
// Make newform 4025.2.a.bb in Magma, downloaded from the LMFDB on 16 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_bb();".
// 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_bb();".
// 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, -3, 386, 193, -1429, -443, 1581, 346, -772, -118, 187, 18, -22, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1719962, -4823059, -22002987, 8180960, 40706850, -5051683, -25174921, 1395439, 6807185, -178425, -833971, 9431, 37911, -134], [208671, -8567221, -7553195, 34327336, 4401504, -35751429, 2054981, 15312112, -1722211, -3046784, 349093, 273594, -22078, -8617], [-136436, -1764856, 746869, 3436136, -3337746, -1876665, 3684384, 41160, -1485425, 208975, 246032, -50538, -14217, 3427], [-670, 476687, -4771335, -21977125, 7989474, 40647488, -4839829, -25128557, 1291991, 6791373, -153367, -831559, 6483, 37777], [-1259865, -9402664, 5168560, 27355117, 449829, -25230657, -5168881, 11524948, 2649746, -2836277, -477956, 353631, 28793, -17266], [3221839, 22678219, 12237873, -72909281, -36175764, 89700733, 29999800, -48372151, -10583948, 12628452, 1647493, -1571588, -92562, 74561], [4568996, 6952400, -3662787, -61333048, -29107302, 87794291, 33005087, -51178406, -12965704, 14235777, 2127977, -1867552, -123405, 92428], [1997443, -23191094, -42632694, 158892526, 81640410, -199755620, -53798933, 99159767, 15754684, -23419830, -2117591, 2641672, 106461, -114541], [1835824, -9671628, -21439823, 53938016, 41000631, -70995548, -30423831, 39148617, 10248993, -10454477, -1563334, 1331797, 87137, -64550], [-2790322, -40794727, -12281418, 175750652, 69984327, -221496586, -72669595, 121398712, 28618406, -32522883, -4742359, 4169738, 277101, -203639]];
    Rf_basisdens := [1, 1, 1, 1, 1242873, 414291, 138097, 1242873, 414291, 1242873, 1242873, 1242873, 414291, 1242873];
