Embedded newform 91.8.f.a.22.17

• Label: 91.8.f.a.22.17
• Level: $91$
• Weight: $8$
• Character: 91.22
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.17
Character	\(\chi\)	\(=\)	91.22
Dual form			91.8.f.a.29.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.98688 - 8.63753i) q^{2} +(13.2496 - 22.9490i) q^{3} +(14.2620 + 24.7026i) q^{4} +86.0031 q^{5} +(-132.148 - 228.888i) q^{6} +(171.500 + 297.047i) q^{7} +1561.13 q^{8} +(742.397 + 1285.87i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{2} + 41 q^{3} - 1472 q^{4} + 526 q^{5} - 519 q^{6} + 8232 q^{7} + 474 q^{8} - 15427 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.98688	−	8.63753i	0.440782	−	0.763457i	−0.556966	−	0.830536i	\(-0.688034\pi\)
							0.997748	+	0.0670785i	\(0.0213678\pi\)
\(3\)	13.2496	−	22.9490i	0.283321	−	0.490726i	−0.688880	−	0.724875i	\(-0.741897\pi\)
							0.972201	+	0.234150i	\(0.0752306\pi\)
\(5\)	86.0031			0.307694			0.153847	−	0.988095i	\(-0.450834\pi\)
							0.153847	+	0.988095i	\(0.450834\pi\)
\(7\)	171.500	+	297.047i	0.188982	+	0.327327i
							\(11\)	−2607.28	+	4515.94i	−0.590627	+	1.02300i	0.403521	+	0.914970i	\(0.367786\pi\)
−0.994148	+	0.108026i	\(0.965547\pi\)
\(13\)	−7315.89	+	3037.48i	−0.923560	+	0.383453i
							\(17\)	5823.11	+	10085.9i	0.287464	+	0.497903i	0.973204	−	0.229944i	\(-0.0738544\pi\)
−0.685740	+	0.727847i	\(0.740521\pi\)
\(19\)	22620.4	+	39179.7i	0.756594	+	1.31046i	0.944578	+	0.328288i	\(0.106472\pi\)
							−0.187983	+	0.982172i	\(0.560195\pi\)
\(23\)	−11290.6	+	19556.0i	−0.193496	+	0.335144i	−0.946406	−	0.322979i	\(-0.895316\pi\)
							0.752911	+	0.658123i	\(0.228649\pi\)
\(29\)	34779.6	−	60240.1i	0.264808	−	0.458662i	−0.702705	−	0.711481i	\(-0.748025\pi\)
							0.967513	+	0.252820i	\(0.0813579\pi\)
\(31\)	−58795.0			−0.354466			−0.177233	−	0.984169i	\(-0.556715\pi\)
							−0.177233	+	0.984169i	\(0.556715\pi\)
\(37\)	167794.	−	290628.i	0.544591	−	0.943258i	−0.454042	−	0.890980i	\(-0.650018\pi\)
							0.998633	−	0.0522782i	\(-0.0166483\pi\)
\(41\)	260084.	−	450478.i	0.589345	−	1.02078i	−0.404974	−	0.914328i	\(-0.632719\pi\)
							0.994318	−	0.106447i	\(-0.0339474\pi\)
\(43\)	−232118.	−	402041.i	−0.445215	−	0.771136i	0.552852	−	0.833280i	\(-0.313540\pi\)
							−0.998067	+	0.0621439i	\(0.980206\pi\)
\(47\)	922147.			1.29556			0.647780	−	0.761827i	\(-0.275697\pi\)
							0.647780	+	0.761827i	\(0.275697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.f.a.22.17	✓	48
13.3	even	3	inner	91.8.f.a.29.17	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.f.a.22.17	✓	48	1.1	even	1	trivial
91.8.f.a.29.17	yes	48	13.3	even	3	inner