
// Make newform 4025.2.a.v in Magma, downloaded from the LMFDB on 18 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_v();".
// 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_v();".
// 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 := [3, 7, -16, -22, 28, 9, -10, -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, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [0, -4, 0, 1, 0, 0, 0, 0], [1, 4, -5, -1, 1, 0, 0, 0], [-7, -7, 17, 6, -8, -1, 1, 0], [3, -5, -17, 21, 8, -9, -1, 1], [3, -10, -21, 27, 9, -10, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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], [1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, -1], [0, 1, 0, 0, -1, 1, 0, -1], [0, 0, 1, 0, -1, -1, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 0, -2, -1, 0, -1, 1], [0, 1, 0, 0, 0, 1, 1, -1], [1, -1, 1, 1, 0, 0, 1, -3], [-1, 0, -3, 0, 3, -1, -1, 1], [0, 0, -2, -1, 0, 2, -2, 1], [3, 0, 0, 0, 2, -1, -2, 2], [2, 2, -1, 0, 1, 0, -1, 0], [1, 1, 3, -1, 0, -2, -2, -1], [1, 0, 0, -2, 1, 0, -1, 0], [4, 0, 1, -1, 2, -1, -1, 3], [3, -1, 0, -1, 0, 0, 3, 0], [4, 1, 1, -1, -1, 2, 1, 0], [-3, 3, 1, -2, -3, 1, 2, -3], [1, 1, -1, -3, -1, 0, -2, 2], [3, 2, 3, 1, -1, -1, 1, 2], [1, -2, -1, -1, -1, 2, 1, 2], [0, 1, -2, -1, 3, 0, -5, 1], [7, 0, 1, 1, 1, -2, 0, 1], [-1, 0, 1, -2, -3, -1, -2, -2], [-1, -2, -4, -4, 4, 1, -2, 5], [4, -3, -2, 3, 1, 2, -1, 1], [6, -4, 1, 1, -2, 0, 0, 0], [2, 0, 0, -2, -3, -1, 1, -1], [3, -1, 0, -4, 1, 1, 1, 0], [-5, 2, 1, 2, -4, 1, 2, -7], [2, 3, 2, 0, -1, 1, 2, -2], [-2, 1, 2, 2, -3, 2, -2, -2], [6, -6, -1, 3, 1, 0, -1, 3], [10, -3, 1, -3, 2, -1, 0, 0], [-7, 4, -3, -3, -3, -1, -2, 3], [-3, -3, -5, 2, 6, 0, -4, 1], [-3, -3, -2, -2, 4, -2, -8, 2], [6, 1, 5, -3, -3, 1, 5, -1], [1, -3, 1, 4, -1, 0, 3, -1], [3, 7, 1, -1, -3, 2, 3, -4], [1, 0, 2, 0, -6, -2, -3, -4], [2, -1, 2, 0, 2, -1, -1, 1], [-5, -3, -3, -2, -4, 4, 4, -2], [10, 3, 2, -1, 5, -1, 3, 2], [-10, -2, -6, -2, -1, 4, -3, 0], [3, -3, -5, 5, 9, 1, -6, 0], [0, -3, 2, -2, -1, -4, 4, 3], [-5, -1, -2, 1, 1, -4, -3, 4], [-4, 2, 6, -1, -3, -3, -3, -1], [0, 6, -6, 5, 2, 2, 2, -2], [7, -3, -6, 4, 4, 5, 4, 3], [6, 3, -3, 2, 8, -4, 0, 2], [-3, -5, 5, -4, -6, 0, -1, 2], [-4, -5, -2, 0, -6, 0, -2, 3], [5, 3, 1, 0, -1, 0, 3, 1], [4, 0, 1, 5, 0, -5, 3, -2], [13, 4, 5, -5, -3, -3, 4, 0], [-4, 7, 0, 2, -4, 1, 0, -8], [3, -6, 3, 1, -8, 0, -2, -1], [1, 1, 7, -1, -7, -3, 2, 