Embedded newform 63.4.g.a.4.4

• Label: 63.4.g.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93332 + 3.34860i) q^{2} +(4.87234 + 1.80563i) q^{3} +(-3.47543 - 6.01962i) q^{4} -13.3562 q^{5} +(-15.4661 + 12.8247i) q^{6} +(1.72288 + 18.4399i) q^{7} -4.05663 q^{8} +(20.4794 + 17.5952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93332	+	3.34860i	−0.683531	+	1.18391i	0.290365	+	0.956916i	\(0.406223\pi\)
							−0.973896	+	0.226994i	\(0.927110\pi\)
\(3\)	4.87234	+	1.80563i	0.937683	+	0.347493i
							\(5\)	−13.3562			−1.19461			−0.597307	−	0.802013i	\(-0.703763\pi\)
−0.597307	+	0.802013i	\(0.703763\pi\)
\(7\)	1.72288	+	18.4399i	0.0930267	+	0.995664i
							\(11\)	−28.7209			−0.787244			−0.393622	−	0.919272i	\(-0.628778\pi\)
−0.393622	+	0.919272i	\(0.628778\pi\)
\(13\)	2.75723	−	4.77566i	0.0588244	−	0.101887i	−0.835114	−	0.550078i	\(-0.814598\pi\)
							0.893938	+	0.448191i	\(0.147931\pi\)
\(17\)	−8.48010	+	14.6880i	−0.120984	+	0.209550i	−0.920156	−	0.391552i	\(-0.871938\pi\)
							0.799172	+	0.601102i	\(0.205272\pi\)
\(19\)	31.6694	+	54.8530i	0.382393	+	0.662323i	0.991404	−	0.130838i	\(-0.0417669\pi\)
							−0.609011	+	0.793162i	\(0.708434\pi\)
\(23\)	134.837			1.22241			0.611204	−	0.791473i	\(-0.290686\pi\)
							0.611204	+	0.791473i	\(0.290686\pi\)
\(29\)	118.224	+	204.770i	0.757022	+	1.31120i	0.944363	+	0.328905i	\(0.106680\pi\)
							−0.187341	+	0.982295i	\(0.559987\pi\)
\(31\)	−46.4658	−	80.4810i	−0.269210	−	0.466285i	0.699448	−	0.714683i	\(-0.253429\pi\)
							−0.968658	+	0.248399i	\(0.920096\pi\)
\(37\)	202.752	+	351.177i	0.900872	+	1.56036i	0.826365	+	0.563135i	\(0.190405\pi\)
							0.0745066	+	0.997221i	\(0.476262\pi\)
\(41\)	−166.014	+	287.545i	−0.632368	+	1.09529i	0.354698	+	0.934981i	\(0.384584\pi\)
							−0.987066	+	0.160313i	\(0.948750\pi\)
\(43\)	−173.016	−	299.673i	−0.613599	−	1.06279i	−0.990629	−	0.136584i	\(-0.956388\pi\)
							0.377029	−	0.926201i	\(-0.376946\pi\)
\(47\)	152.478	−	264.100i	0.473218	−	0.819638i	−0.526312	−	0.850292i	\(-0.676426\pi\)
							0.999530	+	0.0306535i	\(0.00975884\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.4	✓	44
3.2	odd	2		189.4.g.a.172.19		44
7.2	even	3		63.4.h.a.58.19	yes	44
9.2	odd	6		189.4.h.a.46.4		44
9.7	even	3		63.4.h.a.25.19	yes	44
21.2	odd	6		189.4.h.a.37.4		44
63.2	odd	6		189.4.g.a.100.19		44
63.16	even	3	inner	63.4.g.a.16.4	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.4	✓	44	1.1	even	1	trivial
63.4.g.a.16.4	yes	44	63.16	even	3	inner
63.4.h.a.25.19	yes	44	9.7	even	3
63.4.h.a.58.19	yes	44	7.2	even	3
189.4.g.a.100.19		44	63.2	odd	6
189.4.g.a.172.19		44	3.2	odd	2
189.4.h.a.37.4		44	21.2	odd	6
189.4.h.a.46.4		44	9.2	odd	6