Embedded newform 507.2.s.a.5.5

• Label: 507.2.s.a.5.5
• Level: $507$
• Weight: $2$
• Character: 507.5
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $1392$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(58\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.436480 + 2.38179i) q^{2} +(1.15819 + 1.28787i) q^{3} +(-3.61239 - 1.37000i) q^{4} +(-0.777816 - 0.0470492i) q^{5} +(-3.57296 + 2.19643i) q^{6} +(3.90248 + 2.35913i) q^{7} +(2.33436 - 3.86150i) q^{8} +(-0.317208 + 2.98318i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{52}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.436480	+	2.38179i	−0.308638	+	1.68418i	0.354003	+	0.935244i	\(0.384820\pi\)
							−0.662641	+	0.748937i	\(0.730565\pi\)
\(3\)	1.15819	+	1.28787i	0.668679	+	0.743551i
							\(5\)	−0.777816	−	0.0470492i	−0.347850	−	0.0210410i	−0.114470	−	0.993427i	\(-0.536517\pi\)
−0.233380	+	0.972386i	\(0.574979\pi\)
\(7\)	3.90248	+	2.35913i	1.47500	+	0.891668i	0.999931	+	0.0117186i	\(0.00373022\pi\)
							0.475068	+	0.879949i	\(0.342424\pi\)
\(11\)	0.401992	+	2.19360i	0.121205	+	0.661395i	0.986965	+	0.160935i	\(0.0514510\pi\)
							−0.865760	+	0.500460i	\(0.833164\pi\)
\(13\)	−3.07952	+	1.87524i	−0.854106	+	0.520098i
							\(17\)	2.71123	−	0.668259i	0.657570	−	0.162077i	0.103608	−	0.994618i	\(-0.466961\pi\)
0.553962	+	0.832542i	\(0.313115\pi\)
\(19\)	4.51732	−	4.51732i	1.03634	−	1.03634i	0.0370302	−	0.999314i	\(-0.488210\pi\)
							0.999314	−	0.0370302i	\(-0.0117898\pi\)
\(23\)	−4.33705			−0.904338			−0.452169	−	0.891932i	\(-0.649350\pi\)
							−0.452169	+	0.891932i	\(0.649350\pi\)
\(29\)	7.82506	−	5.40125i	1.45308	−	1.00299i	0.459701	−	0.888074i	\(-0.347957\pi\)
							0.993376	−	0.114912i	\(-0.0366587\pi\)
\(31\)	−4.42063	−	3.46334i	−0.793968	−	0.622034i	0.135567	−	0.990768i	\(-0.456714\pi\)
							−0.929535	+	0.368734i	\(0.879791\pi\)
\(37\)	−2.61638	+	3.33957i	−0.430131	+	0.549021i	−0.952833	−	0.303494i	\(-0.901847\pi\)
							0.522703	+	0.852515i	\(0.324924\pi\)
\(41\)	−1.28150	−	4.11248i	−0.200137	−	0.642262i	−0.999161	−	0.0409626i	\(-0.986958\pi\)
							0.799024	−	0.601300i	\(-0.205350\pi\)
\(43\)	0.00737244		0.000895177i	0.00112429		0.000136513i	0.121099	−	0.992640i	\(-0.461358\pi\)
							−0.119975	+	0.992777i	\(0.538281\pi\)
\(47\)	6.36565	+	2.86494i	0.928525	+	0.417895i	0.817504	−	0.575923i	\(-0.195357\pi\)
							0.111021	+	0.993818i	\(0.464588\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.5	✓	1392
3.2	odd	2	inner	507.2.s.a.5.54	yes	1392
169.34	odd	52	inner	507.2.s.a.203.54	yes	1392
507.203	even	52	inner	507.2.s.a.203.5	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.5	✓	1392	1.1	even	1	trivial
507.2.s.a.5.54	yes	1392	3.2	odd	2	inner
507.2.s.a.203.5	yes	1392	507.203	even	52	inner
507.2.s.a.203.54	yes	1392	169.34	odd	52	inner