Embedded newform 91.10.bc.a.6.15

• Label: 91.10.bc.a.6.15
• 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			6.15
Character	\(\chi\)	\(=\)	91.6
Dual form			91.10.bc.a.76.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.37802 - 31.2672i) q^{2} +(-114.682 - 66.2115i) q^{3} +(-464.042 + 267.915i) q^{4} +(1420.94 + 1420.94i) q^{5} +(-1109.44 + 4140.50i) q^{6} +(6248.40 + 1145.02i) q^{7} +(545.436 + 545.436i) q^{8} +(-1073.58 - 1859.49i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.37802	−	31.2672i	−0.370260	−	1.38183i	−0.860149	−	0.510043i	\(-0.829629\pi\)
							0.489889	−	0.871785i	\(-0.337037\pi\)
\(3\)	−114.682	−	66.2115i	−0.817426	−	0.471941i	0.0321021	−	0.999485i	\(-0.489780\pi\)
							−0.849528	+	0.527544i	\(0.823113\pi\)
\(5\)	1420.94	+	1420.94i	1.01674	+	1.01674i	0.999857	+	0.0168867i	\(0.00537547\pi\)
							0.0168867	+	0.999857i	\(0.494625\pi\)
\(7\)	6248.40	+	1145.02i	0.983621	+	0.180248i
							\(11\)	−2482.65	+	665.224i	−0.0511268	+	0.0136994i	−0.284292	−	0.958738i	\(-0.591758\pi\)
0.233165	+	0.972437i	\(0.425092\pi\)
\(13\)	−18822.1	−	101243.i	−0.182778	−	0.983154i
							\(17\)	170830.	+	295886.i	0.496070	+	0.859219i	0.999990	−	0.00453148i	\(-0.00144242\pi\)
−0.503919	+	0.863751i	\(0.668109\pi\)
\(19\)	−141364.	+	527578.i	−0.248856	+	0.928742i	0.722550	+	0.691318i	\(0.242970\pi\)
							−0.971406	+	0.237424i	\(0.923697\pi\)
\(23\)	−621552.	−	358853.i	−0.463129	−	0.267388i	0.250230	−	0.968186i	\(-0.419494\pi\)
							−0.713359	+	0.700799i	\(0.752827\pi\)
\(29\)	−1.19740e6	+	2.07395e6i	−0.314374	+	0.544512i	−0.979304	−	0.202393i	\(-0.935128\pi\)
							0.664930	+	0.746906i	\(0.268461\pi\)
\(31\)	−5.28706e6	−	5.28706e6i	−1.02822	−	1.02822i	−0.999590	−	0.0286320i	\(-0.990885\pi\)
							−0.0286320	−	0.999590i	\(-0.509115\pi\)
\(37\)	−6.96360e6	+	1.86589e6i	−0.610838	+	0.163674i	−0.550959	−	0.834532i	\(-0.685738\pi\)
							−0.0598787	+	0.998206i	\(0.519071\pi\)
\(41\)	−9.98368e6	+	2.67512e6i	−0.551776	+	0.147848i	−0.523926	−	0.851764i	\(-0.675533\pi\)
							−0.0278508	+	0.999612i	\(0.508866\pi\)
\(43\)	3.16356e6	−	1.82648e6i	0.141113	−	0.0814717i	−0.427781	−	0.903882i	\(-0.640705\pi\)
							0.568894	+	0.822411i	\(0.307371\pi\)
\(47\)	−1.45012e7	+	1.45012e7i	−0.433475	+	0.433475i	−0.889809	−	0.456334i	\(-0.849162\pi\)
							0.456334	+	0.889809i	\(0.349162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bc.a.6.15	✓	328
7.6	odd	2	inner	91.10.bc.a.6.16	yes	328
13.11	odd	12	inner	91.10.bc.a.76.16	yes	328
91.76	even	12	inner	91.10.bc.a.76.15	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bc.a.6.15	✓	328	1.1	even	1	trivial
91.10.bc.a.6.16	yes	328	7.6	odd	2	inner
91.10.bc.a.76.15	yes	328	91.76	even	12	inner
91.10.bc.a.76.16	yes	328	13.11	odd	12	inner