    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, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, -1, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0], [1, 1, 0, -1, 0, 0, 0, 1, 0, -2, 0, 0, 1, -1], [0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, -1, -1, 0, 0, 0, -1, 0], [0, 2, -1, -1, 1, -1, 0, 2, 0, -2, 0, -1, 1, 0], [1, -1, 0, 0, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0], [0, 0, -1, 0, 0, 1, -1, 0, -1, -1, 0, 0, 0, 0], [3, 2, -1, -2, -3, 1, 1, 1, 1, -2, 0, 1, 0, -1], [1, 0, 1, -1, 0, -1, 0, 1, -2, -1, -1, 0, 0, -1], [1, -1, 1, 2, 1, -2, -1, 0, 0, 1, -1, -1, 0, 0], [-1, 2, 0, 1, 0, -1, 0, -1, -1, 0, 1, 1, -1, 0], [1, 2, -2, 0, 0, 0, 2, 1, 2, -1, 0, 2, 0, 0], [4, -1, 0, 1, 0, 0, 0, 0, 0, 1, -1, -1, 0, 1], [-1, 4, 0, -2, 1, -1, 1, 2, 0, -2, 0, -1, 3, -1], [1, 2, 0, 1, 2, -2, -1, 1, -1, 0, -1, -2, 2, 0], [-2, 0, 0, 0, 2, 0, 0, -2, -2, 0, 0, -1, 0, 0], [1, 1, -1, 0, -1, 1, 2, -1, 0, 0, 2, 2, -1, 1], [-3, 2, 1, 0, 1, -2, 0, 2, -1, -1, 0, 0, 2, 0], [1, 1, -1, 0, 1, -2, 1, 2, 0, -1, 0, 0, 3, 0], [0, 2, -2, 0, 0, 0, 2, 3, 2, -1, 0, 0, 2, 1], [3, -2, 0, 1, 2, 2, -1, -3, 1, 2, 1, 1, -1, 3], [0, -5, -1, 3, -1, 3, -1, -4, 1, 4, 1, 0, -2, 2], [3, 1, 1, 0, -3, 1, 0, -1, 0, 2, 1, 2, -2, -1], [1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 0, 0, -1, 0], [3, 1, -2, -2, -2, 0, 1, -1, -1, -1, 1, 1, 1, -1], [-2, 3, 2, 0, 1, -1, -2, 1, -3, 1, 1, -3, 0, 2], [2, -1, -1, 0, -3, -1, 2, -2, 0, 1, 2, 1, -1, 0], [-2, -1, 1, 1, 0, 0, 0, -1, 1, 1, 1, 1, 1, 0], [-3, 0, -1, 1, 1, 2, 1, -2, 2, 2, 2, 0, 0, 2], [2, 1, -1, -1, 0, 0, 0, 3, -1, -1, 1, 0, 1, 0], [4, 3, -2, 0, -2, -1, -2, 1, -1, -3, 1, 0, 2, -1], [2, 0, 1, 2, 1, 1, -3, -2, -1, 3, -1, -2, -2, 1], [2, -2, -1, -2, 3, -1, 1, -1, -1, 0, -1, -1, -1, 0], [5, 1, -1, 0, -3, 2, -1, -1, 0, 2, 2, 2, -1, -1], [4, 3, 1, 1, -1, -2, 0, 3, 1, -1, -3, 0, -1, 0], [1, -4, -1, 0, -3, 2, 0, 0, 1, 1, -1, -1, -1, -1], [1, -1, -1, 0, -3, 2, 1, -2, 2, 1, 2, 2, 1, 1], [2, 1, -3, 0, -2, 5, 2, -2, 2, -1, 2, 2, -3, 1], [-7, -3, -1, 1, 4, -1, -1, -3, -1, 3, 0, -2, -1, 1], [-5, -2, -2, 1, 2, 1, -2, -2, -1, 5, 1, -2, 1, 3], [6, 0, 3, 1, -3, 0, 1, 0, 0, 0, -1, 2, -3, 1], [9, -4, 2, 3, 2, -2, -1, 0, -1, 3, -3, -2, -1, -1], [-3, 5, -1, -3, 1, 1, 3, 1, 1, -1, 2, 0, 1, 