Embedded newform 91.10.e.a.53.10

• Label: 91.10.e.a.53.10
• Level: $91$
• Weight: $10$
• Character: 91.53
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			53.10
Character	\(\chi\)	\(=\)	91.53
Dual form			91.10.e.a.79.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.68351 - 16.7723i) q^{2} +(-16.9452 + 29.3499i) q^{3} +(68.4592 - 118.575i) q^{4} +(-732.091 - 1268.02i) q^{5} +656.356 q^{6} +(-5194.70 - 3656.32i) q^{7} -12567.6 q^{8} +(9267.22 + 16051.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 65 q^{2} - 8533 q^{4} - 2118 q^{5} + 17358 q^{6} - 8764 q^{7} - 106980 q^{8} - 215597 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.68351	−	16.7723i	−0.427955	−	0.741239i	0.568737	−	0.822520i	\(-0.307432\pi\)
							−0.996691	+	0.0812804i	\(0.974099\pi\)
\(3\)	−16.9452	+	29.3499i	−0.120782	+	0.209200i	−0.920076	−	0.391740i	\(-0.871873\pi\)
							0.799294	+	0.600940i	\(0.205207\pi\)
\(5\)	−732.091	−	1268.02i	−0.523841	−	0.907320i	−0.999615	−	0.0277520i	\(-0.991165\pi\)
							0.475774	−	0.879568i	\(-0.342168\pi\)
\(7\)	−5194.70	−	3656.32i	−0.817748	−	0.575577i
							\(11\)	−5473.92	+	9481.11i	−0.112728	+	0.195250i	−0.916869	−	0.399188i	\(-0.869292\pi\)
0.804141	+	0.594438i	\(0.202626\pi\)
\(13\)	−28561.0			−0.277350
							\(17\)	−135047.	+	233907.i	−0.392160	+	0.679241i	−0.992734	−	0.120328i	\(-0.961605\pi\)
0.600574	+	0.799569i	\(0.294939\pi\)
\(19\)	125616.	+	217574.i	0.221133	+	0.383014i	0.955152	−	0.296115i	\(-0.0956911\pi\)
							−0.734019	+	0.679129i	\(0.762358\pi\)
\(23\)	−625159.	−	1.08281e6i	−0.465817	−	0.806818i	0.533421	−	0.845850i	\(-0.320906\pi\)
							−0.999238	+	0.0390317i	\(0.987573\pi\)
\(29\)	870623.			0.228581			0.114290	−	0.993447i	\(-0.463541\pi\)
							0.114290	+	0.993447i	\(0.463541\pi\)
\(31\)	−2.67451e6	+	4.63238e6i	−0.520135	+	0.900900i	0.479591	+	0.877492i	\(0.340785\pi\)
							−0.999726	+	0.0234079i	\(0.992548\pi\)
\(37\)	−2.92272e6	−	5.06231e6i	−0.256378	−	0.444059i	0.708891	−	0.705318i	\(-0.249196\pi\)
							−0.965269	+	0.261259i	\(0.915862\pi\)
\(41\)	−5.46139e6			−0.301839			−0.150920	−	0.988546i	\(-0.548223\pi\)
							−0.150920	+	0.988546i	\(0.548223\pi\)
\(43\)	1.64916e7			0.735623			0.367811	−	0.929900i	\(-0.380107\pi\)
							0.367811	+	0.929900i	\(0.380107\pi\)
\(47\)	5.67768e6	+	9.83403e6i	0.169719	+	0.293962i	0.938321	−	0.345765i	\(-0.112381\pi\)
							−0.768602	+	0.639727i	\(0.779047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.e.a.53.10	✓	70
7.2	even	3	inner	91.10.e.a.79.10	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.e.a.53.10	✓	70	1.1	even	1	trivial
91.10.e.a.79.10	yes	70	7.2	even	3	inner