
// Make newform 1008.2.r.i in Magma, downloaded from the LMFDB on 07 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_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_1008_2_r_i();".
// 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_1008_2_r_i();".
// 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, 0, -1, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 1, 1, 0, -1, 0], [0, 0, 0, 0, 1, -1], [0, 1, 0, 0, -1, -1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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_1008_r();"
function MakeCharacter_1008_r()
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    v := [3, 3, 1, 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_1008_r_Hecke();"
function MakeCharacter_1008_r_Hecke(Kf)
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0], [1, 0, 0, 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+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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1], [1, -1, -1, 1, 0, -1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 1, 1, -1], [1, -1, -2, 2, 0, -2], [2, 0, 0, -1, -2, 0], [3, 0, 0, -1, 2, 0], [2, -2, 2, -2, 0, -3], [0, 1, -4, 3, 3, -3], [-3, 3, 3, -3, 0, 1], [-5, 0, 0, 2, -1, 0], [-2, 2, 3, -3, 0, 3], [0, -5, -3, 1, 1, -1], [0, 3, 1, 2, 2, -2], [-2, 0, 0, -1, 1, 0], [1, -1, 0, 0, 0, -7], [0, 4, -4, 3, 3, -3], [-4, 4, 1, -1, 0, 6], [-7, 0, 0, -3, -7, 0], [1, 0, 0, 7, 2, 0], [0, -5, -7, 2, 2, -2], [0, -2, 9, -4, -4, 4], [-6, 0, 0, 2, 5, 0], [0, -5, 2, 9, 9, -9], [0, 3, 7, -8, -8, 8], [-2, 2, -4, 4, 0, -9], [1, 0, 0, -6, -8, 0], [-3, 0, 0, 1, 4, 0], [4, -4, 4, -4, 0, -2], [10, 0, 0, 2, -5, 0], [1, -1, -6, 6, 0, -3], [0, 11, 5, -1, -1, 1], [-1, 1, -3, 3, 0, 10], [6, -6, -11, 11, 0, -3], [0, -1, 6, -7, -7, 7], [-3, 3, -5, 5, 0, 8], [2, 0, 0, 1, 14, 0], [7, -7, 8, -8, 0, 6], [0, 9, 7, -9, -9, 9], [1, 0, 0, 13, -2, 0], [7, 0, 0, 2, -3, 0], [0, -3, 3, -7, -7, 7], [3, -3, 4, -4, 0, 11], [-17, 0, 0, -3, -5, 0], [6, 0, 0, 5, -3, 0], [-7, 7, -8, 8, 0, 2], [0, -1, 10, 2, 2, -2], [0, 11, -2, 9, 9, -9], [15, -15, -10, 10, 0, -4], [3, 0, 0, 8, -3, 0], [13, -13, 8, -8, 0, 4], [0, 4, -4, -3, -3, 3], [-10, 0, 0, 3, -2, 0], [8, -8, 2, -2, 0, -4], [0, 5, -11, 8, 8, -8], [-9, 0, 0, -3, -15, 0], [0, 0, 0, 0, -7, 0], [0, 2, 1, -14, -14, 14], [0, 2, -5, 8, 8, -8], [-5, 5, -1, 1, 0, -13], [11, -11, 4, -4, 0, 3], [-6, 0, 0, 17, 8, 0], [4, -4, -5, 5, 0, 7], [0, 7, -4, -13, -13, 13], [0, 16, -9, 5, 5, -5], [0, -2, -15, -2, -2, 2], [-1, 1, 4, -4, 0, 1], [11, -11, -13, 13, 0, 2], [0, 8, 20, -14, -14, 14], [0, 2, 14, -15, -15, 15], [2, 0, 0, -19, -12, 0], [0, 3, 21, -7, -7, 7], [11, -11, 3, -3, 0, -13], [5, 0, 0, -1, 3, 0], [7, -7, -5, 5, 0, 7], [0, -9, 3, 4, 4, -4], [3, 0, 0, -10, 9, 0], [-12, 12, -11, 11, 0, -13], [8, -8, 10, -10, 0, 0], [0, 0, 12, -12, 0, 7], [0, -3, -9, 14, 14, -14], [7, 0, 0, 9, 19, 0], [-7, 0, 0, 18, 10, 0], [0, -7, 8, -9, -9, 