load "mazur.mg"


========================================================================
d=5


order:=function(a)
if a eq 1 then
   f:=1;
else
   if a^2 eq 1 then
      f:=2;
      else
      f:=4;
      end if;
   end if;
return f;
end function;


Chi:= function(p)
Eps1:=Generators(DirichletGroup(15,CyclotomicField(4)));
Eps1:=Eps1[1]*Eps1[2];
Eps2:=Generators(DirichletGroup(20,CyclotomicField(4)));
Eps2:=Eps2[1]*Eps2[2];
if KroneckerSymbol(-5,p) eq -1 then
   f:=order(Eps1(p));
else
	f:=order(Eps2(p))*2;
end if;
return f;
end function;


G:=DirichletGroup(2^4*3^2*5^2,CyclotomicField(4));                           
eps:=(Elements(G)[50]);
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);
CM:=FormsWithCM(new);

[ 1, 2, 3, 4, 7, 9, 10 ]

for i in [1..#new] do         
if i notin CM then
print(DiscardPlace(5,eps,Chi,new,i,13,50));
end if;
end for;      

{@ 2, 5 @}
{@ 2, 5 @}
Cannot discard the form with parameter:  8
[]
Cannot discard the form with parameter:  11
[]
Cannot discard the form with parameter:  12
[]
Cannot discard the form with parameter:  13
[]
{@ 2, 5 @}
{@ 2, 5 @}


----------------------------------------------------------------------

eps2:=Extend(eps,2^4*3^3*5^2);
M2:=ModularSymbols(eps2,2,1);
S2:=NewSubspace(CuspidalSubspace(M2));
new2:=NewformDecomposition(S2);
CM2:=FormsWithCM(new2);

[ 1, 2, 3, 4, 5, 6 ]

for i in [1..#new2] do         
if i notin CM2 then
print(DiscardPlace(5,eps2,Chi,new2,i,13,50));
end if;
end for;      

Cannot discard the form with parameter:  7
[]
Cannot discard the form with parameter:  8
[]
Cannot discard the form with parameter:  9
[]
Cannot discard the form with parameter:  10
[]
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 3, 5 @}
{@ 2, 5 @}
{@ 2, 3, 5 @}
{@ 2, 3, 5 @}
{@ 2, 3, 5 @}
{@ 2, 3, 7 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 5 @}

P<x> := PolynomialRing(Rationals());
K<a>:=NumberField(x^2+5);
O:=RingOfIntegers(K);
S:=EllipticCurveSearch(2*3^3*5*O,100);

======================================================================

d=7

Chi:= function(p)
if KroneckerSymbol(-11,p) eq -1 then
   f:=1;
else
	f:=KroneckerSymbol(p,3)*KroneckerSymbol(p,7);
    if f eq 1 then
        f:=2;
    else 
        f:=4;
    end if;
end if;
return f;
end function;


----------------------------------------------------------------------

G:=DirichletGroup(2^1*3^1*7^2);
eps:=Generators(G)[1]*Generators(G)[2];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

for i in [1..#new] do         
print(DiscardPlace(7,eps,Chi,new,i,13,23));
end for;      

Cannot discard the form with parameter:  1
[]
{@ 2, 7 @}


------------------------------------------------------------------------
G:=DirichletGroup(2^2*3*7^2);
eps:=Generators(G)[2]*Generators(G)[3];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);


HasCM(new[1]);
true -3

for i in [2..#new] do
for> print(DiscardPlace(7,eps,Chi,new,i,2,20));
for> end for;
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 7 @}

------------------------------------------------------------------------

G:=DirichletGroup(2^2*3^3*7^2);
eps:=Generators(G)[2]*Generators(G)[3];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

FormsWithCM(new);
[ 1, 2, 3 ]

for i in [4..#new] do
for> DiscardPlace(7,eps,Chi,new,i,2,20);             
for> end for;
{@ 5, 2, 3 @}
{@ 3, 7 @}
{@ 2, 7 @}
{@ 2, 7 @}



