
// Make newform 756.2.x.a in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_756_x_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_756_2_x_a();".
// 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_756_2_x_a();".
// 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 := [6561, 0, -2187, 0, -729, 0, -81, 0, 225, 0, -9, 0, -9, 0, -3, 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, 17496, 0, 1944, 0, -891, 0, -423, 0, 153, 0, 72, 0, 15, 0, 1], [9477, 0, -4617, 0, -1782, 0, -153, 0, 369, 0, 18, 0, -12, 0, -5, 0], [0, -9477, 0, -3888, 0, -297, 0, 657, 0, 72, 0, -18, 0, -9, 0, -2], [2187, 0, 0, 0, 81, 0, -225, 0, 9, 0, 9, 0, 3, 0, -1, 0], [-6561, 0, -729, 0, 1134, 0, 369, 0, -99, 0, -45, 0, 3, 0, 2, 0], [-247131, 67797, 6561, 12636, 25272, -5508, 11637, -5175, -3051, 774, -702, 342, -99, 87, 69, -32], [0, -19683, 0, 4617, 0, 1782, 0, 153, 0, -369, 0, -18, 0, 12, 0, 5], [41553, -8748, -2916, -243, -3888, 2268, -2349, 396, 648, -207, 162, -72, 27, 12, -18, 5], [-369603, -61236, -49572, -1701, 25272, 15876, 18819, 2772, -2862, -1449, -1080, -504, -225, 84, 48, 35], [-247131, -67797, 6561, -12636, 25272, 5508, 11637, 5175, -3051, -774, -702, -342, -99, -87, 69, 32], [0, 24057, 0, -9234, 0, -3564, 0, -306, 0, 738, 0, 36, 0, -24, 0, -10], [0, 3645, 0, 2673, 0, 27, 0, 261, 0, -207, 0, -27, 0, -3, 0, 11], [-41553, -8748, 2916, -243, 3888, 2268, 2349, 396, -648, -207, -162, -72, -27, 12, 18, 5], [-535815, 67797, -45927, 12636, 71442, -5508, 34209, -5175, -9261, 774, -3024, 342, -63, 87, 273, -32], [0, 157464, 0, -1215, 0, -18792, 0, -8217, 0, 2574, 0, 711, 0, 30, 0, -61]];
    Rf_basisdens := [1, 15309, 5103, 5103, 729, 729, 30618, 5103, 4374, 30618, 30618, 5103, 5103, 4374, 30618, 15309];
    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_756_x();"
function MakeCharacter_756_x()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [6, 5, 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_756_x_Hecke();"
function MakeCharacter_756_x_Hecke( : Kf := false)
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 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, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 2, 0, -1, 0, -2, 1, 1, -1, 0, -2], [0, -2, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0], [2, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -2, 0, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, -2], [0, 0, -1, 0, -2, -1, 1, 0, 1, 0, 1, 0, 0, -1, 0, 0], [0, -1, 0, 2, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0], [2, 0, 4, 0, 2, 0, -3, 0, -1, -1, -2, 0, 0, 2, 1, 0], [0, -1, 0, 1, 0, 0, -1, -1, 2, 0, 1, -1, 0, 2, 0, 1], [0, 2, 0, 0, 0, 0, 1, -1, -1, 0, -1, -1, -1, -1, 0, -4], [0, 0, 0, 0, -1, 1, 1, 0, 0, 1, -1, 0, 0, -1, -2, 0], [-3, 0, -6, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 1, 1, 0], [0, 2, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, -1], [0, 0, -1, 0, 2, 1, -1, 