
// Make newform 1280.2.f.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1280_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_f_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_2_f_i();".
// 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_2_f_i();".
// 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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-9, 8, -4, -16, 9, -4], [14, -38, -4, -16, 9, -4], [-2, -80, -6, -1, 2, -6], [26, -64, -60, -10, 20, -14], [56, -60, -16, -41, 36, -16]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_f();"
function MakeCharacter_1280_f()
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    v := [2, 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_f_Hecke();"
function MakeCharacter_1280_f_Hecke(Kf)
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, -1, 0, -1], [0, 0, 2, 0, -1, 0], [-2, 0, 0, -1, 0, 1], [0, 0, 2, 0, 0, 0], [0, 0, -2, 2, 1, 2], [0, 0, 1, -1, -2, -1], [0, 0, 0, 0, -1, 0], [4, 0, 0, -2, 0, 2], [-4, -2, 0, 1, 0, -1], [-2, -2, 0, 1, 0, -1], [5, -3, 0, -2, 0, 2], [0, 0, 3, -1, -2, -1], [6, 0, 0, -1, 0, 1], [0, 0, -2, 2, 1, 2], [0, 0, 4, -1, 0, -1], [3, -5, 0, -2, 0, 2], [4, -4, 0, -2, 0, 2], [0, 0, -6, 2, 6, 2], [-4, 0, 0, -2, 0, 2], [-9, -1, 0, -2, 0, 2], [-2, -4, 0, -4, 0, 4], [0, 0, 2, 0, 4, 0], [0, 0, 8, -4, -3, -4], [0, 0, 5, -1, -6, -1], [1, 1, 0, 4, 0, -4], [0, 0, 0, -1, 4, -1], [0, 0, 0, -4, -4, -4], [0, 0, -1, -1, -4, -1], [0, 0, 6, -2, -7, -2], [0, 0, -4, 6, 2, 6], [0, 0, 2, 0, 3, 0], [0, 0, 4, -3, -4, -3], [16, 4, 0, 0, 0, 0], [4, -2, 0, 1, 0, -1], [-9, -1, 0, 0, 0, 0], [0, 0, -3, 3, 2, 3], [-4, 6, 0, 5, 0, -5], [0, 0, 2, -4, -1, -4], [0, 0, 0, 4, 1, 4], [8, 0, 0, 4, 0, -4], [0, 0, -2, 0, -4, 0], [-6, -4, 0, -5, 0, 5], [-12, -4, 0, -2, 0, 2], [0, 0, 6, -2, -7, -2], [0, 0, 7, 3, 4, 3], [-1, 7, 0, 0, 0, 0], [0, 0, -8, 0, 5, 0], [0, 0, -6, 2, 6, 2], [4, 8, 0, 6, 0, -6], [-2, -6, 0, -1, 0, 1], [0, 0, -6, -4, -5, -4], [0, 0, -8, 4, 0, 4], [0, 0, -3, 3, 4, 3], [0, 0, 4, 3, 8, 3], [4, -16, 0, -6, 0, 6], [-4, 6, 0, 5, 0, -5], [6, 2, 0, -3, 0, 3], [9, 1, 0, 0, 0, 0], [0, 2, 0, 1, 0, -1], [3, -5, 0, 2, 0, -2], [8, -4, 0, 0, 0, 0], [0, 0, 4, -6, -2, -6], [-6, -4, 0, -5, 0, 5], [0, 0, 6, -2, -11, -2], [0, 0, -4, 0, 4, 0], [5, 5, 0, 2, 0, -2], [0, 0, 8, -4, -13, -4], [0, 0, -4, 0, 8, 0], [8, 4, 0, 8, 0, -8], [0, 0, 11, -1, -8, -1], [-4, -10, 0, -3, 0, 3], [0, 0, 2, 4, 3, 4], [0, 0, 11, -1, 0, -1], [0, 0, 4, 1, 4, 1], [26, 4, 0, -1, 0, 1], [-6, -4, 0, -4, 0, 4], [-10, -10, 0, -7, 0, 7], [0, 0, -6, 0, -5, 0], [0, 0, -4, 1, 0, 1], [12, 8, 0, 2, 0, -2], [0, 0, -10, 8, 0, 8], [-8, 12, 0, 0, 0, 0], [-15, 1, 0, 0, 0, 0], [-2, 10, 0, 7, 0, -7], [0, 0, -8, -2, 2, -2], [0, 0, 8, 4, -5, 4], [0, 0, -1, -9, -6, -9], [3, 3, 0, 