
// Make newform 1400.2.bh.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1400_bh();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1400_bh_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_1400_2_bh_i();".
// 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_1400_2_bh_i();".
// 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 := [576, 0, 2592, 0, 3780, 0, 2040, 0, 432, 0, 36, 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, 9360, 0, 9564, 0, 5112, 0, 1176, 0, 104, 0, 3], [40464, -28080, 53460, -134868, 17592, -105720, 2688, -25704, 192, -2272, 5, -65], [76176, -9360, 199476, -9564, 126648, -5112, 28560, -1176, 2448, -104, 69, -3], [384, 9936, 0, 25956, 0, 16632, 0, 3744, 0, 320, 0, 9], [-111888, 46800, -345492, 153996, -235704, 115944, -54432, 28056, -4704, 2480, -133, 71], [93312, -110160, 202752, -274500, 110016, -181176, 22176, -41496, 1776, -3576, 48, -101], [-79776, -9360, -203400, -9564, -139728, -5112, -32928, -1176, -2880, -104, -82, -3], [155952, -135504, 402876, -319980, 266376, -191880, 61488, -42000, 5328, -3544, 151, -99], [-171648, -51408, -488448, 14364, -335232, 40488, -77952, 12264, -6768, 1176, -192, 35], [-93312, -110160, -202752, -274500, -110016, -181176, -22176, -41496, -1776, -3576, -48, -101], [33120, 7344, 86400, 36108, 55656, 27708, 12516, 6720, 1068, 594, 30, 17]];
    Rf_basisdens := [1, 2016, 8064, 4032, 768, 8064, 8064, 4032, 8064, 8064, 8064, 1008];
    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_1400_bh();"
function MakeCharacter_1400_bh()
    N := 1400;
    order := 6;
    char_gens := [351, 701, 1177, 801];
    v := [6, 6, 3, 4];
    // 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_1400_bh_Hecke();"
function MakeCharacter_1400_bh_Hecke(Kf)
    N := 1400;
    order := 6;
    char_gens := [351, 701, 1177, 801];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 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, -1, 0, -1, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 1, 0, -2, 0, 0, 0, -1, 0], [0, -2, 2, -3, 0, -2, 0, 1, 2, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0], [0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [-4, 0, 0, 0, 4, 0, 1, 0, 0, 0, 1, 1], [0, -1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0], [3, 0, 0, 0, 0, 0, 1, 0, 0, 1, -2, -1], [0, -1, -1, 2, 0, 1, 0, -1, -2, 0, 0, 0], [0, 1, 2, -3, 0, 0, 0, 3, -1, 0, 0, 0], [0, 2, 0, -1, 0, -2, 0, 4, 3, 0, 0, 0], [8, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 3, 0], [0, -1, 0, -4, 0, -3, 0, 3, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 