
// Make newform 9075.2.a.dy in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9075_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_9075_2_a_dy();".
// 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_9075_2_a_dy();".
// 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 := [-5, 10, 71, -148, -145, 330, 61, -220, 18, 52, -10, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5310, 130, 22435, 7961, -16686, -6883, 5252, 1980, -784, -212, 46, 6], [160, -749, -7088, 3184, 15420, -2450, -8360, 1306, 1720, -336, -122, 29], [604, -4816, -1552, 10780, 8110, -6408, -5734, 2296, 1328, -442, -99, 32], [1432, 14318, 1848, -35136, -16941, 25074, 12983, -8868, -3200, 1538, 251, -102], [-8810, 7863, 60756, -22435, -105562, 21144, 59002, -12385, -12584, 3006, 907, -241], [-2750, -22184, 80505, 57872, -186265, -23357, 123544, -12246, -29046, 5775, 2226, -563], [-8105, -17098, 95120, 31365, -211549, -2241, 139153, -20124, -32638, 7207, 2500, -662], [7516, 6833, -89584, -22736, 204239, -1717, -136527, 21114, 32246, -7313, -2477, 665], [13055, -3298, -124333, 23754, 252421, -48893, -159684, 39481, 36783, -10371, -2788, 849]];
    Rf_basisdens := [1, 1, 1, 1033, 1033, 1033, 1033, 1033, 1033, 1033, 1033, 1033];
    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_9075_a();"
function MakeCharacter_9075_a()
    N := 9075;
    order := 1;
    char_gens := [3026, 727, 5326];
    v := [1, 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_9075_a_Hecke(Kf)
    return MakeCharacter_9075_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 := [[0, -1, 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], [-1, 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, 1, 1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0], [-2, 1, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0], [1, 2, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0], [0, 1, -1, -1, 0, 0, -1, 1, 1, 0, 0, -1], [0, 3, -2, -2, 2, 0, -2, 1, 0, 1, 0, -1], [0, -3, 0, 0, -2, 1, 1, -1, 0, 0, 0, -1], [-2, 2, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0], [-3, -1, 2, 1, -2, 0, 0, -1, 1, -1, -1, 0], [0, -1, 1, 1, 0, -1, -1, 0, 2, -1, 0, 0], [-1, 0, 2, 2, 0, -1, 1, 0, 0, -1, 0, 2], [2, 0, -1, 0, -2, 0, 0, 0, -2, 0, 0, -1], [-1, -1, 0, 0, 0, 0, 0, -1, 0, -1, -2, -1], [0, -3, 2, -1, -1, 1, 0, -1, -1, -2, -1, 0], [4, 4, -3, 0, 3, -1, -1, 2, -1, 2, 1, 0], [-1, 4, -2, -1, 1, -3, -1, 0, 0, 2, 1, -2], [0, -1, -1, 0, 0, -2, 0, 1, 0, 3, 2, -1], [-2, -2, 2, 0, -3, -2, 2, -2, 0, 0, 0, 0], [-2, -1, 1, -1, -4, 2, 0, 1, 0, 0, 0, 0], [-1, 0, -2, -1, -1, 2, 1, 2, -1, 2, 1, -2], [4, -1, 1, -1, 0, -4, -1, 0, -1, 1, 0, 0], [3, -2, 0, 1, -2, 1, 1, 0, 1, -1, -1, 1], [-3, 1, -2, 2, 0, 1, 1, 0, 0, 1, 2, -2], [-3, 0, 2, 2, 0, 2, 1, 0, 0, -2, 0, 2], [-2, 