Embedded newform 91.2.bb.a.5.7

• Label: 91.2.bb.a.5.7
• Level: $91$
• Weight: $2$
• Character: 91.5
• 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			5.7
Character	\(\chi\)	\(=\)	91.5
Dual form			91.2.bb.a.73.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.493585 + 1.84208i) q^{2} +(2.29307 - 1.32391i) q^{3} +(-1.41759 + 0.818448i) q^{4} +(-3.30509 + 0.885596i) q^{5} +(3.57057 + 3.57057i) q^{6} +(-1.12943 - 2.39257i) q^{7} +(0.489646 + 0.489646i) q^{8} +(2.00545 - 3.47355i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 12 q^{3} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.493585	+	1.84208i	0.349017	+	1.30255i	0.887849	+	0.460136i	\(0.152199\pi\)
							−0.538832	+	0.842414i	\(0.681134\pi\)
\(3\)	2.29307	−	1.32391i	1.32391	−	0.764357i	0.339557	−	0.940586i	\(-0.389723\pi\)
							0.984349	+	0.176228i	\(0.0563897\pi\)
\(5\)	−3.30509	+	0.885596i	−1.47808	+	0.396051i	−0.905694	−	0.423932i	\(-0.860649\pi\)
							−0.572388	+	0.819983i	\(0.693983\pi\)
\(7\)	−1.12943	−	2.39257i	−0.426884	−	0.904306i
							\(11\)	−0.445825	+	1.66384i	−0.134421	+	0.501667i	0.865578	+	0.500773i	\(0.166951\pi\)
−1.00000		0.000893225i	\(0.999716\pi\)
\(13\)	−3.57057	−	0.501030i	−0.990298	−	0.138961i
							\(17\)	−1.22596	−	2.12343i	−0.297339	−	0.515006i	0.678187	−	0.734889i	\(-0.262766\pi\)
−0.975526	+	0.219883i	\(0.929433\pi\)
\(19\)	5.03057	−	1.34794i	1.15409	−	0.309238i	0.369488	−	0.929235i	\(-0.379533\pi\)
							0.784604	+	0.619997i	\(0.212866\pi\)
\(23\)	3.97172	+	2.29307i	0.828160	+	0.478139i	0.853222	−	0.521547i	\(-0.174645\pi\)
							−0.0250620	+	0.999686i	\(0.507978\pi\)
\(29\)	0.184063			0.0341796			0.0170898	−	0.999854i	\(-0.494560\pi\)
							0.0170898	+	0.999854i	\(0.494560\pi\)
\(31\)	0.659317	−	2.46060i	0.118417	−	0.441938i	−0.881103	−	0.472924i	\(-0.843198\pi\)
							0.999520	+	0.0309869i	\(0.00986501\pi\)
\(37\)	−0.210190	+	0.0563202i	−0.0345550	+	0.00925899i	−0.276055	−	0.961142i	\(-0.589027\pi\)
							0.241500	+	0.970401i	\(0.422361\pi\)
\(41\)	4.63239	+	4.63239i	0.723458	+	0.723458i	0.969308	−	0.245850i	\(-0.0790671\pi\)
							−0.245850	+	0.969308i	\(0.579067\pi\)
\(43\)		−	0.562412i		−	0.0857671i	−0.999080	−	0.0428835i	\(-0.986346\pi\)
							0.999080	−	0.0428835i	\(-0.0136544\pi\)
\(47\)	0.998090	+	3.72492i	0.145586	+	0.543336i	0.999729	+	0.0232964i	\(0.00741615\pi\)
							−0.854142	+	0.520040i	\(0.825917\pi\)