1], [0, -4, 4, -1, -8, 3, 4, -3], [-4, 2, -4, 3, 5, -5, -1, 3], [-5, -4, -1, 3, -5, 3, 4, -4], [7, 4, 0, -2, -5, 1, 2, 3], [-8, 5, 5, -5, -4, 0, 3, -5], [-3, -3, -2, 1, -6, -2, 0, 3], [-3, 0, 3, 1, -8, 1, 6, -5], [7, -4, 1, -3, 5, -3, -5, 7], [4, 2, -2, 1, 0, 3, -3, -2], [6, -7, 3, 2, 1, 3, 4, 1], [0, -5, -1, -2, -3, 1, -6, -5], [-5, 0, 0, 2, -1, 0, 3, -7], [5, -6, 0, 3, 0, -1, 5, -4], [3, -2, 1, -5, -9, 2, 0, -5], [-5, 2, -4, 2, 10, -1, -1, -2], [2, -9, 2, -4, -3, 3, 0, 4], [0, -7, -6, 5, 0, 6, 1, -2], [-3, -8, -3, 1, 2, 0, -4, 0], [-11, 0, 1, -3, -17, 5, 2, -5], [-2, 2, -7, 6, 10, -5, -6, 2], [6, 3, 1, 5, -7, 0, 8, -2], [11, -7, -2, 6, 1, 0, -6, 6], [13, -6, -2, -1, 4, 2, -1, 9], [8, -1, 9, -2, -4, 0, 6, 0], [6, -3, -2, 1, 0, 6, 3, -2], [4, -3, 5, 1, 1, -2, -1, 3], [10, 11, 3, -2, 3, -2, 3, 5], [6, -10, 7, 6, -3, 3, 0, -4], [14, 5, 6, 2, -1, 0, 5, -9], [-6, 0, -7, -7, -7, 5, -3, 4], [-8, -8, -7, 6, 1, 6, -2, 0], [7, 6, -2, -2, 8, 3, 4, 2], [7, 4, 2, -1, 2, 6, 0, -3], [-3, 7, 9, -1, -3, 0, 4, -5], [2, -4, 1, 3, 4, 2, -3, 2], [-8, -7, 0, 0, 2, 7, -1, -2], [6, 1, -2, -2, -2, 0, 7, -4], [1, 0, -1, 4, 6, 4, -5, 0], [-8, 0, 0, 1, -13, 6, 3, -8], [-1, 7, 2, -7, -2, -6, 0, 3], [3, 0, 15, -3, -9, 1, 2, -1], [7, -5, 8, -1, 4, -8, -1, 1], [6, 2, 2, 4, 7, -2, 5, 0], [-14, -6, 4, -1, 1, -1, -5, 5], [12, 2, 0, 5, 11, -1, 5, 2], [-5, 3, 9, 7, -5, -3, 6, -5], [13, -7, -2, 1, -1, -1, -3, -2], [-15, 6, 0, -9, -17, 3, 0, 3], [6, 5, 9, 4, -5, 1, 13, -6], [10, -9, -9, -1, 2, 4, -6, 2], [1, -4, -2, 2, 8, -5, -3, 2], [5, 2, 3, -1, 14, -5, -7, 11], [-1, 8, 2, -7, -6, 1, 6, -2], [2, 12, 2, 1, -1, 1, 4, -4], [-2, 6, 5, 2, -1, -3, 2, -7], [-10, 8, 3, -4, -10, 4, 7, -8], [3, -6, -2, -3, 9, 1, -6, 12], [-13, 3, 5, 6, -10, 0, -1, -7], [18, 11, 3, -2, -2, -2, 8, 2], [7, -1, 8, -12, 0, -2, 1, -1], [4, -6, -1, 11, -8, -3, 7, -5], [-7, -3, -2, -5, -2, 3, 2, -5], [-3, 8, 7, 1, -5, -10, -2, -4], [1, 4, -2, -7, -4, -3, 2, -4], [2, -1, 0, -2, 0, 1, -7, -7], [-22, 6, -1, -2, 7, -4, -1, -5], [3, 1, 6, -1, 4, 2, 5, 4], [17, 3, -1, -2, 14, 1, 0, 5], [-1, -8, 9, 0, 1, -2, -1, 5], [-3, -11, -15, 1, 0, 5, -5, 4], [13, 5, 4, 3, -4, 0, 3, -10], [4, 3, -5, 12, -2, 0, 2, 0], [-1, -8, 3, 6, -10, -5, -3, 2], [12, -13, -4, 7, 12, -1, -3, 4], [17, 13, 12, -8, -2, 3, 0, -3], [0, 1, 3, 2, -12, 6, -4, -1], [18, 0, 6, 4, 5, -3, 9, 4], [-4, 6, 2, -3, 12, 2, -1, 1], [3, 2, -4, 0, 7, 0, -16, 2], [-9, 4, -10, -7, -1, 5, -2, 4], [9, 10, -4, 