Embedded newform 91.10.i.a.34.7

• Label: 91.10.i.a.34.7
• Level: $91$
• Weight: $10$
• Character: 91.34
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			34.7
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.0602 - 26.0602i) q^{2} -88.5050i q^{3} +846.263i q^{4} +(553.439 - 553.439i) q^{5} +(-2306.45 + 2306.45i) q^{6} +(-5157.40 + 3708.75i) q^{7} +(8710.95 - 8710.95i) q^{8} +11849.9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.0602	−	26.0602i	−1.15171	−	1.15171i	−0.986210	−	0.165497i	\(-0.947077\pi\)
							−0.165497	−	0.986210i	\(-0.552923\pi\)
\(3\)		−	88.5050i		−	0.630844i	−0.948951	−	0.315422i	\(-0.897854\pi\)
							0.948951	−	0.315422i	\(-0.102146\pi\)
\(5\)	553.439	−	553.439i	0.396009	−	0.396009i	−0.480814	−	0.876823i	\(-0.659659\pi\)
							0.876823	+	0.480814i	\(0.159659\pi\)
\(7\)	−5157.40	+	3708.75i	−0.811876	+	0.583830i
							\(11\)	−47557.0	+	47557.0i	−0.979372	+	0.979372i	−0.999792	−	0.0204192i	\(-0.993500\pi\)
0.0204192	+	0.999792i	\(0.493500\pi\)
\(13\)	−81418.1	+	63052.2i	−0.790635	+	0.612288i
							\(17\)	−623623.			−1.81093			−0.905466	−	0.424418i	\(-0.860479\pi\)
−0.905466	+	0.424418i	\(0.860479\pi\)
\(19\)	−297335.	+	297335.i	−0.523426	+	0.523426i	−0.918604	−	0.395179i	\(-0.870683\pi\)
							0.395179	+	0.918604i	\(0.370683\pi\)
\(23\)		−	884450.i		−	0.659019i	−0.944152	−	0.329510i	\(-0.893117\pi\)
							0.944152	−	0.329510i	\(-0.106883\pi\)
\(29\)	4.20755e6			1.10469			0.552343	−	0.833617i	\(-0.313734\pi\)
							0.552343	+	0.833617i	\(0.313734\pi\)
\(31\)	4.01831e6	−	4.01831e6i	0.781477	−	0.781477i	−0.198603	−	0.980080i	\(-0.563640\pi\)
							0.980080	+	0.198603i	\(0.0636404\pi\)
\(37\)	7.47233e6	−	7.47233e6i	0.655463	−	0.655463i	−0.298840	−	0.954303i	\(-0.596600\pi\)
							0.954303	+	0.298840i	\(0.0965997\pi\)
\(41\)	1.21725e7	−	1.21725e7i	0.672750	−	0.672750i	−0.285599	−	0.958349i	\(-0.592193\pi\)
							0.958349	+	0.285599i	\(0.0921926\pi\)
\(43\)			9.52735e6i			0.424976i	0.977164	+	0.212488i	\(0.0681566\pi\)
							−0.977164	+	0.212488i	\(0.931843\pi\)
\(47\)	−7.98733e6	−	7.98733e6i	−0.238760	−	0.238760i	0.577577	−	0.816336i	\(-0.303998\pi\)
							−0.816336	+	0.577577i	\(0.803998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.7	✓	164
7.6	odd	2	inner	91.10.i.a.34.8	yes	164
13.5	odd	4	inner	91.10.i.a.83.7	yes	164
91.83	even	4	inner	91.10.i.a.83.8	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.7	✓	164	1.1	even	1	trivial
91.10.i.a.34.8	yes	164	7.6	odd	2	inner
91.10.i.a.83.7	yes	164	13.5	odd	4	inner
91.10.i.a.83.8	yes	164	91.83	even	4	inner