
// Make newform 8030.2.a.bf in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_bf();".
// 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_8030_2_a_bf();".
// 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 := [8, 46, -626, -1731, 7698, 479, -8875, 957, 4108, -702, -922, 191, 99, -23, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [261843832, 521031646, -8728032188, 9832804595, 6932586289, -10202941704, -1431593699, 3898870430, -160464288, -692885868, 90005878, 56953737, -10239084, -1741279, 366035], [-70626556, 49446534, 2073107780, -3986272579, 29989029, 3235047750, -979688875, -782126404, 359392846, 17357120, -36046464, 13572861, -1195170, -1050335, 221319], [498906736, 1214106778, -7486177226, 3189119675, 4203159601, -4071404154, 448142761, 1857649106, -810364518, -426649374, 218289340, 48433869, -24085422, -2079073, 960341], [246176608, -487849418, -2405458214, 1426926773, 1208212627, -2989722630, 1045234987, 2084627546, -949617366, -618823506, 269290000, 80618787, -32508126, -3706999, 1401323], [60835336, -529393910, 4690178716, -14598872695, 943404775, 19320093438, -5141897405, -9607202860, 2954517636, 2266987038, -718017836, -252534357, 80134722, 10474127, -3305617], [-18158464, 577080874, 2615461640, -13138428591, 1004705713, 14236309866, -2987617231, -6030088596, 1518438146, 1227637336, -333339020, -119493219, 33773790, 4426957, -1280809], [-66381772, -831757504, -1640072710, 16363093441, -1610019898, -19523419710, 3306920312, 9216665638, -1804697133, -2100319149, 453199208, 228472740, -53426214, -9327080, 2319073], [-214919840, -141037466, 11525790424, -25646611771, -14021936813, 36275272134, 4280176927, -19433892424, 780653028, 4911689490, -625755272, -580526073, 106053138, 25183331, -5483185], [23240302, -730702446, -2014418084, 9118578187, -83605584, -10082589522, 1656705742, 4476683578, -965567359, -966740297, 233803794, 100332648, -25941384, -3945448, 1064697], [771923284, -1450048838, -22043262632, 59543491481, 9165309703, -69771551034, 6904612423, 31839705608, -5708071680, -6991802706, 1531273792, 733439763, -178480998, -29036557, 7527395], [-15724668, 249964550, 677370717, -3368581534, -112742041, 3901113249, -460844871, -1809931594, 315281564, 406761742, -84343819, -43697817, 10104105, 1765566, -438707], [-1774460, 2067898252, 6483494065, -29893392208, -1839177515, 35037808089, -3770851553, -16337319754, 2782401546, 3679695978, -759666443, -395775153, 91665171, 16003502, -3990373]];
