Embedded newform 63.4.h.a.25.20

• Label: 63.4.h.a.25.20
• 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			25.20
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+4.21476 q^{2} +(2.84415 + 4.34866i) q^{3} +9.76423 q^{4} +(3.91201 - 6.77580i) q^{5} +(11.9874 + 18.3286i) q^{6} +(-15.9002 - 9.49654i) q^{7} +7.43579 q^{8} +(-10.8216 + 24.7365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.21476			1.49014			0.745072	−	0.666984i	\(-0.232415\pi\)
							0.745072	+	0.666984i	\(0.232415\pi\)
\(3\)	2.84415	+	4.34866i	0.547357	+	0.836900i
							\(5\)	3.91201	−	6.77580i	0.349901	−	0.606046i	−0.636331	−	0.771416i	\(-0.719549\pi\)
0.986232	+	0.165370i	\(0.0528820\pi\)
\(7\)	−15.9002	−	9.49654i	−0.858529	−	0.512765i
							\(11\)	6.42137	+	11.1221i	0.176011	+	0.304859i	0.940511	−	0.339764i	\(-0.110347\pi\)
−0.764500	+	0.644624i	\(0.777014\pi\)
\(13\)	6.74197	+	11.6774i	0.143837	+	0.249134i	0.928939	−	0.370234i	\(-0.120722\pi\)
							−0.785101	+	0.619367i	\(0.787389\pi\)
\(17\)	35.5009	−	61.4893i	0.506484	−	0.877256i	−0.493488	−	0.869753i	\(-0.664278\pi\)
							0.999972	−	0.00750336i	\(-0.00238842\pi\)
\(19\)	−46.2187	−	80.0532i	−0.558069	−	0.966604i	−0.997658	−	0.0684045i	\(-0.978209\pi\)
							0.439589	−	0.898199i	\(-0.355124\pi\)
\(23\)	−1.97063	+	3.41322i	−0.0178654	+	0.0309438i	−0.874820	−	0.484448i	\(-0.839020\pi\)
							0.856954	+	0.515392i	\(0.172354\pi\)
\(29\)	−90.3269	+	156.451i	−0.578389	+	1.00180i	0.417275	+	0.908780i	\(0.362985\pi\)
							−0.995664	+	0.0930193i	\(0.970348\pi\)
\(31\)	270.523			1.56733			0.783667	−	0.621181i	\(-0.213347\pi\)
							0.783667	+	0.621181i	\(0.213347\pi\)
\(37\)	110.157	+	190.798i	0.489452	+	0.847756i	0.999926	−	0.0121369i	\(-0.00386339\pi\)
							−0.510474	+	0.859893i	\(0.670530\pi\)
\(41\)	−33.6506	−	58.2845i	−0.128179	−	0.222012i	0.794792	−	0.606882i	\(-0.207580\pi\)
							−0.922971	+	0.384869i	\(0.874247\pi\)
\(43\)	237.806	−	411.893i	0.843375	−	1.46077i	−0.0436494	−	0.999047i	\(-0.513898\pi\)
							0.887025	−	0.461722i	\(-0.152768\pi\)
\(47\)	−512.858			−1.59166			−0.795829	−	0.605521i	\(-0.792965\pi\)
							−0.795829	+	0.605521i	\(0.792965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.20	yes	44
3.2	odd	2		189.4.h.a.46.3		44
7.2	even	3		63.4.g.a.16.3	yes	44
9.4	even	3		63.4.g.a.4.3	✓	44
9.5	odd	6		189.4.g.a.172.20		44
21.2	odd	6		189.4.g.a.100.20		44
63.23	odd	6		189.4.h.a.37.3		44
63.58	even	3	inner	63.4.h.a.58.20	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.3	✓	44	9.4	even	3
63.4.g.a.16.3	yes	44	7.2	even	3
63.4.h.a.25.20	yes	44	1.1	even	1	trivial
63.4.h.a.58.20	yes	44	63.58	even	3	inner
189.4.g.a.100.20		44	21.2	odd	6
189.4.g.a.172.20		44	9.5	odd	6
189.4.h.a.37.3		44	63.23	odd	6
189.4.h.a.46.3		44	3.2	odd	2