
// Make newform 4025.2.a.y in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4025_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_4025_2_a_y();".
// 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_4025_2_a_y();".
// 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 := [17, 58, -81, -210, 170, 244, -162, -120, 71, 26, -14, -2, 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], [-94, -241, 81, 1297, 537, -1780, -670, 966, 237, -218, -24, 17], [138, 571, -151, -2125, -159, 2367, 171, -1035, -53, 188, 5, -12], [-120, -958, -281, 2767, 1036, -2840, -822, 1277, 239, -253, -22, 18], [-114, -913, -77, 3329, 613, -3539, -575, 1551, 185, -294, -18, 20], [322, 1574, -275, -4736, -806, 4740, 950, -2067, -317, 400, 31, -28], [-508, -1751, 1056, 5468, -523, -5256, -52, 2128, 77, -376, -10, 24], [336, 1592, -785, -5561, 179, 5864, 347, -2578, -182, 488, 21, -33], [-591, -1953, 1569, 6355, -863, -6592, -71, 2852, 128, -529, -17, 35], [1028, 3824, -1414, -12451, -757, 12971, 1468, -5680, -552, 1076, 57, -73]];
    Rf_basisdens := [1, 1, 1, 29, 29, 29, 29, 29, 29, 29, 29, 29];
    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_4025_a();"
function MakeCharacter_4025_a()
    N := 4025;
    order := 1;
    char_gens := [2577, 1151, 3501];
    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_4025_a_Hecke(Kf)
    return MakeCharacter_4025_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], [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], [-1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, -1, 0, -1, 1, 0, 1, 0, 0, 0, 1, -1], [0, 0, 0, 1, -1, -1, 0, -1, 0, 0, -1, 1], [-3, 0, -1, 0, 0, -1, 0, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, -1, -1, -1, 0, 0, -2, -1, -1, -1], [-3, -2, -1, 1, 1, 1, 0, 0, 0, 1, 0, -1], [0, 1, 0, -1, -1, 0, 3, 1, -1, 0, 0, -1], [1, -2, 1, 2, -1, 0, 0, -1, 1, 0, 1, 2], [3, -1, 0, 3, -2, -1, -2, 0, 1, 0, -2, 2], [1, -1, 2, -1, 2, 2, 0, 0, 0, -2, 2, 0], [-3, 1, 1, -3, -1, 0, 1, 0, -2, -2, -2, -1], [-2, 0, 1, -2, 0, 2, -1, -2, 0, -1, 1, 1], [-1, 1, 0, -2, -1, 1, 1, 1, -2, -2, -1, -1], [3, 1, -1, 0, 2, 0, 0, 0, 1, 1, -1, 0], [-2, -1, 2, 0, 2, 0, 0, 2, 1, 0, 2, 2], [-2, -5, 1, -1, 3, 4, 2, 3, 0, -2, 2, 2], [-5, -1, -2, -1, 3, 2, 0, 1, 0, -1, -1, 0], [-1, -1, 0, 2, 0, -1, -1, 0, 3, 1, -1, 1], [0, 1, 3, 3, -1, -2, -3, -2, 0, -4, -4, 2], [0, 0, -4, 1, 1, -2, 2, 0, 2, 4, 2, -1], [-1, -1, 2, 2, 3, 1, -3, 3, 0, -1, 1, 1], [-1, 1, 0, 2, -2, 1, 0, 1, -2, -1, -2, -1], [-1, -6, -4, 1, 1, 3, 4, 4, 2, 2, 5, 2], [-1, -1, -1, 0, 0, 1, 4, 1, -3, 1, -2, -2], [2, 1, 1, -2, -4, -3, 1, -2, -2, 1, -1, -1], [2, 6, 1, -1, -3, -1, -1, -1, -3, 1, -1, 2], [-4, 1, -3, -3, 2, 0, 3, 1, -2, 0, -1, -6], [-4, -3, -4, 1, 2, 0, -1, 