Embedded newform 384.2.w.a.11.4

• Label: 384.2.w.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39869 - 0.208968i) q^{2} +(-0.339181 + 1.69852i) q^{3} +(1.91266 + 0.584564i) q^{4} +(-0.321371 - 0.171776i) q^{5} +(0.829345 - 2.30482i) q^{6} +(-1.36158 - 0.270834i) q^{7} +(-2.55307 - 1.21731i) q^{8} +(-2.76991 - 1.15221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39869	−	0.208968i	−0.989023	−	0.147763i
							\(3\)	−0.339181	+	1.69852i	−0.195826	+	0.980639i
\(5\)	−0.321371	−	0.171776i	−0.143721	−	0.0768207i								0.397969	−	0.917399i	\(-0.369715\pi\)
							−0.541690	+	0.840578i	\(0.682215\pi\)
\(7\)	−1.36158	−	0.270834i	−0.514627	−	0.102366i	−0.0690570	−	0.997613i	\(-0.521999\pi\)
							−0.445570	+	0.895247i	\(0.646999\pi\)
\(11\)	−0.181039	+	0.148575i	−0.0545853	+	0.0447970i	−0.661290	−	0.750130i	\(-0.729991\pi\)
							0.606705	+	0.794927i	\(0.292491\pi\)
\(13\)	−2.85114	+	1.52396i	−0.790763	+	0.422672i	−0.816751	−	0.576991i	\(-0.804227\pi\)
							0.0259877	+	0.999662i	\(0.491727\pi\)
\(17\)	−5.78209	+	2.39502i	−1.40236	+	0.580878i	−0.950364	−	0.311141i	\(-0.899289\pi\)
							−0.451999	+	0.892019i	\(0.649289\pi\)
\(19\)	1.97671	−	6.51633i	0.453487	−	1.49495i	−0.370766	−	0.928726i	\(-0.620905\pi\)
							0.824253	−	0.566221i	\(-0.191595\pi\)
\(23\)	4.20927	+	2.81254i	0.877694	+	0.586456i	0.910732	−	0.412998i	\(-0.135518\pi\)
							−0.0330384	+	0.999454i	\(0.510518\pi\)
\(29\)	−6.21541	−	5.10086i	−1.15417	−	0.947205i	−0.155002	−	0.987914i	\(-0.549538\pi\)
							−0.999171	+	0.0407087i	\(0.987038\pi\)
\(31\)	−4.89115	+	4.89115i	−0.878476	+	0.878476i	−0.993377	−	0.114901i	\(-0.963345\pi\)
							0.114901	+	0.993377i	\(0.463345\pi\)
\(37\)	0.879289	+	2.89863i	0.144554	+	0.476531i	0.999080	−	0.0428770i	\(-0.0136524\pi\)
							−0.854526	+	0.519408i	\(0.826152\pi\)
\(41\)	−0.650342	−	0.434545i	−0.101566	−	0.0678645i	0.503750	−	0.863850i	\(-0.331953\pi\)
							−0.605316	+	0.795985i	\(0.706953\pi\)
\(43\)	0.655730	−	6.65774i	0.0999978	−	1.01530i	−0.805972	−	0.591953i	\(-0.798357\pi\)
							0.905970	−	0.423342i	\(-0.139143\pi\)
\(47\)	−3.86428	+	1.60064i	−0.563663	+	0.233477i	−0.646275	−	0.763105i	\(-0.723674\pi\)
							0.0826114	+	0.996582i	\(0.473674\pi\)