Embedded newform 128.5.l.a.3.19

• Label: 128.5.l.a.3.19
• 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			3.19
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35233 + 3.23520i) q^{2} +(1.64666 - 1.35138i) q^{3} +(-4.93309 - 15.2205i) q^{4} +(31.2429 + 9.47742i) q^{5} +(0.498498 + 8.50617i) q^{6} +(-6.24158 - 31.3785i) q^{7} +(60.8458 + 19.8442i) q^{8} +(-14.9171 + 74.9931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.35233	+	3.23520i	−0.588083	+	0.808801i
							\(3\)	1.64666	−	1.35138i	0.182962	−	0.150153i	−0.538454	−	0.842655i	\(-0.680991\pi\)
0.721416	+	0.692502i	\(0.243491\pi\)
\(5\)	31.2429	+	9.47742i	1.24971	+	0.379097i	0.844713	−	0.535219i	\(-0.179771\pi\)
							0.405001	+	0.914316i	\(0.367271\pi\)
\(7\)	−6.24158	−	31.3785i	−0.127379	−	0.640378i	−0.990738	−	0.135788i	\(-0.956643\pi\)
							0.863359	−	0.504591i	\(-0.168357\pi\)
\(11\)	−22.4545	−	227.984i	−0.185574	−	1.88417i	−0.413525	−	0.910493i	\(-0.635702\pi\)
							0.227951	−	0.973673i	\(-0.426798\pi\)
\(13\)	−13.4564	−	44.3598i	−0.0796236	−	0.262484i	0.908086	−	0.418784i	\(-0.137543\pi\)
							−0.987710	+	0.156300i	\(0.950043\pi\)
\(17\)	23.6566	−	57.1121i	0.0818568	−	0.197620i	−0.877652	−	0.479298i	\(-0.840891\pi\)
							0.959509	+	0.281679i	\(0.0908912\pi\)
\(19\)	197.207	−	368.948i	0.546280	−	1.02202i	−0.445647	−	0.895209i	\(-0.647026\pi\)
							0.991927	−	0.126809i	\(-0.0404736\pi\)
\(23\)	362.906	−	242.486i	0.686022	−	0.458385i	−0.163081	−	0.986613i	\(-0.552143\pi\)
							0.849103	+	0.528227i	\(0.177143\pi\)
\(29\)	−13.1941	+	133.962i	−0.0156886	+	0.159289i	−0.999837	−	0.0180581i	\(-0.994252\pi\)
							0.984148	+	0.177347i	\(0.0567516\pi\)
\(31\)	1037.08	+	1037.08i	1.07916	+	1.07916i	0.996584	+	0.0825801i	\(0.0263160\pi\)
							0.0825801	+	0.996584i	\(0.473684\pi\)
\(37\)	−285.081	−	533.348i	−0.208240	−	0.389590i	0.756020	−	0.654549i	\(-0.227141\pi\)
							−0.964260	+	0.264959i	\(0.914641\pi\)
\(41\)	1982.79	−	1324.86i	1.17953	−	0.788136i	0.198141	−	0.980174i	\(-0.436510\pi\)
							0.981387	+	0.192038i	\(0.0615097\pi\)
\(43\)	1282.80	+	1052.77i	0.693780	+	0.569370i	0.913820	−	0.406121i	\(-0.133119\pi\)
							−0.220040	+	0.975491i	\(0.570619\pi\)
\(47\)	−19.6692	+	47.4857i	−0.00890412	+	0.0214964i	−0.928269	−	0.371909i	\(-0.878703\pi\)
							0.919365	+	0.393406i	\(0.128703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.19	✓	1008
128.43	odd	32	inner	128.5.l.a.43.19	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.19	✓	1008	1.1	even	1	trivial
128.5.l.a.43.19	yes	1008	128.43	odd	32	inner