3, 0, 4, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, -6, 0, -2, 0, 0, 2, -2, 0], [0, 4, 1, -2, 0, -1, 0, 1, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, -6, 0, 0, 0, 0, 0, -1, -2], [0, -1, 3, -1, 0, 0, 0, 2, 1, 0, 0, 0], [0, -3, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0], [2, 0, 0, 0, -2, 0, 3, 0, 0, 0, 0, -3], [0, 4, -4, 8, 0, 4, 0, -4, -8, 0, 0, 0], [0, 4, -6, 11, 0, 6, 0, -5, -10, 0, 0, 0], [0, 0, 0, 0, 3, 0, 1, 0, 0, 3, 5, 0], [0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1], [0, 0, 0, 0, 8, 0, 3, 0, 0, -1, 5, 0], [10, 0, 0, 0, -10, 0, 3, 0, 0, 0, 1, -1], [0, 1, 0, 0, 0, 2, 0, 4, 1, 0, 0, 0], [0, -2, 4, 1, 0, 0, 0, 2, 2, 0, 0, 0], [0, 2, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, -2, 8, 0, 0, 0, -1, -1, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2], [-4, 0, 0, 0, 0, 0, -3, 0, 0, 0, 3, 0], [0, 0, 0, 0, -8, 0, 0, 0, 0, 1, 1, 0], [0, 3, 0, -3, 0, -4, 0, 8, 6, 0, 0, 0], [0, 2, -2, 3, 0, 2, 0, -1, -2, 0, 0, 0], [8, 0, 0, 0, -8, 0, -4, 0, 0, 0, 0, 4], [-7, 0, 0, 0, 0, 0, -2, 0, 0, 3, -1, -3], [0, -3, 4, -10, 0, -4, 0, 6, 12, 0, 0, 0], [0, -1, 0, -7, 0, -8, 0, 4, 6, 0, 0, 0], [0, 0, 0, 0, 16, 0, 2, 0, 0, -2, 2, 0], [0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 1, 0, 0, -3, 2, 3], [7, 0, 0, 0, -7, 0, -5, 0, 0, 0, -2, 1], [3, 0, 0, 0, 0, 0, 3, 0, 0, 0, -3, 0], [0, 4, -8, 6, 0, 0, 0, -4, -4, 0, 0, 0], [0, 8, 0, 7, 0, -3, 0, -1, 1, 0, 0, 0], [2, 0, 0, 0, -2, 0, -3, 0, 0, 0, -2, -1], [0, 0, 0, 0, 0, 0, -4, 0, 0, 4, -4, 0], [0, -7, 0, -11, 0, -4, 0, 4, 4, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 2, 0, 0, 3, -5, -3], [0, -5, 0, -1, 0, 4, 0, -4, -4, 0, 0, 0], [0, -5, -2, 3, 0, 2, 0, -1, -2, 0, 0, 0], [0, 6, -3, 8, 0, 3, 0, -5, -10, 0, 0, 0], [16, 0, 0, 0, -16, 0, 2, 0, 0, 0, 2, 2], [0, -8, 12, -8, 0, 0, 0, 4, 8, 0, 0, 0], [0, 0, -4, 5, 0, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -1, 0, 0, 0, -2, 0], [0, 0, -4, 2, 0, 4, 0, 2, 4, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, -2, 3, 2], [0, -4, 0, -8, 0, -8, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, -6, 0, -2, 0, 0, -1, -5, 0], [0, 5, 0, -1, 0, -3, 0, 9, 6, 0, 0, 0], [0, -2, -4, 2, 0, 0, 0, -6, 2, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 6, 0, 0, -3, -3, 3], [0, 4, -1, 4, 0, 0, 0, 3, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, -4, -2], [0, 0, 8, -5, 0, 0, 0, 8, 0, 0, 0, 0], [0, 0, 0, 0, 15, 0, -2, 0, 0, 2, -2, 0], [4, 0, 0, 0, -4, 0, 6, 0, 0, 0, 1, -4], [-9, 0, 0, 0, 0, 0, -2, 0, 0, 5, -3, -5], [-12, 0, 0, 0, 