3, 8, 0, 1, 3], [10, 2, 0, 3, -8, -1, 10, -1], [7, 13, 1, 0, 10, -5, 9, -3], [-5, -3, -5, 14, 3, 4, 1, 0], [14, -3, -7, 10, 13, -5, -6, 3], [-7, -9, -3, 7, 0, 2, -1, -10], [14, -1, -6, 2, 10, 5, -2, -2], [-5, -2, -10, -8, 0, 10, -6, -1], [-8, -1, 3, 2, -5, 4, 5, 0], [-3, -2, 3, 4, -12, 3, 7, -13], [0, -4, -1, 9, -2, -3, 1, -7], [2, -10, -4, -4, 1, -3, -9, 5], [-9, 12, -9, -6, -9, 4, -1, 9], [2, -2, 2, -10, -5, 6, 3, 1], [7, 5, -9, -1, 17, 2, 3, 7], [-18, 11, -7, -4, -13, 8, -2, -11], [-7, -1, -5, 1, 2, -5, -9, 5], [-4, 6, 11, 0, 1, -9, -3, -7], [13, 14, -4, 0, -5, 1, -2, -4], [4, 8, 1, 2, -9, 7, 0, -2], [-20, 2, -4, -6, -1, 0, -10, 1], [9, -8, -8, 4, 11, 8, -5, 9], [-7, 3, -5, -4, -7, 14, 2, -7], [0, -14, 3, 6, -2, -3, 7, 0], [-3, -5, -3, -12, -6, 4, -12, 8], [-18, 1, 5, 7, -8, -2, 3, -13], [9, -6, -6, -6, 5, -4, -10, 14], [4, 2, 8, -6, -2, 0, -2, 11], [-3, 12, 6, 3, 10, -11, -7, -1], [3, -2, -2, -6, -9, 11, -6, 3], [-10, -4, -4, 2, 0, -6, -4, 1], [1, 6, -11, -1, -5, -1, -11, 3], [-1, 2, -9, 18, 2, 2, 2, 0], [-5, 8, 10, 5, 1, -7, 6, -7], [6, 9, 0, 9, -7, 1, 2, -11], [16, -3, -9, -2, 4, 3, -2, 6], [-1, 3, 6, -6, -12, -8, 1, -3], [6, 9, 9, -3, -1, -13, 3, 1], [-1, -16, -8, 3, 5, 3, -3, 9], [14, 2, -1, 4, 6, -1, 1, -9], [15, -3, 6, -12, -6, -3, -5, 5], [-7, 2, -1, -2, -1, -8, 5, -7], [0, 9, 5, 6, -14, 9, 10, -11], [26, -6, -4, 3, 12, -2, 8, -5], [-16, -3, -13, 4, -13, -1, -2, 2], [-28, 6, -7, 13, -7, 7, -1, -5], [-23, 7, -2, -5, -5, 1, -12, 0], [-10, -8, -6, -1, -10, 7, -1, -2], [-5, 2, 2, -9, 7, -6, -2, 2], [16, 10, 3, -2, -8, -1, 11, 0], [-5, 2, 9, 9, 7, -7, 7, 0], [-3, -7, -3, 11, 7, -5, 2, 7], [-12, -8, 7, 0, -12, 10, 7, -9], [10, 5, -2, -2, 11, -4, -9, 11], [-11, 1, -10, -2, 10, 2, -2, 8], [-2, -5, 12, -3, -5, -7, 12, 3], [-2, -2, 6, -7, -3, -2, -13, 1], [13, 0, 13, 3, -7, 4, 0, -9], [15, -7, -14, 4, 8, 7, 2, 8], [4, 4, -6, -5, -6, -4, 0, 2], [-2, 10, -2, -8, 8, -6, -9, 7], [7, -4, 15, 6, -3, 1, -7, -3], [16, -11, -10, -3, 2, 7, 4, 9], [22, -7, 1, -1, 13, 5, 14, 7], [-14, 17, 1, -2, 3, -8, -3, -5], [-19, 2, -2, 3, -7, 2, -1, -7], [3, -12, -7, 2, 2, 1, -14, -3], [-16, 8, -12, -7, 3, -1, 3, 14], [-7, 15, 0, 5, 7, -4, 0, 2], [-4, -7, -1, 4, -5, -12, -10, -2], [2, -13, -7, -2, 14, -8, -2, 7], [-1, 12, -5, -1, 1, 10, 7, -13], [14, 3, -4, 5, -2, 5, 7, -1], [-4, -5, -2, 18, 8, -2, 3, -3], [7, -1, 10, -8, -11, 11, 7, -9], [14, -9, 9, -4, 14, -5, 0, 1], [-8, -5, -4, 11, 