Embedded newform 304.2.bg.a.3.19

• Label: 304.2.bg.a.3.19
• Level: $304$
• Weight: $2$
• Character: 304.3
• 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			3.19
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.256606 - 1.39074i) q^{2} +(-0.0196774 - 0.00917574i) q^{3} +(-1.86831 + 0.713745i) q^{4} +(-0.252165 - 0.360129i) q^{5} +(-0.00771170 + 0.0297207i) q^{6} +(-1.27538 - 2.20902i) q^{7} +(1.47205 + 2.41517i) q^{8} +(-1.92806 - 2.29777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.256606	−	1.39074i	−0.181448	−	0.983401i
							\(3\)	−0.0196774	−	0.00917574i	−0.0113608	−	0.00529762i	0.416930	−	0.908939i	\(-0.363106\pi\)
−0.428290	+	0.903641i	\(0.640884\pi\)
\(5\)	−0.252165	−	0.360129i	−0.112772	−	0.161055i	0.758798	−	0.651326i	\(-0.225787\pi\)
							−0.871570	+	0.490271i	\(0.836898\pi\)
\(7\)	−1.27538	−	2.20902i	−0.482048	−	0.834931i	0.517740	−	0.855538i	\(-0.326773\pi\)
							−0.999788	+	0.0206070i	\(0.993440\pi\)
\(11\)	1.46776	−	0.393285i	0.442546	−	0.118580i	−0.0306636	−	0.999530i	\(-0.509762\pi\)
							0.473209	+	0.880950i	\(0.343095\pi\)
\(13\)	−3.70051	+	1.72558i	−1.02634	+	0.478589i	−0.861514	−	0.507734i	\(-0.830483\pi\)
							−0.164823	+	0.986323i	\(0.552705\pi\)
\(17\)	−4.81603	−	4.04113i	−1.16806	−	0.980117i	−0.168074	−	0.985774i	\(-0.553755\pi\)
							−0.999984	+	0.00565735i	\(0.998199\pi\)
\(19\)	−4.33636	+	0.442671i	−0.994830	+	0.101556i
							\(23\)	1.27628	+	7.23817i	0.266124	+	1.50926i	0.765816	+	0.643059i	\(0.222335\pi\)
−0.499693	+	0.866203i	\(0.666554\pi\)
\(29\)	0.711123	−	8.12817i	0.132052	−	1.50936i	−0.584250	−	0.811574i	\(-0.698611\pi\)
							0.716302	−	0.697790i	\(-0.245833\pi\)
\(31\)	−0.318732	−	0.552059i	−0.0572459	−	0.0991528i	0.835982	−	0.548756i	\(-0.184899\pi\)
							−0.893228	+	0.449604i	\(0.851565\pi\)
\(37\)	5.67503	+	5.67503i	0.932969	+	0.932969i	0.997890	−	0.0649215i	\(-0.0206797\pi\)
							−0.0649215	+	0.997890i	\(0.520680\pi\)
\(41\)	−6.07718	+	2.21191i	−0.949097	+	0.345443i	−0.769752	−	0.638343i	\(-0.779620\pi\)
							−0.179345	+	0.983786i	\(0.557398\pi\)
\(43\)	9.33151	−	6.53399i	1.42304	−	0.996424i	0.427312	−	0.904104i	\(-0.359461\pi\)
							0.995729	−	0.0923201i	\(-0.0294283\pi\)
\(47\)	2.00973	+	2.39511i	0.293150	+	0.349362i	0.892437	−	0.451172i	\(-0.148994\pi\)
							−0.599287	+	0.800534i	\(0.704549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.19	✓	456
16.11	odd	4	inner	304.2.bg.a.155.38	yes	456
19.13	odd	18	inner	304.2.bg.a.51.38	yes	456
304.203	even	36	inner	304.2.bg.a.203.19	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.19	✓	456	1.1	even	1	trivial
304.2.bg.a.51.38	yes	456	19.13	odd	18	inner
304.2.bg.a.155.38	yes	456	16.11	odd	4	inner
304.2.bg.a.203.19	yes	456	304.203	even	36	inner