Embedded newform 91.8.e.b.53.19

• Label: 91.8.e.b.53.19
• 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			53.19
Character	\(\chi\)	\(=\)	91.53
Dual form			91.8.e.b.79.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.17186 + 3.76177i) q^{2} +(8.57877 - 14.8589i) q^{3} +(54.5660 - 94.5112i) q^{4} +(225.848 + 391.181i) q^{5} +74.5276 q^{6} +(886.944 - 192.023i) q^{7} +1030.04 q^{8} +(946.309 + 1639.06i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.17186	+	3.76177i	0.191967	+	0.332497i	0.945902	−	0.324452i	\(-0.105180\pi\)
							−0.753935	+	0.656949i	\(0.771847\pi\)
\(3\)	8.57877	−	14.8589i	0.183443	−	0.317732i	−0.759608	−	0.650381i	\(-0.774609\pi\)
							0.943051	+	0.332649i	\(0.107942\pi\)
\(5\)	225.848	+	391.181i	0.808019	+	1.39953i	0.914234	+	0.405187i	\(0.132794\pi\)
							−0.106214	+	0.994343i	\(0.533873\pi\)
\(7\)	886.944	−	192.023i	0.977357	−	0.211597i
							\(11\)	323.833	−	560.896i	0.0733579	−	0.127060i	−0.827013	−	0.562183i	\(-0.809962\pi\)
0.900371	+	0.435123i	\(0.143295\pi\)
\(13\)	−2197.00			−0.277350
							\(17\)	−15462.3	+	26781.4i	−0.763312	+	1.32210i	0.177823	+	0.984063i	\(0.443095\pi\)
−0.941134	+	0.338032i	\(0.890239\pi\)
\(19\)	−14116.1	−	24449.9i	−0.472148	−	0.817784i	0.527344	−	0.849652i	\(-0.323188\pi\)
							−0.999492	+	0.0318674i	\(0.989855\pi\)
\(23\)	31899.5	+	55251.6i	0.546685	+	0.946886i	0.998499	+	0.0547734i	\(0.0174437\pi\)
							−0.451814	+	0.892112i	\(0.649223\pi\)
\(29\)	−20794.0			−0.158323			−0.0791617	−	0.996862i	\(-0.525224\pi\)
							−0.0791617	+	0.996862i	\(0.525224\pi\)
\(31\)	162780.	−	281944.i	0.981377	−	1.69980i	0.324335	−	0.945942i	\(-0.394860\pi\)
							0.657043	−	0.753853i	\(-0.271807\pi\)
\(37\)	198268.	+	343411.i	0.643498	+	1.11457i	0.984646	+	0.174562i	\(0.0558510\pi\)
							−0.341148	+	0.940010i	\(0.610816\pi\)
\(41\)	−184732.			−0.418600			−0.209300	−	0.977851i	\(-0.567119\pi\)
							−0.209300	+	0.977851i	\(0.567119\pi\)
\(43\)	−318430.			−0.610766			−0.305383	−	0.952230i	\(-0.598784\pi\)
							−0.305383	+	0.952230i	\(0.598784\pi\)
\(47\)	442975.	+	767255.i	0.622353	+	1.07795i	0.989046	+	0.147605i	\(0.0471564\pi\)
							−0.366694	+	0.930342i	\(0.619510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.e.b.53.19	✓	58
7.2	even	3	inner	91.8.e.b.79.19	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.e.b.53.19	✓	58	1.1	even	1	trivial
91.8.e.b.79.19	yes	58	7.2	even	3	inner