0, 0, 2, 0, 0, -3, 1, 3], [-8, 0, 0, 0, 8, 0, -1, 0, 0, 0, -1, -1], [0, -8, 4, -8, 0, -4, 0, 4, 8, 0, 0, 0], [0, 0, 0, 0, -2, 0, 6, 0, 0, 0, 12, 0], [0, 5, -3, -5, 0, 0, 0, 2, -5, 0, 0, 0], [-21, 0, 0, 0, 0, 0, -1, 0, 0, 3, -2, -3], [0, -2, 0, -8, 0, 0, 0, -2, 2, 0, 0, 0], [18, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4, 0], [0, 10, -1, -1, 0, 1, 0, 2, 4, 0, 0, 0], [0, 3, -5, -4, 0, 0, 0, -2, -3, 0, 0, 0], [-20, 0, 0, 0, 20, 0, 5, 0, 0, 0, 1, -3], [0, 9, 0, 15, 0, 4, 0, -8, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, -3, 8, 3], [0, 0, 0, 0, 8, 0, 0, 0, 0, -3, -3, 0], [0, -8, -1, 2, 0, 1, 0, -1, -2, 0, 0, 0], [0, 0, 0, 0, 4, 0, 1, 0, 0, 2, 4, 0], [17, 0, 0, 0, -17, 0, -7, 0, 0, 0, -6, -5], [0, 2, 0, 17, 0, 0, 0, 2, -2, 0, 0, 0], [0, 0, 0, 0, 2, 0, -1, 0, 0, 6, 4, 0], [0, 6, -3, 10, 0, 3, 0, -7, -14, 0, 0, 0], [0, 3, 0, 14, 0, 10, 0, -12, -11, 0, 0, 0], [0, -7, 0, -8, 0, 5, 0, 7, 1, 0, 0, 0], [0, 0, 0, 0, 11, 0, -9, 0, 0, 3, -15, 0], [-4, 0, 0, 0, 4, 0, -10, 0, 0, 0, -2, 6], [0, 7, 0, 1, 0, -8, 0, 4, 6, 0, 0, 0], [0, -1, 8, -14, 0, -8, 0, 6, 12, 0, 0, 0], [0, 8, -12, 10, 0, 0, 0, -4, -8, 0, 0, 0], [18, 0, 0, 0, -18, 0, -2, 0, 0, 0, 0, 2], [-14, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4, 0], [0, -8, 11, -2, 0, 0, 0, 3, 8, 0, 0, 0], [0, 1, 0, 12, 0, 10, 0, -12, -11, 0, 0, 0], [0, 2, -4, 10, 0, 4, 0, -6, -12, 0, 0, 0], [5, 0, 0, 0, -5, 0, -10, 0, 0, 0, -7, -4], [-32, 0, 0, 0, 0, 0, 1, 0, 0, 3, -4, -3], [-27, 0, 0, 0, 27, 0, -1, 0, 0, 0, 1, 3], [0, 5, -4, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, -1, 0, -3, 0, -7, 0, -3, 2, 0, 0, 0], [0, 5, -6, -3, 0, 0, 0, -1, -5, 0, 0, 0], [-24, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4, 0], [6, 0, 0, 0, -6, 0, -13, 0, 0, 0, -4, 5], [0, -12, 0, -6, 0, 0, 0, 6, 12, 0, 0, 0], [0, 3, -2, 3, 0, 0, 0, 1, -3, 0, 0, 0], [0, 10, 0, 7, 0, -1, 0, 5, 3, 0, 0, 0], [0, 0, 0, 0, 4, 0, 4, 0, 0, -3, 5, 0], [20, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1], [0, 0, 0, 0, -16, 0, -3, 0, 0, 6, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, -3, -3, 0], [0, -8, 7, -15, 0, -7, 0, 8, 16, 0, 0, 0], [0, 11, -6, 7, 0, 0, 0, 5, -11, 0, 0, 0], [-15, 0, 0, 0, 15, 0, 6, 0, 0, 0, 5, 4], [0, -2, 11, -13, 0, -11, 0, 2, 4, 0, 0, 0], [0, 0, 0, 0, -18, 0, -2, 0, 0, 2, -2, 0], [0, 3, 4, -8, 0, -4, 0, 4, 8, 0, 0, 0], [0, 0, 0, 0, 21, 0, -3, 0, 0, 1, -5, 0], [23, 0, 0, 0, 0, 0, 2, 0, 0, 3, -5, -3], [0, 1, 0, -12, 0, -14, 0, 12, 13, 0, 0, 0], [0, -1, 0, 2, 0, 9, 0, 3, -3, 0, 0, 0], [0, 6, 4, -6, 0, -4, 0, 2, 4, 0, 0, 0], [-21, 0, 0, 0, 21, 