
// Make newform 1003.2.a.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1003_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_f();".
// 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_1003_2_a_f();".
// 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 := [4, 2, -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], [-3, -1, 1, 0], [1, -3, -1, 1]];
    Rf_basisdens := [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_1003_a();"
function MakeCharacter_1003_a()
    N := 1003;
    order := 1;
    char_gens := [768, 120];
    v := [1, 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;

function MakeCharacter_1003_a_Hecke(Kf)
    return MakeCharacter_1003_a();
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 := [[-1, 1, 0, 0], [0, 0, 1, 0], [0, -1, -1, 0], [2, 0, 0, 1], [-3, 1, 0, -1], [-2, -1, 0, 0], [-1, 0, 0, 0], [2, 2, 0, -1], [-2, 1, -2, 0], [-2, -1, 0, -1], [-1, -2, -1, 0], [-8, -2, -1, 1], [1, 4, 2, 0], [4, -1, 1, 1], [-8, 1, 1, -1], [3, 2, 4, 0], [-1, 0, 0, 0], [-3, 0, 3, 0], [-2, -2, -1, -1], [-1, -3, -1, 2], [6, 0, 3, 1], [7, 5, 3, -5], [-6, -4, -3, 1], [-4, 5, -1, -1], [-8, 3, 1, 1], [4, -6, -3, -1], [3, 2, 0, 3], [-1, 1, 3, -1], [-4, 2, -2, 0], [5, -2, -3, -2], [-2, -4, -2, 5], [-6, 2, -2, -2], [-1, -2, -2, -6], [14, -2, 3, 3], [-1, 6, 7, -2], [-9, 5, 1, 0], [-8, 1, 4, 0], [3, 5, 3, -2], [-9, -5, -3, 3], [-11, 2, -1, -6], [2, -3, 0, -6], [3, 0, -1, 3], [-7, -6, -2, 5], [1, 10, 4, -6], [-5, 7, 2, 1], [14, -2, -1, 2], [3, 2, 5, -2], [-1, -3, -11, 0], [-15, -1, 1, 4], [-3, -5, 3, -2], [0, 2, 8, 0], [-3, 5, 5, -3], [14, -7, -3, -2], [-7, -1, 1, -3], [7, -6, 0, 6], [-13, 5, 0, 4], [1, -3, 5, 4], [2, -14, -2, 5], [4, 3, -6, 5], [6, -11, 0, 1], [-4, 1, -3, -1], [-16, -1, -1, 6], [10, 2, 8, 5], [10, -8, -2, -1], [-16, 1, -6, -6], [3, -1, -4, -1], [-5, -7, -10, 0], [1, -1, -10, 3], [-2, -6, 8, -6], [-2, -8, -2, 10], [7, 5, -2, 3], [-14, -4, -9, -4], [-3, -1, 0, -5], [0, -2, -1, -1], [-17, 1, -9, -3], [10, -12, -2, 4], [8, 2, 0, -2], [-4, -5, 2, -10], [13, 2, -1, 2], [-1, -16, 2, 1], [9, 4, 6, -5], [-19, 2, -2, -5], [-2, -6, 3, -1], [-16, 7, -2, -3], [3, -3, -10, -5], [-11, 2, 1, 8], [31, -4, -4, -2], [-10, 20, -1, -5], [-4, 8, 11, -5], [17, -8, 7, 4], [11, -11, 2, 1], [-8, 2, -3, -5], [20, 0, -2, 5], [-4, 14, -3, -2], [8, 0, 5, 12], [3, -3, 1, 4], [1, -2, 4, -9], [6, -8, 8, 4], [25, -3, -1, -5], [10, 11, 9, 1], [14, 8, 5, -11], [-2, -17, -3, 12], [2, -12, 3, 5], [18, 8, -3, 3], [3, -11, 2, -1], [-3, -12, -5, 13], [23, 6, 11, -4], [3, 0, 9, -11], [2, -4, 14, 1], [-3, 7, 3, -8], [-1, 3, -2, 6], [2, 7, 7, 1], [5, -4, 7, 5], [-3, 11, 2, 8], [-3, 21, -3, -2], [3, -3, -6, 5], [-24, 0, -12, 1], [9, 9, 14, -4], [-25, 0, -8, 4], [4, -2, -6, 2], [-7, 18, 0, 2], [-5, 4, -5, -8], [-15, -13, -20, 9], [5, -2, -7, 8], [13, -22, -3, 2], [-11, -4, -16, 1], [-10, 11, 9, -6], [3, 10, -4, 7], [-9, 1, 6, -5], [-3, -7, -12, 1], [-3, 0, -2, 7], [-2, 