
// Make newform 2100.2.bo.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2100_bo();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2100_bo_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_2100_2_bo_h();".
// 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_2100_2_bo_h();".
// 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 := [59049, 0, -19683, 0, -11664, 0, 7047, 0, 819, 0, -1104, 0, 91, 0, 87, 0, -16, 0, -3, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2119203, 0, 1981422, 0, -18954, 0, -164367, 0, 13224, 0, -4664, 0, 6063, 0, 3218, 0, -966, 0, -29], [1673055, 0, -1217430, 0, 300834, 0, -63765, 0, -57192, 0, 18808, 0, 1797, 0, -1354, 0, 654, 0, -47, 0], [0, -1541835, 0, 6617862, 0, -966978, 0, -840303, 0, 292920, 0, 64280, 0, -29625, 0, -86, 0, 930, 0, 83], [0, -448335, 0, 477738, 0, 99306, 0, -97623, 0, -528, 0, 13744, 0, -1677, 0, -466, 0, 222, 0, 19], [0, -2512863, 0, 997272, 0, 262764, 0, -275409, 0, 1212, 0, 26636, 0, -3921, 0, -908, 0, 48, 0, -31], [2224179, 0, -10206, 0, -842886, 0, 172431, 0, 100920, 0, -42728, 0, -6303, 0, 4094, 0, -138, 0, -131, 0], [3759453, 0, -166698, 0, -905202, 0, 293625, 0, 74808, 0, -42344, 0, 63, 0, 3098, 0, -366, 0, -53, 0], [1581201, 0, 6561, 0, -339309, 0, 65628, 0, 43206, 0, -10642, 0, -2391, 0, 865, 0, 111, 0, -22, 0], [0, 426465, 0, 437400, 0, 48924, 0, -228033, 0, 30396, 0, 32620, 0, -6897, 0, -2188, 0, 672, 0, 49], [-37179, 0, 251262, 0, -66906, 0, -43047, 0, 18600, 0, 3464, 0, -2841, 0, 322, 0, 138, 0, -37, 0], [4323699, 0, -5259006, 0, 720090, 0, 714735, 0, -303240, 0, -39208, 0, 32097, 0, -4130, 0, -1194, 0, 221, 0], [0, 19683, 0, 11664, 0, -7047, 0, -819, 0, 1104, 0, -91, 0, -87, 0, 16, 0, 3, 0, -1], [137781, 0, 169857, 0, -30861, 0, -15696, 0, -1290, 0, 3134, 0, 309, 0, -527, 0, -33, 0, 38, 0], [1502469, 0, -557442, 0, -142938, 0, 150201, 0, -20952, 0, -9976, 0, 4455, 0, -62, 0, -246, 0, 59, 0], [-4139991, 0, 2576286, 0, 696438, 0, -263691, 0, -57576, 0, 42872, 0, 5187, 0, -3374, 0, -150, 0, 383, 0], [0, -2224179, 0, 10206, 0, 842886, 0, -172431, 0, -100920, 0, 42728, 0, 6303, 0, -4094, 0, 138, 0, 131], [0, -5570289, 0, 868968, 0, 1085076, 0, -339759, 0, -119868, 0, 48628, 0, 3777, 0, -4084, 0, -48, 0, 127], [0, -277749, 0, -37746, 0, 144846, 0, -4269, 0, -33136, 0, 6224, 0, 3073, 0, -902, 0, -118, 0, 65]];
