Embedded newform 304.2.bg.a.211.27

• Label: 304.2.bg.a.211.27
• 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			211.27
Character	\(\chi\)	\(=\)	304.211
Dual form			304.2.bg.a.219.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.690757 - 1.23404i) q^{2} +(0.0466704 + 0.533445i) q^{3} +(-1.04571 - 1.70484i) q^{4} +(-0.114521 + 0.245591i) q^{5} +(0.690530 + 0.310887i) q^{6} +(-2.09917 - 3.63588i) q^{7} +(-2.82618 + 0.112818i) q^{8} +(2.67204 - 0.471152i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.690757	−	1.23404i	0.488439	−	0.872598i
							\(3\)	0.0466704	+	0.533445i	0.0269452	+	0.307985i	0.997693	+	0.0678841i	\(0.0216248\pi\)
−0.970748	+	0.240100i	\(0.922820\pi\)
\(5\)	−0.114521	+	0.245591i	−0.0512154	+	0.109832i	−0.930250	−	0.366926i	\(-0.880410\pi\)
							0.879035	+	0.476758i	\(0.158188\pi\)
\(7\)	−2.09917	−	3.63588i	−0.793413	−	1.37423i	−0.923842	−	0.382774i	\(-0.874969\pi\)
							0.130429	−	0.991458i	\(-0.458365\pi\)
\(11\)	0.0583413	−	0.0156325i	0.0175906	−	0.00471338i	−0.250013	−	0.968242i	\(-0.580435\pi\)
							0.267604	+	0.963529i	\(0.413768\pi\)
\(13\)	0.511883	−	5.85085i	0.141971	−	1.62273i	−0.507018	−	0.861936i	\(-0.669252\pi\)
							0.648988	−	0.760798i	\(-0.275192\pi\)
\(17\)	−0.701778	+	3.97998i	−0.170206	+	0.965287i	0.773327	+	0.634008i	\(0.218591\pi\)
							−0.943533	+	0.331279i	\(0.892520\pi\)
\(19\)	0.386923	−	4.34169i	0.0887663	−	0.996052i
							\(23\)	1.06115	+	0.386227i	0.221265	+	0.0805339i	0.450274	−	0.892890i	\(-0.351326\pi\)
−0.229009	+	0.973424i	\(0.573548\pi\)
\(29\)	−0.382398	−	0.267758i	−0.0710095	−	0.0497214i	0.537533	−	0.843243i	\(-0.319356\pi\)
							−0.608542	+	0.793522i	\(0.708245\pi\)
\(31\)	1.73360	+	3.00268i	0.311364	+	0.539298i	0.978658	−	0.205496i	\(-0.0658809\pi\)
							−0.667294	+	0.744794i	\(0.732548\pi\)
\(37\)	1.67042	+	1.67042i	0.274616	+	0.274616i	0.830955	−	0.556339i	\(-0.187795\pi\)
							−0.556339	+	0.830955i	\(0.687795\pi\)
\(41\)	4.26249	+	3.57665i	0.665688	+	0.558579i	0.911786	−	0.410666i	\(-0.134704\pi\)
							−0.246097	+	0.969245i	\(0.579148\pi\)
\(43\)	−8.34613	−	3.89187i	−1.27277	−	0.593504i	−0.335481	−	0.942047i	\(-0.608899\pi\)
							−0.937293	+	0.348543i	\(0.886676\pi\)
\(47\)	4.28579	−	0.755701i	0.625147	−	0.110230i	0.147904	−	0.989002i	\(-0.452747\pi\)
							0.477243	+	0.878771i	\(0.341636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.211.27	yes	456
16.11	odd	4	inner	304.2.bg.a.59.22	✓	456
19.10	odd	18	inner	304.2.bg.a.67.22	yes	456
304.219	even	36	inner	304.2.bg.a.219.27	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.22	✓	456	16.11	odd	4	inner
304.2.bg.a.67.22	yes	456	19.10	odd	18	inner
304.2.bg.a.211.27	yes	456	1.1	even	1	trivial
304.2.bg.a.219.27	yes	456	304.219	even	36	inner