Embedded newform 507.2.s.a.5.20

• Label: 507.2.s.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.205529 + 1.12154i) q^{2} +(0.653740 - 1.60394i) q^{3} +(0.654428 + 0.248192i) q^{4} +(-3.58890 - 0.217089i) q^{5} +(1.66452 + 1.06285i) q^{6} +(-1.80817 - 1.09308i) q^{7} +(-1.59261 + 2.63450i) q^{8} +(-2.14525 - 2.09712i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.205529	+	1.12154i	−0.145331	+	0.793047i	0.827488	+	0.561483i	\(0.189769\pi\)
							−0.972820	+	0.231564i	\(0.925616\pi\)
\(3\)	0.653740	−	1.60394i	0.377437	−	0.926035i
							\(5\)	−3.58890	−	0.217089i	−1.60501	−	0.0970850i	−0.766598	−	0.642127i	\(-0.778052\pi\)
−0.838409	+	0.545042i	\(0.816514\pi\)
\(7\)	−1.80817	−	1.09308i	−0.683423	−	0.413144i	0.132595	−	0.991170i	\(-0.457669\pi\)
							−0.816018	+	0.578027i	\(0.803823\pi\)
\(11\)	1.03620	+	5.65434i	0.312425	+	1.70485i	0.647010	+	0.762482i	\(0.276019\pi\)
							−0.334585	+	0.942366i	\(0.608596\pi\)
\(13\)	2.04874	+	2.96693i	0.568219	+	0.822877i
							\(17\)	−4.54606	+	1.12050i	−1.10258	+	0.271762i	−0.748289	−	0.663373i	\(-0.769124\pi\)
−0.354292	+	0.935135i	\(0.615278\pi\)
\(19\)	−2.39174	+	2.39174i	−0.548703	+	0.548703i	−0.926066	−	0.377362i	\(-0.876831\pi\)
							0.377362	+	0.926066i	\(0.376831\pi\)
\(23\)	−1.16467			−0.242849			−0.121425	−	0.992601i	\(-0.538746\pi\)
							−0.121425	+	0.992601i	\(0.538746\pi\)
\(29\)	3.68914	−	2.54643i	0.685056	−	0.472860i	−0.173998	−	0.984746i	\(-0.555668\pi\)
							0.859053	+	0.511886i	\(0.171053\pi\)
\(31\)	2.84435	+	2.22841i	0.510861	+	0.400234i	0.838779	−	0.544472i	\(-0.183270\pi\)
							−0.327918	+	0.944706i	\(0.606347\pi\)
\(37\)	−3.98148	+	5.08199i	−0.654551	+	0.835473i	−0.994474	−	0.104980i	\(-0.966522\pi\)
							0.339923	+	0.940453i	\(0.389599\pi\)
\(41\)	−3.02507	−	9.70781i	−0.472437	−	1.51610i	−0.816605	−	0.577197i	\(-0.804147\pi\)
							0.344168	−	0.938908i	\(-0.388161\pi\)
\(43\)	−5.12736	−	0.622574i	−0.781915	−	0.0949417i	−0.280162	−	0.959953i	\(-0.590388\pi\)
							−0.501753	+	0.865011i	\(0.667311\pi\)
\(47\)	3.40765	+	1.53366i	0.497057	+	0.223707i	0.643405	−	0.765526i	\(-0.277521\pi\)
							−0.146348	+	0.989233i	\(0.546752\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.20	✓	1392
3.2	odd	2	inner	507.2.s.a.5.39	yes	1392
169.34	odd	52	inner	507.2.s.a.203.39	yes	1392
507.203	even	52	inner	507.2.s.a.203.20	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.20	✓	1392	1.1	even	1	trivial
507.2.s.a.5.39	yes	1392	3.2	odd	2	inner
507.2.s.a.203.20	yes	1392	507.203	even	52	inner
507.2.s.a.203.39	yes	1392	169.34	odd	52	inner