
// Make newform 210.2.r.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_210_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_210_r_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_210_2_r_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_210_2_r_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 := [729, -972, 891, -864, 576, -360, 237, -120, 64, -32, 11, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [972, -891, 864, -576, 360, -237, 120, -64, 32, -11, 4, -1], [1701, -243, 351, -630, 63, -183, 123, -10, 35, -11, 1, -4], [-6075, 5184, -4158, 3843, -1971, 1290, -810, 343, -191, 74, -16, 7], [-10935, 7695, -8721, 8082, -4005, 3243, -1821, 802, -545, 191, -67, 28], [5832, -4212, 4455, -4302, 2070, -1668, 969, -400, 284, -98, 31, -16], [-8748, 9801, -10071, 8487, -5130, 3495, -1971, 1027, -578, 227, -79, 25], [6885, -4644, 4878, -4635, 2157, -1734, 998, -415, 289, -98, 32, -15], [-2268, 1809, -1917, 1719, -894, 675, -377, 171, -110, 39, -13, 5], [-12636, 9801, -9558, 8766, -4356, 3255, -1920, 812, -538, 193, -56, 26], [-3888, 3240, -3348, 3123, -1638, 1230, -708, 317, -208, 76, -26, 11]];
    Rf_basisdens := [1, 1, 243, 324, 972, 972, 486, 972, 324, 108, 486, 162];
    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_210_r();"
function MakeCharacter_210_r()
    N := 210;
    order := 6;
    char_gens := [71, 127, 31];
    v := [3, 6, 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_210_r_Hecke();"
function MakeCharacter_210_r_Hecke(Kf)
    N := 210;
    order := 6;
    char_gens := [71, 127, 31];
    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, 1, 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, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, -1, -1, -2, -1, 1, -1, 1, 0, -1], [2, 0, -1, -1, 0, 0, 1, -1, 1, 0, -1, 1], [-1, 1, -1, 0, 0, 0, -2, -1, 0, -1, -2, -2], [0, 1, 0, 1, 2, 1, 3, -1, 0, -1, 1, 0], [1, -2, -1, -2, 0, 0, -1, 2, 0, 0, -2, 1], [1, 0, 1, 0, 0, 0, -1, 1, -1, -1, 0, 1], [-2, 1, -1, 0, 0, 0, -4, 2, 0, 2, -1, -1], [-2, 3, 2, 1, 0, -3, -1, 1, -2, 1, -1, -2], [-1, 1, -2, 1, 0, 0, -1, 0, 1, -1, -1, 1], [3, -2, -2, 0, -2, 2, 0, 3, 0, -3, 1, -1], [-2, 2, 2, 1, -2, 2, 0, 0, -1, 0, 0, 0], [0, -4, 2, 0, 3, 6, 0, -1, 1, -1, 4, 2], [6, -1, 2, 0, 0, 0, 3, 0, 0, 0, 3, -2], [0, 0, -2, 0, 4, 2, 0, 0, 2, 2, -2, -2], [-3, 0, 0, 4, 3, 0, 3, -3, -1, -1, 0, 0], [0, -1, 0, -2, -2, -1, -1, 2, 0, 2, -1, 0], [1, -1, 