Embedded newform 403.2.bs.a.4.20

• Label: 403.2.bs.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0550517 - 0.258998i) q^{2} +(-1.89412 - 2.10363i) q^{3} +(1.76304 + 0.784957i) q^{4} -4.39411i q^{5} +(-0.649111 + 0.374764i) q^{6} +(0.553765 - 1.24378i) q^{7} +(0.611633 - 0.841841i) q^{8} +(-0.523995 + 4.98548i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0550517	−	0.258998i	0.0389275	−	0.183139i	−0.954388	−	0.298568i	\(-0.903491\pi\)
							0.993316	+	0.115429i	\(0.0368243\pi\)
\(3\)	−1.89412	−	2.10363i	−1.09357	−	1.21453i	−0.975145	−	0.221569i	\(-0.928882\pi\)
							−0.118424	−	0.992963i	\(-0.537784\pi\)
\(5\)		−	4.39411i		−	1.96511i	−0.185983	−	0.982553i	\(-0.559547\pi\)
							0.185983	−	0.982553i	\(-0.440453\pi\)
\(7\)	0.553765	−	1.24378i	0.209304	−	0.470103i	−0.778137	−	0.628095i	\(-0.783835\pi\)
							0.987441	+	0.157991i	\(0.0505017\pi\)
\(11\)	−0.790074	+	0.0830401i	−0.238216	+	0.0250375i	−0.222884	−	0.974845i	\(-0.571547\pi\)
							−0.0153320	+	0.999882i	\(0.504881\pi\)
\(13\)	1.86721	−	3.08440i	0.517870	−	0.855459i
							\(17\)	−0.311040	+	2.95935i	−0.0754384	+	0.717748i	0.889795	+	0.456360i	\(0.150847\pi\)
−0.965234	+	0.261388i	\(0.915820\pi\)
\(19\)	3.72017	+	3.34966i	0.853466	+	0.768465i	0.974548	−	0.224178i	\(-0.0719697\pi\)
							−0.121082	+	0.992643i	\(0.538636\pi\)
\(23\)	1.46843	−	0.653788i	0.306189	−	0.136324i	−0.247890	−	0.968788i	\(-0.579737\pi\)
							0.554079	+	0.832464i	\(0.313070\pi\)
\(29\)	−7.48597	−	1.59119i	−1.39011	−	0.295477i	−0.548765	−	0.835977i	\(-0.684902\pi\)
							−0.841344	+	0.540500i	\(0.818235\pi\)
\(31\)	0.976804	+	5.48141i	0.175439	+	0.984490i
							\(37\)	−1.72444	−	0.995604i	−0.283496	−	0.163676i	0.351509	−	0.936184i	\(-0.385669\pi\)
−0.635005	+	0.772508i	\(0.719002\pi\)
\(41\)	2.48362	−	11.6845i	0.387876	−	1.82481i	−0.158603	−	0.987342i	\(-0.550699\pi\)
							0.546480	−	0.837472i	\(-0.315968\pi\)
\(43\)	1.12931	−	1.25423i	0.172218	−	0.191268i	−0.650857	−	0.759201i	\(-0.725590\pi\)
							0.823075	+	0.567933i	\(0.192257\pi\)
\(47\)	1.63765	+	0.532106i	0.238876	+	0.0776156i	0.426009	−	0.904719i	\(-0.359919\pi\)
							−0.187132	+	0.982335i	\(0.559919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.20	✓	288
13.10	even	6	inner	403.2.bs.a.283.17	yes	288
31.8	even	5	inner	403.2.bs.a.225.17	yes	288
403.101	even	30	inner	403.2.bs.a.101.20	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.20	✓	288	1.1	even	1	trivial
403.2.bs.a.101.20	yes	288	403.101	even	30	inner
403.2.bs.a.225.17	yes	288	31.8	even	5	inner
403.2.bs.a.283.17	yes	288	13.10	even	6	inner