Embedded newform 507.2.x.b.2.10

• Label: 507.2.x.b.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0409543 + 2.03336i) q^{2} +(0.256039 + 1.71302i) q^{3} +(-2.13452 - 0.0860180i) q^{4} +(-0.267381 + 1.45905i) q^{5} +(-3.49368 + 0.450464i) q^{6} +(0.531912 - 0.804804i) q^{7} +(0.0167307 - 0.276592i) q^{8} +(-2.86889 + 0.877200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0409543	+	2.03336i	−0.0289590	+	1.43781i	0.680339	+	0.732897i	\(0.261832\pi\)
							−0.709298	+	0.704908i	\(0.750988\pi\)
\(3\)	0.256039	+	1.71302i	0.147824	+	0.989014i
							\(5\)	−0.267381	+	1.45905i	−0.119576	+	0.652507i	0.868149	+	0.496304i	\(0.165310\pi\)
−0.987725	+	0.156203i	\(0.950075\pi\)
\(7\)	0.531912	−	0.804804i	0.201044	−	0.304187i	−0.719753	−	0.694230i	\(-0.755745\pi\)
							0.920797	+	0.390043i	\(0.127540\pi\)
\(11\)	4.65244	+	2.56259i	1.40276	+	0.772649i	0.990382	−	0.138362i	\(-0.0441837\pi\)
							0.412383	+	0.911011i	\(0.364697\pi\)
\(13\)	−2.45047	+	2.64484i	−0.679638	+	0.733547i
							\(17\)	−0.262597	−	1.28629i	−0.0636892	−	0.311970i	0.935450	−	0.353460i	\(-0.114995\pi\)
−0.999139	+	0.0414894i	\(0.986790\pi\)
\(19\)	−1.44876	−	5.40683i	−0.332368	−	1.24041i	−0.906695	−	0.421787i	\(-0.861403\pi\)
							0.574327	−	0.818626i	\(-0.305264\pi\)
\(23\)	−1.40336	−	2.43070i	−0.292621	−	0.506835i	0.681807	−	0.731532i	\(-0.261194\pi\)
							−0.974429	+	0.224697i	\(0.927861\pi\)
\(29\)	0.647954	−	0.622368i	0.120322	−	0.115571i	−0.630149	−	0.776474i	\(-0.717006\pi\)
							0.750471	+	0.660903i	\(0.229827\pi\)
\(31\)	4.58769	+	2.06475i	0.823974	+	0.370841i	0.778126	−	0.628108i	\(-0.216170\pi\)
							0.0458476	+	0.998948i	\(0.485401\pi\)
\(37\)	1.44933	−	0.146432i	0.238269	−	0.0240732i	0.0193356	−	0.999813i	\(-0.493845\pi\)
							0.218933	+	0.975740i	\(0.429742\pi\)
\(41\)	5.20036	+	0.737984i	0.812160	+	0.115254i	0.534221	−	0.845345i	\(-0.320605\pi\)
							0.277939	+	0.960599i	\(0.410349\pi\)
\(43\)	−0.861357	+	0.703276i	−0.131356	+	0.107249i	−0.695846	−	0.718191i	\(-0.744970\pi\)
							0.564490	+	0.825440i	\(0.309073\pi\)
\(47\)	2.24178	−	0.698566i	0.326997	−	0.101896i	−0.129610	−	0.991565i	\(-0.541372\pi\)
							0.456607	+	0.889669i	\(0.349065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.10	✓	2784
3.2	odd	2	inner	507.2.x.b.2.49	yes	2784
169.85	odd	156	inner	507.2.x.b.254.49	yes	2784
507.254	even	156	inner	507.2.x.b.254.10	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.10	✓	2784	1.1	even	1	trivial
507.2.x.b.2.49	yes	2784	3.2	odd	2	inner
507.2.x.b.254.10	yes	2784	507.254	even	156	inner
507.2.x.b.254.49	yes	2784	169.85	odd	156	inner