Embedded newform 320.2.bj.a.3.13

• Label: 320.2.bj.a.3.13
• Level: $320$
• Weight: $2$
• Character: 320.3
• 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.bj (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			3.13
Character	\(\chi\)	\(=\)	320.3
Dual form			320.2.bj.a.107.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06436 - 0.931203i) q^{2} +(-0.000769131 - 0.00115109i) q^{3} +(0.265723 + 1.98227i) q^{4} +(1.37433 + 1.76387i) q^{5} +(-0.000253262 + 0.00194139i) q^{6} +(-3.49768 - 1.44879i) q^{7} +(1.56307 - 2.35729i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06436	−	0.931203i	−0.752616	−	0.658460i
							\(3\)	−0.000769131	−	0.00115109i	−0.000444058	−	0.000664580i	0.831248	−	0.555902i	\(-0.187627\pi\)
−0.831692	+	0.555238i	\(0.812627\pi\)
\(5\)	1.37433	+	1.76387i	0.614617	+	0.788826i
							\(7\)	−3.49768	−	1.44879i	−1.32200	−	0.547590i	−0.393637	−	0.919266i	\(-0.628783\pi\)
−0.928362	+	0.371676i	\(0.878783\pi\)
\(11\)	0.865514	+	4.35123i	0.260962	+	1.31195i	0.859617	+	0.510939i	\(0.170702\pi\)
							−0.598655	+	0.801007i	\(0.704298\pi\)
\(13\)	4.20088	+	0.835608i	1.16512	+	0.231756i	0.739505	−	0.673151i	\(-0.235060\pi\)
							0.425610	+	0.904907i	\(0.360060\pi\)
\(17\)	3.20395			0.777071			0.388535	−	0.921434i	\(-0.372981\pi\)
							0.388535	+	0.921434i	\(0.372981\pi\)
\(19\)	4.68388	+	7.00992i	1.07456	+	1.60819i	0.748881	+	0.662705i	\(0.230592\pi\)
							0.325675	+	0.945482i	\(0.394408\pi\)
\(23\)	0.212001	+	0.511815i	0.0442052	+	0.106721i	0.944440	−	0.328684i	\(-0.106605\pi\)
							−0.900235	+	0.435405i	\(0.856605\pi\)
\(29\)	1.21061	−	6.08616i	0.224805	−	1.13017i	−0.689232	−	0.724541i	\(-0.742052\pi\)
							0.914037	−	0.405631i	\(-0.132948\pi\)
\(31\)	3.83217			0.688278			0.344139	−	0.938919i	\(-0.388171\pi\)
							0.344139	+	0.938919i	\(0.388171\pi\)
\(37\)	0.234958	+	1.18121i	0.0386268	+	0.194190i	0.995281	−	0.0970369i	\(-0.0309365\pi\)
							−0.956654	+	0.291227i	\(0.905936\pi\)
\(41\)	−0.783099	+	1.89057i	−0.122300	+	0.295257i	−0.973158	−	0.230138i	\(-0.926082\pi\)
							0.850859	+	0.525395i	\(0.176082\pi\)
\(43\)	−3.83898	−	2.56512i	−0.585439	−	0.391178i	0.227284	−	0.973829i	\(-0.427015\pi\)
							−0.812722	+	0.582651i	\(0.802015\pi\)
\(47\)		−	2.32696i		−	0.339422i	−0.985494	−	0.169711i	\(-0.945717\pi\)
							0.985494	−	0.169711i	\(-0.0542834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.3.13	yes	368
5.2	odd	4		320.2.bd.a.67.36	yes	368
64.43	odd	16		320.2.bd.a.43.36	✓	368
320.107	even	16	inner	320.2.bj.a.107.13	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.36	✓	368	64.43	odd	16
320.2.bd.a.67.36	yes	368	5.2	odd	4
320.2.bj.a.3.13	yes	368	1.1	even	1	trivial
320.2.bj.a.107.13	yes	368	320.107	even	16	inner