
// Make newform 2151.2.a.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2151_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_2151_2_a_h();".
// 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_2151_2_a_h();".
// 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 := [-31, 88, 129, -479, 23, 571, -134, -256, 75, 47, -15, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 51, -18, -143, 118, 3, -103, 45, 26, -13, -2, 1], [-18, 23, 34, -114, 177, 23, -162, 35, 40, -12, -3, 1], [59, -103, -234, 541, -68, -331, 173, 35, -50, 7, 4, -1], [-95, 103, 510, -581, -604, 655, 291, -279, -58, 49, 4, -3], [-371, 409, 2046, -1805, -2662, 1509, 1215, -533, -222, 85, 14, -5], [43, -33, -277, 169, 437, -172, -227, 78, 45, -15, -3, 1], [345, -619, -1954, 2775, 2242, -2423, -933, 839, 162, -127, -10, 7], [-429, 581, 2594, -2723, -3508, 2453, 1653, -917, -310, 151, 20, -9]];
    Rf_basisdens := [1, 1, 1, 1, 4, 2, 4, 4, 8, 1, 8, 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_2151_a();"
function MakeCharacter_2151_a()
    N := 2151;
    order := 1;
    char_gens := [479, 1441];
    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_2151_a_Hecke(Kf)
    return MakeCharacter_2151_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, 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], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, -1, 1, -1, 0, 0, 1, 1, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1], [2, 0, 0, 0, -1, 0, -1, 1, 0, -1, -1, -1], [-2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0], [0, -1, 0, 1, -1, -1, 0, 0, -1, -1, -1, -1], [0, 0, -1, 0, 2, 1, 1, 0, 0, 1, 2, 0], [2, 0, -2, 3, -2, -1, 0, -1, 1, 1, -1, 0], [2, 0, 2, -1, 0, 1, 2, 0, 0, 1, 3, 1], [4, 1, -2, 0, 1, 0, -2, -1, 0, -1, -1, 0], [-2, 2, -2, 1, 1, -1, -1, 0, 1, 2, -1, 0], [0, 1, 1, 0, -2, -1, 2, 0, 2, 0, -1, 0], [1, 2, 0, 0, -1, -2, -1, 0, 0, -1, -4, -1], [2, -2, 1, 1, -3, 1, 1, -1, 1, -2, -1, -2], [5, -1, 1, -1, 1, 0, -1, 1, -1, 1, 1, 0], [-1, 2, -2, 1, 1, 0, 0, -2, 1, -1, -1, -1], [3, -2, -1, -1, 0, 0, -1, 2, -1, 1, -1, 1], [2, -2, 1, -2, 0, 1, 2, 1, -1, 0, 0, 1], [-2, 1, 0, -1, 1, 2, -1, 1, -1, 0, 1, 1], [6, 3, -2, 1, 0, -1, -1, 1, 1, -1, 0, 0], [-1, -1, 0, -1, 2, -1, 2, -1, 0, 3, 4, 2], [2, 0, 0, 0, -2, 0, 1, -1, -1, -2, -1, -2], [1, 2, -4, 2, 1, 0, -2, -2, 0, -1, -3, -2], [-2, -1, 0, -1, -2, 1, 2, 0, 2, 1, 4, 1], [1, -2, 4, -2, 2, 3, 2, 3, -1, 3, 3, 1], [-1, 3, -4, 2, 0, -2, -1, -1, 2, 0, -4, 0], [2, -1, 1, 2, 0, -1, -2, 2, 0, 1, 0, 1], [0, -1, 1, 0, -1, -1, 2, -2, -1, -1, 1, 1], [3, -5, 2, -4, 1, 2, 3, 4, 0, 3, 3, 3], [3, 4, -2, 2, -2, 0, -1, -3, 0, 0, -1, 0], [6, 0, 2, -1, 1, 1, -1, 2, -1, 0, 