Embedded newform 384.2.w.a.11.60

• Label: 384.2.w.a.11.60
• 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			11.60
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.60

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41282 + 0.0626704i) q^{2} +(-1.72977 + 0.0889062i) q^{3} +(1.99214 + 0.177085i) q^{4} +(-1.34559 - 0.719232i) q^{5} +(-2.44943 + 0.0172036i) q^{6} +(2.49948 + 0.497176i) q^{7} +(2.80345 + 0.375038i) q^{8} +(2.98419 - 0.307574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41282	+	0.0626704i	0.999018	+	0.0443147i
							\(3\)	−1.72977	+	0.0889062i	−0.998682	+	0.0513300i
\(5\)	−1.34559	−	0.719232i	−0.601766	−	0.321651i								0.142231	−	0.989833i	\(-0.454572\pi\)
							−0.743997	+	0.668183i	\(0.767072\pi\)
\(7\)	2.49948	+	0.497176i	0.944713	+	0.187915i	0.643331	−	0.765588i	\(-0.277552\pi\)
							0.301382	+	0.953503i	\(0.402552\pi\)
\(11\)	0.433718	−	0.355943i	0.130771	−	0.107321i	−0.566734	−	0.823901i	\(-0.691793\pi\)
							0.697505	+	0.716580i	\(0.254293\pi\)
\(13\)	0.780211	−	0.417031i	0.216392	−	0.115664i	−0.359640	−	0.933091i	\(-0.617101\pi\)
							0.576032	+	0.817427i	\(0.304601\pi\)
\(17\)	4.49075	−	1.86013i	1.08917	−	0.451147i	0.235450	−	0.971886i	\(-0.424344\pi\)
							0.853716	+	0.520739i	\(0.174344\pi\)
\(19\)	−0.781779	+	2.57718i	−0.179352	+	0.591245i	0.820457	+	0.571708i	\(0.193719\pi\)
							−0.999809	+	0.0195368i	\(0.993781\pi\)
\(23\)	2.97104	+	1.98519i	0.619505	+	0.413940i	0.825332	−	0.564648i	\(-0.190988\pi\)
							−0.205826	+	0.978589i	\(0.565988\pi\)
\(29\)	0.665484	+	0.546148i	0.123577	+	0.101417i	0.694152	−	0.719828i	\(-0.255780\pi\)
							−0.570575	+	0.821246i	\(0.693280\pi\)
\(31\)	−0.843718	+	0.843718i	−0.151536	+	0.151536i	−0.778804	−	0.627268i	\(-0.784173\pi\)
							0.627268	+	0.778804i	\(0.284173\pi\)
\(37\)	−1.46960	−	4.84463i	−0.241601	−	0.796453i	−0.990945	−	0.134271i	\(-0.957131\pi\)
							0.749343	−	0.662182i	\(-0.230369\pi\)
\(41\)	−5.55397	−	3.71104i	−0.867384	−	0.579568i	0.0403158	−	0.999187i	\(-0.487164\pi\)
							−0.907700	+	0.419619i	\(0.862164\pi\)
\(43\)	−0.716820	+	7.27799i	−0.109314	+	1.10988i	0.771438	+	0.636305i	\(0.219538\pi\)
							−0.880752	+	0.473579i	\(0.842962\pi\)
\(47\)	−10.0843	+	4.17707i	−1.47095	+	0.609288i	−0.967076	−	0.254488i	\(-0.918093\pi\)
							−0.503876	+	0.863776i	\(0.668093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.60	yes	992
3.2	odd	2	inner	384.2.w.a.11.3	✓	992
128.35	odd	32	inner	384.2.w.a.35.3	yes	992
384.35	even	32	inner	384.2.w.a.35.60	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.3	✓	992	3.2	odd	2	inner
384.2.w.a.11.60	yes	992	1.1	even	1	trivial
384.2.w.a.35.3	yes	992	128.35	odd	32	inner
384.2.w.a.35.60	yes	992	384.35	even	32	inner