70610i	−1.86506	−	2.12639i	−2.44239	+	0.519145i	0.166956	+	5.46025i
35.8	−1.35728	+	0.397241i	−1.69383	+	0.178028i	1.68440	−	1.07833i	−0.635019	−	1.95439i	2.22827	−	0.914490i	−0.118464	+	1.12711i	−1.85784	+	2.13271i	−0.0970887	+	0.0206368i	1.63826	+	2.40039i
35.9	−1.33593	−	0.464004i	2.67160	−	0.280796i	1.56940	+	1.23975i	−0.259301	−	0.798046i	−3.69935	−	0.864509i	0.286196	−	2.72297i	−1.52136	−	2.38442i	4.12414	−	0.876614i	−0.0238895	+	1.18645i
35.10	−1.32463	+	0.495335i	0.111975	−	0.0117691i	1.50929	−	1.31227i	0.137946	+	0.424554i	−0.142496	+	0.0710549i	−0.311356	+	2.96235i	−1.34923	+	2.48587i	−2.92204	+	0.621099i	−0.393024	−	0.494048i
35.11	−1.30497	+	0.545018i	3.13196	−	0.329183i	1.40591	−	1.42247i	−0.288212	−	0.887025i	−3.90772	+	2.13655i	0.347803	−	3.30913i	−1.05941	+	2.62253i	6.76639	−	1.43824i	0.859553	+	1.00046i
35.12	−1.29084	−	0.577698i	−0.134266	+	0.0141119i	1.33253	+	1.49143i	1.07700	+	3.31468i	0.181468	+	0.0593488i	−0.0713495	+	0.678845i	−0.858484	−	2.69500i	−2.91661	+	0.619946i	0.524645	−	4.90090i
35.13	−1.27612	−	0.609521i	−1.34702	+	0.141577i	1.25697	+	1.55564i	−0.804721	−	2.47668i	1.80525	+	0.640366i	−0.453255	+	4.31244i	−0.655846	−	2.75134i	−1.14003	+	0.242321i	−0.482665	+	3.65103i
35.14	−1.26262	+	0.637013i	−2.11369	+	0.222158i	1.18843	−	1.60861i	1.25520	+	3.86310i	2.52728	−	1.62695i	−0.342039	+	3.25429i	−0.475830	+	2.78812i	1.48390	−	0.315412i	−4.04568	−	4.07806i
35.15	−1.14774	+	0.826246i	1.80992	−	0.190230i	0.634634	−	1.89664i	0.746857	+	2.29859i	−1.92014	+	1.71377i	−0.161700	+	1.53847i	0.838694	+	2.70122i	0.305164	−	0.0648647i	−2.75640	−	2.02111i
35.16	−1.14220	−	0.833894i	0.971799	−	0.102140i	0.609240	+	1.90495i	−0.384851	−	1.18445i	−1.19516	−	0.693713i	−0.230566	+	2.19369i	0.892651	−	2.68387i	−2.00048	+	0.425215i	−0.548129	+	1.67380i
35.17	−1.12305	+	0.859505i	0.297649	−	0.0312842i	0.522503	−	1.93054i	−1.00335	−	3.08799i	−0.307388	+	0.290965i	0.422442	−	4.01927i	1.07251	+	2.61720i	−2.84683	+	0.605112i	3.78095	+	2.60559i
35.18	−1.10128	+	0.887234i	−2.91603	+	0.306487i	0.425631	−	1.95418i	−0.642340	−	1.97692i	2.93943	−	2.92472i	0.368936	−	3.51019i	1.26508	+	2.52974i	5.47483	−	1.16371i	2.46138	+	1.60723i
35.19	−1.06158	−	0.934374i	−2.14948	+	0.225919i	0.253889	+	1.98382i	−0.00262404	−	0.00807598i	2.49293	+	1.76858i	0.302942	−	2.88230i	1.58411	−	2.34320i	1.63476	−	0.347479i	−0.00476036	+	0.0110251i
35.20	−1.05823	−	0.938168i	−0.852202	+	0.0895701i	0.239682	+	1.98559i	0.526525	+	1.62048i	0.985854	+	0.704723i	0.121128	−	1.15246i	1.60917	−	2.32606i	−2.21622	+	0.471071i	0.963098	−	2.20880i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.d	odd	10	1	inner
13.c	even	3	1	inner
44.g	even	10	1	inner
52.j	odd	6	1	inner
143.t	odd	30	1	inner
572.bm	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	572.2.bm.a	✓	640
4.b	odd	2	1	inner	572.2.bm.a	✓	640
11.d	odd	10	1	inner	572.2.bm.a	✓	640
13.c	even	3	1	inner	572.2.bm.a	✓	640
44.g	even	10	1	inner	572.2.bm.a	✓	640
52.j	odd	6	1	inner	572.2.bm.a	✓	640
143.t	odd	30	1	inner	572.2.bm.a	✓	640
572.bm	even	30	1	inner	572.2.bm.a	✓	640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
572.2.bm.a	✓	640	1.a	even	1	1	trivial
572.2.bm.a	✓	640	4.b	odd	2	1	inner
572.2.bm.a	✓	640	11.d	odd	10	1	inner
572.2.bm.a	✓	640	13.c	even	3	1	inner
572.2.bm.a	✓	640	44.g	even	10	1	inner
572.2.bm.a	✓	640	52.j	odd	6	1	inner
572.2.bm.a	✓	640	143.t	odd	30	1	inner
572.2.bm.a	✓	640	572.bm	even	30	1	inner

Hecke kernels

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