Embedded newform 384.3.u.a.235.64

• Label: 384.3.u.a.235.64
• 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			235.64
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99805 - 0.0881975i) q^{2} +(1.09880 - 1.33889i) q^{3} +(3.98444 - 0.352447i) q^{4} +(2.10941 + 6.95380i) q^{5} +(2.07738 - 2.77209i) q^{6} +(1.42361 - 7.15697i) q^{7} +(7.93005 - 1.05563i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99805	−	0.0881975i	0.999027	−	0.0440987i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	2.10941	+	6.95380i	0.421882	+	1.39076i								0.868347	+	0.495958i	\(0.165183\pi\)
							−0.446464	+	0.894801i	\(0.647317\pi\)
\(7\)	1.42361	−	7.15697i	0.203373	−	1.02242i	−0.735333	−	0.677706i	\(-0.762974\pi\)
							0.938706	−	0.344719i	\(-0.112026\pi\)
\(11\)	4.59488	+	0.452557i	0.417717	+	0.0411415i	0.304693	−	0.952451i	\(-0.401446\pi\)
							0.113024	+	0.993592i	\(0.463946\pi\)
\(13\)	−9.07385	−	2.75252i	−0.697989	−	0.211733i	−0.0787028	−	0.996898i	\(-0.525078\pi\)
							−0.619286	+	0.785166i	\(0.712578\pi\)
\(17\)	3.09596	+	7.47431i	0.182115	+	0.439666i	0.988402	−	0.151859i	\(-0.0485261\pi\)
							−0.806287	+	0.591525i	\(0.798526\pi\)
\(19\)	1.98240	−	1.05962i	0.104337	−	0.0557693i	−0.418403	−	0.908262i	\(-0.637410\pi\)
							0.522740	+	0.852492i	\(0.324910\pi\)
\(23\)	21.6280	+	14.4513i	0.940346	+	0.628319i	0.928389	−	0.371609i	\(-0.121194\pi\)
							0.0119571	+	0.999929i	\(0.496194\pi\)
\(29\)	−12.2101	+	1.20259i	−0.421039	+	0.0414687i	−0.306318	−	0.951929i	\(-0.599097\pi\)
							−0.114720	+	0.993398i	\(0.536597\pi\)
\(31\)	−10.5345	+	10.5345i	−0.339823	+	0.339823i	−0.856301	−	0.516477i	\(-0.827243\pi\)
							0.516477	+	0.856301i	\(0.327243\pi\)
\(37\)	−38.6194	−	20.6425i	−1.04377	−	0.557905i	−0.141883	−	0.989883i	\(-0.545316\pi\)
							−0.901884	+	0.431978i	\(0.857816\pi\)
\(41\)	−1.18460	−	0.791526i	−0.0288927	−	0.0193055i	0.541040	−	0.840997i	\(-0.318031\pi\)
							−0.569933	+	0.821691i	\(0.693031\pi\)
\(43\)	−22.3865	−	27.2781i	−0.520617	−	0.634374i	0.444630	−	0.895715i	\(-0.353335\pi\)
							−0.965247	+	0.261341i	\(0.915835\pi\)
\(47\)	12.5269	+	30.2426i	0.266530	+	0.643459i	0.999315	−	0.0370001i	\(-0.0117802\pi\)
							−0.732786	+	0.680460i	\(0.761780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.64	yes	1024
128.67	odd	32	inner	384.3.u.a.67.64	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.64	✓	1024	128.67	odd	32	inner
384.3.u.a.235.64	yes	1024	1.1	even	1	trivial