Embedded newform 503.2.c.a.2.4

• Label: 503.2.c.a.2.4
• Level: $503$
• Weight: $2$
• Character: 503.2
• Analytic conductor: $4.016$
• Analytic rank: $0$
• Dimension: $10250$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	503.c (of order \(251\), degree \(250\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.01647522167\)
Analytic rank:	\(0\)
Dimension:	\(10250\)
Relative dimension:	\(41\) over \(\Q(\zeta_{251})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{251}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49378 - 1.83917i) q^{2} +(-0.567738 + 0.491382i) q^{3} +(-0.741049 + 3.53712i) q^{4} +(-0.276099 - 0.871972i) q^{5} +(1.75181 + 0.310147i) q^{6} +(0.530619 - 1.54068i) q^{7} +(3.39955 - 1.75096i) q^{8} +(-0.349454 + 2.41102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/503\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{101}{251}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49378	−	1.83917i	−1.05626	−	1.30049i	−0.952053	−	0.305933i	\(-0.901032\pi\)
							−0.104209	−	0.994555i	\(-0.533231\pi\)
\(3\)	−0.567738	+	0.491382i	−0.327783	+	0.283700i	−0.802764	−	0.596297i	\(-0.796638\pi\)
							0.474981	+	0.879996i	\(0.342455\pi\)
\(5\)	−0.276099	−	0.871972i	−0.123475	−	0.389958i	0.871461	−	0.490464i	\(-0.163173\pi\)
							−0.994936	+	0.100507i	\(0.967954\pi\)
\(7\)	0.530619	−	1.54068i	0.200555	−	0.582323i	−0.799271	−	0.600971i	\(-0.794781\pi\)
							0.999826	+	0.0186478i	\(0.00593611\pi\)
\(11\)	1.06340	−	0.344047i	0.320626	−	0.103734i	−0.143112	−	0.989706i	\(-0.545711\pi\)
							0.463738	+	0.885972i	\(0.346508\pi\)
\(13\)	1.04837	−	0.353742i	0.290766	−	0.0981103i	−0.170542	−	0.985350i	\(-0.554552\pi\)
							0.461308	+	0.887240i	\(0.347380\pi\)
\(17\)	0.703646	+	0.351344i	0.170659	+	0.0852135i	0.530020	−	0.847985i	\(-0.322184\pi\)
							−0.359361	+	0.933199i	\(0.617005\pi\)
\(19\)	2.56369	+	5.75168i	0.588151	+	1.31953i	0.925714	+	0.378224i	\(0.123465\pi\)
							−0.337564	+	0.941303i	\(0.609603\pi\)
\(23\)	1.59661	−	5.51876i	0.332916	−	1.15074i	−0.602773	−	0.797913i	\(-0.705937\pi\)
							0.935688	−	0.352827i	\(-0.114780\pi\)
\(29\)	0.369203	−	3.09038i	0.0685593	−	0.573870i	−0.916297	−	0.400499i	\(-0.868837\pi\)
							0.984857	−	0.173371i	\(-0.0554659\pi\)
\(31\)	−0.367397	−	4.50594i	−0.0659864	−	0.809292i	−0.943849	−	0.330378i	\(-0.892824\pi\)
							0.877862	−	0.478913i	\(-0.158969\pi\)
\(37\)	−1.71657	−	2.16837i	−0.282202	−	0.356477i	0.622427	−	0.782678i	\(-0.286147\pi\)
							−0.904629	+	0.426200i	\(0.859852\pi\)
\(41\)	0.592561	−	3.01598i	0.0925425	−	0.471017i	−0.906288	−	0.422661i	\(-0.861096\pi\)
							0.998830	−	0.0483556i	\(-0.0153981\pi\)
\(43\)	−0.191768	+	1.04502i	−0.0292443	+	0.159365i	−0.994966	−	0.100216i	\(-0.968047\pi\)
							0.965721	+	0.259581i	\(0.0835844\pi\)
\(47\)	1.96389	−	0.0737766i	0.286463	−	0.0107614i	0.106079	−	0.994358i	\(-0.466170\pi\)
							0.180384	+	0.983596i	\(0.442266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.4	✓	10250
503.252	even	251	inner	503.2.c.a.252.4	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.4	✓	10250	1.1	even	1	trivial
503.2.c.a.252.4	yes	10250	503.252	even	251	inner