Embedded newform 116.2.l.b.39.11

• Label: 116.2.l.b.39.11
• Level: $116$
• Weight: $2$
• Character: 116.39
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.l (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			39.11
Character	\(\chi\)	\(=\)	116.39
Dual form			116.2.l.b.3.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.850690 + 1.12975i) q^{2} +(0.490607 - 1.40207i) q^{3} +(-0.552652 + 1.92213i) q^{4} +(1.23888 + 0.282765i) q^{5} +(2.00134 - 0.638469i) q^{6} +(-0.694506 - 1.44216i) q^{7} +(-2.64165 + 1.01078i) q^{8} +(0.620380 + 0.494736i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.850690	+	1.12975i	0.601529	+	0.798851i
							\(3\)	0.490607	−	1.40207i	0.283252	−	0.809487i	−0.711075	−	0.703117i	\(-0.751791\pi\)
0.994327	−	0.106371i	\(-0.0339231\pi\)
\(5\)	1.23888	+	0.282765i	0.554042	+	0.126456i	0.490370	−	0.871514i	\(-0.336862\pi\)
							0.0636720	+	0.997971i	\(0.479719\pi\)
\(7\)	−0.694506	−	1.44216i	−0.262499	−	0.545084i	0.727509	−	0.686098i	\(-0.240678\pi\)
							−0.990008	+	0.141014i	\(0.954964\pi\)
\(11\)	−2.27026	−	0.255797i	−0.684509	−	0.0771256i	−0.237140	−	0.971475i	\(-0.576210\pi\)
							−0.447368	+	0.894350i	\(0.647639\pi\)
\(13\)	−1.12266	+	0.895291i	−0.311370	+	0.248309i	−0.766688	−	0.642019i	\(-0.778097\pi\)
							0.455319	+	0.890329i	\(0.349525\pi\)
\(17\)	−2.82259	−	2.82259i	−0.684579	−	0.684579i	0.276449	−	0.961028i	\(-0.410842\pi\)
							−0.961028	+	0.276449i	\(0.910842\pi\)
\(19\)	1.65808	−	0.580188i	0.380390	−	0.133104i	−0.133309	−	0.991074i	\(-0.542560\pi\)
							0.513699	+	0.857970i	\(0.328275\pi\)
\(23\)	−4.79554	+	1.09455i	−0.999939	+	0.228230i	−0.691002	−	0.722853i	\(-0.742830\pi\)
							−0.308937	+	0.951082i	\(0.599973\pi\)
\(29\)	3.61693	+	3.98972i	0.671647	+	0.740872i
							\(31\)	1.91155	+	3.04222i	0.343325	+	0.546398i	0.973101	−	0.230378i	\(-0.0739964\pi\)
−0.629776	+	0.776776i	\(0.716853\pi\)
\(37\)	−0.666096	−	5.91176i	−0.109505	−	0.971888i	−0.921541	−	0.388280i	\(-0.873069\pi\)
							0.812036	−	0.583608i	\(-0.198359\pi\)
\(41\)	5.73546	−	5.73546i	0.895728	−	0.895728i	−0.0993269	−	0.995055i	\(-0.531669\pi\)
							0.995055	+	0.0993269i	\(0.0316690\pi\)
\(43\)	3.78278	+	2.37688i	0.576869	+	0.362471i	0.788646	−	0.614847i	\(-0.210782\pi\)
							−0.211778	+	0.977318i	\(0.567925\pi\)
\(47\)	−1.33877	+	11.8819i	−0.195280	+	1.73316i	0.388678	+	0.921374i	\(0.372932\pi\)
							−0.583958	+	0.811784i	\(0.698497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.39.11	yes	144
4.3	odd	2	inner	116.2.l.b.39.6	yes	144
29.3	odd	28	inner	116.2.l.b.3.6	✓	144
116.3	even	28	inner	116.2.l.b.3.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.3.6	✓	144	29.3	odd	28	inner
116.2.l.b.3.11	yes	144	116.3	even	28	inner
116.2.l.b.39.6	yes	144	4.3	odd	2	inner
116.2.l.b.39.11	yes	144	1.1	even	1	trivial