Embedded newform 63.4.i.a.5.15

• Label: 63.4.i.a.5.15
• Level: $63$
• Weight: $4$
• Character: 63.5
• 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.i (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			5.15
Character	\(\chi\)	\(=\)	63.5
Dual form			63.4.i.a.38.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.83815i q^{2} +(-1.68860 + 4.91412i) q^{3} +4.62119 q^{4} +(0.207277 + 0.359014i) q^{5} +(-9.03291 - 3.10391i) q^{6} +(5.26194 + 17.7570i) q^{7} +23.1997i q^{8} +(-21.2972 - 16.5960i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.83815i			0.649885i	0.945734	+	0.324943i	\(0.105345\pi\)
							−0.945734	+	0.324943i	\(0.894655\pi\)
\(3\)	−1.68860	+	4.91412i	−0.324972	+	0.945724i
							\(5\)	0.207277	+	0.359014i	0.0185394	+	0.0321112i	0.875146	−	0.483858i	\(-0.160765\pi\)
−0.856607	+	0.515970i	\(0.827432\pi\)
\(7\)	5.26194	+	17.7570i	0.284118	+	0.958789i
							\(11\)	−42.9636	−	24.8051i	−1.17764	−	0.679910i	−0.222172	−	0.975008i	\(-0.571314\pi\)
−0.955467	+	0.295098i	\(0.904648\pi\)
\(13\)	−1.43477	−	0.828367i	−0.0306104	−	0.0176729i	0.484617	−	0.874727i	\(-0.338959\pi\)
							−0.515227	+	0.857054i	\(0.672292\pi\)
\(17\)	20.6496	+	35.7662i	0.294604	+	0.510269i	0.974893	−	0.222675i	\(-0.0714790\pi\)
							−0.680289	+	0.732944i	\(0.738146\pi\)
\(19\)	130.814	+	75.5256i	1.57952	+	0.911935i	0.994926	+	0.100608i	\(0.0320787\pi\)
							0.584592	+	0.811327i	\(0.301255\pi\)
\(23\)	114.013	−	65.8253i	1.03362	−	0.596762i	0.115602	−	0.993296i	\(-0.463120\pi\)
							0.918020	+	0.396534i	\(0.129787\pi\)
\(29\)	−136.539	+	78.8310i	−0.874300	+	0.504777i	−0.868775	−	0.495207i	\(-0.835092\pi\)
							−0.00552512	+	0.999985i	\(0.501759\pi\)
\(31\)			18.4071i			0.106646i	0.998577	+	0.0533229i	\(0.0169813\pi\)
							−0.998577	+	0.0533229i	\(0.983019\pi\)
\(37\)	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	\(-0.812729\pi\)
							0.896555	+	0.442933i	\(0.146062\pi\)
\(43\)	29.5623	+	51.2034i	0.104842	+	0.181592i	0.913674	−	0.406449i	\(-0.133233\pi\)
							−0.808832	+	0.588040i	\(0.799900\pi\)
\(47\)	216.532			0.672011			0.336005	−	0.941860i	\(-0.390924\pi\)
							0.336005	+	0.941860i	\(0.390924\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.15	✓	44	1.1	even	1	trivial
63.4.i.a.38.8	yes	44	63.38	even	6	inner
63.4.s.a.47.8	yes	44	9.2	odd	6
63.4.s.a.59.8	yes	44	7.3	odd	6
189.4.i.a.143.15		44	63.52	odd	6
189.4.i.a.152.8		44	3.2	odd	2
189.4.s.a.17.15		44	21.17	even	6
189.4.s.a.89.15		44	9.7	even	3