2, -1], [-1, -1, -2, 0, -1, 2, 4, 0, 2, 3, 1, 1], [0, 3, -3, 0, 2, -1, 1, -1, 1, 1, 1, -1], [10, 0, 1, -1, -1, 0, 2, 0, -1, -1, 1, 0], [0, 3, -1, 1, -2, -2, -3, -2, 0, -5, -2, -1], [1, 0, 0, 2, -2, -2, 1, 1, 0, 2, -1, 0], [0, 0, -1, 3, -4, -2, -4, 2, 0, -5, -6, -2], [-1, -1, 0, 2, 0, 0, -1, 1, -2, 0, -2, -2], [-1, 0, 3, -4, 4, 1, 5, 1, -2, 4, 4, 1], [2, 3, -1, -2, -2, -1, 6, 0, 2, 3, 0, 1], [-4, -2, 2, -4, 2, 2, 1, 3, -1, 1, -1, 0], [-1, 4, -2, 0, 1, -4, 0, -2, 2, -3, -3, 0], [3, -1, 1, 0, -2, 3, -1, -2, -3, -3, -1, -2], [1, 2, -1, 1, 2, -2, 1, -4, 1, -1, -2, 0], [1, -1, 3, -4, 1, 1, 3, 4, 0, 1, 4, 2], [0, 2, 1, -2, 0, 1, 1, 0, -2, 0, 0, 2], [-2, 4, -3, 2, -3, -1, -1, -5, 2, -1, -5, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 4, 0, 4, -2, 0, -5, -2, -1, -4, -3, -5], [-5, 8, 0, 0, 2, 0, -1, 0, 1, -1, -1, 1], [6, -1, -2, -1, 1, 1, 3, -3, 4, 1, 2, 1], [0, 1, -4, 2, -4, -5, -3, -2, -1, -2, -7, -1], [3, -4, 2, -1, -1, 1, -2, 4, 0, 1, 2, 1], [3, -4, 1, -1, -1, -2, 3, -2, 0, -3, 0, 1], [-2, -3, 2, -5, 5, 5, 2, 2, -3, 2, 7, 3], [4, 3, 4, -3, 1, 1, 2, 1, 0, 2, 3, 0], [6, -3, 5, -4, 0, 3, 1, 3, -3, 1, 3, 3], [-2, 2, -2, 2, -2, -2, -1, -3, -1, -2, -3, -2], [3, -9, 5, -4, 6, 3, 1, 2, -3, 3, 5, 4], [-3, 1, 2, -3, 5, 4, -1, 2, -1, 2, 1, 4], [1, 3, 1, 3, -5, 0, -1, 3, -1, 0, -1, -2], [-3, 3, -3, 3, -3, 0, 4, 0, 2, 0, 1, -3], [4, -2, 3, -1, 2, 2, -2, 1, -5, 1, 0, 3], [0, -2, 1, 0, 0, 2, 0, -1, 2, -2, -2, 0], [-7, -3, 4, 1, -3, 3, 0, 1, -3, -1, -1, 2], [0, 4, 3, 4, -4, -4, -7, 2, -1, -6, -7, -6], [5, 0, -3, 3, -7, -2, -4, -3, 3, -8, -8, -4], [-3, 2, -4, 0, -2, -2, 3, 0, 1, -2, -5, -1], [5, -5, 1, 0, 0, 1, 5, -2, 3, 1, 1, 4], [1, 2, -2, 2, -6, 0, 2, 1, 0, -2, -4, -3], [5, 2, -2, 3, 0, -1, -1, 0, -1, -3, -4, 0], [3, -3, 2, 3, -3, -3, -1, 0, 2, -4, -2, -2], [5, -2, 2, -2, 2, 2, -1, 0, -3, 1, -1, -1], [3, -6, 3, 1, -7, 0, 5, -2, 4, -4, 0, -1], [5, 1, 0, 3, -7, -3, 2, -1, 7, -3, -5, -2], [-8, 8, -3, 3, -3, -1, -3, -3, -3, -8, -5, -2], [5, -2, 2, -4, 6, 2, 0, 3, 0, 1, 8, 1], [-1, 5, -4, 0, -4, -5, 1, -1, 5, -3, -3, 0], [-3, -4, 4, -4, 4, 0, 4, 1, -4, 5, 4, 7], [1, -9, 2, -1, -3, 1, 3, 2, 0, -2, 4, 4], [-6, -2, 2, -3, 5, -3, -1, 4, -1, 6, 1, 0], [-5, -3, -2, -2, 6, -2, 2, 0, 3, 3, 5, 0], [0, -3, -2, 7, -2, -1, 0, -1, -1, -3, -2, -4], [-1, 5, -4, 3, 1, -3, -3, 0, -2, 2, -2, 0], [-1, -6, 7, -3, 1, 6, 4, 5, -3, 4, 8, 2], [-3, 5, -3, 1, 3, 4, -3, -1, 1, -4, 2, 1], [-12, -3, -2, 1, 1, -3, 3, -1, 2, 0, 2, -1], [1, -3, 0, -1, 3, 0, 0, 1, 2, 4, 6, 4], [3, 0, -1, 5, -1, -2, 0, 1, 5, 1, 1, -1], [-4, 4, -3, 3, -10, -2, -4, 0, 2, -5, -6, 0], [-1, 5, 2, -5, 4, 1, 1, -1, 0, 1, 3, -1], [4, 5, -6, 7, -3, -3, -5, -3, 2, -7, -6, -7], [9, -3, 