Newform orbit 380.2.bj.a

• Label: 380.2.bj.a
• Level: $380$
• Weight: $2$
• Character orbit: 380.bj
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.bj (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(56\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 12 q^{2} - 24 q^{5} - 36 q^{6} - 6 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.41421	−	0.00173027i	−2.30458	−	1.61369i	1.99999	+	0.00489393i	−1.01610	+	1.99187i	3.25638	+	2.28608i	0.362300	−	0.0970781i	−2.82841	−	0.0103816i	1.68106	+	4.61866i	1.44042	−	2.81517i
23.2	−1.41416	+	0.0118355i	−0.0933215	−	0.0653444i	1.99972	−	0.0334747i	1.39136	−	1.75046i	0.132745	+	0.0913032i	4.78721	−	1.28273i	−2.82754	+	0.0710065i	−1.02162	−	2.80688i	−1.94689	+	2.49191i
23.3	−1.40256	−	0.181148i	−1.67151	−	1.17040i	1.93437	+	0.508144i	−0.660038	−	2.13643i	2.13238	+	1.94436i	−2.38759	+	0.639752i	−2.62103	−	1.06311i	0.398042	+	1.09361i	0.538734	+	3.11605i
23.4	−1.39453	+	0.235114i	1.39631	+	0.977708i	1.88944	−	0.655749i	−1.14825	−	1.91873i	−2.17707	−	1.03515i	−2.67634	+	0.717123i	−2.48071	+	1.35870i	−0.0322876	−	0.0887093i	2.05239	+	2.40576i
23.5	−1.37359	+	0.336528i	−0.322158	−	0.225578i	1.77350	−	0.924503i	0.973777	+	2.01290i	0.518426	+	0.201436i	−2.45904	+	0.658899i	−2.12494	+	1.86672i	−0.973160	−	2.67373i	−2.01497	−	2.43719i
23.6	−1.37318	+	0.338181i	2.50164	+	1.75167i	1.77127	−	0.928769i	2.16402	−	0.563029i	−4.02760	−	1.55936i	−0.675461	+	0.180989i	−2.11818	+	1.87438i	2.16382	+	5.94503i	−2.78120	+	1.50497i
23.7	−1.34500	−	0.436992i	−0.109637	−	0.0767686i	1.61808	+	1.17551i	−1.98421	+	1.03097i	0.113915	+	0.151165i	2.66363	−	0.713718i	−1.66263	−	2.28816i	−1.01993	−	2.80224i	3.11930	−	0.519577i
23.8	−1.31289	−	0.525665i	1.78355	+	1.24886i	1.44735	+	1.38028i	−1.31769	+	1.80657i	−1.68512	−	2.57716i	−4.05211	+	1.08576i	−1.17465	−	2.57297i	0.595352	+	1.63572i	2.67963	−	1.67916i
23.9	−1.28719	−	0.585784i	1.15005	+	0.805273i	1.31371	+	1.50803i	2.01826	+	0.962606i	−1.00862	−	1.71022i	0.399624	−	0.107079i	−0.807621	−	2.71067i	−0.351912	−	0.966869i	−2.03401	−	2.42132i
23.10	−1.21069	−	0.730902i	−2.10842	−	1.47633i	0.931564	+	1.76980i	2.23455	+	0.0823462i	1.47360	+	3.32843i	−1.24717	+	0.334178i	0.165710	−	2.82357i	1.23982	+	3.40636i	−2.64517	−	1.73293i
23.11	−1.14173	+	0.834541i	−2.50164	−	1.75167i	0.607081	−	1.90564i	2.16402	−	0.563029i	4.31804	−	0.0877964i	0.675461	−	0.180989i	0.897212	+	2.68235i	2.16382	+	5.94503i	−2.00085	+	2.44879i
