
// Make newform 3725.2.a.c in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3725_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_3725_2_a_c();".
// 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_3725_2_a_c();".
// 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 := [-39, 68, 76, -137, -48, 75, 12, -15, -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, 1, 0, 0, 0, 0, 0, 0, 0], [-26, -3, 92, 19, -63, -9, 14, 1, -1], [51, -37, -117, 24, 72, -3, -15, 0, 1], [-91, 31, 185, -12, -94, 1, 17, 0, -1], [-87, 2, 225, 13, -137, -8, 29, 1, -2], [101, -38, -237, 5, 139, 6, -29, -1, 2], [-56, 16, 143, -9, -82, 1, 16, 0, -1], [-180, 49, 418, -13, -233, -5, 46, 1, -3]];
    Rf_basisdens := [1, 1, 4, 4, 4, 4, 4, 2, 4];
    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_3725_a();"
function MakeCharacter_3725_a()
    N := 3725;
    order := 1;
    char_gens := [3577, 151];
    v := [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_3725_a_Hecke(Kf)
    return MakeCharacter_3725_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]];
    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], [-1, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 1, 0, -1, 1], [1, 0, 0, 0, -1, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0, -1, 0, 1], [0, 0, 0, -1, 0, -1, 1, 1, 0], [4, 1, -1, -1, 0, 0, -1, 0, 0], [0, 1, -2, 0, -1, 0, 0, 1, 1], [-2, 0, -1, 0, 0, 0, 1, 0, 1], [3, -2, 0, -1, -1, -1, 1, -1, 0], [3, 1, 0, -2, 1, 3, 2, -3, 2], [1, -1, -1, -2, -1, -1, 2, -1, 0], [0, -1, -1, 0, -1, -3, -1, 0, 1], [1, -1, 1, 2, -1, -3, -2, -1, 0], [0, 0, 1, 2, -2, -2, -1, 0, 1], [5, 2, -1, 0, 1, 1, -1, 0, 0], [-1, -1, -1, 1, -3, -2, -1, 2, 0], [-3, 0, 0, -3, 2, 1, 0, 1, -1], [3, 1, 1, -1, 1, 2, 0, 0, 2], [1, -2, -1, -1, 0, -1, 1, -1, -2], [-2, 3, -1, 1, -3, 0, -2, 4, -1], [1, 1, 1, -1, -2, 2, 3, -2, 1], [-2, 1, 1, 1, -1, 0, 0, -2, -1], [-1, -3, -1, 0, 2, -3, -1, 1, -3], [2, -1, -1, -1, 1, 0, -2, -2, -1], [-1, 3, 1, 2, 0, -1, -1, -1, -1], [-1, -3, -1, 0, -3, -3, -2, 1, 0], [0, 3, -1, 1, -3, -4, -4, 2, -1], [5, -1, -2, -2, 0, -1, 0, 1, -2], [-4, 1, -1, 3, -3, -2, -2, 4, -1], [-3, -2, 2, 2, -1, 0, 2, 1, -2], [-1, -5, 5, 4, -1, -1, 0, -1, -6], [7, 2, -1, -4, 3, 3, 3, 0, 4], [1, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -1, -1, 2, 2, 0, -3, 6], [3, 1, 1, -2, 2, -1, -1, 3, -1], [-3, -1, -2, 1, 2, -1, -1, 4, -1], [3, -3, 2, 1, 3, 3, 0, 0, 