0, 3, 1, -1, 1, -4, 0, -1, 2, 2], [4, 4, -5, 8, -4, -5, -5, 0, 2, -6, -9, -7], [-2, -2, -7, 1, 1, 0, -6, -2, 1, -3, -5, -2], [8, -4, -2, 0, 1, 0, -3, -5, 0, -1, 2, 0], [6, -11, 5, -5, -4, 2, 2, 2, 0, -1, 1, 0], [2, 0, -3, 8, -3, -3, -4, -2, -1, -5, -8, -7], [-2, -6, 0, -6, 2, 5, 7, 1, 2, -1, 6, 2], [12, 1, -4, -2, 2, 2, 0, 1, -3, 3, 5, 1], [-5, 5, 0, 5, 3, -3, 0, 1, -1, 8, 1, 2], [-3, -1, -2, 9, -5, -2, -2, -5, 4, 3, -2, 0], [-2, 0, 0, 3, -1, 2, -1, 4, -2, 0, 2, -2], [7, -2, 6, -6, 4, -2, 2, -3, 0, 0, 1, 6], [3, -8, 2, -8, 6, 3, 1, 4, 2, 1, 2, 1], [1, -9, 0, 1, 2, -3, -3, 2, -1, 3, -1, 2], [-1, 10, -7, 2, -6, -1, -2, -4, 5, -2, -8, -4], [8, 4, -2, 2, -8, -5, -1, -1, 2, -11, -12, -6], [-4, -4, -2, 6, -1, -2, -1, -1, 4, 11, 1, 1], [9, -9, 5, -4, 2, -1, 4, 3, -6, 8, 5, 3], [-3, -3, 1, 0, 4, -1, 0, -1, 4, 1, -4, 2], [4, 1, 2, -2, 2, 0, -1, 0, 2, -1, 8, 5], [3, 15, -9, 11, -5, -8, -4, -4, 2, -3, -6, -6], [-5, -5, 4, -5, -1, 4, 2, -1, 2, 0, 2, -4], [0, -5, 10, -8, 6, 6, 0, 2, -6, 2, 9, 0], [8, -5, 8, 3, -5, -3, -4, 2, -3, -4, -5, -3], [-4, -2, 1, -4, 2, 5, 2, -3, -3, -1, 5, 8], [7, 1, 1, 3, -3, -2, -8, -2, -2, -6, -4, -2], [13, -8, -4, -2, 5, 2, -8, -1, -7, -5, -4, 2], [-5, 5, 0, 3, 2, -1, -1, 2, -1, 5, 1, 2], [-6, -8, 3, -6, 2, 3, 3, 2, -8, 5, 4, 0], [0, 4, 0, -4, 2, 2, 2, -3, 1, 4, 0, -3], [-3, -8, 4, 1, -1, 3, -2, -2, -4, -6, 0, -1], [9, -10, 8, -3, -3, 2, 8, 0, -3, 2, 3, -1], [-6, -9, 12, 0, 0, 2, 0, 1, -1, 2, -1, 1], [3, 2, -5, 0, 4, -1, 0, 2, 1, -1, -3, -1], [-13, -5, 8, -8, 14, 8, 4, 10, -5, 8, 15, 6], [6, -3, -5, 5, -8, -4, 2, -1, 5, -7, -9, -7], [3, 4, 2, -3, 5, -3, 0, 6, -2, 12, 2, 3], [4, -1, -1, 1, 1, -5, -2, 3, -1, 2, -3, -1], [6, 6, -8, 10, -12, -8, 0, -4, 6, -8, -12, -12], [7, -1, 6, -3, -3, -1, 5, -2, 2, 0, 2, 2], [-2, -7, 9, -1, 3, 4, -3, 1, -6, 2, 3, 2], [-1, -4, 1, -4, -1, -1, 0, 0, -2, -3, -3, 4], [1, 2, 4, 2, -4, 0, -5, 0, 3, -1, -3, 3], [1, 0, -4, 1, -2, 5, 2, 2, 5, 1, 4, 2], [1, -5, 3, 3, -2, -2, 1, -8, 1, -7, 4, -3], [4, -1, 2, 0, -1, -2, -3, 2, -1, -1, -6, 2], [-4, -1, -4, 2, 0, -2, -3, 0, -2, -2, -7, 2], [-16, -3, 2, -1, 5, 1, -2, -4, -5, 1, 5, 5], [15, -9, 2, -9, 4, 5, 7, 4, 1, 11, 9, 8], [16, -6, 4, -2, 4, 4, -5, 1, -3, 1, 3, 0], [0, -5, 3, -5, 1, 7, 9, -6, 2, 6, 6, 3], [1, 5, 0, 3, -6, -1, -6, -4, -3, -9, -10, -5], [11, -4, -1, 0, -4, 1, 1, 3, -2, 1, 2, 1], [-8, 2, 5, 0, 0, -1, 1, -4, -8, 2, -4, 2], [5, -1, -5, 0, 2, 0, -5, 2, -2, 2, -6, 0], [5, 4, 1, 1, -5, -2, 8, -2, 4, 0, -1, 0], [-8, -5, -3, 4, -2, -3, -1, 3, -5, 1, -1, 1], [11, -4, -8, -4, 6, -1, -2, 5, -1, -1, -3, -3], [12, 0, 8, -7, 9, 3, 8, 4, -3, 14, 20, 7], [-2, 10, -1, 1, 1, 0, -3, -1, 0, 0, 4, 0], [-1, -3, -3, 