Embedded newform 91.10.bb.a.5.13

• Label: 91.10.bb.a.5.13
• Level: $91$
• Weight: $10$
• Character: 91.5
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			5.13
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.94964 - 33.4005i) q^{2} +(-222.806 + 128.637i) q^{3} +(-592.093 + 341.845i) q^{4} +(-2577.17 + 690.551i) q^{5} +(6290.58 + 6290.58i) q^{6} +(-4593.27 + 4388.10i) q^{7} +(4197.96 + 4197.96i) q^{8} +(23253.6 - 40276.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.94964	−	33.4005i	−0.395522	−	1.47611i	−0.820889	−	0.571087i	\(-0.806522\pi\)
							0.425368	−	0.905021i	\(-0.360145\pi\)
\(3\)	−222.806	+	128.637i	−1.58811	+	0.916898i	−0.594496	+	0.804098i	\(0.702649\pi\)
							−0.993618	+	0.112800i	\(0.964018\pi\)
\(5\)	−2577.17	+	690.551i	−1.84407	+	0.494118i	−0.999168	−	0.0407953i	\(-0.987011\pi\)
							−0.844907	+	0.534913i	\(0.820344\pi\)
\(7\)	−4593.27	+	4388.10i	−0.723071	+	0.690773i
							\(11\)	5843.12	−	21806.8i	0.120331	−	0.449082i	−0.879299	−	0.476270i	\(-0.841989\pi\)
0.999630	+	0.0271880i	\(0.00865526\pi\)
\(13\)	−30354.5	−	98402.8i	−0.294767	−	0.955569i
							\(17\)	119468.	+	206925.i	0.346922	+	0.600887i	0.985701	−	0.168503i	\(-0.0538934\pi\)
−0.638779	+	0.769391i	\(0.720560\pi\)
\(19\)	519755.	−	139268.i	0.914972	−	0.245166i	0.229537	−	0.973300i	\(-0.426279\pi\)
							0.685435	+	0.728134i	\(0.259612\pi\)
\(23\)	−645125.	−	372463.i	−0.480694	−	0.277529i	0.240012	−	0.970770i	\(-0.422849\pi\)
							−0.720706	+	0.693241i	\(0.756182\pi\)
\(29\)	−1.00334e6			−0.263425			−0.131712	−	0.991288i	\(-0.542048\pi\)
							−0.131712	+	0.991288i	\(0.542048\pi\)
\(31\)	−1.77075e6	+	6.60854e6i	−0.344374	+	1.28522i	0.548967	+	0.835844i	\(0.315021\pi\)
							−0.893342	+	0.449378i	\(0.851646\pi\)
\(37\)	−1.20234e7	+	3.22166e6i	−1.05468	+	0.282600i	−0.744183	−	0.667976i	\(-0.767161\pi\)
							−0.310493	+	0.950576i	\(0.600494\pi\)
\(41\)	9.78455e6	+	9.78455e6i	0.540771	+	0.540771i	0.923755	−	0.382984i	\(-0.125103\pi\)
							−0.382984	+	0.923755i	\(0.625103\pi\)
\(43\)		−	1.81291e7i		−	0.808666i	−0.914612	−	0.404333i	\(-0.867504\pi\)
							0.914612	−	0.404333i	\(-0.132496\pi\)
\(47\)	−1.37941e7	−	5.14801e7i	−0.412337	−	1.53886i	−0.790111	−	0.612963i	\(-0.789977\pi\)
							0.377775	−	0.925898i	\(-0.376689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.13	✓	328
7.3	odd	6	inner	91.10.bb.a.31.70	yes	328
13.8	odd	4	inner	91.10.bb.a.47.70	yes	328
91.73	even	12	inner	91.10.bb.a.73.13	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.13	✓	328	1.1	even	1	trivial
91.10.bb.a.31.70	yes	328	7.3	odd	6	inner
91.10.bb.a.47.70	yes	328	13.8	odd	4	inner
91.10.bb.a.73.13	yes	328	91.73	even	12	inner