
// Make newform 504.2.r.d in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_504_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_504_r_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_504_2_r_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_504_2_r_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 := [19, 3, -22, 3, 21, -15, 5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-478, -752, 1011, -1008, 459, -177, 79, -4], [-2193, 1604, 2894, -2687, 260, -395, 164, 35], [-2525, 1610, 6498, -4095, -876, 39, 146, 217], [343, 6122, -5166, 1953, -3630, 1101, -328, 241], [1799, 730, -438, -2019, 1944, -585, 298, -145], [-6265, -6296, 10122, 2181, -6966, 1863, -1016, 701]];
    Rf_basisdens := [1, 1, 933, 1866, 5598, 5598, 1866, 5598];
    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_504_r();"
function MakeCharacter_504_r()
    N := 504;
    order := 3;
    char_gens := [127, 253, 281, 73];
    v := [3, 3, 2, 3];
    // 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_504_r_Hecke();"
function MakeCharacter_504_r_Hecke(Kf)
    N := 504;
    order := 3;
    char_gens := [127, 253, 281, 73];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 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, 0, 0], [0, 0, 0, 0, -1, 0, -1, 0], [1, 1, 0, 2, -1, -1, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, -2, 1, -1, 3, 2, -1, -1], [0, 2, -1, 2, -2, -1, 1, 2], [-5, 3, 2, -1, 1, -1, -2, -3], [-2, 1, 0, -1, 1, -1, 0, -1], [-2, 3, 1, 2, 0, -1, 1, 1], [0, 1, 3, -1, 0, 2, -2, 1], [2, 1, 1, 0, 1, 0, -3, 0], [6, 1, 0, 0, 1, 0, 0, -1], [-4, 1, 6, -2, 7, 1, -2, -4], [0, 0, 3, -2, 2, 4, -5, -1], [0, 3, 0, 2, -5, -4, 1, 4], [-1, 0, 3, 0, -3, 0, -3, 0], [3, -1, -1, 2, -2, -1, -2, -1], [0, 1, -1, -1, 0, 2, -3, -3], [-3, 0, 0, -2, 1, 1, 3, 1], [0, -6, -3, -2, -3, -2, 3, 6], [1, 5, 3, 2, 2, 2, -3, -5], [0, 1, -6, 1, -2, -2, -4, -4], [0, -1, 0, -3, 4, 6, -3, -6], [-2, 1, 1, 3, 0, 3, -1, -1], [0, -7, 1, -1, 8, 2, 4, -1], [0, -1, 4, -3, 4, 6, 2, -2], [-3, 5, 0, 4, -2, -2, 3, 3], [0, 4, 1, 1, 3, 1, -1, -4], [7, -7, -7, 2, 0, 2, 7, 7], [-3, 2, 3, -2, 4, 1, 0, 0], [5, -5, -6, -2, 1, -2, 6, 5], [-2, -5, 0, -8, 4, 4, 2, -1], [0, -6, -1, 2, 4, -4, 5, 3], [6, 4, -8, 6, -11, -3, 2, 9], [2, 3, 0, -2, 1, 1, -2, 4], [0, -3, 9, -4, 7, 8, -1, 1], [-6, 6, -2, 10, -7, -5, 8, 3], [7, 3, 2, -1, 1, -1, -2, -3], [0, -2, -3, 2, -4, -1, 3, 0], [0, -4, -5, 2, 2, -4, 15, -1], [5, 5, 1, 4, 4, 4, -1, -5], [-1, -11, -7, -5, -4, -5, 7, 11], [0, 0, -3, 1, -1, -2, -16, -1], [-12, 3, 8, -4, 10, 2, 4, -3], [-9, -2, -4, 1, 2, 1, 4, 2], [-1, 10, 5, 2, 5, 2, -5, -10], [-8, 6, 3, 6, 0, -3, 5, 0], [0, 2, -8, 1, -3, -2, 2, -6], [0, -3, -7, 0, 3, 0, 11, -7], [-11, -4, 4, -4, 6, 2, 7, -6], [-7, 11, 9, 1, 2, 1, -9, -11], [-5, 0, 0, 0, 0, 0, 5, 0], [0, -3, 3, 2, 1, -4, -4, 7], [10, -1, -3, -2, 2, -2, 3, 1], [0, -8, 2, -8, 6, 4, -2, -6], [0, -1, -8, 5, -4, -10, 2, 2], [0, -2, 1, -4, -3, -4, -1, 2], [8, 1, 5, 5, -4, 5, -5, -1], [0, -1, 6, -1, 2, 2, -1, 4], [0, 1, -2, 0, -1, 0, 8, -2], [-11, 3, 12, -2, 13, 1, -1, -8], [18, 3, 2, 0, 2, 0, -20, 1], [4, 7, 7, 6, 0, 6, -7, -7], [-12, -1, 0, 6, -3, -3, 12, -4], [0, 5, 7, -3, -2, 6, -18, 1], [0, 4, -1, -1, -3, 2, 7, -3], [0, -11, 6, -1, 12, 2, 7, 4], [-17, 5, -3, 6, -6, -3, 20, 5], [9, 0, -3, -4, -1, 2, -6, 5], [0, 2, 2, 2, -4, -4, 14, 6], [0, 3, -9, 4, -7, -8, 0, -1], [-12, 11, 7, -2, 4, -2, -7, -11], [0, 8, -1, 0, -8, 0, 5, -1], [4, 9, -11, 20, -21, -10, 7, 10], [3, -4, -6, -5, 2, -5, 6, 4], [-12, -5, -1, 4, -3, -2, 