
// Make newform 8030.2.a.be in Magma, downloaded from the LMFDB on 18 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_be();".
// 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_be();".
// 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 := [-52, -907, -1239, 3622, 3892, -5157, -4142, 3463, 2074, -1225, -524, 237, 64, -24, -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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5186399506, 3839031984, 27532331027, 2956713916, -39315304221, 12690525605, 20746207779, -13747836174, -4217825796, 4744001082, 67269559, -679696493, 73715727, 34883556, -6135455], [16257697244, 24406922512, -19134815857, -97161579378, -22387758233, 95890554923, 35107273271, -36349354126, -15127942180, 6117736632, 2818120875, -452867377, -241656913, 11327530, 7950717], [-12450340512, -36912682832, -8311173255, 100643975784, 115626370497, -76879409387, -120798745963, 20671511970, 47598412884, -1601443234, -8687620119, -92125045, 748401619, 12188068, -24837945], [17646606370, 75474187418, -21481263848, -131259374745, -41339133295, 55228228592, 72071650321, 7755983846, -36848460158, -8753931606, 8530051604, 1659753197, -950444152, -97335759, 41839165], [8151943214, 77244383618, 67012434018, -156888344263, -189177229519, 74458102334, 164499818719, 7471313626, -64630380466, -11049545846, 12997204796, 2167586419, -1338802868, -129019881, 56714341], [-18624471416, -65954539994, 86714808271, 212376010591, -135509066570, -247774605843, 107882694636, 133503050370, -46132733754, -35752487220, 10745591943, 4591312948, -1288366493, -225208367, 61936144], [-11100190102, -37744064900, 118920622125, 134816092571, -248989059686, -167355932817, 199912233004, 97447776890, -76493019332, -27574646724, 15268939361, 3633751796, -1562396749, -179003743, 65242380], [-7735840732, -104754666984, -29263842569, 391395922546, 84647326467, -460051369589, -42825791741, 238753967290, -1942348058, -61363519446, 5344844255, 7646764207, -1126751873, -368116072, 71064649], [8066141436, 34500882696, -44401968095, -110080563943, 118581520582, 164574576525, -122243760128, -110901532418, 57042850138, 34378751670, -13496443259, -4818142064, 1603739133, 248764393, -76022316], [-6182535442, -14632506249, 89043617457, 117986128256, -193099646255, -202517077432, 161480829438, 138751040662, -67657466690, -43241172145, 