Embedded newform 63.4.s.a.47.15

• Label: 63.4.s.a.47.15
• Level: $63$
• Weight: $4$
• Character: 63.47
• 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.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.15
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.57448 - 0.909026i) q^{2} +(0.874715 + 5.12200i) q^{3} +(-2.34735 + 4.06572i) q^{4} -10.3247 q^{5} +(6.03325 + 7.26934i) q^{6} +(15.1453 + 10.6593i) q^{7} +23.0796i q^{8} +(-25.4697 + 8.96058i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.57448	−	0.909026i	0.556662	−	0.321389i	−0.195143	−	0.980775i	\(-0.562517\pi\)
							0.751805	+	0.659386i	\(0.229184\pi\)
\(3\)	0.874715	+	5.12200i	0.168339	+	0.985729i
							\(5\)	−10.3247			−0.923470			−0.461735	−	0.887018i	\(-0.652773\pi\)
−0.461735	+	0.887018i	\(0.652773\pi\)
\(7\)	15.1453	+	10.6593i	0.817769	+	0.575547i
							\(11\)		−	11.7504i		−	0.322080i	−0.986948	−	0.161040i	\(-0.948515\pi\)
0.986948	−	0.161040i	\(-0.0514848\pi\)
\(13\)	46.7419	−	26.9865i	0.997221	−	0.575746i	0.0897962	−	0.995960i	\(-0.471378\pi\)
							0.907425	+	0.420214i	\(0.138045\pi\)
\(17\)	31.9804	+	55.3917i	0.456258	+	0.790262i	0.998760	−	0.0497928i	\(-0.0158561\pi\)
							−0.542502	+	0.840055i	\(0.682523\pi\)
\(19\)	87.6701	+	50.6164i	1.05857	+	0.611168i	0.925037	−	0.379876i	\(-0.124033\pi\)
							0.133536	+	0.991044i	\(0.457367\pi\)
\(23\)		−	194.655i		−	1.76471i	−0.470583	−	0.882356i	\(-0.655956\pi\)
							0.470583	−	0.882356i	\(-0.344044\pi\)
\(29\)	13.4708	+	7.77735i	0.0862572	+	0.0498006i	0.542508	−	0.840051i	\(-0.317475\pi\)
							−0.456251	+	0.889851i	\(0.650808\pi\)
\(31\)	173.244	+	100.023i	1.00373	+	0.579502i	0.909349	−	0.416034i	\(-0.136580\pi\)
							0.0943785	+	0.995536i	\(0.469914\pi\)
\(37\)	−152.809	+	264.674i	−0.678965	+	1.17600i	0.296327	+	0.955086i	\(0.404238\pi\)
							−0.975293	+	0.220916i	\(0.929095\pi\)
\(41\)	−35.3661	−	61.2559i	−0.134714	−	0.233331i	0.790774	−	0.612108i	\(-0.209678\pi\)
							−0.925488	+	0.378777i	\(0.876345\pi\)
\(43\)	52.4919	−	90.9186i	0.186161	−	0.322441i	−0.757806	−	0.652480i	\(-0.773729\pi\)
							0.943967	+	0.330039i	\(0.107062\pi\)
\(47\)	−3.63684	−	6.29920i	−0.0112870	−	0.0195496i	0.860327	−	0.509743i	\(-0.170260\pi\)
							−0.871614	+	0.490193i	\(0.836926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.15	yes	44
3.2	odd	2		189.4.s.a.89.8		44
7.3	odd	6		63.4.i.a.38.15	yes	44
9.4	even	3		189.4.i.a.152.15		44
9.5	odd	6		63.4.i.a.5.8	✓	44
21.17	even	6		189.4.i.a.143.8		44
63.31	odd	6		189.4.s.a.17.8		44
63.59	even	6	inner	63.4.s.a.59.15	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.8	✓	44	9.5	odd	6
63.4.i.a.38.15	yes	44	7.3	odd	6
63.4.s.a.47.15	yes	44	1.1	even	1	trivial
63.4.s.a.59.15	yes	44	63.59	even	6	inner
189.4.i.a.143.8		44	21.17	even	6
189.4.i.a.152.15		44	9.4	even	3
189.4.s.a.17.8		44	63.31	odd	6
189.4.s.a.89.8		44	3.2	odd	2