
// Make newform 5203.2.a.i in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5203_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_5203_2_a_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_5203_2_a_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, 3, 3, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [1, -4, -2, 1, 0], [0, 1, -4, -2, 1], [2, 3, -5, -2, 1]];
    Rf_basisdens := [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_5203_a();"
function MakeCharacter_5203_a()
    N := 5203;
    order := 1;
    char_gens := [3269, 3873];
    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_5203_a_Hecke(Kf)
    return MakeCharacter_5203_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]];
    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, 0, 0, 1, 0], [-1, 1, 0, 0, -1], [0, -1, 0, 1, 0], [1, 1, 0, 0, -1], [0, 0, 0, 0, 0], [1, 0, -2, 1, -1], [3, 1, 2, 0, 0], [3, -1, 2, 1, 0], [0, 1, 0, -2, -1], [-4, 0, -1, -2, -2], [-4, 0, 1, -1, 1], [-1, -2, 3, -2, 1], [-4, 2, -2, -2, 1], [-1, 0, 0, 0, 0], [-7, -1, 2, -1, -1], [3, 0, -2, 1, 3], [-5, 4, 0, -1, -1], [0, 1, 0, -1, -2], [-5, -2, -2, 2, -4], [-3, 5, 0, 1, -2], [2, -3, 0, -2, 5], [0, 2, 0, 2, -1], [3, 0, 1, 2, -1], [2, -4, -3, -1, -2], [-4, -1, 4, 0, 0], [0, 1, -2, -1, -5], [-8, 3, 3, 0, 0], [3, -3, 5, 0, 3], [10, 0, 3, 5, 1], [6, 2, -1, 0, 0], [10, -5, 3, 0, 1], [-3, -3, -1, 4, 1], [0, 0, 0, 6, -1], [7, -3, 4, -3, 2], [-8, 5, -1, 3, -2], [6, -4, 2, -4, 4], [1, 2, -6, 5, -4], [-5, 2, 1, -2, -3], [-6, 5, 0, -1, -5], [9, -4, -4, -1, 5], [9, -7, 6, -5, 8], [5, 7, 0, 0, 7], [11, 0, -2, 4, 2], [-8, 0, -1, 1, 2], [6, 3, 4, 0, 8], [4, -6, 4, 1, -1], [-5, 0, 0, -8, -3], [-13, -2, -3, -1, -1], [12, -4, -3, -1, 1], [0, -2, -2, 3, 4], [3, 5, 1, -1, -4], [6, -12, 4, 2, 3], [9, -6, 3, 1, 6], [-2, -7, -1, -6, -1], [-2, -3, -4, -7, -7], [-5, 1, 5, 0, -7], [-6, -6, -5, -1, -4], [6, 4, 3, -8, 3], [-10, 8, -2, -1, 4], [-5, -5, -13, 3, -5], [-10, 9, 2, -9, -10], [20, -5, 3, 0, 3], [10, 7, 1, -4, 2], [10, -7, 3, 1, -3], [5, -2, 4, -6, 11], [11, -2, 6, 5, -3], [-7, 5, 0, -1, 2], [1, -4, -5, -2, 5], [12, 3, -1, -2, 8], [0, -1, -1, 6, -2], [18, -7, 12, 4, 8], [-3, 5, -1, -7, -10], [8, -7, 5, 8, 13], [-8, -1, -9, -2, -7], [-1, -10, -2, 1, -6], [-14, 1, -11, -4, -11], [10, -2, 2, -9, 4], [0, -2, -11, 2, -3], [16, -3, 1, 4, 0], [4, -11, 0, -3, 1], [1, -1, -1, 3, -2], [12, 3, 13, -7, 8], [-16, -4, 1, -3, -5], [12, -7, -8, 5, 5], [12, 5, 0, 4, 0], [16, 5, 9, -8, 6], [-8, 2, -3, -12, -11], [5, 1, 0, 0, -14], [2, -6, 2, 11, 6], [8, -4, 9, -8, 12], [29, -7, 0, -2, 3], [-3, 6, -3, -6, -12], [-15, -2, -1, -4, 5], [7, 