Embedded newform 304.2.bg.a.211.11

• Label: 304.2.bg.a.211.11
• Level: $304$
• Weight: $2$
• Character: 304.211
• 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			211.11
Character	\(\chi\)	\(=\)	304.211
Dual form			304.2.bg.a.219.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00576 + 0.994211i) q^{2} +(0.187445 + 2.14250i) q^{3} +(0.0230881 - 1.99987i) q^{4} +(-1.45391 + 3.11793i) q^{5} +(-2.31862 - 1.96847i) q^{6} +(0.904356 + 1.56639i) q^{7} +(1.96507 + 2.03433i) q^{8} +(-1.60076 + 0.282256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00576	+	0.994211i	−0.711177	+	0.703013i
							\(3\)	0.187445	+	2.14250i	0.108221	+	1.23697i	0.835057	+	0.550163i	\(0.185434\pi\)
−0.726836	+	0.686811i	\(0.759010\pi\)
\(5\)	−1.45391	+	3.11793i	−0.650210	+	1.39438i	0.253964	+	0.967214i	\(0.418266\pi\)
							−0.904174	+	0.427165i	\(0.859512\pi\)
\(7\)	0.904356	+	1.56639i	0.341815	+	0.592040i	0.984770	−	0.173864i	\(-0.0556252\pi\)
							−0.642955	+	0.765904i	\(0.722292\pi\)
\(11\)	−0.0590652	+	0.0158265i	−0.0178088	+	0.00477186i	−0.267712	−	0.963499i	\(-0.586268\pi\)
							0.249904	+	0.968271i	\(0.419601\pi\)
\(13\)	−0.0802608	+	0.917385i	−0.0222603	+	0.254437i	0.976908	+	0.213661i	\(0.0685386\pi\)
							−0.999168	+	0.0407763i	\(0.987017\pi\)
\(17\)	0.132515	−	0.751530i	0.0321396	−	0.182273i	−0.964512	−	0.264039i	\(-0.914945\pi\)
							0.996652	+	0.0817666i	\(0.0260562\pi\)
\(19\)	−1.18008	−	4.19612i	−0.270730	−	0.962655i
							\(23\)	2.43974	+	0.887991i	0.508720	+	0.185159i	0.583612	−	0.812033i	\(-0.301639\pi\)
−0.0748919	+	0.997192i	\(0.523861\pi\)
\(29\)	2.36101	+	1.65319i	0.438428	+	0.306990i	0.771870	−	0.635780i	\(-0.219321\pi\)
							−0.333443	+	0.942770i	\(0.608210\pi\)
\(31\)	−5.35505	−	9.27521i	−0.961795	−	1.66588i	−0.717990	−	0.696053i	\(-0.754938\pi\)
							−0.243805	−	0.969824i	\(-0.578396\pi\)
\(37\)	6.90482	+	6.90482i	1.13515	+	1.13515i	0.989309	+	0.145836i	\(0.0465873\pi\)
							0.145836	+	0.989309i	\(0.453413\pi\)
\(41\)	0.428396	+	0.359467i	0.0669042	+	0.0561393i	0.675627	−	0.737244i	\(-0.263873\pi\)
							−0.608722	+	0.793383i	\(0.708318\pi\)
\(43\)	−6.34258	−	2.95760i	−0.967235	−	0.451029i	−0.126197	−	0.992005i	\(-0.540277\pi\)
							−0.841038	+	0.540976i	\(0.818055\pi\)
\(47\)	−7.04533	+	1.24228i	−1.02767	+	0.181205i	−0.661971	−	0.749529i	\(-0.730280\pi\)
							−0.365696	+	0.930735i	\(0.619169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.211.11	yes	456
16.11	odd	4	inner	304.2.bg.a.59.15	✓	456
19.10	odd	18	inner	304.2.bg.a.67.15	yes	456
304.219	even	36	inner	304.2.bg.a.219.11	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.15	✓	456	16.11	odd	4	inner
304.2.bg.a.67.15	yes	456	19.10	odd	18	inner
304.2.bg.a.211.11	yes	456	1.1	even	1	trivial
304.2.bg.a.219.11	yes	456	304.219	even	36	inner