Embedded newform 304.2.bg.a.3.3

• Label: 304.2.bg.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36996 - 0.351025i) q^{2} +(-1.52863 - 0.712812i) q^{3} +(1.75356 + 0.961777i) q^{4} +(2.13072 + 3.04299i) q^{5} +(1.84394 + 1.51311i) q^{6} +(0.591379 + 1.02430i) q^{7} +(-2.06470 - 1.93314i) q^{8} +(-0.0997514 - 0.118879i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36996	−	0.351025i	−0.968706	−	0.248212i
							\(3\)	−1.52863	−	0.712812i	−0.882556	−	0.411542i	−0.0721377	−	0.997395i	\(-0.522982\pi\)
−0.810418	+	0.585852i	\(0.800760\pi\)
\(5\)	2.13072	+	3.04299i	0.952888	+	1.36087i	0.932538	+	0.361073i	\(0.117589\pi\)
							0.0203505	+	0.999793i	\(0.493522\pi\)
\(7\)	0.591379	+	1.02430i	0.223520	+	0.387148i	0.955874	−	0.293775i	\(-0.0949118\pi\)
							−0.732354	+	0.680924i	\(0.761578\pi\)
\(11\)	−6.12271	+	1.64057i	−1.84607	+	0.494652i	−0.999303	−	0.0373196i	\(-0.988118\pi\)
							−0.846762	+	0.531971i	\(0.821451\pi\)
\(13\)	0.768376	−	0.358300i	0.213109	−	0.0993744i	−0.313133	−	0.949709i	\(-0.601378\pi\)
							0.526242	+	0.850335i	\(0.323601\pi\)
\(17\)	0.0293238	+	0.0246056i	0.00711207	+	0.00596773i	0.646337	−	0.763052i	\(-0.276300\pi\)
							−0.639225	+	0.769020i	\(0.720745\pi\)
\(19\)	−3.35310	+	2.78509i	−0.769254	+	0.638943i
							\(23\)	0.511787	+	2.90249i	0.106715	+	0.605210i	0.990521	+	0.137358i	\(0.0438610\pi\)
−0.883807	+	0.467852i	\(0.845028\pi\)
\(29\)	−0.147220	+	1.68274i	−0.0273381	+	0.312476i	0.970198	+	0.242315i	\(0.0779069\pi\)
							−0.997536	+	0.0701610i	\(0.977649\pi\)
\(31\)	−2.04332	−	3.53914i	−0.366991	−	0.635648i	0.622102	−	0.782936i	\(-0.286279\pi\)
							−0.989094	+	0.147288i	\(0.952946\pi\)
\(37\)	4.10116	+	4.10116i	0.674226	+	0.674226i	0.958687	−	0.284462i	\(-0.0918148\pi\)
							−0.284462	+	0.958687i	\(0.591815\pi\)
\(41\)	−11.1501	+	4.05830i	−1.74135	+	0.633800i	−0.999331	−	0.0365757i	\(-0.988355\pi\)
							−0.742021	+	0.670376i	\(0.766133\pi\)
\(43\)	5.24994	−	3.67605i	0.800608	−	0.560592i	−0.100120	−	0.994975i	\(-0.531923\pi\)
							0.900728	+	0.434384i	\(0.143034\pi\)
\(47\)	2.81128	+	3.35036i	0.410068	+	0.488700i	0.931062	−	0.364860i	\(-0.118883\pi\)
							−0.520995	+	0.853560i	\(0.674439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.3	✓	456
16.11	odd	4	inner	304.2.bg.a.155.25	yes	456
19.13	odd	18	inner	304.2.bg.a.51.25	yes	456
304.203	even	36	inner	304.2.bg.a.203.3	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.3	✓	456	1.1	even	1	trivial
304.2.bg.a.51.25	yes	456	19.13	odd	18	inner
304.2.bg.a.155.25	yes	456	16.11	odd	4	inner
304.2.bg.a.203.3	yes	456	304.203	even	36	inner