Embedded newform 91.2.bb.a.47.1

• Label: 91.2.bb.a.47.1
• 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			47.1
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.1

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60347	+	0.697597i	−1.84093	+	0.493276i	−0.998932	−	0.0462053i	\(-0.985287\pi\)
							−0.841998	+	0.539481i	\(0.818621\pi\)
\(3\)	−0.657528	+	0.379624i	−0.379624	+	0.219176i	−0.677655	−	0.735380i	\(-0.737004\pi\)
							0.298031	+	0.954556i	\(0.403670\pi\)
\(5\)	0.654162	+	2.44137i	0.292550	+	1.09181i	0.943143	+	0.332386i	\(0.107854\pi\)
							−0.650593	+	0.759426i	\(0.725480\pi\)
\(7\)	−2.54528	−	0.722189i	−0.962025	−	0.272962i
							\(11\)	−2.08105	−	0.557615i	−0.627459	−	0.168127i	−0.0689427	−	0.997621i	\(-0.521963\pi\)
−0.558516	+	0.829493i	\(0.688629\pi\)
\(13\)	−1.44703	+	3.30244i	−0.401333	+	0.915932i
							\(17\)	0.700866	+	1.21393i	0.169985	+	0.294422i	0.938414	−	0.345512i	\(-0.112295\pi\)
−0.768429	+	0.639935i	\(0.778961\pi\)
\(19\)	0.541814	+	2.02208i	0.124301	+	0.463896i	0.999814	−	0.0192980i	\(-0.00614311\pi\)
							−0.875513	+	0.483194i	\(0.839476\pi\)
\(23\)	1.13887	+	0.657528i	0.237471	+	0.137104i	0.614014	−	0.789295i	\(-0.289554\pi\)
							−0.376543	+	0.926399i	\(0.622887\pi\)
\(29\)	−4.56814			−0.848282			−0.424141	−	0.905596i	\(-0.639424\pi\)
							−0.424141	+	0.905596i	\(0.639424\pi\)
\(31\)	7.03077	+	1.88389i	1.26276	+	0.338357i	0.827254	−	0.561828i	\(-0.189902\pi\)
							0.435509	+	0.900184i	\(0.356568\pi\)
\(37\)	0.591026	+	2.20574i	0.0971640	+	0.362621i	0.997339	−	0.0729080i	\(-0.0232280\pi\)
							−0.900175	+	0.435529i	\(0.856561\pi\)
\(41\)	2.69291	−	2.69291i	0.420562	−	0.420562i	−0.464835	−	0.885397i	\(-0.653886\pi\)
							0.885397	+	0.464835i	\(0.153886\pi\)
\(43\)			0.437721i			0.0667518i	0.999443	+	0.0333759i	\(0.0106258\pi\)
							−0.999443	+	0.0333759i	\(0.989374\pi\)
\(47\)	7.74178	−	2.07440i	1.12926	−	0.302583i	0.354632	−	0.935006i	\(-0.384606\pi\)
							0.774623	+	0.632423i	\(0.217940\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.8	✓	32	13.5	odd	4	inner
91.2.bb.a.31.1	yes	32	91.31	even	12	inner
91.2.bb.a.47.1	yes	32	1.1	even	1	trivial
91.2.bb.a.73.8	yes	32	7.3	odd	6	inner
637.2.i.a.489.15		32	7.2	even	3
637.2.i.a.489.16		32	7.5	odd	6
637.2.i.a.538.15		32	91.44	odd	12
637.2.i.a.538.16		32	91.5	even	12
637.2.bc.b.31.1		32	91.18	odd	12
637.2.bc.b.411.1		32	7.6	odd	2
637.2.bc.b.460.8		32	91.83	even	4
637.2.bc.b.619.8		32	7.4	even	3
819.2.fn.e.73.1		32	21.17	even	6
819.2.fn.e.460.1		32	39.5	even	4
819.2.fn.e.577.8		32	273.122	odd	12
819.2.fn.e.775.8		32	3.2	odd	2