Embedded newform 63.4.s.a.47.8

• Label: 63.4.s.a.47.8
• 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			47.8
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59189 + 0.919076i) q^{2} +(-5.10006 - 0.994690i) q^{3} +(-2.31060 + 4.00207i) q^{4} +0.414554 q^{5} +(9.03291 - 3.10391i) q^{6} +(12.7471 - 13.4355i) q^{7} -23.1997i q^{8} +(25.0212 + 10.1460i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59189	+	0.919076i	−0.562817	+	0.324943i	−0.754275	−	0.656558i	\(-0.772012\pi\)
							0.191458	+	0.981501i	\(0.438678\pi\)
\(3\)	−5.10006	−	0.994690i	−0.981507	−	0.191428i
							\(5\)	0.414554			0.0370788			0.0185394	−	0.999828i	\(-0.494098\pi\)
0.0185394	+	0.999828i	\(0.494098\pi\)
\(7\)	12.7471	−	13.4355i	0.688277	−	0.725448i
							\(11\)		−	49.6101i		−	1.35982i	−0.733296	−	0.679910i	\(-0.762019\pi\)
0.733296	−	0.679910i	\(-0.237981\pi\)
\(13\)	1.43477	−	0.828367i	0.0306104	−	0.0176729i	−0.484617	−	0.874727i	\(-0.661041\pi\)
							0.515227	+	0.857054i	\(0.327708\pi\)
\(17\)	−20.6496	−	35.7662i	−0.294604	−	0.510269i	0.680289	−	0.732944i	\(-0.261854\pi\)
							−0.974893	+	0.222675i	\(0.928521\pi\)
\(19\)	130.814	+	75.5256i	1.57952	+	0.911935i	0.994926	+	0.100608i	\(0.0320787\pi\)
							0.584592	+	0.811327i	\(0.301255\pi\)
\(23\)		−	131.651i		−	1.19352i	−0.802418	−	0.596762i	\(-0.796454\pi\)
							0.802418	−	0.596762i	\(-0.203546\pi\)
\(29\)	−136.539	−	78.8310i	−0.874300	−	0.504777i	−0.00552512	−	0.999985i	\(-0.501759\pi\)
							−0.868775	+	0.495207i	\(0.835092\pi\)
\(31\)	−15.9411	−	9.20357i	−0.0923580	−	0.0533229i	0.453110	−	0.891455i	\(-0.350315\pi\)
							−0.545468	+	0.838132i	\(0.683648\pi\)
\(37\)	173.836	−	301.092i	0.772390	−	1.33782i	−0.163860	−	0.986484i	\(-0.552395\pi\)
							0.936250	−	0.351335i	\(-0.114272\pi\)
\(41\)	−16.9818	−	29.4134i	−0.0646857	−	0.112039i	0.831869	−	0.554972i	\(-0.187271\pi\)
							−0.896555	+	0.442933i	\(0.853938\pi\)
\(43\)	29.5623	−	51.2034i	0.104842	−	0.181592i	−0.808832	−	0.588040i	\(-0.799900\pi\)
							0.913674	+	0.406449i	\(0.133233\pi\)
\(47\)	108.266	+	187.523i	0.336005	+	0.581978i	0.983677	−	0.179941i	\(-0.0575906\pi\)
							−0.647672	+	0.761919i	\(0.724257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.8	yes	44
3.2	odd	2		189.4.s.a.89.15		44
7.3	odd	6		63.4.i.a.38.8	yes	44
9.4	even	3		189.4.i.a.152.8		44
9.5	odd	6		63.4.i.a.5.15	✓	44
21.17	even	6		189.4.i.a.143.15		44
63.31	odd	6		189.4.s.a.17.15		44
63.59	even	6	inner	63.4.s.a.59.8	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.15	✓	44	9.5	odd	6
63.4.i.a.38.8	yes	44	7.3	odd	6
63.4.s.a.47.8	yes	44	1.1	even	1	trivial
63.4.s.a.59.8	yes	44	63.59	even	6	inner
189.4.i.a.143.15		44	21.17	even	6
189.4.i.a.152.8		44	9.4	even	3
189.4.s.a.17.15		44	63.31	odd	6
189.4.s.a.89.15		44	3.2	odd	2