Properties

Label 5040.2.k.f
Level $5040$
Weight $2$
Character orbit 5040.k
Analytic conductor $40.245$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
Defining polynomial: \(x^{8} - 10 x^{4} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{7} ) q^{17} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( 5 - \beta_{4} ) q^{23} + ( 3 \beta_{2} - \beta_{4} ) q^{25} + \beta_{2} q^{29} + ( 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{35} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{41} + ( -5 \beta_{2} + \beta_{5} + \beta_{6} ) q^{43} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{47} + ( -1 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{49} + ( 7 + \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{55} + ( -4 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( -4 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( \beta_{5} + \beta_{6} ) q^{71} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( -7 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{77} + ( -4 + 4 \beta_{4} ) q^{79} + ( -4 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{89} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{91} + ( -3 - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{95} + ( 5 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 40 q^{23} + 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} - 32 q^{79} + 8 q^{85} - 16 q^{91} - 24 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 7 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45 \)\()/36\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45 \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 10 \nu^{3} \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} + 5\)
\(\nu^{5}\)\(=\)\(6 \beta_{3} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{4} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.420861 + 1.68014i
0.420861 1.68014i
−1.68014 0.420861i
−1.68014 + 0.420861i
1.68014 0.420861i
1.68014 + 0.420861i
−0.420861 + 1.68014i
−0.420861 1.68014i
0 0 0 −1.95522 1.08495i 0 −2.37608 1.16372i 0 0 0
1889.2 0 0 0 −1.95522 + 1.08495i 0 −2.37608 + 1.16372i 0 0 0
1889.3 0 0 0 −1.08495 1.95522i 0 0.595188 + 2.57794i 0 0 0
1889.4 0 0 0 −1.08495 + 1.95522i 0 0.595188 2.57794i 0 0 0
1889.5 0 0 0 1.08495 1.95522i 0 −0.595188 2.57794i 0 0 0
1889.6 0 0 0 1.08495 + 1.95522i 0 −0.595188 + 2.57794i 0 0 0
1889.7 0 0 0 1.95522 1.08495i 0 2.37608 + 1.16372i 0 0 0
1889.8 0 0 0 1.95522 + 1.08495i 0 2.37608 1.16372i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.k.f 8
3.b odd 2 1 5040.2.k.e 8
4.b odd 2 1 1260.2.f.a 8
5.b even 2 1 5040.2.k.e 8
7.b odd 2 1 inner 5040.2.k.f 8
12.b even 2 1 1260.2.f.b yes 8
15.d odd 2 1 inner 5040.2.k.f 8
20.d odd 2 1 1260.2.f.b yes 8
20.e even 4 2 6300.2.d.f 16
21.c even 2 1 5040.2.k.e 8
28.d even 2 1 1260.2.f.a 8
35.c odd 2 1 5040.2.k.e 8
60.h even 2 1 1260.2.f.a 8
60.l odd 4 2 6300.2.d.f 16
84.h odd 2 1 1260.2.f.b yes 8
105.g even 2 1 inner 5040.2.k.f 8
140.c even 2 1 1260.2.f.b yes 8
140.j odd 4 2 6300.2.d.f 16
420.o odd 2 1 1260.2.f.a 8
420.w even 4 2 6300.2.d.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.f.a 8 4.b odd 2 1
1260.2.f.a 8 28.d even 2 1
1260.2.f.a 8 60.h even 2 1
1260.2.f.a 8 420.o odd 2 1
1260.2.f.b yes 8 12.b even 2 1
1260.2.f.b yes 8 20.d odd 2 1
1260.2.f.b yes 8 84.h odd 2 1
1260.2.f.b yes 8 140.c even 2 1
5040.2.k.e 8 3.b odd 2 1
5040.2.k.e 8 5.b even 2 1
5040.2.k.e 8 21.c even 2 1
5040.2.k.e 8 35.c odd 2 1
5040.2.k.f 8 1.a even 1 1 trivial
5040.2.k.f 8 7.b odd 2 1 inner
5040.2.k.f 8 15.d odd 2 1 inner
5040.2.k.f 8 105.g even 2 1 inner
6300.2.d.f 16 20.e even 4 2
6300.2.d.f 16 60.l odd 4 2
6300.2.d.f 16 140.j odd 4 2
6300.2.d.f 16 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5040, [\chi])\):

\( T_{11}^{2} + 14 \)
\( T_{13}^{4} - 12 T_{13}^{2} + 8 \)
\( T_{23}^{2} - 10 T_{23} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 + 22 T^{4} + T^{8} \)
$7$ \( 2401 + 196 T^{2} - 10 T^{4} + 4 T^{6} + T^{8} \)
$11$ \( ( 14 + T^{2} )^{4} \)
$13$ \( ( 8 - 12 T^{2} + T^{4} )^{2} \)
$17$ \( ( 8 + 12 T^{2} + T^{4} )^{2} \)
$19$ \( ( 648 + 52 T^{2} + T^{4} )^{2} \)
$23$ \( ( 18 - 10 T + T^{2} )^{4} \)
$29$ \( ( 2 + T^{2} )^{4} \)
$31$ \( ( 72 + 76 T^{2} + T^{4} )^{2} \)
$37$ \( ( 144 + 32 T^{2} + T^{4} )^{2} \)
$41$ \( ( 3528 - 140 T^{2} + T^{4} )^{2} \)
$43$ \( ( 1296 + 128 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2592 + 104 T^{2} + T^{4} )^{2} \)
$53$ \( ( 42 - 14 T + T^{2} )^{4} \)
$59$ \( ( 288 - 152 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( ( 576 + 176 T^{2} + T^{4} )^{2} \)
$71$ \( ( 14 + T^{2} )^{4} \)
$73$ \( ( 72 - 76 T^{2} + T^{4} )^{2} \)
$79$ \( ( -96 + 8 T + T^{2} )^{4} \)
$83$ \( ( 1568 + 168 T^{2} + T^{4} )^{2} \)
$89$ \( ( 72 - 20 T^{2} + T^{4} )^{2} \)
$97$ \( ( 31752 - 364 T^{2} + T^{4} )^{2} \)
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