
// Make newform 1300.2.bs.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1300_bs();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_bs_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_bs_d();".
// 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_1300_2_bs_d();".
// 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 := [9, 0, 666, 0, 6399, 0, 20400, 0, 29001, 0, 21870, 0, 9517, 0, 2460, 0, 371, 0, 30, 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, 0], [99723, 1134948, -2787912, 0, -4645944, 0, -8552045, 0, -14767004, 0, -11423658, 0, -4296019, 0, -834504, 0, -80534, 0, -3047, 0], [-217278, 83635389, -175581882, 718736751, -420626628, 2078820744, -135046284, 2558952720, 285815274, 1610670111, 278388330, 567305219, 105446208, 115467750, 19986924, 13356712, 1878198, 805551, 69462, 19367], [661521, -7543635, 29953629, -88480923, 83549682, -223758558, 135753310, -180768738, 124660699, -29388941, 62402747, 30393363, 17453738, 17694998, 2731870, 3930438, 223669, 402397, 7449, 15705], [15821217, -61004484, 265442769, -453293982, 671275674, -1407545070, 542306214, -2016646506, 88166823, -1522503288, -91180089, -658485308, -53084994, -168552744, -11791314, -25148026, -1207191, -2012742, -47115, -66482], [-15821217, -61004484, -265442769, -453293982, -671275674, -1407545070, -542306214, -2016646506, -88166823, -1522503288, 91180089, -658485308, 53084994, -168552744, 11791314, -25148026, 1207191, -2012742, 47115, -66482], [17017893, 47385108, 231987825, 453293982, 615524346, 1407545070, 439681674, 2016646506, -89037225, 1522503288, -228263985, 658485308, -104637222, 168552744, -21805362, 25148026, -2173599, 2012742, -83679, 66482], [-45388899, -406515213, -1089813825, -2800812015, -4308088248, -6384094374, -6391057812, -7257146256, -4737866193, -4736472213, -1986210429, -1873735009, -492632442, -453746592, -71485608, -65333654, -5590593, -5106855, -181245, -165961], [4472424, 69453771, 99118266, 472806492, 353459730, 1146604077, 534368214, 1347639143, 415161358, 883607113, 181525278, 344915287, 46336394, 81710792, 6830496, 11480801, 537244, 876466, 17400, 27888], [12376215, -114443523, -34492167, -806417469, -297574542, -2213128230, -630401196, -2907434700, -646469163, -2089554123, -358798875, -878530535, -112032558, -220915596, -19533108, -32554318, -1764837, -2581149, -64035, -84623], [-4472424, 69453771, -99118266, 472806492, -353459730, 1146604077, -534368214, 1347639143, -415161358, 883607113, -181525278, 344915287, -46336394, 81710792, -6830496, 