2], [-3, 2, 0, 2, 0, 0, 1, 4, 1], [-3, -1, -1, 0, -2, -7, -1, 5, -3], [-2, 0, 0, -2, -3, -2, 3, 0, -1], [-5, -3, 3, 2, -2, 1, 1, -3, 3], [2, -2, 0, 4, -4, -2, -2, -2, -2], [2, -2, 2, 0, 0, 0, 0, -2, 6], [1, 2, 2, 4, -1, -2, -5, 4, -4], [4, -8, 0, -1, -1, -1, 3, 1, 1], [-5, -3, 3, -3, 3, 2, 5, 3, 2], [2, 5, -3, -6, 0, 1, 0, 2, 4], [8, -1, -2, -1, 2, 6, 2, 0, 3], [2, 1, 1, -5, 0, 6, 1, -6, 2], [0, -1, 3, -4, 6, 1, 3, -3, -2], [0, -1, -3, -3, 0, 4, 5, 2, 0], [5, -2, 0, 3, 0, -1, -2, -3, 1], [-11, 5, 1, 4, 1, 1, -2, 1, 0], [0, 2, -4, 4, -8, -4, -6, 4, 2], [0, -3, 2, -1, 2, 6, 3, -2, 2], [4, 3, 0, 0, 5, 4, 2, -1, 3], [10, 1, 1, -1, 0, 4, -3, 2, 0], [0, 1, 4, 5, 3, 4, -2, -2, -1], [5, -3, -5, -1, 2, 4, 0, -5, 1], [-7, 2, 0, 3, -6, -1, -2, 3, -1], [-4, -6, 2, -2, 2, 4, 4, -2, 4], [0, 3, -3, 2, 1, -3, -4, 7, 1], [-2, 9, -3, -7, 1, 6, 6, 2, 5], [-3, -4, 2, 6, -1, -4, -2, 4, -10], [12, -3, 1, -1, 2, 2, -3, -2, 4], [-3, -2, -3, 5, -2, -3, -2, 5, 1], [-8, 1, 1, 3, 4, -6, -1, 4, -10], [-1, 4, -4, -3, -3, -1, 1, 5, 6], [3, 8, 0, 3, -2, 1, -10, 1, -1], [-1, -4, -2, 3, -5, -3, 1, -1, 2], [9, -4, 4, 3, 2, 5, 0, -5, 5], [1, -3, 1, -4, -3, -5, 2, -7, 2], [6, 0, 2, 4, 3, 8, 3, -6, 1], [-4, -2, 2, 0, 3, 0, 1, 0, -3], [9, 4, 2, -5, 8, 11, 6, -1, -1], [-5, 1, -1, 2, -5, -5, -6, 7, 0], [2, -5, 5, 2, -1, -1, 5, -3, -7], [2, 0, 6, 2, 1, 0, -3, -6, -3], [-6, 5, -3, 1, -4, -2, 5, 0, -2], [-1, -6, 4, -6, 6, 6, 7, -8, -1], [2, 8, -4, -4, -6, -2, -4, -2, 2], [-1, -6, 2, -1, 6, 1, -8, 1, -1], [2, 2, -4, 6, -3, 0, -3, 0, -3], [3, 3, 5, 2, 2, 3, 1, -1, 1], [9, 5, -3, 2, -1, 1, -4, 5, 2], [5, -5, 1, -2, -2, 3, 7, -3, -1], [-7, -2, -2, -1, -3, -11, -5, 7, -2], [-3, -2, 6, -3, -2, 7, 10, -7, 9], [14, -2, 2, -4, 1, 0, 1, -2, -1], [-11, -9, -7, -3, 4, 0, 3, 0, -1], [9, 0, -4, 4, -6, -8, -9, 7, -1], [8, 6, -3, -7, 6, 2, 2, -3, 10], [12, -4, -2, -1, -3, -3, -6, 4, -3], [11, -4, 4, 2, 1, 2, 6, -7, 0], [11, -7, 5, 2, 0, 1, 1, -3, 9], [-4, 4, 2, 6, 1, 2, -1, -2, -9], [-3, 4, 0, -1, 1, 5, -3, -5, -4], [-6, -1, 5, 5, 1, 2, -4, 2, -7], [2, -3, -3, -1, -3, 0, 0, 2, -1], [3, 4, -1, -3, 6, 5, -4, 1, -3], [2, 1, 1, 5, -8, -10, -7, 10, 0], [-8, 0, -2, 0, -4, -4, -8, 2, 2], [-11, 3, 7, 6, -6, -7, 3, 5, -5], [1, 5, -11, -3, -4, -6, 2, 2, -1], [16, 7, 1, -5, 2, 8, -1, -2, 4], [-10, 1, -1, -1, -1, 4, 2, -4, 3], [-2, -7, 5, 8, -4, -1, -6, 0, -2], [0, 8, -4, -9, 6, 9, 1, -3, 4], [12, 5, -3, -4, -7, 3, 6, -3, 