0, 5, 9], [-2, -8, -18, 2], [1, -8, 1, -5], [-19, -16, -21, 9], [-2, 19, 4, -6], [5, 5, 9, -4], [9, 5, 11, -8], [5, -4, 15, 2], [-2, 22, 0, -12], [5, -1, 3, 0], [5, -7, 1, 16], [15, -2, -11, -4], [0, -2, 11, 5], [-13, 20, -1, -9], [-6, 18, 4, 4], [-17, -10, -8, 6], [-6, 3, 5, 13], [-7, -12, 5, 2], [29, 11, 4, -7], [10, -11, 0, 7], [-13, 9, -3, 8], [15, 17, 7, -19], [26, -8, -7, -5], [7, -3, 3, 3], [5, -15, -22, 4], [15, 17, 7, -10], [12, -8, -8, 0], [-28, -12, -8, 10], [12, 1, -4, 6], [-4, 14, 11, 6], [1, -1, -4, 5], [-3, 1, -11, 4], [13, -1, 4, 2], [8, 1, -15, -1], [31, -1, 1, -2], [10, 10, -13, 3], [-22, -11, -20, 3], [14, 0, 6, 12], [-17, -2, -9, -2], [-11, -3, 5, -14], [-5, -14, -5, 18], [20, -14, 3, 0], [-29, 13, -5, 0], [-4, 6, -3, -8], [-7, 22, 7, -3], [-11, -5, -15, -7], [9, 6, -9, -6], [-1, 5, -2, 3], [-17, 8, -10, -14], [-16, 18, -1, -10], [-10, 2, -7, -10], [11, -13, 2, 7], [6, 1, -6, -9], [-2, -6, -7, -7], [7, 4, -12, 11], [-5, 12, -13, -4], [-17, -5, 13, -2], [21, -4, 11, 6], [-35, 0, -11, -8], [2, 7, 18, 8], [-7, 6, 4, -7], [-3, 8, -8, -5], [-20, -10, -13, 11], [14, -18, 6, 17], [-2, 26, 3, -11], [-4, 8, 5, 9], [-2, 1, 12, -14], [31, -4, -2, 7], [-20, -5, 0, 2], [30, 10, 17, -1], [6, 16, -4, 3], [-20, 22, -5, -9], [-1, 0, 9, -10], [-33, 23, 4, -4], [-31, 13, 13, 2], [-17, -10, -7, -6], [-7, 2, -17, 6], [24, 4, -2, -4], [1, -10, 17, -6], [17, 3, 12, -13], [6, -1, -5, 10], [-2, 2, -19, 2], [25, 9, 1, -15], [-22, -10, 6, -5], [-25, -2, -20, -3], [-1, 3, 14, -17], [-5, -7, 10, 9], [-2, 0, -17, 13], [-6, 24, 12, 0], [26, -15, -17, -3], [16, -16, 3, -3], [19, -38, 3, 10], [-4, -12, 17, -10], [9, 18, 5, 6], [-10, 25, 0, 6], [-6, -5, -4, 1], [-12, 4, -7, 19], [-13, -30, 0, 7], [1, -27, -4, 16], [-24, 19, 7, 1], [14, 6, 11, -17], [21, 18, 9, -10], [-2, 28, 5, -3], [20, -9, -9, 15], [-14, -6, -12, -7], [14, 6, 7, -15], [-10, 8, 21, 7], [15, 20, -1, -2], [-13, -1, -7, 4], [2, -12, 21, 2], [5, -27, -1, 12], [-13, -13, -15, 15], [-32, -7, -10, 7], [31, -20, 17, 13], [-30, 4, -12, -6], [-29, -10, 13, 0], [9, 6, 20, -13], [16, -30, -26, -3], [-13, -7, 12, -14], [22, -23, -5, 0], [7, 8, 0, 13], [-10, 11, -12, -12], [2, 1, 11, 4], [-21, -24, -28, 6], [-5, 7, 0, -23], [-23, -10, -8, 8], [11, 15, 3, -8], [-27, 7, -2, 9], [-7, 9, 10, -21], [24, -1, -9, -9], [4, 2, -11, -12], [-16, -19, -11, 18], [-18, -14, -3, 3], [-25, 6, 2, -8], [13, -20, 6, 1], [-28, -17, -8, 9], [4, 15, 8, 12], [-12, -24, -23, 12], [3, -4, -18, 4], [-12, 33, 28, 0], [-31, 15, -16, -9], [-18, 8, -11, 9], [16, -14, -5, 20], [-8, -7, -4, -5], [-5, 3, -4, -1], [-7, -17, -26, 1], [11, 10, 5, -29], [-24, 0, -17, -15], [14, 0, -2, 27], [-44, 18, 8, -4], [-14, 11, -11, -13], [-30, -12, -5, -3], [-2, -14, 1, -11], [8, 2, 24, 8], [-41, 20, 18, 3], [21, -26, 12, 13], [-19, 2, -11, -15], [11, 6, 15, -8], [-10, 15, 6, -21], [12, -40, -6, 4], [5, 35, 15, -16], [-19, -21, -23, 0], [-16, 13, 8, -4], [-33, 4, 