11480801, -537244, 876466, -17400, 27888], [-12376215, -114443523, 34492167, -806417469, 297574542, -2213128230, 630401196, -2907434700, 646469163, -2089554123, 358798875, -878530535, 112032558, -220915596, 19533108, -32554318, 1764837, -2581149, 64035, -84623], [-31334967, 406515213, -738650061, 2800812015, -3466834992, 6384094374, -6120965244, 7257146256, -5309496741, 4736472213, -2542987089, 1873735009, -703524858, 453746592, -111459456, 65333654, -9346989, 5106855, -320169, 165961], [42156486, -20563029, 490895010, 470305035, 1691072568, 3035691966, 2899824276, 5582082666, 2575434930, 4826080581, 1270292058, 2289442933, 361205580, 628204794, 58624560, 98942810, 5018946, 8264211, 174918, 282319], [-8376, -145143, 0, -1984833, 0, -6763380, 0, -9886332, 0, -7574149, 0, -3326801, 0, -864458, 0, -130732, 0, -10585, 0, -353], [24710427, 216209178, 281619531, 2736156870, 1227051090, 8306063952, 1885023858, 11346635628, 1413069225, 8408441922, 592149393, 3650325002, 145593894, 949107816, 20826990, 144498028, 1600971, 11805666, 50979, 397382], [22008474, 54298512, 298541511, 1133458272, 1060506387, 4213523391, 1606404909, 6231655746, 1247438982, 4730874033, 545967756, 2050478141, 139985934, 526330083, 20796687, 78814861, 1652277, 6333330, 54111, 210017], [7414695, 30631614, -30017193, 455334531, -263519850, 1551072951, -487184352, 2196992015, -423532263, 1624006860, -204225755, 690311702, -57243966, 174392772, -9210622, 25759345, -784089, 2045559, -27215, 67147], [-35447103, 364306149, 417021795, 4803350499, 2752673256, 17296540278, 4712105964, 25876910064, 3780968571, 19989440181, 1683203871, 8805945593, 439458966, 2291248896, 66687072, 346829194, 5420187, 28105911, 181527, 938081], [-15698898, -548621685, -16176762, -5742061569, -555775416, -19138877934, -1342717644, -28119598350, -1324902402, -21783539727, -683329482, -9697129103, -198678888, -2554928178, -32618304, -391463830, -2808162, -32065821, -98094, -1080005]];
    Rf_basisdens := [1, 1134948, 13619376, 4539792, 13619376, 13619376, 13619376, 13619376, 2269896, 6809688, 2269896, 6809688, 13619376, 13619376, 16752, 13619376, 6809688, 2269896, 13619376, 13619376];
    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_1300_bs();"
function MakeCharacter_1300_bs()
    N := 1300;
    order := 12;
    char_gens := [651, 677, 301];
    v := [12, 9, 7];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_bs_Hecke();"
function MakeCharacter_1300_bs_Hecke(Kf)
    N := 1300;
    order := 12;