1], [1, -8, -3, 0, -1, 3, 1, -1, 3, 2, -2, 0, -2, 1], [3, 4, -1, -2, -2, -2, 0, 4, -1, -3, -2, 0, 1, -4], [4, -5, 3, 1, 0, 2, -2, 1, 1, 1, -1, 1, 0, 2], [-4, -2, 0, 3, 0, 0, -1, -2, -1, 3, 2, -1, -2, 0], [0, -1, 3, -1, -3, -3, 1, 2, 1, 0, -1, -1, 2, -1], [-2, 1, 1, 0, -2, 0, -1, 3, -2, -2, -1, -2, 2, 0], [-1, -6, 1, 1, -1, 2, -3, 0, 0, 4, 1, 0, 1, 3], [1, 5, -3, -2, 0, -1, 2, 1, -2, -2, 2, -1, 2, -1], [1, 0, 4, -2, -4, -2, 2, 4, 0, -4, -1, 1, 1, -3], [3, -7, -2, 2, -3, 2, 1, 1, 3, -1, 0, 1, 1, 0], [-1, 4, -4, 0, -2, 2, 2, -1, 0, -1, 0, 3, -4, 1], [-2, 2, -2, -2, 0, 2, 2, 0, 0, -2, 4, 3, 0, 1], [-1, -1, -1, -2, -3, -1, 0, 3, -2, -4, 2, -1, 3, -2], [7, -1, 0, 0, 0, -3, 2, 2, 3, 0, -1, 2, 0, 1], [3, -3, -4, 1, -2, 1, -1, 0, 0, 1, -2, 1, -2, 0], [-3, 7, -1, -2, 3, -5, 0, 5, -2, -4, -2, -5, 3, -1], [-2, 6, 2, -1, -2, -2, 3, -2, -1, 3, 0, 1, -2, 0], [8, 0, -1, 2, -6, -2, 0, 1, -1, -4, 0, 0, -2, -2], [-3, -4, 1, 3, 7, -2, -3, 2, 0, 4, -1, -3, 1, 3], [4, -3, 4, 2, -2, -2, -5, -1, -3, 3, -1, -2, 0, 2], [-1, 1, -1, -5, 4, 1, 1, -1, 1, -3, 3, -2, 1, 0], [-3, -4, -1, 4, 2, 1, 1, -5, 3, 4, 3, 1, -1, 2], [2, -1, 3, -1, -3, -2, 0, 3, -3, -3, -3, -2, 2, -6], [13, 6, -2, -1, -6, -1, 2, 2, 1, -5, 3, 0, 1, -2], [4, -1, -1, 3, -1, 1, -1, -2, 5, 0, 2, 3, 1, 1], [2, -1, 1, -1, -3, -1, 1, 2, -3, -2, -4, 1, 1, -3], [14, -5, -2, 3, 0, 0, 2, -2, 4, 1, 1, 2, 0, 2], [1, 3, 3, -4, -1, 2, 3, 0, -2, -3, 1, 0, -4, -1], [2, 0, 3, -3, 1, 0, -1, 1, -2, -1, -2, 0, 1, -2], [-7, -6, -2, -1, -2, 1, -2, -2, -1, 3, 0, -1, 0, -2], [3, 4, -4, -1, -5, 6, -1, 1, 1, -2, 4, 1, 0, 1], [-4, -7, 3, 1, 5, 0, -2, -5, -1, 3, -1, -1, 0, 4], [-4, 0, 4, -1, 0, 1, -4, 1, -1, 0, -2, -2, 0, 3], [5, 3, -2, -1, 0, 0, -2, 3, 0, -2, -3, 3, 0, -3], [8, -1, 2, -3, -10, 1, 1, 3, 0, -2, -1, 1, 0, -1], [-1, -4, 1, 1, 0, 2, -1, 0, 0, 0, -4, 3, -2, -1], [-6, -8, 4, 0, 1, 1, -5, -3, -2, 5, 1, -2, -2, 1], [5, -6, 2, 2, 0, -3, 1, 3, 2, 3, -2, 0, -2, 2], [5, -2, 3, 1, -3, 2, -3, -2, -6, 2, 1, 0, -5, 1], [3, -7, 3, -1, -2, 2, -2, -4, -1, 2, -2, 1, 0, 0], [12, 4, 0, 1, -2, 1, 0, 