2, 2, 1, -2, -3], [0, -1, 3, 3, -2, -1, -6, -1, 1, -1, -3, 0], [-4, 0, -1, 4, -3, 1, 1, 1, -1, -2, -4, 1], [-8, 1, 1, 0, -3, -3, -4, -2, 1, 0, -1, 3], [-4, 1, 0, 2, -2, -2, -3, -2, 1, -4, -2, 4], [3, 0, 5, -1, 1, 1, 0, -3, -2, -1, 2, 3], [3, -2, 5, 1, 1, 0, -1, 2, 0, 0, 3, 1], [1, 6, 0, -2, -3, 0, 1, -4, -4, 0, -5, -5], [-2, 0, -2, -1, 2, 3, 3, 2, -1, -3, -1, -5], [-1, -3, 3, -3, 0, -1, 1, -1, -1, -3, -1, 3], [-4, 3, -7, -1, -2, 1, 7, 1, -1, 3, -1, -6], [9, 2, -1, 5, 0, 0, -3, -2, 2, 1, -2, -1], [-2, 0, -3, 5, -1, -3, -1, -3, 3, 3, -1, -2], [-9, -3, 3, -3, 4, 1, 1, 3, 1, 1, 3, 1], [4, 5, 4, 7, -2, -5, -5, -2, 2, 1, -1, 5], [-1, 0, 4, -1, -3, 0, 0, -3, 0, 2, 0, 2], [-4, -3, -2, -3, 7, 3, -1, 1, 1, -2, 1, 1], [-4, -2, 4, -1, 4, 0, 0, -1, -1, 0, 0, 1], [0, 5, -6, 1, -8, -4, -2, -1, 0, 0, -6, -1], [-7, 4, -1, -4, -1, 0, 1, 2, -2, -2, -4, 0], [-3, -2, -2, -1, -1, 1, 5, 3, -1, 4, 6, 2], [1, 1, -1, 0, -4, -5, -1, -2, 0, 3, -1, 0], [4, -2, 3, 3, -2, -2, -1, 0, -2, 0, -4, 0], [-5, -2, -5, -2, 1, 0, 2, 4, 2, 0, 2, -4], [4, 3, -2, 5, 0, -4, -2, -3, 2, 5, 0, 1], [-7, 7, 2, -1, -7, -2, -1, -8, 1, 0, -4, -2], [2, 1, 2, -5, 3, 3, 3, 2, -2, -2, 6, -1], [-2, 4, 0, -1, 2, -1, -3, 1, -2, 2, 0, -3], [-8, 6, -2, 6, -6, -5, -5, -2, 3, 0, -4, 2], [-3, -2, 4, -6, 4, 5, 1, 1, -1, -7, 4, 1], [-4, 2, -3, 0, -2, 4, 3, -3, -2, 4, 1, 1], [4, -1, 7, -5, 6, 2, -3, -1, -2, -2, 3, 2], [-6, 0, 0, 0, 4, 2, -3, 1, 1, -4, -3, -4], [0, -3, -1, -4, 8, 0, 0, 1, 1, 3, 7, -3], [-6, -1, -1, -2, -1, 3, 8, 2, -3, 2, 3, 3], [3, 7, -1, 1, -5, -4, -7, -6, 0, -1, -2, -1], [10, 5, 5, 6, -6, -4, -6, -5, -1, 1, -3, 5], [4, -7, 1, 2, 10, 7, 1, 2, 1, 3, 6, -1], [5, 3, -5, 8, -2, -2, -4, -2, 4, 6, 4, 2], [-2, 5, -4, 4, -5, -5, -3, -1, 6, 3, 1, 2], [-2, 8, 2, -5, -1, -1, -4, -3, 4, 2, -1, -1], [5, -1, 0, -2, 2, 3, 3, -3, -1, 4, 7, -3], [-5, -2, -9, 6, -1, 2, 1, 0, 5, 6, 3, 2], [1, 4, 2, -2, 4, 0, -2, -5, 2, 1, 4, 5], [0, 0, 2, 4, 2, -2, -4, -2, 2, -2, 0, 4], [-5, 3, 5, -2, -6, -2, -2, -2, 4, 0, -2, 6], [-5, 5, -3, -3, -7, -4, 6, -4, -1, -4, -3, -1], [-2, 3, 2, -3, 2, 2, 2, 3, -2, -1, 4, 3], [1, -7, 6, -6, 9, 5, 3, 3, -2, 0, 7, 3], [-11, -2, -1, -3, -4, 2, 1, -1, -1, -3, -1, 3], [5, -5, 4, 4, 4, -1, -2, -2, -2, 0, 2, -1], [-1, -2, 4, 3, 1, -2, 0, 3, -2, -1, 4, 2], [-6, 3, 6, -10, 6, 4, 5, 0, -6, -5, 5, -2], [-3, 5, 4, 3, -1, -3, -2, -4, 3, 4, -7, 2], [-9, -4, -2, -3, 10, 4, -2, 6, -1, -1, 8, -4], [2, 2, -6, -1, 5, 6, 1, 4, 1, 0, -1, -3], [5, -12, 3, -2, 5, 5, -1, 5, -2, -1, -1, -2], [0, -7, -1, -2, 11, -1, -1, 6, 1, -5, 5, 1], [-6, 0, 4, -3, -1, 2, 5, -1, -3, 2, 5, 3], [-4, 17, 0, -1, -8, -4, -7, -5, -2, -1, -8, -3], [12, 6, 7, -1, -7, 1, -3, -3, -3, -1, -1, 4], [2, 2, 2, -3, 2, 5, 5, 2, -3, 3, 5, 3], [-15, 3, -4, -1, 