Embedded newform 63.4.s.a.59.9

• Label: 63.4.s.a.59.9
• Level: $63$
• Weight: $4$
• Character: 63.59
• 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			59.9
Character	\(\chi\)	\(=\)	63.59
Dual form			63.4.s.a.47.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54833 - 0.893930i) q^{2} +(5.18374 - 0.358957i) q^{3} +(-2.40178 - 4.16000i) q^{4} -16.9434 q^{5} +(-8.34703 - 4.07812i) q^{6} +(-9.83512 - 15.6930i) q^{7} +22.8910i q^{8} +(26.7423 - 3.72148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54833	−	0.893930i	−0.547418	−	0.316052i	0.200662	−	0.979661i	\(-0.435691\pi\)
							−0.748080	+	0.663609i	\(0.769024\pi\)
\(3\)	5.18374	−	0.358957i	0.997611	−	0.0690814i
							\(5\)	−16.9434			−1.51546			−0.757730	−	0.652568i	\(-0.773692\pi\)
−0.757730	+	0.652568i	\(0.773692\pi\)
\(7\)	−9.83512	−	15.6930i	−0.531046	−	0.847343i
							\(11\)		−	7.14600i		−	0.195873i	−0.995193	−	0.0979364i	\(-0.968776\pi\)
0.995193	−	0.0979364i	\(-0.0312242\pi\)
\(13\)	−67.0144	−	38.6908i	−1.42973	−	0.825453i	−0.432627	−	0.901573i	\(-0.642413\pi\)
							−0.997099	+	0.0761201i	\(0.975747\pi\)
\(17\)	1.06481	−	1.84430i	0.0151914	−	0.0263122i	−0.858330	−	0.513098i	\(-0.828498\pi\)
							0.873521	+	0.486786i	\(0.161831\pi\)
\(19\)	104.730	−	60.4661i	1.26457	−	0.730099i	0.290614	−	0.956840i	\(-0.406141\pi\)
							0.973955	+	0.226741i	\(0.0728072\pi\)
\(23\)		−	10.8686i		−	0.0985329i	−0.998786	−	0.0492664i	\(-0.984312\pi\)
							0.998786	−	0.0492664i	\(-0.0156883\pi\)
\(29\)	−110.704	+	63.9149i	−0.708869	+	0.409266i	−0.810642	−	0.585542i	\(-0.800882\pi\)
							0.101773	+	0.994808i	\(0.467548\pi\)
\(31\)	184.393	−	106.459i	1.06832	−	0.616796i	0.140600	−	0.990067i	\(-0.455097\pi\)
							0.927723	+	0.373270i	\(0.121764\pi\)
\(37\)	−12.2062	−	21.1417i	−0.0542346	−	0.0939371i	0.837634	−	0.546233i	\(-0.183939\pi\)
							−0.891868	+	0.452296i	\(0.850605\pi\)
\(41\)	−64.8686	+	112.356i	−0.247092	+	0.427976i	−0.962718	−	0.270508i	\(-0.912808\pi\)
							0.715626	+	0.698484i	\(0.246142\pi\)
\(43\)	86.4878	+	149.801i	0.306727	+	0.531267i	0.977644	−	0.210265i	\(-0.0674327\pi\)
							−0.670917	+	0.741532i	\(0.734099\pi\)
\(47\)	202.426	−	350.613i	0.628232	−	1.08813i	−0.359674	−	0.933078i	\(-0.617112\pi\)
							0.987906	−	0.155052i	\(-0.0495546\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.59.9	yes	44
3.2	odd	2		189.4.s.a.17.14		44
7.5	odd	6		63.4.i.a.5.14	✓	44
9.2	odd	6		63.4.i.a.38.9	yes	44
9.7	even	3		189.4.i.a.143.14		44
21.5	even	6		189.4.i.a.152.9		44
63.47	even	6	inner	63.4.s.a.47.9	yes	44
63.61	odd	6		189.4.s.a.89.14		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.14	✓	44	7.5	odd	6
63.4.i.a.38.9	yes	44	9.2	odd	6
63.4.s.a.47.9	yes	44	63.47	even	6	inner
63.4.s.a.59.9	yes	44	1.1	even	1	trivial
189.4.i.a.143.14		44	9.7	even	3
189.4.i.a.152.9		44	21.5	even	6
189.4.s.a.17.14		44	3.2	odd	2
189.4.s.a.89.14		44	63.61	odd	6