
// Make newform 1638.2.r.y in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_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_1638_2_r_y();".
// 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_r_y();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_r();"
function MakeCharacter_1638_r()
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    v := [3, 3, 1];
    // 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_r_Hecke();"
function MakeCharacter_1638_r_Hecke(Kf)
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    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, 0, 0, 0], [1, 0, 0, 1], [0, 0, -1, 0], [0, 1, 0, 1], [-1, -2, 1, -1], [0, 2, -5, 0], [0, 1, 0, 0], [4, 2, -4, 2], [-1, 2, 1, 2], [0, 0, 0, 0], [5, 1, -5, 1], [-1, 4, 1, 4], [0, 0, 8, 0], [8, 0, 0, 3], [-7, 0, 0, 0], [0, 2, 0, 0], [0, 4, 1, 0], [-4, 2, 4, 2], [0, -6, 4, 0], [-5, 0, 0, -1], [-4, 0, 0, 1], [0, 0, 0, -2], [6, 1, -6, 1], [0, -2, 6, 0], [7, 5, -7, 5], [-4, 0, 0, -4], [-4, -7, 4, -7], [-2, 0, 0, 4], [0, 1, -1, 0], [8, 8, -8, 8], [-4, 0, 0, 4], [0, -1, 15, 0], [0, -1, 8, 0], [0, -1, -13, 0], [-4, 0, 0, 7], [-9, 0, 0, -3], [0, -8, -4, 0], [-8, -8, 8, -8], [0, 0, -18, 0], [-12, -4, 12, -4], [3, 0, 0, -4], [0, -2, 0, 0], [5, 2, -5, 2], [10, -1, -10, -1], [0, 0, 16, 0], [0, 2, 0, 2], [-20, -4, 20, -4], [0, 2, -16, 0], [-6, 0, 0, -9], [14, 0, 0, 6], [-4, 0, 0, 6], [0, -5, 13, 0], [0, -10, 12, 0], [-9, 6, 9, 6], [0, -2, 0, -2], [0, 2, 22, 0], [-8, -12, 8, -12], [0, 1, 17, 0], [5, 0, 0, 7], [20, 0, -20, 0], [0, -7, 11, 0], [8, 0, 0, 7], [-4, 0, 0, 7], [-6, 0, 0, 0], [-15, 0, 0, -1], [0, -6, -4, 0], [7, 0, 0, 8], [0, 9, -8, 0], [18, 0, -18, 0], [-21, 5, 21, 5], [12, 0, 0, 4], [12, 10, -12, 10], [0, -7, 21, 0], [-12, 10, 12, 10], [0, -5, 0, 0], [13, 0, 0, 3], [0, -7, -10, 0], [11, 5, -11, 5], [0, 1, 9, 0], [8, -6, -8, -6], [11, 0, 0, 13], [4, -6, -4, -6], [0, -3, 29, 0], [16, -4, -16, -4], [4, 0, 0, -9], [0, 6, -22, 0], [-11, 7, 11, 7], [0, 9, -17, 0], [20, 0, 0, 3], [-4, 0, 0, -16], [16, 7, -16, 7], [0, -3, -16, 0], [4, 8, -4, 8], [20, 0, 0, -4], [0, -8, -8, 0], [-29, 5, 29, 5], [-31, 0, 0, -2], [4, -5, -4, -5], [-9, 0, 0, -11], [-8, 0, 0, 10], [7, -2, -7, -2], [0, 4, -16, 0], [-22, 0, 22, 0], [-16, 0, 0, 2], [-33, 0, 0, 1], [-8, -4, 8, -4], [-7, 0, 0, -16], [12, 0, 0, 14], [5, 13, -5, 13], [0, 16, 8, 0], [-19, 1, 19, 1], [0, 11, -5, 0], [20, 0, 0, 7], [0, -7, 28, 0], [0, -1, 11, 0], [0, -3, -40, 0], [-32, -9, 32, -9], [18, -1, -18, -1], [0, -7, 20, 0], [-3, -9, 3, -9], [13, 4, -13, 4], [-6, 0, 0, -12], [0, 8, 20, 0], [-8, -16, 8, -16], [6, 0, 0, -13], [0, 13, 5, 0], [0, 5, -12, 0], [40, 0, 0, 0], [7, 0, 0, 14], [0, 6, 0, 6], [16, 8, -16, 8], [28, -5, -28, -5], [-22, 12, 22, 12], [0, 16, 10, 0], [-10, -2, 10, -2], [0, 12, -10, 0], [0, 11, -28, 0], [0, 8, 30, 0], [-9, -7, 9, -7], [24, 0, 0, 0], [-6, -13, 6, -13], [0, 4, 28, 0], [-24, 0, 0, 4], [-15, -6, 15, -6], [0, 4, 8, 0], [15, 0, 0, -6], [-19, 0, 0, -1], [0, 0, 0, -7], [-36, 0, 0, 2], [0, 17, -15, 0], [43, 9, -43, 9], [-20, 0, 0, 