Embedded newform 403.2.s.a.160.10

• Label: 403.2.s.a.160.10
• 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			160.10
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06311 - 0.613789i) q^{2} +(-0.443615 - 0.768363i) q^{3} +(-0.246526 - 0.426995i) q^{4} +(0.237568 - 0.137160i) q^{5} +1.08914i q^{6} +1.55840i q^{7} +3.06042i q^{8} +(1.10641 - 1.91636i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06311	−	0.613789i	−0.751735	−	0.434015i	0.0745853	−	0.997215i	\(-0.476237\pi\)
							−0.826321	+	0.563200i	\(0.809570\pi\)
\(3\)	−0.443615	−	0.768363i	−0.256121	−	0.443615i	0.709078	−	0.705130i	\(-0.249111\pi\)
							−0.965199	+	0.261515i	\(0.915778\pi\)
\(5\)	0.237568	−	0.137160i	0.106244	−	0.0613397i	−0.445937	−	0.895064i	\(-0.647129\pi\)
							0.552180	+	0.833725i	\(0.313796\pi\)
\(7\)			1.55840i			0.589019i	0.955649	+	0.294510i	\(0.0951563\pi\)
							−0.955649	+	0.294510i	\(0.904844\pi\)
\(11\)		−	5.37759i		−	1.62141i	−0.585458	−	0.810703i	\(-0.699085\pi\)
							0.585458	−	0.810703i	\(-0.300915\pi\)
\(13\)	−2.90601	−	2.13428i	−0.805981	−	0.591942i
							\(17\)	−3.69029			−0.895026			−0.447513	−	0.894277i	\(-0.647690\pi\)
−0.447513	+	0.894277i	\(0.647690\pi\)
\(19\)		−	1.80172i		−	0.413342i	−0.978411	−	0.206671i	\(-0.933737\pi\)
							0.978411	−	0.206671i	\(-0.0662630\pi\)
\(23\)	−2.40111	+	4.15885i	−0.500667	+	0.867180i	0.499333	+	0.866410i	\(0.333578\pi\)
							−1.00000		0.000770114i	\(0.999755\pi\)
\(29\)	−0.431591	+	0.747537i	−0.0801444	+	0.138814i	−0.903312	−	0.428985i	\(-0.858871\pi\)
							0.823167	+	0.567799i	\(0.192205\pi\)
\(31\)	−1.09696	+	5.45863i	−0.197019	+	0.980400i
							\(37\)	−2.64696	+	1.52822i	−0.435157	+	0.251238i	−0.701541	−	0.712629i	\(-0.747504\pi\)
0.266384	+	0.963867i	\(0.414171\pi\)
\(41\)		−	10.9659i		−	1.71259i	−0.516491	−	0.856293i	\(-0.672762\pi\)
							0.516491	−	0.856293i	\(-0.327238\pi\)
\(43\)	−3.99327			−0.608968			−0.304484	−	0.952517i	\(-0.598484\pi\)
							−0.304484	+	0.952517i	\(0.598484\pi\)
\(47\)		−	3.07294i		−	0.448235i	−0.974562	−	0.224118i	\(-0.928050\pi\)
							0.974562	−	0.224118i	\(-0.0719499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.10	✓	70
13.10	even	6		403.2.v.a.36.10	yes	70
31.25	even	3		403.2.v.a.56.10	yes	70
403.335	even	6	inner	403.2.s.a.335.10	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.10	✓	70	1.1	even	1	trivial
403.2.s.a.335.10	yes	70	403.335	even	6	inner
403.2.v.a.36.10	yes	70	13.10	even	6
403.2.v.a.56.10	yes	70	31.25	even	3