
// Make newform 1950.2.y.e in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_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_y_e();".
// 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_y_e();".
// 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, 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_1950_y();"
function MakeCharacter_1950_y()
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    v := [6, 3, 5];
    // 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_y_Hecke();"
function MakeCharacter_1950_y_Hecke(Kf)
    N := 1950;
    order := 6;
    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], [0, -1, 0, 0], [0, 0, 0, 0], [0, 0, -3, 0], [-1, -2, -1, 0], [-4, 0, 1, 0], [0, 4, 0, -4], [-2, 4, 1, -4], [-2, 0, -2, 0], [-2, -2, 2, 4], [-4, 0, 8, 2], [1, 4, -1, -8], [0, 4, 0, 0], [0, -6, 0, 6], [-3, 4, 0, -2], [-1, 0, 2, 2], [4, -8, -2, 8], [0, 2, 4, 2], [-2, -2, 2, 4], [-8, 6, 4, -6], [0, -8, 0, 4], [-10, -8, 0, 4], [-6, 4, 0, -2], [7, -2, 7, 0], [0, 6, 2, 6], [-2, 6, 2, -12], [9, 0, -18, -4], [-4, 6, -4, 0], [-6, 0, 12, 0], [0, 12, 0, -12], [-5, 4, -5, 0], [-3, -20, 0, 10], [0, 2, -10, 2], [0, 12, 1, 12], [-24, -2, 12, 2], [-12, 0, 24, -2], [3, 0, -6, 16], [0, 4, 4, 4], [3, 2, -3, -4], [-14, -10, 7, 10], [16, 4, -16, -8], [10, -8, 0, 4], [0, -10, 0, -10], [-4, -8, 4, 16], [-1, 6, 1, -12], [0, 4, -18, 4], [15, 4, -15, -8], [5, -8, -5, 16], [0, -2, -10, -2], [-2, 0, 4, 8], [0, 0, 0, -18], [-2, 0, 4, 0], [-6, 20, 3, -20], [0, -2, -23, -2], [-2, -18, -2, 0], [13, -2, 13, 0], [0, -4, -24, -4], [6, -14, 6, 0], [-18, -4, 9, 4], [4, 0, -8, 0], [6, 4, 6, 0], [0, -14, 5, -14], [4, -28, 0, 14], [12, -8, 0, 4], [10, 0, -20, 16], [-27, 4, 0, -2], [12, -4, -6, 4], [-6, 0, 12, -16], [-12, -6, 6, 6], [-2, -14, -2, 0], [-18, -6, 18, 12], [-4, 0, 8, 6], [10, -4, 10, 0], [-8, -16, 4, 16], [1, -8, 1, 0], [0, -12, -16, -12], [-4, 16, 0, -8], [0, -4, -1, -4], [-7, 10, -7, 0], [22, -20, -11, 20], [-4, -8, 4, 16], [8, 0, -16, 8], [-8, 0, -8, 0], [-24, -12, 12, 12], [4, 10, -4, -20], [8, 0, -16, 6], [14, 6, -7, -6], [-4, 2, 4, -4], [12, 22, -6, -22], [-20, 24, 0, -12], [6, 0, -12, 14], [14, 2, 14, 0], [0, 0, -21, 0], [5, 6, -5, -12], [-10, 0, 20, -16], [2, -2, -1, 2], [-10, 12, -10, 0], [-17, -12, 0, 6], [16, -14, 16, 0], [12, 0, -24, -6], [-10, 0, 20, 12], [-33, -6, 33, 12], [-20, -2, 10, 2], [1, 10, -1, -20], [-3, -16, 0, 8], [-8, -20, 0, 10], [-2, 4, 2, -8], [-24, 24, 0, -12], [-22, 8, 0, -4], [-3, 16, 3, -32], [34, 8, -17, -8], [-45, 0, 45, 0], [0, -2, 32, -2], [-13, 0, 26, 20], [0, 8, 0, -8], [0, 2, -11, 2], [0, -12, -8, -12], [-1, 34, -1, 0], [9, 14, 9, 0], [0, 16, -12, 16], [6, 32, 6, 0], [12, 4, 12, 0], [4, 0, -8, 4], [0, -10, 24, -10], [11, 4, 11, 0], [6, 12, 0, -6], [20, -10, -10, 10], [0, 0, 16, 0], [-5, 0, 10, 4], [7, -16, 0, 8], [-23, -4, -23, 0], [4, -4, -4, 8], [-10, -18, 10, 36], [1, -20, 1, 0], [30, 18, -15, -18], [-16, -12, -16, 0], [0, -6, -43, -6], [0, 2, 2, 2], [-16, -24, 8, 24], [14, 12, -14, -24], [9, 0, -18, 8], [10, 32, 10, 0], [18, 24, -9, -24], [2, 56, 0, -28], [-22, -10, 22, 20], [-36, -12, 18, 12], [10, 40, 0, -20], [-4, 0, 8, -22], [13, 8, 0, -4], [-4, -8, 