
// Make newform 5819.2.a.m in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5819_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_5819_2_a_m();".
// 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_5819_2_a_m();".
// 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, 0, -212, -1386, -1546, 3880, 6165, -4004, -7793, 1876, 4680, -434, -1521, 48, 275, -2, -26, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [80782880, -386313102, -5196886524, -966085854, 16253905664, 2205964352, -19467650535, -1234648727, 11840574073, 99825873, -4000371403, 132713741, 757262509, -49662910, -74806734, 6798802, 2987085, -328565], [-1708753, 4156670, 33106482, 476713730, -290157273, -2288748242, 850929628, 3586612173, -871380852, -2682169304, 383636808, 1049558317, -83002797, -220021595, 8866189, 23362110, -380825, -984466], [-4156670, 251942239, 1891617928, 2947330794, -4341213398, -11385391873, 3255234839, 14187692981, -523451324, -8380600848, -307959515, 2682016110, 138001451, -478773264, -19944604, 44808403, 984466, -1708753], [4156670, -329149154, -1891617928, -2931889411, 4341213398, 11385391873, -3255234839, -14187692981, 523451324, 8380600848, 307959515, -2682016110, -138001451, 478773264, 19944604, -44808403, -984466, 1708753], [-30337775, -251819776, -1828833303, -2960496515, 6988449291, 12735172686, -9228156972, -17193552551, 5410019573, 10772121394, -1632905423, -3595371265, 266273144, 660638610, -22546430, -63051704, 792919, 2438491], [-5197969, -698612190, -4923061708, -2472871832, 17720173953, 15354111473, -22589014791, -22678493595, 13182217747, 14671144024, -4038051971, -4952886002, 676497736, 912564442, -59091504, -87007460, 2126298, 3356951], [91405616, -187869852, -6388668236, -9515391515, 18481156119, 30379415960, -19866447330, -35802286074, 9796025480, 20647108781, -2462826845, -6512315001, 320149209, 1149371424, -19863881, -106536316, 447948, 4031929], [98659328, -223326880, -7131361644, -10878070189, 20578081403, 35696326792, -21839851224, -42649247828, 10389570532, 24657646672, -2457091489, -7750299832, 287865163, 1358800014, -14492291, -124918594, 176628, 4686441], [-35457028, 795093536, 8690966158, 13311164036, -22827491728, -46692538374, 22196901094, 57121722668, -9597425821, -33941636804, 1910126177, 11000611906, -138749586, -1989399210, -3874854, 188325192, 654512, -7253712], [-84822899, -68116818, 10049858231, 21014528326, -25765689603, -63974087386, 24285591682, 74235026830, -9942610192, -42879382418, 1734189918, 13653468626, -58055055, -2439736042, -15576181, 229002906, 1231060, -8765642], [110208007, -260286514, -11538673177, -19756274022, 31691244707, 63299260208, -32312589270, -75796371475, 14848787381, 44737620172, -3289438021, -14471705323, 324472223, 