
// Make newform 8034.2.a.w in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_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_8034_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 := [768, 1768, -352, -2790, -308, 1787, 180, -580, -7, 94, -8, -6, 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], [143896, 207480, -190830, -292064, 120931, 141564, -48976, -27271, 10622, 1400, -874, 81], [165236, 199594, -253616, -254607, 175289, 108598, -64829, -17071, 11894, 276, -835, 103], [-70548, -89922, 123561, 125519, -97565, -56903, 39381, 8967, -7606, -5, 551, -75], [-281980, -276606, 514220, 369489, -389932, -158256, 146843, 21678, -26572, 814, 1827, -268], [131084, 152708, -240617, -211208, 199984, 93363, -82784, -13086, 16048, -551, -1156, 176], [376680, 402794, -702716, -561859, 569936, 247440, -228825, -34366, 43376, -1486, -3081, 464], [-977728, -1045464, 1679930, 1378000, -1267853, -593500, 488408, 85377, -90898, 1920, 6414, -927], [-1227304, -1257836, 2140510, 1662758, -1610725, -714928, 614922, 101765, -113458, 2688, 7952, -1151], [1262856, 1387536, -2177110, -1877140, 1658113, 821112, -645932, -121245, 121322, -1900, -8618, 1227], [-1251440, -1408360, 2233342, 1927456, -1763901, -850752, 704600, 124697, -134594, 2964, 9686, -1415], [1437152, 1485492, -2588182, -1999642, 2017465, 860768, -788534, -118241, 147226, -4912, -10372, 1555]];
    Rf_basisdens := [1, 2936, 1468, 734, 1468, 734, 1468, 2936, 2936, 2936, 2936, 2936];
    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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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 := [[-1, 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, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 0, 1, 1, 1, 0, 0, -1, 1, 0, -2], [0, -1, 0, -1, 1, 0, -1, 0, -1, 1, 0, -1], [-2, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0], [-2, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, 1, 0, 1, -1, 1, 1, -1, 0, 0, 0, -1], [1, 2, 1, 1, -1, 0, 1, -1, 2, -1, -1, 0], [0, 3, 0, 1, -1, -2, 0, -1, 2, -2, 1, 3], [0, -1, 2, 0, 0, -1, -1, -1, 1, 0, -1, 0], [0, 1, -1, -1, -1, -1, -1, -1, 1, 0, -1, 1], [-1, 2, -1, 1, -1, 0, 2, -1, 2, 0, 0, 0], [-2, 1, 2, -2, 0, 1, -1, 0, 1, 0, 0, 0], [-2, 1, 0, 1, -1, 0, 1, 1, 0, -2, 0, 0], [0, 0, -1, -1, 0, 0, 0, 1, -1, 1, 2, -1], [-4, 0, 1, 4, 0, 0, 1, -1, -1, 0, -1, -2], [1, 3, -1, -2, 2, 1, 0, 0, 1, -1, 2, 3], [-2, 2, 0, -2, -2, 0, -2, 2, 1, -1, 1, 1], [-3, -3, -1, 1, 0, 0, 2, -1, -3, 2, -1, -4], [0, -1, 0, 0, 2, 1, 1, -2, -1, 2, 0, -1], [-2, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0], [-4, -1, 1, -2, 0, 2, -3, 2, 0, -1, 3, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -4, 