
// Make newform 1950.2.i.z in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_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_1950_2_i_z();".
// 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_1950_2_i_z();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_1950_i();"
function MakeCharacter_1950_i()
    N := 1950;
    order := 3;
    char_gens := [1301, 1327, 301];
    v := [3, 3, 2];
    // 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_1950_i_Hecke();"
function MakeCharacter_1950_i_Hecke(Kf)
    N := 1950;
    order := 3;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 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 := 2;
    raw_aps := [[-1, 0, 1, 0], [-1, 0, 1, 0], [0, 0, 0, 0], [0, 1, 1, 0], [-2, -1, 2, -1], [-1, 1, 0, -1], [0, 2, -2, 0], [0, -1, 5, 0], [0, -3, 0, -3], [4, 2, -4, 2], [4, 0, 0, 0], [2, 1, -2, 1], [-2, -2, 2, -2], [0, 1, -3, 0], [4, 0, 0, 0], [-6, 0, 0, -4], [0, 4, 0, 0], [0, 3, -6, 0], [1, 5, -1, 5], [0, -1, -10, 0], [9, 0, 0, 0], [-3, 0, 0, 5], [4, 0, 0, -1], [0, 4, 0, 4], [0, 5, -10, 0], [-4, -4, 4, -4], [-7, 0, 0, -1], [10, 6, -10, 6], [10, 0, 0, 5], [0, 0, -4, 0], [-9, 5, 9, 5], [6, 0, 0, 2], [0, -8, 4, 0], [0, -1, -19, 0], [0, 4, 0, 0], [-9, 0, 0, 3], [7, 0, 0, -4], [0, 8, -4, 0], [-16, 3, 16, 3], [0, -2, 14, 0], [8, -3, -8, -3], [5, 0, 0, 8], [0, -7, 2, 0], [-3, 0, 3, 0], [0, 2, 0, 2], [0, -1, 7, 0], [4, 2, -4, 2], [3, 1, -3, 1], [0, -3, -22, 0], [-11, 0, 0, -8], [-10, 0, 0, 6], [4, 0, 0, -9], [0, -1, -13, 0], [0, -7, -6, 0], [-12, -10, 12, -10], [-14, -3, 14, -3], [0, 6, -12, 0], [9, 7, -9, 7], [0, 0, -5, 0], [-12, 0, 0, -4], [-4, 10, 4, 10], [0, -4, -10, 0], [12, 0, 0, -4], [-18, 0, 0, -3], [31, 0, 0, 0], [6, 0, 0, -4], [0, 3, 9, 0], [-15, 0, 0, -9], [0, 5, 12, 0], [9, -4, -9, -4], [8, 4, -8, 4], [-10, 0, 0, -2], [23, 5, -23, 5], [0, 8, -11, 0], [1, -9, -1, -9], [0, -1, 8, 0], [-4, 0, 0, -6], [0, 5, -3, 0], [28, 6, -28, 6], [0, -4, -6, 0], [-6, -7, 6, -7], [21, 0, 0, -4], [-26, -3, 26, -3], [0, 3, 6, 0], [-11, -7, 11, -7], [10, 0, 0, 5], [0, 2, -2, 0], [-25, -8, 25, -8], [0, 6, -26, 0], [15, 0, 0, -5], [-6, 0, 0, -1], [-14, -10, 14, -10], [0, -3, -35, 0], [2, 13, -2, 13], [-17, 0, 0, -9], [0, -3, 32, 0], [6, -4, -6, -4], [4, 0, 0, -12], [-7, 7, 7, 7], [-30, 0, 0, -3], [-13, 0, 0, 7], [6, -10, -6, -10], [0, -5, -8, 0], [32, 8, -32, 8], [-17, 0, 0, -15], [23, 0, 0, 0], [0, -7, 0, -7], [2, 0, 0, 16], [-26, 0, 0, -3], [-7, -5, 7, -5], [0, 6, 24, 0], [-7, -17, 7, -17], [0, -6, -32, 0], [21, 0, 0, 13], [0, -7, -15, 0], [0, -4, -28, 0], [0, -17, 19, 0], [6, 15, -6, 15], [12, -10, -12, -10], [0, 3, -26, 0], [5, -17, -5, -17], [-39, -4, 39, -4], [-14, 0, 0, -10], [0, 1, -18, 0], [33, 11, -33, 11], [-6, 0, 0, -2], [0, -4, 39, 0], [0, -1, 20, 0], [3, 0, 0, 21], [-7, 0, 0, 16], [8, -2, -8, -2], [-6, 2, 6, 2], [-40, 4, 40, 4], [23, -3, -23, -3], [0, 16, -18, 0], [17, 3, -17, 3], [0, 10, 4, 0], [0, 0, 12, 0], [0, 4, 16, 0], [-6, 10, 6, 10], [1, 0, 0, 7], [2, 16, -2, 16], [0, -21, 3, 0], [24, 0, 0, 7], [31, 17, -31, 17], [0, -3, -6, 0], [-22, 0, 0, -16], [-46, 0, 0, 4], [-11, 0, 0, -15], [20, 