Embedded newform 320.2.bf.a.269.20

• Label: 320.2.bf.a.269.20
• Level: $320$
• Weight: $2$
• Character: 320.269
• 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			269.20
Character	\(\chi\)	\(=\)	320.269
Dual form			320.2.bf.a.69.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.146402 - 1.40662i) q^{2} +(1.26531 + 1.89367i) q^{3} +(-1.95713 + 0.411863i) q^{4} +(2.09912 - 0.770508i) q^{5} +(2.47843 - 2.05705i) q^{6} +(1.27045 - 0.526239i) q^{7} +(0.865861 + 2.69264i) q^{8} +(-0.836936 + 2.02054i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.146402	−	1.40662i	−0.103522	−	0.994627i
							\(3\)	1.26531	+	1.89367i	0.730529	+	1.09331i	0.991769	+	0.128040i	\(0.0408684\pi\)
−0.261240	+	0.965274i	\(0.584132\pi\)
\(5\)	2.09912	−	0.770508i	0.938756	−	0.344581i
							\(7\)	1.27045	−	0.526239i	0.480186	−	0.198900i	−0.129442	−	0.991587i	\(-0.541319\pi\)
0.609629	+	0.792687i	\(0.291319\pi\)
\(11\)	0.153255	−	0.229363i	0.0462082	−	0.0691554i	−0.807646	−	0.589667i	\(-0.799259\pi\)
							0.853854	+	0.520512i	\(0.174259\pi\)
\(13\)	−1.03160	+	5.18622i	−0.286115	+	1.43840i	0.523795	+	0.851844i	\(0.324516\pi\)
							−0.809910	+	0.586554i	\(0.800484\pi\)
\(17\)	−2.56304	+	2.56304i	−0.621630	+	0.621630i	−0.945948	−	0.324318i	\(-0.894865\pi\)
							0.324318	+	0.945948i	\(0.394865\pi\)
\(19\)	1.49288	−	7.50520i	0.342489	−	1.72181i	−0.298630	−	0.954369i	\(-0.596530\pi\)
							0.641119	−	0.767441i	\(-0.278470\pi\)
\(23\)	1.01769	−	2.45693i	0.212204	−	0.512305i	−0.781558	−	0.623833i	\(-0.785574\pi\)
							0.993761	+	0.111528i	\(0.0355745\pi\)
\(29\)	3.04176	+	4.55231i	0.564840	+	0.845343i	0.998444	−	0.0557608i	\(-0.0177584\pi\)
							−0.433604	+	0.901103i	\(0.642758\pi\)
\(31\)			0.412032i			0.0740031i	0.999315	+	0.0370015i	\(0.0117806\pi\)
							−0.999315	+	0.0370015i	\(0.988219\pi\)
\(37\)	−0.872863	+	0.173623i	−0.143498	+	0.0285435i	−0.266316	−	0.963886i	\(-0.585807\pi\)
							0.122819	+	0.992429i	\(0.460807\pi\)
\(41\)	−11.5185	−	4.77113i	−1.79889	−	0.745126i	−0.986874	−	0.161495i	\(-0.948368\pi\)
							−0.812019	−	0.583631i	\(-0.801632\pi\)
\(43\)	1.17018	−	1.75130i	0.178451	−	0.267071i	−0.731451	−	0.681894i	\(-0.761156\pi\)
							0.909902	+	0.414823i	\(0.136156\pi\)
\(47\)	−5.32930	+	5.32930i	−0.777359	+	0.777359i	−0.979381	−	0.202022i	\(-0.935249\pi\)
							0.202022	+	0.979381i	\(0.435249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.269.20	yes	368
5.4	even	2	inner	320.2.bf.a.269.27	yes	368
64.5	even	16	inner	320.2.bf.a.69.27	yes	368
320.69	even	16	inner	320.2.bf.a.69.20	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.69.20	✓	368	320.69	even	16	inner
320.2.bf.a.69.27	yes	368	64.5	even	16	inner
320.2.bf.a.269.20	yes	368	1.1	even	1	trivial
320.2.bf.a.269.27	yes	368	5.4	even	2	inner