    char_gens := [651, 677, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, -1, -1, 0, 1, 0, 1, 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, 0, -1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 2, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1], [1, 1, 0, 3, 1, 0, -2, -1, 0, 2, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0], [0, 2, 0, 2, 1, 1, -1, 1, 0, 0, -1, -1, 1, 1, -1, 0, 0, 0, 1, 0], [2, 0, -2, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, 0], [-2, 0, 0, -1, 1, 1, 1, 0, 0, -1, 0, 1, 0, 0, -2, 1, 0, 0, -1, 0], [0, 0, 1, -2, -2, 1, 4, 2, 0, -3, -2, -2, 2, 2, 1, 3, -1, -1, -1, 1], [1, -1, 0, 0, -1, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0], [1, 0, -2, 0, -2, -1, -1, 1, 0, 0, 3, 2, 1, 0, 0, -1, 2, 0, -1, 0], [2, 1, -2, 2, 3, 1, -1, 0, 0, 0, 1, 3, 1, 0, 2, -3, 2, 1, -1, -1], [-2, 0, 1, 0, 2, 0, -1, -1, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 1, 1], [2, -1, 0, 1, 0, 1, -1, -1, 1, 2, 1, 2, 0, -1, 4, -1, 0, 2, 1, 0], [0, 0, 3, 1, -1, 4, 0, -1, -1, 1, -1, 0, 1, -1, 6, -1, 0, 0, 0, 0], [1, -2, -3, -4, -4, -2, 3, 1, 2, -3, 0, -2, -3, 2, 0, 2, -1, -2, -3, 0], [-3, 4, 0, 6, 0, 1, -3, -2, 0, 4, 0, 4, 3, -1, -4, -3, 2, 1, 1, -1], [3, -1, 1, 2, 0, 4, 0, -2, 0, 1, 0, -1, 0, 0, 3, 0, -1, -2, 0, 2], [2, -1, 0, 1, -2, -1, -2, -2, 0, 0, 1, 1, 0, -1, 1, 0, -2, 0, 0, 1], [1, 2, 2, 3, 4, 1, 1, 2, -1, 2, 1, -1, 3, 0, 1, 0, -1, 0, 2, 1], [6, -2, -2, 2, -3, 0, -4, -2, 1, 4, 1, 2, -2, -2, 5, -1, -2, 4, 2, 3], [4, -1, -3, 2, -7, -1, 0, 0, 1, -1, 1, -2, -1, 0, -2, 3, -1, 0, 0, 4], [-4, -1, 1, -4, -3, -4, 4, 1, -2, -7, -2, -2, 1, 2, -1, 2, 2, -2, -2, 1], [-6, 0, 0, -2, 0, 0, 2, 2, 0, -3, 0, -3, 0, 1, -5, 2, -1, -1, 0, 0], [-1, 2, 2, 6, 6, 0, -9, -3, 0, 7, 1, 3, 1, -4, 0, -6, 1, 2, 5, -2], [0, -1, -3, -2, -1, -6, 1, -1, 0, -2, 1, 2, -2, -1, 4, -3, 1, 1, -2, -1], [4, -2, -2, -4, -6, -5, 1, 0, 1, -3, -1, 1, -1, 1, 2, 3, -2, 1, -2, 0], [-2, 3, -1, 3, 3, 2, 1, 1, 0, 1, 1, 2, 1, -1, -1, -3, 2, 1, 0, -2], [1, -1, 5, 4, -2, 4, 2, -2, 0, 4, -1, -1, 3, 1, 6, -1, -1, 0, 3, 2], [-7, 2, -2, 0, 3, -4, 1, 3, 0, -1, 2, 1, 2, 0, 1, -1, 2, 2, 0, -2], [-2, 4, -1, 0, 5, -4, 1, 2, -1, 2, 1, 1, 3, -2, -2, -3, 3, 0, -2, -2], [3, 0, 0, 4, 0, 7, -2, -2, 3, 2, 0, 1, -1, 0, 3, -1, -2, 0, 2, 2], [1, -3, 1, -2, -3, 8, 1, -2, 1, 0, 0, 0, -3, 2, 6, 5, -7, 2, 2, 2], [-3, 1, 2, -4, 1, 1, 5, 3, -2, -4, 0, -4, 2, 2, 1, 3, 1, -2, -2, 0], [-5, -2, 5, -8, -4, 7, 13, 7, 0, -7, -5, -9, 2, 5, -2, 13, -3, -5, 0, 3], [-6, 0, 4, -4, -2, 3, 8, 4, -1, -4, -4, -4, 4, 4, -1, 4, 0, -3, 0, 0], [2, 0, 