Newform orbit 1764.2.j.c

• Label: 1764.2.j.c
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.j
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 2 q^{5} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

589.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

1.50000

−

0.866025i

0

−1.00000

+

1.73205i

0

0

0

1.50000

−

2.59808i

0

1177.1

0

1.50000

+

0.866025i

0

−1.00000

−

1.73205i

0

0

0

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.2.j.c		2
3.b	odd	2	1		5292.2.j.c		2
7.b	odd	2	1		1764.2.j.a		2
7.c	even	3	1		252.2.i.a	✓	2
7.c	even	3	1		252.2.l.a	yes	2
7.d	odd	6	1		1764.2.i.b		2
7.d	odd	6	1		1764.2.l.b		2
9.c	even	3	1	inner	1764.2.j.c		2
9.d	odd	6	1		5292.2.j.c		2
21.c	even	2	1		5292.2.j.b		2
21.g	even	6	1		5292.2.i.b		2
21.g	even	6	1		5292.2.l.b		2
21.h	odd	6	1		756.2.i.a		2
21.h	odd	6	1		756.2.l.a		2
28.g	odd	6	1		1008.2.q.f		2
28.g	odd	6	1		1008.2.t.b		2
63.g	even	3	1		252.2.i.a	✓	2
63.g	even	3	1		2268.2.k.a		2
63.h	even	3	1		252.2.l.a	yes	2
63.h	even	3	1		2268.2.k.a		2
63.i	even	6	1		5292.2.l.b		2
63.j	odd	6	1		756.2.l.a		2
63.j	odd	6	1		2268.2.k.b		2
63.k	odd	6	1		1764.2.i.b		2
63.l	odd	6	1		1764.2.j.a		2
63.n	odd	6	1		756.2.i.a		2
63.n	odd	6	1		2268.2.k.b		2
63.o	even	6	1		5292.2.j.b		2
63.s	even	6	1		5292.2.i.b		2
63.t	odd	6	1		1764.2.l.b		2
84.n	even	6	1		3024.2.q.e		2
84.n	even	6	1		3024.2.t.b		2
252.o	even	6	1		3024.2.q.e		2
252.u	odd	6	1		1008.2.t.b		2
252.bb	even	6	1		3024.2.t.b		2
252.bl	odd	6	1		1008.2.q.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.i.a	✓	2	7.c	even	3	1
252.2.i.a	✓	2	63.g	even	3	1
252.2.l.a	yes	2	7.c	even	3	1
252.2.l.a	yes	2	63.h	even	3	1
756.2.i.a		2	21.h	odd	6	1
756.2.i.a		2	63.n	odd	6	1
756.2.l.a		2	21.h	odd	6	1
756.2.l.a		2	63.j	odd	6	1
1008.2.q.f		2	28.g	odd	6	1
1008.2.q.f		2	252.bl	odd	6	1
1008.2.t.b		2	28.g	odd	6	1
1008.2.t.b		2	252.u	odd	6	1
1764.2.i.b		2	7.d	odd	6	1
1764.2.i.b		2	63.k	odd	6	1
1764.2.j.a		2	7.b	odd	2	1
1764.2.j.a		2	63.l	odd	6	1
1764.2.j.c		2	1.a	even	1	1	trivial
1764.2.j.c		2	9.c	even	3	1	inner
1764.2.l.b		2	7.d	odd	6	1
1764.2.l.b		2	63.t	odd	6	1
2268.2.k.a		2	63.g	even	3	1
2268.2.k.a		2	63.h	even	3	1
2268.2.k.b		2	63.j	odd	6	1
2268.2.k.b		2	63.n	odd	6	1
3024.2.q.e		2	84.n	even	6	1
3024.2.q.e		2	252.o	even	6	1
3024.2.t.b		2	84.n	even	6	1
3024.2.t.b		2	252.bb	even	6	1
5292.2.i.b		2	21.g	even	6	1
5292.2.i.b		2	63.s	even	6	1
5292.2.j.b		2	21.c	even	2	1
5292.2.j.b		2	63.o	even	6	1
5292.2.j.c		2	3.b	odd	2	1
5292.2.j.c		2	9.d	odd	6	1
5292.2.l.b		2	21.g	even	6	1
5292.2.l.b		2	63.i	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T + 3 \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T + 16 \)