Embedded newform 384.3.u.a.235.12

• Label: 384.3.u.a.235.12
• 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			235.12
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67794 + 1.08835i) q^{2} +(1.09880 - 1.33889i) q^{3} +(1.63097 - 3.65239i) q^{4} +(2.48236 + 8.18324i) q^{5} +(-0.386535 + 3.44247i) q^{6} +(1.88120 - 9.45743i) q^{7} +(1.23841 + 7.90357i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67794	+	1.08835i	−0.838971	+	0.544177i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	2.48236	+	8.18324i	0.496472	+	1.63665i								0.744301	+	0.667845i	\(0.232783\pi\)
							−0.247829	+	0.968804i	\(0.579717\pi\)
\(7\)	1.88120	−	9.45743i	0.268743	−	1.35106i	−0.576683	−	0.816968i	\(-0.695653\pi\)
							0.845426	−	0.534093i	\(-0.179347\pi\)
\(11\)	9.05582	+	0.891920i	0.823256	+	0.0810837i	0.500868	−	0.865524i	\(-0.333014\pi\)
							0.322388	+	0.946607i	\(0.395514\pi\)
\(13\)	5.86713	+	1.77977i	0.451317	+	0.136906i	0.507757	−	0.861500i	\(-0.330475\pi\)
							−0.0564395	+	0.998406i	\(0.517975\pi\)
\(17\)	0.464790	+	1.12210i	0.0273406	+	0.0660060i	0.936960	−	0.349436i	\(-0.113627\pi\)
							−0.909620	+	0.415442i	\(0.863627\pi\)
\(19\)	7.80694	−	4.17290i	0.410892	−	0.219626i	−0.252982	−	0.967471i	\(-0.581411\pi\)
							0.663874	+	0.747845i	\(0.268911\pi\)
\(23\)	−4.88473	−	3.26387i	−0.212380	−	0.141907i	0.444831	−	0.895615i	\(-0.353264\pi\)
							−0.657211	+	0.753707i	\(0.728264\pi\)
\(29\)	−17.5350	+	1.72705i	−0.604656	+	0.0595534i	−0.395713	−	0.918374i	\(-0.629502\pi\)
							−0.208943	+	0.977928i	\(0.567002\pi\)
\(31\)	31.2716	−	31.2716i	1.00876	−	1.00876i	0.00879897	−	0.999961i	\(-0.497199\pi\)
							0.999961	−	0.00879897i	\(-0.00280084\pi\)
\(37\)	56.0912	+	29.9814i	1.51598	+	0.810308i	0.998957	−	0.0456644i	\(-0.0145405\pi\)
							0.517022	+	0.855972i	\(0.327040\pi\)
\(41\)	4.58704	+	3.06496i	0.111879	+	0.0747552i	0.610251	−	0.792208i	\(-0.291069\pi\)
							−0.498372	+	0.866963i	\(0.666069\pi\)
\(43\)	39.3475	+	47.9451i	0.915059	+	1.11500i	0.993008	+	0.118045i	\(0.0376628\pi\)
							−0.0779494	+	0.996957i	\(0.524837\pi\)
\(47\)	−27.5232	−	66.4468i	−0.585599	−	1.41376i	−0.887672	−	0.460476i	\(-0.847679\pi\)
							0.302073	−	0.953285i	\(-0.402321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.12	yes	1024
128.67	odd	32	inner	384.3.u.a.67.12	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.12	✓	1024	128.67	odd	32	inner
384.3.u.a.235.12	yes	1024	1.1	even	1	trivial