5], [1, 3, 2, -2, 5, 5, 4, -1, -4], [-7, 7, 1, -2, 2, 11, 5, -1, 5], [9, 1, 3, 6, -2, -7, -3, 7, 1], [3, -8, 2, -7, 2, 1, 6, -1, -1], [-1, 8, 1, -5, 0, 1, 3, -7, 0], [-13, 4, -6, -3, 0, -1, 4, -3, 5], [-11, -5, 5, 7, -1, -6, -3, 9, -12], [1, -2, 0, -3, 4, -5, -6, 1, -1], [-13, -1, -5, 8, -3, -9, -2, 7, -10], [-7, 6, -10, -3, 0, -7, -8, 15, -3], [5, -6, -2, 1, -6, -3, -6, 1, 5], [-5, 4, -2, 1, 1, 3, -3, -7, 0], [-9, 2, -2, -5, 6, -1, 6, 1, -7], [12, 7, 5, 3, 5, 4, 4, -4, -7], [-9, 1, 1, 0, 0, 7, -3, -7, -5], [4, -3, -4, 3, -11, -4, -5, 2, 4], [0, -5, 5, 1, 3, 0, -12, 4, -1], [-4, 10, -7, -3, -5, 0, 3, 7, -1], [-1, -4, -2, -1, -3, -3, 1, 3, 4], [8, -2, 2, 6, 0, -6, -18, 0, -6], [-6, -4, -2, -7, 1, -1, 10, -4, 9], [-1, 9, -9, 4, -8, -7, -1, 9, 1], [-21, 0, 4, 5, 5, 5, 7, 5, -10], [-13, 6, 3, 3, -5, -1, -4, 7, 1], [4, 6, 7, 0, -9, 6, -1, -4, -1], [20, 5, -4, -3, 4, 6, -4, -10, 11], [-12, 9, -3, 13, -14, -12, -9, 12, -4], [8, -2, -2, -6, -1, -2, -1, -6, 9], [-1, -1, 0, -6, 2, -3, -5, 5, -5], [14, -11, 5, 6, 5, -3, 6, -1, -9], [-5, 0, 6, 5, -1, 3, -9, 5, -2], [-2, 3, 0, 4, 0, 0, -9, -7, -8], [6, -5, 11, 3, 8, -2, 3, 0, -4], [3, 3, 7, 6, 4, -3, -3, 5, -3], [21, -6, -1, -10, -2, 3, 6, -8, 9], [-13, 0, -10, 1, -4, -7, -2, 3, -5], [14, 4, 4, 5, -2, -1, -3, -1, -6], [11, 6, 10, 1, 7, 7, 1, 1, 0], [13, 2, 2, 1, 10, 13, -6, -5, 1], [3, 0, 2, 4, -2, -2, -11, 0, -9], [-6, -1, -1, 7, -6, -18, -1, 0, -2], [-15, 3, 3, 5, -2, -10, 1, 6, -3], [-4, 1, -7, -5, -4, -4, -1, 2, 10], [-7, -6, -2, -4, 1, -2, 2, 3, -6], [-13, 5, 3, 2, 3, -1, -2, 11, -4], [-2, 7, -3, -3, -6, -2, 5, -2, 4], [-8, 7, -6, 17, -11, -12, -9, 16, -6], [19, -2, 4, -3, 6, 1, 2, -7, 7], [3, 8, 1, -7, -2, 17, 0, -7, 9], [12, 12, 2, -8, 9, 6, -1, 0, -5], [10, 3, 3, -11, 6, 16, 5, -14, 10], [-7, -1, 5, 12, 0, -1, -9, 5, 5], [16, 3, -1, -6, 0, -5, 1, -1, 12], [9, 3, -1, -2, -1, 3, 2, -5, 4], [-3, 0, 4, 5, -7, -1, -7, 3, -2], [8, 1, 5, 4, 4, 3, -4, 8, -2], [-5, -1, -4, 6, -10, -5, 3, 3, 5], [-10, 5, 11, 5, 8, 4, 9, 2, 4], [-21, -3, -12, -6, 0, 1, 4, 5, 2], [-5, -8, 0, -3, 0, -3, -2, -5, -3], [-3, -1, -6, 2, -2, -5, -8, -1, -2], [-1, -2, -2, -10, 4, 8, 5, -5, 1], [14, 0, -6, 0, 1, -6, -7, -2, -5], [1, 15, -7, -6, 6, 5, 1, 3, 11], [24, 0, 4, 0, 1, 8, 3, -6, -1], [0, -1, -5, 0, 7, 9, 11, 2, -5], [-4, 10, -2, -10, 6, 6, 8, 6, 6], [9, 17, -5, -2, -5, 1, 2, 3, 10], [1, 5, -3, -5, 11, 0, 3, 2, 10], [21, 1, -1, -12, 