
// Make newform 1638.2.bj.e in Magma, downloaded from the LMFDB on 23 April 2024.
// 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_e();".
// 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_e();".
// 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)
    N := 1638;
    order := 6;
    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 := 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, 1, 0, -1], [0, 0, 0, 0], [2, 0, -4, 0], [0, 1, 0, 0], [0, 4, 0, -4], [0, -3, 0, -1], [-3, 2, 3, -4], [2, -4, 2, 0], [0, 1, -4, 1], [0, 0, 8, 0], [-2, 0, 4, -3], [-8, -2, 4, 2], [8, 0, -4, 0], [1, -2, -1, 4], [2, 0, -4, -4], [10, 2, 0, -1], [-1, -8, -1, 0], [0, -3, 0, 6], [6, -2, -3, 2], [-4, -5, -4, 0], [4, 0, -8, 4], [-6, 4, 0, -2], [-1, 0, 2, 0], [2, -10, -1, 10], [6, -2, 6, 0], [0, 0, 0, 0], [-2, 10, 0, -5], [0, 2, 12, 2], [0, 0, 0, -12], [6, 6, -6, -12], [0, 4, 12, 4], [-19, 0, 0, 0], [8, 0, 8, 0], [4, 4, -4, -8], [-2, -15, -2, 0], [0, 0, 0, 12], [4, 8, 0, -4], [-5, -10, -5, 0], [0, 20, 0, -20], [-8, -4, 8, 8], [0, -4, 10, -4], [-4, 24, 0, -12], [-4, 13, 4, -26], [16, -4, -8, 4], [12, 13, -6, -13], [-10, 7, 10, -14], [0, 4, 16, 4], [4, 11, -2, -11], [10, 4, 10, 0], [-4, 0, 8, -13], [10, -4, 0, 2], [-8, 0, 16, 1], [8, -2, 8, 0], [17, 4, -17, -8], [0, 2, 3, 2], [0, 10, 0, 10], [12, -6, -12, 12], [4, -9, -2, 9], [-10, -10, 10, 20], [-6, 0, 12, 8], [0, 2, -8, 2], [2, -30, 2, 0], [0, 0, 0, 22], [-4, -20, 0, 10], [18, 20, 0, -10], [-2, 0, 4, -13], [10, -4, 10, 0], [-6, 0, 0, 0], [-2, -16, 2, 32], [-40, 1, 20, -1], [10, 18, -5, -18], [0, 0, 0, -8], [0, -11, 14, -11], [2, -14, -2, 28], [12, 20, -6, -20], [-10, -10, -10, 0], [-6, 10, 0, -5], [-12, 9, -12, 0], [12, -6, -6, 6], [6, 4, 6, 0], [0, 0, -5, 0], [6, 0, -12, -12], [-16, -9, 8, 9], [-4, -8, 4, 16], [0, -6, -24, -6], [12, -12, 0, 6], [0, 18, 0, 0], [6, 4, -3, -4], [-4, -12, -4, 0], [-14, 0, 28, 16], [-5, 40, 0, -20], [12, 6, -6, -6], [6, 10, 6, 0], [0, -8, -4, -8], [1, 0, -2, -2], [0, 4, 0, -8], [28, -10, -14, 10], [30, 0, 0, 0], [0, 4, -8, 4], [10, 0, -20, 0], [-20, 0, 0, 0], [12, 35, -6, -35], [0, -8, 0, 16], [0, 2, -12, 2], [-7, -12, 0, 6], [-12, 0, 24, 22], [-14, -4, 7, 4], [-1, 0, 2, 6], [0, -22, 0, 11], [0, -14, 0, -14], [10, -3, -10, 6], [12, -24, -6, 24], [8, 8, 8, 0], [-12, 0, 24, -10], [4, -28, 4, 0], [-8, -20, 8, 40], [-14, -4, -14, 0], [0, 10, -20, 10], [0, 9, -22, 9], [12, 6, -12, -12], [32, -23, -16, 23], [0, -20, 5, -20], [-8, -8, 0, 4], [-18, 20, -18, 0], [-36, 10, 18, -10], [10, 14, 0, -7], [-16, -20, -16, 0], [-38, -2, 38, 4], [-10, -38, 0, 19], [8, 0, -16, -9], [2, -6, -1, 6], [-24, -19, 12, 19], [0, -10, 12, -10], [0, 8, -6, 8], [-4, 0, -4, 0], [40, -8, -20, 8], [-4, 6, -4, 0], [-6, -2, -6, 0], [-18, 14, 18, -28], [0, -16, -12, -16], [-8, 0, 16, 16], [-28, 7, 14, -7], [34, 6, -34, -12], [6, 0, -12, 26], [0, 12, 28, 12], [-6, -12, -6, 0], [-20, 0, 40, -13], [-6, 32, 0, -16], [-2, 16, 0, -8], [-4, 0, 8, -36], [16, 