2, -3, 0, 2, 6, 4, 0, -7, 0, -1, 3, 2, 6, 5, 3, -2, -1, 1], [8, -2, 3, 0, -3, 5, 6, 2, 0, -2, -2, -5, 2, 6, 5, 5, -2, -3, 1, 3], [-3, 3, -3, -4, -1, -3, 3, 3, 0, -1, 2, 0, 2, 2, -3, 1, 0, 0, -2, -2], [9, 0, -7, 6, -1, 5, -4, 3, 6, 1, 3, -4, -3, 0, -3, 1, -1, 1, 3, 5], [6, -3, 1, 0, 1, 4, 0, -2, 1, 0, 1, -3, -1, 0, 7, 0, -3, 0, 0, 3], [0, -7, 1, -10, -13, -4, 10, 3, -1, -7, -2, -6, -2, 6, -2, 9, -2, -6, -5, 5], [-3, -2, 5, -2, -4, -4, 5, 0, -6, -3, -3, -5, 3, 0, 1, 2, 3, -4, -1, 2], [0, 1, -2, 1, -1, -2, 0, 1, 0, -2, 0, -1, 0, 0, -5, 1, 2, -3, -2, -1], [-1, -1, 2, 1, 3, 5, 5, 1, 0, -1, 0, 3, 2, 2, 0, 2, 4, -2, -1, 1], [-2, 3, -3, 2, 2, -2, -2, 3, -2, 0, -1, 2, 1, 0, -3, -2, 0, -2, -2, -2], [5, 0, 5, 3, 1, 7, 3, 0, 0, 1, -4, -7, 0, 4, 4, 4, -5, -2, 6, 2], [-4, 2, -1, -1, 2, 1, 2, 4, 1, -5, 2, 2, 1, 0, -1, 2, 5, 0, 1, -1], [3, 1, 8, 2, -5, -8, -3, -4, -4, 2, -4, 0, 0, -4, 2, -4, 2, 0, 2, -2], [-1, -1, 1, -1, -2, -5, -1, -2, -1, -5, -2, 1, 2, 2, 2, -4, 1, -1, -1, -1], [2, -4, 4, 1, -10, 0, 0, -5, 1, 3, -3, -3, 0, 3, 0, 3, -4, -4, 1, 4], [3, -5, 2, 3, -3, 0, 0, 1, 1, -1, 1, -4, -2, 0, 0, 2, -1, 1, 2, 4], [0, 4, 2, 8, 4, 4, -3, 1, -1, 4, 0, -2, 4, -3, -4, -5, 6, -3, 1, 1], [1, -4, 3, -7, -4, 1, 10, 4, -5, -12, -4, -8, 1, 5, 2, 8, 0, -4, -4, 4], [0, -4, -1, -6, -5, -5, 4, 1, -4, -3, 0, -3, 0, 4, 2, 3, -2, 0, -2, 4], [-2, 0, -2, -2, 0, -10, -3, -4, -3, -4, 0, 6, 0, -1, -3, -6, 5, -1, -5, -5], [2, -4, 1, -5, -4, -8, -2, 0, -1, -2, -4, 2, -1, -1, 4, 0, -1, 4, 2, -2], [15, -5, 6, 5, 3, -1, -4, -7, -1, 10, 0, 1, 4, -1, 11, -4, 0, 1, 4, 3], [1, -7, -3, -16, -2, 1, 10, 4, -2, -9, 0, -4, -5, 0, 2, 8, -2, 2, -3, 2], [-1, -1, 3, 3, 2, 3, -5, -5, -1, 3, -4, 0, 2, -1, -3, -2, -2, 4, 2, 1], [-2, -2, -3, -11, -4, -5, 4, 4, -2, -11, -4, -8, -6, 2, -3, 7, -2, -2, -1, 1], [-12, 5, 0, 0, 19, 5, -7, 0, 1, 4, 1, 6, 0, -2, -11, -9, 2, 4, 2, -9], [-10, 1, 7, -6, 9, 2, 5, 1, -6, -5, -6, -5, 2, 3, -7, 0, 0, -6, -3, -3], [4, -3, -3, -2, -1, 6, -3, 0, 5, 4, 0, -1, -5, 0, 9, 0, -10, 5, 3, -2], [-5, -3, -6, -8, -1, -10, -2, 1, 1, 0, -1, -1, -1, -1, -2, 0, 1, 0, -3, -2], [-1, -6, -3, -9, -8, -11, 4, -1, 1, -4, 3, 2, -5, -2, 7, -1, -1, -1, -5, -2], [2, 4, -1, 1, 1, 4, 0, 1, 1, 3, 1, 0, -1, 1, 2, 1, -6, 0, 2, 2], [-7, 1, 3, -1, -4, -7, 5, 1, -4, -4, -3, 1, 5, 1, -7, 0, 4, -4, -5, 1], [-2, -1, -3, -1, 4, -4, -4, 1, 3, -1, 3, 0, -4, -3, 3, 0, 0, 3, 1, 1], [9, 0, -3, -1, 5, -1, 4, 1, 0, -2, 2, 3, 2, 2, 3, 0, 4, -1, -6, 1], [12, 2, -2, 3, -12, -1, 3, 2, 0, -2, 2, -1, 5, 2, 5, 2, 5, -2, -2, 5], [-9, -3, -6, -15, 