
// Make newform 2695.2.a.w in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2695_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_2695_2_a_w();".
// 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_2695_2_a_w();".
// 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 := [1, -22, 31, 116, -86, -92, 56, 24, -13, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-29, -80, 169, 177, -130, -116, 35, 27, -3, -2], [61, 85, 49, -247, -112, 198, 39, -56, -4, 5], [-15, 214, -72, 63, 112, -175, -39, 56, 4, -5], [2, -73, -18, 401, 5, -314, -4, 83, 1, -7], [57, -206, -582, 446, 591, -255, -183, 54, 17, -4], [-91, 320, 681, -800, -538, 487, 136, -108, -11, 8], [120, -355, -850, 646, 668, -371, -171, 81, 14, -6]];
    Rf_basisdens := [1, 1, 1, 23, 23, 23, 23, 23, 23, 23];
    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_2695_a();"
function MakeCharacter_2695_a()
    N := 2695;
    order := 1;
    char_gens := [2157, 1816, 981];
    v := [1, 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_2695_a_Hecke(Kf)
    return MakeCharacter_2695_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, -1, 0], [-1, 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, -1, 0, 0, 0, 0, 0, 1, 0, -1], [-2, -1, 1, -1, 0, -1, 0, 0, 0, -1], [0, -1, 1, 0, 1, -1, 1, 0, 0, -1], [-1, 0, 0, 0, 0, 0, -1, -1, 0, -1], [0, 0, 0, 0, 0, -1, 1, -1, 1, -1], [0, -1, -1, -1, -1, 1, 0, 0, 0, 2], [2, 0, -1, 0, -1, 1, -1, -1, 0, 1], [-4, 1, 1, 0, 1, -1, 2, 0, 2, 0], [3, -2, 1, 1, 1, -1, 0, 2, 1, -1], [-2, 0, -1, 1, 0, 1, 1, -1, 3, 2], [0, 0, -2, 0, 0, 1, 3, -1, 2, 2], [-3, 0, -2, 2, -1, 2, -2, -1, 0, 1], [-2, 0, 2, 1, 2, 0, -1, 2, -2, -1], [2, 0, 0, -2, 0, 1, 1, -1, 0, 0], [-1, 1, 0, 2, -1, 0, -1, 0, 1, 1], [-4, 2, -2, 4, 0, 0, 0, 0, 1, 4], [-1, 0, -3, -2, -2, -1, -1, 0, 0, -1], [-1, 3, -1, 2, -3, 0, 0, 0, 0, 0], [-2, -1, -1, -4, -1, -1, 1, 0, 0, -3], [-1, 3, 1, -2, 1, 2, 1, -2, 0, 0], [-7, 0, -2, 0, -1, 0, 0, -1, -2, 3], [2, -2, 0, -4, -2, 0, -1, -2, 1, 0], [0, -2, 1, -4, -3, 0, -2, 2, 0, 1], [-2, 2, 0, 2, 2, 0, 0, 0, 0, -2], [4, -2, -2, -3, -2, -3, 1, 1, -1, -1], [-1, 2, -1, 2, 1, -1, 4, 2, -2, 0], [-4, 2, 0, 1, -2, 0, -2, 0, -2, 2], [-1, 0, 4, 1, 2, 1, -2, 0, -3, -4], [1, 1, 2, -1, 1, -1, -1, 1, 0, -4], [0, 1, 0, -2, -3, -2, -3, 1, 