2, -2, -7, -2, -1, -8, -5, -6, -2], [14, -6, 0, -3, 5, -1, 0, 0, -3, 2, 6, 3], [3, 11, -1, -1, 1, -1, -4, 0, -5, -4, -3, -6], [-10, -4, 3, -9, 3, 8, 1, 0, 1, -6, -2, -3], [-7, 5, -7, 0, -6, 5, -3, 0, 1, -10, -7, -6], [1, -5, 9, 1, -7, -6, 6, 4, -2, 8, 3, 3], [3, -11, 2, -8, 2, 2, 0, 4, -7, 4, -3, 4], [0, -6, -2, -1, 7, 7, -5, 2, -7, 1, 7, 4], [-3, 6, -2, 8, 2, -4, -10, -3, 0, -10, -3, -4], [-3, -8, 10, 0, 0, 2, 5, 2, -1, -1, 5, 7], [0, 1, 1, 3, -6, -2, -4, -2, 0, -1, 1, 6], [6, -1, -2, -3, 6, 3, -1, -3, -3, -1, 3, 1], [4, -9, -4, -2, 2, 0, 0, -1, 3, 10, -5, 1], [-4, -2, -7, 5, -6, 2, 3, -6, -4, -1, -1, 1], [-23, 10, -7, 6, 2, -5, -3, -9, 2, 2, -6, -1], [-2, -4, 3, 7, -2, -4, 0, -2, 0, 1, -6, -2], [-2, 7, 0, -5, -1, -1, 5, -1, 0, 1, 4, 1], [7, -1, 1, 0, 2, 1, -1, 2, 3, -2, 1, -2], [2, -4, 1, 4, -12, 3, 5, 0, 4, -3, 0, 0], [1, 11, -8, 5, -5, -5, -10, -7, -1, -11, -13, -8], [2, -2, 0, 3, 0, 3, -6, -2, -4, -1, 3, -3], [-1, 10, -5, 9, -9, 0, -5, -1, 5, -2, -8, -10], [-8, 0, 1, -11, 4, 6, -1, 14, 0, 7, 3, 1], [1, 0, -5, 7, -9, -3, 5, -10, 7, -6, 1, 3], [-5, -6, -3, -1, -1, 0, 1, -3, 5, 0, -2, -2], [-1, -12, 10, -4, 4, 9, 3, -2, -4, 1, 4, -3], [1, 6, -8, 0, -5, 2, 2, -3, 1, 1, -3, -1], [2, -4, -3, 3, -3, 0, -3, 3, 0, -4, -1, -3], [-6, 4, -2, 10, -4, -2, -11, 3, -3, -3, -9, -8], [4, -9, -2, 2, 0, 2, 2, 1, -1, -4, -1, 1], [-1, 5, -1, 4, -6, -3, -8, 1, 2, -7, -8, -12], [-1, -2, -6, 2, 2, -4, 1, -6, 1, 8, 6, 2], [9, 1, -4, 1, 4, 7, -5, 0, -3, -3, -1, 2], [-19, 3, -9, 5, -2, -4, -1, -7, 0, -5, -6, 0], [13, 8, 3, 2, 0, 4, 0, 4, 4, 0, 8, -1], [-7, -9, 7, -11, 14, 12, 3, 5, -8, 11, 16, 6], [2, 7, -12, 4, -8, -8, 5, -5, 3, -2, -4, 0], [6, -4, -4, -2, -4, 0, -2, 2, 8, -4, 0, -4], [-9, -2, 7, -1, 2, 6, 2, -5, -4, -7, 6, -1], [0, 1, 2, -3, -1, 3, -6, 3, -8, -8, -3, -4], [-5, 8, 6, -7, 5, 1, 7, -3, -3, 4, 7, -3], [4, -8, 11, -8, 0, -1, 9, 6, -2, 9, 4, 2], [-15, -3, 4, -7, 9, 1, 0, 9, -1, -1, 5, 4], [2, -12, -13, 9, -1, -8, -8, -4, 1, 0, -9, -2], [-26, -3, -6, -2, 0, -8, 6, -7, 5, 1, -3, 3], [2, 14, -5, -2, 2, -1, -4, 7, 3, 5, -1, 4], [-3, 0, -6, 7, 6, 3, -2, -6, -3, -3, 5, 1], [12, -7, -11, -2, 2, 2, -1, 1, 4, 8, -4, 1], [12, -10, -6, 1, -1, 0, 0, -1, -5, 0, -11, -2], [4, -3, -3, 0, 6, -1, -1, -1, -1, 0, -2, 0], [-12, -8, 10, -3, 2, 3, 5, 9, -3, 7, 8, 7], [-10, 12, -2, -1, 5, -11, -7, 6, -1, 6, -3, -2], [2, -6, 8, -6, -4, 2, 1, 5, -7, -4, -10, -5], [4, 5, -2, -4, 2, 4, -6, 1, 3, -8, -3, -1], [-8, 5, -3, 9, -1, -3, -15, 4, -4, -4, -12, -7], [4, 13, -11, 11, 2, -12, -5, 2, 6, 7, -9, -4], [-2, -8, 4, 2, -4, -4, -3, 5, -5, 4, -5, 0], [-8, 2, -7, 5, 1, -2, -2, -2, -1, 4, -3, 2], [5, -2, -4, 5, -3, 1, -2, -4, 0, -7, 4, 