    Rf_basisdens := [1, 1, 1, 85835268, 28611756, 85835268, 85835268, 85835268, 28611756, 42917634, 85835268, 14305878, 85835268, 4768626, 42917634];
    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_8030_a();"
function MakeCharacter_8030_a()
    N := 8030;
    order := 1;
    char_gens := [1607, 2191, 881];
    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_8030_a_Hecke(Kf)
    return MakeCharacter_8030_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, 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, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -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, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, 1, 0, 0], [-1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [-1, 1, 0, -1, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1], [-2, 1, -1, 1, 0, 0, 1, -1, 0, 0, 0, -1, -1, 0, 0], [-2, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 1, 1, 0, 0], [1, -1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 1, -1, 0, 1], [-2, 1, 0, 0, -1, 1, 0, -1, 1, 0, 1, 0, -1, 0, 0], [-1, 0, 0, 1, 1, 0, -2, 0, 1, -1, 0, 0, 0, 1, -1], [-3, 1, -2, 2, 1, -2, 2, 0, 1, 2, 0, 2, 0, 1, 1], [0, 0, 1, -1, -1, 1, -1, -1, 0, 0, 1, -1, 0, 0, 0], [-5, 2, -1, 2, 0, -1, 2, 0, 0, 1, -1, 2, -1, 0, 1], [0, -1, 0, 1, 0, -1, 0, 1, -2, 1, -1, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -1, 1, 1, 0, -1, -1, -2, -1, 0, -1, -1], [-4, 2, -3, 1, 0, -3, 2, 0, 1, 2, 0, 2, 0, 2, 3], [0, -1, 0, 2, 1, -2, 0, -1, 0, 1, 0, -1, 2, 0, 1], [-1, -2, 0, 1, 0, 0, -2, 0, 1, 1, 0, 0, 1, 2, 0], [-2, 0, 0, -2, 0, 0, -1, 2, -1, -1, -1, 1, 1, 1, 1], [-2, 0, 1, -1, 1, 1, -2, -1, 2, 0, 1, 0, 1, -1, 0], [0, -2, 2, -1, 0, 1, 0, -2, 0, -2, 0, -1, 1, -2, -2], [-2, 0, 1, 0, 1, 0, 2, 1, -1, 1, -1, 1, 0, 0, 0], [2, 2, 2, 0, -2, 1, 2, 0, -2, 0, -2, 0, -2, 0, -1], [-1, 1, -1, 2, 1, -1, -2, 1, 1, 0, 1, -1, 1, 2, -2], [-6, 1, 0, -1, -2, 1, 1, -1, 0, 0, 0, -1, -1, -1, 1], [4, -2, 0, 0, 1, -1, 0, -1, -1, 2, 2, 1, -2, -1, 2], [-5, -1, 3, 0, 1, 0, -1, 0, 1, 0, 1, 0, 0, -2, 1], [-3, 1, 0, 0, -1, 0, -1, 0, 1, -1, -1, 1, 1, 0, -3], [1, -1, -1, 1, -1, 0, 3, 0, -2, 3, 0, 2, -4, 1, 2], [-6, 0, 2, 0, 1, 0, 0, -1, 0, -2, 0, -1, 0, -2, 0], [-4, -3, 1, -2, 1, 0, -2, 2, 1, -2, -1, 0, 1, 1, -1], [-2, 0, 0, 0, 0, 2, 0, 0, 0, -2, 1, -2, -2, 0, 0], [-2, -1, -1, 2, 3, -2, 0, -2, 1, 1, 3, 1, 0, 0, 1], [-2, 0, 0, -1, 0, -3, 1, 1, 0, 3, 1, 2, 0, 2, 4], [-2, -2, -2, 1, 1, -1, -1, 0, 1, 2, 1, 0, 1, 1, -1], [0, 1, 1, -3, 0, 2, -1, 0, 1, -3, 