Embedded newform 384.2.w.a.35.10

• Label: 384.2.w.a.35.10
• Level: $384$
• Weight: $2$
• Character: 384.35
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06625543762\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			35.10
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24002 - 0.679964i) q^{2} +(1.09320 + 1.34347i) q^{3} +(1.07530 + 1.68634i) q^{4} +(0.708082 - 0.378478i) q^{5} +(-0.442086 - 2.40927i) q^{6} +(-4.90117 + 0.974904i) q^{7} +(-0.186740 - 2.82226i) q^{8} +(-0.609807 + 2.93737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24002	−	0.679964i	−0.876826	−	0.480807i
							\(3\)	1.09320	+	1.34347i	0.631162	+	0.775651i
\(5\)	0.708082	−	0.378478i	0.316664	−	0.169260i								−0.305406	−	0.952222i	\(-0.598792\pi\)
							0.622070	+	0.782962i	\(0.286292\pi\)
\(7\)	−4.90117	+	0.974904i	−1.85247	+	0.368479i	−0.990391	−	0.138297i	\(-0.955837\pi\)
							−0.862078	+	0.506776i	\(0.830837\pi\)
\(11\)	3.64450	+	2.99097i	1.09886	+	0.901810i	0.995680	−	0.0928524i	\(-0.0295985\pi\)
							0.103179	+	0.994663i	\(0.467098\pi\)
\(13\)	1.28552	+	0.687127i	0.356540	+	0.190575i	0.639936	−	0.768428i	\(-0.278961\pi\)
							−0.283396	+	0.959003i	\(0.591461\pi\)
\(17\)	−4.17380	−	1.72885i	−1.01230	−	0.419307i	−0.186004	−	0.982549i	\(-0.559554\pi\)
							−0.826292	+	0.563242i	\(0.809554\pi\)
\(19\)	1.43426	+	4.72812i	0.329042	+	1.08470i	0.952930	+	0.303191i	\(0.0980521\pi\)
							−0.623888	+	0.781514i	\(0.714448\pi\)
\(23\)	−3.44961	+	2.30495i	−0.719293	+	0.480616i	−0.860556	−	0.509356i	\(-0.829884\pi\)
							0.141263	+	0.989972i	\(0.454884\pi\)
\(29\)	6.17167	−	5.06495i	1.14605	−	0.940538i	0.147229	−	0.989102i	\(-0.452964\pi\)
							0.998820	+	0.0485639i	\(0.0154645\pi\)
\(31\)	−1.73956	−	1.73956i	−0.312434	−	0.312434i	0.533418	−	0.845852i	\(-0.320907\pi\)
							−0.845852	+	0.533418i	\(0.820907\pi\)
\(37\)	−1.79864	+	5.92932i	−0.295695	+	0.974775i	0.675674	+	0.737200i	\(0.263853\pi\)
							−0.971369	+	0.237575i	\(0.923647\pi\)
\(41\)	1.22041	−	0.815454i	0.190597	−	0.127353i	−0.456611	−	0.889667i	\(-0.650937\pi\)
							0.647207	+	0.762314i	\(0.275937\pi\)
\(43\)	0.444604	+	4.51414i	0.0678015	+	0.688400i	0.967667	+	0.252232i	\(0.0811647\pi\)
							−0.899865	+	0.436168i	\(0.856335\pi\)
\(47\)	1.80461	+	0.747495i	0.263230	+	0.109033i	0.510395	−	0.859940i	\(-0.329499\pi\)
							−0.247166	+	0.968973i	\(0.579499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.35.10	yes	992
3.2	odd	2	inner	384.2.w.a.35.53	yes	992
128.11	odd	32	inner	384.2.w.a.11.53	yes	992
384.11	even	32	inner	384.2.w.a.11.10	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.10	✓	992	384.11	even	32	inner
384.2.w.a.11.53	yes	992	128.11	odd	32	inner
384.2.w.a.35.10	yes	992	1.1	even	1	trivial
384.2.w.a.35.53	yes	992	3.2	odd	2	inner