
// Make newform 225.2.k.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_225_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_225_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_225_2_k_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_225_2_k_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 := [81, 0, -441, 0, 2023, 0, -1842, 0, 1167, 0, -406, 0, 102, 0, -12, 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, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7606359, 0, 7466556, 0, -1299110, 0, 709118, 0, -306752, 0, 210280, 0, -28693, 0, 3457, 0], [-18063, 0, 82467, 0, -94895, 0, 38576, 0, -10439, 0, 10, 0, 74, 0, -26, 0], [0, -16863264, 0, 19722086, 0, -20515435, 0, 10468603, 0, -3838362, 0, 784955, 0, -90503, 0, 5917], [-5812992, 0, -7975497, 0, 3626995, 0, -4803841, 0, 1494574, 0, -572135, 0, 69641, 0, -7559, 0], [-814212, 0, 21981813, 0, -19088030, 0, 12811274, 0, -4500371, 0, 1170340, 0, -138574, 0, 11836, 0], [0, 2721243, 0, 467253, 0, -2137205, 0, 2480389, 0, -1090526, 0, 332215, 0, -41414, 0, 3766], [0, 4790637, 0, -21981813, 0, 19088030, 0, -12811274, 0, 4500371, 0, -1170340, 0, 138574, 0, -11836], [-5535918, 0, -1848168, 0, 8453480, 0, -9114389, 0, 4313456, 0, -1314040, 0, 165889, 0, -14896, 0], [0, 11503071, 0, 20660066, 0, -24805735, 0, 17277208, 0, -6027522, 0, 1451855, 0, -168383, 0, 13477], [1468098, 0, -8169474, 0, 6436600, 0, -3846136, 0, 1252588, 0, -324695, 0, 38066, 0, -3323, 0], [0, -10338741, 0, 47402299, 0, -44578415, 0, 26781707, 0, -9130758, 0, 2129695, 0, -245932, 0, 19253], [-212679, 0, 809301, 0, -869185, 0, 579583, 0, -215542, 0, 56330, 0, -6758, 0, 572, 0], [0, -6887691, 0, 89205019, 0, -81131165, 0, 51780332, 0, -17623398, 0, 4432420, 0, -518857, 0, 43568], [0, -564507, 0, -70902062, 0, 67106795, 0, -47712236, 0, 17097154, 0, -4656460, 0, 557311, 0, -49089]];
    Rf_basisdens := [1, 1, 3976425, 29025, 3976425, 3976425, 3976425, 1325475, 3976425, 3976425, 3976425, 795285, 3976425, 92475, 3976425, 3976425];
    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_225_k();"
function MakeCharacter_225_k()
    N := 225;
    order := 6;
    char_gens := [101, 127];
    v := [2, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_225_k_Hecke();"
function MakeCharacter_225_k_Hecke(Kf)
    N := 225;
    order := 6;
