"Source: Text/Geometry/Lseries.text";
"Line: 3617";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/Lseries.text, line: 3617
// Example: H139E46 ()
print "Example: H139E46";
ei := GetEchoInput();
SetEchoInput(true);
K<s7> := QuadraticField(-7);
f := func<x|x^3-35*x-98>;
d := f((1+s7)/2); // -118 - 18*s7, norm 2^6 * 11 * 23
eta := QuadraticCharacter(d);
xi := eta*Grossencharacter(EllipticCurve("49a"));
L := LSeries(xi); // conductor 7^2 * 11 * 23
CentralValue(L); Sign(L);
E := EllipticCurve([K|-35,-98]); // a model for 49a
Q := QuadraticTwist(E,d);
pt := Points(Q,(1+s7)*d/2)[1];
Height(pt); // nonzero
b,G,m := PseudoMordellWeilGroup(Q : SearchBound:=10);
PTS := [m(G.3),m(G.4)]; // G.1 and G.2 are 2-torsion
[Height(P) : P in PTS]; // differ by multiple of 2
fd := Coefficient(LTaylor(L,1,2),1); // L'(1)
OM := Periods(Q);
h := Height(pt : Precision:=10);
[fd/p/h : p in &cat OM];
[Norm(x) : x in $1];
chi := DirichletGroup(122).1;
f := Newforms(CuspForms(chi))[1][1]; // 1st isogeny class
LRatio(ModularSymbols(f),1);
L := LSeries(f : Embedding:=func<x|Conjugates(x)[1]>);
CentralValue(L), Sign(L);
chi := DirichletGroup(61,CyclotomicField(6)).1;
f := Newforms(CuspForms(chi))[1][1]; // 1st isogeny class
LRatio(ModularSymbols(f),1);
L := LSeries(f : Embedding:=func<x|Conjugates(x)[1]>);
CentralValue(L), Sign(L);
SetEchoInput(ei);
