Newform orbit 882.3.c.b

• Label: 882.3.c.b
• Level: $882$
• Weight: $3$
• Character orbit: 882.c
• Analytic conductor: $24.033$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0327593166\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + 2 q^{4} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{5} - 2 \beta_1 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} ) / 2 \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 1 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + 4\nu ) / 2 \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(-1\)	\(1\)

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

−0.707107	+	1.22474i
−0.707107	−	1.22474i
0.707107	+	1.22474i
0.707107	−	1.22474i

−1.41421

0

2.00000

−

8.36308i

0

0

−2.82843

0

11.8272i

685.2

−1.41421

0

2.00000

8.36308i

0

0

−2.82843

0

−

11.8272i

685.3

1.41421

0

2.00000

−

1.43488i

0

0

2.82843

0

−

2.02922i

685.4

1.41421

0

2.00000

1.43488i

0

0

2.82843

0

2.02922i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.3.c.b		4
3.b	odd	2	1		294.3.c.a		4
7.b	odd	2	1	inner	882.3.c.b		4
7.c	even	3	1		126.3.n.a		4
7.c	even	3	1		882.3.n.e		4
7.d	odd	6	1		126.3.n.a		4
7.d	odd	6	1		882.3.n.e		4
12.b	even	2	1		2352.3.f.e		4
21.c	even	2	1		294.3.c.a		4
21.g	even	6	1		42.3.g.a	✓	4
21.g	even	6	1		294.3.g.a		4
21.h	odd	6	1		42.3.g.a	✓	4
21.h	odd	6	1		294.3.g.a		4
28.f	even	6	1		1008.3.cg.h		4
28.g	odd	6	1		1008.3.cg.h		4
84.h	odd	2	1		2352.3.f.e		4
84.j	odd	6	1		336.3.bh.e		4
84.n	even	6	1		336.3.bh.e		4
105.o	odd	6	1		1050.3.p.a		4
105.p	even	6	1		1050.3.p.a		4
105.w	odd	12	2		1050.3.q.a		8
105.x	even	12	2		1050.3.q.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.3.g.a	✓	4	21.g	even	6	1
42.3.g.a	✓	4	21.h	odd	6	1
126.3.n.a		4	7.c	even	3	1
126.3.n.a		4	7.d	odd	6	1
294.3.c.a		4	3.b	odd	2	1
294.3.c.a		4	21.c	even	2	1
294.3.g.a		4	21.g	even	6	1
294.3.g.a		4	21.h	odd	6	1
336.3.bh.e		4	84.j	odd	6	1
336.3.bh.e		4	84.n	even	6	1
882.3.c.b		4	1.a	even	1	1	trivial
882.3.c.b		4	7.b	odd	2	1	inner
882.3.n.e		4	7.c	even	3	1
882.3.n.e		4	7.d	odd	6	1
1008.3.cg.h		4	28.f	even	6	1
1008.3.cg.h		4	28.g	odd	6	1
1050.3.p.a		4	105.o	odd	6	1
1050.3.p.a		4	105.p	even	6	1
1050.3.q.a		8	105.w	odd	12	2
1050.3.q.a		8	105.x	even	12	2
2352.3.f.e		4	12.b	even	2	1
2352.3.f.e		4	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 72T_{5}^{2} + 144 \)

\( T_{23}^{2} - 24T_{23} - 504 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{2} \)
$3$	\( T^{4} \)
$5$	\( T^{4} + 72T^{2} + 144 \)
$7$	\( T^{4} \)
$11$	\( (T + 6)^{4} \)