Embedded newform 63.4.f.b.22.4

• Label: 63.4.f.b.22.4
• Level: $63$
• Weight: $4$
• Character: 63.22
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 3*x^15 + 58*x^14 - 129*x^13 + 2107*x^12 - 4455*x^11 + 42901*x^10 - 76404*x^9 + 599392*x^8 - 1089732*x^7 + 4808401*x^6 - 7939134*x^5 + 26225236*x^4 - 39450864*x^3 + 62254768*x^2 - 39660672*x + 21307456
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.4
Root			\(0.797492 + 1.38130i\) of defining polynomial
Character	\(\chi\)	\(=\)	63.22
Dual form			63.4.f.b.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.797492 - 1.38130i) q^{2} +(5.17737 - 0.441383i) q^{3} +(2.72801 - 4.72505i) q^{4} +(1.27816 - 2.21384i) q^{5} +(-4.73860 - 6.79949i) q^{6} +(3.50000 + 6.06218i) q^{7} -21.4622 q^{8} +(26.6104 - 4.57041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 2 q^{3} - 43 q^{4} - 30 q^{5} + 19 q^{6} + 56 q^{7} + 12 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.797492	−	1.38130i	−0.281956	−	0.488362i	0.689910	−	0.723895i	\(-0.257650\pi\)
							−0.971866	+	0.235532i	\(0.924317\pi\)
\(3\)	5.17737	−	0.441383i	0.996386	−	0.0849442i
							\(5\)	1.27816	−	2.21384i	0.114322	−	0.198012i	−0.803186	−	0.595728i	\(-0.796864\pi\)
0.917509	+	0.397716i	\(0.130197\pi\)
\(7\)	3.50000	+	6.06218i	0.188982	+	0.327327i
							\(11\)	−4.04539	−	7.00681i	−0.110885	−	0.192058i	0.805243	−	0.592945i	\(-0.202035\pi\)
−0.916127	+	0.400888i	\(0.868702\pi\)
\(13\)	13.1187	−	22.7222i	0.279882	−	0.484769i	−0.691473	−	0.722402i	\(-0.743038\pi\)
							0.971355	+	0.237633i	\(0.0763714\pi\)
\(17\)	−69.7407			−0.994977			−0.497489	−	0.867471i	\(-0.665744\pi\)
							−0.497489	+	0.867471i	\(0.665744\pi\)
\(19\)	105.751			1.27689			0.638445	−	0.769667i	\(-0.279578\pi\)
							0.638445	+	0.769667i	\(0.279578\pi\)
\(23\)	−77.1585	+	133.642i	−0.699507	+	1.21158i	0.269131	+	0.963104i	\(0.413263\pi\)
							−0.968638	+	0.248477i	\(0.920070\pi\)
\(29\)	36.3274	+	62.9209i	0.232615	+	0.402900i	0.958577	−	0.284834i	\(-0.0919386\pi\)
							−0.725962	+	0.687735i	\(0.758605\pi\)
\(31\)	−141.400	+	244.911i	−0.819230	+	1.41895i	0.0870211	+	0.996206i	\(0.472265\pi\)
							−0.906251	+	0.422741i	\(0.861068\pi\)
\(37\)	25.7974			0.114623			0.0573117	−	0.998356i	\(-0.481747\pi\)
							0.0573117	+	0.998356i	\(0.481747\pi\)
\(41\)	43.5544	−	75.4385i	0.165904	−	0.287354i	−0.771072	−	0.636748i	\(-0.780279\pi\)
							0.936976	+	0.349394i	\(0.113613\pi\)
\(43\)	−44.5553	−	77.1721i	−0.158014	−	0.273689i	0.776138	−	0.630563i	\(-0.217176\pi\)
							−0.934153	+	0.356874i	\(0.883843\pi\)
\(47\)	−157.309	−	272.467i	−0.488209	−	0.845603i	0.511699	−	0.859165i	\(-0.329016\pi\)
							−0.999908	+	0.0135621i	\(0.995683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.f.b.22.4	✓	16
3.2	odd	2		189.4.f.b.64.5		16
9.2	odd	6		189.4.f.b.127.5		16
9.4	even	3		567.4.a.i.1.5		8
9.5	odd	6		567.4.a.g.1.4		8
9.7	even	3	inner	63.4.f.b.43.4	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.4	✓	16	1.1	even	1	trivial
63.4.f.b.43.4	yes	16	9.7	even	3	inner
189.4.f.b.64.5		16	3.2	odd	2
189.4.f.b.127.5		16	9.2	odd	6
567.4.a.g.1.4		8	9.5	odd	6
567.4.a.i.1.5		8	9.4	even	3