Embedded newform 503.2.c.a.2.12

• Label: 503.2.c.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948756 - 1.16812i) q^{2} +(1.18005 - 1.02135i) q^{3} +(-0.0542683 + 0.259030i) q^{4} +(0.841155 + 2.65653i) q^{5} +(-2.31264 - 0.409440i) q^{6} +(-1.01850 + 2.95726i) q^{7} +(-2.32163 + 1.19577i) q^{8} +(-0.0809476 + 0.558491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948756	−	1.16812i	−0.670872	−	0.825989i	0.321803	−	0.946807i	\(-0.395711\pi\)
							−0.992675	+	0.120818i	\(0.961448\pi\)
\(3\)	1.18005	−	1.02135i	0.681304	−	0.589675i	−0.243600	−	0.969876i	\(-0.578328\pi\)
							0.924904	+	0.380201i	\(0.124145\pi\)
\(5\)	0.841155	+	2.65653i	0.376176	+	1.18803i	0.933756	+	0.357911i	\(0.116511\pi\)
							−0.557580	+	0.830123i	\(0.688270\pi\)
\(7\)	−1.01850	+	2.95726i	−0.384956	+	1.11774i	0.570167	+	0.821529i	\(0.306879\pi\)
							−0.955123	+	0.296211i	\(0.904277\pi\)
\(11\)	1.26893	−	0.410546i	0.382598	−	0.123784i	−0.110241	−	0.993905i	\(-0.535162\pi\)
							0.492839	+	0.870121i	\(0.335959\pi\)
\(13\)	−0.0896291	+	0.0302426i	−0.0248586	+	0.00838780i	−0.332112	−	0.943240i	\(-0.607761\pi\)
							0.307254	+	0.951628i	\(0.400590\pi\)
\(17\)	2.91671	+	1.45637i	0.707406	+	0.353222i	0.764036	−	0.645174i	\(-0.223215\pi\)
							−0.0566304	+	0.998395i	\(0.518036\pi\)
\(19\)	1.08885	+	2.44286i	0.249800	+	0.560431i	0.994230	−	0.107270i	\(-0.0342108\pi\)
							−0.744430	+	0.667701i	\(0.767279\pi\)
\(23\)	−1.00982	+	3.49048i	−0.210561	+	0.727815i	0.783729	+	0.621103i	\(0.213315\pi\)
							−0.994290	+	0.106712i	\(0.965968\pi\)
\(29\)	0.162162	−	1.35736i	0.0301127	−	0.252056i	−0.969852	−	0.243693i	\(-0.921641\pi\)
							0.999965	−	0.00836235i	\(-0.00266185\pi\)
\(31\)	−0.777649	−	9.53750i	−0.139670	−	1.71299i	−0.579556	−	0.814932i	\(-0.696774\pi\)
							0.439886	−	0.898054i	\(-0.355019\pi\)
\(37\)	−3.22248	−	4.07064i	−0.529772	−	0.669209i	0.444186	−	0.895934i	\(-0.353493\pi\)
							−0.973959	+	0.226726i	\(0.927198\pi\)
\(41\)	−1.23271	+	6.27419i	−0.192518	+	0.979864i	0.753917	+	0.656969i	\(0.228162\pi\)
							−0.946435	+	0.322895i	\(0.895344\pi\)
\(43\)	−0.791381	+	4.31257i	−0.120684	+	0.657661i	0.866639	+	0.498937i	\(0.166276\pi\)
							−0.987323	+	0.158724i	\(0.949262\pi\)
\(47\)	4.81688	−	0.180954i	0.702614	−	0.0263948i	0.316163	−	0.948705i	\(-0.397605\pi\)
							0.386451	+	0.922310i	\(0.373701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.12	✓	10250
503.252	even	251	inner	503.2.c.a.252.12	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.12	✓	10250	1.1	even	1	trivial
503.2.c.a.252.12	yes	10250	503.252	even	251	inner