Newform orbit 6336.2.b.a

• Label: 6336.2.b.a
• Level: $6336$
• Weight: $2$
• Character orbit: 6336.b
• Analytic conductor: $50.593$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 198)
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 + 2*b * q^5 + (b - 3) * q^11 - 3*b * q^13 - 6 * q^17 - 3*b * q^19 + b * q^23 - 3 * q^25 + 2 * q^31 + 10 * q^37 - 6 * q^41 + 9*b * q^43 + 7*b * q^47 + 7 * q^49 - 4*b * q^53 + (-6*b - 4) * q^55 - 4*b * q^59 - 9*b * q^61 + 12 * q^65 - 8 * q^67 - 11*b * q^71 - 6*b * q^73 - 6*b * q^79 - 6 * q^83 - 12*b * q^85 - 5*b * q^89 + 12 * q^95 - 4 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{11} - 12 q^{17} - 6 q^{25} + 4 q^{31} + 20 q^{37} - 12 q^{41} + 14 q^{49} - 8 q^{55} + 24 q^{65} - 16 q^{67} - 12 q^{83} + 24 q^{95} - 8 q^{97}+O(q^{100}) \)

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

−

2.82843i

0

0

0

0

0

2177.2

0

0

0

2.82843i

0

0

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.a		2
3.b	odd	2	1		6336.2.b.n		2
4.b	odd	2	1		6336.2.b.i		2
8.b	even	2	1		198.2.b.b	yes	2
8.d	odd	2	1		1584.2.b.a		2
11.b	odd	2	1		6336.2.b.n		2
12.b	even	2	1		6336.2.b.f		2
24.f	even	2	1		1584.2.b.d		2
24.h	odd	2	1		198.2.b.a	✓	2
33.d	even	2	1	inner	6336.2.b.a		2
40.f	even	2	1		4950.2.d.b		2
40.i	odd	4	2		4950.2.f.b		4
44.c	even	2	1		6336.2.b.f		2
72.j	odd	6	2		1782.2.i.g		4
72.n	even	6	2		1782.2.i.b		4
88.b	odd	2	1		198.2.b.a	✓	2
88.g	even	2	1		1584.2.b.d		2
120.i	odd	2	1		4950.2.d.e		2
120.w	even	4	2		4950.2.f.a		4
132.d	odd	2	1		6336.2.b.i		2
264.m	even	2	1		198.2.b.b	yes	2
264.p	odd	2	1		1584.2.b.a		2
440.o	odd	2	1		4950.2.d.e		2
440.t	even	4	2		4950.2.f.a		4
792.w	even	6	2		1782.2.i.b		4
792.be	odd	6	2		1782.2.i.g		4
1320.u	even	2	1		4950.2.d.b		2
1320.bn	odd	4	2		4950.2.f.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
198.2.b.a	✓	2	24.h	odd	2	1
198.2.b.a	✓	2	88.b	odd	2	1
198.2.b.b	yes	2	8.b	even	2	1
198.2.b.b	yes	2	264.m	even	2	1
1584.2.b.a		2	8.d	odd	2	1
1584.2.b.a		2	264.p	odd	2	1
1584.2.b.d		2	24.f	even	2	1
1584.2.b.d		2	88.g	even	2	1
1782.2.i.b		4	72.n	even	6	2
1782.2.i.b		4	792.w	even	6	2
1782.2.i.g		4	72.j	odd	6	2
1782.2.i.g		4	792.be	odd	6	2
4950.2.d.b		2	40.f	even	2	1
4950.2.d.b		2	1320.u	even	2	1
4950.2.d.e		2	120.i	odd	2	1
4950.2.d.e		2	440.o	odd	2	1
4950.2.f.a		4	120.w	even	4	2
4950.2.f.a		4	440.t	even	4	2
4950.2.f.b		4	40.i	odd	4	2
4950.2.f.b		4	1320.bn	odd	4	2
6336.2.b.a		2	1.a	even	1	1	trivial
6336.2.b.a		2	33.d	even	2	1	inner
6336.2.b.f		2	12.b	even	2	1
6336.2.b.f		2	44.c	even	2	1
6336.2.b.i		2	4.b	odd	2	1
6336.2.b.i		2	132.d	odd	2	1
6336.2.b.n		2	3.b	odd	2	1
6336.2.b.n		2	11.b	odd	2	1

Hecke kernels

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

\( T_{5}^{2} + 8 \)

\( T_{7} \)

\( T_{13}^{2} + 18 \)

\( T_{17} + 6 \)

\( T_{31} - 2 \)

\( T_{83} + 6 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 8 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 6T + 11 \)