Embedded newform 63.4.h.a.25.19

• Label: 63.4.h.a.25.19
• Level: $63$
• Weight: $4$
• Character: 63.25
• 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.h (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			25.19
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.86663 q^{2} +(-3.99989 - 3.31676i) q^{3} +6.95086 q^{4} +(6.67810 - 11.5668i) q^{5} +(-15.4661 - 12.8247i) q^{6} +(15.1080 - 10.7120i) q^{7} -4.05663 q^{8} +(4.99821 + 26.5333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.86663			1.36706			0.683531	−	0.729922i	\(-0.260443\pi\)
							0.683531	+	0.729922i	\(0.260443\pi\)
\(3\)	−3.99989	−	3.31676i	−0.769779	−	0.638311i
							\(5\)	6.67810	−	11.5668i	0.597307	−	1.03457i	−0.395910	−	0.918289i	\(-0.629571\pi\)
0.993217	−	0.116277i	\(-0.0370959\pi\)
\(7\)	15.1080	−	10.7120i	0.815757	−	0.578395i
							\(11\)	14.3605	+	24.8730i	0.393622	+	0.681773i	0.992924	−	0.118750i	\(-0.0378886\pi\)
−0.599302	+	0.800523i	\(0.704555\pi\)
\(13\)	2.75723	+	4.77566i	0.0588244	+	0.101887i	0.893938	−	0.448191i	\(-0.147931\pi\)
							−0.835114	+	0.550078i	\(0.814598\pi\)
\(17\)	−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	\(0.708434\pi\)
\(23\)	−67.4183	+	116.772i	−0.611204	+	1.05864i	0.379834	+	0.925055i	\(0.375981\pi\)
							−0.991038	+	0.133582i	\(0.957352\pi\)
\(29\)	118.224	−	204.770i	0.757022	−	1.31120i	−0.187341	−	0.982295i	\(-0.559987\pi\)
							0.944363	−	0.328905i	\(-0.106680\pi\)
\(31\)	92.9315			0.538419			0.269210	−	0.963082i	\(-0.413238\pi\)
							0.269210	+	0.963082i	\(0.413238\pi\)
\(37\)	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	−0.987066	−	0.160313i	\(-0.948750\pi\)
							0.354698	−	0.934981i	\(-0.384584\pi\)
\(43\)	−173.016	+	299.673i	−0.613599	+	1.06279i	0.377029	+	0.926201i	\(0.376946\pi\)
							−0.990629	+	0.136584i	\(0.956388\pi\)
\(47\)	−304.957			−0.946437			−0.473218	−	0.880945i	\(-0.656908\pi\)
							−0.473218	+	0.880945i	\(0.656908\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.4	✓	44	9.4	even	3
63.4.g.a.16.4	yes	44	7.2	even	3
63.4.h.a.25.19	yes	44	1.1	even	1	trivial
63.4.h.a.58.19	yes	44	63.58	even	3	inner
189.4.g.a.100.19		44	21.2	odd	6
189.4.g.a.172.19		44	9.5	odd	6
189.4.h.a.37.4		44	63.23	odd	6
189.4.h.a.46.4		44	3.2	odd	2