Embedded newform 304.2.v.a.125.17

• Label: 304.2.v.a.125.17
• Level: $304$
• Weight: $2$
• Character: 304.125
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(38\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			125.17
Character	\(\chi\)	\(=\)	304.125
Dual form			304.2.v.a.197.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.249474 - 1.39204i) q^{2} +(2.44318 - 0.654649i) q^{3} +(-1.87553 + 0.694553i) q^{4} +(0.771051 + 2.87760i) q^{5} +(-1.52081 - 3.23768i) q^{6} +1.57323i q^{7} +(1.43474 + 2.43753i) q^{8} +(2.94251 - 1.69886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.249474	−	1.39204i	−0.176405	−	0.984318i
							\(3\)	2.44318	−	0.654649i	1.41057	−	0.377962i	0.528443	−	0.848969i	\(-0.322776\pi\)
0.882130	+	0.471007i	\(0.156109\pi\)
\(5\)	0.771051	+	2.87760i	0.344824	+	1.28690i	0.892817	+	0.450419i	\(0.148725\pi\)
							−0.547993	+	0.836483i	\(0.684608\pi\)
\(7\)			1.57323i			0.594623i	0.954780	+	0.297312i	\(0.0960901\pi\)
							−0.954780	+	0.297312i	\(0.903910\pi\)
\(11\)	1.70822	−	1.70822i	0.515047	−	0.515047i	−0.401021	−	0.916069i	\(-0.631345\pi\)
							0.916069	+	0.401021i	\(0.131345\pi\)
\(13\)	0.987480	−	3.68533i	0.273878	−	1.02213i	−0.682712	−	0.730687i	\(-0.739200\pi\)
							0.956590	−	0.291438i	\(-0.0941337\pi\)
\(17\)	2.56530	−	4.44324i	0.622178	−	1.07764i	−0.366902	−	0.930260i	\(-0.619581\pi\)
							0.989079	−	0.147384i	\(-0.0470852\pi\)
\(19\)	−3.77996	+	2.17068i	−0.867183	+	0.497989i
							\(23\)	−3.41703	+	1.97282i	−0.712499	+	0.411362i	−0.811986	−	0.583677i	\(-0.801613\pi\)
0.0994865	+	0.995039i	\(0.468280\pi\)
\(29\)	−1.31938	+	4.92401i	−0.245004	+	0.914366i	0.728378	+	0.685176i	\(0.240275\pi\)
							−0.973382	+	0.229190i	\(0.926392\pi\)
\(31\)	0.363399			0.0652685			0.0326342	−	0.999467i	\(-0.489610\pi\)
							0.0326342	+	0.999467i	\(0.489610\pi\)
\(37\)	−3.71287	+	3.71287i	−0.610392	+	0.610392i	−0.943048	−	0.332656i	\(-0.892055\pi\)
							0.332656	+	0.943048i	\(0.392055\pi\)
\(41\)	−8.01808	−	4.62924i	−1.25221	−	0.722966i	−0.280665	−	0.959806i	\(-0.590555\pi\)
							−0.971549	+	0.236840i	\(0.923888\pi\)
\(43\)	−0.222419	−	0.830078i	−0.0339185	−	0.126586i	0.946891	−	0.321556i	\(-0.104206\pi\)
							−0.980809	+	0.194970i	\(0.937539\pi\)
\(47\)	−1.30951	−	2.26814i	−0.191012	−	0.330842i	0.754574	−	0.656215i	\(-0.227844\pi\)
							−0.945586	+	0.325373i	\(0.894510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.125.17	yes	152
16.5	even	4	inner	304.2.v.a.277.35	yes	152
19.7	even	3	inner	304.2.v.a.45.35	✓	152
304.197	even	12	inner	304.2.v.a.197.17	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.35	✓	152	19.7	even	3	inner
304.2.v.a.125.17	yes	152	1.1	even	1	trivial
304.2.v.a.197.17	yes	152	304.197	even	12	inner
304.2.v.a.277.35	yes	152	16.5	even	4	inner