Embedded newform 320.2.bf.a.309.12

• Label: 320.2.bf.a.309.12
• Level: $320$
• Weight: $2$
• Character: 320.309
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (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			309.12
Character	\(\chi\)	\(=\)	320.309
Dual form			320.2.bf.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.963775 + 1.03496i) q^{2} +(-0.0117336 - 0.0589886i) q^{3} +(-0.142276 - 1.99493i) q^{4} +(-2.09609 + 0.778715i) q^{5} +(0.0723593 + 0.0447080i) q^{6} +(0.480951 - 1.16112i) q^{7} +(2.20179 + 1.77542i) q^{8} +(2.76830 - 1.14667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.963775	+	1.03496i	−0.681492	+	0.731826i
							\(3\)	−0.0117336	−	0.0589886i	−0.00677438	−	0.0340571i	0.977250	−	0.212089i	\(-0.0680268\pi\)
−0.984025	+	0.178032i	\(0.943027\pi\)
\(5\)	−2.09609	+	0.778715i	−0.937401	+	0.348252i
							\(7\)	0.480951	−	1.16112i	0.181782	−	0.438861i	−0.806552	−	0.591164i	\(-0.798669\pi\)
0.988334	+	0.152302i	\(0.0486687\pi\)
\(11\)	−0.377263	+	1.89663i	−0.113749	+	0.571855i	0.881307	+	0.472543i	\(0.156664\pi\)
							−0.995056	+	0.0993113i	\(0.968336\pi\)
\(13\)	2.09734	+	1.40140i	0.581696	+	0.388677i	0.811320	−	0.584602i	\(-0.198749\pi\)
							−0.229624	+	0.973280i	\(0.573749\pi\)
\(17\)	2.30915	+	2.30915i	0.560052	+	0.560052i	0.929322	−	0.369270i	\(-0.120392\pi\)
							−0.369270	+	0.929322i	\(0.620392\pi\)
\(19\)	4.58476	+	3.06344i	1.05181	+	0.702800i	0.956229	−	0.292621i	\(-0.0945274\pi\)
							0.0955864	+	0.995421i	\(0.469527\pi\)
\(23\)	0.939181	−	0.389022i	0.195833	−	0.0811166i	−0.282612	−	0.959234i	\(-0.591201\pi\)
							0.478444	+	0.878118i	\(0.341201\pi\)
\(29\)	1.37026	+	6.88877i	0.254451	+	1.27921i	0.870760	+	0.491708i	\(0.163627\pi\)
							−0.616309	+	0.787505i	\(0.711373\pi\)
\(31\)		−	3.51627i		−	0.631541i	−0.948836	−	0.315770i	\(-0.897737\pi\)
							0.948836	−	0.315770i	\(-0.102263\pi\)
\(37\)	−0.866024	−	1.29610i	−0.142374	−	0.213077i	0.753429	−	0.657529i	\(-0.228398\pi\)
							−0.895803	+	0.444452i	\(0.853398\pi\)
\(41\)	2.28975	+	5.52794i	0.357599	+	0.863320i	0.995637	+	0.0933147i	\(0.0297463\pi\)
							−0.638038	+	0.770005i	\(0.720254\pi\)
\(43\)	0.0677321	−	0.340512i	0.0103291	−	0.0519277i	−0.975277	−	0.220986i	\(-0.929073\pi\)
							0.985606	+	0.169058i	\(0.0540726\pi\)
\(47\)	−6.21269	−	6.21269i	−0.906213	−	0.906213i	0.0897509	−	0.995964i	\(-0.471393\pi\)
							−0.995964	+	0.0897509i	\(0.971393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.309.12	yes	368
5.4	even	2	inner	320.2.bf.a.309.35	yes	368
64.29	even	16	inner	320.2.bf.a.29.35	yes	368
320.29	even	16	inner	320.2.bf.a.29.12	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.12	✓	368	320.29	even	16	inner
320.2.bf.a.29.35	yes	368	64.29	even	16	inner
320.2.bf.a.309.12	yes	368	1.1	even	1	trivial
320.2.bf.a.309.35	yes	368	5.4	even	2	inner