Embedded newform 91.10.u.a.30.2

• Label: 91.10.u.a.30.2
• Level: $91$
• Weight: $10$
• Character: 91.30
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.2
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-37.6617 + 21.7440i) q^{2} +84.7460 q^{3} +(689.605 - 1194.43i) q^{4} +(1069.03 + 617.207i) q^{5} +(-3191.68 + 1842.72i) q^{6} +(664.570 - 6317.59i) q^{7} +37713.2i q^{8} -12501.1 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−37.6617	+	21.7440i	−1.66443	+	0.960959i	−0.693870	+	0.720101i	\(0.744096\pi\)
							−0.970560	+	0.240858i	\(0.922571\pi\)
\(3\)	84.7460			0.604051			0.302026	−	0.953300i	\(-0.402337\pi\)
							0.302026	+	0.953300i	\(0.402337\pi\)
\(5\)	1069.03	+	617.207i	0.764938	+	0.441637i	0.831066	−	0.556174i	\(-0.187731\pi\)
							−0.0661277	+	0.997811i	\(0.521064\pi\)
\(7\)	664.570	−	6317.59i	0.104616	−	0.994513i
							\(11\)		−	35339.1i		−	0.727759i	−0.931446	−	0.363880i	\(-0.881452\pi\)
0.931446	−	0.363880i	\(-0.118548\pi\)
\(13\)	77147.6	−	68211.0i	0.749165	−	0.662383i
							\(17\)	56442.7	−	97761.6i	0.163903	−	0.283889i	−0.772362	−	0.635183i	\(-0.780925\pi\)
0.936265	+	0.351294i	\(0.114258\pi\)
\(19\)		−	547223.i		−	0.963326i	−0.876357	−	0.481663i	\(-0.840033\pi\)
							0.876357	−	0.481663i	\(-0.159967\pi\)
\(23\)	9079.67	+	15726.5i	0.00676542	+	0.0117181i	0.869388	−	0.494130i	\(-0.164513\pi\)
							−0.862623	+	0.505848i	\(0.831180\pi\)
\(29\)	−539806.	+	934972.i	−0.141725	+	0.245475i	−0.928146	−	0.372215i	\(-0.878598\pi\)
							0.786421	+	0.617691i	\(0.211932\pi\)
\(31\)	−5.96452e6	+	3.44362e6i	−1.15997	+	0.669711i	−0.951298	−	0.308274i	\(-0.900249\pi\)
							−0.208676	+	0.977985i	\(0.566915\pi\)
\(37\)	−9.20878e6	+	5.31669e6i	−0.807782	+	0.466373i	−0.846185	−	0.532889i	\(-0.821106\pi\)
							0.0384027	+	0.999262i	\(0.487773\pi\)
\(41\)	294924.	+	170274.i	0.0162998	+	0.00941069i	0.508128	−	0.861282i	\(-0.330338\pi\)
							−0.491828	+	0.870692i	\(0.663671\pi\)
\(43\)	1.26641e7	+	2.19349e7i	0.564895	+	0.978427i	0.997059	+	0.0766319i	\(0.0244166\pi\)
							−0.432165	+	0.901795i	\(0.642250\pi\)
\(47\)	−2.11333e7	−	1.22013e7i	−0.631722	−	0.364725i	0.149696	−	0.988732i	\(-0.452170\pi\)
							−0.781419	+	0.624007i	\(0.785504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.2	yes	164
7.4	even	3		91.10.k.a.4.2	✓	164
13.10	even	6		91.10.k.a.23.81	yes	164
91.88	even	6	inner	91.10.u.a.88.2	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.2	✓	164	7.4	even	3
91.10.k.a.23.81	yes	164	13.10	even	6
91.10.u.a.30.2	yes	164	1.1	even	1	trivial
91.10.u.a.88.2	yes	164	91.88	even	6	inner