Embedded newform 91.8.f.a.22.12

• Label: 91.8.f.a.22.12
• Level: $91$
• Weight: $8$
• Character: 91.22
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.12
Character	\(\chi\)	\(=\)	91.22
Dual form			91.8.f.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.463694 + 0.803142i) q^{2} +(-3.61457 + 6.26061i) q^{3} +(63.5700 + 110.106i) q^{4} -525.610 q^{5} +(-3.35211 - 5.80602i) q^{6} +(171.500 + 297.047i) q^{7} -236.614 q^{8} +(1067.37 + 1848.74i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{2} + 41 q^{3} - 1472 q^{4} + 526 q^{5} - 519 q^{6} + 8232 q^{7} + 474 q^{8} - 15427 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.463694	+	0.803142i	−0.0409852	+	0.0709884i	−0.885790	−	0.464086i	\(-0.846383\pi\)
							0.844805	+	0.535074i	\(0.179716\pi\)
\(3\)	−3.61457	+	6.26061i	−0.0772915	+	0.133873i	−0.902080	−	0.431568i	\(-0.857961\pi\)
							0.824789	+	0.565441i	\(0.191294\pi\)
\(5\)	−525.610			−1.88048			−0.940241	−	0.340511i	\(-0.889400\pi\)
							−0.940241	+	0.340511i	\(0.889400\pi\)
\(7\)	171.500	+	297.047i	0.188982	+	0.327327i
							\(11\)	−3055.38	+	5292.07i	−0.692135	+	1.19881i	0.279002	+	0.960290i	\(0.409996\pi\)
−0.971137	+	0.238522i	\(0.923337\pi\)
\(13\)	−35.4559	−	7921.32i	−0.00447596	−	0.999990i
							\(17\)	6122.12	+	10603.8i	0.302225	+	0.523469i	0.976640	−	0.214884i	\(-0.0689373\pi\)
−0.674415	+	0.738353i	\(0.735604\pi\)
\(19\)	−20346.5	−	35241.3i	−0.680539	−	1.17873i	−0.974817	−	0.223008i	\(-0.928412\pi\)
							0.294277	−	0.955720i	\(-0.404921\pi\)
\(23\)	6796.98	−	11772.7i	0.116485	−	0.201757i	−0.801888	−	0.597475i	\(-0.796171\pi\)
							0.918372	+	0.395718i	\(0.129504\pi\)
\(29\)	−26791.0	+	46403.4i	−0.203984	+	0.353311i	−0.949809	−	0.312832i	\(-0.898722\pi\)
							0.745825	+	0.666142i	\(0.232056\pi\)
\(31\)	−84622.5			−0.510176			−0.255088	−	0.966918i	\(-0.582104\pi\)
							−0.255088	+	0.966918i	\(0.582104\pi\)
\(37\)	138610.	−	240080.i	0.449872	−	0.779201i	−0.548506	−	0.836147i	\(-0.684803\pi\)
							0.998377	+	0.0569463i	\(0.0181364\pi\)
\(41\)	392569.	−	679949.i	0.889554	−	1.54075i	0.0491510	−	0.998791i	\(-0.484348\pi\)
							0.840403	−	0.541962i	\(-0.182318\pi\)
\(43\)	−227062.	−	393282.i	−0.435516	−	0.754336i	0.561822	−	0.827258i	\(-0.310101\pi\)
							−0.997338	+	0.0729225i	\(0.976767\pi\)
\(47\)	−567883.			−0.797841			−0.398920	−	0.916986i	\(-0.630615\pi\)
							−0.398920	+	0.916986i	\(0.630615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.f.a.22.12	✓	48
13.3	even	3	inner	91.8.f.a.29.12	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.f.a.22.12	✓	48	1.1	even	1	trivial
91.8.f.a.29.12	yes	48	13.3	even	3	inner