Embedded newform 384.2.w.a.11.5

• Label: 384.2.w.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37752 + 0.320040i) q^{2} +(-1.61000 - 0.638681i) q^{3} +(1.79515 - 0.881727i) q^{4} +(-0.665070 - 0.355487i) q^{5} +(2.42221 + 0.364535i) q^{6} +(0.505023 + 0.100455i) q^{7} +(-2.19067 + 1.78912i) q^{8} +(2.18417 + 2.05655i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37752	+	0.320040i	−0.974057	+	0.226303i
							\(3\)	−1.61000	−	0.638681i	−0.929532	−	0.368743i
\(5\)	−0.665070	−	0.355487i	−0.297428	−	0.158979i								0.315932	−	0.948782i	\(-0.397683\pi\)
							−0.613361	+	0.789803i	\(0.710183\pi\)
\(7\)	0.505023	+	0.100455i	0.190881	+	0.0379686i	0.289605	−	0.957146i	\(-0.406476\pi\)
							−0.0987240	+	0.995115i	\(0.531476\pi\)
\(11\)	4.34903	−	3.56916i	1.31128	−	1.07614i	0.319585	−	0.947558i	\(-0.396457\pi\)
							0.991699	−	0.128585i	\(-0.0410434\pi\)
\(13\)	−5.75137	+	3.07417i	−1.59514	+	0.852622i	−0.596390	+	0.802695i	\(0.703399\pi\)
							−0.998753	+	0.0499268i	\(0.984101\pi\)
\(17\)	3.41772	−	1.41567i	0.828919	−	0.343349i	0.0724443	−	0.997372i	\(-0.476920\pi\)
							0.756475	+	0.654023i	\(0.226920\pi\)
\(19\)	−0.0735879	+	0.242587i	−0.0168822	+	0.0556532i	−0.964977	−	0.262335i	\(-0.915507\pi\)
							0.948095	+	0.317989i	\(0.103007\pi\)
\(23\)	−2.84698	−	1.90229i	−0.593637	−	0.396656i	0.222146	−	0.975013i	\(-0.428694\pi\)
							−0.815783	+	0.578358i	\(0.803694\pi\)
\(29\)	−7.03105	−	5.77023i	−1.30563	−	1.07151i	−0.992558	−	0.121777i	\(-0.961141\pi\)
							−0.313075	−	0.949728i	\(-0.601359\pi\)
\(31\)	6.56862	−	6.56862i	1.17976	−	1.17976i	0.199955	−	0.979805i	\(-0.435921\pi\)
							0.979805	−	0.199955i	\(-0.0640795\pi\)
\(37\)	−0.907311	−	2.99100i	−0.149161	−	0.491718i	0.850237	−	0.526399i	\(-0.176458\pi\)
							−0.999398	+	0.0346815i	\(0.988958\pi\)
\(41\)	3.08027	+	2.05817i	0.481057	+	0.321432i	0.772345	−	0.635203i	\(-0.219084\pi\)
							−0.291288	+	0.956635i	\(0.594084\pi\)
\(43\)	0.601451	−	6.10664i	0.0917205	−	0.931253i	−0.833676	−	0.552254i	\(-0.813768\pi\)
							0.925397	−	0.379000i	\(-0.123732\pi\)
\(47\)	5.19189	−	2.15055i	0.757314	−	0.313690i	0.0295923	−	0.999562i	\(-0.490579\pi\)
							0.727722	+	0.685872i	\(0.240579\pi\)