0, 0, 0], [8, 8, 0, 0, 0, 0], [0, 0, 13, -5, -4, -5], [0, 0, -2, 2, 5, 2], [0, 0, 10, 0, 3, 0], [0, 0, -11, 7, 10, 7], [0, 0, 0, 0, -13, 0], [6, -8, 0, -4, 0, 4], [-7, -15, 0, -4, 0, 4], [0, 0, 0, 4, 7, 4], [27, 3, 0, -4, 0, 4], [-20, 14, 0, 5, 0, -5], [19, -5, 0, -8, 0, 8], [6, -2, 0, 5, 0, -5], [0, 0, -6, 8, 3, 8], [0, 0, 12, -8, 0, -8], [17, 1, 0, -6, 0, 6], [0, 0, 0, 0, 4, 0], [12, -4, 0, -10, 0, 10], [14, 10, 0, 5, 0, -5], [0, 0, 11, -9, -10, -9], [-6, 20, 0, 3, 0, -3], [0, 0, 6, 6, -2, 6], [0, 0, 14, 2, 1, 2], [0, -12, 0, -8, 0, 8], [22, 10, 0, 3, 0, -3], [3, -13, 0, -10, 0, 10], [0, 0, 5, -1, -4, -1], [26, -4, 0, -1, 0, 1], [0, 0, 2, 0, -13, 0], [0, 0, 8, -11, -4, -11], [0, 0, 10, -8, -4, -8], [6, 0, 0, 3, 0, -3], [5, 5, 0, 6, 0, -6], [0, 0, -10, 10, 5, 10], [0, 0, -16, 7, 8, 7], [0, 0, 0, 4, 5, 4], [28, 0, 0, -2, 0, 2], [0, 0, 5, -5, 4, -5], [0, 10, 0, 1, 0, -1], [0, 0, 10, 4, 3, 4], [0, 0, 5, -1, 4, -1], [0, 0, 0, 4, 0, -4], [-8, -6, 0, -3, 0, 3], [-14, -4, 0, 4, 0, -4], [6, 16, 0, -4, 0, 4], [-2, 0, 0, 3, 0, -3], [7, -1, 0, -10, 0, 10], [-2, 0, 0, -5, 0, 5], [6, 8, 0, -4, 0, 4], [0, 0, 10, -4, 3, -4], [0, 0, 0, 9, 12, 9], [0, 0, 5, 3, -6, 3], [9, 9, 0, 6, 0, -6], [0, 0, 20, -5, -20, -5], [16, 12, 0, 8, 0, -8], [-2, 16, 0, 15, 0, -15], [0, 0, -8, 2, 22, 2], [0, 0, -10, 10, 1, 10], [0, 0, -9, 11, -6, 11], [0, -6, 0, 5, 0, -5], [-2, -18, 0, -5, 0, 5], [-13, 3, 0, 0, 0, 0], [0, 0, -3, -9, -4, -9], [13, 5, 0, -8, 0, 8], [-28, -8, 0, -2, 0, 2], [-4, -4, 0, 6, 0, -6], [14, -14, 0, -5, 0, 5], [0, 0, 10, -2, -10, -2], [0, 0, 0, -12, -5, -12], [-13, 3, 0, 10, 0, -10], [0, 0, -16, 2, -2, 2], [0, 0, 9, 3, 14, 3], [0, 0, 14, -14, 5, -14], [0, 0, 22, 0, -8, 0], [0, 0, 1, -5, 0, -5], [-16, 16, 0, 8, 0, -8], [-34, 0, 0, 3, 0, -3], [6, 16, 0, 12, 0, -12], [10, 12, 0, 7, 0, -7], [0, 0, 10, -4, -25, -4], [0, 0, -24, 4, 11, 4], [-8, 12, 0, 12, 0, -12], [0, 0, -2, -6, 6, -6], [-16, 16, 0, 4, 0, -4], [-6, 16, 0, 12, 0, -12], [0, 0, -14, -4, 3, -4], [0, 0, -8, 0, -7, 0], [0, 0, -19, -1, 10, -1], [0, 0, 8, -9, 16, -9], [0, 0, -13, 11, 0, 11], [0, 0, -10, 2, 25, 2], [26, 4, 0, -13, 0, 13], [0, 0, 14, -10, -18, -10], [0, 0, 11, 3, -10, 3], [0, 0, -8, -4, -7, -4], [4, -10, 0, 1, 0, -1], [-33, -9, 0, -6, 0, 6], [6, -10, 0, -7, 0, 7], [-24, -8, 0, 8, 0, -8], [0, 0, 20, -4, -16, -4], [-19, -3, 0, 8, 0, -8], [0, 0, 6, -6, 9, -6], [0, 0, 24, -5, -8, -5], [-9, 23, 0, 2, 0, -2], [0, 0, -2, -2, -2, -2], [6, 14, 0, 15, 0, -15], [14, 16, 0, 3, 0, -3], [0, 0, 16, 4, 0, 4], [0, 0, -11, 3, -14, 3], [0, 0, -24, 16, 19, 16], [8, 0, 0, 8, 0, -8], [-34, 8, 0, -1, 0, 1], [-2, 4, 0, -8, 0, 8], [0, 0, 6, -6, -23, -6], [-16, -14, 0, 9, 0, -9], [-12, -8, 0, 2, 0, -2], [-9, 23, 0, 18, 0, -18], [10, 0, 0, -4, 0, 4], [0, 0, -18, 10, 25, 