Embedded newform 320.2.bj.a.27.18

• Label: 320.2.bj.a.27.18
• Level: $320$
• Weight: $2$
• Character: 320.27
• 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			27.18
Character	\(\chi\)	\(=\)	320.27
Dual form			320.2.bj.a.83.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.579937 + 1.28983i) q^{2} +(-2.18663 + 0.434947i) q^{3} +(-1.32735 - 1.49605i) q^{4} +(-2.12006 - 0.710868i) q^{5} +(0.707097 - 3.07263i) q^{6} +(0.175581 + 0.423890i) q^{7} +(2.69943 - 0.844442i) q^{8} +(1.82052 - 0.754085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.579937	+	1.28983i	−0.410078	+	0.912051i
							\(3\)	−2.18663	+	0.434947i	−1.26245	+	0.251117i	−0.780541	−	0.625104i	\(-0.785056\pi\)
−0.481909	+	0.876221i	\(0.660056\pi\)
\(5\)	−2.12006	−	0.710868i	−0.948121	−	0.317910i
							\(7\)	0.175581	+	0.423890i	0.0663634	+	0.160215i	0.953582	−	0.301134i	\(-0.0973653\pi\)
−0.887218	+	0.461350i	\(0.847365\pi\)
\(11\)	−1.89467	−	2.83557i	−0.571264	−	0.854957i	0.427533	−	0.904000i	\(-0.359383\pi\)
							−0.998797	+	0.0490430i	\(0.984383\pi\)
\(13\)	4.10050	+	2.73986i	1.13727	+	0.759902i	0.973971	−	0.226674i	\(-0.0727853\pi\)
							0.163303	+	0.986576i	\(0.447785\pi\)
\(17\)	4.14234			1.00467			0.502333	−	0.864674i	\(-0.332475\pi\)
							0.502333	+	0.864674i	\(0.332475\pi\)
\(19\)	−2.46167	+	0.489657i	−0.564746	+	0.112335i	−0.469203	−	0.883090i	\(-0.655459\pi\)
							−0.0955429	+	0.995425i	\(0.530459\pi\)
\(23\)	0.780139	+	0.323144i	0.162670	+	0.0673802i	0.462533	−	0.886602i	\(-0.346941\pi\)
							−0.299862	+	0.953982i	\(0.596941\pi\)
\(29\)	3.73483	−	5.58957i	0.693541	−	1.03796i	−0.302847	−	0.953039i	\(-0.597937\pi\)
							0.996388	−	0.0849180i	\(-0.0270628\pi\)
\(31\)	3.15049			0.565845			0.282922	−	0.959143i	\(-0.408696\pi\)
							0.282922	+	0.959143i	\(0.408696\pi\)
\(37\)	−2.55880	−	3.82952i	−0.420665	−	0.629570i	0.559247	−	0.829001i	\(-0.311090\pi\)
							−0.979912	+	0.199432i	\(0.936090\pi\)
\(41\)	5.12309	−	2.12205i	0.800092	−	0.331409i	0.0550989	−	0.998481i	\(-0.482453\pi\)
							0.744993	+	0.667072i	\(0.232453\pi\)
\(43\)	−2.38608	+	11.9957i	−0.363874	+	1.82932i	0.172092	+	0.985081i	\(0.444947\pi\)
							−0.535966	+	0.844239i	\(0.680053\pi\)
\(47\)		−	6.39434i		−	0.932710i	−0.884598	−	0.466355i	\(-0.845567\pi\)
							0.884598	−	0.466355i	\(-0.154433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.27.18	yes	368
5.3	odd	4		320.2.bd.a.283.40	yes	368
64.19	odd	16		320.2.bd.a.147.40	✓	368
320.83	even	16	inner	320.2.bj.a.83.18	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.40	✓	368	64.19	odd	16
320.2.bd.a.283.40	yes	368	5.3	odd	4
320.2.bj.a.27.18	yes	368	1.1	even	1	trivial
320.2.bj.a.83.18	yes	368	320.83	even	16	inner