5], [-9, 1, -1, -2, 16, -1, -5, 0, -3, 0, 5, 2], [-14, -1, -6, 8, -8, 2, 3, -6, 0, -4, -10, -7], [16, -6, 10, -15, 9, 11, 0, 9, -4, 4, 8, 8], [-5, 6, 4, -7, 8, -1, -3, 5, 0, 3, 4, 5], [-4, 9, 0, -5, 7, -3, 6, 4, -1, 10, 14, 4], [-23, 17, -11, 5, 5, -6, 2, -14, 6, 1, 0, 2], [-15, -3, 9, -9, 1, -2, 12, 6, -4, 15, 8, 6], [3, 20, -12, 5, -11, -3, -5, -8, 4, -14, -15, -8], [14, -1, 2, -4, 14, 2, -7, 2, -8, 8, 13, 2], [16, -12, 14, -10, 4, -1, 1, 7, 0, 2, 8, 8], [-3, 0, -10, 2, -2, 4, -3, 2, 3, -14, -1, -3], [-3, 7, -2, 9, -9, -7, -3, -5, 3, -10, -8, -11], [0, 5, 2, -5, 3, 5, 2, -4, 5, -3, 7, 1], [14, -15, 3, 3, -5, 6, -8, 3, -4, -8, -5, 2], [-4, -4, 6, -5, 7, 6, 4, -3, -7, 1, 7, 2], [10, 6, 10, -4, 1, -2, 2, 0, -1, -1, 3, 4], [-6, 5, 5, -1, 1, 2, -3, -2, 1, -11, -7, -3], [-8, -8, -2, 2, 8, -2, 0, 3, 3, 10, 4, 1], [0, 0, 1, -8, 3, 7, 6, -3, -8, 1, 11, 7], [8, -5, 6, -6, 12, 4, 3, 4, -4, 6, 14, 11], [11, -3, 0, -8, 2, 2, -2, 8, -5, -8, -1, 0], [10, -8, 5, 7, -3, -5, 1, 3, -3, -4, -5, 0], [22, 2, 9, -7, -1, 4, -6, 1, -4, -8, -1, -1], [1, 1, 1, -3, -2, -2, 0, 4, -3, 5, -1, 0], [15, 9, 1, 4, 1, -5, -10, -1, -1, -1, -7, -2], [1, 14, 2, 0, -6, -6, -1, 4, 3, -4, -3, -1], [-12, 2, -4, -2, 4, 4, 3, 4, 6, 6, 5, -1], [-13, 10, -1, -1, 3, 0, 7, -8, 10, 9, 16, 9], [6, -12, -5, 3, -6, -2, 6, 0, -4, -1, -6, 6], [-14, 0, 2, 7, 5, -3, -1, 2, -1, -2, 4, 1], [-6, 1, -6, 5, 5, 9, -5, -1, 0, -6, -4, -3], [-6, 12, 7, -5, 1, -4, 2, -4, 7, -1, 3, 2], [-7, -5, -2, 9, -3, -7, 3, 2, 12, 8, -2, 0], [-5, 10, -12, 10, -12, -8, -7, 0, 5, -10, -13, -7], [11, 5, -8, -3, 8, -3, -11, 4, -9, -5, -13, 0], [-11, 2, -1, 1, -3, -3, -1, -4, -3, 4, -9, -3], [5, 11, 0, 1, 1, -3, 3, -2, 2, 2, -1, 7], [-19, 4, -9, 13, -9, -6, 1, -8, 2, -6, -10, -7], [-4, 1, 11, 0, 0, -3, 1, 3, -1, 8, 1, 5], [-11, 0, -14, 7, 2, 1, -9, -3, -2, -9, -3, -4], [13, -11, 17, -12, 5, 7, 9, 8, 0, 11, 10, 6], [-14, -12, 4, 1, -5, 4, 2, -3, -5, -11, 3, -4], [4, -10, 9, -3, 1, 12, -1, 7, -10, -5, 8, 2], [6, 1, 4, -5, 2, -1, 2, 12, 4, 1, 5, 6], [8, 6, -5, 0, -2, -3, 2, -7, 3, 0, 7, 10], [-31, 6, -1, 2, -6, -3, 4, -1, 10, -2, -3, 0], [-14, -6, 4, 3, -1, -3, 13, -8, 5, 6, 11, 8], [-14, -13, 9, -1, -9, 2, 3, 0, -3, -4, -3, -1], [2, 6, 3, -5, 1, 3, 2, -2, 0, 4, 8, 10], [-23, 3, -4, 5, 1, 3, 7, -2, 6, 4, 10, 0], [-5, 6, 1, -3, 5, -6, -6, -2, -6, 4, -3, 6], [3, -7, 10, -3, 5, -7, -11, 10, -2, 3, 0, 0], [11, 13, -8, 4, -2, -12, -16, 2, -1, -13, -13, -8], [5, 10, -3, 3, 7, -4, 3, 2, 4, 13, 0, 5], [-21, 15, 2, 0, 10, 0, -4, 2, -1, 2, -5, 0], [6, 13, -5, -2, 10, -3, -13, -3, -1, 1, -5, 3], [2, 22, -3, -3, -3, -8, -8, -3, 0, -6, -9, 1], [10, 6, 5, -4, -2, -3, 2, 3, 1, -3, -11, -6], [8, 