Embedded newform 91.8.f.a.22.5

• Label: 91.8.f.a.22.5
• Level: $91$
• Weight: $8$
• Character: 91.22
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.5
Character	\(\chi\)	\(=\)	91.22
Dual form			91.8.f.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.31356 + 14.3995i) q^{2} +(3.55199 - 6.15223i) q^{3} +(-74.2305 - 128.571i) q^{4} -126.313 q^{5} +(59.0594 + 102.294i) q^{6} +(171.500 + 297.047i) q^{7} +340.208 q^{8} +(1068.27 + 1850.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{2} + 41 q^{3} - 1472 q^{4} + 526 q^{5} - 519 q^{6} + 8232 q^{7} + 474 q^{8} - 15427 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.31356	+	14.3995i	−0.734822	+	1.27275i	0.219980	+	0.975504i	\(0.429401\pi\)
							−0.954802	+	0.297244i	\(0.903932\pi\)
\(3\)	3.55199	−	6.15223i	0.0759535	−	0.131555i	−0.825547	−	0.564333i	\(-0.809133\pi\)
							0.901501	+	0.432778i	\(0.142467\pi\)
\(5\)	−126.313			−0.451911			−0.225956	−	0.974138i	\(-0.572550\pi\)
							−0.225956	+	0.974138i	\(0.572550\pi\)
\(7\)	171.500	+	297.047i	0.188982	+	0.327327i
							\(11\)	2330.09	−	4035.84i	0.527836	−	0.914239i	−0.471637	−	0.881793i	\(-0.656337\pi\)
0.999473	−	0.0324465i	\(-0.0103298\pi\)
\(13\)	7707.20	−	1829.65i	0.972959	−	0.230976i
							\(17\)	12861.4	+	22276.6i	0.634916	+	1.09971i	0.986533	+	0.163563i	\(0.0522988\pi\)
−0.351617	+	0.936144i	\(0.614368\pi\)
\(19\)	6757.27	+	11703.9i	0.226013	+	0.391466i	0.956623	−	0.291329i	\(-0.0940974\pi\)
							−0.730610	+	0.682795i	\(0.760764\pi\)
\(23\)	−27754.1	+	48071.4i	−0.475641	+	0.823834i	−0.999611	−	0.0279029i	\(-0.991117\pi\)
							0.523970	+	0.851737i	\(0.324450\pi\)
\(29\)	367.504	−	636.535i	0.00279814	−	0.00484651i	−0.864623	−	0.502421i	\(-0.832443\pi\)
							0.867421	+	0.497575i	\(0.165776\pi\)
\(31\)	67336.7			0.405962			0.202981	−	0.979183i	\(-0.434937\pi\)
							0.202981	+	0.979183i	\(0.434937\pi\)
\(37\)	−146243.	+	253300.i	−0.474644	+	0.822107i	−0.999578	−	0.0290355i	\(-0.990756\pi\)
							0.524935	+	0.851143i	\(0.324090\pi\)
\(41\)	315091.	−	545754.i	0.713991	−	1.23667i	−0.249356	−	0.968412i	\(-0.580219\pi\)
							0.963347	−	0.268257i	\(-0.0864476\pi\)
\(43\)	116159.	+	201194.i	0.222800	+	0.385900i	0.955657	−	0.294482i	\(-0.0951471\pi\)
							−0.732857	+	0.680382i	\(0.761814\pi\)
\(47\)	−1.26684e6			−1.77983			−0.889915	−	0.456126i	\(-0.849237\pi\)
							−0.889915	+	0.456126i	\(0.849237\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.f.a.22.5	✓	48
13.3	even	3	inner	91.8.f.a.29.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.f.a.22.5	✓	48	1.1	even	1	trivial
91.8.f.a.29.5	yes	48	13.3	even	3	inner