2, -1, -1, -1, -2], [0, 0, -1, -2, 1, -1, 0, 2, -1, 1, -2, 3, 0, 3, 1], [-1, 0, 3, 0, -1, 3, -1, 2, 0, -1, 0, -1, 1, 0, -2], [-3, -2, 2, 1, 0, 0, -1, -2, 0, -2, 0, -3, 0, -1, -1], [-2, -1, -1, -1, 1, 0, -2, 3, 3, -1, 0, 2, 3, 1, 0], [-2, 3, 3, 1, 0, 2, 0, -1, -1, -1, 1, -3, -1, -5, 0], [-5, -3, 2, -3, -2, 0, -1, 1, 0, -2, -3, -1, 2, -1, -2], [-4, 0, -1, 2, 0, 0, 1, 1, -2, 0, 0, 2, -2, 0, 0], [5, 0, 0, 1, 0, -1, -2, -2, 4, -1, 0, 0, 1, 1, 0], [-1, -3, -3, 0, 1, -4, 4, 0, -1, 3, 0, 2, 0, 2, 1], [-2, 1, 3, -1, -2, 5, 0, 0, -3, -2, -1, -4, -2, -2, -1], [2, -4, -3, 1, 3, 1, -3, -1, 3, 2, 2, 0, 1, 2, -1], [0, -3, -1, 0, 1, -1, -1, 3, 0, -2, -3, 1, 2, 2, 0], [-2, -1, 1, -1, 1, -2, -2, -4, 1, -1, 0, -1, 2, -1, 1], [3, -1, 0, -1, -1, 1, -1, 0, -1, 0, 1, -2, -2, 0, 0], [-5, -2, -4, -1, 3, -2, -1, 2, 0, 2, 1, 2, 2, 1, 0], [-1, 2, -1, -1, -4, 2, 3, 2, -2, 0, 1, -2, -3, 1, 0], [-2, 2, 0, 2, 2, -1, -1, -2, 1, 0, 3, -2, 0, -1, -1], [2, -3, -2, 0, -1, 1, 3, -2, -2, 3, 1, 0, -3, 0, 0], [-4, 1, -4, 1, -2, -4, 5, 0, -2, 5, -1, 4, -2, 4, 4], [-4, -4, 5, -1, -1, 2, -1, 1, -3, -3, 0, -3, 2, -2, -2], [3, -1, -2, -1, 0, 0, 0, -1, 0, -1, 0, 1, -2, 0, 0], [-5, -1, 2, 0, 0, 0, 1, -3, -1, 1, -2, 0, -1, -3, 1], [-3, -2, 4, -2, -2, 1, 1, 1, -3, -3, -3, -2, -1, -4, -1], [-3, 1, 1, 1, 0, 0, 2, -2, -2, 2, 1, -2, -3, -4, 0], [-4, 0, -4, 1, 2, -3, -1, 2, 6, 0, 0, 4, 2, 2, -1], [3, -3, -1, 2, 3, 0, -2, -1, -1, 1, 1, 1, 1, -1, -1], [-3, 0, -6, 4, 2, 0, -1, -1, 2, 3, 2, -1, 2, 3, -2], [5, 3, 2, 1, 0, 1, -1, 0, 0, -2, 4, -4, -1, -2, 0], [-6, 0, -1, 1, 3, -1, 0, -2, 3, 0, 3, 3, 0, -1, 2], [-4, -1, 2, -3, 0, 3, -3, 0, 1, -3, -1, -1, 1, -3, -1], [4, 2, 3, -2, -2, 2, 0, 2, -3, 0, 0, -1, -3, 0, 2], [-3, 2, -1, 2, -2, -1, 4, -1, 1, -1, -1, -2, -1, -2, -1], [2, -4, 1, -1, 2, 1, -8, 1, 5, -2, 0, 0, 6, 1, -4], [-7, 0, 1, 4, 1, 1, -2, 0, 0, -2, 1, -3, -1, -2, -1], [-2, -2, 3, -1, -1, 2, -1, 2, -4, -1, -3, -1, -2, 0, 1], [7, 0, -4, 1, 4, -3, -2, 1, 2, 2, 1, 3, 0, 5, 3], [-4, 2, -4, -1, -3, 2, 3, 1, 2, 1, 1, 4, -1, 3, 2], [-7, 0, 1, -1, 1, 3, 1, 0, -1, 0, -1, 4, 0, 0, -1], [-4, 5, -1, 3, -2, 1, 1, -3, 2, 1, 0, 1, 1, -2, 1], [-1, -1, -2, 0, -1, -2, 0, -2, 2, 0, -2, 2, 2, 0, -3], [0, 3, -1, 2, 2, 0, 3, -1, -1, 0, 2, -3, -1, -2, 1], [3, -7, 6, -3, 2, 2, -1, -1, -2, -3, 1, 0, -3, -1, 0], [6, 1, 5, -1, -2, 2, -1, -4, -1, -4, 3, -4, -2, -4, -2], [-2, -6, -1, 2, 0, -1, 5, -2, -3, 5, 1, 4, -2, -1, 5], [5, 7, 2, 1, -1, 5, 3, -2, -2, 0, 1, -2, -2, -3, 1], [1, 1, 3, -4, -3, 4, 1, 3, -1, -4, -1, 0, 0, 