
// Make newform 1638.2.j.k in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_j_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_1638_2_j_k();".
// 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_1638_2_j_k();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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_1638_j();"
function MakeCharacter_1638_j()
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    v := [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_1638_j_Hecke();"
function MakeCharacter_1638_j_Hecke(Kf)
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    char_values := [[1, 0, 0, 0], [-1, 0, -1, 0], [1, 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]];
    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, 0, 0, 0], [-1, 1, -1, 0], [0, -1, 0, 0], [0, 1, 2, 1], [1, 0, 0, 0], [0, -1, 4, -1], [-5, 0, -5, 0], [-5, -1, -5, 0], [1, 0, 0, -2], [0, -2, -2, -2], [-3, 1, -3, 0], [5, 0, 0, -1], [0, 0, 0, -2], [3, 0, 3, 0], [0, 0, 3, 0], [0, -3, 0, -3], [4, -3, 4, 0], [0, -4, -3, -4], [-11, 0, 0, -2], [0, 1, -11, 1], [10, 0, 10, 0], [8, 0, 0, 2], [9, -3, 9, 0], [-7, 0, 0, 3], [0, 0, 0, 0], [0, -2, 0, 0], [-12, 0, -12, 0], [0, 0, -4, 0], [-2, 0, 0, -5], [3, 0, 0, -5], [3, -3, 3, 0], [0, -7, 1, -7], [-9, 0, 0, -5], [-15, -3, -15, 0], [0, 3, 14, 3], [0, -1, 0, -1], [-13, -4, -13, 0], [-9, 0, 0, 6], [-17, 2, -17, 0], [0, 0, 6, 0], [12, 0, 0, 1], [0, 0, 0, 0], [0, -3, 11, -3], [3, 0, 0, 9], [0, 5, -9, 5], [-9, 0, 0, 1], [-2, 0, 0, -7], [0, 2, 10, 2], [0, 10, 0, 0], [-2, -1, -2, 0], [-3, 0, 0, 0], [0, 6, 2, 6], [8, 0, 0, 2], [0, 6, 0, 0], [0, -1, 1, -1], [0, 6, 9, 6], [20, -1, 20, 0], [0, -7, 0, -7], [-8, 0, 0, 4], [0, -3, -19, -3], [11, 0, 0, -7], [1, 0, 0, -4], [0, 3, 9, 3], [8, 8, 8, 0], [-8, 2, -8, 0], [-22, -4, -22, 0], [-5, 0, 0, -4], [0, 5, 13, 5], [11, 0, 0, 3], [0, -4, 10, -4], [-6, 0, -6, 0], [0, 4, 10, 4], [-12, 1, -12, 0], [12, 0, 0, -8], [-6, -12, -6, 0], [0, -2, -13, -2], [7, -3, 7, 0], [-5, 5, -5, 0], [0, 3, -19, 3], [-4, 0, 0, -10], [-27, 0, 0, 1], [0, 4, -10, 4], [-23, 0, 0, -4], [-21, 1, -21, 0], [27, 3, 27, 0], [-12, 0, 0, 0], [-19, 5, -19, 0], [10, 0, 0, 10], [4, 0, 0, 8], [-19, 7, -19, 0], [0, -12, 3, -12], [0, 3, -16, 3], [10, 0, 0, -8], [24, -2, 24, 0], [-10, 0, 0, -10], [24, 0, 24, 0], [0, 0, -18, 0], [32, 2, 32, 0], [-39, 1, -39, 0], [-13, 0, 0, -3], [0, 8, 10, 8], [0, -14, -10, -14], [-20, 5, -20, 0], [0, 0, -16, 0], [0, 4, -2, 4], [-24, 0, 0, -3], [25, -7, 25, 0], [0, -9, 9, -9], [-1, 0, 0, 6], [7, 9, 7, 0], [0, -6, 26, -6], [16, 0, 0, -2], [0, 12, -10, 12], [-12, 0, 0, 10], [0, 4, 32, 4], [5, 0, 0, -6], [0, 12, 6, 12], [12, 0, 12, 0], [9, 0, 0, 3], [0, -7, -3, -7], [4, 0, 0, -4], [-1, -8, -1, 0], [0, 2, -14, 2], [3, -14, 3, 0], [-12, 0, 0, 0], [-25, -7, -25, 0], [34, -4, 34, 0], [10, 0, 0, -10], [-1, 5, -1, 0], [0, 2, -12, 2], [25, 0, 0, 10], [12, -8, 12, 0], [24, 0, 0, 1], [2, -2, 2, 0], [3, 0, 0, 1], [0, 16, 2, 16], [0, -18, 5, -18], [30, 0, 0, 6], [0, 9, 0, 9], [38, 0, 0, 0], [-19, -5, -19, 0], [0, -7, -3, -7], [-38, 0, 0, 1], [0, -5, 34, -5], [-21, 0, 0, 