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Properties

Label 5040.2.a.bu
Level $5040$
Weight $2$
Character orbit 5040.a
Self dual yes
Analytic conductor $40.245$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5040,2,Mod(1,5040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5040.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,2,0,0,0,4,0,-4,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + q^{7} + (\beta + 2) q^{11} + (\beta - 2) q^{13} + ( - 2 \beta - 2) q^{17} + \beta q^{19} + ( - 2 \beta + 2) q^{23} + q^{25} - 8 q^{29} - 3 \beta q^{31} - q^{35} - 6 q^{37} + (2 \beta + 2) q^{41} + \cdots + (\beta - 6) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 16 q^{29} - 2 q^{35} - 12 q^{37} + 4 q^{41} + 8 q^{43} - 8 q^{47} + 2 q^{49} - 16 q^{53} - 4 q^{55} - 8 q^{59} + 12 q^{61}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0 0 −1.00000 0 1.00000 0 0 0
1.2 0 0 0 −1.00000 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 5040.2.a.bu 2
3.b odd 2 1 5040.2.a.bx 2
4.b odd 2 1 315.2.a.c ✓ 2
12.b even 2 1 315.2.a.f yes 2
20.d odd 2 1 1575.2.a.u 2
20.e even 4 2 1575.2.d.h 4
28.d even 2 1 2205.2.a.p 2
60.h even 2 1 1575.2.a.m 2
60.l odd 4 2 1575.2.d.j 4
84.h odd 2 1 2205.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.a.c ✓ 2 4.b odd 2 1
315.2.a.f yes 2 12.b even 2 1
1575.2.a.m 2 60.h even 2 1
1575.2.a.u 2 20.d odd 2 1
1575.2.d.h 4 20.e even 4 2
1575.2.d.j 4 60.l odd 4 2
2205.2.a.p 2 28.d even 2 1
2205.2.a.y 2 84.h odd 2 1
5040.2.a.bu 2 1.a even 1 1 trivial
5040.2.a.bx 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} - 8 \) Copy content Toggle raw display
\( T_{29} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 32 \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 92 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
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