Embedded newform 507.2.x.b.2.2

• Label: 507.2.x.b.2.2
• Level: $507$
• Weight: $2$
• Character: 507.2
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $2784$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(2784\)
Relative dimension:	\(58\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0547878 + 2.72020i) q^{2} +(1.34843 + 1.08707i) q^{3} +(-5.39809 - 0.217536i) q^{4} +(-0.502675 + 2.74301i) q^{5} +(-3.03093 + 3.60845i) q^{6} +(-0.957757 + 1.44913i) q^{7} +(0.558940 - 9.24037i) q^{8} +(0.636546 + 2.93169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{156}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0547878	+	2.72020i	−0.0387409	+	1.92347i	0.253871	+	0.967238i	\(0.418296\pi\)
							−0.292612	+	0.956231i	\(0.594524\pi\)
\(3\)	1.34843	+	1.08707i	0.778518	+	0.627622i
							\(5\)	−0.502675	+	2.74301i	−0.224803	+	1.22671i	0.656560	+	0.754273i	\(0.272011\pi\)
−0.881364	+	0.472438i	\(0.843374\pi\)
\(7\)	−0.957757	+	1.44913i	−0.361998	+	0.547718i	−0.969025	−	0.246963i	\(-0.920567\pi\)
							0.607027	+	0.794681i	\(0.292362\pi\)
\(11\)	0.875610	+	0.482290i	0.264006	+	0.145416i	0.608953	−	0.793207i	\(-0.291590\pi\)
							−0.344946	+	0.938622i	\(0.612103\pi\)
\(13\)	2.69664	−	2.39335i	0.747913	−	0.663797i
							\(17\)	−0.249807	−	1.22364i	−0.0605872	−	0.296776i	0.938195	−	0.346106i	\(-0.112496\pi\)
−0.998783	+	0.0493306i	\(0.984291\pi\)
\(19\)	−0.305353	−	1.13959i	−0.0700528	−	0.261440i	0.922013	−	0.387158i	\(-0.126543\pi\)
							−0.992066	+	0.125718i	\(0.959877\pi\)
\(23\)	4.28516	+	7.42211i	0.893517	+	1.54762i	0.835629	+	0.549294i	\(0.185103\pi\)
							0.0578882	+	0.998323i	\(0.481563\pi\)
\(29\)	6.14945	−	5.90663i	1.14192	−	1.09683i	0.147753	−	0.989024i	\(-0.452796\pi\)
							0.994172	−	0.107810i	\(-0.0343837\pi\)
\(31\)	−8.41638	−	3.78790i	−1.51163	−	0.680328i	−0.526189	−	0.850368i	\(-0.676380\pi\)
							−0.985437	+	0.170040i	\(0.945610\pi\)
\(37\)	0.453201	−	0.0457886i	0.0745058	−	0.00752760i	−0.0631986	−	0.998001i	\(-0.520130\pi\)
							0.137704	+	0.990473i	\(0.456028\pi\)
\(41\)	−4.92568	−	0.699003i	−0.769261	−	0.109166i	−0.255164	−	0.966898i	\(-0.582129\pi\)
							−0.514097	+	0.857732i	\(0.671873\pi\)
\(43\)	4.83245	−	3.94557i	0.736941	−	0.601694i	−0.187837	−	0.982200i	\(-0.560148\pi\)
							0.924778	+	0.380506i	\(0.124250\pi\)
\(47\)	−7.60036	+	2.36837i	−1.10863	+	0.345462i	−0.796536	−	0.604591i	\(-0.793337\pi\)
							−0.312090	+	0.950053i	\(0.601029\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.2	✓	2784
3.2	odd	2	inner	507.2.x.b.2.57	yes	2784
169.85	odd	156	inner	507.2.x.b.254.57	yes	2784
507.254	even	156	inner	507.2.x.b.254.2	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.2	✓	2784	1.1	even	1	trivial
507.2.x.b.2.57	yes	2784	3.2	odd	2	inner
507.2.x.b.254.2	yes	2784	507.254	even	156	inner
507.2.x.b.254.57	yes	2784	169.85	odd	156	inner