Newform orbit 1764.2.k.b

• Label: 1764.2.k.b
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.k
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

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:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 - 3*z * q^5 + (3*z - 3) * q^11 - 2 * q^13 + (3*z - 3) * q^17 - z * q^19 + 3*z * q^23 + (4*z - 4) * q^25 + 6 * q^29 + (7*z - 7) * q^31 + z * q^37 + 6 * q^41 - 4 * q^43 + 9*z * q^47 + (-3*z + 3) * q^53 + 9 * q^55 + (9*z - 9) * q^59 - z * q^61 + 6*z * q^65 + (-7*z + 7) * q^67 + (z - 1) * q^73 + 13*z * q^79 + 12 * q^83 + 9 * q^85 - 15*z * q^89 + (3*z - 3) * q^95 + 10 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^5 - 3 * q^11 - 4 * q^13 - 3 * q^17 - q^19 + 3 * q^23 - 4 * q^25 + 12 * q^29 - 7 * q^31 + q^37 + 12 * q^41 - 8 * q^43 + 9 * q^47 + 3 * q^53 + 18 * q^55 - 9 * q^59 - q^61 + 6 * q^65 + 7 * q^67 - q^73 + 13 * q^79 + 24 * q^83 + 18 * q^85 - 15 * q^89 - 3 * q^95 + 20 * q^97

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

−1.50000

−

2.59808i

0

0

0

0

0

1549.1

0

0

0

−1.50000

+

2.59808i

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.2.k.b		2
3.b	odd	2	1		196.2.e.a		2
7.b	odd	2	1		252.2.k.c		2
7.c	even	3	1		1764.2.a.j		1
7.c	even	3	1	inner	1764.2.k.b		2
7.d	odd	6	1		252.2.k.c		2
7.d	odd	6	1		1764.2.a.a		1
12.b	even	2	1		784.2.i.d		2
21.c	even	2	1		28.2.e.a	✓	2
21.g	even	6	1		28.2.e.a	✓	2
21.g	even	6	1		196.2.a.b		1
21.h	odd	6	1		196.2.a.a		1
21.h	odd	6	1		196.2.e.a		2
28.d	even	2	1		1008.2.s.p		2
28.f	even	6	1		1008.2.s.p		2
28.f	even	6	1		7056.2.a.f		1
28.g	odd	6	1		7056.2.a.bw		1
63.i	even	6	1		2268.2.l.h		2
63.k	odd	6	1		2268.2.i.h		2
63.l	odd	6	1		2268.2.i.h		2
63.l	odd	6	1		2268.2.l.a		2
63.o	even	6	1		2268.2.i.a		2
63.o	even	6	1		2268.2.l.h		2
63.s	even	6	1		2268.2.i.a		2
63.t	odd	6	1		2268.2.l.a		2
84.h	odd	2	1		112.2.i.b		2
84.j	odd	6	1		112.2.i.b		2
84.j	odd	6	1		784.2.a.d		1
84.n	even	6	1		784.2.a.g		1
84.n	even	6	1		784.2.i.d		2
105.g	even	2	1		700.2.i.c		2
105.k	odd	4	2		700.2.r.b		4
105.o	odd	6	1		4900.2.a.n		1
105.p	even	6	1		700.2.i.c		2
105.p	even	6	1		4900.2.a.g		1
105.w	odd	12	2		700.2.r.b		4
105.w	odd	12	2		4900.2.e.i		2
105.x	even	12	2		4900.2.e.h		2
168.e	odd	2	1		448.2.i.c		2
168.i	even	2	1		448.2.i.e		2
168.s	odd	6	1		3136.2.a.v		1
168.v	even	6	1		3136.2.a.k		1
168.ba	even	6	1		448.2.i.e		2
168.ba	even	6	1		3136.2.a.h		1
168.be	odd	6	1		448.2.i.c		2
168.be	odd	6	1		3136.2.a.s		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.e.a	✓	2	21.c	even	2	1
28.2.e.a	✓	2	21.g	even	6	1
112.2.i.b		2	84.h	odd	2	1
112.2.i.b		2	84.j	odd	6	1
196.2.a.a		1	21.h	odd	6	1
196.2.a.b		1	21.g	even	6	1
196.2.e.a		2	3.b	odd	2	1
196.2.e.a		2	21.h	odd	6	1
252.2.k.c		2	7.b	odd	2	1
252.2.k.c		2	7.d	odd	6	1
448.2.i.c		2	168.e	odd	2	1
448.2.i.c		2	168.be	odd	6	1
448.2.i.e		2	168.i	even	2	1
448.2.i.e		2	168.ba	even	6	1
700.2.i.c		2	105.g	even	2	1
700.2.i.c		2	105.p	even	6	1
700.2.r.b		4	105.k	odd	4	2
700.2.r.b		4	105.w	odd	12	2
784.2.a.d		1	84.j	odd	6	1
784.2.a.g		1	84.n	even	6	1
784.2.i.d		2	12.b	even	2	1
784.2.i.d		2	84.n	even	6	1
1008.2.s.p		2	28.d	even	2	1
1008.2.s.p		2	28.f	even	6	1
1764.2.a.a		1	7.d	odd	6	1
1764.2.a.j		1	7.c	even	3	1
1764.2.k.b		2	1.a	even	1	1	trivial
1764.2.k.b		2	7.c	even	3	1	inner
2268.2.i.a		2	63.o	even	6	1
2268.2.i.a		2	63.s	even	6	1
2268.2.i.h		2	63.k	odd	6	1
2268.2.i.h		2	63.l	odd	6	1
2268.2.l.a		2	63.l	odd	6	1
2268.2.l.a		2	63.t	odd	6	1
2268.2.l.h		2	63.i	even	6	1
2268.2.l.h		2	63.o	even	6	1
3136.2.a.h		1	168.ba	even	6	1
3136.2.a.k		1	168.v	even	6	1
3136.2.a.s		1	168.be	odd	6	1
3136.2.a.v		1	168.s	odd	6	1
4900.2.a.g		1	105.p	even	6	1
4900.2.a.n		1	105.o	odd	6	1
4900.2.e.h		2	105.x	even	12	2
4900.2.e.i		2	105.w	odd	12	2
7056.2.a.f		1	28.f	even	6	1
7056.2.a.bw		1	28.g	odd	6	1

Hecke kernels

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

\( T_{5}^{2} + 3T_{5} + 9 \)

\( T_{11}^{2} + 3T_{11} + 9 \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 3T + 9 \)