Embedded newform 503.2.c.a.2.9

• Label: 503.2.c.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15193 - 1.41827i) q^{2} +(-2.28946 + 1.98155i) q^{3} +(-0.274449 + 1.30998i) q^{4} +(1.06426 + 3.36113i) q^{5} +(5.44767 + 0.964477i) q^{6} +(0.658051 - 1.91069i) q^{7} +(-1.07463 + 0.553494i) q^{8} +(0.884773 - 6.10441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15193	−	1.41827i	−0.814534	−	1.00287i	−0.999796	−	0.0201790i	\(-0.993576\pi\)
							0.185262	−	0.982689i	\(-0.440687\pi\)
\(3\)	−2.28946	+	1.98155i	−1.32182	+	1.14405i	−0.343001	+	0.939335i	\(0.611444\pi\)
							−0.978822	+	0.204715i	\(0.934373\pi\)
\(5\)	1.06426	+	3.36113i	0.475951	+	1.50314i	0.824504	+	0.565856i	\(0.191454\pi\)
							−0.348553	+	0.937289i	\(0.613327\pi\)
\(7\)	0.658051	−	1.91069i	0.248720	−	0.722172i	−0.749470	−	0.662038i	\(-0.769692\pi\)
							0.998190	−	0.0601342i	\(-0.0191529\pi\)
\(11\)	1.91382	−	0.619191i	0.577039	−	0.186693i	−0.00481244	−	0.999988i	\(-0.501532\pi\)
							0.581851	+	0.813295i	\(0.302329\pi\)
\(13\)	5.67857	−	1.91606i	1.57495	−	0.531420i	0.609675	−	0.792652i	\(-0.291300\pi\)
							0.965276	+	0.261232i	\(0.0841288\pi\)
\(17\)	−4.11455	−	2.05448i	−0.997925	−	0.498284i	−0.128162	−	0.991753i	\(-0.540908\pi\)
							−0.869763	+	0.493470i	\(0.835728\pi\)
\(19\)	3.21287	+	7.20813i	0.737083	+	1.65366i	0.756624	+	0.653850i	\(0.226847\pi\)
							−0.0195410	+	0.999809i	\(0.506220\pi\)
\(23\)	−2.08675	+	7.21297i	−0.435118	+	1.50401i	0.380151	+	0.924924i	\(0.375872\pi\)
							−0.815269	+	0.579083i	\(0.803411\pi\)
\(29\)	0.250374	−	2.09573i	0.0464933	−	0.389168i	−0.950440	−	0.310909i	\(-0.899367\pi\)
							0.996933	−	0.0782592i	\(-0.0249362\pi\)
\(31\)	0.229046	+	2.80914i	0.0411378	+	0.504536i	0.984808	+	0.173648i	\(0.0555554\pi\)
							−0.943670	+	0.330888i	\(0.892652\pi\)
\(37\)	−2.93641	−	3.70928i	−0.482743	−	0.609801i	0.480972	−	0.876736i	\(-0.340284\pi\)
							−0.963715	+	0.266935i	\(0.913989\pi\)
\(41\)	0.452070	−	2.30092i	0.0706015	−	0.359343i	−0.929350	−	0.369200i	\(-0.879632\pi\)
							0.999951	+	0.00985719i	\(0.00313769\pi\)
\(43\)	−0.418503	+	2.28060i	−0.0638211	+	0.347788i	−0.999993	−	0.00375442i	\(-0.998805\pi\)
							0.936172	+	0.351543i	\(0.114343\pi\)
\(47\)	3.23635	−	0.121579i	0.472070	−	0.0177340i	0.199557	−	0.979886i	\(-0.436050\pi\)
							0.272513	+	0.962152i	\(0.412145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.9	✓	10250
503.252	even	251	inner	503.2.c.a.252.9	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.9	✓	10250	1.1	even	1	trivial
503.2.c.a.252.9	yes	10250	503.252	even	251	inner