9], [0, 6, -10, 12, 12, -12], [28, 0, 0, -5, -2, 0], [0, 15, 7, 0, 0, 0], [0, 18, -11, 16, 16, -16], [-18, 18, 6, -6, 0, -7], [-11, 0, 0, 20, 11, 0], [0, -3, 13, -24, -24, 24], [-13, 0, 0, -15, -9, 0], [-1, 1, -13, 13, 0, -13], [-14, 14, -2, 2, 0, 14], [-8, 0, 0, -18, -3, 0], [10, -10, -2, 2, 0, 1], [7, 0, 0, -14, 6, 0], [7, 0, 0, -13, -5, 0], [-1, 0, 0, -24, -14, 0], [0, -6, 0, -5, -5, 5], [3, 0, 0, 17, 17, 0], [18, -18, -5, 5, 0, -5], [0, 1, -22, 22, 22, -22], [23, -23, 12, -12, 0, 9], [18, 0, 0, 1, 5, 0], [0, 26, 9, -9, -9, 9], [-25, 0, 0, -9, -4, 0], [-18, 18, 0, 0, 0, 2], [0, -28, 7, -5, -5, 5], [-1, 1, 20, -20, 0, 18], [7, 0, 0, 6, 6, 0], [0, 0, -19, 19, 0, -11], [0, -6, -4, 5, 5, -5], [-4, 0, 0, -18, -5, 0], [0, -10, -22, 1, 1, -1], [31, -31, 6, -6, 0, -3], [5, 0, 0, 11, -6, 0], [0, 0, -9, 9, 0, 14], [0, -9, 19, -10, -10, 10], [-19, 19, -4, 4, 0, -9], [0, 22, 11, -9, -9, 9], [0, 33, -6, 3, 3, -3], [0, 0, 0, 14, 16, 0], [0, 0, 5, -11, -11, 11], [-26, 0, 0, 2, 12, 0], [0, -2, 24, -2, -2, 2], [-15, 0, 0, 7, -1, 0], [0, 41, 7, -5, -5, 5], [1, -1, 6, -6, 0, 22], [-13, 0, 0, -13, 13, 0], [-2, 2, 23, -23, 0, 4], [-7, 7, 13, -13, 0, 26], [-2, 0, 0, 11, -8, 0], [14, -14, -12, 12, 0, 10], [-18, 18, -24, 24, 0, -31], [3, 0, 0, -22, -4, 0], [15, -15, 17, -17, 0, 23], [32, -32, -3, 3, 0, 4], [4, 0, 0, 25, 24, 0], [-32, 0, 0, -7, -7, 0], [0, -15, -21, 18, 18, -18], [-12, 12, 20, -20, 0, 0], [-15, 0, 0, -3, -15, 0], [17, 0, 0, 5, 4, 0], [0, -21, 10, -8, -8, 8], [0, -3, 11, -2, -2, 2], [0, -14, -10, 1, 1, -1], [-25, 25, 4, -4, 0, 16], [-6, 0, 0, 13, 6, 0], [6, -6, 23, -23, 0, 8], [15, 0, 0, -14, -15, 0], [27, 0, 0, 18, -5, 0], [28, -28, -6, 6, 0, 5], [0, 11, -4, -22, -22, 22], [0, 21, -4, 7, 7, -7], [6, 0, 0, 2, -1, 0], [0, 17, -14, 9, 9, -9], [42, 0, 0, 5, -2, 0], [-3, 3, -1, 1, 0, -17], [0, -26, 20, -15, -15, 15], [-14, 0, 0, 4, -8, 0], [10, -10, 1, -1, 0, -2], [14, 0, 0, 1, -6, 0], [-29, 29, -4, 4, 0, 12], [0, -8, 0, -7, -7, 7], [23, 0, 0, -19, -18, 0], [0, -14, 4, 20, 20, -20], [-14, 0, 0, 21, -2, 0], [31, -31, 13, -13, 0, -10], [0, 9, -8, 3, 3, -3], [-34, 34, 5, -5, 0, -1], [17, -17, -22, 22, 0, -4], [0, -17, -4, -12, -12, 12], [11, -11, -22, 22, 0, 2], [-32, 32, -11, 11, 0, -9], [0, -7, -5, 7, 7, -7], [1, 0, 0, -4, -7, 0], [-32, 0, 0, 1, -11, 0], [0, 19, 15, -11, -11, 11], [0, -32, 10, -5, -5, 5], [0, -7, -21, 8, 8, -8], [-16, 16, 12, -12, 0, 18], [39, 0, 0, 5, -4, 0], [20, -20, -14, 14, 0, -15], [0, 19, 19, -12, -12, 12], [21, 0, 0, -10, 5, 0], [0, 10, -11, 38, 38, -38], [12, -12, 9, -9, 0, 27], [19, 0, 0, -3, -20, 0], [25, 0, 0, 5, 3, 0], [0, 13, 9, -31, -31, 31], [-13, 0, 0, 5, -1, 0], [0, -10, 17, 15, 15, -15], [-1, 0, 0, 12, 13, 0], [-30, 30, 8, -8, 0, 1], [-19, 19, 5, -5, 0, 4], [0, -9, -21, 4, 4, -4], [0, 4, 15, -6, -6, 6], [-23, 0, 0, 2, -2, 0], [13, -13, -22, 22, 0, -4], [0, 19, -24, 26, 26, -26], [-31, 31, -18, 18, 0, -1], [0, 10, 15, 1, 1, -1], [-38, 0, 