Embedded newform 91.2.bb.a.47.4

• Label: 91.2.bb.a.47.4
• 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			47.4
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.746505 + 0.200025i) q^{2} +(-0.421869 + 0.243566i) q^{3} +(-1.21479 + 0.701360i) q^{4} +(0.472319 + 1.76272i) q^{5} +(0.266208 - 0.266208i) q^{6} +(-0.210751 + 2.63734i) q^{7} +(1.85952 - 1.85952i) q^{8} +(-1.38135 + 2.39257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.746505	+	0.200025i	−0.527859	+	0.141439i	−0.512899	−	0.858449i	\(-0.671429\pi\)
							−0.0149595	+	0.999888i	\(0.504762\pi\)
\(3\)	−0.421869	+	0.243566i	−0.243566	+	0.140623i	−0.616815	−	0.787108i	\(-0.711577\pi\)
							0.373248	+	0.927731i	\(0.378244\pi\)
\(5\)	0.472319	+	1.76272i	0.211227	+	0.788311i	0.987461	+	0.157865i	\(0.0504612\pi\)
							−0.776233	+	0.630446i	\(0.782872\pi\)
\(7\)	−0.210751	+	2.63734i	−0.0796563	+	0.996822i
							\(11\)	0.990745	+	0.265469i	0.298721	+	0.0800420i	0.405067	−	0.914287i	\(-0.367248\pi\)
−0.106346	+	0.994329i	\(0.533915\pi\)
\(13\)	−0.266208	−	3.59571i	−0.0738329	−	0.997271i
							\(17\)	2.60029	+	4.50383i	0.630663	+	1.09234i	0.987416	+	0.158142i	\(0.0505503\pi\)
−0.356753	+	0.934199i	\(0.616116\pi\)
\(19\)	−1.36051	−	5.07751i	−0.312123	−	1.16486i	−0.926638	−	0.375954i	\(-0.877315\pi\)
							0.614515	−	0.788905i	\(-0.289352\pi\)
\(23\)	0.730699	+	0.421869i	0.152361	+	0.0879659i	0.574242	−	0.818685i	\(-0.305297\pi\)
							−0.421881	+	0.906651i	\(0.638630\pi\)
\(29\)	10.3454			1.92109			0.960543	−	0.278130i	\(-0.0897147\pi\)
							0.960543	+	0.278130i	\(0.0897147\pi\)
\(31\)	−5.69625	−	1.52630i	−1.02308	−	0.274132i	−0.291993	−	0.956421i	\(-0.594318\pi\)
							−0.731083	+	0.682288i	\(0.760985\pi\)
\(37\)	1.61677	+	6.03388i	0.265796	+	0.991964i	0.961761	+	0.273889i	\(0.0883101\pi\)
							−0.695965	+	0.718075i	\(0.745023\pi\)
\(41\)	−0.0927742	+	0.0927742i	−0.0144889	+	0.0144889i	−0.714314	−	0.699825i	\(-0.753261\pi\)
							0.699825	+	0.714314i	\(0.253261\pi\)
\(43\)			7.36681i			1.12343i	0.827331	+	0.561714i	\(0.189858\pi\)
							−0.827331	+	0.561714i	\(0.810142\pi\)
\(47\)	2.17913	−	0.583897i	0.317859	−	0.0851701i	−0.0963621	−	0.995346i	\(-0.530721\pi\)
							0.414221	+	0.910176i	\(0.364054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.47.4	yes	32
3.2	odd	2		819.2.fn.e.775.5		32
7.2	even	3		637.2.i.a.489.9		32
7.3	odd	6	inner	91.2.bb.a.73.5	yes	32
7.4	even	3		637.2.bc.b.619.5		32
7.5	odd	6		637.2.i.a.489.10		32
7.6	odd	2		637.2.bc.b.411.4		32
13.5	odd	4	inner	91.2.bb.a.5.5	✓	32
21.17	even	6		819.2.fn.e.73.4		32
39.5	even	4		819.2.fn.e.460.4		32
91.5	even	12		637.2.i.a.538.10		32
91.18	odd	12		637.2.bc.b.31.4		32
91.31	even	12	inner	91.2.bb.a.31.4	yes	32
91.44	odd	12		637.2.i.a.538.9		32
91.83	even	4		637.2.bc.b.460.5		32
273.122	odd	12		819.2.fn.e.577.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.5	✓	32	13.5	odd	4	inner
91.2.bb.a.31.4	yes	32	91.31	even	12	inner
91.2.bb.a.47.4	yes	32	1.1	even	1	trivial
91.2.bb.a.73.5	yes	32	7.3	odd	6	inner
637.2.i.a.489.9		32	7.2	even	3
637.2.i.a.489.10		32	7.5	odd	6
637.2.i.a.538.9		32	91.44	odd	12
637.2.i.a.538.10		32	91.5	even	12
637.2.bc.b.31.4		32	91.18	odd	12
637.2.bc.b.411.4		32	7.6	odd	2
637.2.bc.b.460.5		32	91.83	even	4
637.2.bc.b.619.5		32	7.4	even	3
819.2.fn.e.73.4		32	21.17	even	6
819.2.fn.e.460.4		32	39.5	even	4
819.2.fn.e.577.5		32	273.122	odd	12
819.2.fn.e.775.5		32	3.2	odd	2