
// Make newform 82.6.d.b in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_82_d();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_82_6_d_b();".
// 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_82_d();"
function MakeCharacter_82_d()
    N := 82;
    order := 5;
    char_gens := [47];
    v := [4];
    // 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_82_d_Hecke( : Kf := Kf)
    return MakeCharacter_82_d();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_82_6_d_b();".
// 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_82_6_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_82_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![76529180092300001280, -96799398234103480320, 32381803761772916736, 2355436648340946432, -2059152098176759104, -19358427863310480, 35197860348306468, 35859927377322, -276621386397043, 121427468807, 1184385630827, -336448472, -2933423794, -392118, 4186687, 1555, -3188, -1, 1]>],Snew);
    return Vf;
end function;