Embedded newform 128.5.l.a.3.11

• Label: 128.5.l.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.37653 + 2.14454i) q^{2} +(-6.28903 + 5.16127i) q^{3} +(6.80190 - 14.4822i) q^{4} +(17.4865 + 5.30446i) q^{5} +(10.1665 - 30.9143i) q^{6} +(-14.2328 - 71.5529i) q^{7} +(8.09084 + 63.4865i) q^{8} +(-2.88917 + 14.5249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.37653	+	2.14454i	−0.844132	+	0.536135i
							\(3\)	−6.28903	+	5.16127i	−0.698781	+	0.573475i	−0.915285	−	0.402807i	\(-0.868035\pi\)
0.216504	+	0.976282i	\(0.430535\pi\)
\(5\)	17.4865	+	5.30446i	0.699458	+	0.212178i	0.619933	−	0.784655i	\(-0.287160\pi\)
							0.0795248	+	0.996833i	\(0.474660\pi\)
\(7\)	−14.2328	−	71.5529i	−0.290464	−	1.46026i	−0.800097	−	0.599871i	\(-0.795219\pi\)
							0.509633	−	0.860392i	\(-0.329781\pi\)
\(11\)	15.2494	+	154.830i	0.126028	+	1.27958i	0.825286	+	0.564715i	\(0.191014\pi\)
							−0.699258	+	0.714869i	\(0.746486\pi\)
\(13\)	33.3643	+	109.987i	0.197422	+	0.650812i	0.998625	+	0.0524199i	\(0.0166934\pi\)
							−0.801203	+	0.598392i	\(0.795807\pi\)
\(17\)	42.4856	−	102.569i	0.147009	−	0.354911i	−0.833172	−	0.553013i	\(-0.813478\pi\)
							0.980181	+	0.198102i	\(0.0634778\pi\)
\(19\)	87.9258	−	164.498i	0.243562	−	0.455672i	−0.730203	−	0.683231i	\(-0.760574\pi\)
							0.973764	+	0.227559i	\(0.0730743\pi\)
\(23\)	−708.945	+	473.702i	−1.34016	+	0.895467i	−0.999008	−	0.0445404i	\(-0.985818\pi\)
							−0.341154	+	0.940008i	\(0.610818\pi\)
\(29\)	−106.923	+	1085.61i	−0.127138	+	1.29085i	0.693939	+	0.720034i	\(0.255874\pi\)
							−0.821077	+	0.570818i	\(0.806626\pi\)
\(31\)	−609.954	−	609.954i	−0.634708	−	0.634708i	0.314537	−	0.949245i	\(-0.398151\pi\)
							−0.949245	+	0.314537i	\(0.898151\pi\)
\(37\)	−314.484	−	588.357i	−0.229718	−	0.429772i	0.740483	−	0.672075i	\(-0.234597\pi\)
							−0.970200	+	0.242304i	\(0.922097\pi\)
\(41\)	−978.884	+	654.069i	−0.582322	+	0.389095i	−0.811555	−	0.584276i	\(-0.801378\pi\)
							0.229232	+	0.973372i	\(0.426378\pi\)
\(43\)	−913.238	−	749.475i	−0.493909	−	0.405341i	0.354194	−	0.935172i	\(-0.384755\pi\)
							−0.848103	+	0.529831i	\(0.822255\pi\)
\(47\)	894.913	−	2160.51i	0.405121	−	0.978049i	−0.581282	−	0.813703i	\(-0.697448\pi\)
							0.986403	−	0.164346i	\(-0.0525515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.11	✓	1008
128.43	odd	32	inner	128.5.l.a.43.11	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.11	✓	1008	1.1	even	1	trivial
128.5.l.a.43.11	yes	1008	128.43	odd	32	inner