DEGREE 3 WEIGHT 12

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eigenvalues | standard Euler factors | spinor Euler factors | identified eigenforms | special values


DIMENSIONS
     dim M123)= 4
     dim S123) = 1


DETERMINING INDICES
Introduce the notation
    [a b c d e f]=
 
a d e
d b f
e f c
 
We use the following set of determining Fourier coefficient indices.
t1½ [0 0 0 0 0 0]
t2½ [2 0 0 0 0 0]
t3½ [2 2 0 1 0 0]
t4½ [2 2 2 1 1 1]


PULLBACK OF EISENSTEIN SERIES
Let E(6)12 denote the monic Eisenstein series of degree 6 and weight 12.
The rescaled Eisenstein series
     E(6)12 = 411162568697368361/6857444385792000 E(6)12
has rational Fourier coefficients that are near-integers in that their denominator prime factors are at most 23.
Consider the Witt map W3,3 and the pullback W*3,3E(6)12 ∈ M123) ⊗ M123).
The columns of the 4 x 4 matrix [a(ti x tj; W*3,3E(6)12)] are determining truncations of an M123)-basis gt1,...,gt4. Each gt is the cofficient function of e(tr(t,z)) in the Fourier expansion of W*3,3E(6)12(w,z). For convenience let gj = gtj for j=1,...,4. The basis elements g1,...,g4 are not Hecke eigenforms. The matrix [a(ti x tj; W*3,3E(6)12)] is as follows (switch to factored form).
t1t2t3t4
t1 411162568697368361/6857444385792000595025425032371/10466184960014147393381239/568814400237449482847/5745600
t2 595025425032371/10466184960011864814481613/20766240314046090217/1128606067815041/1140
t3 14147393381239/568814400314046090217/112860457379889638/2821510963328374/285
t4 237449482847/57456006067815041/114010963328374/28511130020166/95


INDICES FOR COMPUTING HECKE ACTION
Fourier coefficients for the following indices are needed to determine the action of the Hecke operator T(2) at the determining indices t1,...,t4 and also the action of Ti(4) for i=0,1,2,3 at t4.
u1½ [0 0 0 0 0 0]
u2½ [2 0 0 0 0 0]
u3½ [4 0 0 0 0 0]
u4½ [2 2 0 1 0 0]
u5½ [4 4 0 2 0 0]
u6½ [2 2 2 1 1 1]
u7½ [2 2 2 0 0 0]
u8½ [2 2 4 0 0 0]
u9½ [2 2 6 1 1 1]
u10½ [4 4 4 2 2 2]
u11½ [2 6 6 0 0 2]
u12½ [4 4 4 0 0 0]
u13½ [8 8 8 4 4 4]


EIGENFORM BASIS FROM PULLBACK-GENUS BASIS
The M123) eigenform basis f1,...,f4 comprises integer linear combinations of the rational near-integer non-eigenform basis g1,...,g4 from above. Specifically fi = Σj=14 bi,jgj for each i, where the matrix (bi,j) is as follows.
g1g2g3g4
f1 1000
f2 -6552069100
f3 452088000-71468365437810
f4 -895134240001650869528-19783426321826375


EIGENVALUES
The T(2) and Ti(4) (i=0,1,2,3) eigenvalues of the eigenforms are shown in the following table.
f λ2(f) λ0,4(f) λ1,4(f) λ2,4(f) λ3,4(f)
f1 10774154251154901456286648065197177405427072048115423641616777216
f2 -12619800-68586730325856028067790223872020604557721616777216
f3 1428192731393197056914553077766878055628816777216
f4 -47808-16762429442353397766606028816777216


EIGENFORM FOURIER COEFFICIENTS
The eigenform Fourier coefficients at the indices ui are as follows (switch to factored denominator form).
ua(u; f1)a(u; f2)a(u; f3)a(u; f4)
u1 411162568697368361/6857444385792000000
u2 595025425032371/104661849600107235250318/480700
u3 406402365297109393/34887283200-2573646007632/480700
u4 14147393381239/56881440061277285896/2091895054400000/110
u5 396169456854835717/7584192-1507421233041600/2095275831449600000/110
u6 237449482847/574560018433615272/19156152482560034538279731200
u7 9735428796727/2298249273396066000/194228548585984005664277875916800
u8 11867962602175907/2736006368840643033072/19144466455257395200-34814585969049600
u9 248984066375258719/57456009718117528827144/1930185804566759680045866835483033600
u10 10228284879561304375/229824-237376516795377600/192013655576120934400-4551316349858611200
u11 13754430212387729078789/3024003395356035305291568816/1920234853742916689920060539801937798758400
u12 261844591370961233905247/5745600-126820607424689324928/19381198762208242892800-235430598247081574400
u13 54913000017403178146901052947/1149120-7400374905526772072547840/19119715334595396783308800073635214505935896576000


STANDARD EULER FACTORS
The standard 2-Euler factors Q2(f,  x,   st) are given by the following table (switch to expanded form).
fQ2(f,  x,   st)/(1-x)
f1(1-2048*x)*(1-1024*x)*(1-512*x)*(1-x/512)*(1-x/1024)*(1-x/2048)
f2(1-1024*x)*(1-512*x)*(1-x/512)*(1-x/1024)*(1+(55*x)/32+x^2)
f3(1-512*x)*(1-x/512)*(1+(9*x)/64+x^2/2)*(1+(9*x)/32+2*x^2)
f4(1-(57*x)/128+x^2/2)*(1+(55*x)/32+x^2)*(1-(57*x)/64+2*x^2)


SPINOR EULER FACTORS
The spinor 2-Euler factors Q2(f,  x,   spin) are given by the following table (switch to factored form).
fQ2(f,  x,  spin)
f11-1077415425*x+3948360489766400*x^2-4147368577200016588800*x^3+1252772137538465207836213248*x^4-4453203100883030624888369971200*x^5+4552149716591705120038218786406400*x^6-1333775693551174685167056757928755200*x^7+1329227995784915872903807060280344576*x^8
f21+12619800*x+563417099110400*x^2+20803725603687628800*x^3+1682161801676480732725248*x^4+22337830275699055673946931200*x^5+649575689627587392863438530150400*x^6+15622555707773641993171967882035200*x^7+1329227995784915872903807060280344576*x^8
f31-1428192*x+871555039232*x^2-786046410653958144*x^3+304415950273588582416384*x^4-844010906724234050358214656*x^5+1004834547179036984903500562432*x^6-1768016060587065825568730023723008*x^7+1329227995784915872903807060280344576*x^8
f41+47808*x+1970143232*x^2+31682363129856*x^3-1067313529268207616*x^4+34018678375681930297344*x^5+2271420499328436390626066432*x^6+59183437398155460182377330900992*x^7+1329227995784915872903807060280344576*x^8


IDENTIFIED EIGENFORMS
Based on matching standard and spinor 2-Euler factors we identify the following eigenforms.
  • f1 = E 12 : Basic Siegel Eisenstein series
  • f2 = K1 12 1 : Klingen lift from degree 1
  • f3 = K2 12 1 li : Klingen lift of degree 2 lift
  • f4 = M1 12 1 1 : Miyawaki lift of type 1

  • SPECIAL VALUES
    The special values ci such that W*3,3E(6)12i=14 ci fi⊗fi are given in the following table (switch to factored denominator form).
    c16857444385792000/411162568697368361
    c24807/74099557969738
    c311/1030494576686400000
    c41/753845445268070400000


    MORE COEFFICIENTS
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