Embedded newform 320.2.bf.a.29.16

• Label: 320.2.bf.a.29.16
• Level: $320$
• Weight: $2$
• Character: 320.29
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• 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			29.16
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.635601 + 1.26333i) q^{2} +(0.627776 - 3.15604i) q^{3} +(-1.19202 - 1.60595i) q^{4} +(-1.59499 - 1.56716i) q^{5} +(3.58812 + 2.79908i) q^{6} +(-0.393699 - 0.950472i) q^{7} +(2.78650 - 0.485175i) q^{8} +(-6.79487 - 2.81453i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.635601	+	1.26333i	−0.449438	+	0.893312i
							\(3\)	0.627776	−	3.15604i	0.362447	−	1.82214i	−0.181942	−	0.983309i	\(-0.558238\pi\)
0.544388	−	0.838833i	\(-0.316762\pi\)
\(5\)	−1.59499	−	1.56716i	−0.713302	−	0.700857i
							\(7\)	−0.393699	−	0.950472i	−0.148804	−	0.359245i	0.831848	−	0.555003i	\(-0.187283\pi\)
−0.980652	+	0.195759i	\(0.937283\pi\)
\(11\)	0.793485	+	3.98912i	0.239245	+	1.20276i	0.894399	+	0.447269i	\(0.147603\pi\)
							−0.655155	+	0.755495i	\(0.727397\pi\)
\(13\)	−3.25630	+	2.17579i	−0.903134	+	0.603455i	−0.918063	−	0.396434i	\(-0.870248\pi\)
							0.0149292	+	0.999889i	\(0.495248\pi\)
\(17\)	4.57231	−	4.57231i	1.10895	−	1.10895i	0.115659	−	0.993289i	\(-0.463102\pi\)
							0.993289	−	0.115659i	\(-0.0368978\pi\)
\(19\)	−1.03313	+	0.690319i	−0.237017	+	0.158370i	−0.668407	−	0.743796i	\(-0.733023\pi\)
							0.431389	+	0.902166i	\(0.358023\pi\)
\(23\)	−4.62250	−	1.91470i	−0.963858	−	0.399243i	−0.155436	−	0.987846i	\(-0.549678\pi\)
							−0.808422	+	0.588603i	\(0.799678\pi\)
\(29\)	1.04038	−	5.23033i	0.193193	−	0.971248i	−0.755524	−	0.655121i	\(-0.772618\pi\)
							0.948717	−	0.316127i	\(-0.102382\pi\)
\(31\)		−	5.14150i		−	0.923440i	−0.887026	−	0.461720i	\(-0.847232\pi\)
							0.887026	−	0.461720i	\(-0.152768\pi\)
\(37\)	−0.223761	+	0.334881i	−0.0367860	+	0.0550541i	−0.849407	−	0.527739i	\(-0.823040\pi\)
							0.812621	+	0.582793i	\(0.198040\pi\)
\(41\)	3.90431	−	9.42585i	0.609751	−	1.47207i	−0.253520	−	0.967330i	\(-0.581588\pi\)
							0.863271	−	0.504740i	\(-0.168412\pi\)
\(43\)	−0.307542	−	1.54612i	−0.0468997	−	0.235781i	0.950224	−	0.311566i	\(-0.100854\pi\)
							−0.997124	+	0.0757855i	\(0.975854\pi\)
\(47\)	2.52100	−	2.52100i	0.367726	−	0.367726i	−0.498921	−	0.866647i	\(-0.666270\pi\)
							0.866647	+	0.498921i	\(0.166270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.16	✓	368
5.4	even	2	inner	320.2.bf.a.29.31	yes	368
64.53	even	16	inner	320.2.bf.a.309.31	yes	368
320.309	even	16	inner	320.2.bf.a.309.16	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.16	✓	368	1.1	even	1	trivial
320.2.bf.a.29.31	yes	368	5.4	even	2	inner
320.2.bf.a.309.16	yes	368	320.309	even	16	inner
320.2.bf.a.309.31	yes	368	64.53	even	16	inner