Embedded newform 403.2.bz.a.38.18

• Label: 403.2.bz.a.38.18
• Level: $403$
• Weight: $2$
• Character: 403.38
• 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.bz (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			38.18
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0579861 + 0.0188408i) q^{2} +(1.89240 - 0.402242i) q^{3} +(-1.61503 + 1.17339i) q^{4} +(-2.24387 - 1.29550i) q^{5} +(-0.102154 + 0.0589787i) q^{6} +(-2.03436 + 4.56926i) q^{7} +(0.143216 - 0.197120i) q^{8} +(0.678735 - 0.302192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0579861	+	0.0188408i	−0.0410024	+	0.0133225i	−0.329446	−	0.944174i	\(-0.606862\pi\)
							0.288444	+	0.957497i	\(0.406862\pi\)
\(3\)	1.89240	−	0.402242i	1.09258	−	0.232234i	0.373823	−	0.927500i	\(-0.378047\pi\)
							0.718753	+	0.695266i	\(0.244713\pi\)
\(5\)	−2.24387	−	1.29550i	−1.00349	−	0.579366i	−0.0942118	−	0.995552i	\(-0.530033\pi\)
							−0.909279	+	0.416186i	\(0.863366\pi\)
\(7\)	−2.03436	+	4.56926i	−0.768917	+	1.72702i	−0.0862663	+	0.996272i	\(0.527494\pi\)
							−0.682651	+	0.730745i	\(0.739173\pi\)
\(11\)	−2.27243	+	0.238842i	−0.685164	+	0.0720137i	−0.440713	−	0.897648i	\(-0.645274\pi\)
							−0.244451	+	0.969662i	\(0.578608\pi\)
\(13\)	−1.59774	−	3.23222i	−0.443133	−	0.896456i
							\(17\)	−0.522162	+	4.96804i	−0.126643	+	1.20493i	0.727951	+	0.685629i	\(0.240473\pi\)
−0.854594	+	0.519297i	\(0.826194\pi\)
\(19\)	−3.07559	−	2.76928i	−0.705589	−	0.635316i	0.236127	−	0.971722i	\(-0.424122\pi\)
							−0.941717	+	0.336407i	\(0.890788\pi\)
\(23\)	1.33604	+	0.970690i	0.278583	+	0.202403i	0.718299	−	0.695734i	\(-0.244921\pi\)
							−0.439716	+	0.898137i	\(0.644921\pi\)
\(29\)	2.52474	+	7.77035i	0.468833	+	1.44292i	0.854098	+	0.520112i	\(0.174110\pi\)
							−0.385265	+	0.922806i	\(0.625890\pi\)
\(31\)	2.19426	+	5.11715i	0.394100	+	0.919068i
							\(37\)	3.55709	−	2.05369i	0.584782	−	0.337624i	−0.178249	−	0.983985i	\(-0.557043\pi\)
0.763032	+	0.646361i	\(0.223710\pi\)
\(41\)	−0.298739	+	1.40546i	−0.0466552	+	0.219495i	−0.995299	−	0.0968511i	\(-0.969123\pi\)
							0.948644	+	0.316347i	\(0.102456\pi\)
\(43\)	5.30629	−	5.89323i	0.809202	−	0.898709i	−0.187298	−	0.982303i	\(-0.559973\pi\)
							0.996500	+	0.0835937i	\(0.0266398\pi\)
\(47\)	10.7807	+	3.50285i	1.57252	+	0.510943i	0.960116	−	0.279600i	\(-0.0902020\pi\)
							0.612406	+	0.790544i	\(0.290202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.18	✓	288
13.12	even	2	inner	403.2.bz.a.38.19	yes	288
31.9	even	15	inner	403.2.bz.a.350.19	yes	288
403.350	even	30	inner	403.2.bz.a.350.18	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.18	✓	288	1.1	even	1	trivial
403.2.bz.a.38.19	yes	288	13.12	even	2	inner
403.2.bz.a.350.18	yes	288	403.350	even	30	inner
403.2.bz.a.350.19	yes	288	31.9	even	15	inner