Newform orbit 5070.2.b.p

• Label: 5070.2.b.p
• Level: $5070$
• Weight: $2$
• Character orbit: 5070.b
• Analytic conductor: $40.484$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(40.4841538248\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{3} \)
\(\beta_{2}\)	\(=\)	\( 2\zeta_{12}^{2} - 1 \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\)	\(1691\)	\(1861\)	\(4057\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1351.1

0.866025	+	0.500000i
−0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i

−

1.00000i

−1.00000

−1.00000

1.00000i

1.00000i

3.00000i

1.00000i

1.00000

1.00000

1351.2

−

1.00000i

−1.00000

−1.00000

1.00000i

1.00000i

3.00000i

1.00000i

1.00000

1.00000

1351.3

1.00000i

−1.00000

−1.00000

−

1.00000i

−

1.00000i

−

3.00000i

−

1.00000i

1.00000

1.00000

1351.4

1.00000i

−1.00000

−1.00000

−

1.00000i

−

1.00000i

−

3.00000i

−

1.00000i

1.00000

1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5070.2.b.p		4
13.b	even	2	1	inner	5070.2.b.p		4
13.c	even	3	1		390.2.bb.a	✓	4
13.d	odd	4	1		5070.2.a.ba		2
13.d	odd	4	1		5070.2.a.be		2
13.e	even	6	1		390.2.bb.a	✓	4
39.h	odd	6	1		1170.2.bs.d		4
39.i	odd	6	1		1170.2.bs.d		4
65.l	even	6	1		1950.2.bc.a		4
65.n	even	6	1		1950.2.bc.a		4
65.q	odd	12	1		1950.2.y.d		4
65.q	odd	12	1		1950.2.y.e		4
65.r	odd	12	1		1950.2.y.d		4
65.r	odd	12	1		1950.2.y.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.bb.a	✓	4	13.c	even	3	1
390.2.bb.a	✓	4	13.e	even	6	1
1170.2.bs.d		4	39.h	odd	6	1
1170.2.bs.d		4	39.i	odd	6	1
1950.2.y.d		4	65.q	odd	12	1
1950.2.y.d		4	65.r	odd	12	1
1950.2.y.e		4	65.q	odd	12	1
1950.2.y.e		4	65.r	odd	12	1
1950.2.bc.a		4	65.l	even	6	1
1950.2.bc.a		4	65.n	even	6	1
5070.2.a.ba		2	13.d	odd	4	1
5070.2.a.be		2	13.d	odd	4	1
5070.2.b.p		4	1.a	even	1	1	trivial
5070.2.b.p		4	13.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{2} + 9 \)

\( T_{11}^{4} + 14T_{11}^{2} + 1 \)

\( T_{17} - 4 \)

\( T_{31}^{4} + 104T_{31}^{2} + 1936 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{2} \)
$3$	\( (T + 1)^{4} \)
$5$	\( (T^{2} + 1)^{2} \)
$7$	\( (T^{2} + 9)^{2} \)
$11$	\( T^{4} + 14T^{2} + 1 \)