Embedded newform 128.5.l.a.3.4

• Label: 128.5.l.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.93799 - 0.701605i) q^{2} +(11.4156 - 9.36851i) q^{3} +(15.0155 + 5.52582i) q^{4} +(39.4094 + 11.9547i) q^{5} +(-51.5273 + 28.8839i) q^{6} +(-18.6869 - 93.9456i) q^{7} +(-55.2539 - 32.2956i) q^{8} +(26.7438 - 134.450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.93799	−	0.701605i	−0.984497	−	0.175401i
							\(3\)	11.4156	−	9.36851i	1.26840	−	1.04095i	0.271482	−	0.962444i	\(-0.412486\pi\)
0.996914	−	0.0785018i	\(-0.0250136\pi\)
\(5\)	39.4094	+	11.9547i	1.57637	+	0.478188i	0.952806	−	0.303580i	\(-0.0981821\pi\)
							0.623569	+	0.781768i	\(0.285682\pi\)
\(7\)	−18.6869	−	93.9456i	−0.381366	−	1.91726i	−0.397976	−	0.917396i	\(-0.630287\pi\)
							0.0166099	−	0.999862i	\(-0.494713\pi\)
\(11\)	2.41151	+	24.4844i	0.0199298	+	0.202351i	0.999993	+	0.00366605i	\(0.00116694\pi\)
							−0.980063	+	0.198685i	\(0.936333\pi\)
\(13\)	5.12747	+	16.9030i	0.0303400	+	0.100018i	0.970803	−	0.239879i	\(-0.0771078\pi\)
							−0.940463	+	0.339897i	\(0.889608\pi\)
\(17\)	−132.265	+	319.315i	−0.457664	+	1.10490i	0.511677	+	0.859178i	\(0.329024\pi\)
							−0.969341	+	0.245720i	\(0.920976\pi\)
\(19\)	184.969	−	346.052i	0.512378	−	0.958592i	−0.484122	−	0.875001i	\(-0.660861\pi\)
							0.996500	−	0.0835917i	\(-0.0266391\pi\)
\(23\)	−311.409	+	208.077i	−0.588674	+	0.393340i	−0.813932	−	0.580960i	\(-0.802677\pi\)
							0.225258	+	0.974299i	\(0.427677\pi\)
\(29\)	−29.9606	+	304.195i	−0.0356249	+	0.361706i	0.960801	+	0.277238i	\(0.0894191\pi\)
							−0.996426	+	0.0844679i	\(0.973081\pi\)
\(31\)	134.864	+	134.864i	0.140337	+	0.140337i	0.773785	−	0.633448i	\(-0.218361\pi\)
							−0.633448	+	0.773785i	\(0.718361\pi\)
\(37\)	720.373	+	1347.72i	0.526204	+	0.984458i	0.994878	+	0.101086i	\(0.0322319\pi\)
							−0.468674	+	0.883371i	\(0.655268\pi\)
\(41\)	1377.13	−	920.169i	0.819233	−	0.547394i	−0.0738588	−	0.997269i	\(-0.523531\pi\)
							0.893092	+	0.449875i	\(0.148531\pi\)
\(43\)	−2309.68	−	1895.50i	−1.24915	−	1.02515i	−0.998368	−	0.0571080i	\(-0.981812\pi\)
							−0.250783	−	0.968043i	\(-0.580688\pi\)
\(47\)	−387.364	+	935.180i	−0.175357	+	0.423350i	−0.986982	−	0.160829i	\(-0.948583\pi\)
							0.811625	+	0.584179i	\(0.198583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.4	✓	1008
128.43	odd	32	inner	128.5.l.a.43.4	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.4	✓	1008	1.1	even	1	trivial
128.5.l.a.43.4	yes	1008	128.43	odd	32	inner