    Rf_basisdens := [1, 1, 5038848, 1679616, 5038848, 839808, 1259712, 1679616, 559872, 209952, 1259712, 279936, 1679616, 19683, 209952, 279936, 1679616, 1679616, 1259712, 279936];
    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_2100_bo();"
function MakeCharacter_2100_bo()
    N := 2100;
    order := 6;
    char_gens := [1051, 701, 1177, 1501];
    v := [6, 3, 3, 5];
    // 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_2100_bo_Hecke();"
function MakeCharacter_2100_bo_Hecke(Kf)
    N := 2100;
    order := 6;
    char_gens := [1051, 701, 1177, 1501];
    char_values := [[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, 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, 0, 1, 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], [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, 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, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 1, -2, 0, -1, 0, 2, 0, 0, 0, -2, 0, 0, 1, 0, 0, 0, 1, -1, 0], [0, -1, 1, 0, 1, -1, -1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1], [-2, 0, 0, -1, 0, 0, 0, 2, 0, -1, 0, 1, 2, 0, 0, -1, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0], [0, -1, -3, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0, -1, -2, 1], [-2, 0, 0, 3, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, -3, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -3, 3], [0, 2, -6, 0, 2, 0, 2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4, 3], [0, 2, 0, 0, 2, -1, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 2], [0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, -4, -1, 0, 0, 0, -2, 0, 0, 0], [6, 0, 0, -1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 3, -1, 0, -3, -1, 1, 0, 0, 0, -1, 0, 0, 4, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, -1, -3, 0, 2, 2, 0, 0, 0, 0], [0, -3, -1, 0, -3, -1, 2, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -4, -3, 2], [2, 0, 0, 1, 0, 0, 0, 2, -2, 1, 0, -2, -4, 0, 3, 0, -3, 0, 0, 0], [0, -4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -5, 2], [0, 0, 0, -2, 0, 0, 0, -6, 0, -2, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0], [0, -1, 0, 0, -2, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 2, 0, 0], [-2, 0, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, -2, 0, -1, 0, -2, 0, 0, 0], [0, 2, 3, 0, -2, -4, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2], [0, -2, -3, 0, 4, 0, 3, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 2, 1, 0], [0, 0, 0, 3, 0, 0, 