Newform orbit 450.2.j.f

• Label: 450.2.j.f
• Level: $450$
• Weight: $2$
• Character orbit: 450.j
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b3 - b1) * q^2 + (b7 + b3) * q^3 + (-b2 + 1) * q^4 + (-b5 - b2) * q^6 + (2*b7 - b4 - 2*b3 + 2*b1) * q^7 + b3 * q^8 + (2*b6 - 2*b5 + 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{4} \)
\(\beta_{3}\)	\(=\)	\( \zeta_{24}^{6} \)
\(\beta_{4}\)	\(=\)	\( \zeta_{24}^{7} + \zeta_{24} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24} \)
\(\beta_{6}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \)
\(\beta_{7}\)	\(=\)	\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1 + \beta_{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

−0.965926	−	0.258819i
0.965926	+	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i
0.258819	+	0.965926i
−0.258819	−	0.965926i

−0.866025

+

0.500000i

−1.41421

+

1.00000i

0.500000

−

0.866025i

0

0.724745

−

1.57313i

−0.389270

+

0.224745i

1.00000i

1.00000

−

2.82843i

0

49.2

−0.866025

+

0.500000i

1.41421

+

1.00000i

0.500000

−

0.866025i

0

−1.72474

−

0.158919i

3.85337

−

2.22474i

1.00000i

1.00000

+

2.82843i

0

49.3

0.866025

−

0.500000i

−1.41421

−

1.00000i

0.500000

−

0.866025i

0

−1.72474

−

0.158919i

−3.85337

+

2.22474i

−

1.00000i

1.00000

+

2.82843i

0

49.4

0.866025

−

0.500000i

1.41421

−

1.00000i

0.500000

−

0.866025i

0

0.724745

−

1.57313i

0.389270

−

0.224745i

−

1.00000i

1.00000

−

2.82843i

0

349.1

−0.866025

−

0.500000i

−1.41421

−

1.00000i

0.500000

+

0.866025i

0

0.724745

+

1.57313i

−0.389270

−

0.224745i

−

1.00000i

1.00000

+

2.82843i

0

349.2

−0.866025

−

0.500000i

1.41421

−

1.00000i

0.500000

+

0.866025i

0

−1.72474

+

0.158919i

3.85337

+

2.22474i

−

1.00000i

1.00000

−

2.82843i

0

349.3

0.866025

+

0.500000i

−1.41421

+

1.00000i

0.500000

+

0.866025i

0

−1.72474

+

0.158919i

−3.85337

−

2.22474i

1.00000i

1.00000

−

2.82843i

0

349.4

0.866025

+

0.500000i

1.41421

+

1.00000i

0.500000

+

0.866025i

0

0.724745

+

1.57313i

0.389270

+

0.224745i

1.00000i

1.00000

+

2.82843i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.2.j.f		8
3.b	odd	2	1		1350.2.j.g		8
5.b	even	2	1	inner	450.2.j.f		8
5.c	odd	4	1		450.2.e.l	✓	4
5.c	odd	4	1		450.2.e.m	yes	4
9.c	even	3	1	inner	450.2.j.f		8
9.c	even	3	1		4050.2.c.y		4
9.d	odd	6	1		1350.2.j.g		8
9.d	odd	6	1		4050.2.c.w		4
15.d	odd	2	1		1350.2.j.g		8
15.e	even	4	1		1350.2.e.k		4
15.e	even	4	1		1350.2.e.n		4
45.h	odd	6	1		1350.2.j.g		8
45.h	odd	6	1		4050.2.c.w		4
45.j	even	6	1	inner	450.2.j.f		8
45.j	even	6	1		4050.2.c.y		4
45.k	odd	12	1		450.2.e.l	✓	4
45.k	odd	12	1		450.2.e.m	yes	4
45.k	odd	12	1		4050.2.a.br		2
45.k	odd	12	1		4050.2.a.bu		2
45.l	even	12	1		1350.2.e.k		4
45.l	even	12	1		1350.2.e.n		4
45.l	even	12	1		4050.2.a.bl		2
45.l	even	12	1		4050.2.a.by		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.2.e.l	✓	4	5.c	odd	4	1
450.2.e.l	✓	4	45.k	odd	12	1
450.2.e.m	yes	4	5.c	odd	4	1
450.2.e.m	yes	4	45.k	odd	12	1
450.2.j.f		8	1.a	even	1	1	trivial
450.2.j.f		8	5.b	even	2	1	inner
450.2.j.f		8	9.c	even	3	1	inner
450.2.j.f		8	45.j	even	6	1	inner
1350.2.e.k		4	15.e	even	4	1
1350.2.e.k		4	45.l	even	12	1
1350.2.e.n		4	15.e	even	4	1
1350.2.e.n		4	45.l	even	12	1
1350.2.j.g		8	3.b	odd	2	1
1350.2.j.g		8	9.d	odd	6	1
1350.2.j.g		8	15.d	odd	2	1
1350.2.j.g		8	45.h	odd	6	1
4050.2.a.bl		2	45.l	even	12	1
4050.2.a.br		2	45.k	odd	12	1
4050.2.a.bu		2	45.k	odd	12	1
4050.2.a.by		2	45.l	even	12	1
4050.2.c.w		4	9.d	odd	6	1
4050.2.c.w		4	45.h	odd	6	1
4050.2.c.y		4	9.c	even	3	1
4050.2.c.y		4	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{8} - 20T_{7}^{6} + 396T_{7}^{4} - 80T_{7}^{2} + 16 \)

\( T_{11}^{4} + 24T_{11}^{2} + 576 \)

\( T_{19}^{2} + 10T_{19} + 19 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{2} \)
$3$	\( (T^{4} - 2 T^{2} + 9)^{2} \)
$5$	\( T^{8} \)
$7$	T^8 - 20*T^6 + 396*T^4 - 80*T^2 + 16
$11$	\( (T^{4} + 24 T^{2} + 576)^{2} \)