1, -11, 0, 2, -1, -9, -2, 2, -8, -1, -6, 0, 0, 2, -5, -9], [-4, -4, -4, -8, -7, -2, 4, -1, 1, -4, 0, -1, -6, 1, -4, 2, 0, -1, -6, -1], [6, -4, 4, 1, 1, 0, 3, 3, 1, 2, 0, -2, 1, 1, 7, 0, -7, 0, 5, 2], [0, 0, -5, 2, 0, 17, 0, 6, 5, 1, 0, 0, 2, 1, 4, -1, -1, 1, 2, 2], [3, 0, -6, -5, 4, -13, -3, -2, -3, -2, 3, 5, -1, -2, -3, -4, 5, 0, -4, -3], [-9, 2, 1, -3, 2, -9, 1, -2, -3, -5, 0, 2, 0, 1, 3, -3, 5, 1, -4, -4], [3, 4, -10, -5, 8, -6, 1, 6, 3, -5, 3, -5, -7, 3, -10, 1, 3, -3, -2, -4], [1, 0, -2, -3, 1, 5, 1, 3, 2, -2, 0, -8, -2, -2, 5, 3, -1, 2, 2, 2], [-1, -2, -1, -4, -1, -1, 4, 1, 0, -5, -1, 0, -1, 2, -1, 3, 1, -1, -3, 1], [-11, -1, 9, 2, 8, -3, 2, -2, -6, -1, -2, -2, 3, -2, -8, -2, 6, -4, -1, 0], [-4, -2, 1, 2, -7, -8, -6, -2, 0, 4, 3, 0, -2, 0, -6, 0, -3, 3, 3, 2], [15, -3, -7, 6, 1, 7, -11, -3, 8, 9, 6, 4, -8, -2, 7, -3, -6, 8, 8, 5], [-6, 3, 7, -1, 2, -1, 3, -5, -4, -5, -2, -1, 2, 0, -15, 2, 5, -8, -6, -4], [-9, 4, 12, 0, 1, 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5, -2, 2, 12, -14, -13, -2, -8, 5, 2, 0, -3, -8, -8, -16, 14, -2, 1, -11], [-22, 17, -3, 9, -1, -1, -6, 7, -3, -2, 1, 4, 2, -4, -11, 0, 6, 3, 2, -4], [-13, -1, 2, 2, 7, 12, -3, 0, -2, 2, -1, 3, 3, -2, -6, -1, 5, 1, 3, 1], [4, 5, -2, 5, -2, -5, -10, -4, -1, 3, -4, -2, -1, -4, -4, -4, 3, 3, 6, 1], [-12, 6, 2, -1, 4, -7, -3, -7, -4, 5, -8, 6, 2, -4, -7, -5, 0, -4, -2, -7], [-9, -1, -7, -8, 5, -4, 9, 11, 1, -9, 1, -3, -4, 1, -3, -2, 6, -2, -1, -9], [-18, -4, 5, -6, 11, 5, 0, -3, 0, -2, -3, 6, -2, -3, 2, -3, 1, 3, -4, -9], [-23, 10, -10, 4, 2, 5, -5, 3, 2, -1, 11, 9, -5, -9, -21, -4, 5, -2, -5, -5], [-29, 2, 18, -12, 2, 0, 16, 9, -9, -23, -18, -18, 5, 7, -9, 12, 0, -7, 6, -6], [14, 0, -9, 8, 2, 23, -1, 8, 12, 7, 6, -6, -6, 0, 4, 11, -7, -1, 6, 13], [-2, -4, 17, 8, -15, -6, -4, -15, -10, 9, -10, 7, 10, -3, 4, -3, -2, 6, 3, 2], [3, -6, 3, -8, -8, 9, 18, 7, -7, -10, -2, -18, 1, 7, 7, 18, -7, -2, 3, 6], [19, -1, 0, 9, -4, 0, -10, -2, 3, 6, -7, -5, 4, 3, 4, 1, -4, 7, 11, 7], [12, 0, 1, 1, 17, 11, 4, -4, -3, 6, 2, 6, 4, -2, 14, -10, 3, 5, 3, -3], [-11, -9, 8, -13, -20, 1, 10, 2, -4, -17, -12, -8, 2, 8, 7, 12, -13, 4, 2, 4], [-6, 3, 13, 5, 5, 1, 2, -3, -5, 2, -4, 11, 8, -5, -1, 1, 10, 4, 3, 3], [2, 6, 7, 2, 6, -7, -3, -6, -8, -3, 0, 5, 0, -4, -21, -5, 7, -4, -7, -7], [-14, 8, 5, -3, 14, 1, 7, 9, -1, -13, -4, 1, 9, 1, 0, -1, 11, -4, -6, -8], [-6, -1, -2, -4, 13, -20, -2, -5, -2, -5, 2, 6, -7, -3, -4, -2, 3, 0, -5, -4], [-24, 1, 9, -17, -13, -6, 31, 8, -8, -23, -15, -16, 9, 8, -6, 16, 4, -15, -6, -1], [1, 