Embedded newform 91.8.e.b.53.14

• Label: 91.8.e.b.53.14
• 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			53.14
Character	\(\chi\)	\(=\)	91.53
Dual form			91.8.e.b.79.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65473 - 2.86607i) q^{2} +(-41.7818 + 72.3681i) q^{3} +(58.5238 - 101.366i) q^{4} +(-271.168 - 469.677i) q^{5} +276.550 q^{6} +(-86.3153 - 903.378i) q^{7} -810.974 q^{8} +(-2397.93 - 4153.34i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65473	−	2.86607i	−0.146259	−	0.253327i	0.783583	−	0.621287i	\(-0.213390\pi\)
							−0.929842	+	0.367959i	\(0.880057\pi\)
\(3\)	−41.7818	+	72.3681i	−0.893434	+	1.54747i	−0.0577028	+	0.998334i	\(0.518378\pi\)
							−0.835731	+	0.549139i	\(0.814956\pi\)
\(5\)	−271.168	−	469.677i	−0.970161	−	1.68037i	−0.695058	−	0.718954i	\(-0.744621\pi\)
							−0.275103	−	0.961415i	\(-0.588712\pi\)
\(7\)	−86.3153	−	903.378i	−0.0951141	−	0.995466i
							\(11\)	−1034.70	+	1792.15i	−0.234390	+	0.405976i	−0.959095	−	0.283084i	\(-0.908643\pi\)
0.724705	+	0.689059i	\(0.241976\pi\)
\(13\)	−2197.00			−0.277350
							\(17\)	14289.8	−	24750.7i	0.705433	−	1.22185i	−0.261103	−	0.965311i	\(-0.584086\pi\)
0.966535	−	0.256534i	\(-0.0825806\pi\)
\(19\)	−170.166	−	294.736i	−0.00569160	−	0.00985814i	0.863166	−	0.504921i	\(-0.168478\pi\)
							−0.868857	+	0.495063i	\(0.835145\pi\)
\(23\)	9756.62	+	16899.0i	0.167206	+	0.289609i	0.937436	−	0.348156i	\(-0.113192\pi\)
							−0.770230	+	0.637766i	\(0.779859\pi\)
\(29\)	−92697.9			−0.705792			−0.352896	−	0.935663i	\(-0.614803\pi\)
							−0.352896	+	0.935663i	\(0.614803\pi\)
\(31\)	−22990.6	+	39820.9i	−0.138607	+	0.240074i	−0.926970	−	0.375137i	\(-0.877596\pi\)
							0.788363	+	0.615211i	\(0.210929\pi\)
\(37\)	−244761.	−	423938.i	−0.794394	−	1.37593i	−0.923223	−	0.384264i	\(-0.874455\pi\)
							0.128829	−	0.991667i	\(-0.458878\pi\)
\(41\)	616168.			1.39623			0.698113	−	0.715988i	\(-0.254023\pi\)
							0.698113	+	0.715988i	\(0.254023\pi\)
\(43\)	27374.9			0.0525065			0.0262532	−	0.999655i	\(-0.491642\pi\)
							0.0262532	+	0.999655i	\(0.491642\pi\)
\(47\)	−145879.	−	252670.i	−0.204951	−	0.354986i	0.745166	−	0.666879i	\(-0.232370\pi\)
							−0.950117	+	0.311893i	\(0.899037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.e.b.53.14	✓	58
7.2	even	3	inner	91.8.e.b.79.14	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.e.b.53.14	✓	58	1.1	even	1	trivial
91.8.e.b.79.14	yes	58	7.2	even	3	inner