Embedded newform 304.2.bg.a.243.38

• Label: 304.2.bg.a.243.38
• Level: $304$
• Weight: $2$
• Character: 304.243
• 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			243.38
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41406 + 0.0208333i) q^{2} +(0.999905 - 1.42801i) q^{3} +(1.99913 + 0.0589192i) q^{4} +(0.239149 - 2.73348i) q^{5} +(1.44368 - 1.99846i) q^{6} +(-1.78194 + 3.08641i) q^{7} +(2.82567 + 0.124964i) q^{8} +(-0.0133496 - 0.0366778i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41406	+	0.0208333i	0.999891	+	0.0147314i
							\(3\)	0.999905	−	1.42801i	0.577296	−	0.824464i	−0.419219	−	0.907885i	\(-0.637696\pi\)
0.996514	+	0.0834216i	\(0.0265848\pi\)
\(5\)	0.239149	−	2.73348i	0.106950	−	1.22245i	−0.733168	−	0.680047i	\(-0.761959\pi\)
							0.840119	−	0.542402i	\(-0.182485\pi\)
\(7\)	−1.78194	+	3.08641i	−0.673509	+	1.16655i	0.303393	+	0.952866i	\(0.401881\pi\)
							−0.976902	+	0.213687i	\(0.931453\pi\)
\(11\)	−0.995797	+	3.71636i	−0.300244	+	1.12053i	0.636719	+	0.771096i	\(0.280291\pi\)
							−0.936963	+	0.349430i	\(0.886375\pi\)
\(13\)	−1.39335	−	1.98992i	−0.386447	−	0.551903i	0.578049	−	0.816002i	\(-0.303814\pi\)
							−0.964496	+	0.264099i	\(0.914925\pi\)
\(17\)	−6.72368	−	2.44722i	−1.63073	−	0.593538i	−0.645348	−	0.763889i	\(-0.723288\pi\)
							−0.985383	+	0.170351i	\(0.945510\pi\)
\(19\)	3.15759	−	3.00494i	0.724400	−	0.689380i
							\(23\)	−0.799988	−	0.671270i	−0.166809	−	0.139969i	0.555562	−	0.831475i	\(-0.312503\pi\)
−0.722371	+	0.691506i	\(0.756948\pi\)
\(29\)	−3.76088	+	8.06524i	−0.698378	+	1.49768i	0.161335	+	0.986900i	\(0.448420\pi\)
							−0.859713	+	0.510777i	\(0.829358\pi\)
\(31\)	−1.21187	+	2.09902i	−0.217658	+	0.376995i	−0.954092	−	0.299515i	\(-0.903175\pi\)
							0.736433	+	0.676510i	\(0.236509\pi\)
\(37\)	−4.45728	−	4.45728i	−0.732773	−	0.732773i	0.238395	−	0.971168i	\(-0.423379\pi\)
							−0.971168	+	0.238395i	\(0.923379\pi\)
\(41\)	1.37578	+	7.80242i	0.214860	+	1.21853i	0.881149	+	0.472840i	\(0.156771\pi\)
							−0.666288	+	0.745694i	\(0.732118\pi\)
\(43\)	11.2467	+	0.983957i	1.71510	+	0.150052i	0.901904	−	0.431936i	\(-0.142169\pi\)
							0.813198	+	0.581988i	\(0.197725\pi\)
\(47\)	−2.65598	−	7.29725i	−0.387415	−	1.06441i	−0.968161	−	0.250328i	\(-0.919461\pi\)
							0.580746	−	0.814085i	\(-0.302761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.38	yes	456
16.11	odd	4	inner	304.2.bg.a.91.8	✓	456
19.14	odd	18	inner	304.2.bg.a.147.8	yes	456
304.299	even	36	inner	304.2.bg.a.299.38	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.8	✓	456	16.11	odd	4	inner
304.2.bg.a.147.8	yes	456	19.14	odd	18	inner
304.2.bg.a.243.38	yes	456	1.1	even	1	trivial
304.2.bg.a.299.38	yes	456	304.299	even	36	inner