
// Make newform 3025.2.a.bk in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3025_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_3025_2_a_bk();".
// 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_3025_2_a_bk();".
// 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 := [11, 0, -31, 0, 27, 0, -9, 0, 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], [-2, 0, 1, 0, 0, 0, 0, 0], [4, 0, -5, 0, 1, 0, 0, 0], [0, 5, 0, -13, 0, 7, 0, -1], [-8, 0, 14, 0, -7, 0, 1, 0], [0, -8, 0, 14, 0, -7, 0, 1], [0, 12, 0, -19, 0, 8, 0, -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_3025_a();"
function MakeCharacter_3025_a()
    N := 3025;
    order := 1;
    char_gens := [727, 2301];
    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_3025_a_Hecke(Kf)
    return MakeCharacter_3025_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, -1, 0, 0, 0, 0, -1, -1], [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, 1, 0, 0, 0, 0, 2, 0], [0, 2, 0, 0, 0, 0, 1, 3], [0, 0, -2, 1, 0, 1, 0, 0], [0, -2, 0, 0, 2, 0, -1, 0], [-4, 0, 0, -1, 0, -1, 0, 0], [2, 0, -1, 1, 0, -1, 0, 0], [0, 0, 0, 0, 2, 0, 3, -1], [-2, 0, -3, -2, 0, 5, 0, 0], [0, 3, 0, 0, 0, 0, -2, 2], [0, 1, 0, 0, 3, 0, -1, -1], [0, -3, 0, 0, 1, 0, 3, 2], [0, 0, -5, -3, 0, 2, 0, 0], [-4, 0, 4, 4, 0, -5, 0, 0], [0, -4, 0, 0, -1, 0, -2, -3], [-9, 0, 3, -3, 0, -3, 0, 0], [0, 2, 0, 0, -5, 0, 0, 1], [-6, 0, 4, -1, 0, -5, 0, 0], [0, -1, 0, 0, -1, 0, -3, 1], [-5, 0, -2, -6, 0, -2, 0, 0], [0, -1, 0, 0, 1, 0, 0, 3], [-7, 0, -3, 4, 0, -5, 0, 0], [0, -5, 0, 0, -3, 0, 0, -5], [0, -4, 0, 0, 3, 0, -3, -2], [-1, 0, 5, -5, 0, 3, 0, 0], [0, 3, 0, 0, 1, 0, 1, 5], [0, -7, 0, 0, 4, 0, 1, -3], [-12, 0, 3, -3, 0, -1, 0, 0], [0, 4, 0, 0, 3, 0, -2, 0], [-1, 0, -5, -2, 0, 7, 0, 0], [-2, 0, 8, 3, 0, 2, 0, 0], [-3, 0, 6, 5, 0, 2, 0, 0], [0, 5, 0, 0, -6, 0, 2, 4], [0, -1, 0, 0, -6, 0, -7, 0], [0, -3, 0, 0, -4, 0, -2, -5], [0, 3, 0, 0, 0, 0, 0, 3], [1, 0, -8, -6, 0, 2, 0, 0], [-2, 0, -3, 3, 0, 3, 0, 0], [5, 0, 7, 1, 0, 3, 0, 0], [0, -5, 0, 