4, -3, 0, 4, 7, 5, 11, 3, 2, 0, -4, 5, -1, -8, -1, 4, -1, 1, 1], [4, -4, 7, 0, 7, 9, 3, 3, 0, -1, 5, -3, -1, 5, 9, 4, -3, 0, 1, 12], [-5, -8, 3, -12, -18, -6, 7, 0, -2, -5, -8, -8, -1, 8, -6, 8, -4, -6, -5, 4], [3, 11, 7, 25, 12, 19, -17, -7, 0, 13, 1, 8, 11, -1, -13, -15, 4, 1, 8, -4], [-18, -7, 11, -14, -37, -10, 26, 10, -4, -26, -8, -20, 5, 7, -4, 19, -6, -7, -5, 5], [3, -4, -4, -4, -8, -12, 3, 1, 6, 4, 3, 4, -3, 0, -19, 0, -4, 2, 1, 0], [4, 6, -4, 2, -1, -1, -5, 1, -2, -3, 2, 14, 7, -5, -4, -10, 9, 0, -3, -4], [1, -4, 10, -2, 5, 23, 4, 0, -2, 8, -1, -3, 1, 0, 7, 7, -8, 7, 9, 11], [4, -5, 8, 3, -6, -7, -9, -7, -5, 5, -7, -7, -2, -7, 6, -7, 0, 2, 5, -4], [-17, -8, -6, -22, -21, -6, 20, 10, 7, -7, 0, -9, -10, 0, -7, 18, -4, -7, -6, 4], [-6, 1, 8, -3, 1, 13, 9, 2, -9, 5, 0, -6, 7, 0, 6, 2, -1, 0, 5, 5], [-15, 10, 5, 3, -3, -2, 6, 6, -2, -7, -6, 7, 12, 2, -19, -7, 12, -6, -9, -10], [18, -11, -8, -14, -14, -6, -7, -10, 2, -6, -2, 2, -16, -2, -1, 5, -6, 0, 0, -1], [7, 3, -18, 4, 10, -1, -5, 4, 4, -1, 8, 14, -1, -4, 6, -14, 9, 8, -7, -3], [-14, 14, 14, 16, 14, -2, -5, 2, -15, 0, 0, 2, 12, -3, -11, -10, 11, -3, -3, -3], [3, 0, 4, -4, -10, -17, 8, 5, -1, -8, 1, 2, 2, -2, 23, 8, -4, 1, 2, 2], [2, 7, -3, 17, 1, -7, -18, -5, 4, 15, 3, 6, -1, -3, -7, -9, 2, -1, 0, -2], [-4, 7, -11, 1, -8, 5, -7, -2, 2, -2, 0, -3, -3, 2, -4, 1, 4, 0, -4, -1], [-9, 0, 2, 5, -8, 0, -12, -6, 0, 16, -2, -7, -2, -2, -6, -5, -8, 1, 8, -1], [-32, 12, 9, -5, 8, -10, 12, 18, -9, -20, -2, 2, 21, 9, -12, -2, 17, -2, -10, -12], [5, 6, -22, 9, 28, 24, -6, 7, 14, 5, 14, 7, -6, -1, -10, -7, 2, 2, 1, -8], [2, -6, 3, 6, 5, 15, -4, -1, 10, 8, -1, 6, -2, -1, 10, -1, -9, 11, 8, 5], [-24, 6, 2, 18, 15, -11, -23, -17, -8, 17, 8, 12, -1, -13, -4, -20, 11, 0, 5, -4], [0, -2, -16, -19, -2, -8, 13, 10, 7, -17, 4, -3, -14, 3, -10, 11, 0, -7, -14, -3], [10, -9, 7, -1, -14, -2, 4, -7, 4, 9, 2, -1, -2, 0, 11, 9, -9, 4, 7, 10], [6, -1, 10, 14, -7, 1, 0, -1, 4, 18, 8, 7, 8, 3, 12, -10, 2, -3, 5, -5], [4, 2, 13, 7, -1, 4, -6, -9, -2, 7, -1, -1, 1, 0, 9, 3, -7, 6, 7, 8], [30, -4, 10, 3, -2, 13, 9, 4, 0, -2, -3, -14, -1, 8, 20, 10, -10, -4, 10, 4], [6, 4, 6, 12, 8, 21, 24, 12, 0, 0, 6, -10, 16, 16, 8, 9, -2, -8, -2, 8], [19, 3, -14, -5, 2, -39, -14, 2, 0, 3, 4, 4, -6, -4, 11, -8, 11, 0, -6, -8], [-12, 11, 1, 10, 8, 26, -3, -1, 3, 1, -6, 0, 4, 3, -11, 6, -9, 6, 6, 7], [-5, -2, 10, 5, -2, 23, 8, 4, 3, 12, 0, -11, 0, 3, 24, 16, -20, 3, 15, 15], [-14, 10, 0, 2, 26, -22, 0, 0, -2, 2, 2, 8, 2, -8, -6, -8, 6, 0, -4, -6], [8, -8, -18, -17, -8, -2, 