0, 4, 0, 0, 0, 2, 0], [5, 0, 0, 0, 0, 0, -7, 0, 0, 0, 7, 0], [0, 0, 0, 0, 2, 0, 8, 0, 0, -8, 8, 0], [0, -3, 0, 0, 0, -1, 0, -7, -3, 0, 0, 0], [0, 7, -1, 2, 0, 1, 0, -1, -2, 0, 0, 0], [-2, 0, 0, 0, 2, 0, -8, 0, 0, 0, -4, 0], [-26, 0, 0, 0, 0, 0, -3, 0, 0, 1, 2, -1], [0, 11, -14, 21, 0, 14, 0, -7, -14, 0, 0, 0], [0, -1, 0, -9, 0, -4, 0, 12, 8, 0, 0, 0], [0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0], [0, -12, 9, -9, 0, 0, 0, -3, 12, 0, 0, 0], [0, -7, 10, -20, 0, 0, 0, 3, 7, 0, 0, 0], [-27, 0, 0, 0, 0, 0, 3, 0, 0, 1, -4, -1], [0, -1, -8, 18, 0, 8, 0, -10, -20, 0, 0, 0], [0, 7, -3, 16, 0, 0, 0, 4, -7, 0, 0, 0], [0, -4, 0, -1, 0, 7, 0, 1, -3, 0, 0, 0], [10, 0, 0, 0, 0, 0, 3, 0, 0, -1, -2, 1], [0, 0, 0, 0, -10, 0, 10, 0, 0, -6, 14, 0], [0, 0, 0, 0, -23, 0, 2, 0, 0, -3, 1, 0], [0, -8, 4, -2, 0, -4, 0, -2, -4, 0, 0, 0], [-4, 0, 0, 0, 4, 0, 10, 0, 0, 0, 8, 6], [0, -1, 7, 10, 0, 0, 0, 6, 1, 0, 0, 0], [0, 20, -16, 28, 0, 16, 0, -12, -24, 0, 0, 0], [0, -4, 7, -14, 0, -7, 0, 7, 14, 0, 0, 0], [0, 0, 0, 0, -5, 0, -3, 0, 0, -3, -9, 0], [0, -7, 0, -3, 0, 4, 0, -4, -4, 0, 0, 0], [0, -8, 0, -6, 0, -5, 0, -9, -2, 0, 0, 0], [40, 0, 0, 0, -40, 0, -3, 0, 0, 0, -3, -3], [0, 4, 0, 10, 0, -4, 0, -16, -6, 0, 0, 0], [-11, 0, 0, 0, 0, 0, -6, 0, 0, 4, 2, -4], [0, -7, 6, -6, 0, 0, 0, -1, 7, 0, 0, 0], [20, 0, 0, 0, -20, 0, 6, 0, 0, 0, 2, -2], [-14, 0, 0, 0, 0, 0, -7, 0, 0, 2, 5, -2], [0, 0, 0, 0, -20, 0, 4, 0, 0, 1, 9, 0], [0, 6, 0, 4, 0, 8, 0, 12, 2, 0, 0, 0], [-4, 0, 0, 0, 4, 0, -1, 0, 0, 0, -5, -9], [-12, 0, 0, 0, 0, 0, 2, 0, 0, -9, 7, 9], [1, 0, 0, 0, 0, 0, 2, 0, 0, -1, -1, 1], [-14, 0, 0, 0, 14, 0, -4, 0, 0, 0, -4, -4], [0, 13, -3, 9, 0, 3, 0, -6, -12, 0, 0, 0], [0, 0, 0, 0, 10, 0, 5, 0, 0, 3, 13, 0], [0, 2, -11, -3, 0, 0, 0, -9, -2, 0, 0, 0], [-15, 0, 0, 0, 0, 0, -9, 0, 0, -2, 11, 2], [0, -16, 6, -9, 0, -6, 0, 3, 6, 0, 0, 0], [0, -8, 4, -1, 0, 0, 0, -4, 8, 0, 0, 0], [0, 13, 0, 12, 0, 2, 0, 4, 1, 0, 0, 0], [-4, 0, 0, 0, 4, 0, 12, 0, 0, 0, 1, -10], [0, 10, 0, 3, 0, -6, 0, 8, 7, 0, 0, 0], [0, 3, 0, -1, 0, 0, 0, 8, 4, 0, 0, 0], [0, 0, 0, 0, -7, 0, 6, 0, 0, 1, 13, 0], [8, 0, 0, 0, -8, 0, 9, 0, 0, 0, 1, -7], [0, -12, 8, -14, 0, 0, 0, -4, 12, 0, 0, 0], [0, -7, -8, 16, 0, 8, 0, -8, -16, 0, 0, 0], [0, 0, 0, 0, -11, 0, -5, 0, 0, -6, -16, 0], [0, 0, 0, 0, 34, 0, -3, 0, 0, -1, -7, 0], [0, -5, 0, -13, 0, -8, 0, 8, 8, 0, 0, 0], [0, 19, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 0, 0, 21, 0, 1, 0, 0, 0, 1, 1], [0, -12, 8, -7, 0, 0, 0, -4, 12, 0, 0, 0], [0, 8, 12, -16, 0, -12, 0, 4, 8, 0, 0, 0], [0, -2, -4, 3, 0, 0, 0, -6, 