0, -4, -2, -1, 2, 6, -9, 2, -1, 1, -3], [-13, 2, 5, -9, 7, -2, 12, 3, -1, 8, 13, -1], [-11, -14, 15, -2, 0, 7, 12, 6, 1, 9, 9, 7], [-23, 13, -10, 6, 2, -10, 2, -12, -3, -10, -1, 0], [25, 3, 3, 0, -12, -4, 3, 2, 8, -8, -4, -4], [13, -10, 3, -11, 1, -7, 5, 10, -1, 4, -7, 1], [-12, 1, 3, 2, 8, 7, 1, -5, 5, 6, 15, 7], [6, 1, -8, 4, -4, 0, -2, -9, -1, 1, 1, 3], [6, 7, -12, 9, 3, -3, -8, -9, 4, 4, -1, 4], [-12, -14, 10, -5, 7, 6, 9, 14, 0, 9, 12, 8], [-17, 2, -5, 2, 2, -5, -6, -6, -3, 0, -9, -5], [8, -17, 14, -1, -1, 5, -8, 5, -4, -2, -5, -2], [10, -3, 4, 1, 1, 1, -12, 6, -1, -7, -7, -5], [11, -2, 11, -5, -5, 2, 8, -1, 5, 3, -3, -1], [11, -13, 10, -5, -13, 3, 4, 3, -1, -5, -9, -6], [-25, 7, 5, -5, -2, -4, 5, 6, -3, 3, 2, -3], [8, 2, -4, -6, 2, -4, -4, 6, -2, 9, -8, 4], [-19, 22, -16, 14, -12, -14, -6, -12, 3, -10, -22, -8], [-5, -12, 2, -1, 5, 7, 7, -5, -7, 6, 9, 1], [3, 17, -8, 1, 1, -3, -14, 1, 1, -5, -5, -6], [-15, 8, -1, 7, -4, 4, -4, -9, 6, 3, 2, 1], [-6, 1, -11, 11, -2, -10, -15, 8, -2, -5, -16, -7], [11, 6, -5, 0, -8, -3, -1, 3, -4, -10, -1, -8], [1, -5, -12, 11, -5, -7, -8, -1, 11, -10, -22, -7], [9, 5, 4, -4, -6, 2, 2, -6, -1, -3, 7, 2], [-8, -12, -4, 0, 2, 3, 10, 6, 7, 8, 1, 8], [13, 6, -9, 10, -14, -9, -8, -12, 13, -25, -5, -7], [12, -3, 20, -6, -2, 3, -10, 7, -10, -7, 4, -3], [-6, 5, -4, 1, -1, -1, -7, -3, -6, -4, -8, -3], [-2, 11, -6, 9, -7, -5, 0, -4, 5, 0, 3, -1], [-21, 4, 2, -4, -8, 7, 9, 4, 4, -1, -4, -3], [5, -19, 14, -13, 1, 9, 10, 11, -7, 0, 9, 2], [3, -1, 2, -5, 4, -5, 0, 7, -8, 7, -4, 2], [-13, 1, 5, 4, -2, 1, -5, 8, 3, 6, 1, -2], [-11, 2, 0, 6, 1, 0, 1, -12, -4, -1, -2, -5], [-17, 5, 12, -13, 15, 13, 13, 4, -6, 7, 18, 8], [-6, -3, 5, 1, -1, 4, -3, 0, -7, 2, 3, 3], [2, -10, 6, -8, 9, -6, -2, 6, -1, 7, -1, 6], [2, -2, 16, -11, 5, 3, 4, 3, -4, 3, 12, 4], [3, 3, -2, -3, 3, 7, -3, 2, 2, -1, -8, -6], [1, 13, -6, 10, -4, -14, -15, 3, -6, -8, -23, -11], [-4, -10, -3, -1, 3, 4, 10, 6, 5, 12, 7, 6], [31, -13, 11, -15, 10, 8, 3, 13, -8, 3, 14, 4], [9, 2, 14, -5, -9, -3, 1, 7, -5, 5, -7, -7], [1, -16, 9, -6, 6, 5, 6, 14, -7, 10, 11, 9], [-5, 0, -10, 5, 1, -8, -3, -1, 2, -1, -12, 1], [-4, 2, 3, -5, 11, -2, -7, 13, -2, -2, 7, 3], [-10, -13, 1, -4, 12, -3, -4, 9, -5, 8, 2, 2], [10, -18, 25, -16, 4, 16, 14, 7, -10, 0, 22, 10], [-6, 3, 9, -4, -1, 3, -2, -5, -6, -17, -3, 2], [-9, 4, -10, -3, 10, 9, -1, -3, 4, -1, 3, -4], [-24, 22, -4, 5, -3, -7, -7, -10, -1, -12, -15, -4], [-16, -9, 6, -5, 7, -1, 8, 8, 5, 10, 11, -1], [-6, 1, 3, 0, -1, -3, -6, -6, -1, -5, -4, -2], [12, 4, -4, 2, 4, 4, -2, -8, 0, 6, -1, 5], [-15, 13, -8, 7, -3, -11, 0, -11, 9, -12, -15, -6], [11, 1, 6, 0, 4, 6, 2, 4, -11, 2, 9, -2], [-9, 