Embedded newform 63.4.g.a.4.22

• Label: 63.4.g.a.4.22
• 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			4.22
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.54715 - 4.41178i) q^{2} +(-3.72363 - 3.62416i) q^{3} +(-8.97590 - 15.5467i) q^{4} +6.84046 q^{5} +(-25.4737 + 7.19661i) q^{6} +(16.1584 + 9.05014i) q^{7} -50.6973 q^{8} +(0.730904 + 26.9901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.54715	−	4.41178i	0.900552	−	1.55980i	0.0737725	−	0.997275i	\(-0.476496\pi\)
							0.826779	−	0.562526i	\(-0.190171\pi\)
\(3\)	−3.72363	−	3.62416i	−0.716614	−	0.697470i
							\(5\)	6.84046			0.611829			0.305915	−	0.952059i	\(-0.401038\pi\)
0.305915	+	0.952059i	\(0.401038\pi\)
\(7\)	16.1584	+	9.05014i	0.872474	+	0.488661i
							\(11\)	−41.0518			−1.12523			−0.562617	−	0.826718i	\(-0.690205\pi\)
−0.562617	+	0.826718i	\(0.690205\pi\)
\(13\)	31.8869	−	55.2298i	0.680296	−	1.17831i	−0.294595	−	0.955622i	\(-0.595185\pi\)
							0.974891	−	0.222685i	\(-0.0714820\pi\)
\(17\)	38.0175	−	65.8483i	0.542389	−	0.939445i	−0.456378	−	0.889786i	\(-0.650853\pi\)
							0.998766	−	0.0496584i	\(-0.0158133\pi\)
\(19\)	46.7854	+	81.0346i	0.564911	+	0.978454i	0.997058	+	0.0766508i	\(0.0244227\pi\)
							−0.432147	+	0.901803i	\(0.642244\pi\)
\(23\)	42.1281			0.381927			0.190963	−	0.981597i	\(-0.438839\pi\)
							0.190963	+	0.981597i	\(0.438839\pi\)
\(29\)	21.9258	+	37.9765i	0.140397	+	0.243175i	0.927646	−	0.373460i	\(-0.121829\pi\)
							−0.787249	+	0.616635i	\(0.788495\pi\)
\(31\)	100.068	+	173.322i	0.579765	+	1.00418i	0.995506	+	0.0946989i	\(0.0301888\pi\)
							−0.415741	+	0.909483i	\(0.636478\pi\)
\(37\)	−52.1566	−	90.3379i	−0.231743	−	0.401391i	0.726578	−	0.687084i	\(-0.241110\pi\)
							−0.958321	+	0.285693i	\(0.907776\pi\)
\(41\)	−108.674	+	188.229i	−0.413951	+	0.716984i	−0.995318	−	0.0966575i	\(-0.969185\pi\)
							0.581367	+	0.813642i	\(0.302518\pi\)
\(43\)	236.154	+	409.031i	0.837515	+	1.45062i	0.891966	+	0.452102i	\(0.149326\pi\)
							−0.0544516	+	0.998516i	\(0.517341\pi\)
\(47\)	8.94210	−	15.4882i	0.0277519	−	0.0480677i	−0.851816	−	0.523841i	\(-0.824498\pi\)
							0.879568	+	0.475774i	\(0.157832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.22	✓	44
3.2	odd	2		189.4.g.a.172.1		44
7.2	even	3		63.4.h.a.58.1	yes	44
9.2	odd	6		189.4.h.a.46.22		44
9.7	even	3		63.4.h.a.25.1	yes	44
21.2	odd	6		189.4.h.a.37.22		44
63.2	odd	6		189.4.g.a.100.1		44
63.16	even	3	inner	63.4.g.a.16.22	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.22	✓	44	1.1	even	1	trivial
63.4.g.a.16.22	yes	44	63.16	even	3	inner
63.4.h.a.25.1	yes	44	9.7	even	3
63.4.h.a.58.1	yes	44	7.2	even	3
189.4.g.a.100.1		44	63.2	odd	6
189.4.g.a.172.1		44	3.2	odd	2
189.4.h.a.37.22		44	21.2	odd	6
189.4.h.a.46.22		44	9.2	odd	6