6, 2, 11, 1], [-15, 2, -2, 4, -2], [23, -6, 1, 0, 0], [-3, 2, 1, -5, -7], [-8, 0, -2, 9, -5], [-3, 5, 3, -5, 8], [15, 3, 0, -1, -6], [-21, 16, -11, 1, -8], [-1, 1, 6, -2, 7], [14, 3, 7, 1, -8], [11, 4, -3, -2, 7], [6, 2, 1, -3, 3], [-10, 10, -5, -6, 5], [-20, 0, -2, -11, 6], [7, -3, -1, 1, 13], [11, 9, 6, 5, -1], [9, 0, -5, 1, 4], [-13, 4, -7, -3, -3], [6, -11, -4, -2, 9], [-8, 8, 5, 5, 2], [-17, 12, -9, -7, -8], [-4, 5, 1, -3, 3], [6, 5, 0, 4, -12], [-11, 4, -4, -8, -7], [10, 2, -2, 6, 7], [9, 1, -7, 8, -5], [14, -2, 7, 3, -1], [30, 1, 2, 7, 6], [19, -9, 8, 2, 13], [-10, 6, -2, -2, 1], [-20, 5, -7, 14, -9], [10, 3, 0, -8, -2], [6, 8, 3, 10, 15], [9, -7, 2, -9, -5], [-5, -2, 2, -5, -8], [7, 1, 4, 11, 14], [8, -3, -1, 7, 5], [-20, 13, 1, 2, -8], [-3, 12, 3, -1, -5], [4, -10, -5, -6, -4], [6, 13, -3, 7, 7], [23, -9, -3, -8, -2], [-9, -6, -11, 3, -2], [12, 0, 7, 16, 11], [-3, -8, -3, 5, -19], [20, -1, -4, -2, -4], [-2, 0, 14, -9, 0], [-1, 6, 6, -13, 3], [-4, -11, -12, 9, -3], [-31, 2, -1, -2, -4], [2, -1, -1, -13, 17], [-9, 3, 13, -5, -5], [-2, 14, 0, -6, -15], [14, 0, -2, 0, -21], [-10, 18, -8, -7, -10], [13, -11, -12, 0, 13], [3, 2, 5, 6, 0], [8, -13, 4, -10, 9], [11, -5, 2, -8, 1], [25, -20, -2, -1, 2], [18, -14, 5, -4, 1], [5, 6, 11, 5, -16], [-8, 9, -2, -10, 3], [-7, -10, -19, 7, -5], [2, 3, 0, 0, 11], [18, 5, -6, 11, -6], [11, -2, -2, 2, 0], [6, -3, -2, -5, 8], [-19, -6, -14, 8, -15], [8, 0, 9, 6, -17], [4, 11, -5, 16, 3], [10, 8, 2, 9, 19], [20, 3, -4, 1, 9], [-12, -3, 7, -7, -11], [7, 3, 3, -8, 12], [-9, -11, 1, 9, 7], [-32, 12, -12, 4, -15], [-17, 11, 7, 7, -6], [2, 8, -8, 8, -2], [-4, 14, 2, 4, -19], [-8, 15, 0, -8, -14], [21, 13, 4, -4, -1], [-20, 10, -4, -2, 2], [-21, -5, -9, 0, -8], [-8, -3, -1, -3, -6], [11, -21, 4, 10, 16], [18, -10, 2, 12, 4], [-13, -5, -14, -3, 5], [-9, 12, 13, -1, -5], [32, -1, 2, 17, 1], [23, -10, 18, -5, 21], [14, 6, 17, 5, 15], [-13, 21, -9, -3, -2], [-25, 3, 12, -19, -10], [1, -11, 4, -6, 23], [-14, 13, 3, 10, -17], [15, 2, -18, 4, 8], [1, 10, 5, -2, 7], [-33, 1, 7, -17, -8], [-10, 0, 1, 8, -8], [4, 12, 7, -12, 12], [18, 9, -11, 1, -13], [-37, 2, -2, 8, 3], [18, -7, 2, 8, -11], [-18, -4, -13, -16, 17], [-12, -10, -17, 3, -5], [5, 2, 8, 6, -7], [-18, 4, 9, -21, -3], [13, 6, -4, 10, 19], [-12, 9, 11, -17, -3], [8, 6, 6, -1, 23], [20, -10, -2, 8, 17], [32, -8, 10, 8, 0], [-28, 11, 14, -5, -5], [-18, -14, 8, -2, -10], [-17, 9, -21, 1, -17], [37, -4, 4, -11, 3], [-14, -12, -9, 14, 3], [-6, 9, -5, -8, 9], [-19, 1, -3, -11, 7], [-15, 8, 3, -2, -21], [3, 3, -1, -3, 3], [2, -4, -7, 13, 0], [-9, 7, 1, -3, 2], [13, -13, 3, -4, 