Embedded newform 128.3.l.a.3.18

• Label: 128.3.l.a.3.18
• Level: $128$
• Weight: $3$
• Character: 128.3
• Analytic conductor: $3.488$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.48774738381\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(31\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.18
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.286149 + 1.97942i) q^{2} +(0.798268 - 0.655122i) q^{3} +(-3.83624 + 1.13282i) q^{4} +(0.972729 + 0.295074i) q^{5} +(1.52519 + 1.39265i) q^{6} +(1.76467 + 8.87160i) q^{7} +(-3.34006 - 7.26939i) q^{8} +(-1.54777 + 7.78114i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.286149	+	1.97942i	0.143074	+	0.989712i
							\(3\)	0.798268	−	0.655122i	0.266089	−	0.218374i	−0.491882	−	0.870662i	\(-0.663691\pi\)
0.757971	+	0.652288i	\(0.226191\pi\)
\(5\)	0.972729	+	0.295074i	0.194546	+	0.0590148i	0.386054	−	0.922476i	\(-0.373838\pi\)
							−0.191508	+	0.981491i	\(0.561338\pi\)
\(7\)	1.76467	+	8.87160i	0.252096	+	1.26737i	0.874635	+	0.484783i	\(0.161101\pi\)
							−0.622539	+	0.782589i	\(0.713899\pi\)
\(11\)	0.852321	+	8.65376i	0.0774837	+	0.786705i	0.952890	+	0.303317i	\(0.0980941\pi\)
							−0.875406	+	0.483388i	\(0.839406\pi\)
\(13\)	−0.633852	−	2.08953i	−0.0487579	−	0.160733i	0.929181	−	0.369626i	\(-0.120514\pi\)
							−0.977939	+	0.208893i	\(0.933014\pi\)
\(17\)	−4.15286	+	10.0259i	−0.244286	+	0.589758i	−0.997700	−	0.0677885i	\(-0.978406\pi\)
							0.753414	+	0.657547i	\(0.228406\pi\)
\(19\)	9.45597	−	17.6909i	0.497683	−	0.931099i	−0.500191	−	0.865915i	\(-0.666737\pi\)
							0.997873	−	0.0651833i	\(-0.0207632\pi\)
\(23\)	21.8363	−	14.5905i	0.949402	−	0.634370i	0.0185741	−	0.999827i	\(-0.494087\pi\)
							0.930828	+	0.365457i	\(0.119087\pi\)
\(29\)	2.67563	−	27.1661i	0.0922630	−	0.936762i	−0.831941	−	0.554863i	\(-0.812770\pi\)
							0.924204	−	0.381898i	\(-0.124730\pi\)
\(31\)	13.6182	+	13.6182i	0.439297	+	0.439297i	0.891775	−	0.452479i	\(-0.149460\pi\)
							−0.452479	+	0.891775i	\(0.649460\pi\)
\(37\)	7.77988	+	14.5551i	0.210267	+	0.393382i	0.964841	−	0.262833i	\(-0.0846567\pi\)
							−0.754574	+	0.656214i	\(0.772157\pi\)
\(41\)	−5.21815	+	3.48665i	−0.127272	+	0.0850404i	−0.617577	−	0.786511i	\(-0.711885\pi\)
							0.490305	+	0.871551i	\(0.336885\pi\)
\(43\)	45.8232	+	37.6061i	1.06565	+	0.874560i	0.992437	−	0.122756i	\(-0.0391731\pi\)
							0.0732179	+	0.997316i	\(0.476673\pi\)
\(47\)	27.3501	−	66.0289i	0.581917	−	1.40487i	−0.309156	−	0.951011i	\(-0.600047\pi\)
							0.891073	−	0.453860i	\(-0.149953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.18	✓	496
128.43	odd	32	inner	128.3.l.a.43.18	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.18	✓	496	1.1	even	1	trivial
128.3.l.a.43.18	yes	496	128.43	odd	32	inner