
// Make newform 1520.2.g.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1520_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1520_g_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_1520_2_g_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_1520_2_g_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 := [38416, 0, -56840, 0, 58816, 0, -29570, 0, 10645, 0, -2000, 0, 271, 0, -20, 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], [469265976, 0, 37711184, 0, -38435480, 0, 31135885, 0, -6294330, 0, 1007855, 0, -77314, 0, 4429, 0], [-12772202408, 0, 19612506512, 0, -8702669560, 0, 3614511225, 0, -696224210, 0, 100144915, 0, -7515898, 0, 397497, 0], [81350328808, 0, -109267132432, 0, 77259028920, 0, -27198089825, 0, 5182162850, 0, -682415115, 0, 50395018, 0, -2445217, 0], [1133374508, 0, -1598980868, 0, 968268310, 0, -357807455, 0, 68382760, 0, -9132730, 0, 671313, 0, -32348, 0], [3806333, 0, 2196768, 0, -2238960, 0, 1709600, 0, -366660, 0, 58710, 0, -4537, 0, 258, 0], [0, -6646371368, 0, 19612506512, 0, -8702669560, 0, 3614511225, 0, -696224210, 0, 100144915, 0, -7515898, 0, 397497], [-419116013568, 1260457378040, 688708328448, -784321942448, -223553890560, 362008005480, 107038243200, -106365244795, -20673461760, 19149751990, 3340938720, -2399690025, -258490848, 174764750, 15441888, -8087003], [0, -610375641064, 0, 369343421456, 0, -176800898360, 0, 19983308945, 0, -1681047170, 0, -243837925, 0, 28801046, 0, -3060399], [-419116013568, -1260457378040, 688708328448, 784321942448, -223553890560, -362008005480, 107038243200, 106365244795, -20673461760, -19149751990, 3340938720, 2399690025, -258490848, -174764750, 15441888, 8087003], [0, 21347294920, 0, -12690218704, 0, 5948481240, 0, -388311005, 0, -34220230, 0, 26162145, 0, -2601950, 0, 202691], [0, 17822113400, 0, -24375491888, 0, 5428627560, 0, -38437913515, 0, 9317827030, 0, -1713925305, 0, 139049390, 0, -8503883], [0, -12558478520, 0, 38758828144, 0, -16443903400, 0, 6829705875, 0, -1281113910, 0, 189226225, 0, -14201470, 0, 794819], [0, 217078731800, 0, -344266874864, 0, 284239981000, 0, -118054419375, 0, 23498819390, 0, -3270857125, 0, 245478550, 0, -12433839], [-16824899112, -110570775168, -38553644528, 227361858560, 39294121160, -144779890560, -28618680055, 60131955600, 6434944110, -11841223600, -1030370285, 1666037040, 81780118, -125036448, -4527943, 6388160], [16824899112, -110570775168, 38553644528, 227361858560, -39294121160, -144779890560, 28618680055, 60131955600, -6434944110, -11841223600, 1030370285, 1666037040, -81780118, -125036448, 4527943, 6388160]];
    Rf_basisdens := [1, 125016960, 6125831040, 18377493120, 191432220, 976695, 12251662080, 257284903680, 85761634560, 257284903680, 5593150080, 128642451840, 10720204320, 85761634560, 42880817280, 42880817280];
    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_1520_g();"
function MakeCharacter_1520_g()
    N := 1520;
    order := 2;
    char_gens := [191, 1141, 1217, 401];
    v := [1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1520_g_Hecke();"
function MakeCharacter_1520_g_Hecke(Kf)
    N := 1520;
    order := 2;
    char_gens := [191, 1141, 1217, 401];
    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], [-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]];
    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, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -2, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0], [0, 0, -4, 2, -2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 2, -2, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0], [-5, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 1, 1], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 2, 0, 0], [2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, -2, -4, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -1, 2, 1, 3, 0, 0, 0, 0], [-4, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1], [0, 0, 1, -3, -2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, -2, 2, 2], [0, 0, 0, 2, -4, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 2], [0, 0, 2, 3, 3, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, 0, -2, -2], [0, 0, 0, 0, 0, 0, 0, -3, -4, 3, 2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 3, 0, -5, -5], [0, 0, 0, 0, 0, 0, 0, 2, 3, -2, -2, -6, 0, 0, 0, 0], [8, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 8, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, -3, 7, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, -1, -1], [0, 0, -7, 3, 2, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [15, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, -5, 3, 3], [6, 1, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 4, 2, -4, -4], [0, 0, -3, 5, -4, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -2, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 4, -4, -2, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -3, 1, 5, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 2, 2, 0, -4, 0, 0, 0, 0], [0, 0, 2, -8, -2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, -5, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 2, 4, -2, -2], [0, 0, 2, 4, 6, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -2, 2, 8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -3, 4, 8, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 1, -5, -5], [0, 0, 1, -4, -5, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, -1, 0, -4, -4], [0, 0, 0, 0, 0, 0, 0, 6, 1, -6, -5, -4, 0, 0, 0, 0], [-1, -4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -6, 2, 6, 2, 0, 0, 0, 0], [6, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -3, -4, 0, 0], [0, 0, -5, -2, -1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -6, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 6, 5, 0, 0, 0, 0], [-11, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -7, -1, 2, 2], [-6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 2, -4, -4], [0, 0, 2, 2, 4, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 12, -6, 6, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -2, 4, 12, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 3, -6, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 5], [0, 0, 0, 0, 0, 0, 0, -3, -3, 3, 12, 2, 0, 0, 0, 0], [12, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 3, 4, -3, 6, 6, 0, 0, 0, 0], [0, 0, 14, -4, -6, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 1, -2, 0, 0], [8, -6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 6, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, -4, -3, 2, -2, 0, 0, 0, 0], [0, 0, 6, -1, 7, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, -3, -5, -5], [0, 0, -6, -2, -2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -10, 3, -9, 0, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 4, 2, -7, 6, 0, 0, 0, 0], [0, 0, 0, -4, 8, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, -2, 10, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 4, 0, -2, -2], [10, -5, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 2, -8, -8], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -5, 2, 2, 2], [0, 0, -6, -4, -2, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 8, -3, 3, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -5, 0, -2, -2], [0, 0, 0, 0, 0, 0, 0, 5, 0, -5, -6, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 6, -4, 4, 4], [-4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5], [0, 0, 4, -6, 6, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 3, 1, 7, 7], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 4, -4, -9, -9], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 2], [0, 0, 2, -9, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 9, 7, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -2, 4, -6, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 2, 2, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 4, -4, -13, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2, -6, -6], [19, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 9, -3, 6, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -1, 4, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -3, 4, -8, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -5, 2, 2, 2], [0, 0, 0, -3, -5, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [-18, -3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [12, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [0, 0, 11, 2, -5, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, -12, 6, 9, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 4, 4], [0, 0, -2, -6, -10, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, -4, 5, 5], [-6, -2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [-3, 1, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 0, 0, 0, 0, 0, -6, -3, 6, 1, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, -8, -4, -4], [0, 0, 0, 0, 0, 0, 0, -8, -7, 8, 0, 5, 0, 0, 0, 0], [-1, 6, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [9, -6, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [-6, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, -8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 4, 6, 6], [0, 0, -17, 3, 10, 0, 0, 5, 0, 5, 0, 0, 0, 0, 0, 0], [-8, 6, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -8, 8], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -8, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, -5, 10, -5, 3, 0, 0, 0, 0], [0, 0, -2, 12, 12, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1, 0, 3, 3], [0, 0, 0, 0, 0, 0, 0, -6, 1, 6, 0, 13, 0, 0, 0, 0], [2, 1, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, -4, -2, 4, -8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -7, -1, 2, 2], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, -5, -3, -3], [-12, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 5, -3, 4, 0, 0, -7, 0, -7, 0, 0, 0, 0, 0, 0], [-13, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 3, 2, 0, 0], [0, 0, 9, 7, -8, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -2, 2, -6, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 13, -3, -7, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 8, 2, 0, 0, 0, 0], [-2, -6, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, 0, 11, 9, -11, -2, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 1, 5, -7, -7], [0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -15, 15], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -10, -2, 5, 5], [0, 0, -2, 6, 4, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -6, 0, 5, 5], [-7, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 19, -3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 20, -11, -5, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 3, 6, -2, -2], [-26, -2, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7], [-2, -3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 10, -10], [0, 0, -16, -6, 4, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6], [0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 7, 0, -7, -7], [-24, 3, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3], [0, 0, 0, 0, 0, 0, 0, 10, 12, -10, -15, -8, 0, 0, 0, 0], [11, 6, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, -8, 8], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 7, -4, -2, -14, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -5, 2, 0, 0, 0, 0], [-16, -4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 6, 5, -6, 5, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -1, -6, -12, -12], [0, 0, 2, 12, -10, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -5, 1, 2, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -7, 7, 7, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 2, -4, -14, -8, 0, 0, 0, 0], [-10, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4], [0, 0, -9, 7, -2, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 6, 2, -6, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -7, 5, 14, 18, 0, 0, 0, 0], [11, 10, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -6, 8, -4, -4], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 6, -4, -4], [0, 0, 0, 0, 0, 0, 0, -2, -12, 2, 5, 0, 0, 0, 0, 0], [0, 0, 0, 4, 16, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0], [-28, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, -12, -11, 12, 5, 8, 0, 0, 0, 0], [0, 0, -6, 6, 16, 0, 0, 5, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, -19, 3, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 2, -10, -8, -8], [0, 0, 21, -12, -1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, -5, -11, -11], [17, -6, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 10, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 11, 10, -6, -6], [2, -7, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 14, -14], [0, 0, 0, 0, 0, 0, 0, -6, -10, 6, 14, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 0, 8, 8], [-12, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3], [0, 0, 0, -6, -6, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, 8, -8, 5, -8, 0, 0, 0, 0], [0, 0, -10, 10, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 11, 2, -6, -6], [0, 0, 0, 0, 0, 0, -20, 0, 0, 0, 0, 0, 0, 12, -10, -10], [0, 0, 28, -6, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, -2, 10, 0, 0, 0, 5, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, -12, 6, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 3, -1, -1], [8, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, -12], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -18, -6, 0, 0, 0, 0], [42, -10, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 7, -7], [0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, -5, -1, 8, 8], [0, 0, 12, -4, -18, 0, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 12, 4, 1, 1], [13, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 14, -14], [-14, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -12, 12], [0, 0, 2, 4, 6, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 2, 4, -14, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 9, -6, 0, 0, 0, 0], [14, 1, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, -2, 1, 1], [0, 0, -4, 17, 3, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [-8, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5], [0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 7, 9, -11, -11], [0, 0, 2, 0, -12, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 12, -4, -11, -4, 0, 0, 0, 0], [0, 0, 11, 9, -12, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -7, 5, 13, 7, 0, 0, 0, 0], [0, 0, -2, 10, -6, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -6, -6, -8, -8, 0, 0, 0, 0], [6, -8, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 16, -16], [22, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [0, 0, 0, 0, 0, 0, 0, -6, -4, 6, 5, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, -6, 13, 13], [-7, 8, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4], [0, 0, 0, 0, 0, 0, 0, -4, -4, 4, -1, 2, 0, 0, 0, 0], [0, 0, 6, 12, 18, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, -3, -3], [0, 0, 2, -10, 6, 0, 0, 5, 0, 5, 0, 0, 0, 0, 0, 0], [-22, 2, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2], [0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, -1, 1, 9, 9], [0, 0, 0, 0, 0, 0, 0, -4, -13, 4, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -12, 0, 18, 18], [0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, -7, -5, 10, 10], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -5, -6, 4, 4], [8, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7], [0, 0, 0, 0, 0, 0, 0, 8, 17, -8, -8, -11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -8, 10, -6, -6], [-35, 6, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4], [0, 0, -25, 13, 4, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [-3, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 7, -9, -9], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, -1, -1], [0, 0, -7, -9, -16, 0, 0, -5, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -14, -14], [0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -11, -6, -4, -4], [0, 0, 3, 6, -25, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 1, -5, -17, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 8, -12, -5, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 4, -1, -12, 8, 0, 0, 0, 0], [0, 0, -5, 9, 4, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -4, -18, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -10, -5, -9, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -7, 9, 9], [0, 0, 9, 0, 7, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15, -18, 0, 0, 0, 0], [0, 0, -1, -3, -4, 0, 0, -9, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -10, 2, 8, -2, 0, 0, 0, 0], [0, 0, 2, -13, 7, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 5, -4, -4, -4], [0, 0, 0, 0, 0, 0, 0, 4, -5, -4, 8, 6, 0, 0, 0, 0], [0, 0, 13, -3, -4, 0, 0, -11, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, -8, 2, -22, 0, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0], [14, 14, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5], [0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 10, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -11, 5, 15, -1, 0, 0, 0, 0], [-10, 4, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 7, 4, 2, -14, 0, 0, 0, 0], [4, 3, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 0, 0, -5, 5], [-2, -6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 10, -2, -10, -10], [-9, -16, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, -5, -16, 5, 6, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 4, -6, 2, 2], [0, 0, 0, 0, 0, 0, 0, -12, -19, 12, 5, 4, 0, 0, 0, 0], [0, 0, -5, 6, -1, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1520_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1520_2_g_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_1520_2_g_g(:prec:=16)
    chi := MakeCharacter_1520_g();
    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_1520_2_g_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_1520_2_g_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1520_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![196, 0, 290, 0, 129, 0, 20, 0, 1]>,<7,R![64, 0, -248, 0, 153, 0, -26, 0, 1]>,<31,R![672, -648, 198, -24, 1]>],Snew);
    return Vf;
end function;