0, -2, 0, -1, 2, -2, -2, -2, 2, 0, 0], [-2, 3, 0, -1, 1, -2, 2, -1, 0, 0, -1, 0], [-2, -3, 4, -1, -1, 5, 1, -1, -1, -2, -1, 0], [-2, -1, 1, 1, 3, 1, 0, 0, 0, -1, 1, 2], [-5, -4, 1, -2, -4, 3, -1, -1, -1, -1, -3, 1], [6, 0, 4, 3, 5, -5, 2, -1, 2, -1, 1, -1], [-5, 1, 1, 2, 0, 2, 0, 1, 2, -1, 0, 1], [2, -2, 2, 3, 5, -4, 0, -2, 0, 0, 2, 1], [-6, 1, 0, -4, 1, -2, 1, 1, 2, -1, 0, 0], [-4, 4, -2, -2, 0, 1, 0, 1, 2, 0, 0, -2], [-6, 1, 0, 0, 2, 0, -2, -2, 0, 0, 0, -2], [-8, 2, 0, 0, 0, 8, -1, -2, 1, -2, -2, -1], [0, -7, 6, 2, -4, 3, 2, -3, 0, -5, -2, 1], [0, 0, 2, -1, 2, -2, 1, 1, 1, -3, -2, 0], [-6, -2, 4, -1, 1, -2, 1, 0, 1, -4, -1, -1], [-9, 2, 1, -1, -1, 2, 0, 0, 0, -2, 0, 2], [0, 3, -4, -2, -2, 3, 0, 3, 0, 1, 0, -1], [-6, -5, 0, -2, -4, 3, 1, 2, 1, -2, -1, 0], [2, 2, -2, 0, 4, 3, -1, 4, 2, 2, 1, -2], [-4, -7, 6, 2, -2, 4, 3, -2, 2, -2, -3, 2], [5, -4, 4, 2, 0, 2, -1, 0, 2, -2, -4, 2], [-10, 3, -5, -3, -1, 7, -5, 2, -2, 1, -2, 0], [4, 0, -1, 0, -4, -4, 1, -2, 1, -2, -2, -2], [0, 1, -6, 0, 4, 0, -2, 2, -3, 2, 1, -2], [2, -6, 5, 5, -1, 1, 6, -1, 2, -1, -1, 0], [2, 0, 3, 3, -2, -3, 2, -2, 2, 1, 1, 2], [-6, -1, -2, -2, 3, -1, -3, 1, -4, 2, 1, -2], [2, -3, -1, 0, -4, 0, -2, -1, -2, -4, -2, 2], [-6, -5, 4, 4, -4, 4, 5, 1, 2, -1, 0, 3], [-4, 0, 0, 2, 6, -4, -1, 2, 1, 2, 1, 0], [8, 7, -3, -4, 5, -3, -5, 3, -1, 4, -3, -1], [-6, 3, -4, -2, 0, 4, -4, 3, -4, 0, 1, 1], [-6, 0, 3, -1, 4, -5, 1, 0, 0, 1, 2, 2], [-3, -3, 8, 6, 3, 1, 4, -3, 2, -3, -1, 1], [-4, -5, -3, -1, -1, 3, -1, -1, 1, -2, -1, 1], [-3, 1, 0, 3, 3, 0, 0, 1, 0, 0, 5, -1], [-9, -3, -1, -1, 0, 3, -3, -1, -2, -4, -3, 2], [-3, -7, -1, 0, -6, -4, 1, -1, 2, -5, 2, -3], [-5, 2, 0, -1, -2, -3, 1, 4, 1, 3, 1, -1], [8, -5, 3, 3, -1, -12, -1, 1, 2, 1, 0, -2], [-1, 5, -1, -4, 0, -1, -2, 2, 1, 4, -3, 2], [0, -2, 3, -1, -3, -5, 5, 0, 2, -1, -2, 4], [2, 5, -7, -1, 6, 4, -6, 0, -4, 3, 0, 3], [12, -7, -2, 2, 2, -11, -2, 1, -1, 2, 3, -1], [-19, 8, -2, -5, 2, 8, -2, 0, -1, 3, -2, -1], [0, 4, 2, 0, 4, -2, 1, -2, -2, 2, 2, -2], [-1, 2, -2, 0, 4, -6, -3, 1, -2, 5, 2, 0], [2, 1, 0, -2, 1, -1, 2, -3, -2, 0, -1, 3], [0, 0, -6, -3, -2, -3, -3, 0, -1, 3, -1, -2], [2, 5, -2, -1, 4, -8, 0, 1, 2, 0, 4, -1], [-4, -2, 4, 0, -4, 2, 4, -2, 2, -2, -2, 2], [-10, -7, 6, 2, -4, -4, 1, -2, 1, -3, 0, -1], [7, -3, 4, 2, -2, -3, 3, -2, 1, 4, 1, 0], [4, 3, 0, 2, -4, 0, -1, 3, 1, 2, 0, -2], [1, 3, -6, -4, 4, 1, -5, -3, -4, 2, 2, 0], [-9, 0, 0, -4, 4, -5, -1, -1, 1, -3, -1, 3], [-6, 2, -5, 1, -3, -1, 1, 2, 0, -3, 0, 2], [6, 4, -2, 0, 1, -7, 2, 1, 2, 5, 5, -3], [-4, 6, -2, -2, 0, -2, -5, 0, -1, 0, -5, -2], [4, -1, 6, 8, 5, 1, 4, -4, 0, -1, 3, 2], [2, 1, 4, 4, -1, -12, 5, -3, 0, 3, 6, 2], [5, 2, 3, 7, 6, 2, 5, 0, 0, 4, 5, 0], [0, 0, -1, -1, -4, -6, 2, -4, 0, 0, -2, 1], [-5, -3, 0, -3, 3, 2, 2, -3, -2, 