0, -1], [0, 5, 0, -2, 0, 2, -3, 3, 1, 0, 3, -1, 3, 2, -1], [10, -7, 4, -6, 0, 1, -3, 5, -4, -3, -4, -1, 5, -1, 0], [-4, -1, -2, -1, -2, 1, 0, 1, 1, 0, -1, -2, 0, 0, -1], [-1, -4, -1, 0, 3, -4, 3, -2, 2, 3, 1, 3, -1, 1, 1], [-3, -4, -3, -3, 2, -3, -3, 3, -1, 0, -2, 1, 2, 5, 1], [2, -5, 4, -3, -2, 3, -7, 1, 4, -2, 0, -3, 3, 4, -3], [-2, 4, -1, 1, 0, 3, 3, 1, -1, 3, 0, 1, -3, -1, 2], [0, -4, 0, 1, 3, -1, -5, 3, 2, 3, 1, 5, 2, 4, 0], [13, -5, -1, -4, 4, -1, -7, 1, 4, 1, 1, 3, 6, 4, -4], [-2, 4, -5, 6, 3, 1, 1, -1, -1, 4, 2, 2, 1, 1, -1], [-3, 5, 0, 0, 1, -2, -1, 1, 3, -3, 3, 2, 2, -2, 0], [2, -2, 2, -6, -1, 1, 3, 0, -3, -1, 0, -5, 0, -3, 1], [4, 1, -3, 4, 3, -3, 1, -1, -2, 0, 2, 1, 0, 1, 2], [2, -2, 1, -2, 1, 4, -1, -1, 2, -4, -1, 0, 1, 0, -5], [-4, 3, 2, -6, -2, 0, 5, 1, -5, 2, 1, 0, -4, -5, 4], [-1, 0, -3, 1, -2, -3, 3, 3, -1, 3, -4, 3, 0, 2, 2], [-1, 2, -3, 3, 2, -3, 9, 1, -4, 4, 0, 2, -5, 1, 2], [6, -5, 1, -3, -2, 1, 1, 2, -2, -3, -2, -3, -1, 2, -2], [4, 1, 0, 0, -4, -3, 4, 3, -4, 1, -7, 1, -1, -1, 0], [4, -2, 4, -5, -1, 2, 1, 1, -5, -1, 0, -6, -1, -3, 1], [-10, -6, 1, -1, 2, -3, -2, 0, 3, -1, 2, -2, 4, -1, 0], [0, -2, -3, -1, -3, -1, -1, 0, 1, -3, -1, -2, 1, 4, -2], [-2, 4, -3, -1, 1, 1, 2, 0, 0, -4, 1, 1, 0, -1, -2], [8, 0, 6, -3, -3, 4, 0, -3, -1, -4, 1, -6, 3, -6, -1], [-1, -2, 4, -7, -3, 6, -6, 1, -3, -3, 0, -2, 0, -2, -3], [5, -3, 0, 3, 1, 0, 1, 0, 0, 5, 2, 3, -3, 0, 4], [-1, 7, 2, 4, -2, 3, 2, 1, -2, -1, -1, 2, -1, -1, -1], [-9, 1, 1, 3, -2, 1, -2, -4, 2, -4, 2, -1, 2, -3, -1], [2, -3, -4, 4, 5, -4, -1, -1, 5, 4, 5, 6, 1, 4, 3], [6, -6, 2, -3, -1, 1, -2, 0, -3, 0, -2, 1, -2, -2, -1], [4, -10, 0, -4, 4, 0, -11, 3, 5, -2, 1, 3, 5, 6, -4], [3, -1, 2, -9, -2, 4, -6, 3, 5, -6, 1, -2, 5, 0, -6], [5, 1, 0, 2, -2, -4, 7, -1, -1, 6, -3, 6, -5, 0, 7], [4, 3, -4, 4, 1, -6, 2, 1, -2, 6, -1, 3, 0, -1, 1], [9, -2, 1, 0, 1, 5, -3, 0, 1, -1, 3, -1, 1, -4, -3], [1, -11, -1, 5, 4, -3, -3, -3, 1, 4, 2, 1, 4, 1, -1], [-2, -2, 3, 2, -1, -1, 5, -2, -4, 0, -2, -1, 0, -2, 3], [1, -9, -3, 2, 4, -4, -5, -1, 4, 6, 2, 3, 2, 4, 3], [-10, 0, 4, -1, -4, 0, 1, -3, 0, -1, 3, -4, -4, -3, -1], [-3, -5, 1, 4, 4, -1, -6, -7, 1, -2, 4, -6, 3, -3, -1], [-11, 2, 2, 0, -3, -1, 1, -2, 0, -1, 2, -2, -2, -4, 3], [-8, 5, -4, 4, 3, -3, 1, 1, 1, 1, 1, -1, 4, -2, 2], [-3, 6, -3, 2, -1, 1, 4, 1, 3, 0, 0, -3, -1, -1, -6], [-3, 3, 2, -3, -2, 1, 0, 6, -3, -3, -5, -1, 0, 2, -1], [-2, -4, 1, -2, 3, 4, -10, 4, 3, -4, -2, -1, 8, 3, -7], [6, 7, 1, 0, 2, 2, 2, -1, -6, 2, 2, 2, -4, -3, 3], [-5, 