Embedded newform 91.2.bb.a.47.8

• Label: 91.2.bb.a.47.8
• 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			47.8
Character	\(\chi\)	\(=\)	91.47
Dual form			91.2.bb.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.38839 - 0.639966i) q^{2} +(-1.77043 + 1.02216i) q^{3} +(3.56278 - 2.05697i) q^{4} +(-0.423894 - 1.58199i) q^{5} +(-3.57432 + 3.57432i) q^{6} +(-0.716327 + 2.54693i) q^{7} +(3.69606 - 3.69606i) q^{8} +(0.589612 - 1.02124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.38839	−	0.639966i	1.68884	−	0.452524i	0.718753	−	0.695266i	\(-0.244713\pi\)
							0.970091	+	0.242741i	\(0.0780466\pi\)
\(3\)	−1.77043	+	1.02216i	−1.02216	+	0.590143i	−0.914728	−	0.404070i	\(-0.867595\pi\)
							−0.107429	+	0.994213i	\(0.534262\pi\)
\(5\)	−0.423894	−	1.58199i	−0.189571	−	0.707488i	−0.993606	−	0.112906i	\(-0.963984\pi\)
							0.804035	−	0.594582i	\(-0.202683\pi\)
\(7\)	−0.716327	+	2.54693i	−0.270746	+	0.962651i
							\(11\)	−5.55653	−	1.48887i	−1.67536	−	0.448911i	−0.708811	−	0.705399i	\(-0.750768\pi\)
−0.966547	+	0.256488i	\(0.917435\pi\)
\(13\)	3.57432	+	0.473526i	0.991338	+	0.131333i
							\(17\)	−0.991968	−	1.71814i	−0.240588	−	0.416710i	0.720294	−	0.693669i	\(-0.244007\pi\)
−0.960882	+	0.276959i	\(0.910673\pi\)
\(19\)	−0.246155	−	0.918664i	−0.0564719	−	0.210756i	0.931925	−	0.362652i	\(-0.118129\pi\)
							−0.988396	+	0.151896i	\(0.951462\pi\)
\(23\)	3.06647	+	1.77043i	0.639404	+	0.369160i	0.784385	−	0.620274i	\(-0.212979\pi\)
							−0.144981	+	0.989434i	\(0.546312\pi\)
\(29\)	2.83949			0.527281			0.263640	−	0.964621i	\(-0.415077\pi\)
							0.263640	+	0.964621i	\(0.415077\pi\)
\(31\)	4.33096	+	1.16048i	0.777863	+	0.208428i	0.625842	−	0.779950i	\(-0.284755\pi\)
							0.152021	+	0.988377i	\(0.451422\pi\)
\(37\)	1.00161	+	3.73806i	0.164664	+	0.614534i	0.998083	+	0.0618923i	\(0.0197135\pi\)
							−0.833419	+	0.552642i	\(0.813620\pi\)
\(41\)	−4.02565	+	4.02565i	−0.628701	+	0.628701i	−0.947741	−	0.319040i	\(-0.896640\pi\)
							0.319040	+	0.947741i	\(0.396640\pi\)
\(43\)		−	5.30948i		−	0.809688i	−0.914386	−	0.404844i	\(-0.867326\pi\)
							0.914386	−	0.404844i	\(-0.132674\pi\)
\(47\)	0.448482	−	0.120170i	0.0654178	−	0.0175287i	−0.225962	−	0.974136i	\(-0.572552\pi\)
							0.291380	+	0.956608i	\(0.405886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.bb.a.47.8	yes	32
3.2	odd	2		819.2.fn.e.775.1		32
7.2	even	3		637.2.i.a.489.1		32
7.3	odd	6	inner	91.2.bb.a.73.1	yes	32
7.4	even	3		637.2.bc.b.619.1		32
7.5	odd	6		637.2.i.a.489.2		32
7.6	odd	2		637.2.bc.b.411.8		32
13.5	odd	4	inner	91.2.bb.a.5.1	✓	32
21.17	even	6		819.2.fn.e.73.8		32
39.5	even	4		819.2.fn.e.460.8		32
91.5	even	12		637.2.i.a.538.2		32
91.18	odd	12		637.2.bc.b.31.8		32
91.31	even	12	inner	91.2.bb.a.31.8	yes	32
91.44	odd	12		637.2.i.a.538.1		32
91.83	even	4		637.2.bc.b.460.1		32
273.122	odd	12		819.2.fn.e.577.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.1	✓	32	13.5	odd	4	inner
91.2.bb.a.31.8	yes	32	91.31	even	12	inner
91.2.bb.a.47.8	yes	32	1.1	even	1	trivial
91.2.bb.a.73.1	yes	32	7.3	odd	6	inner
637.2.i.a.489.1		32	7.2	even	3
637.2.i.a.489.2		32	7.5	odd	6
637.2.i.a.538.1		32	91.44	odd	12
637.2.i.a.538.2		32	91.5	even	12
637.2.bc.b.31.8		32	91.18	odd	12
637.2.bc.b.411.8		32	7.6	odd	2
637.2.bc.b.460.1		32	91.83	even	4
637.2.bc.b.619.1		32	7.4	even	3
819.2.fn.e.73.8		32	21.17	even	6
819.2.fn.e.460.8		32	39.5	even	4
819.2.fn.e.577.1		32	273.122	odd	12
819.2.fn.e.775.1		32	3.2	odd	2