Embedded newform 384.2.w.a.11.19

• Label: 384.2.w.a.11.19
• 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			11.19
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.845116 - 1.13392i) q^{2} +(-0.765283 + 1.55382i) q^{3} +(-0.571557 + 1.91659i) q^{4} +(-0.974296 - 0.520772i) q^{5} +(2.40866 - 0.445384i) q^{6} +(-1.27617 - 0.253846i) q^{7} +(2.65630 - 0.971642i) q^{8} +(-1.82869 - 2.37822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.845116	−	1.13392i	−0.597587	−	0.801804i
							\(3\)	−0.765283	+	1.55382i	−0.441836	+	0.897096i
\(5\)	−0.974296	−	0.520772i	−0.435719	−	0.232896i								0.238928	−	0.971037i	\(-0.423204\pi\)
							−0.674647	+	0.738141i	\(0.735704\pi\)
\(7\)	−1.27617	−	0.253846i	−0.482347	−	0.0959447i	−0.0520732	−	0.998643i	\(-0.516583\pi\)
							−0.430274	+	0.902699i	\(0.641583\pi\)
\(11\)	−2.45402	+	2.01397i	−0.739916	+	0.607233i	−0.926978	−	0.375115i	\(-0.877603\pi\)
							0.187062	+	0.982348i	\(0.440103\pi\)
\(13\)	2.24004	−	1.19733i	0.621276	−	0.332079i	−0.130549	−	0.991442i	\(-0.541674\pi\)
							0.751825	+	0.659363i	\(0.229174\pi\)
\(17\)	5.66801	−	2.34777i	1.37469	−	0.569417i	0.431637	−	0.902048i	\(-0.357936\pi\)
							0.943057	+	0.332631i	\(0.107936\pi\)
\(19\)	1.05544	−	3.47933i	0.242136	−	0.798214i	−0.748673	−	0.662940i	\(-0.769308\pi\)
							0.990808	−	0.135274i	\(-0.0431915\pi\)
\(23\)	−7.00650	−	4.68160i	−1.46096	−	0.976180i	−0.995858	−	0.0909212i	\(-0.971019\pi\)
							−0.465099	−	0.885259i	\(-0.653981\pi\)
\(29\)	−2.10632	−	1.72861i	−0.391134	−	0.320995i	0.418188	−	0.908360i	\(-0.362665\pi\)
							−0.809322	+	0.587365i	\(0.800165\pi\)
\(31\)	6.37098	−	6.37098i	1.14426	−	1.14426i	0.156599	−	0.987662i	\(-0.449947\pi\)
							0.987662	−	0.156599i	\(-0.0500530\pi\)
\(37\)	−1.88396	−	6.21057i	−0.309721	−	1.02101i	−0.964254	−	0.264979i	\(-0.914635\pi\)
							0.654534	−	0.756033i	\(-0.272865\pi\)
\(41\)	0.168224	+	0.112403i	0.0262721	+	0.0175545i	0.568637	−	0.822589i	\(-0.307471\pi\)
							−0.542365	+	0.840143i	\(0.682471\pi\)
\(43\)	−1.21655	+	12.3519i	−0.185523	+	1.88364i	0.228741	+	0.973487i	\(0.426539\pi\)
							−0.414264	+	0.910157i	\(0.635961\pi\)
\(47\)	−3.43982	+	1.42482i	−0.501750	+	0.207832i	−0.619179	−	0.785250i	\(-0.712535\pi\)
							0.117429	+	0.993081i	\(0.462535\pi\)