1, 3, 3, 3, 2, -4, 3, -4, -1, -1], [2, -3, 2, 1, 0, 3, 1, 1, 0, -1, 5, -2], [1, -2, -1, -2, -1, -2, 1, -1, -1, 1, 2, 3], [3, 1, 1, 0, 1, -1, 0, 0, 0, 0, 1, -1], [-3, 0, 0, 2, -3, -6, -3, -1, 3, -3, 0, 0], [1, -1, 1, 1, 9, 9, 2, -4, 1, -4, 3, 3], [0, 4, -2, -1, -6, -3, 3, 1, 0, 1, 2, -2], [3, -2, -1, 0, -3, 0, -3, -2, 2, 2, -2, 1], [-4, 2, 6, 4, 6, 0, 4, -3, -1, -1, 2, 4], [0, 1, -2, -2, -12, -6, -2, 2, 0, 2, -1, -2], [-1, 3, -3, -1, -3, -3, -2, 0, -1, 0, 1, 1], [2, 0, 0, 1, -3, 3, 0, -1, -1, 1, 3, -3], [-3, 1, 0, -1, -2, -4, -3, 2, -3, 3, -1, -1], [2, -1, -2, -3, 0, 3, 1, -3, 2, 1, -1, 2], [2, 2, -2, 0, 6, 6, 4, 0, 0, 0, -2, -2], [2, -2, -5, -4, -12, 0, -2, 4, 0, 0, -2, -3], [0, -2, 8, 0, 4, 2, 4, 0, 0, 0, 6, 8], [6, -2, -3, 1, 0, 0, 3, 1, 1, -2, -1, 3], [4, 0, 0, 4, 0, 0, 4, -2, 6, -6, 0, 0], [-9, 1, 1, 1, 0, 0, 0, -2, -1, 2, -1, 1], [-9, 0, -1, 4, 4, 8, -9, 6, -2, 2, 0, 1], [14, 0, -1, 1, 0, -6, 7, 1, -3, 2, -1, 1], [0, -3, 3, 2, 6, 6, 0, 2, 2, 2, 1, 1], [-1, -4, -5, -2, 3, 0, 1, 3, -1, -1, -4, -1], [0, 1, -6, -1, 2, 1, 1, 1, 0, 1, -5, -6], [-1, -1, 1, -6, 0, 0, -2, 4, -6, 4, -1, -1], [-4, 4, -2, 0, 0, -6, -2, 0, -6, 6, -6, 2], [7, -7, -7, 0, -6, 6, 0, 6, 0, -6, 1, -1], [4, -4, 4, 2, 18, 18, 8, -2, 2, -2, 6, 6], [0, 4, 2, 6, 4, 2, 6, -6, 2, -4, 6, 2], [7, 0, -5, 0, -15, 0, -7, 3, -3, -3, 0, -5], [0, 2, -4, 2, 6, 0, 0, -4, 2, 2, 2, -6], [4, -2, 2, -4, -6, -6, 8, 4, -4, 4, -2, -2], [0, 4, -2, -3, 0, -3, 0, -3, 2, 1, -6, 2], [-9, 5, 5, 4, 6, -6, 0, -3, -4, 3, -2, 2], [0, 2, -2, 2, 4, 8, 0, 0, 2, -2, -2, 0], [-6, 11, 8, -4, 0, -3, -3, -4, 2, 2, -3, -8], [0, 0, 4, 1, 10, 5, 3, -1, -2, -3, 4, 4], [4, 4, -2, -4, 0, 0, -4, 4, 0, 0, 4, -6], [0, 0, 4, 4, 4, 2, 2, -4, 0, -4, 4, 4], [-2, -6, 6, 3, -3, -3, -4, -1, 3, -1, 1, 1], [4, -4, -2, 2, 0, -12, 2, 2, -2, 0, 2, 2], [-3, 1, 1, -2, -4, 4, 0, 0, 2, 0, -3, 3], [-6, 2, -2, -5, -6, -6, -12, 2, -5, 2, -4, -4], [0, 2, 2, 0, -12, -6, -6, 0, 2, 2, 4, 2], [2, 6, 8, 2, 6, 0, -2, -2, 0, 0, 6, 2], [-8, -4, -2, 0, 0, 0, 8, -2, 2, 2, -4, 2], [-1, -4, 5, -2, 6, 12, -1, -6, 4, -4, 4, -1], [13, -1, -1, -3, 1, -1, 0, -4, 3, 4, -3, 3], [4, -4, -2, 3, 0, -12, 2, 3, -8, 5, 2, 2], [0, 1, -1, -6, -6, -6, 0, 4, -6, 4, -1, -1], [0, -8, 4, -2, 0, 0, 0, 2, -4, -2, -4, 4], [-1, 0, -3, 2, -15, 0, 1, 1, -3, -3, 0, -3], [-14, 0, 1, 2, 0, 0, -7, 2, -3, 1, 1, -1], [4, 2, 2, 2, -6, -12, 4, 6, -4, 4, -2, -4], [-5, -7, -7, 4, -2, 2, 0, 2, -4, -2, -3, 