Embedded newform 320.2.bd.a.43.15

• Label: 320.2.bd.a.43.15
• Level: $320$
• Weight: $2$
• Character: 320.43
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			43.15
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.828734 + 1.14595i) q^{2} +(1.54280 + 1.03087i) q^{3} +(-0.626401 - 1.89937i) q^{4} +(-2.19943 + 0.403103i) q^{5} +(-2.45989 + 0.913659i) q^{6} +(1.23320 + 2.97720i) q^{7} +(2.69571 + 0.856252i) q^{8} +(0.169501 + 0.409211i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.828734	+	1.14595i	−0.586003	+	0.810309i
							\(3\)	1.54280	+	1.03087i	0.890737	+	0.595172i	0.914518	−	0.404546i	\(-0.132570\pi\)
−0.0237806	+	0.999717i	\(0.507570\pi\)
\(5\)	−2.19943	+	0.403103i	−0.983617	+	0.180273i
							\(7\)	1.23320	+	2.97720i	0.466105	+	1.12528i	0.965849	+	0.259104i	\(0.0834274\pi\)
−0.499744	+	0.866173i	\(0.666573\pi\)
\(11\)	−1.06676	+	5.36294i	−0.321639	+	1.61699i	0.394408	+	0.918935i	\(0.370950\pi\)
							−0.716047	+	0.698052i	\(0.754050\pi\)
\(13\)	−0.965962	−	4.85622i	−0.267910	−	1.34687i	−0.846993	−	0.531605i	\(-0.821589\pi\)
							0.579083	−	0.815269i	\(-0.303411\pi\)
\(17\)			5.17739i			1.25570i	0.778334	+	0.627850i	\(0.216065\pi\)
							−0.778334	+	0.627850i	\(0.783935\pi\)
\(19\)	−2.66610	+	3.99010i	−0.611646	+	0.915393i	−0.999981	−	0.00612553i	\(-0.998050\pi\)
							0.388336	+	0.921518i	\(0.373050\pi\)
\(23\)	−0.967974	−	0.400948i	−0.201836	−	0.0836034i	0.279475	−	0.960153i	\(-0.409839\pi\)
							−0.481312	+	0.876549i	\(0.659839\pi\)
\(29\)	−0.191365	−	0.962057i	−0.0355356	−	0.178649i	0.958941	−	0.283604i	\(-0.0915302\pi\)
							−0.994477	+	0.104955i	\(0.966530\pi\)
\(31\)	2.51719			0.452101			0.226051	−	0.974116i	\(-0.427419\pi\)
							0.226051	+	0.974116i	\(0.427419\pi\)
\(37\)	−2.14618	−	0.426902i	−0.352830	−	0.0701823i	0.0154932	−	0.999880i	\(-0.495068\pi\)
							−0.368324	+	0.929698i	\(0.620068\pi\)
\(41\)	0.740094	+	1.78674i	0.115583	+	0.279043i	0.971075	−	0.238773i	\(-0.0767453\pi\)
							−0.855492	+	0.517816i	\(0.826745\pi\)
\(43\)	4.23741	+	6.34173i	0.646198	+	0.967104i	0.999501	+	0.0315950i	\(0.0100587\pi\)
							−0.353302	+	0.935509i	\(0.614941\pi\)
\(47\)	9.14833			1.33442			0.667211	−	0.744869i	\(-0.267488\pi\)
							0.667211	+	0.744869i	\(0.267488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.15	✓	368
5.2	odd	4		320.2.bj.a.107.10	yes	368
64.3	odd	16		320.2.bj.a.3.10	yes	368
320.67	even	16	inner	320.2.bd.a.67.15	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.15	✓	368	1.1	even	1	trivial
320.2.bd.a.67.15	yes	368	320.67	even	16	inner
320.2.bj.a.3.10	yes	368	64.3	odd	16
320.2.bj.a.107.10	yes	368	5.2	odd	4