0, 4, 6, 8, 0, -5, 0, 0], [-1, 11, 2, -1, 3, -2, -4, -2, -4, -1, 3, -2], [10, -1, -2, 3, -6, -1, 3, -1, -1, 4, -4, 5], [3, 3, 1, 7, -10, -7, -7, -5, 1, 3, -5, 1], [6, 0, 0, 3, 9, 0, -4, 4, 2, 6, 6, -1], [-4, 10, -5, -1, -3, 4, 11, -2, -1, 7, 4, -4], [10, 1, 0, 7, -1, -1, -3, 0, 2, -2, -6, 1], [3, -1, -3, -5, 4, 0, 5, 7, -7, -2, -2, -1], [-1, -1, 4, 2, -10, -6, -3, -3, -1, 1, 4, 5], [-4, -5, 4, -9, 7, 5, 2, -4, 0, -5, 6, -1], [-3, -3, -2, 8, -6, -5, -2, -2, 6, -2, -4, 5], [3, -1, -9, 0, 0, 2, 8, 2, -2, 6, 0, -8], [-4, -2, -8, 2, -2, 4, -1, 2, 0, 2, 2, -2], [-11, 0, -1, 4, -1, 2, -2, 3, 4, -4, -2, 2], [-5, -7, 0, -8, -6, 1, 7, 0, -5, 0, 3, -1], [12, -13, 0, 5, 8, -1, 3, 7, 3, 4, 8, 1], [-9, 2, 1, -1, 1, 0, -1, 0, -4, -2, -1, -7], [4, 7, -6, -5, -1, 4, 7, 1, -3, -1, -4, -7], [-8, -2, -1, -2, 9, 11, 5, 6, -7, 0, 4, -3], [-6, 3, 5, 0, -3, 0, -3, -1, -3, -4, 2, 1], [-10, 3, -5, 3, 3, -3, 4, -1, 1, -2, -6, -2], [-3, 3, -12, 0, -4, -4, 0, -2, 0, 2, -9, -7], [-1, 10, -8, -1, -9, -2, -1, -4, -3, -1, -5, 2], [2, -1, 0, -6, 1, 8, 8, -5, 0, 0, 4, -2], [7, 5, -2, 3, -2, -2, -1, 1, 3, 4, -1, 2], [14, -13, 4, -1, 10, 8, 1, 4, 0, 0, 4, 3], [4, -16, 5, 0, 5, 2, -4, 3, 2, -1, 4, 5], [5, 1, 4, -1, -7, -5, -8, -6, -1, 0, -9, -6], [-5, 0, 0, -1, -1, 1, 6, -2, 4, 5, 5, 0], [3, 5, -6, -1, 4, 0, 1, -1, -7, -2, -7, -10], [-7, -1, -2, -3, 13, 3, -2, 6, -4, 3, 10, -8], [13, 4, -6, -1, -7, -7, 0, -4, 2, 5, 1, 4], [-4, 14, -10, 9, -13, -9, 0, -7, 2, 7, -7, -5], [-10, -16, -4, -3, 10, 2, 3, 7, 8, 6, 11, -1], [3, -3, 4, 8, -2, -5, -10, -3, 0, -5, -8, 5], [-15, 4, -2, -6, -6, -2, 1, 2, 1, -3, -3, -3], [9, 3, 4, -3, 0, -4, 0, -6, -4, 1, -4, -8], [-8, -3, -3, -3, 10, 9, 0, 1, -4, -2, 0, -6], [-6, 9, -3, 4, -10, -4, -2, -8, 3, 3, -13, -1], [3, -6, 13, 11, -2, -2, -9, -2, 1, -4, -5, 5], [9, -8, -14, 11, 3, 0, 4, 2, 9, 9, 3, 0], [-4, 3, 2, -6, -6, 0, 4, 1, -2, -4, -7, 0], [-6, -13, 0, -8, 22, 14, 6, 15, -4, -2, 11, -6], [-2, -6, 8, 2, 0, -2, -6, 0, 6, -2, 8, 10], [-7, 8, -4, 2, -8, -6, 4, 3, -2, 1, -8, 1], [-13, -7, -6, -10, 8, 5, 7, 8, 1, -4, 7, -3], [-3, -8, -3, -9, -1, 4, 1, -4, -1, 4, 8, -3], [3, 3, -3, -2, 1, 2, 9, 2, 3, -2, 4, 0], [-4, -9, 0, -9, 9, 5, -3, 2, 2, -6, 0, 3], [6, 5, 1, -7, 3, 1, -8, 4, -1, 4, 4, 0], [2, 2, -4, 2, 0, -1, -9, -4, 7, 3, 4, 0], [9, 2, 6, 6, -7, -1, -3, 2, -4, -2, -6, 1], [-2, 3, 3, -9, 1, -2, 5, 7, -10, 0, 8, -6], [14, -15, -1, 1, 7, 10, 10, 5, 0, 2, 6, -2], [-18, 9, -3, 1, -8, -3, 4, 1, 7, 5, 1, -2], [11, -10, 11, -1, 14, 6, 1, 0, -2, -3, 4, 3], [4, 11, 1, -2, -1, -5, 1, -6, -1, -1, -7, 3], [1, -9, 5, -4, 2, 1, 15, 6, -6, -7, 1, 0], [6, -5, -7, 6, -4, 8, 0, -1, 1, 3, -5, 3], [2, -2, 15, 2, -2, 4, -10, -5, -2, -4, -5, 9], [-6, -9, 0, 8, -7, -6, -2, 0, 3, 3, 