0, -1], [-2, -4, 1, 2, 0, -2, -2, 2, 0, 0], [0, -3, 4, -2, -2, 0, -2, -1, 1, 1], [-1, -2, 0, -3, 0, 1, -4, 0, 1, 0], [-2, 0, 3, 1, 4, 3, -3, 1, 0, -4], [-9, 2, -1, 0, -1, 0, 0, 1, 2, 3], [-4, 7, -6, 0, -3, 5, -2, -4, 3, 4], [-1, 0, 0, -1, 1, 4, -3, -1, 2, 2], [1, -2, 2, -2, 2, -1, 1, 4, -4, -3], [8, -4, 2, 5, 2, 0, -2, 4, -5, -1], [4, -4, 2, -5, -2, -2, 0, 0, 1, 1], [1, -1, 1, 2, 0, 3, 1, 1, 2, -2], [-2, -4, 0, 0, 2, 1, 3, 1, -1, -3], [7, -5, 2, 3, 1, -1, 1, 3, 1, -2], [-4, 3, -5, -3, 2, -1, 0, 0, 1, 1], [-2, 2, -1, 2, -3, 2, -3, 0, -6, -1], [7, -2, 1, 2, 3, -2, -1, 3, -4, -4], [-6, 3, 2, 2, 5, 2, -1, 1, -2, 3], [1, 2, -2, 1, -7, 0, 0, -1, 0, 3], [-7, 6, 2, -3, 1, 0, -2, 1, 0, -5], [-12, 3, -2, 4, 4, 2, 1, -3, -1, -1], [-7, 0, -2, 4, 4, 1, 0, -2, 2, 5], [-10, 3, -3, 0, -1, 1, -1, 2, 2, 5], [-4, 8, -5, 2, -1, -4, 3, 0, -2, 3], [9, 4, -1, 1, -1, -3, 0, -2, 3, -1], [-1, -4, 4, -4, 0, 1, 1, 0, -2, -5], [-5, -2, 1, 4, 1, 2, -6, 1, 1, -1], [-4, 3, -1, -1, 4, 1, 3, 2, 0, -1], [3, 0, 0, 1, 1, 1, 4, -4, 5, 3], [2, -2, 6, -5, 4, 0, 1, 2, -4, -3], [-5, 2, -1, -4, -1, 2, -2, -5, 1, 1], [-3, 2, -1, 7, -3, 0, -3, -3, -1, 5], [-3, -2, 4, -2, 6, -1, 3, 0, 2, 3], [1, 4, 0, 1, 0, 3, 2, 2, 1, 4], [-1, 4, 0, 11, 2, -1, 4, 2, -3, 4], [-6, 0, -7, -1, 1, 0, -2, -2, -4, 0], [4, 9, -5, 3, -4, -3, 2, -2, 3, 3], [-2, 5, 3, 2, 1, -6, 1, 3, 2, -1], [0, 2, -1, -3, 6, -3, 4, 1, 3, 0], [5, 4, 1, 2, 3, 4, -1, -3, 0, 0], [-5, 6, -4, -6, 0, 3, 1, -4, 0, 1], [-8, 3, -1, -5, 0, 1, -4, -2, -2, -3], [0, 8, 2, 4, 0, -1, -3, 3, -9, -1], [5, 4, 2, 7, 7, -1, 0, 2, 1, -1], [-8, -2, 4, 2, 2, 2, 0, 0, -8, -2], [2, -6, -2, 0, -8, -4, 3, 2, 0, -1], [1, 3, -2, 3, -5, -1, 0, -5, 4, 3], [-3, -9, 1, -6, -1, -1, 2, 3, 4, -2], [10, -10, 1, -4, 2, 1, 1, 1, -3, -7], [3, -4, 5, 6, 3, -4, 3, 1, 5, 3], [-6, 4, -5, 2, -5, 6, -3, -4, 0, 5], [1, -4, -3, 1, 1, 1, 2, 4, 5, -3], [0, 1, -3, -8, 1, 1, 4, 0, 3, 0], [-7, 4, -6, -6, 0, 0, -5, -5, -2, -1], [0, 1, 6, 0, 0, -1, -3, 4, -8, -5], [5, 0, 1, -1, -3, -3, 2, 0, -5, 1], [-6, -8, 6, 2, 4, -2, 1, 2, -5, -10], [-5, 0, 2, -3, 3, -2, -1, -1, -4, -8], [-5, 8, 0, 10, 0, -2, -1, -1, 6, 5], [-10, 3, -3, 2, -3, 4, -1, -5, 6, 7], [4, 3, 6, -8, -3, -2, 1, 1, -2, -3], [-8, -1, -2, 2, 0, 4, -2, -1, 1, 7], [0, -12, 4, 4, 0, 0, 2, -2, 0, -2], [-1, -6, -4, 