
// Make newform 592.3.z.a in Magma, downloaded from the LMFDB on 20 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_592_z();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_592_z_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_592_3_z_a();".
// 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_592_3_z_a();".
// 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;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 4, 0, 4], [0, 8, 0, -4]];
    Rf_basisdens := [1, 1, 1, 1];
    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_592_z();"
function MakeCharacter_592_z()
    N := 592;
    order := 6;
    char_gens := [223, 149, 113];
    v := [3, 6, 4];
    // 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_592_z_Hecke();"
function MakeCharacter_592_z_Hecke(Kf)
    N := 592;
    order := 6;
    char_gens := [223, 149, 113];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [0, -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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, -3, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -10, 0, 0], [-15, 15, 2, -2], [0, 0, 0, 0], [0, 0, 0, 0], [-21, 0, 0, -5], [0, 0, 0, 0], [0, 35, -1, 2], [0, 9, -10, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-90, 90, 0, 0], [0, 0, 0, 0], [0, -11, 15, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-110, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-39, 39, 20, -20], [-65, 0, 0, 18], [99, 0, 0, 5], [0, 0, 0, 0], [0, 0, 0, 0], [-91, 91, -15, 15], [30, -30, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-105, 0, 0, -22], [0, 0, 0, 0], [51, 0, 0, 35], [0, 0, 0, 0], [85, -85, -33, 33], [0, 0, 0, 0], [0, 0, 0, 0], [165, -165, 13, -13], [0, 0, 0, 0], [0, -19, 45, 0], [0, 0, 0, 0], [95, 0, 0, 42], [-195, 195, -7, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 221, -15, 0], [-105, 0, 0, 52], [0, 0, 0, 0], [0, -418, 0, 0], [0, 0, 0, 0], [-255, 255, -8, 8], [0, 0, 0, 0], [138, 0, 0, 0], [0, 0, 0, 0], [0, -115, 63, 0], [-231, 231, -40, 40], [0, 0, 0, 0], [0, 285, -17, 0], [0, 0, 0, 0], [0, 0, 0, 0], [25, -25, -78, 78], [-75, 75, 77, -77], [0, 0, 0, 0], [0, -175, -72, 0], [0, 0, 0, 0], [-299, 299, 45, -45], [225, -225, 68, -68], [0, 0, 0, 0], [0, 0, 0, 0], [0, -275, 63, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 189, 85, 0], [-325, 0, 0, 57], [-798, 0, 0, 0], [-391, 391, 30, -30], [0, 0, 0, 0], [-29, 0, 0, -105], [0, 0, 0, 0], [-145, 0, 0, -102], [0, 0, 0, 0], [0, 0, 0, 0], [0, 702, 0, 0], [0, 425, -42, 0], [-522, 522, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-459, 459, -55, 55], [0, 558, 0, 0], [0, 0, 0, 0], [-341, 0, 0, 105], [0, 0, 0, 0], [165, -165, -133, 133], [0, 0, 0, 0], [231, 0, 0, -130], [0, 0, 0, 0], [1150, -1150, 0, 0], [0, 0, 0, 0], [-465, 0, 0, 92], [0, 0, 0, 0], [-551, 551, -60, 60], [0, 0, 0, 0], [-35, 35, -153, 153], [-210, 210, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [609, -609, 50, -50], [0, 0, 0, 0], [0, 0, 0, 0], [0, -315, 143, 0], [0, 0, 0, 0], [0, 589, 75, 0], [0, -770, 0, 0], [675, 0, 0, -13], [0, 0, 0, 0], [0, 0, 0, 0], [1302, -1302, 0, 0], [-518, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1450, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-595, 595, 117, -117], [-39, 39, 190, -190], [962, 0, 0, 0], [0, -195, -187, 0], [0, 0, 0, 0], [1110, -1110, 0, 0], [0, 1518, 0, 0], [0, 0, 0, 0], [0, -858, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1258, 1258, 0, 0], [0, 0, 0, 0], [205, -205, 207, -207], [-825, 0, 0, -58], [0, 0, 0, 0], [0, 0, 0, 0], [-805, 0, 0, -87], [0, 369, -200, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 129, 230, 0], [-215, 215, 228, -228], [741, -741, 145, -145], [0, 0, 0, 0], [1230, -1230, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1890, 1890, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1850, 1850, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_592_z_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_592_3_z_a();".
// 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_592_3_z_a(:prec:=4)
    chi := MakeCharacter_592_z();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_592_3_z_a();".
// 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_592_3_z_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_592_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;