3, 3, 8, -1, 2], [-4, 7, -5, -1, -5, -6, 4, 2, 3], [-21, 9, 3, 0, 1, 5, 10, -5, 0], [14, 1, -2, 5, 2, 6, -5, -6, -2], [0, -3, 1, -8, 5, 7, 1, -16, -5], [28, -5, -1, -1, -12, -4, -3, -6, 6], [13, 1, 5, 2, 8, 13, 1, -9, -7], [-2, 2, 6, 0, 18, 8, 4, 2, -6], [-5, -1, 3, -11, 13, 8, 9, -2, -6], [-8, -1, 3, 5, 6, 4, 1, 2, -6], [-1, 10, -4, -10, 0, -4, -1, 4, 3], [-6, -14, 6, 2, 1, 2, 15, -4, 5], [2, 3, -7, -10, 2, 1, 11, -3, -4], [9, -12, -2, -3, 8, 5, -2, -7, 1], [-11, 1, 5, 10, 3, -3, -2, 11, -4], [8, 0, -3, -4, 10, 14, 6, -2, -4], [-1, 2, -9, -7, 8, 9, 5, -3, 4], [36, -4, 2, 2, 2, -2, -4, -4, 10], [3, -12, -3, 5, -3, -1, 2, -3, 1], [14, 7, 9, 5, 3, 4, -10, -2, 1], [-30, -7, -4, 11, 4, -16, -10, 14, -17], [-15, 14, 0, 11, -3, 1, -3, 13, -8], [10, -1, -4, 3, -7, -6, 3, 12, 8], [-3, 0, -2, 1, 1, -11, -11, 9, -4], [-5, 17, 7, -6, 0, 13, 11, 3, 7], [9, 13, 4, 10, -7, -7, 1, 5, 3], [20, -4, 9, -1, -1, 0, 3, -5, -9], [11, 9, -1, -4, 6, 11, 9, -11, 15], [4, -10, 6, 2, 2, 8, 10, -16, 10], [22, -8, 1, 5, 0, 4, -4, -9, 14], [26, 6, -4, -6, 2, -4, 6, 10, 0], [1, 9, 1, -6, 2, 7, 3, 9, 9], [22, 9, 4, 6, -7, 2, -8, 3, 1], [5, -14, 8, -10, 16, 10, 10, -14, 5], [-14, -4, 4, 8, -4, 2, -8, 2, 10], [-15, -2, 9, 9, -1, -3, 4, 7, 3], [4, 2, 2, -1, 4, 1, -4, 1, -6], [-12, 2, -2, -1, 1, -3, -10, 5, 9], [17, -2, -8, -9, 1, 11, 5, -5, 14], [-41, -6, 4, 11, -1, -11, -3, 15, -14], [7, -1, -9, -4, 0, 1, -9, -3, 11], [-4, -11, 17, 14, 1, -3, -13, -4, -9], [13, -9, -8, 2, 5, 7, 4, -7, -2], [-9, 7, -1, 8, 1, 11, 0, 5, 4], [6, -9, 15, -13, 14, 16, 15, -12, 0], [0, -7, -4, -7, 7, 10, 18, -4, 1], [-19, -11, 5, 10, -1, 3, 2, -1, -8], [-6, 6, -4, -9, 5, 15, 11, -11, 1], [-6, -3, 0, -4, -4, 0, 13, -5, 0], [0, 9, -15, -3, -9, 0, 0, 12, 11], [13, -4, 14, -4, 11, 2, 0, -2, -6], [12, 2, -4, 2, -3, 12, 1, -16, 11], [-7, -18, 0, 3, -8, -7, 4, 7, 1], [-12, 15, 5, 14, -2, -11, -12, 15, -16], [-12, 0, 0, 0, -1, 10, -3, -12, 5], [-10, -5, 4, 2, -3, -10, -2, -7, -7], [-1, -7, -2, 6, 3, -5, 0, -3, -12], [1, -19, -9, -6, 1, -1, 0, 1, 4], [-4, 3, 3, 1, 5, 0, 8, 4, -23], [22, 3, 11, 5, -1, 0, 4, -8, 1], [-1, -7, -9, 0, -6, -9, -1, 15, -11], [19, -6, 6, 0, 1, 6, 9, -16, 12], [9, -9, 17, -6, 9, 15, 16, -13, -14], [13, -3, 5, 2, -13, -7, 4, 1, 14], [15, -8, -9, -5, 5, 1, -11, -5, -2], [-11, 7, 3, -4, 8, 5, 11, -5, -9], [-20, -3, 5, 8, 2, -17, -2, 10, -10], [-2, -12, 8, -2, 6, -8, 8, -2, -16], [20, -9, 16, 