------------------------------------------------------------------------
G:=DirichletGroup(2*3^3*7^2);
eps:=Generators(G)[1]*Generators(G)[2];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

for i in [1..#new] do
for> DiscardPlace(7,eps,Chi,new,i,2,13);
for> end for;
Cannot discard the form with parameter:  1
[]
Cannot discard the form with parameter:  2
[]
Cannot discard the form with parameter:  3
[]
{@ 2, 3, 7 @}
{@ 2, 7 @}
{@ 2, 7 @}


==========================================================================================
d=11

Chi:= function(p)
if KroneckerSymbol(-11,p) eq -1 then
   f:=(3-KroneckerSymbol(p,33)) div 2;
else
	f:= 2;
end if;
return f;
end function;

eps:=Elements(DirichletGroup(1))[1];


M3:=ModularSymbols(2^2*3^3*11^2,2,1);
S3:=NewSubspace(CuspidalSubspace(M3));
new3:=NewformDecomposition(S3);

Output:

CM3:=FormsWithCM(new3);

[ 2, 3, 4, 8, 9, 10, 11, 12, 13, 18, 19 ]

for i in [1..#new3] do         
if i notin CM3 then
print(DiscardPlace(11,eps,Chi,new3,i,13,50));
end if;
end for;      

{@ 13, 2, 3 @}
{@ 2 @}
{@ 2 @}
{@ 13, 2, 3 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 11, 2, 3 @}
{@ 3 @}
{@ 3 @}
{@ 11, 2, 3 @}
{@ 3 @}
{@ 3 @}
{@ 2, 3, 7 @}
{@ 2, 3, 5 @}
{@ 2, 3, 5 @}
{@ 2, 3, 5 @}
{@ 3, 19 @}
{@ 3, 19 @}
{@ @}
{@ 11, 5 @}
{@ @}
{@ 11, 5 @}
{@ @}
{@ 11 @}
{@ @}
{@ 11 @}

----------------------------------------------------------------------


M2:=ModularSymbols(2^2*3^2*11^2,2,1);
S2:=NewSubspace(CuspidalSubspace(M2));
new2:=NewformDecomposition(S2);
eps:=Elements(DirichletGroup(1))[1];
CM2:=FormsWithCM(new2);

[ 1, 2, 3, 13, 14, 19 ]

for i in [1..#new2] do         
if i notin CM2  then
print(DiscardPlace(11,eps,Chi,new2,i,5,40));
end if;
end for;      

{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2 @}
{@ 2 @}
Cannot discard the form with parameter:  15
{@ 11 @}
{@ 11 @}
{@ 11 @}
Cannot discard the form with parameter:  20
{@ 11 @}
{@ 11 @}
{@ 11 @}
{@ 2, 3, 5, 7 @}
{@ 11, 5 @}
{@ 11, 5 @}

M:=ModularSymbols(2^2*11^2,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);
P1:=PowerSeries(new2[20],100);
P2:=PowerSeries(new[4],100);
bol,map:=IsIsomorphic(Parent(Coefficient(P1,11)),Parent(Coefficient(P2,11)));

[map(Coefficient(P1,g)^2) eq Coefficient(P2,g)^2 : g in PrimesInInterval(7,9\
9)];
[ true, true, true, true, true, true, true, true, true, true, true, true, true, 
true, true, true, true, true, true, true, true, true ]



==================================================================================================================
d=13

Chi:=function(p)
if KroneckerSymbol(-13,p) eq -1 then
    f:=1;
else
    f:=KroneckerSymbol(-1,p)*KroneckerSymbol(p,3)*KroneckerSymbol(p,13);
    if f eq 1 then
        f:=2;
    else
        f:=4;
    end if;
end if;
return f;
end function;

G:=DirichletGroup(2^4*3*13^2);
eps:=Elements(G)[14];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