3], [-3, 6, -3, -10, 0, 0, -3, -3, -7, 7, -6, -3], [-14, -3, 0, 3, 0, -3, -7, 3, -2, -1, 3, 0], [-4, 2, 4, 6, 18, 0, 4, -6, 0, 0, 2, 2], [-5, -1, 1, -3, 6, 0, 5, -2, 5, 5, -1, 2], [2, -2, 2, 3, 0, 3, 1, 3, 6, -9, 4, -2], [-15, 1, 1, -4, 4, -4, 0, -3, 4, 3, -2, 2], [-19, 6, 6, 9, 3, -3, 0, 2, -9, -2, 6, -6], [-8, 2, 2, 0, 0, 0, -4, 0, -2, 2, 0, -2], [-2, 6, -6, 4, -6, -6, -4, -4, 4, -4, -6, -6], [-3, 2, 3, -2, 3, 0, 3, 1, 1, 1, 2, 1], [-9, 6, 1, 4, 15, 0, 9, 4, -8, -8, 6, -5], [1, -2, 2, 4, 24, 24, 2, -3, 4, -3, 3, 3], [-4, -6, 2, 6, 10, 20, -4, 6, 0, 0, 6, 4], [15, -5, -5, 7, -3, 3, 0, -2, -7, 2, -1, 1], [-13, 9, 9, 3, 10, -10, 0, -9, -3, 9, -4, 4], [-12, 2, 2, -2, 0, 0, -12, 5, -7, 7, -2, -4], [0, -3, 6, -3, 18, 9, -3, 3, 4, 7, 3, 6], [0, -4, 4, 2, -16, -8, 4, -2, 4, 2, 0, 4], [6, 4, -4, -8, -18, -18, 12, 4, -8, 4, -8, -8], [8, -4, -4, 8, 0, 0, 8, 2, 6, -6, 4, 8], [9, -7, -7, -4, -11, 11, 0, 2, 4, -2, 3, -3], [-9, -4, -2, 2, -5, -10, -9, 7, -5, 5, 4, 6], [0, 5, 5, 0, 6, 3, 6, 0, -5, -5, 10, 5], [1, 0, -3, -2, 15, 0, -1, 1, 1, 1, 0, -3], [16, -4, 1, -6, 4, 8, 16, 0, -6, 6, 4, 3], [11, -3, -3, -10, -3, 3, 0, -8, 10, 8, -3, 3], [2, 7, -2, 4, 0, -6, 1, 4, 0, -4, -9, 2], [0, 3, -6, 4, -18, -9, -3, -4, -6, -10, -3, -6], [-16, 7, 11, 9, -9, 0, 16, -13, 4, 4, 7, 4], [0, -2, 0, 0, 8, 4, -16, 0, -4, -4, -2, 0], [-6, 5, 2, -5, 0, 3, -3, -5, 8, -3, -3, -2], [21, -5, -5, -11, 1, -1, 0, -2, 11, 2, 1, -1], [18, -10, 0, 6, 14, 28, 18, -8, 14, -14, 10, 10], [-4, 6, -4, -4, 0, 0, -2, -4, 4, 0, -10, 4], [10, -2, 2, 4, -12, -12, 20, -8, 4, -8, 2, 2], [-5, -3, -6, -1, -18, 0, 5, 3, -2, -2, -3, -3], [0, 4, 1, 13, 0, 0, 17, -13, 7, -6, 5, 1], [5, 1, -1, 9, 9, 9, 10, -2, 9, -2, -3, -3], [14, -5, -2, 0, 0, 6, 7, 0, -2, 2, 3, 2], [-7, 3, 3, -5, 3, -3, 0, 0, 5, 0, 3, -3], [2, -1, -1, 1, 0, -12, 1, 1, 3, -4, 0, 1], [0, 0, 0, -10, -18, -18, 0, 10, -10, 10, -2, -2], [0, 7, -6, 9, 8, 4, -3, -9, 2, -7, 1, -6], [-7, 2, 11, -4, 18, 0, 7, -2, 6, 6, 2, 9], [2, 0, 0, 6, 18, 18, 4, -8, 6, -8, 4, 4], [-10, 2, 2, -7, 0, -9, -5, -7, -2, 9, 0, -2], [19, -11, -11, -3, -5, 5, 0, 6, 3, -6, 1, -1], [-3, 0, 3, 4, -12, -24, -3, 8, -4, 4, 0, -3], [42, -19, -8, -2, 0, 15, 21, -2, 10, -8, 11, 8], [-6, -4, 6, 0, 24, 0, 6, -2, 2, 2, -4, 10], [-7, -2, 2, -5, 3, 3, -14, -4, -5, -4, 2, 2], [-4, -2, -3, 8, 10, 20, -4, 6, 2, -2, 2, 5], [-3, -12, 7, -2, -7, -14, -3, 3, -5, 5, 12, 5], [-7, -1, 1, -9, -24, -24, -14, 