Newform orbit 1470.2.i.j

• Label: 1470.2.i.j
• Level: $1470$
• Weight: $2$
• Character orbit: 1470.i
• Analytic conductor: $11.738$
• Analytic rank: $0$
• Dimension: $2$
• 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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7380090971\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} - q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{3} - q^{4} + q^{5} - 2 q^{6} + 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(491\)	\(1081\)	\(1177\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

0.500000

+

0.866025i

−1.00000

0

1.00000

−0.500000

−

0.866025i

0.500000

−

0.866025i

961.1

−0.500000

+

0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

0.500000

−

0.866025i

−1.00000

0

1.00000

−0.500000

+

0.866025i

0.500000

+

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1470.2.i.j		2
7.b	odd	2	1		1470.2.i.a		2
7.c	even	3	1		1470.2.a.j		1
7.c	even	3	1	inner	1470.2.i.j		2
7.d	odd	6	1		210.2.a.e	✓	1
7.d	odd	6	1		1470.2.i.a		2
21.g	even	6	1		630.2.a.a		1
21.h	odd	6	1		4410.2.a.t		1
28.f	even	6	1		1680.2.a.j		1
35.i	odd	6	1		1050.2.a.c		1
35.j	even	6	1		7350.2.a.w		1
35.k	even	12	2		1050.2.g.g		2
56.j	odd	6	1		6720.2.a.j		1
56.m	even	6	1		6720.2.a.bq		1
84.j	odd	6	1		5040.2.a.k		1
105.p	even	6	1		3150.2.a.bp		1
105.w	odd	12	2		3150.2.g.q		2
140.s	even	6	1		8400.2.a.ce		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.2.a.e	✓	1	7.d	odd	6	1
630.2.a.a		1	21.g	even	6	1
1050.2.a.c		1	35.i	odd	6	1
1050.2.g.g		2	35.k	even	12	2
1470.2.a.j		1	7.c	even	3	1
1470.2.i.a		2	7.b	odd	2	1
1470.2.i.a		2	7.d	odd	6	1
1470.2.i.j		2	1.a	even	1	1	trivial
1470.2.i.j		2	7.c	even	3	1	inner
1680.2.a.j		1	28.f	even	6	1
3150.2.a.bp		1	105.p	even	6	1
3150.2.g.q		2	105.w	odd	12	2
4410.2.a.t		1	21.h	odd	6	1
5040.2.a.k		1	84.j	odd	6	1
6720.2.a.j		1	56.j	odd	6	1
6720.2.a.bq		1	56.m	even	6	1
7350.2.a.w		1	35.j	even	6	1
8400.2.a.ce		1	140.s	even	6	1

Hecke kernels

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

\( T_{11}^{2} - 4T_{11} + 16 \)

\( T_{13} - 2 \)

\( T_{17}^{2} - 2T_{17} + 4 \)

\( T_{19}^{2} - 4T_{19} + 16 \)

\( T_{31} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T + 16 \)