
// Make newform 1638.2.r.z in Magma, downloaded from the LMFDB on 15 December 2025.
// 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_z();".
// 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_z();".
// 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 := false)
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := [[0, 0, 1, 0], [0, 0, 0, 0], [1, 0, 0, -1], [1, 0, -1, 0], [0, 1, 0, 0], [0, 2, -1, 1], [-5, 0, 5, 0], [0, 3, 0, 3], [0, -2, -4, 0], [0, 0, -1, 0], [0, 0, 0, -2], [0, 5, -3, 0], [0, 2, -1, 0], [8, 2, -8, 2], [0, 0, 0, -3], [9, 0, 0, 2], [8, -2, -8, -2], [1, 0, -1, 0], [0, -2, 4, 0], [-4, 0, 4, 0], [-5, 0, 0, -1], [4, 0, 0, 1], [0, 0, 0, -4], [0, -3, -2, 0], [-2, 4, 2, 4], [0, -1, -17, 0], [4, 0, 0, 6], [0, 1, -12, 0], [6, 0, 0, 2], [-1, -7, 1, -7], [0, -4, 8, 0], [-4, 0, 0, -8], [7, 5, -7, 5], [0, -3, 0, -3], [-5, 5, 5, 5], [-20, 0, 0, -1], [7, 0, 0, 5], [-4, -2, 4, -2], [0, 8, 0, 0], [-10, -2, 10, -2], [0, -8, 4, 0], [3, 0, 0, -4], [16, 4, -16, 4], [0, 6, 5, 0], [0, -1, 2, 0], [-8, -10, 8, -10], [0, -2, 8, 0], [0, 8, -4, 0], [8, -2, -8, -2], [10, 0, 0, -5], [6, 0, 0, 0], [-4, 0, 0, 0], [-19, -3, 19, -3], [-12, -8, 12, -8], [0, 4, -25, 0], [0, -8, 0, 0], [-10, -2, 10, -2], [0, -2, -8, 0], [1, -1, -1, -1], [5, 0, 0, 5], [0, -8, 12, 0], [3, 3, -3, 3], [-16, 0, 0, 7], [-20, 0, 0, 1], [2, 0, 0, -10], [9, 0, 0, -1], [20, -6, -20, -6], [-9, 0, 0, 8], [-8, -3, 8, -3], [0, 8, -14, 0], [0, -5, 3, 0], [20, 0, 0, 6], [0, -6, 4, 0], [-19, 1, 19, 1], [0, -8, 20, 0], [16, 3, -16, 3], [-11, 0, 0, 3], [6, 3, -6, 3], [0, 11, -13, 0], [33, 1, -33, 1], [0, 12, -8, 0], [11, 0, 0, 1], [0, -12, 12, 0], [13, 3, -13, 3], [0, 8, 8, 0], [-4, 0, 0, -15], [18, 0, -18, 0], [0, -3, 21, 0], [7, -9, -7, -9], [-4, 0, 0, 3], [-28, 0, 0, 0], [0, -1, 0, 0], [-16, 3, 16, 3], [0, 4, -12, 0], [-12, 0, 0, 6], [0, -8, 0, -8], [0, -3, 35, 0], [17, 0, 0, 0], [0, 5, -28, 0], [-1, 0, 0, -1], [8, 0, 0, 12], [0, 12, -9, 0], [-16, -6, 16, -6], [0, 10, -14, 0], [24, 0, 0, 4], [39, 0, 0, 3], [0, 6, 16, 0], [-7, 0, 0, 10], [28, 0, 0, 2], [0, -7, -19, 0], [-16, 2, 16, 2], [0, -13, -3, 0], [11, 15, -11, 15], [12, 0, 0, 3], [-36, -5, 36, -5], [19, -7, -19, -7], [24, -1, -24, -1], [0, -5, -8, 0], [0, -5, -6, 0], [-36, -3, 36, -3], [0, 5, -3, 0], [0, 4, -19, 0], [34, 0, 0, 10], [4, 0, -4, 0], [0, 8, 8, 0], [14, 0, 0, 9], [13, 13, -13, 13], [4, -7, -4, -7], [40, 0, 0, -4], [7, 0, 0, -10], [0, -6, -32, 0], [0, -6, 24, 0], [0, 5, -12, 0], [0, 2, 2, 0], [-22, 8, 22, 8], [0, 16, -18, 0], [6, -8, -6, -8], [12, 5, -12, 5], [22, 18, -22, 18], [0, -7, 7, 0], [16, 0, 0, 4], [0, 7, -30, 0], [-4, 4, 4, 4], [32, 0, 0, 8], [0, -18, 17, 0], [8, 0, -8, 0], [15, 0, 0, 6], [-11, 0, 0, -13], [-24, 0, 0, 5], [20, 0, 0, 18], [25, 13, -25, 13], [0, 11, 19, 0], [12, 