0, -7, 0, -5, -1], [0, 5, 0, 0, 7, 0, 5, -4], [3, 0, 8, 1, 0, -5, 0, 0], [-9, 0, 9, 1, 0, -7, 0, 0], [0, 6, 0, 0, -2, 0, 6, -2], [0, -8, 0, 0, -5, 0, -7, -2], [-6, 0, -9, -6, 0, 2, 0, 0], [0, -2, 0, 0, -1, 0, -6, 0], [-17, 0, -1, 4, 0, -4, 0, 0], [16, 0, -2, 10, 0, 13, 0, 0], [10, 0, 0, 11, 0, 6, 0, 0], [0, 3, 0, 0, -1, 0, -5, -9], [0, 0, 0, 0, 5, 0, 8, -5], [10, 0, -9, -5, 0, 13, 0, 0], [-1, 0, -1, -9, 0, 8, 0, 0], [0, 7, 0, 0, 3, 0, 0, 8], [-13, 0, 6, -5, 0, -7, 0, 0], [0, 1, 0, 0, 2, 0, -2, -11], [0, -2, 0, 0, 1, 0, 2, 8], [0, -5, 0, 0, -5, 0, -7, -8], [-8, 0, -1, 0, 0, -2, 0, 0], [0, 5, 0, 0, -4, 0, -7, 4], [0, 8, 0, 0, -2, 0, -3, -2], [16, 0, 4, 4, 0, 6, 0, 0], [0, 2, 0, 0, 2, 0, 16, -1], [0, 1, 0, 0, -6, 0, -2, -3], [2, 0, 7, 7, 0, -9, 0, 0], [0, 4, 0, 0, 1, 0, 10, -6], [-13, 0, -6, 6, 0, -4, 0, 0], [0, 0, 0, 0, -1, 0, 5, 5], [0, 1, 0, 0, 4, 0, 5, 5], [-4, 0, 14, -4, 0, -5, 0, 0], [0, 15, 0, 0, -3, 0, 3, 8], [-9, 0, 10, 0, 0, -22, 0, 0], [0, 4, 0, 0, -2, 0, 10, 3], [16, 0, 3, 4, 0, 7, 0, 0], [9, 0, -24, 0, 0, 3, 0, 0], [6, 0, 1, 8, 0, 7, 0, 0], [3, 0, -5, -7, 0, -2, 0, 0], [-3, 0, -19, -8, 0, 21, 0, 0], [0, -2, 0, 0, 7, 0, -5, -6], [-3, 0, -14, 5, 0, 7, 0, 0], [0, -1, 0, 0, -2, 0, -10, -3], [-5, 0, 6, 17, 0, -4, 0, 0], [0, -12, 0, 0, -8, 0, -2, 7], [-9, 0, -13, -14, 0, 2, 0, 0], [0, -3, 0, 0, -4, 0, -5, -5], [0, 5, 0, 0, -13, 0, -6, 2], [-25, 0, 1, -6, 0, 6, 0, 0], [0, 0, 0, 0, 1, 0, 11, -12], [11, 0, 1, 9, 0, 7, 0, 0], [15, 0, -10, 15, 0, 10, 0, 0], [0, -6, 0, 0, 4, 0, -9, -10], [-1, 0, 15, -3, 0, 0, 0, 0], [-3, 0, -9, 1, 0, -16, 0, 0], [0, -1, 0, 0, 6, 0, -5, 10], [14, 0, 0, -8, 0, -5, 0, 0], [0, 5, 0, 0, 14, 0, 3, -10], [0, 2, 0, 0, 15, 0, 3, -1], [0, -8, 0, 0, 1, 0, -1, -11], [-24, 0, -1, -8, 0, -4, 0, 0], [10, 0, -1, 1, 0, -9, 0, 0], [0, -14, 0, 0, -2, 0, -3, 4], [0, 14, 0, 0, -5, 0, -1, 13], [0, -1, 0, 0, -6, 0, -13, -6], [12, 0, -9, -4, 0, 20, 0, 0], [0, 0, -13, -15, 0, -7, 0, 0], [0, -6, 0, 0, -12, 0, -19, -12], [0, 1, 0, 0, -1, 0, 9, 13], [0, 13, 0, 0, 9, 0, 1, 