Embedded newform 63.4.g.a.4.10

• Label: 63.4.g.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.491847 + 0.851904i) q^{2} +(-0.512079 + 5.17086i) q^{3} +(3.51617 + 6.09019i) q^{4} -18.7142 q^{5} +(-4.15321 - 2.97951i) q^{6} +(10.0203 - 15.5754i) q^{7} -14.7872 q^{8} +(-26.4756 - 5.29577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.491847	+	0.851904i	−0.173894	+	0.301194i	−0.939778	−	0.341785i	\(-0.888968\pi\)
							0.765884	+	0.642979i	\(0.222302\pi\)
\(3\)	−0.512079	+	5.17086i	−0.0985496	+	0.995132i
							\(5\)	−18.7142			−1.67385			−0.836926	−	0.547317i	\(-0.815649\pi\)
−0.836926	+	0.547317i	\(0.815649\pi\)
\(7\)	10.0203	−	15.5754i	0.541046	−	0.840993i
							\(11\)	23.4959			0.644025			0.322013	−	0.946735i	\(-0.395641\pi\)
0.322013	+	0.946735i	\(0.395641\pi\)
\(13\)	−23.8730	+	41.3493i	−0.509322	+	0.882172i	0.490619	+	0.871374i	\(0.336771\pi\)
							−0.999942	+	0.0107981i	\(0.996563\pi\)
\(17\)	−47.7799	+	82.7573i	−0.681667	+	1.18068i	0.292805	+	0.956172i	\(0.405411\pi\)
							−0.974472	+	0.224510i	\(0.927922\pi\)
\(19\)	28.4637	+	49.3006i	0.343686	+	0.595281i	0.985114	−	0.171902i	\(-0.0549912\pi\)
							−0.641428	+	0.767183i	\(0.721658\pi\)
\(23\)	32.6236			0.295760			0.147880	−	0.989005i	\(-0.452755\pi\)
							0.147880	+	0.989005i	\(0.452755\pi\)
\(29\)	81.3794	+	140.953i	0.521096	+	0.902564i	0.999699	+	0.0245330i	\(0.00780988\pi\)
							−0.478603	+	0.878031i	\(0.658857\pi\)
\(31\)	20.2272	+	35.0345i	0.117190	+	0.202980i	0.918653	−	0.395065i	\(-0.129278\pi\)
							−0.801463	+	0.598045i	\(0.795945\pi\)
\(37\)	−127.944	−	221.605i	−0.568482	−	0.984639i	−0.996716	−	0.0809715i	\(-0.974198\pi\)
							0.428235	−	0.903667i	\(-0.359136\pi\)
\(41\)	−35.9678	+	62.2981i	−0.137006	+	0.237301i	−0.926362	−	0.376635i	\(-0.877081\pi\)
							0.789356	+	0.613935i	\(0.210415\pi\)
\(43\)	237.322	+	411.054i	0.841658	+	1.45779i	0.888492	+	0.458892i	\(0.151753\pi\)
							−0.0468344	+	0.998903i	\(0.514913\pi\)
\(47\)	132.436	−	229.386i	0.411017	−	0.711902i	−0.583984	−	0.811765i	\(-0.698507\pi\)
							0.995001	+	0.0998630i	\(0.0318404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.10	✓	44
3.2	odd	2		189.4.g.a.172.13		44
7.2	even	3		63.4.h.a.58.13	yes	44
9.2	odd	6		189.4.h.a.46.10		44
9.7	even	3		63.4.h.a.25.13	yes	44
21.2	odd	6		189.4.h.a.37.10		44
63.2	odd	6		189.4.g.a.100.13		44
63.16	even	3	inner	63.4.g.a.16.10	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.10	✓	44	1.1	even	1	trivial
63.4.g.a.16.10	yes	44	63.16	even	3	inner
63.4.h.a.25.13	yes	44	9.7	even	3
63.4.h.a.58.13	yes	44	7.2	even	3
189.4.g.a.100.13		44	63.2	odd	6
189.4.g.a.172.13		44	3.2	odd	2
189.4.h.a.37.10		44	21.2	odd	6
189.4.h.a.46.10		44	9.2	odd	6