1, 4], [1, 1, -1, 12, -13, -4, -9, -5, 5, 4, -10, 2], [-1, 6, -1, 8, -7, -2, -1, 0, -5, 0, -7, 0], [0, 1, -2, 3, -6, 0, -2, -3, 4, 7, 10, 1], [-3, 5, -2, -4, 0, -4, -5, -5, -3, -3, -12, -5], [14, 7, 2, 9, -2, -6, -7, -7, 8, 5, 4, 7], [7, -6, 1, 5, 4, -10, -9, 4, 6, -5, -4, 3], [-10, 7, -5, -10, 4, -3, -1, 2, 0, 3, 5, -7], [-10, 2, -3, -1, -2, 2, 10, 4, -7, 3, -5, -4], [-7, 2, 0, -12, 11, 13, 7, 8, -6, -6, 8, -3], [-7, -14, -16, 4, 5, 9, 5, 7, 12, 15, 14, -3], [10, -15, 0, 3, 9, -1, 4, 10, 4, 10, 8, -5], [-13, -1, 7, -6, 5, 4, 3, -4, -3, -2, 10, 4], [-4, -6, -6, 1, 4, 2, 5, 5, -2, 4, 7, -5], [-1, -11, 0, 1, -2, 1, -1, -3, 6, 4, 9, -2], [13, -16, -1, 0, 11, 4, 7, 5, 6, 10, 10, -2], [18, -14, -1, -2, 8, 0, 8, 7, -5, 5, 6, -5], [-9, -16, -8, -7, 19, 3, -1, 4, 6, 4, 7, -10], [-6, 13, 1, -1, -5, 3, 8, -5, -1, -2, -2, -4], [-4, -6, 9, -2, 2, 7, -3, -3, 8, 0, 10, 11], [-16, -5, -12, 10, -6, -5, 2, 4, 5, 8, -5, -4], [-4, 3, -13, -7, -5, 2, 14, 5, -5, 1, 3, -4], [-13, 0, 2, -4, -2, 1, 3, -4, -4, -6, -5, -9], [25, 2, 11, -10, 1, 4, -5, -2, -2, -2, 2, 4], [3, -7, -2, 3, 3, -5, 5, 2, 1, 7, 5, 0], [-4, 9, 3, -1, -1, 4, 1, 0, -4, -8, -10, -2], [5, -8, 7, -7, 13, 5, -10, 7, 2, -3, 4, -3], [11, -4, 4, 3, 5, 5, 1, 5, -5, -6, 0, 4], [-1, 5, 0, 7, 6, 0, -1, 3, -1, 0, -5, 0], [11, -11, 7, 0, -2, 6, 3, 2, -4, -2, 0, -2], [0, -4, 18, -6, 9, 2, -4, 1, 1, -6, 12, 8], [4, 11, 1, -2, -9, -10, 0, 2, -2, 1, 1, -9], [3, 6, -6, 2, -7, -3, -4, 1, -1, 0, 1, 9], [2, -6, 8, -4, -2, -1, -5, -8, 5, 2, -2, 2], [-3, 6, 7, 0, -9, 0, 1, -2, -2, 0, -10, 2], [-6, 2, -4, -1, 8, 2, -13, -1, 2, -4, 1, 3], [21, -2, 8, 0, -1, -4, -8, -8, 2, 4, 9, 5], [2, -5, -11, -8, 3, 3, 7, 0, 1, 7, 3, -9], [-5, 3, 10, -5, -1, -4, 4, -1, -8, -5, 5, 0], [12, 6, 8, -6, -2, -3, 3, -4, 3, 3, 4, -6], [3, 2, -4, -3, -2, -5, -10, -5, -3, -1, -13, -14], [-11, 2, -4, -6, -2, 2, 7, 8, -3, -3, 1, 3], [-6, 10, -5, 1, -10, -6, -11, 0, -1, 5, -11, -10], [-4, -12, -5, 0, -8, -1, 7, 9, -8, -8, -6, -5], [-2, 8, 2, 5, -21, -10, -8, -10, 4, 0, -8, 7], [16, -17, 3, -2, 5, 2, -3, 8, -2, -6, 3, 5], [-7, -2, -1, 3, -1, -4, 0, 1, -1, 8, 4, -7], [-11, 9, -3, 6, -1, -3, -10, -5, 2, -6, -18, -2], [-3, 8, 0, -4, -1, -4, -10, 2, 0, -2, -5, -9], [0, -24, -2, -7, 26, 10, 7, 10, 6, 7, 13, -3], [0, -1, -7, 7, -7, -14, -4, -3, 13, 7, -5, 6], [2, -3, 3, 9, -5, -8, -12, 1, 0, 4, -4, 2], [-15, 4, -4, -4, 3, 9, -3, 4, -8, 0, 8, -7], [2, -2, -3, -5, -3, 1, -7, -1, 1, -1, -5, -12], [5, -12, 0, 2, -2, 1, -9, 4, -2, 0, -5, -3], [-10, 1, 1, -3, 4, -3, -2, 7, -5, 5, 7, -4], [15, -6, 7, -3, 14, 6, -1, 0, -4, -11, 0, 5], [-20, 1, 5, -4, 6, 8, -4, 0, -4, -10, -6, 3], [8, 0, 5, -7, 8, 9, -6, -8, -8, -9, 1, 0], [-20, -5, -18, -2, -8, -1, 11, 5, 3, 6, 3, -8], [-3, 23, -1, 