Embedded newform 63.4.g.a.4.9

• Label: 63.4.g.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.667800 + 1.15666i) q^{2} +(5.07151 + 1.13126i) q^{3} +(3.10809 + 5.38337i) q^{4} +9.00469 q^{5} +(-4.69524 + 5.11058i) q^{6} +(-14.2234 - 11.8615i) q^{7} -18.9871 q^{8} +(24.4405 + 11.4744i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.667800	+	1.15666i	−0.236103	+	0.408942i	−0.959593	−	0.281393i	\(-0.909204\pi\)
							0.723490	+	0.690335i	\(0.242537\pi\)
\(3\)	5.07151	+	1.13126i	0.976013	+	0.217710i
							\(5\)	9.00469			0.805404			0.402702	−	0.915331i	\(-0.368071\pi\)
0.402702	+	0.915331i	\(0.368071\pi\)
\(7\)	−14.2234	−	11.8615i	−0.767990	−	0.640462i
							\(11\)	29.3524			0.804554			0.402277	−	0.915518i	\(-0.368219\pi\)
0.402277	+	0.915518i	\(0.368219\pi\)
\(13\)	−21.1758	+	36.6776i	−0.451779	+	0.782504i	−0.998497	−	0.0548133i	\(-0.982544\pi\)
							0.546718	+	0.837317i	\(0.315877\pi\)
\(17\)	−2.56927	+	4.45010i	−0.0366552	+	0.0634888i	−0.883771	−	0.467920i	\(-0.845004\pi\)
							0.847116	+	0.531408i	\(0.178337\pi\)
\(19\)	−71.2462	−	123.402i	−0.860263	−	1.49002i	−0.871675	−	0.490084i	\(-0.836966\pi\)
							0.0114121	−	0.999935i	\(-0.496367\pi\)
\(23\)	178.083			1.61447			0.807235	−	0.590230i	\(-0.200963\pi\)
							0.807235	+	0.590230i	\(0.200963\pi\)
\(29\)	−109.202	−	189.143i	−0.699249	−	1.21114i	−0.968727	−	0.248129i	\(-0.920184\pi\)
							0.269478	−	0.963007i	\(-0.413149\pi\)
\(31\)	−73.9272	−	128.046i	−0.428314	−	0.741861i	0.568410	−	0.822745i	\(-0.307559\pi\)
							−0.996723	+	0.0808847i	\(0.974225\pi\)
\(37\)	−21.2781	−	36.8547i	−0.0945431	−	0.163753i	0.814875	−	0.579637i	\(-0.196806\pi\)
							−0.909418	+	0.415884i	\(0.863472\pi\)
\(41\)	83.7600	−	145.077i	0.319051	−	0.552613i	−0.661239	−	0.750175i	\(-0.729969\pi\)
							0.980290	+	0.197562i	\(0.0633024\pi\)
\(43\)	121.454	+	210.365i	0.430734	+	0.746054i	0.996937	−	0.0782125i	\(-0.0249213\pi\)
							−0.566202	+	0.824266i	\(0.691588\pi\)
\(47\)	38.2567	−	66.2626i	0.118730	−	0.205647i	−0.800535	−	0.599287i	\(-0.795451\pi\)
							0.919265	+	0.393640i	\(0.128784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.9	✓	44
3.2	odd	2		189.4.g.a.172.14		44
7.2	even	3		63.4.h.a.58.14	yes	44
9.2	odd	6		189.4.h.a.46.9		44
9.7	even	3		63.4.h.a.25.14	yes	44
21.2	odd	6		189.4.h.a.37.9		44
63.2	odd	6		189.4.g.a.100.14		44
63.16	even	3	inner	63.4.g.a.16.9	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.9	✓	44	1.1	even	1	trivial
63.4.g.a.16.9	yes	44	63.16	even	3	inner
63.4.h.a.25.14	yes	44	9.7	even	3
63.4.h.a.58.14	yes	44	7.2	even	3
189.4.g.a.100.14		44	63.2	odd	6
189.4.g.a.172.14		44	3.2	odd	2
189.4.h.a.37.9		44	21.2	odd	6
189.4.h.a.46.9		44	9.2	odd	6