Newform orbit 882.2.f.d

• Label: 882.2.f.d
• Level: $882$
• Weight: $2$
• Character orbit: 882.f
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + (\zeta_{6} - 2) q^{6} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{3} - q^{4} - 3 q^{6} + 2 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

295.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

1.50000

+

0.866025i

−0.500000

−

0.866025i

0

−1.50000

+

0.866025i

0

1.00000

1.50000

+

2.59808i

0

589.1

−0.500000

−

0.866025i

1.50000

−

0.866025i

−0.500000

+

0.866025i

0

−1.50000

−

0.866025i

0

1.00000

1.50000

−

2.59808i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.2.f.d		2
3.b	odd	2	1		2646.2.f.g		2
7.b	odd	2	1		18.2.c.a	✓	2
7.c	even	3	1		882.2.e.g		2
7.c	even	3	1		882.2.h.b		2
7.d	odd	6	1		882.2.e.i		2
7.d	odd	6	1		882.2.h.c		2
9.c	even	3	1	inner	882.2.f.d		2
9.c	even	3	1		7938.2.a.x		1
9.d	odd	6	1		2646.2.f.g		2
9.d	odd	6	1		7938.2.a.i		1
21.c	even	2	1		54.2.c.a		2
21.g	even	6	1		2646.2.e.b		2
21.g	even	6	1		2646.2.h.h		2
21.h	odd	6	1		2646.2.e.c		2
21.h	odd	6	1		2646.2.h.i		2
28.d	even	2	1		144.2.i.c		2
35.c	odd	2	1		450.2.e.i		2
35.f	even	4	2		450.2.j.e		4
56.e	even	2	1		576.2.i.a		2
56.h	odd	2	1		576.2.i.g		2
63.g	even	3	1		882.2.e.g		2
63.h	even	3	1		882.2.h.b		2
63.i	even	6	1		2646.2.h.h		2
63.j	odd	6	1		2646.2.h.i		2
63.k	odd	6	1		882.2.e.i		2
63.l	odd	6	1		18.2.c.a	✓	2
63.l	odd	6	1		162.2.a.c		1
63.n	odd	6	1		2646.2.e.c		2
63.o	even	6	1		54.2.c.a		2
63.o	even	6	1		162.2.a.b		1
63.s	even	6	1		2646.2.e.b		2
63.t	odd	6	1		882.2.h.c		2
84.h	odd	2	1		432.2.i.b		2
105.g	even	2	1		1350.2.e.c		2
105.k	odd	4	2		1350.2.j.a		4
168.e	odd	2	1		1728.2.i.f		2
168.i	even	2	1		1728.2.i.e		2
252.s	odd	6	1		432.2.i.b		2
252.s	odd	6	1		1296.2.a.f		1
252.bi	even	6	1		144.2.i.c		2
252.bi	even	6	1		1296.2.a.g		1
315.z	even	6	1		1350.2.e.c		2
315.z	even	6	1		4050.2.a.v		1
315.bg	odd	6	1		450.2.e.i		2
315.bg	odd	6	1		4050.2.a.c		1
315.cb	even	12	2		450.2.j.e		4
315.cb	even	12	2		4050.2.c.c		2
315.cf	odd	12	2		1350.2.j.a		4
315.cf	odd	12	2		4050.2.c.r		2
504.be	even	6	1		576.2.i.a		2
504.be	even	6	1		5184.2.a.o		1
504.bn	odd	6	1		576.2.i.g		2
504.bn	odd	6	1		5184.2.a.r		1
504.cc	even	6	1		1728.2.i.e		2
504.cc	even	6	1		5184.2.a.q		1
504.co	odd	6	1		1728.2.i.f		2
504.co	odd	6	1		5184.2.a.p		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.2.c.a	✓	2	7.b	odd	2	1
18.2.c.a	✓	2	63.l	odd	6	1
54.2.c.a		2	21.c	even	2	1
54.2.c.a		2	63.o	even	6	1
144.2.i.c		2	28.d	even	2	1
144.2.i.c		2	252.bi	even	6	1
162.2.a.b		1	63.o	even	6	1
162.2.a.c		1	63.l	odd	6	1
432.2.i.b		2	84.h	odd	2	1
432.2.i.b		2	252.s	odd	6	1
450.2.e.i		2	35.c	odd	2	1
450.2.e.i		2	315.bg	odd	6	1
450.2.j.e		4	35.f	even	4	2
450.2.j.e		4	315.cb	even	12	2
576.2.i.a		2	56.e	even	2	1
576.2.i.a		2	504.be	even	6	1
576.2.i.g		2	56.h	odd	2	1
576.2.i.g		2	504.bn	odd	6	1
882.2.e.g		2	7.c	even	3	1
882.2.e.g		2	63.g	even	3	1
882.2.e.i		2	7.d	odd	6	1
882.2.e.i		2	63.k	odd	6	1
882.2.f.d		2	1.a	even	1	1	trivial
882.2.f.d		2	9.c	even	3	1	inner
882.2.h.b		2	7.c	even	3	1
882.2.h.b		2	63.h	even	3	1
882.2.h.c		2	7.d	odd	6	1
882.2.h.c		2	63.t	odd	6	1
1296.2.a.f		1	252.s	odd	6	1
1296.2.a.g		1	252.bi	even	6	1
1350.2.e.c		2	105.g	even	2	1
1350.2.e.c		2	315.z	even	6	1
1350.2.j.a		4	105.k	odd	4	2
1350.2.j.a		4	315.cf	odd	12	2
1728.2.i.e		2	168.i	even	2	1
1728.2.i.e		2	504.cc	even	6	1
1728.2.i.f		2	168.e	odd	2	1
1728.2.i.f		2	504.co	odd	6	1
2646.2.e.b		2	21.g	even	6	1
2646.2.e.b		2	63.s	even	6	1
2646.2.e.c		2	21.h	odd	6	1
2646.2.e.c		2	63.n	odd	6	1
2646.2.f.g		2	3.b	odd	2	1
2646.2.f.g		2	9.d	odd	6	1
2646.2.h.h		2	21.g	even	6	1
2646.2.h.h		2	63.i	even	6	1
2646.2.h.i		2	21.h	odd	6	1
2646.2.h.i		2	63.j	odd	6	1
4050.2.a.c		1	315.bg	odd	6	1
4050.2.a.v		1	315.z	even	6	1
4050.2.c.c		2	315.cb	even	12	2
4050.2.c.r		2	315.cf	odd	12	2
5184.2.a.o		1	504.be	even	6	1
5184.2.a.p		1	504.co	odd	6	1
5184.2.a.q		1	504.cc	even	6	1
5184.2.a.r		1	504.bn	odd	6	1
7938.2.a.i		1	9.d	odd	6	1
7938.2.a.x		1	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5} \)

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

\( T_{13}^{2} - 2T_{13} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} - 3T + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 3T + 9 \)