12], [-2, 0, 18, 15, 0, -26, 0, 0], [5, 0, 8, 12, 0, -10, 0, 0], [13, 0, -4, 4, 0, 10, 0, 0], [0, 4, 0, 0, 9, 0, 15, 15], [0, -2, 0, 0, -6, 0, 2, 3], [0, -1, 0, 0, 13, 0, 21, -1], [-14, 0, 13, -3, 0, -12, 0, 0], [-3, 0, -4, -5, 0, 13, 0, 0], [0, 10, 0, 0, 1, 0, 1, -16], [0, 5, 0, 0, -5, 0, 10, 1], [0, -5, 0, 0, 3, 0, -5, 9], [-11, 0, -4, -3, 0, 18, 0, 0], [12, 0, -3, 22, 0, 5, 0, 0], [16, 0, 6, 13, 0, 17, 0, 0], [6, 0, 3, 4, 0, -14, 0, 0], [0, -21, 0, 0, 5, 0, 10, -5], [0, 5, 0, 0, 13, 0, 14, -12], [13, 0, 10, 11, 0, -5, 0, 0], [0, 2, 0, 0, 10, 0, 6, -8], [3, 0, 18, 24, 0, -3, 0, 0], [0, -8, 0, 0, -2, 0, -6, 8], [10, 0, -1, -8, 0, 14, 0, 0], [3, 0, 10, 8, 0, 6, 0, 0], [0, -13, 0, 0, 0, 0, -8, -9], [0, -13, 0, 0, 11, 0, -16, -8], [0, -23, 0, 0, 4, 0, 3, -9], [7, 0, -16, -6, 0, 28, 0, 0], [-15, 0, -5, -12, 0, -19, 0, 0], [12, 0, -11, 3, 0, -8, 0, 0], [0, -5, 0, 0, 17, 0, 22, -4], [0, -13, 0, 0, 13, 0, 9, 2], [-34, 0, 9, -14, 0, -13, 0, 0], [15, 0, -2, 7, 0, 25, 0, 0], [0, -12, 0, 0, 9, 0, -1, -13], [0, -7, 0, 0, 12, 0, -1, -19], [10, 0, 5, -4, 0, -16, 0, 0], [0, -12, 0, 0, -2, 0, 8, -9], [0, 5, 0, 0, -25, 0, -10, 5], [13, 0, -2, 1, 0, 34, 0, 0], [0, 10, 0, 0, 9, 0, 14, -6], [0, -5, 0, 0, -14, 0, -21, -13], [0, -9, 0, 0, 10, 0, 0, -6], [6, 0, -2, -16, 0, 16, 0, 0], [2, 0, 6, 30, 0, 0, 0, 0], [11, 0, -12, -5, 0, -4, 0, 0], [0, 12, 0, 0, -16, 0, -19, -6], [-3, 0, -14, -7, 0, 1, 0, 0], [0, -7, 0, 0, -18, 0, -16, -7], [0, -5, 0, 0, -6, 0, -27, -11], [0, -18, 0, 0, 1, 0, -16, -2], [-31, 0, 23, -7, 0, -16, 0, 0], [0, -13, 0, 0, 11, 0, -1, -14], [0, 18, 0, 0, 10, 0, 8, 7], [-6, 0, -1, 2, 0, -24, 0, 0], [0, -2, 0, 0, 5, 0, -6, -10], [-3, 0, 16, 1, 0, -19, 0, 0], [0, 9, 0, 0, -6, 0, 11, 14], [-31, 0, 6, -12, 0, -21, 0, 0], [8, 0, 12, 7, 0, -17, 0, 0], [-27, 0, 14, -9, 0, 1, 0, 0], [0, -16, 0, 0, 13, 0, -1, -10], [4, 0, -31, -5, 0, 13, 0, 0], [19, 0, -29, 9, 0, 4, 0, 0], [-9, 0, 28, 19, 0, -11, 0, 0], [-5, 0, 14, -17, 0, -4, 0, 0], [0, -25, 0, 0, 3, 0, -6, -19], [-10, 0, 19, 20, 0, -7, 0, 0], [0, 