15534144122, 6110728165, -1872097191, -318996982, 91585352], [-5408793724, 72667782626, 192306033104, -341590452637, -420061369229, 510133874052, 275813864357, -318204584738, -65335095990, 94117009160, 1743712172, -13031879625, 1226139356, 678967023, -112912991], [10284133320, 18858709056, -63563333451, -144657019529, 168410118860, 277012186727, -177527508028, -196128090616, 85918496500, 61551327578, -21227439485, -8681786756, 2633495927, 450924867, -129594252]];
    Rf_basisdens := [1, 1, 1, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354, 2177153354];
    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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 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], [2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, 1, 1, 0, 0, -1, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1], [1, -1, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0], [3, 0, 3, -1, 1, -1, 1, -1, 1, 0, 0, 1, 1, 0, 2], [1, 0, 0, 2, 0, 0, -1, 0, -2, 2, 1, -1, -1, 0, -1], [-1, 2, -1, 0, -1, -1, -1, 1, 1, -1, 1, 0, 0, 0, 1], [1, -1, 2, 1, 1, 0, 1, -1, 0, 0, -1, 0, 0, -1, 1], [-3, -1, 0, 1, -1, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1], [3, 2, -1, -1, -2, 1, 0, 1, 1, -1, -1, 0, 0, 0, 0], [1, 1, 1, -1, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 2], [0, -3, 1, 1, -1, 0, 2, -2, -2, 1, -1, 0, 0, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 0, 0, 1], [0, 0, -4, 0, -2, 0, -1, 1, 1, -1, -2, -1, -1, 1, -2], [0, 1, 1, -2, 0, -1, 1, 0, 0, 0, 1, 1, 2, 0, 0], [3, -4, 2, 1, 3, 0, 1, 0, 0, 3, 1, 3, 0, 0, 0], [0, 1, -3, 0, -2, 0, 0, 2, 0, -2, -1, -1, -1, 0, 0], [0, -3, 0, 1, 1, 0, 0, 0, -3, 1, 0, 1, 0, 0, -1], [1, 0, 1, 1, 0, 0, 0, -1, 0, 1, -1, -1, -1, -2, 1], [2, 2, 0, -1, 1, 0, 0, -1, 2, -2, -1, -1, 2, 0, 2], [0, 0, -1, 2, -1, 0, 0, 1, -1, 1, 1, -1, -2, 0, -1], [4, 1, 1, -2, 0, -2, 0, 0, 0, 0, 2, 1, 1, 1, 1], [-1, 1, -2, 1, 1, -1, -1, 1, -2, 0, 1, 0, -2, 1, -2], [4, -1, 2, -2, 0, 1, 1, 1, 3, -1, -1, 2, 1, 1, 0], [2, -1, 1, 1, 0, 2, 0, -1, 2, -1, -2, -1, 1, 0, 2], [4, 3, 0, -2, 0, 0, -1, 1, 0, 0, 1, -2, 0, 0, 1], [-4, 1, -2, 1, -3, -1, -1, 0, -1, -1, 0, 1, -1, -2, 0], [0, -1, -1, 0, -1, 0, -1, 0, -1, -1, 0, -1, 0, -2, 2], [3, -2, 0, 0, 2, -1, -1, -1, 2, -1, -1, 2, 0, 0, 1], [0, 2, -4, -1, -1, 1, -1, 1, 1, -4, 0, -2, -1, 0, -1], [4, 1, 