Embedded newform 63.4.g.a.4.15

• Label: 63.4.g.a.4.15
• Level: $63$
• Weight: $4$
• Character: 63.4
• 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.g (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			4.15
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18941 - 2.06012i) q^{2} +(2.07198 - 4.76518i) q^{3} +(1.17060 + 2.02754i) q^{4} +18.4675 q^{5} +(-7.35242 - 9.93628i) q^{6} +(-16.1997 + 8.97605i) q^{7} +24.5999 q^{8} +(-18.4138 - 19.7467i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18941	−	2.06012i	0.420521	−	0.728363i	−0.575470	−	0.817823i	\(-0.695181\pi\)
							0.995990	+	0.0894600i	\(0.0285141\pi\)
\(3\)	2.07198	−	4.76518i	0.398752	−	0.917059i
							\(5\)	18.4675			1.65178			0.825891	−	0.563830i	\(-0.190673\pi\)
0.825891	+	0.563830i	\(0.190673\pi\)
\(7\)	−16.1997	+	8.97605i	−0.874702	+	0.484661i
							\(11\)	−56.7601			−1.55580			−0.777901	−	0.628387i	\(-0.783715\pi\)
−0.777901	+	0.628387i	\(0.783715\pi\)
\(13\)	3.61910	−	6.26847i	0.0772121	−	0.133735i	−0.824834	−	0.565375i	\(-0.808731\pi\)
							0.902046	+	0.431640i	\(0.142065\pi\)
\(17\)	−42.2739	+	73.2205i	−0.603113	+	1.04462i	0.389234	+	0.921139i	\(0.372740\pi\)
							−0.992347	+	0.123483i	\(0.960594\pi\)
\(19\)	−1.77532	−	3.07495i	−0.0214362	−	0.0371286i	0.855108	−	0.518449i	\(-0.173491\pi\)
							−0.876544	+	0.481321i	\(0.840157\pi\)
\(23\)	90.6857			0.822142			0.411071	−	0.911603i	\(-0.365155\pi\)
							0.411071	+	0.911603i	\(0.365155\pi\)
\(29\)	25.6893	+	44.4952i	0.164496	+	0.284915i	0.936476	−	0.350731i	\(-0.114067\pi\)
							−0.771980	+	0.635647i	\(0.780734\pi\)
\(31\)	−78.2320	−	135.502i	−0.453254	−	0.785059i	0.545332	−	0.838220i	\(-0.316404\pi\)
							−0.998586	+	0.0531611i	\(0.983070\pi\)
\(37\)	28.9522	+	50.1467i	0.128641	+	0.222813i	0.923150	−	0.384439i	\(-0.125605\pi\)
							−0.794509	+	0.607252i	\(0.792272\pi\)
\(41\)	−79.8397	+	138.286i	−0.304119	+	0.526749i	−0.977065	−	0.212943i	\(-0.931695\pi\)
							0.672946	+	0.739692i	\(0.265029\pi\)
\(43\)	−20.6544	−	35.7745i	−0.0732505	−	0.126874i	0.827074	−	0.562094i	\(-0.190004\pi\)
							−0.900324	+	0.435220i	\(0.856671\pi\)
\(47\)	−72.4925	+	125.561i	−0.224981	+	0.389679i	−0.956314	−	0.292342i	\(-0.905565\pi\)
							0.731333	+	0.682021i	\(0.238899\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.15	✓	44
3.2	odd	2		189.4.g.a.172.8		44
7.2	even	3		63.4.h.a.58.8	yes	44
9.2	odd	6		189.4.h.a.46.15		44
9.7	even	3		63.4.h.a.25.8	yes	44
21.2	odd	6		189.4.h.a.37.15		44
63.2	odd	6		189.4.g.a.100.8		44
63.16	even	3	inner	63.4.g.a.16.15	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.15	✓	44	1.1	even	1	trivial
63.4.g.a.16.15	yes	44	63.16	even	3	inner
63.4.h.a.25.8	yes	44	9.7	even	3
63.4.h.a.58.8	yes	44	7.2	even	3
189.4.g.a.100.8		44	63.2	odd	6
189.4.g.a.172.8		44	3.2	odd	2
189.4.h.a.37.15		44	21.2	odd	6
189.4.h.a.46.15		44	9.2	odd	6