
// Make newform 1008.2.r.g in Magma, downloaded from the LMFDB on 08 December 2025.
// 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_g();".
// 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_g();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [3, -2, 1, 0, 0, 0], [-12, 18, -20, 11, -4, 1], [9, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2], [6, -13, 9, -8, 2, -1]];
    Rf_basisdens := [1, 1, 3, 3, 3, 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 := false)
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, 0], [-1, 0, 1, 1, 1, 0], [0, -1, 0, 1, 1, 1], [1, 0, 0, -1, 0, 0], [-2, 1, 1, 2, 0, 0], [0, 0, 0, 1, 0, 0], [0, -1, 3, 1, 1, 1], [-1, -1, 1, 1, -1, -1], [1, 0, -2, -3, -2, 1], [6, -1, 0, -5, -2, 1], [0, -3, 0, -1, 3, 3], [-1, 0, -3, 0, -3, -3], [0, 1, 0, 2, -1, -1], [3, -1, 1, -1, -4, 2], [-4, -2, -1, 5, -2, 1], [-6, -1, 0, 1, -2, -2], [1, 2, -2, -2, -4, -1], [3, 1, 2, -2, -2, 1], [1, -1, -2, -3, -1, 2], [5, 1, -4, -1, -2, -2], [3, -1, 1, 1, -1, -1], [0, -4, -5, -1, 2, -1], [-5, 0, 1, 6, -2, 1], [2, 2, -1, -2, 3, 3], [6, 1, 2, -5, -2, 1], [-2, 0, 1, 3, -2, 1], [-3, 0, 6, 11, 6, -3], [5, 0, 2, 0, 2, 2], [-3, -1, 1, 1, -1, -1], [2, -4, -4, -2, 0, 6], [-8, 0, 3, 0, 3, 3], [-1, -4, 2, -8, 6, 3], [9, 1, 3, -7, -4, 2], [-3, 3, 6, 2, 3, -6], [2, 3, -4, 0, -7, -1], [-4, 3, 0, 1, 6, -3], [-3, -3, 6, -2, 9, 0], [12, 1, -1, -1, 1, 1], [0, 2, 0, -17, -2, -2], [-13, 1, -1, 11, 4, -2], [11, -1, -1, 1, -3, -3], [-5, -2, 5, 2, 1, 1], [11, -5, -1, -7, -8, 4], [-1, 4, 2, 0, -2, -5], [-9, -3, 12, 3, 6, 6], [-4, -1, -2, 1, -4, -4], [4, -4, -8, -3, -4, 8], [1, -2, -4, -3, 4, -2], [-2, 7, 4, -1, 6, -3], [2, 4, -4, -3, -8, -2], [11, 2, -7, -2, -3, -3], [2, 0, -4, -9, -4, 2], [-7, -1, -2, 6, 2, -1], [2, 1, -1, -1, 1, 1], [-2, -2, 4, -10, 6, 0], [0, 2, 3, 1, -2, 1], [13, 1, -2, -1, 0, 0], [22, 0, 3, 0, 3, 3], [-3, 4, -1, -2, 10, -5], [1, -2, 1, 2, -6, 3], [2, 1, -4, -17, -5, 1], [-1, 8, 2, -2, -6, -9], [-10, -1, 7, 1, 5, 5], [0, 3, 0, 18, -3, -3], [-6, -5, -4, 7, -2, 1], [-2, -1, 1, 4, -4, 2], [-11, -2, -1, 12, -2, 1], [-3, 0, 6, 22, 6, -3], [-3, 9, 6, 0, -3, -12], [-10, 2, 4, 12, -4, 2], [-23, 3, 4, 24, -2, 1], [-16, -3, 5, 3, -1, -1], [1, -5, -1, 3, -8, 4], [2, 1, -4, -3, -5, 1], [5, -1, 4, 1, 2, 2], [2, -7, -4, -5, 3, 9], [18, -2, -3, -19, 2, -1], [3, 2, -11, -2, -7, -7], [-4, -9, 8, -12, 17, 5], [2, -2, -4, -6, -2, 4], [1, 4, -2, -7, -6, -3], [-2, 0, -3, -1, 6, -3], [7, -3, 10, 3, 4, 4], [-15, 2, -2, -2, 2, 2], [-16, -3, -6, 13, 6, -3], [8, 4, 2, -10, 4, -2], [8, 1, -7, -1, -5, -5], [-22, 2, 1, 21, 2, -1], [-13, -6, -7, 12, 2, -1], [1, -4, -2, -3, 2, 5], [15, 2, -9, -2, -5, -5], [16, -8, -5, -13, -6, 3], [11, 