Newform orbit 6336.2.b.k

• Label: 6336.2.b.k
• Level: $6336$
• Weight: $2$
• Character orbit: 6336.b
• Analytic conductor: $50.593$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(50.5932147207\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{5} - 3 \beta q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).

\(n\)	\(1729\)	\(3521\)	\(4159\)	\(4357\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2177.1

	−	1.41421i
		1.41421i

0

0

0

−

1.41421i

0

4.24264i

0

0

0

2177.2

0

0

0

1.41421i

0

−

4.24264i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6336.2.b.k		2
3.b	odd	2	1		6336.2.b.h		2
4.b	odd	2	1		6336.2.b.b		2
8.b	even	2	1		396.2.b.a	✓	2
8.d	odd	2	1		1584.2.b.c		2
11.b	odd	2	1		6336.2.b.h		2
12.b	even	2	1		6336.2.b.o		2
24.f	even	2	1		1584.2.b.b		2
24.h	odd	2	1		396.2.b.b	yes	2
33.d	even	2	1	inner	6336.2.b.k		2
40.f	even	2	1		9900.2.d.a		2
40.i	odd	4	2		9900.2.n.a		4
44.c	even	2	1		6336.2.b.o		2
72.j	odd	6	2		3564.2.q.b		4
72.n	even	6	2		3564.2.q.c		4
88.b	odd	2	1		396.2.b.b	yes	2
88.g	even	2	1		1584.2.b.b		2
120.i	odd	2	1		9900.2.d.b		2
120.w	even	4	2		9900.2.n.b		4
132.d	odd	2	1		6336.2.b.b		2
264.m	even	2	1		396.2.b.a	✓	2
264.p	odd	2	1		1584.2.b.c		2
440.o	odd	2	1		9900.2.d.b		2
440.t	even	4	2		9900.2.n.b		4
792.w	even	6	2		3564.2.q.c		4
792.be	odd	6	2		3564.2.q.b		4
1320.u	even	2	1		9900.2.d.a		2
1320.bn	odd	4	2		9900.2.n.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
396.2.b.a	✓	2	8.b	even	2	1
396.2.b.a	✓	2	264.m	even	2	1
396.2.b.b	yes	2	24.h	odd	2	1
396.2.b.b	yes	2	88.b	odd	2	1
1584.2.b.b		2	24.f	even	2	1
1584.2.b.b		2	88.g	even	2	1
1584.2.b.c		2	8.d	odd	2	1
1584.2.b.c		2	264.p	odd	2	1
3564.2.q.b		4	72.j	odd	6	2
3564.2.q.b		4	792.be	odd	6	2
3564.2.q.c		4	72.n	even	6	2
3564.2.q.c		4	792.w	even	6	2
6336.2.b.b		2	4.b	odd	2	1
6336.2.b.b		2	132.d	odd	2	1
6336.2.b.h		2	3.b	odd	2	1
6336.2.b.h		2	11.b	odd	2	1
6336.2.b.k		2	1.a	even	1	1	trivial
6336.2.b.k		2	33.d	even	2	1	inner
6336.2.b.o		2	12.b	even	2	1
6336.2.b.o		2	44.c	even	2	1
9900.2.d.a		2	40.f	even	2	1
9900.2.d.a		2	1320.u	even	2	1
9900.2.d.b		2	120.i	odd	2	1
9900.2.d.b		2	440.o	odd	2	1
9900.2.n.a		4	40.i	odd	4	2
9900.2.n.a		4	1320.bn	odd	4	2
9900.2.n.b		4	120.w	even	4	2
9900.2.n.b		4	440.t	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6336, [\chi])\):

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

\( T_{7}^{2} + 18 \)

\( T_{13} \)

\( T_{17} + 6 \)

\( T_{31} - 8 \)

\( T_{83} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2 \)
$7$	\( T^{2} + 18 \)
$11$	\( T^{2} - 6T + 11 \)