Embedded newform 91.2.bb.a.47.2

• Label: 91.2.bb.a.47.2
• Level: $91$
• Weight: $2$
• Character: 91.47
• 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			47.2
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.2

$q$-expansion

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

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84208	+	0.493585i	−1.30255	+	0.349017i	−0.842414	−	0.538832i	\(-0.818866\pi\)
							−0.460136	+	0.887849i	\(0.652199\pi\)
\(3\)	2.29307	−	1.32391i	1.32391	−	0.764357i	0.339557	−	0.940586i	\(-0.389723\pi\)
							0.984349	+	0.176228i	\(0.0563897\pi\)
\(5\)	−0.885596	−	3.30509i	−0.396051	−	1.47808i	−0.819983	−	0.572388i	\(-0.806017\pi\)
							0.423932	−	0.905694i	\(-0.360649\pi\)
\(7\)	−2.39257	+	1.12943i	−0.904306	+	0.426884i
							\(11\)	1.66384	+	0.445825i	0.501667	+	0.134421i	0.500773	−	0.865578i	\(-0.333049\pi\)
0.000893225		1.00000i	\(0.499716\pi\)
\(13\)	3.57057	−	0.501030i	0.990298	−	0.138961i
							\(17\)	1.22596	+	2.12343i	0.297339	+	0.515006i	0.975526	−	0.219883i	\(-0.0705675\pi\)
−0.678187	+	0.734889i	\(0.737234\pi\)
\(19\)	1.34794	+	5.03057i	0.309238	+	1.15409i	0.929235	+	0.369488i	\(0.120467\pi\)
							−0.619997	+	0.784604i	\(0.712866\pi\)
\(23\)	−3.97172	−	2.29307i	−0.828160	−	0.478139i	0.0250620	−	0.999686i	\(-0.492022\pi\)
							−0.853222	+	0.521547i	\(0.825355\pi\)
\(29\)	0.184063			0.0341796			0.0170898	−	0.999854i	\(-0.494560\pi\)
							0.0170898	+	0.999854i	\(0.494560\pi\)
\(31\)	2.46060	+	0.659317i	0.441938	+	0.118417i	0.472924	−	0.881103i	\(-0.343198\pi\)
							−0.0309869	+	0.999520i	\(0.509865\pi\)
\(37\)	0.0563202	+	0.210190i	0.00925899	+	0.0345550i	0.970401	−	0.241500i	\(-0.0776394\pi\)
							−0.961142	+	0.276055i	\(0.910973\pi\)
\(41\)	−4.63239	+	4.63239i	−0.723458	+	0.723458i	−0.969308	−	0.245850i	\(-0.920933\pi\)
							0.245850	+	0.969308i	\(0.420933\pi\)
\(43\)			0.562412i			0.0857671i	0.999080	+	0.0428835i	\(0.0136544\pi\)
							−0.999080	+	0.0428835i	\(0.986346\pi\)
\(47\)	3.72492	−	0.998090i	0.543336	−	0.145586i	0.0232964	−	0.999729i	\(-0.492584\pi\)
							0.520040	+	0.854142i	\(0.325917\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.7	✓	32	13.5	odd	4	inner
91.2.bb.a.31.2	yes	32	91.31	even	12	inner
91.2.bb.a.47.2	yes	32	1.1	even	1	trivial
91.2.bb.a.73.7	yes	32	7.3	odd	6	inner
637.2.i.a.489.13		32	7.5	odd	6
637.2.i.a.489.14		32	7.2	even	3
637.2.i.a.538.13		32	91.5	even	12
637.2.i.a.538.14		32	91.44	odd	12
637.2.bc.b.31.2		32	91.18	odd	12
637.2.bc.b.411.2		32	7.6	odd	2
637.2.bc.b.460.7		32	91.83	even	4
637.2.bc.b.619.7		32	7.4	even	3
819.2.fn.e.73.2		32	21.17	even	6
819.2.fn.e.460.2		32	39.5	even	4
819.2.fn.e.577.7		32	273.122	odd	12
819.2.fn.e.775.7		32	3.2	odd	2