2, -3, 3, -3, 1, -5, 2], [5, -2, 2, 1, -6, 3, 0, 0, 1, -1, 2, 2, -2, 0], [0, 2, -6, 4, 0, 2, 0, -4, 2, 0, 3, 2, -3, 3], [14, -4, 0, -2, -3, 2, -2, 2, -2, 0, -2, 1, 1, 1], [-4, -5, 2, 5, 3, -1, 1, -3, 2, 2, -3, 2, -1, -1], [2, -11, 1, 2, 4, 2, -3, -7, -2, 4, 5, 1, -6, 2], [6, -4, -3, 3, -3, 1, 0, 0, 2, -4, -2, 2, -2, 1], [5, -3, 1, -1, -4, 2, 2, -2, 3, 0, 4, 2, -3, 1], [-12, -4, -1, -1, 1, -1, -2, -2, 0, 4, 3, 0, 1, -1], [5, 1, -2, 0, 2, 2, 1, -1, 1, -1, 4, -1, -1, 1], [-10, -1, -3, -4, 6, 1, 0, 2, 2, -3, 0, -1, 5, 0], [-5, 4, 2, -6, 3, -1, 1, 2, 0, -4, -1, -4, 4, -3], [5, 3, 0, 2, 2, -2, 3, -1, -1, 1, -3, 2, -4, -1], [-5, -4, 2, -2, 2, 2, 0, 5, 2, -3, -3, 1, 1, -2], [6, 1, -1, -3, -5, -1, -1, 2, -1, -4, 4, 0, 3, -2], [-9, -9, -2, 1, 0, 2, -2, -3, 2, 2, 3, -3, 2, 3], [3, 3, 1, -5, 8, -3, -1, 3, -1, -5, 0, -4, 3, -2], [0, 7, -5, -5, 6, -3, 3, 2, -1, -6, 1, -4, 3, -1], [-4, -2, 0, -1, -4, 2, -1, 2, -3, 1, 0, -1, 2, 2], [4, -8, -1, -1, -11, 3, 2, -3, 2, -1, 1, 5, -1, -1], [-5, 5, -4, -3, -2, -1, 3, 2, 2, -1, 0, 3, 0, -2], [6, 8, 4, 1, -4, -2, 3, 2, 1, -3, 3, 2, -1, -1], [9, -9, 6, 2, 2, -1, 0, -2, -3, 2, -6, -2, -1, 0], [2, 2, -2, 2, 4, 3, 1, -3, 0, 5, -3, 2, -5, 4], [-1, -7, 4, 4, 0, -1, -2, -4, 1, 6, 0, -3, 1, 0], [-9, -4, 8, -1, 2, -1, -2, -2, -5, 1, -2, -3, 0, 0], [0, 4, -5, 2, 3, 1, 3, -2, 1, 1, 1, -1, 1, 2], [14, -5, 2, 4, -6, 2, -1, 0, 3, 2, -3, 1, 2, 0], [3, 5, 0, -2, 2, -4, 3, 5, 1, -1, -4, -1, 6, 0], [-8, -1, 1, 1, 1, 0, -4, -1, -5, 3, -5, -3, 1, 0], [7, -8, -5, 3, -5, 7, -4, -3, -2, 5, -1, -2, -3, 2], [-6, -2, 0, 3, 8, 1, 0, -1, -3, 2, -3, -2, -1, 2], [-11, 8, 0, -5, 5, -5, 2, 4, -5, -5, 2, 0, 0, -2], [-9, 0, 1, 1, -3, 0, 3, -2, 2, -4, 1, 2, -3, -5], [-2, -2, 0, 0, -6, 2, 0, 0, -2, 4, 0, 1, -5, -1], [-13, 0, 3, 2, 1, 1, 3, -4, 3, 3, 1, -1, 1, 2], [-14, 8, -5, 1, 1, 3, -2, 2, 2, 0, 4, 2, 2, 4], [10, 15, -4, -8, 2, -6, 3, 8, -1, -12, 2, -3, 7, -5], [-2, 5, 0, -4, -2, -4, 5, 6, 5, -8, 3, -1, 6, 0], [11, -4, 0, 9, 4, -1, -4, 0, 1, 1, -1, -2, 1, 1], [8, -4, 1, 1, -5, 2, -3, 1, 0, -3, -4, 1, 0, -2], [7, -2, 