Embedded newform 304.2.v.a.125.34

• Label: 304.2.v.a.125.34
• 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			125.34
Character	\(\chi\)	\(=\)	304.125
Dual form			304.2.v.a.197.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32483 + 0.494789i) q^{2} +(0.514836 - 0.137950i) q^{3} +(1.51037 + 1.31103i) q^{4} +(0.425723 + 1.58882i) q^{5} +(0.750327 + 0.0719746i) q^{6} -0.191074i q^{7} +(1.35230 + 2.48421i) q^{8} +(-2.35205 + 1.35796i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32483	+	0.494789i	0.936799	+	0.349869i
							\(3\)	0.514836	−	0.137950i	0.297240	−	0.0796453i	−0.107117	−	0.994246i	\(-0.534162\pi\)
0.404357	+	0.914601i	\(0.367495\pi\)
\(5\)	0.425723	+	1.58882i	0.190389	+	0.710541i	0.993412	+	0.114594i	\(0.0365567\pi\)
							−0.803023	+	0.595947i	\(0.796777\pi\)
\(7\)		−	0.191074i		−	0.0722191i	−0.999348	−	0.0361096i	\(-0.988503\pi\)
							0.999348	−	0.0361096i	\(-0.0114965\pi\)
\(11\)	−2.27261	+	2.27261i	−0.685218	+	0.685218i	−0.961171	−	0.275953i	\(-0.911007\pi\)
							0.275953	+	0.961171i	\(0.411007\pi\)
\(13\)	1.44896	−	5.40759i	0.401869	−	1.49980i	−0.407889	−	0.913031i	\(-0.633735\pi\)
							0.809758	−	0.586764i	\(-0.199598\pi\)
\(17\)	3.39065	−	5.87278i	0.822353	−	1.42436i	−0.0815718	−	0.996667i	\(-0.525994\pi\)
							0.903925	−	0.427690i	\(-0.140673\pi\)
\(19\)	1.41338	−	4.12339i	0.324252	−	0.945971i
							\(23\)	−0.00874195	+	0.00504717i	−0.00182282	+	0.00105241i	−0.500911	−	0.865499i	\(-0.667002\pi\)
0.499088	+	0.866551i	\(0.333668\pi\)
\(29\)	−1.55646	+	5.80879i	−0.289027	+	1.07867i	0.656818	+	0.754049i	\(0.271902\pi\)
							−0.945846	+	0.324616i	\(0.894765\pi\)
\(31\)	−6.57597			−1.18108			−0.590539	−	0.807009i	\(-0.701085\pi\)
							−0.590539	+	0.807009i	\(0.701085\pi\)
\(37\)	0.615729	−	0.615729i	0.101225	−	0.101225i	−0.654681	−	0.755906i	\(-0.727197\pi\)
							0.755906	+	0.654681i	\(0.227197\pi\)
\(41\)	5.18775	+	2.99515i	0.810191	+	0.467764i	0.847022	−	0.531557i	\(-0.178393\pi\)
							−0.0368312	+	0.999322i	\(0.511726\pi\)
\(43\)	−1.12939	−	4.21495i	−0.172231	−	0.642774i	−0.997007	−	0.0773142i	\(-0.975366\pi\)
							0.824776	−	0.565459i	\(-0.191301\pi\)
\(47\)	−6.37448	−	11.0409i	−0.929814	−	1.61049i	−0.783630	−	0.621228i	\(-0.786634\pi\)
							−0.146184	−	0.989257i	\(-0.546699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.125.34	yes	152
16.5	even	4	inner	304.2.v.a.277.9	yes	152
19.7	even	3	inner	304.2.v.a.45.9	✓	152
304.197	even	12	inner	304.2.v.a.197.34	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.9	✓	152	19.7	even	3	inner
304.2.v.a.125.34	yes	152	1.1	even	1	trivial
304.2.v.a.197.34	yes	152	304.197	even	12	inner
304.2.v.a.277.9	yes	152	16.5	even	4	inner