Newform orbit 380.3.h.a

• Label: 380.3.h.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.h
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 4 q^{5} + 324 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
39.1	−1.99911	−	0.0595768i	−2.57497			3.99290	+	0.238201i	2.22773	−	4.47629i	5.14765	+	0.153408i	−1.63196			−7.96807	−	0.714075i	−2.36954			−4.72017	+	8.81589i
39.2	−1.99911	+	0.0595768i	−2.57497			3.99290	−	0.238201i	2.22773	+	4.47629i	5.14765	−	0.153408i	−1.63196			−7.96807	+	0.714075i	−2.36954			−4.72017	−	8.81589i
39.3	−1.99455	−	0.147607i	−0.578687			3.95642	+	0.588817i	−3.52665	−	3.54440i	1.15422	+	0.0854181i	11.0384			−7.80436	−	1.75842i	−8.66512			6.51089	+	7.59002i
39.4	−1.99455	+	0.147607i	−0.578687			3.95642	−	0.588817i	−3.52665	+	3.54440i	1.15422	−	0.0854181i	11.0384			−7.80436	+	1.75842i	−8.66512			6.51089	−	7.59002i
39.5	−1.94809	−	0.452727i	2.05284			3.59008	+	1.76390i	4.88304	−	1.07515i	−3.99911	−	0.929375i	−5.89695			−6.19521	−	5.06155i	−4.78585			−9.99932	−	0.116190i
39.6	−1.94809	+	0.452727i	2.05284			3.59008	−	1.76390i	4.88304	+	1.07515i	−3.99911	+	0.929375i	−5.89695			−6.19521	+	5.06155i	−4.78585			−9.99932	+	0.116190i
39.7	−1.94008	−	0.485899i	4.64902			3.52780	+	1.88537i	2.66799	+	4.22869i	−9.01947	−	2.25896i	9.90681			−5.92811	−	5.37191i	12.6134			−3.12138	−	9.50037i
39.8	−1.94008	+	0.485899i	4.64902			3.52780	−	1.88537i	2.66799	−	4.22869i	−9.01947	+	2.25896i	9.90681			−5.92811	+	5.37191i	12.6134			−3.12138	+	9.50037i
39.9	−1.91201	−	0.586697i	−5.34961			3.31157	+	2.24354i	−3.58581	+	3.48454i	10.2285	+	3.13860i	0.878631			−5.01548	−	6.23257i	19.6184			8.90047	−	4.55869i
39.10	−1.91201	+	0.586697i	−5.34961			3.31157	−	2.24354i	−3.58581	−	3.48454i	10.2285	−	3.13860i	0.878631			−5.01548	+	6.23257i	19.6184			8.90047	+	4.55869i
39.11	−1.86011	−	0.734843i	2.97071			2.92001	+	2.73378i	−0.632199	+	4.95987i	−5.52585	−	2.18301i	−5.57969			−3.42265	−	7.23087i	−0.174871			4.82068	−	8.76134i
39.12	−1.86011	+	0.734843i	2.97071			2.92001	−	2.73378i	−0.632199	−	4.95987i	−5.52585	+	2.18301i	−5.57969			−3.42265	+	7.23087i	−0.174871			4.82068	+	8.76134i
39.13	−1.85933	−	0.736815i	−2.04408			2.91421	+	2.73996i	−4.55181	−	2.06907i	3.80062	+	1.50611i	−11.4458			−3.39962	−	7.24172i	−4.82174			6.93879	+	7.20092i
39.14	−1.85933	+	0.736815i	−2.04408			2.91421	−	2.73996i	−4.55181	+	2.06907i	3.80062	−	1.50611i	−11.4458			−3.39962	+	7.24172i	−4.82174			6.93879	−	7.20092i
39.15	−1.75001	−	0.968228i	−4.24708			2.12507	+	3.38882i	4.77157	+	1.49403i	7.43242	+	4.11214i	7.86306			−0.437742	−	7.98801i	9.03765			−6.90374	−	7.23453i
39.16	−1.75001	+	0.968228i	−4.24708			2.12507	−	3.38882i	4.77157	−	1.49403i	7.43242	−	4.11214i	7.86306			−0.437742	+	7.98801i	9.03765			−6.90374	+	7.23453i
39.17	−1.70952	−	1.03804i	1.20444			1.84495	+	3.54911i	3.17819	−	3.85994i	−2.05903	−	1.25026i	6.44137			0.530130	−	7.98242i	−7.54932			−9.43996	+	3.29958i
39.18	−1.70952	+	1.03804i	1.20444			1.84495	−	3.54911i	3.17819	+	3.85994i	−2.05903	+	1.25026i	6.44137			0.530130	+	7.98242i	−7.54932			−9.43996	−	3.29958i
39.19	−1.68898	−	1.07114i	5.94620			1.70531	+	3.61828i	1.71987	−	4.69490i	−10.0430	−	6.36923i	−9.59523			0.995461	−	7.93782i	26.3573			−7.93373	+	6.08736i
39.20	−1.68898	+	1.07114i	5.94620			1.70531	−	3.61828i	1.71987	+	4.69490i	−10.0430	+	6.36923i	−9.59523			0.995461	+	7.93782i	26.3573			−7.93373	−	6.08736i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.h.a	✓	108
4.b	odd	2	1	inner	380.3.h.a	✓	108
5.b	even	2	1	inner	380.3.h.a	✓	108
20.d	odd	2	1	inner	380.3.h.a	✓	108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.h.a	✓	108	1.a	even	1	1	trivial
380.3.h.a	✓	108	4.b	odd	2	1	inner
380.3.h.a	✓	108	5.b	even	2	1	inner
380.3.h.a	✓	108	20.d	odd	2	1	inner

Hecke kernels

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