23.12	−1.14072	+	0.835915i	0.322158	+	0.225578i	0.602493	−	1.90709i	0.973777	+	2.01290i	−0.556057	+	0.0119755i	2.45904	−	0.658899i	0.906891	+	2.67909i	−0.973160	−	2.67373i	−2.79342	−	1.48216i
23.13	−1.12405	−	0.858201i	2.71916	+	1.90398i	0.526984	+	1.92932i	−1.53955	−	1.62166i	−1.42248	−	4.47376i	3.91237	−	1.04832i	1.06339	−	2.62092i	2.74265	+	7.53538i	0.338817	+	3.14407i
23.14	−1.07650	+	0.917145i	−1.39631	−	0.977708i	0.317689	−	1.97461i	−1.14825	−	1.91873i	2.39982	−	0.228122i	2.67634	−	0.717123i	1.46901	+	2.41702i	−0.0322876	−	0.0887093i	2.99584	+	1.01239i
23.15	−0.986683	−	1.01314i	−0.566781	−	0.396864i	−0.0529140	+	1.99930i	−1.91076	−	1.16147i	0.157153	+	0.965809i	0.335634	−	0.0899329i	2.07778	−	1.91907i	−0.862321	−	2.36921i	0.708583	+	3.08187i
23.16	−0.918074	+	1.07570i	0.0933215	+	0.0653444i	−0.314282	−	1.97515i	1.39136	−	1.75046i	−0.155967	+	0.0403954i	−4.78721	+	1.28273i	2.41321	+	1.47526i	−1.02162	−	2.80688i	0.605609	+	3.10375i
23.17	−0.907713	+	1.08446i	2.30458	+	1.61369i	−0.352115	−	1.96876i	−1.01610	+	1.99187i	−3.84188	+	1.03447i	−0.362300	+	0.0970781i	2.45466	+	1.40521i	1.68106	+	4.61866i	−1.23778	−	2.90997i
23.18	−0.895026	−	1.09496i	0.336022	+	0.235285i	−0.397856	+	1.96003i	1.19920	−	1.88731i	−0.0431218	−	0.578516i	−1.38682	+	0.371598i	2.50224	−	1.31864i	−0.968509	−	2.66096i	−3.13983	+	0.376122i
23.19	−0.762783	+	1.19087i	1.67151	+	1.17040i	−0.836324	−	1.81674i	−0.660038	−	2.13643i	−2.66880	+	1.09778i	2.38759	−	0.639752i	2.80143	+	0.389833i	0.398042	+	1.09361i	3.04767	+	0.843619i
23.20	−0.631329	−	1.26547i	−2.52409	−	1.76739i	−1.20285	+	1.59786i	−2.23416	+	0.0923380i	−0.643051	+	4.30997i	1.52327	−	0.408158i	2.78144	+	0.513399i	2.22131	+	6.10300i	1.52734	+	2.76898i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
19.e	even	9	1	inner
20.e	even	4	1	inner
76.l	odd	18	1	inner
95.q	odd	36	1	inner
380.bj	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.2.bj.a	✓	672
4.b	odd	2	1	inner	380.2.bj.a	✓	672
5.c	odd	4	1	inner	380.2.bj.a	✓	672
19.e	even	9	1	inner	380.2.bj.a	✓	672
20.e	even	4	1	inner	380.2.bj.a	✓	672
76.l	odd	18	1	inner	380.2.bj.a	✓	672
95.q	odd	36	1	inner	380.2.bj.a	✓	672
380.bj	even	36	1	inner	380.2.bj.a	✓	672

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.bj.a	✓	672	1.a	even	1	1	trivial
380.2.bj.a	✓	672	4.b	odd	2	1	inner
380.2.bj.a	✓	672	5.c	odd	4	1	inner
380.2.bj.a	✓	672	19.e	even	9	1	inner
380.2.bj.a	✓	672	20.e	even	4	1	inner
380.2.bj.a	✓	672	76.l	odd	18	1	inner
380.2.bj.a	✓	672	95.q	odd	36	1	inner
380.2.bj.a	✓	672	380.bj	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(380, [\chi])\).