Properties

Label 5040.2.t.t
Level $5040$
Weight $2$
Character orbit 5040.t
Analytic conductor $40.245$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{2} q^{7} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{13} - 2 \beta_{2} q^{17} + ( - \beta_{3} + \beta_1 + 4) q^{19} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + (2 \beta_{3} - 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{35} + 2 \beta_{2} q^{37} + ( - 2 \beta_{3} + 2 \beta_1 + 6) q^{41} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{47} - q^{49} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{53} + ( - 6 \beta_{2} - 4 \beta_1 + 6) q^{55} + ( - \beta_{3} + \beta_1 + 4) q^{59} + ( - \beta_{3} + \beta_1 + 6) q^{61} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 1) q^{65} + 8 \beta_{2} q^{67} + (2 \beta_{3} - 2 \beta_1 - 6) q^{71} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{73} + (2 \beta_{3} + 2 \beta_1) q^{77} + (2 \beta_{3} - 2 \beta_1 + 2) q^{79} + ( - \beta_{3} - \beta_1) q^{83} + (2 \beta_{2} - 2 \beta_1 - 2) q^{85} - 10 q^{89} + (\beta_{3} - \beta_1 + 2) q^{91} + ( - 4 \beta_{3} - 7 \beta_{2} - 2 \beta_1 - 1) q^{95} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 16 q^{19} + 8 q^{29} - 16 q^{31} + 4 q^{35} + 24 q^{41} - 4 q^{49} + 24 q^{55} + 16 q^{59} + 24 q^{61} + 4 q^{65} - 24 q^{71} + 8 q^{79} - 8 q^{85} - 40 q^{89} + 8 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.22474 1.22474i
0 0 0 −2.22474 0.224745i 0 1.00000i 0 0 0
1009.2 0 0 0 −2.22474 + 0.224745i 0 1.00000i 0 0 0
1009.3 0 0 0 0.224745 2.22474i 0 1.00000i 0 0 0
1009.4 0 0 0 0.224745 + 2.22474i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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.t 4
3.b odd 2 1 560.2.g.e 4
4.b odd 2 1 630.2.g.g 4
5.b even 2 1 inner 5040.2.t.t 4
12.b even 2 1 70.2.c.a 4
15.d odd 2 1 560.2.g.e 4
15.e even 4 1 2800.2.a.bl 2
15.e even 4 1 2800.2.a.bm 2
20.d odd 2 1 630.2.g.g 4
20.e even 4 1 3150.2.a.bs 2
20.e even 4 1 3150.2.a.bt 2
24.f even 2 1 2240.2.g.j 4
24.h odd 2 1 2240.2.g.i 4
60.h even 2 1 70.2.c.a 4
60.l odd 4 1 350.2.a.g 2
60.l odd 4 1 350.2.a.h 2
84.h odd 2 1 490.2.c.e 4
84.j odd 6 2 490.2.i.f 8
84.n even 6 2 490.2.i.c 8
120.i odd 2 1 2240.2.g.i 4
120.m even 2 1 2240.2.g.j 4
420.o odd 2 1 490.2.c.e 4
420.w even 4 1 2450.2.a.bl 2
420.w even 4 1 2450.2.a.bq 2
420.ba even 6 2 490.2.i.c 8
420.be odd 6 2 490.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 12.b even 2 1
70.2.c.a 4 60.h even 2 1
350.2.a.g 2 60.l odd 4 1
350.2.a.h 2 60.l odd 4 1
490.2.c.e 4 84.h odd 2 1
490.2.c.e 4 420.o odd 2 1
490.2.i.c 8 84.n even 6 2
490.2.i.c 8 420.ba even 6 2
490.2.i.f 8 84.j odd 6 2
490.2.i.f 8 420.be odd 6 2
560.2.g.e 4 3.b odd 2 1
560.2.g.e 4 15.d odd 2 1
630.2.g.g 4 4.b odd 2 1
630.2.g.g 4 20.d odd 2 1
2240.2.g.i 4 24.h odd 2 1
2240.2.g.i 4 120.i odd 2 1
2240.2.g.j 4 24.f even 2 1
2240.2.g.j 4 120.m even 2 1
2450.2.a.bl 2 420.w even 4 1
2450.2.a.bq 2 420.w even 4 1
2800.2.a.bl 2 15.e even 4 1
2800.2.a.bm 2 15.e even 4 1
3150.2.a.bs 2 20.e even 4 1
3150.2.a.bt 2 20.e even 4 1
5040.2.t.t 4 1.a even 1 1 trivial
5040.2.t.t 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13}^{4} + 20T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 10 \) Copy content Toggle raw display
\( T_{29}^{2} - 4T_{29} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 8 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 264T^{2} + 3600 \) Copy content Toggle raw display
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