Embedded newform 128.2.k.a.77.15

• Label: 128.2.k.a.77.15
• Level: $128$
• Weight: $2$
• Character: 128.77
• Analytic conductor: $1.022$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.02208514587\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(15\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			77.15
Character	\(\chi\)	\(=\)	128.77
Dual form			128.2.k.a.5.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37613 + 0.325966i) q^{2} +(-1.74451 + 0.529190i) q^{3} +(1.78749 + 0.897147i) q^{4} +(2.07613 - 0.204481i) q^{5} +(-2.57317 + 0.159587i) q^{6} +(-0.655035 + 0.980329i) q^{7} +(2.16739 + 1.81726i) q^{8} +(0.268850 - 0.179640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{31}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37613	+	0.325966i	0.973074	+	0.230493i
							\(3\)	−1.74451	+	0.529190i	−1.00719	+	0.305528i	−0.750432	−	0.660947i	\(-0.770155\pi\)
−0.256759	+	0.966476i	\(0.582655\pi\)
\(5\)	2.07613	−	0.204481i	0.928473	−	0.0914466i	0.377537	−	0.925994i	\(-0.376771\pi\)
							0.550936	+	0.834548i	\(0.314271\pi\)
\(7\)	−0.655035	+	0.980329i	−0.247580	+	0.370529i	−0.934357	−	0.356339i	\(-0.884025\pi\)
							0.686777	+	0.726868i	\(0.259025\pi\)
\(11\)	−0.641887	−	1.20089i	−0.193536	−	0.362081i	0.766371	−	0.642398i	\(-0.222060\pi\)
							−0.959907	+	0.280317i	\(0.909560\pi\)
\(13\)	0.356345	−	3.61804i	0.0988324	−	1.00346i	−0.810036	−	0.586380i	\(-0.800553\pi\)
							0.908869	−	0.417083i	\(-0.136947\pi\)
\(17\)	−0.871613	−	0.361034i	−0.211397	−	0.0875636i	0.274472	−	0.961595i	\(-0.411497\pi\)
							−0.485869	+	0.874031i	\(0.661497\pi\)
\(19\)	−3.46157	−	4.21794i	−0.794139	−	0.967661i	0.205801	−	0.978594i	\(-0.434020\pi\)
							−0.999940	+	0.0109329i	\(0.996520\pi\)
\(23\)	−0.0751658	+	0.377884i	−0.0156731	+	0.0787942i	−0.987830	−	0.155536i	\(-0.950290\pi\)
							0.972157	+	0.234330i	\(0.0752896\pi\)
\(29\)	−8.81508	−	4.71176i	−1.63692	−	0.874952i	−0.994044	−	0.108982i	\(-0.965241\pi\)
							−0.642876	−	0.765970i	\(-0.722259\pi\)
\(31\)	−0.693719	+	0.693719i	−0.124596	+	0.124596i	−0.766655	−	0.642059i	\(-0.778080\pi\)
							0.642059	+	0.766655i	\(0.278080\pi\)
\(37\)	5.99840	+	4.92276i	0.986131	+	0.809297i	0.981650	−	0.190693i	\(-0.0610736\pi\)
							0.00448115	+	0.999990i	\(0.498574\pi\)
\(41\)	8.66836	+	1.72424i	1.35377	+	0.269282i	0.818066	−	0.575124i	\(-0.195046\pi\)
							0.535704	+	0.844406i	\(0.320046\pi\)
\(43\)	8.68394	+	2.63425i	1.32429	+	0.401719i	0.871717	−	0.490009i	\(-0.163007\pi\)
							0.452572	+	0.891728i	\(0.350507\pi\)
\(47\)	−3.44487	+	8.31666i	−0.502486	+	1.21311i	0.445639	+	0.895213i	\(0.352976\pi\)
							−0.948125	+	0.317896i	\(0.897024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.77.15	yes	240
4.3	odd	2		512.2.k.a.145.12		240
128.5	even	32	inner	128.2.k.a.5.15	✓	240
128.123	odd	32		512.2.k.a.113.12		240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.5.15	✓	240	128.5	even	32	inner
128.2.k.a.77.15	yes	240	1.1	even	1	trivial
512.2.k.a.113.12		240	128.123	odd	32
512.2.k.a.145.12		240	4.3	odd	2