Embedded newform 128.3.l.a.3.2

• Label: 128.3.l.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98170 - 0.269964i) q^{2} +(-2.40785 + 1.97607i) q^{3} +(3.85424 + 1.06997i) q^{4} +(4.69198 + 1.42330i) q^{5} +(5.30509 - 3.26594i) q^{6} +(-0.744763 - 3.74418i) q^{7} +(-7.34908 - 3.16086i) q^{8} +(0.137064 - 0.689066i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98170	−	0.269964i	−0.990848	−	0.134982i
							\(3\)	−2.40785	+	1.97607i	−0.802615	+	0.658689i	−0.943513	−	0.331335i	\(-0.892501\pi\)
0.140898	+	0.990024i	\(0.455001\pi\)
\(5\)	4.69198	+	1.42330i	0.938396	+	0.284659i	0.722195	−	0.691689i	\(-0.243133\pi\)
							0.216201	+	0.976349i	\(0.430633\pi\)
\(7\)	−0.744763	−	3.74418i	−0.106395	−	0.534882i	−0.996815	−	0.0797438i	\(-0.974590\pi\)
							0.890421	−	0.455138i	\(-0.150410\pi\)
\(11\)	−0.301062	−	3.05673i	−0.0273693	−	0.277885i	−0.999122	−	0.0418853i	\(-0.986664\pi\)
							0.971753	−	0.236000i	\(-0.0758364\pi\)
\(13\)	6.89065	+	22.7154i	0.530050	+	1.74734i	0.655454	+	0.755235i	\(0.272477\pi\)
							−0.125404	+	0.992106i	\(0.540023\pi\)
\(17\)	−8.37067	+	20.2086i	−0.492392	+	1.18874i	0.461107	+	0.887345i	\(0.347452\pi\)
							−0.953499	+	0.301396i	\(0.902548\pi\)
\(19\)	−2.02166	+	3.78226i	−0.106403	+	0.199067i	−0.929488	−	0.368852i	\(-0.879751\pi\)
							0.823085	+	0.567918i	\(0.192251\pi\)
\(23\)	−22.9906	+	15.3618i	−0.999592	+	0.667906i	−0.943793	−	0.330536i	\(-0.892770\pi\)
							−0.0557986	+	0.998442i	\(0.517770\pi\)
\(29\)	0.374150	−	3.79881i	0.0129017	−	0.130993i	−0.986576	−	0.163301i	\(-0.947786\pi\)
							0.999478	+	0.0323079i	\(0.0102857\pi\)
\(31\)	28.4440	+	28.4440i	0.917547	+	0.917547i	0.996851	−	0.0793034i	\(-0.0252696\pi\)
							−0.0793034	+	0.996851i	\(0.525270\pi\)
\(37\)	0.682562	+	1.27698i	0.0184476	+	0.0345131i	0.890976	−	0.454050i	\(-0.150021\pi\)
							−0.872529	+	0.488563i	\(0.837521\pi\)
\(41\)	−6.74262	+	4.50527i	−0.164454	+	0.109885i	−0.635074	−	0.772451i	\(-0.719030\pi\)
							0.470620	+	0.882336i	\(0.344030\pi\)
\(43\)	49.2969	+	40.4569i	1.14644	+	0.940858i	0.998838	−	0.0481888i	\(-0.0153449\pi\)
							0.147601	+	0.989047i	\(0.452845\pi\)
\(47\)	−6.63363	+	16.0150i	−0.141141	+	0.340745i	−0.978605	−	0.205748i	\(-0.934037\pi\)
							0.837464	+	0.546493i	\(0.184037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.2	✓	496
128.43	odd	32	inner	128.3.l.a.43.2	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.2	✓	496	1.1	even	1	trivial
128.3.l.a.43.2	yes	496	128.43	odd	32	inner