Embedded newform 63.4.s.a.59.10

• Label: 63.4.s.a.59.10
• 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			59.10
Character	\(\chi\)	\(=\)	63.59
Dual form			63.4.s.a.47.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.998155 - 0.576285i) q^{2} +(-1.29916 - 5.03112i) q^{3} +(-3.33579 - 5.77776i) q^{4} +0.274718 q^{5} +(-1.60260 + 5.77053i) q^{6} +(-9.15344 + 16.1001i) q^{7} +16.9100i q^{8} +(-23.6244 + 13.0725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.998155	−	0.576285i	−0.352901	−	0.203748i	0.313061	−	0.949733i	\(-0.398646\pi\)
							−0.665962	+	0.745985i	\(0.731979\pi\)
\(3\)	−1.29916	−	5.03112i	−0.250024	−	0.968240i
							\(5\)	0.274718			0.0245716			0.0122858	−	0.999925i	\(-0.496089\pi\)
0.0122858	+	0.999925i	\(0.496089\pi\)
\(7\)	−9.15344	+	16.1001i	−0.494239	+	0.869326i
							\(11\)		−	26.2387i		−	0.719205i	−0.933106	−	0.359602i	\(-0.882912\pi\)
0.933106	−	0.359602i	\(-0.117088\pi\)
\(13\)	−39.6128	−	22.8704i	−0.845123	−	0.487932i	0.0138792	−	0.999904i	\(-0.495582\pi\)
							−0.859002	+	0.511972i	\(0.828915\pi\)
\(17\)	30.2340	−	52.3669i	0.431343	−	0.747108i	−0.565646	−	0.824648i	\(-0.691373\pi\)
							0.996989	+	0.0775399i	\(0.0247065\pi\)
\(19\)	−38.8827	+	22.4489i	−0.469490	+	0.271060i	−0.716026	−	0.698074i	\(-0.754041\pi\)
							0.246536	+	0.969134i	\(0.420708\pi\)
\(23\)		−	127.767i		−	1.15832i	−0.815214	−	0.579159i	\(-0.803381\pi\)
							0.815214	−	0.579159i	\(-0.196619\pi\)
\(29\)	58.2779	−	33.6468i	0.373170	−	0.215450i	−0.301672	−	0.953412i	\(-0.597545\pi\)
							0.674843	+	0.737962i	\(0.264212\pi\)
\(31\)	85.6986	−	49.4781i	0.496513	−	0.286662i	−0.230759	−	0.973011i	\(-0.574121\pi\)
							0.727273	+	0.686349i	\(0.240788\pi\)
\(37\)	−123.714	−	214.278i	−0.549686	−	0.952085i	−0.998296	−	0.0583567i	\(-0.981414\pi\)
							0.448610	−	0.893728i	\(-0.351919\pi\)
\(41\)	134.234	−	232.500i	0.511312	−	0.885618i	−0.488602	−	0.872507i	\(-0.662493\pi\)
							0.999914	−	0.0131112i	\(-0.00417355\pi\)
\(43\)	−72.4990	−	125.572i	−0.257116	−	0.445338i	0.708352	−	0.705859i	\(-0.249439\pi\)
							−0.965468	+	0.260521i	\(0.916106\pi\)
\(47\)	−250.874	+	434.526i	−0.778589	+	1.34856i	0.154166	+	0.988045i	\(0.450731\pi\)
							−0.932755	+	0.360511i	\(0.882602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.59.10	yes	44
3.2	odd	2		189.4.s.a.17.13		44
7.5	odd	6		63.4.i.a.5.13	✓	44
9.2	odd	6		63.4.i.a.38.10	yes	44
9.7	even	3		189.4.i.a.143.13		44
21.5	even	6		189.4.i.a.152.10		44
63.47	even	6	inner	63.4.s.a.47.10	yes	44
63.61	odd	6		189.4.s.a.89.13		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.13	✓	44	7.5	odd	6
63.4.i.a.38.10	yes	44	9.2	odd	6
63.4.s.a.47.10	yes	44	63.47	even	6	inner
63.4.s.a.59.10	yes	44	1.1	even	1	trivial
189.4.i.a.143.13		44	9.7	even	3
189.4.i.a.152.10		44	21.5	even	6
189.4.s.a.17.13		44	3.2	odd	2
189.4.s.a.89.13		44	63.61	odd	6