Newform orbit 1764.3.d.b

• Label: 1764.3.d.b
• Level: $1764$
• Weight: $3$
• Character orbit: 1764.d
• Analytic conductor: $48.066$
• 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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(48.0655186332\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 15 \beta q^{13} + 5 \beta q^{19} + 25 q^{25} + 11 \beta q^{31} - 47 q^{37} - 83 q^{43} - 56 \beta q^{61} - 109 q^{67} + 17 \beta q^{73} + 131 q^{79} - 112 \beta q^{97} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{25} - 94 q^{37} - 166 q^{43} - 218 q^{67} + 262 q^{79}+O(q^{100}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

685.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

0

0

0

0

685.2

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.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.3.d.b		2
3.b	odd	2	1	CM	1764.3.d.b		2
7.b	odd	2	1	inner	1764.3.d.b		2
7.c	even	3	1		252.3.z.c	✓	2
7.c	even	3	1		1764.3.z.c		2
7.d	odd	6	1		252.3.z.c	✓	2
7.d	odd	6	1		1764.3.z.c		2
21.c	even	2	1	inner	1764.3.d.b		2
21.g	even	6	1		252.3.z.c	✓	2
21.g	even	6	1		1764.3.z.c		2
21.h	odd	6	1		252.3.z.c	✓	2
21.h	odd	6	1		1764.3.z.c		2
28.f	even	6	1		1008.3.cg.e		2
28.g	odd	6	1		1008.3.cg.e		2
84.j	odd	6	1		1008.3.cg.e		2
84.n	even	6	1		1008.3.cg.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.3.z.c	✓	2	7.c	even	3	1
252.3.z.c	✓	2	7.d	odd	6	1
252.3.z.c	✓	2	21.g	even	6	1
252.3.z.c	✓	2	21.h	odd	6	1
1008.3.cg.e		2	28.f	even	6	1
1008.3.cg.e		2	28.g	odd	6	1
1008.3.cg.e		2	84.j	odd	6	1
1008.3.cg.e		2	84.n	even	6	1
1764.3.d.b		2	1.a	even	1	1	trivial
1764.3.d.b		2	3.b	odd	2	1	CM
1764.3.d.b		2	7.b	odd	2	1	inner
1764.3.d.b		2	21.c	even	2	1	inner
1764.3.z.c		2	7.c	even	3	1
1764.3.z.c		2	7.d	odd	6	1
1764.3.z.c		2	21.g	even	6	1
1764.3.z.c		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5} \)

\( T_{11} \)

Hecke characteristic polynomials

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