Embedded newform 384.2.w.a.251.42

• Label: 384.2.w.a.251.42
• Level: $384$
• Weight: $2$
• Character: 384.251
• 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			251.42
Character	\(\chi\)	\(=\)	384.251
Dual form			384.2.w.a.179.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.726644 + 1.21326i) q^{2} +(-0.594224 + 1.62693i) q^{3} +(-0.943978 + 1.76321i) q^{4} +(-0.0821559 + 0.834143i) q^{5} +(-2.40567 + 0.461252i) q^{6} +(0.978435 - 0.653770i) q^{7} +(-2.82516 + 0.135938i) q^{8} +(-2.29380 - 1.93352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.726644	+	1.21326i	0.513815	+	0.857901i
							\(3\)	−0.594224	+	1.62693i	−0.343075	+	0.939308i
\(5\)	−0.0821559	+	0.834143i	−0.0367412	+	0.373040i								0.959179	+	0.282800i	\(0.0912634\pi\)
							−0.995920	+	0.0902398i	\(0.971237\pi\)
\(7\)	0.978435	−	0.653770i	0.369814	−	0.247102i	−0.356751	−	0.934199i	\(-0.616116\pi\)
							0.726565	+	0.687098i	\(0.241116\pi\)
\(11\)	−1.45799	+	2.72770i	−0.439599	+	0.822433i	−0.999983	−	0.00580043i	\(-0.998154\pi\)
							0.560384	+	0.828233i	\(0.310654\pi\)
\(13\)	0.457931	+	4.64945i	0.127007	+	1.28953i	0.821576	+	0.570099i	\(0.193095\pi\)
							−0.694569	+	0.719426i	\(0.744405\pi\)
\(17\)	−2.49467	−	6.02266i	−0.605046	−	1.46071i	−0.868328	−	0.495991i	\(-0.834805\pi\)
							0.263282	−	0.964719i	\(-0.415195\pi\)
\(19\)	3.16631	+	2.59852i	0.726402	+	0.596142i	0.923208	−	0.384300i	\(-0.125557\pi\)
							−0.196807	+	0.980442i	\(0.563057\pi\)
\(23\)	0.368921	+	1.85469i	0.0769253	+	0.386730i	0.999998	+	0.00210361i	\(0.000669600\pi\)
							−0.923072	+	0.384626i	\(0.874330\pi\)
\(29\)	−3.90338	−	7.30270i	−0.724839	−	1.35608i	−0.926377	−	0.376597i	\(-0.877094\pi\)
							0.201538	−	0.979481i	\(-0.435406\pi\)
\(31\)	0.0671033	−	0.0671033i	0.0120521	−	0.0120521i	−0.701055	−	0.713107i	\(-0.747287\pi\)
							0.713107	+	0.701055i	\(0.247287\pi\)
\(37\)	−0.480316	+	0.394185i	−0.0789635	+	0.0648037i	−0.673047	−	0.739600i	\(-0.735015\pi\)
							0.594084	+	0.804403i	\(0.297515\pi\)
\(41\)	1.26074	+	6.33816i	0.196894	+	0.989854i	0.945198	+	0.326498i	\(0.105868\pi\)
							−0.748304	+	0.663356i	\(0.769132\pi\)
\(43\)	3.03253	+	9.99691i	0.462456	+	1.52451i	0.809760	+	0.586762i	\(0.199597\pi\)
							−0.347303	+	0.937753i	\(0.612903\pi\)
\(47\)	4.56889	+	11.0303i	0.666441	+	1.60893i	0.787521	+	0.616288i	\(0.211364\pi\)
							−0.121080	+	0.992643i	\(0.538636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.251.42	yes	992
3.2	odd	2	inner	384.2.w.a.251.21	yes	992
128.51	odd	32	inner	384.2.w.a.179.21	✓	992
384.179	even	32	inner	384.2.w.a.179.42	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.179.21	✓	992	128.51	odd	32	inner
384.2.w.a.179.42	yes	992	384.179	even	32	inner
384.2.w.a.251.21	yes	992	3.2	odd	2	inner
384.2.w.a.251.42	yes	992	1.1	even	1	trivial