15, -9, 15, -4, -4], [-3, -1, 6, 9, -12, 0, 3, -2, -7, -7, -1, 7], [0, -3, 0, 4, -12, -6, -1, -4, -2, -6, -3, 0], [5, -7, 7, -9, 6, 6, 10, 16, -9, 16, 3, 3], [-4, 6, -6, -6, -6, -12, -4, -6, 0, 0, -6, 0], [10, 0, 0, 0, -2, 2, 0, 4, 0, -4, 8, -8], [-12, -3, 7, 1, -1, -2, -12, 3, -2, 2, 3, -4], [6, -4, 4, 3, 9, 9, 12, -6, 3, -6, 2, 2], [0, 5, -4, -4, -24, -12, 21, 4, 4, 8, 1, -4], [-38, 15, 10, 4, 0, -15, -19, 4, -12, 8, -5, -10], [12, -8, -8, -8, -4, 4, 0, 0, 8, 0, 0, 0], [2, 3, 9, -3, 0, -6, 1, -3, 5, -2, 6, -9], [-1, -5, 5, 10, 6, 6, -2, 2, 10, 2, 3, 3], [-8, -2, -6, -6, -6, 0, 8, -2, 8, 8, -2, -4], [0, -13, 6, -10, 22, 11, -1, 10, -2, 8, -7, 6], [-3, 1, -1, -6, -3, -3, -6, 6, -6, 6, -5, -5], [-18, 7, 4, 1, 0, 9, -9, 1, 4, -5, -3, -4], [3, 1, 1, -1, 3, -3, 0, 2, 1, -2, 5, -5], [-13, -1, 1, 2, -6, -6, -26, 9, 2, 9, -2, -2], [0, -2, -6, 2, -16, -8, 6, -2, 2, 0, -8, -6], [-4, 4, 12, 0, -6, 0, 4, -6, 6, 6, 4, 8], [15, -8, -5, -16, 6, 0, -15, 5, 11, 11, -8, 3], [1, -1, -6, 3, -4, -8, 1, -2, 5, -5, 1, 7], [-3, 10, 10, 8, 2, -2, 0, 5, -8, -5, 3, -3], [-11, 11, 11, -3, 5, -5, 0, -4, 3, 4, 5, -5], [15, 2, -7, -6, -3, -6, 15, -8, 2, -2, -2, 5], [0, 3, -10, -12, -10, -5, -23, 12, -6, 6, -7, -10], [2, 14, -14, 10, 0, 0, 4, -10, 10, -10, -4, -4], [-13, 12, -1, 4, 1, 2, -13, 5, -1, 1, -12, -11], [15, -5, 1, -3, -12, -24, 15, -2, -1, 1, 5, 4], [5, 5, -5, 1, -15, -15, 10, -14, 1, -14, 3, 3], [0, -1, -12, -9, -26, -13, -3, 9, -2, 7, -13, -12], [11, -2, -7, -4, -3, 0, -11, 4, 0, 0, -2, -5], [-6, -2, 2, 14, -18, -18, -12, -4, 14, -4, -2, -2], [-1, -5, -5, 8, 17, -17, 0, -2, -8, 2, -3, 3], [-15, 5, 0, -9, 2, 4, -15, 2, -11, 11, -5, -5], [0, 2, 4, -2, 0, -18, 0, -2, -8, 10, 2, -4], [0, 5, 6, -3, -8, -4, -21, 3, -2, 1, 11, 6], [0, 13, -6, -5, 2, 1, -11, 5, -6, -1, 7, -6], [16, -8, -4, 2, 0, 24, 8, 2, 16, -18, 4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_210_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_210_2_r_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_210_2_r_b(:prec:=12)
    chi := MakeCharacter_210_r();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_210_2_r_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_210_2_r_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_210_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![1296, -15552, 75132, -155088, 168265, -99516, 30782, -3192, -645, 120, 38, -12, 1]>],Snew);
    return Vf;
end function;