CM:=FormsWithCM(new);
[ 1, 2, 3, 4, 5, 8, 9 ]

for i in [1..27] do         
if i notin CM then
print(DiscardPlace(13,eps,Chi,new,i,5,40));
end if;
end for;    

{@ 2, 13 @}
{@ 2, 13 @}
{@ 11, 2 @}
{@ 11, 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 11, 2 @}
{@ 2 @}
{@ 11, 2 @}
{@ 2, 5, 7 @}
{@ 2 @}
{@ 2 @}
{@ 2, 5 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2, 13 @}
{@ 2, 13 @}

print(DiscardPlace(13,eps,Chi,new,28,5,10));
{@ 13 @}

print(DiscardPlace(13,eps,Chi,new,29,5,10));
{@ 2, 13, 7 @}



------------------------------------------------------------------------------

G:=DirichletGroup(2^4*3^3*13^2);
eps:=Elements(G)[14];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

CM:=FormsWithCM(new);
[ 1, 2, 3, 4, 5, 6, 7, 8 ]

for i in [1..57] do         
if i notin CM then
print(DiscardPlace(13,eps,Chi,new,i,5,40));
end if;
end for;    

{@ 11, 2, 13 @}
{@ 11, 23, 2, 13 @}
{@ 11, 2, 13 @}
{@ 11, 2, 13 @}
{@ 11, 2, 13 @}
{@ 11, 2, 13 @}
{@ 11, 2 @}
{@ 11, 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2, 3, 5 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 5, 7 @}
{@ 5, 7 @}
{@ @}
{@ @}
{@ @}
{@ @}
{@ 2 @}
{@ 2 @}
{@ @}
{@ 2 @}
{@ @}
{@ 2 @}                            
{@ 2 @}
{@ 2, 7 @}
{@ 2, 7, 29 @}
{@ 2, 7 @}
{@ 2, 7, 29 @}
{@ 2, 13 @}
{@ @}
{@ 2 @}

for i in [58..68] do         
if i notin CM then
print(DiscardPlace(13,eps,Chi,new,i,5,9));
end if;
end for;    

{@ 2, 13, 3, 5, 7 @}
{@ 2, 13, 3, 5, 7 @}
{@ 2, 7 @}
{@ 2, 7 @}
{@ 2, 13 @}
{@ 13, 7 @}
{@ 13, 7 @}
{@ 13 @}
{@ 13 @}
{@ 13, 7 @}
{@ 13, 7 @}


==================================================================================================================
d=15

order:=function(a)
if a eq 1 then
   f:=1;
else
   if a^2 eq 1 then
      f:=2;
      else
      f:=4;
      end if;
   end if;
return f;
end function;


G:=DirichletGroup(2*3^5*5^2,CyclotomicField(4));
eps:=Elements(G)[4];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);


Chi:= function(p)

G:=DirichletGroup(2*3^5*5^2,CyclotomicField(4));
eps:=Elements(G)[4];

if KroneckerSymbol(-15,p) eq -1 then
   f:=order(eps(p)*KroneckerSymbol(p,5));
else
	f:=order(eps(p)*KroneckerSymbol(p,5))*2;
end if;
return f;
end function;




> for i in [7..#new] do
for>  print(DiscardPlace(15,eps,Chi,new,i,5,40));
for> end for;
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 3, 7 @}
{@ 2, 3, 7 @}
{@ 11, 3, 7 @}
{@ 3, 5 @}
{@ 3, 5 @}
{@ 3 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 3 @}
{@ 3 @}
{@ 3 @}
{@ @}



---------------------------------------------------------------

G:=DirichletGroup(2^2*3^5*5^2,CyclotomicField(4));
eps:=Elements(G)[7];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);

Chi:= function(p)
G:=DirichletGroup(2^2*3^5*5^2,CyclotomicField(4));
eps:=Elements(G)[7];

if KroneckerSymbol(-15,p) eq -1 then
   f:=order(eps(p)*KroneckerSymbol(p,5));
else
	f:=order(eps(p)*KroneckerSymbol(p,5))*2;
end if;
return f;
end function;