8], [-3, -10, -8, -1, -11], [6, 3, 5, 15, 19], [32, -5, 7, -2, 14], [29, -12, 1, -5, 8], [-13, 7, -9, -14, -14], [-28, 8, -6, -3, -11], [22, -18, -7, 6, 22], [14, -8, 22, -8, 17], [19, -19, 16, 4, 21], [-15, 13, 10, -1, 0], [17, -1, 5, 1, 16], [29, 10, -10, 9, 11], [29, -14, 21, -1, 16], [19, 3, 11, 7, -19], [7, -7, 6, 9, -10], [27, -9, 3, 0, -3], [-32, 7, 9, -4, -2], [17, -10, 4, -22, 11], [-2, -16, -19, -1, 7], [29, -19, 3, -12, 15], [45, -25, 9, 6, 4], [34, -27, -3, 1, 3], [26, 2, 5, -9, -7], [2, -8, 2, -1, 15], [-18, -15, -2, 5, 14], [-8, -11, 17, -2, 9], [29, -4, 1, 26, 2], [0, 20, 2, -1, -4], [-9, 1, 9, 6, 13], [-4, -10, -1, -13, 6], [-18, 8, -7, 2, 1], [-9, -11, -4, 6, 3], [15, -1, 3, -18, 23], [-4, -20, 0, 17, 9], [13, 0, -12, 5, 2], [4, 0, 3, 14, 10], [-18, -8, -27, 5, -20], [-27, 20, 3, -16, -8], [14, -3, -5, 0, 10], [-2, -1, -19, 2, -20], [20, 13, 6, 6, 0], [3, 7, 7, -11, -10], [14, 10, -4, 14, 21], [-35, -2, 5, 5, -7], [-9, -8, -3, 11, -24], [-16, 20, -9, -3, -22], [-18, -12, -4, -8, -19], [26, 10, -16, 3, 6], [4, 12, 0, 0, -16], [-2, -8, -21, 15, -5], [29, -24, -8, -3, 9], [33, -21, 1, 0, 14], [-20, 8, 7, 7, -5], [-8, 2, 15, -5, 4], [-19, 6, 0, -10, 3], [-32, 9, -14, 10, -20], [-22, 4, -9, 1, -4], [32, 11, -4, 19, 16], [18, 6, 15, 3, 13], [-4, 4, 12, 15, 22], [-3, -2, -8, -6, 26], [10, -10, 5, 3, -5], [28, 13, -3, -2, -18], [-2, 2, -10, 21, -8], [16, 2, 17, 6, 7], [-5, -8, -3, 6, -20], [19, 0, -2, 13, 25], [-5, -1, 6, -8, 7], [0, -12, -4, 9, -13], [22, 0, 19, 5, 29], [-14, -3, -18, -2, -20], [-1, 2, -2, -14, -4], [38, -32, 0, -6, 11], [-1, 11, -7, 14, 5], [2, -2, -9, -11, -10], [-11, 12, -1, -8, 6], [16, 6, -11, 1, -5], [23, -5, 17, -13, 24], [13, 3, 0, -20, 13], [-13, 10, -2, -25, 3], [-9, 2, -17, -1, -17], [-22, 19, -9, 4, -5], [-21, 11, -2, 19, 3], [-23, 13, -10, 1, 8], [-34, -10, 12, -9, -14], [45, -20, 17, 10, 20], [6, 11, -25, 4, -36], [-37, -1, -12, 11, -16], [16, 16, 5, -6, 16], [-15, -13, 2, 0, -11], [-18, -11, -6, -16, -5], [30, -9, -11, -2, 0], [-10, 10, 21, -18, 2], [-19, -7, -8, -2, -2], [-26, 2, -15, -14, -20], [-3, -6, -3, -3, -36], [27, 0, -3, 10, -10], [-13, 23, -1, 18, -6], [11, -6, 35, -7, 22], [6, 12, -9, 0, 15], [-29, 24, -20, -6, -2], [-2, 19, 20, 5, 10], [-14, -8, -5, -13, 11], [-4, -3, -2, 10, 3], [-12, 1, 0, -7, -14], [-38, 3, -4, -15, -12], [6, -7, 15, -21, 10], [-38, -7, -8, 13, -15], [24, 3, 12, 0, -4], [37, -7, 10, -8, 15], [14, -3, 0, 20, -4], [-2, -7, 8, -21, 11], [-37, 7, 1, -13, 4], [44, 0, 9, -11, 5], [10, 13, -10, -2, -13], [2, -8, -5, 6, 3], [-18, -14, -2, -4, 24], [15, -6, -5, 26, -2], [1, 7, 12, 9, -23], [-6, 0, 