0, 0, 1, -3, 0, 0, -1, 0, 3, 0, 0, 0, 0, 0], [0, 2, -4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 1], [0, -5, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, -1, -3, 1], [-2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 2, 4, 0, -2, 2, 2, 0, 0, 0], [0, -3, -1, 0, -3, 1, 1, 0, 0, 0, -1, 0, 0, 5, 0, 0, 0, 2, -2, -2], [2, 0, 0, 2, 0, 0, 0, -2, 0, 5, 0, -2, -2, 0, -3, 0, -2, 0, 0, 0], [-3, 0, 0, -4, 0, 0, 0, 1, 1, 4, 0, 5, 0, 0, -1, -3, -1, 0, 0, 0], [-3, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, -5, -1, 0, -1, 0, -3, 0, 0, 0], [0, 2, 4, 0, 2, -1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 4, 2], [0, -2, -6, 0, -2, 0, 6, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, -6, 1], [0, 4, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, -4, 0, 0, 0, 0, -5, 0], [0, 0, -4, 0, 2, 0, 4, 0, 0, 0, -4, 0, 0, -2, 0, 0, 0, -2, -4, -2], [6, 0, 0, 0, 0, 0, 0, 6, -2, 0, 0, -2, -2, 0, 0, 4, 0, 0, 0, 0], [5, 0, 0, 1, 0, 0, 0, -5, 0, 0, 0, 0, -5, 0, 1, 5, -1, 0, 0, 0], [12, 0, 0, -3, 0, 0, 0, -6, 0, 3, 0, -2, 0, 0, -5, 1, -5, 0, 0, 0], [0, -3, 5, 0, 3, 2, -2, 0, 0, 0, -1, 0, 0, -5, 0, 0, 0, 2, 3, 0], [0, 8, 2, 0, -3, -2, -2, 0, 0, 0, 2, 0, 0, 8, 0, 0, 0, 3, 1, -2], [-1, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 3, 1, 0, 1, -6, -1, 0, 0, 0], [6, 0, 0, 1, 0, 0, 0, 1, -2, 1, 0, -5, -1, 0, 2, 6, 1, 0, 0, 0], [0, 0, 12, 0, 0, -3, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 6, -3], [0, -3, 11, 0, 3, -2, -9, 0, 0, 0, 7, 0, 0, -3, 0, 0, 0, 0, 2, 0], [5, 0, 0, -1, 0, 0, 0, -2, -1, 1, 0, 5, 0, 0, 2, -2, 2, 0, 0, 0], [0, -2, 0, 0, 2, 4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 4, -2], [0, 0, 0, 4, 0, 0, 0, 6, 0, -2, 0, -4, -6, 0, 6, 2, -4, 0, 0, 0], [3, 0, 0, 2, 0, 0, 0, 3, -1, 1, 0, 3, -1, 0, 1, -6, 1, 0, 0, 0], [-7, 0, 0, 6, 0, 0, 0, 1, -2, -3, 0, 2, -1, 0, 3, -1, 6, 0, 0, 0], [0, -5, 1, 0, -4, 0, 7, 0, 0, 0, -3, 0, 0, -1, 0, 0, 0, -1, -3, 5], [0, 1, 0, 0, 1, -1, -4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -2, 4, 2], [-6, 0, 0, -3, 0, 0, 0, 6, -5, 1, 0, 5, 5, 0, -1, 0, 2, 0, 0, 0], [-5, 0, 0, 6, 0, 0, 0, 0, 5, -6, 0, -1, 0, 0, 5, -2, 5, 0, 0, 0], [0, -1, -1, 0, 1, 5, 2, 0, 0, 0, -3, 0, 0, -1, 0, 0, 0, 0, 1, 0], [4, 0, 0, 0, 0, 0, 0, -12, 0, -4, 0, 2, 4, 0, 4, -2, 0, 0, 0, 0], [0, 5, 1, 0, 5, -2, 2, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 2, -1, 4], [0, 0, 0, 0, 2, 2, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 