2617183371, -7939320, -247977222, -385571, 9563456], [89594943, -248260572, -9749703239, -20345633821, 25649810407, 68473878674, -24758601853, -84168836957, 9842184522, 50383575776, -1437833488, -16436891789, -53136443, 2989568132, 31402566, -284424442, -2007514, 11002028], [122423289, -1149765246, -17163803758, -28310221737, 44645146380, 94110012567, -42709724478, -112217527001, 17823484054, 65386365591, -3267037758, -20849973571, 161705996, 3720004421, 20000541, -348268130, -1834850, 13293155], [-96308443, 917621891, 15520161893, 28478833165, -39834600503, -94600692198, 37831780847, 113707233537, -15598295529, -67045707598, 2735442949, 21655895520, -91865171, -3911804263, -24703438, 370325372, 1950452, -14272726], [194187329, -1351084498, -28860163169, -53194236794, 76978387382, 178346726247, -74541512956, -216619460777, 30322834420, 128163794271, -4926948245, -41384133034, 28103421, 7461970829, 69618017, -704855985, -4885552, 27105813]];
    Rf_basisdens := [1, 1, 1, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383, 15441383];
    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_5819_a();"
function MakeCharacter_5819_a()
    N := 5819;
    order := 1;
    char_gens := [530, 3708];
    v := [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_5819_a_Hecke( : Kf := Kf)
    return MakeCharacter_5819_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]];
    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], [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, -1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, -1], [2, 0, 0, 0, 1, 0, -1, 0, -1, 1, -1, -2, 0, 0, 0, -1, 0, 1], [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, -1, 0, 0, 1, 0, -1, -1, 0, 1, 0, -1], [-1, 1, 0, 0, 1, 1, -1, -1, 0, 2, -1, 0, 2, 1, -1, 0, 0, 0], [2, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -1, -2, 0, 1], [1, 0, 0, 0, 1, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0], [1, 1, -1, 1, 3, 0, -1, 1, -1, 0, 0, 0, 2, 1, 0, -1, 0, 0], [1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 1, 0, 2, 0], [2, -1, 1, -1, -2, 0, 0, 0, 0, -2, 1, -2, -1, 0, 0, 2, 2, 1], [2, 2, 1, 3, 3, 2, 1, 2, -2, 2, -3, -2, 2, -2, 0, -2, -2, 1], [3, -1, -1, -1, -1, -2, 1, -1, 0, -1, 0, 0, -2, 1, 1, 0, 1, 0], [-2, 1, 1, -1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, -1, 0, 2, 0], [1, -1, -1, 1, 0, 0, 2, 1, 1, 0, 0, -2, -1, 0, -1, -1, 1, 0], [-1, -1, 1, -1, 2, 2, 1, 2, 0, 0, 1, -2, 0, 1, -2, -1, 2, 0], [3, -1, 2, -1, 0, 1, 1, 0, -1, 0, 2, 1, 0, -1, 0, 1, 1, -1], [-2, 0, 0, 0, 0, 0, -1, -2, 0, 2, -1, 0, -2, -2, -1, 0, 1, 1], [1, 1, 0, -2, -2, 0, -1, -1, 1, -1, 2, 2, -1, 0, 0, 0, -2, -3], [5, 2, -1, 1, 1, -1, -1, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0], [1, 3, 1, -1, -2, -1, -1, 0, -1, -2, 2, 2, 0, 1, -1, 