%I #15 Mar 27 2018 17:09:36

%S 45,469872,9215455916,222628516313454,5992746995783064438,

%T 172421735348939185816992,5190295355475474691505991096,

%U 161438078857853783804701774555530,5147432171207325946295914462556988257,167348179641043887251862298724868852231264

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

%C 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 R_1 = -t_1*t_2+t_3;

%C 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 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 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 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 R_6 = -6*t_2*t_6;

%C R_7 = -18*t_1^2+1/2*t_4^2;

%C 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 For more details see the Movasati & Nikdelan link Section 8.3.

%H 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, (1/2)*t_5.

%H H. Movasati, <a href="http://w3.impa.br/~hossein/foliation-allversions/foliation.lib">Foliation.lib</a>.

%o (SINGULAR)

%o // This program has to be compiled in SINGULAR. By changing "int iter" you can

%o // calculate more coefficients. Note that this program is using a library calling

%o // "foliation.lib" written by H. Movasati, which is available in the link given in

%o // LINKS section as Foliation.lib.

%o LIB "linalg.lib"; LIB "foliation.lib";

%o ring r=0, (t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,q),dp;

%o 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 poly dis=t_1^6-t_6;

%o poly dt1=dis*(-t_1*t_2+t_3);

%o poly dt2=(1296*c4*t_3^2*t_4*t_8-t_1^6*t_2^2+t_2^2*t_6);

%o 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 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 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 poly dt6=dis*(-6*t_2*t_6);

%o poly dt7=dis*((1296*c4*t_4^2-t_1^2)/(2592*c4));

%o 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 list pose;

%o 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 list vecfield=dt1,dt2,dt3,dt4,dt5,dt6,dt7,dt8;

%o list denomv=dis,dis,dis,dis,dis,dis,dis,dis;

%o intvec upto=1,1,1,1,1,1,1,1;intvec whichpow;

%o int iter=20;

%o int n;

%o 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 1/2*pose[5]+1/144;

%Y Cf. A300194, A300195, A300196, A300197, A300199, A300200, A300201.

%K nonn

%O 1,1

%A _Younes Nikdelan_, Mar 22 2018