0, -6, -2, 3, -11, 8], [-1, 5, 1, -5, 0, -6, -7, 14, -15], [-15, -4, 6, 9, -4, -1, 6, -5, 1], [-5, 1, -7, -10, -9, -5, 10, -1, 0], [-12, 8, -2, -7, 8, -5, -2, 7, 2], [8, -19, 0, 7, 5, 4, -3, -4, 2], [3, -12, 5, 9, 6, -7, -4, -3, 1], [-25, -7, 1, 6, 6, -5, -7, 5, -9], [-3, -11, -1, 6, -4, -5, 1, -1, -7], [-3, -12, -2, 6, 7, 4, 5, -6, 0], [-4, 10, -14, -14, 4, 6, 0, 2, 4], [8, 7, -5, 11, -13, -16, -22, 14, -1], [-17, 1, 7, -8, 0, 7, 13, -5, 5], [-13, 9, -7, -4, 1, 3, 10, 5, 0], [5, -3, -3, -1, 0, 12, 0, -8, 15], [0, -3, -1, -4, -2, 3, 12, 11, -6], [-11, 15, 5, 6, 13, 7, 0, -1, -8], [33, -6, -6, -9, -3, 15, 11, -9, 14], [-5, -12, 14, 1, 1, -3, -5, -1, -14], [-28, -12, -3, 8, -3, -16, -8, 4, -12], [10, 10, -6, 1, 2, -5, 2, 7, -6], [-13, -10, 10, 3, 4, 11, 4, -15, 3], [12, -14, 14, -10, 11, 16, 11, -10, 1], [19, 4, 7, -10, 16, 17, 4, -14, 9], [-18, -2, 6, 7, 5, -1, 2, 5, -23], [-19, 7, -5, -8, -3, -7, 2, -1, -12], [-18, 10, -6, 0, -3, -4, -1, -2, 13], [2, -7, 9, 1, 4, -4, 9, -6, -14], [-5, -9, -9, 2, -6, -3, 3, 15, -1], [-7, 4, 11, 1, 1, 3, 1, 1, -8], [16, 3, -9, -1, -7, -4, 2, -2, 5], [21, -6, 1, 8, -8, -5, -4, -2, 15], [7, -6, 2, -11, 14, 13, 22, -9, 5], [3, -9, -2, -3, -10, -11, 1, 12, -3], [-30, 12, 1, 7, -6, 0, -8, 9, 12], [8, 0, 8, 7, -11, -3, 0, -9, 3], [-2, -1, -9, -1, -6, 6, 1, 0, 0], [-7, -2, -2, -11, 8, 9, 0, -5, -3], [2, 18, 3, 0, 12, 20, 6, -4, 4], [-9, -2, 6, -1, -9, -3, -7, -5, 8], [7, -7, 3, 5, 3, 6, 6, 0, 0], [-10, -6, 0, 2, -3, 6, 5, -12, 9], [-8, 0, -12, 1, -11, -17, -10, 1, 3], [-4, 12, -10, 10, -5, -6, -15, 14, -1], [-38, -11, -5, 9, -17, -18, 2, 12, -3], [-1, 1, -1, 4, -10, 11, 1, -11, 7], [21, -9, -2, -2, 2, 15, 18, -15, 8], [-13, 6, -6, -1, -1, -5, 15, -3, 0], [13, 8, 8, -2, 0, 14, -1, -9, -3], [7, 8, -2, 6, 1, -2, -12, -8, 0], [-13, 9, 5, 0, -10, 5, -1, 3, 1], [-15, 9, -5, 5, -5, -6, -13, -1, -2], [-2, -1, 5, -7, 11, 10, 12, -12, -3], [-17, -6, 0, 9, 1, -15, 5, 5, -12], [-1, -14, -7, 4, -7, -13, -9, 0, -2], [-4, 6, -9, 3, 11, 8, -1, 1, -1], [-27, -2, 11, 8, -5, -3, 5, 8, -10], [-15, 19, -21, 0, -11, -3, -4, 7, 8], [11, -1, 5, 5, 0, -16, -6, -11, -3], [-17, -2, -8, 13, 2, -13, -8, 21, -9], [-6, -6, 2, -2, 0, -4, 0, 8, -16], [-14, -4, -19, 4, -3, -16, -5, 14, -17], [-16, -8, 11, 9, 8, -2, 0, -1, -20], [-2, -5, -3, 0, -7, 5, 9, -5, 11], [-2, 0, -3, 4, -1, 12, 1, 10, 13], [-6, -9, 3, -5, 0, -18, 7, 4, -6], [28, -10, 12, 8, 1, -6, -5, 