1, 2, 1, -3, -3, 5, 3, 6, 0, -2, 3, 2], [6, -5, -2, 1, -6, 3, -1, -1, 2, -4, 0, 0, -3, 0], [2, 7, 2, 4, 0, -5, 4, -1, -3, 1, -2, -1, -3, -3], [0, -1, 4, -7, 6, -4, 0, 4, 0, -5, -2, -4, 5, -4], [-3, -2, 7, -4, -1, 1, -1, -5, -3, 6, -2, 0, -3, 0], [3, 10, -11, -3, -1, -2, 7, -2, 0, -4, 3, 2, -1, -1], [-7, 0, 2, 0, 4, 2, 2, -1, 2, 3, 1, 2, 1, 3], [4, 2, -5, 1, -3, 2, 3, -1, 2, 1, 1, 5, -3, -2], [4, 3, 4, -2, -2, -4, 3, 10, -1, -8, -3, 1, 4, -7], [13, -2, 0, -7, -4, 3, -2, 5, -1, -2, -2, -2, 0, -2], [-4, -8, 5, 1, 0, -1, -4, 4, -2, 2, -5, -2, 0, 3], [3, -1, 3, -5, -1, -2, 2, 3, 3, -5, -2, 0, -1, -1], [8, 4, 0, 0, 3, -4, 2, 6, 0, 2, -2, -2, 2, 4], [2, 2, -2, 1, -6, -1, 4, 1, -3, 0, -3, 0, -5, -2], [7, -8, 2, 4, 4, -1, -3, -5, -4, 3, -4, -2, -6, -2], [-15, -3, -3, 4, 3, -1, 0, -1, 0, 4, -4, -3, -1, 7], [1, 7, 3, -5, -7, -1, 1, 3, 1, -3, 2, 3, 1, 1], [-1, -8, -4, 9, 6, 4, -5, -5, 3, 10, 6, 1, -1, 8], [1, 3, 4, -3, 6, -3, -3, -4, -4, 1, -3, -2, 1, 0], [-4, -2, 9, -3, -2, 4, 3, 1, 0, -1, -2, 1, -4, -3], [2, -6, 0, 3, 2, 0, 3, -1, 5, 2, -2, 1, -4, -1], [-13, 4, 5, 0, 1, -3, -1, -3, -5, -2, 1, 0, -3, -3], [11, 0, -3, 1, 1, 2, 1, -3, 4, 1, 5, 2, 1, 1], [-12, 5, -5, -1, 6, 2, 0, -1, -1, -3, 6, 1, 1, 2], [8, -5, -2, -3, -4, 2, 2, 2, -2, 1, -5, 1, -2, -3], [4, 17, -1, -4, -3, -1, 4, 4, -2, -9, -1, 4, 0, -7], [10, 7, -4, 0, -6, 1, 8, -3, 3, 3, 6, 5, -3, -1], [3, -6, 2, 2, -8, 3, 1, -6, 0, 8, 0, 1, -3, 0], [-1, -12, 0, -1, 2, 1, 0, -2, -3, 3, 3, -1, -3, 0], [-2, -4, 3, -1, -3, 1, -2, 0, -2, 0, -2, 5, -6, 3], [-4, -6, 5, 0, 7, 0, -8, 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1, -2, -7, 8, 2, 0, -3, 2], [-27, -28, 9, 9, 17, -5, -6, -11, -8, 17, -5, -4, -6, 5], [-1, -10, 1, 0, 2, 9, -7, -3, 1, 2, 13, -2, -2, 1], [-1, -4, -2, 1, 0, -2, 1, 1, 1, 2, -7, -1, 3, 5], [-5, 18, 0, -7, -5, 1, 4, 2, 1, -5, -1, 1, 5, -8], [6, 13, -5, -11, 7, 5, -5, 10, -1, -6, 4, -5, 5, -5], [-9, 9, -6, -4, 14, 7, -2, -2, -3, 2, -1, -5, 2, 0], [-24, -2, -2, 2, -1, 2, 2, 0, 10, -4, 2, 1, 8, -3], [33, -4, -4, -5, 0, 1, 4, -9, -3, 0, -3, 