Embedded newform 507.2.s.a.5.10

• Label: 507.2.s.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.378713 + 2.06657i) q^{2} +(-1.55175 - 0.769460i) q^{3} +(-2.25726 - 0.856068i) q^{4} +(2.99779 + 0.181333i) q^{5} +(2.17781 - 2.91540i) q^{6} +(-2.77544 - 1.67781i) q^{7} +(0.450143 - 0.744627i) q^{8} +(1.81586 + 2.38802i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.378713	+	2.06657i	−0.267791	+	1.46129i	0.524639	+	0.851325i	\(0.324200\pi\)
							−0.792430	+	0.609963i	\(0.791184\pi\)
\(3\)	−1.55175	−	0.769460i	−0.895904	−	0.444248i
							\(5\)	2.99779	+	0.181333i	1.34065	+	0.0810946i	0.715065	−	0.699058i	\(-0.246397\pi\)
0.625589	+	0.780153i	\(0.284859\pi\)
\(7\)	−2.77544	−	1.67781i	−1.04902	−	0.634153i	−0.115729	−	0.993281i	\(-0.536920\pi\)
							−0.933288	+	0.359128i	\(0.883074\pi\)
\(11\)	−0.732370	−	3.99642i	−0.220818	−	1.20496i	−0.887920	−	0.459998i	\(-0.847850\pi\)
							0.667102	−	0.744967i	\(-0.267535\pi\)
\(13\)	3.27744	−	1.50278i	0.908999	−	0.416797i
							\(17\)	1.33823	−	0.329845i	0.324569	−	0.0799992i	−0.0736645	−	0.997283i	\(-0.523469\pi\)
0.398234	+	0.917284i	\(0.369623\pi\)
\(19\)	2.35729	−	2.35729i	0.540799	−	0.540799i	−0.382964	−	0.923763i	\(-0.625097\pi\)
							0.923763	+	0.382964i	\(0.125097\pi\)
\(23\)	4.94668			1.03145			0.515727	−	0.856753i	\(-0.327522\pi\)
							0.515727	+	0.856753i	\(0.327522\pi\)
\(29\)	2.98264	−	2.05877i	0.553862	−	0.382303i	−0.258007	−	0.966143i	\(-0.583066\pi\)
							0.811869	+	0.583840i	\(0.198450\pi\)
\(31\)	6.99197	+	5.47786i	1.25580	+	0.983853i	0.999853	+	0.0171606i	\(0.00546267\pi\)
							0.255942	+	0.966692i	\(0.417614\pi\)
\(37\)	−5.27437	+	6.73224i	−0.867102	+	1.10677i	0.126289	+	0.991993i	\(0.459693\pi\)
							−0.993391	+	0.114780i	\(0.963384\pi\)
\(41\)	−2.06500	−	6.62682i	−0.322499	−	1.03494i	−0.963571	−	0.267454i	\(-0.913818\pi\)
							0.641072	−	0.767481i	\(-0.278490\pi\)
\(43\)	−6.48251	−	0.787120i	−0.988574	−	0.120035i	−0.389755	−	0.920919i	\(-0.627440\pi\)
							−0.598819	+	0.800884i	\(0.704363\pi\)
\(47\)	7.77446	+	3.49900i	1.13402	+	0.510382i	0.888439	−	0.458994i	\(-0.151790\pi\)
							0.245582	+	0.969376i	\(0.421021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.10	✓	1392
3.2	odd	2	inner	507.2.s.a.5.49	yes	1392
169.34	odd	52	inner	507.2.s.a.203.49	yes	1392
507.203	even	52	inner	507.2.s.a.203.10	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.10	✓	1392	1.1	even	1	trivial
507.2.s.a.5.49	yes	1392	3.2	odd	2	inner
507.2.s.a.203.10	yes	1392	507.203	even	52	inner
507.2.s.a.203.49	yes	1392	169.34	odd	52	inner