0, -1, -2], [-1, -2, 1, -1, -3, -1, -1, -1, 1, 0, 1, 4, 0, 3, 1, 2, 0, -2], [-1, -1, -1, -1, -1, -1, -2, -1, 1, -2, 1, 2, 0, 1, 1, 2, 2, -3], [6, 2, 1, 0, 2, 2, -2, 1, -2, 0, 0, 1, 0, 0, 2, 1, 0, -3], [-5, -2, 0, -1, 1, -3, -3, -1, 1, 2, 0, 1, 0, 1, -3, -1, 0, 1], [1, -1, -1, 0, 4, 1, 1, 3, -3, -1, 1, -3, 0, 1, -1, -2, 0, 0], [0, -1, -3, 0, -1, -2, 1, -2, -1, -1, 0, 1, 0, 1, 1, 0, -1, 0], [-1, 3, 1, -2, -4, 0, -1, -2, 1, -1, 1, 6, 0, 1, 2, 5, 0, -3], [2, 0, -1, 1, -3, 0, -1, -1, 1, 0, -1, -3, -2, -3, 2, 0, 1, 2], [2, -1, -1, 0, 3, -2, -2, 2, -2, 0, -2, -2, 0, 1, 1, -1, 1, 1], [4, -1, 0, 0, -1, -2, 0, -2, -1, -2, 1, -1, -1, 1, 2, 0, 1, 1], [6, 4, -2, 0, 5, 1, 1, 3, 0, 1, -1, -1, 1, 0, -2, -2, -2, -1], [4, 1, 1, 2, 0, 1, 1, 3, -2, 0, -1, -2, 1, 2, 3, 0, 0, 2], [1, -1, -2, -3, 2, -1, -1, 0, 0, -2, 1, 2, -1, 1, 1, 1, 4, -1], [-4, 1, -1, -1, -3, 0, 0, -1, 2, -1, 3, 2, 2, 1, 0, 1, -1, -2], [-2, -1, 0, -1, 2, 0, 0, -3, 1, 2, -2, -1, -2, -2, -3, -1, 1, 2], [7, -1, 2, 0, -2, 0, 3, 3, 1, 1, 2, 0, -1, 0, 0, 0, 0, 1], [6, -1, -3, 3, 3, -1, 2, 0, 0, 3, -5, -2, 0, -1, 0, -2, 1, 3], [6, 7, -2, 1, -1, 1, -2, 1, -2, -3, -1, 0, 1, 1, 3, 1, 1, -1], [-3, -1, 1, -4, -3, -1, 1, -1, 2, -5, 4, 5, -1, 2, 2, 6, 2, -5], [5, 3, 1, 2, -1, 3, 2, 3, 1, 0, 0, 3, 5, 1, 0, -1, -6, 0], [-4, -1, 2, -4, -4, -1, -3, -2, 2, -1, 5, 3, -2, 2, -1, 4, -1, -4], [1, 4, 2, 3, 2, 0, 3, 2, -2, 2, -1, -4, 1, -1, 0, 0, 0, 1], [3, -1, -1, 1, -1, 0, -1, 0, -1, 0, -4, -2, 0, -1, 2, 1, 2, 2], [8, 2, -1, -2, 4, 0, -1, 2, -2, 0, 1, 1, 3, 4, 1, -1, 1, -3], [2, -1, 1, 0, 1, 2, -6, 4, -2, -2, 0, -1, 0, 1, 0, -1, 0, 0], [3, -1, 1, 1, 0, 0, 1, 2, 0, 0, 1, -4, 2, -1, 0, -3, -2, 0], [4, -1, 4, 2, 0, 2, 2, -1, 0, 0, -2, -3, 0, -3, -1, 0, -1, 0], [-7, -4, 2, 1, -1, 1, 3, 0, 1, 4, 0, -4, -1, -2, -3, -5, -2, 2], [-2, -1, -1, 2, -3, -2, 1, -1, 4, -3, 2, 2, 1, -3, -2, 0, -5, 1], [0, 0, -1, 2, 5, 1, 0, 3, -2, 4, -2, -3, 2, -4, -2, -4, -1, 3], [3, 0, -3, -2, -3, -2, 4, 0, 1, -3, 1, 8, -2, 2, 1, 1, 0, -4], [1, 1, 1, 0, 1, -2, -2, 1, -2, -2, 1, -1, -3, 2, 4, 2, 6, -1], [-2, 4, -1, 0, 1, -1, -4, 0, -1, 2, -1, 0, 0, -3, -2, 0, 2, 2], [7, 2, -1, -1, 0, -3, 2, -2, 0, -1, 3, 1, 0, 0, -1, 1, -2, -1], [-2, 0, -4, 0, -4, -2, 3, -3, 4, 0, 1, 2, -2, -2, -1, 0, -5, -1], [4, 3, -2, 1, -3, -2, -1, 0, 2, 0, -1, 2, -2, -1, 0, 3, -3, 0], [-3, -1, -2, -4, -3, 0, -4, -5, 1, -2, 0, 5, 4, 4, 2, 1, -1, -2], [8, -2, 2, 2, 0, 0, 0, 4, -4, -2, -2, -5, -2, -2, 0, 0, 0, 2], [-1, -3, 1, -2, -3, -3, -2, -4, 1, 0, -2, -2, -2, 0, -4, 3, 2, -1], [-5, -1, 1, -1, -1, -3, -4, -1, 1, -1, 3, 5, -1, -1, -2, 3, 0, -3], [-4, -6, -1, -1, -6, 1, 3, -4, 3, -2, 1, 4, -1, 0, 0, 3, -1, 0], [-5, 2, -2, 1, 4, 2, -5, 0, -1, 2, -2, -2, 4, -2, -3, 0, -3, -1], [10, 1, -1, -2, 0, -1, 0, 0, 2, -2, 4, 5, 1, 2, 0, -2, -5, -5], [5, 