Embedded newform 304.2.bi.a.61.6

• Label: 304.2.bi.a.61.6
• Level: $304$
• Weight: $2$
• Character: 304.61
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bi (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			61.6
Character	\(\chi\)	\(=\)	304.61
Dual form			304.2.bi.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29576 + 0.566568i) q^{2} +(0.898567 - 1.28329i) q^{3} +(1.35800 - 1.46828i) q^{4} +(0.294709 - 3.36854i) q^{5} +(-0.437260 + 2.17193i) q^{6} +(3.47194 + 2.00453i) q^{7} +(-0.927768 + 2.67194i) q^{8} +(0.186659 + 0.512841i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29576	+	0.566568i	−0.916242	+	0.400624i
							\(3\)	0.898567	−	1.28329i	0.518788	−	0.740906i	−0.471205	−	0.882024i	\(-0.656181\pi\)
0.989993	+	0.141118i	\(0.0450697\pi\)
\(5\)	0.294709	−	3.36854i	0.131798	−	1.50646i	−0.586060	−	0.810268i	\(-0.699322\pi\)
							0.717858	−	0.696190i	\(-0.245123\pi\)
\(7\)	3.47194	+	2.00453i	1.31227	+	0.757640i	0.982472	−	0.186412i	\(-0.0596858\pi\)
							0.329799	+	0.944051i	\(0.393019\pi\)
\(11\)	−1.04889	−	0.281049i	−0.316251	−	0.0847393i	0.0972017	−	0.995265i	\(-0.469011\pi\)
							−0.413453	+	0.910525i	\(0.635677\pi\)
\(13\)	1.54884	−	1.08451i	0.429572	−	0.300790i	−0.338722	−	0.940886i	\(-0.609995\pi\)
							0.768294	+	0.640097i	\(0.221106\pi\)
\(17\)	−2.40935	−	0.876933i	−0.584354	−	0.212687i	0.0328903	−	0.999459i	\(-0.489529\pi\)
							−0.617244	+	0.786771i	\(0.711751\pi\)
\(19\)	1.42265	−	4.12020i	0.326378	−	0.945239i
							\(23\)	−1.77884	+	2.11994i	−0.370914	+	0.442038i	−0.918925	−	0.394433i	\(-0.870941\pi\)
0.548011	+	0.836471i	\(0.315385\pi\)
\(29\)	−1.74292	−	0.812738i	−0.323653	−	0.150922i	0.254000	−	0.967204i	\(-0.418254\pi\)
							−0.577653	+	0.816282i	\(0.696031\pi\)
\(31\)	−1.98792	+	3.44318i	−0.357041	+	0.618413i	−0.987465	−	0.157838i	\(-0.949548\pi\)
							0.630424	+	0.776251i	\(0.282881\pi\)
\(37\)	−8.54634	+	8.54634i	−1.40501	+	1.40501i	−0.621961	+	0.783048i	\(0.713664\pi\)
							−0.783048	+	0.621961i	\(0.786336\pi\)
\(41\)	2.00456	−	0.353458i	0.313060	−	0.0552009i	−0.0149109	−	0.999889i	\(-0.504746\pi\)
							0.327970	+	0.944688i	\(0.393635\pi\)
\(43\)	0.981760	−	11.2216i	0.149717	−	1.71127i	−0.435328	−	0.900272i	\(-0.643368\pi\)
							0.585045	−	0.811001i	\(-0.301077\pi\)
\(47\)	9.49300	−	3.45517i	1.38470	−	0.503988i	0.461097	−	0.887350i	\(-0.347456\pi\)
							0.923598	+	0.383361i	\(0.125234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.61.6	yes	456
16.5	even	4	inner	304.2.bi.a.213.14	yes	456
19.5	even	9	inner	304.2.bi.a.157.14	yes	456
304.5	even	36	inner	304.2.bi.a.5.6	✓	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.6	✓	456	304.5	even	36	inner
304.2.bi.a.61.6	yes	456	1.1	even	1	trivial
304.2.bi.a.157.14	yes	456	19.5	even	9	inner
304.2.bi.a.213.14	yes	456	16.5	even	4	inner