Newform orbit 1045.2.a.f

• Label: 1045.2.a.f
• Level: $1045$
• Weight: $2$
• Character orbit: 1045.a
• Self dual: yes
• Analytic conductor: $8.344$
• Analytic rank: $1$
• Dimension: $6$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.34436701122\)
Analytic rank:	\(1\)
Dimension:	\(6\)
Coefficient field:	6.6.7281497.1
Defining polynomial:	\( x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} + \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} - q^{5} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 2\nu - 1 \)
\(\beta_{4}\)	\(=\)	\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 5\nu - 1 \)
\(\beta_{5}\)	\(=\)	\( -\nu^{5} + 3\nu^{4} + 3\nu^{3} - 11\nu^{2} + 4 \)

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

0.497517
−1.79049
−0.748369
1.77015
2.59744
−0.326248

−2.51246

0.895533

4.31247

−1.00000

−2.24999

−0.393051

−5.80998

−2.19802

2.51246

1.2

−2.23198

−1.34246

2.98176

−1.00000

2.99635

4.13295

−2.19127

−1.19780

2.23198

1.3

−0.412130

1.67805

−1.83015

−1.00000

−0.691575

0.0703171

1.57852

−0.184142

0.412130

1.4

0.205229

−3.10246

−1.95788

−1.00000

−0.636714

2.33231

−0.812271

6.62527

−0.205229

1.5

1.21244

1.77266

−0.529980

−1.00000

2.14925

−3.37010

−3.06746

0.142317

−1.21244

1.6

1.73890

−0.901323

1.02379

−1.00000

−1.56731

2.22757

−1.69754

−2.18762

−1.73890

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(11\)	\(1\)
\(19\)	\(1\)

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	1045.2.a.f	✓	6
3.b	odd	2	1		9405.2.a.z		6
5.b	even	2	1		5225.2.a.l		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1045.2.a.f	✓	6	1.a	even	1	1	trivial
5225.2.a.l		6	5.b	even	2	1
9405.2.a.z		6	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 8T_{2}^{3} + 11T_{2}^{2} + 3T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 + 2*T^5 - 6*T^4 - 8*T^3 + 11*T^2 + 3*T - 1
$3$	T^6 + T^5 - 9*T^4 - 2*T^3 + 19*T^2 + T - 10
$5$	\( (T + 1)^{6} \)
$7$	T^6 - 5*T^5 - 7*T^4 + 58*T^3 - 53*T^2 - 25*T + 2
$11$	\( (T + 1)^{6} \)