5, -14, -5, -4, -4], [0, -1, 0, 1, 1, 1], [0, -6, 0, 2, 6, 6], [2, -2, 5, 2, 1, 1], [-1, 0, 2, -3, 2, -1], [27, 2, -6, -2, -2, -2], [1, 3, 0, -3, 6, 6], [-1, 0, 6, 0, 6, 6], [-7, 3, 6, 10, -6, 3], [-13, -3, 2, 3, -4, -4], [0, 3, 0, 24, -3, -3], [-23, 2, -2, 19, 8, -4], [1, 2, -2, 6, -4, -1], [8, -3, 0, 3, -6, -6], [-24, 1, -3, 20, 8, -4], [29, -1, -1, 1, -3, -3], [-6, -12, 12, 0, 24, 6], [2, -13, -11, 0, -4, 2], [0, 6, 0, -1, -6, -6], [-13, 6, -12, -6, 0, 0], [2, 11, -4, -2, -15, -9], [5, 3, 0, -8, 6, -3], [-2, 0, -9, 0, -9, -9], [-5, -15, -14, 6, -2, 1], [3, 0, -6, 14, -6, 3], [-19, -1, -7, 1, -9, -9], [1, 1, -2, -1, -3, 0], [-6, -8, -3, 11, -10, 5], [3, 0, -6, 16, -6, 3], [-18, 1, 5, 22, -8, 4], [18, 3, 6, -15, -6, 3], [4, -6, 10, 6, -2, -2], [-20, -5, -7, 18, 4, -2], [6, 6, -6, -6, 6, 6], [20, -10, -8, -18, -4, 2], [7, -5, 4, 5, -6, -6], [-1, 9, 9, 1, 0, 0], [4, -4, -8, -7, -4, 8], [-9, 7, -22, -7, -8, -8], [-3, 5, 6, -29, 1, -8], [3, 3, -6, 20, -9, 0], [-14, 1, -1, -1, 1, 1], [2, 0, -4, -36, -4, 2], [5, 10, -10, 15, -20, -5], [-1, 6, -16, -6, -4, -4], [4, 11, -8, -15, -19, -7], [7, -3, -14, 3, -11, 10], [-24, -1, -3, 1, -5, -5], [10, 3, -6, -3, 0, 0], [14, 4, 5, -13, -2, 1], [-4, 4, 8, -8, 4, -8], [21, -3, 0, 3, -6, -6], [-29, 1, 5, -1, 7, 7], [-13, 8, 8, 13, 0, 0], [26, 2, 1, -27, 2, -1], [-11, 3, 4, 12, -2, 1], [2, 10, -4, 1, -14, -8], [4, 1, 7, -1, 9, 9], [-5, 5, 10, 11, 5, -10], [15, -6, 21, 6, 9, 9], [-27, -2, -1, 2, -5, -5], [-3, 10, 6, -19, -4, -13], [-13, 12, 12, 13, 0, 0], [-6, -11, -6, 11, -10, 5], [28, 0, 3, 0, 3, 3], [4, 5, 10, 1, -10, 5], [-8, 1, -7, -1, -5, -5], [0, 15, 0, -3, -15, -15], [-17, 5, -2, 10, 14, -7], [-10, 4, -16, -4, -8, -8], [-3, -9, 6, 11, 15, 6], [6, -1, -3, 1, -5, -5], [-8, -4, 16, 19, 20, -4], [-1, -9, -4, 6, -10, 5], [13, 3, 3, -3, 9, 9], [-26, 0, 0, 26, 0, 0], [-26, 7, -13, -7, 1, 1], [-3, -1, 6, -8, 7, -2], [18, 5, 6, -17, -2, 1], [-2, -13, 4, -16, 17, 11], [0, -2, 0, 29, 2, 2], [-5, 12, 12, 5, 0, 0], [-2, -4, 4, -3, 8, 2], [-2, 3, 4, 18, 1, -5], [1, -11, -13, -3, 4, -2], [5, -6, 17, 6, 5, 5], [-30, -5, 14, 5, 4, 4], [-1, -1, -5, -3, 8, -4], [-15, -13, -8, 20, -10, 5], [-10, 8, 5, 7, 6, -3], [6, 0, -12, -14, -12, 6], [5, -7, 17, 7, 3, 3], [1, 8, -2, -29, -10, -7], [6, -8, -15, -13, 14, -7], [29, 0, 3, 0, 3, 3], [17, 6, 9, -14, -6, 3], [0, 3, 0, 16, -3, -3], [-17, -3, 13, 3, 7, 7], [7, 1, -4, -1, -2, -2], [-7, -1, 5, 13, -12, 6], [3, 1, -10, -1, -8, -8], [6, 3, 3, -6, 0, 0], [13, 0, 7, 0, 7, 7], [9, -6, -18, -9, -12, 15], [1, -1, -2, 32, -1, 2], [-10, -10, -11, 9, 2, -1], [21, -16, -9, -14, -14, 7], [1, 4, -14, -4, -6, -6], [2, -2, -4, 35, -2, 4], [-1, 6, 6, 1, 0, 0], [-7, 10, 14, 6, 4, -17], [10, -3, -3, -10, 0, 