0, 1, -3, 2, -3, -2, 0, 2, -1, 2, 4, 5, 1, 4], [2, 1, 3, -5, 0, -3, 6, 7, -6, -2, -4, 1, 1, -4, 3], [-1, 2, 1, -6, -6, 5, -1, -4, 4, -4, 0, 0, -3, 1, -1], [9, 0, 0, 2, 0, 3, -3, -6, 6, -3, 1, -2, -1, 0, -3], [-3, 1, -7, -2, 1, -4, 2, 4, 0, 2, 0, 3, 5, 2, 7], [1, -4, -2, 7, 3, -3, -2, -4, 7, -1, 1, -2, 4, -1, -2], [-3, 2, 1, -2, 1, 3, -4, 0, -1, 1, 4, -3, 1, -2, -3], [-10, -4, -2, 3, 0, -7, 1, 0, 3, -2, -3, 2, 4, 0, -2], [5, 3, 4, -3, -2, 4, 1, -2, -3, -1, -1, -3, -2, -3, 4], [-16, -2, -2, 2, -2, -2, 6, -2, 0, 1, -2, 1, -1, 0, 2], [-7, 4, -3, -3, -3, -3, 9, 0, 2, 3, -2, 4, -4, 0, 2], [-2, -3, -1, -2, 3, -3, -9, -3, 6, 1, 2, -2, 5, 1, -2], [-5, 1, -4, 5, 0, 3, -3, -1, 6, -1, 4, -4, 3, -1, -5], [-1, 5, 1, 6, 1, -4, 1, -1, 1, 1, 4, 3, -1, 0, 1], [7, 8, 1, -8, -5, 2, 1, 3, 0, -6, -4, 2, 0, 1, 4], [8, -1, 7, -4, -4, 1, 1, 5, -3, -7, 0, 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9, 0, 6, -5, 1, 7, -5, -1, -1, -2, -5, 0, -4, 2], [7, -4, 3, -8, -7, 2, 3, -1, 0, -5, 3, -5, -5, -3, 0], [-6, -3, 5, -7, -3, 1, 8, 5, -10, 2, -3, -2, -5, -6, 5], [-32, 4, -5, 4, -2, -7, 14, 3, -5, 8, -5, 6, -4, 1, 8], [-1, -2, 2, 1, -2, 2, -10, 1, 9, -7, -3, -2, 7, 1, -9], [-14, -1, 0, -6, 4, -4, -2, 0, 2, -3, 0, 5, 9, 1, 3], [12, 4, 4, -5, -3, 4, 3, -5, 4, -4, 5, -6, -3, -5, -6], [-6, -12, 11, -9, 2, 2, -12, 3, 0, -8, 1, -4, 12, -2, -6], [6, 5, 5, 3, -3, 5, 8, -3, -2, 2, 2, 1, -6, -6, 7], [-8, -3, -4, -5, 1, 2, -4, 9, -2, -3, -2, 6, 1, 4, -6], [5, 2, 0, -9, -2, 3, -2, 5, -3, 0, 2, 2, -1, 6, 0], [1, -8, 1, -4, -3, -3, -4, 5, 3, -3, -5, 4, 6, -1, 0], [-11, 5, -3, 8, -4, -6, 6, -2, 0, 5, -1, 4, -7, 6, 4], [8, 8, 0, 1, 0, -5, 13, -2, -4, 4, -2, 2, -2, -10, 9], [-15, -2, 4, 0, -8, 2, -3, 0, 0, -9, -4, -11, 0, -7, 0], [-2, -3, 4, -8, 2, 2, -8, 2, -3, -4, -3, 3, 1, 5, 5], [3, 12, 2, -3, -1, 5, 7, -2, -6, -3, -5, -5, -7, -7, -4], [-11, 18, -3, -5, -10, 2, 13, -1, -2, 1, 0, 0, -3, -2, 7], [-7, 0, 1, -6, -4, 6, -2, -6, 1, -5, -2, -6, -1, -5, 0], [-7, -6, 11, -2, -2, -2, -5, 5, -5, 2, -2, 0, -1, 2, 4], [-3, 10, -3, 6, 2, 0, 4, 0, -4, 3, -7, 5, 0, -2, 2], [-6, -3, 10, 2, 1, 0, -5, 1, 2, -3, -4, -1, 0, 0, -6], [5, -1, 6, -3, 1, -5, 2, 7, -10, -1, -10, 1, 3, 2, 7], [-2, 4, 6, -6, -2, 3, -1, -2, -9, -1, -6, -3, -1, 0, 5], [9, 5, 1, -2, 1, 0, 2, 0, 1, 2, 2, 2, -2, -3, 1], [20, -3, 2, -4, 4, 0, 2, -5, -4, 3, 9, -1, -5, -3, 1], [12, -5, 15, -9, 0, 4, -11, -1, -8, -7, -4, -8, 0, -5, 2], [8, 3, 0, -5, -9, 3, 2, -7, 4, -6, 5, -4, -1, -3, -3], [3, 11, -6, 5, 3, -2, 4, 2, 0, 5, 5, 