4, 0, 0, -10, 0, -9, -2], [-14, 0, 20, 25, 0, -8, 0, 0], [0, 10, 0, 0, -9, 0, 1, 20], [0, 11, 0, 0, 13, 0, 14, -1], [0, -16, 0, 0, -2, 0, -4, 9], [-14, 0, -2, -10, 0, -18, 0, 0], [0, 5, 0, 0, -4, 0, -12, 13], [0, 2, 0, 0, -8, 0, -22, 2], [9, 0, -25, 12, 0, 1, 0, 0], [-44, 0, 7, 2, 0, -9, 0, 0], [0, 9, 0, 0, 10, 0, 4, 5], [0, 8, 0, 0, -17, 0, -25, -5], [-10, 0, 10, -1, 0, 3, 0, 0], [-44, 0, 10, 0, 0, -13, 0, 0], [0, -25, 0, 0, 12, 0, 18, -2], [0, 6, 0, 0, 5, 0, 19, 16], [22, 0, -5, 14, 0, 0, 0, 0], [0, 12, 0, 0, -5, 0, 8, 5], [0, -7, 0, 0, 5, 0, 2, 6], [0, 24, 0, 0, 4, 0, -1, 18], [-3, 0, 18, 1, 0, 19, 0, 0], [-50, 0, 14, -14, 0, -14, 0, 0], [0, 23, 0, 0, -14, 0, -10, 5], [8, 0, -22, -11, 0, 1, 0, 0], [-6, 0, 7, 13, 0, -8, 0, 0], [0, -15, 0, 0, -6, 0, 4, -10], [-42, 0, 23, -14, 0, -25, 0, 0], [0, 2, 0, 0, 1, 0, -10, -3], [3, 0, 2, -8, 0, 11, 0, 0], [26, 0, -1, -13, 0, 35, 0, 0], [0, -1, 0, 0, -1, 0, -3, 8], [35, 0, 5, 6, 0, 17, 0, 0], [0, -11, 0, 0, -2, 0, -11, -18], [0, 6, 0, 0, -4, 0, -2, 8], [-19, 0, 10, 18, 0, -11, 0, 0], [11, 0, 0, 19, 0, -10, 0, 0], [0, 10, 0, 0, 8, 0, 9, 26], [-11, 0, -3, 1, 0, 26, 0, 0], [0, 1, 0, 0, 12, 0, 8, -8], [0, -3, 0, 0, -13, 0, -15, -12], [-11, 0, -11, -14, 0, -29, 0, 0], [-31, 0, 7, -11, 0, -29, 0, 0], [-25, 0, -3, -7, 0, 4, 0, 0], [0, 10, 0, 0, -3, 0, -19, 1], [0, 21, 0, 0, 16, 0, 2, 12], [-20, 0, 14, -13, 0, 4, 0, 0], [0, 27, 0, 0, -12, 0, 2, 2], [-6, 0, 2, 19, 0, -24, 0, 0], [0, 13, 0, 0, -22, 0, 11, 10], [-13, 0, 23, 29, 0, 3, 0, 0], [0, 16, 0, 0, -20, 0, 12, 20], [35, 0, 11, -6, 0, 22, 0, 0], [35, 0, -22, 18, 0, 23, 0, 0], [-43, 0, 20, -10, 0, -47, 0, 0], [0, 22, 0, 0, -26, 0, 0, 10], [0, 15, 0, 0, 13, 0, 4, 3], [29, 0, 13, -4, 0, 25, 0, 0], [0, 24, 0, 0, -19, 0, 9, 11], [-7, 0, -11, 5, 0, 18, 0, 0], [14, 0, -1, 21, 0, 12, 0, 0], [0, 4, 0, 0, -16, 0, -30, 1], [-39, 0, 16, 2, 0, -1, 0, 0], [0, 15, 0, 0, 24, 0, 19, 3], [15, 0, 21, 29, 0, -24, 0, 0], [0, -30, 0, 0, 9, 0, -10, -7], [-37, 0, 19, 7, 0, -10, 0, 0], [0, 4, 0, 0, -1, 0, 1, 18], [-6, 