
// Make newform 810.3.j.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_810_j();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_810_3_j_h();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_810_j();"
function MakeCharacter_810_j()
    N := 810;
    order := 6;
    char_gens := [731, 487];
    v := [1, 3];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_810_j_Hecke(Kf)
    return MakeCharacter_810_j();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_810_3_j_h();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_810_3_j_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_810_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![8386816, 33709440, 34859232, -41415120, 8604420, 3380928, -139180, -97800, 5598, 1368, -66, -12, 1]>,<17,R![1460109722500, 0, -1040615269800, 0, 23433607236, 0, -199784716, 0, 793137, 0, -1464, 0, 1]>],Snew);
    return Vf;
end function;