DEGREE 3 WEIGHT 12

Scroll through or jump to chosen results (a jump may take some time due to page reload)
 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.
 t1 ½ [0 0 0 0 0 0] t2 ½ [2 0 0 0 0 0] t3 ½ [2 2 0 1 0 0] t4 ½ [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).
 t1 t2 t3 t4 t1 411162568697368361/6857444385792000 595025425032371/104661849600 14147393381239/568814400 237449482847/5745600 t2 595025425032371/104661849600 11864814481613/20766240 314046090217/112860 6067815041/1140 t3 14147393381239/568814400 314046090217/112860 457379889638/28215 10963328374/285 t4 237449482847/5745600 6067815041/1140 10963328374/285 11130020166/95

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.
 u1 ½ [0 0 0 0 0 0] u2 ½ [2 0 0 0 0 0] u3 ½ [4 0 0 0 0 0] u4 ½ [2 2 0 1 0 0] u5 ½ [4 4 0 2 0 0] u6 ½ [2 2 2 1 1 1] u7 ½ [2 2 2 0 0 0] u8 ½ [2 2 4 0 0 0] u9 ½ [2 2 6 1 1 1] u10 ½ [4 4 4 2 2 2] u11 ½ [2 6 6 0 0 2] u12 ½ [4 4 4 0 0 0] u13 ½ [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.
 g1 g2 g3 g4 f1 1 0 0 0 f2 -65520 691 0 0 f3 452088000 -7146836 543781 0 f4 -89513424000 1650869528 -197834263 21826375

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 1077415425 1154901456286648065 1971774054270720 481154236416 16777216 f2 -12619800 -685867303258560 280677902238720 206045577216 16777216 f3 1428192 731393197056 91455307776 68780556288 16777216 f4 -47808 -1676242944 235339776 66060288 16777216

EIGENFORM FOURIER COEFFICIENTS
The eigenform Fourier coefficients at the indices ui are as follows (switch to factored denominator form).
 u a(u; f1) a(u; f2) a(u; f3) a(u; f4) u1 411162568697368361/6857444385792000 0 0 0 u2 595025425032371/104661849600 107235250318/4807 0 0 u3 406402365297109393/34887283200 -2573646007632/4807 0 0 u4 14147393381239/568814400 61277285896/209 1895054400000/11 0 u5 396169456854835717/7584192 -1507421233041600/209 5275831449600000/11 0 u6 237449482847/5745600 18433615272/19 1561524825600 34538279731200 u7 9735428796727/229824 9273396066000/19 422854858598400 5664277875916800 u8 11867962602175907/273600 6368840643033072/19 144466455257395200 -34814585969049600 u9 248984066375258719/5745600 9718117528827144/19 301858045667596800 45866835483033600 u10 10228284879561304375/229824 -237376516795377600/19 2013655576120934400 -4551316349858611200 u11 13754430212387729078789/302400 3395356035305291568816/19 202348537429166899200 60539801937798758400 u12 261844591370961233905247/5745600 -126820607424689324928/19 381198762208242892800 -235430598247081574400 u13 54913000017403178146901052947/1149120 -7400374905526772072547840/19 1197153345953967833088000 73635214505935896576000

STANDARD EULER FACTORS
The standard 2-Euler factors Q2(f,  x,   st) are given by the following table (switch to expanded form).
 f Q2(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).
 f Q2(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).
 c1 6857444385792000/411162568697368361 c2 4807/74099557969738 c3 11/1030494576686400000 c4 1/753845445268070400000

MORE COEFFICIENTS