    char_gens := [101, 127];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, -1, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 0, 2, 0, -2, 0], [0, -1, 0, 0, -1, 0, 0, 3, 1, 0, -1, 0, 3, 0, -1, 2], [-1, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 1], [0, 0, -2, -1, 0, 2, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [-1, 0, -2, 1, 0, -2, 1, 0, 0, -1, 0, -2, 0, 2, 0, 0], [0, -3, 0, 0, -2, 0, 0, -1, 3, 0, -2, 0, -1, 0, 1, 1], [-2, 0, -1, 2, 0, 1, 3, 0, 0, -2, 0, 1, 0, 1, 0, 0], [0, 4, 0, 0, -2, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0], [0, -2, 0, 0, 2, 0, 0, -6, 0, 0, -1, 0, 0, 0, -1, -1], [0, 1, 0, 0, -3, 0, 0, -3, -1, 0, -3, 0, -3, 0, 0, 3], [-2, 0, -3, 2, 0, 0, 6, 0, 0, -2, 0, 0, 0, 3, 0, 0], [4, 0, 3, 1, 0, -3, -4, 0, 0, 0, 0, 1, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -1, 0, -2, 0, 0, -1], [-1, 0, 3, 0, 0, 0, 0, 0, 0, -1, 0, 3, 0, 3, 0, 0], [0, -1, 0, 0, 1, 0, 0, -2, 1, 0, 1, 0, -2, 0, 2, -3], [-1, 0, 2, 1, 0, -2, 1, 0, 0, 0, 0, -1, 0, 3, 0, 0], [0, 3, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0], [0, 3, 0, 0, -1, 0, 0, 7, 0, 0, -1, 0, 0, 0, 2, 2], [3, 0, 2, -3, 0, -2, -3, 0, 0, 0, 0, -2, 0, 4, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, -3, 0, -1, 0, -1, 0, 2, -1], [0, 6, 0, 0, 1, 0, 0, 6, -6, 0, 1, 0, 6, 0, 1, -2], [7, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, -5, 0, 1, 0, 3, 0, 1, 1], [0, 1, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 2, 0, 2, -2], [0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, -3, 0, 0], [0, 3, 0, 0, -4, 0, 0, 0, 0, 0, -1, 0, 0, 0, 5, 5], [-6, 0, 0, 6, 0, -3, 5, 0, 0, -6, 0, -3, 0, 0, 0, 0], [5, 0, 1, -5, 0, -1, -9, 0, 0, 5, 0, -1, 0, -1, 0, 0], [-5, 0, -6, 4, 0, 6, 5, 0, 0, 0, 0, 1, 0, -7, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 3, 0, -2, 0, -2, 0, 1, -2], [0, -6, 0, 0, -5, 0, 0, 4, 6, 0, -5, 0, 4, 0, -2, 7], [0, 0, 0, 0, -1, 0, 0, 0, -6, 0, -1, 0, -6, 0, -1, -1], [0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 1], [-4, 0, -6, 0, 0, 0, 0, 0, 0, 2, 0, -6, 0, -6, 0, 0], [-4, 0, 4, 0, 0, 2, 0, 0, 0, -3, 0, 2, 0, 2, 0, 0], [-9, 0, 0, 10, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 0, -4, 0, 1, -2], [0, -6, 0, 0, -1, 0, 0, 3, 6, 0, -1, 0, 3, 0, -1, 2], [-4, 0, -1, 0, 0, 5, 0, 0, 0, 7, 0, -6, 0, -6, 0, 0], [3, 0, -4, -3, 0, 1, 1, 0, 0, 3, 0, 1, 0, 4, 0, 0], [0, -6, 0, 0, 0, 0, 0, 2, 0, 0, -3, 0, 0, 0, 3, 3], [0, -1, 0, 0, 1, 0, 0, -12, 0, 0, 1, 0, 0, 0, -2, -2], [-2, 0, -1, 2, 0, 5, -1, 0, 0, -2, 0, 5, 0, 1, 0, 0], [0, -3, 