Embedded newform 91.2.bb.a.47.3

• Label: 91.2.bb.a.47.3
• 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			47.3
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58070 + 0.423548i) q^{2} +(-1.71590 + 0.990678i) q^{3} +(0.587174 - 0.339005i) q^{4} +(-0.742954 - 2.77274i) q^{5} +(2.29273 - 2.29273i) q^{6} +(1.81015 - 1.92960i) q^{7} +(1.52975 - 1.52975i) q^{8} +(0.462886 - 0.801742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58070	+	0.423548i	−1.11772	+	0.299493i	−0.769962	−	0.638089i	\(-0.779725\pi\)
							−0.347762	+	0.937583i	\(0.613058\pi\)
\(3\)	−1.71590	+	0.990678i	−0.990678	+	0.571968i	−0.905477	−	0.424395i	\(-0.860487\pi\)
							−0.0852012	+	0.996364i	\(0.527153\pi\)
\(5\)	−0.742954	−	2.77274i	−0.332259	−	1.24001i	−0.906810	−	0.421539i	\(-0.861490\pi\)
							0.574551	−	0.818469i	\(-0.305177\pi\)
\(7\)	1.81015	−	1.92960i	0.684172	−	0.729321i
							\(11\)	−3.33916	−	0.894725i	−1.00679	−	0.269770i	−0.282505	−	0.959266i	\(-0.591165\pi\)
−0.724290	+	0.689496i	\(0.757832\pi\)
\(13\)	−2.29273	−	2.78269i	−0.635890	−	0.771780i
							\(17\)	−3.22444	−	5.58489i	−0.782040	−	1.35453i	−0.930751	−	0.365653i	\(-0.880846\pi\)
0.148711	−	0.988881i	\(-0.452488\pi\)
\(19\)	0.786677	+	2.93592i	0.180476	+	0.673546i	0.995554	+	0.0941943i	\(0.0300275\pi\)
							−0.815078	+	0.579352i	\(0.803306\pi\)
\(23\)	2.97203	+	1.71590i	0.619712	+	0.357791i	0.776757	−	0.629801i	\(-0.216863\pi\)
							−0.157045	+	0.987591i	\(0.550197\pi\)
\(29\)	−4.40371			−0.817748			−0.408874	−	0.912591i	\(-0.634079\pi\)
							−0.408874	+	0.912591i	\(0.634079\pi\)
\(31\)	3.24019	+	0.868206i	0.581955	+	0.155934i	0.537774	−	0.843089i	\(-0.319265\pi\)
							0.0441812	+	0.999024i	\(0.485932\pi\)
\(37\)	−1.00744	−	3.75982i	−0.165622	−	0.618110i	−0.997960	−	0.0638414i	\(-0.979665\pi\)
							0.832338	−	0.554268i	\(-0.187002\pi\)
\(41\)	3.03989	−	3.03989i	0.474751	−	0.474751i	−0.428697	−	0.903448i	\(-0.641027\pi\)
							0.903448	+	0.428697i	\(0.141027\pi\)
\(43\)			4.48958i			0.684654i	0.939581	+	0.342327i	\(0.111215\pi\)
							−0.939581	+	0.342327i	\(0.888785\pi\)
\(47\)	−5.51561	+	1.47790i	−0.804535	+	0.215574i	−0.637574	−	0.770389i	\(-0.720062\pi\)
							−0.166961	+	0.985964i	\(0.553395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.47.3	yes	32
3.2	odd	2		819.2.fn.e.775.6		32
7.2	even	3		637.2.i.a.489.11		32
7.3	odd	6	inner	91.2.bb.a.73.6	yes	32
7.4	even	3		637.2.bc.b.619.6		32
7.5	odd	6		637.2.i.a.489.12		32
7.6	odd	2		637.2.bc.b.411.3		32
13.5	odd	4	inner	91.2.bb.a.5.6	✓	32
21.17	even	6		819.2.fn.e.73.3		32
39.5	even	4		819.2.fn.e.460.3		32
91.5	even	12		637.2.i.a.538.12		32
91.18	odd	12		637.2.bc.b.31.3		32
91.31	even	12	inner	91.2.bb.a.31.3	yes	32
91.44	odd	12		637.2.i.a.538.11		32
91.83	even	4		637.2.bc.b.460.6		32
273.122	odd	12		819.2.fn.e.577.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.6	✓	32	13.5	odd	4	inner
91.2.bb.a.31.3	yes	32	91.31	even	12	inner
91.2.bb.a.47.3	yes	32	1.1	even	1	trivial
91.2.bb.a.73.6	yes	32	7.3	odd	6	inner
637.2.i.a.489.11		32	7.2	even	3
637.2.i.a.489.12		32	7.5	odd	6
637.2.i.a.538.11		32	91.44	odd	12
637.2.i.a.538.12		32	91.5	even	12
637.2.bc.b.31.3		32	91.18	odd	12
637.2.bc.b.411.3		32	7.6	odd	2
637.2.bc.b.460.6		32	91.83	even	4
637.2.bc.b.619.6		32	7.4	even	3
819.2.fn.e.73.3		32	21.17	even	6
819.2.fn.e.460.3		32	39.5	even	4
819.2.fn.e.577.6		32	273.122	odd	12
819.2.fn.e.775.6		32	3.2	odd	2