Embedded newform 116.2.l.b.3.11

• Label: 116.2.l.b.3.11
• Level: $116$
• Weight: $2$
• Character: 116.3
• 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			3.11
Character	\(\chi\)	\(=\)	116.3
Dual form			116.2.l.b.39.11

$q$-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 \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{28}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.3.6	✓	144	4.3	odd	2	inner
116.2.l.b.3.11	yes	144	1.1	even	1	trivial
116.2.l.b.39.6	yes	144	29.10	odd	28	inner
116.2.l.b.39.11	yes	144	116.39	even	28	inner