
// Make newform 1280.3.e.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1280_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_3_e_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1280_3_e_d();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1280_e();"
function MakeCharacter_1280_e()
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    v := [1, 1, 1];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_e_Hecke();"
function MakeCharacter_1280_e_Hecke(Kf)
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    char_values := [[-1, 0], [-1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, 0], [0, 0], [4, 3], [0, 0], [0, 0], [24, 0], [0, -16], [0, 0], [0, 0], [0, -42], [0, 0], [-24, 0], [18, 0], [0, 0], [0, 0], [56, 0], [0, 0], [0, -22], [0, 0], [0, 0], [0, 96], [0, 0], [0, 0], [78, 0], [0, 144], [0, -198], [0, 0], [0, 0], [0, -182], [0, 224], [0, 0], [0, 0], [0, 176], [0, 0], [0, 102], [0, 0], [264, 0], [0, 0], [0, 0], [104, 0], [0, 0], [0, -38], [0, 0], [0, -336], [56, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 442], [0, 416], [0, 0], [-418, 0], [0, 0], [0, 64], [0, 0], [0, 138], [0, 0], [-504, 0], [-462, 0], [0, 0], [136, 0], [0, 0], [0, 0], [0, -624], [-616, 0], [0, 0], [0, -576], [0, 0], [0, 598], [0, 544], [0, 0], [0, 0], [-504, 0], [0, 0], [0, 0], [0, -378], [-456, 0], [-798, 0], [-782, 0], [0, 0], [0, -58], [0, 0], [0, -816], [0, 0], [0, 0], [702, 0], [0, 336], [0, -522], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 918], [558, 0], [0, 0], [0, -682], [0, 0], [1064, 0], [0, 0], [-462, 0], [0, 0], [0, -96], [0, 0], [0, -736], [0, 0], [-1102, 0], [0, 0], [-1224, 0], [0, 1216], [0, 0], [0, 0], [-1218, 0], [0, 0], [0, 0], [1144, 0], [0, 0], [0, -1178], [0, 1104], [-104, 0], [0, 0], [0, 0], [0, -1302], [0, -518], [0, 0], [0, 0], [-216, 0], [0, 0], [0, 0], [0, 0], [936, 0], [78, 0], [962, 0], [1496, 0], [0, 0], [1144, 0], [1518, 0], [0, 0], [0, -858], [0, 0], [0, 0], [0, 1258], [0, 0], [1656, 0], [0, -464], [0, 0], [0, 0], [-696, 0], [-738, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-258, 0], [0, -1824], [0, -1482], [0, 0], [0, 1456], [0, 0], [0, 0], [0, -496], [0, 0], [0, 0], [-744, 0], [-1118, 0], [-2024, 0], [0, 0], [0, 1558], [0, 0], [0, -384], [0, 0], [1998, 0], [0, 0], [0, 1722], [0, 0], [0, -1462], [0, 0], [0, 0], [-264, 0], [0, 1856], [0, 0], [0, 282], [2184, 0], [0, 0], [658, 0], [0, 0], [0, -1056], [0, 0], [0, 0], [0, -2262], [0, 0], [0, -1664], [-98, 0], [-2376, 0], [0, 1984], [0, 0], [0, -2442], [0, 0], [-1224, 0], [-1598, 0], [0, 0], [-1496, 0], [0, 0], [0, 0], [-2322, 0], [0, 0], [0, 144], [0, 102], [0, 0], [0, 0], [0, 0], [-2542, 0], [0, 0], [1122, 0], [0, 0], [-296, 0], [0, 1862], [0, 0], [-318, 0], [0, 0], [0, 0], [0, 742], [0, -1184], [0, 0], [0, 0], [0, 0], [-456, 0], [0, 0], [0, 0], [1938, 0], [0, 0], [0, 0], [-1378, 0], [-1064, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -1802], [0, 2944], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-2856, 0], [-3198, 0], [0, 0], [-3182, 0], [-1976, 0], [0, 0], [0, 2842], [0, 0], [-3224, 0], [0, 2736], [0, 0], [0, 0], [0, -2438], [2664, 0], [0, -656], [0, 0], [0, -1482], [2958, 0], [0, 0], [-2584, 0], [0, 118], [0, 0], [0, 3456], [0, 0], [0, -2496], [0, 0], [0, 0], [0, 3478], [1298, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 122], [0, 0], [0, 0], [0, -3696], [2296, 0], [0, 0], [2622, 0], [0, 1098], [0, 0], [0, 1376], [0, 0], [2184, 0], [0, 3498], [0, 0], [3496, 0], [0, 0], [0, 0], [0, -2064], [3944, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1280_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_3_e_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1280_3_e_d(:prec:=2)
    chi := MakeCharacter_1280_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1280_3_e_d();".
// 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_1280_3_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1280_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<11,R![0, 1]>,<13,R![-24, 1]>],Snew);
    return Vf;
end function;