2, -4, 3, 1, -3, 0, 0, 2, 1, 0], [-3, -1, -2, 2, 0, 0, -1, 2, -2, 0, 0, -2], [-2, -1, 0, 0, 1, -1, -2, 1, 0, 0, -1, 1], [-4, -2, 2, -1, 0, 1, -2, 2, -1, 0, -1, -1], [-3, -3, 0, 2, 1, 0, 3, -1, 0, 1, -1, -2], [2, 0, 1, 0, 0, -2, 2, -2, 2, 1, 0, 1], [-2, -2, -1, 0, 1, -2, -1, 0, -1, 0, -2, -2], [-2, -4, -2, -1, 2, -1, -1, 0, -1, 2, -2, -2], [4, -2, 1, -4, -2, 0, 2, 1, 1, 0, 0, 1], [1, -1, -1, 3, -1, -1, 1, -1, 0, -1, -1, 1], [0, 3, 1, 2, 1, 1, 2, 2, 1, 0, 3, 1], [1, 0, 0, -4, 0, 0, -5, 3, 0, 0, 1, 3], [-5, 4, 3, 0, -1, 1, 1, 0, 2, -4, 4, 3], [-2, 0, 1, -5, 1, 2, -5, 0, 2, -3, 5, 5], [-2, -1, 2, -3, 0, -1, -1, 2, 3, 1, 0, 2], [-4, 3, 0, 1, -4, -2, 2, -1, 1, -3, 0, 1], [2, -3, -3, -2, 1, 1, 0, 2, -1, 2, 1, -4], [2, 0, -2, -3, -1, 1, 2, 0, -2, 1, -2, -2], [-4, 4, 4, 0, -1, 2, 4, -2, 4, 0, 0, -1], [2, -2, 2, -3, 1, -2, -1, 2, 2, 1, 1, 2], [3, -2, -1, -4, -1, -2, 2, -1, 0, 0, -2, 0], [-7, -1, -1, 1, 3, 1, 3, -3, -3, 4, -2, -5], [2, -4, -1, -3, -3, 1, -5, 5, -1, -1, 1, 0], [-5, 0, -1, 1, 1, -1, -1, -2, 0, 3, 0, -3], [-3, 7, 2, 4, 0, -3, 1, -5, 3, -6, 0, 6], [-5, -2, 2, 0, 1, 5, -4, 5, -2, 3, 2, -2], [-2, -3, -4, -5, -1, -1, -3, 3, -2, 0, -2, 2], [-1, 1, 3, -4, 1, 0, 0, 2, 1, -3, -2, 3], [-6, 3, -3, -1, 2, 0, 1, -2, 1, -2, 4, 1], [-2, 0, -1, -8, 2, -1, -6, 0, 0, -1, 0, 3], [4, -4, 0, 4, 2, -2, 0, -2, -3, -1, -1, 0], [2, 2, -2, -3, 1, -1, -5, 2, 0, 3, 1, 6], [0, 1, 5, -1, -3, 1, -1, 0, 2, -5, -1, 4], [-1, -3, -3, 3, -2, -2, 1, -4, -1, -2, -3, 1], [-2, -1, 2, -2, 0, -2, -6, 3, 3, -2, 2, 6], [-3, -1, 2, 5, 1, -2, 2, -1, -2, 4, 0, -4], [-7, -6, -3, -3, -1, -1, 3, 2, -2, 3, -2, -3], [3, -4, 3, -2, -3, 0, -1, 4, -3, 2, -1, 2], [2, -3, -3, -1, 0, 1, -1, 1, -3, 0, 1, 3], [0, -5, -3, 4, -3, 0, 1, 3, -3, 1, -5, -5], [-1, -4, -2, -5, 3, -1, -1, 0, 1, 4, -1, 3], [-8, 3, 0, 7, -2, -3, 6, -4, -2, -5, -2, -2], [6, -4, 0, 4, 0, -2, -3, 0, -1, -3, -4, 1], [-1, 3, 1, -5, -3, 0, -3, 0, 7, -4, 3, 7], [-3, 5, 2, 0, 2, 0, 2, -8, 4, -5, -1, 4], [6, 1, 0, -3, -3, -2, 0, 0, 0, 2, -2, 3], [-6, 4, 0, 1, 1, 3, 5, -3, 2, -1, 2, -2], [-2, 6, -1, -2, -1, -1, -4, 1, 3, -6, 4, 8], [-5, -4, -2, -1, 4, -3, -8, 3, 0, 3, 0, 1], [-4, -2, 2, -6, 0, 1, -6, 1, 4, 1, 3, 6], [9, 3, 1, -6, -1, -1, -6, -1, 5, -1, 3, 10], [8, 7, 1, -4, -4, -3, -1, 0, 3, -5, 4, 11], [-10, 0, 5, 9, 1, 7, 3, 3, -2, 4, 0, -9], [-6, -8, -9, 0, 2, -4, -1, 3, -7, 5, -1, -5], [-4, 0, 4, 6, 1, 5, 5, -4, -6, 2, -1, -4], [1, -2, 2, 1, 0, -5, -4, 2, 1, -2, 0, 1], [8, -4, 1, -10, -2, -3, -6, 4, 2, 1, 4, 7], [-2, 10, 0, 7, 0, 1, 5, -2, 5, 0, 2, 6], [-8, 1, 1, 4, 2, 2, 4, -2, 3, 3, 1, -2], [0, -10, 5, 9, -2, 3, 2, 2, -6, 4, -6, -12], [7, 2, -2, -2, 2, -3, 0, 0, 2, 1, -3, 7], [-6, 2, -1, 4, 2, 2, 1, -2, 0, -3, 