
// Make newform 1520.2.g.f 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_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_1520_2_g_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 := [484, 0, 2992, 0, 7352, 0, 7824, 0, 1904, 0, -1752, 0, 380, 0, -32, 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, 1434400396, 0, 2314047260, 0, 2175968394, 0, 463305394, 0, -465108276, 0, 102633488, 0, -8704991, 0, 273231], [-18646288, 0, -69461172, 0, -87304466, 0, -12340648, 0, 15542327, 0, -3506282, 0, 299253, 0, -9418, 0], [-977505188, 0, -3078554636, 0, -3869146772, 0, -546462062, 0, 689037518, 0, -155512084, 0, 13275704, 0, -417883, 0], [574684088, 0, 2555420376, 0, 3885107176, 0, 1745285738, 0, -1184238544, 0, 240757978, 0, -19751712, 0, 607847, 0], [-1394895040, 0, -5287396490, 0, -6645946230, 0, -940027588, 0, 1182820446, 0, -266747237, 0, 22762217, 0, -716283, 0], [-566216376, 0, -3190168300, 0, -5797901006, 0, -2755021050, 0, 1818589610, 0, -366750576, 0, 29979829, 0, -920545, 0], [-215694204, 0, -1222168758, 0, -2228864424, 0, -1060054206, 0, 699541134, 0, -141053133, 0, 11529208, 0, -353982, 0], [0, -11340893344, 0, -53288307768, 0, -82991136530, 0, -37894771792, 0, 25531213262, 0, -5180449592, 0, 424638381, 0, -13061002], [0, -1093754068, 0, -5054759548, 0, -7742464084, 0, -3487097926, 0, 2363301958, 0, -480280280, 0, 39394852, 0, -1212203], [0, -162272, 0, -655920, 0, -910798, 0, -225338, 0, 201430, 0, -43270, 0, 3627, 0, -113], [0, 828700840, 0, 2878061792, 0, 3855966728, 0, 943754254, 0, -850654440, 0, 183103598, 0, -15361992, 0, 478877], [552888116, 819989720, 2499339172, 5328528790, 3857356908, 10563796026, 1741812188, 5663689412, -1179063414, -3575281818, 239522302, 712771771, -19643140, -57983887, 604356, 1775295], [552888116, -819989720, 2499339172, -5328528790, 3857356908, -10563796026, 1741812188, -5663689412, -1179063414, 3575281818, 239522302, -712771771, -19643140, 57983887, 604356, -1775295], [0, 7996450550, 0, 51378699528, 0, 101208945526, 0, 54072323630, 0, -34181649619, 0, 6817424848, 0, -554706168, 0, 16985579], [0, 3174857356, 0, 13062002142, 0, 18205941992, 0, 4509531406, 0, -4027203638, 0, 864912425, 0, -72492744, 0, 2258416]];
    Rf_basisdens := [1, 371999496, 768594, 33818136, 33818136, 16909068, 16909068, 5636356, 371999496, 33818136, 2904, 11272712, 33818136, 33818136, 185999748, 16909068];
    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, 1, 0, 0, 0, 0, 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, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 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, 0, 0, 0, 0, -1, 0, 0, 1, -1, 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, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 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, 0, 0, 1, 0, 0, -1, 1, 0, 0], [3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, -2, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 1, 0, 0, 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, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 4, 0, 0], [-3, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -1, 1, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, -1, 0, 0], [0, 0, 0, 3, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, 0, 0, 0, 3, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 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, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 3, 0, 0], [0, 0, 0, 5, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 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, 4, 4, 0, 0], [0, -15, 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, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -4, 0], [0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 2, 2, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2, 2, 0, 0], [0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0], [-20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0], [0, 0, 0, 0, 3, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [18, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 5, -5, 0, 0], [0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1], [0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 5, 5, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, -5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 6, 6, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 4, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, -9, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -4, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 4, 0], [0, -18, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2], [0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -8, -8, 0, 0], [-15, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -8, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -5, 0], [0, -9, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -3, -3, 0, 0], [0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, -1, 1, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0], [3, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [19, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 19, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, -4, 4, 0, 0], [18, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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, -4, 0, 0, 0, 0, 0, -4, -4, 0, 0], [0, 0, 0, 0, -3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 0, -25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 15, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 5, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0], [-12, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 36, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 4, -4, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 5, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -5], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 2, 2, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, -1, 1, 0, 0], [-21, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [21, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -21, 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, -8, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 2, -2, 0, 0], [0, 0, 0, 3, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -3, 0], [-12, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 9, 9, 0, 0], [0, 0, 0, -3, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -10, 0], [0, 0, 0, -7, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0], [0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 2, -2, 0, 0], [0, 0, 0, 3, 0, -2, 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], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 2, 2, 0, 0], [0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 8, 8, 0, 0], [0, 0, 0, 0, -3, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -7, 7, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0], [27, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 11, 0], [0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, -7, 7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 2], [0, 0, 0, 0, -15, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -12, 0], [0, 0, 0, 0, -27, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 11, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, -9, 9, 0, 0], [0, 0, 0, 0, -13, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, -5, -5, 0, 0], [0, 12, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 11, 0], [-9, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -37, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 12, 12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 4, -4, 0, 0], [0, 0, 0, 0, 21, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -10, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2], [0, 0, 0, -9, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -15, -15, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 4, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 12, -12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -3, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0], [-42, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, -3], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0], [30, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -15, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0], [-31, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -15, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 8, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2], [33, 0, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3], [0, 0, 0, 0, -25, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 12, 12, 0, 0], [0, 0, 0, -13, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, -12, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -10, -10, 0, 0], [0, 0, 0, 0, -3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, -8, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -4, -4, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, 6, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 4], [-42, 0, -14, 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, 1, 0], [0, 0, 0, 9, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -1, 1, 0, 0], [0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-33, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 0, 0, 2, -2, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 3, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 11, 0], [0, -33, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -10, -10, 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_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_1520_2_g_f(: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_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_1520_2_g_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1520_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![22, 0, 10, 0, 1]>,<7,R![52, 0, -16, 0, 1]>,<31,R![792, 0, -60, 0, 1]>],Snew);
    return Vf;
end function;