Embedded newform 304.2.bi.a.157.14

• Label: 304.2.bi.a.157.14
• Level: $304$
• Weight: $2$
• Character: 304.157
• 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.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			157.14
Character	\(\chi\)	\(=\)	304.157
Dual form			304.2.bi.a.213.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.628429 + 1.26692i) q^{2} +(-1.28329 + 0.898567i) q^{3} +(-1.21016 - 1.59233i) q^{4} +(3.36854 - 0.294709i) q^{5} +(-0.331955 - 2.19050i) q^{6} +(-3.47194 + 2.00453i) q^{7} +(2.77785 - 0.532498i) q^{8} +(-0.186659 + 0.512841i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.628429	+	1.26692i	−0.444366	+	0.895845i
							\(3\)	−1.28329	+	0.898567i	−0.740906	+	0.518788i	−0.882024	−	0.471205i	\(-0.843819\pi\)
0.141118	+	0.989993i	\(0.454930\pi\)
\(5\)	3.36854	−	0.294709i	1.50646	−	0.131798i	0.696190	−	0.717858i	\(-0.254877\pi\)
							0.810268	+	0.586060i	\(0.199322\pi\)
\(7\)	−3.47194	+	2.00453i	−1.31227	+	0.757640i	−0.982472	−	0.186412i	\(-0.940314\pi\)
							−0.329799	+	0.944051i	\(0.606981\pi\)
\(11\)	0.281049	+	1.04889i	0.0847393	+	0.316251i	0.995265	−	0.0972017i	\(-0.0309892\pi\)
							−0.910525	+	0.413453i	\(0.864323\pi\)
\(13\)	−1.08451	+	1.54884i	−0.300790	+	0.429572i	−0.940886	−	0.338722i	\(-0.890005\pi\)
							0.640097	+	0.768294i	\(0.278894\pi\)
\(17\)	−2.40935	+	0.876933i	−0.584354	+	0.212687i	−0.617244	−	0.786771i	\(-0.711751\pi\)
							0.0328903	+	0.999459i	\(0.489529\pi\)
\(19\)	−4.12020	+	1.42265i	−0.945239	+	0.326378i
							\(23\)	1.77884	+	2.11994i	0.370914	+	0.442038i	0.918925	−	0.394433i	\(-0.129059\pi\)
−0.548011	+	0.836471i	\(0.684615\pi\)
\(29\)	−0.812738	−	1.74292i	−0.150922	−	0.323653i	0.816282	−	0.577653i	\(-0.196031\pi\)
							−0.967204	+	0.254000i	\(0.918254\pi\)
\(31\)	−1.98792	−	3.44318i	−0.357041	−	0.618413i	0.630424	−	0.776251i	\(-0.282881\pi\)
							−0.987465	+	0.157838i	\(0.949548\pi\)
\(37\)	−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	\(-0.495254\pi\)
							−0.327970	+	0.944688i	\(0.606365\pi\)
\(43\)	11.2216	−	0.981760i	1.71127	−	0.149717i	0.811001	−	0.585045i	\(-0.198923\pi\)
							0.900272	+	0.435328i	\(0.143368\pi\)
\(47\)	9.49300	+	3.45517i	1.38470	+	0.503988i	0.923598	−	0.383361i	\(-0.125234\pi\)
							0.461097	+	0.887350i	\(0.347456\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.6	✓	456	16.5	even	4	inner
304.2.bi.a.61.6	yes	456	19.4	even	9	inner
304.2.bi.a.157.14	yes	456	1.1	even	1	trivial
304.2.bi.a.213.14	yes	456	304.213	even	36	inner