0, -1, 1], [2, -3, -4, -3, 1, 11, -4, 0, -2, 1, 1, -4], [-1, -1, 2, 2, 2, 4, 2, 1, 2, 1, 2, -5], [-5, 3, -1, -3, -2, 8, 0, 4, 2, 2, 2, 2], [-10, -1, -4, 1, -2, 10, -4, 3, 0, 0, 0, 1], [-4, 3, -1, -1, 7, 3, 0, 1, -4, 6, 5, 2], [9, 7, -2, -1, 5, -11, -2, 1, -5, 4, -1, -1], [3, -4, -3, -2, -12, 4, -3, -2, -4, -2, -2, 2], [1, 2, 4, 3, -2, 3, 4, 2, 0, 3, 4, 1], [-13, 6, 0, 0, 6, -2, 0, -3, -4, 1, 1, -2], [-3, -6, 6, -2, -4, 8, -1, -3, -2, -7, -2, 2], [8, 9, -5, 0, 6, -8, 1, 0, -1, 1, 6, 0], [-2, 1, -3, -2, 6, -2, -2, 4, -1, 2, 3, 2], [9, 0, -2, -3, -6, -3, -1, 2, -1, -1, 1, -1], [8, 9, -9, -7, 4, -6, -7, 5, -2, -1, -6, -2], [-7, -1, 6, 0, -7, 10, 1, 1, 0, -3, -4, -4], [12, -3, -3, 1, 1, -5, 1, 2, 1, 5, 1, -2], [8, -1, -7, -4, 0, -11, -3, 4, -1, -2, -1, -2], [-1, -6, 2, 0, 0, 1, -2, -4, -1, 2, -4, -3], [-6, -10, 6, 1, -10, 2, 0, -6, 1, 1, -4, 0], [0, 12, -8, -4, 9, 0, 0, 2, -4, 6, 4, -2], [5, -2, -2, 0, 2, -6, 0, 0, 0, 4, 2, -6], [-5, 2, -4, 2, 2, 2, -5, -3, -3, 1, -1, -1], [-6, -1, -4, 3, -8, 0, 1, 1, -3, 0, 6, 0], [-5, 6, 2, -3, -6, 11, 0, -2, 0, 3, 0, 1], [-4, 2, 3, -1, 2, 4, 1, -1, 4, -1, -6, -4], [8, -2, -3, 1, -5, -1, -4, 2, 2, -1, -1, 0], [10, -9, 1, 4, -5, 3, -3, -3, -1, 2, 1, 1], [8, -10, 1, 0, 0, -9, 0, 1, -1, 1, -1, 5], [2, 2, -4, -1, 1, -11, -3, -4, 2, 2, -3, -7], [-8, 1, -6, -4, 1, -6, -1, 0, -4, 0, 1, 0], [2, -6, -1, 7, 1, -4, 4, -1, 0, -1, 3, -4], [6, 0, -6, -2, 0, -5, -3, 3, -3, 3, -3, -5], [6, -8, 9, 4, -4, 4, 8, -6, 2, -2, -2, 5], [-5, -1, 0, 4, 6, -11, -2, 0, 3, 0, -1, 2], [10, -4, 2, 6, 10, -8, 4, 3, 4, 1, 4, -3], [-5, 9, 0, 4, 6, 3, 4, 5, 1, 4, 5, 6], [-4, -4, 6, 7, -3, -6, 6, -2, 4, 2, 0, 1], [-1, 6, -3, 2, 0, -6, -1, 3, 7, -3, 3, -3], [4, 0, 1, -5, -6, -1, -3, 0, 2, 1, -6, 0], [-6, 1, 4, 6, 2, 0, 3, 1, 4, 1, 4, 3], [0, 3, -2, -3, -2, 1, -1, -1, -1, 5, 2, 1], [12, -5, 3, 5, -5, -7, 4, -4, 2, -3, 1, 3], [-1, -5, 9, 6, 4, 1, 4, 3, 4, 3, 4, 1], [-12, -8, 3, -3, -2, 9, -1, -2, 2, 1, 0, 2], [2, 1, 10, 10, 1, -4, 6, 3, 3, -1, 3, 6], [5, 7, 4, 2, 2, 2, -2, -1, 4, -3, -3, 4], [14, -3, -8, -9, -4, -4, -2, 6, -6, -1, 2, -4], [-3, 13, -6, -4, 2, -4, -1, -1, -2, 5, 2, -7], [8, -19, 5, 2, -14, 8, 5, -7, 5, -10, -6, -3], [-11, -3, 2, -4, -7, 4, -4, -3, -1, -5, -5, -1], [-5, 1, -4, 2, 2, -14, 2, -3, -6, 3, 7, 2], [4, 1, -2, 4, 10, 0, -6, 3, 0, 3, 4, 3], [6, -8, 0, -2, -7, 5, -3, 1, 5, 1, 3, -2], [-11, 3, 10, 10, 7, -6, 10, -7, 5, -1, 5, 7], [-2, 3, -9, -7, 0, 0, -5, 3, 2, 3, 0, -6], [-6, -11, 3, 0, -2, 7, -3, 0, 9, -10, -9, -2], [8, -11, 3, 1, 1, 12, -5, -1, -4, -1, -6, 0], [-14, -1, -10, -2, 1, 11, -1, 5, -5, -3, 4, -2], [-16, -13, 4, -4, -13, 9, 5, -2, 1, -7, -7, 7], [5, 6, 4, 4, 5, -9, 5, -2, -3, 1, 11, 4], [-4, 1, 2, 6, 8, -12, -1, 0, -2, 0, 7, 4], [-4, 5, -2, 2, -3, 3, 2, -5, -2, 1, 3, 0], [4, -16, 