3, -3, 2, 3, 3, 2, -4], [-8, 9, 0, -8, 0, 0, 4, -3, 3, 3], [-2, -11, 4, 0, -1, -4, -3, 3, -2, -1], [-7, 0, 4, -2, 6, 1, 4, -4, 8, 1], [1, 2, -3, -7, 3, 3, 6, -4, 3, 7], [-2, -2, -2, 0, -4, -4, 2, 0, 6, 2], [3, -4, 2, 4, 0, -2, 6, 3, 3, 10], [-2, 6, 0, 12, 4, -1, 5, 1, 5, -3], [-10, 0, -1, 3, 0, 1, -5, -3, -4, 4], [10, -6, 0, -10, -6, -4, -4, 4, -1, 0], [-17, 2, 4, -5, 3, 3, 3, 0, 2, 3], [9, 1, 4, 12, 7, -2, 7, 4, -5, 5], [-4, 1, -3, 3, 1, 5, -2, 0, 8, 0], [5, -12, 8, -1, -3, -9, 1, 6, 1, -7], [10, 4, 8, 9, 4, -6, 0, 4, -5, -3], [-8, -4, -1, 4, 3, 3, -9, 1, 2, 5], [12, -8, -4, -5, 4, -2, -1, 0, 4, 1], [-12, 2, -2, -2, -4, 0, 0, -4, -8, 6], [-7, -12, 5, -6, 2, 4, -4, 1, -5, -4], [18, -2, 3, 11, -4, -5, 6, 5, 1, -2], [-8, 5, 4, 12, 2, 8, -9, -1, 0, 5], [6, -10, 9, 5, 1, -1, 1, 3, -7, -6], [-11, -4, 3, -6, 0, 2, -8, -5, -1, -4], [3, -2, 4, -8, 3, -4, 5, -5, 6, -2], [0, 0, -3, 4, 1, 3, 5, -1, 0, 5], [-11, 5, -7, 4, -3, 10, 1, -6, 3, 12], [-4, 2, -8, -7, -2, 4, 4, 2, -7, -1], [2, 4, 1, 10, -1, 0, -2, -2, 12, 2], [-6, 10, 0, -4, -2, 4, -6, -4, -8, 6], [4, -5, 6, -8, -7, -5, 0, 4, -11, -12], [12, 2, -1, -5, -3, -6, 10, 2, -6, 2], [9, 3, 3, -4, 3, -6, 13, -2, 11, 0], [0, -3, -9, -1, -4, 7, -9, -2, -4, 5], [-15, 5, 6, 12, 5, -3, -2, -1, 0, 2], [3, 0, -3, -5, -5, 4, -3, -3, -7, 1], [-7, -4, 0, 2, 2, 6, 0, -1, 3, 4], [6, 4, -1, -13, -1, 3, 1, -3, -11, -4], [-15, -3, 3, 15, 6, -1, -7, 3, -2, 2], [-8, -3, 2, 8, 0, -1, -1, 0, 6, -1], [3, -5, -5, -4, -3, 0, 4, 2, -2, -2], [-4, -2, 8, -10, 4, 0, 3, -4, 4, -2], [-17, -10, 4, -11, 5, -1, 3, 0, 3, -1], [-2, -6, 6, 2, -2, -4, 6, 4, 2, -2], [9, -12, -7, 2, 0, 0, 2, -3, 10, 7], [5, -2, 2, 14, 4, -5, 3, 2, -4, -3], [-5, 2, 4, 12, -2, 0, -3, -1, -4, -5], [14, 2, 9, -2, 3, 1, 1, 1, -8, 1], [5, 5, -8, 1, -7, 1, -7, -3, 4, 8], [-13, -11, 1, 7, 0, -1, 1, 3, -2, -4], [-1, 4, -8, -5, -5, -1, -4, 0, -6, -1], [-22, 3, -1, 5, -6, 1, -8, -2, -6, -3], [16, 1, -5, 0, -3, -5, -6, 6, 4, -4], [1, 0, 5, 1, 3, 3, -2, -2, -7, -5], [1, 6, -5, 6, 3, 6, -1, -1, -4, 2], [-16, 3, 3, 1, 1, 1, -7, 2, -4, 1], [-2, 0, 1, 5, -1, -4, 4, -4, 5, -2], [10, -10, 2, -4, -8, -8, 5, 6, -6, -6], [13, -4, -8, 4, -2, -2, -1, -3, -6, 1], [7, 9, -12, 6, -13, -2, 1, -4, 5, 11], [-1, 7, -2, -10, -1, 8, -3, -8, 5, 9], [-15, -6, 