
// Make newform 1500.2.o.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1500_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1500_o_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_1500_2_o_b();".
// 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_1500_2_o_b();".
// 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 := [256, 0, 320, 0, 96, 0, -80, 0, -79, 0, -20, 0, 6, 0, 5, 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, 0, 0, 0, 0, 0, 0, 0], [0, -192, 0, -288, 0, -108, 0, -1, 0, 44, 0, 14, 0, -1, 0, -1], [64, 0, 48, 0, 76, 0, 63, 0, 12, 0, -10, 0, -9, 0, -1, 0], [0, -192, 0, 256, 0, 364, 0, 65, 0, -76, 0, -46, 0, 1, 0, 1], [-128, 0, 32, 0, 204, 0, 95, 0, -4, 0, -26, 0, -9, 0, -1, 0], [0, -960, 0, -720, 0, -112, 0, 175, 0, 100, 0, 10, 0, -5, 0, -1], [0, 8, 0, 18, 0, 23, 0, 8, 0, -4, 0, -3, 0, -1, 0, 0], [192, 0, 320, 0, 180, 0, 93, 0, -12, 0, -30, 0, -11, 0, -3, 0], [-960, 0, -752, 0, -184, 0, 83, 0, 68, 0, 26, 0, 7, 0, 3, 0], [512, 0, 512, 0, 96, 0, -139, 0, -100, 0, -18, 0, 9, 0, 5, 0], [0, 224, 0, 96, 0, -21, 0, -58, 0, -22, 0, 3, 0, 5, 0, 2], [-200, 0, -162, 0, -14, 0, 51, 0, 30, 0, 2, 0, -3, 0, -1, 0], [0, -832, 0, -192, 0, 392, 0, 363, 0, 84, 0, -38, 0, -33, 0, -5], [0, -224, 0, -216, 0, 26, 0, 147, 0, 72, 0, -4, 0, -15, 0, -5], [0, -448, 0, -508, 0, -103, 0, 113, 0, 86, 0, 15, 0, -6, 0, -3], [2304, 0, 2016, 0, 96, 0, -707, 0, -356, 0, -2, 0, 49, 0, 13, 0]];
    Rf_basisdens := [1, 384, 192, 384, 192, 384, 48, 192, 192, 192, 96, 24, 384, 192, 96, 192];
    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_1500_o();"
function MakeCharacter_1500_o()
    N := 1500;
    order := 10;
    char_gens := [751, 1001, 877];
    v := [10, 10, 7];
    // 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_1500_o_Hecke();"
function MakeCharacter_1500_o_Hecke(Kf)
    N := 1500;
    order := 10;
    char_gens := [751, 1001, 877];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, -1, 0], [2, 0, -2, 0, -2, 0, 0, -1, -2, 1, 0, 1, 0, 0, 0, 0], [0, -2, 0, 0, 0, -2, -2, 0, 0, 0, -2, 0, 1, -2, 1, 0], [0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1], [0, 2, 0, -1, 0, -1, -1, 0, 0, 0, -1, 0, 2, -1, 0, 0], [0, 0, 1, 0, -1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, -2, 0, 0, 0, -3, 1, 0, -1, 0, 0, 0, -3], [0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0], [-1, 0, 3, 0, 3, 0, 0, 4, 0, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 3, -2, 0], [0, -4, 0, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0, -4, 1, 0], [0, 5, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, -1, 0], [5, 0, 3, 0, 3, 0, 0, -1, 0, 5, 0, 0, 0, 0, 0, 1], [6, 0, -6, 0, 3, 0, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0], [0, 1, 0, 2, 0, -1, 0, 0, 0, 0, 2, 0, 2, -8, -2, 0], [0, 0, -7, 0, -4, 0, 0, -1, -1, 3, 0, 0, 0, 0, 0, -1], [0, 2, 0, 2, 0, -3, -3, 0, 0, 0, 2, 0, -2, 2, 0, 0], [0, 0, -1, 0, 0, 0, 0, -4, -3, -4, 0, -1, 0, 0, 0, -4], [0, 10, 0, 6, 0, 4, 0, 0, 0, 0, 2, 0, 6, 9, -2, 0], [9, 0, -9, 0, -11, 0, 0, -3, -4, 3, 0, 3, 0, 0, 0, 0], [0, -6, 0, -4, 0, 0, 3, 0, 0, 0, 0, 0, 0, -6, 4, 0], [-8, 0, 7, 0, 2, 0, 0, 2, 2, -6, 0, 1, 0, 0, 0, 1], [0, -7, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, 0, -7, -1, 0], [0, 4, 0, 4, 0, -2, 3, 0, 0, 0, 4, 0, 0, 7, -4, 0], [-10, 0, 7, 0, 7, 0, 0, -2, 0, -10, 0, 0, 0, 0, 0, -5], [0, -11, 0, 0, 0, -13, -11, 0, 0, 0, -2, 0, 3, -13, 3, 0], [0, 0, 0, 6, 0, 2, 2, 0, 0, 0, 6, 0, 1, 6, 0, 0], [10, 0, 0, 0, -5, 0, 0, 0, 5, 8, 0, 2, 0, 0, 0, 5], [0, 10, 0, 0, 0, 1, 10, 0, 0, 0, -4, 0, 3, 1, 3, 0], [6, 0, -6, 0, -8, 0, 0, 4, 6, -4, 0, -4, 0, 0, 0, 0], [-4, 0, -7, 0, 3, 0, 0, 3, 3, 7, 0, 0, 0, 0, 0, 0], [-9, 0, 5, 0, -2, 0, 0, -2, -2, -9, 0, -4, 0, 0, 0, -4], [0, -4, 0, -4, 0, 9, 6, 0, 0, 0, 1, 0, -5, 2, 4, 0], [0, -3, 0, 0, 0, -2, -3, 0, 0, 0, 3, 0, -6, -2, -6, 0], [0, -4, 0, -1, 0, -3, 0, 0, 0, 0, 2, 0, -1, -2, -2, 0], [0, 3, 0, 0, 0, -6, -6, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 7, 0, -4, 0, 0, 1, -4, 5, 0, 5, 0, 0, 0, 1], [3, 0, 0, 0, -8, 0, 0, 0, -5, 5, 0, -3, 0, 0, 0, -5], [0, 0, 7, 0, 7, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 3], [0, -5, 0, -5, 0, 0, 8, 0, 0, 0, -2, 0, -3, 3, 5, 0], [0, 1, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 1, -6, 0], [-11, 0, -2, 0, -1, 0, 0, -1, -1, 1, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 5, 0, 0, -3, -6, 3, 0, 3, 0, 0, 0, 0], [0, -1, 0, 1, 0, 6, 6, 0, 0, 0, 1, 0, -7, 1, 0, 0], [0, -8, 0, -6, 0, -8, -8, 0, 0, 0, -6, 0, 1, -6, 0, 0], [0, 0, 11, 0, 16, 0, 0, 1, 8, -15, 0, -7, 0, 0, 0, 1], [0, -7, 0, -6, 0, -1, 0, 0, 0, 0, -3, 0, -6, -8, 3, 0], [8, 0, -8, 0, -11, 0, 0, 2, -1, -2, 0, -2, 0, 0, 0, 0], [5, 0, -13, 0, -13, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -7], [-18, 0, 12, 0, -2, 0, 0, -2, -2, -13, 0, -1, 0, 0, 0, -1], [0, -7, 0, -7, 0, 2, 1, 0, 0, 0, -10, 0, 3, -6, 7, 0], [0, -2, 0, 0, 0, -7, -2, 0, 0, 0, 2, 0, 2, -7, 