5, 2], [0, -1, 0, 3, 1, 0, 3, 2, 0, 2, -1, -5], [-8, 0, 6, 3, 1, 4, 3, -3, 0, 0, -2, -6], [3, 0, 1, 1, 2, 1, 4, -2, -5, -3, -4, -1], [9, 1, -1, -10, -1, -2, -4, 2, 5, 1, 0, 10], [-10, 4, -3, 3, -2, -3, 5, -2, 1, -5, 0, 5], [0, -7, -2, -6, 0, 2, -5, 7, -3, 0, -3, -1], [-5, 4, 2, 1, -1, -2, 2, 0, 0, -2, 1, -3], [-6, 8, -1, 2, -3, -2, 9, -7, 3, -2, 1, 0], [-4, -5, 2, 13, 2, 0, 6, -1, -5, 4, -4, -9], [0, 0, -2, 5, -4, 3, 2, 3, -2, 1, 3, -4], [-2, -2, -2, -7, 6, 5, -4, 1, -3, 10, 4, -3], [-4, -6, 4, -3, -1, 2, -3, 1, -4, 3, 0, -4], [6, -8, -5, 3, 1, 1, -5, 3, -6, 6, -2, -6], [-4, 3, 1, 13, 3, 1, 5, -6, 0, 1, -3, -5], [-4, 5, 2, 6, 1, 2, -2, -8, 2, -6, -4, 5], [-6, 6, -1, 3, 1, 0, 1, 4, -2, 4, 3, -3], [3, -6, -7, 1, 2, -1, -2, 5, -8, 3, 0, -2], [0, 7, 5, 8, -6, 0, 5, -2, 5, -7, -4, -1], [-5, -8, -2, -4, 4, 2, -6, 9, -1, 9, 1, -8], [-10, -3, -1, -9, 4, 4, -8, 2, -3, 2, 3, -2], [8, 0, -3, -7, 1, -2, -1, 2, 3, 5, -2, 8], [-1, 2, -1, 5, 2, -1, 3, -7, 3, -1, 2, -3], [2, -5, -2, 6, 3, -2, -2, -4, -6, -1, -6, -1], [-1, -1, -3, 1, 2, 4, 1, 1, -2, 5, 1, -3], [10, 1, -4, -5, -1, -5, -5, 3, 2, 0, 4, 10], [6, -11, -3, -5, 3, 1, 0, -2, -4, 6, -2, -4], [3, -1, -1, 1, 1, -4, 0, 4, -4, 1, 3, 2], [2, 4, -2, -4, 1, 1, 1, -2, 4, 5, 3, 2], [-8, 0, -5, -2, -3, 4, 7, 4, -5, 5, 0, -6], [-5, 4, 2, -2, -2, -1, 2, 0, -4, 1, 7, 0], [4, 12, -5, -5, -2, -3, -1, -6, 6, -12, 3, 14], [-1, 7, 6, 9, -2, 0, 0, -4, 5, -9, 0, 2], [0, 2, -1, -6, -3, -1, -10, 4, 5, -8, -1, 15], [5, 1, -2, 0, -5, -5, 0, -4, 6, -5, -9, 1], [1, 0, 3, -7, -1, 1, -5, 2, 6, -2, 4, 4], [-4, -4, -1, 7, 3, -3, 2, -1, -4, 4, -1, -5], [-9, 6, -2, -1, 0, -1, -8, 4, 4, 1, 3, 4], [14, 3, -3, -3, -3, -3, 5, 2, 0, -3, -3, 7], [-12, -9, 0, 5, 0, -1, -1, 0, -6, -2, -5, -11], [3, -2, 3, -3, 5, 1, -2, -6, 6, 4, 3, 6], [3, 4, 7, -5, -5, 0, 2, -1, 3, -10, 5, 4], [3, -9, 3, -3, -1, -1, -2, 5, -6, -4, 0, -6], [0, 6, -3, 2, -2, -3, 5, -6, 4, -5, -1, -1], [-9, -5, 0, -4, 1, 2, 1, 3, -6, 3, 1, -2], [5, -6, 5, 0, 3, -3, -6, -1, -5, 2, -5, 3], [-3, 10, 0, 10, -5, -3, 7, -5, 0, -4, 3, -1], [-2, 6, 5, -1, 1, 6, 1, -6, 3, -6, 4, 9], [0, -3, -3, 4, -5, -5, 7, 1, -4, 6, -4, -9], [-3, 0, 7, 12, 0, 1, 0, -3, -3, -1, -3, -14], [8, -1, -4, -4, 2, -2, -4, 11, -2, 1, 5, 10], [-11, 2, -2, 4, -3, 2, 7, 0, 0, 6, 0, -12], [5, 6, 2, 6, 3, 4, 3, -4, -1, 1, -3, 0], [-3, -13, -1, 5, -1, 3, -2, 10, -4, 5, -3, -12], [-8, -5, -1, 6, -2, 5, 1, 8, -2, 6, -3, -10], [11, 11, -3, 1, -6, -9, 1, -7, 8, -5, -3, 11], [-1, 12, 3, -10, -1, -1, 1, -2, 5, -6, 3, 14], [5, 5, -8, -7, -2, 0, -3, 4, 1, -5, 3, 6], [-1, 0, 3, 1, 1, 2, -4, -1, 5, 3, 3, 5], [-21, -8, 4, 5, 5, 10, -5, 4, -5, 6, 5, -13], [2, -4, 3, -10, 4, -5, -7, -5, 6, -5, 0, 10], [3, -1, 