6, -2], [0, 1, 2, 0, -4, -3, -2, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 4, 1, -3], [2, 0, 0, -4, 0, 0, 0, -2, -5, 2, 0, 0, 5, 0, -2, 0, 2, 0, 0, 0], [0, 1, 3, 0, 4, 2, -3, 0, 0, 0, -2, 0, 0, -5, 0, 0, 0, 5, 2, -1], [0, 3, -9, 0, -2, -2, 9, 0, 0, 0, -3, 0, 0, -1, 0, 0, 0, 1, 3, 1], [-1, 0, 0, 1, 0, 0, 0, 6, 1, 1, 0, 1, 2, 0, 2, -10, -2, 0, 0, 0], [0, -5, 0, 0, -5, 3, 4, 0, 0, 0, 4, 0, 0, 5, 0, 0, 0, -5, 3, -3], [0, 7, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -2], [-3, 0, 0, 3, 0, 0, 0, 13, 0, 0, 0, -3, -7, 0, 3, 4, -3, 0, 0, 0], [0, 3, 3, 0, -3, -6, -1, 0, 0, 0, -1, 0, 0, 3, 0, 0, 0, 0, 2, 0], [-13, 0, 0, 5, 0, 0, 0, 5, 3, -5, 0, -5, 0, 0, 3, 1, 3, 0, 0, 0], [0, 3, -1, 0, 3, 2, -1, 0, 0, 0, -1, 0, 0, -4, 0, 0, 0, -1, 0, -4], [0, -1, 9, 0, 3, 0, -5, 0, 0, 0, 7, 0, 0, -4, 0, 0, 0, -4, 7, -2], [0, 0, 0, 5, 0, 0, 0, 1, -2, -5, 0, -2, -1, 0, 0, 3, 5, 0, 0, 0], [0, -3, 6, 0, -4, 0, -6, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 1, 6, -6], [-2, 0, 0, -3, 0, 0, 0, -2, 3, -2, 0, -1, 3, 0, -2, 2, -1, 0, 0, 0], [0, 5, -5, 0, -10, 8, 5, 0, 0, 0, -2, 0, 0, 5, 0, 0, 0, -5, 2, -4], [1, 0, 0, -5, 0, 0, 0, 1, -3, 5, 0, -9, 0, 0, -2, 6, -2, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 2, -2, -1, 0, 0, 0], [14, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 4, 0, 0, 0, -4, 2, 0, 0, 0], [-1, 0, 0, 9, 0, 0, 0, -3, 6, -3, 0, -1, 3, 0, 6, -2, 9, 0, 0, 0], [-4, 0, 0, -1, 0, 0, 0, -4, -2, 4, 0, -1, -2, 0, 4, 2, -5, 0, 0, 0], [0, -11, 2, 0, 9, -1, -2, 0, 0, 0, 2, 0, 0, -11, 0, 0, 0, -9, 1, -1], [3, 0, 0, -1, 0, 0, 0, -2, 1, 1, 0, -3, 0, 0, -6, 1, -6, 0, 0, 0], [0, 5, -2, 0, -6, 8, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, -4], [-1, 0, 0, -3, 0, 0, 0, -5, 0, 3, 0, 6, 7, 0, -6, -1, 3, 0, 0, 0], [0, 3, 1, 0, 11, 0, -1, 0, 0, 0, 1, 0, 0, -8, 0, 0, 0, -8, 1, 0], [13, 0, 0, 4, 0, 0, 0, -1, 2, 1, 0, 10, 1, 0, 5, -11, 4, 0, 0, 0], [0, 1, 0, 0, 4, 5, -2, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0, 4, -1, -5], [0, 5, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3], [-6, 0, 0, -2, 0, 0, 0, 6, 0, 4, 0, 2, 0, 0, -4, 0, -2, 0, 0, 0], [0, -10, 1, 0, 10, 1, -3, 0, 0, 0, 5, 0, 0, -1, 0, 0, 0, -9, -2, 0], [4, 0, 0, -2, 0, 0, 0, -8, 0, -10, 0, 4, 0, 0, 8, 4, 2, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 6, -1, 0, 0, -1, -2, 0, 3, 1, -3, 0, 0, 0], [0, 8, 0, 0, 1, -8, -4, 0, 0, 0, -4, 0, 0, -8, 0, 0, 0, 1, 0, 