A Borcherds Product in S2(K(349))+

(Filename: BP-2-349-1-0-1-hs.html)

DEFINITION OF ψ AND BORCH(ψ)
The weakly holomorphic Jacobi form ψ of weight 0 and index 349, is defined as
    ψ = Σj=122 cj (φj|V2)/φj
where the coefficients cj and theta blocks φj are given as follows. Note each φj ∈ J2,349cusp with ν(φj) = 1.
cjφj
-1
TB(2;1,2,3,7,8,9,10,10,11,13)
-1
TB(2;1,1,2,3,7,10,10,11,12,13)
1
TB(2;1,1,2,7,8,9,9,10,11,14)
1
TB(2;1,2,3,5,6,9,9,11,12,14)
1
TB(2;1,2,4,4,6,8,10,11,12,14)
-1
TB(2;1,2,4,5,6,7,9,11,13,14)
-1
TB(2;1,4,5,5,6,7,10,10,11,15)
1
TB(2;1,3,5,6,7,8,8,9,12,15)
-2
TB(2;1,2,5,5,6,7,8,10,13,15)
1
TB(2;1,2,3,5,5,7,8,10,14,15)
-2
TB(2;1,2,5,5,6,7,9,10,11,16)
1
TB(2;1,3,4,4,7,7,9,10,11,16)
1
TB(2;1,1,2,4,5,7,9,11,12,16)
-1
TB(2;1,1,2,3,5,7,8,8,15,16)
2
TB(2;1,2,3,5,6,7,8,10,11,17)
-1
TB(2;1,2,4,5,6,6,7,11,11,17)
-1
TB(2;2,2,3,5,5,7,7,10,12,17)
1
TB(2;1,2,3,4,5,7,7,10,11,18)
1
TB(2;1,1,1,2,6,7,8,9,10,19)
1
TB(2;1,1,2,4,5,6,6,7,13,19)
-1
TB(2;1,1,2,3,4,5,6,7,14,19)
-1
TB(2;1,2,3,3,4,5,7,8,11,20)
Note that BORCH(ψ) has leading Jacobi coefficient the theta block φ ∈ J2,349cusp, with ν(φ) = 1,
     φ = TB(2;1,1,2,2,−4,−6,7,8,9,9,10,10,11,12)
= 12224−16−1718192102111121
= η(τ)4(θ(τ,z)/η(τ))2(θ(τ,2z)/η(τ))2(θ(τ,4z)/η(τ))−1(θ(τ,6z)/η(τ))−1(θ(τ,7z)/η(τ))(θ(τ,8z)/η(τ))(θ(τ,9z)/η(τ))2(θ(τ,10z)/η(τ))2(θ(τ,11z)/η(τ))(θ(τ,12z)/η(τ))
and so BORCH(ψ) begins with Jacobi coefficient #1.

PROPERTIES OF ψ AND BORCH(ψ)
See this page for properties of ψ and BORCH(ψ), such as Humbert divisors of BORCH(ψ) and some Fourier coefficients of BORCH(ψ).