2, 0], [0, 0, 0, 0, 8, 0, 0, 0, 8, -6, 0, 6, 0, 0, 0, 8], [0, 0, -5, 0, 4, 0, 0, 0, 7, -4, 0, -7, 0, 0, 0, 0], [0, -22, 0, -9, 0, -14, -14, 0, 0, 0, -9, 0, 5, -9, 0, 0], [6, 0, 0, 0, -3, 0, 0, 0, 3, 22, 0, 7, 0, 0, 0, 3], [0, -11, 0, -7, 0, -4, 0, 0, 0, 0, 3, 0, -7, 7, -3, 0], [0, -2, 0, -2, 0, 18, 16, 0, 0, 0, 0, 0, -2, 14, 2, 0], [0, 0, 0, 0, 0, -7, 6, 0, 0, 0, -3, 0, 3, 6, 0, 0], [3, 0, -3, 0, -1, 0, 0, 2, 8, -2, 0, -2, 0, 0, 0, 0], [0, 8, 0, 0, 0, -1, 8, 0, 0, 0, 6, 0, -2, -1, -2, 0], [0, 1, 0, -1, 0, 2, 0, 0, 0, 0, -6, 0, -1, 13, 6, 0], [15, 0, 0, 0, -10, 0, 0, 0, 5, 11, 0, -2, 0, 0, 0, 5], [0, 14, 0, 0, 0, 12, 14, 0, 0, 0, 3, 0, -1, 12, -1, 0], [0, -10, 0, 2, 0, 0, -23, 0, 0, 0, 0, 0, 0, -10, 1, 0], [-5, 0, -6, 0, 6, 0, 0, 6, 6, 5, 0, -1, 0, 0, 0, -1], [0, 1, 0, -1, 0, 0, -3, 0, 0, 0, 0, 0, 0, 1, 4, 0], [13, 0, -6, 0, -6, 0, 0, 3, 0, 13, 0, 0, 0, 0, 0, 3], [0, 2, 0, -1, 0, 3, 0, 0, 0, 0, 3, 0, -1, -8, -3, 0], [0, -15, 0, -9, 0, -8, -8, 0, 0, 0, -9, 0, 2, -9, 0, 0], [0, 0, 7, 0, 22, 0, 0, 8, 11, -14, 0, -3, 0, 0, 0, 8], [0, -5, 0, -2, 0, -3, 0, 0, 0, 0, 4, 0, -2, 4, -4, 0], [6, 0, -6, 0, 4, 0, 0, 4, 15, -4, 0, -4, 0, 0, 0, 0], [0, -1, 0, -7, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, -1, 0], [0, 0, 12, 0, -1, 0, 0, -1, -1, -5, 0, 7, 0, 0, 0, 7], [-3, 0, -5, 0, -5, 0, 0, -5, 0, -3, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 7, 0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0], [0, 0, 11, 0, 11, 0, 0, -6, -3, -17, 0, -3, 0, 0, 0, -6], [-5, 0, 0, 0, -2, 0, 0, 0, -7, 11, 0, -3, 0, 0, 0, -7], [0, 9, 0, 2, 0, 7, 0, 0, 0, 0, -3, 0, 2, 9, 3, 0], [1, 0, -1, 0, -5, 0, 0, -4, -8, 4, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 15, -1, 0, 0, 0, -4, 0, 4, -1, 0, 0], [2, 0, -11, 0, 4, 0, 0, 4, 4, 12, 0, 1, 0, 0, 0, 1], [0, -3, 0, -3, 0, 0, 18, 0, 0, 0, 2, 0, -5, 15, 3, 0], [13, 0, -13, 0, -26, 0, 0, -6, -4, 6, 0, 6, 0, 0, 0, 0], [0, -9, 0, 0, 0, -11, -9, 0, 0, 0, 2, 0, -2, -11, -2, 0], [0, -8, 0, -1, 0, -7, 0, 0, 0, 0, -2, 0, -1, 5, 2, 0], [0, 0, 12, 0, 13, 0, 0, -2, 12, -15, 0, -14, 0, 0, 0, -2], [0, -12, 0, 0, 0, -17, -12, 0, 0, 0, -2, 0, 3, -17, 3, 0], [-7, 0, 10, 0, 10, 0, 0, -2, 0, -7, 0, 0, 0, 0, 0, -7], [-18, 0, -4, 0, 6, 0, 0, 6, 6, 0, 0, -4, 0, 0, 0, -4], [0, 3, 0, -10, 0, 0, 5, 0, 0, 0, 0, 0, 0, 3, -6, 0], [-7, 0, 11, 0, 11, 0, 0, 3, 0, -7, 0, 0, 0, 0, 0, -3], [0, 0, -7, 0, 15, 0, 0, 8, 3, -7, 0, 5, 0, 0, 0, 8], [0, -1, 0, -5, 0, 0, 0, 0, 0, 0, -5, 0, 2, -5, 0, 