Embedded newform 384.2.w.a.107.9

• Label: 384.2.w.a.107.9
• Level: $384$
• Weight: $2$
• Character: 384.107
• 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			107.9
Character	\(\chi\)	\(=\)	384.107
Dual form			384.2.w.a.323.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26399 - 0.634293i) q^{2} +(1.25131 + 1.19759i) q^{3} +(1.19534 + 1.60348i) q^{4} +(0.160457 - 0.0486740i) q^{5} +(-0.822018 - 2.30744i) q^{6} +(-0.313859 + 1.57788i) q^{7} +(-0.493828 - 2.78498i) q^{8} +(0.131546 + 2.99711i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26399	−	0.634293i	−0.893776	−	0.448513i
							\(3\)	1.25131	+	1.19759i	0.722443	+	0.691430i
\(5\)	0.160457	−	0.0486740i	0.0717585	−	0.0217677i								−0.254201	−	0.967151i	\(-0.581813\pi\)
							0.325960	+	0.945384i	\(0.394313\pi\)
\(7\)	−0.313859	+	1.57788i	−0.118628	+	0.596381i	0.875043	+	0.484046i	\(0.160833\pi\)
							−0.993670	+	0.112336i	\(0.964167\pi\)
\(11\)	0.127776	−	1.29733i	0.0385259	−	0.391160i	−0.956514	−	0.291686i	\(-0.905784\pi\)
							0.995040	−	0.0994745i	\(-0.0317162\pi\)
\(13\)	5.48845	+	1.66490i	1.52222	+	0.461761i	0.937064	−	0.349158i	\(-0.113532\pi\)
							0.585158	+	0.810919i	\(0.301032\pi\)
\(17\)	−3.05865	+	1.26694i	−0.741832	+	0.307277i	−0.721404	−	0.692514i	\(-0.756503\pi\)
							−0.0204280	+	0.999791i	\(0.506503\pi\)
\(19\)	0.517836	−	0.276789i	0.118800	−	0.0634998i	−0.410926	−	0.911669i	\(-0.634795\pi\)
							0.529726	+	0.848169i	\(0.322295\pi\)
\(23\)	0.750420	−	1.12308i	0.156473	−	0.234179i	−0.744947	−	0.667123i	\(-0.767525\pi\)
							0.901421	+	0.432944i	\(0.142525\pi\)
\(29\)	0.813539	+	8.26000i	0.151070	+	1.53384i	0.709746	+	0.704458i	\(0.248810\pi\)
							−0.558676	+	0.829386i	\(0.688690\pi\)
\(31\)	0.245451	−	0.245451i	0.0440844	−	0.0440844i	−0.684721	−	0.728805i	\(-0.740076\pi\)
							0.728805	+	0.684721i	\(0.240076\pi\)
\(37\)	1.71818	+	0.918386i	0.282467	+	0.150982i	0.606546	−	0.795048i	\(-0.292554\pi\)
							−0.324079	+	0.946030i	\(0.605054\pi\)
\(41\)	3.17024	−	4.74459i	0.495108	−	0.740981i	−0.496811	−	0.867859i	\(-0.665496\pi\)
							0.991918	+	0.126878i	\(0.0404956\pi\)
\(43\)	−3.82631	−	4.66238i	−0.583507	−	0.711006i	0.394659	−	0.918827i	\(-0.370863\pi\)
							−0.978167	+	0.207822i	\(0.933363\pi\)
\(47\)	8.52881	−	3.53275i	1.24405	−	0.515304i	0.339075	−	0.940759i	\(-0.389886\pi\)
							0.904980	+	0.425455i	\(0.139886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.107.9	✓	992
3.2	odd	2	inner	384.2.w.a.107.54	yes	992
128.67	odd	32	inner	384.2.w.a.323.54	yes	992
384.323	even	32	inner	384.2.w.a.323.9	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.107.9	✓	992	1.1	even	1	trivial
384.2.w.a.107.54	yes	992	3.2	odd	2	inner
384.2.w.a.323.9	yes	992	384.323	even	32	inner
384.2.w.a.323.54	yes	992	128.67	odd	32	inner