Embedded newform 304.2.bg.a.211.21

• Label: 304.2.bg.a.211.21
• 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			211.21
Character	\(\chi\)	\(=\)	304.211
Dual form			304.2.bg.a.219.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.143757 + 1.40689i) q^{2} +(-0.287050 - 3.28100i) q^{3} +(-1.95867 + 0.404499i) q^{4} +(-1.61983 + 3.47374i) q^{5} +(4.57474 - 0.875514i) q^{6} +(0.0566174 + 0.0980642i) q^{7} +(-0.850657 - 2.69748i) q^{8} +(-7.72815 + 1.36268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.143757	+	1.40689i	0.101651	+	0.994820i
							\(3\)	−0.287050	−	3.28100i	−0.165729	−	1.89429i	−0.391608	−	0.920132i	\(-0.628081\pi\)
0.225880	−	0.974155i	\(-0.427474\pi\)
\(5\)	−1.61983	+	3.47374i	−0.724410	+	1.55350i	0.104755	+	0.994498i	\(0.466594\pi\)
							−0.829165	+	0.559004i	\(0.811184\pi\)
\(7\)	0.0566174	+	0.0980642i	0.0213994	+	0.0370648i	0.876527	−	0.481353i	\(-0.159855\pi\)
							−0.855127	+	0.518418i	\(0.826521\pi\)
\(11\)	−1.61353	+	0.432345i	−0.486498	+	0.130357i	−0.493727	−	0.869617i	\(-0.664366\pi\)
							0.00722850	+	0.999974i	\(0.497699\pi\)
\(13\)	−0.319010	+	3.64630i	−0.0884774	+	1.01130i	0.814086	+	0.580744i	\(0.197238\pi\)
							−0.902563	+	0.430557i	\(0.858317\pi\)
\(17\)	−0.522266	+	2.96192i	−0.126668	+	0.718370i	0.853635	+	0.520872i	\(0.174393\pi\)
							−0.980303	+	0.197499i	\(0.936718\pi\)
\(19\)	−3.21725	+	2.94097i	−0.738088	+	0.674705i
							\(23\)	−0.261848	−	0.0953047i	−0.0545990	−	0.0198724i	0.314576	−	0.949232i	\(-0.398138\pi\)
−0.369175	+	0.929360i	\(0.620360\pi\)
\(29\)	0.312234	+	0.218629i	0.0579805	+	0.0405984i	0.602207	−	0.798340i	\(-0.294288\pi\)
							−0.544227	+	0.838938i	\(0.683177\pi\)
\(31\)	−3.83547	−	6.64323i	−0.688871	−	1.19316i	−0.972204	−	0.234137i	\(-0.924774\pi\)
							0.283333	−	0.959022i	\(-0.408560\pi\)
\(37\)	−3.15790	−	3.15790i	−0.519155	−	0.519155i	0.398161	−	0.917316i	\(-0.369649\pi\)
							−0.917316	+	0.398161i	\(0.869649\pi\)
\(41\)	3.11603	+	2.61466i	0.486643	+	0.408342i	0.852821	−	0.522203i	\(-0.174890\pi\)
							−0.366179	+	0.930545i	\(0.619334\pi\)
\(43\)	−7.30327	−	3.40557i	−1.11374	−	0.519345i	−0.223438	−	0.974718i	\(-0.571728\pi\)
							−0.890301	+	0.455373i	\(0.849506\pi\)
\(47\)	5.19223	−	0.915529i	0.757364	−	0.133544i	0.218385	−	0.975863i	\(-0.429921\pi\)
							0.538979	+	0.842319i	\(0.318810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.211.21	yes	456
16.11	odd	4	inner	304.2.bg.a.59.25	✓	456
19.10	odd	18	inner	304.2.bg.a.67.25	yes	456
304.219	even	36	inner	304.2.bg.a.219.21	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.25	✓	456	16.11	odd	4	inner
304.2.bg.a.67.25	yes	456	19.10	odd	18	inner
304.2.bg.a.211.21	yes	456	1.1	even	1	trivial
304.2.bg.a.219.21	yes	456	304.219	even	36	inner