
// Make newform 675.2.k.c in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_675_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_675_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_675_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], [0, 4310244, 0, -3474, 0, 15890, 0, 208132, 0, 8108, 0, -2470, 0, 943, 0, -28], [18063, 0, -82467, 0, 94895, 0, -38576, 0, 10439, 0, -10, 0, -74, 0, 26, 0], [0, 2158308, 0, 191070, 0, -873950, 0, 1153589, 0, -445940, 0, 135850, 0, -16519, 0, 1540], [-126387, 0, 33003, 0, -150955, 0, 148849, 0, -77026, 0, 23465, 0, -2999, 0, 266, 0], [814212, 0, -21981813, 0, 19088030, 0, -12811274, 0, 4500371, 0, -1170340, 0, 138574, 0, -11836, 0], [713286, 0, -26841054, 0, 16721965, 0, -11223247, 0, 3257143, 0, -1025270, 0, 121397, 0, -12338, 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], [-12812391, 0, 58685109, 0, -59345540, 0, 31813907, 0, -10392128, 0, 2012770, 0, -220432, 0, 13873, 0], [-5535918, 0, -1848168, 0, 8453480, 0, -9114389, 0, 4313456, 0, -1314040, 0, 165889, 0, -14896, 0], [14978844, 0, 2357109, 0, -10781365, 0, 13209112, 0, -5501278, 0, 1675895, 0, -206837, 0, 18998, 0], [0, -10338741, 0, 47402299, 0, -44578415, 0, 26781707, 0, -9130758, 0, 2129695, 0, -245932, 0, 19253], [0, -13412214, 0, 61524806, 0, -57068185, 0, 35467228, 0, -12331002, 0, 3087680, 0, -363428, 0, 30232], [0, -26067942, 0, 119626108, 0, -105967505, 0, 70028009, 0, -24700761, 0, 6504640, 0, -773434, 0, 66701]];
    Rf_basisdens := [1, 1, 795285, 29025, 795285, 92475, 3976425, 3976425, 1325475, 3976425, 3976425, 3976425, 3976425, 3976425, 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_675_k();"
function MakeCharacter_675_k()
    N := 675;
    order := 6;
    char_gens := [326, 352];
    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_675_k_Hecke();"
function MakeCharacter_675_k_Hecke(Kf)
    N := 675;
    order := 6;
    char_gens := [326, 352];
    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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 2, -1, -2, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0], [0, 0, -1, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, -2, -1], [-2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 5, 1, -1], [-2, 0, 0, -2, 0, 1, -2, -1, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 2, -1, -2, 0, 0, 2, 1, 2, 0, 0, 0], [0, 4, -1, 0, 1, 0, 0, 0, 3, -4, 0, 0, 0, 3, 1, -1], [1, 0, 0, 1, 0, -1, 1, -1, 0, 0, -1, 1, -1, 0, 0, 0], [0, -2, -2, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -1, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -3, 0, 0, 0, -6, -1, 0, 0, 0, -6, -3, 0], [-1, 0, 0, -1, 0, -3, 3, 0, 0, 0, -3, -1, -3, 0, 0, 0], [1, 0, 0, -2, 0, 2, 1, -2, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, -1], [4, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 3, -2, 0, -3, 0, 0, 0, 1, -3, 0, 0, 0, 1, -3, -2], [-3, 0, 0, 0, 0, 3, -3, -3, 0, 0, 1, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0], [0, -2, -1, 0, 1, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -5, 0, -4, -1, 4, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -1], [0, 5, 1, 0, 2, 0, 0, 0, 7, -5, 0, 0, 0, 7, 2, 1], [6, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 5, 1, -1], [0, 1, -2, 0, -2, 0, 0, 0, -2, -1, 0, 0, 0, -2, -2, -2], [3, 0, 0, 3, 0, 3, 6, 0, 0, 0, 3, 3, 3, 0, 0, 0], [0, -1, 4, 0, -1, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, -6, 0, 0, -8, -3, 0, 0, 0, -6, 0, 0, 0, 0], [-4, 0, 0, -4, 0, 1, -7, 1, 0, 0, 1, -4, 1, 0, 0, 0], [1, 0, 0, -5, 0, -7, 1, 7, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 3, -1, -2], [0, 4, 2, 0, 7, 0, 0, 0, 1, -4, 0, 0, 0, 1, 7, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -8, -1, 