10], [0, 0, -26, 8, 24, 8], [0, 0, -12, 5, 12, 5], [0, 0, 9, 3, 0, 3], [9, 1, 0, -2, 0, 2], [12, -20, 0, -14, 0, 14], [-26, -2, 0, -3, 0, 3], [0, 0, -5, -13, 0, -13], [10, 28, 0, 4, 0, -4], [0, 0, -31, 3, 14, 3], [-30, 12, 0, 3, 0, -3], [0, 0, 16, -15, -8, -15], [-8, -20, 0, -8, 0, 8], [-22, -20, 0, -4, 0, 4], [0, 0, 7, 3, 20, 3], [11, 11, 0, 2, 0, -2], [0, 0, 16, -15, -12, -15], [0, 0, -26, 14, 18, 14], [-4, -32, 0, -6, 0, 6], [0, 0, -3, 3, 34, 3], [0, 0, -2, 2, 5, 2], [-6, 12, 0, 15, 0, -15], [0, 0, 18, -8, -1, -8], [-24, 32, 0, 12, 0, -12], [22, -6, 0, 1, 0, -1], [45, 5, 0, -2, 0, 2], [0, 0, 19, -9, 0, -9], [-26, -14, 0, 7, 0, -7], [18, -4, 0, -1, 0, 1], [0, 0, -2, -2, -15, -2], [-48, -12, 0, 4, 0, -4], [-9, -25, 0, -10, 0, 10], [0, 0, 18, 0, -5, 0], [0, 0, 1, 3, -16, 3], [0, 0, -8, 0, 11, 0], [0, 0, 12, 4, -4, 4], [48, 12, 0, -4, 0, 4], [0, 0, -13, 7, -16, 7], [0, 0, -6, 16, 11, 16], [0, 0, -18, 2, 5, 2], [0, 0, 7, 3, 26, 3], [58, 4, 0, 3, 0, -3], [30, -28, 0, -8, 0, 8], [0, 0, 33, -13, -24, -13], [-26, 18, 0, 5, 0, -5], [40, 2, 0, -3, 0, 3], [0, 0, 14, -14, -19, -14], [0, 0, -24, 0, 5, 0], [-19, -11, 0, -14, 0, 14], [12, 22, 0, 9, 0, -9], [0, 0, -16, -2, -10, -2], [0, 0, -13, 3, -2, 3], [-1, -17, 0, -8, 0, 8], [0, 0, 8, -4, -3, -4], [-24, -22, 0, -7, 0, 7], [0, 0, -26, 4, 12, 4], [0, 0, 2, 8, 19, 8], [0, 0, 8, -16, 7, -16], [-22, 16, 0, 12, 0, -12], [29, 5, 0, 12, 0, -12], [28, -10, 0, -7, 0, 7], [0, 0, 16, 0, -5, 0], [51, 3, 0, 8, 0, -8], [0, 0, 18, -22, 2, -22], [-4, -8, 0, 6, 0, -6], [0, 0, -20, 8, 20, 8], [0, 0, 13, -9, -28, -9], [17, -23, 0, -16, 0, 16], [0, 0, 8, -16, -21, -16], [22, -14, 0, -3, 0, 3], [0, 0, -22, 12, 19, 12], [0, 0, 7, -17, 2, -17], [-12, -12, 0, -6, 0, 6], [0, 0, 1, -5, -2, -5], [0, 0, -24, 12, 17, 12], [-35, 13, 0, 6, 0, -6], [-8, 8, 0, 12, 0, -12], [0, 0, -2, 12, -8, 12], [-10, 0, 0, 7, 0, -7], [-44, 4, 0, -2, 0, 2], [14, 14, 0, 11, 0, -11], [0, 0, 4, -1, -4, -1], [-13, -5, 0, 10, 0, -10], [0, 0, 6, -14, -6, -14], [0, 0, -2, -2, -3, -2], [-16, 34, 0, 13, 0, -13], [0, 0, 4, 3, -4, 3], [8, -40, 0, -24, 0, 24], [-16, -30, 0, 5, 0, -5], [0, 0, -18, 6, -7, 6], [-21, -13, 0, 12, 0, -12], [0, 0, -4, 10, 18, 10], [28, 14, 0, -11, 0, 11], [-24, -16, 0, 8, 0, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1280_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_2_f_i();".
// 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_2_f_i(:prec:=6)
    chi := MakeCharacter_1280_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_f_i();".
// 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_2_f_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1280_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -4, 2, 1]>,<13,R![-16, 8, 8, 1]>],Snew);
    return Vf;
end function;