0, 0, -2, 0, 0, 0, 2, 1, 0], [0, -2, 0, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, 0, 0, 0], [0, 0, 0, 3, 0, 0, 2, 3, -1, 0, -2, 1, 1, -1, 0, 0], [0, -1, 0, -1, 0, 0, -1, 1, 1, 0, 1, 2, 1, 1, 0, 2], [0, 3, 0, 0, 0, 0, -2, 1, -2, 0, 2, 1, -1, -2, 0, 0], [0, 2, 0, 1, 0, 0, -1, 4, 0, 0, 1, 1, -3, 0, 0, 2], [-1, 0, -2, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, -1, -2, 0], [-1, 0, 0, 0, 1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, 0], [-4, 0, 4, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, -2, 0, 0], [-2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, -3, 0, 5, 0, 0, -1, -3, 2, 0, 1, -5, 2, 2, 0, 3], [-10, 0, -5, 0, 0, -1, 1, 0, -1, 0, 1, 0, 0, 1, 0, 0], [0, 3, 0, -1, 0, 0, -4, 0, 0, 0, 4, -1, -1, 0, 0, 3], [-2, 0, 2, 0, 1, 1, 2, 0, 4, -2, 2, 0, 0, -2, 0, 0], [0, 0, 1, 0, 0, 0, 2, 0, 3, -2, 3, 0, 0, -1, 1, 0], [0, -1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, -1], [-4, 0, 0, 0, 1, 2, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0], [0, -5, 0, 0, 0, 0, -1, 1, 2, 0, 1, 0, 1, 2, 0, 5], [0, 3, 0, 2, 0, 0, -2, 0, -2, 0, 2, 2, 0, -2, 0, 0], [1, 0, 2, 0, -3, 0, 1, 0, 1, -1, 2, 0, 0, 0, 1, 0], [0, 6, 0, -4, 0, 0, 0, 4, -1, 0, 0, 0, 0, -1, 0, -3], [-2, 0, -1, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, -1, 0, 0], [-1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 3, 0, 0, 1, 2, 0], [-5, 0, -10, 0, 0, 0, -3, 0, -2, 1, -4, 0, 0, 1, -1, 0], [0, 8, 0, -3, 0, 0, 0, 3, -4, 0, 0, 2, -2, -4, 0, -4], [-3, 0, -3, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 3, 4, -3, 0, -3, 3, 4, -3, 0, -2], [0, -9, 0, 3, 0, 0, 3, -3, 3, 0, -3, 0, 3, 3, 0, 0], [0, 1, 0, 2, 0, 0, 3, 3, 0, 0, -3, 2, -1, 0, 0, 1], [-3, 0, -6, 0, 3, 0, -2, 0, 0, -2, 0, 0, 0, 2, 2, 0], [-9, 0, 9, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0], [0, 0, 0, 2, 0, 0, 1, 1, -1, 0, -1, -1, 1, -1, 0, 0], [0, 0, 0, -2, 0, 0, -2, -2, 1, 0, 2, 1, 1, 1, 0, -2], [0, 0, 0, 4, 0, 0, -1, -4, 2, 0, 1, -4, 0, 2, 0, 0], [-6, 0, -3, 0, 0, 5, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 2, 0, 0, 4, 2, -2, 0, -4, 3, 3, -2, 0, -1], [0, 4, 0, 1, 0, 0, 0, -1, -1, 0, 0, 3, -3, -1, 0, -2], [0, 0, -2, 0, -2, -1, -2, 0, -1, -2, -1, 0, 0, 3, 1, 0], [16, 0, 8, 0, 0, -1, -2, 0, 1, 0, -1, 0, 0, -1, 1, 0], [0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, -1], [0, -1, 0, -3, 0, 0, -1, 1, 2, 0, 1, 3, -2, 2, 0, 1], [0, -12, 0, 6, 0, 0, 0, -6, 3, 0, 0, -5, 5, 3, 0, 6], [0, 4, 0, -4, 0, 0, 4, 1, -8, 0, -4, 4, -3, -8, 0, -4], [0, 3, 0, -1, 0, 0, -1, -3, 1, 0, 1, -2, -3, 1, 0, -6], [12, 0, 6, 0, 0, -5, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, -2, 0, -2, -1, -2, 0, -3, 2, -3, 0, 0, 1, -1, 0], [-1, 0, -1, 0, 0, 0, -1, 0, 0, 1, -3, 0, 0, -1, -2, 0], [5, 0, -5, 0, -3, -3, -2, 0, -4, 5, -2, 0, 0, -1, 0, 0], [0, 0, 0, -6, 0, 0, 1, -2, -1, 0, -1, 4, -2, -1, 0, 0], [0, 2, 0, 2, 0, 0, 1, -3, 1, 0, -1, -1, 3, 1, 0, 0], [2, 0, 4, 0, 