5, 3, -2, 4, -4, 3, -2, 1], [-26, -4, 7, -1, 6, 3, -8, -3, -9, -10], [-3, -5, 2, -3, 3, 5, 4, -1, 2, 3], [1, -2, -3, 1, 1, 1, 6, -2, 1, -7], [5, 4, 1, 3, 3, -9, 6, 2, -3, -7], [4, -8, 10, -2, -4, -3, -5, 1, -10, -10], [-3, -3, -3, -5, 2, 1, -7, -5, 0, -4], [5, -8, 4, -6, -6, 1, -6, 6, -6, -11], [5, -7, 15, 4, 9, 0, 8, 4, -3, -8], [-12, 5, 2, -6, 3, 1, -6, -2, -1, -8], [9, 8, 1, 2, 7, 0, 1, -1, -10, -5], [6, -3, 8, -6, 3, -6, 5, -3, -2, -5], [16, -4, 5, 5, -2, -1, 3, 5, 4, -6], [-13, 1, 4, -12, 1, 2, -3, -4, 3, -3], [-9, 2, -2, -2, 5, 2, -5, -3, 8, 0], [8, 3, 3, -8, 7, 5, 8, 4, 7, -2], [7, -6, 7, -8, -3, 0, -1, -1, -4, -2], [4, 2, 11, -1, -1, -2, -2, 2, -4, -4], [-15, 1, 3, 3, 6, 5, 8, -3, 0, -3], [-4, -7, -1, 8, -3, 2, 5, 5, 4, 7], [8, 4, -1, -6, -2, -8, 8, -2, -4, -6], [0, 2, -5, -11, -2, -7, 0, 1, 6, -4], [-8, 9, -5, 3, -1, 7, -6, -6, 0, 4], [10, 0, -1, -2, -11, -5, 1, 5, -2, 1], [-6, -7, 1, -9, -3, -1, 3, -2, -4, 5], [21, -12, 9, -2, -1, -7, 0, 10, -2, -14], [-6, 1, 0, -8, -4, 4, 7, 3, -2, 13], [-7, 6, -1, -14, 7, -6, 1, 3, 6, -4], [-14, 5, -3, -3, -3, -1, -11, -4, 0, 3], [6, 8, -1, 10, -3, -3, -7, 3, 4, 1], [8, 6, -5, 5, -4, 1, -9, -1, -11, -6], [-15, 6, -4, 18, 0, 5, -1, -4, 4, -1], [1, 0, 7, 3, 6, 4, 8, -5, -6, -5], [-8, 9, -15, -7, -12, 9, -2, -10, 5, 9], [13, 0, 0, 7, 6, 0, -3, 7, 5, -6], [8, -5, 10, -14, 5, -5, 10, 0, 1, -10], [-5, -2, -4, -11, -7, 0, -13, 1, 4, 2], [24, 0, 4, -2, 0, -9, 5, 1, 0, 4], [-18, 4, -11, -9, -14, 3, -8, -3, -10, 10], [-12, 2, -8, -10, 4, 7, -9, -7, -3, 3], [12, -6, 10, 6, 10, -3, 11, -3, 10, -4], [-7, -4, 8, 1, -5, 5, -8, 0, 4, 5], [-4, 7, -11, -3, -6, 7, 3, -2, 1, 9], [1, -6, -8, 6, 2, 3, -3, -6, 4, 11], [-16, 11, -5, 7, 1, 1, -1, 4, -10, 3], [-8, 10, 0, 6, 0, 2, -5, 0, -6, 4], [-31, 1, -4, 9, -1, 7, -6, -5, 10, 5], [6, 1, -7, 4, -1, 1, 8, 2, 4, 2], [-2, 8, 2, 2, -2, -4, -9, -2, -4, 0], [-3, 0, -6, -5, -1, 6, -2, -3, -4, 1], [-8, -8, -3, 5, -1, 1, 5, -1, 3, 6], [-16, -5, -8, 2, -1, 0, 3, 3, 2, 5], [15, 6, 6, -4, -11, -6, 6, 3, -2, -5], [11, 2, 8, -4, 6, 4, 5, 1, 8, -9], [-12, -14, 7, 11, 5, -4, 1, -2, 8, -1], [-14, 10, -4, 4, -4, 11, -9, -5, -2, 16], [7, -3, 15, -6, 7, 4, -5, 4, -13, -10], [-10, 7, 1, -16, 1, 1, -5, -4, 10, -3], [20, 5, 3, -8, -1, -7, 13, -2, 4, -9], [1, 