Embedded newform 320.2.bf.a.29.5

• Label: 320.2.bf.a.29.5
• 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			29.5
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33571 - 0.464641i) q^{2} +(0.488587 - 2.45629i) q^{3} +(1.56822 + 1.24125i) q^{4} +(0.609350 + 2.15144i) q^{5} +(-1.79390 + 3.05386i) q^{6} +(-0.627512 - 1.51495i) q^{7} +(-1.51794 - 2.38660i) q^{8} +(-3.02301 - 1.25217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33571	−	0.464641i	−0.944486	−	0.328551i
							\(3\)	0.488587	−	2.45629i	0.282086	−	1.41814i	−0.536570	−	0.843856i	\(-0.680280\pi\)
0.818656	−	0.574284i	\(-0.194720\pi\)
\(5\)	0.609350	+	2.15144i	0.272510	+	0.962153i
							\(7\)	−0.627512	−	1.51495i	−0.237177	−	0.572596i	0.759812	−	0.650143i	\(-0.225291\pi\)
−0.996989	+	0.0775471i	\(0.975291\pi\)
\(11\)	−0.948438	−	4.76812i	−0.285965	−	1.43764i	−0.810242	−	0.586095i	\(-0.800664\pi\)
							0.524277	−	0.851548i	\(-0.324336\pi\)
\(13\)	−2.02466	+	1.35283i	−0.561538	+	0.375208i	−0.803706	−	0.595026i	\(-0.797142\pi\)
							0.242168	+	0.970234i	\(0.422142\pi\)
\(17\)	5.52592	−	5.52592i	1.34023	−	1.34023i	0.444406	−	0.895826i	\(-0.353415\pi\)
							0.895826	−	0.444406i	\(-0.146585\pi\)
\(19\)	4.06903	−	2.71884i	0.933500	−	0.623745i	0.00697395	−	0.999976i	\(-0.497780\pi\)
							0.926526	+	0.376231i	\(0.122780\pi\)
\(23\)	−7.89691	−	3.27101i	−1.64662	−	0.682052i	−0.649678	−	0.760209i	\(-0.725096\pi\)
							−0.996941	+	0.0781572i	\(0.975096\pi\)
\(29\)	−0.377363	+	1.89713i	−0.0700745	+	0.352288i	−0.999876	−	0.0157741i	\(-0.994979\pi\)
							0.929801	+	0.368062i	\(0.119979\pi\)
\(31\)			1.19435i			0.214512i	0.994231	+	0.107256i	\(0.0342064\pi\)
							−0.994231	+	0.107256i	\(0.965794\pi\)
\(37\)	0.545247	−	0.816020i	0.0896381	−	0.134153i	−0.783941	−	0.620836i	\(-0.786793\pi\)
							0.873579	+	0.486683i	\(0.161793\pi\)
\(41\)	−0.637284	+	1.53854i	−0.0995270	+	0.240280i	−0.965798	−	0.259294i	\(-0.916510\pi\)
							0.866271	+	0.499574i	\(0.166510\pi\)
\(43\)	2.23569	+	11.2396i	0.340940	+	1.71402i	0.647413	+	0.762139i	\(0.275851\pi\)
							−0.306473	+	0.951879i	\(0.599149\pi\)
\(47\)	−1.84035	+	1.84035i	−0.268443	+	0.268443i	−0.828473	−	0.560029i	\(-0.810790\pi\)
							0.560029	+	0.828473i	\(0.310790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.5	✓	368
5.4	even	2	inner	320.2.bf.a.29.42	yes	368
64.53	even	16	inner	320.2.bf.a.309.42	yes	368
320.309	even	16	inner	320.2.bf.a.309.5	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.5	✓	368	1.1	even	1	trivial
320.2.bf.a.29.42	yes	368	5.4	even	2	inner
320.2.bf.a.309.5	yes	368	320.309	even	16	inner
320.2.bf.a.309.42	yes	368	64.53	even	16	inner