Embedded newform 403.2.bs.a.4.6

• Label: 403.2.bs.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.431958 + 2.03220i) q^{2} +(1.72231 + 1.91282i) q^{3} +(-2.11617 - 0.942181i) q^{4} +3.56131i q^{5} +(-4.63120 + 2.67382i) q^{6} +(1.47626 - 3.31574i) q^{7} +(0.386433 - 0.531879i) q^{8} +(-0.378938 + 3.60536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.431958	+	2.03220i	−0.305441	+	1.43699i	0.511012	+	0.859573i	\(0.329271\pi\)
							−0.816453	+	0.577412i	\(0.804063\pi\)
\(3\)	1.72231	+	1.91282i	0.994375	+	1.10437i	0.994539	+	0.104366i	\(0.0332812\pi\)
							−0.000163734		1.00000i	\(0.500052\pi\)
\(5\)			3.56131i			1.59267i	0.604857	+	0.796334i	\(0.293230\pi\)
							−0.604857	+	0.796334i	\(0.706770\pi\)
\(7\)	1.47626	−	3.31574i	0.557974	−	1.25323i	−0.385782	−	0.922590i	\(-0.626068\pi\)
							0.943757	−	0.330641i	\(-0.107265\pi\)
\(11\)	−1.56108	+	0.164076i	−0.470684	+	0.0494709i	−0.336902	−	0.941540i	\(-0.609379\pi\)
							−0.133782	+	0.991011i	\(0.542712\pi\)
\(13\)	2.76209	−	2.31751i	0.766066	−	0.642762i
							\(17\)	−0.247176	+	2.35172i	−0.0599489	+	0.570376i	0.922780	+	0.385326i	\(0.125911\pi\)
−0.982729	+	0.185049i	\(0.940756\pi\)
\(19\)	−0.0995353	−	0.0896220i	−0.0228350	−	0.0205607i	0.657634	−	0.753337i	\(-0.271557\pi\)
							−0.680469	+	0.732777i	\(0.738224\pi\)
\(23\)	8.30014	−	3.69546i	1.73070	−	0.770557i	0.734985	−	0.678084i	\(-0.237189\pi\)
							0.995715	−	0.0924734i	\(-0.0294773\pi\)
\(29\)	−7.06762	−	1.50227i	−1.31242	−	0.278964i	−0.502030	−	0.864850i	\(-0.667413\pi\)
							−0.810393	+	0.585886i	\(0.800747\pi\)
\(31\)	1.85916	+	5.24819i	0.333915	+	0.942603i
							\(37\)	1.38441	+	0.799291i	0.227596	+	0.131403i	0.609463	−	0.792815i	\(-0.291385\pi\)
−0.381866	+	0.924218i	\(0.624719\pi\)
\(41\)	−0.652874	+	3.07153i	−0.101962	+	0.479693i	0.897305	+	0.441411i	\(0.145522\pi\)
							−0.999267	+	0.0382820i	\(0.987811\pi\)
\(43\)	6.22115	−	6.90929i	0.948717	−	1.05366i	−0.0497754	−	0.998760i	\(-0.515851\pi\)
							0.998492	−	0.0548962i	\(-0.0174828\pi\)
\(47\)	−5.64720	−	1.83489i	−0.823728	−	0.267646i	−0.133327	−	0.991072i	\(-0.542566\pi\)
							−0.690401	+	0.723427i	\(0.742566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.6	✓	288
13.10	even	6	inner	403.2.bs.a.283.31	yes	288
31.8	even	5	inner	403.2.bs.a.225.31	yes	288
403.101	even	30	inner	403.2.bs.a.101.6	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.6	✓	288	1.1	even	1	trivial
403.2.bs.a.101.6	yes	288	403.101	even	30	inner
403.2.bs.a.225.31	yes	288	31.8	even	5	inner
403.2.bs.a.283.31	yes	288	13.10	even	6	inner