Newform orbit 1050.3.h.a

• Label: 1050.3.h.a
• Level: $1050$
• Weight: $3$
• Character orbit: 1050.h
• Analytic conductor: $28.610$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.6104277578\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{7} q^{6} + (\beta_{5} - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} + 2 \beta_1 q^{8} + 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \)

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(451\)	\(701\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

349.1

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

−

1.41421i

−1.73205

−2.00000

0

2.44949i

−6.63103

+

2.24264i

2.82843i

3.00000

0

349.2

−

1.41421i

−1.73205

−2.00000

0

2.44949i

3.16693

+

6.24264i

2.82843i

3.00000

0

349.3

−

1.41421i

1.73205

−2.00000

0

−

2.44949i

−3.16693

+

6.24264i

2.82843i

3.00000

0

349.4

−

1.41421i

1.73205

−2.00000

0

−

2.44949i

6.63103

+

2.24264i

2.82843i

3.00000

0

349.5

1.41421i

−1.73205

−2.00000

0

−

2.44949i

−6.63103

−

2.24264i

−

2.82843i

3.00000

0

349.6

1.41421i

−1.73205

−2.00000

0

−

2.44949i

3.16693

−

6.24264i

−

2.82843i

3.00000

0

349.7

1.41421i

1.73205

−2.00000

0

2.44949i

−3.16693

−

6.24264i

−

2.82843i

3.00000

0

349.8

1.41421i

1.73205

−2.00000

0

2.44949i

6.63103

−

2.24264i

−

2.82843i

3.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1050.3.h.a		8
5.b	even	2	1	inner	1050.3.h.a		8
5.c	odd	4	1		42.3.c.a	✓	4
5.c	odd	4	1		1050.3.f.a		4
7.b	odd	2	1	inner	1050.3.h.a		8
15.e	even	4	1		126.3.c.b		4
20.e	even	4	1		336.3.f.c		4
35.c	odd	2	1	inner	1050.3.h.a		8
35.f	even	4	1		42.3.c.a	✓	4
35.f	even	4	1		1050.3.f.a		4
35.k	even	12	1		294.3.g.b		4
35.k	even	12	1		294.3.g.c		4
35.l	odd	12	1		294.3.g.b		4
35.l	odd	12	1		294.3.g.c		4
40.i	odd	4	1		1344.3.f.f		4
40.k	even	4	1		1344.3.f.e		4
60.l	odd	4	1		1008.3.f.g		4
105.k	odd	4	1		126.3.c.b		4
105.w	odd	12	1		882.3.n.a		4
105.w	odd	12	1		882.3.n.d		4
105.x	even	12	1		882.3.n.a		4
105.x	even	12	1		882.3.n.d		4
140.j	odd	4	1		336.3.f.c		4
280.s	even	4	1		1344.3.f.f		4
280.y	odd	4	1		1344.3.f.e		4
420.w	even	4	1		1008.3.f.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.3.c.a	✓	4	5.c	odd	4	1
42.3.c.a	✓	4	35.f	even	4	1
126.3.c.b		4	15.e	even	4	1
126.3.c.b		4	105.k	odd	4	1
294.3.g.b		4	35.k	even	12	1
294.3.g.b		4	35.l	odd	12	1
294.3.g.c		4	35.k	even	12	1
294.3.g.c		4	35.l	odd	12	1
336.3.f.c		4	20.e	even	4	1
336.3.f.c		4	140.j	odd	4	1
882.3.n.a		4	105.w	odd	12	1
882.3.n.a		4	105.x	even	12	1
882.3.n.d		4	105.w	odd	12	1
882.3.n.d		4	105.x	even	12	1
1008.3.f.g		4	60.l	odd	4	1
1008.3.f.g		4	420.w	even	4	1
1050.3.f.a		4	5.c	odd	4	1
1050.3.f.a		4	35.f	even	4	1
1050.3.h.a		8	1.a	even	1	1	trivial
1050.3.h.a		8	5.b	even	2	1	inner
1050.3.h.a		8	7.b	odd	2	1	inner
1050.3.h.a		8	35.c	odd	2	1	inner
1344.3.f.e		4	40.k	even	4	1
1344.3.f.e		4	280.y	odd	4	1
1344.3.f.f		4	40.i	odd	4	1
1344.3.f.f		4	280.s	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 12T_{11} + 18 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{4} \)
$3$	\( (T^{2} - 3)^{4} \)
$5$	\( T^{8} \)
$7$	T^8 - 20*T^6 + 294*T^4 - 48020*T^2 + 5764801
$11$	\( (T^{2} + 12 T + 18)^{4} \)