1, 0, 1, 0, 2, -3, 4, 0, 0, 1, 3, 0], [0, 1, 0, -2, 0, 0, -4, -2, 4, 0, 4, 0, -2, 4, 0, -2], [7, 0, 7, 0, -1, 1, 1, 0, 0, 2, -3, 0, 0, -2, -4, 0], [7, 0, 0, 0, -2, -4, -1, 0, 0, 1, 0, 0, 0, -1, 1, 0], [0, 5, 0, 0, 0, 0, 4, 2, -8, 0, -4, 0, 2, -8, 0, -5], [-18, 0, -9, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, -14, 0, 6, 0, 0, 0, -6, 9, 0, 0, -4, 4, 9, 0, 7], [-4, 0, 4, 0, 3, 3, -2, 0, -4, -2, -2, 0, 0, 6, 0, 0], [0, 0, 0, 4, 0, 0, -1, 0, 0, 0, 1, 4, 4, 0, 0, 0], [0, 10, 0, -1, 0, 0, 3, 2, -6, 0, -3, 1, 1, -6, 0, -10], [0, 0, -9, 0, 2, 1, 3, 0, 4, -2, 4, 0, 0, -2, 1, 0], [5, 0, 10, 0, 2, 0, 7, 0, 2, 3, 4, 0, 0, -5, -3, 0], [0, 2, 0, -3, 0, 0, 0, 3, 0, 0, 0, -1, 1, 0, 0, -1], [0, 3, 0, -3, 0, 0, 9, 1, -9, 0, -9, 4, 1, -9, 0, -6], [4, 0, 2, 0, 0, -2, -5, 0, 1, 0, -1, 0, 0, -1, 4, 0], [-10, 0, -20, 0, -3, 0, 3, 0, 2, -1, 4, 0, 0, -1, 1, 0], [0, 0, -11, 0, -6, -3, 1, 0, 0, 2, 0, 0, 0, -2, -1, 0], [0, -8, 0, -1, 0, 0, 2, 1, 2, 0, -2, 0, -1, 2, 0, 0], [10, 0, 10, 0, 0, 0, 4, 0, 0, -1, 6, 0, 0, 1, 2, 0], [0, 0, 0, 0, 0, 0, 2, 0, -1, 0, -2, 4, 4, -1, 0, -9], [0, 13, 0, -6, 0, 0, -2, 7, -2, 0, 2, 1, -7, -2, 0, 0], [-3, 0, 0, 0, -2, -4, -1, 0, -2, 1, 0, 0, 0, 1, 1, 0], [-9, 0, 9, 0, 3, 3, -3, 0, -6, 0, -3, 0, 0, 6, 0, 0], [4, 0, 4, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, -2, 3, 1, 0, 2, 5, 5, 1, 0, 8], [0, 2, 0, -3, 0, 0, 1, 2, -2, 0, -1, 3, -1, -2, 0, -2], [0, 0, 0, 1, 0, 0, -8, 1, 4, 0, 8, 1, 1, 4, 0, 5], [0, 10, 0, 0, 0, 0, 0, 0, -4, 0, 0, 1, -1, -4, 0, -5], [-13, 0, 0, 0, -2, -4, 0, 0, 3, 0, 0, 0, 0, -3, 0, 0], [0, 0, -4, 0, -8, -4, -1, 0, 0, -2, 0, 0, 0, 2, 1, 0], [13, 0, 26, 0, -4, 0, -1, 0, -2, 3, -4, 0, 0, -1, -3, 0], [0, 0, 0, -2, 0, 0, 2, -1, -4, 0, -2, 2, -3, -4, 0, 0], [6, 0, 3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, -5, 0, 0, 0, 5, 0, 0, 1, 3, 0, 0, -1, -2, 0], [0, -4, 0, 1, 0, 0, 0, -1, -8, 0, 0, -2, 2, -8, 0, 2], [0, -12, 0, 7, 0, 0, 4, -10, 4, 0, -4, -3, 10, 4, 0, 0], [0, 0, 0, -3, 0, 0, -6, -3, 3, 0, 6, -6, -6, 3, 0, 5], [-2, 0, 2, 0, 8, 8, -1, 0, -2, -2, -1, 0, 0, 4, 0, 0], [0, -4, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 8], [0, -3, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, -3], [-3, 0, 0, 0, 2, 4, 0, 0, 5, 0, 0, 0, 0, -5, 0, 0], [-6, 0, 6, 0, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -2, 0, 0, 5, 3, -5, 0, -5, 5, 3, -5, 0, -8], [-4, 0, 0, 0, 3, 6, 0, 0, -3, 0, 0, 0, 0, 3, 0, 0], [-28, 0, -14, 0, 0, 4, -1, 0, 2, 0, -2, 0, 0, -2, -1, 0], [0, 3, 0, 5, 0, 0, 3, 3, 0, 0, -3, 5, 2, 0, 0, 3], [0, 0, 0, -3, 0, 0, 2, -3, -1, 0, -2, -2, -2, -1, 0, 3], [-6, 0, 6, 0, -5, -5, 3, 0, 6, -3, 3, 0, 0, -3, 0, 0], [18, 0, 9, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 7, 0, -1, 0, 0, 1, 3, 0, 0, -1, -1, -4, 0, 0, 7], [0, 0, 12, 0, -2, -1, 0, 0, 2, -4, 2, 0, 0, 2, 2, 0], [0, -3, 0, -5, 0, 0, -3, 1, -3, 0, 3, -4, -1, -3, 0, 0], [0, 0, 0, 0, 2, 0, -4, 0, 0, -4, 0, 0, 0, 4, 4, 