0, -1, 2, 3, -2, -1, 3, -1, 1, -1, 0, -3, -2, 2, -1, 0, -1], [-1, -1, 4, 0, -1, 1, 1, -3, 2, 0, 1, 0, 0, -3, -3, 2, 2, 0], [6, -3, -1, 2, 0, 0, -1, 0, -2, -2, -1, -2, 2, -3, 0, 1, -2, 2], [-3, 0, 0, -3, -3, 1, 1, -3, 3, 0, 2, 7, 0, 1, -2, 1, 0, 1], [-1, -6, -1, 1, 1, 0, 1, -2, 3, 2, -2, -3, -2, -2, -5, -1, 3, 2], [4, 2, 3, 0, -3, -1, -2, 0, 2, -1, 3, 1, 0, 3, 2, 3, -2, -1], [-3, -8, -1, -7, -6, -4, -2, -4, 5, -3, 3, 7, -3, 5, 0, 2, 0, -3], [2, 4, 5, 2, 0, 2, 0, 2, -2, -2, 1, -1, 2, 2, 4, -1, -2, -3], [-1, 1, -3, 1, 4, 2, 0, 3, -1, -2, -2, -2, 7, 4, 1, -3, -3, 0], [3, 4, -1, 3, 2, 2, -3, 4, -1, -1, -1, -1, 2, -1, 0, 2, -1, 1], [-4, -1, 2, -2, -5, -2, 1, 0, 0, -1, 5, -1, -5, 1, 3, 3, 6, -1], [-2, -2, -3, -2, -2, -2, 1, -3, 6, -4, 2, 4, 1, 1, -1, 5, -2, -2], [7, 0, -1, 3, 5, 1, 2, 4, -5, 2, -2, -9, -1, -4, 2, -8, 2, 5], [1, 1, -1, -1, -4, -1, -3, -3, 0, -3, 0, 6, 0, 1, 1, 3, 2, -2], [5, 0, -2, -1, 0, -2, 5, 0, -1, -3, 2, 0, 0, 1, -1, 3, 1, 2], [0, 1, 2, 1, 6, 1, 1, 5, -6, -2, 0, -4, 0, 0, 1, 0, 0, 0], [0, 1, -4, 3, 1, -1, 2, 3, -2, -2, -4, -4, -1, 2, -1, -2, -2, 1], [0, -1, -1, 2, -2, -1, -5, 1, 0, 4, -3, -1, 1, -2, -1, -2, -2, 6], [-5, -9, -3, -4, -1, 0, -5, -5, 3, 2, -2, 1, -2, -1, -2, -2, 1, 1], [0, -7, -3, -3, -2, -1, 3, -5, 3, 0, 3, 2, -1, 0, 1, 5, 5, -3], [-7, 0, -4, -1, -4, -4, -7, -4, 3, -5, 2, 3, 0, -2, -1, 1, 0, 1], [5, -3, -1, -2, 1, 1, 6, -1, 5, -2, 3, 1, 1, -1, 2, 1, 0, -2], [-2, 4, 5, 1, 2, 7, 4, 3, -2, 4, -2, 2, 2, 0, 0, -1, -2, 0], [7, 1, 1, 1, 0, -2, -6, 1, -2, -1, 3, -5, -2, 1, 2, -2, 1, 2], [-2, -2, 4, 1, 5, 0, -2, 1, -5, 3, -6, -3, -3, -5, -3, 0, 3, 5], [0, 3, 3, -4, -4, 1, -1, 1, -2, -2, 5, 4, 3, 2, 3, 4, 2, -2], [3, 0, 4, 0, 1, 3, 0, 1, 0, -3, -3, -3, -2, -2, 3, 1, -3, -3], [-8, -3, -2, -3, -1, 2, 2, -3, 4, -3, -1, 5, -1, 2, 2, 4, 3, -4], [1, -3, 4, 0, 2, 4, 5, 3, -1, 1, 2, 2, 5, 2, 0, 0, -4, -5], [10, 7, 1, 1, -1, 0, 8, 1, -2, 2, 0, 3, 0, 1, 3, 0, 2, 2], [3, 1, -1, 0, 4, 3, -4, -3, 0, -2, 1, 5, 2, 1, 4, 1, -1, -4], [-7, 0, -2, 2, -2, 4, 7, -1, 1, 1, 1, 2, 1, -1, -1, -2, -3, -1], [5, 4, -4, 0, -3, 1, -3, -2, 3, -2, 4, 3, 2, 2, 5, -2, -3, 0], [-6, 6, -6, 4, -1, 1, 7, 5, 1, 1, -3, 0, 0, -2, 2, 0, -1, 3], [-1, -3, -5, -1, 3, 0, 6, -1, 2, 3, 3, -2, 0, 1, 0, -2, 4, 1], [1, 3, 0, 4, 5, 1, -5, 0, -5, 4, -4, -4, 2, -1, 2, -4, 2, 2], [-3, -1, -1, -1, -3, -1, 10, -3, 6, 0, 4, 1, 2, 0, -3, 0, -4, -1], [-4, 0, 3, 1, -1, 2, 1, 0, 2, 6, -1, 5, 4, 2, -2, -2, -2, -3], [1, -7, -2, 0, -2, -2, 9, 1, 1, -1, -1, -5, -1, -1, -2, -2, 0, 3], [-3, -7, -1, -3, -2, -1, -2, -1, 4, 0, 0, 7, 0, 1, 3, 0, -2, -1], [6, 5, 2, 6, 3, 2, 2, 4, -7, 6, -7, -11, 1, -1, 2, -6, -3, 5], [-5, -3, 3, -2, 1, -4, 3, -1, 3, -1, 3, 1, -1, 4, -1, -3, 5, 1], [1, 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-5, -7, 4], [-6, -2, 