5, -11, -2, -5, -6, 7, 5, -6, 7, 11], [-11, 2, -2, 3, -3, 2, 3, -2, 4, 0, 2, -7], [2, 2, 1, -6, -2, -2, -2, -1, 5, -6, 8, 6], [10, 9, -4, -3, -6, 1, 0, -2, 4, -6, -2, 10], [-3, -4, 5, 4, 2, -3, -6, -1, 0, 2, 2, -3], [-11, -5, -1, 3, 3, -2, 1, 8, -6, 7, 2, -5], [-16, 2, 2, -1, 7, 8, 3, -5, -2, 7, 2, -4], [-10, 1, -2, 2, 7, -4, -3, -7, 2, -9, 2, 0], [-19, -4, -4, 4, 3, 1, 1, 6, -5, 7, 1, -11], [8, -5, -5, -8, 4, -1, -3, 3, -5, 5, -1, 7], [16, 5, -9, -12, 1, -3, -10, 4, 3, -1, 5, 13], [4, -1, 6, -2, -4, 3, 2, -1, -2, -1, -1, -9], [-5, -7, -4, 2, 2, -4, -3, -2, 0, -1, -8, -3], [0, 2, 3, 5, -3, 0, 6, 0, -2, -6, 0, -1], [1, 14, -1, 3, -2, -3, 10, -13, 7, -5, 1, 6], [10, 10, 1, -6, -4, 1, 5, 1, 5, -4, 1, 5], [9, -8, 0, -10, 6, 3, -11, 2, -1, 7, 1, 6], [6, 13, 5, 11, -5, 1, 13, -6, 8, 1, -1, 5], [-11, 3, -2, 10, 5, 5, 4, -4, -12, -1, 1, -5], [-19, 5, 5, 8, -5, 5, 14, -3, 2, 2, -6, -14], [-23, 0, 4, 13, 2, 6, 11, -1, -2, 6, -5, -17], [-6, -1, -3, 7, 7, 6, -2, -5, -4, 3, 3, 0], [-1, -4, -1, 2, -2, 0, -7, 6, -4, -2, -4, -1], [-6, -6, -3, -3, 3, 5, -8, -3, -10, 4, 5, -4], [-9, 3, -6, 8, -3, 2, 9, -7, -5, 1, -2, -4], [6, 1, -4, -12, -2, -5, -5, 2, 4, -7, 7, 11], [-13, -17, 6, 0, 8, 6, 0, 2, -7, 8, -2, -9], [2, 3, 5, 15, -3, 8, 11, -5, -3, 2, 1, -13], [0, -7, -3, -9, 3, -2, -1, -1, -3, 4, -3, -1], [-10, 2, -9, 1, -3, 5, 8, 3, -7, 5, -4, -5], [1, -18, -2, -6, 4, 1, -7, 8, -15, 5, -5, -6], [-6, -4, 3, 7, -3, 6, 2, 10, -7, 3, 1, -10], [0, 3, -1, 2, -2, 1, 15, -7, -2, -1, -2, -6], [-10, -7, 4, 6, 2, 10, 2, 4, -5, 6, 3, -13], [7, -6, 1, 0, 1, -1, -1, 4, -2, 7, -4, -6], [-8, -9, -2, 5, 3, 2, -10, 2, -3, 1, -3, -3], [-20, 7, 6, -6, -4, -3, -4, 0, 10, -5, 4, 8], [-6, -5, -3, 12, 1, 3, 10, -2, -5, 7, -7, -13], [10, 0, -6, -3, -2, 1, 6, 5, -1, -1, -6, -1], [9, -2, 4, -3, 5, -2, -3, 0, 4, -3, 2, 3], [9, 8, -4, 5, -7, -5, 8, -3, 7, 0, -7, 4], [10, 6, 4, 12, -6, -4, 0, -2, 3, -7, 3, 3], [-8, -2, 1, -11, 5, 0, -6, 8, 5, -1, 3, 11], [8, 10, -6, -4, -2, -4, 7, -6, 5, -7, 0, 12], [9, -14, 2, -11, -2, 5, -2, 10, -3, 10, -2, -8], [0, 10, 3, -4, 0, 2, 0, 3, -4, -1, -1, 5], [0, -1, 3, -3, -4, -2, 3, 2, 4, 1, -9, -4], [4, 3, -4, -13, -5, -3, -10, 5, 4, -2, 3, 7], [1, 6, 10, -1, -1, 7, -6, -7, 2, -5, 6, 9], [2, 10, 2, -1, -1, -7, 3, -10, 5, -12, -2, 14], [-2, -7, -1, 0, -1, -2, -1, 3, -5, 1, -5, -4], [-12, 0, -1, 1, 5, 9, -1, 1, -3, 5, -1, -7], [0, 0, 0, 4, -8, -2, 0, 8, 2, 2, 2, -6], [0, 9, 4, -8, 1, 4, 7, -6, 6, -1, 3, 8], [-6, 11, 4, 9, 2, 3, 8, -12, 7, 3, -1, -1], [9, -8, -9, -12, -1, 0, 1, 9, -5, 12, 2, -4], [13, 4, -2, 9, -6, -4, 2, 3, 2, -5, -2, 6], [6, 1, -1, 1, -9, -8, -1, 1, 3, -8, -7, 3], [2, 5, 3, 6, -6, 0, -2, -4, 4, -8, -2, 1], [-2, 1, -8, 3, -1, -4, 0, 2, -1, 