Embedded newform 503.2.c.a.2.19

• Label: 503.2.c.a.2.19
• 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			2.19
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.225034 - 0.277066i) q^{2} +(-0.186491 + 0.161409i) q^{3} +(0.383983 - 1.83280i) q^{4} +(0.927472 + 2.92913i) q^{5} +(0.0866877 + 0.0153475i) q^{6} +(-0.576054 + 1.67260i) q^{7} +(-1.22886 + 0.632932i) q^{8} +(-0.421597 + 2.90877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.225034	−	0.277066i	−0.159123	−	0.195915i	0.690459	−	0.723372i	\(-0.257409\pi\)
							−0.849582	+	0.527457i	\(0.823146\pi\)
\(3\)	−0.186491	+	0.161409i	−0.107670	+	0.0931897i	−0.706605	−	0.707608i	\(-0.749774\pi\)
							0.598935	+	0.800798i	\(0.295591\pi\)
\(5\)	0.927472	+	2.92913i	0.414778	+	1.30995i	0.900127	+	0.435627i	\(0.143473\pi\)
							−0.485349	+	0.874320i	\(0.661308\pi\)
\(7\)	−0.576054	+	1.67260i	−0.217728	+	0.632185i	0.782230	+	0.622990i	\(0.214082\pi\)
							−0.999958	+	0.00919480i	\(0.997073\pi\)
\(11\)	−3.22302	+	1.04276i	−0.971777	+	0.314405i	−0.750547	−	0.660817i	\(-0.770210\pi\)
							−0.221229	+	0.975222i	\(0.571007\pi\)
\(13\)	−2.34736	+	0.792046i	−0.651040	+	0.219674i	−0.625771	−	0.780007i	\(-0.715216\pi\)
							−0.0252692	+	0.999681i	\(0.508044\pi\)
\(17\)	−5.70024	−	2.84624i	−1.38251	−	0.690315i	−0.407652	−	0.913137i	\(-0.633652\pi\)
							−0.974859	+	0.222822i	\(0.928473\pi\)
\(19\)	2.32308	+	5.21188i	0.532952	+	1.19569i	0.957011	+	0.290052i	\(0.0936727\pi\)
							−0.424059	+	0.905635i	\(0.639395\pi\)
\(23\)	0.793009	−	2.74108i	0.165354	−	0.571554i	−0.834437	−	0.551103i	\(-0.814207\pi\)
							0.999791	−	0.0204505i	\(-0.00651006\pi\)
\(29\)	0.610455	−	5.10976i	0.113359	−	0.948858i	−0.815585	−	0.578637i	\(-0.803585\pi\)
							0.928944	−	0.370221i	\(-0.120718\pi\)
\(31\)	0.577153	+	7.07851i	0.103660	+	1.27134i	0.819502	+	0.573076i	\(0.194250\pi\)
							−0.715842	+	0.698262i	\(0.753957\pi\)
\(37\)	6.67682	+	8.43416i	1.09766	+	1.38657i	0.915030	+	0.403387i	\(0.132167\pi\)
							0.182633	+	0.983181i	\(0.441538\pi\)
\(41\)	1.21801	−	6.19934i	0.190221	−	0.968174i	−0.758407	−	0.651781i	\(-0.774022\pi\)
							0.948628	−	0.316393i	\(-0.102472\pi\)
\(43\)	−0.335582	+	1.82873i	−0.0511758	+	0.278879i	−0.999230	−	0.0392294i	\(-0.987510\pi\)
							0.948054	+	0.318108i	\(0.103048\pi\)
\(47\)	3.35408	−	0.126001i	0.489243	−	0.0183792i	0.208224	−	0.978081i	\(-0.433232\pi\)
							0.281020	+	0.959702i	\(0.409327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.19	✓	10250
503.252	even	251	inner	503.2.c.a.252.19	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.19	✓	10250	1.1	even	1	trivial
503.2.c.a.252.19	yes	10250	503.252	even	251	inner