Embedded newform 384.2.w.a.107.53

• Label: 384.2.w.a.107.53
• Level: $384$
• Weight: $2$
• Character: 384.107
• 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			107.53
Character	\(\chi\)	\(=\)	384.107
Dual form			384.2.w.a.323.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24308 - 0.674360i) q^{2} +(-1.63002 - 0.585693i) q^{3} +(1.09048 - 1.67656i) q^{4} +(-4.12559 + 1.25148i) q^{5} +(-2.42121 + 0.371158i) q^{6} +(-0.422304 + 2.12307i) q^{7} +(0.224941 - 2.81947i) q^{8} +(2.31393 + 1.90938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24308	−	0.674360i	0.878988	−	0.476844i
							\(3\)	−1.63002	−	0.585693i	−0.941092	−	0.338150i
\(5\)	−4.12559	+	1.25148i	−1.84502	+	0.559680i								−0.845342	+	0.534226i	\(0.820603\pi\)
							−0.999676	+	0.0254543i	\(0.991897\pi\)
\(7\)	−0.422304	+	2.12307i	−0.159616	+	0.802444i	0.815156	+	0.579242i	\(0.196651\pi\)
							−0.974772	+	0.223203i	\(0.928349\pi\)
\(11\)	−0.302177	+	3.06805i	−0.0911097	+	0.925052i	0.835616	+	0.549314i	\(0.185111\pi\)
							−0.926726	+	0.375738i	\(0.877389\pi\)
\(13\)	0.171337	+	0.0519744i	0.0475202	+	0.0144151i	0.313955	−	0.949438i	\(-0.398346\pi\)
							−0.266435	+	0.963853i	\(0.585846\pi\)
\(17\)	−4.66446	+	1.93208i	−1.13130	+	0.468599i	−0.868222	−	0.496176i	\(-0.834737\pi\)
							−0.263076	+	0.964775i	\(0.584737\pi\)
\(19\)	−3.10645	+	1.66043i	−0.712668	+	0.380929i	−0.787540	−	0.616264i	\(-0.788645\pi\)
							0.0748713	+	0.997193i	\(0.476145\pi\)
\(23\)	−1.45869	+	2.18308i	−0.304158	+	0.455205i	−0.951791	−	0.306748i	\(-0.900759\pi\)
							0.647633	+	0.761953i	\(0.275759\pi\)
\(29\)	−0.0508006	−	0.515787i	−0.00943343	−	0.0957792i	0.989315	−	0.145793i	\(-0.0465734\pi\)
							−0.998749	+	0.0500140i	\(0.984073\pi\)
\(31\)	−6.38649	+	6.38649i	−1.14705	+	1.14705i	−0.159918	+	0.987130i	\(0.551123\pi\)
							−0.987130	+	0.159918i	\(0.948877\pi\)
\(37\)	−6.84520	−	3.65884i	−1.12534	−	0.601509i	−0.199682	−	0.979861i	\(-0.563991\pi\)
							−0.925662	+	0.378352i	\(0.876491\pi\)
\(41\)	6.61203	−	9.89560i	1.03263	−	1.54543i	0.209311	−	0.977849i	\(-0.432878\pi\)
							0.823315	−	0.567584i	\(-0.192122\pi\)
\(43\)	−1.77061	−	2.15749i	−0.270015	−	0.329014i	0.620295	−	0.784368i	\(-0.287013\pi\)
							−0.890310	+	0.455354i	\(0.849513\pi\)
\(47\)	1.93563	−	0.801763i	0.282340	−	0.116949i	−0.237019	−	0.971505i	\(-0.576171\pi\)
							0.519360	+	0.854556i	\(0.326171\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.107.53	yes	992
3.2	odd	2	inner	384.2.w.a.107.10	✓	992
128.67	odd	32	inner	384.2.w.a.323.10	yes	992
384.323	even	32	inner	384.2.w.a.323.53	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.107.10	✓	992	3.2	odd	2	inner
384.2.w.a.107.53	yes	992	1.1	even	1	trivial
384.2.w.a.323.10	yes	992	128.67	odd	32	inner
384.2.w.a.323.53	yes	992	384.323	even	32	inner