Embedded newform 304.2.bi.a.157.36

• Label: 304.2.bi.a.157.36
• Level: $304$
• Weight: $2$
• Character: 304.157
• 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			157.36
Character	\(\chi\)	\(=\)	304.157
Dual form			304.2.bi.a.213.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36801 + 0.358547i) q^{2} +(-1.11291 + 0.779268i) q^{3} +(1.74289 + 0.980990i) q^{4} +(-3.17036 + 0.277371i) q^{5} +(-1.80187 + 0.667014i) q^{6} +(-0.306437 + 0.176922i) q^{7} +(2.03255 + 1.96691i) q^{8} +(-0.394751 + 1.08457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36801	+	0.358547i	0.967327	+	0.253531i
							\(3\)	−1.11291	+	0.779268i	−0.642539	+	0.449910i	−0.848891	−	0.528567i	\(-0.822729\pi\)
0.206353	+	0.978478i	\(0.433841\pi\)
\(5\)	−3.17036	+	0.277371i	−1.41783	+	0.124044i	−0.770147	−	0.637867i	\(-0.779817\pi\)
							−0.647682	+	0.761911i	\(0.724262\pi\)
\(7\)	−0.306437	+	0.176922i	−0.115822	+	0.0668701i	−0.556792	−	0.830652i	\(-0.687968\pi\)
							0.440970	+	0.897522i	\(0.354635\pi\)
\(11\)	1.06944	+	3.99120i	0.322448	+	1.20339i	0.916853	+	0.399226i	\(0.130721\pi\)
							−0.594405	+	0.804166i	\(0.702612\pi\)
\(13\)	−2.57703	+	3.68038i	−0.714740	+	1.02075i	0.283352	+	0.959016i	\(0.408553\pi\)
							−0.998092	+	0.0617387i	\(0.980335\pi\)
\(17\)	5.28840	−	1.92482i	1.28263	−	0.466838i	0.391326	−	0.920252i	\(-0.372016\pi\)
							0.891300	+	0.453414i	\(0.149794\pi\)
\(19\)	−0.299449	−	4.34860i	−0.0686983	−	0.997637i
							\(23\)	−3.70189	−	4.41174i	−0.771897	−	0.919911i	0.226641	−	0.973978i	\(-0.427226\pi\)
−0.998537	+	0.0540679i	\(0.982781\pi\)
\(29\)	3.03766	+	6.51428i	0.564079	+	1.20967i	0.956997	+	0.290098i	\(0.0936881\pi\)
							−0.392917	+	0.919574i	\(0.628534\pi\)
\(31\)	−1.77768	−	3.07904i	−0.319281	−	0.553011i	0.661057	−	0.750336i	\(-0.270108\pi\)
							−0.980338	+	0.197324i	\(0.936775\pi\)
\(37\)	4.96110	−	4.96110i	0.815600	−	0.815600i	−0.169867	−	0.985467i	\(-0.554334\pi\)
							0.985467	+	0.169867i	\(0.0543337\pi\)
\(41\)	8.60993	+	1.51816i	1.34465	+	0.237097i	0.799208	−	0.601055i	\(-0.205253\pi\)
							0.545437	+	0.838152i	\(0.316364\pi\)
\(43\)	7.15757	−	0.626206i	1.09152	−	0.0954956i	0.472847	−	0.881145i	\(-0.343226\pi\)
							0.618672	+	0.785649i	\(0.287671\pi\)
\(47\)	8.85921	+	3.22449i	1.29225	+	0.470340i	0.894464	−	0.447139i	\(-0.147557\pi\)
							0.397784	+	0.917479i	\(0.369779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.157.36	yes	456
16.5	even	4	inner	304.2.bi.a.5.29	✓	456
19.4	even	9	inner	304.2.bi.a.61.29	yes	456
304.213	even	36	inner	304.2.bi.a.213.36	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.29	✓	456	16.5	even	4	inner
304.2.bi.a.61.29	yes	456	19.4	even	9	inner
304.2.bi.a.157.36	yes	456	1.1	even	1	trivial
304.2.bi.a.213.36	yes	456	304.213	even	36	inner