Embedded newform 507.2.s.a.5.16

• Label: 507.2.s.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.287822 + 1.57060i) q^{2} +(1.61828 - 0.617401i) q^{3} +(-0.513895 - 0.194895i) q^{4} +(-0.563686 - 0.0340967i) q^{5} +(0.503912 + 2.71936i) q^{6} +(2.05136 + 1.24009i) q^{7} +(-1.19811 + 1.98191i) q^{8} +(2.23763 - 1.99825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.287822	+	1.57060i	−0.203521	+	1.11058i	0.710127	+	0.704073i	\(0.248637\pi\)
							−0.913648	+	0.406505i	\(0.866747\pi\)
\(3\)	1.61828	−	0.617401i	0.934312	−	0.356457i
							\(5\)	−0.563686	−	0.0340967i	−0.252088	−	0.0152485i	−0.0661488	−	0.997810i	\(-0.521071\pi\)
−0.185939	+	0.982561i	\(0.559533\pi\)
\(7\)	2.05136	+	1.24009i	0.775340	+	0.468709i	0.848882	−	0.528583i	\(-0.177276\pi\)
							−0.0735416	+	0.997292i	\(0.523430\pi\)
\(11\)	−0.235347	−	1.28425i	−0.0709598	−	0.387215i	−0.999858	−	0.0168788i	\(-0.994627\pi\)
							0.928898	−	0.370336i	\(-0.120758\pi\)
\(13\)	1.58427	+	3.23884i	0.439398	+	0.898293i
							\(17\)	2.94651	−	0.726251i	0.714635	−	0.176142i	0.134797	−	0.990873i	\(-0.456962\pi\)
0.579838	+	0.814732i	\(0.303116\pi\)
\(19\)	0.734211	−	0.734211i	0.168439	−	0.168439i	−0.617854	−	0.786293i	\(-0.711998\pi\)
							0.786293	+	0.617854i	\(0.211998\pi\)
\(23\)	−2.68577			−0.560023			−0.280011	−	0.959997i	\(-0.590338\pi\)
							−0.280011	+	0.959997i	\(0.590338\pi\)
\(29\)	−3.25401	+	2.24608i	−0.604255	+	0.417087i	−0.830528	−	0.556977i	\(-0.811961\pi\)
							0.226273	+	0.974064i	\(0.427346\pi\)
\(31\)	6.78244	+	5.31370i	1.21816	+	0.954369i	0.999769	−	0.0214924i	\(-0.00684176\pi\)
							0.218393	+	0.975861i	\(0.429919\pi\)
\(37\)	4.81493	−	6.14580i	0.791569	−	1.01036i	−0.207912	−	0.978148i	\(-0.566667\pi\)
							0.999481	−	0.0322158i	\(-0.0102564\pi\)
\(41\)	3.47479	+	11.1510i	0.542671	+	1.74149i	0.662862	+	0.748741i	\(0.269341\pi\)
							−0.120192	+	0.992751i	\(0.538351\pi\)
\(43\)	−1.84364	−	0.223858i	−0.281152	−	0.0341380i	−0.0212539	−	0.999774i	\(-0.506766\pi\)
							−0.259898	+	0.965636i	\(0.583689\pi\)
\(47\)	−5.07004	−	2.28184i	−0.739541	−	0.332840i	0.00538688	−	0.999985i	\(-0.498285\pi\)
							−0.744928	+	0.667145i	\(0.767516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.16	✓	1392
3.2	odd	2	inner	507.2.s.a.5.43	yes	1392
169.34	odd	52	inner	507.2.s.a.203.43	yes	1392
507.203	even	52	inner	507.2.s.a.203.16	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.16	✓	1392	1.1	even	1	trivial
507.2.s.a.5.43	yes	1392	3.2	odd	2	inner
507.2.s.a.203.16	yes	1392	507.203	even	52	inner
507.2.s.a.203.43	yes	1392	169.34	odd	52	inner