
// Make newform 1100.3.e.b in Magma, downloaded from the LMFDB on 04 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1100_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1100_e_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_1100_3_e_b();".
// 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_1100_3_e_b();".
// 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 := [92416, 0, 633600, 0, 1098976, 0, 712960, 0, 215801, 0, 32605, 0, 2456, 0, 85, 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], [87809107120, 0, 223762937264, 0, 245214629400, 0, 107624995343, 0, 20113525535, 0, 1709306672, 0, 63432595, 0, 773771, 0], [-11318737840, 0, 17074353904, 0, 87679994552, 0, 50103813459, 0, 10228121691, 0, 903498960, 0, 34184767, 0, 420887, 0], [146115456, 0, 741283184, 0, 948559480, 0, 417992347, 0, 77544527, 0, 6556704, 0, 242607, 0, 2955, 0], [788722334496, 0, 10327727592240, 0, 15782973299102, 0, 7394418669415, 0, 1407174092933, 0, 120417116030, 0, 4482938619, 0, 54764265, 0], [1644909664896, 0, 9747912720944, 0, 11289579585016, 0, 4646063351261, 0, 827610903545, 0, 68283056336, 0, 2488784233, 0, 30071229, 0], [217449787648, 0, 11736694331280, 0, 27194153811216, 0, 13956691689695, 0, 2733874494699, 0, 236701101560, 0, 8861050787, 0, 108529015, 0], [-3688089010224, 0, -17066354067520, 0, -22836618279013, 0, -10264379658455, 0, -1917452946702, 0, -162514662625, 0, -6017693411, 0, -73311950, 0], [0, 318422512960, 0, 912957479376, 0, 665827087724, 0, 212849379951, 0, 33247815327, 0, 2558264900, 0, 89743059, 0, 1063643], [0, -234099532992, 0, 2019119346832, 0, 3734792446476, 0, 1806713378339, 0, 346608435459, 0, 29739874116, 0, 1108359607, 0, 13546063], [0, -70628062332352, 0, -179100262348400, 0, -179298328842724, 0, -72756023280605, 0, -13043236262221, 0, -1084368718700, 0, -39753406153, 0, -481943745], [0, 706369897354176, 0, 2139005148509936, 0, 1567031304003332, 0, 475224202404597, 0, 69836137709413, 0, 5114270628428, 0, 173225094529, 0, 2012005689], [0, -44068563980224, 0, -120875189308560, 0, -103672714920340, 0, -37117786495311, 0, -6189843210903, 0, -494121743628, 0, -17700040563, 0, -212050611], [7071920116480, -16795308146272, 69578652083952, 30871740246656, 95283820863800, 160131002106446, 42053341239379, 92006955532482, 7779204044295, 18827971472604, 656451986336, 1665824863158, 24261360775, 63102564522, 295333283, 777460604], [14143840232960, 461762444393728, 139157304167904, 994246798944496, 190567641727600, 803867609885136, 84106682478758, 294450462971877, 15558408088590, 50396816656649, 1312903972672, 4096094703048, 48522721550, 148403486737, 590666566, 1788796429], [0, -31300004371616, 0, -217552478598192, 0, -281091958187582, 0, -123137012459979, 0, -22737316869673, 0, -1916537290046, 0, -70769220879, 0, -861011933]];
    Rf_basisdens := [1, 5355322600, 1071064520, 5824960, 85685161600, 34274064640, 85685161600, 42842580800, 15597777920, 119487564800, 3256036140800, 13024144563200, 651207228160, 1628018070400, 3256036140800, 814009035200];
    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_1100_e();"
function MakeCharacter_1100_e()
    N := 1100;
    order := 2;
    char_gens := [551, 177, 101];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1100_e_Hecke();"
function MakeCharacter_1100_e_Hecke(Kf)
    N := 1100;
    order := 2;
    char_gens := [551, 177, 101];
    char_values := [[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, 0, 0], [-1, 0, 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]];
    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 := 3;
    raw_aps := [[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, 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, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 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, -2, 4, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [0, -1, -5, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -8, 7, 0, 0, 0, 0, 0], [0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -1, 1], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 14, -5, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 6, 5, -2, 0, 0, 0, 0], [0, -3, -4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, -5, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -5, 15, 3, -5, 0, 0, 0, 0], [-24, -6, 7, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, -2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1, -5], [0, 0, 0, 0, -8, -4, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-26, 14, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 9, -17, -1, 0, 0, 0, 0], [0, -3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, -5, -1, -6], [0, 0, 0, 0, 0, 0, 0, 0, 17, -14, 18, 4, 0, 0, 0, 0], [0, 0, 0, 0, 8, -4, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, -2, 8], [0, 0, 0, 0, 0, 0, 0, 0, 14, 43, 12, -4, 0, 0, 0, 0], [0, 0, 0, 0, 4, -4, -5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -5, -3, -1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 35, 5, -13, 3, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -4, -1, 3, 0], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 2, 13], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -3, 3, -8], [0, 0, 0, 0, 0, 0, 0, 0, -25, 30, -53, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -24, -23, -5, -6, 0, 0, 0, 0], [0, 0, 0, 0, 5, -6, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 4, -1, 13, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -14, -18, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [98, -29, -10, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [78, 13, -6, -41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13, 2, 5, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 2, 2, -7, 0, 0, 0, 0, 0, 0, 0, 0], [4, 8, 15, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 5, -1, -3, 12], [0, 0, 0, 0, 0, 0, 0, 0, 25, 14, 12, 12, 0, 0, 0, 0], [0, 0, 0, 0, -14, -2, 2, 13, 0, 0, 0, 0, 0, 0, 0, 0], [48, 13, -26, -26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 2, 3, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 7, -15, -1, -5], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 7, -1, 5, 13], [198, 6, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -26, -11, 14, -16, 0, 0, 0, 0], [0, 0, 0, 0, -11, 6, 6, 8, 0, 0, 0, 0, 0, 0, 0, 0], [134, -14, 16, -42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -5, 14, -6, 5], [0, 0, 0, 0, -10, 14, -9, -9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 16, 10, -2, 8], [0, 0, 0, 0, -10, 6, -6, -11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 14, -2, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 6, -7, 0, 0, 0, 0, 0, 0, 0, 0], [-28, -60, 15, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -7, -61, -5, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 29, -106, 5, -12, 0, 0, 0, 0], [-36, 50, -10, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 20, 7, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -10, -5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 0, -15, 0, 0, 0, 0, 0, 0, 0, 0, -20, 28, 2, 8], [0, 0, 0, 0, 0, 0, 0, 0, 17, 29, 21, -5, 0, 0, 0, 0], [0, -10, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -4, -30, 10, 2], [0, 0, 0, 0, 0, 0, 0, 0, -15, -98, 2, -10, 0, 0, 0, 0], [0, 0, 0, 0, 4, 16, -10, -11, 0, 0, 0, 0, 0, 0, 0, 0], [-118, 4, -6, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -13, -94, -36, -22, 0, 0, 0, 0], [-102, 10, -2, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 77, -174, 68, 24, 0, 0, 0, 0], [98, -60, -3, 69, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, -26, 25, -1, 6], [150, -47, -38, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-162, -45, 12, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 20, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0, -22, 25, 15, 0], [0, 0, 0, 0, 0, 0, 0, 0, -68, -11, -162, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 37, -1, 1, -7], [0, 0, 0, 0, 0, 0, 0, 0, -11, 135, 71, 11, 0, 0, 0, 0], [-102, 15, 18, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 16, -9, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 13, -5, -21], [0, 0, 0, 0, 0, 0, 0, 0, 20, 110, -47, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -72, 203, -69, -26, 0, 0, 0, 0], [0, -18, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, -1, -27, -9, -19], [0, 0, 0, 0, 0, 0, 0, 0, 26, 110, -77, 5, 0, 0, 0, 0], [0, 8, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, -21, 3, 13, 28], [102, 74, 18, -60, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -20, 0, 17, 15, 0, 0, 0, 0, 0, 0, 0, 0], [-18, -12, -18, -50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [150, 19, 50, 57, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 52, 2, -1, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 7, -15, 7], [0, 0, 0, 0, -10, 14, -11, 27, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -56, 6, -8, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, -22, 12, -37, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 8, 21, -17, 18], [0, -16, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 37, -17, -15, 18], [0, 0, 0, 0, 0, 0, 0, 0, 2, -119, -20, -20, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 56, 107, 57, -22, 0, 0, 0, 0], [0, 0, 0, 0, 18, -14, 9, 27, 0, 0, 0, 0, 0, 0, 0, 0], [216, -39, -55, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -30, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 44, -64, 4, -18], [0, 0, 0, 0, -41, 2, -6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -38, 10, 7, 11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 23, 245, 49, -5, 0, 0, 0, 0], [74, 29, 48, 61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [364, 12, 39, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-526, -28, -71, -39, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 146, 97, -35, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -74, -64, -29, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -101, -16, -117, -26, 0, 0, 0, 0], [0, -26, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 1, -36, -16, -28], [-372, -15, -92, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 17, -36, -21, -20, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, -20, -1, -29, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 64, 309, -59, -2, 0, 0, 0, 0], [648, 13, 14, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, -18, -11, -1, 25], [242, -43, -60, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-176, 15, -65, -83, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 41, -98, -68, 10, 0, 0, 0, 0], [0, 0, 0, 0, -8, -6, -24, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 0, -15, 0, 0, 0, 0, 0, 0, 0, 0, 28, 45, -15, 36], [0, 0, 0, 0, 31, -32, 4, -24, 0, 0, 0, 0, 0, 0, 0, 0], [-648, -52, -43, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -117, -30, -150, -18, 0, 0, 0, 0], [0, -40, 0, 40, 0, 0, 0, 0, 0, 0, 0, 0, 67, -77, -3, -37], [0, 44, 0, -44, 0, 0, 0, 0, 0, 0, 0, 0, -46, 73, 15, 10], [0, 0, 0, 0, 0, 0, 0, 0, -121, 154, 35, 36, 0, 0, 0, 0], [0, 0, 0, 0, -54, 28, -2, -9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, -212, 76, 24, 0, 0, 0, 0], [0, -32, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 46, -49, -15, 12], [0, -24, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, -49, -45, -3, -36], [0, 3, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -36, 5, 1, 24], [0, 0, 0, 0, 0, 0, 0, 0, -34, -248, -199, -7, 0, 0, 0, 0], [0, 0, 0, 0, 6, -24, -28, 11, 0, 0, 0, 0, 0, 0, 0, 0], [214, -103, 14, -51, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-334, 12, 68, -130, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 10, 30, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 71, -20, -1, 22, 0, 0, 0, 0, 0, 0, 0, 0], [54, -53, 144, 69, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 150, -122, 79, -17, 0, 0, 0, 0], [0, 0, 0, 0, -20, 10, -31, -39, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -80, 44, 92, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 78, 255, 47, 26, 0, 0, 0, 0], [0, 0, 0, 0, -38, -18, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -200, -57, -35, 10, 0, 0, 0, 0], [168, 3, 103, 39, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 64, 0, -64, 0, 0, 0, 0, 0, 0, 0, 0, -17, 107, 21, -1], [170, -83, 35, -48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -28, 8, 27, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 40, 0, -40, 0, 0, 0, 0, 0, 0, 0, 0, -58, 73, 7, 9], [0, 0, 0, 0, 0, 0, 0, 0, -220, 173, -319, 4, 0, 0, 0, 0], [0, 0, 0, 0, -81, -12, 15, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -69, 32, -16, -40, 0, 0, 0, 0, 0, 0, 0, 0], [-706, 61, -8, 121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -163, -151, -39, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -15, -460, 6, 46, 0, 0, 0, 0], [10, -157, -18, 107, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -56, 30, -38, 35, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 27, -28, 24, -61], [0, 0, 0, 0, 0, 0, 0, 0, 113, 256, 