0, -14, -1, 0, 7, 0, 0], [3, 0, 14, 10, 0, -38, 0, 0], [0, -6, 0, 0, -12, 0, -2, 12], [0, -15, 0, 0, -7, 0, -10, 9], [-21, 0, 2, 10, 0, -27, 0, 0], [0, -5, 0, 0, 1, 0, 10, 22], [20, 0, -12, 13, 0, 16, 0, 0], [0, -6, 0, 0, 3, 0, 5, 5], [-19, 0, 0, -32, 0, 5, 0, 0], [18, 0, -17, -13, 0, 16, 0, 0], [0, 2, 0, 0, -9, 0, 11, -3], [0, -6, 0, 0, -16, 0, 9, -13], [0, 2, 0, 0, -22, 0, -30, -12], [0, 17, 0, 0, -5, 0, 4, 18], [0, -4, 0, 0, 12, 0, 15, 15], [28, 0, -17, 7, 0, 36, 0, 0], [0, 16, 0, 0, 17, 0, 40, 28], [0, -7, 0, 0, 17, 0, 14, 5], [-40, 0, 19, -16, 0, -14, 0, 0], [15, 0, 5, -1, 0, -9, 0, 0], [13, 0, -1, -8, 0, 12, 0, 0], [0, 24, 0, 0, -24, 0, 2, 16], [0, 21, 0, 0, 12, 0, 25, 32], [9, 0, 18, -4, 0, 14, 0, 0], [0, 19, 0, 0, -1, 0, 20, 18], [0, 0, 0, 0, -20, 0, -15, 1], [16, 0, -16, 9, 0, -17, 0, 0], [0, 20, 0, 0, -8, 0, -16, -7], [0, 6, 0, 0, -16, 0, -36, -4], [0, 6, 0, 0, 2, 0, -3, -26], [17, 0, 23, 29, 0, -18, 0, 0], [-3, 0, -16, -46, 0, 15, 0, 0], [-16, 0, -33, -15, 0, 29, 0, 0], [0, -13, 0, 0, 19, 0, 28, 11], [-34, 0, 19, 2, 0, -51, 0, 0], [0, 24, 0, 0, -6, 0, 21, 1], [-5, 0, -30, -23, 0, -6, 0, 0], [0, -3, 0, 0, 7, 0, 25, 12], [-3, 0, -21, 4, 0, -8, 0, 0], [0, 7, 0, 0, -29, 0, -6, 3], [0, 8, 0, 0, -6, 0, -7, -10], [20, 0, -30, 16, 0, 9, 0, 0], [29, 0, 1, 20, 0, 23, 0, 0], [-9, 0, -9, -22, 0, -23, 0, 0], [0, -27, 0, 0, 16, 0, 3, -18], [0, -10, 0, 0, 31, 0, 7, -16], [-35, 0, 10, 0, 0, -22, 0, 0], [0, -20, 0, 0, 11, 0, 29, 0], [14, 0, -10, 5, 0, -7, 0, 0], [15, 0, -12, -27, 0, 7, 0, 0], [0, 18, 0, 0, -9, 0, 31, 11], [-69, 0, 14, -11, 0, -30, 0, 0], [0, -26, 0, 0, 15, 0, -4, -5], [0, 12, 0, 0, 15, 0, 37, 3], [0, -6, 0, 0, 2, 0, 15, -4], [-6, 0, 8, -19, 0, -4, 0, 0], [0, -30, 0, 0, 6, 0, 4, -21], [13, 0, 9, 0, 0, 18, 0, 0], [0, 26, 0, 0, -14, 0, 7, 4], [0, -6, 0, 0, -23, 0, -10, 21], [-16, 0, 17, 12, 0, -46, 0, 0], [-4, 0, 22, 17, 0, -19, 0, 0], [0, 25, 0, 0, 19, 0, 3, -4], [0, -17, 0, 0, 12, 0, 22, 14], [-13, 0, 29, 4, 0, 25, 0, 0], [35, 0, -4, 2, 0, 38, 0, 0], [0, 11, 0, 0, 21, 0, 