2, -6, -3, -5, -2, -2, 2, 1, 3, 8, 0, 5, -1, 2, 1, -1], [1, 2, 0, 3, -8, 0, 0, 2, -1, -5, 1, 2, -4, -5, 5, 5, -1, 0], [4, -2, 3, 2, -2, 2, 3, 1, -2, -2, -1, 7, 4, 0, -1, 1, -3, -2], [0, -4, 3, 0, -3, 1, -4, -2, 1, 0, 3, 8, 2, 3, 0, 2, -4, -4], [-7, -11, 2, -4, -7, -6, -1, -8, 9, 0, 4, 10, 2, -3, -10, 1, -7, 0], [5, -2, -4, 1, -2, 0, 0, -1, -1, -1, -5, -1, -3, -6, 6, 3, 3, 3], [6, 2, -2, 0, -8, 2, 2, 1, 2, -4, 6, 1, 1, 1, 4, 2, -2, -5], [-8, 1, 9, 0, 0, 2, -4, 2, -3, 2, 1, -7, 5, 4, -1, -1, 2, 1], [5, -4, 4, -2, -2, 5, 0, 1, 0, 3, 1, 1, 6, 2, -1, 5, 3, -1], [-1, -5, 0, -4, -2, -2, -3, -3, 1, -1, 1, 4, -2, 4, 6, 6, 3, -3], [-3, -5, -5, 0, -4, -6, 4, 2, 3, -4, 1, 5, 0, 7, -1, -2, -4, 0], [-15, -4, -7, 5, 3, 1, 5, -1, -1, 2, -7, -10, -4, -5, 0, -2, 4, 8], [1, -2, -2, -8, -7, -5, 0, 1, 2, -4, 3, 4, -8, 4, -4, 4, 1, -4], [-5, 1, 0, 6, 14, 5, 8, 1, -3, 10, -8, -13, 7, -7, 0, -6, 2, 7], [1, -6, 3, -1, 2, -6, -5, -6, 2, 2, -3, -7, -2, -5, -5, -1, 3, 2], [-8, 7, -1, 0, -6, -1, 9, -3, -1, -3, -2, 11, 0, 1, 4, 1, -2, 0], [4, 9, 8, -6, -5, 0, -2, 0, 0, -4, 8, 3, 2, 7, 5, 2, 1, -4], [-2, -4, 4, 4, 1, 0, 0, 6, -2, 0, 2, -5, 0, -1, -2, -4, -2, 4], [6, -1, 3, -1, 0, -4, -8, -1, -2, 4, 0, -3, -1, 1, -4, -6, -5, 0], [10, 1, -6, 1, 0, -2, 0, -3, 2, 3, -1, 4, 4, 0, 1, 0, -2, 2], [6, -8, -2, 1, -2, 0, 3, 3, 1, 6, -3, -7, -2, -2, 1, -3, 4, 8], [15, 0, 6, 3, 6, 5, -5, 12, -5, 2, -1, -13, 0, 2, 2, -11, 1, 2], [0, -4, -2, -8, -11, -4, -8, -8, 3, -3, 1, 3, -1, 0, -1, 1, 0, 0], [1, 0, -6, 5, 5, 0, 11, 4, -2, 0, -7, -13, -2, -2, 0, -8, -4, 8], [-8, -7, 1, -3, -7, -7, 2, -10, 8, 3, -4, 4, -10, -5, -5, 8, 3, -1], [5, 7, 2, 4, 2, 0, -3, -1, -2, 5, -4, -3, 4, 2, 0, -8, -7, 5], [-12, -3, -5, 8, 3, 0, 0, -2, 2, 6, -10, -9, -8, -11, 0, -5, 0, 8], [5, -10, -4, -5, 0, -8, -8, -2, 5, -9, 9, 6, -1, 5, -3, 1, -2, -6], [7, -6, 5, -7, -7, -3, 2, -1, 1, -4, 6, 10, 0, 3, 0, 2, 2, -2], [0, 8, -3, 5, -1, 8, 2, 3, -5, -3, -2, 8, 0, -2, 7, 5, 0, 0], [9, -2, 3, 0, 2, 8, -2, 4, -3, 3, -3, -5, 10, 1, -5, -7, -6, 2], [8, 3, 5, 0, 5, 6, -5, 4, 0, 2, -5, -6, 0, -5, 0, -3, 2, 2], [5, 6, 1, 4, 3, 8, 7, 5, -2, 1, 0, -4, 4, 0, 4, -1, -7, 1], [2, -13, -3, -4, -11, -6, -1, -5, 7, -4, 3, 7, -2, 4, 3, 7, -5, -3], [-11, -2, 4, -6, -3, -5, 4, -2, 3, 0, 3, 7, -2, 2, 2, 4, 4, -3], [4, -7, -7, -3, -3, -3, -4, -5, 3, 1, -4, -2, -4, 3, 5, -4, 0, 0], [5, 10, -4, 5, -4, 0, 2, 1, 1, -1, 0, -11, 3, 2, 2, -5, -1, 1], [-4, 3, -4, 1, -3, -1, -2, -1, 4, -1, 4, 8, 1, -2, 1, 7, 1, -3], [9, 8, -3, 3, 2, -4, -2, 5, -5, -5, -5, 8, 1, 2, 5, 0, -9, -2], [6, -6, 0, -2, 0, 0, 2, -6, -3, 3, -3, -9, 1, -3, 1, -3, 2, 6], [-7, 10, 7, 5, 2, 12, -3, 8, -4, 4, -5, -7, 4, -5, 5, -3, 1, 4], [4, -10, 0, -4, 9, -6, -6, -2, -4, 2, -8, -2, -4, 0, -4, 0, 4, 2], [-3, -2, 3, 0, 6, 6, 6, 6, -2, 0, 4, -8, 1, -2, 