Newform orbit 1764.2.k.g

• Label: 1764.2.k.g
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.k
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not 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:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
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.

\(f(q)\)	\(=\)	\( q+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

0

0

0

0

1549.1

0

0

0

0

0

0

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	1764.2.k.g		2
3.b	odd	2	1	CM	1764.2.k.g		2
7.b	odd	2	1		1764.2.k.h		2
7.c	even	3	1		1764.2.a.e		1
7.c	even	3	1	inner	1764.2.k.g		2
7.d	odd	6	1		36.2.a.a	✓	1
7.d	odd	6	1		1764.2.k.h		2
21.c	even	2	1		1764.2.k.h		2
21.g	even	6	1		36.2.a.a	✓	1
21.g	even	6	1		1764.2.k.h		2
21.h	odd	6	1		1764.2.a.e		1
21.h	odd	6	1	inner	1764.2.k.g		2
28.f	even	6	1		144.2.a.a		1
28.g	odd	6	1		7056.2.a.bb		1
35.i	odd	6	1		900.2.a.g		1
35.k	even	12	2		900.2.d.b		2
56.j	odd	6	1		576.2.a.e		1
56.m	even	6	1		576.2.a.f		1
63.i	even	6	1		324.2.e.c		2
63.k	odd	6	1		324.2.e.c		2
63.s	even	6	1		324.2.e.c		2
63.t	odd	6	1		324.2.e.c		2
77.i	even	6	1		4356.2.a.g		1
84.j	odd	6	1		144.2.a.a		1
84.n	even	6	1		7056.2.a.bb		1
91.s	odd	6	1		6084.2.a.i		1
91.bb	even	12	2		6084.2.b.f		2
105.p	even	6	1		900.2.a.g		1
105.w	odd	12	2		900.2.d.b		2
112.v	even	12	2		2304.2.d.a		2
112.x	odd	12	2		2304.2.d.q		2
140.s	even	6	1		3600.2.a.e		1
140.x	odd	12	2		3600.2.f.m		2
168.ba	even	6	1		576.2.a.e		1
168.be	odd	6	1		576.2.a.f		1
231.k	odd	6	1		4356.2.a.g		1
252.n	even	6	1		1296.2.i.h		2
252.r	odd	6	1		1296.2.i.h		2
252.bj	even	6	1		1296.2.i.h		2
252.bn	odd	6	1		1296.2.i.h		2
273.ba	even	6	1		6084.2.a.i		1
273.cb	odd	12	2		6084.2.b.f		2
336.bo	even	12	2		2304.2.d.q		2
336.br	odd	12	2		2304.2.d.a		2
420.be	odd	6	1		3600.2.a.e		1
420.br	even	12	2		3600.2.f.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.a.a	✓	1	7.d	odd	6	1
36.2.a.a	✓	1	21.g	even	6	1
144.2.a.a		1	28.f	even	6	1
144.2.a.a		1	84.j	odd	6	1
324.2.e.c		2	63.i	even	6	1
324.2.e.c		2	63.k	odd	6	1
324.2.e.c		2	63.s	even	6	1
324.2.e.c		2	63.t	odd	6	1
576.2.a.e		1	56.j	odd	6	1
576.2.a.e		1	168.ba	even	6	1
576.2.a.f		1	56.m	even	6	1
576.2.a.f		1	168.be	odd	6	1
900.2.a.g		1	35.i	odd	6	1
900.2.a.g		1	105.p	even	6	1
900.2.d.b		2	35.k	even	12	2
900.2.d.b		2	105.w	odd	12	2
1296.2.i.h		2	252.n	even	6	1
1296.2.i.h		2	252.r	odd	6	1
1296.2.i.h		2	252.bj	even	6	1
1296.2.i.h		2	252.bn	odd	6	1
1764.2.a.e		1	7.c	even	3	1
1764.2.a.e		1	21.h	odd	6	1
1764.2.k.g		2	1.a	even	1	1	trivial
1764.2.k.g		2	3.b	odd	2	1	CM
1764.2.k.g		2	7.c	even	3	1	inner
1764.2.k.g		2	21.h	odd	6	1	inner
1764.2.k.h		2	7.b	odd	2	1
1764.2.k.h		2	7.d	odd	6	1
1764.2.k.h		2	21.c	even	2	1
1764.2.k.h		2	21.g	even	6	1
2304.2.d.a		2	112.v	even	12	2
2304.2.d.a		2	336.br	odd	12	2
2304.2.d.q		2	112.x	odd	12	2
2304.2.d.q		2	336.bo	even	12	2
3600.2.a.e		1	140.s	even	6	1
3600.2.a.e		1	420.be	odd	6	1
3600.2.f.m		2	140.x	odd	12	2
3600.2.f.m		2	420.br	even	12	2
4356.2.a.g		1	77.i	even	6	1
4356.2.a.g		1	231.k	odd	6	1
6084.2.a.i		1	91.s	odd	6	1
6084.2.a.i		1	273.ba	even	6	1
6084.2.b.f		2	91.bb	even	12	2
6084.2.b.f		2	273.cb	odd	12	2
7056.2.a.bb		1	28.g	odd	6	1
7056.2.a.bb		1	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5} \)

\( T_{11} \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

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