3, -4, -6, -2], [0, 1, 4, -8, -10, 0, -6, 4], [12, -10, -5, 9, 1, 13, 4, 2], [25, 9, 6, 6, 4, -5, 7, -8], [-12, 15, -7, -8, -12, 6, 1, 4], [-6, 16, 3, -1, 13, -1, -4, 2], [-3, 10, 5, -14, -18, 3, 8, 3], [-13, -15, 6, 4, -2, 6, -3, -9], [20, -12, 3, 11, -3, -2, 1, 10], [5, 21, 11, -1, -18, 2, 9, -5], [18, -6, -3, 2, 17, -2, 1, 13], [-14, 14, -3, 4, -11, 3, -1, -15], [-12, -9, 2, 7, -18, -2, 6, 2], [-15, 12, -1, -17, -4, 2, 4, 10], [6, -2, -20, 6, 12, 8, -6, -4], [22, 4, 9, -2, -6, 1, 1, -10], [9, -11, 3, 14, 8, 3, -7, -15], [-20, -7, 3, -8, -14, 5, 0, -8], [-17, 10, -1, -11, -4, -3, 5, 15], [9, 3, -5, 4, 0, 6, -3, -11], [-5, -19, -5, 1, 3, 1, 0, 7], [-3, 16, -1, -9, -19, 6, -2, -6], [25, -13, -4, -5, -13, 8, 2, -1], [3, -8, -1, 7, -9, 7, 6, -9], [7, -12, -7, -4, -5, 2, -12, 11], [-10, -22, 2, 7, -2, -6, 2, 3], [31, 11, 6, 0, -7, -1, 2, -3], [-1, 6, 6, 3, -14, -6, -2, 2], [-23, 9, 17, -3, -17, -9, -1, -4], [5, -2, 6, -2, 8, -2, 3, 5], [3, 10, -4, -2, -16, 4, 1, 6], [3, -7, -5, 3, 1, -5, 5, 3], [-11, -7, -1, -7, -18, -2, 0, 13], [6, 12, 16, -20, -14, -4, 9, -4], [-30, -11, -14, 7, -10, 5, -5, 1], [-10, -16, -8, 3, 18, 8, -5, 9], [11, -11, 14, -3, 2, -4, 9, 7], [-2, 6, 0, -14, -1, -1, 6, 9], [-10, 9, 0, 2, 14, -17, 1, 5], [5, 17, 0, 9, -9, -5, 1, -11], [25, 12, -15, 5, 8, 5, 6, -8], [5, 13, -1, -1, 5, 13, 7, -10], [14, -21, -3, 2, 8, -1, -2, -7], [15, -7, -4, 1, 14, -11, 4, 5], [11, -16, -4, 10, 9, 2, 2, 4], [-8, -11, 3, 1, -10, -5, -3, 5], [20, 8, 6, 2, -6, 5, 3, -7], [9, 8, 8, -9, 7, 4, 2, -9], [11, -8, -4, 2, 13, -7, 4, -4], [5, -19, -1, 11, 2, 4, -2, -1], [-18, 2, -21, 0, 16, -7, 0, 8], [11, 10, 8, -3, -11, -1, 6, 7], [-32, 5, -11, 9, -9, -6, -11, -12], [19, -4, 3, 8, 2, -2, 17, -4], [-6, 4, -11, 2, 22, 0, -6, 8], [2, -11, 2, 6, -25, 3, 14, -8], [11, 5, 3, -4, 8, -17, 5, 7], [-1, 14, 10, -10, 5, 5, 0, 4], [-6, 4, -7, 10, 10, -7, 5, -3], [20, -18, 2, -10, -7, -5, -5, 4], [17, 2, -9, -2, -8, -3, 3, -1], [6, -1, 0, 9, 11, -6, 6, 12], [-20, 19, -8, -2, 9, 4, 6, -5], [33, 8, 8, -5, -8, -1, 3, -5], [-3, 1, -5, -6, -21, 2, -2, 3], [-4, 10, -6, -9, 10, 6, -14, -2], [-13, 2, -6, 2, -2, 5, 7, -12], [-15, 11, 3, 3, 1, -5, -5, 13], [-13, -4, -2, -10, 5, -1, -2, 0], [-3, 7, -3, 2, 6, -6, -13, 5], [9, -29, 0, -2, 15, -9, -6, 20], [-24, -10, 11, -5, -9, -6, -2, -13], [6, -15, 6, 7, 8, -13, -2, 9], [23, -24, -12, 0, 2, 4, 6, 1], [-6, -9, 12, 5, -17, 0, -5, -5], [15, 1, -2, -3, 17, 5, 0, -6], [32, 12, 