Embedded newform 63.4.s.a.47.14

• Label: 63.4.s.a.47.14
• Level: $63$
• Weight: $4$
• Character: 63.47
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.14
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.958607 - 0.553452i) q^{2} +(4.58314 + 2.44844i) q^{3} +(-3.38738 + 5.86712i) q^{4} +12.4738 q^{5} +(5.74852 - 0.189454i) q^{6} +(-18.2811 - 2.96677i) q^{7} +16.3542i q^{8} +(15.0103 + 22.4431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.958607	−	0.553452i	0.338919	−	0.195675i	−0.320875	−	0.947122i	\(-0.603977\pi\)
							0.659794	+	0.751447i	\(0.270644\pi\)
\(3\)	4.58314	+	2.44844i	0.882025	+	0.471203i
							\(5\)	12.4738			1.11569			0.557844	−	0.829946i	\(-0.311629\pi\)
0.557844	+	0.829946i	\(0.311629\pi\)
\(7\)	−18.2811	−	2.96677i	−0.987086	−	0.160190i
							\(11\)			3.72145i			0.102005i	0.998699	+	0.0510027i	\(0.0162417\pi\)
−0.998699	+	0.0510027i	\(0.983758\pi\)
\(13\)	68.0440	−	39.2852i	1.45169	−	0.838135i	0.453115	−	0.891452i	\(-0.350313\pi\)
							0.998578	+	0.0533171i	\(0.0169794\pi\)
\(17\)	−56.8448	−	98.4582i	−0.810994	−	1.40468i	−0.912169	−	0.409814i	\(-0.865594\pi\)
							0.101175	−	0.994869i	\(-0.467740\pi\)
\(19\)	33.4526	+	19.3138i	0.403923	+	0.233205i	0.688175	−	0.725544i	\(-0.258412\pi\)
							−0.284252	+	0.958750i	\(0.591745\pi\)
\(23\)			37.6101i			0.340967i	0.985361	+	0.170483i	\(0.0545329\pi\)
							−0.985361	+	0.170483i	\(0.945467\pi\)
\(29\)	−144.984	−	83.7067i	−0.928376	−	0.535998i	−0.0420787	−	0.999114i	\(-0.513398\pi\)
							−0.886298	+	0.463116i	\(0.846731\pi\)
\(31\)	−183.916	−	106.184i	−1.06556	−	0.615201i	−0.138595	−	0.990349i	\(-0.544259\pi\)
							−0.926965	+	0.375148i	\(0.877592\pi\)
\(37\)	132.003	−	228.635i	0.586516	−	1.01588i	−0.408169	−	0.912907i	\(-0.633832\pi\)
							0.994685	−	0.102969i	\(-0.0328342\pi\)
\(41\)	190.215	+	329.461i	0.724549	+	1.25496i	0.959159	+	0.282866i	\(0.0912853\pi\)
							−0.234610	+	0.972090i	\(0.575381\pi\)
\(43\)	90.2450	−	156.309i	0.320052	−	0.554346i	−0.660447	−	0.750873i	\(-0.729633\pi\)
							0.980498	+	0.196527i	\(0.0629663\pi\)
\(47\)	77.0348	+	133.428i	0.239078	+	0.414096i	0.960450	−	0.278452i	\(-0.0898214\pi\)
							−0.721372	+	0.692548i	\(0.756488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.14	yes	44
3.2	odd	2		189.4.s.a.89.9		44
7.3	odd	6		63.4.i.a.38.14	yes	44
9.4	even	3		189.4.i.a.152.14		44
9.5	odd	6		63.4.i.a.5.9	✓	44
21.17	even	6		189.4.i.a.143.9		44
63.31	odd	6		189.4.s.a.17.9		44
63.59	even	6	inner	63.4.s.a.59.14	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.9	✓	44	9.5	odd	6
63.4.i.a.38.14	yes	44	7.3	odd	6
63.4.s.a.47.14	yes	44	1.1	even	1	trivial
63.4.s.a.59.14	yes	44	63.59	even	6	inner
189.4.i.a.143.9		44	21.17	even	6
189.4.i.a.152.14		44	9.4	even	3
189.4.s.a.17.9		44	63.31	odd	6
189.4.s.a.89.9		44	3.2	odd	2