Embedded newform 63.4.h.a.58.2

• Label: 63.4.h.a.58.2
• 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			58.2
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-5.07998 q^{2} +(-4.95329 - 1.57000i) q^{3} +17.8062 q^{4} +(-9.23499 - 15.9955i) q^{5} +(25.1626 + 7.97558i) q^{6} +(-8.82618 - 16.2818i) q^{7} -49.8155 q^{8} +(22.0702 + 15.5533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.07998			−1.79605			−0.898023	−	0.439949i	\(-0.854996\pi\)
							−0.898023	+	0.439949i	\(0.854996\pi\)
\(3\)	−4.95329	−	1.57000i	−0.953261	−	0.302147i
							\(5\)	−9.23499	−	15.9955i	−0.826003	−	1.43068i	−0.901151	−	0.433506i	\(-0.857276\pi\)
0.0751481	−	0.997172i	\(-0.476057\pi\)
\(7\)	−8.82618	−	16.2818i	−0.476569	−	0.879137i
							\(11\)	−8.59062	+	14.8794i	−0.235470	+	0.407846i	−0.959409	−	0.282018i	\(-0.908996\pi\)
0.723939	+	0.689864i	\(0.242330\pi\)
\(13\)	−27.2834	+	47.2562i	−0.582080	+	1.00819i	0.413152	+	0.910662i	\(0.364428\pi\)
							−0.995233	+	0.0975305i	\(0.968906\pi\)
\(17\)	−7.61880	−	13.1962i	−0.108696	−	0.188267i	0.806546	−	0.591171i	\(-0.201334\pi\)
							−0.915242	+	0.402904i	\(0.868001\pi\)
\(19\)	43.7138	−	75.7145i	0.527823	−	0.914216i	−0.471651	−	0.881785i	\(-0.656342\pi\)
							0.999474	−	0.0324308i	\(-0.0103249\pi\)
\(23\)	48.3456	+	83.7371i	0.438294	+	0.759147i	0.997558	−	0.0698421i	\(-0.0222496\pi\)
							−0.559264	+	0.828990i	\(0.688916\pi\)
\(29\)	57.3750	+	99.3765i	0.367389	+	0.636336i	0.989156	−	0.146866i	\(-0.0469185\pi\)
							−0.621768	+	0.783202i	\(0.713585\pi\)
\(31\)	50.0365			0.289897			0.144949	−	0.989439i	\(-0.453698\pi\)
							0.144949	+	0.989439i	\(0.453698\pi\)
\(37\)	3.58236	−	6.20483i	0.0159172	−	0.0275694i	−0.857957	−	0.513721i	\(-0.828267\pi\)
							0.873874	+	0.486152i	\(0.161600\pi\)
\(41\)	−87.4719	+	151.506i	−0.333191	+	0.577103i	−0.983136	−	0.182878i	\(-0.941459\pi\)
							0.649945	+	0.759981i	\(0.274792\pi\)
\(43\)	−77.4059	−	134.071i	−0.274519	−	0.475480i	0.695495	−	0.718531i	\(-0.255185\pi\)
							−0.970014	+	0.243051i	\(0.921852\pi\)
\(47\)	−117.733			−0.365386			−0.182693	−	0.983170i	\(-0.558481\pi\)
							−0.182693	+	0.983170i	\(0.558481\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.58.2	yes	44
3.2	odd	2		189.4.h.a.37.21		44
7.4	even	3		63.4.g.a.4.21	✓	44
9.2	odd	6		189.4.g.a.100.2		44
9.7	even	3		63.4.g.a.16.21	yes	44
21.11	odd	6		189.4.g.a.172.2		44
63.11	odd	6		189.4.h.a.46.21		44
63.25	even	3	inner	63.4.h.a.25.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.21	✓	44	7.4	even	3
63.4.g.a.16.21	yes	44	9.7	even	3
63.4.h.a.25.2	yes	44	63.25	even	3	inner
63.4.h.a.58.2	yes	44	1.1	even	1	trivial
189.4.g.a.100.2		44	9.2	odd	6
189.4.g.a.172.2		44	21.11	odd	6
189.4.h.a.37.21		44	3.2	odd	2
189.4.h.a.46.21		44	63.11	odd	6