6, 9, 12, -8, 0, 7, -5, 5, 0, 4, -8, 5, -3, -3], [-23, 9, -7, -4, -3, 0, -9, 0, -5, -4, -1, 5, -4, -5, -4, -5, 11, 1, -3, -9], [7, 7, -14, 4, -1, -5, -17, -8, 9, 15, 3, 16, -6, -6, -5, -11, -4, 9, 6, -1], [26, -8, -12, -3, -25, 1, 1, 0, 10, -5, -2, -3, 2, 10, 6, 17, -2, 2, -3, 12], [-36, 2, -9, -17, 13, -15, 6, 5, 0, -18, 1, 8, -7, -12, -17, 2, 2, 6, -13, -6], [15, -2, -12, 6, 0, 2, -12, -12, 5, 19, 8, 9, -7, -5, 10, -9, -6, 8, 9, 2], [18, -9, 14, -2, -11, 1, 11, -8, -9, -2, -9, -12, 4, 6, 27, 5, -10, -12, -6, 9], [18, -5, 1, 9, -8, 4, 1, -7, 8, 9, 4, -3, -4, 0, 0, 1, -9, -8, 2, 12], [7, 8, -9, -5, 4, 4, 6, 8, 7, -5, 8, -13, -8, 8, -2, 8, -3, -1, 3, 3], [-5, 1, 9, 1, 12, 6, 13, 2, -1, -4, -1, 0, 3, 0, 10, -8, -4, 1, 3, -4], [1, 15, -5, 15, 19, 12, -7, -1, 1, 10, 2, 2, 8, 3, -8, -5, 0, -3, 3, -3], [-19, 2, 6, -16, -4, -18, 15, 3, -2, -2, -1, 8, 1, 0, -7, 10, 2, -8, -9, -10], [-20, -8, 24, -1, 22, 1, 1, -5, -3, 4, -5, -2, -8, -5, -2, -5, -4, 2, 7, -6], [-15, 10, 6, 12, -8, -4, -7, -5, -3, 6, -6, -3, 8, -3, -11, -7, -5, 3, 5, -5], [-32, 4, -1, -12, -9, -10, 15, 14, -9, -13, -2, -2, 5, 2, -7, 2, 6, 7, -3, -6], [9, 5, -4, 13, 9, -17, -18, -8, -8, 13, 7, -2, 3, -8, -11, -18, 4, -7, -2, -5], [-24, -3, -1, -31, 1, 1, 19, 18, -4, -17, -6, -5, 6, 6, -13, 4, 7, -2, -7, -7], [4, -9, -4, 2, -24, -5, -4, -7, 8, 6, 0, -9, -17, -8, -12, 9, -12, 0, 7, 9], [7, -6, -6, -4, -6, 22, 1, 7, 3, -8, 0, -7, 0, 11, 9, 7, -3, 11, 3, 3], [-24, 7, -1, 2, 7, 6, 6, 10, 3, 3, 0, -4, 6, -2, 15, -4, 7, -2, 1, 1], [2, -5, -14, 8, 18, 10, -17, -9, 8, 14, 1, -1, -14, -7, 10, -3, -11, 8, 14, 1], [-10, -5, 7, 3, 5, 30, 0, -3, 3, 0, -3, 9, 5, -3, 26, -3, 6, 6, -2, 0], [5, 1, 15, 13, 37, 20, -4, -7, -2, 10, -4, 9, 8, -6, 19, -11, 4, 6, 6, -6], [13, -1, -17, 5, 5, 15, -1, 6, 7, -5, 9, 10, 3, 2, 15, -4, 9, 7, -3, 0], [13, 2, -17, 1, -9, -2, 11, 13, 15, -9, 15, -2, -3, 3, 14, 8, 7, -6, -3, -2], [14, 2, -4, 6, 7, -11, -13, -11, -2, 9, 2, 6, -1, -5, -2, -12, 9, 0, 1, -2], [14, -13, -11, -6, -11, 6, 2, 1, 14, 1, 7, -3, -7, 0, 17, 8, -1, 12, 8, 4], [-10, 2, -3, -4, -8, -5, -3, -1, 0, -7, 0, 3, 4, -4, -10, 4, 2, 4, -5, 5], [-13, 1, -5, -2, -3, -10, 2, 7, 5, -11, 1, -6, 1, -4, -10, 1, 10, 5, -1, -1], [0, -8, 4, -14, -22, -3, 21, 9, 0, -18, -2, -3, 0, 2, 6, 11, -2, -1, -14, 1], [22, 12, 3, 33, 0, 9, -12, -10, 5, 21, 10, 24, 17, -12, 27, -18, 9, 12, 3, -3], [6, 7, -1, 1, -13, -13, 7, 2, 0, -3, 2, 3, 10, 2, -7, 2, 5, -2, -9, 3], [-7, 8, 1, -7, -4, -3, 6, 1, -5, -4, -7, 0, 7, -2, -3, -1, 