2, -2, -11, -4, 3, -8, -8, -5, -6], [-14, 7, 5, 10, 7, 5, 0, 2, 8, 8], [5, -2, 14, -3, -2, -7, -8, 0, -1, -6], [-6, 4, -11, 3, -7, 3, -5, 1, 5, 6], [-8, -5, -6, 2, -5, 5, 6, -6, 11, 14], [13, -6, 6, -7, -1, 2, 1, -5, 0, 6], [3, -12, 5, 6, 6, -4, 6, 1, 7, 0], [-10, -10, 0, -25, -4, -6, 6, -6, 3, -7], [-14, 7, 6, -2, -6, -6, 9, -7, 1, 5], [-27, 1, -8, 6, 5, 3, 0, -7, 16, 4], [1, -8, 7, 0, -3, -12, 1, -1, -7, -13], [-8, -5, -2, -18, -1, -5, 10, 0, 3, 0], [-14, 10, -6, 22, 0, 4, -1, -6, 4, 14], [13, -7, 5, -4, -5, -5, 2, 5, 4, -4], [2, 12, -2, -14, -8, 9, -7, -9, 16, 6], [0, 8, 9, 5, 1, -2, -7, -4, 4, -1], [4, -3, 2, -6, -3, -11, 6, 4, -1, 2], [-17, 5, -4, -12, 1, 7, -10, -1, 2, 6], [0, -5, -10, -6, -8, -4, -7, 1, -8, 3], [5, -1, 3, 4, 3, -2, 2, -4, 10, 0], [15, -13, 4, -17, -5, -5, -1, 3, -6, -2], [6, 18, 0, 7, -2, -2, -4, -4, -3, -1], [9, -2, 0, 18, 6, 9, 1, -4, 8, -3], [17, 1, 8, -19, 3, 3, 0, 1, -6, -9], [-15, 0, 5, -7, 3, 1, 2, 12, -7, 1], [11, -7, 7, 1, 4, 3, 2, 7, -14, -5], [-2, -4, 1, 15, 6, -1, -1, 9, -7, -4], [-5, 17, -9, 10, -1, -6, 11, 2, -7, 9], [-19, -2, -10, 9, -3, -1, -2, 4, -2, 13], [9, -2, 12, -6, -5, -12, -5, 7, -8, -14], [27, -1, 5, -6, -3, 10, -7, 0, -11, -4], [1, -9, -4, -11, -3, 5, 6, 5, -6, 7], [-4, -1, -2, 10, 9, -3, -4, 2, 1, 4], [-15, -4, -11, 14, -1, 7, -8, 6, -4, 2], [23, 0, -1, -10, -13, -6, 5, 1, -2, 8], [-1, -7, 8, -9, -11, -1, -4, 1, 4, 5], [14, 0, 3, -13, -3, 0, 10, -2, 9, -2], [-16, -1, -9, 5, 7, -1, 11, 4, -6, 1], [11, -7, 10, -15, 7, 3, 4, 9, -1, -8], [-1, -8, -12, 12, 0, 6, -14, 3, -5, 4], [6, -11, -3, 6, 5, -2, 9, 1, 6, -5], [-6, 3, -8, -16, -4, 4, -7, -1, 11, 5], [-23, 13, -2, 1, -7, 5, -5, -5, -20, 4], [2, 9, -12, 2, -8, -8, -5, 3, -10, -3], [-29, 6, 0, 3, -8, 5, -14, -2, -9, 4], [-2, 12, -6, 16, 8, -2, 10, -4, 6, 4], [24, -1, 13, -2, 13, 3, 12, 2, -8, -6], [-7, 6, 7, 2, 7, -4, 9, 5, 2, 1], [0, -22, 7, -1, 8, -9, 10, 5, -2, -14], [22, 2, 8, -14, 8, -8, 5, 0, -4, -9], [2, -5, 9, -2, -5, -2, 9, -3, 4, 1], [-33, 11, -4, -10, 9, 5, 1, -1, 12, 8], [10, -2, 15, -4, 1, -15, 1, 5, 2, -11], [-6, -2, -12, 6, -4, 3, -7, -1, 6, -12], [-11, 17, -13, -3, -2, 3, 8, -5, 2, 7], [19, 12, -3, 5, 7, -5, 14, 4, 7, 5], [-10, 8, -10, -6, 2, 8, 9, 2, 2, 8], [-2, -18, 8, -2, 0, 0, -2, 0, -6, -12], [-3, -8, -5, -8, 