2, -5, -5, 6, 2, -5, -2, 3, -9, 1, -1], [-7, 16, -8, 5, -1, -11, -7, -6, 6, -4, 1, 4], [-29, -10, -1, -4, -6, 3, 7, 1, 4, -5, -6, -3], [14, 12, -15, 8, -4, -5, -14, 1, 1, -14, -15, -8], [7, -18, 4, -8, 2, 2, 13, -4, -1, -2, 9, 9], [5, 6, -4, -3, -3, -5, -1, 1, -15, -4, -5, -1], [-2, -15, 1, 7, -3, 4, 11, -6, 3, 2, 1, -5], [-1, -9, -2, 8, -6, 0, 8, -6, -7, 1, -3, -6], [-12, 14, -3, 1, -6, -14, 6, 2, 4, -1, -6, 2], [-10, 13, 1, -7, -5, -8, -4, 7, -6, 3, -8, -3], [5, -4, -13, 8, -6, -8, -3, 1, 3, -8, -11, -1], [-13, 19, 5, -9, 8, 0, -2, -3, -2, 1, 3, -7], [-3, -5, 15, 1, -6, 4, 8, 5, -2, 13, 11, 7], [-7, -2, -1, -3, -1, 6, 15, -10, -6, 3, 4, -1], [11, 5, -18, 8, -6, 0, -2, 2, 11, 2, -3, -4], [11, 3, 6, -1, 7, 1, -5, 0, 4, -2, 8, -4], [1, 6, -2, -2, 3, -2, -6, 1, -11, 1, -15, 3], [15, -11, 14, -4, 6, 10, 6, 6, -5, -8, 13, -6], [37, -4, 0, -1, 1, 8, 8, 4, 1, 10, 7, 9], [3, 7, -8, 2, 0, 0, 0, 7, 0, 2, 11, -1], [-4, 13, -3, 0, 4, 1, 0, 4, 0, 11, 10, -1], [-9, 7, 6, -1, -5, 8, 7, -6, -5, 5, 11, 2], [4, 13, 1, -6, 6, -1, -2, -7, -7, 0, 6, -2], [-26, 10, 1, 6, -10, -15, -3, 4, -2, -5, -10, -6], [2, -12, 11, -6, -4, -1, 13, 2, 2, 4, 14, 0], [25, -20, 8, -14, 6, 12, 5, 14, -11, 0, 15, 7], [7, 10, -10, -1, -7, -4, -7, -1, -10, -5, -14, -1], [9, 4, -8, 1, -1, -5, -9, -1, 1, 0, -5, 1], [1, -3, -2, 9, -1, 1, -3, 4, -8, 9, -2, -2], [-11, -9, 6, -6, 0, 7, 10, -12, 4, -2, 8, 6], [-5, 22, -7, 11, -10, -10, 6, -13, 12, -9, 2, 1], [-12, -18, 14, -7, 7, 13, 10, 9, -6, 13, 10, 6], [16, 18, -19, 15, -10, -12, -5, -11, 11, -9, -19, -14], [0, -9, 2, 1, 0, 3, 8, -2, -2, 1, 8, 11], [-3, -8, 2, 11, -6, -5, -1, -8, 9, -9, 0, -4], [15, 3, -10, 8, -10, -5, -7, -1, 5, -11, -3, -6], [-14, -15, 22, 4, -16, -2, 13, 6, 2, 4, 0, 1], [-15, -1, 8, 4, -2, -6, -1, 1, 4, -6, 2, 0], [-16, -7, 15, -2, -1, 1, 8, 4, -3, 13, 3, 9], [-2, 18, -9, -11, 13, -6, 1, -1, 6, 8, 4, 6], [-8, 12, -17, 13, -1, -2, -11, -3, 2, -6, -17, -9], [1, -3, 10, -2, 4, 4, 5, 11, 2, 8, 10, 0], [-6, -19, 15, -4, 1, -1, 1, -1, -8, -3, 3, 0], [-17, 6, -2, -2, 10, 6, 0, -7, -2, 4, 4, 7], [-3, 6, -7, 5, -6, 0, -5, -2, 7, 3, -3, 5], [2, -11, 2, -16, 12, 10, -11, 16, -6, -8, -8, -5], [-7, 13, -12, 18, -2, -16, -15, -5, 6, -3, -18, -14], [15, 5, -8, -6, 3, 4, 2, 4, 8, 17, -6, 2], [4, 3, -3, 2, 6, 3, 7, -6, 4, 8, 7, -4], [10, 0, 0, 8, 3, -6, -12, -10, -3, -1, -9, 0], [-26, 8, -8, 5, 18, -1, -8, -9, -5, 7, 0, 5], [-19, -3, 2, -6, 4, 0, -3, -1, 2, 13, 2, 8], [0, 7, -3, -4, 10, -5, -13, 1, 7, -17, -1, 1], [-1, -7, 13, 0, -3, 7, 14, 3, 11, 21, 20, 5], [-6, 7, -8, -7, 11, 5, 7, 1, -2, 14, 10, 7], [6, -3, -8, 14, -4, 1, -6, -1, 2, -6, -2, -11], [-28, -3, -7, 4, -6, 1, -11, 0, -4, -10, -7, -10], [-33, 