Embedded newform 304.2.bg.a.243.19

• Label: 304.2.bg.a.243.19
• 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			243.19
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.116605 - 1.40940i) q^{2} +(1.65703 - 2.36649i) q^{3} +(-1.97281 - 0.328685i) q^{4} +(-0.173966 + 1.98844i) q^{5} +(-3.14211 - 2.61136i) q^{6} +(2.42106 - 4.19341i) q^{7} +(-0.693286 + 2.74214i) q^{8} +(-1.82845 - 5.02362i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.116605	−	1.40940i	0.0824520	−	0.996595i
							\(3\)	1.65703	−	2.36649i	0.956688	−	1.36629i	0.0263250	−	0.999653i	\(-0.491620\pi\)
0.930363	−	0.366639i	\(-0.119492\pi\)
\(5\)	−0.173966	+	1.98844i	−0.0777998	+	0.889256i	0.852565	+	0.522621i	\(0.175046\pi\)
							−0.930365	+	0.366635i	\(0.880510\pi\)
\(7\)	2.42106	−	4.19341i	0.915076	−	1.58496i	0.108286	−	0.994120i	\(-0.465464\pi\)
							0.806790	−	0.590838i	\(-0.201203\pi\)
\(11\)	−1.21643	+	4.53979i	−0.366769	+	1.36880i	0.498239	+	0.867040i	\(0.333980\pi\)
							−0.865007	+	0.501759i	\(0.832686\pi\)
\(13\)	−2.70332	−	3.86074i	−0.749766	−	1.07078i	−0.994858	−	0.101277i	\(-0.967707\pi\)
							0.245093	−	0.969500i	\(-0.421182\pi\)
\(17\)	0.240204	+	0.0874270i	0.0582579	+	0.0212042i	0.370985	−	0.928639i	\(-0.379020\pi\)
							−0.312727	+	0.949843i	\(0.601242\pi\)
\(19\)	3.06158	+	3.10269i	0.702376	+	0.711806i
							\(23\)	−0.364261	−	0.305651i	−0.0759537	−	0.0637327i	0.604020	−	0.796969i	\(-0.293565\pi\)
−0.679974	+	0.733236i	\(0.738009\pi\)
\(29\)	−1.47014	+	3.15272i	−0.272998	+	0.585445i	−0.994233	−	0.107245i	\(-0.965797\pi\)
							0.721235	+	0.692691i	\(0.243575\pi\)
\(31\)	−0.365134	+	0.632430i	−0.0655800	+	0.113588i	−0.896951	−	0.442130i	\(-0.854223\pi\)
							0.831371	+	0.555718i	\(0.187556\pi\)
\(37\)	4.69895	+	4.69895i	0.772503	+	0.772503i	0.978543	−	0.206041i	\(-0.0660579\pi\)
							−0.206041	+	0.978543i	\(0.566058\pi\)
\(41\)	0.869666	+	4.93212i	0.135819	+	0.770268i	0.974286	+	0.225315i	\(0.0723412\pi\)
							−0.838467	+	0.544952i	\(0.816548\pi\)
\(43\)	2.48739	+	0.217618i	0.379324	+	0.0331865i	0.275223	−	0.961380i	\(-0.411248\pi\)
							0.104100	+	0.994567i	\(0.466804\pi\)
\(47\)	−2.37530	−	6.52607i	−0.346472	−	0.951925i	−0.983472	−	0.181061i	\(-0.942047\pi\)
							0.637000	−	0.770864i	\(-0.280175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.19	yes	456
16.11	odd	4	inner	304.2.bg.a.91.28	✓	456
19.14	odd	18	inner	304.2.bg.a.147.28	yes	456
304.299	even	36	inner	304.2.bg.a.299.19	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.28	✓	456	16.11	odd	4	inner
304.2.bg.a.147.28	yes	456	19.14	odd	18	inner
304.2.bg.a.243.19	yes	456	1.1	even	1	trivial
304.2.bg.a.299.19	yes	456	304.299	even	36	inner