131, -68, 0, 0, 0, 0], [0, 18, 0, -18, 0, 0, 0, 0, 0, 0, 0, 0, 27, 42, -6, 32], [648, 102, 26, -101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 22, 0, -22, 0, 0, 0, 0, 0, 0, 0, 0, 4, 49, -5, -36], [0, 0, 0, 0, -59, -14, 9, -14, 0, 0, 0, 0, 0, 0, 0, 0], [-102, -41, 26, 337, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-186, -151, -27, -36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 23, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 6, 14, 32, -10], [236, 71, 46, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -35, 34, -52, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -16, 34, -30, -17], [0, 0, 0, 0, 0, 0, 0, 0, 39, -26, -174, -92, 0, 0, 0, 0], [0, 24, 0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 53, 49, -1, -36], [0, 0, 0, 0, 0, 0, 0, 0, 71, -248, -171, -46, 0, 0, 0, 0], [0, 0, 0, 0, 81, -42, 27, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -97, 732, -70, -24, 0, 0, 0, 0], [-612, 72, -14, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 26, 44, 15, 57, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 99, -357, -13, 0, 0, 0, 0], [0, -6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 65, -4, -8, 49], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -54, 23, -19, 50], [0, 0, 0, 0, 0, 0, 0, 0, 6, -561, 226, -40, 0, 0, 0, 0], [0, 0, 0, 0, 96, 12, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0], [440, 73, 18, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1468, 53, -12, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 80, 28, 10, -63, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -240, 351, -344, -18, 0, 0, 0, 0], [0, -48, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, -38, -78, -18, 24], [0, 0, 0, 0, 0, 0, 0, 0, 283, 158, -108, 76, 0, 0, 0, 0], [0, 0, 0, 0, -56, -28, 37, 11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -57, 28, -72, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -32, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 10, -74, 10, 41], [0, 52, 0, -52, 0, 0, 0, 0, 0, 0, 0, 0, -61, 114, -10, 63], [0, 0, 0, 0, 0, 0, 0, 0, 77, 86, 111, -80, 0, 0, 0, 0], [0, 38, 0, -38, 0, 0, 0, 0, 0, 0, 0, 0, -39, 84, -8, -11], [-940, 70, 12, -179, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 328, -99, 4, 0, 0, 0, 0], [-906, 126, -28, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 30, 24, 25, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -40, 0, 40, 0, 0, 0, 0, 0, 0, 0, 0, 95, -90, 10, -61], [516, 66, 170, 241, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, -2, 33, -40, 0, 0, 0, 0, 0, 0, 0, 0], [-664, 136, -124, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 173, -350, -56, 52, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 244, -191, 27, 100, 0, 0, 0, 0], [0, 58, 0, -58, 0, 0, 0, 0, 0, 0, 0, 0, 11, 111, 5, -17], [-382, -91, 61, 228, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 129, 52, 20, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 42, -58, 22, -60], [0, 0, 0, 0, 0, 0, 0, 0, 213, 400, 122, -44, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 23, -622, -11, -40, 0, 0, 0, 0], [0, 47, 0, -47, 0, 0, 0, 0, 0, 0, 0, 0, 48, 119, -25, 52], [0, 60, 0, -60, 0, 0, 0, 0, 0, 0, 0, 0, 12, 124, -4, -4], [-806, -119, -193, -80, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 139, -446, -164, -70, 0, 0, 0, 0], [0, 0, 0, 0, 38, -4, -39, -39, 0, 0, 0, 0, 0, 0, 0, 0], [0, -54, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, -24, -66, -42, -11], [0, 0, 0, 0, 0, 0, 0, 0, 63, 65, 373, 13, 0, 0, 0, 0], [452, -10, 25, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 30, -25, 27, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 24, -22, 24, -96], [0, 0, 0, 0, 0, 0, 0, 0, -131, -50, 142, 38, 0, 0, 0, 0], [0, 6, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -39, 14, -2, -36], [0, 46, 0, -46, 0, 0, 0, 0, 0, 0, 0, 0, 50, 45, 47, 2], [0, 14, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, -126, 20, 8, 96], [0, 0, 0, 0, 0, 0, 0, 0, -19, -395, -139, -39, 0, 0, 0, 0], [0, 0, 0, 0, 39, 10, -12, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-594, -104, 61, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -190, 64, -17, -19, 0, 0, 0, 0, 0, 0, 0, 0], [-334, -58, -112, 156, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [460, -69, 69, -329, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -124, 