Embedded newform 63.2.i.a.5.1

• Label: 63.2.i.a.5.1
• Level: $63$
• Weight: $2$
• Character: 63.5
• Analytic conductor: $0.503$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	63.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.503057532734\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	63.5
Dual form			63.2.i.a.38.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.73205i q^{3} -1.00000 q^{4} +(-1.50000 - 2.59808i) q^{5} -3.00000 q^{6} +(2.00000 - 1.73205i) q^{7} +1.73205i q^{8} -3.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 3 q^{5} - 6 q^{6} + 4 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.73205i			1.22474i	0.790569	+	0.612372i	\(0.209785\pi\)
							−0.790569	+	0.612372i	\(0.790215\pi\)
\(3\)			1.73205i			1.00000i
							\(5\)	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	\(-0.932609\pi\)
0.306851	−	0.951757i	\(-0.400725\pi\)
\(7\)	2.00000	−	1.73205i	0.755929	−	0.654654i
							\(11\)	1.50000	+	0.866025i	0.452267	+	0.261116i	0.708787	−	0.705422i	\(-0.249243\pi\)
−0.256520	+	0.966539i	\(0.582576\pi\)
\(13\)	1.50000	+	0.866025i	0.416025	+	0.240192i	0.693375	−	0.720577i	\(-0.256123\pi\)
							−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	\(-0.285189\pi\)
							−0.988583	+	0.150675i	\(0.951855\pi\)
\(19\)	−4.50000	−	2.59808i	−1.03237	−	0.596040i	−0.114708	−	0.993399i	\(-0.536593\pi\)
							−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	4.50000	−	2.59808i	0.938315	−	0.541736i	0.0488832	−	0.998805i	\(-0.484434\pi\)
							0.889432	+	0.457068i	\(0.151100\pi\)
\(29\)	−4.50000	+	2.59808i	−0.835629	+	0.482451i	−0.855776	−	0.517346i	\(-0.826920\pi\)
							0.0201471	+	0.999797i	\(0.493587\pi\)
\(31\)			3.46410i			0.622171i	0.950382	+	0.311086i	\(0.100693\pi\)
							−0.950382	+	0.311086i	\(0.899307\pi\)
\(37\)	−3.50000	+	6.06218i	−0.575396	+	0.996616i	0.420602	+	0.907245i	\(0.361819\pi\)
							−0.995998	+	0.0893706i	\(0.971514\pi\)
\(41\)	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	\(-0.908600\pi\)
							0.724797	+	0.688963i	\(0.241934\pi\)
\(43\)	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	\(-0.190961\pi\)
							−0.901629	+	0.432511i	\(0.857628\pi\)
\(47\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.2.i.a.5.1	✓	2
3.2	odd	2		189.2.i.a.152.1		2
4.3	odd	2		1008.2.ca.a.257.1		2
7.2	even	3		441.2.o.a.293.1		2
7.3	odd	6		63.2.s.a.59.1	yes	2
7.4	even	3		441.2.s.a.374.1		2
7.5	odd	6		441.2.o.b.293.1		2
7.6	odd	2		441.2.i.a.68.1		2
9.2	odd	6		63.2.s.a.47.1	yes	2
9.4	even	3		567.2.p.a.404.1		2
9.5	odd	6		567.2.p.b.404.1		2
9.7	even	3		189.2.s.a.89.1		2
12.11	even	2		3024.2.ca.a.2609.1		2
21.2	odd	6		1323.2.o.b.881.1		2
21.5	even	6		1323.2.o.a.881.1		2
21.11	odd	6		1323.2.s.a.962.1		2
21.17	even	6		189.2.s.a.17.1		2
21.20	even	2		1323.2.i.a.1097.1		2
28.3	even	6		1008.2.df.a.689.1		2
36.7	odd	6		3024.2.df.a.1601.1		2
36.11	even	6		1008.2.df.a.929.1		2
63.2	odd	6		441.2.o.b.146.1		2
63.11	odd	6		441.2.i.a.227.1		2
63.16	even	3		1323.2.o.a.440.1		2
63.20	even	6		441.2.s.a.362.1		2
63.25	even	3		1323.2.i.a.521.1		2
63.31	odd	6		567.2.p.b.80.1		2
63.34	odd	6		1323.2.s.a.656.1		2
63.38	even	6	inner	63.2.i.a.38.1	yes	2
63.47	even	6		441.2.o.a.146.1		2
63.52	odd	6		189.2.i.a.143.1		2
63.59	even	6		567.2.p.a.80.1		2
63.61	odd	6		1323.2.o.b.440.1		2
84.59	odd	6		3024.2.df.a.17.1		2
252.115	even	6		3024.2.ca.a.2033.1		2
252.227	odd	6		1008.2.ca.a.353.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.i.a.5.1	✓	2	1.1	even	1	trivial
63.2.i.a.38.1	yes	2	63.38	even	6	inner
63.2.s.a.47.1	yes	2	9.2	odd	6
63.2.s.a.59.1	yes	2	7.3	odd	6
189.2.i.a.143.1		2	63.52	odd	6
189.2.i.a.152.1		2	3.2	odd	2
189.2.s.a.17.1		2	21.17	even	6
189.2.s.a.89.1		2	9.7	even	3
441.2.i.a.68.1		2	7.6	odd	2
441.2.i.a.227.1		2	63.11	odd	6
441.2.o.a.146.1		2	63.47	even	6
441.2.o.a.293.1		2	7.2	even	3
441.2.o.b.146.1		2	63.2	odd	6
441.2.o.b.293.1		2	7.5	odd	6
441.2.s.a.362.1		2	63.20	even	6
441.2.s.a.374.1		2	7.4	even	3
567.2.p.a.80.1		2	63.59	even	6
567.2.p.a.404.1		2	9.4	even	3
567.2.p.b.80.1		2	63.31	odd	6
567.2.p.b.404.1		2	9.5	odd	6
1008.2.ca.a.257.1		2	4.3	odd	2
1008.2.ca.a.353.1		2	252.227	odd	6
1008.2.df.a.689.1		2	28.3	even	6
1008.2.df.a.929.1		2	36.11	even	6
1323.2.i.a.521.1		2	63.25	even	3
1323.2.i.a.1097.1		2	21.20	even	2
1323.2.o.a.440.1		2	63.16	even	3
1323.2.o.a.881.1		2	21.5	even	6
1323.2.o.b.440.1		2	63.61	odd	6
1323.2.o.b.881.1		2	21.2	odd	6
1323.2.s.a.656.1		2	63.34	odd	6
1323.2.s.a.962.1		2	21.11	odd	6
3024.2.ca.a.2033.1		2	252.115	even	6
3024.2.ca.a.2609.1		2	12.11	even	2
3024.2.df.a.17.1		2	84.59	odd	6
3024.2.df.a.1601.1		2	36.7	odd	6