5, -13, 10, -5, 5, 2], [20, -5, 12, 11, 18, -4, -7, -10], [-15, -7, -10, -13, 13, -8, -20, 29], [3, -11, -23, 0, 18, 5, -11, 10], [-22, -8, -16, 3, -6, 7, -1, 7], [-2, -9, -3, 10, -6, 4, 7, -11], [17, -10, -4, 17, 17, -1, -16, 5], [17, 8, -13, -7, -2, 9, -8, 14], [30, 4, -4, -3, 14, 3, 14, -5], [15, 10, -7, 1, 11, -5, -3, -1], [-30, 13, -12, -1, -12, 12, 5, -7], [3, -5, -10, -11, 27, -2, -12, 24], [-28, -5, 2, 10, 9, -1, -7, -11], [16, 4, -9, -3, 6, 18, -1, -3], [-1, 13, 14, 5, 4, -12, -9, -1], [22, 5, -3, 6, 5, 3, 9, 2], [-3, 14, -7, -7, -4, 5, 8, 10], [12, -9, -4, 14, -2, -6, 14, -8], [30, 7, -7, -8, 24, -3, -16, 25], [-21, -11, -8, 11, 6, 4, 0, 7], [-2, -2, -12, 10, 5, -3, -6, 2], [-6, 13, -3, -17, -6, -3, -5, -4], [29, 6, -5, 9, 9, -2, 20, -7], [-48, -7, -19, 5, -6, 6, -11, -7], [-2, -1, -5, -10, -17, 3, 8, 1], [6, 12, 8, 8, -9, 5, 24, -15], [4, 0, -7, 9, 2, 3, 12, 8], [-3, 13, -13, 1, 8, 4, 0, 3], [-1, 6, -4, 0, -24, 5, 11, -23], [18, -18, -2, -9, -6, 4, 10, 18], [-38, 21, 4, -17, -5, -12, -11, -3], [6, -4, -2, -6, 4, -6, 14, -5], [11, -4, -2, 17, 2, 1, 12, -4], [-6, -10, 6, -6, -11, -2, 9, 13], [-3, 27, 8, 1, 7, -4, 8, -13], [-45, 0, 2, -3, -23, 10, -3, -6], [6, 12, 18, -12, -7, -3, -6, -1], [16, -6, 4, 0, 1, -1, 1, 7], [18, -1, 12, -7, 9, -11, -8, 5], [12, -1, 23, 1, -14, 4, -1, -1], [-13, 2, -6, 9, 27, -3, 1, 6], [-16, -3, -1, -13, -8, 12, -6, -6], [33, -11, 12, 11, 23, -15, -1, -3], [-2, -4, -15, 1, -11, 1, 1, -2], [-5, -3, 11, -6, -6, -6, -9, -1], [-28, 13, -7, -10, -6, 8, -11, -7], [13, 9, -6, -2, -2, -1, 22, -5], [12, 6, -5, -3, -13, 0, 20, 0], [-13, 26, 0, -11, 13, -3, -12, 3], [-9, 8, 18, 9, 16, -5, 2, -10], [10, 3, 8, 11, 5, -8, 10, 4], [3, 2, -11, 16, -9, 11, 18, 1], [36, -21, 7, 17, 4, 7, 3, -11], [-1, 20, 9, 0, -9, 5, 12, 5], [20, 4, -12, 5, 0, 12, -8, 3], [-9, -1, 10, 20, -9, -4, 11, -16], [-7, -5, -7, 14, 0, 5, -7, -1], [38, -6, -8, 0, -4, 0, -1, 0], [17, -2, 5, 9, 2, -1, 8, 5], [1, -15, -4, -10, 14, 6, -1, 24], [16, 9, -12, -12, 7, 8, -2, 17], [2, -10, -13, 19, 5, 7, -7, 1], [-3, 2, -10, 15, 17, -4, 0, -4], [-12, -10, -4, 3, 1, 7, 6, -4], [-23, -3, -6, -2, -20, 7, -5, 3], [-8, -10, -5, 3, -2, 11, -15, 11], [-7, -20, -14, -6, 9, 8, -14, 15], [-10, -9, -13, 5, 14, -14, -14, 13], [-19, 7, 2, -8, -21, 7, 25, -4], [20, -20, -8, 5, 16, 8, 2, 2], [-9, -9, -22, 8, 11, -10, -2, 1], [13, -21, -1, -6, 4, -8, -13, 20], [-34, 4, -9, -4, 4, -5, -10, -5], [32, -3, 5, 