2, 5, -4, 1, 1, 3, -1, 4, 0], [-4, 13, 4, 2, -20, 0, 2, -1, -4, -10, -7, 4], [-1, 3, -4, -7, -8, -8, -9, -7, 5, -8, 1, 2], [-1, 6, -2, -15, 3, 5, 5, 0, -2, -8, -3, -4], [2, -3, -12, -10, 11, 0, 8, 8, 1, 3, 11, -6], [7, -6, -5, 3, 5, 1, -6, 3, 4, -3, -2, 3], [17, -13, 7, 7, 19, 7, -6, 10, 2, 1, 7, 9], [-24, 0, -7, -1, 11, 8, 3, 5, 6, -2, 11, -2], [-5, -14, -3, 8, 19, 6, -11, 12, 6, 0, 10, 0], [13, -23, 5, 14, 10, 4, -10, 10, 9, -1, 7, 10], [4, -5, 10, -12, 11, 2, 2, -1, -10, 0, 8, -2], [-15, -4, -8, -6, 4, 1, 13, 10, 0, 8, -1, -9], [-12, -6, -4, 6, 2, 2, -8, 4, 6, 0, -6, 2], [-8, 9, -3, 10, -11, -3, -12, -8, 7, 8, 1, 11], [-1, 7, 2, 2, -10, -14, -3, -7, 5, 11, -8, 1], [-5, 5, 16, 2, -10, 1, 1, -8, -3, -12, -15, 11], [-3, 15, -1, 12, -7, -11, -10, -14, 0, 3, -18, -2], [-1, 13, 6, -1, -5, -13, -7, -12, 7, -2, 1, 8], [-7, 9, 13, 10, -17, -12, -5, -7, -3, -2, -2, 10], [14, -2, 9, -14, -1, 11, 12, -4, -4, 6, 15, -1], [-15, 2, -8, 1, -3, 1, 2, 2, 2, 9, 9, 4], [25, -3, 0, 6, 2, 6, -5, 3, -3, 3, 0, 7], [-7, -15, -1, -2, 14, 1, 6, 5, 7, 8, 8, -10], [-1, 7, 18, -10, 4, 9, 5, -3, -9, -17, 3, 1], [4, 0, -11, -1, 3, 8, 7, -4, -3, -3, 4, -6], [2, 13, -7, -3, -12, 1, 14, 1, -11, 1, -9, -10], [-4, 7, 8, 12, -20, -8, -14, -5, 6, -9, -11, 10], [13, 0, 12, 10, -3, 1, -4, -1, 6, 6, 12, 17], [0, 5, 17, 10, -1, -6, -5, -6, 4, 2, -1, 7], [14, 11, -1, 6, -15, -1, -2, -16, 1, -4, -15, 3], [-18, 4, 3, -3, 18, 6, -2, 6, 8, 6, 8, -2], [13, 0, -6, -4, 7, 10, 5, 1, -7, -1, -6, -15], [-2, 1, -17, 4, -1, 12, 9, 0, 4, 6, 9, -3], [-5, -7, 5, 14, -8, -7, -8, -1, 1, -8, -14, 4], [-13, -30, 8, -1, 24, 6, 3, 4, 4, 2, 11, 2], [-2, 2, -2, -4, -2, -2, 2, -2, 10, 5, 12, 16], [3, -1, 9, 1, 0, -4, -5, 3, -7, -10, 0, 1], [-7, 9, 4, 10, -9, -5, -19, -7, 6, -4, -3, 17], [-16, 1, -10, -4, -1, 0, 8, 6, -1, -10, -9, -4], [-24, 5, -3, -15, -3, -2, 16, 1, -9, -5, -3, 0], [24, 4, 15, 7, -8, -2, -2, -4, -8, -4, -6, 4], [-3, 15, -6, -13, 14, 6, 4, -2, -8, -9, 8, -4], [-1, 8, 20, -5, 3, -3, -14, -8, -10, -21, -15, 4], [-12, -30, -6, -14, 20, 19, 13, 18, -1, -2, 14, -2], [-2, -14, -14, -1, 8, 10, 2, 1, 6, 8, 5, -5], [-8, 1, 9, 1, 1, -2, -8, -9, 8, 4, 0, 0], [0, 10, 5, 1, -1, -1, 4, -9, -5, 3, -5, -4], [-4, 14, 5, -1, 6, 1, -6, -4, 0, -1, -1, -2], [5, 19, -6, -10, 5, 8, -2, -3, -3, 0, 3, -1], [-21, 1, 6, 1, -7, -4, -1, -3, -5, -9, 0, 8], [-8, 12, 1, 3, -20, -12, 2, -18, 4, -8, -18, 2], [-26, 17, -5, -12, -13, 0, 0, -10, 0, -3, -13, -7], [12, -24, -1, 7, -1, 0, 8, 6, 9, 10, 4, 2], [5, 9, -1, 1, 2, 7, 14, 0, -4, 2, -2, 1], [-3, 17, -5, -12, -12, -2, -6, -6, -1, -4, -13, -2], [-8, 20, -3, 3, -4, -4, 0, -4, -11, -7, -13, -8], [-19, 0, -16, 3, -5, -14, 6, 2, 7, 3, -11, -2], [2, -13, -5, -9, 18, 13, 14, 11, -14, 5, 