0, 0, 1, 0, 0, 3, 3, 0, 1, 0, 3, 0, -2, 1], [-8, 0, -5, 8, 0, -7, -9, 0, 0, -8, 0, -7, 0, 5, 0, 0], [-5, 0, 4, 7, 0, -4, 5, 0, 0, 0, 0, 0, 0, 4, 0, 0], [0, 0, 3, 0, 0, -6, 0, 0, 0, -6, 0, 9, 0, 9, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 3, 0, 3, 0, -12, 0, 6, 3], [0, -8, 0, 0, 8, 0, 0, 3, 0, 0, -7, 0, 0, 0, -1, -1], [14, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0], [4, 0, -7, 0, 0, 0, 0, 0, 0, -1, 0, -7, 0, -7, 0, 0], [0, 4, 0, 0, 1, 0, 0, -2, 0, 0, 11, 0, 0, 0, -12, -12], [-6, 0, -2, -3, 0, 2, 6, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, -3, 0, 0, 0, 7, 0, 7, 0, -4, 0, -3, 7], [0, 0, 0, 0, -5, 0, 0, 0, 3, 0, -2, 0, -3, 0, -5, -2], [0, 1, 0, 0, -5, 0, 0, -13, -1, 0, -5, 0, -13, 0, -3, 8], [-9, 0, 2, 9, 0, 7, -9, 0, 0, -9, 0, 7, 0, -2, 0, 0], [0, 6, 0, 0, -5, 0, 0, -7, 0, 0, 1, 0, 0, 0, 4, 4], [0, 2, 0, 0, -7, 0, 0, 0, 0, 0, 2, 0, 0, 0, 5, 5], [4, 0, -7, 2, 0, 7, -4, 0, 0, 0, 0, -6, 0, -1, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 4, 0, 1, 0, -8, 0, -6, 1], [0, 0, 0, 0, 13, 0, 0, 0, 2, 0, -2, 0, -9, 0, 13, -2], [-7, 0, 1, -1, 0, -1, 7, 0, 0, 0, 0, 7, 0, -6, 0, 0], [0, 3, 0, 0, -2, 0, 0, -9, 0, 0, -5, 0, 0, 0, 7, 7], [-3, 0, -1, 0, 0, 4, 0, 0, 0, -3, 0, -5, 0, -5, 0, 0], [0, 7, 0, 0, -2, 0, 0, 10, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, -4, 0, -6, 0, 2, 0, 4, -6], [11, 0, 5, 0, 0, -1, 0, 0, 0, -5, 0, 6, 0, 6, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, -9, 0, -8, 0, 9, 0, 4, -8], [-15, 0, 8, 3, 0, -8, 15, 0, 0, 0, 0, 1, 0, 7, 0, 0], [0, -8, 0, 0, 6, 0, 0, -4, 8, 0, 6, 0, -4, 0, -7, 1], [0, 0, -7, 0, 0, -5, -12, 0, 0, 0, 0, -5, 0, 7, 0, 0], [0, 0, -5, 0, 0, -1, 17, 0, 0, 0, 0, -1, 0, 5, 0, 0], [8, 0, 4, -8, 0, 5, 9, 0, 0, 8, 0, 5, 0, -4, 0, 0], [1, 0, 0, -8, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0], [-9, 0, -8, 0, 0, -1, 0, 0, 0, 6, 0, -7, 0, -7, 0, 0], [0, 2, 0, 0, -5, 0, 0, 4, -2, 0, -5, 0, 4, 0, -1, 6], [-4, 0, -6, 8, 0, 6, 4, 0, 0, 0, 0, 2, 0, -8, 0, 0], [0, 3, 0, 0, 2, 0, 0, -3, 0, 0, 5, 0, 0, 0, -7, -7], [14, 0, -3, 0, 0, -6, 0, 0, 0, -4, 0, 3, 0, 3, 0, 0], [0, -7, 0, 0, 1, 0, 0, -20, 0, 0, -6, 0, 0, 0, 5, 5], [-27, 0, -13, 15, 0, 13, 27, 0, 0, 0, 0, -8, 0, -5, 0, 0], [0, 0, 0, 0, -8, 0, 0, 0, -8, 0, 5, 0, 8, 0, -8, 5], [0, -8, 0, 0, 2, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, -4], [-15, 0, -2, 18, 0, 2, 15, 0, 0, 0, 0, -4, 0, 2, 0, 0], [0, 0, 0, 0, 6, 0, 0, 11, 0, 0, 6, 