Embedded newform 403.2.e.a.191.10

• Label: 403.2.e.a.191.10
• Level: $403$
• Weight: $2$
• Character: 403.191
• 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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			191.10
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.852641 + 1.47682i) q^{2} +(0.937886 + 1.62447i) q^{3} +(-0.453992 - 0.786337i) q^{4} +(1.48433 + 2.57094i) q^{5} -3.19872 q^{6} +0.0782312 q^{7} -1.86219 q^{8} +(-0.259260 + 0.449051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.852641	+	1.47682i	−0.602908	+	1.04427i	0.389470	+	0.921039i	\(0.372658\pi\)
							−0.992378	+	0.123228i	\(0.960675\pi\)
\(3\)	0.937886	+	1.62447i	0.541489	+	0.937886i	0.998819	+	0.0485889i	\(0.0154724\pi\)
							−0.457330	+	0.889297i	\(0.651194\pi\)
\(5\)	1.48433	+	2.57094i	0.663813	+	1.14976i	0.979606	+	0.200930i	\(0.0643963\pi\)
							−0.315792	+	0.948828i	\(0.602270\pi\)
\(7\)	0.0782312			0.0295686			0.0147843	−	0.999891i	\(-0.495294\pi\)
							0.0147843	+	0.999891i	\(0.495294\pi\)
\(11\)	−1.79880			−0.542359			−0.271179	−	0.962529i	\(-0.587414\pi\)
							−0.271179	+	0.962529i	\(0.587414\pi\)
\(13\)	−3.56024	+	0.569800i	−0.987434	+	0.158034i
							\(17\)	6.75495			1.63832			0.819158	−	0.573568i	\(-0.194441\pi\)
0.819158	+	0.573568i	\(0.194441\pi\)
\(19\)	2.50687			0.575116			0.287558	−	0.957763i	\(-0.407157\pi\)
							0.287558	+	0.957763i	\(0.407157\pi\)
\(23\)	−3.04167	+	5.26833i	−0.634233	+	1.09852i	0.352445	+	0.935833i	\(0.385351\pi\)
							−0.986677	+	0.162690i	\(0.947983\pi\)
\(29\)	1.41480	−	2.45050i	0.262721	−	0.455047i	−0.704243	−	0.709959i	\(-0.748713\pi\)
							0.966964	+	0.254913i	\(0.0820467\pi\)
\(31\)	−5.11875	−	2.19053i	−0.919354	−	0.393432i
							\(37\)	−1.40636	−	2.43588i	−0.231204	−	0.400457i	0.726959	−	0.686681i	\(-0.240933\pi\)
−0.958163	+	0.286224i	\(0.907600\pi\)
\(41\)	8.33213			1.30126			0.650630	−	0.759395i	\(-0.274505\pi\)
							0.650630	+	0.759395i	\(0.274505\pi\)
\(43\)	6.72446			1.02547			0.512736	−	0.858547i	\(-0.328632\pi\)
							0.512736	+	0.858547i	\(0.328632\pi\)
\(47\)	−4.01751			−0.586014			−0.293007	−	0.956110i	\(-0.594656\pi\)
							−0.293007	+	0.956110i	\(0.594656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.10	✓	70
13.3	even	3		403.2.g.a.315.10	yes	70
31.25	even	3		403.2.g.a.87.10	yes	70
403.211	even	3	inner	403.2.e.a.211.10	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.10	✓	70	1.1	even	1	trivial
403.2.e.a.211.10	yes	70	403.211	even	3	inner
403.2.g.a.87.10	yes	70	31.25	even	3
403.2.g.a.315.10	yes	70	13.3	even	3