Embedded newform 384.2.w.a.11.18

• Label: 384.2.w.a.11.18
• 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			11.18
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.917171 + 1.07647i) q^{2} +(-0.843817 - 1.51260i) q^{3} +(-0.317594 - 1.97462i) q^{4} +(-3.45356 - 1.84597i) q^{5} +(2.40220 + 0.478969i) q^{6} +(-1.46677 - 0.291758i) q^{7} +(2.41692 + 1.46919i) q^{8} +(-1.57594 + 2.55272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.917171	+	1.07647i	−0.648538	+	0.761182i
							\(3\)	−0.843817	−	1.51260i	−0.487178	−	0.873303i
\(5\)	−3.45356	−	1.84597i	−1.54448	−	0.825542i								−0.544567	−	0.838718i	\(-0.683306\pi\)
							−0.999913	+	0.0131759i	\(0.995806\pi\)
\(7\)	−1.46677	−	0.291758i	−0.554386	−	0.110274i	−0.0900563	−	0.995937i	\(-0.528705\pi\)
							−0.464330	+	0.885662i	\(0.653705\pi\)
\(11\)	0.819471	−	0.672522i	0.247080	−	0.202773i	−0.502699	−	0.864461i	\(-0.667660\pi\)
							0.749779	+	0.661688i	\(0.230160\pi\)
\(13\)	2.93804	−	1.57041i	0.814865	−	0.435554i	−0.0106442	−	0.999943i	\(-0.503388\pi\)
							0.825509	+	0.564389i	\(0.190888\pi\)
\(17\)	−2.84513	+	1.17849i	−0.690045	+	0.285826i	−0.700019	−	0.714124i	\(-0.746825\pi\)
							0.00997400	+	0.999950i	\(0.496825\pi\)
\(19\)	−1.56815	+	5.16949i	−0.359758	+	1.18596i	0.571193	+	0.820816i	\(0.306481\pi\)
							−0.930950	+	0.365146i	\(0.881019\pi\)
\(23\)	4.50863	+	3.01257i	0.940115	+	0.628165i	0.928327	−	0.371766i	\(-0.121247\pi\)
							0.0117886	+	0.999931i	\(0.496247\pi\)
\(29\)	−4.19303	−	3.44113i	−0.778626	−	0.639002i	0.158751	−	0.987319i	\(-0.449253\pi\)
							−0.937377	+	0.348317i	\(0.886753\pi\)
\(31\)	−2.35727	+	2.35727i	−0.423379	+	0.423379i	−0.886365	−	0.462987i	\(-0.846778\pi\)
							0.462987	+	0.886365i	\(0.346778\pi\)
\(37\)	−0.347870	−	1.14677i	−0.0571895	−	0.188528i	0.923692	−	0.383137i	\(-0.125156\pi\)
							−0.980881	+	0.194609i	\(0.937656\pi\)
\(41\)	2.55512	+	1.70728i	0.399043	+	0.266632i	0.738859	−	0.673860i	\(-0.235365\pi\)
							−0.339816	+	0.940492i	\(0.610365\pi\)
\(43\)	−0.727253	+	7.38393i	−0.110905	+	1.12604i	0.765158	+	0.643842i	\(0.222661\pi\)
							−0.876063	+	0.482196i	\(0.839839\pi\)
\(47\)	−9.75123	+	4.03909i	−1.42236	+	0.589162i	−0.955453	−	0.295143i	\(-0.904633\pi\)
							−0.466909	+	0.884305i	\(0.654633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.18	✓	992
3.2	odd	2	inner	384.2.w.a.11.45	yes	992
128.35	odd	32	inner	384.2.w.a.35.45	yes	992
384.35	even	32	inner	384.2.w.a.35.18	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.18	✓	992	1.1	even	1	trivial
384.2.w.a.11.45	yes	992	3.2	odd	2	inner
384.2.w.a.35.18	yes	992	384.35	even	32	inner
384.2.w.a.35.45	yes	992	128.35	odd	32	inner