1], [0, -5, 5, 0, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [-8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -5, -2, 0, 0, 0], [9, 0, 0, 10, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 5, -1, -2], [0, -5, -1, 0, -2, 0, 0, 0, 2, 5, 0, 0, 0, 2, -2, -1], [-3, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 2, -5, 0, 0, 0], [7, 0, 0, 7, 0, 4, 4, 1, 0, 0, 4, 7, 4, 0, 0, 0], [0, 6, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 1, 7, 5, 0, 0, 1, -1, 1, 0, 0, 0], [0, -1, -2, 0, -1, 0, 0, 0, 4, 1, 0, 0, 0, 4, -1, -2], [3, 0, 0, 3, 0, -5, -7, 7, 0, 0, -5, 3, -5, 0, 0, 0], [-9, 0, 0, -7, 0, 4, -9, -4, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 6, -3], [0, 0, -8, 0, -7, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [11, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, -5, 1, 0, -11, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, -4, 0, 1, 4, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, -10, -5, 2], [0, -4, 3, 0, 8, 0, 0, 0, 18, 4, 0, 0, 0, 18, 8, 3], [11, 0, 0, 11, 0, 2, -14, -7, 0, 0, 2, 11, 2, 0, 0, 0], [0, -1, -5, 0, -1, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0], [0, -5, 7, 0, 2, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [11, 0, 0, 4, 0, -1, 11, 1, 0, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 13, 6, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 13, 2], [-8, 0, 0, -6, 0, -6, -8, 6, 0, 0, -7, 0, 0, 0, 0, 0], [0, 1, 2, 0, -5, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 7, 4, 0, 0, 0], [0, -5, -2, 0, -2, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -4, -6], [6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 5, 4, 8], [23, 0, 0, 4, 0, -7, 23, 7, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 7, 0, 1, 0, 0, 0, -2, -1, 0, 0, 0, -2, 1, 7], [-7, 0, 0, -7, 0, -7, -14, 5, 0, 0, -7, -7, -7, 0, 0, 0], [5, 0, 0, 5, 0, 5, -13, -1, 0, 0, 5, 5, 5, 0, 0, 0], [-4, 0, 0, -4, 0, 4, 8, -5, 0, 0, 4, -4, 4, 0, 0, 0], [1, 0, 0, 7, 0, -1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -7, -1, 0, 0, 0], [0, -3, 1, 0, 6, 0, 0, 0, 1, 3, 0, 0, 0, 1, 6, 1], [2, 0, 0, -10, 0, -8, 2, 8, 0, 0, -2, 0, 0, 0, 0, 0], [0, 5, -2, 0, 5, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [-17, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, -6, 0, 0, 0], [0, 6, 1, 0, 6, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0], [14, 0, 0, 7, 0, 5, 14, -5, 0, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, -5, 8, 5], [0, -10, 2, 0, 4, 0, 0, 0, 2, 10, 0, 0, 0, 2, 4, 2], [13, 0, 0, 14, 0, -2, 13, 2, 0, 0, -4, 0, 0, 0, 0, 0], [0, -6, 6, 0, 0, 0, 0, 0, -17, 6, 0, 0, 0, -17, 0, 6], [-9, 0, 0, -9, 0, 4, -4, -11, 0, 0, 4, -9, 4, 0, 0, 0], [9, 0, 0, 9, 0, -2, 17, 6, 0, 0, -2, 9, -2, 0, 0, 0], [0, 2, 3, 0, 15, 0, 0, 0, -6, -2, 0, 0, 0, -6, 15, 3], [13, 0, 0, 13, 0, 8, -2, -4, 0, 0, 8, 13, 8, 0, 0, 0], [-17, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 6, 0, 0, 0], [0, -11, 0, 0, -2, 0, 0, 0, -24, 11, 0, 0, 0, -24, -2, 0], [2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -5, -6, 0, 0, 0], [0, -3, -2, 0, 12, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, -7, 7, 0, 5, 0, 0, 0, -8, 7, 0, 0, 0, -8, 5, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 10, 8, 4], [8, 0, 0, -5, 0, 8, 8, -8, 0, 0, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -16, 6, 0, 0, -3, 0, -3, 0, 0, 0], [0, 10, -10, 0, -5, 0, 0, 0, 18, -10, 0, 0, 0, 18, -5, -10], [0, 5, -5, 0, 8, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [0, 5, 9, 0, 3, 0, 0, 0, 9, -5, 0, 0, 0, 9, 3, 9], [-4, 0, 0, -4, 0, 1, -7, 1, 0, 0, 1, -4, 1, 0, 0, 0], [-4, 0, 0, 2, 0, 1, -4, -1, 0, 0, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 8, 2, 8], [0, 4, 9, 0, 1, 0, 0, 0, 6, -4, 