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 nonfactored denominator form).
ua(u; f1)a(u; f2)a(u; f3)a(u; f4)
u1 131*283*593*617*691*43867/(2^13*3^7*5^3*7^2*11*13*19*23)000
u2 131*283*593*617*43867/(2^9*3^5*5^2*7*11*19*23)2*7*283*617*43867/(11*19*23)00
u3 131*283*593*617*683*43867/(2^9*3^4*5^2*7*11*19*23)-2^4*3*7*283*617*43867/(11*19*23)00
u4 283*617*1847*43867/(2^6*3^5*5^2*7*11*19)2^3*283*617*43867/(11*19)2^9*3^3*5^5*43867/(11)0
u5 41*283*617*683*1847*43867/(2^6*3^4*7*11*19)-2^6*3*5^2*41*283*617*43867/(11*19)2^14*3^4*5^5*29*43867/(11)0
u6 31*283*617*43867/(2^6*3^3*5^2*7*19)2^3*3*17509*43867/(19)2^9*3^3*5^2*103*438672^16*3^11*5^2*7*17
u7 31*41*283*617*43867/(2^6*3^3*7*19)2^4*3*5^3*11*3203*43867/(19)2^11*3^3*5^2*19*367*438672^18*3^11*5^2*7*17*41
u8 31*151*283*331*617*43867/(2^6*3^2*5^2*19)2^4*3^2*257*43867*3923077/(19)2^12*3^4*5^2*23*61*283*43867-2^20*3^13*5^2*7^2*17
u9 17*31*283*617*43867*61681/(2^6*3^3*5^2*7*19)2^3*3*47*3463*43867*56713/(19)2^9*3^3*5^2*167*43867*1192272^20*3^11*5^2*7*17*83
u10 5^4*31*41^3*283*617*43867/(2^6*3^3*7*19)-2^6*3*5^2*563*43867*2002397/(19)2^14*3^3*5^2*19*43867*218459-2^22*3^11*5^2*7*17*29*71
u11 31*79*283*617*43867*733236853/(2^6*3^3*5^2*7)2^4*3*43867*1612523857771451/(19)2^14*3^3*5^2*43867*4170983032^24*3^11*5^2*7*17*41*167
u12 31*283*617*25117*43867*43904053/(2^6*3^3*5^2*7*19)-2^7*3*2557*43867*2944353443/(19)2^16*3^3*5^2*31*43867*6336773-2^24*3^11*5^2*7*17*26627
u13 13*31*283*617*43867*17789370917002177/(2^6*3^3*5*7*19)-2^9*3*5*7*311*16649*43867*606049343/(19)2^19*3^3*5^3*7*4261*43867*5170812^28*3^11*5^3*7*17*19*5479


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 expanded form).
fQ2(f,  x,  spin)
f1(1-1073741824*x)*(1-2097152*x)*(1-1048576*x)*(1-524288*x)*(1-2048*x)*(1-1024*x)*(1-512*x)*(1-x)
f2(1+24*x+2048*x^2)*(1+12288*x+536870912*x^2)*(1+24576*x+2147483648*x^2)*(1+12582912*x+562949953421312*x^2)
f3(1-1048576*x)*(1-524288*x)*(1-2048*x)*(1-1024*x)*(1+288*x+2097152*x^2)*(1+147456*x+549755813888*x^2)
f4(1+12288*x+536870912*x^2)*(1+24576*x+2147483648*x^2)*(1+10944*x-1419640832*x^2+11751030521856*x^3+1152921504606846976*x^4)


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


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