0, 1, 0, 1, -1, 0, -2, 2, 2, -1, -1, 0, -2], [-3, -1, -4, 1, 0, 2, 0, 0, 0, 0, 0, -1, 0, 2, -2], [6, -1, 2, 1, 1, -2, 0, 1, -1, 2, 3, 1, 0, 0, 0], [-4, 2, -1, 2, -1, 0, -1, 2, -3, 1, 1, 0, -2, 1, -2], [6, -4, 3, 0, 2, 0, 3, -4, 1, 2, -3, 0, 2, -1, 2], [-1, 0, -3, 2, 0, 1, -1, 1, 1, 1, 0, -1, -2, 0, -1], [0, 4, 0, 1, 0, -2, 0, 1, 0, -1, 2, 0, 0, 0, 2], [-1, -2, -1, 2, 2, 0, 1, -3, -2, 2, 0, 0, -2, -1, -2], [2, -2, -1, 0, 0, -1, 1, -3, 0, -1, -2, 0, 0, 0, 2], [-3, -3, -3, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 1, -1], [3, 2, 0, 2, 0, -1, -1, -2, -3, 2, 1, -4, -1, -1, 0], [4, -1, 0, 0, 0, 0, 1, -1, -3, 3, 2, -2, -1, 1, -1], [1, 2, -3, -1, 1, 0, -2, 3, 0, -4, 1, 1, 0, 1, -1], [4, -1, -1, -2, 1, 1, 1, -1, 1, 0, 0, 1, 0, 2, 0], [-5, -6, 1, 2, 2, 2, 0, -2, -3, 4, 1, 2, 0, 1, -1], [3, 0, 1, 0, 3, 0, 1, 1, 3, 0, -2, 1, 1, 0, 0], [-2, 5, -8, 0, -6, 0, -2, 3, -1, -3, -1, -5, -2, 0, -3], [-2, 0, 2, 0, 2, 1, 1, -2, 2, 0, -2, -2, 1, -1, 1], [2, 0, -1, 0, -1, 0, 1, 2, 2, -1, -3, 1, 0, 1, 0], [6, -2, -1, -1, -1, -1, 2, 2, 0, 1, -1, 2, 0, 1, -2], [0, 2, -3, 0, -1, -3, -3, -1, -1, 0, 3, -1, -2, 0, 0], [6, 3, 5, -4, 1, -2, 2, 0, 1, 1, 3, 3, 1, 1, 4], [4, 5, 1, -3, -1, 1, 0, 4, 3, -1, 1, -1, 3, 1, 0], [6, 1, 7, 0, 3, -1, 3, -4, 3, 3, -3, 0, 3, -2, 3], [-8, -1, 0, -2, 2, -1, 1, -2, 1, 0, 1, -1, 1, 0, 0], [7, 1, 0, -2, -3, 0, 1, 0, -1, -2, -2, -4, -1, 0, -1], [4, 0, 0, -4, 1, -2, -1, -1, 1, -1, -1, 3, 2, 0, 1], [-1, -2, 2, -1, 1, 2, 1, -2, 0, 1, -4, -1, 0, -1, 1], [6, 1, 3, 1, 1, -1, 1, -3, -4, 4, 2, -3, -2, -1, -1], [8, -6, 6, 2, 4, -1, 3, -3, -3, 3, 0, 2, -1, 0, -2], [2, 7, -3, -2, 1, 2, -3, 3, 3, -5, 3, 0, 1, 2, -1], [-2, 0, -3, 4, 1, 0, 0, -2, -3, 3, 4, -1, -1, 0, -4], [-3, 3, -1, -3, 0, -3, -1, 1, -1, -1, 2, -1, 1, -1, 0], [-3, 2, -5, 3, -6, 1, -3, 2, -3, 2, -1, -4, -5, -2, -1], [7, 1, 3, -3, 0, -1, 2, -2, 0, 0, -3, 0, 0, -1, 1], [-8, -3, -5, 3, -3, 1, -4, 0, 0, -3, -2, -2, -4, 0, -2], [-4, -3, -3, -1, -6, -1, 1, -2, 0, -2, -2, 0, 0, -1, 2], [-4, 3, -2, -3, -1, 0, -1, -1, 1, -4, 1, -1, 3, 1, 0], [7, -1, 5, -1, 6, 0, 1, -1, 0, 4, 1, -1, 0, -2, -1], [-2, 0, 0, -2, 1, 1, -2, 4, 2, -3, 1, 0, 2, 1, 2], [4, 0, 4, 0, -2, -1, 3, -2, -2, 3, 2, 2, -2, 1, 3], [-4, -1, 4, 3, 2, 0, 1, -1, -4, 3, 0, 2, -1, -2, -2], [7, -3, 4, -2, 3, -1, 3, -1, 1, -2, -1, 3, 4, -1, -2], [-4, -4, 3, 4, 2, 0, 0, 3, -1, 3, 2, 3, 1, -1, -3], [-1, 