Embedded newform 507.2.x.b.2.3

• Label: 507.2.x.b.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0543226 + 2.69710i) q^{2} +(-0.327883 - 1.70073i) q^{3} +(-5.27300 - 0.212495i) q^{4} +(0.410508 - 2.24007i) q^{5} +(4.60485 - 0.791944i) q^{6} +(-1.50010 + 2.26971i) q^{7} +(0.533803 - 8.82481i) q^{8} +(-2.78499 + 1.11528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0543226	+	2.69710i	−0.0384119	+	1.90714i	0.281299	+	0.959620i	\(0.409235\pi\)
							−0.319711	+	0.947515i	\(0.603586\pi\)
\(3\)	−0.327883	−	1.70073i	−0.189303	−	0.981919i
							\(5\)	0.410508	−	2.24007i	0.183585	−	1.00179i	−0.754738	−	0.656027i	\(-0.772236\pi\)
0.938322	−	0.345762i	\(-0.112380\pi\)
\(7\)	−1.50010	+	2.26971i	−0.566983	+	0.857868i	−0.999032	−	0.0439818i	\(-0.985996\pi\)
							0.432050	+	0.901850i	\(0.357791\pi\)
\(11\)	5.27437	+	2.90515i	1.59028	+	0.875935i	0.997507	+	0.0705680i	\(0.0224812\pi\)
							0.592777	+	0.805367i	\(0.298032\pi\)
\(13\)	0.352941	+	3.58824i	0.0978883	+	0.995197i
							\(17\)	−0.394632	−	1.93303i	−0.0957123	−	0.468830i	−0.999192	−	0.0401983i	\(-0.987201\pi\)
0.903479	−	0.428631i	\(-0.141004\pi\)
\(19\)	1.11791	+	4.17210i	0.256467	+	0.957146i	0.967269	+	0.253754i	\(0.0816654\pi\)
							−0.710802	+	0.703392i	\(0.751668\pi\)
\(23\)	1.59632	+	2.76492i	0.332857	+	0.576525i	0.983071	−	0.183226i	\(-0.0586541\pi\)
							−0.650214	+	0.759751i	\(0.725321\pi\)
\(29\)	−4.50484	+	4.32695i	−0.836527	+	0.803495i	−0.982591	−	0.185780i	\(-0.940519\pi\)
							0.146064	+	0.989275i	\(0.453339\pi\)
\(31\)	−2.18208	−	0.982073i	−0.391912	−	0.176385i	0.204869	−	0.978789i	\(-0.434323\pi\)
							−0.596781	+	0.802404i	\(0.703554\pi\)
\(37\)	2.56340	−	0.258989i	0.421420	−	0.0425776i	0.112468	−	0.993655i	\(-0.464124\pi\)
							0.308952	+	0.951078i	\(0.400022\pi\)
\(41\)	−5.86012	−	0.831611i	−0.915198	−	0.129876i	−0.333003	−	0.942926i	\(-0.608062\pi\)
							−0.582194	+	0.813050i	\(0.697806\pi\)
\(43\)	−3.43702	+	2.80624i	−0.524140	+	0.427947i	−0.857219	−	0.514952i	\(-0.827810\pi\)
							0.333079	+	0.942899i	\(0.391912\pi\)
\(47\)	9.60743	−	2.99380i	1.40139	−	0.436690i	0.498621	−	0.866820i	\(-0.333840\pi\)
							0.902767	+	0.430130i	\(0.141532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.3	✓	2784
3.2	odd	2	inner	507.2.x.b.2.56	yes	2784
169.85	odd	156	inner	507.2.x.b.254.56	yes	2784
507.254	even	156	inner	507.2.x.b.254.3	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.3	✓	2784	1.1	even	1	trivial
507.2.x.b.2.56	yes	2784	3.2	odd	2	inner
507.2.x.b.254.3	yes	2784	507.254	even	156	inner
507.2.x.b.254.56	yes	2784	169.85	odd	156	inner