Embedded newform 63.4.h.a.25.10

• Label: 63.4.h.a.25.10
• Level: $63$
• Weight: $4$
• Character: 63.25
• 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			25.10
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.10

$q$-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 \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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	−0.999949	−	0.0100861i	\(-0.996789\pi\)
0.508709	+	0.860938i	\(0.330123\pi\)
\(7\)	−16.7104	+	7.98524i	−0.902274	+	0.431162i
							\(11\)	−22.8924	−	39.6508i	−0.627483	−	1.08683i	−0.988055	−	0.154101i	\(-0.950752\pi\)
0.360572	−	0.932731i	\(-0.382581\pi\)
\(13\)	37.1957	+	64.4248i	0.793556	+	1.37448i	0.923752	+	0.382991i	\(0.125106\pi\)
							−0.130196	+	0.991488i	\(0.541561\pi\)
\(17\)	−40.4681	+	70.0928i	−0.577351	+	1.00000i	0.418431	+	0.908248i	\(0.362580\pi\)
							−0.995782	+	0.0917521i	\(0.970753\pi\)
\(19\)	8.05953	+	13.9595i	0.0973149	+	0.168554i	0.910572	−	0.413350i	\(-0.135641\pi\)
							−0.813258	+	0.581904i	\(0.802308\pi\)
\(23\)	67.0601	−	116.152i	0.607957	−	1.05301i	−0.383620	−	0.923491i	\(-0.625323\pi\)
							0.991577	−	0.129521i	\(-0.0413439\pi\)
\(29\)	114.686	−	198.643i	0.734370	−	1.27197i	−0.220629	−	0.975358i	\(-0.570811\pi\)
							0.954999	−	0.296609i	\(-0.0958557\pi\)
\(31\)	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	0.878542	−	0.477664i	\(-0.158517\pi\)
							−0.852941	+	0.522008i	\(0.825183\pi\)
\(41\)	56.3705	+	97.6366i	0.214722	+	0.371909i	0.953187	−	0.302383i	\(-0.0977822\pi\)
							−0.738465	+	0.674292i	\(0.764449\pi\)
\(43\)	−248.834	+	430.992i	−0.882483	+	1.52851i	−0.0339110	+	0.999425i	\(0.510796\pi\)
							−0.848572	+	0.529080i	\(0.822537\pi\)
\(47\)	−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.25.10	yes	44
3.2	odd	2		189.4.h.a.46.13		44
7.2	even	3		63.4.g.a.16.13	yes	44
9.4	even	3		63.4.g.a.4.13	✓	44
9.5	odd	6		189.4.g.a.172.10		44
21.2	odd	6		189.4.g.a.100.10		44
63.23	odd	6		189.4.h.a.37.13		44
63.58	even	3	inner	63.4.h.a.58.10	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.13	✓	44	9.4	even	3
63.4.g.a.16.13	yes	44	7.2	even	3
63.4.h.a.25.10	yes	44	1.1	even	1	trivial
63.4.h.a.58.10	yes	44	63.58	even	3	inner
189.4.g.a.100.10		44	21.2	odd	6
189.4.g.a.172.10		44	9.5	odd	6
189.4.h.a.37.13		44	63.23	odd	6
189.4.h.a.46.13		44	3.2	odd	2