5, -15], [0, -17, 0, 0, -8, 0, 0, -25], [21, 0, 18, 19, 0, -2, 0, 0], [16, 0, -7, -16, 0, 17, 0, 0], [-70, 0, 7, -33, 0, -38, 0, 0], [0, -9, 0, 0, 30, 0, 0, -8], [-31, 0, 58, 2, 0, -38, 0, 0], [-15, 0, 5, 35, 0, -20, 0, 0], [0, 42, 0, 0, 1, 0, 9, 17], [-24, 0, -3, -25, 0, -34, 0, 0], [0, -25, 0, 0, 10, 0, 12, -11], [0, -12, 0, 0, 7, 0, -25, 5], [43, 0, -7, 0, 0, 15, 0, 0], [45, 0, 7, 8, 0, 42, 0, 0], [0, 12, 0, 0, 8, 0, -1, 27], [0, 17, 0, 0, 7, 0, 24, -9], [0, -8, 0, 0, -30, 0, -11, 24], [67, 0, -8, 5, 0, 28, 0, 0], [0, 10, 0, 0, -6, 0, 4, 16], [-17, 0, 9, -19, 0, 15, 0, 0], [0, 48, 0, 0, -6, 0, -5, 9], [25, 0, 23, 32, 0, 12, 0, 0], [0, -48, 0, 0, 6, 0, 0, -9], [-11, 0, -29, 19, 0, 7, 0, 0], [0, 0, 0, 0, 6, 0, 9, -7], [-29, 0, 41, -5, 0, -21, 0, 0], [0, 16, 0, 0, 22, 0, 16, 2], [0, -8, 0, 0, 13, 0, -1, -9], [0, 10, 0, 0, 8, 0, -7, 28], [30, 0, -33, 25, 0, 8, 0, 0], [-8, 0, 35, 11, 0, -15, 0, 0], [0, -1, 0, 0, -34, 0, -16, -3], [24, 0, -22, -27, 0, 28, 0, 0], [-31, 0, -1, 23, 0, -19, 0, 0], [0, 33, 0, 0, -4, 0, 6, -1], [-25, 0, -22, 4, 0, 3, 0, 0], [0, -15, 0, 0, 17, 0, 19, -5], [75, 0, -39, 22, 0, 48, 0, 0], [0, -18, 0, 0, 11, 0, 4, -12], [36, 0, -46, -15, 0, -7, 0, 0], [0, 1, 0, 0, -24, 0, -19, 4], [31, 0, -51, -17, 0, 15, 0, 0], [0, -14, 0, 0, 2, 0, 0, -5], [-8, 0, 20, 25, 0, -51, 0, 0], [-7, 0, -1, -1, 0, 36, 0, 0], [0, -7, 0, 0, 14, 0, 13, 5], [0, 31, 0, 0, 7, 0, -7, 8], [0, 2, 0, 0, 15, 0, -7, -24], [15, 0, -19, -27, 0, 0, 0, 0], [0, -14, 0, 0, 1, 0, -2, 31], [22, 0, -23, -23, 0, -14, 0, 0], [0, 1, 0, 0, -24, 0, -47, -18], [0, 14, 0, 0, -22, 0, 12, 10], [0, 0, 0, 0, -15, 0, -6, -3], [0, 19, 0, 0, -23, 0, -3, 21], [48, 0, 7, 11, 0, -15, 0, 0], [2, 0, -15, -38, 0, 37, 0, 0], [35, 0, -9, -8, 0, 72, 0, 0], [0, -3, 0, 0, 10, 0, 16, 3], [-1, 0, -36, -28, 0, -1, 0, 0], [15, 0, 13, 31, 0, -34, 0, 0], [0, 18, 0, 0, 9, 0, 6, -29], [-40, 0, -8, -13, 0, -49, 0, 0], [-16, 0, -3, -36, 0, 0, 0, 0], [0, -27, 0, 0, 2, 0, -30, -40], [-14, 0, 23, -19, 0, 5, 0, 0], [0, -5, 0, 0, -26, 0, 6, -7], [7, 