0], [-5, 0, 2, 0, 2, 0, 0, 2, 0, -5, 0, 0, 0, 0, 0, 2], [0, -1, 0, 4, 0, 0, -16, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 11, 0, 11, 0, 10, -2, 0, 0, 0, 6, 0, 5, 9, -11, 0], [0, 7, 0, 0, 0, -8, 7, 0, 0, 0, 0, 0, -2, -8, -2, 0], [-11, 0, 0, 0, 9, 0, 0, 0, -2, -22, 0, 0, 0, 0, 0, -2], [0, 0, -5, 0, -1, 0, 0, 5, -2, 6, 0, 7, 0, 0, 0, 5], [0, -14, 0, -3, 0, 6, 6, 0, 0, 0, -3, 0, -3, -3, 0, 0], [0, 13, 0, 0, 0, 1, 13, 0, 0, 0, -2, 0, 8, 1, 8, 0], [0, 1, 0, 1, 0, 20, 23, 0, 0, 0, 4, 0, -3, 24, -1, 0], [-11, 0, 13, 0, -4, 0, 0, -4, -4, -12, 0, 1, 0, 0, 0, 1], [6, 0, 1, 0, 4, 0, 0, 4, 4, 3, 0, 4, 0, 0, 0, 4], [0, 4, 0, 4, 0, -19, -20, 0, 0, 0, 2, 0, 2, -16, -4, 0], [0, 10, 0, 0, 0, 13, 10, 0, 0, 0, 0, 0, 3, 13, 3, 0], [0, -9, 0, 2, 0, -11, 0, 0, 0, 0, 11, 0, 2, 3, -11, 0], [17, 0, 0, 0, -13, 0, 0, 0, 4, 11, 0, 6, 0, 0, 0, 4], [16, 0, 0, 0, -15, 0, 0, 0, 1, -9, 0, -1, 0, 0, 0, 1], [25, 0, -19, 0, -19, 0, 0, -7, 0, 25, 0, 0, 0, 0, 0, 0], [0, 7, 0, 7, 0, -13, -18, 0, 0, 0, 1, 0, 6, -11, -7, 0], [0, -27, 0, -5, 0, 0, -18, 0, 0, 0, 0, 0, 0, -27, -3, 0], [0, 1, 0, 1, 0, 0, -27, 0, 0, 0, 0, 0, 0, 1, -4, 0], [15, 0, -8, 0, -8, 0, 0, -13, 0, 15, 0, 0, 0, 0, 0, -7], [-12, 0, 12, 0, 24, 0, 0, 8, 8, -8, 0, -8, 0, 0, 0, 0], [0, 21, 0, 6, 0, -11, -11, 0, 0, 0, 6, 0, 9, 6, 0, 0], [0, 17, 0, -2, 0, 1, 1, 0, 0, 0, -2, 0, 3, -2, 0, 0], [0, -13, 0, -4, 0, -9, 0, 0, 0, 0, -5, 0, -4, -5, 5, 0], [-3, 0, 6, 0, 6, 0, 0, 10, 0, -3, 0, 0, 0, 0, 0, 9], [12, 0, 6, 0, 4, 0, 0, 4, 4, -16, 0, -10, 0, 0, 0, -10], [1, 0, 4, 0, 4, 0, 0, -10, 0, 1, 0, 0, 0, 0, 0, 8], [2, 0, 0, 0, -12, 0, 0, 0, -10, 1, 0, -4, 0, 0, 0, -10], [0, 1, 0, 6, 0, -12, -12, 0, 0, 0, 6, 0, 9, 6, 0, 0], [0, 22, 0, 12, 0, 10, 0, 0, 0, 0, 7, 0, 12, 35, -7, 0], [7, 0, -7, 0, 4, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 2, 0, 24, 7, 0, 0, 0, 4, 0, -2, 9, -2, 0], [3, 0, 1, 0, 3, 0, 0, 3, 3, 11, 0, 12, 0, 0, 0, 12], [0, 6, 0, 6, 0, -6, -9, 0, 0, 0, -6, 0, 12, -3, -6, 0], [-2, 0, 2, 0, -15, 0, 0, 2, -9, -2, 0, -2, 0, 0, 0, 0], [10, 0, 0, 0, -15, 0, 0, 0, -5, 24, 0, -8, 0, 0, 0, -5], [0, -21, 0, -9, 0, -2, -2, 0, 0, 0, -9, 0, -8, -9, 0, 0], [0, -23, 0, 0, 0, -10, -23, 0, 0, 0, -2, 0, 0, -10, 0, 0], [0, -5, 0, 11, 0, 0, -12, 0, 0, 0, 0, 0, 0, -5, -2, 0], [12, 0, 4, 0, 4, 0, 0, 16, 0, 12, 0, 0, 0, 0, 0, 4], [-1, 0, 1, 0, -1, 0, 0, -1, -10, 1, 0, 1, 0, 0, 0, 0], [0, 0, 30, 0, 10, 0, 0, 5, 11, -5, 0, -6, 0, 0, 0, 5], [0, 31, 0, 8, 0, 7, 7, 0, 0, 0, 8, 0, 2, 8, 