Embedded newform 91.8.e.b.53.2

• Label: 91.8.e.b.53.2
• 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			53.2
Character	\(\chi\)	\(=\)	91.53
Dual form			91.8.e.b.79.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.9318 - 18.9344i) q^{2} +(-36.9978 + 64.0821i) q^{3} +(-175.008 + 303.123i) q^{4} +(-66.9050 - 115.883i) q^{5} +1617.81 q^{6} +(901.581 - 103.412i) q^{7} +4854.07 q^{8} +(-1644.18 - 2847.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.9318	−	18.9344i	−0.966243	−	1.67358i	−0.706238	−	0.707974i	\(-0.749609\pi\)
							−0.260005	−	0.965607i	\(-0.583724\pi\)
\(3\)	−36.9978	+	64.0821i	−0.791137	+	1.37029i	0.134126	+	0.990964i	\(0.457177\pi\)
							−0.925263	+	0.379326i	\(0.876156\pi\)
\(5\)	−66.9050	−	115.883i	−0.239366	−	0.414595i	0.721166	−	0.692762i	\(-0.243606\pi\)
							−0.960533	+	0.278167i	\(0.910273\pi\)
\(7\)	901.581	−	103.412i	0.993486	−	0.113954i
							\(11\)	−2423.63	+	4197.85i	−0.549024	+	0.950938i	0.449317	+	0.893372i	\(0.351667\pi\)
−0.998342	+	0.0575658i	\(0.981666\pi\)
\(13\)	−2197.00			−0.277350
							\(17\)	7854.86	−	13605.0i	0.387764	−	0.671627i	−0.604385	−	0.796693i	\(-0.706581\pi\)
0.992148	+	0.125066i	\(0.0399143\pi\)
\(19\)	−13436.7	−	23273.0i	−0.449421	−	0.778421i	0.548927	−	0.835870i	\(-0.315036\pi\)
							−0.998348	+	0.0574496i	\(0.981703\pi\)
\(23\)	12134.3	+	21017.3i	0.207954	+	0.360188i	0.951070	−	0.308975i	\(-0.0999861\pi\)
							−0.743116	+	0.669163i	\(0.766653\pi\)
\(29\)	132189.			1.00647			0.503237	−	0.864148i	\(-0.332142\pi\)
							0.503237	+	0.864148i	\(0.332142\pi\)
\(31\)	111299.	−	192775.i	0.671002	−	1.16221i	−0.306619	−	0.951832i	\(-0.599198\pi\)
							0.977620	−	0.210377i	\(-0.0674690\pi\)
\(37\)	241482.	+	418259.i	0.783752	+	1.35750i	0.929742	+	0.368212i	\(0.120030\pi\)
							−0.145989	+	0.989286i	\(0.546637\pi\)
\(41\)	95424.4			0.216230			0.108115	−	0.994138i	\(-0.465519\pi\)
							0.108115	+	0.994138i	\(0.465519\pi\)
\(43\)	−672675.			−1.29022			−0.645112	−	0.764088i	\(-0.723189\pi\)
							−0.645112	+	0.764088i	\(0.723189\pi\)
\(47\)	191643.	+	331936.i	0.269247	+	0.466349i	0.968668	−	0.248361i	\(-0.0798919\pi\)
							−0.699421	+	0.714710i	\(0.746559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.e.b.53.2	✓	58
7.2	even	3	inner	91.8.e.b.79.2	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.e.b.53.2	✓	58	1.1	even	1	trivial
91.8.e.b.79.2	yes	58	7.2	even	3	inner