6, 7], [-16, 16, -12, 8, -10, -20, -18, 14, -6], [15, -9, -12, -11, -5, -5, -4, 6, 2], [-3, -6, -8, -15, 16, 7, 18, -9, 5], [15, 0, -4, 3, -3, -13, -19, -1, -2], [10, 2, -10, 0, -10, -10, -6, 14, -4], [-3, 7, -1, 6, 3, -3, 2, 5, -6], [-5, 0, -8, 1, 8, 3, -6, 1, -1], [-24, -1, -1, 3, -6, -12, -5, 2, -2], [5, -20, 2, -6, 0, -8, -8, 0, -3], [-13, 4, 1, 7, 0, 1, 3, 17, -8], [-9, 6, 4, 1, 6, -1, -2, -3, -3], [-13, 15, 0, 3, 11, -5, -10, 8, -14], [-9, 8, 12, 10, -9, 6, -11, 0, 2], [-39, -7, -9, -4, 1, 3, 6, 1, -4], [33, 5, 1, -11, 12, 14, 6, -6, 7], [0, 16, 10, 0, -2, 16, 2, -8, 6], [7, -8, 10, -9, -1, 5, -9, -9, 12], [14, 0, 10, -10, -2, 6, 10, -8, 24], [29, -6, -11, -20, 3, 21, 11, -4, 14], [4, 12, -18, -10, 6, 6, -8, 14, 10], [14, 16, 2, 7, -6, 5, 5, 0, 22], [-8, 12, 8, 4, 3, -2, -5, -2, -15], [-55, -3, 9, 0, 0, 3, 3, -7, 1], [8, 26, -2, -5, 4, 17, 5, 3, 16], [-23, -10, 0, 2, -1, -10, 11, 4, 0], [4, 7, 15, 9, 2, -8, -1, 6, -8], [15, -29, 11, 10, -10, -9, 7, -7, -5], [24, -17, -7, -1, 12, -4, -9, -4, 0], [-4, -3, -1, -5, -3, 0, -4, -8, 3], [-7, 16, 0, 11, -10, -11, -14, 23, -13], [-29, 22, 0, -13, 1, -3, 1, 7, -4], [2, -1, 7, -3, 5, -8, -4, 4, 7], [-17, 7, -7, 4, -16, -7, -15, 17, 1], [6, 0, -14, -6, -6, -4, -2, 10, 0], [-24, -9, -3, 7, -13, -14, 4, 18, -1], [30, 4, -2, -13, 2, 9, 1, 2, 4], [-7, 0, 14, -6, 6, 12, 9, -14, 13], [-4, -5, -5, 17, -12, -18, -15, 0, -10], [-7, 3, -11, 6, -3, -15, -14, 11, -14], [15, 4, 3, 7, 0, -3, -9, 7, 16], [5, 3, -5, -6, 0, -1, 17, -5, -5], [5, -2, -2, -14, 17, 24, 8, -13, -6], [-10, -4, 10, -10, 16, 30, 24, -12, -8], [-2, -12, 21, -6, 12, 20, 24, -20, 8], [-28, 7, 1, 1, -7, 0, 2, 18, 7], [11, -3, 6, -2, 5, 5, -6, 3, -4], [-15, 2, -6, 5, 2, 7, 8, 5, 5], [-38, -6, 4, 4, -2, 4, 6, 4, -2], [-29, -12, 13, 10, 1, -1, -1, -4, -2], [-22, 18, -10, 9, 4, -5, -16, 15, -12], [30, 5, -13, -25, 20, 16, 19, -6, 16], [-20, 14, -6, -4, -19, -10, 11, 2, -5], [-6, 22, 8, 0, -1, 4, 3, 4, -1], [-1, -10, -2, -3, -9, -9, 3, 23, 18], [2, -18, 16, 14, -1, 0, -5, 0, -9], [21, 8, -4, -7, -4, -5, -10, -1, -13], [12, 11, 0, -6, 8, 2, 11, 3, -6], [-2, -7, 0, 19, -15, -22, 0, 16, 3], [3, 3, -7, -11, -2, -8, 8, -5, -15], [-24, -10, -1, -10, -2, -8, 16, 0, 6], [13, 11, 1, -14, 18, 15, 5, -5, 9], [1, 3, -12, -7, -19, -11, 0, 22, 14], [-9, 18, -6, 5, -7, 5, -11, -1, 8], [-17, 2, 0, 5, 8, 1, -16, -3, -17], [-6, 9, 5, -1, 12, 6, 5, 0, -8], [-2, -5, 9, 