Newform orbit 882.2.g.d

• Label: 882.2.g.d
• Level: $882$
• Weight: $2$
• Character orbit: 882.g
• 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.g (of order \(3\), degree \(2\), not 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 14)
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.

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

0

0

0

1.00000

0

0

667.1

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

0

0

0

1.00000

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	882.2.g.d		2
3.b	odd	2	1		98.2.c.a		2
7.b	odd	2	1		882.2.g.c		2
7.c	even	3	1		882.2.a.i		1
7.c	even	3	1	inner	882.2.g.d		2
7.d	odd	6	1		126.2.a.b		1
7.d	odd	6	1		882.2.g.c		2
12.b	even	2	1		784.2.i.i		2
21.c	even	2	1		98.2.c.b		2
21.g	even	6	1		14.2.a.a	✓	1
21.g	even	6	1		98.2.c.b		2
21.h	odd	6	1		98.2.a.a		1
21.h	odd	6	1		98.2.c.a		2
28.f	even	6	1		1008.2.a.h		1
28.g	odd	6	1		7056.2.a.bd		1
35.i	odd	6	1		3150.2.a.i		1
35.k	even	12	2		3150.2.g.j		2
56.j	odd	6	1		4032.2.a.w		1
56.m	even	6	1		4032.2.a.r		1
63.i	even	6	1		1134.2.f.l		2
63.k	odd	6	1		1134.2.f.f		2
63.s	even	6	1		1134.2.f.l		2
63.t	odd	6	1		1134.2.f.f		2
84.h	odd	2	1		784.2.i.c		2
84.j	odd	6	1		112.2.a.c		1
84.j	odd	6	1		784.2.i.c		2
84.n	even	6	1		784.2.a.b		1
84.n	even	6	1		784.2.i.i		2
105.o	odd	6	1		2450.2.a.t		1
105.p	even	6	1		350.2.a.f		1
105.w	odd	12	2		350.2.c.d		2
105.x	even	12	2		2450.2.c.c		2
168.s	odd	6	1		3136.2.a.e		1
168.v	even	6	1		3136.2.a.z		1
168.ba	even	6	1		448.2.a.g		1
168.be	odd	6	1		448.2.a.a		1
231.k	odd	6	1		1694.2.a.e		1
273.ba	even	6	1		2366.2.a.j		1
273.cb	odd	12	2		2366.2.d.b		2
336.bo	even	12	2		1792.2.b.c		2
336.br	odd	12	2		1792.2.b.g		2
357.s	even	6	1		4046.2.a.f		1
399.s	odd	6	1		5054.2.a.c		1
420.be	odd	6	1		2800.2.a.g		1
420.br	even	12	2		2800.2.g.h		2
483.o	odd	6	1		7406.2.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.2.a.a	✓	1	21.g	even	6	1
98.2.a.a		1	21.h	odd	6	1
98.2.c.a		2	3.b	odd	2	1
98.2.c.a		2	21.h	odd	6	1
98.2.c.b		2	21.c	even	2	1
98.2.c.b		2	21.g	even	6	1
112.2.a.c		1	84.j	odd	6	1
126.2.a.b		1	7.d	odd	6	1
350.2.a.f		1	105.p	even	6	1
350.2.c.d		2	105.w	odd	12	2
448.2.a.a		1	168.be	odd	6	1
448.2.a.g		1	168.ba	even	6	1
784.2.a.b		1	84.n	even	6	1
784.2.i.c		2	84.h	odd	2	1
784.2.i.c		2	84.j	odd	6	1
784.2.i.i		2	12.b	even	2	1
784.2.i.i		2	84.n	even	6	1
882.2.a.i		1	7.c	even	3	1
882.2.g.c		2	7.b	odd	2	1
882.2.g.c		2	7.d	odd	6	1
882.2.g.d		2	1.a	even	1	1	trivial
882.2.g.d		2	7.c	even	3	1	inner
1008.2.a.h		1	28.f	even	6	1
1134.2.f.f		2	63.k	odd	6	1
1134.2.f.f		2	63.t	odd	6	1
1134.2.f.l		2	63.i	even	6	1
1134.2.f.l		2	63.s	even	6	1
1694.2.a.e		1	231.k	odd	6	1
1792.2.b.c		2	336.bo	even	12	2
1792.2.b.g		2	336.br	odd	12	2
2366.2.a.j		1	273.ba	even	6	1
2366.2.d.b		2	273.cb	odd	12	2
2450.2.a.t		1	105.o	odd	6	1
2450.2.c.c		2	105.x	even	12	2
2800.2.a.g		1	420.be	odd	6	1
2800.2.g.h		2	420.br	even	12	2
3136.2.a.e		1	168.s	odd	6	1
3136.2.a.z		1	168.v	even	6	1
3150.2.a.i		1	35.i	odd	6	1
3150.2.g.j		2	35.k	even	12	2
4032.2.a.r		1	56.m	even	6	1
4032.2.a.w		1	56.j	odd	6	1
4046.2.a.f		1	357.s	even	6	1
5054.2.a.c		1	399.s	odd	6	1
7056.2.a.bd		1	28.g	odd	6	1
7406.2.a.a		1	483.o	odd	6	1

Hecke kernels

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

\( T_{5} \)

\( T_{11} \)

\( T_{13} - 4 \)

Hecke characteristic polynomials

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