Embedded newform 384.2.w.a.11.13

• Label: 384.2.w.a.11.13
• 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			11.13
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11732 - 0.866940i) q^{2} +(1.27976 - 1.16714i) q^{3} +(0.496829 + 1.93731i) q^{4} +(-2.95304 - 1.57843i) q^{5} +(-2.44175 + 0.194595i) q^{6} +(-1.29330 - 0.257253i) q^{7} +(1.12441 - 2.59532i) q^{8} +(0.275578 - 2.98732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11732	−	0.866940i	−0.790068	−	0.613019i
							\(3\)	1.27976	−	1.16714i	0.738871	−	0.673847i
\(5\)	−2.95304	−	1.57843i	−1.32064	−	0.705896i								−0.347841	−	0.937554i	\(-0.613085\pi\)
							−0.972797	+	0.231658i	\(0.925585\pi\)
\(7\)	−1.29330	−	0.257253i	−0.488821	−	0.0972326i	−0.0554750	−	0.998460i	\(-0.517667\pi\)
							−0.433346	+	0.901227i	\(0.642667\pi\)
\(11\)	−0.793729	+	0.651397i	−0.239318	+	0.196403i	−0.746406	−	0.665491i	\(-0.768222\pi\)
							0.507087	+	0.861895i	\(0.330722\pi\)
\(13\)	−0.787862	+	0.421121i	−0.218513	+	0.116798i	−0.577022	−	0.816729i	\(-0.695785\pi\)
							0.358509	+	0.933526i	\(0.383285\pi\)
\(17\)	−2.18009	+	0.903022i	−0.528749	+	0.219015i	−0.631055	−	0.775738i	\(-0.717378\pi\)
							0.102306	+	0.994753i	\(0.467378\pi\)
\(19\)	−0.960720	+	3.16707i	−0.220404	+	0.726576i	0.775098	+	0.631841i	\(0.217700\pi\)
							−0.995503	+	0.0947349i	\(0.969800\pi\)
\(23\)	−3.39191	−	2.26640i	−0.707263	−	0.472578i	0.149187	−	0.988809i	\(-0.452335\pi\)
							−0.856449	+	0.516231i	\(0.827335\pi\)
\(29\)	−5.01064	−	4.11212i	−0.930452	−	0.763602i	0.0414128	−	0.999142i	\(-0.486814\pi\)
							−0.971865	+	0.235540i	\(0.924314\pi\)
\(31\)	−0.756547	+	0.756547i	−0.135880	+	0.135880i	−0.771775	−	0.635895i	\(-0.780631\pi\)
							0.635895	+	0.771775i	\(0.280631\pi\)
\(37\)	−0.944277	−	3.11287i	−0.155238	−	0.511752i	0.844478	−	0.535591i	\(-0.179911\pi\)
							−0.999716	+	0.0238390i	\(0.992411\pi\)
\(41\)	9.51538	+	6.35797i	1.48605	+	0.992948i	0.992369	+	0.123307i	\(0.0393499\pi\)
							0.493684	+	0.869642i	\(0.335650\pi\)
\(43\)	1.19133	−	12.0958i	0.181676	−	1.84459i	−0.283016	−	0.959115i	\(-0.591335\pi\)
							0.464692	−	0.885472i	\(-0.346165\pi\)
\(47\)	0.478143	−	0.198053i	0.0697443	−	0.0288890i	−0.347539	−	0.937666i	\(-0.612982\pi\)
							0.417283	+	0.908777i	\(0.362982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.13	✓	992
3.2	odd	2	inner	384.2.w.a.11.50	yes	992
128.35	odd	32	inner	384.2.w.a.35.50	yes	992
384.35	even	32	inner	384.2.w.a.35.13	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.13	✓	992	1.1	even	1	trivial
384.2.w.a.11.50	yes	992	3.2	odd	2	inner
384.2.w.a.35.13	yes	992	384.35	even	32	inner
384.2.w.a.35.50	yes	992	128.35	odd	32	inner