Embedded newform 507.2.x.b.2.15

• Label: 507.2.x.b.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0337489 + 1.67562i) q^{2} +(-1.44454 + 0.955666i) q^{3} +(-0.808183 - 0.0325687i) q^{4} +(0.693384 - 3.78367i) q^{5} +(-1.55258 - 2.45276i) q^{6} +(-0.672208 + 1.01708i) q^{7} +(-0.120536 + 1.99269i) q^{8} +(1.17340 - 2.76100i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0337489	+	1.67562i	−0.0238641	+	1.18484i	0.793445	+	0.608642i	\(0.208285\pi\)
							−0.817309	+	0.576200i	\(0.804535\pi\)
\(3\)	−1.44454	+	0.955666i	−0.834007	+	0.551754i
							\(5\)	0.693384	−	3.78367i	0.310091	−	1.69211i	−0.346651	−	0.937994i	\(-0.612681\pi\)
0.656741	−	0.754116i	\(-0.271934\pi\)
\(7\)	−0.672208	+	1.01708i	−0.254071	+	0.384419i	−0.938847	−	0.344334i	\(-0.888105\pi\)
							0.684777	+	0.728753i	\(0.259900\pi\)
\(11\)	2.88205	+	1.58745i	0.868971	+	0.478633i	0.853406	−	0.521246i	\(-0.174533\pi\)
							0.0155647	+	0.999879i	\(0.495045\pi\)
\(13\)	−2.88054	+	2.16852i	−0.798918	+	0.601440i
							\(17\)	1.34190	+	6.57306i	0.325458	+	1.59420i	0.732484	+	0.680784i	\(0.238361\pi\)
−0.407026	+	0.913417i	\(0.633434\pi\)
\(19\)	1.00298	+	3.74317i	0.230099	+	0.858743i	0.980297	+	0.197529i	\(0.0632917\pi\)
							−0.750198	+	0.661214i	\(0.770042\pi\)
\(23\)	3.05558	+	5.29243i	0.637133	+	1.10355i	0.986059	+	0.166397i	\(0.0532133\pi\)
							−0.348925	+	0.937150i	\(0.613453\pi\)
\(29\)	5.09498	−	4.89379i	0.946113	−	0.908754i	−0.0498282	−	0.998758i	\(-0.515867\pi\)
							0.995941	+	0.0900037i	\(0.0286879\pi\)
\(31\)	1.93141	+	0.869255i	0.346891	+	0.156123i	0.576366	−	0.817192i	\(-0.304470\pi\)
							−0.229475	+	0.973314i	\(0.573701\pi\)
\(37\)	−2.93716	+	0.296752i	−0.482866	+	0.0487858i	−0.338951	−	0.940804i	\(-0.610072\pi\)
							−0.143916	+	0.989590i	\(0.545969\pi\)
\(41\)	6.84315	+	0.971112i	1.06872	+	0.151662i	0.652642	−	0.757667i	\(-0.273661\pi\)
							0.416078	+	0.909329i	\(0.363404\pi\)
\(43\)	5.60024	−	4.57245i	0.854028	−	0.697293i	−0.100653	−	0.994922i	\(-0.532093\pi\)
							0.954682	+	0.297629i	\(0.0961958\pi\)
\(47\)	−2.06572	+	0.643705i	−0.301317	+	0.0938940i	−0.444434	−	0.895812i	\(-0.646595\pi\)
							0.143117	+	0.989706i	\(0.454287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.15	✓	2784
3.2	odd	2	inner	507.2.x.b.2.44	yes	2784
169.85	odd	156	inner	507.2.x.b.254.44	yes	2784
507.254	even	156	inner	507.2.x.b.254.15	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.15	✓	2784	1.1	even	1	trivial
507.2.x.b.2.44	yes	2784	3.2	odd	2	inner
507.2.x.b.254.15	yes	2784	507.254	even	156	inner
507.2.x.b.254.44	yes	2784	169.85	odd	156	inner