0, -7, -5, -1, -5, 0, -3, 0, -3, 5], [-14, 8, -8, -8, 0, 3, -8, 5, 3, 1, -5, 1], [10, 3, -3, -2, 9, 3, -2, 5, 4, 6, 3, 0], [-4, -8, 4, -1, 0, -5, -1, 2, -1, -7, -1, 4], [6, 8, -11, -1, 3, -5, 2, -1, -2, 9, 7, -4], [2, -13, 16, 10, 1, 1, 1, -5, 0, -8, -1, 2], [-7, 4, 0, 6, 7, 2, 5, 3, 0, 1, 4, 2], [-2, 11, -12, -5, 5, -3, -8, 5, 1, 6, 5, -2], [6, -2, 0, -6, -4, 12, -4, 0, -2, 4, -6, -2], [-4, 15, -6, 0, 5, 9, -4, 3, -6, 8, 4, -4], [-7, -4, 8, 8, 4, -1, 6, -4, 6, -5, -5, -2], [0, -8, -1, -2, -2, 10, -2, 0, -8, -2, 0, 4], [-13, 4, -10, -3, 0, 2, -11, 8, -2, 3, 3, 3], [-9, -7, 9, 6, 0, 4, 12, 2, 9, -8, -1, -2], [-5, 6, -2, -10, 2, 1, -8, 2, 2, 1, -1, -4], [0, 1, 0, 1, -7, 17, 1, 0, -1, -5, 3, 5], [-2, -12, 6, 4, -6, 9, 1, -8, 2, -2, -6, 4], [4, 4, 1, -1, 3, -3, -3, -1, 1, -7, -5, 3], [-4, 3, 2, 5, 4, 0, 4, -1, 7, -1, -1, 5], [-6, -8, -1, -2, 0, -8, 3, -5, -3, 1, 7, 3], [8, -6, -8, -11, 0, -12, -4, 1, 0, 1, -4, 1], [6, 5, -8, -6, 2, -2, -6, 0, -11, 6, 1, 2], [12, -17, 3, 5, -7, -7, 5, -2, 1, -3, 1, 4], [15, -9, -4, 0, -5, -3, -1, 1, 5, 1, 4, -5], [-14, -10, 7, 6, -4, 18, -1, -1, -3, -3, -7, 1], [-17, -8, 6, 0, -2, 14, 8, -4, -2, 0, 2, 1], [-6, 0, -4, -1, -3, -11, 1, 0, -1, -4, -1, 5], [-7, 2, 6, -1, -1, -2, 2, -4, -2, 2, 6, 2], [-8, 10, -6, -2, 6, 6, 1, -2, -4, 4, 1, 6], [12, 1, 6, -2, 0, -12, 0, 2, 2, 0, -2, 4], [-12, 1, -2, -2, 2, 4, 1, 2, 1, -1, 0, 7], [-17, 2, -12, -6, -2, 9, -3, 4, -6, -2, -1, -7], [4, 2, -8, -6, -3, 4, -9, -4, -2, 5, -7, -9], [11, -6, 19, 10, 8, -17, 4, -3, 7, -3, 1, -1], [-4, -3, -4, 7, 2, 10, 2, 1, -4, 4, 2, -1], [12, 1, -2, 1, 8, 7, -6, 7, 4, 1, -3, 3], [17, 5, 1, 1, 11, 6, 4, -2, 4, -2, -4, -4], [1, 4, -2, -6, 2, 1, -3, 3, 3, 1, 3, -3], [0, 0, 2, 1, -3, -3, 4, 0, 6, -8, -7, 6], [-5, -12, 3, -1, -5, 3, 3, -7, -4, 2, -3, 2], [18, 9, -6, 4, 9, -26, 4, 2, 0, 2, 6, 4], [2, -9, 9, 0, -2, -2, 5, -5, -6, 1, 10, 1], [-6, 5, -2, 1, 6, 2, -1, 3, 10, -5, -4, -3], [-14, -5, -7, -1, -8, 3, 2, -4, 0, -1, -1, -4], [7, 16, 2, 4, 9, -7, 3, 2, 0, -1, 2, 0], [8, 7, -13, 1, 3, 13, -8, 3, -2, 6, -7, -2], [-14, 4, -3, 0, 0, -8, 1, 3, -3, 1, -1, 1], [17, -3, -2, 2, 8, -5, -1, 9, 0, 4, 6, 6], [-4, 9, -2, 0, -10, 7, 0, -1, 6, -3, -6, -5], [2, -17, 9, 4, -5, 9, 3, -8, -3, -13, -3, 4], [-16, -5, 9, -1, -13, 16, 4, -1, 8, -3, -5, 2], [8, 5, -9, 6, 4, -21, 1, -1, 0, 5, 12, -3], [-13, -3, -6, -4, -7, 5, 0, -6, 0, -5, -5, 0], [0, 15, -15, -13, 3, -7, -5, 7, -1, 4, -3, -1], [8, 3, 3, 0, 4, -8, 3, -3, 4, 3, 0, -3], [-5, -5, 2, 7, 1, 18, 3, 1, -3, 0, 2, -1], [4, 0, 2, 0, 10, -10, -10, 3, -6, 3, -4, 5], [5, 7, 4, 2, 0, 5, 3, 1, 2, -6, -2, -2], [-19, -8, 4, 4, -3, 6, 3, 1, 8, -9, 0, -2], [10, 6, 4, 5, 7, -9, 0, -7, 6, 1, -1, -1], [2, 7, -3, 0, 0, -2, -5, 0, 3, 2, 3, 2], [-18, 6, -6, -1, -2, 6, -6, -4, -1, 5, 0, -6], [2, 