0, 0], [0, 15, 0, 7, 0, 21, 21, 0, 0, 0, 7, 0, 10, 7, 0, 0], [0, 0, -9, 0, -27, 0, 0, -11, -6, 16, 0, -5, 0, 0, 0, -11], [7, 0, -7, 0, 16, 0, 0, 3, 7, -3, 0, -3, 0, 0, 0, 0], [0, -11, 0, -9, 0, 0, -26, 0, 0, 0, 0, 0, 0, -11, -12, 0], [0, -7, 0, -7, 0, 5, -1, 0, 0, 0, 12, 0, -19, -8, 7, 0], [1, 0, -13, 0, -13, 0, 0, 10, 0, 1, 0, 0, 0, 0, 0, -7], [0, -21, 0, 0, 0, -5, -21, 0, 0, 0, -3, 0, -3, -5, -3, 0], [0, 7, 0, 1, 0, -19, -19, 0, 0, 0, 1, 0, -3, 1, 0, 0], [-6, 0, 0, 0, 2, 0, 0, 0, -4, -4, 0, 0, 0, 0, 0, -4], [0, 4, 0, 5, 0, -1, 0, 0, 0, 0, 4, 0, 5, 0, -4, 0], [0, 47, 0, 0, 0, 44, 47, 0, 0, 0, 9, 0, -5, 44, -5, 0], [0, 13, 0, 13, 0, 5, 0, 0, 0, 0, 14, 0, -1, 13, -13, 0], [3, 0, -3, 0, 1, 0, 0, -14, -17, 14, 0, 14, 0, 0, 0, 0], [7, 0, 0, 0, 1, 0, 0, 0, 8, 30, 0, 10, 0, 0, 0, 8], [0, 0, 1, 0, -16, 0, 0, -4, 7, 12, 0, -11, 0, 0, 0, -4], [0, -17, 0, 0, 0, -13, -17, 0, 0, 0, 5, 0, -7, -13, -7, 0], [-27, 0, 9, 0, 9, 0, 0, 5, 0, -27, 0, 0, 0, 0, 0, -2], [0, 1, 0, -17, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 5, 0], [0, -4, 0, 6, 0, 0, -16, 0, 0, 0, 0, 0, 0, -4, 12, 0], [0, 7, 0, 18, 0, -11, 0, 0, 0, 0, 1, 0, 18, 5, -1, 0], [0, 0, -4, 0, 7, 0, 0, 10, 13, 3, 0, -3, 0, 0, 0, 10], [0, 5, 0, 11, 0, -4, -4, 0, 0, 0, 11, 0, -1, 11, 0, 0], [0, -11, 0, -2, 0, -9, 0, 0, 0, 0, -1, 0, -2, -19, 1, 0], [-16, 0, 7, 0, 7, 0, 0, -2, 0, -16, 0, 0, 0, 0, 0, -8], [0, 17, 0, 8, 0, 0, 28, 0, 0, 0, 0, 0, 0, 17, -4, 0], [1, 0, 10, 0, 10, 0, 0, 10, 0, 1, 0, 0, 0, 0, 0, 8], [0, -9, 0, 0, 0, 7, -9, 0, 0, 0, 7, 0, -14, 7, -14, 0], [-9, 0, 0, 0, 16, 0, 0, 0, 7, -18, 0, -2, 0, 0, 0, 7], [0, 0, 24, 0, 3, 0, 0, 0, 12, -3, 0, -12, 0, 0, 0, 0], [-13, 0, 0, 0, 16, 0, 0, 0, 3, -26, 0, -2, 0, 0, 0, 3], [0, 20, 0, 12, 0, 8, 0, 0, 0, 0, 16, 0, 12, 11, -16, 0], [-7, 0, 7, 0, 1, 0, 0, -11, 5, 11, 0, 11, 0, 0, 0, 0], [20, 0, 15, 0, -1, 0, 0, -1, -1, -12, 0, 3, 0, 0, 0, 3], [0, 0, -20, 0, -1, 0, 0, -1, -1, 24, 0, 4, 0, 0, 0, 4], [10, 0, -10, 0, -20, 0, 0, 6, 2, -6, 0, -6, 0, 0, 0, 0], [0, -24, 0, 0, 0, -12, -24, 0, 0, 0, -15, 0, 13, -12, 13, 0], [15, 0, 0, 0, -2, 0, 0, 0, 13, 10, 0, -1, 0, 0, 0, 13], [0, -6, 0, 0, 0, -9, -6, 0, 0, 0, -21, 0, 6, -9, 6, 0], [19, 0, -2, 0, -2, 0, 0, -13, 0, 19, 0, 0, 0, 0, 0, -10], [0, 11, 0, 11, 0, -8, -1, 0, 0, 0, 12, 0, -1, 10, -11, 0], [0, -30, 0, 6, 0, 0, -22, 0, 0, 0, 0, 0, 0, -30, -4, 0], [0, 18, 0, 2, 0, 0, 24, 0, 0, 0, 0, 0, 0, 18, 6, 0], [-27, 0, 6, 0, 6, 0, 0, -16, 0, -27, 0, 0, 0, 0, 0, -8], [0, 