Embedded newform 63.4.s.a.47.6

• Label: 63.4.s.a.47.6
• 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			47.6
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00186 + 1.73312i) q^{2} +(2.62001 + 4.48727i) q^{3} +(2.00744 - 3.47699i) q^{4} +18.0540 q^{5} +(-15.6419 - 8.92935i) q^{6} +(12.2090 + 13.9263i) q^{7} -13.8134i q^{8} +(-13.2711 + 23.5133i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00186	+	1.73312i	−1.06132	+	0.612752i	−0.925796	−	0.378023i	\(-0.876604\pi\)
							−0.135521	+	0.990774i	\(0.543271\pi\)
\(3\)	2.62001	+	4.48727i	0.504220	+	0.863575i
							\(5\)	18.0540			1.61480			0.807400	−	0.590004i	\(-0.200874\pi\)
0.807400	+	0.590004i	\(0.200874\pi\)
\(7\)	12.2090	+	13.9263i	0.659223	+	0.751947i
							\(11\)		−	11.7949i		−	0.323300i	−0.986848	−	0.161650i	\(-0.948318\pi\)
0.986848	−	0.161650i	\(-0.0516816\pi\)
\(13\)	−45.9746	+	26.5435i	−0.980851	+	0.566295i	−0.902527	−	0.430633i	\(-0.858290\pi\)
							−0.0783244	+	0.996928i	\(0.524957\pi\)
\(17\)	−5.93967	−	10.2878i	−0.0847401	−	0.146774i	0.820540	−	0.571589i	\(-0.193673\pi\)
							−0.905280	+	0.424814i	\(0.860339\pi\)
\(19\)	−12.1984	−	7.04273i	−0.147289	−	0.0850376i	0.424544	−	0.905407i	\(-0.360434\pi\)
							−0.571834	+	0.820370i	\(0.693768\pi\)
\(23\)		−	161.271i		−	1.46206i	−0.682344	−	0.731031i	\(-0.739040\pi\)
							0.682344	−	0.731031i	\(-0.260960\pi\)
\(29\)	131.443	+	75.8889i	0.841670	+	0.485939i	0.857832	−	0.513931i	\(-0.171811\pi\)
							−0.0161612	+	0.999869i	\(0.505145\pi\)
\(31\)	−78.0806	−	45.0799i	−0.452377	−	0.261180i	0.256457	−	0.966556i	\(-0.417445\pi\)
							−0.708834	+	0.705376i	\(0.750778\pi\)
\(37\)	185.189	−	320.757i	0.822835	−	1.42519i	−0.0807273	−	0.996736i	\(-0.525724\pi\)
							0.903563	−	0.428456i	\(-0.140942\pi\)
\(41\)	129.206	+	223.791i	0.492161	+	0.852447i	0.999959	−	0.00902843i	\(-0.00287388\pi\)
							−0.507798	+	0.861476i	\(0.669541\pi\)
\(43\)	−173.288	+	300.144i	−0.614563	+	1.06446i	0.375897	+	0.926661i	\(0.377334\pi\)
							−0.990461	+	0.137794i	\(0.955999\pi\)
\(47\)	−35.8309	−	62.0610i	−0.111202	−	0.192607i	0.805053	−	0.593202i	\(-0.202137\pi\)
							−0.916255	+	0.400596i	\(0.868803\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.6	yes	44
3.2	odd	2		189.4.s.a.89.17		44
7.3	odd	6		63.4.i.a.38.6	yes	44
9.4	even	3		189.4.i.a.152.6		44
9.5	odd	6		63.4.i.a.5.17	✓	44
21.17	even	6		189.4.i.a.143.17		44
63.31	odd	6		189.4.s.a.17.17		44
63.59	even	6	inner	63.4.s.a.59.6	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.17	✓	44	9.5	odd	6
63.4.i.a.38.6	yes	44	7.3	odd	6
63.4.s.a.47.6	yes	44	1.1	even	1	trivial
63.4.s.a.59.6	yes	44	63.59	even	6	inner
189.4.i.a.143.17		44	21.17	even	6
189.4.i.a.152.6		44	9.4	even	3
189.4.s.a.17.17		44	63.31	odd	6
189.4.s.a.89.17		44	3.2	odd	2