Embedded newform 63.4.g.a.4.16

• Label: 63.4.g.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32738 - 2.29909i) q^{2} +(-4.04989 + 3.25551i) q^{3} +(0.476130 + 0.824682i) q^{4} +7.35561 q^{5} +(2.10896 + 13.6324i) q^{6} +(16.6607 + 8.08840i) q^{7} +23.7661 q^{8} +(5.80329 - 26.3690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32738	−	2.29909i	0.469299	−	0.812850i	−0.530085	−	0.847945i	\(-0.677840\pi\)
							0.999384	+	0.0350944i	\(0.0111732\pi\)
\(3\)	−4.04989	+	3.25551i	−0.779403	+	0.626523i
							\(5\)	7.35561			0.657906			0.328953	−	0.944346i	\(-0.393304\pi\)
0.328953	+	0.944346i	\(0.393304\pi\)
\(7\)	16.6607	+	8.08840i	0.899591	+	0.436733i
							\(11\)	11.3432			0.310919			0.155460	−	0.987842i	\(-0.450314\pi\)
0.155460	+	0.987842i	\(0.450314\pi\)
\(13\)	−30.2941	+	52.4710i	−0.646314	+	1.11945i	0.337683	+	0.941260i	\(0.390357\pi\)
							−0.983996	+	0.178188i	\(0.942976\pi\)
\(17\)	−5.29282	+	9.16743i	−0.0755116	+	0.130790i	−0.901309	−	0.433178i	\(-0.857392\pi\)
							0.825797	+	0.563967i	\(0.190726\pi\)
\(19\)	−41.6545	−	72.1477i	−0.502958	−	0.871148i	−0.999994	−	0.00341841i	\(-0.998912\pi\)
							0.497037	−	0.867730i	\(-0.334421\pi\)
\(23\)	87.6234			0.794380			0.397190	−	0.917736i	\(-0.369985\pi\)
							0.397190	+	0.917736i	\(0.369985\pi\)
\(29\)	−67.6917	−	117.246i	−0.433450	−	0.750757i	0.563718	−	0.825967i	\(-0.309370\pi\)
							−0.997168	+	0.0752104i	\(0.976037\pi\)
\(31\)	−26.7514	−	46.3349i	−0.154990	−	0.268451i	0.778065	−	0.628184i	\(-0.216201\pi\)
							−0.933056	+	0.359732i	\(0.882868\pi\)
\(37\)	−149.868	−	259.578i	−0.665894	−	1.15336i	−0.979042	−	0.203658i	\(-0.934717\pi\)
							0.313148	−	0.949704i	\(-0.398616\pi\)
\(41\)	−194.038	+	336.083i	−0.739112	+	1.28018i	0.213784	+	0.976881i	\(0.431421\pi\)
							−0.952896	+	0.303298i	\(0.901912\pi\)
\(43\)	−215.569	−	373.377i	−0.764511	−	1.32417i	−0.940505	−	0.339781i	\(-0.889647\pi\)
							0.175993	−	0.984391i	\(-0.443686\pi\)
\(47\)	255.582	−	442.681i	0.793202	−	1.37387i	−0.130773	−	0.991412i	\(-0.541746\pi\)
							0.923975	−	0.382454i	\(-0.124921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.16	✓	44
3.2	odd	2		189.4.g.a.172.7		44
7.2	even	3		63.4.h.a.58.7	yes	44
9.2	odd	6		189.4.h.a.46.16		44
9.7	even	3		63.4.h.a.25.7	yes	44
21.2	odd	6		189.4.h.a.37.16		44
63.2	odd	6		189.4.g.a.100.7		44
63.16	even	3	inner	63.4.g.a.16.16	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.16	✓	44	1.1	even	1	trivial
63.4.g.a.16.16	yes	44	63.16	even	3	inner
63.4.h.a.25.7	yes	44	9.7	even	3
63.4.h.a.58.7	yes	44	7.2	even	3
189.4.g.a.100.7		44	63.2	odd	6
189.4.g.a.172.7		44	3.2	odd	2
189.4.h.a.37.16		44	21.2	odd	6
189.4.h.a.46.16		44	9.2	odd	6