0], [9, 0, 0, -11], [0, 3, -6, 3], [-18, 10, -18, 0], [0, -12, 0, 0], [31, -9, 31, 0], [16, 0, 0, -2], [-37, 0, 0, 3], [10, -4, 10, 0], [0, -5, -17, -5], [6, 0, 0, 12], [-32, 6, -32, 0], [18, 12, 18, 0], [-5, 0, 0, -10], [0, -8, -4, -8], [8, 7, 8, 0], [-4, 0, 0, 11], [-20, 0, 0, 11], [9, -3, 9, 0], [0, -6, -36, -6], [0, -14, -7, -14], [0, 4, 16, 4], [0, 5, 12, 5], [0, 0, 0, -8], [13, 2, 13, 0], [0, 1, 32, 1], [37, 0, 0, -1], [-3, -3, -3, 0], [0, 1, 19, 1], [0, 13, 25, 13], [12, 0, 0, 3], [-29, 0, 0, 8], [0, -3, -21, -3], [11, 0, 0, 9], [4, -3, 4, 0], [-2, 15, -2, 0], [-59, 0, 0, 1], [26, 0, 0, -3], [-47, 5, -47, 0], [0, -6, -39, -6], [0, 0, -30, 0], [0, -5, -14, -5], [0, -4, 48, -4], [37, 3, 37, 0], [0, 4, 17, 4], [21, 16, 21, 0], [-40, 0, 0, -7], [38, 8, 38, 0], [-6, 0, -6, 0], [0, -8, -34, -8], [0, 9, 18, 9], [0, 3, -31, 3], [20, -16, 20, 0], [41, 0, 0, -7], [-9, 15, -9, 0], [0, -6, -18, -6], [2, 0, 0, -6], [-43, 5, -43, 0], [0, -4, -12, -4], [-4, 0, 0, -16], [0, 4, -4, 4], [-6, 1, -6, 0], [18, 0, 18, 0], [24, 0, 0, -6], [0, 0, 50, 0], [-3, -8, -3, 0], [7, 0, 0, 4], [18, 0, 0, -2], [-10, -5, -10, 0], [0, 4, -34, 4], [41, 5, 41, 0], [0, -13, -33, -13], [0, 9, -18, 9], [9, 0, 9, 0], [16, 0, 0, -2], [6, -5, 6, 0], [-46, 0, 0, 6], [-27, 15, -27, 0], [-8, -12, -8, 0], [27, 0, 0, 3], [38, 0, 0, 3], [-48, 0, -48, 0], [0, 8, 4, 8], [58, 0, 58, 0], [26, 7, 26, 0], [0, 11, 12, 11], [0, -13, 27, -13], [20, 0, 0, 15], [0, -19, -2, -19], [-12, 0, 0, -2], [0, 0, 12, 0], [-8, -18, -8, 0], [-49, 7, -49, 0], [-4, 0, 0, -22], [-19, 0, 0, 11], [0, -17, 2, -17], [-30, -2, -30, 0], [0, 0, 32, 0], [34, 0, 34, 0], [-16, 0, 0, 8], [-18, 12, -18, 0], [12, 0, 0, -11], [0, 15, 6, 15], [0, 6, 17, 6], [-4, 0, 0, -4], [-31, 0, 0, 3], [6, -6, 6, 0], [0, -14, -19, -14], [5, 0, 0, 12], [0, 14, -17, 14], [44, 4, 44, 0], [0, 19, -26, 19], [0, -16, -15, -16], [-37, 0, 0, 8], [-7, 17, -7, 0], [0, -13, -27, -13], [-15, 0, 0, 21], [0, 8, -6, 8], [-42, 0, 0, -8], [0, -5, 17, -5], [-39, -11, -39, 0], [-6, 0, 0, -6], [45, 0, 0, -3], [4, 0, 0, -16], [0, -2, 5, -2], [40, 12, 40, 0], [0, 24, 5, 24], [0, 8, -48, 8], [47, -7, 47, 0], [26, 0, 0, -6], [-30, -11, -30, 0], [6, 9, 6, 0], [0, 0, 56, 0], [17, -7, 17, 0], [35, -17, 35, 0], [0, 2, 28, 2], [0, 2, -18, 2], [-9, 0, 0, -15], [-48, 0, 0, 10], [9, 10, 9, 0], [32, 4, 32, 0], [0, 6, 23, 6], [36, 0, 0, 18], [0, 11, 48, 11], [-30, 0, 0, -15], [0, 8, -17, 8], [0, -3, 57, -3], [-15, 15, -15, 0], [40, 0, 0, 13], [-76, 0, 0, 0], [0, -8, -31, -8], [26, -22, 26, 0], [21, 0, 0, -18], [5, -17, 5, 0], [-30, 0, 0, 4], [45, 10, 45, 0], [-48, -12, -48, 0], [0, 1, 61, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_j_k();".
// 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_1638_2_j_k(:prec:=4)
    chi := MakeCharacter_1638_j();
    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_1638_2_j_k();".
// 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_1638_2_j_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, -12, 10, 2, 1]>,<11,R![9, -12, 19, 4, 1]>,<17,R![81, 72, 55, 8, 1]>],Snew);
    return Vf;
end function;