Embedded newform 320.2.bf.a.29.14

• Label: 320.2.bf.a.29.14
• 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			29.14
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.878389 - 1.10835i) q^{2} +(0.572105 - 2.87617i) q^{3} +(-0.456865 + 1.94712i) q^{4} +(1.03679 - 1.98118i) q^{5} +(-3.69032 + 1.89230i) q^{6} +(-0.591145 - 1.42715i) q^{7} +(2.55939 - 1.20396i) q^{8} +(-5.17340 - 2.14289i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.878389	−	1.10835i	−0.621115	−	0.783719i
							\(3\)	0.572105	−	2.87617i	0.330305	−	1.66056i	−0.356938	−	0.934128i	\(-0.616179\pi\)
0.687243	−	0.726428i	\(-0.258821\pi\)
\(5\)	1.03679	−	1.98118i	0.463668	−	0.886009i
							\(7\)	−0.591145	−	1.42715i	−0.223432	−	0.539412i	0.771920	−	0.635720i	\(-0.219297\pi\)
−0.995352	+	0.0963078i	\(0.969297\pi\)
\(11\)	0.342375	+	1.72123i	0.103230	+	0.518972i	0.997452	+	0.0713473i	\(0.0227299\pi\)
							−0.894222	+	0.447624i	\(0.852270\pi\)
\(13\)	1.29453	−	0.864976i	0.359038	−	0.239901i	−0.362948	−	0.931809i	\(-0.618230\pi\)
							0.721986	+	0.691908i	\(0.243230\pi\)
\(17\)	−3.31231	+	3.31231i	−0.803354	+	0.803354i	−0.983618	−	0.180264i	\(-0.942305\pi\)
							0.180264	+	0.983618i	\(0.442305\pi\)
\(19\)	6.57237	−	4.39151i	1.50780	−	1.00748i	0.519588	−	0.854417i	\(-0.326085\pi\)
							0.988216	−	0.153066i	\(-0.0489146\pi\)
\(23\)	3.06370	+	1.26902i	0.638825	+	0.264610i	0.678497	−	0.734603i	\(-0.262632\pi\)
							−0.0396727	+	0.999213i	\(0.512632\pi\)
\(29\)	−1.89053	+	9.50433i	−0.351062	+	1.76491i	0.252503	+	0.967596i	\(0.418746\pi\)
							−0.603566	+	0.797313i	\(0.706254\pi\)
\(31\)			2.92048i			0.524534i	0.964995	+	0.262267i	\(0.0844701\pi\)
							−0.964995	+	0.262267i	\(0.915530\pi\)
\(37\)	−3.97091	+	5.94288i	−0.652813	+	0.977004i	0.346429	+	0.938076i	\(0.387394\pi\)
							−0.999242	+	0.0389278i	\(0.987606\pi\)
\(41\)	−0.553436	+	1.33611i	−0.0864322	+	0.208666i	−0.961186	−	0.275903i	\(-0.911023\pi\)
							0.874753	+	0.484568i	\(0.161023\pi\)
\(43\)	−1.22894	−	6.17828i	−0.187411	−	0.942178i	−0.953947	−	0.299977i	\(-0.903021\pi\)
							0.766536	−	0.642202i	\(-0.221979\pi\)
\(47\)	5.85658	−	5.85658i	0.854270	−	0.854270i	−0.136386	−	0.990656i	\(-0.543549\pi\)
							0.990656	+	0.136386i	\(0.0435488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.14	✓	368
5.4	even	2	inner	320.2.bf.a.29.33	yes	368
64.53	even	16	inner	320.2.bf.a.309.33	yes	368
320.309	even	16	inner	320.2.bf.a.309.14	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.14	✓	368	1.1	even	1	trivial
320.2.bf.a.29.33	yes	368	5.4	even	2	inner
320.2.bf.a.309.14	yes	368	320.309	even	16	inner
320.2.bf.a.309.33	yes	368	64.53	even	16	inner