Embedded newform 91.10.bb.a.5.5

• Label: 91.10.bb.a.5.5
• 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			5.5
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.6545 - 39.7631i) q^{2} +(-221.572 + 127.925i) q^{3} +(-1024.18 + 591.312i) q^{4} +(1246.07 - 333.883i) q^{5} +(7447.41 + 7447.41i) q^{6} +(4766.00 + 4199.86i) q^{7} +(19520.9 + 19520.9i) q^{8} +(22887.9 - 39642.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.6545	−	39.7631i	−0.470867	−	1.75730i	−0.636667	−	0.771139i	\(-0.719688\pi\)
							0.165801	−	0.986159i	\(-0.446979\pi\)
\(3\)	−221.572	+	127.925i	−1.57932	+	0.911818i	−0.584360	+	0.811494i	\(0.698654\pi\)
							−0.994955	+	0.100323i	\(0.968012\pi\)
\(5\)	1246.07	−	333.883i	0.891614	−	0.238907i	0.216203	−	0.976348i	\(-0.430633\pi\)
							0.675412	+	0.737441i	\(0.263966\pi\)
\(7\)	4766.00	+	4199.86i	0.750262	+	0.661141i
							\(11\)	−4396.22	+	16406.9i	−0.0905341	+	0.337878i	−0.996305	−	0.0858908i	\(-0.972626\pi\)
0.905770	+	0.423769i	\(0.139293\pi\)
\(13\)	43554.1	+	93314.2i	0.422945	+	0.906155i
							\(17\)	99997.1	+	173200.i	0.290380	+	0.502953i	0.973900	−	0.226979i	\(-0.0728849\pi\)
−0.683519	+	0.729932i	\(0.739552\pi\)
\(19\)	46586.3	−	12482.8i	0.0820100	−	0.0219745i	−0.217581	−	0.976042i	\(-0.569817\pi\)
							0.299591	+	0.954068i	\(0.403150\pi\)
\(23\)	−919590.	−	530925.i	−0.685202	−	0.395602i	0.116610	−	0.993178i	\(-0.462797\pi\)
							−0.801812	+	0.597576i	\(0.796131\pi\)
\(29\)	5.36115e6			1.40756			0.703780	−	0.710418i	\(-0.251494\pi\)
							0.703780	+	0.710418i	\(0.251494\pi\)
\(31\)	2.12576e6	−	7.93346e6i	0.413416	−	1.54289i	−0.374571	−	0.927198i	\(-0.622210\pi\)
							0.787987	−	0.615692i	\(-0.211123\pi\)
\(37\)	−1.67622e7	+	4.49141e6i	−1.47036	+	0.393981i	−0.903054	−	0.429528i	\(-0.858680\pi\)
							−0.567303	+	0.823509i	\(0.692013\pi\)
\(41\)	1.57377e7	+	1.57377e7i	0.869790	+	0.869790i	0.992449	−	0.122659i	\(-0.0391421\pi\)
							−0.122659	+	0.992449i	\(0.539142\pi\)
\(43\)		−	2.48787e7i		−	1.10974i	−0.831938	−	0.554869i	\(-0.812768\pi\)
							0.831938	−	0.554869i	\(-0.187232\pi\)
\(47\)	1.50040e6	+	5.59957e6i	0.0448505	+	0.167384i	0.984719	−	0.174154i	\(-0.0557189\pi\)
							−0.939868	+	0.341538i	\(0.889052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.5	✓	328
7.3	odd	6	inner	91.10.bb.a.31.78	yes	328
13.8	odd	4	inner	91.10.bb.a.47.78	yes	328
91.73	even	12	inner	91.10.bb.a.73.5	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.5	✓	328	1.1	even	1	trivial
91.10.bb.a.31.78	yes	328	7.3	odd	6	inner
91.10.bb.a.47.78	yes	328	13.8	odd	4	inner
91.10.bb.a.73.5	yes	328	91.73	even	12	inner