Embedded newform 63.4.h.a.58.14

• Label: 63.4.h.a.58.14
• 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			58.14
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.33560 q^{2} +(-3.51545 + 3.82643i) q^{3} -6.21617 q^{4} +(-4.50235 - 7.79829i) q^{5} +(-4.69524 + 5.11058i) q^{6} +(-3.16069 - 18.2486i) q^{7} -18.9871 q^{8} +(-2.28318 - 26.9033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33560			0.472206			0.236103	−	0.971728i	\(-0.424130\pi\)
							0.236103	+	0.971728i	\(0.424130\pi\)
\(3\)	−3.51545	+	3.82643i	−0.676549	+	0.736397i
							\(5\)	−4.50235	−	7.79829i	−0.402702	−	0.697501i	0.591349	−	0.806416i	\(-0.298596\pi\)
−0.994051	+	0.108915i	\(0.965262\pi\)
\(7\)	−3.16069	−	18.2486i	−0.170661	−	0.985330i
							\(11\)	−14.6762	+	25.4199i	−0.402277	+	0.696764i	−0.994000	−	0.109377i	\(-0.965114\pi\)
0.591724	+	0.806141i	\(0.298448\pi\)
\(13\)	−21.1758	+	36.6776i	−0.451779	+	0.782504i	−0.998497	−	0.0548133i	\(-0.982544\pi\)
							0.546718	+	0.837317i	\(0.315877\pi\)
\(17\)	−2.56927	−	4.45010i	−0.0366552	−	0.0634888i	0.847116	−	0.531408i	\(-0.178337\pi\)
							−0.883771	+	0.467920i	\(0.845004\pi\)
\(19\)	−71.2462	+	123.402i	−0.860263	+	1.49002i	0.0114121	+	0.999935i	\(0.496367\pi\)
							−0.871675	+	0.490084i	\(0.836966\pi\)
\(23\)	−89.0414	−	154.224i	−0.807235	−	1.39817i	−0.914772	−	0.403972i	\(-0.867629\pi\)
							0.107536	−	0.994201i	\(-0.465704\pi\)
\(29\)	−109.202	−	189.143i	−0.699249	−	1.21114i	−0.968727	−	0.248129i	\(-0.920184\pi\)
							0.269478	−	0.963007i	\(-0.413149\pi\)
\(31\)	147.854			0.856627			0.428314	−	0.903630i	\(-0.359108\pi\)
							0.428314	+	0.903630i	\(0.359108\pi\)
\(37\)	−21.2781	+	36.8547i	−0.0945431	+	0.163753i	−0.909418	−	0.415884i	\(-0.863472\pi\)
							0.814875	+	0.579637i	\(0.196806\pi\)
\(41\)	83.7600	−	145.077i	0.319051	−	0.552613i	−0.661239	−	0.750175i	\(-0.729969\pi\)
							0.980290	+	0.197562i	\(0.0633024\pi\)
\(43\)	121.454	+	210.365i	0.430734	+	0.746054i	0.996937	−	0.0782125i	\(-0.0249213\pi\)
							−0.566202	+	0.824266i	\(0.691588\pi\)
\(47\)	−76.5135			−0.237460			−0.118730	−	0.992927i	\(-0.537882\pi\)
							−0.118730	+	0.992927i	\(0.537882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.58.14	yes	44
3.2	odd	2		189.4.h.a.37.9		44
7.4	even	3		63.4.g.a.4.9	✓	44
9.2	odd	6		189.4.g.a.100.14		44
9.7	even	3		63.4.g.a.16.9	yes	44
21.11	odd	6		189.4.g.a.172.14		44
63.11	odd	6		189.4.h.a.46.9		44
63.25	even	3	inner	63.4.h.a.25.14	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.9	✓	44	7.4	even	3
63.4.g.a.16.9	yes	44	9.7	even	3
63.4.h.a.25.14	yes	44	63.25	even	3	inner
63.4.h.a.58.14	yes	44	1.1	even	1	trivial
189.4.g.a.100.14		44	9.2	odd	6
189.4.g.a.172.14		44	21.11	odd	6
189.4.h.a.37.9		44	3.2	odd	2
189.4.h.a.46.9		44	63.11	odd	6