CM:=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];


> for i in [1..23] do
for> if i notin CM then
for|if> print(DiscardPlace(15,eps,Chi,new,i,5,40));
for|if> end if;
for> end for;
{@ 2, 3 @}
{@ 11, 2, 3 @}
{@ 2, 3 @}
{@ 11, 2, 3 @}
{@ 2, 3, 7 @}
{@ 2, 3, 7 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 3 @}
{@ 2, 3, 5 @}



> for i in [24..#new] do
for> print(DiscardPlace(15,eps,Chi,new,i,5,23));
for> end for;
{@ 2, 3 @}
{@ 11, 3, 5 @}
{@ 2, 3, 7 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 2, 5 @}
{@ 3 @}
{@ 3 @}
{@ 3 @}
{@ @}


===================================================================================
d=17

G:=DirichletGroup(2^4*3^2*17^2,CyclotomicField(16));
eps:=Elements(G)[18];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);



======================================================================================================
d=19

Chi:= function(p)
if KroneckerSymbol(-19,p) eq -1 then
   f:=1;
else
	f:= KroneckerSymbol(p,3)*KroneckerSymbol(p,19);
      if f eq 1 then
            f:=2;
      else 
            f:=4;
      end if;
end if;
return f;
end function;



G:=DirichletGroup(2^2*3*19^2);
eps:=Elements(G)[7];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);


CM:=FormsWithCM(new);
[1,2,7];

for i in [1..#new] do
if i notin CM then
for|if> print(DiscardPlace(19,eps,Chi,new,i,2,30));
for|if> end if;
for> end for;
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2 @}
{@ 2, 19 @}
{@ 3 @}
{@ 2, 19 @}


------------------------------------------------------------------------------

G:=DirichletGroup(2^2*3^3*19^2);
eps:=Elements(G)[7];
M:=ModularSymbols(eps,2,1);
S:=NewSubspace(CuspidalSubspace(M));
new:=NewformDecomposition(S);



HasCM(new[1]);
true -3

HasCM(new[2]);
true -3

for i in [3..6] do
for> print(DiscardPlace(19,eps,Chi,new,i,2,50));
for> end for;
{@ 2, 7 @}
{@ 2, 3, 5 @}
{@ 2, 3, 5 @}

DiscardPlace(19,eps,Chi,new,6,2,70);
{@ 2, 7 @}

HasCM(new[7]);
true -3

for i in [8..15] do
for> print(DiscardPlace(19,eps,Chi,new,i,2,50));
for> end for;
{@ 3, 17 @}
{@ 3, 17 @}
{@ 2, 5 @}
{@ @}
{@ 3 @}
{@ 5 @}
{@ 2, 5, 19 @}
{@ 3 @}


for i in [17..#new] do
for> print(DiscardPlace(19,eps,Chi,new,i,2,13));
for> end for;
{@ 5 @}
{@ 5, 19 @}

To discard new[16] we use Pari/GP, since Magma has problems to compute the norm of a big number:

Magma:

f:=PowerSeries(new[16],40);
Write("coef5.txt",Coefficient(f,5));
Write("coef11.txt",Coefficient(f,11));
Write("pol5.txt",Parent(Coefficient(f,5)));

> ApCand(19,7);
{ -3, 0, 3 }

> ApCand(19,11);
{ -6, -3, 0, 3, 6 }


Pari/GP:

P=read("pol5.txt");
coef5=read("coef5.txt");
coef11=read("coef11.txt");
P=subst(P,x,a);

n5=norm(coef5*Mod(1,P))*norm(coef5^2*Mod(1,P)+9);
n11=norm(coef11*Mod(1,P))*norm(coef11^2*Mod(1,P)+9)*norm(coef11^2*Mod(1,P)+36);

factor(gcd(n5,n11))
[3 60]