3, -7, -6], [-18, 13, 2, -4, 2, -1, -6, 5, -3, 3, -5, 1, 0, 4], [-6, 7, -7, -13, -1, 7, 1, 4, 1, -12, 9, -1, -2, -5], [-7, 11, 3, -11, 1, -5, 3, 5, -5, -11, 4, -2, 11, -10], [16, -11, -9, 9, 1, -2, -4, -5, 5, 5, -2, -1, 2, 1], [-18, -21, 0, 18, 6, 2, -9, -8, -1, 22, 1, -5, -4, 10], [12, -4, 3, 4, 5, 1, -3, 6, 3, -9, -1, 0, -1, -5], [15, -12, 7, 6, -10, 2, 0, -2, 5, 3, 1, 7, 2, 4], [4, 11, -9, -3, 8, 5, -1, 0, -5, -2, 6, -1, 2, 1], [-8, 5, 6, -2, 3, 3, -4, 5, 5, 3, -6, -7, 2, 6], [-11, -7, -10, 0, 2, 11, 0, -14, 9, 0, 2, 5, 0, 4], [8, 16, 0, -1, -10, 1, 0, 7, -1, -2, 0, -3, -2, 1], [0, 6, 11, -1, -4, 1, 6, 0, 2, -2, 0, 6, -1, -1], [13, 6, -11, -5, -7, 11, 2, -9, 0, -1, 15, 3, -7, 3], [-21, 3, -6, -4, 2, 1, 2, 0, -1, -2, 2, -7, 1, -4], [8, 9, -9, 0, -7, 4, 7, -1, 6, 2, 5, 3, -1, 6], [-7, 7, 0, -5, 0, 7, -9, -4, -10, -3, 6, -2, -1, -3], [0, -8, 2, -3, -4, 7, -10, 3, -3, 0, -1, -6, 4, -2], [3, 23, -2, -10, 12, -4, 7, 15, -7, -13, 2, 3, 7, 0], [-18, -11, -4, 2, 2, 6, -7, -4, 1, 6, 5, -1, -2, -1], [-2, 4, -3, -5, 19, -7, 2, 0, 2, -4, 0, -1, 4, -4], [3, 17, -8, 0, 0, -3, 8, 14, 9, -4, 0, -4, 9, 4], [0, 9, -8, 0, -2, 2, 3, -2, 1, -4, 1, 3, 0, -2], [0, 22, -8, -2, 5, -5, 7, -1, -4, -11, 2, -3, 1, -7], [2, -7, -4, 3, 2, 6, -4, -3, -2, 4, 2, -4, -1, 11], [3, -1, 5, -5, -9, 5, 1, 1, -1, -13, 1, 4, -6, -5], [1, 2, 1, 4, -11, 3, 3, 3, 3, -8, -2, 0, 2, -2], [1, -8, 8, 4, -5, 3, -3, -2, 6, 8, -2, -2, 2, 4], [-19, -12, 7, 0, -2, -3, 1, -3, -5, 8, 1, -7, -2, -2], [11, -13, 9, 11, -5, -5, -7, -9, -9, 11, -6, -5, -3, -3], [13, 4, -1, 8, 11, 2, -2, -10, 3, 3, -2, 1, -6, 1], [-3, 14, 14, -6, 2, -3, -9, 6, -10, -6, -1, -6, 3, -7], [6, 11, 1, -5, 5, -2, 2, 7, 7, -9, -6, -2, 13, 0], [-16, 20, 9, -6, 13, -4, 4, -1, -7, -4, 3, -2, 5, 4], [30, -18, 4, 12, -2, 1, -13, -7, 4, 13, 7, -2, -1, 5], [-3, -17, -1, 3, -3, 5, -3, -13, 1, 15, -2, -5, -4, 3], [7, -24, -8, 3, 2, 7, -2, -1, 9, 4, -7, 4, 3, 6], [-13, -13, 1, 4, -1, 2, -3, 4, 10, 7, -7, 0, -2, 2], [-11, 9, 4, -11, -5, -6, -2, 3, -2, -4, 4, -5, 6, 3], [-11, -1, 2, -1, -1, 6, 0, 1, 6, 10, 3, 