3, 0, 0, 0, 3, 0, 0, 0, 0, -6, 0, 0, 6, 6, 0], [0, -7, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, -7, 0, 0, 0], [0, 0, 14, 0, -4, 0, 0, 12, 0, 16, 0, 12, 0, 0, 0, 12], [15, 0, 6, 0, 12, 0, 0, 12, 12, -2, 0, 4, 0, 0, 0, 4], [0, 10, 0, -5, 0, 0, 22, 0, 0, 0, 0, 0, 0, 10, 7, 0], [0, 3, 0, 0, 0, -9, 3, 0, 0, 0, 14, 0, 6, -9, 6, 0], [0, 0, 24, 0, -4, 0, 0, 17, 4, 21, 0, 13, 0, 0, 0, 17], [-24, 0, 0, 0, 27, 0, 0, 0, 3, 6, 0, -6, 0, 0, 0, 3], [0, -3, 0, 0, 0, 28, -3, 0, 0, 0, -7, 0, 2, 28, 2, 0], [0, 1, 0, 1, 0, -3, 38, 0, 0, 0, 2, 0, -1, 39, -1, 0], [-1, 0, -2, 0, -2, 0, 0, -2, -2, -2, 0, -4, 0, 0, 0, -4], [0, 19, 0, 0, 0, 39, 19, 0, 0, 0, 3, 0, -5, 39, -5, 0], [0, -20, 0, -2, 0, -18, 0, 0, 0, 0, -19, 0, -2, 8, 19, 0], [0, -34, 0, -12, 0, 7, 7, 0, 0, 0, -12, 0, -12, -12, 0, 0], [0, 0, 10, 0, -1, 0, 0, 11, 13, 12, 0, -2, 0, 0, 0, 11], [-3, 0, 0, 0, 17, 0, 0, 0, 14, -3, 0, 2, 0, 0, 0, 14], [0, -18, 0, 0, 0, 15, -18, 0, 0, 0, -6, 0, -2, 15, -2, 0], [29, 0, 15, 0, -2, 0, 0, -2, -2, -11, 0, 4, 0, 0, 0, 4], [20, 0, 12, 0, 12, 0, 0, 9, 0, 20, 0, 0, 0, 0, 0, 5], [0, 10, 0, -3, 0, 23, 23, 0, 0, 0, -3, 0, -4, -3, 0, 0], [0, 0, -53, 0, -5, 0, 0, 5, 8, 10, 0, -3, 0, 0, 0, 5], [0, -42, 0, -11, 0, -31, 0, 0, 0, 0, -14, 0, -11, -41, 14, 0], [-31, 0, 31, 0, 24, 0, 0, 2, -2, -2, 0, -2, 0, 0, 0, 0], [-14, 0, 13, 0, 13, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 13], [0, -11, 0, -8, 0, 0, -2, 0, 0, 0, 0, 0, 0, -11, -1, 0], [8, 0, -24, 0, -16, 0, 0, -16, -16, 18, 0, -6, 0, 0, 0, -6], [0, 12, 0, -7, 0, 0, 13, 0, 0, 0, 0, 0, 0, 12, 11, 0], [0, -15, 0, -15, 0, -19, -22, 0, 0, 0, -13, 0, -2, -37, 15, 0], [-8, 0, 0, 0, 28, 0, 0, 0, 20, -4, 0, 12, 0, 0, 0, 20], [0, 0, 36, 0, 12, 0, 0, 1, 1, -11, 0, 0, 0, 0, 0, 1], [0, -3, 0, -18, 0, -11, -11, 0, 0, 0, -18, 0, 1, -18, 0, 0], [-7, 0, 7, 0, -32, 0, 0, -13, 0, 13, 0, 13, 0, 0, 0, 0], [0, -37, 0, -15, 0, -22, 0, 0, 0, 0, 5, 0, -15, 2, -5, 0], [0, -11, 0, -3, 0, 35, 35, 0, 0, 0, -3, 0, -12, -3, 0, 0], [-13, 0, 0, 0, 15, 0, 0, 0, 2, -22, 0, -1, 0, 0, 0, 2], [-17, 0, -4, 0, -6, 0, 0, -6, -6, 16, 0, 12, 0, 0, 0, 12], [21, 0, -18, 0, -18, 0, 0, 6, 0, 21, 0, 0, 0, 0, 0, 15], [0, -4, 0, -11, 0, -11, -11, 0, 0, 0, -11, 0, -11, -11, 0, 0], [0, 12, 0, -8, 0, -13, -13, 0, 0, 0, -8, 0, -11, -8, 0, 0], [0, 0, -17, 0, -19, 0, 0, 6, 17, 25, 0, -11, 0, 0, 0, 6], [0, 26, 0, 15, 0, 11, 0, 0, 0, 0, -9, 0, 15, -11, 9, 0], [-7, 0, 7, 0, 21, 0, 0, 10, 1, -10, 0, -10, 0, 0, 0, 0], [0, 16, 