Newform orbit 1458.2.a.g

• Label: 1458.2.a.g
• Level: $1458$
• Weight: $2$
• Character orbit: 1458.a
• Self dual: yes
• Analytic conductor: $11.642$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.6421886147\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.6.6357609.1
Defining polynomial:	\( x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + (\beta_{4} + 1) q^{5} + (\beta_1 + 1) q^{7} + q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{5} - 2\nu^{4} + 9\nu^{3} + 19\nu^{2} - 16\nu - 24 ) / 4 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} + 4\nu^{4} - 9\nu^{3} - 29\nu^{2} + 10\nu + 24 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 13\nu^{2} + 16\nu - 12 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} + \nu^{4} - 7\nu^{3} - 10\nu^{2} + 3\nu + 10 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} - \nu^{4} - 7\nu^{3} + 6\nu^{2} + 9\nu - 8 ) / 2 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

1.25119
2.63764
−2.37441
0.842316
−1.59848
−0.758254

1.00000

0

1.00000

−4.04750

0

−0.153630

1.00000

0

−4.04750

1.2

1.00000

0

1.00000

−1.02159

0

2.66680

1.00000

0

−1.02159

1.3

1.00000

0

1.00000

−0.740720

0

4.13282

1.00000

0

−0.740720

1.4

1.00000

0

1.00000

2.08802

0

−4.01220

1.00000

0

2.08802

1.5

1.00000

0

1.00000

3.16812

0

3.68572

1.00000

0

3.16812

1.6

1.00000

0

1.00000

3.55368

0

−0.319501

1.00000

0

3.55368

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1458.2.a.g		6
3.b	odd	2	1		1458.2.a.f		6
9.c	even	3	2		1458.2.c.f		12
9.d	odd	6	2		1458.2.c.g		12
27.e	even	9	2		54.2.e.b	✓	12
27.e	even	9	2		486.2.e.f		12
27.e	even	9	2		486.2.e.h		12
27.f	odd	18	2		162.2.e.b		12
27.f	odd	18	2		486.2.e.e		12
27.f	odd	18	2		486.2.e.g		12
108.j	odd	18	2		432.2.u.b		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.2.e.b	✓	12	27.e	even	9	2
162.2.e.b		12	27.f	odd	18	2
432.2.u.b		12	108.j	odd	18	2
486.2.e.e		12	27.f	odd	18	2
486.2.e.f		12	27.e	even	9	2
486.2.e.g		12	27.f	odd	18	2
486.2.e.h		12	27.e	even	9	2
1458.2.a.f		6	3.b	odd	2	1
1458.2.a.g		6	1.a	even	1	1	trivial
1458.2.c.f		12	9.c	even	3	2
1458.2.c.g		12	9.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3T_{5}^{5} - 18T_{5}^{4} + 57T_{5}^{3} + 36T_{5}^{2} - 108T_{5} - 72 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1458))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{6} \)
$3$	\( T^{6} \)
$5$	T^6 - 3*T^5 - 18*T^4 + 57*T^3 + 36*T^2 - 108*T - 72
$7$	T^6 - 6*T^5 - 9*T^4 + 101*T^3 - 114*T^2 - 72*T - 8
$11$	T^6 - 3*T^5 - 27*T^4 + 57*T^3 + 171*T^2 - 81*T + 9