0, 0, -2, 0], [17, -8, 4, -2, 5, -3, -1, -1, -2, -3, 2, -3, -2, 6, -2, -1, 2, 0], [-4, -9, 5, 2, 6, 5, 5, 5, -5, 3, -2, -11, 2, -1, -4, -3, 2, 0], [-18, 1, 0, 8, 9, 3, 3, 4, -4, 6, -7, -9, 4, -5, -2, -7, -3, 9], [6, -4, -1, 1, -2, 2, 2, 5, 1, -1, 0, -12, 0, -4, -3, -3, 2, 4], [-8, -2, 0, -4, -5, 5, -2, -8, 6, 5, -1, 3, -4, 1, 3, 2, 3, -3], [-21, -8, -6, -5, -8, -4, -9, -11, 11, 3, -2, 11, -4, -2, -6, 5, 0, -1], [0, -1, 0, -5, 1, -3, -2, -5, 2, 1, 3, 0, 1, 1, 3, -3, -1, -5], [-2, -2, 3, -3, 0, 3, 7, 4, 2, 3, 6, 3, 2, -1, -5, -2, -1, 4], [-27, -1, 3, -4, -4, -2, -3, -4, 1, 0, 6, 5, -2, 1, -2, 10, 5, -1], [28, 10, 0, 10, 4, 6, 3, 13, -7, 3, -3, -8, 6, 0, 4, -9, -4, 5], [7, -2, -2, 3, 1, -3, 3, -4, -1, 2, 1, 0, 0, -7, 0, 3, -3, -2], [-11, 2, -9, 5, -3, -6, 1, -5, 0, -1, -2, -5, 0, 0, 2, 0, 0, 0], [2, -9, 0, 1, 6, 0, -2, 1, -1, 0, -6, -1, -4, -4, 1, 1, 3, 6], [12, 3, 5, 4, 4, 7, 1, 12, -5, 3, 0, -19, 3, 6, 1, -9, 6, 7], [-8, -5, -6, -1, -3, -9, 4, -4, 6, 1, -2, 13, -2, -1, -3, 5, -1, -2], [-7, -5, 7, -6, -6, -2, -6, -5, 5, 3, 3, 7, -7, -4, -8, 7, 6, -2], [9, -3, 4, -2, 3, 0, -8, 5, -5, -2, 2, -11, -2, 4, 6, -5, 11, 2], [-1, 4, 1, -1, -6, -4, -1, -5, -3, -3, 2, 6, 1, 0, -2, 0, -5, -4], [-9, -2, 1, 2, 4, 2, 14, 3, 3, 8, -4, -8, 0, -3, 2, -8, -1, 8], [-4, -4, -2, 2, 0, 2, 7, 2, 3, 7, -4, -4, 2, -2, -1, -1, -1, 6], [7, 17, 8, 4, 2, 10, 3, 13, -7, 0, 1, -3, 9, 1, 7, -7, -7, -3], [-12, -10, -3, -7, -5, -7, -4, -9, 5, -1, -6, 5, -6, -1, 0, 8, 5, 0], [-1, -2, -7, 1, 2, 0, -8, -7, 3, 7, -7, -4, 8, 0, -5, -8, -4, 4], [4, 8, -3, 5, 11, 6, 1, 2, 3, 1, 4, 1, 3, -1, 1, 1, 0, -1], [6, 12, -4, 3, 0, 6, 1, 3, -5, -3, 3, -1, 10, 6, 6, -2, -3, -4], [0, 2, 4, 0, -6, -6, 0, -5, 4, 0, 4, 9, 1, 1, 4, 2, 0, -1], [1, 7, 6, 2, 1, 3, -6, 9, -6, -5, -1, -12, -4, -5, -4, -7, 0, 5], [8, -11, 1, -5, 3, -5, -5, -1, -2, 1, 3, -7, 4, 2, -1, -2, 6, 2], [-6, 4, -1, -2, 3, 1, 8, -1, -1, 5, 0, -1, 0, 6, -7, -5, 0, -1], [-2, 9, 4, 6, 4, 2, 6, 8, -6, 0, -2, -3, 6, -2, 3, -4, -5, 4], [2, -5, 7, 0, -1, 0, -5, 2, -10, -4, 1, -8, -2, -1, 2, 1, 4, 0], [-6, 8, -2, 1, 7, 1, -4, -1, -6, 9, -11, -11, -5, -3, 0, -1, 6, 3], [5, -5, 3, 1, 3, 4, 4, 5, -6, 3, 1, -6, 0, 0, 2, -6, -3, 4], [14, 5, -3, 3, 3, -3, -9, 4, -5, -11, 1, -1, 2, 1, 13, 0, -1, 1], [-5, -9, 6, 0, 1, 5, -5, 3, -1, 0, 0, -4, 1, -1, 2, -4, 4, 1], [7, 5, 4, -2, -8, -7, -9, 1, -1, -5, 8, 2, -3, 5, -2, 3, 4, 0], [12, 11, 7, 1, 0, 8, -2, 5, -4, -4, 9, 1, 3, 5, 6, 2, -1, -4], [-10, -4, 8, -5, -11, -4, 5, -3, -2, -6, 2, -1, -5, 3, -2, 7, 6, -1], [5, 5, -8, 2, 12, -7, 0, 1, -5, -2, 0, -5, -1, 1, 0, -2, 4, 3], [16, 2, -2, -6, -18, -1, 4, -4, 6, -9, 13, 12, -2, 2, 9, 8, -1, -5], [12, 6, 1, 6, 6, 9, 0, 6, -8, 6, -12, -8, 3, 2, 7, -8, 0, 3], [-4, 10, 3, 1, 2, 6, -5, 4, 4, 8, 1, -10, 2, -3, -6, -6, -4, 