71, -43, -16, 0, 0, 0, 0], [0, 57, 0, -57, 0, 0, 0, 0, 0, 0, 0, 0, 44, 144, -30, -24], [0, 0, 0, 0, 0, 0, 0, 0, 252, 116, 51, -25, 0, 0, 0, 0], [-304, 134, -18, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 90, -70, -39, -41, 0, 0, 0, 0, 0, 0, 0, 0], [0, -58, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 68, -90, -26, 18], [0, 0, 0, 0, 0, 0, 0, 0, -39, 82, 154, 102, 0, 0, 0, 0], [116, -51, -24, 176, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 65, -26, -10, -2], [0, 0, 0, 0, 101, 0, 52, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, -22, 33, 43, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, -4, -70, 54, -32], [0, 0, 0, 0, 0, 0, 0, 0, 112, -720, 209, 17, 0, 0, 0, 0], [206, -3, 177, 456, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 118, 6, -71, -37, 0, 0, 0, 0, 0, 0, 0, 0], [0, 42, 0, -42, 0, 0, 0, 0, 0, 0, 0, 0, 33, 134, -50, 56], [76, 18, -116, -189, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -134, 20, -75, 57, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 119, -306, -38, -38, 0, 0, 0, 0], [0, 0, 0, 0, -63, -90, -57, -58, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -88, 32, 77, 27, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 14, -4, -24, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 44, 0, -44, 0, 0, 0, 0, 0, 0, 0, 0, -20, 94, -6, 53], [0, 0, 0, 0, 164, -30, -38, 19, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -40, 323, -216, 88, 0, 0, 0, 0], [360, -21, 94, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1838, -139, 108, 205, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-154, -17, -155, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -140, -6, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 38, 26, 39, 67, 0, 0, 0, 0, 0, 0, 0, 0], [-310, 118, 58, -244, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 348, -755, 253, 40, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 46, -143, 264, 42, 0, 0, 0, 0], [0, -50, 0, 50, 0, 0, 0, 0, 0, 0, 0, 0, 43, -77, -23, -5], [0, 0, 0, 0, -47, -36, 49, 16, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -180, 506, -141, -27, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 177, 151, -175, 25, 0, 0, 0, 0], [0, -29, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 56, -36, -22, -28], [0, -140, 0, 140, 0, 0, 0, 0, 0, 0, 0, 0, 83, -233, -47, -27], [0, -82, 0, 82, 0, 0, 0, 0, 0, 0, 0, 0, -15, -205, 41, -40], [0, 0, 0, 0, -31, -76, -52, -6, 0, 0, 0, 0, 0, 0, 0, 0], [1298, -53, -218, -137, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 128, 0, 41, -25, 0, 0, 0, 0, 0, 0, 0, 0], [0, -60, 0, 60, 0, 0, 0, 0, 0, 0, 0, 0, -50, -125, 5, 111], [0, 0, 0, 0, 76, -76, 71, 19, 0, 0, 0, 0, 0, 0, 0, 0], [-488, 67, 55, -73, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 333, -195, 609, -45, 0, 0, 0, 0], [0, 0, 0, 0, 24, -2, 84, 47, 0, 0, 0, 0, 0, 0, 0, 0], [-1048, 117, 5, 241, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -79, -32, 12, -93], [-466, 68, 166, 260, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -95, 443, -355, -15, 0, 0, 0, 0], [0, 0, 0, 0, 129, -82, 77, -74, 0, 0, 0, 0, 0, 0, 0, 0], [0, -22, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, -69, -31, -13, 0], [0, 0, 0, 0, -62, -78, -41, -69, 0, 0, 0, 0, 0, 0, 0, 0], [0, 49, 0, -49, 0, 0, 0, 0, 0, 0, 0, 0, -124, 118, -20, 72], [-22, 172, -68, -252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 247, 478, -181, 0, 0, 0, 0, 0], [0, -79, 0, 79, 0, 0, 0, 0, 0, 0, 0, 0, 86, -177, 19, -42], [0, 0, 0, 0, 286, -18, -9, 23, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -161, -92, -3, 70, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -134, 76, 11, 93, 0, 0, 0, 0, 0, 0, 0, 0], [0, 14, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, -34, 48, -20, 156]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1100_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1100_3_e_b();".
// 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_1100_3_e_b(:prec:=16)
    chi := MakeCharacter_1100_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1100_3_e_b();".
// 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_1100_3_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1100_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1936, 0, 4200, 0, 793, 0, 50, 0, 1]>],Snew);
    return Vf;
end function;