Embedded newform 384.2.w.a.131.33

• Label: 384.2.w.a.131.33
• Level: $384$
• Weight: $2$
• Character: 384.131
• 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			131.33
Character	\(\chi\)	\(=\)	384.131
Dual form			384.2.w.a.299.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0871908 - 1.41152i) q^{2} +(1.16547 - 1.28128i) q^{3} +(-1.98480 - 0.246144i) q^{4} +(-0.108259 + 0.356883i) q^{5} +(-1.70693 - 1.75681i) q^{6} +(-0.537756 - 2.70348i) q^{7} +(-0.520493 + 2.78012i) q^{8} +(-0.283343 - 2.98659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0871908	−	1.41152i	0.0616532	−	0.998098i
							\(3\)	1.16547	−	1.28128i	0.672886	−	0.739746i
\(5\)	−0.108259	+	0.356883i	−0.0484151	+	0.159603i								−0.977815	−	0.209472i	\(-0.932825\pi\)
							0.929400	+	0.369075i	\(0.120325\pi\)
\(7\)	−0.537756	−	2.70348i	−0.203253	−	1.02182i	−0.938830	−	0.344380i	\(-0.888089\pi\)
							0.735577	−	0.677441i	\(-0.236911\pi\)
\(11\)	−2.68099	+	0.264054i	−0.808349	+	0.0796154i	−0.493744	−	0.869607i	\(-0.664372\pi\)
							−0.314605	+	0.949223i	\(0.601872\pi\)
\(13\)	−0.608731	−	2.00672i	−0.168832	−	0.556564i	−1.00000		0.000542392i	\(-0.999827\pi\)
							0.831168	−	0.556021i	\(-0.187673\pi\)
\(17\)	1.76035	+	0.729163i	0.426949	+	0.176848i	0.585802	−	0.810454i	\(-0.300780\pi\)
							−0.158853	+	0.987302i	\(0.550780\pi\)
\(19\)	0.122089	−	0.228412i	0.0280090	−	0.0524012i	−0.867527	−	0.497389i	\(-0.834292\pi\)
							0.895537	+	0.444988i	\(0.146792\pi\)
\(23\)	−3.72397	−	5.57332i	−0.776502	−	1.16212i	−0.982988	−	0.183667i	\(-0.941203\pi\)
							0.206487	−	0.978449i	\(-0.433797\pi\)
\(29\)	4.75894	+	0.468715i	0.883713	+	0.0870381i	0.529683	−	0.848196i	\(-0.322311\pi\)
							0.354030	+	0.935234i	\(0.384811\pi\)
\(31\)	−0.807701	−	0.807701i	−0.145067	−	0.145067i	0.630843	−	0.775910i	\(-0.282709\pi\)
							−0.775910	+	0.630843i	\(0.782709\pi\)
\(37\)	−0.189028	−	0.353647i	−0.0310761	−	0.0581392i	0.865904	−	0.500210i	\(-0.166744\pi\)
							−0.896980	+	0.442071i	\(0.854244\pi\)
\(41\)	1.36610	+	2.04451i	0.213349	+	0.319300i	0.922673	−	0.385583i	\(-0.126000\pi\)
							−0.709324	+	0.704883i	\(0.751000\pi\)
\(43\)	7.31434	+	6.00273i	1.11543	+	0.915407i	0.996988	−	0.0775541i	\(-0.0247110\pi\)
							0.118439	+	0.992961i	\(0.462211\pi\)
\(47\)	7.57642	+	3.13825i	1.10513	+	0.457761i	0.859259	−	0.511540i	\(-0.170925\pi\)
							0.245875	+	0.969302i	\(0.420925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.131.33	yes	992
3.2	odd	2	inner	384.2.w.a.131.30	✓	992
128.43	odd	32	inner	384.2.w.a.299.30	yes	992
384.299	even	32	inner	384.2.w.a.299.33	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.131.30	✓	992	3.2	odd	2	inner
384.2.w.a.131.33	yes	992	1.1	even	1	trivial
384.2.w.a.299.30	yes	992	128.43	odd	32	inner
384.2.w.a.299.33	yes	992	384.299	even	32	inner