2, -1, -4, -1, -1, 2, -2, -1, -2, -1, -1, 1, 1, 0], [2, 5, -5, 0, 1, 3, -2, 4, -2, -3, 1, -4, -1, 2, -3], [-3, 2, -5, 0, -2, -1, -2, 3, 4, -1, -2, 1, 0, 2, -1], [2, -4, -4, 2, -2, 3, 0, 2, 3, 0, -2, -1, -2, 1, -1], [8, -8, 3, 1, 1, -1, 4, -2, -4, 5, -1, 1, 0, -1, -2], [-2, -1, 5, 1, 2, -3, -2, -2, 1, 3, 2, 4, 0, -1, 0], [0, -1, -2, -1, -1, 2, 0, 2, 0, -1, 2, 4, -2, 1, -1], [0, -4, 2, 5, 0, -1, -2, -3, -1, 3, 2, -3, -4, -1, 1], [-5, 0, -6, 4, -4, 0, -2, 3, -4, -1, 3, -3, -3, -1, -2], [8, -3, -4, -1, 0, 0, 1, 1, 3, -1, -1, 0, -1, 3, 0], [0, -7, 9, 2, 5, 1, 3, -5, 0, 4, 1, 2, 2, 1, 2], [-2, -3, -3, 2, 0, -2, 1, 0, -7, 4, 2, 5, -2, 2, -4], [9, -5, 2, -2, 0, -1, 1, -3, 0, -3, -4, 2, 3, 1, 1], [-8, 0, -5, 3, -4, -1, 0, 0, -2, 2, -1, -2, -3, -3, -2], [1, 0, 5, -3, -4, -1, 4, -3, 0, -1, -1, -2, 2, 0, 6], [4, -5, 2, -1, 5, 1, 0, -2, 0, -1, 4, 0, 4, -1, 0], [-1, -5, -3, 6, -2, -2, -2, 0, -3, 4, 4, -1, -2, 0, -1], [2, -3, -5, 4, -3, 0, -3, 1, -3, -2, -1, -2, -2, 0, 1], [6, 0, -8, 0, -1, 0, -4, 5, -1, -2, 1, 0, -2, 3, -5], [-7, -9, -8, 5, -1, 4, -1, -1, -4, 0, -5, 1, -4, 1, -5], [10, 0, 2, 0, 0, 0, -1, 0, -1, -2, 2, -3, 3, -2, 2], [0, 0, 4, -1, -3, -2, 2, 0, 2, 1, -1, -1, 0, 0, 5], [6, -8, -1, -3, -1, 0, -1, -1, 1, 0, -4, -2, 0, 0, 0], [-2, -1, 0, 6, 2, 1, -5, 0, -5, 2, 6, -1, -3, 1, -1], [8, 0, 1, -2, 4, -2, 1, 1, -2, 3, 0, 3, -1, 4, -1], [-2, -6, 2, -1, -2, 3, 0, -1, 0, -1, -3, -1, 2, 0, 1], [14, 2, 4, -4, 0, -1, -3, 0, 2, -4, 2, -4, 1, -2, 5], [17, 4, 11, -3, -1, -4, 5, -4, 2, -1, 1, 2, 2, -3, 6], [1, -1, -4, 0, -2, -2, -3, 0, 2, 0, 2, 1, 0, 0, 2], [0, 0, -3, -2, -4, -2, 0, 1, 5, -3, -1, 2, 3, 1, 1], [8, -3, -2, -3, 2, 1, 1, 2, -1, 0, -2, 0, 1, 1, 0], [-3, -7, 3, 6, 0, 3, 2, -3, -5, 5, -4, -2, -4, 0, -3], [3, -8, 9, 0, 6, 3, 1, -2, 3, 4, -1, -1, 3, 0, 2], [1, -9, 2, -2, 3, 1, 1, -3, 3, 0, 0, 4, 2, 2, -1], [5, 5, -2, 2, 2, 0, -1, 1, -3, -4, 6, 2, -1, 0, -2], [-5, -1, -1, -5, 0, 1, 4, -1, 0, -3, -1, 2, 2, 0, 2], [1, 1, -1, -2, 2, 1, -3, 2, 2, 1, 1, 1, 4, 0, 4], [3, -1, -7, -1, -2, 3, 0, 4, 1, -2, -1, -1, -4, 2, -1], [9, -2, 6, -3, 1, 2, 3, -5, 4, 0, -7, 0, 2, -1, 4], [-5, -6, -7, -1, -1, 1, -2, -2, -1, -3, -3, 0, -3, -1, -1], [13, 3, 12, -6, 7, -2, 3, -5, 4, 4, 0, 3, 4, -1, 2], [2, 11, -4, -3, -3, -1, -2, 7, 3, -4, 1, -4, 0, 1, -4], [10, -3, -2, 0, -1, -1, -1, 2, 1, -1, 1, -1, 3, -1, 3], [-3, -12, 2, 4, 6, -1, 