Embedded newform 63.4.h.a.25.2

• Label: 63.4.h.a.25.2
• 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			25.2
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.2

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−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	0.0751481	+	0.997172i	\(0.476057\pi\)
−0.901151	+	0.433506i	\(0.857276\pi\)
\(7\)	−8.82618	+	16.2818i	−0.476569	+	0.879137i
							\(11\)	−8.59062	−	14.8794i	−0.235470	−	0.407846i	0.723939	−	0.689864i	\(-0.242330\pi\)
−0.959409	+	0.282018i	\(0.908996\pi\)
\(13\)	−27.2834	−	47.2562i	−0.582080	−	1.00819i	−0.995233	−	0.0975305i	\(-0.968906\pi\)
							0.413152	−	0.910662i	\(-0.364428\pi\)
\(17\)	−7.61880	+	13.1962i	−0.108696	+	0.188267i	−0.915242	−	0.402904i	\(-0.868001\pi\)
							0.806546	+	0.591171i	\(0.201334\pi\)
\(19\)	43.7138	+	75.7145i	0.527823	+	0.914216i	0.999474	+	0.0324308i	\(0.0103249\pi\)
							−0.471651	+	0.881785i	\(0.656342\pi\)
\(23\)	48.3456	−	83.7371i	0.438294	−	0.759147i	−0.559264	−	0.828990i	\(-0.688916\pi\)
							0.997558	+	0.0698421i	\(0.0222496\pi\)
\(29\)	57.3750	−	99.3765i	0.367389	−	0.636336i	−0.621768	−	0.783202i	\(-0.713585\pi\)
							0.989156	+	0.146866i	\(0.0469185\pi\)
\(31\)	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	0.873874	−	0.486152i	\(-0.161600\pi\)
							−0.857957	+	0.513721i	\(0.828267\pi\)
\(41\)	−87.4719	−	151.506i	−0.333191	−	0.577103i	0.649945	−	0.759981i	\(-0.274792\pi\)
							−0.983136	+	0.182878i	\(0.941459\pi\)
\(43\)	−77.4059	+	134.071i	−0.274519	+	0.475480i	−0.970014	−	0.243051i	\(-0.921852\pi\)
							0.695495	+	0.718531i	\(0.255185\pi\)
\(47\)	−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.25.2	yes	44
3.2	odd	2		189.4.h.a.46.21		44
7.2	even	3		63.4.g.a.16.21	yes	44
9.4	even	3		63.4.g.a.4.21	✓	44
9.5	odd	6		189.4.g.a.172.2		44
21.2	odd	6		189.4.g.a.100.2		44
63.23	odd	6		189.4.h.a.37.21		44
63.58	even	3	inner	63.4.h.a.58.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.21	✓	44	9.4	even	3
63.4.g.a.16.21	yes	44	7.2	even	3
63.4.h.a.25.2	yes	44	1.1	even	1	trivial
63.4.h.a.58.2	yes	44	63.58	even	3	inner
189.4.g.a.100.2		44	21.2	odd	6
189.4.g.a.172.2		44	9.5	odd	6
189.4.h.a.37.21		44	63.23	odd	6
189.4.h.a.46.21		44	3.2	odd	2