Embedded newform 384.2.w.a.11.15

• Label: 384.2.w.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10845 + 0.878258i) q^{2} +(-0.117171 - 1.72808i) q^{3} +(0.457325 - 1.94701i) q^{4} +(3.15748 + 1.68771i) q^{5} +(1.64758 + 1.81259i) q^{6} +(4.43843 + 0.882860i) q^{7} +(1.20306 + 2.55982i) q^{8} +(-2.97254 + 0.404962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10845	+	0.878258i	−0.783793	+	0.621022i
							\(3\)	−0.117171	−	1.72808i	−0.0676486	−	0.997709i
\(5\)	3.15748	+	1.68771i	1.41207	+	0.754767i								0.988518	−	0.151105i	\(-0.0482832\pi\)
							0.423552	+	0.905872i	\(0.360783\pi\)
\(7\)	4.43843	+	0.882860i	1.67757	+	0.333690i	0.939894	−	0.341465i	\(-0.110923\pi\)
							0.737676	+	0.675155i	\(0.235923\pi\)
\(11\)	0.254277	−	0.208679i	0.0766673	−	0.0629192i	−0.595279	−	0.803519i	\(-0.702959\pi\)
							0.671946	+	0.740600i	\(0.265459\pi\)
\(13\)	−0.958114	+	0.512123i	−0.265733	+	0.142037i	−0.598883	−	0.800837i	\(-0.704389\pi\)
							0.333150	+	0.942874i	\(0.391889\pi\)
\(17\)	−3.78622	+	1.56830i	−0.918293	+	0.380369i	−0.791225	−	0.611525i	\(-0.790556\pi\)
							−0.127067	+	0.991894i	\(0.540556\pi\)
\(19\)	−0.753264	+	2.48318i	−0.172811	+	0.569680i	0.827160	+	0.561967i	\(0.189955\pi\)
							−0.999970	+	0.00771328i	\(0.997545\pi\)
\(23\)	3.81700	+	2.55044i	0.795900	+	0.531803i	0.885756	−	0.464151i	\(-0.153641\pi\)
							−0.0898560	+	0.995955i	\(0.528641\pi\)
\(29\)	−0.997059	−	0.818265i	−0.185149	−	0.151948i	0.537253	−	0.843421i	\(-0.319462\pi\)
							−0.722402	+	0.691473i	\(0.756962\pi\)
\(31\)	4.20593	−	4.20593i	0.755408	−	0.755408i	−0.220075	−	0.975483i	\(-0.570630\pi\)
							0.975483	+	0.220075i	\(0.0706302\pi\)
\(37\)	−0.965656	−	3.18334i	−0.158753	−	0.523338i	0.841093	−	0.540890i	\(-0.181913\pi\)
							−0.999846	+	0.0175522i	\(0.994413\pi\)
\(41\)	−7.88010	−	5.26531i	−1.23066	−	0.822304i	−0.241685	−	0.970355i	\(-0.577700\pi\)
							−0.988980	+	0.148051i	\(0.952700\pi\)
\(43\)	1.17417	−	11.9215i	0.179059	−	1.81801i	−0.315562	−	0.948905i	\(-0.602193\pi\)
							0.494621	−	0.869109i	\(-0.335307\pi\)
\(47\)	−0.396717	+	0.164326i	−0.0578671	+	0.0239693i	−0.411429	−	0.911442i	\(-0.634970\pi\)
							0.353562	+	0.935411i	\(0.384970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.15	✓	992
3.2	odd	2	inner	384.2.w.a.11.48	yes	992
128.35	odd	32	inner	384.2.w.a.35.48	yes	992
384.35	even	32	inner	384.2.w.a.35.15	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.15	✓	992	1.1	even	1	trivial
384.2.w.a.11.48	yes	992	3.2	odd	2	inner
384.2.w.a.35.15	yes	992	384.35	even	32	inner
384.2.w.a.35.48	yes	992	128.35	odd	32	inner