6, -4], [-9, -2, -8, -6, 4, 10, 0, -5, -4, -7, -8, -5], [-16, 0, 3, -10, 8, 6, -6, 5, 3, -3, 10, 1], [-22, 3, -11, -11, 1, 2, 14, 3, 8, 10, 6, -2], [-32, 5, -9, -22, 11, 18, 19, 14, -6, -6, 7, -11], [18, -14, -4, 10, 11, 7, 3, 19, 2, 10, 20, 6], [-13, -9, -3, 4, 1, 4, 3, 4, 1, 10, 4, 2], [6, -17, -4, 4, 21, 2, 11, 1, 10, 13, 14, -4], [-8, -19, -12, -6, 10, 4, 11, 4, 5, 13, 12, -10], [-20, 16, -2, 0, -28, -20, -9, -14, 4, 0, -12, 0], [-18, 7, -15, -14, 1, 0, 5, -4, -3, 12, 10, -11], [5, 5, 22, -9, -3, 4, 8, 4, -16, -21, -7, 6], [0, 5, -6, 8, -2, -6, -3, 2, 1, 7, 2, 10], [-7, 4, 8, -1, 1, -6, 4, 7, -4, -4, -6, -4], [6, 14, 2, 0, -12, -6, -8, -8, 2, 12, -2, -6], [12, -6, -7, -4, 2, -1, -1, -3, -6, -6, -14, -7], [5, 3, 2, 4, -8, -3, 15, -7, -3, -4, -11, -3], [-27, 13, -2, 7, -17, -5, -2, -2, 6, -3, -8, 12], [6, -24, 9, 11, 2, -4, -2, -7, 4, 3, 0, 4], [-2, -6, 5, 4, 8, -1, -20, 1, 15, -5, 3, 9], [23, 2, 14, 9, 1, 4, 1, 0, -7, -5, -5, 8], [6, 18, -16, -14, -7, -1, 13, -3, -14, 4, -8, -16], [-11, 11, -12, 4, -13, 1, 3, -7, 10, 4, 3, 3], [8, -2, -13, -7, -3, 3, 21, 3, -5, 9, 1, -8], [14, -9, 8, -8, 10, 12, 19, 4, -5, 3, 18, 0], [-22, 13, 7, 0, -17, -16, -10, -10, -4, 3, -9, 1], [4, 6, -16, 3, 0, -12, 4, -5, 10, 20, 13, -1], [-3, -18, 10, 0, -5, -3, -8, -9, 0, 8, 2, 1], [-17, -14, 5, 9, 13, 0, -6, 10, 6, -8, 11, 7], [-21, 28, -14, -12, -20, -9, 6, -12, -4, -5, -17, -3], [-11, 22, -9, -7, -26, -8, 1, -18, -3, -3, -13, -1], [-16, -1, 3, -6, 7, 0, -9, 8, 5, -4, 14, 1], [-23, -2, -1, 9, -4, 10, 15, 4, 0, -1, 2, 7], [-4, -10, 5, 1, 1, 7, 2, 4, -1, -4, 3, -4], [29, 23, -4, 1, -11, -5, 2, -12, -3, 4, -9, -2], [-4, -16, -8, 12, 19, 5, 5, -1, 10, 6, 8, 2], [20, 17, 11, 7, -11, -10, -6, -15, -3, -7, -21, -8], [-4, -23, 1, -2, 18, 4, -6, 14, -1, 3, 5, -9], [9, -3, 9, 10, 6, -4, -3, 1, 8, 9, 14, 6], [17, -8, 1, 1, -9, -4, -6, -2, -4, -8, -7, 9], [9, -3, -11, 12, 11, -1, -19, 1, 1, 4, 1, 2], [5, -1, -19, 9, -15, 8, 12, -4, 7, 8, -13, -7], [0, 2, -1, 1, 10, 14, -2, 4, 5, -11, 7, 6], [-13, 4, -15, 16, -3, -12, -2, 0, 6, 6, 0, -8], [-15, 4, 19, 7, -6, 3, -2, -1, -4, -6, -3, 3], [17, -24, 2, 10, 17, 3, 0, 11, 14, 16, 14, 1], [-7, 0, 3, 2, 11, -2, -5, -2, 4, 0, 4, -4], [-9, -11, 2, -8, 14, 2, 2, 6, 0, 5, 5, -1], [-2, -1, 8, -12, -16, -2, -2, -6, -7, -4, 2, 2], [-4, -6, 4, -8, -10, -6, 5, -1, -3, 13, 11, 6], [-6, -4, -1, 0, 7, -5, -5, 4, -1, -3, -2, -7], [4, -5, -11, 6, 12, -1, 5, -2, 9, 3, 0, 1], [-1, -1, -5, 2, 4, -2, 3, 7, 2, -3, -4, -4], [-13, 9, 12, 0, 8, -1, 1, 7, -3, -4, 11, -3], [-8, 4, 0, -11, 4, 8, 1, 2, 6, -5, -1, -1], [-9, -9, -2, -6, 11, -1, 13, 12, -6, 9, 3, -7], [15, 2, -3, 9, -3, 0, -12, 0, 6, 0, 5, 5], [7, -7, -4, -1, -8, -3, 6, 12, -13, -5, 0, -2], [7, -17, -8, 4, 4, -5, 