2, 16, 0], [0, 26, 13, 26], [-17, -20, 0, 10], [0, -8, -2, -8], [0, 2, -15, 2], [4, 36, 0, -18], [26, 0, -26, 0], [7, 18, 7, 0], [-38, 8, 0, -4], [-6, 0, 12, 30], [-28, 6, 14, -6], [-30, 0, 30, 0], [-2, 0, 4, 20], [17, -4, -17, 8], [16, -12, -8, 12], [-4, 0, 8, 26], [0, -8, -14, -8], [-8, 15, 8, -30], [-7, 0, 14, 4], [-18, 4, 0, -2], [20, 0, -40, 20], [-28, -6, -28, 0], [-24, -18, 24, 36], [-5, 0, -5, 0], [-32, -12, 0, 6], [31, 10, -31, -20], [18, 2, -9, -2], [2, 0, -4, -57], [0, 3, -38, 3], [0, 35, 4, 35], [0, 0, 0, 12], [-9, -56, 0, 28], [20, 36, 0, -18], [5, 0, -10, 42], [-16, 25, 8, -25], [-12, 0, 12, 0], [24, -4, 0, 2], [4, 0, -8, 42], [0, 12, 0, -12], [-22, -6, -22, 0], [8, 22, -8, -44], [4, -2, 4, 0], [-19, -12, 0, 6], [-44, 28, 22, -28], [-46, -6, 46, 12], [0, 16, -6, 16], [8, 0, -16, -8], [8, -26, -8, 52], [9, 0, -18, 6], [-46, -4, 0, 2], [10, 5, 10, 0], [-30, -17, 30, 34], [-32, -34, 16, 34], [6, 60, 0, -30], [-20, -24, 10, 24], [0, 12, -8, 12], [-28, 0, 56, 20], [-8, 2, 8, -4], [40, 10, -20, -10], [-10, 8, 10, -16], [0, -8, 5, -8], [14, -40, 0, 20], [0, -12, 20, -12], [3, 16, 3, 0], [-12, -16, -12, 0], [-4, 0, 8, -16], [32, -4, 0, 2], [-23, 10, 23, -20], [32, 8, -16, -8], [12, 0, -24, 26], [0, 0, 40, 0], [4, 0, -8, 20], [2, 0, -4, 30], [8, 56, 8, 0], [0, -8, 13, -8], [-32, 44, 0, -22], [0, 6, -23, 6], [12, -17, -12, 34], [-8, 18, 8, -36], [-6, 0, 12, -18], [0, 22, -24, 22], [0, -14, 2, -14], [-8, -40, 4, 40], [32, 8, 0, -4], [10, -44, 0, 22], [10, 0, -20, 20], [-32, 20, -32, 0], [68, 3, -34, -3], [-12, 30, 12, -60], [0, -24, -12, -24], [44, -22, -22, 22], [0, 2, -50, 2], [-10, -15, 10, 30], [12, -22, -6, 22], [13, -6, 13, 0], [-20, 16, 0, -8], [20, 0, 20, 0], [36, 24, -18, -24], [-23, -14, -23, 0], [0, 6, -32, 6], [20, 20, -10, -20], [0, 4, 0, -4], [24, 0, -48, -1], [0, 24, 22, 24], [38, -16, 0, 8], [38, 6, 38, 0], [34, -14, -34, 28], [8, -18, -4, 18], [-8, -44, 0, 22], [32, -16, 32, 0], [-8, 8, 0, -4], [0, 8, -2, 8], [-8, 0, 16, 34], [0, 23, -16, 23], [13, 54, 13, 0], [18, -28, -18, 56], [-10, -5, -10, 0], [-21, 0, 42, 34], [11, 38, 11, 0], [6, 11, -6, -22], [36, -38, 0, 19], [-15, 0, 30, 50], [4, -18, -2, 18], [-44, 4, 44, -8], [22, 2, -22, -4], [32, -48, -16, 48], [-4, -42, -4, 0], [-16, 0, 32, -36], [13, -4, 13, 0], [-1, 8, 1, -16], [0, -12, 48, -12], [-8, 42, 4, -42], [-24, 4, 0, -2], [-32, 34, 16, -34], [-22, 0, 44, 12], [-8, -54, 0, 27], [-69, 0, 0, 0], [-4, 0, 8, 18], [2, 17, 2, 0], [-51, -6, 51, 12], [0, -7, 10, -7], [-40, -20, 40, 40], [-12, -20, 6, 20], [-19, 36, -19, 0], [8, 14, -8, -28], [26, -4, 0, 2], [0, -76, 0, 38], [0, -40, -14, -40], [0, -8, -3, -8], [4, 60, -2, -60], [8, -30, -8, 60], [26, 0, -52, -37], [0, 6, 32, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_bj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_bj_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12, 0, 1]>,<11,R![256, 0, -16, 0, 1]>],Snew);
    return Vf;
end function;