Newform orbit 1470.2.g.h

• Label: 1470.2.g.h
• Level: $1470$
• Weight: $2$
• Character orbit: 1470.g
• Analytic conductor: $11.738$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7380090971\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.29160000.2
Defining polynomial:	\( x^{6} - 20x^{3} + 125 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 20x^{3} + 125 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + 20\nu^{2} ) / 25 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} - 10 ) / 5 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} - 10\nu ) / 5 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{5} - 15\nu^{2} ) / 25 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\)	\(491\)	\(1081\)	\(1177\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

589.1

2.20942	+	0.344208i
−0.806615	−	2.08551i
−1.40280	+	1.74131i
2.20942	−	0.344208i
−0.806615	+	2.08551i
−1.40280	−	1.74131i

−

1.00000i

−

1.00000i

−1.00000

−2.20942

+

0.344208i

−1.00000

0

1.00000i

−1.00000

0.344208

+

2.20942i

589.2

−

1.00000i

−

1.00000i

−1.00000

0.806615

−

2.08551i

−1.00000

0

1.00000i

−1.00000

−2.08551

−

0.806615i

589.3

−

1.00000i

−

1.00000i

−1.00000

1.40280

+

1.74131i

−1.00000

0

1.00000i

−1.00000

1.74131

−

1.40280i

589.4

1.00000i

1.00000i

−1.00000

−2.20942

−

0.344208i

−1.00000

0

−

1.00000i

−1.00000

0.344208

−

2.20942i

589.5

1.00000i

1.00000i

−1.00000

0.806615

+

2.08551i

−1.00000

0

−

1.00000i

−1.00000

−2.08551

+

0.806615i

589.6

1.00000i

1.00000i

−1.00000

1.40280

−

1.74131i

−1.00000

0

−

1.00000i

−1.00000

1.74131

+

1.40280i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1470.2.g.h		6
5.b	even	2	1	inner	1470.2.g.h		6
5.c	odd	4	1		7350.2.a.do		3
5.c	odd	4	1		7350.2.a.dp		3
7.b	odd	2	1		1470.2.g.i		6
7.c	even	3	2		1470.2.n.j		12
7.d	odd	6	2		210.2.n.b	✓	12
21.g	even	6	2		630.2.u.f		12
28.f	even	6	2		1680.2.di.c		12
35.c	odd	2	1		1470.2.g.i		6
35.f	even	4	1		7350.2.a.dn		3
35.f	even	4	1		7350.2.a.dq		3
35.i	odd	6	2		210.2.n.b	✓	12
35.j	even	6	2		1470.2.n.j		12
35.k	even	12	2		1050.2.i.u		6
35.k	even	12	2		1050.2.i.v		6
105.p	even	6	2		630.2.u.f		12
140.s	even	6	2		1680.2.di.c		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.2.n.b	✓	12	7.d	odd	6	2
210.2.n.b	✓	12	35.i	odd	6	2
630.2.u.f		12	21.g	even	6	2
630.2.u.f		12	105.p	even	6	2
1050.2.i.u		6	35.k	even	12	2
1050.2.i.v		6	35.k	even	12	2
1470.2.g.h		6	1.a	even	1	1	trivial
1470.2.g.h		6	5.b	even	2	1	inner
1470.2.g.i		6	7.b	odd	2	1
1470.2.g.i		6	35.c	odd	2	1
1470.2.n.j		12	7.c	even	3	2
1470.2.n.j		12	35.j	even	6	2
1680.2.di.c		12	28.f	even	6	2
1680.2.di.c		12	140.s	even	6	2
7350.2.a.dn		3	35.f	even	4	1
7350.2.a.do		3	5.c	odd	4	1
7350.2.a.dp		3	5.c	odd	4	1
7350.2.a.dq		3	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{3} - 3T_{11}^{2} - 27T_{11} + 49 \)

\( T_{17}^{6} + 132T_{17}^{4} + 5568T_{17}^{2} + 73984 \)

\( T_{19}^{3} - 3T_{19}^{2} - 12T_{19} + 24 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{3} \)
$3$	\( (T^{2} + 1)^{3} \)
$5$	\( T^{6} + 20T^{3} + 125 \)
$7$	\( T^{6} \)
$11$	\( (T^{3} - 3 T^{2} - 27 T + 49)^{2} \)