Embedded newform 384.3.u.a.235.57

• Label: 384.3.u.a.235.57
• 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			235.57
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88062 + 0.680636i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(3.07347 + 2.56004i) q^{4} +(-1.13524 - 3.74238i) q^{5} +(-2.97773 + 1.77007i) q^{6} +(1.55808 - 7.83300i) q^{7} +(4.03757 + 6.90637i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88062	+	0.680636i	0.940310	+	0.340318i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	−1.13524	−	3.74238i	−0.227048	−	0.748475i								−0.994251	−	0.107072i	\(-0.965852\pi\)
							0.767204	−	0.641404i	\(-0.221648\pi\)
\(7\)	1.55808	−	7.83300i	0.222583	−	1.11900i	−0.694252	−	0.719732i	\(-0.744264\pi\)
							0.916834	−	0.399268i	\(-0.130736\pi\)
\(11\)	8.20162	+	0.807789i	0.745602	+	0.0734354i	0.463681	−	0.886002i	\(-0.346528\pi\)
							0.281921	+	0.959438i	\(0.409028\pi\)
\(13\)	15.8401	+	4.80505i	1.21847	+	0.369619i	0.833089	−	0.553139i	\(-0.186570\pi\)
							0.385381	+	0.922758i	\(0.374070\pi\)
\(17\)	−1.56195	−	3.77088i	−0.0918794	−	0.221816i	0.871259	−	0.490824i	\(-0.163304\pi\)
							−0.963138	+	0.269008i	\(0.913304\pi\)
\(19\)	12.6007	−	6.73519i	0.663192	−	0.354484i	−0.105211	−	0.994450i	\(-0.533552\pi\)
							0.768403	+	0.639966i	\(0.221052\pi\)
\(23\)	−2.67064	−	1.78447i	−0.116115	−	0.0775855i	0.496157	−	0.868233i	\(-0.334744\pi\)
							−0.612272	+	0.790647i	\(0.709744\pi\)
\(29\)	3.60622	−	0.355182i	0.124353	−	0.0122477i	−0.0356494	−	0.999364i	\(-0.511350\pi\)
							0.160002	+	0.987117i	\(0.448850\pi\)
\(31\)	−25.7916	+	25.7916i	−0.831988	+	0.831988i	−0.987789	−	0.155800i	\(-0.950204\pi\)
							0.155800	+	0.987789i	\(0.450204\pi\)
\(37\)	25.0889	+	13.4103i	0.678078	+	0.362440i	0.774210	−	0.632928i	\(-0.218147\pi\)
							−0.0961324	+	0.995369i	\(0.530647\pi\)
\(41\)	3.25035	+	2.17182i	0.0792769	+	0.0529711i	0.594577	−	0.804039i	\(-0.297320\pi\)
							−0.515300	+	0.857010i	\(0.672320\pi\)
\(43\)	−49.9703	−	60.8889i	−1.16210	−	1.41602i	−0.891317	−	0.453381i	\(-0.850218\pi\)
							−0.270782	−	0.962641i	\(-0.587282\pi\)
\(47\)	24.7537	+	59.7607i	0.526675	+	1.27150i	0.933690	+	0.358083i	\(0.116570\pi\)
							−0.407015	+	0.913421i	\(0.633430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.57	yes	1024
128.67	odd	32	inner	384.3.u.a.67.57	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.57	✓	1024	128.67	odd	32	inner
384.3.u.a.235.57	yes	1024	1.1	even	1	trivial