Embedded newform 91.10.bc.a.6.20

• Label: 91.10.bc.a.6.20
• Level: $91$
• Weight: $10$
• Character: 91.6
• 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.bc (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			6.20
Character	\(\chi\)	\(=\)	91.6
Dual form			91.10.bc.a.76.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.75438 - 25.2077i) q^{2} +(153.838 + 88.8186i) q^{3} +(-146.400 + 84.5241i) q^{4} +(-1293.94 - 1293.94i) q^{5} +(1199.83 - 4477.82i) q^{6} +(2844.58 - 5679.96i) q^{7} +(-6328.59 - 6328.59i) q^{8} +(5935.98 + 10281.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.75438	−	25.2077i	−0.298504	−	1.11403i	−0.938394	−	0.345566i	\(-0.887687\pi\)
							0.639890	−	0.768466i	\(-0.278980\pi\)
\(3\)	153.838	+	88.8186i	1.09653	+	0.633079i	0.935306	−	0.353839i	\(-0.115124\pi\)
							0.161219	+	0.986919i	\(0.448457\pi\)
\(5\)	−1293.94	−	1293.94i	−0.925868	−	0.925868i	0.0715682	−	0.997436i	\(-0.477200\pi\)
							−0.997436	+	0.0715682i	\(0.977200\pi\)
\(7\)	2844.58	−	5679.96i	0.447793	−	0.894137i
							\(11\)	−56214.1	+	15062.5i	−1.15765	+	0.310192i	−0.786028	−	0.618191i	\(-0.787866\pi\)
−0.371624	+	0.928383i	\(0.621199\pi\)
\(13\)	−24109.1	−	100116.i	−0.234119	−	0.972208i
							\(17\)	88908.9	+	153995.i	0.258182	+	0.447184i	0.965755	−	0.259456i	\(-0.0835434\pi\)
−0.707573	+	0.706640i	\(0.750210\pi\)
\(19\)	−52977.1	+	197713.i	−0.0932603	+	0.348052i	−0.996750	−	0.0805577i	\(-0.974330\pi\)
							0.903490	+	0.428610i	\(0.140997\pi\)
\(23\)	2.17328e6	+	1.25474e6i	1.61935	+	0.934930i	0.987089	+	0.160170i	\(0.0512044\pi\)
							0.632256	+	0.774759i	\(0.282129\pi\)
\(29\)	−1.87726e6	+	3.25150e6i	−0.492870	+	0.853676i	−0.999966	−	0.00821356i	\(-0.997386\pi\)
							0.507096	+	0.861889i	\(0.330719\pi\)
\(31\)	−2.69911e6	−	2.69911e6i	−0.524919	−	0.524919i	0.394134	−	0.919053i	\(-0.371045\pi\)
							−0.919053	+	0.394134i	\(0.871045\pi\)
\(37\)	−263808.	+	70687.3i	−0.0231409	+	0.00620059i	−0.270371	−	0.962756i	\(-0.587146\pi\)
							0.247230	+	0.968957i	\(0.420480\pi\)
\(41\)	6.48502e6	−	1.73766e6i	0.358413	−	0.0960365i	−0.0751191	−	0.997175i	\(-0.523934\pi\)
							0.433532	+	0.901138i	\(0.357267\pi\)
\(43\)	−2.61837e7	+	1.51171e7i	−1.16794	+	0.674313i	−0.953195	−	0.302356i	\(-0.902227\pi\)
							−0.214750	+	0.976669i	\(0.568894\pi\)
\(47\)	−2.62294e7	+	2.62294e7i	−0.784059	+	0.784059i	−0.980513	−	0.196454i	\(-0.937057\pi\)
							0.196454	+	0.980513i	\(0.437057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bc.a.6.20	yes	328
7.6	odd	2	inner	91.10.bc.a.6.19	✓	328
13.11	odd	12	inner	91.10.bc.a.76.19	yes	328
91.76	even	12	inner	91.10.bc.a.76.20	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bc.a.6.19	✓	328	7.6	odd	2	inner
91.10.bc.a.6.20	yes	328	1.1	even	1	trivial
91.10.bc.a.76.19	yes	328	13.11	odd	12	inner
91.10.bc.a.76.20	yes	328	91.76	even	12	inner