Embedded newform 403.2.bs.a.4.9

• Label: 403.2.bs.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.350844 + 1.65059i) q^{2} +(-1.46122 - 1.62285i) q^{3} +(-0.774264 - 0.344725i) q^{4} -3.64318i q^{5} +(3.19131 - 1.84250i) q^{6} +(-1.03720 + 2.32958i) q^{7} +(-1.14309 + 1.57333i) q^{8} +(-0.184888 + 1.75909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.350844	+	1.65059i	−0.248084	+	1.16714i	0.660953	+	0.750427i	\(0.270152\pi\)
							−0.909037	+	0.416716i	\(0.863181\pi\)
\(3\)	−1.46122	−	1.62285i	−0.843634	−	0.936950i	0.155067	−	0.987904i	\(-0.450441\pi\)
							−0.998701	+	0.0509536i	\(0.983774\pi\)
\(5\)		−	3.64318i		−	1.62928i	−0.579968	−	0.814640i	\(-0.696935\pi\)
							0.579968	−	0.814640i	\(-0.303065\pi\)
\(7\)	−1.03720	+	2.32958i	−0.392023	+	0.880498i	0.604455	+	0.796639i	\(0.293391\pi\)
							−0.996478	+	0.0838584i	\(0.973276\pi\)
\(11\)	2.31391	−	0.243201i	0.697669	−	0.0733280i	0.250947	−	0.968001i	\(-0.419258\pi\)
							0.446722	+	0.894673i	\(0.352591\pi\)
\(13\)	−2.98240	+	2.02616i	−0.827168	+	0.561955i
							\(17\)	0.442452	−	4.20965i	0.107310	−	1.02099i	−0.799849	−	0.600202i	\(-0.795087\pi\)
0.907159	−	0.420788i	\(-0.138246\pi\)
\(19\)	−5.78743	−	5.21103i	−1.32773	−	1.19549i	−0.964589	−	0.263757i	\(-0.915038\pi\)
							−0.363139	−	0.931735i	\(-0.618295\pi\)
\(23\)	−6.56907	+	2.92474i	−1.36975	+	0.609850i	−0.954048	−	0.299653i	\(-0.903129\pi\)
							−0.415697	+	0.909503i	\(0.636462\pi\)
\(29\)	0.840220	+	0.178594i	0.156025	+	0.0331641i	0.285262	−	0.958450i	\(-0.407919\pi\)
							−0.129237	+	0.991614i	\(0.541253\pi\)
\(31\)	−5.10999	−	2.21089i	−0.917781	−	0.397087i
							\(37\)	4.99486	+	2.88378i	0.821150	+	0.474091i	0.850813	−	0.525469i	\(-0.176110\pi\)
−0.0296630	+	0.999560i	\(0.509443\pi\)
\(41\)	0.567361	−	2.66922i	0.0886069	−	0.416863i	−0.911379	−	0.411568i	\(-0.864981\pi\)
							0.999986	−	0.00529480i	\(-0.00168540\pi\)
\(43\)	0.967319	−	1.07432i	0.147515	−	0.163832i	−0.664859	−	0.746969i	\(-0.731508\pi\)
							0.812374	+	0.583137i	\(0.198175\pi\)
\(47\)	−1.41773	−	0.460649i	−0.206797	−	0.0671926i	0.203787	−	0.979015i	\(-0.434675\pi\)
							−0.410584	+	0.911823i	\(0.634675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.9	✓	288
13.10	even	6	inner	403.2.bs.a.283.28	yes	288
31.8	even	5	inner	403.2.bs.a.225.28	yes	288
403.101	even	30	inner	403.2.bs.a.101.9	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.9	✓	288	1.1	even	1	trivial
403.2.bs.a.101.9	yes	288	403.101	even	30	inner
403.2.bs.a.225.28	yes	288	31.8	even	5	inner
403.2.bs.a.283.28	yes	288	13.10	even	6	inner