0], [-12, 6, -9, -6, 3, 3], [-26, -2, 1, 2, -3, -3], [2, 2, 4, -2, 8, 8], [5, 2, -10, -23, -12, 3], [-22, 6, -4, 12, 20, -10], [-6, -3, 12, 26, 15, -3], [-10, 6, -21, -6, -9, -9], [7, -15, -14, 0, 1, 22], [-4, 18, 18, 4, 0, 0], [-32, -1, -5, 28, 8, -4], [11, 1, -22, -13, -23, 10], [-12, -5, -4, 13, -2, 1], [-3, 6, 6, -19, 0, -9], [-25, 5, 5, 25, 0, 0], [-18, -1, -15, 1, -17, -17], [9, 16, -18, 2, -34, -7], [-9, 0, 18, 10, 18, -9], [0, 0, 0, 50, 0, 0], [-4, 12, 8, 12, -4, -16], [-21, -2, 11, 2, 7, 7], [2, -12, -4, 12, 8, 14], [-39, -8, -7, 40, -2, 1], [-24, -3, 3, 30, -12, 6], [10, -9, 10, 9, -8, -8], [6, -7, -5, -4, -4, 2], [27, -9, -3, -21, -12, 6], [2, 7, -4, -33, -11, -5], [30, -2, 8, 2, 4, 4], [-8, 2, 16, 24, 14, -10], [3, -15, -6, 12, 9, 18], [-28, -24, -18, 34, -12, 6], [-1, -3, -1, 3, -4, 2], [-4, 10, 8, 6, -2, -14], [-30, 1, 6, -1, 8, 8], [0, -19, 0, -5, 19, 19], [5, 1, 11, -1, 13, 13], [12, -13, -3, -2, -20, 10], [-27, 1, 2, -1, 4, 4], [0, -21, 0, 14, 21, 21], [-19, -9, 18, 9, 0, 0], [3, 4, -6, -13, -10, -1], [25, 2, 1, -26, 2, -1], [23, 5, -2, -5, 8, 8], [0, 9, 0, 12, -9, -9], [3, 9, -6, 14, -15, -6], [-7, 4, -4, -4, 4, 4], [-6, 9, 12, -14, 3, -15], [26, 2, 2, -2, 6, 6], [-5, 2, 10, 25, 8, -7], [13, 0, 9, -4, -18, 9], [24, 16, 15, -25, 2, -1], [-2, -5, 19, 5, 9, 9], [5, 9, -3, -9, 15, 15], [-8, -2, -13, -3, 22, -11], [-29, -4, 13, 4, 5, 5], [-15, -10, 16, 10, -4, -4], [-15, -25, -20, 20, -10, 5], [-29, 0, -5, 24, 10, -5], [8, 19, -16, -6, -35, -11], [-23, -8, 32, 8, 16, 16], [4, 8, -8, -8, -16, -4], [8, 24, 18, -14, 12, -6], [-19, 1, -13, -1, -11, -11], [-39, -7, -3, 43, -8, 4], [-4, 19, 8, -8, -11, -23], [-3, 6, 6, 21, 0, -9], [3, 3, -6, -17, -9, 0], [9, 1, -13, -1, -11, -11], [47, -10, -2, -39, -16, 8], [-3, 15, 6, -31, -9, -18], [7, -10, -14, -10, -4, 17], [25, -6, 24, 6, 12, 12], [-5, 24, 10, -3, -14, -29], [8, -22, -23, -9, 2, -1], [-14, 10, -22, -10, -2, -2], [15, -15, -9, -9, -12, 6], [-2, -10, 4, 25, 14, 8], [6, 9, -12, -4, -21, -3], [-5, 15, 16, 6, -2, 1], [16, -9, -6, -13, -6, 3], [-3, -3, 6, 5, 9, 0], [-31, -3, -1, 3, -7, -7], [29, 3, -3, -3, 3, 3], [1, -12, -2, 6, 10, 13], [-4, -3, 0, 7, -6, 3], [46, -1, 7, 1, 5, 5], [-11, -15, -14, 12, -2, 1], [-14, 0, 4, 0, 4, 4], [-1, 3, 2, 36, -1, -4], [-1, 3, 2, 42, -1, -4], [-5, -15, -6, 14, -18, 9], [0, -3, 0, 51, 3, 3], [-18, 5, 4, 17, 2, -1], [8, 6, 8, -6, -4, 2], [-23, 4, 1, -4, 9, 9], [-37, 12, 6, 31, 12, -6], [2, 4, -4, 21, -8, -2], [-50, 3, -8, -3, -2, -2], [-33, 7, -10, -7, 4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_r_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, -54, 63, 0, 15, -3, 1]>,<11,R![81, -27, 63, 36, 33, 6, 1]>],Snew);
    return Vf;
end function;