Embedded newform 91.8.e.b.53.1

• Label: 91.8.e.b.53.1
• 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			53.1
Character	\(\chi\)	\(=\)	91.53
Dual form			91.8.e.b.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.1978 - 19.3951i) q^{2} +(34.3559 - 59.5061i) q^{3} +(-186.780 + 323.513i) q^{4} +(189.321 + 327.914i) q^{5} -1538.84 q^{6} +(-105.148 - 901.381i) q^{7} +5499.45 q^{8} +(-1267.15 - 2194.77i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.1978	−	19.3951i	−0.989752	−	1.71430i	−0.618540	−	0.785754i	\(-0.712275\pi\)
							−0.371213	−	0.928548i	\(-0.621058\pi\)
\(3\)	34.3559	−	59.5061i	0.734643	−	1.27244i	−0.220236	−	0.975447i	\(-0.570683\pi\)
							0.954880	−	0.296993i	\(-0.0959838\pi\)
\(5\)	189.321	+	327.914i	0.677335	+	1.17318i	0.975780	+	0.218753i	\(0.0701988\pi\)
							−0.298445	+	0.954427i	\(0.596468\pi\)
\(7\)	−105.148	−	901.381i	−0.115867	−	0.993265i
							\(11\)	2988.77	−	5176.71i	0.677047	−	1.17268i	−0.298819	−	0.954310i	\(-0.596593\pi\)
0.975866	−	0.218370i	\(-0.0700739\pi\)
\(13\)	−2197.00			−0.277350
							\(17\)	791.356	−	1370.67i	0.0390661	−	0.0676645i	−0.845831	−	0.533451i	\(-0.820895\pi\)
0.884897	+	0.465786i	\(0.154228\pi\)
\(19\)	−11062.5	−	19160.9i	−0.370013	−	0.640881i	0.619554	−	0.784954i	\(-0.287313\pi\)
							−0.989567	+	0.144073i	\(0.953980\pi\)
\(23\)	31225.9	+	54084.8i	0.535140	+	0.926890i	0.999157	+	0.0410633i	\(0.0130745\pi\)
							−0.464016	+	0.885827i	\(0.653592\pi\)
\(29\)	−105461.			−0.802973			−0.401486	−	0.915865i	\(-0.631506\pi\)
							−0.401486	+	0.915865i	\(0.631506\pi\)
\(31\)	123679.	−	214218.i	0.745642	−	1.29149i	−0.204252	−	0.978918i	\(-0.565476\pi\)
							0.949894	−	0.312572i	\(-0.101190\pi\)
\(37\)	−144569.	−	250402.i	−0.469213	−	0.812701i	0.530167	−	0.847893i	\(-0.322129\pi\)
							−0.999381	+	0.0351918i	\(0.988796\pi\)
\(41\)	−263998.			−0.598215			−0.299107	−	0.954219i	\(-0.596689\pi\)
							−0.299107	+	0.954219i	\(0.596689\pi\)
\(43\)	−213974.			−0.410413			−0.205206	−	0.978719i	\(-0.565787\pi\)
							−0.205206	+	0.978719i	\(0.565787\pi\)
\(47\)	−309399.	−	535895.i	−0.434687	−	0.752900i	0.562583	−	0.826741i	\(-0.309808\pi\)
							−0.997270	+	0.0738411i	\(0.976474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.e.b.53.1	✓	58
7.2	even	3	inner	91.8.e.b.79.1	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.e.b.53.1	✓	58	1.1	even	1	trivial
91.8.e.b.79.1	yes	58	7.2	even	3	inner