0, -10, -13, 0], [-26, 0, 0, 14, -1, 0], [10, 0, 0, 18, 18, 0], [-46, 46, -8, 8, 0, -9], [0, -28, 8, 14, 14, -14], [-22, 22, -3, 3, 0, -25], [-18, 0, 0, 16, 23, 0], [25, -25, -1, 1, 0, -30], [0, -24, -2, 26, 26, -26], [0, 18, 15, -3, -3, 3], [22, -22, 5, -5, 0, -2], [0, 13, 8, -29, -29, 29], [42, -42, 4, -4, 0, -1], [0, -21, 1, -21, -21, 21], [-8, 0, 0, 23, 38, 0], [3, -3, 12, -12, 0, -11], [25, -25, 12, -12, 0, 11], [14, -14, -20, 20, 0, -20], [-16, 16, 8, -8, 0, -20], [25, 0, 0, 14, -1, 0], [30, -30, -12, 12, 0, 2], [0, 36, 3, -2, -2, 2], [0, 22, -11, -15, -15, 15], [38, 0, 0, -3, 3, 0], [0, -6, -1, 15, 15, -15], [0, -21, -11, 25, 25, -25], [-15, 15, -3, 3, 0, -7], [-14, 0, 0, -6, 30, 0], [-12, 12, -10, 10, 0, -34], [-33, 33, -13, 13, 0, -16], [0, 22, 22, 4, 4, -4], [0, -25, 1, -1, -1, 1], [22, -22, 18, -18, 0, -6], [10, 0, 0, -27, -23, 0], [5, -5, 29, -29, 0, 37], [25, 0, 0, -11, 6, 0], [0, 32, 5, -17, -17, 17], [27, 0, 0, -7, 15, 0], [20, -20, -25, 25, 0, -5], [-35, 0, 0, -9, -21, 0], [12, -12, 26, -26, 0, -23], [0, 34, 21, -2, -2, 2], [-19, 0, 0, -11, -13, 0], [-18, 18, 29, -29, 0, -7], [15, -15, -21, 21, 0, 0], [-35, 0, 0, 2, -6, 0], [40, -40, -9, 9, 0, -9], [6, 0, 0, 30, 20, 0], [-40, 40, -8, 8, 0, -7], [0, 6, -31, 26, 26, -26], [0, 11, 25, -12, -12, 12], [40, 0, 0, -5, 12, 0], [5, 0, 0, -5, 2, 0], [0, 19, -31, 42, 42, -42], [-21, 0, 0, -32, -13, 0], [1, 0, 0, -22, -4, 0], [0, 4, -10, 15, 15, -15], [0, -6, 9, 12, 12, -12], [-14, 14, -39, 39, 0, -19], [13, 0, 0, -24, -6, 0], [-24, 24, -4, 4, 0, 16], [0, 14, -2, -8, -8, 8], [29, 0, 0, 15, 24, 0], [0, -15, -3, 7, 7, -7], [-46, 46, -1, 1, 0, 5], [2, -2, 30, -30, 0, -17], [36, -36, 19, -19, 0, 22], [-33, 0, 0, 3, -8, 0], [0, 3, 40, -28, -28, 28], [-32, 32, -7, 7, 0, 6], [-1, 1, -2, 2, 0, -3], [17, 0, 0, -4, -10, 0], [0, 0, -22, 22, 0, -25], [0, -11, 11, -12, -12, 12], [26, 0, 0, 26, 52, 0], [0, -21, -35, 7, 7, -7], [-53, 53, 4, -4, 0, -8], [-28, 28, 13, -13, 0, 29], [0, 16, -4, -11, -11, 11], [0, -1, -10, 3, 3, -3], [46, -46, 1, -1, 0, 12], [35, 0, 0, 27, -16, 0], [-27, 0, 0, 9, 12, 0], [31, -31, 16, -16, 0, 5], [0, -17, -14, -9, -9, 9], [-24, 0, 0, -13, 10, 0], [0, -24, -30, 9, 9, -9], [10, 0, 0, -24, -12, 0], [-49, 49, 29, -29, 0, 10], [5, -5, -7, 7, 0, 16], [0, 42, -14, 29, 29, -29], [11, -11, -35, 35, 0, 1], [0, -27, -54, 15, 15, -15], [0, 2, 4, -26, -26, 26], [-29, 0, 0, -10, -1, 0], [0, 5, -8, 16, 16, -16], [43, -43, -22, 22, 0, -20], [6, 0, 0, 7, 17, 0], [19, 0, 0, -13, 5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_r_i();".
// 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_1008_2_r_i(:prec:=6)
    chi := MakeCharacter_1008_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_1008_2_r_i();".
// 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_1008_2_r_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 0, 3, -2, 9, -3, 1]>,<11,R![81, 81, 81, 18, 9, 0, 1]>],Snew);
    return Vf;
end function;