Embedded newform 503.2.c.a.2.17

• Label: 503.2.c.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.448047 - 0.551642i) q^{2} +(2.42081 - 2.09524i) q^{3} +(0.306545 - 1.46318i) q^{4} +(-0.150619 - 0.475684i) q^{5} +(-2.24046 - 0.396660i) q^{6} +(-0.834374 + 2.42265i) q^{7} +(-2.20809 + 1.13729i) q^{8} +(1.04000 - 7.17539i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.448047	−	0.551642i	−0.316817	−	0.390070i	0.592916	−	0.805264i	\(-0.297977\pi\)
							−0.909733	+	0.415194i	\(0.863714\pi\)
\(3\)	2.42081	−	2.09524i	1.39766	−	1.20969i	0.448940	−	0.893562i	\(-0.351802\pi\)
							0.948718	−	0.316124i	\(-0.102382\pi\)
\(5\)	−0.150619	−	0.475684i	−0.0673590	−	0.212732i	0.913719	−	0.406348i	\(-0.133198\pi\)
							−0.981078	+	0.193615i	\(0.937979\pi\)
\(7\)	−0.834374	+	2.42265i	−0.315364	+	0.915676i	0.669571	+	0.742748i	\(0.266478\pi\)
							−0.984934	+	0.172928i	\(0.944677\pi\)
\(11\)	−0.445595	+	0.144166i	−0.134352	+	0.0434677i	−0.374234	−	0.927334i	\(-0.622094\pi\)
							0.239882	+	0.970802i	\(0.422891\pi\)
\(13\)	−1.75071	+	0.590724i	−0.485559	+	0.163837i	−0.551817	−	0.833965i	\(-0.686065\pi\)
							0.0662578	+	0.997803i	\(0.478894\pi\)
\(17\)	3.92999	+	1.96232i	0.953163	+	0.475933i	0.854651	−	0.519203i	\(-0.173771\pi\)
							0.0985119	+	0.995136i	\(0.468592\pi\)
\(19\)	−3.18127	−	7.13723i	−0.729833	−	1.63739i	−0.769875	−	0.638195i	\(-0.779681\pi\)
							0.0400414	−	0.999198i	\(-0.487251\pi\)
\(23\)	1.76279	−	6.09319i	0.367568	−	1.27052i	−0.536780	−	0.843722i	\(-0.680360\pi\)
							0.904348	−	0.426795i	\(-0.140358\pi\)
\(29\)	0.111865	−	0.936356i	0.0207728	−	0.173877i	−0.978739	−	0.205108i	\(-0.934245\pi\)
							0.999512	+	0.0312308i	\(0.00994270\pi\)
\(31\)	0.748569	+	9.18085i	0.134447	+	1.64893i	0.627272	+	0.778801i	\(0.284172\pi\)
							−0.492825	+	0.870129i	\(0.664036\pi\)
\(37\)	5.69235	+	7.19058i	0.935816	+	1.18212i	0.983068	+	0.183240i	\(0.0586587\pi\)
							−0.0472523	+	0.998883i	\(0.515046\pi\)
\(41\)	−0.404281	+	2.05768i	−0.0631380	+	0.321356i	−0.999565	−	0.0295078i	\(-0.990606\pi\)
							0.936427	+	0.350864i	\(0.114112\pi\)
\(43\)	−1.87460	+	10.2155i	−0.285873	+	1.55785i	0.457648	+	0.889133i	\(0.348692\pi\)
							−0.743522	+	0.668712i	\(0.766846\pi\)
\(47\)	8.14420	−	0.305950i	1.18795	−	0.0446273i	0.563788	−	0.825920i	\(-0.309344\pi\)
							0.624165	+	0.781292i	\(0.285439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.17	✓	10250
503.252	even	251	inner	503.2.c.a.252.17	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.17	✓	10250	1.1	even	1	trivial
503.2.c.a.252.17	yes	10250	503.252	even	251	inner