Embedded newform 384.2.w.a.11.16

• Label: 384.2.w.a.11.16
• 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			11.16
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07112 + 0.923423i) q^{2} +(1.49221 - 0.879386i) q^{3} +(0.294582 - 1.97819i) q^{4} +(-1.88241 - 1.00617i) q^{5} +(-0.786282 + 2.31986i) q^{6} +(2.66028 + 0.529163i) q^{7} +(1.51117 + 2.39089i) q^{8} +(1.45336 - 2.62445i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07112	+	0.923423i	−0.757394	+	0.652958i
							\(3\)	1.49221	−	0.879386i	0.861526	−	0.507714i
\(5\)	−1.88241	−	1.00617i	−0.841839	−	0.449973i								−0.00667811	−	0.999978i	\(-0.502126\pi\)
							−0.835161	+	0.550005i	\(0.814626\pi\)
\(7\)	2.66028	+	0.529163i	1.00549	+	0.200005i	0.670256	−	0.742130i	\(-0.266184\pi\)
							0.335235	+	0.942135i	\(0.391184\pi\)
\(11\)	−2.44223	+	2.00429i	−0.736360	+	0.604315i	−0.925993	−	0.377540i	\(-0.876770\pi\)
							0.189633	+	0.981855i	\(0.439270\pi\)
\(13\)	0.359681	−	0.192253i	0.0997575	−	0.0533215i	−0.420763	−	0.907170i	\(-0.638238\pi\)
							0.520521	+	0.853849i	\(0.325738\pi\)
\(17\)	7.42890	−	3.07715i	1.80177	−	0.746319i	0.816042	−	0.577993i	\(-0.196164\pi\)
							0.985731	−	0.168326i	\(-0.0538362\pi\)
\(19\)	2.07770	−	6.84926i	0.476657	−	1.57133i	−0.307993	−	0.951389i	\(-0.599657\pi\)
							0.784650	−	0.619939i	\(-0.212843\pi\)
\(23\)	0.519431	+	0.347073i	0.108309	+	0.0723697i	0.608545	−	0.793519i	\(-0.291753\pi\)
							−0.500236	+	0.865889i	\(0.666753\pi\)
\(29\)	−5.63366	−	4.62343i	−1.04614	−	0.858549i	−0.0559768	−	0.998432i	\(-0.517827\pi\)
							−0.990168	+	0.139883i	\(0.955327\pi\)
\(31\)	0.985204	−	0.985204i	0.176948	−	0.176948i	−0.613076	−	0.790024i	\(-0.710068\pi\)
							0.790024	+	0.613076i	\(0.210068\pi\)
\(37\)	2.05688	+	6.78064i	0.338150	+	1.11473i	0.946943	+	0.321401i	\(0.104154\pi\)
							−0.608793	+	0.793329i	\(0.708346\pi\)
\(41\)	−0.894207	−	0.597490i	−0.139652	−	0.0933122i	0.483782	−	0.875189i	\(-0.339263\pi\)
							−0.623434	+	0.781876i	\(0.714263\pi\)
\(43\)	−0.827086	+	8.39754i	−0.126129	+	1.28061i	0.698775	+	0.715341i	\(0.253729\pi\)
							−0.824905	+	0.565272i	\(0.808771\pi\)
\(47\)	6.66824	−	2.76208i	0.972663	−	0.402890i	0.160960	−	0.986961i	\(-0.448541\pi\)
							0.811703	+	0.584071i	\(0.198541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.16	✓	992
3.2	odd	2	inner	384.2.w.a.11.47	yes	992
128.35	odd	32	inner	384.2.w.a.35.47	yes	992
384.35	even	32	inner	384.2.w.a.35.16	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.16	✓	992	1.1	even	1	trivial
384.2.w.a.11.47	yes	992	3.2	odd	2	inner
384.2.w.a.35.16	yes	992	384.35	even	32	inner
384.2.w.a.35.47	yes	992	128.35	odd	32	inner