Embedded newform 384.2.w.a.11.20

• Label: 384.2.w.a.11.20
• Level: $384$
• Weight: $2$
• Character: 384.11
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.w (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06625543762\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			11.20
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.813493 + 1.15682i) q^{2} +(-1.73071 - 0.0681515i) q^{3} +(-0.676460 - 1.88213i) q^{4} +(1.66341 + 0.889109i) q^{5} +(1.48676 - 1.94668i) q^{6} +(-2.12405 - 0.422500i) q^{7} +(2.72758 + 0.748555i) q^{8} +(2.99071 + 0.235901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} + O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.813493	+	1.15682i	−0.575226	+	0.817994i
							\(3\)	−1.73071	−	0.0681515i	−0.999226	−	0.0393473i
\(5\)	1.66341	+	0.889109i	0.743898	+	0.397622i								0.799370	−	0.600838i	\(-0.205166\pi\)
							−0.0554724	+	0.998460i	\(0.517666\pi\)
\(7\)	−2.12405	−	0.422500i	−0.802816	−	0.159690i	−0.223405	−	0.974726i	\(-0.571717\pi\)
							−0.579411	+	0.815036i	\(0.696717\pi\)
\(11\)	−2.38724	+	1.95916i	−0.719780	+	0.590708i	−0.921335	−	0.388769i	\(-0.872900\pi\)
							0.201555	+	0.979477i	\(0.435400\pi\)
\(13\)	−1.07693	+	0.575629i	−0.298686	+	0.159651i	−0.613932	−	0.789359i	\(-0.710413\pi\)
							0.315246	+	0.949010i	\(0.397913\pi\)
\(17\)	−3.03653	+	1.25777i	−0.736466	+	0.305054i	−0.719206	−	0.694797i	\(-0.755494\pi\)
							−0.0172602	+	0.999851i	\(0.505494\pi\)
\(19\)	0.0115783	−	0.0381685i	0.00265624	−	0.00875645i	−0.955602	−	0.294660i	\(-0.904794\pi\)
							0.958258	+	0.285903i	\(0.0922937\pi\)
\(23\)	−4.68419	−	3.12987i	−0.976720	−	0.652624i	−0.0387171	−	0.999250i	\(-0.512327\pi\)
							−0.938003	+	0.346627i	\(0.887327\pi\)
\(29\)	−2.81564	−	2.31074i	−0.522852	−	0.429093i	0.335587	−	0.942009i	\(-0.391065\pi\)
							−0.858439	+	0.512916i	\(0.828565\pi\)
\(31\)	−0.683005	+	0.683005i	−0.122671	+	0.122671i	−0.765777	−	0.643106i	\(-0.777645\pi\)
							0.643106	+	0.765777i	\(0.277645\pi\)
\(37\)	1.24167	+	4.09324i	0.204129	+	0.672924i	0.997904	+	0.0647123i	\(0.0206130\pi\)
							−0.793775	+	0.608212i	\(0.791887\pi\)
\(41\)	−8.68478	−	5.80298i	−1.35633	−	0.906273i	−0.356724	−	0.934210i	\(-0.616106\pi\)
							−0.999610	+	0.0279362i	\(0.991106\pi\)
\(43\)	−0.213016	+	2.16279i	−0.0324847	+	0.329822i	0.965184	+	0.261573i	\(0.0842413\pi\)
							−0.997668	+	0.0682491i	\(0.978259\pi\)
\(47\)	−3.02989	+	1.25502i	−0.441955	+	0.183064i	−0.592553	−	0.805531i	\(-0.701880\pi\)
							0.150598	+	0.988595i	\(0.451880\pi\)