Embedded newform 91.2.bb.a.73.7

• Label: 91.2.bb.a.73.7
• Level: $91$
• Weight: $2$
• Character: 91.73
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.7
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.493585 - 1.84208i) q^{2} +(2.29307 + 1.32391i) q^{3} +(-1.41759 - 0.818448i) q^{4} +(-3.30509 - 0.885596i) q^{5} +(3.57057 - 3.57057i) q^{6} +(-1.12943 + 2.39257i) q^{7} +(0.489646 - 0.489646i) q^{8} +(2.00545 + 3.47355i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.493585	−	1.84208i	0.349017	−	1.30255i	−0.538832	−	0.842414i	\(-0.681134\pi\)
							0.887849	−	0.460136i	\(-0.152199\pi\)
\(3\)	2.29307	+	1.32391i	1.32391	+	0.764357i	0.984349	−	0.176228i	\(-0.0563897\pi\)
							0.339557	+	0.940586i	\(0.389723\pi\)
\(5\)	−3.30509	−	0.885596i	−1.47808	−	0.396051i	−0.572388	−	0.819983i	\(-0.693983\pi\)
							−0.905694	+	0.423932i	\(0.860649\pi\)
\(7\)	−1.12943	+	2.39257i	−0.426884	+	0.904306i
							\(11\)	−0.445825	−	1.66384i	−0.134421	−	0.501667i	−1.00000		0.000893225i	\(-0.999716\pi\)
0.865578	−	0.500773i	\(-0.166951\pi\)
\(13\)	−3.57057	+	0.501030i	−0.990298	+	0.138961i
							\(17\)	−1.22596	+	2.12343i	−0.297339	+	0.515006i	−0.975526	−	0.219883i	\(-0.929433\pi\)
0.678187	+	0.734889i	\(0.262766\pi\)
\(19\)	5.03057	+	1.34794i	1.15409	+	0.309238i	0.784604	−	0.619997i	\(-0.212866\pi\)
							0.369488	+	0.929235i	\(0.379533\pi\)
\(23\)	3.97172	−	2.29307i	0.828160	−	0.478139i	−0.0250620	−	0.999686i	\(-0.507978\pi\)
							0.853222	+	0.521547i	\(0.174645\pi\)
\(29\)	0.184063			0.0341796			0.0170898	−	0.999854i	\(-0.494560\pi\)
							0.0170898	+	0.999854i	\(0.494560\pi\)
\(31\)	0.659317	+	2.46060i	0.118417	+	0.441938i	0.999520	−	0.0309869i	\(-0.00986501\pi\)
							−0.881103	+	0.472924i	\(0.843198\pi\)
\(37\)	−0.210190	−	0.0563202i	−0.0345550	−	0.00925899i	0.241500	−	0.970401i	\(-0.422361\pi\)
							−0.276055	+	0.961142i	\(0.589027\pi\)
\(41\)	4.63239	−	4.63239i	0.723458	−	0.723458i	−0.245850	−	0.969308i	\(-0.579067\pi\)
							0.969308	+	0.245850i	\(0.0790671\pi\)
\(43\)			0.562412i			0.0857671i	0.999080	+	0.0428835i	\(0.0136544\pi\)
							−0.999080	+	0.0428835i	\(0.986346\pi\)
\(47\)	0.998090	−	3.72492i	0.145586	−	0.543336i	−0.854142	−	0.520040i	\(-0.825917\pi\)
							0.999729	−	0.0232964i	\(-0.00741615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.73.7	yes	32
3.2	odd	2		819.2.fn.e.73.2		32
7.2	even	3		637.2.bc.b.411.2		32
7.3	odd	6		637.2.i.a.489.14		32
7.4	even	3		637.2.i.a.489.13		32
7.5	odd	6	inner	91.2.bb.a.47.2	yes	32
7.6	odd	2		637.2.bc.b.619.7		32
13.5	odd	4	inner	91.2.bb.a.31.2	yes	32
21.5	even	6		819.2.fn.e.775.7		32
39.5	even	4		819.2.fn.e.577.7		32
91.5	even	12	inner	91.2.bb.a.5.7	✓	32
91.18	odd	12		637.2.i.a.538.13		32
91.31	even	12		637.2.i.a.538.14		32
91.44	odd	12		637.2.bc.b.460.7		32
91.83	even	4		637.2.bc.b.31.2		32
273.5	odd	12		819.2.fn.e.460.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.7	✓	32	91.5	even	12	inner
91.2.bb.a.31.2	yes	32	13.5	odd	4	inner
91.2.bb.a.47.2	yes	32	7.5	odd	6	inner
91.2.bb.a.73.7	yes	32	1.1	even	1	trivial
637.2.i.a.489.13		32	7.4	even	3
637.2.i.a.489.14		32	7.3	odd	6
637.2.i.a.538.13		32	91.18	odd	12
637.2.i.a.538.14		32	91.31	even	12
637.2.bc.b.31.2		32	91.83	even	4
637.2.bc.b.411.2		32	7.2	even	3
637.2.bc.b.460.7		32	91.44	odd	12
637.2.bc.b.619.7		32	7.6	odd	2
819.2.fn.e.73.2		32	3.2	odd	2
819.2.fn.e.460.2		32	273.5	odd	12
819.2.fn.e.577.7		32	39.5	even	4
819.2.fn.e.775.7		32	21.5	even	6