4, -4, 0, 8, -2, 6, -3, -2, -1], [0, 9, -5, -3, -1, 0, -1, 4, 1, -1, -1, 2, 2, 2, -3], [8, 1, 5, -2, 2, 2, 2, -2, 5, -2, -2, -3, 1, -1, 4], [-3, -1, -4, 5, 2, 3, -4, 0, 0, 0, -3, -5, -3, -4, -1], [-7, -2, 2, 4, 4, 1, -1, -2, -3, 6, 7, 1, -2, -1, 0], [-3, -1, -3, -1, 6, 2, -4, 2, 1, -2, 3, 0, 0, -2, 4], [8, 1, 0, 2, -2, 0, -4, 6, -3, 2, 2, -1, -4, 0, -6], [19, -2, 2, 0, 2, 2, 2, -2, 5, -1, -3, 1, 4, 0, 5], [-6, -12, 0, 3, 3, 2, 4, -1, 1, 1, -7, 0, -1, -1, -2], [14, 0, -2, 1, -3, 3, 2, -1, 0, -1, -1, -9, -1, 0, 3], [5, 1, 4, 3, 1, -5, 1, 1, 2, 4, 2, 3, 0, 1, 2], [5, -4, -8, 4, -4, 2, 0, -1, -4, -3, -3, -5, -3, 1, -2], [8, 0, 4, -4, 3, -4, 0, 2, 4, 1, 3, 2, 4, -1, 6], [2, 0, -3, -4, 0, -1, 3, 1, 4, -5, -5, 5, -2, 2, -2], [5, -7, -4, -3, -3, 4, 2, 2, -3, -1, -6, -3, -1, 1, -6], [9, 0, -2, -2, 2, 0, -5, 4, 0, -5, 3, -4, 1, -2, -1], [-3, 4, 4, -1, -3, 0, -2, 4, -1, -2, -1, -2, -2, -4, 4], [-3, -6, -2, 8, 6, 0, -1, -4, -6, 6, 3, 4, 0, 0, -3], [6, -1, 5, 2, 3, -5, 2, -2, 1, 5, -2, 4, -3, 1, -5], [-5, -4, 7, 1, 4, 0, 3, -3, -5, 5, 3, 4, 1, -1, 2], [5, -5, -4, 1, 2, 0, -2, 4, 1, 0, 1, 2, -1, 3, 0], [-7, -8, -4, 9, -4, 2, 2, 0, -4, 4, -5, 2, -2, 1, -4], [14, 2, 1, -3, 1, 0, 4, -1, 6, 0, -3, 3, 2, 3, -2], [-3, -8, 2, -1, 3, -2, 1, 2, 0, 5, 6, 4, 2, 1, -2], [20, -9, 8, 3, 5, -1, 4, -6, -2, 5, 0, -2, 2, -3, 4], [-4, 6, -1, 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-13, 7, -1, -2, -10, 3, -6, 1, 7, -6, -8, 0, -8], [1, 10, 9, -4, 8, -8, -1, -3, 6, -3, 3, 4, 3, 0, 8], [-16, -17, -3, 13, 0, 2, 4, -7, -5, 13, -6, 4, -9, -5, -4], [-7, 9, -7, 1, -13, 3, 0, 3, -2, -8, 1, -13, -4, 1, 0], [-8, -1, 0, -1, 2, 3, 0, 5, -2, -1, 1, -4, -3, -5, 3], [-4, 1, -2, 1, 2, 3, -2, 2, -5, 0, 6, -2, -4, -1, -2], [-6, -1, -5, 5, 0, 0, -4, -3, 6, 0, -2, -1, 2, 2, 0], [17, -7, 10, -2, -2, 4, 4, -4, 7, 6, -12, -5, -1, -1, 3], [8, 6, 11, -7, 4, 0, 4, -2, 2, 0, 3, -2, 3, 1, 9], [-8, -6, -8, 10, -9, -5, -4, -1, -9, 0, 2, -9, -6, -7, -2], [-13, 2, -10, -1, -4, -2, -3, 4, -8, 0, 4, -8, -3, -2, -4], [-13, -13, -14, 3, -6, 3, 1, 4, 4, -6, -6, 1, 0, -1, -1], [-9, 1, -1, 9, -3, 1, -3, 4, -10, 9, 9, -5, -7, -4, -1], [4, -9, 6, -10, -1, 0, 5, -6, 4, -1, -6, -1, 5, 3, 7], [-2, -6, -2, -5, 1, 7, 1, 3, 3, 0, -2, 3, 5, 4, 5], [-19, 0, 0, 1, 4, -5, 3, 2, 0, 4, 9, 4, 0, 0, 1], [-9, -1, -1, 0, 4, 0, 2, 1, -7, 3, 1, -1, -2, -4, -10], [17, 6, -8, -1, 