1, -3, -9], [-11, 1, -1, 7, 4, 6, 12, -9, -5, 10, -5, -14], [8, -1, 7, -4, -5, -3, -11, 3, 6, -2, 6, 13], [8, 5, -12, 0, 3, -4, -2, -8, -3, -5, 1, 4], [12, 15, -6, -7, -3, -1, -9, 0, 6, -2, 5, 14], [10, -2, 6, 2, 0, 6, 5, 2, 2, 0, -3, -4], [-14, 13, 0, -11, -4, -4, -3, -7, 9, -9, 3, 18], [-4, -16, -3, -12, 4, 1, -6, 6, -3, 10, -7, -7], [-11, -11, -2, 15, 8, 1, 8, -1, -12, 10, 0, -20], [11, 4, -2, 1, -5, -2, -3, -4, -5, -1, -5, -1], [6, 5, 5, -2, -4, -2, -3, 6, 9, 5, 2, 1], [10, -11, 2, -1, 5, 9, 2, 4, -9, 11, -1, -10], [13, 2, -7, 2, 4, -3, 3, -7, 3, 6, 4, -3], [-3, -4, -2, -5, 1, -4, 2, -6, -2, 0, -1, -1], [15, 0, -1, 8, 4, -1, 5, -7, 3, 0, -6, 2], [-13, 5, 3, -4, -3, 5, 2, -1, 4, -2, 4, 3], [-7, 5, -3, 12, -3, -2, 0, 2, 0, -4, 3, 6], [-1, 4, -8, 2, -4, 1, 4, 8, -8, 0, 5, -2], [3, -8, 1, 4, -2, 3, -8, 9, -8, 0, -6, -9], [-11, 0, 1, 15, 7, 5, 8, -6, -14, 5, -1, -14], [5, 9, -2, -1, 1, 5, -5, 0, 1, 1, 7, 9], [-7, 1, 12, -4, 7, 5, 0, -10, 4, 1, 1, 1], [-3, -4, -6, 2, 9, -5, 2, -11, -4, 2, -6, -5], [6, 4, -2, 11, 8, 1, 4, -5, -4, 5, -3, -8], [-1, -15, -2, 3, 3, 0, -2, -7, -6, 10, -2, -9], [-2, -3, -8, -3, 2, 1, 2, 8, -4, 3, -4, -4], [13, 5, 2, -7, -1, -3, -4, 0, 6, 0, -1, 15], [13, -2, 0, 4, -15, -1, 1, 14, -2, -4, 6, 7], [-7, -19, -7, 6, 3, -3, 0, 7, -19, 4, -3, -18], [21, -1, 8, -5, 4, -9, -4, -1, 8, 0, -4, 3], [-1, 5, -4, 8, 7, 5, 14, -8, -9, 4, -8, -8], [11, 8, -2, -13, -5, 2, -6, 12, 4, -3, 5, 1], [9, -2, -1, 2, 5, 0, 8, -3, 1, 2, 3, -7], [-10, 3, 2, 2, -4, -2, 10, -3, 9, 0, -2, -5], [-16, -19, -1, 11, 8, 1, -1, 0, -9, 10, -8, -19], [-5, 4, -7, -1, -2, 0, 0, -4, 1, -11, 0, 8], [-10, -14, 0, 16, 6, 4, 10, 0, -18, 12, -8, -22], [-4, -8, 0, 2, -6, -6, 6, 6, -6, 2, -10, -6], [17, -3, 3, 2, -1, -5, 0, -9, -1, -3, -11, -4], [12, 5, 7, 9, -9, 1, 9, 0, 8, -7, -1, -3], [-12, 8, 8, 21, -2, -3, 10, -9, -2, -4, -9, -4], [-8, -13, 0, -9, 6, 2, -15, 9, -6, 13, 4, -3], [15, -1, -2, 1, -1, -6, 4, -1, -2, 0, -4, -7], [10, -3, 7, -6, 8, 11, -3, 4, -2, 8, 9, -10], [8, 21, 12, 0, -12, -2, -3, 1, 15, -16, 7, 27], [1, -9, -6, -12, -5, -7, -1, 8, 4, 1, -4, 0], [1, 3, -2, 1, -8, -10, 4, -2, -5, -3, -4, -3], [-1, 0, 8, 1, -3, 5, 6, 0, -1, -2, 4, -11], [10, -1, 7, 11, 1, -5, 4, -9, 1, -2, -2, -1], [-7, -3, -2, -13, 2, 8, -12, 12, -1, 7, 4, 3], [10, 3, 2, 9, -9, -3, 8, 1, 1, -5, -6, -6], [-19, -8, -10, -9, 8, 10, -5, 7, -10, 13, 11, -13], [8, 1, 14, 3, 1, 5, -4, -1, 10, 0, -1, -5], [18, -12, -4, -14, 2, -6, -14, 4, -6, -1, -2, 2], [0, -6, -9, 5, 6, -5, 4, -6, -6, 13, -4, -16], [-1, 18, -1, 18, -8, 1, 17, -1, 9, -10, 4, -1], [9, 12, -6, -4, 1, 1, -5, -8, -2, -16, 2, 16], [4, 5, -1, 2, 0, 0, 2, 3, -5, -5, 2, -12], [-5, 17, 6, 13, 3, 2, -1, -11, 14, -10, 5, 6], [11, 