Embedded newform 63.4.h.a.58.10

• Label: 63.4.h.a.58.10
• Level: $63$
• Weight: $4$
• Character: 63.58
• 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.h (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			58.10
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.590775 q^{2} +(4.35408 - 2.83584i) q^{3} -7.65099 q^{4} +(-5.49223 - 9.51282i) q^{5} +(-2.57228 + 1.67534i) q^{6} +(-16.7104 - 7.98524i) q^{7} +9.24621 q^{8} +(10.9161 - 24.6949i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.590775			−0.208870			−0.104435	−	0.994532i	\(-0.533303\pi\)
							−0.104435	+	0.994532i	\(0.533303\pi\)
\(3\)	4.35408	−	2.83584i	0.837943	−	0.545757i
							\(5\)	−5.49223	−	9.51282i	−0.491240	−	0.850852i	0.508709	−	0.860938i	\(-0.330123\pi\)
−0.999949	+	0.0100861i	\(0.996789\pi\)
\(7\)	−16.7104	−	7.98524i	−0.902274	−	0.431162i
							\(11\)	−22.8924	+	39.6508i	−0.627483	+	1.08683i	0.360572	+	0.932731i	\(0.382581\pi\)
−0.988055	+	0.154101i	\(0.950752\pi\)
\(13\)	37.1957	−	64.4248i	0.793556	−	1.37448i	−0.130196	−	0.991488i	\(-0.541561\pi\)
							0.923752	−	0.382991i	\(-0.125106\pi\)
\(17\)	−40.4681	−	70.0928i	−0.577351	−	1.00000i	−0.995782	−	0.0917521i	\(-0.970753\pi\)
							0.418431	−	0.908248i	\(-0.362580\pi\)
\(19\)	8.05953	−	13.9595i	0.0973149	−	0.168554i	−0.813258	−	0.581904i	\(-0.802308\pi\)
							0.910572	+	0.413350i	\(0.135641\pi\)
\(23\)	67.0601	+	116.152i	0.607957	+	1.05301i	0.991577	+	0.129521i	\(0.0413439\pi\)
							−0.383620	+	0.923491i	\(0.625323\pi\)
\(29\)	114.686	+	198.643i	0.734370	+	1.27197i	0.954999	+	0.296609i	\(0.0958557\pi\)
							−0.220629	+	0.975358i	\(0.570811\pi\)
\(31\)	91.6652			0.531083			0.265541	−	0.964099i	\(-0.414449\pi\)
							0.265541	+	0.964099i	\(0.414449\pi\)
\(37\)	5.76196	−	9.98001i	0.0256016	−	0.0443434i	−0.852941	−	0.522008i	\(-0.825183\pi\)
							0.878542	+	0.477664i	\(0.158517\pi\)
\(41\)	56.3705	−	97.6366i	0.214722	−	0.371909i	−0.738465	−	0.674292i	\(-0.764449\pi\)
							0.953187	+	0.302383i	\(0.0977822\pi\)
\(43\)	−248.834	−	430.992i	−0.882483	−	1.52851i	−0.848572	−	0.529080i	\(-0.822537\pi\)
							−0.0339110	−	0.999425i	\(-0.510796\pi\)
\(47\)	−82.9547			−0.257451			−0.128725	−	0.991680i	\(-0.541089\pi\)
							−0.128725	+	0.991680i	\(0.541089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.58.10	yes	44
3.2	odd	2		189.4.h.a.37.13		44
7.4	even	3		63.4.g.a.4.13	✓	44
9.2	odd	6		189.4.g.a.100.10		44
9.7	even	3		63.4.g.a.16.13	yes	44
21.11	odd	6		189.4.g.a.172.10		44
63.11	odd	6		189.4.h.a.46.13		44
63.25	even	3	inner	63.4.h.a.25.10	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.13	✓	44	7.4	even	3
63.4.g.a.16.13	yes	44	9.7	even	3
63.4.h.a.25.10	yes	44	63.25	even	3	inner
63.4.h.a.58.10	yes	44	1.1	even	1	trivial
189.4.g.a.100.10		44	9.2	odd	6
189.4.g.a.172.10		44	21.11	odd	6
189.4.h.a.37.13		44	3.2	odd	2
189.4.h.a.46.13		44	63.11	odd	6