3, -6, 13, -3, 14, 3], [15, -7, 0, 3, -5, -7, 4, 1, -3, 4], [7, 10, -7, 4, -5, 2, 3, 1, 6, 18], [-17, 8, -2, -10, 10, 4, 4, 9, -1, 2], [-10, 7, 4, 0, 3, 6, -3, -9, 0, -11], [16, 4, 0, 6, 0, 2, -6, -6, 2, 10], [-9, 2, 12, 5, 5, -11, -4, -2, 7, -5], [18, -4, 6, 24, 8, 0, -4, 6, -2, -6], [-7, 0, -4, -1, 11, 1, -4, 2, 11, 1], [-19, -7, 3, -10, 0, 5, -7, -1, -2, 10], [-2, 0, -1, 9, -6, -1, 2, -5, -2, -2], [19, -6, 4, 0, -10, -1, -13, 8, -12, -9], [19, -10, 14, -17, 2, -13, 4, 6, -3, -20], [23, -14, 8, -2, -8, 2, -7, 3, -6, 5], [26, 4, -13, -10, -5, -6, -4, 4, 0, -10], [-6, 4, 10, 9, 4, -6, -3, 8, -12, -7], [-4, -4, 1, -8, -4, -1, -1, 3, -5, 9], [22, -9, 3, 5, -4, -9, 0, 6, -6, -9], [-16, 0, -5, 3, 9, 0, 2, 0, 1, -14], [8, -4, 3, 21, 5, -4, -1, 6, -10, -5], [14, -6, -1, -21, -5, 6, -10, 8, -13, -2], [-9, -18, -3, -18, 2, 6, -1, -3, -6, -4], [18, 1, -5, 2, -5, -11, -2, 6, -6, -6], [-3, 2, 2, 9, 13, -1, 3, -6, 5, -5], [-15, 13, 0, 10, 3, 5, -1, -11, 16, 6], [3, 4, -17, 2, -3, 4, -1, 7, 1, 3], [-1, 5, 1, -4, 3, 8, 3, 6, -3, 8], [-3, -10, 9, 1, 5, 1, -8, -2, 13, -3], [-1, -17, 11, 12, 3, -5, 0, 13, -2, -4], [-10, 10, 4, 15, -2, 2, -8, 6, -5, 5], [-8, -4, 6, 6, 16, 2, 12, -8, 6, -10], [-13, -6, -13, 11, 5, 3, -6, 0, 3, 13], [14, 0, -5, -16, -5, 11, -3, 3, -4, 5], [-2, 20, 7, -1, -3, 6, -7, -4, 4, -5], [-13, 8, 8, -2, 6, 6, -6, 5, -11, -10], [-8, 0, 0, 22, 6, -7, 13, 1, 4, 4], [9, -22, -5, -6, -3, -8, 7, 3, 13, 7], [-17, -14, 2, 0, -4, 5, 11, -6, 12, 7], [0, 11, -18, 16, -11, 2, -9, -1, -4, 9], [-12, -16, 9, 9, 3, 2, 4, -8, 7, 2], [2, 2, 6, 10, 0, 3, 1, -5, -10, -16], [-18, 1, 6, 19, 12, -9, 7, 4, 8, -1], [-5, -3, -5, 17, -4, 5, -3, 5, 0, 14], [17, -2, 12, -8, 12, -3, 3, 2, -6, -11], [-21, 22, -3, -2, -1, 0, 0, -3, 1, -7], [-13, -6, 2, -4, 2, -6, 0, 1, 3, -4], [9, -9, -4, -19, 1, -5, 3, 3, -8, -14], [3, 12, -15, 12, -3, 3, -6, 0, 4, 12], [7, -13, 3, 8, -2, -15, 1, 7, -2, -20], [-15, -4, -8, -21, -2, 5, -6, -6, -3, 2], [10, 16, -13, -9, 1, 0, 12, -6, 5, 2], [-15, 11, 12, 1, -3, -1, -1, 1, -8, -6], [-27, 1, -6, -6, -1, 3, 4, 1, 4, 14], [12, 8, 4, 8, 6, -2, -8, -2, -7, -16], [1, -8, 2, 19, 0, -1, 4, 4, 7, 6], [-34, 10, 5, -9, -1, 2, 4, 2, -3, 2], [-15, 15, -8, 12, 7, -8, 15, 0, 1, 13], [14, 4, -3, -4, -1, -1, 1, 3, 14, -5], [23, -6, 