31, -15, -22, -14, 16, -5], [-18, 7, 1, 2, 2, 7, -12, 3, 0], [-16, 24, -12, -2, -6, 4, -2, 8, 0], [-7, 2, 3, -1, 9, -1, -9, 17, 4], [30, -5, 1, 2, -3, -15, 4, 3, -5], [-2, 12, -6, 8, -9, -2, -9, 16, 7], [-5, -13, -8, 7, -12, -23, -5, 22, -9], [1, -5, -12, -16, 10, 7, 11, 3, 17], [-7, -5, 13, 6, 1, 3, 2, 3, 2], [-8, 14, -8, -2, 8, 10, -12, 2, -10], [-16, -21, 5, -14, 7, 1, 17, -11, 1], [18, -5, -13, -23, 7, 10, 14, -10, 9], [13, 1, -15, -2, -6, -5, -9, -9, 5], [21, -15, -9, -18, 3, 13, 20, -15, -6], [0, -1, 11, -1, -12, -20, -1, 20, 0], [-24, 21, -14, 9, -16, -2, -10, 14, 17], [13, 5, -13, -18, -6, -15, 5, 11, 5], [-19, 13, -2, 16, -13, -13, -8, 9, 4], [-11, 16, -2, -1, -8, -7, 6, 1, 1], [14, -10, 4, -20, 23, 18, 19, -18, 1], [8, 4, 1, 6, 4, 10, -7, -4, 11], [4, -1, 3, -13, -10, -2, 5, -4, 18], [12, 12, -22, -12, -8, -8, -10, 20, -4], [4, 15, -22, -17, 11, 8, 8, 14, 15], [-4, -21, -7, -6, 4, 11, 4, -5, 8], [3, 12, -5, -14, -7, -1, -3, 0, 14], [-24, 12, 4, 4, -6, 0, 8, 8, -2], [8, -1, 6, -5, 5, 0, 10, -2, 21], [-55, 8, -4, 15, -10, -17, -20, 11, -7], [-4, 1, -9, 14, -4, -17, -1, 23, -8], [5, -6, 0, 1, 4, 9, -8, -7, 13], [4, -5, -16, -13, 1, 4, 2, -6, 13], [0, 11, -4, 12, -4, 6, -7, 1, 2], [3, -19, -5, 16, -4, -5, -3, -11, 5], [-16, -8, 11, 10, 7, 6, -7, 12, -9], [-13, 16, -10, -18, 1, 8, 6, -8, 10], [-7, -20, 8, 14, 8, -18, -25, -3, -21], [-3, -4, -4, 7, -10, -15, -4, -5, 15], [1, -4, -2, 11, -12, -23, -2, 17, -1], [-20, -18, 4, -1, 3, -21, -8, 14, -27], [25, 5, 1, -20, 1, 7, -8, -9, -2], [1, 1, 6, 4, 16, -13, -8, 17, 0], [-29, 12, -8, 1, 4, 3, 0, -7, 1], [-6, -2, 1, -1, 14, 0, -10, 3, -20], [2, -10, 22, 2, 1, 0, 9, -16, -25], [11, -9, -3, -8, -14, 3, -3, -11, -1], [-9, -18, 18, 11, 9, 7, 1, -9, -12], [-30, 11, 11, 5, 13, 18, 20, 2, -13], [54, 9, -1, 1, -5, -8, -10, 4, -5], [-43, -12, 6, 21, 7, -11, -3, 25, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3725_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3725_2_a_c();".
// 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_3725_2_a_c(:prec:=9)
    chi := MakeCharacter_3725_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_3725_2_a_c();".
// 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_3725_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3725_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-39, 68, 76, -137, -48, 75, 12, -15, -1, 1]>],Snew);
    return Vf;
end function;