7, -12, -1, 5, -5, -6, 5, 4, 2, 5, -1], [14, -4, -1, 1, 4, -15, -1, 0, -2, 7, 8, -6], [15, 1, 1, 3, -3, -1, 5, -3, 2, 8, 5, 2], [-22, -9, 5, -6, -2, -6, 1, 0, -1, -10, 1, 4], [-6, 9, 12, 6, 4, -6, 8, -4, -3, 4, 1, 4], [-14, -19, 17, 4, -13, 9, 9, -8, 3, -9, -3, 8], [10, 0, 8, 4, -1, -21, 0, 2, 6, 1, -4, 5], [-4, -8, -3, 3, -9, 14, 1, 5, 6, -7, -4, -2], [6, 12, 6, 4, 2, -15, 2, 7, 5, 3, 3, -3], [-5, -8, 9, 5, -12, 0, 6, 0, -2, 0, 0, 4], [12, 0, 6, -4, -9, -3, -2, -4, 0, 2, -7, 0], [7, -10, 8, 2, 0, -8, 2, -6, 2, -12, -2, 1], [8, -3, -6, 6, 9, -5, -6, 6, 2, 1, 5, -2], [-12, -4, -4, 1, -13, 14, -2, 2, -4, -6, -2, 7], [-7, 11, -8, -2, 6, -2, 3, -5, -2, 7, 0, -3], [-8, 1, 5, 4, 2, -7, 3, 2, -5, 4, 1, 4], [-6, 30, -16, -12, 8, 6, -6, 6, -4, 10, -4, -4], [-8, 19, -20, -8, 7, 2, -10, 9, -9, 7, 3, -4], [10, 6, 3, 3, -7, -13, -3, 0, 2, 3, -10, -6], [-22, 13, -7, -10, 2, 13, 2, -1, -4, -1, -2, -1], [-17, 4, -5, -1, -7, 4, 3, 6, -2, 2, -3, -6], [12, -1, 5, 5, 6, 10, 0, 4, 0, 5, 2, 1], [-2, -1, -6, 0, 9, -19, -1, 0, -3, 9, 9, 5], [-2, 6, -2, 0, 1, 5, 5, 3, 0, 2, 7, 8], [-18, 6, -5, -2, -6, 7, -9, 5, -1, 5, -3, 3], [11, -4, 12, 7, 5, -18, 10, -4, 2, -6, 0, 6], [-21, -5, -3, -6, -6, 20, 2, -3, 2, 1, -6, -5], [7, 11, -18, -6, 5, -20, 3, 3, -4, 9, 6, -2], [24, 14, -11, -6, 7, -7, -3, 7, -5, 7, -3, -3], [9, 3, -2, -2, -3, 10, 1, -5, 0, -3, -6, -3], [22, 2, 6, 4, 5, -9, 1, 0, -1, 2, 11, 7], [3, -5, 13, -2, 0, -4, 1, 1, 0, -5, -6, 5], [8, 12, -19, -3, 0, -4, -4, 9, -6, -1, 3, -2], [-4, -15, 4, -4, -15, -12, -1, 1, -4, -5, -2, 0], [12, -12, 11, 7, -3, -3, 4, -4, 6, -16, -7, -5], [3, 17, -12, -2, 6, 2, 2, 1, 0, 1, 9, -4], [10, 2, 3, -4, -10, -2, 5, 3, 3, -5, -5, 5], [-13, -8, 9, 5, -16, 30, 0, 0, 4, -4, -4, 4], [2, -23, 13, 7, -5, 1, 8, -12, 2, -5, 1, 9], [6, 16, -2, 0, 16, -9, -8, 3, -7, 1, 9, 1], [0, -18, 10, 3, -19, 14, 2, -4, 12, -12, -16, -3], [-13, -8, -1, -5, -5, 11, 4, 3, -4, 0, 4, 14], [7, 1, 0, 0, -3, -1, 0, -1, 0, 9, -7, -5], [19, -13, -4, -8, -7, 2, -1, 3, -4, -1, -12, -5], [-4, 16, 0, 4, 14, -7, 1, 3, 2, 8, 9, 2], [-12, -7, 8, -4, -12, 9, -4, 4, 5, -4, -5, 6], [-5, -5, 4, 5, -6, 9, 6, -3, -2, -1, -5, -2], [-30, 10, -6, -8, 5, 12, -1, 0, 2, 6, 5, 2], [10, -2, -6, -4, 0, 7, -4, 3, 1, -1, 1, -5], [0, 3, 9, 16, -6, -24, 6, -8, 0, -3, 0, 7], [14, 7, -5, -7, 3, -13, 0, -7, -4, 4, 1, -2], [-3, 9, 4, 7, 5, 9, 5, -5, 4, -2, 8, -1], [-33, -15, 9, 3, -12, 24, -1, -8, 4, -8, -7, 4], [-19, 4, 2, -2, -8, 7, -6, -7, -3, -5, -11, -3], [-10, 8, -4, 0, 0, 7, 4, 4, -7, -6, 0, 4], [-5, 7, -4, 7, -4, -4, 0, 1, -1, 3, -5, -10], [-5, -19, 11, 0, -18, 2, 1, -12, -3, -8, -7, 6], [22, 18, -10, -8, 19, -15, -7, 12, 7, 9, -3, 1], [-18, 1, -8, -6, -12, 29, 2, -3, -1, -4, 5, 1], [5, 6, 1, -3, 1, 7, -2, -5, -6, 12, 6, -4], [23, -6, 4, -2, -4, -11, 5, 