Embedded newform 507.2.x.b.2.8

• Label: 507.2.x.b.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0447349 + 2.22107i) q^{2} +(1.12900 - 1.31353i) q^{3} +(-2.93279 - 0.118187i) q^{4} +(-0.425593 + 2.32239i) q^{5} +(2.86694 + 2.56635i) q^{6} +(-1.87109 + 2.83104i) q^{7} +(0.125436 - 2.07370i) q^{8} +(-0.450722 - 2.96595i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0447349	+	2.22107i	−0.0316324	+	1.57054i	0.603012	+	0.797732i	\(0.293967\pi\)
							−0.634645	+	0.772804i	\(0.718854\pi\)
\(3\)	1.12900	−	1.31353i	0.651828	−	0.758367i
							\(5\)	−0.425593	+	2.32239i	−0.190331	+	1.03860i	0.740199	+	0.672387i	\(0.234731\pi\)
−0.930530	+	0.366215i	\(0.880654\pi\)
\(7\)	−1.87109	+	2.83104i	−0.707207	+	1.07003i	0.286506	+	0.958078i	\(0.407506\pi\)
							−0.993713	+	0.111955i	\(0.964289\pi\)
\(11\)	0.0342399	+	0.0188595i	0.0103237	+	0.00568635i	0.487613	−	0.873060i	\(-0.337868\pi\)
							−0.477289	+	0.878746i	\(0.658381\pi\)
\(13\)	−2.98678	−	2.01969i	−0.828384	−	0.560160i
							\(17\)	0.951830	+	4.66237i	0.230853	+	1.13079i	0.915625	+	0.402034i	\(0.131697\pi\)
−0.684772	+	0.728757i	\(0.740098\pi\)
\(19\)	0.425622	+	1.58844i	0.0976443	+	0.364413i	0.997407	−	0.0719659i	\(-0.0229273\pi\)
							−0.899763	+	0.436379i	\(0.856261\pi\)
\(23\)	−0.180682	−	0.312950i	−0.0376747	−	0.0652546i	0.846573	−	0.532272i	\(-0.178662\pi\)
							−0.884248	+	0.467018i	\(0.845328\pi\)
\(29\)	−3.63732	+	3.49369i	−0.675433	+	0.648762i	−0.949798	−	0.312863i	\(-0.898712\pi\)
							0.274365	+	0.961626i	\(0.411532\pi\)
\(31\)	6.78425	+	3.05334i	1.21849	+	0.548396i	0.914619	−	0.404318i	\(-0.132491\pi\)
							0.303868	+	0.952714i	\(0.401722\pi\)
\(37\)	2.55498	−	0.258139i	0.420036	−	0.0424378i	0.111761	−	0.993735i	\(-0.464351\pi\)
							0.308275	+	0.951297i	\(0.400248\pi\)
\(41\)	0.859269	+	0.121939i	0.134195	+	0.0190437i	0.207277	−	0.978282i	\(-0.433540\pi\)
							−0.0730818	+	0.997326i	\(0.523283\pi\)
\(43\)	−1.05578	+	0.862021i	−0.161006	+	0.131457i	−0.709523	−	0.704682i	\(-0.751090\pi\)
							0.548518	+	0.836139i	\(0.315192\pi\)
\(47\)	−11.8268	+	3.68539i	−1.72512	+	0.537569i	−0.990086	−	0.140461i	\(-0.955142\pi\)
							−0.735034	+	0.678030i	\(0.762834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.8	✓	2784
3.2	odd	2	inner	507.2.x.b.2.51	yes	2784
169.85	odd	156	inner	507.2.x.b.254.51	yes	2784
507.254	even	156	inner	507.2.x.b.254.8	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.8	✓	2784	1.1	even	1	trivial
507.2.x.b.2.51	yes	2784	3.2	odd	2	inner
507.2.x.b.254.8	yes	2784	507.254	even	156	inner
507.2.x.b.254.51	yes	2784	169.85	odd	156	inner