Newform orbit 380.2.d.b

• Label: 380.2.d.b
• Level: $380$
• Weight: $2$
• Character orbit: 380.d
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 8 q^{6} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
379.1	−1.39672	−	0.221721i		−	2.04365i	1.90168	+	0.619368i	0.665544	−	2.13473i	−0.453121	+	2.85442i	2.07881			−2.51879	−	1.28673i	−1.17650			−1.40290	+	2.83406i
379.2	−1.39672	−	0.221721i		−	2.04365i	1.90168	+	0.619368i	0.665544	+	2.13473i	−0.453121	+	2.85442i	−2.07881			−2.51879	−	1.28673i	−1.17650			−0.456267	−	3.12919i
379.3	−1.39672	+	0.221721i			2.04365i	1.90168	−	0.619368i	0.665544	−	2.13473i	−0.453121	−	2.85442i	−2.07881			−2.51879	+	1.28673i	−1.17650			−0.456267	+	3.12919i
379.4	−1.39672	+	0.221721i			2.04365i	1.90168	−	0.619368i	0.665544	+	2.13473i	−0.453121	−	2.85442i	2.07881			−2.51879	+	1.28673i	−1.17650			−1.40290	−	2.83406i
379.5	−1.27059	−	0.620975i			0.701989i	1.22878	+	1.57800i	−1.10449	−	1.94425i	0.435918	−	0.891938i	1.18642			−0.581371	−	2.76803i	2.50721			0.196019	+	3.15620i
379.6	−1.27059	−	0.620975i			0.701989i	1.22878	+	1.57800i	−1.10449	+	1.94425i	0.435918	−	0.891938i	−1.18642			−0.581371	−	2.76803i	2.50721			2.61068	−	1.78448i
379.7	−1.27059	+	0.620975i		−	0.701989i	1.22878	−	1.57800i	−1.10449	−	1.94425i	0.435918	+	0.891938i	−1.18642			−0.581371	+	2.76803i	2.50721			2.61068	+	1.78448i
379.8	−1.27059	+	0.620975i		−	0.701989i	1.22878	−	1.57800i	−1.10449	+	1.94425i	0.435918	+	0.891938i	1.18642			−0.581371	+	2.76803i	2.50721			0.196019	−	3.15620i
379.9	−0.948256	−	1.04920i		−	1.76151i	−0.201622	+	1.98981i	−1.85854	−	1.24332i	−1.84816	+	1.67036i	−4.29252			2.27889	−	1.67531i	−0.102903			0.457886	+	3.12895i
379.10	−0.948256	−	1.04920i		−	1.76151i	−0.201622	+	1.98981i	−1.85854	+	1.24332i	−1.84816	+	1.67036i	4.29252			2.27889	−	1.67531i	−0.102903			3.06685	+	0.770986i
379.11	−0.948256	+	1.04920i			1.76151i	−0.201622	−	1.98981i	−1.85854	−	1.24332i	−1.84816	−	1.67036i	4.29252			2.27889	+	1.67531i	−0.102903			3.06685	−	0.770986i
379.12	−0.948256	+	1.04920i			1.76151i	−0.201622	−	1.98981i	−1.85854	+	1.24332i	−1.84816	−	1.67036i	−4.29252			2.27889	+	1.67531i	−0.102903			0.457886	−	3.12895i
379.13	−0.584780	−	1.28765i			2.81267i	−1.31606	+	1.50598i	0.390055	−	2.20178i	3.62172	−	1.64479i	−4.13073			2.70878	+	0.813956i	−4.91111			−3.06322	+	0.785308i
379.14	−0.584780	−	1.28765i			2.81267i	−1.31606	+	1.50598i	0.390055	+	2.20178i	3.62172	−	1.64479i	4.13073			2.70878	+	0.813956i	−4.91111			2.60702	−	1.78981i
379.15	−0.584780	+	1.28765i		−	2.81267i	−1.31606	−	1.50598i	0.390055	−	2.20178i	3.62172	+	1.64479i	4.13073			2.70878	−	0.813956i	−4.91111			2.60702	+	1.78981i
379.16	−0.584780	+	1.28765i		−	2.81267i	−1.31606	−	1.50598i	0.390055	+	2.20178i	3.62172	+	1.64479i	−4.13073			2.70878	−	0.813956i	−4.91111			−3.06322	−	0.785308i
379.17	−0.440015	−	1.34402i		−	0.562756i	−1.61277	+	1.18278i	1.40743	−	1.73757i	−0.756355	+	0.247621i	3.12767			2.29932	+	1.64716i	2.68331			−2.95462	−	1.12705i
379.18	−0.440015	−	1.34402i		−	0.562756i	−1.61277	+	1.18278i	1.40743	+	1.73757i	−0.756355	+	0.247621i	−3.12767			2.29932	+	1.64716i	2.68331			1.71604	−	2.65617i
379.19	−0.440015	+	1.34402i			0.562756i	−1.61277	−	1.18278i	1.40743	−	1.73757i	−0.756355	−	0.247621i	−3.12767			2.29932	−	1.64716i	2.68331			1.71604	+	2.65617i
379.20	−0.440015	+	1.34402i			0.562756i	−1.61277	−	1.18278i	1.40743	+	1.73757i	−0.756355	−	0.247621i	3.12767			2.29932	−	1.64716i	2.68331			−2.95462	+	1.12705i

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
19.b	odd	2	1	inner
20.d	odd	2	1	inner
76.d	even	2	1	inner
95.d	odd	2	1	inner
380.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.2.d.b	✓	40
4.b	odd	2	1	inner	380.2.d.b	✓	40
5.b	even	2	1	inner	380.2.d.b	✓	40
19.b	odd	2	1	inner	380.2.d.b	✓	40
20.d	odd	2	1	inner	380.2.d.b	✓	40
76.d	even	2	1	inner	380.2.d.b	✓	40
95.d	odd	2	1	inner	380.2.d.b	✓	40
380.d	even	2	1	inner	380.2.d.b	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.d.b	✓	40	1.a	even	1	1	trivial
380.2.d.b	✓	40	4.b	odd	2	1	inner
380.2.d.b	✓	40	5.b	even	2	1	inner
380.2.d.b	✓	40	19.b	odd	2	1	inner
380.2.d.b	✓	40	20.d	odd	2	1	inner
380.2.d.b	✓	40	76.d	even	2	1	inner
380.2.d.b	✓	40	95.d	odd	2	1	inner
380.2.d.b	✓	40	380.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 16T_{3}^{8} + 83T_{3}^{6} + 162T_{3}^{4} + 94T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).