0, 4], [0, 16, 6, 16], [-1, 22, 1, -44], [10, 0, -20, -4], [1, -54, 1, 0], [4, -14, 4, 0], [-8, -24, 0, 12], [0, 30, 4, 30], [-16, 44, 8, -44], [-16, 0, 32, 6], [-28, 0, 56, -12], [24, -6, -24, 12], [-20, 36, 10, -36], [15, 16, 0, -8], [0, -6, 31, -6], [-2, 2, 2, -4], [15, -36, 0, 18], [-8, 18, 8, -36], [-30, 36, 15, -36], [-19, 0, 38, -20], [-24, 0, 48, -12], [10, 0, -20, 28], [-24, 26, 12, -26], [0, 16, -6, 16], [0, -6, 48, -6], [6, -32, 0, 16], [0, -22, -3, -22], [-6, -40, -6, 0], [-4, 0, 8, 48], [-2, 0, -2, 0], [-8, -18, 8, 36], [-28, -16, 0, 8], [-7, -4, 0, 2], [11, 0, -22, -16], [-4, 64, 0, -32], [15, -10, -15, 20], [0, -12, 18, -12], [-1, 0, 2, -20], [-26, -28, 0, 14], [5, -32, 5, 0], [8, -18, -4, 18], [64, -2, -32, 2], [0, 18, 14, 18], [-39, 16, 0, -8], [-4, -42, -4, 0], [4, -12, -2, 12], [-4, 6, -4, 0], [0, 0, 0, 52], [-26, 44, 13, -44], [-12, 8, 0, -4], [-18, 0, 36, -8], [-4, 12, 2, -12], [0, -14, -16, -14], [-26, 0, 26, 0], [-21, -16, 0, 8], [-15, 22, -15, 0], [-8, -24, -8, 0], [0, 0, 0, -56], [4, 34, -2, -34], [25, 10, 25, 0], [0, 4, -49, 4], [14, 38, 14, 0], [-12, -36, 0, 18], [1, 40, 1, 0], [0, 0, 34, 0], [-8, -18, 4, 18], [25, 0, -50, 24], [-2, 0, 4, -12], [0, 18, -13, 18], [-37, 2, 37, -4], [-33, -28, 0, 14], [22, 14, -22, -28], [24, 0, -48, 26], [-15, 0, 30, -22], [0, 16, -20, 16], [-10, -36, -10, 0], [30, 4, 0, -2], [-2, -60, -2, 0], [0, 0, -10, 0], [-28, -28, 14, 28], [-9, 0, 18, 22], [-13, -12, -13, 0], [41, 0, -41, 0], [-18, 42, -18, 0], [-5, 4, 0, -2], [16, 0, -32, 4], [35, -20, 0, 10], [0, 32, 0, -32], [30, 20, -30, -40], [0, -10, -17, -10], [34, 12, -34, -24], [-2, 16, 2, -32], [16, 12, -16, -24], [26, 20, -13, -20], [-12, -34, -12, 0], [0, 8, -42, 8], [-18, -52, 0, 26], [0, 16, 16, 16], [5, 38, 5, 0], [-42, -28, 21, 28], [3, 18, 3, 0], [-18, 12, 18, -24], [11, 18, 11, 0], [-9, -68, 0, 34], [26, -12, -26, 24], [3, 0, -6, 46], [14, -54, -7, 54], [0, 6, -20, 6], [16, 18, -16, -36], [3, 0, -6, -30], [0, -24, 2, -24], [-25, 0, 50, -4], [16, -44, 16, 0], [12, 0, -24, 28], [4, 48, 4, 0], [0, -12, -6, -12], [0, 8, -8, 8], [-4, 28, 2, -28], [11, 0, -22, -2], [0, 14, 2, 14], [-16, -48, 8, 48], [42, -32, 0, 16], [-34, 44, 0, -22], [14, 10, -14, -20], [0, 14, 22, 14], [24, -56, -12, 56], [-20, 20, 20, -40], [0, -4, -42, -4], [38, 0, -76, 8], [32, -20, -16, 20], [0, -8, -4, -8], [5, -62, 5, 0], [-6, -26, -6, 0], [6, 0, -12, 16], [-12, -28, -12, 0], [-28, 12, 0, -6], [36, 12, 0, -6], [32, 0, -64, -18], [-13, -68, 0, 34], [-4, 24, 2, -24], [0, -2, -67, -2], [28, -12, -28, 24], [-8, 12, 4, -12], [-32, 12, 32, -24], [-34, 38, 17, -38], [64, -8, -32, 8], [-10, 72, 0, -36], [-2, -16, 0, 8], [-11, -46, -11, 0], [-8, -36, 8, 72], [26, 10, -26, -20], [8, -44, -4, 44], [-14, 40, 0, -20], [0, 16, 0, -32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_e();".
// 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_y_e(:prec:=4)
    chi := MakeCharacter_1950_y();
    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_y_e();".
// 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_y_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![9, 3, 1]>],Snew);
    return Vf;
end function;