8, -2, 4], [-22, 4, -6, -5, -2, 3, 2, 5, 3, -2, 5, 5, -1, -4], [25, -7, 1, 6, 8, -3, -6, -3, 6, 6, -1, 1, -3, -2], [-17, -5, 3, -9, 8, 1, -9, 7, -1, -3, -4, -7, 4, -7], [-7, -5, 7, 5, -7, 1, -7, -1, -7, 9, -4, -9, 4, 4], [28, 3, 3, 4, 9, 1, -4, -8, -4, 5, -1, 7, -4, -3], [11, -4, 8, 0, 20, -7, -7, 6, 0, 0, -12, -5, 10, 1], [8, -6, -4, 3, 5, 4, 3, -2, 5, -3, -2, 6, -6, 2], [-9, 5, 3, 0, 15, -2, -5, 11, 8, -10, -7, -4, 6, -4], [4, -9, -8, -2, 3, 6, 1, 0, 7, 4, 0, -1, -3, 0], [17, -8, -9, 1, -3, 5, 2, -13, 6, -1, 12, 8, -8, 6], [-12, -20, 9, 8, 5, 1, -11, 6, -1, 11, -5, -8, -1, 1], [0, 0, 2, 6, -6, -3, 1, -11, -6, 1, 4, 7, -9, -3], [-12, -15, 0, -2, 9, 4, -3, -2, -7, 0, -8, -8, -2, 1], [15, 9, 4, -8, -2, 3, -2, 4, 1, -2, -5, 0, 0, 2], [-15, 3, -9, 10, 3, 2, 5, -6, 10, -1, 4, 10, -3, 5], [2, -16, 1, 6, 7, 3, -7, 0, 1, 7, 9, -6, -1, 1], [2, 6, 11, 5, -7, -8, -3, 3, -2, -1, 0, -6, 3, 3], [-8, 0, 2, -1, 16, -8, 1, 8, 9, -7, -6, 2, 4, -1], [-4, 6, 6, -11, -14, 1, 4, 13, 5, -18, 9, 5, 7, -6], [-8, -9, -13, 6, -7, 7, -4, -11, -2, 8, 9, -1, -4, 1], [-9, -5, -2, -1, 3, 11, 1, -10, -4, -1, 4, -3, -11, 5], [4, -3, -6, 10, 2, -2, 7, -6, 5, -2, -4, 5, -1, -5], [-5, 7, 4, -8, -2, -3, -8, 2, -3, -4, 3, -1, 5, -1], [5, 5, -1, 5, -1, -4, 2, -12, -5, 8, -5, 4, -11, -4], [3, 14, 1, 10, 5, -10, 2, 5, 3, 8, -1, -3, 5, 4], [3, -9, -12, 6, 4, 6, -1, -15, 1, 5, 9, 1, -8, 5], [-14, 5, 11, 5, 21, -9, 1, 0, -3, 2, 2, -5, 5, 5], [-13, 2, 6, -2, -2, 5, -5, 0, -2, 0, 2, -3, 0, 2], [5, -11, 4, -10, 0, 2, 3, 0, -3, -4, 4, 3, -3, -7], [-17, 5, 11, -7, -6, 1, 1, -3, -9, -3, 9, -1, -5, -6], [7, 3, 9, -10, 3, 2, -1, 4, -4, -5, -7, 0, 2, 1], [1, -14, 1, 0, -11, 11, -7, -11, -11, 12, 6, 1, -8, 3], [-10, 6, -2, -9, 4, 6, -3, -4, 1, -9, 0, 1, -2, -2], [1, 14, -9, -9, -3, -6, -1, 16, -10, -6, -8, -5, 10, -5], [14, -9, 15, 6, 3, -12, 5, 7, -4, -2, -12, -2, 1, -9], [8, -16, 0, 16, 14, -3, -3, -5, 6, 9, 3, -9, 11, 8], [6, -4, -4, 9, -8, 2, -1, -8, 1, 9, -6, 1, -10, 0], [15, 4, -9, -4, -12, 5, 13, -5, 11, -8, 9, 8, -1, 2], [12, -10, 7, 13, -10, 1, -18, -4, -4, 