1], [-7, -6, 2, -7, -6, -12, -8, -14, 5, 3, 2, 8, 1, 3, -3, 5, 4, -4], [-11, 1, 0, 2, 2, 0, 1, 8, -1, 3, 2, -5, 1, -1, 0, -8, -1, 6], [3, -18, 2, -6, -3, -13, -12, -11, 2, -4, -1, 6, -10, 2, -2, 6, 11, 0], [16, -2, 0, -4, 2, -6, 1, -3, 4, -2, 8, -1, -2, 1, -1, -4, -5, -3], [1, 2, 1, -2, -15, 0, 13, -1, 7, -3, 5, 6, -7, 2, 6, 9, 2, -1], [2, -4, 1, -2, -3, 3, -8, -2, -1, -10, 3, 4, 0, -1, 6, 6, 4, -4], [-3, 10, 1, 1, 1, -3, -1, -2, -3, -2, -5, 1, 4, 4, 2, 2, -1, 3], 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4, -3, 7, 12, 12, 27, 7, -6, 3, -2, 3, 8, 2, 0, -8, -6, 0], [-6, -3, -8, 9, 7, 5, 0, 9, -2, 6, -2, -16, 11, 10, -1, -8, 6, 7], [9, 3, 8, 2, 1, 15, 3, 5, 8, -2, 9, 3, 10, 7, 1, -4, -14, -12], [-4, 9, 7, -7, 2, 7, 12, 3, 4, 7, 1, 0, 7, 0, -2, 4, -4, -9], [15, -2, 4, -4, -1, 6, -4, 6, -1, -8, 17, 7, 2, 0, 6, 1, -11, -13], [-3, -4, -3, -4, -4, 0, 1, -5, -4, 3, -10, 0, -8, 0, 4, 1, 7, -1], [-10, 14, 4, -6, -14, 4, -16, -6, 3, -17, 13, 28, 5, 7, 11, 19, -4, -20], [22, -4, 0, 4, 4, -1, -4, -2, 2, 3, -5, -12, -4, -4, -1, -4, 3, 15], [15, 1, -5, 10, 9, -1, 7, 5, -2, 5, -8, 1, 8, -5, -5, -6, -13, 1], [10, 17, -7, 10, 15, 2, 3, 5, -10, -8, 3, 2, 6, 1, 4, -5, -14, -4], [6, 11, 3, 2, 7, 9, 10, -3, 0, 0, 6, 10, 8, 1, -1, 2, -8, -7], [-18, -16, 11, -4, 1, -1, -15, -10, 0, 11, -5, -14, 4, 1, -21, -5, -3, 8], [12, -1, -4, -5, -15, -1, 12, -8, 8, -3, 1, 9, -3, 7, 6, 7, -2, -2], [1, -8, 9, 9, 19, 9, 9, 14, -5, 8, 1, -5, 7, 2, -3, -16, -4, 3], [-3, 22, 1, 20, 0, 7, 7, 11, -11, 7, -13, -8, -2, -3, -4, -10, -9, 8], [-9, -6, 10, -7, -1, 1, -9, -4, -3, 2, 6, 3, -8, -8, -9, -1, -1, 2], [29, 12, -5, 2, 1, 2, -12, 11, -5, -16, 5, 10, 12, 12, 22, 4, -3, -6], [-1, -7, -9, 3, -11, -4, -2, 1, 5, 8, 2, -5, -3, -5, -7, -9, -9, 6], [-12, -15, -7, 4, 10, -1, 7, 1, 2, 14, -9, -12, -9, -6, -11, -12, 0, 14], [4, -18, 0, -10, -4, -7, -8, -14, 12, -1, 9, 25, -2, 4, -11, 8, -5, -9], [12, -6, 3, -11, -6, 1, -10, -8, 8, 2, 0, 4, 2, 0, -4, 7, -2, -8], [-7, 1, -2, 2, -4, 4, 1, -6, 3, 7, 2, 1, 5, 3, 0, 2, -5, 0], [19, -2, -1, -1, 1, -7, -7, 2, -5, 8, -5, -13, -5, -2, 5, -2, 6, 5], [-19, -3, 1, 1, 9, 1, -16, -1, -3, 15, -7, 1, 7, 5, -6, -11, 0, 5], [20, 8, 0, 6, 3, 1, 6, 16, -1, -4, 10, 0, 7, -2, -12, -7, -15, -3], [9, -8, -10, 3, -1, -13, 9, -4, -1, 4, -5, 1, 2, 3, -2, -5, -5, 2], [14, 5, 3, -5, 8, -2, -8, 5, -8, -8, 3, 1, 6, 7, 4, 4, -8, -12], [26, -5, -18, 5, 5, -8, 3, 6, 0, 0, -3, 2, 8, 1, -12, -13, -15, 8], [4, -9, 3, 5, -2, 2, -3, 9, -1, -3, 9, 3, -8, 1, 9, -3, 6, -3], [6, -4, 7, -4, 8, -4, 5, 16, -4, 0, 4, -5, -3, 6, -5, -5, 5, 5], [5, 18, -4, -1, -5, -3, -1, -1, 5, -4, 7, 17, -1, -1, -2, -1, -7, -5], [-11, -12, 0, 3, 6, -4, 1, 0, -5, 2, -2, -19, -7, -3, 5, -8, 0, 3], [-17, 11, 6, 3, -2, 10, 18, -2, 4, 9, 0, -4, 3, -13, 1, 7, 4, 7], [-21, -6, 