Embedded newform 384.3.u.a.235.52

• Label: 384.3.u.a.235.52
• 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			235.52
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.55815 - 1.25386i) q^{2} +(1.09880 - 1.33889i) q^{3} +(0.855686 - 3.90740i) q^{4} +(0.955896 + 3.15117i) q^{5} +(0.0333210 - 3.46394i) q^{6} +(1.49796 - 7.53074i) q^{7} +(-3.56603 - 7.16124i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55815	−	1.25386i	0.779077	−	0.626928i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	0.955896	+	3.15117i	0.191179	+	0.630234i								0.999158	+	0.0410223i	\(0.0130615\pi\)
							−0.807979	+	0.589211i	\(0.799439\pi\)
\(7\)	1.49796	−	7.53074i	0.213994	−	1.07582i	−0.713121	−	0.701041i	\(-0.752719\pi\)
							0.927114	−	0.374778i	\(-0.122281\pi\)
\(11\)	−1.39474	−	0.137370i	−0.126794	−	0.0124882i	0.0344209	−	0.999407i	\(-0.489041\pi\)
							−0.161215	+	0.986919i	\(0.551541\pi\)
\(13\)	21.8099	+	6.61596i	1.67769	+	0.508920i	0.978540	−	0.206059i	\(-0.0660638\pi\)
							0.699146	+	0.714979i	\(0.253564\pi\)
\(17\)	−6.88174	−	16.6140i	−0.404808	−	0.977293i	−0.986482	−	0.163871i	\(-0.947602\pi\)
							0.581674	−	0.813422i	\(-0.302398\pi\)
\(19\)	−16.6064	+	8.87628i	−0.874019	+	0.467173i	−0.846463	−	0.532448i	\(-0.821272\pi\)
							−0.0275556	+	0.999620i	\(0.508772\pi\)
\(23\)	−21.7684	−	14.5452i	−0.946453	−	0.632400i	−0.0164158	−	0.999865i	\(-0.505226\pi\)
							−0.930037	+	0.367466i	\(0.880226\pi\)
\(29\)	25.8018	−	2.54125i	0.889717	−	0.0876295i	0.357179	−	0.934036i	\(-0.383739\pi\)
							0.532538	+	0.846406i	\(0.321239\pi\)
\(31\)	−18.8885	+	18.8885i	−0.609307	+	0.609307i	−0.942765	−	0.333458i	\(-0.891784\pi\)
							0.333458	+	0.942765i	\(0.391784\pi\)
\(37\)	41.6370	+	22.2554i	1.12532	+	0.601498i	0.925656	−	0.378365i	\(-0.123514\pi\)
							0.199668	+	0.979864i	\(0.436014\pi\)
\(41\)	16.3445	+	10.9210i	0.398646	+	0.266366i	0.738693	−	0.674042i	\(-0.235443\pi\)
							−0.340048	+	0.940408i	\(0.610443\pi\)
\(43\)	12.7293	+	15.5106i	0.296029	+	0.360713i	0.899691	−	0.436528i	\(-0.143792\pi\)
							−0.603661	+	0.797241i	\(0.706292\pi\)
\(47\)	1.73940	+	4.19927i	0.0370084	+	0.0893462i	0.941303	−	0.337563i	\(-0.109602\pi\)
							−0.904295	+	0.426909i	\(0.859602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.52	yes	1024
128.67	odd	32	inner	384.3.u.a.67.52	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.52	✓	1024	128.67	odd	32	inner
384.3.u.a.235.52	yes	1024	1.1	even	1	trivial