Embedded newform 128.3.l.a.3.9

• Label: 128.3.l.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39308 + 1.43504i) q^{2} +(0.447688 - 0.367408i) q^{3} +(-0.118660 - 3.99824i) q^{4} +(-2.47385 - 0.750435i) q^{5} +(-0.0964209 + 1.15428i) q^{6} +(0.987061 + 4.96229i) q^{7} +(5.90292 + 5.39958i) q^{8} +(-1.69038 + 8.49810i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39308	+	1.43504i	−0.696540	+	0.717518i
							\(3\)	0.447688	−	0.367408i	0.149229	−	0.122469i	−0.556816	−	0.830636i	\(-0.687977\pi\)
0.706046	+	0.708166i	\(0.250477\pi\)
\(5\)	−2.47385	−	0.750435i	−0.494771	−	0.150087i	0.0330316	−	0.999454i	\(-0.489484\pi\)
							−0.527802	+	0.849367i	\(0.676984\pi\)
\(7\)	0.987061	+	4.96229i	0.141009	+	0.708899i	0.985002	+	0.172545i	\(0.0551989\pi\)
							−0.843993	+	0.536354i	\(0.819801\pi\)
\(11\)	0.355541	+	3.60986i	0.0323219	+	0.328169i	0.997725	+	0.0674091i	\(0.0214733\pi\)
							−0.965404	+	0.260760i	\(0.916027\pi\)
\(13\)	6.50922	+	21.4580i	0.500709	+	1.65062i	0.734711	+	0.678380i	\(0.237318\pi\)
							−0.234002	+	0.972236i	\(0.575182\pi\)
\(17\)	9.43380	−	22.7752i	0.554929	−	1.33972i	−0.358808	−	0.933412i	\(-0.616817\pi\)
							0.913737	−	0.406306i	\(-0.133183\pi\)
\(19\)	−13.6843	+	25.6016i	−0.720229	+	1.34745i	0.208988	+	0.977918i	\(0.432983\pi\)
							−0.929217	+	0.369535i	\(0.879517\pi\)
\(23\)	−6.73776	+	4.50203i	−0.292946	+	0.195740i	−0.693355	−	0.720596i	\(-0.743868\pi\)
							0.400409	+	0.916337i	\(0.368868\pi\)
\(29\)	0.212124	−	2.15373i	0.00731461	−	0.0742665i	−0.990834	−	0.135082i	\(-0.956870\pi\)
							0.998149	+	0.0608157i	\(0.0193702\pi\)
\(31\)	36.6146	+	36.6146i	1.18112	+	1.18112i	0.979455	+	0.201663i	\(0.0646345\pi\)
							0.201663	+	0.979455i	\(0.435366\pi\)
\(37\)	−10.2620	−	19.1989i	−0.277352	−	0.518889i	0.704220	−	0.709982i	\(-0.251297\pi\)
							−0.981572	+	0.191093i	\(0.938797\pi\)
\(41\)	63.1162	−	42.1729i	1.53942	−	1.02861i	0.559592	−	0.828768i	\(-0.310958\pi\)
							0.979829	−	0.199839i	\(-0.0640421\pi\)
\(43\)	−35.7546	−	29.3430i	−0.831501	−	0.682395i	0.119076	−	0.992885i	\(-0.462007\pi\)
							−0.950577	+	0.310490i	\(0.899507\pi\)
\(47\)	−15.2774	+	36.8829i	−0.325051	+	0.784743i	0.673894	+	0.738828i	\(0.264620\pi\)
							−0.998945	+	0.0459151i	\(0.985380\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.9	✓	496
128.43	odd	32	inner	128.3.l.a.43.9	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.9	✓	496	1.1	even	1	trivial
128.3.l.a.43.9	yes	496	128.43	odd	32	inner