0, 1, 0, 0, 15, 0, 0, 0, 0, 0, 0, 16, 11, 0], [-14, 0, 10, 0, -8, 0, 0, -8, -8, -12, 0, -2, 0, 0, 0, -2], [0, -29, 0, -3, 0, 0, -36, 0, 0, 0, 0, 0, 0, -29, 1, 0], [-23, 0, 11, 0, 11, 0, 0, 4, 0, -23, 0, 0, 0, 0, 0, 0], [0, 0, -33, 0, -16, 0, 0, -14, -14, 2, 0, 0, 0, 0, 0, -14], [1, 0, 0, 0, 6, 0, 0, 0, 7, 23, 0, 13, 0, 0, 0, 7], [0, -25, 0, -4, 0, -21, 0, 0, 0, 0, -2, 0, -4, -5, 2, 0], [0, 9, 0, 0, 0, 43, 9, 0, 0, 0, -3, 0, 4, 43, 4, 0], [-16, 0, 16, 0, 13, 0, 0, -12, -19, 12, 0, 12, 0, 0, 0, 0], [0, -19, 0, -19, 0, 22, 32, 0, 0, 0, -11, 0, -8, 13, 19, 0], [-12, 0, 25, 0, 10, 0, 0, 10, 10, -25, 0, 0, 0, 0, 0, 0], [-6, 0, 6, 0, 0, 0, 0, 12, -3, -12, 0, -12, 0, 0, 0, 0], [0, 26, 0, 14, 0, 4, 4, 0, 0, 0, 14, 0, -1, 14, 0, 0], [-15, 0, 0, 0, 10, 0, 0, 0, -5, -42, 0, -5, 0, 0, 0, -5], [0, -15, 0, -15, 0, -24, 8, 0, 0, 0, -5, 0, -10, -7, 15, 0], [-17, 0, 22, 0, 14, 0, 0, 14, 14, -26, 0, -4, 0, 0, 0, -4], [0, 3, 0, -4, 0, 0, -17, 0, 0, 0, 0, 0, 0, 3, -12, 0], [-20, 0, 12, 0, 12, 0, 0, 1, 0, -20, 0, 0, 0, 0, 0, 6], [0, 27, 0, 17, 0, 10, 0, 0, 0, 0, -6, 0, 17, 27, 6, 0], [0, 0, -34, 0, -21, 0, 0, 2, 12, 23, 0, -10, 0, 0, 0, 2], [0, 0, 6, 0, 37, 0, 0, 19, 14, -18, 0, 5, 0, 0, 0, 19], [0, -21, 0, 1, 0, -22, 0, 0, 0, 0, -1, 0, 1, -9, 1, 0], [0, -13, 0, -5, 0, 0, 13, 0, 0, 0, 0, 0, 0, -13, -15, 0], [23, 0, 2, 0, 17, 0, 0, 17, 17, 3, 0, 5, 0, 0, 0, 5], [0, 4, 0, 4, 0, -10, -20, 0, 0, 0, 4, 0, 0, -16, -4, 0], [-4, 0, -25, 0, -25, 0, 0, 5, 0, -4, 0, 0, 0, 0, 0, 0], [0, -38, 0, 0, 0, -5, -38, 0, 0, 0, -8, 0, 7, -5, 7, 0], [6, 0, 0, 0, -16, 0, 0, 0, -10, 27, 0, 2, 0, 0, 0, -10], [0, 0, 13, 0, -32, 0, 0, -12, -1, 20, 0, -11, 0, 0, 0, -12], [0, 5, 0, -5, 0, 42, 42, 0, 0, 0, -5, 0, -10, -5, 0, 0], [0, -33, 0, 0, 0, -30, -33, 0, 0, 0, -9, 0, -3, -30, -3, 0], [0, 1, 0, 1, 0, 21, 23, 0, 0, 0, 13, 0, -12, 24, -1, 0], [0, -8, 0, 0, 0, -6, -8, 0, 0, 0, 4, 0, -7, -6, -7, 0], [0, -21, 0, 5, 0, -26, 0, 0, 0, 0, -17, 0, 5, -3, 17, 0], [-37, 0, 0, 0, 29, 0, 0, 0, -8, -32, 0, -9, 0, 0, 0, -8], [0, 15, 0, 15, 0, -33, -6, 0, 0, 0, 15, 0, 0, 9, -15, 0], [0, 13, 0, -9, 0, 0, 13, 0, 0, 0, 0, 0, 0, 13, 3, 0], [-2, 0, 27, 0, 6, 0, 0, 6, 6, -40, 0, -13, 0, 0, 0, -13], [18, 0, -36, 0, -36, 0, 0, 4, 0, 18, 0, 0, 0, 0, 0, 8], [0, 0, -26, 0, -27, 0, 0, -14, -6, 13, 0, -8, 0, 0, 0, -14], [0, 29, 0, -2, 0, 40, 40, 0, 0, 0, -2, 0, 4, -2, 0, 0], [0, -5, 0, 2, 0, -7, 0, 0, 0, 0, 7, 0, 2, -5, -7, 0], [-26, 0, -12, 0, -12, 0, 0, -4, 0, -26, 0, 0, 0, 0, 