Embedded newform 128.3.l.a.3.16

• Label: 128.3.l.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0383871 - 1.99963i) q^{2} +(-4.01639 + 3.29616i) q^{3} +(-3.99705 + 0.153520i) q^{4} +(6.07117 + 1.84167i) q^{5} +(6.74529 + 7.90476i) q^{6} +(-2.47965 - 12.4660i) q^{7} +(0.460419 + 7.98674i) q^{8} +(3.51086 - 17.6503i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0383871	−	1.99963i	−0.0191935	−	0.999816i
							\(3\)	−4.01639	+	3.29616i	−1.33880	+	1.09872i	−0.352132	+	0.935950i	\(0.614543\pi\)
−0.986664	+	0.162771i	\(0.947957\pi\)
\(5\)	6.07117	+	1.84167i	1.21423	+	0.368334i	0.831500	−	0.555525i	\(-0.187483\pi\)
							0.382734	+	0.923859i	\(0.374983\pi\)
\(7\)	−2.47965	−	12.4660i	−0.354236	−	1.78086i	−0.588301	−	0.808642i	\(-0.700203\pi\)
							0.234065	−	0.972221i	\(-0.424797\pi\)
\(11\)	−0.135226	−	1.37297i	−0.0122932	−	0.124815i	0.987079	−	0.160232i	\(-0.0512242\pi\)
							−0.999373	+	0.0354166i	\(0.988724\pi\)
\(13\)	−2.48659	−	8.19719i	−0.191276	−	0.630553i	−0.999151	−	0.0411991i	\(-0.986882\pi\)
							0.807875	−	0.589354i	\(-0.200618\pi\)
\(17\)	6.61713	−	15.9752i	0.389243	−	0.939715i	−0.600858	−	0.799356i	\(-0.705174\pi\)
							0.990101	−	0.140359i	\(-0.0448258\pi\)
\(19\)	10.1410	−	18.9724i	0.533734	−	0.998547i	−0.460121	−	0.887856i	\(-0.652194\pi\)
							0.993855	−	0.110690i	\(-0.0353061\pi\)
\(23\)	−1.97041	+	1.31659i	−0.0856700	+	0.0572429i	−0.597668	−	0.801744i	\(-0.703906\pi\)
							0.511997	+	0.858987i	\(0.328906\pi\)
\(29\)	−3.67760	+	37.3393i	−0.126814	+	1.28756i	0.695499	+	0.718527i	\(0.255183\pi\)
							−0.822313	+	0.569035i	\(0.807317\pi\)
\(31\)	−8.68999	−	8.68999i	−0.280322	−	0.280322i	0.552915	−	0.833238i	\(-0.313515\pi\)
							−0.833238	+	0.552915i	\(0.813515\pi\)
\(37\)	−8.33614	−	15.5958i	−0.225301	−	0.421508i	0.743718	−	0.668493i	\(-0.233060\pi\)
							−0.969019	+	0.246984i	\(0.920560\pi\)
\(41\)	38.6647	−	25.8349i	0.943041	−	0.630120i	0.0139232	−	0.999903i	\(-0.495568\pi\)
							0.929118	+	0.369783i	\(0.120568\pi\)
\(43\)	−2.21765	−	1.81998i	−0.0515733	−	0.0423251i	0.608254	−	0.793743i	\(-0.291870\pi\)
							−0.659827	+	0.751418i	\(0.729370\pi\)
\(47\)	−21.9633	+	53.0241i	−0.467304	+	1.12817i	0.498031	+	0.867159i	\(0.334057\pi\)
							−0.965335	+	0.261013i	\(0.915943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.16	✓	496
128.43	odd	32	inner	128.3.l.a.43.16	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.16	✓	496	1.1	even	1	trivial
128.3.l.a.43.16	yes	496	128.43	odd	32	inner