5, 5, -3, 3, -5], [-2, 3, -10, -4, 0, 11, -6, 8, -3, 0, -1, 2], [-8, 8, -6, -3, -2, 16, 7, 9, 7, 3, 6, 2], [20, -20, 7, 1, -5, 3, 2, -10, -2, -1, 5, 2], [-10, 12, 0, 0, -2, 0, 6, 6, 4, -6, -4, 10], [-6, 0, 1, -1, 5, 10, 6, 3, -6, -7, 1, 4], [-2, -10, 8, 6, 10, 7, -5, -8, 0, -8, -2, 0], [5, -1, 0, 7, 5, 9, -7, -3, -6, 2, -8, 9], [-16, -4, 7, 4, -5, 21, 6, -2, -2, -6, 5, 7], [-25, 22, -10, -4, 13, 9, -4, 8, -8, 3, 2, 2], [-17, -3, -10, -12, -5, 27, -6, 4, -6, 3, -1, -10], [-9, 7, 4, 0, 0, -10, -5, -4, 5, -6, -5, -8], [-4, -8, -2, -5, -8, 12, -5, -3, -9, -7, 2, 6], [-16, 7, -4, 2, 0, 10, 3, -4, 3, 5, 8, -1], [-10, -12, -4, -8, -9, 1, -9, 3, -14, 0, 1, 2], [10, -3, -1, 3, 3, 13, 6, -2, -6, 1, 5, 6], [-2, -7, 1, -8, -3, 11, 0, -5, -6, 6, 7, -4], [-3, 22, -8, 0, 11, -27, 4, 6, 4, 11, 12, -6], [26, 9, -6, 6, 21, -29, 6, 9, 0, 8, 13, 1], [-4, 9, -6, -10, 0, 0, 3, -2, -5, 6, -5, 0], [11, 15, 2, -4, 10, -9, -3, 2, -6, -5, 2, 4], [-8, 17, -10, -6, 13, -1, 2, 4, 0, 3, 9, 4], [-5, 0, 0, -6, 2, 0, -1, 2, -7, 0, -5, 1], [12, -17, 10, 2, -6, -21, 3, -1, -2, 0, 4, -1], [7, 7, -6, -4, 7, -14, -10, -3, -4, 9, -9, 6], [-28, -13, -7, -6, -14, 23, 2, -9, -10, -1, -2, -3], [-12, 11, -10, -8, 4, 18, -2, 2, 0, -2, 6, -2], [18, 2, -13, 2, 4, -18, -2, 9, -1, 5, 11, -1], [4, -11, 14, 9, -15, -4, 2, 1, 12, -8, -5, 4], [4, -9, 10, 6, -12, 3, 4, -7, -2, -8, -8, -6], [-26, 5, -7, 6, -2, 7, 6, 4, 7, -8, -3, -4], [-20, 10, 1, -3, -1, 17, 3, 4, -5, -2, 3, 4], [-24, 10, -8, -4, -2, 9, 6, 4, 11, -2, -4, 0], [41, 8, -8, 0, 5, -5, -3, 4, -6, 5, 4, 0], [0, 22, -14, 1, 6, -4, 1, 9, -5, 7, 6, -4], [12, -6, 7, 2, -5, 1, 8, 2, 2, 0, 5, 3], [-28, 22, -6, -6, 0, 0, -3, 4, -3, 6, -1, 2], [-9, -9, 11, -7, -5, 14, 4, 0, -2, -2, 0, -2], [-5, 0, 6, 2, 6, 10, 9, -7, -3, 5, 3, 15], [12, -7, -11, -7, 1, -5, -5, 4, -5, 1, -5, -4], [16, 0, -10, -6, 2, -8, -8, 1, -10, 1, -2, 5], [-21, -21, 14, 11, -16, 19, -1, -3, 3, -11, -8, -2], [14, -6, 8, 14, 7, -25, 8, -2, 8, -4, 5, 2], [-7, -17, 7, 0, -12, 20, 0, -1, 12, -11, -6, -3], [-5, -10, -4, 10, -7, -14, 18, 2, 3, -1, 6, 0], [-14, -11, 17, 12, -1, 9, 8, -13, 10, -10, -5, 8], [-20, 19, -2, 3, 16, 14, 5, -7, 0, 1, -2, 3], [-1, 7, -1, 1, 19, -14, -6, 4, -4, 0, -2, 8], [-28, -9, 6, 0, -14, 19, 3, -3, -4, 1, -2, 5], [-2, 10, -4, -8, -10, 0, 2, 3, -1, -3, -5, -3], [25, -2, 16, 5, -1, -6, 9, -2, 1, -4, -7, -2], [-19, 0, 10, 12, -6, -8, 14, -3, 6, -9, 5, 6], [15, 6, 6, 0, 11, -26, -1, 3, 14, 1, -6, -4], [25, 15, -13, -12, -2, -11, 1, 4, -5, 12, 1, 0], [7, -8, 12, -2, 2, -9, -4, -8, -4, 5, -3, 2], [-2, 6, -5, -2, 18, -7, -8, -1, -3, 7, 13, -1], [24, -5, 9, -9, 2, -9, 4, 0, -6, 5, -5, 12], [-9, -15, 0, 0, -28, -1, 10, -3, 8, -7, 0, 1], [-5, 1, -5, -5, 3, 13, -4, -3, 6, -2, -10, -10], [-5, 7, 8, 0, 14, 1, 2, -4, 7, -8, -3, 