Embedded newform 128.4.k.a.5.4

• Label: 128.4.k.a.5.4
• Level: $128$
• Weight: $4$
• Character: 128.5
• Analytic conductor: $7.552$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.k (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.55224448073\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(47\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	128.5
Dual form			128.4.k.a.77.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.72237 - 0.767264i) q^{2} +(-3.00383 - 0.911201i) q^{3} +(6.82261 + 4.17756i) q^{4} +(1.47948 + 0.145716i) q^{5} +(7.47840 + 4.78536i) q^{6} +(3.74690 + 5.60763i) q^{7} +(-15.3684 - 16.6076i) q^{8} +(-14.2570 - 9.52622i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 752 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.72237	−	0.767264i	−0.962504	−	0.271269i
							\(3\)	−3.00383	−	0.911201i	−0.578087	−	0.175361i	−0.0123173	−	0.999924i	\(-0.503921\pi\)
−0.565770	+	0.824563i	\(0.691421\pi\)
\(5\)	1.47948	+	0.145716i	0.132328	+	0.0130332i	0.163964	−	0.986466i	\(-0.447572\pi\)
							−0.0316361	+	0.999499i	\(0.510072\pi\)
\(7\)	3.74690	+	5.60763i	0.202314	+	0.302784i	0.918728	−	0.394892i	\(-0.129218\pi\)
							−0.716414	+	0.697675i	\(0.754218\pi\)
\(11\)	12.5287	−	23.4395i	0.343412	−	0.642479i	−0.649628	−	0.760252i	\(-0.725075\pi\)
							0.993041	+	0.117773i	\(0.0375754\pi\)
\(13\)	1.37746	+	13.9856i	0.0293875	+	0.298377i	0.998633	+	0.0522788i	\(0.0166484\pi\)
							−0.969245	+	0.246098i	\(0.920852\pi\)
\(17\)	−18.3924	+	7.61840i	−0.262401	+	0.108690i	−0.510006	−	0.860171i	\(-0.670357\pi\)
							0.247604	+	0.968861i	\(0.420357\pi\)
\(19\)	−101.737	+	123.967i	−1.22842	+	1.49684i	−0.420785	+	0.907160i	\(0.638245\pi\)
							−0.807638	+	0.589678i	\(0.799255\pi\)
\(23\)	25.7045	+	129.225i	0.233033	+	1.17153i	0.903165	+	0.429293i	\(0.141237\pi\)
							−0.670133	+	0.742241i	\(0.733763\pi\)
\(29\)	−109.176	+	58.3560i	−0.699088	+	0.373670i	−0.782334	−	0.622859i	\(-0.785971\pi\)
							0.0832465	+	0.996529i	\(0.473471\pi\)
\(31\)	60.2900	+	60.2900i	0.349303	+	0.349303i	0.859850	−	0.510547i	\(-0.170557\pi\)
							−0.510547	+	0.859850i	\(0.670557\pi\)
\(37\)	−176.163	+	144.573i	−0.782732	+	0.642371i	−0.938444	−	0.345431i	\(-0.887733\pi\)
							0.155713	+	0.987802i	\(0.450233\pi\)
\(41\)	66.0586	−	13.1399i	0.251625	−	0.0500513i	−0.0676660	−	0.997708i	\(-0.521555\pi\)
							0.319291	+	0.947657i	\(0.396555\pi\)
\(43\)	−262.505	+	79.6299i	−0.930967	+	0.282406i	−0.719111	−	0.694895i	\(-0.755451\pi\)
							−0.211856	+	0.977301i	\(0.567951\pi\)
\(47\)	84.3693	+	203.685i	0.261841	+	0.632140i	0.999052	−	0.0435222i	\(-0.0138579\pi\)
							−0.737212	+	0.675662i	\(0.763858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.k.a.5.4	✓	752
128.77	even	32	inner	128.4.k.a.77.4	yes	752

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.k.a.5.4	✓	752	1.1	even	1	trivial
128.4.k.a.77.4	yes	752	128.77	even	32	inner