Embedded newform 384.2.w.a.11.11

• Label: 384.2.w.a.11.11
• 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			11.11
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15643 - 0.814046i) q^{2} +(-0.824531 - 1.52320i) q^{3} +(0.674657 + 1.88277i) q^{4} +(-0.611958 - 0.327098i) q^{5} +(-0.286446 + 2.43268i) q^{6} +(2.75895 + 0.548789i) q^{7} +(0.752472 - 2.72650i) q^{8} +(-1.64030 + 2.51186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15643	−	0.814046i	−0.817719	−	0.575618i
							\(3\)	−0.824531	−	1.52320i	−0.476043	−	0.879422i
\(5\)	−0.611958	−	0.327098i	−0.273676	−	0.146283i								0.328850	−	0.944382i	\(-0.393339\pi\)
							−0.602526	+	0.798099i	\(0.705839\pi\)
\(7\)	2.75895	+	0.548789i	1.04278	+	0.207423i	0.686632	−	0.727005i	\(-0.259088\pi\)
							0.356152	+	0.934428i	\(0.384088\pi\)
\(11\)	2.48753	−	2.04146i	0.750017	−	0.615523i	−0.179732	−	0.983716i	\(-0.557523\pi\)
							0.929750	+	0.368192i	\(0.120023\pi\)
\(13\)	3.00482	−	1.60611i	0.833386	−	0.445454i	0.00123256	−	0.999999i	\(-0.499608\pi\)
							0.832154	+	0.554545i	\(0.187108\pi\)
\(17\)	−0.432628	+	0.179200i	−0.104928	+	0.0434624i	−0.434530	−	0.900657i	\(-0.643085\pi\)
							0.329602	+	0.944120i	\(0.393085\pi\)
\(19\)	1.08049	−	3.56189i	0.247881	−	0.817154i	−0.741392	−	0.671073i	\(-0.765834\pi\)
							0.989273	−	0.146081i	\(-0.0466662\pi\)
\(23\)	−0.151876	−	0.101480i	−0.0316683	−	0.0211601i	0.539635	−	0.841899i	\(-0.318562\pi\)
							−0.571304	+	0.820739i	\(0.693562\pi\)
\(29\)	0.581600	+	0.477307i	0.108000	+	0.0886337i	0.686844	−	0.726805i	\(-0.258996\pi\)
							−0.578843	+	0.815439i	\(0.696496\pi\)
\(31\)	−3.42580	+	3.42580i	−0.615293	+	0.615293i	−0.944320	−	0.329028i	\(-0.893279\pi\)
							0.329028	+	0.944320i	\(0.393279\pi\)
\(37\)	−2.06718	−	6.81459i	−0.339843	−	1.12031i	−0.945782	−	0.324801i	\(-0.894703\pi\)
							0.605939	−	0.795511i	\(-0.292797\pi\)
\(41\)	−9.94330	−	6.64390i	−1.55288	−	1.03760i	−0.975196	−	0.221343i	\(-0.928956\pi\)
							−0.577686	−	0.816259i	\(-0.696044\pi\)
\(43\)	−0.410961	+	4.17256i	−0.0626710	+	0.636310i	0.911623	+	0.411028i	\(0.134830\pi\)
							−0.974294	+	0.225281i	\(0.927670\pi\)
\(47\)	11.3232	−	4.69022i	1.65166	−	0.684138i	0.654261	−	0.756269i	\(-0.272980\pi\)
							0.997395	+	0.0721308i	\(0.0229799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.11	✓	992
3.2	odd	2	inner	384.2.w.a.11.52	yes	992
128.35	odd	32	inner	384.2.w.a.35.52	yes	992
384.35	even	32	inner	384.2.w.a.35.11	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.11	✓	992	1.1	even	1	trivial
384.2.w.a.11.52	yes	992	3.2	odd	2	inner
384.2.w.a.35.11	yes	992	384.35	even	32	inner
384.2.w.a.35.52	yes	992	128.35	odd	32	inner