Embedded newform 91.8.e.b.53.11

• Label: 91.8.e.b.53.11
• Level: $91$
• Weight: $8$
• Character: 91.53
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			53.11
Character	\(\chi\)	\(=\)	91.53
Dual form			91.8.e.b.79.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.21209 - 7.29555i) q^{2} +(-9.89272 + 17.1347i) q^{3} +(28.5167 - 49.3923i) q^{4} +(32.7323 + 56.6940i) q^{5} +166.676 q^{6} +(-264.708 - 868.028i) q^{7} -1558.75 q^{8} +(897.768 + 1554.98i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.21209	−	7.29555i	−0.372299	−	0.644841i	0.617620	−	0.786477i	\(-0.288097\pi\)
							−0.989919	+	0.141636i	\(0.954764\pi\)
\(3\)	−9.89272	+	17.1347i	−0.211540	+	0.366397i	−0.952197	−	0.305486i	\(-0.901181\pi\)
							0.740657	+	0.671883i	\(0.234514\pi\)
\(5\)	32.7323	+	56.6940i	0.117107	+	0.202835i	0.918620	−	0.395142i	\(-0.129305\pi\)
							−0.801513	+	0.597977i	\(0.795971\pi\)
\(7\)	−264.708	−	868.028i	−0.291692	−	0.956512i
							\(11\)	1402.23	−	2428.73i	0.317646	−	0.550179i	−0.662350	−	0.749194i	\(-0.730441\pi\)
0.979996	+	0.199015i	\(0.0637742\pi\)
\(13\)	−2197.00			−0.277350
							\(17\)	2606.66	−	4514.86i	0.128680	−	0.222881i	−0.794485	−	0.607284i	\(-0.792259\pi\)
0.923166	+	0.384402i	\(0.125593\pi\)
\(19\)	−10675.2	−	18490.0i	−0.357058	−	0.618442i	0.630410	−	0.776262i	\(-0.282887\pi\)
							−0.987468	+	0.157820i	\(0.949553\pi\)
\(23\)	455.030	+	788.135i	0.00779817	+	0.0135068i	0.869898	−	0.493231i	\(-0.164184\pi\)
							−0.862100	+	0.506738i	\(0.830851\pi\)
\(29\)	−173102.			−1.31798			−0.658991	−	0.752151i	\(-0.729016\pi\)
							−0.658991	+	0.752151i	\(0.729016\pi\)
\(31\)	−83841.0	+	145217.i	−0.505464	+	0.875490i	0.494516	+	0.869169i	\(0.335345\pi\)
							−0.999980	+	0.00632135i	\(0.997988\pi\)
\(37\)	157292.	+	272438.i	0.510506	+	0.884223i	0.999926	+	0.0121746i	\(0.00387538\pi\)
							−0.489419	+	0.872048i	\(0.662791\pi\)
\(41\)	−41264.4			−0.0935045			−0.0467522	−	0.998907i	\(-0.514887\pi\)
							−0.0467522	+	0.998907i	\(0.514887\pi\)
\(43\)	−988592.			−1.89617			−0.948085	−	0.318017i	\(-0.896983\pi\)
							−0.948085	+	0.318017i	\(0.896983\pi\)
\(47\)	579195.	+	1.00320e6i	0.813734	+	1.40943i	0.910233	+	0.414095i	\(0.135902\pi\)
							−0.0964995	+	0.995333i	\(0.530765\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.e.b.53.11	✓	58
7.2	even	3	inner	91.8.e.b.79.11	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.e.b.53.11	✓	58	1.1	even	1	trivial
91.8.e.b.79.11	yes	58	7.2	even	3	inner