Embedded newform 116.2.l.b.19.10

• Label: 116.2.l.b.19.10
• Level: $116$
• Weight: $2$
• Character: 116.19
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• 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			19.10
Character	\(\chi\)	\(=\)	116.19
Dual form			116.2.l.b.55.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01985 - 0.979750i) q^{2} +(0.981476 + 0.343433i) q^{3} +(0.0801814 - 1.99839i) q^{4} +(-1.24581 - 0.284348i) q^{5} +(1.33744 - 0.611351i) q^{6} +(0.960687 + 1.99489i) q^{7} +(-1.87615 - 2.11661i) q^{8} +(-1.50015 - 1.19633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01985	−	0.979750i	0.721142	−	0.692788i
							\(3\)	0.981476	+	0.343433i	0.566655	+	0.198281i	0.598376	−	0.801215i	\(-0.295813\pi\)
−0.0317205	+	0.999497i	\(0.510099\pi\)
\(5\)	−1.24581	−	0.284348i	−0.557143	−	0.127164i	−0.0653279	−	0.997864i	\(-0.520809\pi\)
							−0.491815	+	0.870700i	\(0.663666\pi\)
\(7\)	0.960687	+	1.99489i	0.363106	+	0.753996i	0.999854	−	0.0170750i	\(-0.00543541\pi\)
							−0.636749	+	0.771071i	\(0.719721\pi\)
\(11\)	−0.611136	+	5.42398i	−0.184264	+	1.63539i	0.472525	+	0.881317i	\(0.343343\pi\)
							−0.656789	+	0.754074i	\(0.728086\pi\)
\(13\)	3.14757	−	2.51011i	0.872979	−	0.696178i	−0.0807854	−	0.996732i	\(-0.525743\pi\)
							0.953765	+	0.300554i	\(0.0971714\pi\)
\(17\)	−3.76942	+	3.76942i	−0.914220	+	0.914220i	−0.996601	−	0.0823812i	\(-0.973747\pi\)
							0.0823812	+	0.996601i	\(0.473747\pi\)
\(19\)	−0.666071	−	1.90352i	−0.152807	−	0.436698i	0.841877	−	0.539669i	\(-0.181451\pi\)
							−0.994684	+	0.102971i	\(0.967165\pi\)
\(23\)	3.45095	−	0.787657i	0.719573	−	0.164238i	0.152972	−	0.988230i	\(-0.451115\pi\)
							0.566601	+	0.823993i	\(0.308258\pi\)
\(29\)	5.25014	+	1.19835i	0.974926	+	0.222527i
							\(31\)	−2.15705	+	1.35537i	−0.387418	+	0.243431i	−0.711631	−	0.702553i	\(-0.752043\pi\)
0.324213	+	0.945984i	\(0.394900\pi\)
\(37\)	2.49407	−	0.281015i	0.410023	−	0.0461985i	0.0954558	−	0.995434i	\(-0.469569\pi\)
							0.314567	+	0.949235i	\(0.398141\pi\)
\(41\)	−0.390752	−	0.390752i	−0.0610253	−	0.0610253i	0.675936	−	0.736961i	\(-0.263740\pi\)
							−0.736961	+	0.675936i	\(0.763740\pi\)
\(43\)	1.05847	−	1.68454i	0.161415	−	0.256891i	−0.756218	−	0.654320i	\(-0.772955\pi\)
							0.917633	+	0.397430i	\(0.130098\pi\)
\(47\)	−4.37990	−	0.493496i	−0.638874	−	0.0719838i	−0.213415	−	0.976962i	\(-0.568459\pi\)
							−0.425459	+	0.904978i	\(0.639887\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.19.10	yes	144
4.3	odd	2	inner	116.2.l.b.19.2	✓	144
29.26	odd	28	inner	116.2.l.b.55.2	yes	144
116.55	even	28	inner	116.2.l.b.55.10	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.19.2	✓	144	4.3	odd	2	inner
116.2.l.b.19.10	yes	144	1.1	even	1	trivial
116.2.l.b.55.2	yes	144	29.26	odd	28	inner
116.2.l.b.55.10	yes	144	116.55	even	28	inner