Newform orbit 1764.2.e.b

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 2 q^{4} + \beta q^{5} - 2 \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1079.1

	−	1.41421i
		1.41421i

−

1.41421i

0

−2.00000

−

1.41421i

0

0

2.82843i

0

−2.00000

1079.2

1.41421i

0

−2.00000

1.41421i

0

0

−

2.82843i

0

−2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
3.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.2.e.b		2
3.b	odd	2	1	inner	1764.2.e.b		2
4.b	odd	2	1	CM	1764.2.e.b		2
7.b	odd	2	1		36.2.b.a	✓	2
12.b	even	2	1	inner	1764.2.e.b		2
21.c	even	2	1		36.2.b.a	✓	2
28.d	even	2	1		36.2.b.a	✓	2
35.c	odd	2	1		900.2.e.b		2
35.f	even	4	2		900.2.h.a		4
56.e	even	2	1		576.2.c.b		2
56.h	odd	2	1		576.2.c.b		2
63.l	odd	6	2		324.2.h.c		4
63.o	even	6	2		324.2.h.c		4
84.h	odd	2	1		36.2.b.a	✓	2
105.g	even	2	1		900.2.e.b		2
105.k	odd	4	2		900.2.h.a		4
112.j	even	4	2		2304.2.f.d		4
112.l	odd	4	2		2304.2.f.d		4
140.c	even	2	1		900.2.e.b		2
140.j	odd	4	2		900.2.h.a		4
168.e	odd	2	1		576.2.c.b		2
168.i	even	2	1		576.2.c.b		2
252.s	odd	6	2		324.2.h.c		4
252.bi	even	6	2		324.2.h.c		4
336.v	odd	4	2		2304.2.f.d		4
336.y	even	4	2		2304.2.f.d		4
420.o	odd	2	1		900.2.e.b		2
420.w	even	4	2		900.2.h.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.b.a	✓	2	7.b	odd	2	1
36.2.b.a	✓	2	21.c	even	2	1
36.2.b.a	✓	2	28.d	even	2	1
36.2.b.a	✓	2	84.h	odd	2	1
324.2.h.c		4	63.l	odd	6	2
324.2.h.c		4	63.o	even	6	2
324.2.h.c		4	252.s	odd	6	2
324.2.h.c		4	252.bi	even	6	2
576.2.c.b		2	56.e	even	2	1
576.2.c.b		2	56.h	odd	2	1
576.2.c.b		2	168.e	odd	2	1
576.2.c.b		2	168.i	even	2	1
900.2.e.b		2	35.c	odd	2	1
900.2.e.b		2	105.g	even	2	1
900.2.e.b		2	140.c	even	2	1
900.2.e.b		2	420.o	odd	2	1
900.2.h.a		4	35.f	even	4	2
900.2.h.a		4	105.k	odd	4	2
900.2.h.a		4	140.j	odd	4	2
900.2.h.a		4	420.w	even	4	2
1764.2.e.b		2	1.a	even	1	1	trivial
1764.2.e.b		2	3.b	odd	2	1	inner
1764.2.e.b		2	4.b	odd	2	1	CM
1764.2.e.b		2	12.b	even	2	1	inner
2304.2.f.d		4	112.j	even	4	2
2304.2.f.d		4	112.l	odd	4	2
2304.2.f.d		4	336.v	odd	4	2
2304.2.f.d		4	336.y	even	4	2

Hecke kernels

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

\( T_{5}^{2} + 2 \)

\( T_{13} - 4 \)

Hecke characteristic polynomials

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