Embedded newform 63.4.g.a.4.17

• Label: 63.4.g.a.4.17
• Level: $63$
• Weight: $4$
• Character: 63.4
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.17
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46729 - 2.54141i) q^{2} +(4.92390 + 1.65989i) q^{3} +(-0.305860 - 0.529765i) q^{4} -3.69711 q^{5} +(11.4432 - 10.0781i) q^{6} +(12.3500 - 13.8013i) q^{7} +21.6814 q^{8} +(21.4895 + 16.3463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46729	−	2.54141i	0.518764	−	0.898526i	−0.480998	−	0.876722i	\(-0.659726\pi\)
							0.999762	−	0.0218042i	\(-0.00694105\pi\)
\(3\)	4.92390	+	1.65989i	0.947604	+	0.319446i
							\(5\)	−3.69711			−0.330679			−0.165340	−	0.986237i	\(-0.552872\pi\)
−0.165340	+	0.986237i	\(0.552872\pi\)
\(7\)	12.3500	−	13.8013i	0.666838	−	0.745203i
							\(11\)	−65.7074			−1.80105			−0.900524	−	0.434807i	\(-0.856816\pi\)
−0.900524	+	0.434807i	\(0.856816\pi\)
\(13\)	2.73952	−	4.74498i	0.0584465	−	0.101232i	−0.835322	−	0.549761i	\(-0.814719\pi\)
							0.893768	+	0.448529i	\(0.148052\pi\)
\(17\)	−25.1343	+	43.5340i	−0.358587	+	0.621090i	−0.987725	−	0.156203i	\(-0.950075\pi\)
							0.629138	+	0.777293i	\(0.283408\pi\)
\(19\)	0.769884	+	1.33348i	0.00929597	+	0.0161011i	0.870636	−	0.491928i	\(-0.163708\pi\)
							−0.861340	+	0.508029i	\(0.830374\pi\)
\(23\)	−120.016			−1.08804			−0.544021	−	0.839072i	\(-0.683099\pi\)
							−0.544021	+	0.839072i	\(0.683099\pi\)
\(29\)	−39.2967	−	68.0640i	−0.251628	−	0.435833i	0.712346	−	0.701829i	\(-0.247633\pi\)
							−0.963974	+	0.265995i	\(0.914299\pi\)
\(31\)	151.666	+	262.693i	0.878709	+	1.52197i	0.852758	+	0.522305i	\(0.174928\pi\)
							0.0259506	+	0.999663i	\(0.491739\pi\)
\(37\)	96.6335	+	167.374i	0.429363	+	0.743679i	0.996817	−	0.0797263i	\(-0.0254046\pi\)
							−0.567453	+	0.823406i	\(0.692071\pi\)
\(41\)	196.671	−	340.645i	0.749144	−	1.29756i	−0.199090	−	0.979981i	\(-0.563798\pi\)
							0.948233	−	0.317574i	\(-0.102868\pi\)
\(43\)	−138.067	−	239.139i	−0.489651	−	0.848101i	0.510278	−	0.860010i	\(-0.329543\pi\)
							−0.999929	+	0.0119087i	\(0.996209\pi\)
\(47\)	126.380	−	218.896i	0.392221	−	0.679348i	−0.600521	−	0.799609i	\(-0.705040\pi\)
							0.992742	+	0.120262i	\(0.0383733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.17	✓	44
3.2	odd	2		189.4.g.a.172.6		44
7.2	even	3		63.4.h.a.58.6	yes	44
9.2	odd	6		189.4.h.a.46.17		44
9.7	even	3		63.4.h.a.25.6	yes	44
21.2	odd	6		189.4.h.a.37.17		44
63.2	odd	6		189.4.g.a.100.6		44
63.16	even	3	inner	63.4.g.a.16.17	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.17	✓	44	1.1	even	1	trivial
63.4.g.a.16.17	yes	44	63.16	even	3	inner
63.4.h.a.25.6	yes	44	9.7	even	3
63.4.h.a.58.6	yes	44	7.2	even	3
189.4.g.a.100.6		44	63.2	odd	6
189.4.g.a.172.6		44	3.2	odd	2
189.4.h.a.37.17		44	21.2	odd	6
189.4.h.a.46.17		44	9.2	odd	6