
// Make newform 465.2.k.a in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_465_k();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_465_2_k_a();".
// 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_465_k();"
function MakeCharacter_465_k()
    N := 465;
    order := 4;
    char_gens := [311, 187, 406];
    v := [2, 1, 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_465_k_Hecke( : Kf := Kf)
    return MakeCharacter_465_k();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_465_2_k_a();".
// 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_465_2_k_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_465_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![65610000, 0, 0, 0, 6109200144, 0, 0, 0, 77287627240, 0, 0, 0, 377625787644, 0, 0, 0, 876501842597, 0, 0, 0, 1033256523597, 0, 0, 0, 633699796078, 0, 0, 0, 221772161300, 0, 0, 0, 47229846671, 0, 0, 0, 6342035721, 0, 0, 0, 544936449, 0, 0, 0, 29870203, 0, 0, 0, 1021100, 0, 0, 0, 20710, 0, 0, 0, 225, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;