Embedded newform 384.2.w.a.11.36

• Label: 384.2.w.a.11.36
• 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			11.36
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.243011 + 1.39318i) q^{2} +(-0.729538 + 1.57092i) q^{3} +(-1.88189 + 0.677116i) q^{4} +(1.50312 + 0.803437i) q^{5} +(-2.36585 - 0.634626i) q^{6} +(-3.65245 - 0.726518i) q^{7} +(-1.40066 - 2.45726i) q^{8} +(-1.93555 - 2.29208i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.243011	+	1.39318i	0.171835	+	0.985126i
							\(3\)	−0.729538	+	1.57092i	−0.421199	+	0.906968i
\(5\)	1.50312	+	0.803437i	0.672218	+	0.359308i								0.771929	−	0.635709i	\(-0.219292\pi\)
							−0.0997115	+	0.995016i	\(0.531792\pi\)
\(7\)	−3.65245	−	0.726518i	−1.38050	−	0.274598i	−0.551652	−	0.834074i	\(-0.686002\pi\)
							−0.828846	+	0.559476i	\(0.811002\pi\)
\(11\)	−0.268604	+	0.220438i	−0.0809873	+	0.0664645i	−0.674016	−	0.738717i	\(-0.735432\pi\)
							0.593029	+	0.805181i	\(0.297932\pi\)
\(13\)	−3.02112	+	1.61482i	−0.837909	+	0.447872i	−0.833765	−	0.552120i	\(-0.813819\pi\)
							−0.00414434	+	0.999991i	\(0.501319\pi\)
\(17\)	1.39276	−	0.576900i	0.337794	−	0.139919i	−0.207338	−	0.978269i	\(-0.566480\pi\)
							0.545131	+	0.838351i	\(0.316480\pi\)
\(19\)	−1.97368	+	6.50636i	−0.452794	+	1.49266i	0.372541	+	0.928016i	\(0.378487\pi\)
							−0.825334	+	0.564644i	\(0.809013\pi\)
\(23\)	−0.914344	−	0.610945i	−0.190654	−	0.127391i	0.456580	−	0.889682i	\(-0.349074\pi\)
							−0.647234	+	0.762292i	\(0.724074\pi\)
\(29\)	4.31999	+	3.54533i	0.802203	+	0.658351i	0.943410	−	0.331629i	\(-0.107598\pi\)
							−0.141207	+	0.989980i	\(0.545098\pi\)
\(31\)	−3.43730	+	3.43730i	−0.617357	+	0.617357i	−0.944853	−	0.327496i	\(-0.893795\pi\)
							0.327496	+	0.944853i	\(0.393795\pi\)
\(37\)	−0.563982	−	1.85920i	−0.0927181	−	0.305651i	0.898299	−	0.439384i	\(-0.144803\pi\)
							−0.991017	+	0.133734i	\(0.957303\pi\)
\(41\)	2.77860	+	1.85660i	0.433944	+	0.289952i	0.753293	−	0.657685i	\(-0.228464\pi\)
							−0.319349	+	0.947637i	\(0.603464\pi\)
\(43\)	0.0702048	−	0.712801i	0.0107061	−	0.108701i	−0.988346	−	0.152221i	\(-0.951357\pi\)
							0.999053	+	0.0435203i	\(0.0138573\pi\)
\(47\)	−10.4552	+	4.33067i	−1.52504	+	0.631693i	−0.978594	−	0.205801i	\(-0.934020\pi\)
							−0.546447	+	0.837494i	\(0.684020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.36	yes	992
3.2	odd	2	inner	384.2.w.a.11.27	✓	992
128.35	odd	32	inner	384.2.w.a.35.27	yes	992
384.35	even	32	inner	384.2.w.a.35.36	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.27	✓	992	3.2	odd	2	inner
384.2.w.a.11.36	yes	992	1.1	even	1	trivial
384.2.w.a.35.27	yes	992	128.35	odd	32	inner
384.2.w.a.35.36	yes	992	384.35	even	32	inner