14, 6, 2, 8, 7, 12, -8, -3, -5, 6, 6], [9, 0, -1, -12, 3, -5, -9, 4, -3, 2, 1, 9], [-19, -16, -11, 4, 10, 4, 10, -4, -11, 10, -6, -22], [-1, 4, 3, 8, 1, 1, 4, 5, -1, -4, -3, -7], [3, -17, -10, -3, 11, 2, -2, -1, -13, 13, 1, -23], [15, -12, 5, -8, 5, 2, -3, 3, -11, 6, -2, -8], [-2, 15, -9, 7, -9, -8, 12, -5, 8, -11, -11, 6], [-6, -3, 1, 4, -11, 8, 6, 10, -2, 2, 2, -14], [6, 1, 6, -21, 1, 0, -9, 8, 5, -4, 8, 13], [-7, -12, -3, -2, 1, -1, 6, 2, -6, 3, -5, -2], [-20, -10, -1, -1, 0, -3, -5, -2, -4, 2, -5, -14], [28, 5, -5, -1, -1, -2, 4, -3, 5, 2, 2, 1], [-1, 0, 0, 13, 1, 4, 5, -12, 5, 1, -5, -15], [6, -3, 7, 9, -8, -6, 14, 0, 8, -2, -4, -1], [0, 2, 3, -3, -9, 9, 4, 13, -1, -3, -4, -2], [-26, 8, 8, 7, 2, -2, 2, -12, 1, -2, 1, 1], [-6, -8, 12, 5, -3, -3, -3, -2, 7, 4, -8, -3], [1, 7, 0, -1, 5, 3, -3, -10, 12, -3, -4, 6], [11, 14, -1, -7, -2, -2, 1, -4, 12, -10, 0, 8], [18, 3, -2, 2, -7, -5, -5, 3, -8, -5, -2, 8], [8, 15, 2, 19, -8, 3, 13, -5, -1, -15, -1, -6], [3, -14, 2, 4, 8, 5, -7, 0, -4, 6, 0, -5], [-7, 3, 9, 1, -4, -2, -3, 3, 8, 1, 1, 4], [-5, 4, 1, -7, 4, 7, -5, -3, -6, 2, 5, -6], [5, -6, -2, -12, 7, 2, -10, 8, -8, 2, 1, -7], [4, 9, 4, -9, 5, 11, -8, 2, 5, -1, 7, -2], [-2, 2, 4, -2, -1, -1, -4, 10, -6, 0, -6, -2], [-8, 2, 3, 8, -2, -4, 13, -10, 0, -7, -9, -2], [4, 3, 8, -7, -6, 5, -7, 6, 2, -11, 13, 7], [0, 7, 4, -2, 0, 2, -6, 1, 5, 10, 8, 5], [5, 0, -1, 3, 4, 11, -2, 1, 6, -2, 6, -3], [1, 5, -10, 2, 7, 9, 9, -6, -9, 4, -1, -13], [17, 7, 1, 4, -3, 3, 3, -1, 3, -3, 6, 4], [3, -1, 6, 2, -6, -9, 1, -6, 6, -9, -10, 7], [32, 22, -5, 2, -10, 1, 1, 6, 8, -13, 5, 13], [-8, -5, -1, -9, 0, -3, -1, 9, 2, 3, 8, 4], [14, 0, -1, -10, 1, -1, -7, 5, 3, 7, 7, 16], [22, -1, 0, 1, -9, 0, 0, 5, -4, 2, 5, 3], [11, 5, 5, 17, -4, -9, 5, 0, 3, -10, -9, 3], [12, -12, -8, 2, 0, -4, -2, -6, -7, -3, -13, -1], [-9, -9, 3, 16, -1, -1, 2, 2, -5, 5, -6, -11], [23, -3, 2, -11, -1, -5, 0, -2, 7, -3, 2, 10], [8, -9, -3, 13, 3, 5, -1, 4, -12, 5, -5, -19], [-24, -11, -6, -7, 5, 4, -3, 12, -7, 3, 8, -11], [2, -7, 3, -5, 8, -3, -1, -1, -2, 5, -6, -2], [4, 6, -4, 1, -3, 7, 1, 12, -3, 0, 10, 1], [17, 10, 5, 11, -2, 1, 6, -7, 6, -6, -8, 5], [8, 14, 9, -4, -2, -9, -7, -11, 14, -15, -2, 19], [-3, -11, 2, -25, 3, 2, -13, 18, 1, 13, 3, 5], [-9, 11, -6, -6, -2, -5, -1, 2, 3, -8, 11, 21], [0, -7, 6, -5, -7, -1, -14, 21, 6, 4, 5, 2], [1, 13, -4, 8, -4, -3, 10, -7, 0, -9, 2, 11], [6, -5, 2, 21, 0, 2, 12, -7, -10, 5, -13, -28], [11, -11, 0, -2, 7, 1, -7, 1, -5, 2, -10, -9], [7, 7, -1, 10, -1, 8, 5, 8, -12, 5, 4, -10], [12, -5, 9, 8, 0, 11, 1, 5, -4, -2, 0, -10], [-3, -2, -1, 4, -6, 7, 9, 7, 1, -4, 8, -9], [-14, -6, 3, 8, 8, 8, -4, 1, -11, -2, 8, -6], [-4, 8, -3, 9, -9, 2, -1, 2, 