Embedded newform 304.2.bg.a.243.33

• Label: 304.2.bg.a.243.33
• 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			243.33
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20866 + 0.734268i) q^{2} +(0.156460 - 0.223447i) q^{3} +(0.921700 + 1.77496i) q^{4} +(0.245178 - 2.80240i) q^{5} +(0.353176 - 0.155188i) q^{6} +(1.67190 - 2.89581i) q^{7} +(-0.189276 + 2.82209i) q^{8} +(1.00061 + 2.74916i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20866	+	0.734268i	0.854649	+	0.519206i
							\(3\)	0.156460	−	0.223447i	0.0903319	−	0.129007i	−0.771442	−	0.636299i	\(-0.780464\pi\)
0.861774	+	0.507292i	\(0.169353\pi\)
\(5\)	0.245178	−	2.80240i	0.109647	−	1.25327i	−0.719600	−	0.694388i	\(-0.755675\pi\)
							0.829247	−	0.558882i	\(-0.188769\pi\)
\(7\)	1.67190	−	2.89581i	0.631917	−	1.09451i	−0.355242	−	0.934774i	\(-0.615602\pi\)
							0.987159	−	0.159739i	\(-0.0510651\pi\)
\(11\)	−0.324344	+	1.21047i	−0.0977932	+	0.364969i	−0.997428	−	0.0716736i	\(-0.977166\pi\)
							0.899635	+	0.436643i	\(0.143833\pi\)
\(13\)	−2.59910	−	3.71190i	−0.720861	−	1.02950i	−0.997647	−	0.0685640i	\(-0.978158\pi\)
							0.276786	−	0.960932i	\(-0.410731\pi\)
\(17\)	5.19160	+	1.88959i	1.25915	+	0.458292i	0.883483	−	0.468464i	\(-0.155192\pi\)
							0.375664	+	0.926756i	\(0.377415\pi\)
\(19\)	−3.37106	+	2.76333i	−0.773373	+	0.633951i
							\(23\)	−2.66387	−	2.23525i	−0.555455	−	0.466082i	0.321328	−	0.946968i	\(-0.395871\pi\)
−0.876783	+	0.480886i	\(0.840315\pi\)
\(29\)	−2.11977	+	4.54587i	−0.393632	+	0.844146i	0.605212	+	0.796065i	\(0.293089\pi\)
							−0.998843	+	0.0480816i	\(0.984689\pi\)
\(31\)	−3.99525	+	6.91998i	−0.717568	+	1.24286i	0.244392	+	0.969676i	\(0.421411\pi\)
							−0.961961	+	0.273188i	\(0.911922\pi\)
\(37\)	4.00121	+	4.00121i	0.657795	+	0.657795i	0.954858	−	0.297063i	\(-0.0960071\pi\)
							−0.297063	+	0.954858i	\(0.596007\pi\)
\(41\)	−1.22248	−	6.93302i	−0.190919	−	1.08276i	−0.918112	−	0.396322i	\(-0.870286\pi\)
							0.727193	−	0.686434i	\(-0.240825\pi\)
\(43\)	−8.74013	−	0.764662i	−1.33286	−	0.116610i	−0.601647	−	0.798762i	\(-0.705489\pi\)
							−0.731210	+	0.682152i	\(0.761044\pi\)
\(47\)	−1.80876	−	4.96952i	−0.263834	−	0.724879i	−0.998900	−	0.0468833i	\(-0.985071\pi\)
							0.735066	−	0.677995i	\(-0.237151\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.33	yes	456
16.11	odd	4	inner	304.2.bg.a.91.4	✓	456
19.14	odd	18	inner	304.2.bg.a.147.4	yes	456
304.299	even	36	inner	304.2.bg.a.299.33	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.4	✓	456	16.11	odd	4	inner
304.2.bg.a.147.4	yes	456	19.14	odd	18	inner
304.2.bg.a.243.33	yes	456	1.1	even	1	trivial
304.2.bg.a.299.33	yes	456	304.299	even	36	inner