Embedded newform 384.2.w.a.227.23

• Label: 384.2.w.a.227.23
• Level: $384$
• Weight: $2$
• Character: 384.227
• 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			227.23
Character	\(\chi\)	\(=\)	384.227
Dual form			384.2.w.a.203.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.608636 + 1.27654i) q^{2} +(1.73205 - 0.00196270i) q^{3} +(-1.25912 - 1.55390i) q^{4} +(1.34399 + 2.51442i) q^{5} +(-1.05168 + 2.21223i) q^{6} +(1.32386 - 0.263331i) q^{7} +(2.74997 - 0.661564i) q^{8} +(2.99999 - 0.00679899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.608636	+	1.27654i	−0.430371	+	0.902652i
							\(3\)	1.73205	−	0.00196270i	0.999999	−	0.00113317i
\(5\)	1.34399	+	2.51442i	0.601049	+	1.12448i								0.979987	+	0.199061i	\(0.0637892\pi\)
							−0.378939	+	0.925422i	\(0.623711\pi\)
\(7\)	1.32386	−	0.263331i	0.500371	−	0.0995299i	0.0615489	−	0.998104i	\(-0.480396\pi\)
							0.438822	+	0.898574i	\(0.355396\pi\)
\(11\)	0.564988	−	0.688440i	0.170350	−	0.207572i	−0.680836	−	0.732436i	\(-0.738384\pi\)
							0.851187	+	0.524863i	\(0.175884\pi\)
\(13\)	2.06537	−	3.86404i	0.572831	−	1.07169i	−0.414052	−	0.910253i	\(-0.635887\pi\)
							0.986883	−	0.161438i	\(-0.0516133\pi\)
\(17\)	−1.46378	−	0.606319i	−0.355020	−	0.147054i	0.198044	−	0.980193i	\(-0.436541\pi\)
							−0.553063	+	0.833139i	\(0.686541\pi\)
\(19\)	−7.79157	+	2.36355i	−1.78751	+	0.542235i	−0.997487	−	0.0708451i	\(-0.977430\pi\)
							−0.790021	+	0.613080i	\(0.789930\pi\)
\(23\)	−1.85057	+	1.23651i	−0.385870	+	0.257830i	−0.733341	−	0.679860i	\(-0.762040\pi\)
							0.347472	+	0.937691i	\(0.387040\pi\)
\(29\)	1.46313	+	1.78283i	0.271696	+	0.331063i	0.890927	−	0.454146i	\(-0.150056\pi\)
							−0.619231	+	0.785209i	\(0.712556\pi\)
\(31\)	3.09035	+	3.09035i	0.555044	+	0.555044i	0.927892	−	0.372848i	\(-0.121619\pi\)
							−0.372848	+	0.927892i	\(0.621619\pi\)
\(37\)	7.34656	+	2.22855i	1.20777	+	0.366372i	0.829068	−	0.559148i	\(-0.188872\pi\)
							0.378699	+	0.925520i	\(0.376372\pi\)
\(41\)	−6.44363	+	4.30550i	−1.00633	+	0.672406i	−0.945459	−	0.325740i	\(-0.894387\pi\)
							−0.0608670	+	0.998146i	\(0.519387\pi\)
\(43\)	−0.929911	+	0.0915883i	−0.141810	+	0.0139671i	−0.168673	−	0.985672i	\(-0.553948\pi\)
							0.0268630	+	0.999639i	\(0.491448\pi\)
\(47\)	−12.4539	−	5.15859i	−1.81660	−	0.752458i	−0.978285	−	0.207265i	\(-0.933544\pi\)
							−0.838310	−	0.545193i	\(-0.816456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.227.23	yes	992
3.2	odd	2	inner	384.2.w.a.227.40	yes	992
128.75	odd	32	inner	384.2.w.a.203.40	yes	992
384.203	even	32	inner	384.2.w.a.203.23	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.203.23	✓	992	384.203	even	32	inner
384.2.w.a.203.40	yes	992	128.75	odd	32	inner
384.2.w.a.227.23	yes	992	1.1	even	1	trivial
384.2.w.a.227.40	yes	992	3.2	odd	2	inner