Embedded newform 384.3.u.a.235.28

• Label: 384.3.u.a.235.28
• Level: $384$
• Weight: $3$
• Character: 384.235
• Analytic conductor: $10.463$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4632421514\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			235.28
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.427237 - 1.95383i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(-3.63494 + 1.66950i) q^{4} +(-0.959685 - 3.16366i) q^{5} +(3.08542 + 1.57485i) q^{6} +(1.80886 - 9.09377i) q^{7} +(4.81491 + 6.38879i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.427237	−	1.95383i	−0.213619	−	0.976917i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	−0.959685	−	3.16366i	−0.191937	−	0.632731i								−0.999101	−	0.0424035i	\(-0.986499\pi\)
							0.807164	−	0.590328i	\(-0.201001\pi\)
\(7\)	1.80886	−	9.09377i	0.258409	−	1.29911i	−0.605655	−	0.795728i	\(-0.707089\pi\)
							0.864064	−	0.503382i	\(-0.167911\pi\)
\(11\)	6.36898	+	0.627290i	0.578998	+	0.0570264i	0.383277	−	0.923633i	\(-0.374795\pi\)
							0.195721	+	0.980660i	\(0.437295\pi\)
\(13\)	0.536354	+	0.162701i	0.0412580	+	0.0125155i	0.310846	−	0.950460i	\(-0.399388\pi\)
							−0.269588	+	0.962976i	\(0.586888\pi\)
\(17\)	−0.655373	−	1.58221i	−0.0385514	−	0.0930713i	0.903431	−	0.428733i	\(-0.141040\pi\)
							−0.941982	+	0.335662i	\(0.891040\pi\)
\(19\)	−10.8534	+	5.80126i	−0.571231	+	0.305329i	−0.731607	−	0.681727i	\(-0.761229\pi\)
							0.160376	+	0.987056i	\(0.448729\pi\)
\(23\)	−24.7908	−	16.5647i	−1.07786	−	0.720203i	−0.115864	−	0.993265i	\(-0.536964\pi\)
							−0.961996	+	0.273062i	\(0.911964\pi\)
\(29\)	−45.2596	+	4.45768i	−1.56068	+	0.153713i	−0.841169	−	0.540772i	\(-0.818132\pi\)
							−0.719507	+	0.694485i	\(0.755632\pi\)
\(31\)	−33.7255	+	33.7255i	−1.08792	+	1.08792i	−0.0921776	+	0.995743i	\(0.529383\pi\)
							−0.995743	+	0.0921776i	\(0.970617\pi\)
\(37\)	−45.9375	−	24.5541i	−1.24155	−	0.663625i	−0.285941	−	0.958247i	\(-0.592306\pi\)
							−0.955614	+	0.294622i	\(0.904806\pi\)
\(41\)	−35.2106	−	23.5270i	−0.858794	−	0.573828i	0.0463534	−	0.998925i	\(-0.485240\pi\)
							−0.905148	+	0.425097i	\(0.860240\pi\)
\(43\)	51.2998	+	62.5090i	1.19302	+	1.45370i	0.859637	+	0.510905i	\(0.170690\pi\)
							0.333382	+	0.942792i	\(0.391810\pi\)
\(47\)	−8.36702	−	20.1998i	−0.178022	−	0.429782i	0.809530	−	0.587079i	\(-0.199722\pi\)
							−0.987551	+	0.157297i	\(0.949722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.28	yes	1024
128.67	odd	32	inner	384.3.u.a.67.28	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.28	✓	1024	128.67	odd	32	inner
384.3.u.a.235.28	yes	1024	1.1	even	1	trivial