3, -11, 2, -8], [9, 13, 6, -16, 6, -3, -5, 3, 12, -8, 5, 21], [-4, -10, -5, -14, -1, 0, -9, 6, -8, 5, 11, -2], [8, -22, -10, -13, 10, -1, -2, 3, -15, 10, -4, -1], [14, 8, -3, 3, -1, 0, -5, -6, -1, -19, 4, 10], [28, -9, 4, -2, 5, 0, -7, 0, 4, 5, -7, -4], [-11, -1, 3, -8, 1, 4, -10, 2, -3, -1, 6, -5], [3, 3, 0, 2, -5, 5, 0, 4, 8, -13, 1, 5], [-9, 3, 0, 11, 6, 2, 6, -10, -4, -4, -1, -9], [3, 4, 1, -1, -4, -5, 7, -5, 7, -1, 2, 6], [10, 0, -1, 10, -10, -10, 3, 9, 9, -8, -3, 7], [-5, 3, 14, 6, -5, 4, 5, -5, 5, -6, 2, 0], [9, 13, 1, 10, -1, 0, 10, -4, 5, 1, -4, -11], [11, 2, -9, -8, -1, -10, 8, -9, 4, 1, -14, 0], [9, -16, -7, -1, -7, 3, -8, 15, -12, 6, -4, -14], [11, -14, -8, -8, 5, 6, -8, 14, -7, 11, -2, -3], [-5, 24, 6, 21, -4, 1, 23, -12, 9, -6, -7, 12], [-3, -11, -4, -14, 9, 7, -8, 4, -8, 7, 3, -10], [-19, -12, 1, 0, 6, 4, -9, 0, -2, 9, -12, -4], [7, -10, -5, -5, -2, 4, -1, 16, -12, 8, 2, -15], [11, 4, 7, 2, -3, -6, -1, -9, 2, -13, -7, 3], [22, 0, -12, 1, -7, -6, 7, 7, 1, -2, -5, 5], [7, 5, -6, 5, -7, 4, -4, 7, -6, -5, 6, -1], [-30, -3, -4, 13, -3, 2, 1, 3, -9, 3, -5, -8], [19, -7, 1, -1, 0, -5, -1, -6, -4, -3, -10, -14], [3, 6, -2, 8, 2, -4, -7, -1, -3, -4, 0, -2], [-3, -1, 2, 23, 0, 14, 14, 4, -10, 6, -3, -28], [5, 6, -3, 10, -3, -5, 7, -8, 3, -10, 1, -7], [14, 2, -8, 6, -4, -13, 1, -9, 8, -14, -8, 13], [8, 13, -5, 10, -6, 2, 16, -6, -2, -8, 0, -1], [4, 16, 15, 17, -6, 0, 5, -15, 8, -9, 2, 0], [-10, 2, 3, -14, 6, 6, -2, 5, 7, 6, 10, 10], [-12, -1, -4, -2, -9, -5, 0, 5, 1, -4, -6, 3], [21, -5, -8, -10, 0, 1, -18, 5, -3, 7, 7, 5], [-2, 16, -15, -6, 2, -6, 13, -6, 8, -3, 1, 7], [-2, 14, 2, 5, 7, -8, 1, -17, -3, -12, 1, 7], [-4, -3, -3, 9, -6, 8, 4, 16, -9, 2, -1, -22], [-3, 4, -15, 17, -6, 1, 14, 1, -4, 6, 2, -13], [-6, 8, 6, 28, -9, 5, 17, -9, 0, -2, -4, -11], [0, 8, 9, -9, -5, -4, -8, 6, 1, 1, 3, 7], [11, 21, 10, 0, 2, 5, 0, -4, 8, -17, 5, 6], [10, -1, -6, -7, -7, -9, -8, -3, 5, -9, -4, 11], [-25, -7, -15, 5, -2, -4, 0, 5, -8, 9, 0, 0], [8, 2, 3, 7, 9, 6, 8, 1, -9, -3, 0, -5], [-8, -18, -4, -9, 2, 2, 1, 8, -7, 2, 2, 2], [0, 7, 2, 1, 1, 4, -4, -2, 0, -10, -2, -1], [14, -5, 2, 0, 6, -2, -7, -7, -5, 0, -3, 7], [16, -5, 1, -15, 6, 1, -7, 3, 1, 11, -1, 9], [-17, -9, -5, 16, 7, 2, 7, -4, -3, 17, -3, -14], [-2, -15, 9, 2, -8, 1, 0, 13, -3, 3, -12, -12], [-2, 17, 7, 8, -14, 0, 6, 0, 3, -7, -1, 8], [3, -4, -2, -6, 0, 3, -2, 16, -7, 6, 4, -17], [10, -9, -2, -5, 7, -1, -1, 7, -8, 10, -4, -8], [19, -17, -17, -3, 11, 3, -1, 3, -18, 9, -3, -7], [-4, -21, -3, 6, 12, 0, 11, -6, -15, 19, -4, -19], [5, 8, 4, 19, -3, -12, 2, -6, 2, -17, -7, 7], [12, -16, -4, -23, -1, 0, -7, 16, 6, 1, 5, 10], [-13, -10, -8, -3, 3, -3, 9, -3, 2, 1, -3, -2], [-8, -15, -1, -9, 