Newform orbit 572.2.bk.a

• Label: 572.2.bk.a
• Level: $572$
• Weight: $2$
• Character orbit: 572.bk
• 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.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 6 q^{2} - 12 q^{5} - 14 q^{6} - 6 q^{8} + 120 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.41408	+	0.0192463i	0.0675205	+	0.0929340i	1.99926	−	0.0544317i	3.09570	+	1.57734i	−0.0972682	−	0.130117i	0.303959	−	0.0481424i	−2.82607	+	0.115449i	0.922973	−	2.84062i	−4.40794	−	2.17091i
31.2	−1.40755	−	0.137128i	−0.993999	−	1.36812i	1.96239	+	0.386029i	−1.79157	−	0.912850i	1.21150	+	2.06201i	3.11515	−	0.493392i	−2.70923	−	0.812454i	0.0433265	−	0.133345i	2.39655	+	1.53056i
31.3	−1.39732	+	0.217955i	0.585304	+	0.805601i	1.90499	−	0.609104i	−0.214144	−	0.109112i	−0.993440	−	0.998111i	−4.72480	+	0.748335i	−2.52912	+	1.26631i	0.620638	−	1.91013i	0.323009	+	0.105790i
31.4	−1.39464	+	0.234483i	0.111085	+	0.152895i	1.89004	−	0.654039i	−2.01367	−	1.02602i	−0.190775	−	0.187186i	3.86875	−	0.612750i	−2.48256	+	1.35533i	0.916014	−	2.81920i	3.04893	+	0.958750i
31.5	−1.38260	+	0.297351i	−1.11684	−	1.53720i	1.82316	−	0.822235i	1.61342	+	0.822078i	2.00123	+	1.79324i	−1.26651	+	0.200595i	−2.27621	+	1.67894i	−0.188596	+	0.580439i	−2.47516	−	0.656853i
31.6	−1.37530	−	0.329478i	1.80222	+	2.48055i	1.78289	+	0.906260i	1.79602	+	0.915119i	−1.66131	−	4.00528i	−0.175107	+	0.0277342i	−2.15341	−	1.83380i	−1.97806	+	6.08783i	−2.16855	−	1.85031i
31.7	−1.37479	+	0.331572i	1.89320	+	2.60576i	1.78012	−	0.911688i	−3.02502	−	1.54132i	−3.46676	−	2.95466i	−0.922158	+	0.146055i	−2.14501	+	1.84362i	−2.27875	+	7.01328i	4.66983	+	1.11599i
31.8	−1.35684	−	0.398721i	−1.82403	−	2.51057i	1.68204	+	1.08200i	−2.67101	−	1.36095i	1.47391	+	4.13373i	−2.81984	+	0.446619i	−1.85085	−	2.13877i	−2.04880	+	6.30555i	3.08150	+	2.91158i
31.9	−1.33748	−	0.459517i	0.531376	+	0.731377i	1.57769	+	1.22919i	0.313818	+	0.159898i	−0.374624	−	1.22238i	−2.84457	+	0.450536i	−1.54529	−	2.36898i	0.674500	−	2.07590i	−0.346248	−	0.358064i
31.10	−1.33474	+	0.467394i	−1.68703	−	2.32199i	1.56309	−	1.24770i	1.23417	+	0.628841i	3.33703	+	2.31076i	−0.224881	+	0.0356177i	−1.50315	+	2.39594i	−1.61854	+	4.98136i	−1.94122	−	0.262499i
31.11	−1.29838	−	0.560535i	−0.722183	−	0.993999i	1.37160	+	1.45558i	2.78387	+	1.41845i	0.380499	+	1.69540i	1.85332	−	0.293537i	−0.964963	−	2.65873i	0.460564	−	1.41747i	−2.81944	−	3.40216i
31.12	−1.26887	+	0.624477i	−0.732346	−	1.00799i	1.22006	−	1.58476i	−3.57348	−	1.82078i	1.55872	+	0.821671i	−3.95380	+	0.626220i	−0.558450	+	2.77275i	0.447343	−	1.37678i	5.67132	+	0.0787760i
