
// Make newform 882.2.h.m in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_h_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_882_2_h_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_882_2_h_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 := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]];
    Rf_basisdens := [1, 1, 6, 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_882_h();"
function MakeCharacter_882_h()
    N := 882;
    order := 3;
    char_gens := [785, 199];
    v := [1, 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_882_h_Hecke();"
function MakeCharacter_882_h_Hecke(Kf)
    N := 882;
    order := 3;
    char_gens := [785, 199];
    char_values := [[0, 0, -1, 0], [-1, 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 := [[0, 0, 1, 0], [0, -1, 0, 0], [2, -1, -1, -1], [0, 0, 0, 0], [1, 1, 1, 1], [0, 0, -2, 0], [-1, 2, 2, -1], [-5, 0, 5, 0], [-4, -1, -1, -1], [2, 2, -4, -4], [-2, 0, 2, 0], [-2, 0, 2, 0], [1, -2, -8, 1], [-2, 3, -1, -6], [0, 0, 0, 0], [2, -4, 2, 2], [0, 3, -3, -6], [-3, 6, 7, -3], [-8, 3, 5, -6], [0, 3, 3, 3], [-3, 6, -2, -3], [3, -6, -5, 3], [-4, -4, 8, 8], [-8, -2, 10, 4], [-2, 3, -1, -6], [4, 1, 1, 1], [-10, 0, 0, 0], [4, 1, -5, -2], [0, 0, -14, 0], [1, -2, 1, 1], [-4, -3, -3, -3], [-4, -1, -1, -1], [13, 1, 1, 1], [-6, 12, 7, -6], [8, 2, 2, 2], [2, -3, -3, -3], [1, -3, 2, 6], [-2, 6, -4, -12], [4, -8, -8, 4], [0, 0, 6, 0], [2, -4, -10, 2], [8, 3, 3, 3], [3, -6, 9, 3], [7, 0, -7, 0], [-6, 0, 0, 0], [0, 0, 10, 0], [0, 0, 16, 0], [4, 0, -4, 0], [-17, -2, -2, -2], [-10, -3, -3, -3], [7, -2, -5, 4], [-5, 10, 7, -5], [11, -3, -3, -3], [-9, 0, 0, 0], [5, 5, 5, 5], [10, 1, 1, 1], [-1, 2, 5, -1], [4, -6, 2, 12], [2, 6, 6, 6], [-13, -1, 14, 2], [-17, -3, 20, 6], [1, -2, 7, 1], [-13, 0, 0, 0], [12, -6, -6, 12], [3, -6, 10, 3], [0, 0, 6, 0], [-6, 12, -2, -6], [3, -6, -2, 3], [0, 3, -3, -6], [22, 0, -22, 0], [-11, 1, 1, 1], [-5, -5, 10, 10], [2, 6, 6, 6], [-10, 0, 0, 0], [-1, 3, 3, 3], [-28, 2, 2, 2], [-8, -8, -8, -8], [22, 0, -22, 0], [-7, 2, 2, 2], [-8, 9, -1, -18], [7, -14, -11, 7], [-14, -6, 20, 12], [-4, 8, 20, -4], [-13, 3, 3, 3], [0, 0, -8, 0], [3, -6, 30, 3], [33, 0, 0, 0], [12, -24, -5, 12], [-19, 5, 14, -10], [3, -6, 13, 3], [4, -11, 7, 22], [28, -2, -2, -2], [-9, 18, -5, -9], [1, -2, 22, 1], [11, -3, -3, -3], [-18, 6, 6, 6], [-10, 8, 8, 8], [5, -10, 8, 5], [25, 3, -28, -6], [-14, 6, 8, -12], [-8, -3, 11, 6], [-4, 8, -16, -4], [-3, 0, 3, 0], [-3, 6, -6, -3], [-8, 9, -1, -18], [3, -6, -20, 3], [-1, -4, 5, 8], [-8, -2, 10, 4], [-24, 0, 24, 0], [-14, -3, 17, 6], [8, -6, -6, -6], [6, -12, 16, 6], [-5, 10, -2, -5], [17, -12, -12, -12], [-16, 9, 9, 9], [1, -14, -14, -14], [-9, 18, 4, -9], [-4, 8, 8, -4], [-22, 2, 2, 2], [14, 2, -16, -4], [-17, -3, 20, 6], [13, -3, -10, 6], [-12, 24, 6, -12], [-13, 26, 14, -13], [3, -6, 13, 3], [-24, 6, 6, 6], [0, 0, -44, 0], [4, -2, -2, 4], [16, 6, -22, -12], [-34, 3, 3, 3], [3, -6, -2, 3], [10, -20, -20, 10], [-10, 3, 3, 3], [32, -6, -6, -6], [26, -4, -4, -4], [0, 0, 10, 0], [5, -10, 23, 5], [4, 0, -4, 0], [-5, 10, 25, -5], [5, -10, -10, 5], [35, -3, -3, -3], [-6, 12, 24, -6], [-8, 6, 2, -12], [12, 0, 0, 0], [-6, 12, 34, -6], [8, -4, -4, 8], [25, 3, -28, -6], [-2, 16, 16, 16], [41, 3, 3, 3], [7, -5, -2, 10], [-16, 18, 18, 18], [-6, 6, 6, 6], [-19, 9, 9, 9], [2, -10, -10, -10], [-19, 3, 3, 3], [41, -1, -40, 2], [37, 3, -40, -6], [-6, 12, -24, -6], [2, -12, -12, -12], [-9, -3, 12, 6], [-3, 6, -24, -3], [21, -3, -3, -3], [3, -6, -41, 3], [-5, 1, 4, -2], [-30, -3, 33, 6], [-44, -2, -2, -2], [-12, 24, -8, -12], [2, 3, 3, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_2_h_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_882_2_h_m(:prec:=4)
    chi := MakeCharacter_882_h();
    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_882_2_h_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_882_2_h_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-6, -3, 1]>,<11,R![-6, -3, 1]>,<13,R![4, 2, 1]>],Snew);
    return Vf;
end function;