Embedded newform 128.5.l.a.3.8

• Label: 128.5.l.a.3.8
• Level: $128$
• Weight: $5$
• Character: 128.3
• Analytic conductor: $13.231$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2313552747\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(63\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.86190 - 1.04198i) q^{2} +(1.98252 - 1.62701i) q^{3} +(13.8286 + 8.04803i) q^{4} +(-2.07933 - 0.630759i) q^{5} +(-9.35161 + 4.21762i) q^{6} +(-2.78288 - 13.9905i) q^{7} +(-45.0187 - 45.4898i) q^{8} +(-14.5191 + 72.9924i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.86190	−	1.04198i	−0.965475	−	0.260495i
							\(3\)	1.98252	−	1.62701i	0.220280	−	0.180779i	−0.517781	−	0.855513i	\(-0.673242\pi\)
0.738061	+	0.674734i	\(0.235742\pi\)
\(5\)	−2.07933	−	0.630759i	−0.0831733	−	0.0252304i	0.248424	−	0.968651i	\(-0.420087\pi\)
							−0.331597	+	0.943421i	\(0.607587\pi\)
\(7\)	−2.78288	−	13.9905i	−0.0567935	−	0.285520i	0.941943	−	0.335773i	\(-0.108997\pi\)
							−0.998737	+	0.0502525i	\(0.983997\pi\)
\(11\)	−20.7258	−	210.433i	−0.171288	−	1.73912i	−0.569650	−	0.821887i	\(-0.692921\pi\)
							0.398362	−	0.917228i	\(-0.369579\pi\)
\(13\)	57.8185	+	190.602i	0.342122	+	1.12782i	0.944196	+	0.329384i	\(0.106841\pi\)
							−0.602075	+	0.798440i	\(0.705659\pi\)
\(17\)	134.591	−	324.932i	0.465714	−	1.12433i	−0.500302	−	0.865851i	\(-0.666778\pi\)
							0.966016	−	0.258482i	\(-0.0832222\pi\)
\(19\)	60.6637	−	113.494i	0.168044	−	0.314387i	−0.783777	−	0.621042i	\(-0.786710\pi\)
							0.951821	+	0.306655i	\(0.0992096\pi\)
\(23\)	−235.782	+	157.545i	−0.445713	+	0.297816i	−0.758099	−	0.652139i	\(-0.773872\pi\)
							0.312386	+	0.949955i	\(0.398872\pi\)
\(29\)	83.9142	−	851.996i	0.0997791	−	1.01307i	−0.806739	−	0.590908i	\(-0.798770\pi\)
							0.906518	−	0.422167i	\(-0.138730\pi\)
\(31\)	−982.076	−	982.076i	−1.02193	−	1.02193i	−0.999754	−	0.0221776i	\(-0.992940\pi\)
							−0.0221776	−	0.999754i	\(-0.507060\pi\)
\(37\)	−837.434	−	1566.73i	−0.611712	−	1.14443i	−0.976940	−	0.213515i	\(-0.931509\pi\)
							0.365228	−	0.930918i	\(-0.380991\pi\)
\(41\)	−2418.51	+	1616.00i	−1.43873	+	0.961331i	−0.440765	+	0.897622i	\(0.645293\pi\)
							−0.997969	+	0.0637090i	\(0.979707\pi\)
\(43\)	−1692.85	−	1389.29i	−0.915549	−	0.751372i	0.0534153	−	0.998572i	\(-0.482989\pi\)
							−0.968964	+	0.247201i	\(0.920489\pi\)
\(47\)	1334.40	−	3221.52i	0.604074	−	1.45836i	−0.265279	−	0.964172i	\(-0.585464\pi\)
							0.869353	−	0.494192i	\(-0.164536\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.8	✓	1008
128.43	odd	32	inner	128.5.l.a.43.8	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.8	✓	1008	1.1	even	1	trivial
128.5.l.a.43.8	yes	1008	128.43	odd	32	inner