8], [1, 0, 0, 5, 0, 0, 0, -3, -1, 5, 0, -1, -2, 0, 0, 6, 0, 0, 0, 0], [20, 0, 0, -6, 0, 0, 0, -20, -2, 6, 0, 2, 2, 0, -6, 0, 0, 0, 0, 0], [0, 3, -7, 0, -3, 2, 7, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1], [2, 0, 0, 0, 0, 0, 0, -11, -2, 0, 0, -2, -4, 0, 4, 0, -4, 0, 0, 0], [0, -4, 0, 0, 1, -1, -6, 0, 0, 0, -6, 0, 0, 4, 0, 0, 0, 1, 6, 1], [0, 2, -2, 0, 2, 5, -2, 0, 0, 0, -2, 0, 0, 6, 0, 0, 0, 8, 0, -10], [0, -1, 0, 0, 1, -3, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, -6, 0, 0], [-5, 0, 0, -5, 0, 0, 0, 1, 3, 5, 0, -9, 0, 0, -5, 3, -5, 0, 0, 0], [-16, 0, 0, 3, 0, 0, 0, 16, 0, -14, 0, 2, 0, 0, 14, 0, 11, 0, 0, 0], [0, -2, 1, 0, -2, -3, 3, 0, 0, 0, 1, 0, 0, -3, 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-5, 3, -5, 0, 0, 0], [0, 10, 10, 0, -4, 0, -6, 0, 0, 0, 8, 0, 0, 14, 0, 0, 0, 14, 8, -5], [0, 0, 0, 0, -5, 10, 10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -5, 8, -10], [21, 0, 0, -14, 0, 0, 0, 10, -20, 9, 0, -6, -10, 0, -5, 16, -14, 0, 0, 0], [0, -4, -4, 0, -8, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 10], [12, 0, 0, 9, 0, 0, 0, 12, 10, 0, 0, 1, 10, 0, 0, -2, 9, 0, 0, 0], [0, -4, -5, 0, 6, 4, 5, 0, 0, 0, -14, 0, 0, -2, 0, 0, 0, 2, 14, -2], [-5, 0, 0, 3, 0, 0, 0, 5, -5, -3, 0, 3, 0, 0, 1, 1, 1, 0, 0, 0], [0, -9, -3, 0, 9, 13, -2, 0, 0, 0, 7, 0, 0, -3, 0, 0, 0, -6, -5, 0], [-18, 0, 0, -13, 0, 0, 0, -18, 10, -4, 0, 1, 10, 0, -4, -2, -9, 0, 0, 0], [8, 0, 0, -4, 0, 0, 0, 4, -8, 13, 0, -4, -4, 0, 9, 8, -4, 0, 0, 0], [16, 0, 0, -10, 0, 0, 0, 16, -2, -3, 0, 2, -2, 0, -3, -4, -7, 0, 0, 0], [0, 6, -30, 0, -8, -1, 19, 0, 0, 0, -19, 0, 0, 6, 0, 0, 0, 8, -15, -1], [0, 2, -9, 0, -2, 15, 11, 0, 0, 0, -13, 0, 0, 4, 0, 0, 0, -2, 2, 0], [14, 0, 0, -9, 0, 0, 0, -3, -8, 9, 0, -6, 0, 0, -13, 7, -13, 0, 0, 0], [0, 3, -9, 0, -6, -18, 9, 0, 0, 0, 5, 0, 0, 3, 0, 0, 0, -3, -5, 9], [0, 0, 0, -6, 0, 0, 0, -20, 0, -1, 0, 9, 20, 0, -5, -11, 6, 0, 0, 0], [-5, 0, 0, -2, 0, 0, 0, -1, 2, 4, 0, -5, 1, 0, 2, 4, -2, 0, 0, 0], [0, 3, -7, 0, 3, 12, -1, 0, 0, 0, -7, 0, 0, 1, 0, 0, 0, 4, -6, -24], [6, 0, 0, -2, 0, 0, 0, 4, -14, 2, 0, -4, 0, 0, 2, 9, 2, 0, 0, 0], [0, 16, 1, 0, 16, -3, -5, 0, 0, 0, 1, 0, 0, -13, 0, 0, 0, 3, 6, 6], [0, 0, 0, -2, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 8, 9, -8, 0, 0, 0], [0, -4, 0, 0, -4, 4, 2, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, -4, 0, -4], [12, 0, 0, 1, 0, 0, 0, 20, -12, 1, 0, -12, -24, 0, 3, 13, -3, 0, 