0, 0, 7], [0, -8, -23, 0], [-44, 2, 44, 2], [24, 0, 0, -4], [26, -6, -26, -6], [-32, -14, 32, -14], [32, 0, 0, -3], [0, 3, -35, 0], [0, 14, 14, 0], [-1, 0, 0, 4], [50, 0, 0, 6], [2, 19, -2, 19], [0, 4, -2, 0], [32, 0, 0, 6], [0, 6, -30, 0], [36, -4, -36, -4], [-12, 0, 0, 4], [43, 9, -43, 9], [0, -19, 21, 0], [-11, 0, 0, -4], [2, 0, 0, -22], [26, 0, 0, 14], [0, -12, -27, 0], [0, 4, -40, 0], [0, 0, -10, 0], [11, 0, 0, 9], [0, -6, 38, 0], [24, 16, -24, 16], [20, 0, 0, -2], [-16, -10, 16, -10], [13, 20, -13, 20], [5, 0, 0, 13], [-2, 0, 0, 13], [38, 0, 0, -8], [18, 0, 0, 12], [-6, -26, 6, -26], [0, 6, 8, 0], [-22, 0, 0, -7], [-33, 0, 0, -11], [-2, 19, 2, 19], [0, -9, 14, 0], [0, -21, 7, 0], [0, -9, -44, 0], [7, 0, 0, -7], [-8, 2, 8, 2], [0, 21, -22, 0], [-22, 4, 22, 4], [3, 0, 0, -7], [0, 12, 11, 0], [42, 0, 0, 12], [2, 0, 0, 25], [0, 6, -14, 0], [0, -19, 13, 0], [-29, -11, 29, -11], [-19, 0, 0, -1], [-14, 6, 14, 6], [-8, 12, 8, 12], [-4, 0, 0, -10], [0, -5, 54, 0], [26, 0, -26, 0], [0, 1, 47, 0], [14, 17, -14, 17], [-34, 0, 0, 0], [9, -15, -9, -15], [0, 11, -28, 0], [0, -14, -26, 0], [2, 0, 0, -5], [-24, 0, 0, -12], [0, 12, -16, 0], [36, 23, -36, 23], [-22, 0, 0, -24], [55, 4, -55, 4], [-16, 0, 0, -20], [22, 0, 0, 14], [0, 19, -25, 0], [-60, 3, 60, 3], [-13, 0, 0, -8], [6, 4, -6, 4], [0, -20, 8, 0], [0, 0, -24, 0], [22, 0, 0, -10], [-26, -21, 26, -21], [-53, -3, 53, -3], [33, -5, -33, -5], [12, 0, 0, 10], [29, 0, 0, -5], [-6, 0, 0, -19], [0, 3, 7, 0], [10, 28, -10, 28], [0, -13, -14, 0], [-8, -31, 8, -31], [-4, -3, 4, -3], [-17, -15, 17, -15], [0, 7, 1, 0], [26, -11, -26, -11], [0, -14, -6, 0], [10, 0, 0, -5], [0, 11, 27, 0], [-10, -14, 10, -14], [0, 1, 33, 0], [-30, 14, 30, 14], [11, -20, -11, -20], [-8, -30, 8, -30], [-28, 0, 0, -19], [-5, -16, 5, -16], [-36, 0, 0, -24], [0, -1, 62, 0], [0, -32, 11, 0], [12, -6, -12, -6], [-2, 0, 0, -12], [0, -9, -14, 0], [-11, 0, 0, -1], [24, 27, -24, 27], [1, 0, 0, -17], [25, -8, -25, -8], [0, -20, -4, 0], [0, 21, -29, 0], [0, 20, 16, 0], [16, 0, 0, -18], [0, -2, 4, 0], [0, 20, -18, 0], [-65, 0, 0, 1], [59, 0, 0, -5], [30, 8, -30, 8], [0, 17, -53, 0], [0, 16, -26, 0], [-21, 21, 21, 21], [0, -2, -58, 0], [-1, 0, 0, 17], [0, 8, -11, 0], [0, 11, -6, 0], [46, 14, -46, 14], [-8, -10, 8, -10], [-22, 0, 0, 15], [70, 1, -70, 1], [-17, 0, 0, 7], [-4, 0, 0, -7], [2, 0, 0, 24], [14, 0, 0, 6], [0, -1, 25, 0], [0, -4, -28, 0], [-28, 10, 28, 10], [0, -3, 60, 0], [-24, -34, 24, -34], [0, -20, 48, 0], [0, 7, 30, 0], [-70, 0, 0, 0], [61, 0, 0, -1], [-44, 6, 44, 6], [20, 27, -20, 27], [27, 1, -27, 1], [0, 5, 45, 0], [2, 0, 0, -6], [47, -5, -47, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_i_z();".
// 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_1950_2_i_z(:prec:=4)
    chi := MakeCharacter_1950_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_1950_2_i_z();".
// 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_1950_2_i_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 6, 11, -3, 1]>,<11,R![4, -6, 11, 3, 1]>,<17,R![256, -32, 20, 2, 1]>,<19,R![256, -144, 65, -9, 1]>],Snew);
    return Vf;
end function;