Embedded newform 304.2.bg.a.147.12

• Label: 304.2.bg.a.147.12
• Level: $304$
• Weight: $2$
• Character: 304.147
• 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			147.12
Character	\(\chi\)	\(=\)	304.147
Dual form			304.2.bg.a.91.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.733488 + 1.20913i) q^{2} +(-2.46787 + 1.72802i) q^{3} +(-0.923990 - 1.77377i) q^{4} +(3.64398 - 0.318807i) q^{5} +(-0.279248 - 4.25146i) q^{6} +(1.89078 + 3.27492i) q^{7} +(2.82245 + 0.183813i) q^{8} +(2.07826 - 5.70997i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.733488	+	1.20913i	−0.518655	+	0.854984i
							\(3\)	−2.46787	+	1.72802i	−1.42482	+	0.997673i	−0.429300	+	0.903162i	\(0.641240\pi\)
−0.995524	+	0.0945109i	\(0.969871\pi\)
\(5\)	3.64398	−	0.318807i	1.62964	−	0.142575i	0.764676	−	0.644415i	\(-0.222899\pi\)
							0.864961	+	0.501840i	\(0.167343\pi\)
\(7\)	1.89078	+	3.27492i	0.714646	+	1.23780i	0.963096	+	0.269158i	\(0.0867455\pi\)
							−0.248450	+	0.968645i	\(0.579921\pi\)
\(11\)	3.77001	−	1.01017i	1.13670	−	0.304578i	0.359077	−	0.933308i	\(-0.383091\pi\)
							0.777624	+	0.628730i	\(0.216425\pi\)
\(13\)	2.26715	+	1.58747i	0.628794	+	0.440286i	0.844045	−	0.536273i	\(-0.180168\pi\)
							−0.215251	+	0.976559i	\(0.569057\pi\)
\(17\)	−1.60383	+	0.583747i	−0.388986	+	0.141579i	−0.529107	−	0.848555i	\(-0.677473\pi\)
							0.140121	+	0.990134i	\(0.455251\pi\)
\(19\)	−2.00439	−	3.87071i	−0.459838	−	0.888003i
							\(23\)	−0.956517	+	0.802613i	−0.199448	+	0.167356i	−0.737041	−	0.675848i	\(-0.763778\pi\)
0.537594	+	0.843204i	\(0.319333\pi\)
\(29\)	−1.49529	+	0.697266i	−0.277669	+	0.129479i	−0.556465	−	0.830871i	\(-0.687843\pi\)
							0.278796	+	0.960350i	\(0.410065\pi\)
\(31\)	0.632105	+	1.09484i	0.113529	+	0.196639i	0.917191	−	0.398448i	\(-0.130451\pi\)
							−0.803662	+	0.595087i	\(0.797118\pi\)
\(37\)	−1.77674	−	1.77674i	−0.292095	−	0.292095i	0.545812	−	0.837907i	\(-0.316221\pi\)
							−0.837907	+	0.545812i	\(0.816221\pi\)
\(41\)	−1.13097	+	6.41407i	−0.176628	+	1.00171i	0.759619	+	0.650368i	\(0.225385\pi\)
							−0.936247	+	0.351341i	\(0.885726\pi\)
\(43\)	−0.245521	−	2.80632i	−0.0374416	−	0.427959i	−0.991663	−	0.128858i	\(-0.958869\pi\)
							0.954221	−	0.299101i	\(-0.0966868\pi\)
\(47\)	−3.53614	+	9.71546i	−0.515799	+	1.41715i	0.359309	+	0.933219i	\(0.383012\pi\)
							−0.875108	+	0.483928i	\(0.839210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.147.12	yes	456
16.11	odd	4	inner	304.2.bg.a.299.35	yes	456
19.15	odd	18	inner	304.2.bg.a.243.35	yes	456
304.91	even	36	inner	304.2.bg.a.91.12	✓	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.12	✓	456	304.91	even	36	inner
304.2.bg.a.147.12	yes	456	1.1	even	1	trivial
304.2.bg.a.243.35	yes	456	19.15	odd	18	inner
304.2.bg.a.299.35	yes	456	16.11	odd	4	inner