16, -2, -4, -7, 1], [26, -1, 1, 2, 1, -4, 5, 5, 8, -6, -4, 0, 7, 2], [6, -2, 4, 2, -8, -2, 4, 2, -8, 0, -3, -3, -3, 3], [-21, 3, 5, -6, 4, 6, -1, -4, 4, -3, 14, -1, 8, 1], [-1, 1, 7, 3, 1, -9, 5, 9, -11, -1, -2, -3, -2, -7], [20, 7, -4, 1, -4, 3, 5, -1, 4, -6, 9, 3, 0, 2], [6, -11, -3, 4, 4, -2, 11, -3, 8, 0, 2, 11, 3, 2], [-18, -7, -1, 4, -1, 2, 7, -11, 8, 0, 8, 1, -1, 6], [25, 1, -12, 5, 2, 10, 2, -6, 14, 1, 0, 5, -5, 2], [-20, 19, -3, -3, -2, 5, 3, 0, 1, 2, 12, 0, 0, 9], [-43, -3, -14, 6, -1, 3, 0, -14, -1, 14, 0, -3, -6, 10], [-16, 5, 14, 1, 6, -5, -1, -3, -6, -4, -2, -8, 7, 0], [18, 19, 5, -8, 1, -1, 2, 2, -10, 1, -2, 2, -5, -4], [1, 4, 5, -4, 3, 2, 2, 2, 3, -7, -3, 2, 3, -3], [-6, 5, -7, 3, 13, 0, 2, 3, -1, 7, 5, -1, -1, 6], [20, -17, -7, 1, 1, 12, 6, -17, 3, 3, 6, 10, -9, 3], [34, -3, 4, -4, 0, -2, -1, 7, 9, -3, -4, 0, 3, 1], [17, -15, 8, -3, -10, 0, -12, 3, 4, -4, -3, 0, 2, -7], [22, 12, -1, 2, -13, 2, 2, 1, 5, -8, 3, 6, 3, 6], [9, 12, 6, -2, 4, -6, 6, 15, 10, -7, 0, 5, 12, 2], [-9, 34, 8, -16, 6, -12, 6, 15, -8, -13, -1, -3, 11, -10], [16, 3, -13, 3, -13, 3, 9, 3, 1, 1, -3, 5, -4, 2], [-15, 1, 0, 4, 2, -7, 8, 5, -1, -3, -2, -1, 3, 9], [5, -15, 1, 1, 4, 5, 3, -9, 5, 3, -2, -1, -2, 9], [-13, -8, 11, 1, 4, 6, -9, -4, -4, 8, -6, -1, -7, 7], [14, 4, -2, 11, 0, -8, -5, -2, -5, 13, 4, -3, 2, 10], [-6, 10, -10, 2, 4, 0, 6, -1, 14, -3, 7, 7, 5, 0], [10, 9, -11, 3, -8, 6, 4, -3, 9, -1, 9, 9, -6, 3], [-7, 5, -4, 2, 0, 3, -10, -8, -3, 0, 9, 4, -1, -5]];
    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_bb();".
// 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_bb(:prec:=14)
    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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -3, 386, 193, -1429, -443, 1581, 346, -772, -118, 187, 18, -22, -1, 1]>,<3,R![405, -135, -3258, 2694, 5298, -4860, -3000, 3128, 632, -909, -8, 121, -12, -6, 1]>,<11,R![-30528, 1189152, -1593248, -4537424, -1262144, 1283456, 483981, -150514, -61324, 8946, 3589, -265, -98, 3, 1]>],Snew);
    return Vf;
end function;