Embedded newform 91.2.bb.a.73.8

• Label: 91.2.bb.a.73.8
• 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			73.8
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.697597 - 2.60347i) q^{2} +(-0.657528 - 0.379624i) q^{3} +(-4.55935 - 2.63234i) q^{4} +(2.44137 + 0.654162i) q^{5} +(-1.44703 + 1.44703i) q^{6} +(0.722189 + 2.54528i) q^{7} +(-6.22207 + 6.22207i) q^{8} +(-1.21177 - 2.09885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.697597	−	2.60347i	0.493276	−	1.84093i	−0.0462053	−	0.998932i	\(-0.514713\pi\)
							0.539481	−	0.841998i	\(-0.318621\pi\)
\(3\)	−0.657528	−	0.379624i	−0.379624	−	0.219176i	0.298031	−	0.954556i	\(-0.403670\pi\)
							−0.677655	+	0.735380i	\(0.737004\pi\)
\(5\)	2.44137	+	0.654162i	1.09181	+	0.292550i	0.759426	−	0.650593i	\(-0.225480\pi\)
							0.332386	+	0.943143i	\(0.392146\pi\)
\(7\)	0.722189	+	2.54528i	0.272962	+	0.962025i
							\(11\)	0.557615	+	2.08105i	0.168127	+	0.627459i	0.997621	+	0.0689427i	\(0.0219626\pi\)
−0.829493	+	0.558516i	\(0.811371\pi\)
\(13\)	1.44703	−	3.30244i	0.401333	−	0.915932i
							\(17\)	−0.700866	+	1.21393i	−0.169985	+	0.294422i	−0.938414	−	0.345512i	\(-0.887705\pi\)
0.768429	+	0.639935i	\(0.221039\pi\)
\(19\)	2.02208	+	0.541814i	0.463896	+	0.124301i	0.483194	−	0.875513i	\(-0.339476\pi\)
							−0.0192980	+	0.999814i	\(0.506143\pi\)
\(23\)	−1.13887	+	0.657528i	−0.237471	+	0.137104i	−0.614014	−	0.789295i	\(-0.710446\pi\)
							0.376543	+	0.926399i	\(0.377113\pi\)
\(29\)	−4.56814			−0.848282			−0.424141	−	0.905596i	\(-0.639424\pi\)
							−0.424141	+	0.905596i	\(0.639424\pi\)
\(31\)	1.88389	+	7.03077i	0.338357	+	1.26276i	0.900184	+	0.435509i	\(0.143432\pi\)
							−0.561828	+	0.827254i	\(0.689902\pi\)
\(37\)	−2.20574	−	0.591026i	−0.362621	−	0.0971640i	0.0729080	−	0.997339i	\(-0.476772\pi\)
							−0.435529	+	0.900175i	\(0.643439\pi\)
\(41\)	−2.69291	+	2.69291i	−0.420562	+	0.420562i	−0.885397	−	0.464835i	\(-0.846114\pi\)
							0.464835	+	0.885397i	\(0.346114\pi\)
\(43\)			0.437721i			0.0667518i	0.999443	+	0.0333759i	\(0.0106258\pi\)
							−0.999443	+	0.0333759i	\(0.989374\pi\)
\(47\)	2.07440	−	7.74178i	0.302583	−	1.12926i	−0.632423	−	0.774623i	\(-0.717940\pi\)
							0.935006	−	0.354632i	\(-0.115394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.73.8	yes	32
3.2	odd	2		819.2.fn.e.73.1		32
7.2	even	3		637.2.bc.b.411.1		32
7.3	odd	6		637.2.i.a.489.15		32
7.4	even	3		637.2.i.a.489.16		32
7.5	odd	6	inner	91.2.bb.a.47.1	yes	32
7.6	odd	2		637.2.bc.b.619.8		32
13.5	odd	4	inner	91.2.bb.a.31.1	yes	32
21.5	even	6		819.2.fn.e.775.8		32
39.5	even	4		819.2.fn.e.577.8		32
91.5	even	12	inner	91.2.bb.a.5.8	✓	32
91.18	odd	12		637.2.i.a.538.16		32
91.31	even	12		637.2.i.a.538.15		32
91.44	odd	12		637.2.bc.b.460.8		32
91.83	even	4		637.2.bc.b.31.1		32
273.5	odd	12		819.2.fn.e.460.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.8	✓	32	91.5	even	12	inner
91.2.bb.a.31.1	yes	32	13.5	odd	4	inner
91.2.bb.a.47.1	yes	32	7.5	odd	6	inner
91.2.bb.a.73.8	yes	32	1.1	even	1	trivial
637.2.i.a.489.15		32	7.3	odd	6
637.2.i.a.489.16		32	7.4	even	3
637.2.i.a.538.15		32	91.31	even	12
637.2.i.a.538.16		32	91.18	odd	12
637.2.bc.b.31.1		32	91.83	even	4
637.2.bc.b.411.1		32	7.2	even	3
637.2.bc.b.460.8		32	91.44	odd	12
637.2.bc.b.619.8		32	7.6	odd	2
819.2.fn.e.73.1		32	3.2	odd	2
819.2.fn.e.460.1		32	273.5	odd	12
819.2.fn.e.577.8		32	39.5	even	4
819.2.fn.e.775.8		32	21.5	even	6