Embedded newform 320.2.bj.a.267.8

• Label: 320.2.bj.a.267.8
• Level: $320$
• Weight: $2$
• Character: 320.267
• 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			267.8
Character	\(\chi\)	\(=\)	320.267
Dual form			320.2.bj.a.163.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26804 + 0.626160i) q^{2} +(-1.86701 - 1.24749i) q^{3} +(1.21585 - 1.58799i) q^{4} +(1.96476 + 1.06757i) q^{5} +(3.14857 + 0.412825i) q^{6} +(-2.27242 + 0.941268i) q^{7} +(-0.547402 + 2.77495i) q^{8} +(0.781421 + 1.88652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26804	+	0.626160i	−0.896639	+	0.442762i
							\(3\)	−1.86701	−	1.24749i	−1.07792	−	0.720241i	−0.115908	−	0.993260i	\(-0.536978\pi\)
−0.962009	+	0.273019i	\(0.911978\pi\)
\(5\)	1.96476	+	1.06757i	0.878668	+	0.477434i
							\(7\)	−2.27242	+	0.941268i	−0.858895	+	0.355766i	−0.768275	−	0.640120i	\(-0.778885\pi\)
−0.0906194	+	0.995886i	\(0.528885\pi\)
\(11\)	1.49166	+	0.296709i	0.449752	+	0.0894611i	0.414769	−	0.909927i	\(-0.363862\pi\)
							0.0349826	+	0.999388i	\(0.488862\pi\)
\(13\)	−0.00734329	−	0.0369172i	−0.00203666	−	0.0102390i	0.979754	−	0.200207i	\(-0.0641615\pi\)
							−0.981790	+	0.189968i	\(0.939162\pi\)
\(17\)	4.38412			1.06331			0.531653	−	0.846962i	\(-0.321571\pi\)
							0.531653	+	0.846962i	\(0.321571\pi\)
\(19\)	1.60124	+	1.06991i	0.367349	+	0.245455i	0.725520	−	0.688201i	\(-0.241599\pi\)
							−0.358171	+	0.933656i	\(0.616599\pi\)
\(23\)	1.78201	−	4.30215i	0.371575	−	0.897061i	−0.621909	−	0.783089i	\(-0.713643\pi\)
							0.993484	−	0.113972i	\(-0.0363573\pi\)
\(29\)	4.61385	−	0.917751i	0.856770	−	0.170422i	0.252887	−	0.967496i	\(-0.418620\pi\)
							0.603882	+	0.797074i	\(0.293620\pi\)
\(31\)	6.75639			1.21348			0.606741	−	0.794899i	\(-0.292476\pi\)
							0.606741	+	0.794899i	\(0.292476\pi\)
\(37\)	9.33189	+	1.85623i	1.53415	+	0.305162i	0.888646	−	0.458593i	\(-0.151646\pi\)
							0.645506	+	0.763755i	\(0.276646\pi\)
\(41\)	−1.85581	−	4.48033i	−0.289829	−	0.699709i	0.710162	−	0.704039i	\(-0.248622\pi\)
							−0.999991	+	0.00432950i	\(0.998622\pi\)
\(43\)	2.57974	+	3.86086i	0.393407	+	0.588776i	0.974314	−	0.225195i	\(-0.0723018\pi\)
							−0.580906	+	0.813970i	\(0.697302\pi\)
\(47\)		−	11.4592i		−	1.67149i	−0.549115	−	0.835747i	\(-0.685035\pi\)
							0.549115	−	0.835747i	\(-0.314965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.267.8	yes	368
5.3	odd	4		320.2.bd.a.203.30	✓	368
64.35	odd	16		320.2.bd.a.227.30	yes	368
320.163	even	16	inner	320.2.bj.a.163.8	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.203.30	✓	368	5.3	odd	4
320.2.bd.a.227.30	yes	368	64.35	odd	16
320.2.bj.a.163.8	yes	368	320.163	even	16	inner
320.2.bj.a.267.8	yes	368	1.1	even	1	trivial