A Borcherds Product in S₂(K(1174))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(1174/4) is
  q^-1(1)
  +(4 + ζ^(-21) + ζ^(-18) + ζ^(-16) + ζ^(-15) + ζ^(-14) + ζ^(-13) + ζ^(-12) + ζ^(-11) + ζ^(-10) + ζ^(-9) + ζ^(-8) + 2/ζ^7 + ζ^(-6) + 2/ζ^5 + ζ^(-4) + 2/ζ^3 + 2/ζ^2 + ζ^(-1) + ζ + 2*ζ^2 + 2*ζ^3 + ζ^4 + 2*ζ^5 + ζ^6 + 2*ζ^7 + ζ^8 + ζ^9 + ζ^10 + ζ^11 + ζ^12 + ζ^13 + ζ^14 + ζ^15 + ζ^16 + ζ^18 + ζ^21)
  +q(ζ^(-71) + ζ^(-70) + ζ^(-69) + ζ^69 + ζ^70 + ζ^71)
  +q^2(2/ζ^97 + 2*ζ^97)
  +q^3(ζ^(-120) + ζ^(-119) + ζ^119 + ζ^120)
  +q^4(ζ^(-138) + ζ^138)
  +q^5(2/ζ^154 + 2*ζ^154)
  +q^6(2/ζ^168 + 2*ζ^168)
  +q^7(ζ^(-182) + ζ^182)
  +q^8(ζ^(-194) + ζ^194)
  +q^9(ζ^(-206) + ζ^206)
  +q^10(ζ^(-217) + ζ^217)
  +q^18(ζ^(-291) + ζ^291)
  +q^19(ζ^(-299) + ζ^299)
  +q^22(ζ^(-322) + ζ^322)
  +q^23(ζ^(-329) + ζ^329)
  +q^24(ζ^(-336) + ζ^336)
  +q^25(ζ^(-343) + ζ^343)
  +q^28(ζ^(-363) + ζ^363)
  +q^33(ζ^(-394) + ζ^394)
  +q^37(ζ^(-417) + ζ^417)
  +q^39(2/ζ^428 + 2*ζ^428)
  +q^41(ζ^(-439) + ζ^439)
  +q^45(ζ^(-460) + ζ^460)
  +q^46(ζ^(-465) + ζ^465)
  +q^47(ζ^(-470) + ζ^470)
  +q^53(ζ^(-499) + ζ^499)
  +q^60(ζ^(-531) + ζ^531)
  +q^63(ζ^(-544) + ζ^544)
  +q^67(2/ζ^561 + 2*ζ^561)
  +q^80(ζ^(-613) + ζ^613)
  +q^85(ζ^(-632) + ζ^632)
  +q^88(ζ^(-643) + ζ^643)
  +q^91(ζ^(-654) + ζ^654)
  +q^93(ζ^(-661) + ζ^661)
  +q^95(2/ζ^668 + 2*ζ^668)
  +q^97(ζ^(-675) + ζ^675)
  +q^99(ζ^(-682) + ζ^682)
  +q^104(ζ^(-699) + ζ^699)
  +q^111(ζ^(-722) + ζ^722)
  +q^115(ζ^(-735) + ζ^735)
  +q^121(ζ^(-754) + ζ^754)
  +q^122(ζ^(-757) + ζ^757)
  +q^139(2/ζ^808 + 2*ζ^808)
  +q^160(ζ^(-867) + ζ^867)
  +q^163(ζ^(-875) + ζ^875)
  +q^166(2/ζ^883 + 2*ζ^883)
  +q^169(ζ^(-891) + ζ^891)
  +q^172(ζ^(-899) + ζ^899)
  +q^174(ζ^(-904) + ζ^904)
  +q^177(ζ^(-912) + ζ^912)
  +q^179(ζ^(-917) + ζ^917)
  +q^181(2/ζ^922 + 2*ζ^922)
  +q^195(ζ^(-957) + ζ^957)
  +q^197(ζ^(-962) + ζ^962)
  +q^202(ζ^(-974) + ζ^974)
  +q^207(ζ^(-986) + ζ^986)
  +q^209(ζ^(-991) + ζ^991)
  +q^215(ζ^(-1005) + ζ^1005)
  +q^221(ζ^(-1019) + ζ^1019)
  +q^225(ζ^(-1028) + ζ^1028)
  +q^228(ζ^(-1035) + ζ^1035)
  +q^237(2/ζ^1055 + 2*ζ^1055)
  +q^241(ζ^(-1064) + ζ^1064)
  +q^247(ζ^(-1077) + ζ^1077)
  +q^253(2/ζ^1090 + 2*ζ^1090)
  +q^260(ζ^(-1105) + ζ^1105)
  +q^269(ζ^(-1124) + ζ^1124)
  +q^285(ζ^(-1157) + ζ^1157)
  +q^286(2/ζ^1159 + 2*ζ^1159)
  +q^287(ζ^(-1161) + ζ^1161)
  +q^288(ζ^(-1163) + ζ^1163)
  +q^289(ζ^(-1165) + ζ^1165)
  +q^290(ζ^(-1167) + ζ^1167)
  +q^291(ζ^(-1169) + ζ^1169)
  +q^292(2/ζ^1171 + 2*ζ^1171)
  +q^293(2/ζ^1173 + 2*ζ^1173)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(2;1,2,2,3,3,4,5,5,6,7,7,8,9,10,11,12,13,14,15,16,18,21)
  		= 112232415261728191101111121131141151161181211
  		= η(τ)4(θ(τ,z)/η(τ))(θ(τ,2z)/η(τ))2(θ(τ,3z)/η(τ))2(θ(τ,4z)/η(τ))(θ(τ,5z)/η(τ))2(θ(τ,6z)/η(τ))(θ(τ,7z)/η(τ))2(θ(τ,8z)/η(τ))(θ(τ,9z)/η(τ))(θ(τ,10z)/η(τ))(θ(τ,11z)/η(τ))(θ(τ,12z)/η(τ))(θ(τ,13z)/η(τ))(θ(τ,14z)/η(τ))(θ(τ,15z)/η(τ))(θ(τ,16z)/η(τ))(θ(τ,18z)/η(τ))(θ(τ,21z)/η(τ))

