Embedded newform 63.4.g.a.4.6

• Label: 63.4.g.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67658 + 2.90391i) q^{2} +(0.949950 - 5.10858i) q^{3} +(-1.62181 - 2.80905i) q^{4} -8.70652 q^{5} +(13.2422 + 11.3235i) q^{6} +(-14.4030 - 11.6428i) q^{7} -15.9489 q^{8} +(-25.1952 - 9.70580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67658	+	2.90391i	−0.592759	+	1.02669i	0.401100	+	0.916034i	\(0.368628\pi\)
							−0.993859	+	0.110654i	\(0.964705\pi\)
\(3\)	0.949950	−	5.10858i	0.182818	−	0.983147i
							\(5\)	−8.70652			−0.778735			−0.389367	−	0.921083i	\(-0.627306\pi\)
−0.389367	+	0.921083i	\(0.627306\pi\)
\(7\)	−14.4030	−	11.6428i	−0.777689	−	0.628649i
							\(11\)	−11.7974			−0.323367			−0.161684	−	0.986843i	\(-0.551692\pi\)
−0.161684	+	0.986843i	\(0.551692\pi\)
\(13\)	26.5809	−	46.0395i	0.567094	−	0.982235i	−0.429758	−	0.902944i	\(-0.641401\pi\)
							0.996852	−	0.0792910i	\(-0.0252656\pi\)
\(17\)	−22.5463	+	39.0513i	−0.321663	+	0.557137i	−0.980831	−	0.194858i	\(-0.937575\pi\)
							0.659168	+	0.751996i	\(0.270909\pi\)
\(19\)	−13.0906	−	22.6736i	−0.158062	−	0.273772i	0.776107	−	0.630601i	\(-0.217191\pi\)
							−0.934170	+	0.356828i	\(0.883858\pi\)
\(23\)	−58.1880			−0.527523			−0.263762	−	0.964588i	\(-0.584963\pi\)
							−0.263762	+	0.964588i	\(0.584963\pi\)
\(29\)	36.7015	+	63.5688i	0.235010	+	0.407049i	0.959276	−	0.282472i	\(-0.0911544\pi\)
							−0.724266	+	0.689521i	\(0.757821\pi\)
\(31\)	157.403	+	272.629i	0.911947	+	1.57954i	0.811311	+	0.584615i	\(0.198754\pi\)
							0.100636	+	0.994923i	\(0.467912\pi\)
\(37\)	−189.337	−	327.941i	−0.841263	−	1.45711i	−0.888827	−	0.458242i	\(-0.848479\pi\)
							0.0475640	−	0.998868i	\(-0.484854\pi\)
\(41\)	181.180	−	313.812i	0.690134	−	1.19535i	−0.281659	−	0.959515i	\(-0.590885\pi\)
							0.971794	−	0.235833i	\(-0.0757819\pi\)
\(43\)	58.7824	+	101.814i	0.208470	+	0.361081i	0.951233	−	0.308474i	\(-0.0998182\pi\)
							−0.742762	+	0.669555i	\(0.766485\pi\)
\(47\)	−114.110	+	197.644i	−0.354142	+	0.613392i	−0.986971	−	0.160900i	\(-0.948560\pi\)
							0.632829	+	0.774292i	\(0.281894\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.6	✓	44
3.2	odd	2		189.4.g.a.172.17		44
7.2	even	3		63.4.h.a.58.17	yes	44
9.2	odd	6		189.4.h.a.46.6		44
9.7	even	3		63.4.h.a.25.17	yes	44
21.2	odd	6		189.4.h.a.37.6		44
63.2	odd	6		189.4.g.a.100.17		44
63.16	even	3	inner	63.4.g.a.16.6	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.6	✓	44	1.1	even	1	trivial
63.4.g.a.16.6	yes	44	63.16	even	3	inner
63.4.h.a.25.17	yes	44	9.7	even	3
63.4.h.a.58.17	yes	44	7.2	even	3
189.4.g.a.100.17		44	63.2	odd	6
189.4.g.a.172.17		44	3.2	odd	2
189.4.h.a.37.6		44	21.2	odd	6
189.4.h.a.46.6		44	9.2	odd	6