Embedded newform 503.2.c.a.2.14

• Label: 503.2.c.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.766891 - 0.944209i) q^{2} +(-2.44234 + 2.11387i) q^{3} +(0.106699 - 0.509291i) q^{4} +(-0.375743 - 1.18667i) q^{5} +(3.86894 + 0.684973i) q^{6} +(-1.33365 + 3.87232i) q^{7} +(-2.72550 + 1.40379i) q^{8} +(1.06626 - 7.35660i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.766891	−	0.944209i	−0.542274	−	0.667657i	0.429638	−	0.903001i	\(-0.358641\pi\)
							−0.971912	+	0.235345i	\(0.924378\pi\)
\(3\)	−2.44234	+	2.11387i	−1.41009	+	1.22044i	−0.468591	+	0.883415i	\(0.655238\pi\)
							−0.941495	+	0.337028i	\(0.890578\pi\)
\(5\)	−0.375743	−	1.18667i	−0.168037	−	0.530693i	0.831660	−	0.555285i	\(-0.187391\pi\)
							−0.999698	+	0.0245914i	\(0.992172\pi\)
\(7\)	−1.33365	+	3.87232i	−0.504071	+	1.46360i	0.346657	+	0.937992i	\(0.387317\pi\)
							−0.850728	+	0.525606i	\(0.823839\pi\)
\(11\)	1.91069	−	0.618177i	0.576093	−	0.186387i	−0.00533361	−	0.999986i	\(-0.501698\pi\)
							0.581427	+	0.813599i	\(0.302495\pi\)
\(13\)	−4.28835	+	1.44697i	−1.18937	+	0.401318i	−0.843586	−	0.536994i	\(-0.819560\pi\)
							−0.345788	+	0.938313i	\(0.612388\pi\)
\(17\)	−0.395118	−	0.197290i	−0.0958303	−	0.0478500i	0.398170	−	0.917311i	\(-0.369645\pi\)
							−0.494001	+	0.869462i	\(0.664466\pi\)
\(19\)	1.39795	+	3.13633i	0.320712	+	0.719524i	0.999896	−	0.0144451i	\(-0.00459817\pi\)
							−0.679183	+	0.733969i	\(0.737666\pi\)
\(23\)	1.91271	−	6.61140i	0.398829	−	1.37857i	−0.469654	−	0.882851i	\(-0.655621\pi\)
							0.868482	−	0.495721i	\(-0.165096\pi\)
\(29\)	0.00817092	−	0.0683939i	0.00151730	−	0.0127004i	−0.992160	−	0.124973i	\(-0.960116\pi\)
							0.993677	+	0.112272i	\(0.0358129\pi\)
\(31\)	−0.470297	−	5.76797i	−0.0844679	−	1.03596i	−0.893758	−	0.448550i	\(-0.851941\pi\)
							0.809290	−	0.587409i	\(-0.199852\pi\)
\(37\)	0.0289454	+	0.0365638i	0.00475859	+	0.00601105i	0.786430	−	0.617680i	\(-0.211927\pi\)
							−0.781671	+	0.623691i	\(0.785632\pi\)
\(41\)	1.21709	−	6.19468i	0.190078	−	0.967447i	−0.758685	−	0.651458i	\(-0.774158\pi\)
							0.948763	−	0.315989i	\(-0.102336\pi\)
\(43\)	2.16961	−	11.8231i	0.330863	−	1.80301i	−0.227899	−	0.973685i	\(-0.573186\pi\)
							0.558762	−	0.829328i	\(-0.311277\pi\)
\(47\)	10.8232	−	0.406591i	1.57873	−	0.0593074i	0.766344	−	0.642430i	\(-0.222074\pi\)
							0.812384	+	0.583123i	\(0.198169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.14	✓	10250
503.252	even	251	inner	503.2.c.a.252.14	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.14	✓	10250	1.1	even	1	trivial
503.2.c.a.252.14	yes	10250	503.252	even	251	inner