Embedded newform 128.3.l.a.3.8

• Label: 128.3.l.a.3.8
• 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			3.8
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52087 - 1.29883i) q^{2} +(3.39289 - 2.78447i) q^{3} +(0.626076 + 3.95070i) q^{4} +(-2.17916 - 0.661042i) q^{5} +(-8.77669 - 0.171976i) q^{6} +(-1.02400 - 5.14798i) q^{7} +(4.17911 - 6.82166i) q^{8} +(2.00260 - 10.0677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52087	−	1.29883i	−0.760434	−	0.649416i
							\(3\)	3.39289	−	2.78447i	1.13096	−	0.928157i	0.132947	−	0.991123i	\(-0.457556\pi\)
0.998016	+	0.0629660i	\(0.0200560\pi\)
\(5\)	−2.17916	−	0.661042i	−0.435833	−	0.132208i	0.0647415	−	0.997902i	\(-0.479378\pi\)
							−0.500574	+	0.865694i	\(0.666878\pi\)
\(7\)	−1.02400	−	5.14798i	−0.146285	−	0.735426i	−0.982388	−	0.186853i	\(-0.940171\pi\)
							0.836103	−	0.548573i	\(-0.184829\pi\)
\(11\)	−1.58823	−	16.1256i	−0.144385	−	1.46596i	−0.745343	−	0.666681i	\(-0.767714\pi\)
							0.600959	−	0.799280i	\(-0.294786\pi\)
\(13\)	3.26819	+	10.7738i	0.251399	+	0.828751i	0.988270	+	0.152718i	\(0.0488025\pi\)
							−0.736871	+	0.676033i	\(0.763697\pi\)
\(17\)	0.474036	−	1.14443i	0.0278845	−	0.0673191i	−0.909324	−	0.416089i	\(-0.863400\pi\)
							0.937208	+	0.348770i	\(0.113400\pi\)
\(19\)	2.81029	−	5.25768i	0.147910	−	0.276720i	−0.797045	−	0.603919i	\(-0.793605\pi\)
							0.944955	+	0.327199i	\(0.106105\pi\)
\(23\)	8.00349	−	5.34776i	0.347978	−	0.232511i	−0.369282	−	0.929318i	\(-0.620396\pi\)
							0.717259	+	0.696806i	\(0.245396\pi\)
\(29\)	−2.14766	+	21.8056i	−0.0740573	+	0.751917i	0.884409	+	0.466712i	\(0.154562\pi\)
							−0.958467	+	0.285205i	\(0.907938\pi\)
\(31\)	11.2675	+	11.2675i	0.363466	+	0.363466i	0.865087	−	0.501621i	\(-0.167263\pi\)
							−0.501621	+	0.865087i	\(0.667263\pi\)
\(37\)	18.0191	+	33.7114i	0.487003	+	0.911118i	0.998651	+	0.0519234i	\(0.0165352\pi\)
							−0.511648	+	0.859195i	\(0.670965\pi\)
\(41\)	55.3151	−	36.9603i	1.34915	−	0.901472i	0.349773	−	0.936834i	\(-0.386259\pi\)
							0.999375	+	0.0353628i	\(0.0112587\pi\)
\(43\)	48.4484	+	39.7605i	1.12671	+	0.924664i	0.997754	−	0.0669871i	\(-0.0213386\pi\)
							0.128952	+	0.991651i	\(0.458839\pi\)
\(47\)	−14.5449	+	35.1144i	−0.309465	+	0.747115i	0.690258	+	0.723564i	\(0.257497\pi\)
							−0.999723	+	0.0235509i	\(0.992503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.8	✓	496
128.43	odd	32	inner	128.3.l.a.43.8	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.8	✓	496	1.1	even	1	trivial
128.3.l.a.43.8	yes	496	128.43	odd	32	inner