5, 12, 16, -2, 6, 7, 16, -7], [-4, 12, -6, 2, -2, 12, -17, -2, -10, 0], [-1, -21, 15, 0, -3, -1, -4, 3, -8, -2], [15, -4, 4, -15, 5, 5, -10, 0, 4, -3], [-24, 7, 11, -10, 7, 2, 9, -1, 22, 1], [0, -9, -12, 24, -4, 6, 1, -3, 2, 21], [16, 12, 2, 3, 0, 4, -6, -12, 0, -6], [18, -8, 5, -5, -12, -3, 3, -1, 2, -14], [-21, -2, 14, 2, 14, 5, -4, 8, -10, -1], [-10, -12, -1, -4, -9, -5, -11, -5, -4, -7], [-7, 11, -9, 7, -12, -3, -1, -9, 12, 16], [1, -7, -16, 8, 1, 4, -3, 0, 7, -1], [16, 4, 10, 4, 6, -6, 6, -2, 12, 8], [-24, 13, -16, 12, -1, -5, 12, -6, 5, 14], [25, 2, 10, 3, 1, -8, -5, 3, 2, -18], [23, -14, -1, -4, 7, -6, -5, 1, -12, -14], [7, 4, 4, 21, 5, -8, 17, 5, 8, -2], [4, -7, -7, -11, -12, 3, -6, 4, -9, -9], [10, -13, 5, -3, -1, 1, -3, 2, -10, -7], [16, 8, -13, -10, -5, 1, -5, -3, 14, 5], [13, -9, 8, -8, 5, -8, 1, -2, -7, -11], [23, 4, 4, 15, 19, -5, 24, 2, 3, -5], [-14, -3, 0, 20, 3, -3, -2, 8, 5, 6], [-2, -16, 5, -7, -8, 9, -12, -5, -1, -10], [24, -2, -1, -16, -7, -6, -2, 0, 0, 4], [21, -17, 3, -7, 0, -5, 6, -1, -12, -9], [-19, -6, 0, -2, -2, -10, 5, 9, 4, 7], [-1, -4, -8, 7, -10, -3, 16, -4, 7, 18], [5, 4, -11, -8, -11, 0, -1, -11, 14, -4], [-12, 21, -1, -15, 3, 9, -12, -12, -14, -10], [9, -2, 0, 1, 0, 1, 8, -2, -9, -2], [31, -8, 5, -22, -4, -5, 1, 2, -14, -19], [3, 20, -16, -13, -5, 5, 1, -10, 2, 13], [13, -4, 19, 19, 11, -1, 10, 10, -9, -7], [8, -14, -8, -4, -8, 4, 4, 0, 13, 8], [32, -14, 1, 9, -5, -1, -7, -3, -7, -12], [22, -1, -2, 12, 1, 5, 8, -12, 15, -4], [-11, 0, 0, -2, 2, 5, 5, 6, 4, -3], [9, 16, 5, 9, 20, 10, 7, 1, -2, 2], [-10, 10, -3, 7, -13, -7, -7, -1, -15, -6], [-1, -13, -4, -6, -13, 11, -4, -1, 0, 8], [-3, 12, 4, 6, 1, 12, -3, 1, -2, 16], [11, -4, 3, 5, -1, -13, -4, 18, -7, -11], [17, -21, -3, -21, 0, -9, 9, -1, 4, -4], [2, 10, -9, 18, 2, -2, 8, -6, -12, 8], [20, 10, 2, -4, 4, -6, -12, -2, 5, -10], [-3, -15, -10, -8, -3, 5, -8, -1, 6, -7], [-21, 15, -6, -3, -13, -1, -15, 1, -5, 6], [-5, -1, 3, -9, -8, -9, 6, -9, 2, -3], [-33, 4, 1, 13, -3, 7, 8, -8, 11, 5], [9, 14, 1, 5, 3, 3, 0, 2, -1, 13], [-6, -10, 1, 4, 15, 3, 3, -1, 0, 7], [-9, 24, -14, 6, -13, -2, -3, -15, 12, 14], [25, 9, -12, 5, 3, 1, -6, 3, 7, 10], [19, -7, 0, 6, -1, -6, 5, -2, -7, -5], [21, -2, 11, 6, -5, 6, -20, 1, -1, -13], [-17, -8, 2, 0, -12, 