Newform orbit 1764.1.y.c

• Label: 1764.1.y.c
• Level: $1764$
• Weight: $1$
• Character orbit: 1764.y
• Analytic conductor: $0.880$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{2}$
• CM/RM discs: -7, -84, 12
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1764.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.880350682285\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{3}, \sqrt{-7})\)
Artin image:	$D_4:C_6$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \)

Character values

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

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{12}^{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

667.1

0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

−0.866025

−

0.500000i

0

0.500000

+

0.866025i

0

0

0

−

1.00000i

0

0

667.2

0.866025

+

0.500000i

0

0.500000

+

0.866025i

0

0

0

1.00000i

0

0

1243.1

−0.866025

+

0.500000i

0

0.500000

−

0.866025i

0

0

0

1.00000i

0

0

1243.2

0.866025

−

0.500000i

0

0.500000

−

0.866025i

0

0

0

−

1.00000i

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
12.b	even	2	1	RM by \(\Q(\sqrt{3}) \)
84.h	odd	2	1	CM by \(\Q(\sqrt{-21}) \)
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
21.c	even	2	1	inner
21.g	even	6	1	inner
21.h	odd	6	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner
84.j	odd	6	1	inner
84.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.1.y.c		4
3.b	odd	2	1	inner	1764.1.y.c		4
4.b	odd	2	1	inner	1764.1.y.c		4
7.b	odd	2	1	CM	1764.1.y.c		4
7.c	even	3	1		1764.1.g.c	✓	2
7.c	even	3	1	inner	1764.1.y.c		4
7.d	odd	6	1		1764.1.g.c	✓	2
7.d	odd	6	1	inner	1764.1.y.c		4
12.b	even	2	1	RM	1764.1.y.c		4
21.c	even	2	1	inner	1764.1.y.c		4
21.g	even	6	1		1764.1.g.c	✓	2
21.g	even	6	1	inner	1764.1.y.c		4
21.h	odd	6	1		1764.1.g.c	✓	2
21.h	odd	6	1	inner	1764.1.y.c		4
28.d	even	2	1	inner	1764.1.y.c		4
28.f	even	6	1		1764.1.g.c	✓	2
28.f	even	6	1	inner	1764.1.y.c		4
28.g	odd	6	1		1764.1.g.c	✓	2
28.g	odd	6	1	inner	1764.1.y.c		4
84.h	odd	2	1	CM	1764.1.y.c		4
84.j	odd	6	1		1764.1.g.c	✓	2
84.j	odd	6	1	inner	1764.1.y.c		4
84.n	even	6	1		1764.1.g.c	✓	2
84.n	even	6	1	inner	1764.1.y.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1764.1.g.c	✓	2	7.c	even	3	1
1764.1.g.c	✓	2	7.d	odd	6	1
1764.1.g.c	✓	2	21.g	even	6	1
1764.1.g.c	✓	2	21.h	odd	6	1
1764.1.g.c	✓	2	28.f	even	6	1
1764.1.g.c	✓	2	28.g	odd	6	1
1764.1.g.c	✓	2	84.j	odd	6	1
1764.1.g.c	✓	2	84.n	even	6	1
1764.1.y.c		4	1.a	even	1	1	trivial
1764.1.y.c		4	3.b	odd	2	1	inner
1764.1.y.c		4	4.b	odd	2	1	inner
1764.1.y.c		4	7.b	odd	2	1	CM
1764.1.y.c		4	7.c	even	3	1	inner
1764.1.y.c		4	7.d	odd	6	1	inner
1764.1.y.c		4	12.b	even	2	1	RM
1764.1.y.c		4	21.c	even	2	1	inner
1764.1.y.c		4	21.g	even	6	1	inner
1764.1.y.c		4	21.h	odd	6	1	inner
1764.1.y.c		4	28.d	even	2	1	inner
1764.1.y.c		4	28.f	even	6	1	inner
1764.1.y.c		4	28.g	odd	6	1	inner
1764.1.y.c		4	84.h	odd	2	1	CM
1764.1.y.c		4	84.j	odd	6	1	inner
1764.1.y.c		4	84.n	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5} \)

\( T_{11}^{4} - 4T_{11}^{2} + 16 \)

\( T_{29} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - T^{2} + 1 \)
$3$	\( T^{4} \)
$5$	\( T^{4} \)
$7$	\( T^{4} \)
$11$	\( T^{4} - 4T^{2} + 16 \)