9, 19, -18, 9, -2], [-17, -25, 18, 0, -9, -7, 4, 7], [-12, 5, -9, -2, 12, -12, -20, 13], [-4, 1, 0, 1, 7, -3, -13, 3], [16, -8, 0, -10, 13, 12, 11, 11], [-1, 9, 8, -11, -10, 12, -8, -4], [-11, -4, 15, 6, 12, 4, 5, -4], [-25, -7, 3, 14, -20, 0, -6, -2], [-3, 15, 7, 14, -19, 10, 23, -33], [20, -24, 8, 6, 5, -2, -7, 19], [15, -3, 17, -18, -3, -4, -10, -4], [-11, -4, 3, 10, -4, -12, -9, 13], [12, 17, -9, 7, 18, -12, -12, 15], [25, -3, -7, 2, 29, -4, -8, 4], [-30, -25, 8, 9, -23, 4, 1, -9], [29, -2, -5, 7, 11, 8, 16, -8], [-6, -2, 5, 2, 12, -12, -3, -5], [-14, -13, -8, 14, 11, 4, -19, 12], [-22, -8, -5, 14, -14, -7, -6, 4], [6, 17, 8, 4, 7, -3, -15, -9], [-6, -11, -13, 8, 17, 5, -6, 1], [-8, 7, 3, -1, 0, -13, -6, 20], [4, -2, -7, -19, 17, -6, -10, 22], [14, -28, -15, -13, 9, 11, 6, 20], [-7, -28, -4, 5, 0, 7, -3, -12], [-42, 2, -10, -9, 4, 4, -8, -11], [4, 2, -2, 13, 7, -11, -7, 11], [31, 23, 1, -1, 20, -8, -4, 12], [0, 1, 15, -20, -6, 0, -2, 9], [-11, -14, -8, 4, -8, -6, -15, 11], [-10, -22, 2, -15, -17, 21, 9, -7], [12, 1, -2, -15, 9, -8, 4, 2], [-24, -11, -5, -5, 12, 6, -9, 11], [6, 28, 13, -7, -4, 4, 6, 0], [9, 27, 20, -12, -10, -4, 8, -11], [-19, -7, -1, -9, -6, -10, 2, 15], [27, 12, 10, -7, 9, 4, 8, -2], [-42, -19, -5, 1, 6, -1, -18, 7], [2, 5, 22, -9, -8, 3, 14, -9], [-1, 21, 4, 5, 5, 3, -2, -5], [-23, 18, -12, 3, 3, 10, 13, -11], [-26, 27, 5, -11, -6, -3, -5, -7], [19, -26, -4, 0, 6, 1, 5, 5], [-5, 11, 17, 6, -24, 7, 26, -27], [14, 16, 13, 0, 14, -4, 9, -7], [23, 2, 2, 15, -4, 6, 23, 0], [-24, -8, -9, 4, -11, 13, -2, -5], [-15, -22, 8, 3, 10, -2, 3, 5], [-16, -19, 4, -3, 3, 1, -5, -1], [4, 3, 30, 1, 4, -10, 12, -10], [-12, 1, -8, -16, 1, -2, -14, -3], [34, 11, -19, 8, 12, 7, 13, -7], [0, -26, 4, -9, 3, 2, 15, 6], [-32, 11, 14, -8, -25, 0, 3, -1], [-6, 2, 11, 0, -11, 4, 13, -7], [-20, 16, 2, 3, -8, -2, 6, 8], [-10, -8, 8, 8, -10, -11, 9, -1], [-14, 8, 6, -5, 1, -1, 13, -1], [21, 4, -4, -11, -13, -4, 11, -10], [-9, 4, 8, 0, -23, 2, 5, -5], [-11, -13, 4, 2, 11, -9, -5, 2]];
    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_v();".
// 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_v(:prec:=8)
    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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 7, -16, -22, 28, 9, -10, -1, 1]>,<3,R![-1, 5, 15, -47, -3, 27, -4, -4, 1]>,<11,R![-183, -908, -982, -28, 297, 43, -30, -3, 1]>],Snew);
    return Vf;
end function;