6], [2, -6, 2, -12, 0, 11, -9, -5, 2, -4, -3, -8], [-3, -3, -2, 2, -4, 22, 2, 1, -2, -9, 8, 3], [16, 23, -6, -4, 0, -24, 8, 1, -2, 12, 5, -5], [14, 17, -12, 5, -5, -28, -1, -3, -9, 6, 2, -2], [-8, -13, 3, -5, -15, 27, 6, -12, -4, -1, -5, 2], [20, -11, 9, 7, -9, 3, -2, 3, 6, -10, -5, -8], [1, 1, 8, -2, 10, 5, 5, -1, 4, -1, -4, 13], [2, 9, -7, 7, 20, -2, -3, -1, 4, 5, -4, -4], [20, -1, -1, 6, 8, 6, -4, 3, 6, 4, 8, -8], [-7, -7, 5, 8, -6, 12, -2, -10, 1, -10, -7, -6], [23, 7, -8, -10, -9, -18, -5, 3, -2, -5, -2, -7], [6, -2, 5, -7, -6, 4, 6, 3, -4, 11, 4, 10], [0, -6, 4, 2, 4, -4, 10, -8, 2, 0, -4, 0], [-20, 12, -4, 2, 12, 8, 0, 2, -6, 6, 0, -6], [-21, 8, -10, -6, -9, -1, -3, -10, -1, 3, 1, -10], [-2, -6, 11, 12, 0, 8, 5, -8, -5, -4, 0, 0], [13, -5, 0, 5, 0, 6, 0, -1, -13, 9, 5, -4], [20, -10, 5, 2, -4, 8, 5, 0, -2, -2, 2, -10], [-6, -13, -1, 7, -6, 11, -4, 10, 2, 5, 5, 0], [-14, -10, 6, -1, -5, 25, -8, -2, -2, -10, -13, 4], [22, 13, -16, -13, 14, 0, -15, 3, -3, 8, -6, -6], [0, 0, -20, -6, -2, 4, -6, 10, -4, 4, 2, -10], [15, -3, 3, -5, -9, 2, -5, -2, 8, -12, -9, -8], [-32, 3, -14, -16, 4, 7, -20, -1, -7, 4, -3, -5], [24, -1, 1, -5, -6, 4, 9, 6, 1, 3, -4, 4], [8, -31, 5, 5, -9, 10, 0, -5, -10, -11, -1, 8], [40, 12, -6, -2, 11, -5, -10, 5, 0, 7, -5, -3], [3, -16, 10, -2, -13, 6, -2, 4, -1, -9, -16, 0], [-2, -1, 14, 11, 4, 10, 14, -13, -5, -7, 5, 7], [-10, 0, -6, -4, 4, -16, -7, -10, -2, 0, 1, 0], [-4, -5, -2, 0, -2, 23, 0, 7, -1, -8, -5, -1], [13, -5, -2, 6, 16, 5, 4, -1, 2, 1, 4, 5], [2, 4, 9, -3, 2, 6, 0, -4, 8, -6, -4, -13], [-19, 5, 2, 14, 4, -13, 16, 1, 8, -9, 6, 5], [-1, -1, -1, 7, 8, -8, -9, -6, -2, -4, 1, 0], [-6, -14, 10, 0, -4, 18, 6, 10, 12, -12, -4, 6], [-40, 1, 15, 8, -2, 27, 6, -4, 11, -6, -5, -2], [5, -5, 22, 12, -2, 20, 9, -3, 5, -2, -8, -2], [2, 23, 1, 4, 13, -9, -8, -1, -2, 8, -1, 4], [10, -23, 5, 1, -11, 13, -3, -3, 11, -10, -5, -3], [-7, -11, 2, -6, -8, -12, -6, 3, -4, -12, -5, 0], [-14, -8, 10, -5, -26, 10, 4, -9, -6, -7, -8, 5], [8, 16, -12, 2, 15, 3, -1, 4, -5, 17, 11, 5], [-4, -5, 9, 7, 7, -15, 4, 1, 10, -8, 7, -2], [-9, -3, -8, 2, 4, 5, -7, 14, 2, -3, 2, 0], [4, -9, 2, 8, 14, -5, -11, -7, 2, -2, -6, 3], [-14, -14, 5, -3, -17, 18, -1, -3, -2, -7, -6, 0], [29, 7, -5, -6, 0, -16, -11, 2, 3, 12, 1, -4], [29, 2, -8, -4, 8, -1, -3, 16, 4, 15, 4, -6], [6, 16, 0, -8, 2, -19, -7, 3, 5, 5, -9, 5], [-20, 29, -30, -17, 5, 3, -6, 9, -15, 12, 9, 0], [-1, -10, -2, -5, 1, -8, 2, -4, 4, -12, -2, -4], [19, -3, 1, -7, -9, -27, 2, -3, 6, -2, 0, -6], [-19, 7, 2, 2, -4, 23, -2, 1, 0, 7, 2, -5], [13, -8, -3, -5, -13, 17, -8, 7, -6, -10, -10, -6], [-22, 2, 1, 7, 10, -4, 4, -6, 8, 2, 2, 1], [8, -2, 10, 4, -11, 15, 11, 0, 7, -1, 7, 7], [-12, -9, -2, 1, -9, -5, -6, 8, -4, -3, 5, 10], [-19, -14, -10, -6, -3, 28, -14, 