Embedded newform 503.2.c.a.2.5

• Label: 503.2.c.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47995 - 1.82214i) q^{2} +(1.66985 - 1.44527i) q^{3} +(-0.719831 + 3.43585i) q^{4} +(0.408432 + 1.28990i) q^{5} +(-5.10478 - 0.903771i) q^{6} +(0.254080 - 0.737735i) q^{7} +(3.15213 - 1.62353i) q^{8} +(0.269269 - 1.85780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47995	−	1.82214i	−1.04648	−	1.28845i	−0.956258	−	0.292526i	\(-0.905504\pi\)
							−0.0902253	−	0.995921i	\(-0.528759\pi\)
\(3\)	1.66985	−	1.44527i	0.964089	−	0.834428i	−0.0221500	−	0.999755i	\(-0.507051\pi\)
							0.986239	+	0.165327i	\(0.0528679\pi\)
\(5\)	0.408432	+	1.28990i	0.182656	+	0.576863i	1.00000		0.000712307i	\(0.000226734\pi\)
							−0.817344	+	0.576151i	\(0.804554\pi\)
\(7\)	0.254080	−	0.737735i	0.0960332	−	0.278838i	−0.887144	−	0.461493i	\(-0.847314\pi\)
							0.983177	+	0.182656i	\(0.0584693\pi\)
\(11\)	4.53328	−	1.46668i	1.36683	−	0.442221i	0.469251	−	0.883065i	\(-0.344524\pi\)
							0.897584	+	0.440844i	\(0.145321\pi\)
\(13\)	1.71146	−	0.577482i	0.474675	−	0.160165i	−0.0721809	−	0.997392i	\(-0.522996\pi\)
							0.546856	+	0.837227i	\(0.315825\pi\)
\(17\)	−4.79680	−	2.39514i	−1.16340	−	0.580906i	−0.242284	−	0.970205i	\(-0.577897\pi\)
							−0.921111	+	0.389299i	\(0.872717\pi\)
\(19\)	−2.36419	−	5.30410i	−0.542382	−	1.21684i	−0.952424	−	0.304777i	\(-0.901418\pi\)
							0.410042	−	0.912067i	\(-0.365514\pi\)
\(23\)	0.150566	−	0.520438i	0.0313951	−	0.108519i	−0.943376	−	0.331725i	\(-0.892369\pi\)
							0.974771	+	0.223206i	\(0.0716523\pi\)
\(29\)	0.808959	−	6.77132i	0.150220	−	1.25740i	−0.693436	−	0.720518i	\(-0.743904\pi\)
							0.843656	−	0.536884i	\(-0.180399\pi\)
\(31\)	0.360096	+	4.41640i	0.0646751	+	0.793209i	0.946723	+	0.322049i	\(0.104372\pi\)
							−0.882048	+	0.471160i	\(0.843835\pi\)
\(37\)	1.88535	+	2.38158i	0.309950	+	0.391529i	0.914199	−	0.405267i	\(-0.132821\pi\)
							−0.604248	+	0.796796i	\(0.706527\pi\)
\(41\)	1.08316	−	5.51299i	0.169161	−	0.860985i	−0.797184	−	0.603736i	\(-0.793678\pi\)
							0.966345	−	0.257249i	\(-0.0828160\pi\)
\(43\)	−0.402936	+	2.19577i	−0.0614472	+	0.334852i	−0.999946	−	0.0104316i	\(-0.996679\pi\)
							0.938498	+	0.345284i	\(0.112217\pi\)
\(47\)	−5.94387	+	0.223291i	−0.867002	+	0.0325703i	−0.467325	−	0.884086i	\(-0.654782\pi\)
							−0.399677	+	0.916656i	\(0.630878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.5	✓	10250
503.252	even	251	inner	503.2.c.a.252.5	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.5	✓	10250	1.1	even	1	trivial
503.2.c.a.252.5	yes	10250	503.252	even	251	inner