0, 0], [0, -8, 18, 0, 2, -7, -7, 0, 0, 0, 7, 0, 0, -8, 0, 0, 0, -2, 9, -7], [-6, 0, 0, -15, 0, 0, 0, 6, -1, 20, 0, 13, 1, 0, -20, 0, -5, 0, 0, 0], [-5, 0, 0, -4, 0, 0, 0, 5, -3, 1, 0, 15, 3, 0, -1, 0, 3, 0, 0, 0], [0, 2, 4, 0, -8, -13, 2, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 8, 2, -13], [0, -9, 0, 0, 7, 4, 2, 0, 0, 0, 2, 0, 0, 9, 0, 0, 0, 7, -7, -4], [-5, 0, 0, -7, 0, 0, 0, 3, 5, -7, 0, 5, 10, 0, 0, 12, 0, 0, 0, 0], [0, 1, 10, 0, 1, 4, -12, 0, 0, 0, 10, 0, 0, 7, 0, 0, 0, 8, 22, -8], [1, 0, 0, 7, 0, 0, 0, -9, 0, 3, 0, -3, 7, 0, 4, -10, -7, 0, 0, 0], [0, 10, 16, 0, -10, -16, -12, 0, 0, 0, 8, 0, 0, 10, 0, 0, 0, 0, 4, 0], [17, 0, 0, -6, 0, 0, 0, -11, 5, 6, 0, 15, 0, 0, -6, -10, -6, 0, 0, 0], [-38, 0, 0, 0, 0, 0, 0, 38, -3, -9, 0, 7, 3, 0, 9, 0, 9, 0, 0, 0], [0, 9, 5, 0, 9, 2, -1, 0, 0, 0, 5, 0, 0, -8, 0, 0, 0, 1, 6, -4], [0, 8, 0, 0, 11, -8, 4, 0, 0, 0, 4, 0, 0, -8, 0, 0, 0, 11, -7, 8], [0, -14, 8, 0, 12, 10, -8, 0, 0, 0, 10, 0, 0, 2, 0, 0, 0, -2, -10, -5], [0, -16, 3, 0, 16, 14, -6, 0, 0, 0, 9, 0, 0, -12, 0, 0, 0, -4, -3, 0], [0, -6, 20, 0, -5, -5, -2, 0, 0, 0, 2, 0, 0, -6, 0, 0, 0, 5, 10, -5], [-1, 0, 0, 8, 0, 0, 0, -1, -1, -5, 0, -8, -1, 0, -5, 16, 13, 0, 0, 0], [0, 3, 30, 0, -1, -8, -26, 0, 0, 0, 26, 0, 0, 3, 0, 0, 0, 1, 15, -8], [0, -6, -4, 0, 6, 4, -2, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, -6, -6, 0], [1, 0, 0, -4, 0, 0, 0, -4, 7, 4, 0, -9, 0, 0, 1, 1, 1, 0, 0, 0], [3, 0, 0, 14, 0, 0, 0, 7, 0, 1, 0, -20, -13, 0, 13, -7, -14, 0, 0, 0], [16, 0, 0, -12, 0, 0, 0, 10, -20, 14, 0, 4, -10, 0, 2, 6, -12, 0, 0, 0], [0, -1, 0, 0, -9, 5, 3, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, -9, 4, -5], [0, 9, 5, 0, 9, 7, 1, 0, 0, 0, 5, 0, 0, -13, 0, 0, 0, -4, 4, -14], [1, 0, 0, -7, 0, 0, 0, 7, -15, 7, 0, 21, 0, 0, 0, -3, 0, 0, 0, 0], [0, 7, 0, 0, -7, 8, 3, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 7, 3, 0], [0, -9, 5, 0, -9, -4, -2, 0, 0, 0, 5, 0, 0, 1, 0, 0, 0, -8, 7, 8], [1, 0, 0, -1, 0, 0, 0, 8, -1, -1, 0, -1, -2, 0, -10, 8, 10, 0, 0, 0], [0, 10, 20, 0, -10, 1, -12, 0, 0, 0, 12, 0, 0, 10, 0, 0, 0, 10, 10, 1], [0, -14, 8, 0, 18, -6, -8, 0, 0, 0, 2, 0, 0, -4, 0, 0, 0, 4, -2, 3], [0, -13, -5, 0, 2, -8, 5, 0, 0, 0, -9, 0, 0, 11, 0, 0, 0, -11, 9, 4], [7, 0, 0, 2, 0, 0, 0, -7, -6, -4, 0, -4, 6, 0, 4, 0, 2, 0, 0, 0], [3, 0, 0, 10, 0, 0, 0, -3, -3, 10, 0, -3, -6, 0, 1, 0, -1, 0, 0, 0], [-6, 0, 0, -5, 0, 0, 0, -30, 6, -5, 0, 6, 12, 0, -1, -11, 