Embedded newform 403.2.bs.a.4.15

• Label: 403.2.bs.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.117135 + 0.551075i) q^{2} +(1.94238 + 2.15723i) q^{3} +(1.53713 + 0.684373i) q^{4} -0.526119i q^{5} +(-1.41631 + 0.817710i) q^{6} +(0.000659214 - 0.00148062i) q^{7} +(-1.21949 + 1.67849i) q^{8} +(-0.567221 + 5.39675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.117135	+	0.551075i	−0.0828266	+	0.389669i	−0.999960	−	0.00891107i	\(-0.997163\pi\)
							0.917134	+	0.398580i	\(0.130497\pi\)
\(3\)	1.94238	+	2.15723i	1.12143	+	1.24548i	0.966254	+	0.257591i	\(0.0829287\pi\)
							0.155179	+	0.987886i	\(0.450405\pi\)
\(5\)		−	0.526119i		−	0.235288i	−0.993056	−	0.117644i	\(-0.962466\pi\)
							0.993056	−	0.117644i	\(-0.0375341\pi\)
\(7\)	0.000659214	−	0.00148062i	0.000249160	−	0.000559622i	−0.913421	−	0.407016i	\(-0.866569\pi\)
							0.913670	+	0.406457i	\(0.133236\pi\)
\(11\)	−4.57631	+	0.480989i	−1.37981	+	0.145024i	−0.765192	−	0.643802i	\(-0.777356\pi\)
							−0.614616	+	0.788826i	\(0.710689\pi\)
\(13\)	−2.33508	−	2.74726i	−0.647633	−	0.761952i
							\(17\)	0.708584	−	6.74172i	0.171857	−	1.63511i	−0.480357	−	0.877073i	\(-0.659493\pi\)
0.652214	−	0.758035i	\(-0.273840\pi\)
\(19\)	4.55789	+	4.10394i	1.04565	+	0.941508i	0.998353	−	0.0573724i	\(-0.0182722\pi\)
							0.0472982	+	0.998881i	\(0.484939\pi\)
\(23\)	5.24119	−	2.33353i	1.09286	−	0.486574i	0.220478	−	0.975392i	\(-0.429238\pi\)
							0.872386	+	0.488818i	\(0.162572\pi\)
\(29\)	0.908907	+	0.193194i	0.168780	+	0.0358752i	0.291526	−	0.956563i	\(-0.405837\pi\)
							−0.122747	+	0.992438i	\(0.539170\pi\)
\(31\)	−1.79726	−	5.26971i	−0.322798	−	0.946468i
							\(37\)	1.81185	+	1.04607i	0.297867	+	0.171974i	0.641484	−	0.767136i	\(-0.278319\pi\)
−0.343617	+	0.939110i	\(0.611652\pi\)
\(41\)	1.48744	−	6.99787i	0.232299	−	1.09288i	−0.695130	−	0.718884i	\(-0.744653\pi\)
							0.927429	−	0.373999i	\(-0.122014\pi\)
\(43\)	−7.41867	+	8.23926i	−1.13134	+	1.25648i	−0.168703	+	0.985667i	\(0.553958\pi\)
							−0.962633	+	0.270809i	\(0.912709\pi\)
\(47\)	−4.40118	−	1.43003i	−0.641978	−	0.208591i	−0.0301044	−	0.999547i	\(-0.509584\pi\)
							−0.611874	+	0.790955i	\(0.709584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.15	✓	288
13.10	even	6	inner	403.2.bs.a.283.22	yes	288
31.8	even	5	inner	403.2.bs.a.225.22	yes	288
403.101	even	30	inner	403.2.bs.a.101.15	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.15	✓	288	1.1	even	1	trivial
403.2.bs.a.101.15	yes	288	403.101	even	30	inner
403.2.bs.a.225.22	yes	288	31.8	even	5	inner
403.2.bs.a.283.22	yes	288	13.10	even	6	inner