0, -7], [0, 24, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 24, 2, 0], [0, -11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -11, 5, 0], [20, 0, -13, 0, -13, 0, 0, -27, 0, 20, 0, 0, 0, 0, 0, 8], [0, 9, 0, -2, 0, -14, -14, 0, 0, 0, -2, 0, 5, -2, 0, 0], [0, 3, 0, -9, 0, 12, 0, 0, 0, 0, -4, 0, -9, -25, 4, 0], [0, 3, 0, 0, 0, 9, 3, 0, 0, 0, 16, 0, -9, 9, -9, 0], [-4, 0, 4, 0, 28, 0, 0, -1, -10, 1, 0, 1, 0, 0, 0, 0], [-11, 0, -18, 0, 5, 0, 0, 5, 5, 17, 0, -1, 0, 0, 0, -1], [-11, 0, 11, 0, 0, 0, 0, -14, 5, 14, 0, 14, 0, 0, 0, 0], [0, 12, 0, -10, 0, -3, -3, 0, 0, 0, -10, 0, -6, -10, 0, 0], [0, 0, 0, 0, -11, 0, 0, 0, -11, -21, 0, -23, 0, 0, 0, -11], [0, -30, 0, -1, 0, 0, -9, 0, 0, 0, 0, 0, 0, -30, 17, 0], [9, 0, -9, 0, -13, 0, 0, -9, -31, 9, 0, 9, 0, 0, 0, 0], [0, -58, 0, -15, 0, -43, 0, 0, 0, 0, -8, 0, -15, -40, 8, 0], [0, 0, 12, 0, 17, 0, 0, 17, 21, 0, 0, -4, 0, 0, 0, 17], [0, -58, 0, 1, 0, -28, -28, 0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 31, 0, -9, 0, 20, 20, 0, 0, 0, -9, 0, 15, -9, 0, 0], [0, 0, -10, 0, -13, 0, 0, -12, 16, 1, 0, -28, 0, 0, 0, -12], [-4, 0, 4, 0, 41, 0, 0, -6, -17, 6, 0, 6, 0, 0, 0, 0], [48, 0, -9, 0, -15, 0, 0, -15, -15, 9, 0, 0, 0, 0, 0, 0], [0, 4, 0, 4, 0, 10, -18, 0, 0, 0, 9, 0, -5, -14, -4, 0], [0, 4, 0, 0, 0, 23, 4, 0, 0, 0, -1, 0, 6, 23, 6, 0], [-22, 0, 0, 0, 15, 0, 0, 0, -7, -35, 0, 1, 0, 0, 0, -7], [0, -16, 0, 4, 0, -20, 0, 0, 0, 0, -7, 0, 4, -18, 7, 0], [-7, 0, -18, 0, -16, 0, 0, -16, -16, 20, 0, 2, 0, 0, 0, 2], [16, 0, -46, 0, -5, 0, 0, -5, -5, 37, 0, -9, 0, 0, 0, -9], [0, 19, 0, 24, 0, 16, 16, 0, 0, 0, 24, 0, 16, 24, 0, 0], [0, 0, -23, 0, -23, 0, 0, 1, -17, 24, 0, 18, 0, 0, 0, 1], [0, 13, 0, 0, 0, 8, 13, 0, 0, 0, 26, 0, -19, 8, -19, 0], [0, 14, 0, 14, 0, 16, 4, 0, 0, 0, 15, 0, -1, 18, -14, 0], [0, 10, 0, 10, 0, 0, 24, 0, 0, 0, 0, 0, 0, 10, -13, 0], [1, 0, 7, 0, 0, 0, 0, 0, 0, -27, 0, -20, 0, 0, 0, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1500_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1500_2_o_b();".
// 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_1500_2_o_b(:prec:=16)
    chi := MakeCharacter_1500_o();
    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_1500_2_o_b();".
// 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_1500_2_o_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1500_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 129, 0, 106, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;