31.13	−1.26761	+	0.627035i	0.675560	+	0.929829i	1.21365	−	1.58967i	1.46176	+	0.744806i	−1.43938	−	0.755058i	1.82475	−	0.289012i	−0.541659	+	2.77608i	0.518851	−	1.59686i	−2.31996	−	0.0275439i
31.14	−1.25415	−	0.653531i	0.738802	+	1.01687i	1.14579	+	1.63925i	−3.61425	−	1.84155i	−0.262011	−	1.75814i	−0.636645	+	0.100835i	−0.365696	−	2.80469i	0.438847	−	1.35063i	3.32931	+	4.67161i
31.15	−1.19654	+	0.753844i	1.53446	+	2.11200i	0.863437	−	1.80402i	3.64099	+	1.85518i	−3.42817	−	1.37036i	−0.0406960	+	0.00644561i	0.326806	+	2.80948i	−1.17893	+	3.62839i	−5.75512	+	0.524938i
31.16	−1.19169	−	0.761498i	1.13752	+	1.56566i	0.840242	+	1.81494i	−1.23921	−	0.631408i	−0.163321	−	2.73200i	3.50929	−	0.555817i	0.380764	−	2.80268i	−0.230299	+	0.708787i	0.995933	+	1.69609i
31.17	−1.13998	−	0.836921i	−1.62851	−	2.24145i	0.599125	+	1.90815i	2.91084	+	1.48315i	−0.0194442	+	3.91816i	−4.55254	+	0.721051i	0.913981	−	2.67668i	−1.44502	+	4.44730i	−2.07704	−	4.12691i
31.18	−1.09010	+	0.900938i	1.25644	+	1.72935i	0.376620	−	1.96422i	−0.781284	−	0.398084i	−2.92768	−	0.753176i	2.81330	−	0.445583i	1.35909	+	2.48050i	−0.484940	+	1.49249i	1.21032	−	0.269939i
31.19	−1.00757	+	0.992369i	−0.167099	−	0.229991i	0.0304084	−	1.99977i	−2.43805	−	1.24225i	0.396600	+	0.0659099i	0.467841	−	0.0740988i	1.95387	+	2.04509i	0.902077	−	2.77631i	3.68928	−	1.16779i
31.20	−0.986038	−	1.01377i	−0.445909	−	0.613742i	−0.0554586	+	1.99923i	−0.483456	−	0.246333i	−0.182509	+	1.05722i	−2.49202	+	0.394697i	2.08144	−	1.91510i	0.749207	−	2.30582i	0.226981	+	0.733007i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.c	even	5	1	inner
13.d	odd	4	1	inner
44.h	odd	10	1	inner
52.f	even	4	1	inner
143.r	odd	20	1	inner
572.bk	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	572.2.bk.a	✓	640
4.b	odd	2	1	inner	572.2.bk.a	✓	640
11.c	even	5	1	inner	572.2.bk.a	✓	640
13.d	odd	4	1	inner	572.2.bk.a	✓	640
44.h	odd	10	1	inner	572.2.bk.a	✓	640
52.f	even	4	1	inner	572.2.bk.a	✓	640
143.r	odd	20	1	inner	572.2.bk.a	✓	640
572.bk	even	20	1	inner	572.2.bk.a	✓	640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
572.2.bk.a	✓	640	1.a	even	1	1	trivial
572.2.bk.a	✓	640	4.b	odd	2	1	inner
572.2.bk.a	✓	640	11.c	even	5	1	inner
572.2.bk.a	✓	640	13.d	odd	4	1	inner
572.2.bk.a	✓	640	44.h	odd	10	1	inner
572.2.bk.a	✓	640	52.f	even	4	1	inner
572.2.bk.a	✓	640	143.r	odd	20	1	inner
572.2.bk.a	✓	640	572.bk	even	20	1	inner

Hecke kernels

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