Embedded newform 115.2.j.a.9.2

• Label: 115.2.j.a.9.2
• Level: $115$
• Weight: $2$
• Character: 115.9
• Analytic conductor: $0.918$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.918279623245\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(10\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	115.9
Dual form			115.2.j.a.64.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.651256 - 2.21797i) q^{2} +(0.917405 - 0.794936i) q^{3} +(-2.81277 + 1.80765i) q^{4} +(1.79096 - 1.33883i) q^{5} +(-2.36061 - 1.51707i) q^{6} +(-1.70619 - 0.779189i) q^{7} +(2.34716 + 2.03383i) q^{8} +(-0.217236 + 1.51091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 14 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.651256	−	2.21797i	−0.460507	−	1.56834i	−0.783156	−	0.621825i	\(-0.786391\pi\)
							0.322649	−	0.946519i	\(-0.395427\pi\)
\(3\)	0.917405	−	0.794936i	0.529664	−	0.458956i	−0.348505	−	0.937307i	\(-0.613311\pi\)
							0.878169	+	0.478351i	\(0.158765\pi\)
\(5\)	1.79096	−	1.33883i	0.800941	−	0.598743i
							\(7\)	−1.70619	−	0.779189i	−0.644878	−	0.294506i	0.0660019	−	0.997819i	\(-0.478976\pi\)
−0.710880	+	0.703314i	\(0.751703\pi\)
\(11\)	2.86226	+	0.840435i	0.863004	+	0.253401i	0.683137	−	0.730290i	\(-0.260615\pi\)
							0.179867	+	0.983691i	\(0.442433\pi\)
\(13\)	−1.22583	+	0.559818i	−0.339984	+	0.155265i	−0.578089	−	0.815974i	\(-0.696201\pi\)
							0.238105	+	0.971240i	\(0.423474\pi\)
\(17\)	−1.91813	+	2.98466i	−0.465214	+	0.723887i	−0.992018	−	0.126100i	\(-0.959754\pi\)
							0.526804	+	0.849987i	\(0.323390\pi\)
\(19\)	2.14666	−	1.37958i	0.492478	−	0.316496i	−0.270723	−	0.962657i	\(-0.587263\pi\)
							0.763201	+	0.646161i	\(0.223626\pi\)
\(23\)	4.31364	−	2.09582i	0.899457	−	0.437010i
							\(29\)	−3.30718	−	2.12540i	−0.614129	−	0.394677i	0.196275	−	0.980549i	\(-0.437116\pi\)
−0.810403	+	0.585872i	\(0.800752\pi\)
\(31\)	5.50738	−	6.35585i	0.989154	−	1.14155i	−0.000777292	−	1.00000i	\(-0.500247\pi\)
							0.989932	−	0.141545i	\(-0.0452071\pi\)
\(37\)	−3.81551	−	0.548587i	−0.627266	−	0.0901872i	−0.178651	−	0.983912i	\(-0.557173\pi\)
							−0.448615	+	0.893725i	\(0.648082\pi\)
\(41\)	1.67437	+	11.6455i	0.261492	+	1.81872i	0.521659	+	0.853154i	\(0.325313\pi\)
							−0.260167	+	0.965564i	\(0.583777\pi\)
\(43\)	−9.31512	+	8.07159i	−1.42054	+	1.23091i	−0.486531	+	0.873664i	\(0.661738\pi\)
							−0.934011	+	0.357243i	\(0.883717\pi\)
\(47\)			2.86157i			0.417403i	0.977979	+	0.208702i	\(0.0669238\pi\)
							−0.977979	+	0.208702i	\(0.933076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.j.a.9.2	✓	100
5.2	odd	4		575.2.k.g.101.9		100
5.3	odd	4		575.2.k.g.101.2		100
5.4	even	2	inner	115.2.j.a.9.9	yes	100
23.18	even	11	inner	115.2.j.a.64.9	yes	100
115.18	odd	44		575.2.k.g.501.2		100
115.64	even	22	inner	115.2.j.a.64.2	yes	100
115.87	odd	44		575.2.k.g.501.9		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.j.a.9.2	✓	100	1.1	even	1	trivial
115.2.j.a.9.9	yes	100	5.4	even	2	inner
115.2.j.a.64.2	yes	100	115.64	even	22	inner
115.2.j.a.64.9	yes	100	23.18	even	11	inner
575.2.k.g.101.2		100	5.3	odd	4
575.2.k.g.101.9		100	5.2	odd	4
575.2.k.g.501.2		100	115.18	odd	44
575.2.k.g.501.9		100	115.87	odd	44