Newform orbit 63.2.e.a

• Label: 63.2.e.a
• Level: $63$
• Weight: $2$
• Character orbit: 63.e
• Analytic conductor: $0.503$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.503057532734\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - q^{7}+O(q^{10}) \)

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

1.00000

+

1.73205i

0

0

−0.500000

−

2.59808i

0

0

0

46.1

0

0

1.00000

−

1.73205i

0

0

−0.500000

+

2.59808i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.2.e.a	✓	2
3.b	odd	2	1	CM	63.2.e.a	✓	2
4.b	odd	2	1		1008.2.s.j		2
7.b	odd	2	1		441.2.e.c		2
7.c	even	3	1	inner	63.2.e.a	✓	2
7.c	even	3	1		441.2.a.d		1
7.d	odd	6	1		441.2.a.e		1
7.d	odd	6	1		441.2.e.c		2
9.c	even	3	1		567.2.g.d		2
9.c	even	3	1		567.2.h.c		2
9.d	odd	6	1		567.2.g.d		2
9.d	odd	6	1		567.2.h.c		2
12.b	even	2	1		1008.2.s.j		2
21.c	even	2	1		441.2.e.c		2
21.g	even	6	1		441.2.a.e		1
21.g	even	6	1		441.2.e.c		2
21.h	odd	6	1	inner	63.2.e.a	✓	2
21.h	odd	6	1		441.2.a.d		1
28.f	even	6	1		7056.2.a.bf		1
28.g	odd	6	1		1008.2.s.j		2
28.g	odd	6	1		7056.2.a.y		1
63.g	even	3	1		567.2.h.c		2
63.h	even	3	1		567.2.g.d		2
63.j	odd	6	1		567.2.g.d		2
63.n	odd	6	1		567.2.h.c		2
84.j	odd	6	1		7056.2.a.bf		1
84.n	even	6	1		1008.2.s.j		2
84.n	even	6	1		7056.2.a.y		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.e.a	✓	2	1.a	even	1	1	trivial
63.2.e.a	✓	2	3.b	odd	2	1	CM
63.2.e.a	✓	2	7.c	even	3	1	inner
63.2.e.a	✓	2	21.h	odd	6	1	inner
441.2.a.d		1	7.c	even	3	1
441.2.a.d		1	21.h	odd	6	1
441.2.a.e		1	7.d	odd	6	1
441.2.a.e		1	21.g	even	6	1
441.2.e.c		2	7.b	odd	2	1
441.2.e.c		2	7.d	odd	6	1
441.2.e.c		2	21.c	even	2	1
441.2.e.c		2	21.g	even	6	1
567.2.g.d		2	9.c	even	3	1
567.2.g.d		2	9.d	odd	6	1
567.2.g.d		2	63.h	even	3	1
567.2.g.d		2	63.j	odd	6	1
567.2.h.c		2	9.c	even	3	1
567.2.h.c		2	9.d	odd	6	1
567.2.h.c		2	63.g	even	3	1
567.2.h.c		2	63.n	odd	6	1
1008.2.s.j		2	4.b	odd	2	1
1008.2.s.j		2	12.b	even	2	1
1008.2.s.j		2	28.g	odd	6	1
1008.2.s.j		2	84.n	even	6	1
7056.2.a.y		1	28.g	odd	6	1
7056.2.a.y		1	84.n	even	6	1
7056.2.a.bf		1	28.f	even	6	1
7056.2.a.bf		1	84.j	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + T + 7 \)
$11$	\( T^{2} \)