Embedded newform 128.4.k.a.5.6

• Label: 128.4.k.a.5.6
• Level: $128$
• Weight: $4$
• Character: 128.5
• Analytic conductor: $7.552$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.k (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.55224448073\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(47\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	128.5
Dual form			128.4.k.a.77.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60432 - 1.10341i) q^{2} +(9.60431 + 2.91344i) q^{3} +(5.56498 + 5.74726i) q^{4} +(-17.8404 - 1.75713i) q^{5} +(-21.7980 - 18.1850i) q^{6} +(10.3895 + 15.5490i) q^{7} +(-8.15141 - 21.1082i) q^{8} +(61.3050 + 40.9627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 752 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60432	−	1.10341i	−0.920767	−	0.390114i
							\(3\)	9.60431	+	2.91344i	1.84835	+	0.560691i	0.999485	+	0.0320825i	\(0.0102139\pi\)
0.848866	+	0.528609i	\(0.177286\pi\)
\(5\)	−17.8404	−	1.75713i	−1.59570	−	0.157162i	−0.739261	−	0.673419i	\(-0.764825\pi\)
							−0.856434	+	0.516256i	\(0.827325\pi\)
\(7\)	10.3895	+	15.5490i	0.560979	+	0.839565i	0.998211	−	0.0597841i	\(-0.0190412\pi\)
							−0.437232	+	0.899349i	\(0.644041\pi\)
\(11\)	−13.8754	+	25.9591i	−0.380327	+	0.711542i	−0.997141	−	0.0755672i	\(-0.975923\pi\)
							0.616813	+	0.787109i	\(0.288423\pi\)
\(13\)	0.708302	+	7.19151i	0.0151114	+	0.153428i	0.999779	−	0.0210114i	\(-0.00668862\pi\)
							−0.984668	+	0.174440i	\(0.944189\pi\)
\(17\)	68.7543	−	28.4789i	0.980903	−	0.406304i	0.166143	−	0.986102i	\(-0.446869\pi\)
							0.814760	+	0.579798i	\(0.196869\pi\)
\(19\)	−91.1351	+	111.048i	−1.10041	+	1.34086i	−0.165404	+	0.986226i	\(0.552893\pi\)
							−0.935007	+	0.354629i	\(0.884607\pi\)
\(23\)	−5.64740	−	28.3914i	−0.0511985	−	0.257392i	0.946706	−	0.322100i	\(-0.104389\pi\)
							−0.997904	+	0.0647074i	\(0.979389\pi\)
\(29\)	−16.0918	+	8.60124i	−0.103040	+	0.0550762i	−0.522112	−	0.852877i	\(-0.674856\pi\)
							0.419072	+	0.907953i	\(0.362356\pi\)
\(31\)	25.7343	+	25.7343i	0.149097	+	0.149097i	0.777715	−	0.628618i	\(-0.216379\pi\)
							−0.628618	+	0.777715i	\(0.716379\pi\)
\(37\)	118.590	−	97.3247i	0.526923	−	0.432435i	−0.332948	−	0.942945i	\(-0.608044\pi\)
							0.859872	+	0.510510i	\(0.170544\pi\)
\(41\)	235.580	−	46.8598i	0.897352	−	0.178494i	0.275196	−	0.961388i	\(-0.411257\pi\)
							0.622156	+	0.782894i	\(0.286257\pi\)
\(43\)	−6.74138	+	2.04498i	−0.0239082	+	0.00725246i	−0.302216	−	0.953239i	\(-0.597726\pi\)
							0.278308	+	0.960492i	\(0.410226\pi\)
\(47\)	−46.0541	−	111.184i	−0.142929	−	0.345062i	0.836162	−	0.548482i	\(-0.184794\pi\)
							−0.979091	+	0.203420i	\(0.934794\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.k.a.5.6	✓	752
128.77	even	32	inner	128.4.k.a.77.6	yes	752

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.k.a.5.6	✓	752	1.1	even	1	trivial
128.4.k.a.77.6	yes	752	128.77	even	32	inner