Embedded newform 304.2.v.a.197.10

• Label: 304.2.v.a.197.10
• Level: $304$
• Weight: $2$
• Character: 304.197
• 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			197.10
Character	\(\chi\)	\(=\)	304.197
Dual form			304.2.v.a.125.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10167 + 0.886746i) q^{2} +(1.26758 + 0.339646i) q^{3} +(0.427363 - 1.95381i) q^{4} +(0.194470 - 0.725771i) q^{5} +(-1.69763 + 0.749840i) q^{6} -4.39418i q^{7} +(1.26172 + 2.53142i) q^{8} +(-1.10669 - 0.638945i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10167	+	0.886746i	−0.779000	+	0.627024i
							\(3\)	1.26758	+	0.339646i	0.731836	+	0.196095i	0.605447	−	0.795886i	\(-0.292995\pi\)
0.126389	+	0.991981i	\(0.459661\pi\)
\(5\)	0.194470	−	0.725771i	0.0869695	−	0.324575i	−0.908710	−	0.417427i	\(-0.862932\pi\)
							0.995680	+	0.0928527i	\(0.0295986\pi\)
\(7\)		−	4.39418i		−	1.66084i	−0.557135	−	0.830422i	\(-0.688099\pi\)
							0.557135	−	0.830422i	\(-0.311901\pi\)
\(11\)	−0.263502	−	0.263502i	−0.0794490	−	0.0794490i	0.666266	−	0.745715i	\(-0.267892\pi\)
							−0.745715	+	0.666266i	\(0.767892\pi\)
\(13\)	−1.50268	−	5.60808i	−0.416769	−	1.55540i	−0.781266	−	0.624198i	\(-0.785426\pi\)
							0.364497	−	0.931204i	\(-0.381241\pi\)
\(17\)	−0.276435	−	0.478800i	−0.0670454	−	0.116126i	0.830554	−	0.556938i	\(-0.188024\pi\)
							−0.897599	+	0.440812i	\(0.854691\pi\)
\(19\)	−2.96036	+	3.19942i	−0.679152	+	0.733997i
							\(23\)	1.94905	+	1.12528i	0.406404	+	0.234637i	0.689243	−	0.724530i	\(-0.257943\pi\)
−0.282839	+	0.959167i	\(0.591276\pi\)
\(29\)	1.39036	+	5.18889i	0.258183	+	0.963552i	0.966292	+	0.257449i	\(0.0828819\pi\)
							−0.708109	+	0.706103i	\(0.750451\pi\)
\(31\)	3.74971			0.673468			0.336734	−	0.941600i	\(-0.390678\pi\)
							0.336734	+	0.941600i	\(0.390678\pi\)
\(37\)	7.35722	+	7.35722i	1.20952	+	1.20952i	0.971183	+	0.238337i	\(0.0766022\pi\)
							0.238337	+	0.971183i	\(0.423398\pi\)
\(41\)	9.93548	−	5.73625i	1.55166	−	0.895852i	0.553654	−	0.832747i	\(-0.313233\pi\)
							0.998007	−	0.0631045i	\(-0.0201002\pi\)
\(43\)	−1.12639	+	4.20374i	−0.171773	+	0.641064i	0.825306	+	0.564685i	\(0.191002\pi\)
							−0.997079	+	0.0763787i	\(0.975664\pi\)
\(47\)	−2.00093	+	3.46571i	−0.291865	+	0.505526i	−0.974251	−	0.225467i	\(-0.927609\pi\)
							0.682385	+	0.730993i	\(0.260943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.197.10	yes	152
16.13	even	4	inner	304.2.v.a.45.34	✓	152
19.11	even	3	inner	304.2.v.a.277.34	yes	152
304.125	even	12	inner	304.2.v.a.125.10	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.34	✓	152	16.13	even	4	inner
304.2.v.a.125.10	yes	152	304.125	even	12	inner
304.2.v.a.197.10	yes	152	1.1	even	1	trivial
304.2.v.a.277.34	yes	152	19.11	even	3	inner