9, 5, 11, 12, 7, -7], [5, -3, 5, 17, -3, -9, -6, 8, -2, 1, -11, -3], [-4, 8, -1, -3, -6, 8, -6, 0, 3, -3, -7, 2], [-17, 20, 9, 1, -15, -2, -1, -10, -3, -10, -8, 5], [22, 13, -1, -2, -19, -8, 15, -10, -10, 0, -7, -7], [-7, -21, 12, -7, 24, 8, -1, 21, -4, -8, 12, -6], [23, 12, 7, 1, -19, -18, 2, -6, -6, 4, -1, -5], [25, -29, 7, 11, 4, -2, -9, 14, -5, -5, 2, -3], [-12, -14, -4, 13, -5, -6, -9, -7, 5, 2, -13, -7], [8, 13, -9, 0, -24, -10, 12, -6, 2, 6, -8, -9], [7, 12, 2, -8, -16, 2, -3, -6, -13, -3, -13, 3], [7, -10, -19, -1, -1, 1, 8, 7, -2, 1, 10, -9], [3, 21, 4, 0, -19, -9, -1, -16, -10, -15, -25, -3], [3, 9, 9, -3, -13, -12, -4, 4, 3, -11, -9, -3], [-33, 0, -6, -4, -2, 3, 0, 5, -4, 0, 5, -9], [7, -14, 6, 2, 10, 7, 17, 4, -1, -7, 8, 7], [4, -11, 0, -1, 8, 1, 13, 4, 8, 1, 9, 1], [-3, 8, -2, 11, -7, 3, -9, -14, 8, 8, -9, 8], [11, 7, -3, 4, 0, -2, 3, -3, 0, 9, 16, -4], [-4, 5, -4, 7, -2, 0, 14, 4, 1, 6, 1, -1], [-3, -16, 12, -3, 7, 14, 4, 3, 2, -7, 6, 14], [-14, 19, -14, 6, -20, -17, -8, -14, 5, 8, -21, -8], [8, 15, -4, 5, -26, -18, -2, -13, 2, 6, -16, 1], [-10, 11, -4, -1, -12, -17, -13, -6, 0, -5, -13, -3], [-17, -4, 0, -10, -4, 7, 22, 12, -14, 7, 1, -7], [7, -3, 12, -1, -5, 1, 4, -3, -4, 9, 16, 16], [-11, -2, 5, 12, -1, -2, -16, 3, 6, -6, -10, 2], [-10, -11, -7, -14, -3, 0, 6, 0, -2, 7, 7, -3], [-8, -10, -8, 1, 8, 10, 10, 9, 9, 10, 18, 7], [-8, 17, 4, 5, -18, -6, -6, -11, 0, -2, -2, 3], [5, 1, 27, -2, 9, -5, -10, -2, -10, -19, -4, 10], [2, -12, 1, -18, 4, 11, -5, 1, 8, 11, 24, -7], [19, -4, 8, -3, -17, -5, -13, -12, -2, 2, -15, 4], [-20, 11, -2, -3, -17, -11, 9, -2, -2, 0, -12, 11], [17, -1, -5, 0, 17, 10, 3, 3, 11, 18, 6, -2], [-8, -1, -16, 2, 5, 6, 4, 3, -4, 4, -12, -22], [14, -29, -14, 3, 23, 9, 9, 16, 12, 24, 26, -11], [37, 5, -2, 20, 14, 0, -19, -1, 14, 4, 5, 7], [13, 0, -3, 6, -7, 0, 4, 0, 8, 5, -6, -4], [22, -2, 1, 8, -1, -9, -11, 7, -5, -4, -8, 5], [13, -15, -10, 15, 18, 4, -2, 6, 16, 21, 24, 0], [-10, 4, 0, 3, 0, -9, 2, 7, -7, -9, -1, 11], [18, 10, 7, 11, -7, -5, -3, 1, 1, -7, -19, 4], [17, 13, 13, 7, -15, -8, -24, -6, -3, -6, -3, 15], [0, -9, -11, -1, 11, 12, 12, 3, 11, 7, 1, -6], [2, -7, 12, 7, 0, -10, 0, 1, 0, -2, 10, 3], [-4, -7, 8, 0, 6, 15, 8, -2, 5, -8, 7, 2], [10, -18, 5, -10, 18, 14, 8, 17, -13, -7, 0, -7], [-6, -11, 28, -13, 21, 7, -1, 1, -11, -10, 11, 3], [18, -25, -5, 4, 23, 22, 17, 23, -3, 12, 20, 3], [12, -16, -16, 13, 6, -8, -16, 5, 14, 10, 9, 7], [9, 24, 13, -12, -6, -5, -8, -12, -5, -16, 3, 0], [3, 7, 6, 6, -5, 1, 1, 4, 0, 7, 11, 1], [-23, 8, -16, -14, -1, -1, 34, 9, 5, 12, 5, -15], [21, 2, 9, -15, 4, 0, -3, -4, 2, 5, 2, -5], [11, 3, 10, 23, -13, -9, -23, -8, 11, -2, -5, 24], [-16, -32, -10, -12, 16, 10, 19, 14, 6, -8, 20, 8], [-22, -9, -10, 