
// Make newform 1728.2.i.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_i_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_1728_2_i_m();".
// 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_1728_2_i_m();".
// 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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 1], [0, 4, 0, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_1728_i();"
function MakeCharacter_1728_i()
    N := 1728;
    order := 3;
    char_gens := [703, 325, 1217];
    v := [3, 3, 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_1728_i_Hecke();"
function MakeCharacter_1728_i_Hecke(Kf)
    N := 1728;
    order := 3;
    char_gens := [703, 325, 1217];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 1, 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, 0, 0], [1, -1, 0, 0], [0, 1, -1, 0], [0, -1, -1, 0], [1, -1, 2, -2], [0, 0, 0, 2], [4, 0, 0, 0], [3, -3, 1, -1], [0, -5, -2, 0], [5, -5, -1, 1], [-4, 0, 0, -2], [-7, 7, -2, 2], [0, -5, -3, 0], [0, -1, -3, 0], [4, 0, 0, -2], [7, -7, 3, -3], [0, -3, 2, 0], [-5, 5, -3, 3], [-2, 0, 0, -4], [0, 0, 0, 2], [0, 11, -1, 0], [0, -3, -1, 0], [8, 0, 0, -2], [0, -1, -2, 0], [0, -3, 4, 0], [15, -15, -1, 1], [-4, 0, 0, -4], [4, 0, 0, 2], [-1, 1, 4, -4], [-8, 0, 0, 0], [-1, 1, -5, 5], [0, -15, -2, 0], [1, -1, 1, -1], [-9, 9, 2, -2], [0, 3, -5, 0], [-1, 1, -4, 4], [4, 0, 0, 4], [-7, 7, 1, -1], [0, -3, 0, 0], [12, 0, 0, 0], [4, 0, 0, 6], [0, 15, 1, 0], [-9, 9, 2, -2], [2, 0, 0, 8], [-2, 0, 0, -4], [-13, 13, 1, -1], [0, 1, 7, 0], [0, 15, 3, 0], [-1, 1, 0, 0], [6, 0, 0, 0], [3, -3, -7, 7], [0, -7, 0, 0], [-12, 0, 0, 4], [-1, 1, -4, 4], [0, -3, 1, 0], [12, 0, 0, 2], [0, 0, 0, 12], [0, -5, 4, 0], [0, -15, -2, 0], [1, -1, 1, -1], [9, -9, -4, 4], [-12, 0, 0, 4], [3, -3, 5, -5], [0, 17, 0, 0], [0, -21, 2, 0], [0, 1, 5, 0], [1, -1, 4, -4], [-11, 11, 7, -7], [0, -5, 4, 0], [0, 1, 6, 0], [8, 0, 0, 4], [0, 3, 3, 0], [17, -17, 6, -6], [-26, 0, 0, 0], [-31, 31, 1, -1], [0, -3, -4, 0], [-4, 0, 0, -6], [1, -1, -6, 6], [-7, 7, -8, 8], [-19, 19, -5, 5], [0, 5, 2, 0], [24, 0, 0, 4], [16, 0, 0, 10], [0, -29, 3, 0], [0, 21, -1, 0], [-16, 0, 0, -2], [0, 7, -10, 0], [0, -21, 6, 0], [-19, 19, -5, 5], [-20, 0, 0, 8], [0, -3, 9, 0], [8, 0, 0, -4], [23, -23, -5, 5], [9, -9, -3, 3], [-30, 0, 0, 4], [-9, 9, 10, -10], [18, 0, 0, 0], [-6, 0, 0, -8], [-10, 0, 0, 8], [0, 9, 1, 0], [-4, 0, 0, 6], [-3, 3, 3, -3], [0, -1, 0, 0], [11, -11, -3, 3], [-24, 0, 0, -2], [0, -3, 11, 0], [-24, 0, 0, 2], [-31, 31, 1, -1], [0, -25, -6, 0], [-1, 1, 11, -11], [12, 0, 0, -6], [9, -9, 6, -6], [0, 9, -7, 0], [16, 0, 0, 4], [0, 7, -8, 0], [3, -3, -3, 3], [0, 0, 0, -4], [17, -17, 0, 0], [0, 15, 3, 0], [-1, 1, -12, 12], [0, 7, 14, 0], [0, 5, 12, 0], [-12, 0, 0, -12], [0, 11, -11, 0], [12, 0, 0, 10], [0, -5, 0, 0], [-8, 0, 0, -16], [0, -21, -1, 0], [-7, 7, -2, 2], [-14, 0, 0, 0], [33, -33, 1, -1], [-1, 1, -17, 17], [42, 0, 0, 0], [9, -9, -14, 14], [17, -17, 8, -8], [-12, 0, 0, 10], [9, -9, 5, -5], [-33, 33, -6, 6], [8, 0, 0, -10], [28, 0, 0, 8], [0, 19, 14, 0], [13, -13, -1, 1], [28, 0, 0, 0], [28, 