0], [0, -4, 0, -1, 0, 0, -6, 2, 6, 0, 6, 3, 2, 6, 0, 8], [-12, 0, -24, 0, -2, 0, -2, 0, 0, -2, 0, 0, 0, 2, 2, 0], [0, 0, 4, 0, 8, 4, -5, 0, -3, -4, -3, 0, 0, 7, 2, 0], [0, 0, 0, 2, 0, 0, -8, 2, 4, 0, 8, -3, -3, 4, 0, 17], [0, 3, 0, 2, 0, 0, 6, 0, -6, 0, -6, -2, 0, -6, 0, -6], [0, 1, 0, -2, 0, 0, -3, -7, 0, 0, 3, -2, 5, 0, 0, 1], [-1, 0, 0, 0, 4, 8, -1, 0, -6, 1, 0, 0, 0, 5, 1, 0], [-16, 0, 16, 0, 1, 1, 1, 0, 2, -3, 1, 0, 0, 1, 0, 0], [15, 0, 15, 0, 1, -1, -8, 0, 0, -3, -2, 0, 0, 3, 6, 0], [10, 0, 0, 0, 5, 10, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0], [0, -10, 0, 0, 0, 0, -1, -2, 2, 0, 1, 0, -2, 2, 0, 10], [0, 4, 0, -5, 0, 0, -1, 4, 0, 0, 1, -5, -9, 0, 0, 4], [0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, -3, -3, 0, 0, 1], [0, 1, 0, -4, 0, 0, -8, -2, 0, 0, 8, -4, -2, 0, 0, 1], [0, 6, 0, 3, 0, 0, -4, -5, 8, 0, 4, -3, -2, 8, 0, -6], [8, 0, 16, 0, 5, 0, -3, 0, 2, -7, 4, 0, 0, 5, 7, 0], [0, -8, 0, 7, 0, 0, 0, -7, 0, 0, 0, -2, 2, 0, 0, 4], [14, 0, 7, 0, 0, 5, 5, 0, 2, 0, -2, 0, 0, -2, -7, 0], [4, 0, 4, 0, -2, 2, -1, 0, 0, -2, 3, 0, 0, 2, 4, 0], [-7, 0, -14, 0, 2, 0, -3, 0, -4, 5, -8, 0, 0, -1, -5, 0], [0, 2, 0, -3, 0, 0, 0, 3, -1, 0, 0, 2, -2, -1, 0, -1], [0, 5, 0, 3, 0, 0, -5, 4, -5, 0, 5, 7, -4, -5, 0, 0], [0, 1, 0, -4, 0, 0, 2, -1, -2, 0, -2, 3, -1, -2, 0, -2], [0, -10, 0, 4, 0, 0, 3, -11, 3, 0, -3, -7, 11, 3, 0, 0], [0, -3, 0, 2, 0, 0, 7, -3, 0, 0, -7, 2, 5, 0, 0, -3], [8, 0, 16, 0, -7, 0, -1, 0, 2, -5, 4, 0, 0, 3, 5, 0], [4, 0, 4, 0, 1, -1, -7, 0, 0, 1, -9, 0, 0, -1, -2, 0], [0, 0, 0, 2, 0, 0, -6, 2, 3, 0, 6, -2, -2, 3, 0, 15], [-11, 0, 0, 0, -3, -6, -4, 0, 0, 4, 0, 0, 0, -4, 4, 0], [0, 4, 0, -9, 0, 0, -1, 4, 2, 0, 1, 9, -5, 2, 0, -4], [0, 0, -5, 0, -4, -2, 10, 0, 6, 8, 6, 0, 0, -14, -4, 0], [22, 0, 11, 0, 0, -6, -8, 0, 1, 0, -1, 0, 0, -1, 7, 0], [-22, 0, 0, 0, -3, -6, -2, 0, -1, 2, 0, 0, 0, -1, 2, 0], [0, 5, 0, -7, 0, 0, -1, 7, -1, 0, 1, 0, -7, -1, 0, 0], [0, -12, 0, 4, 0, 0, 0, -4, 13, 0, 0, -1, 1, 13, 0, 6], [0, 11, 0, -4, 0, 0, 2, 0, -4, 0, -2, 4, -4, -4, 0, -11], [16, 0, 8, 0, 0, 2, 7, 0, -2, 0, 2, 0, 0, 2, -5, 0], [-2, 0, -4, 0, 4, 0, 0, 0, -2, 4, -4, 0, 0, -2, -4, 0], [-12, 0, -12, 0, 1, -1, 3, 0, 0, 5, -7, 0, 0, -5, -10, 0], [0, 0, 0, -6, 0, 0, 10, -6, -5, 0, -10, 5, 5, -5, 0, -16], [21, 0, -21, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, -2, 0, 0, -1, 3, -1, 0, 1, 1, -3, -1, 0, 0], [7, 0, 0, 0, 1, 2, -5, 0, -3, 5, 0, 0, 0, -2, 5, 0], [0, -6, 0, 2, 0, 0, -9, -6, 9, 0, 9, -8, -6, 9, 0, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_756_x_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_756_2_x_a();".
// 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_756_2_x_a(:prec:=16)
    chi := MakeCharacter_756_x();
    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_756_2_x_a();".
// 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_756_2_x_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;