0], [-32, -9, 32, -9], [-4, -2, 4, -2], [12, 0, 0, 10], [0, 7, 28, 0], [0, 4, -13, 0], [-37, 0, 0, -1], [26, 0, 0, 6], [56, -1, -56, -1], [0, -2, 18, 0], [-8, 0, 0, 8], [0, 10, 16, 0], [-13, 11, 13, 11], [12, 0, 0, 12], [-20, -19, 20, -19], [0, 16, 1, 0], [34, 0, 0, 4], [9, 0, 0, -15], [24, 0, 0, 17], [0, -2, -39, 0], [0, -6, 28, 0], [0, 13, -31, 0], [28, 0, 0, 8], [0, -16, -13, 0], [16, 2, -16, 2], [-35, 0, 0, -16], [-12, -2, 12, -2], [-50, 5, 50, 5], [8, 0, 0, -8], [0, 0, 0, 0], [51, 0, 0, 9], [34, 0, 0, -3], [8, 8, -8, 8], [0, -7, -9, 0], [-2, 0, 0, -24], [-40, 0, 0, -1], [21, 15, -21, 15], [0, 1, -24, 0], [0, -15, -7, 0], [0, 8, -20, 0], [4, 0, 0, 26], [-26, -2, 26, -2], [0, 16, 4, 0], [62, -1, -62, -1], [-33, 0, 0, 10], [0, -11, -27, 0], [-43, 0, 0, 3], [12, 0, 0, 3], [0, 22, -9, 0], [0, -6, 20, 0], [49, 11, -49, 11], [-9, 0, 0, 1], [16, -12, -16, -12], [23, 5, -23, 5], [-32, 0, 0, -4], [0, 4, 16, 0], [27, -3, -27, -3], [0, -23, 24, 0], [5, 0, -5, 0], [26, 0, 0, 0], [-60, 1, 60, 1], [0, -4, 12, 0], [0, 5, 12, 0], [-9, 0, 0, 7], [8, 0, 0, 7], [0, -10, 19, 0], [40, -6, -40, -6], [-66, 0, 0, -3], [-39, 5, 39, 5], [-28, 0, 0, -6], [-11, 0, 0, 19], [0, -31, 16, 0], [36, 8, -36, 8], [-21, 0, 0, -13], [-53, -6, 53, -6], [0, 10, 12, 0], [0, 14, 20, 0], [-12, 0, 0, -32], [-23, -21, 23, -21], [44, 1, -44, 1], [-16, -35, 16, -35], [37, 0, 0, 21], [-4, 0, 0, -3], [36, 0, 0, -11], [0, 8, -34, 0], [11, -14, -11, -14], [0, -4, 40, 0], [12, 9, -12, 9], [0, -9, 0, -9], [0, -9, 0, -9], [0, -10, -28, 0], [25, -11, -25, -11], [0, -14, -1, 0], [-12, 0, 0, 23], [0, -14, 20, 0], [-4, -12, 4, -12], [0, 2, -32, 0], [40, 14, -40, 14], [26, -18, -26, -18], [22, 27, -22, 27], [4, 0, 0, -2], [-3, 13, 3, 13], [-46, 0, 0, 0], [0, -15, -24, 0], [0, -18, 18, 0], [-16, 0, 16, 0], [10, 0, 0, 28], [0, -5, 41, 0], [0, 0, 0, -9], [-16, 11, 16, 11], [-49, 0, 0, 2], [-23, -28, 23, -28], [0, 6, -37, 0], [0, 20, 0, 0], [0, -21, 26, 0], [-39, 0, 0, -5], [0, -6, 12, 0], [0, -4, -41, 0], [19, 0, 0, 33], [-52, 0, 0, -8], [-11, -11, 11, -11], [0, -1, -56, 0], [0, 26, 6, 0], [44, 10, -44, 10], [0, 9, 40, 0], [-41, 0, 0, -13], [0, -24, 37, 0], [0, -14, -44, 0], [-20, -8, 20, -8], [60, 12, -60, 12], [24, 0, 0, -7], [-39, 0, 39, 0], [20, 0, 0, -8], [-32, 0, 0, 10], [59, 0, 0, 0], [9, 0, 0, -10], [0, -10, 24, 0], [0, 3, 35, 0], [-22, -19, 22, -19], [0, 4, 48, 0], [-5, 13, 5, 13], [0, -22, 20, 0], [0, -14, 26, 0], [-7, 0, 0, -15], [-52, 0, 0, -16], [23, 5, -23, 5], [-36, -10, 36, -10], [-20, 17, 20, 17], [0, 7, 17, 0], [-39, 0, 0, 8], [20, 39, -20, 39]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_r_y();".
// 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_r_y(:prec:=4)
    chi := MakeCharacter_1638_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(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_r_y();".
// 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_r_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, -1, 1]>,<11,R![16, -4, 5, 1, 1]>,<17,R![1, -8, 65, 8, 1]>],Snew);
    return Vf;
end function;