Newform 6040.2.a.l

• Label: 6040.2.a.l
• Level: 6040
• Weight: 2
• Character orbit: 6040.a
• Self dual: Yes
• Analytic conductor: 48.230
• Analytic rank: 1
• Dimension: 9
• CM: No
• Inner twists: 1

Newspace parameters

Level:	\( N \)	=	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	6040.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	\(48.2296428209\)
Analytic rank:	\(1\)
Dimension:	\(9\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(11\) \(x^{7}\mathstrut +\mathstrut \) \(9\) \(x^{6}\mathstrut +\mathstrut \) \(32\) \(x^{5}\mathstrut -\mathstrut \) \(17\) \(x^{4}\mathstrut -\mathstrut \) \(27\) \(x^{3}\mathstrut +\mathstrut \) \(10\) \(x^{2}\mathstrut +\mathstrut \) \(3\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\((\)\( 3 \nu^{8} + 4 \nu^{7} - 41 \nu^{6} - 47 \nu^{5} + 164 \nu^{4} + 141 \nu^{3} - 181 \nu^{2} - 89 \nu + 31 \)\()/13\)
\(\beta_{3}\)	\(=\)	\((\)\( 7 \nu^{8} - 8 \nu^{7} - 74 \nu^{6} + 68 \nu^{5} + 192 \nu^{4} - 100 \nu^{3} - 106 \nu^{2} + 22 \nu - 23 \)\()/13\)
\(\beta_{4}\)	\(=\)	\( -\nu^{8} + \nu^{7} + 11 \nu^{6} - 9 \nu^{5} - 32 \nu^{4} + 17 \nu^{3} + 27 \nu^{2} - 10 \nu - 3 \)
\(\beta_{5}\)	\(=\)	\((\)\( 16 \nu^{8} - 9 \nu^{7} - 184 \nu^{6} + 70 \nu^{5} + 580 \nu^{4} - 67 \nu^{3} - 532 \nu^{2} - 24 \nu + 70 \)\()/13\)
\(\beta_{6}\)	\(=\)	\((\)\( -25 \nu^{8} + 23 \nu^{7} + 268 \nu^{6} - 189 \nu^{5} - 734 \nu^{4} + 242 \nu^{3} + 555 \nu^{2} - 34 \nu - 59 \)\()/13\)
\(\beta_{7}\)	\(=\)	\((\)\( -27 \nu^{8} + 16 \nu^{7} + 304 \nu^{6} - 123 \nu^{5} - 917 \nu^{4} + 122 \nu^{3} + 771 \nu^{2} - 18 \nu - 71 \)\()/13\)
\(\beta_{8}\)	\(=\)	\((\)\( 57 \nu^{8} - 41 \nu^{7} - 636 \nu^{6} + 329 \nu^{5} + 1894 \nu^{4} - 389 \nu^{3} - 1606 \nu^{2} + 38 \nu + 160 \)\()/13\)

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

0.390645
2.02556
−1.10984
−0.357360
−1.41589
2.68790
0.289144
1.10659
−2.61675

0

−1.96251

0

1.00000

0

−2.16922

0

0.851433

0

1.2

0

−1.75822

0

1.00000

0

1.53187

0

0.0913304

0

1.3

0

−1.57672

0

1.00000

0

−0.208807

0

−0.513952

0

1.4

0

−1.13091

0

1.00000

0

2.44094

0

−1.72105

0

1.5

0

0.0722232

0

1.00000

0

−0.709626

0

−2.99478

0

1.6

0

0.325171

0

1.00000

0

2.31586

0

−2.89426

0

1.7

0

1.17504

0

1.00000

0

−3.16934

0

−1.61928

0

1.8

0

2.35518

0

1.00000

0

0.202920

0

2.54689

0

1.9

0

2.50073

0

1.00000

0

−2.23459

0

3.25367

0

Inner twists

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

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(-1\)
\(151\)	\(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{9} - \cdots\)

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

\(T_{11}^{9} + \cdots\)