Embedded newform 91.2.bb.a.73.5

• Label: 91.2.bb.a.73.5
• 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			73.5
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.5

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.200025	−	0.746505i	0.141439	−	0.527859i	−0.858449	−	0.512899i	\(-0.828571\pi\)
							0.999888	−	0.0149595i	\(-0.00476194\pi\)
\(3\)	−0.421869	−	0.243566i	−0.243566	−	0.140623i	0.373248	−	0.927731i	\(-0.378244\pi\)
							−0.616815	+	0.787108i	\(0.711577\pi\)
\(5\)	1.76272	+	0.472319i	0.788311	+	0.211227i	0.630446	−	0.776233i	\(-0.282872\pi\)
							0.157865	+	0.987461i	\(0.449539\pi\)
\(7\)	−2.63734	+	0.210751i	−0.996822	+	0.0796563i
							\(11\)	−0.265469	−	0.990745i	−0.0800420	−	0.298721i	0.914287	−	0.405067i	\(-0.132752\pi\)
−0.994329	+	0.106346i	\(0.966085\pi\)
\(13\)	0.266208	+	3.59571i	0.0738329	+	0.997271i
							\(17\)	−2.60029	+	4.50383i	−0.630663	+	1.09234i	0.356753	+	0.934199i	\(0.383884\pi\)
−0.987416	+	0.158142i	\(0.949450\pi\)
\(19\)	−5.07751	−	1.36051i	−1.16486	−	0.312123i	−0.375954	−	0.926638i	\(-0.622685\pi\)
							−0.788905	+	0.614515i	\(0.789352\pi\)
\(23\)	−0.730699	+	0.421869i	−0.152361	+	0.0879659i	−0.574242	−	0.818685i	\(-0.694703\pi\)
							0.421881	+	0.906651i	\(0.361370\pi\)
\(29\)	10.3454			1.92109			0.960543	−	0.278130i	\(-0.0897147\pi\)
							0.960543	+	0.278130i	\(0.0897147\pi\)
\(31\)	−1.52630	−	5.69625i	−0.274132	−	1.02308i	−0.956421	−	0.291993i	\(-0.905682\pi\)
							0.682288	−	0.731083i	\(-0.260985\pi\)
\(37\)	−6.03388	−	1.61677i	−0.991964	−	0.265796i	−0.273889	−	0.961761i	\(-0.588310\pi\)
							−0.718075	+	0.695965i	\(0.754977\pi\)
\(41\)	0.0927742	−	0.0927742i	0.0144889	−	0.0144889i	−0.699825	−	0.714314i	\(-0.746739\pi\)
							0.714314	+	0.699825i	\(0.246739\pi\)
\(43\)			7.36681i			1.12343i	0.827331	+	0.561714i	\(0.189858\pi\)
							−0.827331	+	0.561714i	\(0.810142\pi\)
\(47\)	0.583897	−	2.17913i	0.0851701	−	0.317859i	−0.910176	−	0.414221i	\(-0.864054\pi\)
							0.995346	+	0.0963621i	\(0.0307207\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.5	✓	32	91.5	even	12	inner
91.2.bb.a.31.4	yes	32	13.5	odd	4	inner
91.2.bb.a.47.4	yes	32	7.5	odd	6	inner
91.2.bb.a.73.5	yes	32	1.1	even	1	trivial
637.2.i.a.489.9		32	7.3	odd	6
637.2.i.a.489.10		32	7.4	even	3
637.2.i.a.538.9		32	91.31	even	12
637.2.i.a.538.10		32	91.18	odd	12
637.2.bc.b.31.4		32	91.83	even	4
637.2.bc.b.411.4		32	7.2	even	3
637.2.bc.b.460.5		32	91.44	odd	12
637.2.bc.b.619.5		32	7.6	odd	2
819.2.fn.e.73.4		32	3.2	odd	2
819.2.fn.e.460.4		32	273.5	odd	12
819.2.fn.e.577.5		32	39.5	even	4
819.2.fn.e.775.5		32	21.5	even	6