0, 0, 0, 6, 1, 9], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 2, -14, -4], [-8, 0, 0, -5, 0, 11, -8, -11, 0, 0, -5, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 17, 6, 0, 0, 0], [14, 0, 0, -14, 0, -13, 14, 13, 0, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -15, -3, 1], [0, 6, 2, 0, 1, 0, 0, 0, 32, -6, 0, 0, 0, 32, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, -20, -10, 1], [-27, 0, 0, -11, 0, 6, -27, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 0, -9, 0, -4, 19, 0, 0, 0, -4, -9, -4, 0, 0, 0], [0, 8, -1, 0, -10, 0, 0, 0, -29, 0, 0, 0, 0, 0, 0, 0], [0, -3, 9, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -9, 3, 0, 3, 0, 0, 0, -6, 9, 0, 0, 0, -6, 3, 3], [6, 0, 0, 11, 0, 1, 6, -1, 0, 0, -6, 0, 0, 0, 0, 0], [-9, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0], [12, 0, 0, -5, 0, -7, 12, 7, 0, 0, -9, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, -6, 3, 0, 0, 0], [0, 9, 0, 0, 15, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, -5, -9, 0], [-5, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 10, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 15, 6, 3], [2, 0, 0, 2, 0, -2, 14, -7, 0, 0, -2, 2, -2, 0, 0, 0], [0, -20, -5, 0, -5, 0, 0, 0, -17, 20, 0, 0, 0, -17, -5, -5], [1, 0, 0, 1, 0, -8, -4, 13, 0, 0, -8, 1, -8, 0, 0, 0], [4, 0, 0, 4, 0, 10, 27, -8, 0, 0, 10, 4, 10, 0, 0, 0], [0, -10, 4, 0, -7, 0, 0, 0, -11, 10, 0, 0, 0, -11, -7, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, -9, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 7, 5, 10], [-1, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -16, -6, 0, 0, 0], [26, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 4, 3, 0, 0, 0], [4, 0, 0, -4, 0, 10, 4, -10, 0, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 15, -11, -1], [0, 7, -11, 0, -13, 0, 0, 0, -8, -7, 0, 0, 0, -8, -13, -11], [-10, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 10, -1, 0, 0, 0], [-2, 0, 0, -5, 0, -5, -2, 5, 0, 0, -7, 0, 0, 0, 0, 0], [0, 6, 4, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 10, -12, 0, -6, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 7, 0, -4, 7, 16, 0, 0, -4, 7, -4, 0, 0, 0], [0, 2, -12, 0, 0, 0, 0, 0, -21, -2, 0, 0, 0, -21, 0, -12], [0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 18, -3, -7], [20, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0], [0, -18, 0, 0, 9, 0, 0, 0, 11, 18, 0, 0, 0, 11, 9, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, -18, 3, 6], [0, 13, -5, 0, -17, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 0, 1, 0, 13, -11, -13, 0, 0, 5, 0, 0, 0, 0, 0], [12, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0], [13, 0, 0, 10, 0, 4, 13, -4, 0, 0, 5, 0, 0, 0, 0, 0], [0, -15, -5, 0, 0, 0, 0, 0, 3, 15, 0, 0, 0, 3, 0, -5], [-5, 0, 0, -5, 0, 5, -29, -10, 0, 0, 5, -5, 5, 0, 0, 0], [0, -8, -7, 0, -17, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, -4, -13, 0, -14, 0, 0, 0, -1, 4, 0, 0, 0, -1, -14, -13], [0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 23, 0, -3], [25, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, -7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 22, 5, -11], [0, -3, -3, 0, -3, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 1, -8, 0, 0, 0], [0, -1, -3, 0, -1, 0, 0, 0, 40, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_675_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_675_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_675_2_k_c(:prec:=16)
    chi := MakeCharacter_675_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_675_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_675_2_k_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_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;