0, 16, 8, 14, 14, -1, -4, 4, -8], [-7, 29, -11, -12, -18, -14, -3, -4, -14, -10, -26, -12], [29, -22, 8, 1, 15, -3, 4, 6, 8, 4, 7, 4], [2, -6, 2, -8, 4, -13, 8, 11, -4, -2, 15, -6], [-5, 25, 14, 6, -26, -18, -30, -14, -8, -6, -23, 3], [-16, 4, -2, -13, 12, -6, 3, 3, 4, 0, 21, 5], [3, 25, -16, 3, -14, -2, 2, -8, 2, -1, -6, 4], [-7, -27, 4, -15, 20, 18, 19, 2, -10, -4, 12, -4], [0, -4, -5, 0, 1, 4, -1, -3, -9, -10, -15, -13], [25, -4, -5, 3, 6, -8, -13, -6, 5, 1, -1, 5], [-31, 13, 0, 5, -11, 1, 6, 3, 0, -7, -10, 8], [2, 2, -7, -9, 13, 6, 19, 3, -8, 8, 15, 6], [22, -21, 11, -4, 15, 12, -1, 4, 11, 2, 6, 17], [1, 28, 5, 3, -9, -17, -18, -18, 7, -5, -13, -1], [21, 10, 5, 2, -5, -6, -4, -15, 3, 14, 5, -6], [4, 10, -19, -11, -8, 6, 18, -2, -3, 7, 5, -6], [5, 7, -12, 5, 2, -5, 2, 0, 5, 7, 14, -6], [-23, 8, 4, -12, -10, -12, 4, 3, -4, 5, 4, -5], [-20, 11, -18, -15, -4, 0, -8, 5, -10, -7, -4, 5], [-26, -9, 0, 10, 14, 3, -8, 0, 11, -4, 3, 12], [5, -13, -8, 4, 1, -1, 11, -2, 0, -5, 7, -1], [19, -36, -3, 10, 27, 8, 13, 18, 10, 18, 26, 0], [-1, 5, -4, 3, 1, -4, 0, -2, -2, 7, 13, -8], [19, 3, -8, 11, 1, -2, -5, 0, 3, 8, -10, -16], [-8, -20, 0, 3, 0, 2, 8, 3, 0, 4, -13, 7], [-4, -7, 28, -11, 10, 6, -2, 3, -14, -14, 16, 7], [11, 20, -4, 15, -30, -25, 0, -17, 11, 10, -13, 0], [-3, 6, -2, 7, -7, -6, 4, 14, 8, 1, -4, 18], [14, 8, -3, 12, -7, 1, -5, -6, 7, -12, -18, -5], [26, -7, -14, 1, 2, -2, 2, -5, -2, -5, -2, -9], [-8, 14, 0, 4, 2, 2, -10, -8, 4, -8, -14, -4], [26, -11, 8, 16, 1, -4, -11, -3, 16, 10, 10, 22], [-23, -17, -2, -2, 12, 3, 2, 5, 10, -3, 14, -7], [15, 15, -4, -8, 2, 1, -10, 5, -8, -5, -16, -5], [1, 6, 12, 18, -8, -18, -10, -15, 2, -8, -20, 7], [4, -7, 6, -25, 35, 16, 13, 10, 3, 4, 26, -11], [-11, -21, 15, 8, 29, 9, 2, 10, 6, 3, 34, 16], [-25, 3, -12, -9, 3, 13, 6, -6, 2, -3, 14, 4], [-23, 12, -25, -2, -13, 2, 5, -3, -6, 6, -10, -16], [8, 6, -5, 1, 14, 2, 3, 10, -4, 4, 12, 2], [-6, -9, -19, 11, 14, 11, 8, 1, 8, 0, -6, 6], [13, -6, -5, 1, -17, 3, -6, -8, 11, 5, -17, 7], [-23, 22, 6, -8, 15, -1, -8, -3, 0, 2, -2, -1], [14, 5, 14, 6, -18, -3, 8, -10, -1, -4, -7, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4025_2_a_y();".
// 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_4025_2_a_y(:prec:=12)
    chi := MakeCharacter_4025_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_4025_2_a_y();".
// 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_4025_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![17, 58, -81, -210, 170, 244, -162, -120, 71, 26, -14, -2, 1]>,<3,R![-8, -60, 75, 672, 669, -372, -581, 68, 174, -4, -22, 0, 1]>,<11,R![-7392, -28480, 21040, 45752, -6026, -19422, 3, 3456, 188, -274, -25, 8, 1]>],Snew);
    return Vf;
end function;