Embedded newform 91.10.bb.a.5.11

• Label: 91.10.bb.a.5.11
• Level: $91$
• Weight: $10$
• Character: 91.5
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.bb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.41397 - 35.1334i) q^{2} +(55.9682 - 32.3132i) q^{3} +(-702.328 + 405.489i) q^{4} +(-1552.01 + 415.860i) q^{5} +(-1662.16 - 1662.16i) q^{6} +(-5033.95 - 3874.65i) q^{7} +(7689.56 + 7689.56i) q^{8} +(-7753.21 + 13429.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.41397	−	35.1334i	−0.416042	−	1.55269i	−0.782738	−	0.622351i	\(-0.786178\pi\)
							0.366696	−	0.930341i	\(-0.380489\pi\)
\(3\)	55.9682	−	32.3132i	0.398929	−	0.230322i	−0.287093	−	0.957903i	\(-0.592689\pi\)
							0.686022	+	0.727581i	\(0.259355\pi\)
\(5\)	−1552.01	+	415.860i	−1.11053	+	0.297565i	−0.767045	−	0.641593i	\(-0.778274\pi\)
							−0.343484	+	0.939159i	\(0.611607\pi\)
\(7\)	−5033.95	−	3874.65i	−0.792443	−	0.609947i
							\(11\)	9760.57	−	36426.9i	0.201006	−	0.750163i	−0.789625	−	0.613590i	\(-0.789725\pi\)
0.990630	−	0.136573i	\(-0.0436087\pi\)
\(13\)	101722.	+	16033.7i	0.987804	+	0.155700i
							\(17\)	270138.	+	467892.i	0.784450	+	1.35871i	0.929327	+	0.369257i	\(0.120388\pi\)
−0.144877	+	0.989450i	\(0.546279\pi\)
\(19\)	−777903.	+	208438.i	−1.36941	+	0.366933i	−0.867264	−	0.497849i	\(-0.834123\pi\)
							−0.502148	+	0.864782i	\(0.667457\pi\)
\(23\)	−528313.	−	305021.i	−0.393655	−	0.227277i	0.290088	−	0.957000i	\(-0.406316\pi\)
							−0.683743	+	0.729723i	\(0.739649\pi\)
\(29\)	443011.			0.116312			0.0581559	−	0.998308i	\(-0.481478\pi\)
							0.0581559	+	0.998308i	\(0.481478\pi\)
\(31\)	1.39963e6	−	5.22347e6i	0.272197	−	1.01585i	−0.685499	−	0.728074i	\(-0.740416\pi\)
							0.957696	−	0.287781i	\(-0.0929175\pi\)
\(37\)	−775219.	+	207719.i	−0.0680012	+	0.0182209i	−0.292659	−	0.956217i	\(-0.594540\pi\)
							0.224658	+	0.974438i	\(0.427873\pi\)
\(41\)	−7.66655e6	−	7.66655e6i	−0.423714	−	0.423714i	0.462767	−	0.886480i	\(-0.346857\pi\)
							−0.886480	+	0.462767i	\(0.846857\pi\)
\(43\)		−	875567.i		−	0.0390554i	−0.999809	−	0.0195277i	\(-0.993784\pi\)
							0.999809	−	0.0195277i	\(-0.00621626\pi\)
\(47\)	−1.51047e7	−	5.63714e7i	−0.451514	−	1.68507i	−0.698140	−	0.715962i	\(-0.745989\pi\)
							0.246626	−	0.969111i	\(-0.420678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.11	✓	328
7.3	odd	6	inner	91.10.bb.a.31.72	yes	328
13.8	odd	4	inner	91.10.bb.a.47.72	yes	328
91.73	even	12	inner	91.10.bb.a.73.11	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.11	✓	328	1.1	even	1	trivial
91.10.bb.a.31.72	yes	328	7.3	odd	6	inner
91.10.bb.a.47.72	yes	328	13.8	odd	4	inner
91.10.bb.a.73.11	yes	328	91.73	even	12	inner