Embedded newform 304.2.bg.a.3.17

• Label: 304.2.bg.a.3.17
• Level: $304$
• Weight: $2$
• Character: 304.3
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(38\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.372821 - 1.36419i) q^{2} +(2.26327 + 1.05538i) q^{3} +(-1.72201 + 1.01720i) q^{4} +(2.33412 + 3.33347i) q^{5} +(0.595940 - 3.48099i) q^{6} +(-1.68527 - 2.91898i) q^{7} +(2.02965 + 1.96991i) q^{8} +(2.08020 + 2.47909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.372821	−	1.36419i	−0.263625	−	0.964625i
							\(3\)	2.26327	+	1.05538i	1.30670	+	0.609324i	0.946244	−	0.323452i	\(-0.104844\pi\)
0.360455	+	0.932777i	\(0.382621\pi\)
\(5\)	2.33412	+	3.33347i	1.04385	+	1.49078i	0.861643	+	0.507515i	\(0.169436\pi\)
							0.182209	+	0.983260i	\(0.441675\pi\)
\(7\)	−1.68527	−	2.91898i	−0.636973	−	1.10327i	−0.986094	−	0.166191i	\(-0.946853\pi\)
							0.349121	−	0.937078i	\(-0.386480\pi\)
\(11\)	−4.07347	+	1.09148i	−1.22820	+	0.329094i	−0.813877	−	0.581037i	\(-0.802647\pi\)
							−0.414320	+	0.910131i	\(0.635980\pi\)
\(13\)	2.73157	−	1.27375i	0.757602	−	0.353275i	−0.00513313	−	0.999987i	\(-0.501634\pi\)
							0.762735	+	0.646711i	\(0.223856\pi\)
\(17\)	2.69653	+	2.26266i	0.654005	+	0.548776i	0.908283	−	0.418356i	\(-0.137393\pi\)
							−0.254278	+	0.967131i	\(0.581838\pi\)
\(19\)	4.35187	+	0.247357i	0.998389	+	0.0567475i
							\(23\)	0.469826	+	2.66452i	0.0979655	+	0.555590i	0.993798	+	0.111199i	\(0.0354690\pi\)
−0.895833	+	0.444392i	\(0.853420\pi\)
\(29\)	0.336591	−	3.84726i	0.0625035	−	0.714418i	−0.898698	−	0.438569i	\(-0.855485\pi\)
							0.961201	−	0.275849i	\(-0.0889589\pi\)
\(31\)	−4.63670	−	8.03099i	−0.832775	−	1.44241i	−0.895829	−	0.444399i	\(-0.853417\pi\)
							0.0630539	−	0.998010i	\(-0.479916\pi\)
\(37\)	−5.99723	−	5.99723i	−0.985939	−	0.985939i	0.0139634	−	0.999903i	\(-0.495555\pi\)
							−0.999903	+	0.0139634i	\(0.995555\pi\)
\(41\)	−2.68621	+	0.977700i	−0.419515	+	0.152691i	−0.543148	−	0.839637i	\(-0.682768\pi\)
							0.123633	+	0.992328i	\(0.460546\pi\)
\(43\)	−2.52287	+	1.76653i	−0.384733	+	0.269393i	−0.749903	−	0.661548i	\(-0.769900\pi\)
							0.365170	+	0.930941i	\(0.381011\pi\)
\(47\)	−1.33546	−	1.59154i	−0.194797	−	0.232150i	0.659801	−	0.751441i	\(-0.270641\pi\)
							−0.854598	+	0.519290i	\(0.826196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.17	✓	456
16.11	odd	4	inner	304.2.bg.a.155.37	yes	456
19.13	odd	18	inner	304.2.bg.a.51.37	yes	456
304.203	even	36	inner	304.2.bg.a.203.17	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.17	✓	456	1.1	even	1	trivial
304.2.bg.a.51.37	yes	456	19.13	odd	18	inner
304.2.bg.a.155.37	yes	456	16.11	odd	4	inner
304.2.bg.a.203.17	yes	456	304.203	even	36	inner