Newform orbit 380.3.z.a

• Label: 380.3.z.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.z
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 12 q^{3} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
21.1		0		−3.50098	+	4.17230i		0		2.10122	−	0.764780i		0		0.0977372	+	0.169286i		0		−3.58843	−	20.3510i		0
21.2		0		−3.27512	+	3.90313i		0		−2.10122	+	0.764780i		0		−0.665072	−	1.15194i		0		−2.94521	−	16.7031i		0
21.3		0		−2.44690	+	2.91610i		0		−2.10122	+	0.764780i		0		5.45730	+	9.45231i		0		−0.953496	−	5.40754i		0
21.4		0		−2.04576	+	2.43804i		0		2.10122	−	0.764780i		0		−0.451980	−	0.782852i		0		−0.196085	−	1.11205i		0
21.5		0		−0.604813	+	0.720789i		0		2.10122	−	0.764780i		0		−4.26578	−	7.38855i		0		1.40910	+	7.99138i		0
21.6		0		−0.517041	+	0.616185i		0		−2.10122	+	0.764780i		0		0.802077	+	1.38924i		0		1.45048	+	8.22608i		0
21.7		0		−0.0384606	+	0.0458355i		0		−2.10122	+	0.764780i		0		−2.10463	−	3.64533i		0		1.56221	+	8.85974i		0
21.8		0		0.456529	−	0.544071i		0		2.10122	−	0.764780i		0		1.03563	+	1.79376i		0		1.47524	+	8.36650i		0
21.9		0		1.05737	−	1.26013i		0		2.10122	−	0.764780i		0		5.34752	+	9.26218i		0		1.09295	+	6.19841i		0
21.10		0		1.91968	−	2.28779i		0		−2.10122	+	0.764780i		0		3.16201	+	5.47676i		0		0.0140376	+	0.0796112i		0
21.11		0		2.26023	−	2.69364i		0		−2.10122	+	0.764780i		0		−5.06039	−	8.76485i		0		−0.584207	−	3.31320i		0
21.12		0		2.59038	−	3.08709i		0		2.10122	−	0.764780i		0		0.0686008	+	0.118820i		0		−1.25725	−	7.13021i		0
21.13		0		3.46602	−	4.13064i		0		2.10122	−	0.764780i		0		−2.44904	−	4.24185i		0		−3.48607	−	19.7705i		0
21.14		0		3.51635	−	4.19063i		0		−2.10122	+	0.764780i		0		6.19626	+	10.7322i		0		−3.63378	−	20.6082i		0
41.1		0		−1.99434	−	5.47940i		0		0.388289	−	2.20210i		0		−6.25923	−	10.8413i		0		−19.1520	+	16.0705i		0
41.2		0		−1.53916	−	4.22881i		0		0.388289	−	2.20210i		0		6.48559	+	11.2334i		0		−8.61945	+	7.23258i		0
41.3		0		−1.50108	−	4.12417i		0		−0.388289	+	2.20210i		0		−3.56862	−	6.18103i		0		−7.86117	+	6.59631i		0
41.4		0		−1.40162	−	3.85092i		0		−0.388289	+	2.20210i		0		3.85846	+	6.68304i		0		−5.97063	+	5.00995i		0
41.5		0		−1.26907	−	3.48674i		0		−0.388289	+	2.20210i		0		0.121430	+	0.210324i		0		−3.65243	+	3.06475i		0
41.6		0		−0.965305	−	2.65215i		0		0.388289	−	2.20210i		0		0.726831	+	1.25891i		0		0.792298	−	0.664817i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.z.a	✓	84
19.f	odd	18	1	inner	380.3.z.a	✓	84

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.z.a	✓	84	1.a	even	1	1	trivial
380.3.z.a	✓	84	19.f	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).