"Source: Text/Ring/Char.text";
"Line: 1058";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/Char.text, line: 1058
// Example: H43E11 ()
print "Example: H43E11";
ei := GetEchoInput();
SetEchoInput(true);
K := QuadraticField(-39);
I := 39*IntegerRing(K);
F := &*[f[1] : f in Factorization(I)]; // ideal of norm 39
H := HeckeCharacterGroup(F); H;
Norm(Conductor(H.1)); // H.1 is a Hilbert character of norm 1
assert $1 eq 1;
GR := Grossencharacter(H.0, [[3,0]]); // third power
L0 := LSeries(AssociatedPrimitiveGrossencharacter(GR));
L1 := LSeries(AssociatedPrimitiveGrossencharacter(GR*H.1));
L2 := LSeries(AssociatedPrimitiveGrossencharacter(GR*H.1^2));
L3 := LSeries(AssociatedPrimitiveGrossencharacter(GR*H.1^3));
Ls := [ L0, L1, L2, L3 ]; for L in Ls do LSetPrecision(L, 10); end for;
for L in Ls do [CentralValue(L), Sign(L)]; end for;
K`Hip; // extension of infinite place of K
IP := InfinitePlaces(K`extension_field); IP;
for ip in IP do K`Hip := [ ip ]; // change ip, but use same GR
       L := LSeries(AssociatedPrimitiveGrossencharacter(GR));
       LSetPrecision(L, 10); [CentralValue(L), Sign(L)]; end for;
Ls := [ LSeries(AssociatedPrimitiveCharacter(H.1^k)) : k in [1..4] ];
[ Sign(L) where _:=CFENew(L) : L in Ls ]; // force Sign computation
SetEchoInput(ei);
