Embedded newform 507.2.s.a.5.13

• Label: 507.2.s.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.315367 + 1.72090i) q^{2} +(1.06756 - 1.36393i) q^{3} +(-0.992015 - 0.376222i) q^{4} +(3.73990 + 0.226222i) q^{5} +(2.01052 + 2.26731i) q^{6} +(2.39720 + 1.44916i) q^{7} +(-0.849938 + 1.40597i) q^{8} +(-0.720612 - 2.91217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.315367	+	1.72090i	−0.222998	+	1.21686i	0.661366	+	0.750063i	\(0.269977\pi\)
							−0.884364	+	0.466798i	\(0.845408\pi\)
\(3\)	1.06756	−	1.36393i	0.616359	−	0.787466i
							\(5\)	3.73990	+	0.226222i	1.67253	+	0.101170i	0.869232	−	0.494405i	\(-0.164614\pi\)
0.803300	+	0.595574i	\(0.203075\pi\)
\(7\)	2.39720	+	1.44916i	0.906057	+	0.547730i	0.891932	−	0.452170i	\(-0.149350\pi\)
							0.0141248	+	0.999900i	\(0.495504\pi\)
\(11\)	0.260858	+	1.42346i	0.0786517	+	0.429188i	0.999252	+	0.0386812i	\(0.0123157\pi\)
							−0.920600	+	0.390507i	\(0.872300\pi\)
\(13\)	0.121358	−	3.60351i	0.0336587	−	0.999433i
							\(17\)	−7.89137	+	1.94505i	−1.91394	+	0.471743i	−0.916514	+	0.400003i	\(0.869009\pi\)
−0.997423	+	0.0717403i	\(0.977145\pi\)
\(19\)	3.41908	−	3.41908i	0.784392	−	0.784392i	−0.196177	−	0.980569i	\(-0.562853\pi\)
							0.980569	+	0.196177i	\(0.0628527\pi\)
\(23\)	−4.72662			−0.985569			−0.492785	−	0.870151i	\(-0.664021\pi\)
							−0.492785	+	0.870151i	\(0.664021\pi\)
\(29\)	1.84858	−	1.27598i	0.343272	−	0.236943i	−0.383938	−	0.923359i	\(-0.625432\pi\)
							0.727210	+	0.686416i	\(0.240817\pi\)
\(31\)	−4.32331	−	3.38710i	−0.776489	−	0.608340i	0.148247	−	0.988950i	\(-0.452637\pi\)
							−0.924736	+	0.380610i	\(0.875714\pi\)
\(37\)	−7.17389	+	9.15679i	−1.17938	+	1.50537i	−0.359202	+	0.933260i	\(0.616951\pi\)
							−0.820179	+	0.572108i	\(0.806126\pi\)
\(41\)	1.12563	+	3.61227i	0.175794	+	0.564141i	0.999998	−	0.00215155i	\(-0.000684859\pi\)
							−0.824204	+	0.566293i	\(0.808377\pi\)
\(43\)	4.10862	+	0.498877i	0.626558	+	0.0760780i	0.427656	−	0.903941i	\(-0.359339\pi\)
							0.198902	+	0.980019i	\(0.436262\pi\)
\(47\)	4.04344	+	1.81980i	0.589796	+	0.265446i	0.683258	−	0.730177i	\(-0.260562\pi\)
							−0.0934616	+	0.995623i	\(0.529793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.13	✓	1392
3.2	odd	2	inner	507.2.s.a.5.46	yes	1392
169.34	odd	52	inner	507.2.s.a.203.46	yes	1392
507.203	even	52	inner	507.2.s.a.203.13	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.13	✓	1392	1.1	even	1	trivial
507.2.s.a.5.46	yes	1392	3.2	odd	2	inner
507.2.s.a.203.13	yes	1392	507.203	even	52	inner
507.2.s.a.203.46	yes	1392	169.34	odd	52	inner