Embedded newform 403.2.bs.a.4.11

• Label: 403.2.bs.a.4.11
• Level: $403$
• Weight: $2$
• Character: 403.4
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bs (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.319792 + 1.50450i) q^{2} +(-0.741178 - 0.823162i) q^{3} +(-0.334172 - 0.148783i) q^{4} -1.09342i q^{5} +(1.47547 - 0.851865i) q^{6} +(1.19871 - 2.69235i) q^{7} +(-1.47745 + 2.03354i) q^{8} +(0.185335 - 1.76335i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.319792	+	1.50450i	−0.226127	+	1.06384i	0.707809	+	0.706404i	\(0.249684\pi\)
							−0.933936	+	0.357440i	\(0.883650\pi\)
\(3\)	−0.741178	−	0.823162i	−0.427920	−	0.475253i	0.490169	−	0.871627i	\(-0.336935\pi\)
							−0.918089	+	0.396374i	\(0.870268\pi\)
\(5\)		−	1.09342i		−	0.488994i	−0.969650	−	0.244497i	\(-0.921377\pi\)
							0.969650	−	0.244497i	\(-0.0786229\pi\)
\(7\)	1.19871	−	2.69235i	0.453070	−	1.01761i	−0.532202	−	0.846618i	\(-0.678635\pi\)
							0.985272	−	0.170995i	\(-0.0546981\pi\)
\(11\)	−6.28754	+	0.660847i	−1.89576	+	0.199253i	−0.979517	−	0.201362i	\(-0.935463\pi\)
							−0.916247	+	0.400615i	\(0.868797\pi\)
\(13\)	0.607479	−	3.55401i	0.168484	−	0.985704i
							\(17\)	0.0196837	−	0.187278i	0.00477401	−	0.0454216i	−0.991876	−	0.127212i	\(-0.959397\pi\)
0.996650	+	0.0817904i	\(0.0260638\pi\)
\(19\)	−4.97125	−	4.47613i	−1.14048	−	1.02690i	−0.999319	−	0.0369096i	\(-0.988249\pi\)
							−0.141165	−	0.989986i	\(-0.545085\pi\)
\(23\)	1.06409	−	0.473761i	0.221877	−	0.0987860i	−0.292788	−	0.956177i	\(-0.594583\pi\)
							0.514665	+	0.857391i	\(0.327916\pi\)
\(29\)	5.10745	+	1.08562i	0.948429	+	0.201595i	0.656058	−	0.754710i	\(-0.272222\pi\)
							0.292371	+	0.956305i	\(0.405556\pi\)
\(31\)	5.55998	+	0.294333i	0.998602	+	0.0528639i
							\(37\)	−3.34860	−	1.93331i	−0.550506	−	0.317835i	0.198820	−	0.980036i	\(-0.436289\pi\)
−0.749326	+	0.662201i	\(0.769622\pi\)
\(41\)	1.19896	−	5.64068i	0.187247	−	0.880926i	−0.779741	−	0.626103i	\(-0.784649\pi\)
							0.966987	−	0.254824i	\(-0.0820175\pi\)
\(43\)	−2.02263	+	2.24636i	−0.308449	+	0.342567i	−0.877361	−	0.479832i	\(-0.840698\pi\)
							0.568912	+	0.822398i	\(0.307364\pi\)
\(47\)	10.4789	+	3.40479i	1.52850	+	0.496640i	0.948176	−	0.317746i	\(-0.102926\pi\)
							0.580324	+	0.814385i	\(0.302926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.11	✓	288
13.10	even	6	inner	403.2.bs.a.283.26	yes	288
31.8	even	5	inner	403.2.bs.a.225.26	yes	288
403.101	even	30	inner	403.2.bs.a.101.11	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.11	✓	288	1.1	even	1	trivial
403.2.bs.a.101.11	yes	288	403.101	even	30	inner
403.2.bs.a.225.26	yes	288	31.8	even	5	inner
403.2.bs.a.283.26	yes	288	13.10	even	6	inner