5, -5, -7, -1], [-5, -5, -3, -9, -5, -14, -1, -9, 7, -8, 0, 6, -14, 3, 10, 7, -8, 3, -5, -5], [27, -13, -3, 13, -2, 5, -17, -11, 4, 19, 7, -4, -17, -4, 12, 4, -18, 7, 9, 13], [11, 2, 18, 36, 11, 19, -23, -8, -5, 30, -6, -10, 15, -5, 14, -20, -3, 6, 28, 10], [-26, 1, 0, -19, -6, -5, 21, 17, 1, -7, -5, -7, 4, 5, -6, 9, 4, -6, -5, -4], [-13, -9, -1, -12, -6, -16, 10, 4, -2, -20, -1, -15, -7, 2, 8, 15, -2, -1, 1, 9], [8, 5, 2, 4, 7, 3, -17, -8, -3, 8, -3, 0, -2, -2, 1, -3, -8, 4, 2, 7], [0, 6, 6, 6, -2, 15, 3, -3, -9, -11, -9, -8, 9, 3, -11, 17, 3, -6, -3, 11], [3, 1, 14, 1, -7, -5, 11, 1, -4, 1, 1, 3, 6, 1, 5, 1, -2, -5, -5, 2], [-18, 0, -12, -24, -17, 2, 11, 15, 0, -7, 0, -4, 3, -1, -3, 9, -7, 0, -5, -3], [21, -11, -6, 0, -19, -3, -19, -18, -1, -1, -3, -2, -6, 2, -11, 7, -12, 1, -6, 9], [5, -9, -19, -5, 17, 1, -8, 3, 8, 3, 16, 2, -22, -9, -2, -7, -3, 9, 2, -2], [0, 3, -20, -5, 13, 13, -10, 9, 16, 7, 8, 18, -8, 0, -1, -2, -7, 19, 4, -11], [-9, 2, -1, 8, 6, -6, -10, -6, 4, 5, -3, 13, 0, -7, -6, -10, 5, 4, 0, -16], [-14, 3, -1, 9, 31, -8, -18, -12, 0, 13, 4, 12, -12, -18, -9, -17, 3, 9, 2, -9], [25, 0, -6, 13, -6, 6, -26, -14, 12, 9, -1, 13, -13, -13, 6, -13, 4, 12, 13, -2], [20, -13, 8, 6, -2, 8, -8, -14, 0, 14, -6, -6, 5, 6, 8, -9, -5, 0, 5, 5], [9, 2, 18, 31, 1, 37, -1, -4, -2, 18, 2, -15, 9, 10, 18, 5, -13, 0, 26, 18], [6, 0, 7, 6, -3, 10, -5, 4, 1, 12, -1, -7, 3, -6, 0, 1, -7, 0, 6, 0], [-16, 19, 12, 32, 19, 22, -21, -13, -9, 21, -3, 0, 12, -9, -11, -18, -3, 3, 11, 3], [-14, 8, -19, 0, 17, -13, -15, -10, 0, 6, 8, 13, -16, -14, -18, -13, 6, 7, -4, -7], [-8, -1, 6, 3, -15, 9, 3, -4, 0, -3, -10, -5, -2, 8, -5, 6, -1, -4, 8, 4], [3, -7, 12, 7, 9, 10, 3, -3, -6, -4, -1, 5, 3, 5, 10, 0, 4, -6, -3, 6], [18, 2, 0, 3, -6, -16, -15, -6, -6, 9, -6, 4, -3, -4, 34, -11, -5, 8, 4, -2], [-14, 9, 0, -2, 15, -4, -1, 3, -6, -18, -3, -7, 3, 0, -7, -14, 18, -5, -8, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1300_bs_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_bs_d();".
// 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_1300_2_bs_d(:prec:=20)
    chi := MakeCharacter_1300_bs();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1300_2_bs_d();".
// 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_1300_2_bs_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1300_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![21904, 210160, 701300, 937744, 690777, 374306, 46432, -76600, -9730, -43502, 14776, -5410, 2867, 1366, -500, 106, -26, -28, 8, -2, 1]>],Snew);
    return Vf;
end function;