2, 0, 0, 0], [-6, 0, 0, 0, 6, 0, 6, 0, 0, 0, -3, -12], [10, 0, 0, 0, 0, 0, 7, 0, 0, 3, -10, -3], [0, 14, -12, 20, 0, 0, 0, 2, -14, 0, 0, 0], [-15, 0, 0, 0, 15, 0, 2, 0, 0, 0, 2, 2], [-12, 0, 0, 0, 0, 0, -5, 0, 0, -7, 12, 7], [0, 0, 0, 13, 0, 18, 0, -8, -13, 0, 0, 0], [0, 0, 0, 0, -20, 0, -8, 0, 0, -4, -20, 0], [0, -2, -8, 13, 0, 0, 0, -10, 2, 0, 0, 0], [51, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-35, 0, 0, 0, 35, 0, -1, 0, 0, 0, 2, 5], [0, -8, 0, -14, 0, 0, 0, -8, 8, 0, 0, 0], [-22, 0, 0, 0, 0, 0, -2, 0, 0, 7, -5, -7], [0, 1, 5, -8, 0, -5, 0, 3, 6, 0, 0, 0], [0, 3, -3, -27, 0, 0, 0, 0, -3, 0, 0, 0], [-14, 0, 0, 0, 14, 0, 1, 0, 0, 0, -1, -3], [0, 0, 0, 0, 3, 0, 7, 0, 0, 3, 17, 0], [0, 18, 0, 22, 0, 11, 0, 3, -4, 0, 0, 0], [31, 0, 0, 0, -31, 0, -2, 0, 0, 0, 0, 2], [0, -5, 6, 5, 0, 0, 0, 1, 5, 0, 0, 0], [0, -3, 2, -5, 0, -2, 0, 3, 6, 0, 0, 0], [0, 0, 0, 0, -14, 0, 1, 0, 0, -1, 1, 0], [32, 0, 0, 0, 0, 0, -1, 0, 0, -3, 4, 3], [0, 0, 0, 0, 19, 0, 6, 0, 0, 6, 18, 0], [0, -11, 14, 1, 0, 0, 0, 3, 11, 0, 0, 0], [0, -3, -13, 24, 0, 13, 0, -11, -22, 0, 0, 0], [2, 0, 0, 0, 0, 0, 8, 0, 0, -3, -5, 3], [0, -2, 4, -4, 0, 0, 0, 2, 2, 0, 0, 0], [38, 0, 0, 0, -38, 0, -3, 0, 0, 0, 3, 9], [0, 10, -15, 21, 0, 0, 0, -5, -10, 0, 0, 0], [0, 0, 0, 0, 16, 0, -4, 0, 0, 7, -1, 0], [0, 3, 0, 12, 0, 2, 0, -16, -9, 0, 0, 0], [-27, 0, 0, 0, 27, 0, 11, 0, 0, 0, 2, -7], [-34, 0, 0, 0, 0, 0, -3, 0, 0, -3, 6, 3], [11, 0, 0, 0, -11, 0, 5, 0, 0, 0, -3, -11], [0, -17, -1, 2, 0, 1, 0, -1, -2, 0, 0, 0], [0, -23, 0, -28, 0, -9, 0, 1, 5, 0, 0, 0], [0, 0, 0, 0, 12, 0, 7, 0, 0, -12, 2, 0], [0, -5, 18, -13, 0, 0, 0, 13, 5, 0, 0, 0], [-7, 0, 0, 0, 0, 0, -8, 0, 0, -7, 15, 7], [18, 0, 0, 0, 0, 0, -13, 0, 0, 7, 6, -7], [0, -9, 0, -17, 0, -12, 0, 4, 8, 0, 0, 0], [0, 0, 0, 0, 17, 0, -5, 0, 0, -1, -11, 0], [0, 1, 0, -1, 0, -4, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, -8, 0, 1, 0, 0, 8, 10, 0], [0, 0, 0, -10, 0, 0, 0, 10, 20, 0, 0, 0], [0, 0, 0, 0, 2, 0, -6, 0, 0, 1, -11, 0], [0, -9, -3, -1, 0, 3, 0, 4, 8, 0, 0, 0], [7, 0, 0, 0, -7, 0, -18, 0, 0, 0, -5, 8], [4, 0, 0, 0, -4, 0, -6, 0, 0, 0, -2, 2], [0, 16, -22, 31, 0, 22, 0, -9, -18, 0, 0, 0], [0, 20, -16, 34, 0, 16, 0, -18, -36, 0, 0, 0], [0, 0, 0, 0, 5, 0, -4, 0, 0, 2, -6, 0], [0, 11, 0, -3, 0, -7, 0, 21, 14, 0, 0, 0], [-11, 0, 0, 0, 0, 0, 9, 0, 0, 3, -12, -3], [0, -11, 0, -19, 0, 4, 0, 20, 8, 0, 0, 0], [0, 0, 0, 0, -33, 0, 5, 0, 0, -5, 5, 0], [-8, 0, 0, 0, 8, 0, 2, 0, 0, 0, -1, -4], [0, 18, 0, 11, 0, -13, 0, 1, 7, 0, 0, 0], [0, 22, -6, 5, 0, 6, 0, 1, 2, 0, 0, 0], [0, 14, -12, -2, 