Newform orbit 63.3.m.c

• Label: 63.3.m.c
• Level: $63$
• Weight: $3$
• Character orbit: 63.m
• Analytic conductor: $1.717$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.71662566547\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (7 \zeta_{6} - 7) q^{7} + 7 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 9 q^{5} - 7 q^{7} + 14 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(\zeta_{6}\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

10.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

+

0.866025i

0

1.50000

−

2.59808i

4.50000

−

2.59808i

0

−3.50000

+

6.06218i

7.00000

0

4.50000

+

2.59808i

19.1

0.500000

−

0.866025i

0

1.50000

+

2.59808i

4.50000

+

2.59808i

0

−3.50000

−

6.06218i

7.00000

0

4.50000

−

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.3.m.c		2
3.b	odd	2	1		21.3.f.b	✓	2
4.b	odd	2	1		1008.3.cg.g		2
7.b	odd	2	1		441.3.m.e		2
7.c	even	3	1		441.3.d.b		2
7.c	even	3	1		441.3.m.e		2
7.d	odd	6	1	inner	63.3.m.c		2
7.d	odd	6	1		441.3.d.b		2
12.b	even	2	1		336.3.bh.a		2
15.d	odd	2	1		525.3.o.g		2
15.e	even	4	2		525.3.s.c		4
21.c	even	2	1		147.3.f.c		2
21.g	even	6	1		21.3.f.b	✓	2
21.g	even	6	1		147.3.d.b		2
21.h	odd	6	1		147.3.d.b		2
21.h	odd	6	1		147.3.f.c		2
28.f	even	6	1		1008.3.cg.g		2
84.j	odd	6	1		336.3.bh.a		2
84.j	odd	6	1		2352.3.f.d		2
84.n	even	6	1		2352.3.f.d		2
105.p	even	6	1		525.3.o.g		2
105.w	odd	12	2		525.3.s.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.3.f.b	✓	2	3.b	odd	2	1
21.3.f.b	✓	2	21.g	even	6	1
63.3.m.c		2	1.a	even	1	1	trivial
63.3.m.c		2	7.d	odd	6	1	inner
147.3.d.b		2	21.g	even	6	1
147.3.d.b		2	21.h	odd	6	1
147.3.f.c		2	21.c	even	2	1
147.3.f.c		2	21.h	odd	6	1
336.3.bh.a		2	12.b	even	2	1
336.3.bh.a		2	84.j	odd	6	1
441.3.d.b		2	7.c	even	3	1
441.3.d.b		2	7.d	odd	6	1
441.3.m.e		2	7.b	odd	2	1
441.3.m.e		2	7.c	even	3	1
525.3.o.g		2	15.d	odd	2	1
525.3.o.g		2	105.p	even	6	1
525.3.s.c		4	15.e	even	4	2
525.3.s.c		4	105.w	odd	12	2
1008.3.cg.g		2	4.b	odd	2	1
1008.3.cg.g		2	28.f	even	6	1
2352.3.f.d		2	84.j	odd	6	1
2352.3.f.d		2	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 9T + 27 \)
$7$	\( T^{2} + 7T + 49 \)
$11$	\( T^{2} + 11T + 121 \)