6, 1, 8, 5, 11, 2, 9, 16, -1, 3, 0, -10, -16, -3, -1, 4], [-5, -4, -6, 4, -15, 0, 3, 0, 8, -3, 4, 3, -6, 2, 4, 2, 7, 0], [1, 27, 6, 8, 9, 13, -7, 8, -3, 10, -3, 3, 10, 2, 4, -8, -8, 2], [-9, 10, -14, 4, 3, -4, 1, 7, 0, -3, 0, 12, -6, -4, -2, -8, -7, 8], [-3, -8, 6, -4, 4, -8, 8, -1, 4, 0, 13, 6, 1, 13, -8, 1, 0, -7], [-18, -2, -11, 2, -2, 0, -5, 7, -2, -8, 5, 2, -1, -1, 6, 3, -2, -6], [-21, -3, -2, 2, 16, -3, -11, 5, -6, 7, 0, -3, -1, 3, -5, -17, 1, 7], [-10, -5, 1, 1, 6, 4, 6, 6, -4, 6, -8, -12, -2, 0, -1, -12, 7, 5], [14, -13, 9, -6, -6, -4, -1, 7, -3, 4, -3, -9, -2, 7, -11, -7, 7, 9], [38, 18, 6, 15, 13, 12, -2, 17, -9, 11, -16, -17, 7, -13, -3, -16, -13, 9], [-7, 1, 16, -2, -8, 5, -11, -7, 0, 3, 0, 5, -3, -7, 3, -3, -9, -3], [-1, 8, -1, -3, 1, -3, 7, -7, 1, -4, 9, 2, -11, -11, -1, 8, -3, -6], [-18, 8, -5, 2, 8, -4, -2, -7, -1, 11, -4, -7, 10, 2, -7, -2, -5, 3], [-13, 0, -1, -9, -2, 3, -10, -10, 8, 9, -9, 13, 2, -6, -9, 1, -12, -7], [-9, -19, -13, -3, 0, -13, 0, -11, 17, -1, -7, 5, -5, -12, -10, 1, -8, 7], [-18, -21, 0, -8, -8, -3, 9, -6, 14, 1, 9, 15, 6, 0, -7, 2, -3, -6], [8, 15, 3, 8, 5, 10, 13, 7, 2, 9, -10, -7, 9, -3, 0, -8, -1, 11], [1, -12, 6, 11, 9, 1, -21, 10, -17, 2, 4, -20, 2, 0, -2, -13, 5, 10], [4, -1, -3, 0, -8, 15, 3, 9, 6, 4, -7, -20, 0, -3, 0, -10, -7, 14], [7, -5, 14, -10, -21, -9, -19, -15, 9, -6, 7, 18, -3, -1, -5, 18, -4, -15], [39, 11, 2, 4, 1, 4, -7, 6, 0, 1, -2, 0, 2, -3, -5, -2, -6, 2], [-20, -5, 0, 1, -19, -3, 2, -1, 11, -1, 3, 6, -4, -5, -4, 7, -6, 7], [-21, -15, -3, -9, -4, -15, -18, -14, 0, -15, 0, 8, -10, 13, 6, 11, 10, -4], [6, 0, 0, -14, -22, -15, -16, -6, 7, -22, 20, 15, -11, 11, 4, 15, 9, -13], [21, 6, 7, -8, -4, -6, -20, -6, -11, -13, 7, -5, 4, 13, 15, 11, 12, -8], [7, -7, -3, 8, 12, -4, -1, 3, -9, 1, -4, -11, 5, -10, 2, -7, -2, 2], [2, -6, 17, -10, -2, -2, -6, 0, 7, -6, 11, 16, -1, 8, 2, 11, 5, -11], [14, 5, 0, 5, -2, 12, -2, 11, -4, -3, 1, -16, 1, -7, 8, -2, 3, -2], [0, -10, -7, -4, -7, -1, 7, 4, 4, -3, 3, 0, 2, 15, -1, 1, -9, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5819_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5819_2_a_m();".
// 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_5819_2_a_m(:prec:=18)
    chi := MakeCharacter_5819_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_5819_2_a_m();".
// 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_5819_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5819_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -212, -1386, -1546, 3880, 6165, -4004, -7793, 1876, 4680, -434, -1521, 48, 275, -2, -26, 0, 1]>,<5,R![-8192, -45056, -11520, 213952, 153934, -320296, -213051, 246328, 117334, -108296, -30050, 27464, 3258, -3880, -23, 280, -20, -8, 1]>],Snew);
    return Vf;
end function;