Embedded newform 320.2.bf.a.29.35

• Label: 320.2.bf.a.29.35
• Level: $320$
• Weight: $2$
• Character: 320.29
• 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			29.35
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.963775 + 1.03496i) q^{2} +(0.0117336 - 0.0589886i) q^{3} +(-0.142276 + 1.99493i) q^{4} +(1.52158 + 1.63854i) q^{5} +(0.0723593 - 0.0447080i) q^{6} +(-0.480951 - 1.16112i) q^{7} +(-2.20179 + 1.77542i) q^{8} +(2.76830 + 1.14667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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	\(-0.931973\pi\)
0.984025	+	0.178032i	\(0.0569732\pi\)
\(5\)	1.52158	+	1.63854i	0.680471	+	0.732775i
							\(7\)	−0.480951	−	1.16112i	−0.181782	−	0.438861i	0.806552	−	0.591164i	\(-0.201331\pi\)
−0.988334	+	0.152302i	\(0.951331\pi\)
\(11\)	−0.377263	−	1.89663i	−0.113749	−	0.571855i	−0.995056	−	0.0993113i	\(-0.968336\pi\)
							0.881307	−	0.472543i	\(-0.156664\pi\)
\(13\)	−2.09734	+	1.40140i	−0.581696	+	0.388677i	−0.811320	−	0.584602i	\(-0.801251\pi\)
							0.229624	+	0.973280i	\(0.426251\pi\)
\(17\)	−2.30915	+	2.30915i	−0.560052	+	0.560052i	−0.929322	−	0.369270i	\(-0.879608\pi\)
							0.369270	+	0.929322i	\(0.379608\pi\)
\(19\)	4.58476	−	3.06344i	1.05181	−	0.702800i	0.0955864	−	0.995421i	\(-0.469527\pi\)
							0.956229	+	0.292621i	\(0.0945274\pi\)
\(23\)	−0.939181	−	0.389022i	−0.195833	−	0.0811166i	0.282612	−	0.959234i	\(-0.408799\pi\)
							−0.478444	+	0.878118i	\(0.658799\pi\)
\(29\)	1.37026	−	6.88877i	0.254451	−	1.27921i	−0.616309	−	0.787505i	\(-0.711373\pi\)
							0.870760	−	0.491708i	\(-0.163627\pi\)
\(31\)			3.51627i			0.631541i	0.948836	+	0.315770i	\(0.102263\pi\)
							−0.948836	+	0.315770i	\(0.897737\pi\)
\(37\)	0.866024	−	1.29610i	0.142374	−	0.213077i	−0.753429	−	0.657529i	\(-0.771602\pi\)
							0.895803	+	0.444452i	\(0.146602\pi\)
\(41\)	2.28975	−	5.52794i	0.357599	−	0.863320i	−0.638038	−	0.770005i	\(-0.720254\pi\)
							0.995637	−	0.0933147i	\(-0.0297463\pi\)
\(43\)	−0.0677321	−	0.340512i	−0.0103291	−	0.0519277i	0.975277	−	0.220986i	\(-0.0709274\pi\)
							−0.985606	+	0.169058i	\(0.945927\pi\)
\(47\)	6.21269	−	6.21269i	0.906213	−	0.906213i	−0.0897509	−	0.995964i	\(-0.528607\pi\)
							0.995964	+	0.0897509i	\(0.0286071\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.12	✓	368	5.4	even	2	inner
320.2.bf.a.29.35	yes	368	1.1	even	1	trivial
320.2.bf.a.309.12	yes	368	64.53	even	16	inner
320.2.bf.a.309.35	yes	368	320.309	even	16	inner