0, 0, 0, 2, -14, 0, 0, 0], [0, -8, 0, -22, 0, -13, 0, 15, 14, 0, 0, 0], [0, -18, 13, -14, 0, -13, 0, 1, 2, 0, 0, 0], [-30, 0, 0, 0, 30, 0, 16, 0, 0, 0, 8, 0], [0, 1, 0, -8, 0, 0, 0, 8, 16, 0, 0, 0], [0, 8, 0, -2, 0, -4, 0, 16, 10, 0, 0, 0], [0, 0, 0, 0, -41, 0, 13, 0, 0, -6, 20, 0], [32, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 2], [-3, 0, 0, 0, 0, 0, -6, 0, 0, -3, 9, 3], [0, 5, -1, 10, 0, 1, 0, -9, -18, 0, 0, 0], [0, 10, 0, 13, 0, 10, 0, 4, -3, 0, 0, 0], [0, 0, 0, 0, -50, 0, 3, 0, 0, 0, 6, 0], [0, -1, 0, -9, 0, -12, 0, 4, 8, 0, 0, 0], [0, -6, 0, 12, 0, 12, 0, -24, -18, 0, 0, 0], [0, 0, 0, 0, 14, 0, -10, 0, 0, 4, -16, 0], [0, -13, 4, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, -21, 14, -23, 0, 0, 0, -7, 21, 0, 0, 0], [0, 6, -16, 13, 0, 0, 0, -10, -6, 0, 0, 0], [32, 0, 0, 0, -32, 0, -4, 0, 0, 0, 1, 6], [0, 0, 0, 0, -17, 0, 8, 0, 0, -3, 13, 0], [0, 0, 0, 0, -15, 0, -1, 0, 0, 3, 1, 0], [0, 11, 0, 13, 0, 9, 0, 5, -2, 0, 0, 0], [24, 0, 0, 0, -24, 0, 0, 0, 0, 0, -4, -8], [0, -18, 14, -23, 0, -14, 0, 9, 18, 0, 0, 0], [10, 0, 0, 0, 0, 0, -1, 0, 0, 8, -7, -8], [0, 3, -3, 5, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 26, 0, 6, 0, 0, 2, 14, 0], [0, 0, 0, 4, 0, 8, 0, 0, -4, 0, 0, 0], [0, -5, -2, 5, 0, 2, 0, -3, -6, 0, 0, 0], [-18, 0, 0, 0, 18, 0, -15, 0, 0, 0, -9, -3], [15, 0, 0, 0, 0, 0, -6, 0, 0, 7, -1, -7], [20, 0, 0, 0, -20, 0, 15, 0, 0, 0, 2, -11], [0, -1, 0, -14, 0, -3, 0, 23, 13, 0, 0, 0], [0, 20, -16, 21, 0, 0, 0, 4, -20, 0, 0, 0], [-27, 0, 0, 0, 0, 0, -5, 0, 0, 2, 3, -2], [0, 13, -10, 21, 0, 10, 0, -11, -22, 0, 0, 0], [-18, 0, 0, 0, 18, 0, 13, 0, 0, 0, 6, -1], [0, 0, 0, 0, -14, 0, 6, 0, 0, -6, 6, 0], [0, -16, 16, -26, 0, -16, 0, 10, 20, 0, 0, 0], [0, 0, 0, 0, 7, 0, 1, 0, 0, 8, 10, 0], [0, -1, 8, -14, 0, -8, 0, 6, 12, 0, 0, 0], [0, -6, 4, -23, 0, 0, 0, -2, 6, 0, 0, 0], [0, 9, -14, 16, 0, 0, 0, -5, -9, 0, 0, 0], [-40, 0, 0, 0, 40, 0, -11, 0, 0, 0, -7, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1400_bh_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1400_2_bh_i();".
// 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_1400_2_bh_i(:prec:=12)
    chi := MakeCharacter_1400_bh();
    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_1400_2_bh_i();".
// 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_1400_2_bh_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1400_bh();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1296, 0, -2916, 0, 5913, 0, -1386, 0, 243, 0, -18, 0, 1]>,<11,R![1936, 1056, 708, 16, 33, 3, 1]>],Snew);
    return Vf;
end function;