0, 0, -2], [0, -1, 8, 0], [0, 2, -28, 0], [36, 0, 0, 12], [-4, 1, 4, 1], [19, 10, -19, 10], [11, 0, 0, -1], [-38, 0, 0, 6], [0, 3, -8, 0], [-22, 12, 22, 12], [16, 0, 0, 8], [-16, -2, 16, -2], [0, 9, -21, 0], [-4, 0, 0, 20], [0, -1, 4, 0], [17, 0, -17, 0], [18, 0, 0, -16], [49, 0, 0, 5], [8, 0, 0, -13], [-7, -26, 7, -26], [-36, -6, 36, -6], [-15, 3, 15, 3], [-4, 0, 0, 2], [3, 18, -3, 18], [0, 10, -16, 0], [-35, 0, 0, -18], [0, 12, -20, 0], [0, -5, -18, 0], [-16, 0, 0, 16], [16, 0, 0, -8], [-13, 0, 0, -11], [10, 0, 0, -5], [0, 2, -16, 0], [15, 19, -15, 19], [14, 0, 0, -4], [-8, 0, 0, 11], [0, -3, 21, 0], [-16, -3, 16, -3], [-7, 15, 7, 15], [20, 8, -20, 8], [4, 0, 0, -6], [0, 6, -42, 0], [12, 0, -12, 0], [0, -1, -42, 0], [47, 0, 0, -2], [21, 19, -21, 19], [-19, 0, 0, 3], [-12, 0, 0, -3], [-41, 4, 41, 4], [-60, 2, 60, 2], [0, 13, 17, 0], [-17, 0, 0, -13], [0, 8, -48, 0], [0, -21, 31, 0], [40, 0, 0, -8], [40, 12, -40, 12], [0, -1, -61, 0], [-8, -17, 8, -17], [0, 12, -11, 0], [-30, 0, 0, -18], [0, -9, 20, 0], [-52, -12, 52, -12], [36, -3, -36, -3], [31, 0, 0, 1], [-48, 0, 0, -13], [-13, -16, 13, -16], [0, 8, -32, 0], [6, 0, 0, -5], [0, -17, -7, 0], [-12, 0, 0, -30], [53, 0, 0, 5], [16, 11, -16, 11], [0, 10, -4, 0], [-13, 0, 0, 17], [0, -12, 43, 0], [-44, 0, 44, 0], [-12, 6, 12, 6], [-28, 0, 0, 0], [0, 1, -55, 0], [0, 15, 12, 0], [0, 3, 40, 0], [-27, 0, 0, -9], [4, 0, 0, 29], [12, 0, 0, 27], [-2, -16, 2, -16], [0, -8, 59, 0], [-24, 0, 24, 0], [0, -19, 12, 0], [0, 11, 0, 0], [0, -15, 24, 0], [4, -6, -4, -6], [0, -9, 41, 0], [-65, -8, 65, -8], [-12, 0, 0, -19], [36, 8, -36, 8], [0, -28, 28, 0], [-16, -20, 16, -20], [0, -18, 8, 0], [0, 22, -22, 0], [0, -9, -34, 0], [-12, 0, 0, 22], [0, 19, -3, 0], [2, 0, 0, 4], [32, 17, -32, 17], [-22, -12, 22, -12], [0, 28, -16, 0], [10, 0, 0, 28], [-31, -5, 31, -5], [16, 0, 0, 23], [0, 27, -24, 0], [-1, 0, 0, -18], [0, 4, 57, 0], [-37, -20, 37, -20], [-40, -4, 40, -4], [-46, -9, 46, -9], [33, 0, 0, -9], [-12, 12, 12, 12], [55, -6, -55, -6], [3, 0, 0, 17], [-36, 0, 0, -24], [0, 5, -3, 0], [-24, -15, 24, -15], [-2, 0, 2, 0], [0, 16, 12, 0], [24, 5, -24, 5], [7, 0, 0, -17], [-27, -8, 27, -8], [44, 20, -44, 20], [0, 24, 12, 0], [0, -4, -4, 0], [-24, 0, 0, -13], [0, -24, -23, 0], [20, 0, 0, 20], [-16, 0, 0, 18], [11, 0, 0, 24], [41, 0, 0, 16], [24, -22, -24, -22], [-5, -15, 5, -15], [0, -19, 18, 0], [-8, -18, 8, -18], [0, -3, -37, 0], [52, -8, -52, -8], [-14, 8, 14, 8], [33, 0, 0, 17], [20, 0, 0, 34], [0, -7, -25, 0], [0, 24, -28, 0], [0, -5, -12, 0], [-39, -29, 39, -29], [25, 0, 0, 38], [0, -15, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_r_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, -3, 1]>,<11,R![16, 4, 5, -1, 1]>,<17,R![25, 5, 1]>],Snew);
    return Vf;
end function;