Embedded newform 115.2.j.a.64.2

• Label: 115.2.j.a.64.2
• Level: $115$
• Weight: $2$
• Character: 115.64
• 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			64.2
Character	\(\chi\)	\(=\)	115.64
Dual form			115.2.j.a.9.2

$q$-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 \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{11}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.j.a.9.2	✓	100	115.9	even	22	inner
115.2.j.a.9.9	yes	100	23.9	even	11	inner
115.2.j.a.64.2	yes	100	1.1	even	1	trivial
115.2.j.a.64.9	yes	100	5.4	even	2	inner
575.2.k.g.101.2		100	115.32	odd	44
575.2.k.g.101.9		100	115.78	odd	44
575.2.k.g.501.2		100	5.2	odd	4
575.2.k.g.501.9		100	5.3	odd	4