5, -3, -3, 4, 0, -5, 9, 0, 0, 0, -5], [13, 9, 4, 6, -4, 4, -1, 0, -2, 0, -10, -12, -4, -1, 1], [-12, -14, -10, 0, -2, 2, -7, 1, 2, -3, -6, -1, -5, 3, -3], [-17, 10, -9, -8, -9, -2, -6, 4, 7, -3, 2, -4, -2, 1, 5], [-3, 13, -1, 2, -1, 1, 3, 0, -7, -1, 7, 6, -1, -3, -2], [-2, -4, 4, -3, 6, 1, 3, -1, 7, -3, 3, 11, 1, 2, 3], [8, 0, -10, -10, -2, -2, 4, 4, 4, -4, -4, 1, 0, 6, 2], [-19, 0, -18, -4, 1, 2, -4, 7, 0, -4, 3, 0, -3, 4, -10], [1, -8, 6, -8, 12, -1, 1, 0, 8, -5, -2, 7, 7, 0, 2], [-4, -4, -4, 2, 6, 4, -1, -4, -5, 0, -7, -6, -1, -1, -4], [-2, 4, 11, 1, 7, -5, 6, 2, -1, 5, -4, 11, 2, -2, -4], [13, -1, -8, -7, 3, 0, -5, 3, 4, -8, 3, 6, 2, 1, 1], [24, 1, 16, 0, 10, -1, 3, -5, -1, 9, 2, 0, 5, -2, 3], [-12, -6, 6, 7, -4, -4, 0, -6, -1, 4, -4, 5, 0, -7, 11], [-4, -13, -9, -4, -5, 0, 1, 11, -4, -6, 2, 7, -2, 5, -5], [4, 3, -10, -4, -5, 1, 2, 12, -10, -5, 1, -1, 2, 1, -9], [4, -9, 4, -7, 4, -1, -3, -1, 3, 5, 2, 4, 1, 3, -1], [8, 2, 10, -1, 3, -3, 4, -9, 0, 6, 8, -8, 2, -2, 5], [2, -13, -2, 7, -5, -1, 2, -6, -9, 5, -7, -4, -2, 0, 0], [1, 1, -3, 6, 2, 1, -3, 0, -4, -2, 3, -4, -5, 4, -4], [13, 7, 3, -2, 7, 4, -2, -5, -1, 2, -3, -2, -2, -1, 2], [0, 1, -6, -1, -3, 3, -2, 2, 5, -7, 1, -7, 0, 1, -4], [-3, -7, -12, 3, -11, 2, 5, -4, -2, 3, -10, -3, -4, -1, -10], [10, 10, -2, -8, -5, -3, 5, 1, 3, -12, -3, -7, 3, 2, 2], [3, 7, 2, 0, 1, 0, -9, 3, -3, -2, 4, -5, -4, 3, -3], [-2, 13, -4, -7, -7, -6, 0, 8, 13, -9, 1, 0, 2, 1, 8], [-18, -1, 7, 10, 5, 7, -1, -3, -6, 6, 5, 3, 6, -3, -2], [-8, -6, 0, -3, -2, -4, 6, -3, 3, 3, -10, 2, -6, -4, 0], [-3, -6, 5, -2, 1, 0, 5, -8, 5, 11, -1, 6, 3, 2, -5], [14, 0, 13, -3, 9, -3, 0, 1, 13, 2, 7, 7, 6, -2, 5], [-15, 17, -1, -10, 2, 3, -5, -1, 8, -10, 0, -9, 6, 0, 7], [-14, -9, -8, -4, -3, 8, -3, 8, 3, -8, 0, -4, -2, 1, -2], [7, -16, 5, 9, 11, 4, 5, -3, -9, 10, -3, 11, 0, 0, -11], [-12, 3, -9, -4, 1, 7, 1, 4, -4, -11, 0, -6, 2, 1, -1], [-8, 0, 3, 0, 5, -4, 1, 2, -2, 7, 7, 3, -6, -1, -4], [-5, -7, -7, 10, 1, 5, 2, 6, -3, -1, -2, 6, -4, 6, -8], [1, -14, 3, 6, 8, -3, 6, -9, -8, 11, 1, 3, -6, 5, -3], [14, 2, 17, 3, 8, 2, -4, 0, 1, 0, 7, 1, 5, -4, 1], [-5, -4, -14, 4, -9, 3, -10, 8, -7, -7, -3, -5, 2, -2, -2], [17, 2, 9, -3, 2, -4, 10, 1, -1, -3, 3, 6, 6, 3, -3], [-26, -15, -8, 13, -5, 1, -4, -2, -11, 5, 1, 5, -4, 0, -10], [15, -1, 16, -1, 2, 0, 2, -5, 4, 7, 1, -4, 2, -4, 3], [0, -3, -15, 1, -15, 