13, 9], [17, -20, 6, 11], [-24, 2, -2, 22], [12, 5, 1, -16], [9, 5, 28, -9], [9, -8, 7, -2], [-6, -1, -7, 10], [-21, 24, 4, 5], [4, 18, 3, 7], [-20, 18, -3, -1], [-10, 6, 10, -3], [-3, 4, 14, -13], [-25, 15, -5, 5], [6, -2, 0, 12], [-24, 12, 12, 0], [-28, -31, -16, 18], [17, -27, -16, 0], [41, 4, -8, 2], [39, -4, 10, 12], [-9, 23, -12, -17], [-21, -9, -12, -8], [38, -29, 1, 13], [-13, 24, 8, -5], [18, 4, 9, 24], [3, -3, -14, 1], [-18, 8, -4, 16], [-19, 23, -3, 2], [-22, -11, -12, 4], [-2, -13, 24, -3], [38, 4, 20, -1], [-2, 13, 7, -16], [-7, -12, 15, -2], [9, -8, 9, 14], [2, 18, -5, -9], [-29, 6, 22, -5], [5, 21, -15, 8], [43, -2, -15, 4], [3, 1, 20, 3], [45, 12, 19, -12], [3, 13, -11, -5], [11, -23, 14, 7], [-33, -5, -14, 2], [-5, 14, 14, 4], [-19, -3, 21, -12], [4, -4, -21, -9], [-8, -18, -11, 0], [7, 3, 16, -5], [14, -11, -12, -16], [-24, 11, 20, 2], [-2, -10, -23, 17], [-27, -14, -34, 7], [-11, -19, -15, 26], [-20, 3, -8, 14], [-10, 34, 1, -20], [-5, -38, -22, 15], [-23, -12, -7, -10], [-49, 5, -18, 1], [35, -14, 8, -2], [30, -21, 6, 17], [17, 4, -9, -10], [-2, -26, -39, -5], [-3, -28, -19, 12], [-16, 25, -3, 7], [-6, -10, 0, 3], [16, 5, 16, -11], [21, 39, 16, -22], [37, 21, 22, -13], [3, 2, -9, 9], [-9, 27, 5, -11], [-9, -1, -12, -1], [38, 24, 17, -23], [-34, 0, 9, -5], [-24, -16, 13, -15], [-30, -2, -8, -3], [36, 16, 1, -13], [-8, 24, 6, 2], [-62, 1, -7, -9], [36, -12, -10, 12], [14, 4, -6, 30], [12, -21, 12, 12], [-2, -7, 6, -13], [11, 24, 10, -19], [-18, 23, 3, -17], [23, 23, 19, -4], [-18, 18, -5, -31], [51, 2, -2, 8], [21, 2, 4, 20], [-13, 28, 13, -2], [-6, 0, 5, -19], [-2, -10, -10, -12], [10, -14, -7, -8], [22, -11, -9, -7], [-8, 0, 8, -5], [-21, 25, 17, -2], [-18, 4, -3, -27], [30, -19, 22, 22], [-45, 1, 8, -5], [-36, 3, -8, -2], [-2, -13, -14, 12], [20, 12, 13, 1], [-41, -23, -12, 29], [51, -9, 5, -2], [-17, 1, -1, -27], [-4, 3, -7, -2], [-55, -3, -1, 3], [-14, -1, -21, -20], [-28, -18, -6, 28], [-8, -2, 10, -20], [-35, -9, 11, -3], [11, -11, 25, -2], [-5, -12, 19, 11], [-15, -31, -4, -5], [58, 15, 10, -16], [-31, -20, 9, 5], [25, 45, 12, -23], [32, 34, 30, -14], [-1, -14, -23, -13], [14, 18, 19, 9], [21, 12, 45, 2], [-17, -15, -8, 12], [-11, -14, -2, 32], [-25, -18, -6, -12], [-6, 36, -7, -19], [23, -12, -7, 4], [-19, 19, -5, -12], [40, 6, 27, -2], [37, 8, -17, 14], [-12, 22, 7, 19], [-52, 18, -17, -19], [20, -4, -14, 15], [11, 16, 42, -6], [-75, 0, 2, -2], [3, -11, -9, -14], [-19, -11, -3, 20], [35, 13, 37, -1], [7, -5, 14, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_f();".
// 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_1003_2_a_f(:prec:=4)
    chi := MakeCharacter_1003_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1003_2_a_f();".
// 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_1003_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -7, -2, 3, 1]>,<3,R![4, -6, -5, 2, 1]>],Snew);
    return Vf;
end function;