Embedded newform 91.2.bb.a.73.6

• Label: 91.2.bb.a.73.6
• 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			73.6
Character	\(\chi\)	\(=\)	91.73
Dual form			91.2.bb.a.5.6

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.423548	−	1.58070i	0.299493	−	1.11772i	−0.638089	−	0.769962i	\(-0.720275\pi\)
							0.937583	−	0.347762i	\(-0.113058\pi\)
\(3\)	−1.71590	−	0.990678i	−0.990678	−	0.571968i	−0.0852012	−	0.996364i	\(-0.527153\pi\)
							−0.905477	+	0.424395i	\(0.860487\pi\)
\(5\)	−2.77274	−	0.742954i	−1.24001	−	0.332259i	−0.421539	−	0.906810i	\(-0.638510\pi\)
							−0.818469	+	0.574551i	\(0.805177\pi\)
\(7\)	1.92960	−	1.81015i	0.729321	−	0.684172i
							\(11\)	0.894725	+	3.33916i	0.269770	+	1.00679i	0.959266	+	0.282505i	\(0.0911654\pi\)
−0.689496	+	0.724290i	\(0.742168\pi\)
\(13\)	2.29273	+	2.78269i	0.635890	+	0.771780i
							\(17\)	3.22444	−	5.58489i	0.782040	−	1.35453i	−0.148711	−	0.988881i	\(-0.547512\pi\)
0.930751	−	0.365653i	\(-0.119154\pi\)
\(19\)	2.93592	+	0.786677i	0.673546	+	0.180476i	0.579352	−	0.815078i	\(-0.303306\pi\)
							0.0941943	+	0.995554i	\(0.469973\pi\)
\(23\)	−2.97203	+	1.71590i	−0.619712	+	0.357791i	−0.776757	−	0.629801i	\(-0.783137\pi\)
							0.157045	+	0.987591i	\(0.449803\pi\)
\(29\)	−4.40371			−0.817748			−0.408874	−	0.912591i	\(-0.634079\pi\)
							−0.408874	+	0.912591i	\(0.634079\pi\)
\(31\)	0.868206	+	3.24019i	0.155934	+	0.581955i	0.999024	+	0.0441812i	\(0.0140679\pi\)
							−0.843089	+	0.537774i	\(0.819265\pi\)
\(37\)	3.75982	+	1.00744i	0.618110	+	0.165622i	0.554268	−	0.832338i	\(-0.312998\pi\)
							0.0638414	+	0.997960i	\(0.479665\pi\)
\(41\)	−3.03989	+	3.03989i	−0.474751	+	0.474751i	−0.903448	−	0.428697i	\(-0.858973\pi\)
							0.428697	+	0.903448i	\(0.358973\pi\)
\(43\)			4.48958i			0.684654i	0.939581	+	0.342327i	\(0.111215\pi\)
							−0.939581	+	0.342327i	\(0.888785\pi\)
\(47\)	−1.47790	+	5.51561i	−0.215574	+	0.804535i	0.770389	+	0.637574i	\(0.220062\pi\)
							−0.985964	+	0.166961i	\(0.946605\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.6	✓	32	91.5	even	12	inner
91.2.bb.a.31.3	yes	32	13.5	odd	4	inner
91.2.bb.a.47.3	yes	32	7.5	odd	6	inner
91.2.bb.a.73.6	yes	32	1.1	even	1	trivial
637.2.i.a.489.11		32	7.3	odd	6
637.2.i.a.489.12		32	7.4	even	3
637.2.i.a.538.11		32	91.31	even	12
637.2.i.a.538.12		32	91.18	odd	12
637.2.bc.b.31.3		32	91.83	even	4
637.2.bc.b.411.3		32	7.2	even	3
637.2.bc.b.460.6		32	91.44	odd	12
637.2.bc.b.619.6		32	7.6	odd	2
819.2.fn.e.73.3		32	3.2	odd	2
819.2.fn.e.460.3		32	273.5	odd	12
819.2.fn.e.577.6		32	39.5	even	4
819.2.fn.e.775.6		32	21.5	even	6