Properties

Label 5040.2.t.y
Level $5040$
Weight $2$
Character orbit 5040.t
Analytic conductor $40.245$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{5} ) q^{5} -\beta_{4} q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{5} ) q^{5} -\beta_{4} q^{7} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} + ( \beta_{2} + \beta_{3} ) q^{35} + 6 \beta_{4} q^{37} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{47} - q^{49} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{53} + ( 2 - 3 \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} ) q^{55} + ( -2 + 3 \beta_{1} + 3 \beta_{2} ) q^{59} + ( 8 - \beta_{1} - \beta_{2} ) q^{61} + ( 5 + \beta_{1} - \beta_{2} + 4 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 2 + 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{73} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{79} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} + 6 \beta_{5} ) q^{83} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{85} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{91} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 7 \beta_{4} + 3 \beta_{5} ) q^{95} + ( \beta_{1} - \beta_{2} - 4 \beta_{4} - 3 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q + 14q^{11} + 8q^{19} - 10q^{25} + 6q^{29} - 20q^{31} - 2q^{35} - 36q^{41} - 6q^{49} + 12q^{55} - 12q^{59} + 48q^{61} + 22q^{65} + 8q^{71} + 34q^{79} + 14q^{85} - 10q^{91} - 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.32001 1.32001i
1.32001 + 1.32001i
0.432320 + 0.432320i
0.432320 0.432320i
−1.75233 + 1.75233i
−1.75233 1.75233i
0 0 0 −1.32001 1.80487i 0 1.00000i 0 0 0
1009.2 0 0 0 −1.32001 + 1.80487i 0 1.00000i 0 0 0
1009.3 0 0 0 −0.432320 2.19388i 0 1.00000i 0 0 0
1009.4 0 0 0 −0.432320 + 2.19388i 0 1.00000i 0 0 0
1009.5 0 0 0 1.75233 1.38900i 0 1.00000i 0 0 0
1009.6 0 0 0 1.75233 + 1.38900i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.t.y 6
3.b odd 2 1 560.2.g.f 6
4.b odd 2 1 2520.2.t.g 6
5.b even 2 1 inner 5040.2.t.y 6
12.b even 2 1 280.2.g.b 6
15.d odd 2 1 560.2.g.f 6
15.e even 4 1 2800.2.a.bq 3
15.e even 4 1 2800.2.a.br 3
20.d odd 2 1 2520.2.t.g 6
24.f even 2 1 2240.2.g.l 6
24.h odd 2 1 2240.2.g.m 6
60.h even 2 1 280.2.g.b 6
60.l odd 4 1 1400.2.a.s 3
60.l odd 4 1 1400.2.a.t 3
84.h odd 2 1 1960.2.g.c 6
120.i odd 2 1 2240.2.g.m 6
120.m even 2 1 2240.2.g.l 6
420.o odd 2 1 1960.2.g.c 6
420.w even 4 1 9800.2.a.cd 3
420.w even 4 1 9800.2.a.cg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 12.b even 2 1
280.2.g.b 6 60.h even 2 1
560.2.g.f 6 3.b odd 2 1
560.2.g.f 6 15.d odd 2 1
1400.2.a.s 3 60.l odd 4 1
1400.2.a.t 3 60.l odd 4 1
1960.2.g.c 6 84.h odd 2 1
1960.2.g.c 6 420.o odd 2 1
2240.2.g.l 6 24.f even 2 1
2240.2.g.l 6 120.m even 2 1
2240.2.g.m 6 24.h odd 2 1
2240.2.g.m 6 120.i odd 2 1
2520.2.t.g 6 4.b odd 2 1
2520.2.t.g 6 20.d odd 2 1
2800.2.a.bq 3 15.e even 4 1
2800.2.a.br 3 15.e even 4 1
5040.2.t.y 6 1.a even 1 1 trivial
5040.2.t.y 6 5.b even 2 1 inner
9800.2.a.cd 3 420.w even 4 1
9800.2.a.cg 3 420.w even 4 1

Hecke kernels

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

\( T_{11}^{3} - 7 T_{11}^{2} + 8 T_{11} + 8 \)
\( T_{13}^{6} + 69 T_{13}^{4} + 1544 T_{13}^{2} + 11236 \)
\( T_{17}^{6} + 49 T_{17}^{4} + 536 T_{17}^{2} + 400 \)
\( T_{19}^{3} - 4 T_{19}^{2} - 14 T_{19} - 8 \)
\( T_{29}^{3} - 3 T_{29}^{2} - 72 T_{29} + 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 125 + 25 T^{2} - 8 T^{3} + 5 T^{4} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 8 + 8 T - 7 T^{2} + T^{3} )^{2} \)
$13$ \( 11236 + 1544 T^{2} + 69 T^{4} + T^{6} \)
$17$ \( 400 + 536 T^{2} + 49 T^{4} + T^{6} \)
$19$ \( ( -8 - 14 T - 4 T^{2} + T^{3} )^{2} \)
$23$ \( 18496 + 2416 T^{2} + 92 T^{4} + T^{6} \)
$29$ \( ( 108 - 72 T - 3 T^{2} + T^{3} )^{2} \)
$31$ \( ( -80 + 8 T + 10 T^{2} + T^{3} )^{2} \)
$37$ \( ( 36 + T^{2} )^{3} \)
$41$ \( ( -88 + 68 T + 18 T^{2} + T^{3} )^{2} \)
$43$ \( 4096 + 1600 T^{2} + 80 T^{4} + T^{6} \)
$47$ \( 53824 + 6480 T^{2} + 177 T^{4} + T^{6} \)
$53$ \( 222784 + 11376 T^{2} + 188 T^{4} + T^{6} \)
$59$ \( ( 44 - 78 T + 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( -440 + 182 T - 24 T^{2} + T^{3} )^{2} \)
$67$ \( 262144 + 14336 T^{2} + 228 T^{4} + T^{6} \)
$71$ \( ( 64 - 20 T - 4 T^{2} + T^{3} )^{2} \)
$73$ \( 18496 + 2416 T^{2} + 92 T^{4} + T^{6} \)
$79$ \( ( 548 - 32 T - 17 T^{2} + T^{3} )^{2} \)
$83$ \( 678976 + 39940 T^{2} + 428 T^{4} + T^{6} \)
$89$ \( ( -464 - 172 T + T^{3} )^{2} \)
$97$ \( 1936 + 1048 T^{2} + 113 T^{4} + T^{6} \)
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