Embedded newform 503.2.c.a.2.2

• Label: 503.2.c.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72191 - 2.12004i) q^{2} +(1.69918 - 1.47066i) q^{3} +(-1.11950 + 5.34354i) q^{4} +(-0.0846785 - 0.267431i) q^{5} +(-6.04369 - 1.07000i) q^{6} +(-1.43085 + 4.15454i) q^{7} +(8.40007 - 4.32651i) q^{8} +(0.294060 - 2.02884i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72191	−	2.12004i	−1.21757	−	1.49910i	−0.810539	−	0.585684i	\(-0.800826\pi\)
							−0.407034	−	0.913413i	\(-0.633437\pi\)
\(3\)	1.69918	−	1.47066i	0.981022	−	0.849084i	−0.00752864	−	0.999972i	\(-0.502396\pi\)
							0.988551	+	0.150888i	\(0.0482132\pi\)
\(5\)	−0.0846785	−	0.267431i	−0.0378694	−	0.119599i	0.932538	−	0.361071i	\(-0.117589\pi\)
							−0.970408	+	0.241472i	\(0.922370\pi\)
\(7\)	−1.43085	+	4.15454i	−0.540809	+	1.57027i	0.256402	+	0.966570i	\(0.417463\pi\)
							−0.797212	+	0.603700i	\(0.793693\pi\)
\(11\)	−3.38244	+	1.09434i	−1.01984	+	0.329957i	−0.769805	−	0.638279i	\(-0.779647\pi\)
							−0.250039	+	0.968236i	\(0.580443\pi\)
\(13\)	−4.38145	+	1.47839i	−1.21520	+	0.410031i	−0.852921	−	0.522040i	\(-0.825171\pi\)
							−0.362275	+	0.932071i	\(0.618000\pi\)
\(17\)	0.274557	+	0.137092i	0.0665898	+	0.0332496i	0.479712	−	0.877426i	\(-0.340741\pi\)
							−0.413122	+	0.910676i	\(0.635562\pi\)
\(19\)	3.22255	+	7.22985i	0.739304	+	1.65864i	0.752382	+	0.658728i	\(0.228905\pi\)
							−0.0130772	+	0.999914i	\(0.504163\pi\)
\(23\)	1.27673	−	4.41307i	0.266216	−	0.920188i	−0.710148	−	0.704053i	\(-0.751372\pi\)
							0.976363	−	0.216135i	\(-0.0693453\pi\)
\(29\)	−0.386109	+	3.23189i	−0.0716987	+	0.600147i	0.910662	+	0.413152i	\(0.135572\pi\)
							−0.982361	+	0.186995i	\(0.940125\pi\)
\(31\)	0.503576	+	6.17612i	0.0904449	+	1.10926i	0.873340	+	0.487110i	\(0.161949\pi\)
							−0.782895	+	0.622153i	\(0.786258\pi\)
\(37\)	−0.424820	−	0.536633i	−0.0698400	−	0.0882219i	0.747894	−	0.663818i	\(-0.231065\pi\)
							−0.817734	+	0.575596i	\(0.804770\pi\)
\(41\)	−0.732815	+	3.72983i	−0.114446	+	0.582502i	0.879796	+	0.475351i	\(0.157679\pi\)
							−0.994243	+	0.107151i	\(0.965827\pi\)
\(43\)	0.198085	−	1.07945i	0.0302077	−	0.164614i	−0.965023	−	0.262166i	\(-0.915563\pi\)
							0.995230	+	0.0975515i	\(0.0311011\pi\)
\(47\)	−4.90276	+	0.184180i	−0.715141	+	0.0268654i	−0.392625	−	0.919699i	\(-0.628433\pi\)
							−0.322516	+	0.946564i	\(0.604529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.2	✓	10250
503.252	even	251	inner	503.2.c.a.252.2	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.2	✓	10250	1.1	even	1	trivial
503.2.c.a.252.2	yes	10250	503.252	even	251	inner