Embedded newform 63.4.s.a.47.13

• Label: 63.4.s.a.47.13
• 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			47.13
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.725355 - 0.418784i) q^{2} +(0.141982 - 5.19421i) q^{3} +(-3.64924 + 6.32067i) q^{4} -21.9169 q^{5} +(-2.07226 - 3.82711i) q^{6} +(2.19637 - 18.3896i) q^{7} +12.8135i q^{8} +(-26.9597 - 1.47497i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.725355	−	0.418784i	0.256452	−	0.148062i	−0.366263	−	0.930511i	\(-0.619363\pi\)
							0.622715	+	0.782449i	\(0.286030\pi\)
\(3\)	0.141982	−	5.19421i	0.0273245	−	0.999627i
							\(5\)	−21.9169			−1.96030			−0.980152	−	0.198249i	\(-0.936475\pi\)
−0.980152	+	0.198249i	\(0.936475\pi\)
\(7\)	2.19637	−	18.3896i	0.118593	−	0.992943i
							\(11\)			15.4606i			0.423777i	0.977294	+	0.211889i	\(0.0679614\pi\)
−0.977294	+	0.211889i	\(0.932039\pi\)
\(13\)	33.3594	−	19.2600i	0.711709	−	0.410906i	−0.0999842	−	0.994989i	\(-0.531879\pi\)
							0.811694	+	0.584083i	\(0.198546\pi\)
\(17\)	−34.8682	−	60.3936i	−0.497458	−	0.861623i	0.502537	−	0.864555i	\(-0.332400\pi\)
							−0.999996	+	0.00293248i	\(0.999067\pi\)
\(19\)	−55.4679	−	32.0244i	−0.669748	−	0.386679i	0.126233	−	0.992001i	\(-0.459711\pi\)
							−0.795981	+	0.605322i	\(0.793045\pi\)
\(23\)			69.8230i			0.633005i	0.948592	+	0.316502i	\(0.102509\pi\)
							−0.948592	+	0.316502i	\(0.897491\pi\)
\(29\)	−168.609	−	97.3466i	−1.07965	−	0.623338i	−0.148851	−	0.988860i	\(-0.547557\pi\)
							−0.930803	+	0.365521i	\(0.880891\pi\)
\(31\)	68.0764	+	39.3039i	0.394415	+	0.227716i	0.684071	−	0.729415i	\(-0.260208\pi\)
							−0.289656	+	0.957131i	\(0.593541\pi\)
\(37\)	3.34533	−	5.79428i	0.0148640	−	0.0257453i	−0.858498	−	0.512817i	\(-0.828602\pi\)
							0.873362	+	0.487072i	\(0.161935\pi\)
\(41\)	9.21172	+	15.9552i	0.0350885	+	0.0607751i	0.883036	−	0.469305i	\(-0.155495\pi\)
							−0.847948	+	0.530080i	\(0.822162\pi\)
\(43\)	−12.2255	+	21.1752i	−0.0433574	+	0.0750973i	−0.886890	−	0.461981i	\(-0.847139\pi\)
							0.843532	+	0.537078i	\(0.180472\pi\)
\(47\)	−138.270	−	239.491i	−0.429124	−	0.743264i	0.567672	−	0.823255i	\(-0.307844\pi\)
							−0.996796	+	0.0799909i	\(0.974511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.13	yes	44
3.2	odd	2		189.4.s.a.89.10		44
7.3	odd	6		63.4.i.a.38.13	yes	44
9.4	even	3		189.4.i.a.152.13		44
9.5	odd	6		63.4.i.a.5.10	✓	44
21.17	even	6		189.4.i.a.143.10		44
63.31	odd	6		189.4.s.a.17.10		44
63.59	even	6	inner	63.4.s.a.59.13	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.10	✓	44	9.5	odd	6
63.4.i.a.38.13	yes	44	7.3	odd	6
63.4.s.a.47.13	yes	44	1.1	even	1	trivial
63.4.s.a.59.13	yes	44	63.59	even	6	inner
189.4.i.a.143.10		44	21.17	even	6
189.4.i.a.152.13		44	9.4	even	3
189.4.s.a.17.10		44	63.31	odd	6
189.4.s.a.89.10		44	3.2	odd	2