1, 0, 0, 0], [0, 9, 12, 0, -9, -6, -6, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 18, 6, 0], [-44, 0, 0, -4, 0, 0, 0, 44, -10, 10, 0, 17, 10, 0, -10, 0, -6, 0, 0, 0], [0, -6, 0, 0, 0, 1, -6, 0, 0, 0, -6, 0, 0, 6, 0, 0, 0, 0, -8, -1], [-1, 0, 0, 9, 0, 0, 0, 13, 0, -12, 0, -14, -11, 0, 21, -3, -9, 0, 0, 0], [0, 11, -14, 0, -14, -8, 14, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -3, 0, 4], [-27, 0, 0, -1, 0, 0, 0, 19, -11, 1, 0, 5, 0, 0, 11, 3, 11, 0, 0, 0], [0, -15, 25, 0, 15, 12, -14, 0, 0, 0, 3, 0, 0, -7, 0, 0, 0, -8, 11, 0], [0, 4, -14, 0, 5, 3, 2, 0, 0, 0, -2, 0, 0, 4, 0, 0, 0, -5, -7, 3], [-8, 0, 0, -4, 0, 0, 0, -8, 10, 0, 0, 11, 10, 0, 0, -22, -4, 0, 0, 0], [7, 0, 0, 4, 0, 0, 0, 5, -10, -5, 0, -4, -5, 0, -1, 9, 4, 0, 0, 0], [16, 0, 0, -5, 0, 0, 0, 16, -7, 5, 0, 3, -7, 0, 5, -6, -10, 0, 0, 0], [0, 5, 24, 0, -5, -21, -18, 0, 0, 0, 12, 0, 0, 15, 0, 0, 0, -10, 6, 0], [0, -1, 15, 0, 14, -18, -15, 0, 0, 0, -9, 0, 0, -13, 0, 0, 0, 13, 9, 9], [3, 0, 0, -12, 0, 0, 0, 5, -10, 14, 0, 19, -5, 0, 2, -14, -12, 0, 0, 0], [0, -4, 0, 0, -3, 10, 11, 0, 0, 0, 11, 0, 0, 4, 0, 0, 0, -3, 11, -10], [14, 0, 0, -3, 0, 0, 0, -14, -3, -1, 0, 11, 3, 0, 1, 0, 4, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, -10, 20, -1, 0, -12, 0, 0, -13, -4, -13, 0, 0, 0], [0, -2, 0, 0, 3, 18, -6, 0, 0, 0, -6, 0, 0, 2, 0, 0, 0, 3, 3, -18], [-7, 0, 0, 0, 0, 0, 0, 9, 7, 0, 0, 7, 14, 0, -2, -12, 2, 0, 0, 0], [0, -18, -10, 0, 21, -4, 11, 0, 0, 0, -11, 0, 0, -18, 0, 0, 0, -21, -5, -4], [0, 2, -8, 0, -4, 8, 8, 0, 0, 0, -8, 0, 0, 2, 0, 0, 0, -2, 8, -4], [0, 4, -2, 0, -14, 10, 2, 0, 0, 0, -6, 0, 0, 10, 0, 0, 0, -10, 6, -5], [-15, 0, 0, 3, 0, 0, 0, 15, -1, -15, 0, 16, 1, 0, 15, 0, 12, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2100_bo_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2100_2_bo_h();".
// 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_2100_2_bo_h(:prec:=20)
    chi := MakeCharacter_2100_bo();
    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_2100_2_bo_h();".
// 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_2100_2_bo_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2100_bo();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![1728, 7488, 14128, 14352, 8156, 2280, 84, -96, -4, 6, 1]>,<13,R![-3888, 0, 65704, 0, -22251, 0, 2183, 0, -81, 0, 1]>,<19,R![192, -960, 1864, -1320, -213, 519, 152, -45, -12, 3, 1]>],Snew);
    return Vf;
end function;