Embedded newform 91.2.bb.a.47.7

• Label: 91.2.bb.a.47.7
• 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			47.7
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.7

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53492	−	0.411280i	1.08535	−	0.290819i	0.328565	−	0.944482i	\(-0.393435\pi\)
							0.756786	+	0.653663i	\(0.226768\pi\)
\(3\)	0.436133	−	0.251802i	0.251802	−	0.145378i	−0.368787	−	0.929514i	\(-0.620227\pi\)
							0.620589	+	0.784136i	\(0.286894\pi\)
\(5\)	−0.0206096	−	0.0769162i	−0.00921691	−	0.0343980i	0.961164	−	0.275977i	\(-0.0890014\pi\)
							−0.970381	+	0.241579i	\(0.922335\pi\)
\(7\)	−2.16435	−	1.52171i	−0.818046	−	0.575153i
							\(11\)	4.08225	+	1.09384i	1.23084	+	0.329804i	0.814908	−	0.579590i	\(-0.196787\pi\)
0.415936	+	0.909394i	\(0.363454\pi\)
\(13\)	−0.565867	−	3.56087i	−0.156943	−	0.987608i
							\(17\)	−2.90357	−	5.02912i	−0.704218	−	1.21974i	−0.966973	−	0.254879i	\(-0.917964\pi\)
0.262755	−	0.964863i	\(-0.415369\pi\)
\(19\)	1.36980	+	5.11216i	0.314254	+	1.17281i	0.924682	+	0.380740i	\(0.124331\pi\)
							−0.610429	+	0.792071i	\(0.709003\pi\)
\(23\)	−0.755405	−	0.436133i	−0.157513	−	0.0909400i	0.419172	−	0.907907i	\(-0.362320\pi\)
							−0.576685	+	0.816967i	\(0.695654\pi\)
\(29\)	0.362759			0.0673627			0.0336814	−	0.999433i	\(-0.489277\pi\)
							0.0336814	+	0.999433i	\(0.489277\pi\)
\(31\)	−1.34748	−	0.361057i	−0.242015	−	0.0648477i	0.135773	−	0.990740i	\(-0.456648\pi\)
							−0.377787	+	0.925892i	\(0.623315\pi\)
\(37\)	1.00978	+	3.76857i	0.166008	+	0.619549i	0.997910	+	0.0646262i	\(0.0205855\pi\)
							−0.831902	+	0.554923i	\(0.812748\pi\)
\(41\)	7.70995	−	7.70995i	1.20409	−	1.20409i	0.231182	−	0.972911i	\(-0.425741\pi\)
							0.972911	−	0.231182i	\(-0.0742592\pi\)
\(43\)		−	2.65570i		−	0.404990i	−0.979283	−	0.202495i	\(-0.935095\pi\)
							0.979283	−	0.202495i	\(-0.0649050\pi\)
\(47\)	−2.79467	+	0.748829i	−0.407644	+	0.109228i	−0.456813	−	0.889563i	\(-0.651009\pi\)
							0.0491692	+	0.998790i	\(0.484343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.47.7	yes	32
3.2	odd	2		819.2.fn.e.775.2		32
7.2	even	3		637.2.i.a.489.4		32
7.3	odd	6	inner	91.2.bb.a.73.2	yes	32
7.4	even	3		637.2.bc.b.619.2		32
7.5	odd	6		637.2.i.a.489.3		32
7.6	odd	2		637.2.bc.b.411.7		32
13.5	odd	4	inner	91.2.bb.a.5.2	✓	32
21.17	even	6		819.2.fn.e.73.7		32
39.5	even	4		819.2.fn.e.460.7		32
91.5	even	12		637.2.i.a.538.3		32
91.18	odd	12		637.2.bc.b.31.7		32
91.31	even	12	inner	91.2.bb.a.31.7	yes	32
91.44	odd	12		637.2.i.a.538.4		32
91.83	even	4		637.2.bc.b.460.2		32
273.122	odd	12		819.2.fn.e.577.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.2	✓	32	13.5	odd	4	inner
91.2.bb.a.31.7	yes	32	91.31	even	12	inner
91.2.bb.a.47.7	yes	32	1.1	even	1	trivial
91.2.bb.a.73.2	yes	32	7.3	odd	6	inner
637.2.i.a.489.3		32	7.5	odd	6
637.2.i.a.489.4		32	7.2	even	3
637.2.i.a.538.3		32	91.5	even	12
637.2.i.a.538.4		32	91.44	odd	12
637.2.bc.b.31.7		32	91.18	odd	12
637.2.bc.b.411.7		32	7.6	odd	2
637.2.bc.b.460.2		32	91.83	even	4
637.2.bc.b.619.2		32	7.4	even	3
819.2.fn.e.73.7		32	21.17	even	6
819.2.fn.e.460.7		32	39.5	even	4
819.2.fn.e.577.2		32	273.122	odd	12
819.2.fn.e.775.2		32	3.2	odd	2