  This theta block has q-order of vanishing 2. This theta block has index 1*1174, and so Borch(ψ) begins with Jacobi coefficient #1.
• From the truncated singular part of ψ, we compute that Borch(ψ) is an antisymmetric form, and so the Fricke sign of Borch(ψ) is −1.
• We need to check that certain Fourier coefficients of Borch(ψ) are zero to determine that Borch(ψ) is a cusp form. For this weight and level and Fricke sign, an expansion of Borch(ψ) to 1 Fourier-Jacobi coefficients was needed to show that Borch(ψ) is a cusp form.
• From the truncated singular part of ψ, we can compute that all Humbert divisors of Borch(ψ) have nonnegative multiplicities, and hence Borch(ψ) is holomorphic.

  (D,r)	multiplicity of Hum(D,r)
  (1,1)	22
  (1,1173)	22
  (4,2)	10
  (9,3)	8
  (9,1171)	8
  (12,1090)	8
  (16,4)	4
  (17,97)	4
  (17,1077)	4
  (25,5)	4
  (25,1169)	4
  (28,722)	3
  (36,6)	3
  (40,428)	3
  (48,168)	3
  (49,7)	4
  (49,1167)	2
  (64,8)	2
  (65,69)	2
  (65,1105)	2
  (68,194)	1
  (73,119)	1
  (73,1055)	2
  (81,9)	2
  (81,1165)	2
  (84,974)	1
  (88,544)	1
  (89,561)	2
  (89,613)	1
  (100,10)	1
  (104,668)	2
  (108,922)	2
  (112,904)	1
  (113,499)	1
  (113,675)	1
  (120,808)	2
  (121,11)	1
  (121,1163)	1
  (124,986)	1
  (129,217)	1
  (129,957)	1
  (137,417)	1
  (137,757)	1
  (144,12)	1
  (152,1124)	1
  (153,291)	1
  (153,883)	2
  (169,13)	1
  (169,1161)	1
  (172,206)	1
  (177,299)	1
  (177,875)	1
  (184,1028)	1
  (185,439)	1
  (185,735)	1
  (188,470)	1
  (192,336)	1
  (193,661)	1
  (196,14)	1
  (201,531)	1
  (201,643)	1
  (204,70)	1
  (209,465)	1
  (217,699)	1
  (220,682)	1
  (225,15)	1
  (225,1159)	2
  (233,329)	1
  (236,154)	2
  (249,343)	1
  (252,182)	1
  (256,16)	1
  (257,891)	1
  (260,138)	1
  (264,632)	1
  (268,394)	1
  (280,460)	1
  (281,363)	1
  (289,1157)	1
  (300,754)	1
  (305,917)	1
  (312,120)	1
  (324,18)	1
  (329,867)	1
  (332,962)	1
  (345,71)	1
  (360,1064)	1
  (372,322)	1
  (380,654)	1
  (385,1005)	1
  (441,21)	1
  (489,899)	1
  (537,1035)	1
  (545,1019)	1
  (552,912)	1
  (617,991)	1
  (4696,0)	1

• We compute that the valuation Ord(ψ - (φ|V₂)/φ) is -0.323893, and hence we need to multiply (ψ - (φ|V₂)/φ) by Δ^1 to get a Jacobi cusp form. The definition of ψ is
      ψ = (φ|V₂)/φ + c · θ/ Δ^1.
  Here c is a vector of coefficients, θ is a basis of of J_12,1174^cusp, and Δ is the weight 12 elliptic cusp form. Both c and θ will be given in tables.