Embedded newform 63.4.h.a.25.13

• Label: 63.4.h.a.25.13
• 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			25.13
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.983694 q^{2} +(-4.22206 + 3.02890i) q^{3} -7.03235 q^{4} +(9.35711 - 16.2070i) q^{5} +(-4.15321 + 2.97951i) q^{6} +(-18.4989 - 0.890133i) q^{7} -14.7872 q^{8} +(8.65150 - 25.5764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.983694			0.347788			0.173894	−	0.984764i	\(-0.444365\pi\)
							0.173894	+	0.984764i	\(0.444365\pi\)
\(3\)	−4.22206	+	3.02890i	−0.812535	+	0.582913i
							\(5\)	9.35711	−	16.2070i	0.836926	−	1.44960i	−0.0555274	−	0.998457i	\(-0.517684\pi\)
0.892453	−	0.451140i	\(-0.148983\pi\)
\(7\)	−18.4989	−	0.890133i	−0.998844	−	0.0480626i
							\(11\)	−11.7479	−	20.3480i	−0.322013	−	0.557742i	0.658890	−	0.752239i	\(-0.271026\pi\)
−0.980903	+	0.194496i	\(0.937693\pi\)
\(13\)	−23.8730	−	41.3493i	−0.509322	−	0.882172i	−0.999942	−	0.0107981i	\(-0.996563\pi\)
							0.490619	−	0.871374i	\(-0.336771\pi\)
\(17\)	−47.7799	+	82.7573i	−0.681667	+	1.18068i	0.292805	+	0.956172i	\(0.405411\pi\)
							−0.974472	+	0.224510i	\(0.927922\pi\)
\(19\)	28.4637	+	49.3006i	0.343686	+	0.595281i	0.985114	−	0.171902i	\(-0.0549912\pi\)
							−0.641428	+	0.767183i	\(0.721658\pi\)
\(23\)	−16.3118	+	28.2528i	−0.147880	+	0.256136i	−0.930444	−	0.366435i	\(-0.880578\pi\)
							0.782564	+	0.622571i	\(0.213912\pi\)
\(29\)	81.3794	−	140.953i	0.521096	−	0.902564i	−0.478603	−	0.878031i	\(-0.658857\pi\)
							0.999699	−	0.0245330i	\(-0.00780988\pi\)
\(31\)	−40.4543			−0.234381			−0.117190	−	0.993109i	\(-0.537389\pi\)
							−0.117190	+	0.993109i	\(0.537389\pi\)
\(37\)	−127.944	−	221.605i	−0.568482	−	0.984639i	−0.996716	−	0.0809715i	\(-0.974198\pi\)
							0.428235	−	0.903667i	\(-0.359136\pi\)
\(41\)	−35.9678	−	62.2981i	−0.137006	−	0.237301i	0.789356	−	0.613935i	\(-0.210415\pi\)
							−0.926362	+	0.376635i	\(0.877081\pi\)
\(43\)	237.322	−	411.054i	0.841658	−	1.45779i	−0.0468344	−	0.998903i	\(-0.514913\pi\)
							0.888492	−	0.458892i	\(-0.151753\pi\)
\(47\)	−264.872			−0.822033			−0.411017	−	0.911628i	\(-0.634826\pi\)
							−0.411017	+	0.911628i	\(0.634826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.13	yes	44
3.2	odd	2		189.4.h.a.46.10		44
7.2	even	3		63.4.g.a.16.10	yes	44
9.4	even	3		63.4.g.a.4.10	✓	44
9.5	odd	6		189.4.g.a.172.13		44
21.2	odd	6		189.4.g.a.100.13		44
63.23	odd	6		189.4.h.a.37.10		44
63.58	even	3	inner	63.4.h.a.58.13	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.10	✓	44	9.4	even	3
63.4.g.a.16.10	yes	44	7.2	even	3
63.4.h.a.25.13	yes	44	1.1	even	1	trivial
63.4.h.a.58.13	yes	44	63.58	even	3	inner
189.4.g.a.100.13		44	21.2	odd	6
189.4.g.a.172.13		44	9.5	odd	6
189.4.h.a.37.10		44	63.23	odd	6
189.4.h.a.46.10		44	3.2	odd	2