14, -5, 6, 0, -10, -2, -6, 2, -2, 2, -5], [38, -23, -4, -5, -3, 1, 5, 13, -6, 8, 4, 7], [-4, 2, -2, -1, 3, 0, 6, 7, 1, 16, 3, -2], [0, 14, -9, 0, 0, 4, -13, 2, -9, -10, -3, 0], [1, 13, 2, -4, 8, 8, -5, 3, -4, 9, 12, 8], [20, 5, 2, -11, 8, 1, -11, 5, 1, 5, -3, -3], [7, 20, -1, 5, -4, -18, 1, 1, 2, 7, -9, -6], [29, -16, 12, -3, -2, 1, -1, 9, -10, 1, 14, 5], [14, 6, 5, -3, -2, 0, 2, -2, 0, 3, 2, 0], [-4, 5, 10, -8, -4, -4, 2, 1, 9, -6, 5, -3], [23, 1, -2, -6, -4, 0, 7, -3, -6, -7, -2, 2], [-14, 0, 17, -5, 3, -2, 11, 7, 2, 19, 20, 14], [14, 4, -19, 0, 4, 5, -5, -8, 0, -2, 7, 1], [10, -18, 6, -3, 3, 4, 7, 0, 2, 7, 0, -2], [10, 0, -10, 13, 3, 3, -5, -8, 3, -5, 9, 10], [-19, -2, -6, 13, 3, -1, -1, -12, -2, -2, 7, 2], [-14, -7, 16, -12, 5, 10, 10, -4, -3, 9, 12, 14], [-11, 5, -14, 14, 6, -4, 5, -15, 2, 1, 4, -2], [-14, 12, -5, 9, -5, -6, -7, -5, 6, 5, -16, -2], [8, 23, -4, 4, -1, -10, -9, 0, 1, -5, -11, -3], [8, 2, 9, -2, -1, 5, 22, -3, 4, 9, 17, 7], [27, 1, 7, -5, 13, 0, -1, 7, -3, 3, 15, 4], [2, -5, -8, 0, 3, -4, -5, -2, -7, -7, -5, 3], [-3, 19, -8, 5, -7, 1, 7, -12, 6, -10, 2, -4], [10, -3, 0, 3, -11, -1, -9, 0, -5, 2, -6, -10], [-15, 14, 5, 2, -3, -5, -9, -9, -5, -3, 7, 1], [5, 4, -1, -10, 10, 1, -10, 2, -7, -2, -4, -4], [9, -19, 10, -10, 1, 2, 10, 8, -4, 17, 4, 2], [24, -20, 8, 7, 1, -1, 6, 3, 2, 10, 0, 4], [18, 0, 8, -12, 8, -2, 0, 10, -2, 5, 14, 8], [0, -18, 0, 4, -5, 8, 0, 0, -5, -5, 10, 1], [11, 1, -14, 9, -3, -1, -6, -5, 11, -7, -13, -4], [18, -7, 15, -12, 1, 1, -7, 13, 0, -3, 2, 7], [-35, 4, 7, -13, 0, 6, 7, 10, -11, 3, 0, 2], [19, 9, -11, 15, -7, -6, -11, -1, 5, -7, -15, -16], [-9, -21, 10, -13, 13, 15, 17, 0, -8, 18, 18, 10], [-11, -13, 2, 4, -6, 4, 6, -4, 5, -11, -5, 0], [3, -27, 7, -13, 17, 9, 12, 10, 7, 16, 21, 4], [19, 4, 4, 3, -12, 5, -2, -1, -8, -11, -15, -12], [10, 17, 2, -6, 10, -6, -12, 5, -3, 0, 5, -5], [-8, -7, 0, 1, 7, 7, -17, -1, -4, -5, 6, 9], [1, 5, 5, -1, 9, -2, -15, 5, -11, -2, 2, 1], [-5, -5, -7, -6, 8, 9, 11, -6, 5, 26, 7, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2151_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2151_2_a_h();".
// 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_2151_2_a_h(:prec:=12)
    chi := MakeCharacter_2151_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_2151_2_a_h();".
// 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_2151_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2151_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-31, -88, 129, 479, 23, -571, -134, 256, 75, -47, -15, 3, 1]>,<5,R![1520, 2064, -6360, -3384, 7563, 1629, -3247, -339, 546, 31, -39, -1, 1]>],Snew);
    return Vf;
end function;