Embedded newform 403.2.s.a.160.15

• Label: 403.2.s.a.160.15
• Level: $403$
• Weight: $2$
• Character: 403.160
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			160.15
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.700842 - 0.404632i) q^{2} +(-0.535618 - 0.927717i) q^{3} +(-0.672547 - 1.16488i) q^{4} +(3.37908 - 1.95091i) q^{5} +0.866912i q^{6} -2.38491i q^{7} +2.70706i q^{8} +(0.926227 - 1.60427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.700842	−	0.404632i	−0.495570	−	0.286118i	0.231312	−	0.972880i	\(-0.425698\pi\)
							−0.726882	+	0.686762i	\(0.759032\pi\)
\(3\)	−0.535618	−	0.927717i	−0.309239	−	0.535618i	0.668957	−	0.743301i	\(-0.266741\pi\)
							−0.978196	+	0.207683i	\(0.933408\pi\)
\(5\)	3.37908	−	1.95091i	1.51117	−	0.872474i	0.511254	−	0.859430i	\(-0.329181\pi\)
							0.999915	−	0.0130443i	\(-0.00415224\pi\)
\(7\)		−	2.38491i		−	0.901412i	−0.892673	−	0.450706i	\(-0.851172\pi\)
							0.892673	−	0.450706i	\(-0.148828\pi\)
\(11\)			0.632995i			0.190855i	0.995436	+	0.0954276i	\(0.0304218\pi\)
							−0.995436	+	0.0954276i	\(0.969578\pi\)
\(13\)	3.54252	−	0.671249i	0.982517	−	0.186171i
							\(17\)	2.68281			0.650678			0.325339	−	0.945597i	\(-0.394522\pi\)
0.325339	+	0.945597i	\(0.394522\pi\)
\(19\)			3.62791i			0.832299i	0.909296	+	0.416150i	\(0.136621\pi\)
							−0.909296	+	0.416150i	\(0.863379\pi\)
\(23\)	−2.45825	+	4.25782i	−0.512581	+	0.887817i	0.487312	+	0.873228i	\(0.337977\pi\)
							−0.999894	+	0.0145892i	\(0.995356\pi\)
\(29\)	−3.82172	+	6.61941i	−0.709675	+	1.22919i	0.255303	+	0.966861i	\(0.417825\pi\)
							−0.964978	+	0.262332i	\(0.915508\pi\)
\(31\)	0.0308301	−	5.56768i	0.00553725	−	0.999985i
							\(37\)	−5.75843	+	3.32463i	−0.946680	+	0.546566i	−0.892048	−	0.451941i	\(-0.850732\pi\)
−0.0546316	+	0.998507i	\(0.517398\pi\)
\(41\)		−	6.32039i		−	0.987079i	−0.869723	−	0.493539i	\(-0.835703\pi\)
							0.869723	−	0.493539i	\(-0.164297\pi\)
\(43\)	−2.94792			−0.449554			−0.224777	−	0.974410i	\(-0.572165\pi\)
							−0.224777	+	0.974410i	\(0.572165\pi\)
\(47\)		−	2.05460i		−	0.299695i	−0.988709	−	0.149847i	\(-0.952122\pi\)
							0.988709	−	0.149847i	\(-0.0478782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.15	✓	70
13.10	even	6		403.2.v.a.36.15	yes	70
31.25	even	3		403.2.v.a.56.15	yes	70
403.335	even	6	inner	403.2.s.a.335.15	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.15	✓	70	1.1	even	1	trivial
403.2.s.a.335.15	yes	70	403.335	even	6	inner
403.2.v.a.36.15	yes	70	13.10	even	6
403.2.v.a.56.15	yes	70	31.25	even	3