Embedded newform 384.2.w.a.11.40

• Label: 384.2.w.a.11.40
• 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			11.40
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.651583 - 1.25517i) q^{2} +(-1.52646 + 0.818494i) q^{3} +(-1.15088 - 1.63569i) q^{4} +(-0.995339 - 0.532020i) q^{5} +(0.0327322 + 2.44927i) q^{6} +(0.237974 + 0.0473360i) q^{7} +(-2.80295 + 0.378758i) q^{8} +(1.66014 - 2.49879i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.651583	−	1.25517i	0.460739	−	0.887536i
							\(3\)	−1.52646	+	0.818494i	−0.881300	+	0.472558i
\(5\)	−0.995339	−	0.532020i	−0.445129	−	0.237927i								0.233574	−	0.972339i	\(-0.424958\pi\)
							−0.678703	+	0.734412i	\(0.737458\pi\)
\(7\)	0.237974	+	0.0473360i	0.0899457	+	0.0178913i	0.239858	−	0.970808i	\(-0.422899\pi\)
							−0.149912	+	0.988699i	\(0.547899\pi\)
\(11\)	−3.62046	+	2.97124i	−1.09161	+	0.895862i	−0.995041	−	0.0994645i	\(-0.968287\pi\)
							−0.0965696	+	0.995326i	\(0.530787\pi\)
\(13\)	−2.36270	+	1.26289i	−0.655296	+	0.350263i	−0.765305	−	0.643667i	\(-0.777412\pi\)
							0.110009	+	0.993931i	\(0.464912\pi\)
\(17\)	−1.41189	+	0.584826i	−0.342435	+	0.141841i	−0.547272	−	0.836955i	\(-0.684334\pi\)
							0.204837	+	0.978796i	\(0.434334\pi\)
\(19\)	−1.07254	+	3.53568i	−0.246057	+	0.811142i	0.743717	+	0.668495i	\(0.233061\pi\)
							−0.989774	+	0.142647i	\(0.954439\pi\)
\(23\)	−2.30884	−	1.54272i	−0.481427	−	0.321679i	0.291065	−	0.956703i	\(-0.405990\pi\)
							−0.772493	+	0.635024i	\(0.780990\pi\)
\(29\)	−0.329207	−	0.270173i	−0.0611321	−	0.0501698i	0.603331	−	0.797491i	\(-0.293840\pi\)
							−0.664463	+	0.747321i	\(0.731340\pi\)
\(31\)	−2.76795	+	2.76795i	−0.497139	+	0.497139i	−0.910546	−	0.413407i	\(-0.864339\pi\)
							0.413407	+	0.910546i	\(0.364339\pi\)
\(37\)	−0.220229	−	0.725997i	−0.0362054	−	0.119353i	0.936975	−	0.349396i	\(-0.113613\pi\)
							−0.973180	+	0.230043i	\(0.926113\pi\)
\(41\)	−0.703789	−	0.470257i	−0.109913	−	0.0734418i	0.499399	−	0.866372i	\(-0.333554\pi\)
							−0.609312	+	0.792931i	\(0.708554\pi\)
\(43\)	0.314033	−	3.18843i	0.0478896	−	0.486232i	−0.941087	−	0.338163i	\(-0.890194\pi\)
							0.988977	−	0.148068i	\(-0.0473056\pi\)
\(47\)	6.23714	−	2.58351i	0.909781	−	0.376843i	0.121808	−	0.992554i	\(-0.461131\pi\)
							0.787973	+	0.615710i	\(0.211131\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.40	yes	992
3.2	odd	2	inner	384.2.w.a.11.23	✓	992
128.35	odd	32	inner	384.2.w.a.35.23	yes	992
384.35	even	32	inner	384.2.w.a.35.40	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.23	✓	992	3.2	odd	2	inner
384.2.w.a.11.40	yes	992	1.1	even	1	trivial
384.2.w.a.35.23	yes	992	128.35	odd	32	inner
384.2.w.a.35.40	yes	992	384.35	even	32	inner