0, 11, 0, -6, 0], [13, 0, 4, -13, 0, 11, 3, 0, 0, 13, 0, 11, 0, -4, 0, 0], [11, 0, 2, -11, 0, 6, -13, 0, 0, 11, 0, 6, 0, -2, 0, 0], [0, 5, 0, 0, 12, 0, 0, -18, -5, 0, 12, 0, -18, 0, 3, -15], [-5, 0, 8, 5, 0, 4, -6, 0, 0, -5, 0, 4, 0, -8, 0, 0], [14, 0, -3, 0, 0, 6, 0, 0, 0, -4, 0, -9, 0, -9, 0, 0], [0, 11, 0, 0, 2, 0, 0, 22, -11, 0, 2, 0, 22, 0, 0, -2], [9, 0, 7, 0, 0, 6, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0], [0, 5, 0, 0, -2, 0, 0, 19, 0, 0, -12, 0, 0, 0, 14, 14], [0, 0, 0, 0, -2, 0, 0, -6, 0, 0, -2, 0, -6, 0, 7, -5], [0, 0, 0, 0, 8, 0, 0, 0, 9, 0, -4, 0, 6, 0, 8, -4], [-9, 0, -1, -12, 0, 1, 9, 0, 0, 0, 0, 7, 0, -8, 0, 0], [3, 0, 3, -3, 0, 6, 19, 0, 0, 3, 0, 6, 0, -3, 0, 0], [0, 0, 0, 0, -5, 0, 0, -13, 0, 0, -5, 0, -13, 0, 10, -5], [0, 0, 0, 0, 5, 0, 0, -3, 0, 0, 8, 0, 0, 0, -13, -13], [0, 14, 0, 0, -6, 0, 0, 15, -14, 0, -6, 0, 15, 0, 9, -3], [5, 0, 1, -5, 0, -1, -9, 0, 0, 5, 0, -1, 0, -1, 0, 0], [7, 0, 11, -12, 0, -11, -7, 0, 0, 0, 0, 10, 0, 1, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 22, 0, 8, 0, -14, 0, -2, 8], [0, -13, 0, 0, 8, 0, 0, -14, 13, 0, 8, 0, -14, 0, -9, 1], [0, 0, 0, 0, -14, 0, 0, 0, 3, 0, 4, 0, 12, 0, -14, 4], [8, 0, 16, 0, 0, -16, -8, 0, 0, 0, 0, 5, 0, 11, 0, 0], [1, 0, 2, 0, 0, -6, 0, 0, 0, 11, 0, 8, 0, 8, 0, 0], [-9, 0, 5, -6, 0, -5, 9, 0, 0, 0, 0, -8, 0, 13, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 11, 0, 3, 1], [0, 8, 0, 0, -1, 0, 0, 33, -8, 0, -1, 0, 33, 0, 2, -1], [0, 0, 0, 0, -10, 0, 0, 0, -7, 0, -1, 0, -9, 0, -10, -1], [21, 0, -6, -11, 0, 6, -21, 0, 0, 0, 0, 0, 0, -6, 0, 0], [-5, 0, 4, 5, 0, 0, -23, 0, 0, -5, 0, 0, 0, -4, 0, 0], [0, -7, 0, 0, -1, 0, 0, 20, 0, 0, 10, 0, 0, 0, -9, -9], [0, 6, 0, 0, -9, 0, 0, -6, 0, 0, 15, 0, 0, 0, -6, -6], [0, -6, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, -6, 0, 3, -3], [13, 0, 7, -17, 0, -7, -13, 0, 0, 0, 0, 6, 0, 1, 0, 0], [20, 0, 11, 0, 0, 1, 0, 0, 0, 8, 0, 10, 0, 10, 0, 0], [14, 0, 2, -4, 0, -2, -14, 0, 0, 0, 0, 9, 0, -7, 0, 0], [9, 0, 18, 0, 0, 3, 0, 0, 0, 9, 0, 15, 0, 15, 0, 0], [0, -9, 0, 0, 0, 0, 0, -5, 0, 0, -15, 0, 0, 0, 15, 15], [0, 0, 0, 0, 9, 0, 0, 0, 21, 0, 0, 0, -4, 0, 9, 0], [-4, 0, 1, 0, 0, 3, 0, 0, 0, 13, 0, -2, 0, -2, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, -5, 0, -3, 0, 12, 0, 6, -3], [4, 0, 2, -4, 0, -7, -23, 0, 0, 4, 0, -7, 0, -2, 0, 0], [0, 25, 0, 0, 0, 0, 0, 17, -25, 0, 0, 0, 17, 0, 5, -5], [-9, 0, -8, 9, 0, -13, -9, 0, 0, -9, 