Embedded newform 384.2.w.a.275.10

• Label: 384.2.w.a.275.10
• Level: $384$
• Weight: $2$
• Character: 384.275
• 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			275.10
Character	\(\chi\)	\(=\)	384.275
Dual form			384.2.w.a.155.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22228 - 0.711353i) q^{2} +(0.885840 + 1.48838i) q^{3} +(0.987954 + 1.73895i) q^{4} +(1.29917 + 1.06620i) q^{5} +(-0.0239808 - 2.44937i) q^{6} +(-1.87737 + 2.80968i) q^{7} +(0.0294477 - 2.82827i) q^{8} +(-1.43058 + 2.63694i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22228	−	0.711353i	−0.864285	−	0.503003i
							\(3\)	0.885840	+	1.48838i	0.511440	+	0.859319i
\(5\)	1.29917	+	1.06620i	0.581005	+	0.476818i								0.878393	−	0.477938i	\(-0.158616\pi\)
							−0.297389	+	0.954756i	\(0.596116\pi\)
\(7\)	−1.87737	+	2.80968i	−0.709578	+	1.06196i	0.285057	+	0.958510i	\(0.407987\pi\)
							−0.994635	+	0.103448i	\(0.967013\pi\)
\(11\)	1.89823	+	0.575822i	0.572338	+	0.173617i	0.563170	−	0.826341i	\(-0.309582\pi\)
							0.00916808	+	0.999958i	\(0.497082\pi\)
\(13\)	−1.57868	+	1.29559i	−0.437846	+	0.359331i	−0.827343	−	0.561696i	\(-0.810149\pi\)
							0.389497	+	0.921028i	\(0.372649\pi\)
\(17\)	1.97199	−	4.76080i	0.478278	−	1.15466i	−0.482138	−	0.876095i	\(-0.660140\pi\)
							0.960416	−	0.278569i	\(-0.0898602\pi\)
\(19\)	−6.12255	−	0.603019i	−1.40461	−	0.138342i	−0.632859	−	0.774267i	\(-0.718119\pi\)
							−0.771751	+	0.635925i	\(0.780619\pi\)
\(23\)	0.931034	+	0.185194i	0.194134	+	0.0386157i	0.291200	−	0.956662i	\(-0.405946\pi\)
							−0.0970655	+	0.995278i	\(0.530946\pi\)
\(29\)	−6.17442	+	1.87299i	−1.14656	+	0.347806i	−0.805709	−	0.592311i	\(-0.798216\pi\)
							−0.340852	+	0.940117i	\(0.610716\pi\)
\(31\)	6.66710	+	6.66710i	1.19745	+	1.19745i	0.974929	+	0.222517i	\(0.0714273\pi\)
							0.222517	+	0.974929i	\(0.428573\pi\)
\(37\)	10.3399	−	1.01839i	1.69987	−	0.167422i	0.798945	−	0.601404i	\(-0.205392\pi\)
							0.900922	+	0.433981i	\(0.142892\pi\)
\(41\)	0.0648898	+	0.0129074i	0.0101341	+	0.00201579i	0.200155	−	0.979764i	\(-0.435856\pi\)
							−0.190021	+	0.981780i	\(0.560856\pi\)
\(43\)	3.83013	+	7.16566i	0.584089	+	1.09275i	0.984329	+	0.176343i	\(0.0564267\pi\)
							−0.400240	+	0.916410i	\(0.631073\pi\)
\(47\)	−0.424005	+	1.02364i	−0.0618476	+	0.149313i	−0.951782	−	0.306775i	\(-0.900750\pi\)
							0.889934	+	0.456089i	\(0.150750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.275.10	yes	992
3.2	odd	2	inner	384.2.w.a.275.53	yes	992
128.27	odd	32	inner	384.2.w.a.155.53	yes	992
384.155	even	32	inner	384.2.w.a.155.10	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.10	✓	992	384.155	even	32	inner
384.2.w.a.155.53	yes	992	128.27	odd	32	inner
384.2.w.a.275.10	yes	992	1.1	even	1	trivial
384.2.w.a.275.53	yes	992	3.2	odd	2	inner