Embedded newform 128.3.l.a.3.10

• Label: 128.3.l.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31385 + 1.50791i) q^{2} +(-2.45029 + 2.01090i) q^{3} +(-0.547604 - 3.96234i) q^{4} +(9.14318 + 2.77355i) q^{5} +(0.187046 - 6.33684i) q^{6} +(0.716485 + 3.60202i) q^{7} +(6.69433 + 4.38017i) q^{8} +(0.204383 - 1.02750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31385	+	1.50791i	−0.656924	+	0.753957i
							\(3\)	−2.45029	+	2.01090i	−0.816763	+	0.670300i	−0.947018	−	0.321180i	\(-0.895920\pi\)
0.130255	+	0.991481i	\(0.458420\pi\)
\(5\)	9.14318	+	2.77355i	1.82864	+	0.554711i	0.999987	−	0.00505841i	\(-0.00161015\pi\)
							0.828649	+	0.559769i	\(0.189110\pi\)
\(7\)	0.716485	+	3.60202i	0.102355	+	0.514574i	0.997615	+	0.0690287i	\(0.0219900\pi\)
							−0.895260	+	0.445545i	\(0.853010\pi\)
\(11\)	0.982186	+	9.97230i	0.0892897	+	0.906573i	0.930604	+	0.366027i	\(0.119282\pi\)
							−0.841315	+	0.540546i	\(0.818218\pi\)
\(13\)	−6.88056	−	22.6822i	−0.529274	−	1.74478i	−0.657917	−	0.753090i	\(-0.728562\pi\)
							0.128644	−	0.991691i	\(-0.458938\pi\)
\(17\)	−5.22340	+	12.6104i	−0.307259	+	0.741788i	0.692533	+	0.721386i	\(0.256495\pi\)
							−0.999792	+	0.0204023i	\(0.993505\pi\)
\(19\)	−11.3592	+	21.2516i	−0.597854	+	1.11851i	0.382998	+	0.923749i	\(0.374892\pi\)
							−0.980852	+	0.194757i	\(0.937608\pi\)
\(23\)	6.68053	−	4.46378i	0.290458	−	0.194078i	−0.401801	−	0.915727i	\(-0.631616\pi\)
							0.692259	+	0.721649i	\(0.256616\pi\)
\(29\)	−0.792725	+	8.04867i	−0.0273353	+	0.277540i	0.971794	+	0.235830i	\(0.0757808\pi\)
							−0.999130	+	0.0417106i	\(0.986719\pi\)
\(31\)	3.89963	+	3.89963i	0.125795	+	0.125795i	0.767201	−	0.641407i	\(-0.221649\pi\)
							−0.641407	+	0.767201i	\(0.721649\pi\)
\(37\)	17.2810	+	32.3304i	0.467053	+	0.873796i	0.999624	+	0.0274101i	\(0.00872600\pi\)
							−0.532571	+	0.846385i	\(0.678774\pi\)
\(41\)	−11.4822	+	7.67219i	−0.280055	+	0.187127i	−0.687662	−	0.726031i	\(-0.741363\pi\)
							0.407608	+	0.913157i	\(0.366363\pi\)
\(43\)	−16.0981	−	13.2114i	−0.374374	−	0.307241i	0.428325	−	0.903625i	\(-0.359104\pi\)
							−0.802700	+	0.596384i	\(0.796604\pi\)
\(47\)	26.8031	−	64.7085i	0.570280	−	1.37678i	−0.331038	−	0.943618i	\(-0.607399\pi\)
							0.901317	−	0.433159i	\(-0.142601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.10	✓	496
128.43	odd	32	inner	128.3.l.a.43.10	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.10	✓	496	1.1	even	1	trivial
128.3.l.a.43.10	yes	496	128.43	odd	32	inner