Embedded newform 91.10.u.a.30.20

• Label: 91.10.u.a.30.20
• Level: $91$
• Weight: $10$
• Character: 91.30
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.20
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.2527 + 12.8476i) q^{2} +176.414 q^{3} +(74.1218 - 128.383i) q^{4} +(-1048.20 - 605.176i) q^{5} +(-3925.69 + 2266.50i) q^{6} +(-1476.29 + 6178.53i) q^{7} -9346.80i q^{8} +11439.0 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.2527	+	12.8476i	−0.983440	+	0.567789i	−0.903307	−	0.428995i	\(-0.858868\pi\)
							−0.0801328	+	0.996784i	\(0.525534\pi\)
\(3\)	176.414			1.25744			0.628721	−	0.777631i	\(-0.283579\pi\)
							0.628721	+	0.777631i	\(0.283579\pi\)
\(5\)	−1048.20	−	605.176i	−0.750028	−	0.433029i	0.0756761	−	0.997132i	\(-0.475888\pi\)
							−0.825704	+	0.564104i	\(0.809222\pi\)
\(7\)	−1476.29	+	6178.53i	−0.232397	+	0.972621i
							\(11\)		−	27979.4i		−	0.576196i	−0.957601	−	0.288098i	\(-0.906977\pi\)
0.957601	−	0.288098i	\(-0.0930230\pi\)
\(13\)	47945.1	−	91136.0i	0.465585	−	0.885003i
							\(17\)	−73872.3	+	127951.i	−0.214517	+	0.371554i	−0.953123	−	0.302583i	\(-0.902151\pi\)
0.738606	+	0.674137i	\(0.235484\pi\)
\(19\)		−	194922.i		−	0.343139i	−0.985172	−	0.171569i	\(-0.945116\pi\)
							0.985172	−	0.171569i	\(-0.0548837\pi\)
\(23\)	1.22386e6	+	2.11980e6i	0.911923	+	1.57950i	0.811345	+	0.584568i	\(0.198736\pi\)
							0.100579	+	0.994929i	\(0.467931\pi\)
\(29\)	231538.	−	401035.i	0.0607898	−	0.105291i	−0.834029	−	0.551721i	\(-0.813971\pi\)
							0.894819	+	0.446430i	\(0.147305\pi\)
\(31\)	1.85106e6	−	1.06871e6i	0.359992	−	0.207841i	−0.309085	−	0.951034i	\(-0.600023\pi\)
							0.669077	+	0.743193i	\(0.266690\pi\)
\(37\)	9.39945e6	−	5.42677e6i	0.824507	−	0.476030i	−0.0274609	−	0.999623i	\(-0.508742\pi\)
							0.851968	+	0.523593i	\(0.175409\pi\)
\(41\)	1.33819e7	+	7.72607e6i	0.739592	+	0.427003i	0.821921	−	0.569602i	\(-0.192903\pi\)
							−0.0823292	+	0.996605i	\(0.526236\pi\)
\(43\)	4.65943e6	+	8.07037e6i	0.207838	+	0.359986i	0.951033	−	0.309089i	\(-0.100024\pi\)
							−0.743195	+	0.669075i	\(0.766691\pi\)
\(47\)	−1.76839e7	−	1.02098e7i	−0.528612	−	0.305194i	0.211839	−	0.977305i	\(-0.432055\pi\)
							−0.740451	+	0.672110i	\(0.765388\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.20	yes	164
7.4	even	3		91.10.k.a.4.20	✓	164
13.10	even	6		91.10.k.a.23.63	yes	164
91.88	even	6	inner	91.10.u.a.88.20	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.20	✓	164	7.4	even	3
91.10.k.a.23.63	yes	164	13.10	even	6
91.10.u.a.30.20	yes	164	1.1	even	1	trivial
91.10.u.a.88.20	yes	164	91.88	even	6	inner