1, -3, 2, 2], [-9, 6, 3, -7, 0, 1, 2, 1, 0, -10, 3, -9, -2, -4, 0], [4, -12, 1, 8, 1, -3, -5, -1, -1, 2, 1, -2, -1, 2, -1], [-1, 1, 5, 3, 0, 1, -9, 1, 7, -4, 5, 3, 8, 0, -1], [-2, 7, -12, 1, 1, -4, 4, -6, 0, 3, -1, 2, -1, 1, 5], [3, -5, -3, 5, 1, 2, -1, -6, 1, 4, 1, -5, 7, -2, -2], [0, 1, -1, 10, 0, 5, 3, -3, -3, -1, 2, -12, -6, 1, -8], [10, 7, -1, -4, 2, 6, -9, 1, 12, -5, 0, -1, 5, 4, -9], [3, 4, -8, 0, -3, -3, 10, 7, -4, 3, -9, 5, -5, 3, 10], [1, 1, -7, 5, 3, 0, 3, 7, 0, 3, -6, 7, -3, 1, -8], [20, -9, 1, -8, 0, 5, -7, 1, -3, -1, 2, 2, 2, -6, 0], [-4, 2, -2, -3, 4, 5, -4, 3, 8, -3, 4, 2, -2, 3, -10], [0, 2, 9, -5, 0, -5, -4, 6, 0, -3, -1, -4, 0, 2, -2], [-10, 7, -3, -8, -1, -7, 2, 8, 8, -2, -4, 14, 3, 0, 1], [3, -4, 0, -14, -3, 9, -8, 5, 5, -5, -4, -2, 6, 1, -7], [-3, -12, 2, 7, 7, 3, -3, -2, -5, -1, -3, -6, -2, 0, 0], [-9, -6, 9, 5, -5, 9, -1, -3, 1, -2, 0, -2, -6, -8, -2], [-9, 6, 3, -5, -6, 3, -1, 0, 1, 1, -3, -9, 0, 0, 0], [11, -7, 3, -9, 3, -9, -2, 4, -5, -5, -6, 0, 12, -2, 1], [10, -1, -3, -8, 3, 1, -6, 1, 0, 1, 7, 1, -2, -1, -6], [2, -5, 5, -7, -2, 4, -1, 0, -3, -2, 1, 4, -6, 4, 2], [2, -1, 5, -3, 2, 5, -15, 6, 8, -12, 6, -4, 12, 4, -10], [17, -19, 6, -2, 4, -5, -7, 2, 0, -7, -8, -3, 12, -1, -4], [-19, 5, -10, 4, 4, -8, 12, -2, -10, 8, 0, 4, -11, 2, 7], [-18, -1, -2, -1, 0, 2, 1, 1, 1, -6, -6, 2, 0, -7, -8], [-7, -6, -2, 0, 4, -4, -6, 2, 3, -6, -5, -3, 14, 6, -6], [0, -2, -3, 8, 0, -1, -2, 1, -1, -1, -6, -1, 5, 6, -2], [-19, 5, -2, -2, -4, 2, 15, 1, -1, 0, -5, 0, -7, 2, 3], [-7, -1, -3, 3, -2, -3, 6, -1, 0, 5, 7, -3, -4, -6, 5], [2, 5, -5, -3, 2, 1, 8, 0, -3, -4, -7, -4, 6, 0, -1], [-3, 15, -11, 8, 1, -2, -4, 7, 9, -1, -1, 11, 3, 10, 3], [9, -15, -2, -11, 5, -2, -7, 2, -3, 1, 2, 10, 4, 7, 1], [-5, -2, -9, 3, 1, -7, -1, 10, 10, 5, -1, 14, 6, 10, 6], [-20, 12, -4, -3, 3, -3, 12, -4, -8, 5, 1, 5, -10, -7, 12], [11, -6, -7, 0, -3, -1, -2, 13, -4, -2, -4, 4, 0, 9, -7], [-11, -5, 5, 8, -4, 1, 3, -7, 0, 3, 0, -7, -2, -6, 2], [-7, 6, -4, 1, 0, -3, 4, -7, 8, 5, 7, 5, -6, 6, -1], [16, -5, 4, 5, -1, 5, 6, -2, -12, 4, -1, 1, -10, -4, 10], [-6, -6, -4, -5, 5, -5, -13, -2, 14, 0, 6, 6, 7, 3, 0], [-2, -2, 0, -7, -1, -8, 4, 7, -3, -2, -3, 7, 2, -1, 9], [5, -1, 6, -7, -1, 9, -4, -2, -6, 3, 8, -11, -5, -4, 1], [-25, 16, -8, 3, 1, -2, 1, 12, 0, 2, -3, 12, -2, 5, -2], [9, -10, 14, -15, -10, 9, -6, 3, -3, -11, -6, -7, 6, -1, -4], [4, 2, -5, 2, 8, -1, -2, -1, -2, 5, 9, 0, -2, 7, 1], [3, 3, -5, -7, -5, 1, 3, 0, -2, -1, -3, 1, 1, -5, 1], [5, -6, -6, -3, -6, -5, 