Embedded newform 503.2.c.a.2.8

• Label: 503.2.c.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31051 - 1.61352i) q^{2} +(-0.0751196 + 0.0650167i) q^{3} +(-0.475908 + 2.27157i) q^{4} +(-1.09154 - 3.44730i) q^{5} +(0.203351 + 0.0360021i) q^{6} +(-1.53778 + 4.46504i) q^{7} +(0.592995 - 0.305425i) q^{8} +(-0.428907 + 2.95921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31051	−	1.61352i	−0.926671	−	1.14093i	−0.989734	−	0.142923i	\(-0.954350\pi\)
							0.0630631	−	0.998010i	\(-0.479913\pi\)
\(3\)	−0.0751196	+	0.0650167i	−0.0433703	+	0.0375374i	−0.675847	−	0.737042i	\(-0.736222\pi\)
							0.632476	+	0.774580i	\(0.282039\pi\)
\(5\)	−1.09154	−	3.44730i	−0.488153	−	1.54168i	−0.805043	−	0.593216i	\(-0.797858\pi\)
							0.316891	−	0.948462i	\(-0.397361\pi\)
\(7\)	−1.53778	+	4.46504i	−0.581228	+	1.68763i	0.135933	+	0.990718i	\(0.456597\pi\)
							−0.717160	+	0.696908i	\(0.754558\pi\)
\(11\)	2.37064	−	0.766988i	0.714774	−	0.231256i	0.0721027	−	0.997397i	\(-0.477029\pi\)
							0.642672	+	0.766142i	\(0.277826\pi\)
\(13\)	1.74765	−	0.589692i	0.484711	−	0.163551i	−0.0667200	−	0.997772i	\(-0.521253\pi\)
							0.551431	+	0.834221i	\(0.314082\pi\)
\(17\)	4.26130	+	2.12775i	1.03352	+	0.516055i	0.881431	−	0.472312i	\(-0.156581\pi\)
							0.152085	+	0.988367i	\(0.451401\pi\)
\(19\)	−1.60335	−	3.59714i	−0.367834	−	0.825241i	−0.998778	−	0.0494132i	\(-0.984265\pi\)
							0.630945	−	0.775828i	\(-0.282667\pi\)
\(23\)	−1.56627	+	5.41389i	−0.326590	+	1.12887i	0.613993	+	0.789311i	\(0.289562\pi\)
							−0.940583	+	0.339563i	\(0.889721\pi\)
\(29\)	0.186631	−	1.56218i	0.0346565	−	0.290089i	−0.964960	−	0.262397i	\(-0.915487\pi\)
							0.999616	−	0.0276922i	\(-0.00881584\pi\)
\(31\)	0.313942	+	3.85035i	0.0563857	+	0.691544i	0.962986	+	0.269553i	\(0.0868758\pi\)
							−0.906600	+	0.421991i	\(0.861331\pi\)
\(37\)	2.99947	+	3.78893i	0.493109	+	0.622896i	0.966099	−	0.258172i	\(-0.0831201\pi\)
							−0.472990	+	0.881068i	\(0.656825\pi\)
\(41\)	1.82147	−	9.27080i	0.284466	−	1.44786i	−0.520158	−	0.854070i	\(-0.674127\pi\)
							0.804624	−	0.593785i	\(-0.202367\pi\)
\(43\)	−1.04780	+	5.70990i	−0.159788	+	0.870752i	0.802076	+	0.597223i	\(0.203729\pi\)
							−0.961864	+	0.273530i	\(0.911809\pi\)
\(47\)	12.4531	−	0.467819i	1.81646	−	0.0682383i	0.892575	−	0.450898i	\(-0.148896\pi\)
							0.923889	+	0.382660i	\(0.124992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.8	✓	10250
503.252	even	251	inner	503.2.c.a.252.8	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.8	✓	10250	1.1	even	1	trivial
503.2.c.a.252.8	yes	10250	503.252	even	251	inner