Embedded newform 304.2.bi.a.149.24

• Label: 304.2.bi.a.149.24
• Level: $304$
• Weight: $2$
• Character: 304.149
• 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			149.24
Character	\(\chi\)	\(=\)	304.149
Dual form			304.2.bi.a.253.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.481244 + 1.32981i) q^{2} +(0.747948 - 1.60398i) q^{3} +(-1.53681 + 1.27993i) q^{4} +(-0.193859 + 0.135742i) q^{5} +(2.49294 + 0.222725i) q^{6} +(2.84654 - 1.64345i) q^{7} +(-2.44165 - 1.42771i) q^{8} +(-0.0849609 - 0.101252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.481244	+	1.32981i	0.340291	+	0.940320i
							\(3\)	0.747948	−	1.60398i	0.431828	−	0.926058i	−0.563187	−	0.826330i	\(-0.690425\pi\)
0.995015	−	0.0997283i	\(-0.0317974\pi\)
\(5\)	−0.193859	+	0.135742i	−0.0866964	+	0.0607055i	−0.616121	−	0.787652i	\(-0.711297\pi\)
							0.529425	+	0.848357i	\(0.322408\pi\)
\(7\)	2.84654	−	1.64345i	1.07589	−	0.621166i	0.146106	−	0.989269i	\(-0.453326\pi\)
							0.929785	+	0.368103i	\(0.119993\pi\)
\(11\)	3.79539	−	1.01697i	1.14435	−	0.306628i	0.363653	−	0.931535i	\(-0.381530\pi\)
							0.780700	+	0.624906i	\(0.214863\pi\)
\(13\)	−0.356053	+	0.166030i	−0.0987513	+	0.0460485i	−0.471366	−	0.881938i	\(-0.656239\pi\)
							0.372615	+	0.927986i	\(0.378461\pi\)
\(17\)	0.544325	+	0.456743i	0.132018	+	0.110776i	0.706406	−	0.707807i	\(-0.250315\pi\)
							−0.574388	+	0.818583i	\(0.694760\pi\)
\(19\)	−2.40925	+	3.63256i	−0.552719	+	0.833367i
							\(23\)	−1.06801	+	0.188319i	−0.222696	+	0.0392672i	−0.283882	−	0.958859i	\(-0.591623\pi\)
0.0611869	+	0.998126i	\(0.480511\pi\)
\(29\)	0.468727	−	5.35757i	0.0870404	−	0.994876i	−0.819590	−	0.572950i	\(-0.805799\pi\)
							0.906630	−	0.421926i	\(-0.138646\pi\)
\(31\)	0.498421	+	0.863291i	0.0895191	+	0.155052i	0.907308	−	0.420467i	\(-0.138134\pi\)
							−0.817789	+	0.575518i	\(0.804800\pi\)
\(37\)	−6.86952	−	6.86952i	−1.12934	−	1.12934i	−0.990284	−	0.139058i	\(-0.955592\pi\)
							−0.139058	−	0.990284i	\(-0.544408\pi\)
\(41\)	−0.594831	−	1.63428i	−0.0928970	−	0.255232i	0.884538	−	0.466468i	\(-0.154474\pi\)
							−0.977435	+	0.211235i	\(0.932251\pi\)
\(43\)	−7.42786	+	5.20105i	−1.13274	+	0.793152i	−0.980757	−	0.195231i	\(-0.937454\pi\)
							−0.151981	+	0.988383i	\(0.548565\pi\)
\(47\)	−5.79835	+	4.86539i	−0.845776	+	0.709690i	−0.958855	−	0.283896i	\(-0.908373\pi\)
							0.113079	+	0.993586i	\(0.463929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.149.24	yes	456
16.13	even	4	inner	304.2.bi.a.301.37	yes	456
19.6	even	9	inner	304.2.bi.a.101.37	✓	456
304.253	even	36	inner	304.2.bi.a.253.24	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.101.37	✓	456	19.6	even	9	inner
304.2.bi.a.149.24	yes	456	1.1	even	1	trivial
304.2.bi.a.253.24	yes	456	304.253	even	36	inner
304.2.bi.a.301.37	yes	456	16.13	even	4	inner