3, -9, -24], [-8, -22, -14, -4, -5], [17, -17, 9, -5, 17], [-36, -8, -17, -7, -16], [-10, -1, -6, -21, -14], [18, 21, 14, -20, 9], [-11, -14, -3, 29, 0], [9, -7, -24, 7, -1], [7, 1, -11, -10, 9], [7, 2, -7, 12, 2], [0, 19, -8, -9, 14], [-6, -7, -26, 4, -4], [1, -5, -20, -6, -33], [-27, 26, 21, -11, -16], [-2, 12, -14, -3, -29], [-47, 2, -7, 1, -1], [40, 16, 10, 3, 5], [-16, 23, -12, -7, -34], [-39, 24, -14, 3, -10], [-57, 10, 0, -2, -9], [-6, 17, -11, -11, -16], [11, -17, -21, 1, -9], [12, 0, 12, -4, -14], [-11, -13, -18, -5, -11], [31, -9, -4, 9, 4], [-34, 8, -3, -21, -1], [34, 14, 27, 2, 26], [20, 8, -2, 11, -25], [-47, 7, -25, -5, -4], [12, -20, -7, -9, -21], [-8, 4, 27, -11, 19], [-7, 3, -22, -1, -17], [-21, 22, -20, 7, -13], [4, -9, -16, 3, 9], [-44, -4, 0, -5, 22], [11, -1, -2, -2, 10], [-23, -7, -6, -19, -8], [16, 5, 21, 5, 0], [-13, 2, 8, -1, 0], [-10, 10, -14, -8, -17], [-16, 4, -12, -6, -15], [12, -17, -13, 25, 16], [16, -37, 3, 20, 14], [-20, 1, 5, 6, 2], [-28, 2, 17, -19, 0], [14, -15, -13, -6, -2], [20, 6, -13, 10, 1], [-14, 24, -13, 26, -19], [18, 11, 8, 0, 11], [-17, -12, -31, 11, -8], [-21, 30, 5, -14, -18], [-3, -14, 8, -6, 3], [-5, -10, 6, 5, 5], [34, -37, 9, -5, 10], [2, 0, 1, -28, 6], [14, 23, 15, -2, 19], [-22, 26, -10, 4, -3], [33, 0, 19, 21, 12], [-36, 23, 25, -9, -15], [35, -4, 34, -4, 18], [3, 4, -22, 21, 0], [-15, -5, -17, -17, -34], [21, 13, -25, 10, 1], [23, 10, 5, -17, -1], [-11, 9, -22, 13, -21], [-18, -9, 4, -15, 38], [-6, 26, 10, -3, -8], [-20, 23, -27, 23, -23], [-17, 15, -17, -3, 5], [-13, 20, 17, 3, -11], [-13, 4, -26, 16, -26], [32, 5, 38, 1, 18], [-47, 7, -8, -21, -20], [-1, 10, 2, 35, -6], [-10, 2, -3, -16, -9], [26, -1, 31, 1, 29], [-23, -3, -11, 0, -11], [-42, 20, 0, 1, 12], [-20, -8, -9, -1, -16], [48, -2, 22, 7, 15], [-48, -11, 2, -9, -19], [45, 5, 24, 1, 8], [12, -2, 13, -23, 12], [9, 17, 24, -20, 16], [1, 4, -19, 16, -8], [-27, -5, -19, -10, 10], [-3, -5, -27, 23, -7], [-9, -23, -11, -26, -18], [10, 7, -4, -7, -1], [-10, -19, -14, 13, -2], [31, 10, -23, 6, -8], [-34, -6, 7, -5, -25], [-13, -19, 24, -6, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5203_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5203_2_a_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_5203_2_a_i(:prec:=5)
    chi := MakeCharacter_5203_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_5203_2_a_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_5203_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5203_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, 3, 13, -4, -3, 1]>,<3,R![1, 2, -7, -9, 1, 1]>],Snew);
    return Vf;
end function;