5, 5, 1, 1, -4, 5, -2, 10, 4], [-18, -3, -3, 2, 11, -4, -1, -6, 6, -3, 7, 5, 2, -1, 4], [2, 5, -3, -12, -2, 4, -3, 11, 2, -4, -2, 3, 1, 3, -2], [-7, 2, -7, 12, 11, -8, 1, -3, 2, 3, 4, 0, 0, -5, -4], [3, -3, -3, 3, 2, 0, 5, 2, 0, 3, 1, 6, -7, 8, 2], [4, -2, 1, -10, 5, 4, -3, 7, -11, -5, 1, -9, -2, 1, 4], [0, -17, -2, 4, 4, -5, 11, -10, -7, 10, 2, 1, -7, -2, 6], [1, -10, 6, -16, -1, 3, -16, 6, 10, -19, -9, -3, 17, 4, -11], [16, -5, 10, -10, -3, 8, -10, -6, 3, -5, 0, -13, 2, 0, -4], [-1, 6, -2, 0, 1, -4, 6, -4, -2, 5, 1, -6, 2, 1, 4], [12, -2, 4, -8, -9, 9, -4, 4, -6, -7, -5, -7, 2, -6, -4], [-8, -10, 6, 5, 4, 2, -8, -8, 5, 3, 7, 8, 0, -2, 3], [-14, 4, -3, 7, 6, -5, -4, 7, -5, 4, -4, 2, 10, 4, -6], [-1, 3, -2, -5, 3, 1, -3, 8, 3, 4, -3, 1, 5, 5, -1], [22, -13, 3, -6, 0, 2, 1, -4, 0, 0, 9, -7, 3, -6, -3], [-19, 13, 3, -3, -4, 2, 10, 4, -5, 3, -4, 4, -3, -3, 1], [2, 7, -7, 6, 8, -7, 3, -3, 0, 3, 6, 2, 0, 2, 10], [8, 3, 6, 4, 0, 0, -5, -10, 4, -7, 6, -9, -1, -5, -1], [1, -6, 7, -6, -5, 3, 8, -4, -4, -3, 0, -4, -12, -1, -3], [21, -2, 11, -5, -4, 2, -4, 1, 4, -11, 4, -5, 6, -7, -11], [-25, 5, -6, -4, -7, 1, 6, 2, 5, -2, 0, 2, -2, 1, 0], [-9, 2, -8, 7, 0, -2, 5, -7, 9, 1, 4, -7, 0, 2, -3], [6, -8, 1, 6, -2, -6, 2, -7, 3, -2, 2, -4, 6, 4, 1], [-12, -2, 8, 1, -8, 12, -4, -7, -1, -9, -1, -9, -3, -15, -5], [-9, -3, -2, -1, 4, 3, 2, -5, -8, -1, 2, -4, -3, 2, -5], [11, -8, 3, 3, 8, 1, -17, -1, 16, -6, 5, 5, 8, 10, 0], [0, 4, -5, 8, 2, 1, 3, -10, -1, 5, 10, -8, -2, 0, 5], [-9, -7, 4, -1, 1, 4, 0, -4, -5, 3, 7, 3, -1, -9, 3], [0, 8, -9, 6, -1, -2, 3, 0, 8, 6, 8, 6, 1, 6, 4], [11, 2, -5, -2, 2, -3, 1, 4, 2, 2, -2, 4, -1, 3, 5], [21, -1, -7, 11, 7, -1, -1, -7, 6, 4, 10, 5, 3, 0, 1], [4, -6, 5, -3, -3, -3, 7, 6, -6, -7, -5, -3, 2, -9, -4], [-17, 7, 8, -2, -8, 16, -1, 1, -4, -6, 2, -9, -6, -5, -2], [2, 6, 5, -5, -7, 9, -1, -4, 0, -4, 4, -15, 2, -6, -5], [-14, -15, -18, -1, 13, -13, -8, 0, 7, 11, 3, 12, 3, 15, 5], [21, 0, 11, -12, -9, 16, 3, 3, -6, -2, 1, -8, -4, -11, -2], [12, -4, 6, -14, -11, 11, -7, 8, -1, -7, -10, 3, 8, 9, -5], [1, -24, -7, 5, 8, -5, -9, -2, 7, 10, 2, 2, 5, 10, -2], [-3, -4, -4, 5, -3, -9, 15, 3, -7, -1, -16, 4, -2, 1, 4], [5, 4, -1, 8, 6, 5, -11, -2, 1, -8, 6, -6, 11, -2, -9], [-11, -8, 2, 0, -1, -4, -11, -1, 8, -9, -3, -6, 12, 4, 0], [-1, 8, -5, 2, -3, 1, 4, 6, -2, 0, 1, -1, -7, 8, 1], [-3, 4, 8, 0, -7, 4, 10, -6, -2, -1, 5, -12, -4, -11, 4], [0, 11, -3, 9, -4, 3, 7, -8, -1, 8, 0, 0, -12, -3, 5], [8, -2, 9, -5, -5, 6, 1, 