0, -13, 0, 8, 0, 0], [-6, 0, -10, 6, 0, -8, -25, 0, 0, -6, 0, -8, 0, 10, 0, 0], [0, -6, 0, 0, -11, 0, 0, 0, 6, 0, -11, 0, 0, 0, 4, 7], [0, 0, 0, 0, 9, 0, 0, 0, 4, 0, 4, 0, 4, 0, 9, 4], [0, 0, 0, 0, 5, 0, 0, 0, -14, 0, -10, 0, 12, 0, 5, -10], [1, 0, 0, 0, 0, -6, 0, 0, 0, 10, 0, 6, 0, 6, 0, 0], [18, 0, -8, 0, 0, -3, 0, 0, 0, 1, 0, -5, 0, -5, 0, 0], [-18, 0, -14, 0, 0, 14, 18, 0, 0, 0, 0, -4, 0, -10, 0, 0], [0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -1, 0, -25, 0, 11, -1], [0, -4, 0, 0, -2, 0, 0, -6, 4, 0, -2, 0, -6, 0, -11, 13], [-3, 0, 7, 0, 0, 1, 0, 0, 0, 11, 0, 6, 0, 6, 0, 0], [0, 0, -2, 2, 0, 2, 0, 0, 0, 0, 0, -7, 0, 5, 0, 0], [0, -10, 0, 0, 4, 0, 0, -4, 0, 0, -6, 0, 0, 0, 2, 2], [0, -2, 0, 0, 12, 0, 0, -21, 0, 0, -6, 0, 0, 0, -6, -6], [11, 0, 4, -11, 0, 16, 5, 0, 0, 11, 0, 16, 0, -4, 0, 0], [0, -10, 0, 0, 12, 0, 0, -33, 10, 0, 12, 0, -33, 0, -12, 0], [0, 0, 0, 0, 3, 0, 0, 0, 2, 0, -7, 0, -14, 0, 3, -7], [-29, 0, -9, 0, 0, 0, 0, 0, 0, -2, 0, -9, 0, -9, 0, 0], [0, 18, 0, 0, -9, 0, 0, -2, -18, 0, -9, 0, -2, 0, 0, 9], [0, 0, 0, 0, 3, 0, 0, 0, 12, 0, -6, 0, -15, 0, 3, -6], [0, -8, 0, 0, -5, 0, 0, -2, 0, 0, 17, 0, 0, 0, -12, -12], [3, 0, -8, -4, 0, 8, -3, 0, 0, 0, 0, 5, 0, -13, 0, 0], [5, 0, -7, 0, 0, 2, 0, 0, 0, 4, 0, -9, 0, -9, 0, 0], [-12, 0, 1, 5, 0, -1, 12, 0, 0, 0, 0, 5, 0, -4, 0, 0], [0, 20, 0, 0, -5, 0, 0, 2, -20, 0, -5, 0, 2, 0, 5, 0], [10, 0, 5, -10, 0, 10, -24, 0, 0, 10, 0, 10, 0, -5, 0, 0], [0, -15, 0, 0, 7, 0, 0, 15, 0, 0, -17, 0, 0, 0, 10, 10], [0, -17, 0, 0, -1, 0, 0, 0, 17, 0, -1, 0, 0, 0, -13, 14], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -3, 0, -20, 0, 0, -3], [-33, 0, -8, 0, 0, -7, 0, 0, 0, -9, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 5, 0, 0, 0, -4, 0, 11, 0, 6, 0, 5, 11], [0, -6, 0, 0, 3, 0, 0, 24, 0, 0, -3, 0, 0, 0, 0, 0], [-4, 0, -2, 0, 0, 8, 0, 0, 0, 9, 0, -10, 0, -10, 0, 0], [0, 4, 0, 0, -3, 0, 0, -38, 0, 0, 1, 0, 0, 0, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_225_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_225_2_k_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_225_2_k_c(:prec:=16)
    chi := MakeCharacter_225_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_225_2_k_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_225_2_k_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 0, -441, 0, 2023, 0, -1842, 0, 1167, 0, -406, 0, 102, 0, -12, 0, 1]>],Snew);
    return Vf;
end function;