Newform orbit 1764.1.g.c

• Label: 1764.1.g.c
• Level: $1764$
• Weight: $1$
• Character orbit: 1764.g
• Analytic conductor: $0.880$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -7, -84, 12
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1764.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.880350682285\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
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:	$D_4:C_2$
Artin field:	Galois closure of 8.0.1372257936.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - i q^{2} - q^{4} + i q^{8} + 2 i q^{11} + q^{16} + 2 q^{22} + 2 i q^{23} - q^{25} - i q^{32} + 2 q^{37} - 2 i q^{44} + 2 q^{46} + i q^{50} - q^{64} - 2 i q^{71} - 2 i q^{74} - 2 q^{88} - 2 i q^{92} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{16} + 4 q^{22} - 2 q^{25} + 4 q^{37} + 4 q^{46} - 2 q^{64} - 4 q^{88}+O(q^{100}) \)

Character values

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

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

883.1

		1.00000i
	−	1.00000i

−

1.00000i

0

−1.00000

0

0

0

1.00000i

0

0

883.2

1.00000i

0

−1.00000

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
21.c	even	2	1	inner
28.d	even	2	1	inner

Twists

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

    

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

Hecke kernels

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

\( T_{5} \)

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

\( T_{29} \)

Hecke characteristic polynomials

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