13, -6], [0, -3, -6, 4, -1, -8, 17, 2], [-14, -1, -6, -6, 5, -6, 6, 1], [9, -3, -1, -4, 1, 2, -8, 0], [-6, -4, 8, -12, 14, 6, -2, -6], [4, 9, -15, 14, -22, -7, 11, 17], [0, -5, 8, -8, 13, 16, -9, -8], [3, -2, 2, 5, -4, 5, -2, 2], [-14, 2, 7, -1, -5, -1, -7, -2], [0, -5, 1, 3, 2, -6, 8, 7], [0, -2, 12, -6, 8, 12, -8, 0], [-20, 13, 7, -2, 6, -2, -7, -13], [0, -1, -12, 3, -2, -6, 8, -6], [0, 7, -2, 2, -9, -4, 2, 2], [-2, -1, -11, 2, -12, -1, 13, 9], [18, -3, 0, 2, -3, 2, 0, 3], [0, -7, 6, -6, 13, 12, 11, -6], [-2, -12, -1, -4, -11, -4, 1, 12], [-12, -1, 7, -4, 9, 2, 5, -6], [-4, -10, -10, -8, -6, 4, 14, 4], [-2, -5, 1, -1, -6, -1, -1, 5], [15, 1, -4, 8, -8, -4, -11, 1], [24, -18, -13, 3, -5, 3, 13, 18], [-4, 2, 7, 6, -5, 6, -7, -2], [4, 4, -7, -1, 11, -1, 7, -4], [0, 1, 3, -4, 3, 8, -4, -5], [13, -8, 0, 5, -8, 5, 0, 8], [5, -17, -5, -12, 1, 6, 0, -6], [0, -2, 4, -5, 7, 10, -28, -6], [-29, -1, 21, -10, 26, 5, 8, -17], [-21, 2, 5, 4, -3, 4, -5, -2], [0, 10, -11, 8, -18, -16, -14, 5], [15, -5, 1, -8, -6, -8, -1, 5], [-10, 6, 10, -4, 12, 2, 0, -2], [0, -2, -3, -5, 7, 10, 11, -13], [2, -6, -1, -2, 0, 1, -1, -4], [-20, 22, 15, -5, 7, -5, -15, -22], [6, 3, 2, 2, 1, -1, -8, 0], [0, -15, 11, -7, 22, 14, 1, -3], [5, -5, -4, -8, -1, -8, 4, 5], [0, -7, -7, 3, 4, -6, 15, -1], [41, -1, -3, 2, -4, -1, -38, 1], [9, -5, 7, 0, -12, 0, -7, 5], [13, 8, -8, 14, -15, -7, -5, 9], [0, 13, -4, 2, -15, -4, -19, 0], [-14, -1, 12, -12, 18, 6, 2, -7], [0, 0, -9, 8, -8, -16, 2, 7], [0, -1, 3, 5, -4, -10, 12, 13], [-12, 18, 10, 4, 8, 4, -10, -18], [0, 8, -5, 8, -16, -16, 0, 11], [4, 2, 4, 2, -2, 2, -4, -2], [0, -6, -2, -6, 12, 12, -20, -14], [-27, 12, 4, 1, 8, 1, -4, -12], [0, 0, -11, 4, -4, -8, 27, -3], [17, 14, 8, 12, 2, -6, -25, 0], [7, 6, 6, 15, 0, 15, -6, -6], [-23, 6, 6, 6, 3, -3, 17, -3], [-8, 0, 4, -2, 5, 1, 4, -3], [-17, 9, 4, 1, 5, 1, -4, -9], [12, -22, -20, -8, -16, 4, 8, 2], [17, -3, 0, 4, -2, -2, -17, -5], [-35, 6, 4, 2, 2, 2, -4, -6], [8, -1, -7, -4, -5, 2, -1, 8], [6, -10, 9, -12, 15, 6, -15, -13], [21, -1, -2, 1, 1, 1, 2, 1], [-39, 10, 1, -6, 9, -6, -1, -10], [0, -1, -16, 10, -9, -20, 13, 4], [4, 16, 4, 16, -4, -8, -8, 4], [-9, 12, 6, 9, 6, 9, -6, -12], [23, -19, -11, 6, -8, 6, 11, 19], [0, -12, 6, -3, 15, 6, -18, 0], [0, 17, -2, 13, -30, -26, 16, 24], [0, -13, -1, 0, 13, 0, 4, -1], [-10, -12, 5, -14, 12, 7, 5, -10], [8, 5, 5, 12, 0, 12, -5, -5], [1, -6, 8, -26, 21, 13, -9, -1], [4, 7, 8, 16, -1, 16, -8, -7], [-2, 23, 16, 4, 7, 4, -16, -23], [-21, 3, 12, -8, 16, 4, 9, -5], [0, 8, 4, 3, -11, -6, 12, 10], [0, -3, -5, 8, -5, -16, 13, 11], [43, -5, -4, 4, -1, 4, 4, 5], [0, 9, 1, 3, -12, -6, -2, 7], [-3, -11, -6, 1, -5, 1, 6, 11], [9, -7, 4, -6, 7, 3, -13, -8], [0, 15, -7, 4, -19, -8, -12, 1], [22, -4, 0, 4, -4, 4, 0, 4], [-6, -12, 11, -20, 21, 10, -5, -13], [22, 5, 3, -2, 2, -2, -3, -5], [-12, -8, -7, 2, -8, -1, 19, -2], [0, 3, -19, 12, -15, -24, 22, 5], [-33, -11, -7, -4, -4, -4, 7, 11], [0, -14, -6, 3, 11, -6, 29, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_504_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_504_2_r_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_504_2_r_d(:prec:=8)
    chi := MakeCharacter_504_r();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_504_2_r_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_504_2_r_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_504_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![961, 899, 1120, -13, 166, -22, 25, -4, 1]>],Snew);
    return Vf;
end function;