5, -10, -10, 2, 3], [9, -12, -9, -10, -13, 6, -7, -1, -8, 5], [0, 2, -3, 6, 4, 8, -4, -6, -18, -4], [-12, 1, 6, 24, 6, -12, 6, 15, -9, -11], [1, 16, -14, 2, -12, 8, -12, -7, -15, -4], [21, 19, -6, 3, 5, 3, -4, -5, 9, 12], [25, 3, -16, -12, -9, 0, 1, -10, -3, -1], [4, -4, -4, -2, 0, -9, 13, -1, 30, 6], [6, -4, 9, 3, 3, -6, 9, 2, 4, -5], [-4, 8, -4, 2, -2, -6, -8, 0, -2, -14], [-11, -3, -8, -12, -21, -1, 2, -5, 4, 8], [6, 15, 2, 4, -5, -9, -8, 4, -5, -6], [-1, 8, 11, -16, 12, 4, 11, -11, 14, 2], [-3, 6, -21, 3, -9, -2, -13, -7, 9, 13], [8, 16, -16, -10, 2, 2, 20, -4, -4, -2], [-5, -8, -5, -6, -1, -10, 8, 1, -7, -3], [21, -10, 6, -8, 13, -4, 6, -3, -2, 5], [-23, 9, 4, 6, 7, 13, -5, -7, 10, -2], [8, -22, -2, -12, -8, -2, -6, -4, 12, -14], [18, 0, -4, -10, -4, 15, -21, 1, -7, -9], [39, 9, 6, 9, -7, 1, -4, 5, -9, -4], [8, -4, 8, -10, 8, 8, -1, 8, 12, -11], [19, 26, -6, 1, 3, -7, 1, -2, 10, -3], [19, 0, -15, 5, 1, 7, 12, 4, 3, 9], [-41, 2, 6, 0, 10, 2, -1, 3, 10, 1], [13, 4, 3, 16, -16, -7, 3, 2, 0, 1], [-5, -8, -12, -23, -8, -1, 8, 0, -5, 4], [-5, 12, -8, 4, 9, 6, 1, -1, 2, 10], [22, -26, 4, -12, 8, -10, 4, -6, 16, -6], [25, 18, -13, -6, -13, 4, -2, -3, -2, -5], [-14, 15, -3, -29, 8, -3, -5, 0, 1, -9], [5, 8, -15, 0, -7, 0, -8, 7, -7, -7], [17, 6, 0, -6, 12, 0, 22, 3, 11, 4], [-16, 3, 11, -5, 4, -3, -2, 4, 8, 3], [-7, 14, -7, 19, 5, -3, 0, -4, -11, 11], [-24, -9, -8, 9, 8, -9, 2, 2, 12, 2], [0, 2, 12, 24, 4, -8, 13, 6, -10, -6], [-4, 17, -4, 22, 14, 2, 16, 1, -3, 9], [-13, -4, 5, -15, -5, 5, -4, 2, 3, 17], [10, -12, -4, -4, 10, 3, 3, 9, -1, -3], [1, 4, 12, 18, -1, -6, -18, 3, -8, -7], [-11, 6, 1, -2, 8, -6, 4, 1, -2, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2695_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2695_2_a_w();".
// 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_2695_2_a_w(:prec:=10)
    chi := MakeCharacter_2695_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_2695_2_a_w();".
// 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_2695_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2695_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -22, 31, 116, -86, -92, 56, 24, -13, -2, 1]>,<3,R![-14, -4, 182, 68, -308, -12, 130, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;