3, 5, 0, -6, 1, -3, -9, -6, 5, -6], [-19, 1, 11, 3, -6, 0, 7, -8, 0, 3, -3, -7, 8, -6, 10], [21, -1, 0, -7, -3, -9, -3, -4, -2, -1, 9, 0, -4, 1, 9], [12, 9, 7, -13, 4, 4, 4, 10, 1, -8, 0, 7, 5, -2, -3], [-18, 4, -5, 6, -5, -7, -3, 0, -6, 3, 2, -1, 2, -3, -6], [-7, -11, 3, -5, 1, -5, 4, -5, 7, 0, -2, 4, 2, -4, 2], [-6, 0, 5, -10, -2, 1, -1, -2, 8, -2, -4, -5, 8, -2, 3], [-14, -4, 1, -2, 2, -5, 3, 1, -1, -1, 5, 11, 6, 2, 4], [-6, -7, -15, -1, -3, 1, -5, -2, -4, -5, -7, -3, -6, 5, -10], [6, 10, 2, -5, -7, 1, 2, 3, 7, 0, -6, -3, 4, 1, 3], [-18, -13, 9, -1, 8, -3, 3, -3, 1, -1, 1, 6, 4, -7, 4], [-2, 11, -9, -1, 0, 6, -7, 4, 1, -6, 6, -2, -4, 6, -3], [5, -18, 1, 2, 1, -1, -3, 2, -3, 5, 0, 0, -3, 1, -4], [-26, 6, -1, -1, -5, -2, 0, -5, -5, 0, -5, -1, -5, 0, 4], [4, 0, 1, 2, -8, -2, 8, -6, -5, 6, 0, -12, 1, 2, 2], [3, -12, 3, 0, -9, -1, 5, -2, 6, -3, -14, -2, 3, -4, 9], [10, 7, 10, 4, 8, 3, -5, -1, 4, 4, 4, -1, 4, -4, -2], [6, -1, -9, 10, 1, 5, -3, 3, 2, 2, -3, -1, 4, 1, -10], [11, -1, 5, 8, 6, 4, 1, -4, -10, -1, 0, -9, 2, -2, -5], [-8, 8, 7, -11, -6, -2, 1, -1, 2, -3, 0, -8, 10, -4, 4], [-7, 12, -3, -7, -8, 0, 0, 5, 8, -9, -4, -1, -6, 2, -3], [-6, -4, 4, 2, -3, -8, -1, -7, 1, 6, -2, 3, 4, -4, 3], [-3, 13, 11, -12, -2, -1, 3, 5, -1, -3, 7, -2, 1, 3, 3], [6, 2, 12, 3, 9, -3, 2, -1, 3, 0, 3, 5, 8, -5, 11], [6, 14, -4, -10, -5, -1, -2, 4, 9, -13, 0, -1, 10, -2, 8], [-22, -1, -9, 3, -6, 3, -2, -3, 5, -1, -7, -9, -3, -1, 7], [4, 23, -3, -8, 0, 1, -8, 0, 13, -12, 2, -6, -1, 4, 2], [5, 9, -12, 11, -10, 4, -7, 10, -9, -4, 7, -5, -12, 4, -8], [2, 2, 2, 1, 3, -3, -1, 3, 10, -2, -3, -3, -5, 3, 3], [11, 6, 5, -1, -1, -8, 1, 0, 6, -3, 3, 10, 5, 1, 14], [16, 3, 7, -13, 5, 2, 9, -1, 5, -4, -6, -3, 8, -6, 0], [-3, -1, 2, 12, 8, -1, -6, -2, -1, 12, 4, -6, -7, -4, -8], [15, -19, 0, 5, 1, 7, -1, 4, 2, 3, 3, 0, 6, 1, -3], [-4, -7, 9, -1, 0, 2, 4, 3, 0, -2, 1, 10, 2, 4, -2], [-17, 13, -14, -3, -4, 1, -2, -3, -4, -7, -5, -7, 1, 0, -4], [-7, 14, -6, 4, 3, 7, -3, 3, 14, -8, 2, -4, 0, 3, 3], [-1, 6, 11, 0, 0, 1, 3, 2, -1, 2, 0, -3, 3, -6, 2], [23, -8, 9, 10, 13, 5, 5, -11, 0, 10, -3, 3, -1, 3, 1], [9, -13, -3, -13, 9, -3, 2, -5, 9, -2, -2, 8, 4, 4, -2], [18, 8, 0, -3, 0, -4, 2, 2, -2, -4, -3, 0, 7, 2, -1], [4, -16, 12, 2, 8, -1, 5, -5, -2, 13, -3, 0, -5, 1, -4], [-6, 5, -10, -2, -8, 2, -4, 6, -12, -10, 5, -15, -4, 3, -10], [-7, 