4, -7, -7, -10, 2], [-7, -6, -17, -2, 16, -7, -9, 2, -6, -2, 10, 6], [2, -9, 17, 6, -11, 13, 18, -3, 12, -10, -3, 6], [-4, -6, 5, 2, -2, 29, -4, 4, -6, -2, 4, 4], [20, 7, -9, -11, 2, -26, -13, -3, -13, 0, 0, 3], [10, 15, -8, -6, 12, -6, -8, 10, -7, 2, -1, 0], [-14, 1, -12, -6, 19, -6, -2, 6, -4, 10, 14, -2], [5, -30, 6, 6, -9, -10, 10, -8, 7, -5, -2, 4], [14, 13, -19, -1, 11, -17, -7, 6, 2, 5, 0, -6], [-2, -15, 3, -6, -19, 23, 3, -10, -1, -3, -9, 2], [-22, -6, 9, 16, 2, 7, 14, -6, 8, -4, -6, 6], [-11, 5, 0, 1, 0, 23, 2, 7, 4, 1, 13, 6], [18, 6, 6, 8, 13, -9, -3, 9, 3, 13, 9, 8], [-12, -1, 4, -1, -8, -6, -8, -3, -3, -3, -5, 13], [6, 4, 2, 1, 3, 19, 4, -8, -2, -8, -5, 10], [-19, 11, 0, 1, 4, 19, 0, -3, -2, 9, -3, 2], [-4, -9, 3, 1, 6, -14, 0, 1, 6, 5, -1, -6], [-6, 6, -4, 2, 20, -2, -5, 10, 8, 0, -1, -8], [-16, 12, -3, 8, 8, 2, 2, 0, 6, 0, 18, 4], [-24, 18, -4, -7, -6, 1, 1, 4, 7, 9, -5, -2], [12, 3, 0, 12, 20, -12, -1, -5, -1, 0, 8, 2], [-14, 7, -4, 0, 2, 2, 1, -5, -1, 0, 6, 8], [-11, 3, -10, -4, 8, -5, 1, 3, 10, -6, -4, -4], [23, -20, 10, 8, -7, -10, 13, 5, 6, -5, 2, 4], [12, 10, -8, 0, 9, -39, -5, -1, 0, 8, 3, 12], [-4, 10, 6, 1, 3, 1, 3, -3, -3, -5, -7, 10], [-22, 8, -26, -15, -5, 1, -9, 4, -10, 8, -3, -5], [8, -4, 7, -1, -6, 10, 6, 1, -2, 3, -3, -10], [10, 13, 11, 8, 10, 0, -6, 5, 6, 2, 2, 8], [5, 20, 0, -4, 11, -19, -5, -8, -3, 3, -3, -10], [15, -3, -7, -3, -7, -3, -1, 3, -6, -2, 1, -2], [22, 5, 7, 13, 21, -19, 0, 0, 0, -1, -3, 0], [-2, 10, 1, -5, -3, 3, 1, -2, -9, 4, -1, -2], [2, 8, -18, -4, -4, -2, -3, -7, -9, 3, 3, -13], [20, 9, -7, -7, 4, -22, -7, 10, 5, -1, -2, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9075_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9075_2_a_dy();".
// 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_9075_2_a_dy(:prec:=12)
    chi := MakeCharacter_9075_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_9075_2_a_dy();".
// 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_9075_2_a_dy( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9075_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -10, 71, 148, -145, -330, 61, 220, 18, -52, -10, 4, 1]>,<7,R![-1089, 438, 7573, -484, -13299, -6282, 3233, 2248, -135, -240, -16, 8, 1]>,<13,R![-21101, -64010, 11122, 195208, 225555, 95756, 2239, -11538, -3561, -186, 93, 18, 1]>,<17,R![-66481, -348248, -554099, -228418, 143429, 146596, 30964, -7728, -4133, -380, 79, 18, 1]>,<19,R![734525, 1279690, 51211, -947592, -402597, 137110, 106375, 10358, -4470, -942, 16, 16, 1]>,<23,R![41299351, -17065710, -22549211, 3184940, 3688110, -210060, -268150, 5890, 9560, -60, -161, 0, 1]>,<37,R![1618831, -11311680, 22684779, -14616160, 1725088, 1388010, -372590, -34920, 15718, 190, -231, 0, 1]>],Snew);
    return Vf;
end function;