Embedded newform 320.2.bf.a.29.20

• Label: 320.2.bf.a.29.20
• 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			29.20
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.424621 - 1.34896i) q^{2} +(-0.454817 + 2.28652i) q^{3} +(-1.63939 + 1.14559i) q^{4} +(-0.548285 - 2.16781i) q^{5} +(3.27755 - 0.357373i) q^{6} +(0.363715 + 0.878084i) q^{7} +(2.24148 + 1.72504i) q^{8} +(-2.24966 - 0.931842i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.424621	−	1.34896i	−0.300252	−	0.953860i
							\(3\)	−0.454817	+	2.28652i	−0.262588	+	1.32012i	0.594143	+	0.804359i	\(0.297491\pi\)
−0.856732	+	0.515762i	\(0.827509\pi\)
\(5\)	−0.548285	−	2.16781i	−0.245201	−	0.969472i
							\(7\)	0.363715	+	0.878084i	0.137471	+	0.331885i	0.977590	−	0.210517i	\(-0.0675149\pi\)
−0.840119	+	0.542402i	\(0.817515\pi\)
\(11\)	0.418548	+	2.10418i	0.126197	+	0.634435i	0.991168	+	0.132613i	\(0.0423367\pi\)
							−0.864971	+	0.501822i	\(0.832663\pi\)
\(13\)	−4.56835	+	3.05247i	−1.26703	+	0.846604i	−0.993341	−	0.115213i	\(-0.963245\pi\)
							−0.273692	+	0.961817i	\(0.588245\pi\)
\(17\)	−2.76569	+	2.76569i	−0.670779	+	0.670779i	−0.957896	−	0.287117i	\(-0.907303\pi\)
							0.287117	+	0.957896i	\(0.407303\pi\)
\(19\)	−3.46361	+	2.31431i	−0.794606	+	0.530939i	−0.885344	−	0.464936i	\(-0.846077\pi\)
							0.0907383	+	0.995875i	\(0.471077\pi\)
\(23\)	4.29321	+	1.77831i	0.895196	+	0.370802i	0.782371	−	0.622813i	\(-0.214010\pi\)
							0.112825	+	0.993615i	\(0.464010\pi\)
\(29\)	−0.637950	+	3.20719i	−0.118464	+	0.595561i	0.875255	+	0.483661i	\(0.160693\pi\)
							−0.993720	+	0.111899i	\(0.964307\pi\)
\(31\)			4.99085i			0.896383i	0.893938	+	0.448192i	\(0.147932\pi\)
							−0.893938	+	0.448192i	\(0.852068\pi\)
\(37\)	4.11108	−	6.15266i	0.675857	−	1.01149i	−0.322043	−	0.946725i	\(-0.604370\pi\)
							0.997900	−	0.0647668i	\(-0.0206303\pi\)
\(41\)	−1.75203	+	4.22979i	−0.273622	+	0.660582i	−0.999633	−	0.0271016i	\(-0.991372\pi\)
							0.726011	+	0.687683i	\(0.241372\pi\)
\(43\)	−1.34029	−	6.73811i	−0.204393	−	1.02755i	−0.937645	−	0.347595i	\(-0.886998\pi\)
							0.733252	−	0.679957i	\(-0.238002\pi\)
\(47\)	6.27247	−	6.27247i	0.914934	−	0.914934i	−0.0817216	−	0.996655i	\(-0.526042\pi\)
							0.996655	+	0.0817216i	\(0.0260418\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.20	✓	368
5.4	even	2	inner	320.2.bf.a.29.27	yes	368
64.53	even	16	inner	320.2.bf.a.309.27	yes	368
320.309	even	16	inner	320.2.bf.a.309.20	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.20	✓	368	1.1	even	1	trivial
320.2.bf.a.29.27	yes	368	5.4	even	2	inner
320.2.bf.a.309.20	yes	368	320.309	even	16	inner
320.2.bf.a.309.27	yes	368	64.53	even	16	inner