
// Make newform 1638.2.bj.b in Magma, downloaded from the LMFDB on 10 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_bj_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_bj_b();".
// 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_bj_b();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_bj();"
function MakeCharacter_1638_bj()
    N := 1638;
    order := 6;
    char_gens := [911, 703, 379];
    v := [6, 6, 5];
    // 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_bj_Hecke();"
function MakeCharacter_1638_bj_Hecke( : Kf := false)
    N := 1638;
    order := 6;
    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 := 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, 1, 0, 0], [0, 0, 0, 0], [1, 0, -2, 1], [0, 1, 0, -1], [-1, 0, -1, 0], [0, 3, -2, -3], [0, -1, -2, -1], [0, 1, 0, -1], [0, 2, 0, -4], [3, 2, -3, -4], [-3, 0, 6, -3], [1, 5, 1, 0], [0, -3, 0, 0], [0, -1, 5, -1], [2, 0, -4, 1], [-7, 0, 0, 0], [-8, 6, 4, -6], [0, -3, 0, -3], [4, 2, 4, 0], [6, -3, -3, 3], [2, 0, -4, -2], [9, -8, 0, 4], [3, 0, -6, 5], [2, 7, 2, 0], [10, 7, -5, -7], [-10, 4, 10, -8], [-13, -2, 0, 1], [-3, 8, 3, -16], [-1, 0, 2, 3], [0, -1, 17, -1], [-12, -2, 12, 4], [0, 16, 0, -8], [-12, -4, 6, 4], [0, -9, 4, -9], [-16, 0, 8, 0], [-3, 0, 6, 0], [-4, 4, 0, -2], [2, 1, -1, -1], [8, 0, 8, 0], [0, 7, -3, 7], [-8, -2, 8, 4], [-6, 10, 0, -5], [0, 7, 11, 7], [-3, 4, -3, 0], [-5, -18, -5, 0], [0, -3, -11, -3], [-26, 0, 26, 0], [6, 14, 6, 0], [-12, -8, 6, 8], [4, 0, -8, -13], [3, -14, 0, 7], [1, 0, -2, 1], [0, -8, 0, 8], [0, -7, -1, -7], [-12, -5, 12, 10], [-3, -7, 3, 14], [0, -8, 4, -8], [-11, -9, -11, 0], [0, 10, -8, 10], [5, 0, -10, 9], [4, -8, -4, 16], [-8, 24, 4, -24], [6, 0, -12, 19], [6, -2, 0, 1], [-9, 26, 0, -13], [8, 0, -16, 2], [16, 14, -8, -14], [-19, 0, 0, 0], [0, 6, -5, 6], [-16, 4, -16, 0], [2, 6, 2, 0], [3, 0, -6, 9], [-8, -10, 8, 20], [0, -5, -1, -5], [17, -1, 17, 0], [-4, 9, 2, -9], [-2, 12, 0, -6], [-8, 15, 4, -15], [12, 18, 12, 0], [-2, -37, 1, 37], [-1, 1, 1, -2], [-6, 0, 12, 12], [3, 9, 3, 0], [0, 4, -34, 4], [6, -18, -6, 36], [-1, 40, 0, -20], [-18, -5, 9, 5], [8, -2, 8, 0], [8, 20, -4, -20], [-1, 0, 2, -8], [0, -16, 0, 8], [2, 29, 2, 0], [-10, -8, 5, 8], [-32, -2, 32, 4], [13, 0, -26, 11], [0, 4, 0, 4], [-5, 31, -5, 0], [12, -22, 0, 11], [26, -7, -26, 14], [-5, 0, 10, -33], [-3, 2, 0, -1], [-11, 6, -11, 0], [0, 3, 31, 3], [9, 17, -9, -34], [9, -30, 0, 15], [-5, 0, 10, -19], [5, -21, 5, 0], [2, 0, -4, 3], [-6, 40, 0, -20], [21, 1, -21, -2], [0, 13, 9, 13], [0, 4, 0, 0], [-6, 3, 3, -3], [4, 0, -8, 29], [-6, 22, 3, -22], [0, -18, 0, -18], [16, 19, -8, -19], [4, -23, -4, 46], [19, 16, -19, -32], [0, -6, 29, -6], [6, 2, 6, 0], [-5, -16, 5, 32], [-35, -2, 0, 1], [12, -40, -6, 40], [16, 16, 16, 0], [15, -16, 0, 8], [-22, -13, 11, 13], [0, -13, -4, -13], [2, 20, 0, -10], [18, 0, -36, -17], [18, -16, 18, 0], [11, -5, 11, 0], [-33, 8, 33, -16], [-15, 9, 15, -18], [-20, 2, 10, -2], [25, -1, 25, 0], [36, 16, -18, -16], [-20, 21, 10, -21], [0, -7, 15, -7], [23, -17, -23, 34], [-2, 0, 4, -28], [23, 12, 23, 0], [0, -12, 14, -12], [-10, 0, 20, -10], [-26, 5, 26, -10], [44, -4, -22, 4], [4, 0, -8, -1], [-36, 4, 0, -2], [-4, 54, 0, -27], [-4, 0, 