0, -11, -7, -6, -2, 1, -6, -7], [0, 12, 7, -7, -2, 8, -1, -5, -5, -4, 4, -2, 0, -8, 3], [-28, 11, -2, -1, 4, 0, -1, -2, 10, 7, 6, 8, -2, 5, 6], [10, -8, -9, 14, 10, -15, 10, -8, 0, 16, 2, 8, -2, 0, 9], [-9, -6, 13, 4, -2, 8, -5, 0, -2, -8, 6, -11, -4, -9, -4], [9, -19, -4, -8, -1, 1, -16, -4, 15, -6, 8, 2, 4, 4, 0], [-7, 11, -7, 3, 7, 1, -2, -4, 7, 6, 11, -3, -3, -2, 3], [23, -3, -6, 1, 7, 6, -6, 1, -2, -5, -2, -1, 1, 0, -4], [2, -2, 22, -8, -8, 20, -5, -7, -2, -8, 6, -12, -8, -13, -2], [-11, 4, 8, -1, -3, 9, -5, 1, 1, -8, 0, -15, 1, 0, -12], [-15, -10, 9, -6, 4, 5, -7, -2, 13, -9, 2, -2, 7, -7, -2], [-17, 0, 2, 3, -1, -6, 5, 4, -1, -3, -2, -2, 0, -5, -3], [-1, -15, -1, 6, 7, -2, -8, -9, 17, 0, 7, 0, 4, -2, 1], [7, 4, 8, -5, 0, 5, -3, 2, -8, -3, -9, -5, -2, 0, -6], [-14, -3, 3, 9, -4, -10, 4, -7, 3, 2, -1, 0, 6, -3, 8], [5, -16, 16, 0, -7, 7, 11, -5, -14, -3, -7, -5, -13, -15, 3], [4, -4, -1, 11, -2, -1, 6, 3, -6, 5, 2, -1, -5, -1, 4], [11, -2, -2, -7, -1, 8, 0, 8, -13, -1, -3, -4, -2, -2, 1], [18, -7, -6, 3, 14, 7, -13, 2, 3, 6, 10, 2, 0, 4, -3], [14, 6, 7, -12, -8, 10, 13, 8, -15, 0, -9, 4, -10, -2, 2], [-10, 19, -2, -6, -3, 4, 8, 1, -3, 3, 6, -3, -8, 1, 11], [32, -10, 14, -16, -6, 4, -8, 2, -2, -9, -8, -3, 5, -4, 0], [13, -4, 0, 8, 5, -5, 8, -1, -2, 2, 5, -1, -3, 2, 5], [-9, 22, 0, 9, 3, -4, 9, 6, -13, 8, -6, -1, -1, 2, 1], [4, 1, -7, 1, -4, -3, -3, 5, 10, 4, -2, 4, 8, 4, -3], [8, -6, 4, 3, 0, 3, -6, -3, 6, -5, 3, -6, 2, -8, -2], [0, 9, 0, -6, -8, 10, 6, -4, -7, -4, 5, -9, -9, -9, 1], [0, -3, 5, -7, -3, 9, -13, -1, -3, -5, 5, -11, 4, -5, -5], [16, -5, -1, 0, 6, -12, -1, 2, -2, 5, -3, 5, 7, -1, 7], [13, -11, -2, -6, 0, 0, 7, -1, -1, 1, 1, 5, -6, 6, -1], [-18, 8, -10, 7, 5, -1, 2, -5, 5, 5, 6, -2, 5, -5, 4], [-5, 3, -1, 2, -2, -2, 12, 1, -12, 1, 0, 5, -10, -3, 1], [19, -3, -8, -12, 4, -9, -6, 4, 8, 2, -2, 10, 5, 7, 6], [-1, 6, 2, 0, -3, 6, 12, 0, -16, 5, -6, 2, -8, -2, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_bf();".
// 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_8030_2_a_bf(:prec:=15)
    chi := MakeCharacter_8030_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_8030_2_a_bf();".
// 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_8030_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 46, 626, -1731, -7698, 479, 8875, 957, -4108, -702, 922, 191, -99, -23, 4, 1]>,<7,R![69248, 251648, 61632, -462110, -277794, 235137, 187376, -35712, -46448, -193, 5342, 485, -290, -40, 6, 1]>],Snew);
    return Vf;
end function;