Embedded newform 403.3.o.a.243.13

• Label: 403.3.o.a.243.13
• Level: $403$
• Weight: $3$
• Character: 403.243
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $146$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	403.o (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			243.13
Character	\(\chi\)	\(=\)	403.243
Dual form			403.3.o.a.68.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45014 + 2.51172i) q^{2} +(2.97982 - 1.72040i) q^{3} +(-2.20584 - 3.82062i) q^{4} +(-1.72058 - 2.98013i) q^{5} +9.97932i q^{6} -4.01551 q^{7} +1.19398 q^{8} +(1.41955 - 2.45874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 146 q - 2 q^{2} - 146 q^{4} - 2 q^{5} - 32 q^{7} - 10 q^{8} + 211 q^{9}+O(q^{10}) \)

Character values

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

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

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	\(0.908473\pi\)
\(3\)	2.97982	−	1.72040i	0.993274	−	0.573467i	0.0870223	−	0.996206i	\(-0.472265\pi\)
							0.906251	+	0.422740i	\(0.138932\pi\)
\(5\)	−1.72058	−	2.98013i	−0.344115	−	0.596025i	0.641077	−	0.767476i	\(-0.278488\pi\)
							−0.985193	+	0.171451i	\(0.945155\pi\)
\(7\)	−4.01551			−0.573644			−0.286822	−	0.957984i	\(-0.592599\pi\)
							−0.286822	+	0.957984i	\(0.592599\pi\)
\(11\)			3.22995i			0.293631i	0.989164	+	0.146816i	\(0.0469024\pi\)
							−0.989164	+	0.146816i	\(0.953098\pi\)
\(13\)	−2.30846	+	12.7934i	−0.177574	+	0.984107i
							\(17\)			12.3615i			0.727147i	0.931566	+	0.363573i	\(0.118443\pi\)
−0.931566	+	0.363573i	\(0.881557\pi\)
\(19\)	−1.94590			−0.102416			−0.0512080	−	0.998688i	\(-0.516307\pi\)
							−0.0512080	+	0.998688i	\(0.516307\pi\)
\(23\)	−23.1328	−	13.3557i	−1.00577	−	0.580683i	−0.0958218	−	0.995399i	\(-0.530548\pi\)
							−0.909951	+	0.414715i	\(0.863881\pi\)
\(29\)	−22.1157	−	12.7685i	−0.762610	−	0.440293i	0.0676218	−	0.997711i	\(-0.478459\pi\)
							−0.830232	+	0.557418i	\(0.811792\pi\)
\(31\)	−25.3827	+	17.7965i	−0.818798	+	0.574082i
							\(37\)	−53.1035	+	30.6593i	−1.43523	+	0.828630i	−0.997513	−	0.0704838i	\(-0.977546\pi\)
−0.437716	+	0.899113i	\(0.644212\pi\)
\(41\)	−46.9302			−1.14464			−0.572319	−	0.820031i	\(-0.693956\pi\)
							−0.572319	+	0.820031i	\(0.693956\pi\)
\(43\)			59.3075i			1.37924i	0.724169	+	0.689622i	\(0.242223\pi\)
							−0.724169	+	0.689622i	\(0.757777\pi\)
\(47\)	−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.o.a.243.13	yes	146
13.3	even	3		403.3.n.a.367.13	yes	146
31.6	odd	6		403.3.n.a.347.13	✓	146
403.68	odd	6	inner	403.3.o.a.68.13	yes	146

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.3.n.a.347.13	✓	146	31.6	odd	6
403.3.n.a.367.13	yes	146	13.3	even	3
403.3.o.a.68.13	yes	146	403.68	odd	6	inner
403.3.o.a.243.13	yes	146	1.1	even	1	trivial