8, -10], [-2, 25, 1, -25], [-16, -4, 16, 8], [-11, 22, 0, -11], [14, -17, -14, 34], [-12, 6, 12, -12], [11, 46, 0, -23], [0, -8, -19, -8], [-20, 15, 10, -15], [8, -8, 0, 4], [6, 0, -12, 22], [13, 10, 13, 0], [0, -3, -29, -3], [16, 0, -32, 28], [0, -22, -16, -22], [6, 6, 6, 0], [-8, 0, 16, 22], [35, 14, -35, -28], [0, 11, 2, 11], [0, 0, 0, -2], [-24, 40, 0, -20], [-23, 0, 46, 18], [16, 21, -8, -21], [0, -6, 24, -6], [-16, -12, 8, 12], [-39, -22, 0, 11], [0, 25, 0, 25], [20, -6, 20, 0], [5, 0, -10, -10], [19, 1, -19, -2], [38, 11, -38, -22], [-4, 0, 8, -48], [-20, -16, 0, 8], [-28, 12, 0, -6], [-16, 0, 32, -11], [1, 9, 1, 0], [0, -16, -24, -16], [0, -68, 0, 34], [12, 0, -24, -15], [0, 38, 0, 0], [-56, 9, 28, -9], [0, -10, -16, -10], [24, 4, -12, -4], [2, -8, 0, 4], [0, 12, 0, 0], [0, -8, -28, -8], [20, 3, -20, -6], [27, 0, -54, -2], [0, -10, 20, -10], [2, 0, -4, 54], [26, 6, 0, -3], [58, 16, -29, -16], [0, -8, 6, -8], [4, -32, 4, 0], [-7, -18, 0, 9], [-2, -18, -2, 0], [44, -14, -44, 28], [26, 0, -52, 14], [0, 24, 2, 24], [4, -2, 4, 0], [0, 3, 22, 3], [-35, -18, 35, 36], [9, -10, 0, 5], [-39, -4, 39, 8], [-44, -6, 22, 6], [-20, 5, 10, -5], [-23, 0, 46, -19], [-25, -52, 0, 26], [0, -33, 0, -33], [-9, -27, -9, 0], [13, 0, -26, 30], [38, 6, -38, -12], [-12, 0, 24, -14], [-3, 0, 6, -39], [-6, 38, 3, -38], [-23, 9, 23, -18], [29, -10, 0, 5], [-18, -9, 18, 18], [0, 1, -7, 1], [0, -8, 18, -8], [8, 0, -16, 36], [18, 0, -18, 0], [10, 1, -10, -2], [-13, 28, -13, 0], [-17, -62, 0, 31], [2, -74, 0, 37], [-2, 0, 4, -13], [-52, -8, 26, 8], [-13, -42, -13, 0], [0, -22, -12, -22], [-8, 3, 8, -6], [-13, -6, -13, 0], [-70, -3, 70, 6], [0, 34, 4, 34], [4, 48, 4, 0], [-28, -23, 14, 23], [-44, -22, 0, 11], [-46, -3, 23, 3], [-2, 22, -2, 0], [14, -11, -7, 11], [-8, -14, 8, 28], [-18, 38, -18, 0], [18, 37, 18, 0], [-22, 0, 44, -20], [-46, -6, 46, 12], [-6, -64, 0, 32], [-14, 8, 7, -8], [0, 29, 11, 29], [-6, 20, -6, 0], [-12, -20, 0, 10], [-6, -65, 3, 65], [37, 24, 0, -12], [-9, 16, 9, -32], [12, 0, -24, 41], [10, -29, -10, 58], [-44, -21, 22, 21], [0, 6, 24, 6], [6, -44, -3, 44], [-16, 0, 32, -42], [6, -7, -3, 7], [0, 16, -39, 16], [32, 12, 0, -6], [14, 0, -28, -46], [21, 19, 21, 0], [0, -12, 35, -12], [0, -1, 23, -1], [-13, -21, -13, 0], [-18, -62, 9, 62], [14, 0, -28, 18], [-14, -32, 7, 32], [0, -19, -5, -19], [0, 2, 0, -4], [22, -36, 22, 0], [12, 22, 0, -11], [14, 11, 14, 0], [-18, 0, 36, -8], [-42, -32, 0, 16], [15, 16, 0, -8], [39, 0, -78, -2], [-28, -8, 14, 8], [0, 6, 36, 6], [-71, 2, 71, -4], [0, 7, -49, 7], [9, -25, 9, 0], [38, 19, -19, -19], [0, -45, -7, -45], [46, -20, 0, 10], [-51, -30, 0, 15], [-17, 21, 17, -42], [9, -5, -9, 10], [-28, -21, -28, 0], [0, -17, -5, -17], [-39, 0, 78, -2], [-1, 40, 1, -80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_bj_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_bj_b();".
// 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_bj_b(:prec:=4)
    chi := MakeCharacter_1638_bj();
    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_bj_b();".
// 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_bj_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 0, 8, 0, 1]>,<11,R![3, 3, 1]>],Snew);
    return Vf;
end function;