18, -6, -1, -5, -3, -3, 5, -1, -2, -4, -4, -6, -5, -2], [-2, -2, -7, 1, 8, 3, -5, 2, -2, 8, -2, 4, 0, 4, -16], [14, 7, -12, -2, -2, 9, -6, 7, 0, -3, 0, -7, -8, 6, -12], [4, 2, -10, 10, -1, 7, -3, -4, -2, 0, 1, -8, -4, 3, -5], [6, -8, -14, 3, -9, -3, 0, 3, -1, 2, -9, 8, 0, 1, -7], [6, 6, 2, 10, -2, -8, -9, -1, -6, 9, 7, -6, -11, 1, -1], [-12, 4, -2, 1, 5, 4, 3, -2, 4, -3, -10, 1, 3, -7, 3], [17, -8, -21, 9, -14, 6, -6, 7, -5, -5, -4, -14, -4, -5, -7], [-11, 17, 3, -11, -11, 0, 2, 6, 6, -1, -2, -1, 0, 4, 2], [10, 2, 4, 12, -9, -7, 4, -9, -1, 8, -10, 1, -10, -7, 0], [24, 23, 3, -14, 0, 4, -3, -1, 2, -5, -1, -11, 0, 3, 5], [0, 3, 5, 2, 11, 2, -2, 1, 2, 8, 2, 9, 1, -1, -6], [18, 8, -4, -8, -3, -6, -3, 4, 1, -5, 4, 2, 9, -4, -2], [-9, 19, -3, -4, -5, -1, 0, 4, -3, -15, 6, 4, 6, -1, 3], [14, 11, 6, -3, -3, -5, 4, 3, 1, 0, 2, 8, 2, 9, -1], [-9, 4, 19, -5, 15, -2, 0, 0, 12, 2, 6, 9, 5, 0, 2], [33, 5, 2, -12, -5, -3, -4, 9, 1, -10, 8, 3, 9, 1, -1], [-12, -6, -5, 4, 2, 2, 1, -4, 4, 0, 2, -6, 6, -5, -1], [20, -7, 6, -3, 11, 4, 5, -3, 3, 2, -6, -2, -5, 7, 0], [-11, 4, -8, -7, -13, -8, -6, 6, 0, -7, 8, -4, 3, 1, 2], [-8, -1, -6, 8, 2, 6, -12, 11, -3, 1, 2, 1, -6, 4, -7], [14, 5, 3, -2, 3, 10, 0, 3, 0, -10, -7, -4, 1, -2, -3], [3, 2, 15, 6, 12, 4, 9, -5, 0, 9, -1, 3, 3, -4, 3], [-6, 3, -4, 6, -3, -5, -4, 4, -9, 1, 2, -5, -6, -2, -3], [0, 2, -8, 4, 0, -2, -1, 1, -2, -7, -2, -11, -9, -3, -5], [-21, 12, 10, 1, 9, -2, -5, -1, -5, 4, 12, -1, -6, -6, 3], [-12, 12, -17, -7, -1, 6, -8, 7, 5, -7, -2, -6, -2, 4, -8], [5, -6, 2, -12, -4, 1, 8, -3, 12, -1, -7, 8, 9, 4, 1], [-7, -20, -4, 11, 1, -3, -1, -10, -11, 13, -3, 5, -4, -5, 1], [-18, 20, -15, 2, -9, -2, -12, 11, -4, -8, 10, -1, -5, 1, 1], [-3, -8, -3, 5, -1, 3, 0, 2, -2, 3, -9, 7, -3, -4, 5], [-1, -3, 7, -4, -5, -6, 3, -6, -14, 3, 1, 5, 1, -6, -5]];
    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_be();".
// 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_be(: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_be();".
// 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_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-52, -907, -1239, 3622, 3892, -5157, -4142, 3463, 2074, -1225, -524, 237, 64, -24, -3, 1]>,<7,R![22560, -50896, -233456, -56504, 265100, 83668, -138026, -29259, 38891, 2566, -5551, 254, 345, -38, -7, 1]>],Snew);
    return Vf;
end function;