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A300197 Coefficients of non-constant terms of a Calabi-Yau modular form attached to 4-dimensional Dwork family.

%I A300197 #15 Mar 27 2018 17:09:40
%S A300197 16,193131,3904146832,95619949713765,2594164605185043648,
%T A300197 75018247757143686903060,2266261629414347188622815776,
%U A300197 70674869456542669855003845042969,2258019930744219211729662533571321808,73528348542628960335141142217651558123754
%N A300197 Coefficients of non-constant terms of a Calabi-Yau modular form attached to 4-dimensional Dwork family.
%C A300197 The 8-tuple (1/36 + 20*A300194, -1 + 216*A300195, -1/36 + 14*A300196, -1/6 + 24*A300197, -1/72 + 2*A300198, -1/46656 * A300199, 1/36 - 2*A300200, -1/7776 + 7/18 * A300201) gives a solution of the modular vector field R = Sum_{i=1..8} R_i d/dt_i on the enhanced moduli space arising from 4-dimensional Dwork family, where d/dt_i's give the standard basis of the tangent space in the chart (t_1,t_2,...,t_8) and
%C A300197 R_1 = -t_1*t_2+t_3;
%C A300197 R_2 = (-t_1^6*t_2^2+1/36*t_3^2*t_4*t_8+t_2^2*t_6)/(t_1^6-t_6);
%C A300197 R_3 = (-3*t_1^6*t_2*t_3+1/36*t_3^2*t_5*t_8+3*t_2*t_3*t_6)/(t_1^6-t_6);
%C A300197 R_4 = (-t_1^6*t_2*t_4-1/36*t_3^2*t_7*t_8+t_2*t_4*t_6)/(t_1^6-t_6);
%C A300197 R_5 = (-2*t_1^6*t_3*t_4-4*t_1^6*t_2*t_5+5*t_1^4*t_3*t_8+1/36*t_3*t_5^2*t_8+ 2*t_3*t_4*t_6+4*t_2*t_5*t_6)/(2*(t_1^6-t_6));
%C A300197 R_6 = -6*t_2*t_6;
%C A300197 R_7 = -18*t_1^2+1/2*t_4^2;
%C A300197 R_8 = (-3*t_1^6*t_2*t_8+3*t_1^5*t_3*t_8+3*t_2*t_6*t_8)/(t_1^6-t_6);
%C A300197 For more details see the Movasati & Nikdelan link Section 8.3.
%H A300197 H. Movasati, Y. Nikdelan, <a href="http://arxiv.org/abs/1603.09411">Gauss-Manin Connection in Disguise: Dwork Family</a>, arXiv:1603.09411 [math.AG], 2016-2017. See Table 2, (1/24)*t_4.
%H A300197 H. Movasati, <a href="http://w3.impa.br/~hossein/foliation-allversions/foliation.lib">Foliation.lib</a>.
%o A300197 (SINGULAR)
%o A300197 // This program has to be compiled in SINGULAR. By changing "int iter" you can
%o A300197 // calculate more coefficients. Note that this program is using a library calling
%o A300197 // "foliation.lib" written by H. Movasati, which is available in the link given in
%o A300197 // LINKS section as Foliation.lib.
%o A300197 LIB "linalg.lib"; LIB "foliation.lib";
%o A300197 ring r=0, (t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,q),dp;
%o A300197 int pm=1;number t10=1/36;number ko=1/216;number c4=ko^2;number t20=-1;number t81=49/18;number a=-6*t20;
%o A300197 poly dis=t_1^6-t_6;
%o A300197 poly dt1=dis*(-t_1*t_2+t_3);
%o A300197 poly dt2=(1296*c4*t_3^2*t_4*t_8-t_1^6*t_2^2+t_2^2*t_6);
%o A300197 poly dt3=(1296*c4*t_3^2*t_5*t_8-3*t_1^6*t_2*t_3+3*t_2*t_3*t_6);
%o A300197 poly dt4=(-1296*c4*t_3^2*t_7*t_8-t_1^6*t_2*t_4+t_2*t_4*t_6);
%o A300197 poly dt5=(1296*c4*t_3*t_5^2*t_8-4*t_1^6*t_2*t_5-2*t_1^6*t_3*t_4+5*t_1^4*t_3*t_8+4*t_2*t_5*t_6+2*t_3*t_4*t_6)/(2);
%o A300197 poly dt6=dis*(-6*t_2*t_6);
%o A300197 poly dt7=dis*((1296*c4*t_4^2-t_1^2)/(2592*c4));
%o A300197 poly dt8=(-3*t_1^6*t_2*t_8+3*t_1^5*t_3*t_8+3*t_2*t_6*t_8);
%o A300197 list pose;
%o A300197 pose=(60*ko)/(49*t10^2)*t81*q+(t10),(-162*t20*ko)/(49*t10^3)*t81*q+(t20),(-66*t20*ko)/(7*t10^2)*t81*q+(t10*t20),16/(147*t10^2)*t81*q+(-t10)/(36*ko),45/(49*t10)*t81*q+(-t10^2)/(12*ko),(3888*t10^3*ko)/49*t81*q,1/(1512*t10*t20*ko)*t81*q+(-t10^2)/(1296*t20*ko^2),t81*q+(-t10^3)/(36*ko);
%o A300197 list vecfield=dt1,dt2,dt3,dt4,dt5,dt6,dt7,dt8;
%o A300197 list denomv=dis,dis,dis,dis,dis,dis,dis,dis;
%o A300197 intvec upto=1,1,1,1,1,1,1,1;intvec whichpow;
%o A300197 int iter=20;
%o A300197 int n;
%o A300197 for (n=2; n<=iter;n=n+1){upto=n,n,n,n,n,n,n,n; whichpow=upto;pose=qexpansion(vecfield,denomv,pose,upto,upto,a); n;}
%o A300197 1/24*pose[4]+1/144;
%Y A300197 Cf. A300194, A300195, A300196, A300198, A300199, A300200, A300201.
%K A300197 nonn
%O A300197 1,1
%A A300197 _Younes Nikdelan_, Mar 22 2018