Embedded newform 128.3.l.a.3.14

• Label: 128.3.l.a.3.14
• Level: $128$
• Weight: $3$
• Character: 128.3
• Analytic conductor: $3.488$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.48774738381\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(31\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.494075 - 1.93801i) q^{2} +(0.211016 - 0.173176i) q^{3} +(-3.51178 + 1.91505i) q^{4} +(7.43419 + 2.25514i) q^{5} +(-0.439875 - 0.323389i) q^{6} +(2.23153 + 11.2187i) q^{7} +(5.44647 + 5.85969i) q^{8} +(-1.74128 + 8.75398i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.494075	−	1.93801i	−0.247038	−	0.969006i
							\(3\)	0.211016	−	0.173176i	0.0703386	−	0.0577254i	−0.598567	−	0.801073i	\(-0.704263\pi\)
0.668906	+	0.743347i	\(0.266763\pi\)
\(5\)	7.43419	+	2.25514i	1.48684	+	0.451028i	0.926200	−	0.377033i	\(-0.123056\pi\)
							0.560639	+	0.828061i	\(0.310556\pi\)
\(7\)	2.23153	+	11.2187i	0.318790	+	1.60267i	0.724909	+	0.688845i	\(0.241882\pi\)
							−0.406119	+	0.913820i	\(0.633118\pi\)
\(11\)	−2.14399	−	21.7683i	−0.194908	−	1.97894i	−0.203971	−	0.978977i	\(-0.565385\pi\)
							0.00906259	−	0.999959i	\(-0.497115\pi\)
\(13\)	2.88076	+	9.49658i	0.221597	+	0.730506i	0.995290	+	0.0969456i	\(0.0309073\pi\)
							−0.773693	+	0.633561i	\(0.781593\pi\)
\(17\)	3.47754	−	8.39551i	0.204561	−	0.493854i	−0.787989	−	0.615689i	\(-0.788878\pi\)
							0.992550	+	0.121835i	\(0.0388779\pi\)
\(19\)	−1.26463	+	2.36595i	−0.0665594	+	0.124524i	−0.913000	−	0.407959i	\(-0.866241\pi\)
							0.846441	+	0.532482i	\(0.178741\pi\)
\(23\)	0.199055	−	0.133005i	0.00865458	−	0.00578281i	−0.551235	−	0.834350i	\(-0.685843\pi\)
							0.559890	+	0.828567i	\(0.310843\pi\)
\(29\)	2.56354	−	26.0280i	0.0883978	−	0.897518i	−0.844062	−	0.536245i	\(-0.819842\pi\)
							0.932460	−	0.361273i	\(-0.117658\pi\)
\(31\)	−16.5391	−	16.5391i	−0.533520	−	0.533520i	0.388098	−	0.921618i	\(-0.373132\pi\)
							−0.921618	+	0.388098i	\(0.873132\pi\)
\(37\)	24.9841	+	46.7419i	0.675245	+	1.26329i	0.953095	+	0.302670i	\(0.0978781\pi\)
							−0.277850	+	0.960624i	\(0.589622\pi\)
\(41\)	2.87378	−	1.92020i	0.0700923	−	0.0468342i	−0.520030	−	0.854148i	\(-0.674079\pi\)
							0.590123	+	0.807314i	\(0.299079\pi\)
\(43\)	−29.9204	−	24.5550i	−0.695822	−	0.571047i	0.218597	−	0.975815i	\(-0.429852\pi\)
							−0.914419	+	0.404769i	\(0.867352\pi\)
\(47\)	6.18328	−	14.9278i	0.131559	−	0.317612i	−0.844349	−	0.535794i	\(-0.820013\pi\)
							0.975908	+	0.218182i	\(0.0700125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.14	✓	496
128.43	odd	32	inner	128.3.l.a.43.14	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.14	✓	496	1.1	even	1	trivial
128.3.l.a.43.14	yes	496	128.43	odd	32	inner