Embedded newform 503.2.c.a.2.18

• Label: 503.2.c.a.2.18
• 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			2.18
Character	\(\chi\)	\(=\)	503.2
Dual form			503.2.c.a.252.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.380517 - 0.468498i) q^{2} +(-1.24266 + 1.07553i) q^{3} +(0.335410 - 1.60096i) q^{4} +(-0.464781 - 1.46787i) q^{5} +(0.976735 + 0.172925i) q^{6} +(-0.519527 + 1.50848i) q^{7} +(-1.95081 + 1.00478i) q^{8} +(-0.0428949 + 0.295950i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10250 q - 249 q^{2} - 249 q^{3} - 285 q^{4} - 245 q^{5} - 237 q^{6} - 247 q^{7} - 239 q^{8} - 280 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.380517	−	0.468498i	−0.269066	−	0.331278i	0.623814	−	0.781572i	\(-0.285582\pi\)
							−0.892880	+	0.450294i	\(0.851319\pi\)
\(3\)	−1.24266	+	1.07553i	−0.717448	+	0.620958i	−0.934816	−	0.355132i	\(-0.884436\pi\)
							0.217369	+	0.976090i	\(0.430253\pi\)
\(5\)	−0.464781	−	1.46787i	−0.207856	−	0.656450i	−0.998995	−	0.0448136i	\(-0.985731\pi\)
							0.791139	−	0.611637i	\(-0.209489\pi\)
\(7\)	−0.519527	+	1.50848i	−0.196363	+	0.570150i	−0.999677	−	0.0254048i	\(-0.991913\pi\)
							0.803314	+	0.595555i	\(0.203068\pi\)
\(11\)	−5.11452	+	1.65474i	−1.54209	+	0.498921i	−0.951394	−	0.307975i	\(-0.900349\pi\)
							−0.590693	+	0.806897i	\(0.701145\pi\)
\(13\)	3.94054	−	1.32962i	1.09291	−	0.368769i	0.285270	−	0.958447i	\(-0.407917\pi\)
							0.807638	+	0.589678i	\(0.200745\pi\)
\(17\)	4.81167	+	2.40256i	1.16700	+	0.582707i	0.922142	−	0.386852i	\(-0.126438\pi\)
							0.244859	+	0.969559i	\(0.421258\pi\)
\(19\)	2.02548	+	4.54420i	0.464677	+	1.04251i	0.982370	+	0.186945i	\(0.0598586\pi\)
							−0.517693	+	0.855566i	\(0.673209\pi\)
\(23\)	−0.872219	+	3.01487i	−0.181870	+	0.628644i	0.816792	+	0.576932i	\(0.195750\pi\)
							−0.998662	+	0.0517112i	\(0.983532\pi\)
\(29\)	−0.530662	+	4.44186i	−0.0985415	+	0.824833i	0.853968	+	0.520325i	\(0.174189\pi\)
							−0.952510	+	0.304508i	\(0.901508\pi\)
\(31\)	0.327377	+	4.01512i	0.0587986	+	0.721137i	0.958585	+	0.284806i	\(0.0919293\pi\)
							−0.899787	+	0.436330i	\(0.856278\pi\)
\(37\)	−1.52699	−	1.92890i	−0.251036	−	0.317108i	0.642339	−	0.766420i	\(-0.277964\pi\)
							−0.893375	+	0.449312i	\(0.851669\pi\)
\(41\)	−0.633660	+	3.22516i	−0.0989610	+	0.503686i	0.898891	+	0.438173i	\(0.144374\pi\)
							−0.997852	+	0.0655127i	\(0.979132\pi\)
\(43\)	−0.931336	+	5.07525i	−0.142027	+	0.773968i	0.833235	+	0.552919i	\(0.186486\pi\)
							−0.975263	+	0.221049i	\(0.929052\pi\)
\(47\)	−0.230256	+	0.00864994i	−0.0335863	+	0.00126172i	−0.0543280	−	0.998523i	\(-0.517302\pi\)
							0.0207416	+	0.999785i	\(0.493397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.c.a.2.18	✓	10250
503.252	even	251	inner	503.2.c.a.252.18	yes	10250

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.c.a.2.18	✓	10250	1.1	even	1	trivial
503.2.c.a.252.18	yes	10250	503.252	even	251	inner