Embedded newform 91.2.bb.a.5.2

• Label: 91.2.bb.a.5.2
• Level: $91$
• Weight: $2$
• Character: 91.5
• 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			5.2
Character	\(\chi\)	\(=\)	91.5
Dual form			91.2.bb.a.73.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.411280 - 1.53492i) q^{2} +(0.436133 - 0.251802i) q^{3} +(-0.454770 + 0.262561i) q^{4} +(-0.0769162 + 0.0206096i) q^{5} +(-0.565867 - 0.565867i) q^{6} +(1.52171 - 2.16435i) q^{7} +(-1.65723 - 1.65723i) q^{8} +(-1.37319 + 2.37844i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.411280	−	1.53492i	−0.290819	−	1.08535i	−0.944482	−	0.328565i	\(-0.893435\pi\)
							0.653663	−	0.756786i	\(-0.273232\pi\)
\(3\)	0.436133	−	0.251802i	0.251802	−	0.145378i	−0.368787	−	0.929514i	\(-0.620227\pi\)
							0.620589	+	0.784136i	\(0.286894\pi\)
\(5\)	−0.0769162	+	0.0206096i	−0.0343980	+	0.00921691i	−0.275977	−	0.961164i	\(-0.589001\pi\)
							0.241579	+	0.970381i	\(0.422335\pi\)
\(7\)	1.52171	−	2.16435i	0.575153	−	0.818046i
							\(11\)	−1.09384	+	4.08225i	−0.329804	+	1.23084i	0.579590	+	0.814908i	\(0.303213\pi\)
−0.909394	+	0.415936i	\(0.863454\pi\)
\(13\)	0.565867	−	3.56087i	0.156943	−	0.987608i
							\(17\)	2.90357	+	5.02912i	0.704218	+	1.21974i	0.966973	+	0.254879i	\(0.0820357\pi\)
−0.262755	+	0.964863i	\(0.584631\pi\)
\(19\)	5.11216	−	1.36980i	1.17281	−	0.314254i	0.380740	−	0.924682i	\(-0.375669\pi\)
							0.792071	+	0.610429i	\(0.209003\pi\)
\(23\)	0.755405	+	0.436133i	0.157513	+	0.0909400i	0.576685	−	0.816967i	\(-0.304346\pi\)
							−0.419172	+	0.907907i	\(0.637680\pi\)
\(29\)	0.362759			0.0673627			0.0336814	−	0.999433i	\(-0.489277\pi\)
							0.0336814	+	0.999433i	\(0.489277\pi\)
\(31\)	−0.361057	+	1.34748i	−0.0648477	+	0.242015i	−0.990740	−	0.135773i	\(-0.956648\pi\)
							0.925892	+	0.377787i	\(0.123315\pi\)
\(37\)	−3.76857	+	1.00978i	−0.619549	+	0.166008i	−0.554923	−	0.831902i	\(-0.687252\pi\)
							−0.0646262	+	0.997910i	\(0.520586\pi\)
\(41\)	−7.70995	−	7.70995i	−1.20409	−	1.20409i	−0.972911	−	0.231182i	\(-0.925741\pi\)
							−0.231182	−	0.972911i	\(-0.574259\pi\)
\(43\)			2.65570i			0.404990i	0.979283	+	0.202495i	\(0.0649050\pi\)
							−0.979283	+	0.202495i	\(0.935095\pi\)
\(47\)	−0.748829	−	2.79467i	−0.109228	−	0.407644i	0.889563	−	0.456813i	\(-0.151009\pi\)
							−0.998790	+	0.0491692i	\(0.984343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.5.2	✓	32
3.2	odd	2		819.2.fn.e.460.7		32
7.2	even	3		637.2.i.a.538.4		32
7.3	odd	6	inner	91.2.bb.a.31.7	yes	32
7.4	even	3		637.2.bc.b.31.7		32
7.5	odd	6		637.2.i.a.538.3		32
7.6	odd	2		637.2.bc.b.460.2		32
13.8	odd	4	inner	91.2.bb.a.47.7	yes	32
21.17	even	6		819.2.fn.e.577.2		32
39.8	even	4		819.2.fn.e.775.2		32
91.34	even	4		637.2.bc.b.411.7		32
91.47	even	12		637.2.i.a.489.3		32
91.60	odd	12		637.2.bc.b.619.2		32
91.73	even	12	inner	91.2.bb.a.73.2	yes	32
91.86	odd	12		637.2.i.a.489.4		32
273.164	odd	12		819.2.fn.e.73.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.2	✓	32	1.1	even	1	trivial
91.2.bb.a.31.7	yes	32	7.3	odd	6	inner
91.2.bb.a.47.7	yes	32	13.8	odd	4	inner
91.2.bb.a.73.2	yes	32	91.73	even	12	inner
637.2.i.a.489.3		32	91.47	even	12
637.2.i.a.489.4		32	91.86	odd	12
637.2.i.a.538.3		32	7.5	odd	6
637.2.i.a.538.4		32	7.2	even	3
637.2.bc.b.31.7		32	7.4	even	3
637.2.bc.b.411.7		32	91.34	even	4
637.2.bc.b.460.2		32	7.6	odd	2
637.2.bc.b.619.2		32	91.60	odd	12
819.2.fn.e.73.7		32	273.164	odd	12
819.2.fn.e.460.7		32	3.2	odd	2
819.2.fn.e.577.2		32	21.17	even	6
819.2.fn.e.775.2		32	39.8	even	4