0, -25, 1, 0, 5, 0, 0], [0, -23, 0, 0, -23, 0, -28, -13], [0, 20, 0, 0, 15, 0, 6, 5], [0, -23, 0, 0, -4, 0, -35, -8], [63, 0, -11, -7, 0, 3, 0, 0], [0, 4, 0, 0, -12, 0, -1, 16], [-45, 0, 15, -2, 0, -7, 0, 0], [0, 21, 0, 0, -2, 0, -15, -3], [0, -8, 0, 0, -16, 0, 6, 32], [0, 14, 0, 0, -2, 0, 30, 2], [10, 0, -37, 1, 0, -1, 0, 0], [0, 9, 0, 0, -2, 0, -27, 7], [8, 0, 21, -28, 0, 17, 0, 0], [0, 37, 0, 0, 11, 0, 9, 16], [9, 0, 18, -1, 0, -42, 0, 0], [0, -16, 0, 0, 23, 0, 26, 9], [6, 0, 2, -26, 0, 34, 0, 0], [-16, 0, 14, -16, 0, 6, 0, 0], [-76, 0, 21, -15, 0, -18, 0, 0], [54, 0, -21, 2, 0, 13, 0, 0], [-10, 0, 30, 19, 0, -57, 0, 0], [0, -26, 0, 0, -6, 0, -11, 2], [0, 34, 0, 0, 3, 0, 14, 31], [0, 5, 0, 0, -11, 0, 17, 17], [62, 0, -21, 2, 0, 14, 0, 0], [-18, 0, 2, 10, 0, -39, 0, 0], [0, 35, 0, 0, 8, 0, -7, 7], [30, 0, -6, 19, 0, -10, 0, 0], [0, 19, 0, 0, -25, 0, -8, -5], [-42, 0, 5, 13, 0, -29, 0, 0], [0, -1, 0, 0, -30, 0, -32, -9], [0, 17, 0, 0, -15, 0, -8, 18], [0, -13, 0, 0, -26, 0, -1, -1], [-21, 0, 9, -26, 0, -22, 0, 0], [0, -19, 0, 0, -5, 0, -26, -9], [16, 0, -35, -2, 0, 54, 0, 0], [-3, 0, 39, 26, 0, 18, 0, 0], [0, 20, 0, 0, -17, 0, -32, 3], [0, -36, 0, 0, 19, 0, -17, -3], [0, -21, 0, 0, 33, 0, 31, -4], [36, 0, 8, -7, 0, 9, 0, 0], [0, 3, 0, 0, 2, 0, 20, -4], [0, 47, 0, 0, -29, 0, 25, 22], [28, 0, -4, 16, 0, 18, 0, 0], [0, 15, 0, 0, 6, 0, -6, 16], [0, -7, 0, 0, 2, 0, 6, 9], [0, 45, 0, 0, -11, 0, 1, 5], [-13, 0, 49, -10, 0, -23, 0, 0], [-9, 0, -21, -12, 0, -9, 0, 0], [-40, 0, -15, -4, 0, -6, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3025_2_a_bk();".
// 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_3025_2_a_bk(:prec:=8)
    chi := MakeCharacter_3025_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_3025_2_a_bk();".
// 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_3025_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![11, 0, -31, 0, 27, 0, -9, 0, 1]>,<3,R![11, 0, -91, 0, 62, 0, -14, 0, 1]>,<19,R![11, -39, -4, 6, 1]>],Snew);
    return Vf;
end function;