7, 4, -4, -2, 1, 6, -7, -9], [-7, -7, -2, 2, 5, 2, 0, -7, -14, 5, 6, -6], [16, 9, 9, 7, 7, 6, 4, -16, 9, -4, -5, 6], [-26, -13, -6, 12, 10, 5, -2, -8, -14, 13, -4, -17], [3, -7, 8, 3, 0, 0, 10, 2, 3, 11, 0, -17], [13, -9, -10, -10, 12, -2, -10, 2, -10, 1, 5, 7], [-29, 2, -3, -2, 7, 3, -14, 4, -3, -10, 12, 5], [14, 14, 4, -9, 7, 1, -16, -6, 5, -15, 11, 11], [19, -10, -7, -6, 16, -3, -12, 1, -14, 1, 0, 7], [-23, -3, 3, 16, 6, -4, -2, 1, -8, 0, 6, 0], [12, 5, 10, 4, -5, 2, 11, -14, 10, -5, 1, 6], [-1, 0, 5, 7, -6, 0, -3, -4, 0, 0, -4, -4], [0, 25, -7, -2, -12, -9, -5, -9, 12, -16, 4, 28], [-9, 9, 1, 1, 5, 2, 11, 2, 7, -1, 7, 8], [0, -4, -3, 6, -4, -9, 7, 6, -4, 9, -11, -14], [-19, 5, 5, 5, 6, -1, -1, 0, 13, 2, 3, -3], [29, 6, 8, -4, -3, -6, 1, 1, 6, -6, -5, 12], [-16, -2, -7, -6, 6, 9, -6, -2, -11, 10, 9, -5], [19, 11, -4, -7, -9, -1, -7, 8, -2, 1, -4, 10], [11, -13, -14, 6, 1, 0, 4, 5, -7, 4, -7, -7], [23, 25, -3, 2, -16, -10, 12, -9, 11, -11, -9, 9], [7, -8, 5, 5, -7, -5, -7, 3, 2, 2, -9, -5], [-1, -19, -7, 0, 5, -1, -13, 1, -14, 14, -1, -5], [-26, -16, -3, 3, 0, 2, -3, 1, -2, 6, -2, -16], [-18, -8, 9, -3, -6, 4, -3, 11, -5, -3, 9, -6], [10, 14, 5, -5, 1, 5, 0, -5, 4, -8, 7, -7], [6, 1, 9, -9, 3, 0, -1, -1, 1, 10, 7, 8], [16, -19, 7, -2, -7, 0, -2, 1, -2, -2, -7, -4], [13, -1, 7, 3, 10, 6, 4, 1, -2, 1, -4, -15], [-2, -6, 1, 1, -11, -5, 9, -1, 8, -4, -2, -1], [6, -14, -1, -26, -8, -5, -8, 13, -5, 2, 0, 10], [-19, 6, -5, 1, 3, 4, -13, -1, -3, 1, 16, 8], [5, -10, 7, 5, 6, -5, 1, 3, 6, 5, -1, -16], [-32, 0, -3, 20, -1, 9, 11, -5, -3, 12, -5, -12], [3, 2, -9, 2, 0, 7, 2, -2, -8, 9, -7, -7], [13, -21, -2, -16, 9, 2, -2, 6, -9, 12, 6, 4], [12, -6, 5, -21, 0, 5, -10, 0, 12, 1, 6, 7], [12, 6, -1, -12, -4, -8, -9, -1, -1, 0, -3, 17], [-17, 12, 0, 5, 1, 6, 6, 4, 0, -9, 3, -13], [5, 5, 2, -6, 15, 11, -2, -1, -3, -6, 7, 5], [-17, -8, 8, 20, 1, 10, 3, 8, -15, 12, 3, -19], [10, -18, -16, 7, 6, 5, 5, 11, -22, 12, -12, -28], [-8, -3, -8, 14, 4, 0, -4, -7, -5, -4, 8, 4], [2, 9, -4, 5, 5, 0, 3, -8, 3, -6, 6, 1], [-24, -12, -6, -5, -2, -9, -3, 7, 4, 13, -4, -10], [-3, 6, 12, 3, -10, 0, 6, 5, 5, 6, -7, 2], [-32, -9, 10, 8, 3, 3, 4, -3, 3, -6, 2, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_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_8034_2_a_w(:prec:=12)
    chi := MakeCharacter_8034_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_8034_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_8034_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4096, -6656, 4208, 9852, -638, -5293, -585, 1255, 235, -127, -29, 4, 1]>,<7,R![48, 848, -11911, -3397, 14923, 1175, -5271, -144, 761, 6, -47, 0, 1]>],Snew);
    return Vf;
end function;