Embedded newform 91.10.bc.a.6.12

• Label: 91.10.bc.a.6.12
• 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			6.12
Character	\(\chi\)	\(=\)	91.6
Dual form			91.10.bc.a.76.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.78129 - 32.7722i) q^{2} +(159.314 + 91.9801i) q^{3} +(-553.502 + 319.564i) q^{4} +(-74.9438 - 74.9438i) q^{5} +(1615.41 - 6028.78i) q^{6} +(-4384.56 + 4596.65i) q^{7} +(3049.93 + 3049.93i) q^{8} +(7079.17 + 12261.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.78129	−	32.7722i	−0.388082	−	1.44834i	−0.833251	−	0.552894i	\(-0.813523\pi\)
							0.445170	−	0.895446i	\(-0.353143\pi\)
\(3\)	159.314	+	91.9801i	1.13556	+	0.655614i	0.945326	−	0.326126i	\(-0.105743\pi\)
							0.190230	+	0.981740i	\(0.439077\pi\)
\(5\)	−74.9438	−	74.9438i	−0.0536254	−	0.0536254i	0.679786	−	0.733411i	\(-0.262073\pi\)
							−0.733411	+	0.679786i	\(0.762073\pi\)
\(7\)	−4384.56	+	4596.65i	−0.690216	+	0.723604i
							\(11\)	56251.7	−	15072.6i	1.15843	−	0.310399i	0.372089	−	0.928197i	\(-0.378641\pi\)
0.786337	+	0.617797i	\(0.211975\pi\)
\(13\)	−27194.4	−	99322.5i	−0.264080	−	0.964501i
							\(17\)	168599.	+	292022.i	0.489592	+	0.847998i	0.999928	−	0.0119768i	\(-0.00381242\pi\)
−0.510336	+	0.859975i	\(0.670479\pi\)
\(19\)	−39733.0	+	148286.i	−0.0699455	+	0.261040i	−0.992040	−	0.125925i	\(-0.959810\pi\)
							0.922094	+	0.386965i	\(0.126477\pi\)
\(23\)	310043.	+	179003.i	0.231018	+	0.133379i	0.611042	−	0.791598i	\(-0.290751\pi\)
							−0.380023	+	0.924977i	\(0.624084\pi\)
\(29\)	42806.8	−	74143.5i	0.0112388	−	0.0194662i	−0.860351	−	0.509702i	\(-0.829756\pi\)
							0.871590	+	0.490235i	\(0.163089\pi\)
\(31\)	4.00352e6	+	4.00352e6i	0.778600	+	0.778600i	0.979593	−	0.200993i	\(-0.0644167\pi\)
							−0.200993	+	0.979593i	\(0.564417\pi\)
\(37\)	1.54505e7	−	4.13995e6i	1.35530	−	0.363151i	0.493211	−	0.869909i	\(-0.335823\pi\)
							0.862088	+	0.506758i	\(0.169156\pi\)
\(41\)	9.37166e6	−	2.51113e6i	0.517952	−	0.138785i	0.00963269	−	0.999954i	\(-0.496934\pi\)
							0.508319	+	0.861169i	\(0.330267\pi\)
\(43\)	2.97597e7	−	1.71818e7i	1.32746	−	0.766408i	0.342552	−	0.939499i	\(-0.388709\pi\)
							0.984906	+	0.173091i	\(0.0553753\pi\)
\(47\)	1.92099e7	−	1.92099e7i	0.574228	−	0.574228i	−0.359079	−	0.933307i	\(-0.616909\pi\)
							0.933307	+	0.359079i	\(0.116909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bc.a.6.12	yes	328
7.6	odd	2	inner	91.10.bc.a.6.11	✓	328
13.11	odd	12	inner	91.10.bc.a.76.11	yes	328
91.76	even	12	inner	91.10.bc.a.76.12	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bc.a.6.11	✓	328	7.6	odd	2	inner
91.10.bc.a.6.12	yes	328	1.1	even	1	trivial
91.10.bc.a.76.11	yes	328	13.11	odd	12	inner
91.10.bc.a.76.12	yes	328	91.76	even	12	inner