Embedded newform 403.3.n.a.347.13

• Label: 403.3.n.a.347.13
• Level: $403$
• Weight: $3$
• Character: 403.347
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $146$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	403.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9809546537\)
Analytic rank:	\(0\)
Dimension:	\(146\)
Relative dimension:	\(73\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			347.13
Character	\(\chi\)	\(=\)	403.347
Dual form			403.3.n.a.367.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45014 + 2.51172i) q^{2} -3.44080i q^{3} +(-2.20584 - 3.82062i) q^{4} +(-1.72058 + 2.98013i) q^{5} +(8.64234 + 4.98966i) q^{6} +(2.00775 - 3.47753i) q^{7} +1.19398 q^{8} -2.83911 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 146 q - 2 q^{2} - 146 q^{4} - 2 q^{5} + 12 q^{6} + 16 q^{7} - 10 q^{8} - 422 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45014	+	2.51172i	−0.725072	+	1.25586i	0.233872	+	0.972267i	\(0.424860\pi\)
							−0.958944	+	0.283595i	\(0.908473\pi\)
\(3\)		−	3.44080i		−	1.14693i	−0.819229	−	0.573467i	\(-0.805598\pi\)
							0.819229	−	0.573467i	\(-0.194402\pi\)
\(5\)	−1.72058	+	2.98013i	−0.344115	+	0.596025i	−0.985193	−	0.171451i	\(-0.945155\pi\)
							0.641077	+	0.767476i	\(0.278488\pi\)
\(7\)	2.00775	−	3.47753i	0.286822	−	0.496790i	−0.686227	−	0.727387i	\(-0.740734\pi\)
							0.973049	+	0.230597i	\(0.0740678\pi\)
\(11\)	−2.79722	+	1.61497i	−0.254292	+	0.146816i	−0.621728	−	0.783233i	\(-0.713569\pi\)
							0.367436	+	0.930049i	\(0.380236\pi\)
\(13\)	−12.2336	+	4.39751i	−0.941049	+	0.338270i
							\(17\)	10.7054	+	6.18075i	0.629728	+	0.363573i	0.780647	−	0.624973i	\(-0.214890\pi\)
−0.150919	+	0.988546i	\(0.548223\pi\)
\(19\)	0.972952	−	1.68520i	0.0512080	−	0.0886949i	−0.839285	−	0.543691i	\(-0.817026\pi\)
							0.890493	+	0.454997i	\(0.150360\pi\)
\(23\)	23.1328	+	13.3557i	1.00577	+	0.580683i	0.909951	−	0.414715i	\(-0.136119\pi\)
							0.0958218	+	0.995399i	\(0.469452\pi\)
\(29\)	22.1157	+	12.7685i	0.762610	+	0.440293i	0.830232	−	0.557418i	\(-0.188208\pi\)
							−0.0676218	+	0.997711i	\(0.521541\pi\)
\(31\)	−25.3827	−	17.7965i	−0.818798	−	0.574082i
							\(37\)			61.3186i			1.65726i	0.559797	+	0.828630i	\(0.310879\pi\)
−0.559797	+	0.828630i	\(0.689121\pi\)
\(41\)	23.4651	+	40.6427i	0.572319	+	0.991286i	0.996327	+	0.0856273i	\(0.0272894\pi\)
							−0.424008	+	0.905658i	\(0.639377\pi\)
\(43\)	51.3618	+	29.6538i	1.19446	+	0.689622i	0.959315	−	0.282338i	\(-0.0911099\pi\)
							0.235146	+	0.971960i	\(0.424443\pi\)
\(47\)	−83.2085			−1.77039			−0.885197	−	0.465217i	\(-0.845976\pi\)
							−0.885197	+	0.465217i	\(0.845976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.3.n.a.347.13	✓	146
13.3	even	3		403.3.o.a.68.13	yes	146
31.26	odd	6		403.3.o.a.243.13	yes	146
403.367	odd	6	inner	403.3.n.a.367.13	yes	146

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.3.n.a.347.13	✓	146	1.1	even	1	trivial
403.3.n.a.367.13	yes	146	403.367	odd	6	inner
403.3.o.a.68.13	yes	146	13.3	even	3
403.3.o.a.243.13	yes	146	31.26	odd	6