Embedded newform 384.2.w.a.11.9

• Label: 384.2.w.a.11.9
• Level: $384$
• Weight: $2$
• Character: 384.11
• 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			11.9
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27198 + 0.618111i) q^{2} +(-1.15591 + 1.28991i) q^{3} +(1.23588 - 1.57245i) q^{4} +(1.21911 + 0.651630i) q^{5} +(0.672992 - 2.35522i) q^{6} +(1.31145 + 0.260863i) q^{7} +(-0.600063 + 2.76404i) q^{8} +(-0.327733 - 2.98204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27198	+	0.618111i	−0.899427	+	0.437071i
							\(3\)	−1.15591	+	1.28991i	−0.667366	+	0.744730i
\(5\)	1.21911	+	0.651630i	0.545204	+	0.291418i								0.720914	−	0.693024i	\(-0.243722\pi\)
							−0.175710	+	0.984442i	\(0.556222\pi\)
\(7\)	1.31145	+	0.260863i	0.495681	+	0.0985970i	0.436599	−	0.899656i	\(-0.356183\pi\)
							0.0590815	+	0.998253i	\(0.481183\pi\)
\(11\)	1.62501	−	1.33361i	0.489960	−	0.402100i	−0.356712	−	0.934214i	\(-0.616102\pi\)
							0.846672	+	0.532114i	\(0.178602\pi\)
\(13\)	4.29625	−	2.29639i	1.19156	−	0.636904i	0.248215	−	0.968705i	\(-0.420156\pi\)
							0.943350	+	0.331801i	\(0.107656\pi\)
\(17\)	1.96375	−	0.813410i	0.476278	−	0.197281i	−0.131613	−	0.991301i	\(-0.542016\pi\)
							0.607891	+	0.794020i	\(0.292016\pi\)
\(19\)	−0.802644	+	2.64596i	−0.184139	+	0.607025i	0.815463	+	0.578809i	\(0.196483\pi\)
							−0.999602	+	0.0282157i	\(0.991017\pi\)
\(23\)	2.52503	+	1.68717i	0.526504	+	0.351799i	0.790241	−	0.612796i	\(-0.209955\pi\)
							−0.263737	+	0.964595i	\(0.584955\pi\)
\(29\)	1.95084	+	1.60101i	0.362262	+	0.297301i	0.797863	−	0.602839i	\(-0.205964\pi\)
							−0.435601	+	0.900140i	\(0.643464\pi\)
\(31\)	−3.06285	+	3.06285i	−0.550104	+	0.550104i	−0.926471	−	0.376366i	\(-0.877173\pi\)
							0.376366	+	0.926471i	\(0.377173\pi\)
\(37\)	2.30379	+	7.59457i	0.378740	+	1.24854i	0.914644	+	0.404260i	\(0.132471\pi\)
							−0.535904	+	0.844279i	\(0.680029\pi\)
\(41\)	3.25710	+	2.17633i	0.508674	+	0.339885i	0.783277	−	0.621672i	\(-0.213547\pi\)
							−0.274603	+	0.961558i	\(0.588547\pi\)
\(43\)	−0.725241	+	7.36349i	−0.110598	+	1.12292i	0.766379	+	0.642389i	\(0.222057\pi\)
							−0.876977	+	0.480533i	\(0.840443\pi\)
\(47\)	−6.67745	+	2.76589i	−0.974007	+	0.403447i	−0.812202	−	0.583376i	\(-0.801731\pi\)
							−0.161805	+	0.986823i	\(0.551731\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.9	✓	992
3.2	odd	2	inner	384.2.w.a.11.54	yes	992
128.35	odd	32	inner	384.2.w.a.35.54	yes	992
384.35	even	32	inner	384.2.w.a.35.9	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.9	✓	992	1.1	even	1	trivial
384.2.w.a.11.54	yes	992	3.2	odd	2	inner
384.2.w.a.35.9	yes	992	384.35	even	32	inner
384.2.w.a.35.54	yes	992	128.35	odd	32	inner