0, 0, 2], [0, 23, -3, 0], [0, -3, 2, 0], [0, -23, 2, 0], [19, -19, 1, -1], [-8, 0, 0, 4], [-23, 23, -2, 2], [0, 0, 0, -10], [20, 0, 0, -8], [3, -3, 13, -13], [0, -13, 5, 0], [0, -3, -19, 0], [0, 0, 0, -4], [0, 33, -6, 0], [32, 0, 0, -10], [-7, 7, 0, 0], [0, -33, -5, 0], [-48, 0, 0, -2], [-3, 3, -5, 5], [10, 0, 0, 0], [23, -23, -4, 4], [0, -27, -3, 0], [-8, 0, 0, -12], [0, 3, -12, 0], [-32, 0, 0, 6], [47, -47, -2, 2], [0, 21, 3, 0], [9, -9, 6, -6], [-5, 5, -19, 19], [0, -25, 14, 0], [31, -31, -5, 5], [-17, 17, -4, 4], [0, 19, -11, 0], [-12, 0, 0, -2], [-16, 0, 0, 16], [0, 19, 12, 0], [0, -7, 7, 0], [0, 29, -5, 0], [31, -31, 0, 0], [16, 0, 0, 10], [33, -33, 5, -5], [0, 35, 2, 0], [-30, 0, 0, 0], [0, -53, -3, 0], [25, -25, -12, 12], [-48, 0, 0, 0], [16, 0, 0, 10], [0, -1, -1, 0], [20, 0, 0, 12], [0, -43, 4, 0], [-28, 0, 0, 0], [-9, 9, 20, -20], [23, -23, -14, 14], [0, -29, 4, 0], [0, 23, 4, 0], [40, 0, 0, 0], [33, -33, 4, -4], [0, 41, 3, 0], [-1, 1, 4, -4], [0, -15, -12, 0], [28, 0, 0, 8], [36, 0, 0, 2], [-16, 0, 0, -12], [-17, 17, -1, 1], [0, 7, -8, 0], [-47, 47, -7, 7], [32, 0, 0, -2], [15, -15, 22, -22], [0, 33, 3, 0], [0, 15, 3, 0], [-7, 7, -7, 7], [0, 9, -12, 0], [5, -5, -9, 9], [0, 1, 14, 0], [-16, 0, 0, 12], [-9, 9, -18, 18], [-55, 55, -6, 6], [-49, 49, 3, -3], [9, -9, 2, -2], [-40, 0, 0, 12], [-57, 57, -5, 5], [0, 11, -12, 0], [0, 39, 8, 0], [-24, 0, 0, 4], [0, -5, 3, 0], [0, 23, -17, 0], [55, -55, -4, 4], [20, 0, 0, -20], [13, -13, 23, -23], [-9, 9, -8, 8], [0, 1, 17, 0], [0, 7, 9, 0], [-17, 17, 6, -6], [-36, 0, 0, 10], [-33, 33, -13, 13], [32, 0, 0, 0], [0, -3, 11, 0], [12, 0, 0, 16], [39, -39, 3, -3], [-12, 0, 0, 2], [1, -1, -30, 30], [0, 7, 25, 0], [-40, 0, 0, -8], [21, -21, -1, 1], [35, -35, 13, -13], [-8, 0, 0, 4], [-15, 15, 2, -2], [40, 0, 0, -2], [-21, 21, -7, 7], [0, 33, -8, 0], [0, 37, 8, 0], [18, 0, 0, 8], [36, 0, 0, -6], [0, -31, 5, 0], [18, 0, 0, -8], [-8, 0, 0, -26], [0, -29, 15, 0], [0, -3, 15, 0], [23, -23, -12, 12], [20, 0, 0, -2], [9, -9, 6, -6], [0, -7, -7, 0], [-20, 0, 0, 18], [0, -25, -20, 0], [-29, 29, -3, 3], [-9, 9, -14, 14], [-31, 31, 6, -6], [-44, 0, 0, -8], [0, 1, 0, 0], [-33, 33, 7, -7], [23, -23, -6, 6], [-24, 0, 0, -16], [-25, 25, -17, 17], [0, -19, -6, 0], [16, 0, 0, -14], [0, 7, 11, 0], [-15, 15, 17, -17], [-1, 1, -13, 13], [0, 5, 5, 0], [0, 27, 24, 0], [-7, 7, 9, -9], [54, 0, 0, 4], [-32, 0, 0, -14], [1, -1, -12, 12], [0, -23, 15, 0], [14, 0, 0, 0], [0, 27, -2, 0], [-30, 0, 0, -8], [39, -39, 8, -8], [13, -13, -1, 1], [0, 27, -24, 0], [-39, 39, 4, -4], [0, -5, -9, 0], [0, 3, -6, 0], [-12, 0, 0, 4], [0, 19, 13, 0], [41, -41, -12, 12], [-28, 0, 0, -6], [64, 0, 0, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_2_i_m();".
// 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_1728_2_i_m(:prec:=4)
    chi := MakeCharacter_1728_i();
    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_1728_2_i_m();".
// 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_1728_2_i_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -1, 1]>,<7,R![25, 10, 9, -2, 1]>],Snew);
    return Vf;
end function;