
// Make newform 210.2.n.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_210_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_210_n_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_n_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_n_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 := [64, -192, 288, -112, 84, -216, 368, -108, 21, -14, 18, -6, 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], [11668096, -7541936, -2644000, 4101028, 25802216, -5356872, 192316, 26475, 1200782, -320538, 32246, 10967], [15302400, -11214000, -6514400, -8799372, 23519360, -10333816, 1776760, -841061, 1179630, -554210, 124978, -11057], [4417824, -19968496, -8958456, -68948, 10615144, -11800108, 2503214, -480425, 611754, -569522, 172184, -25665], [6873568, -17161936, 16388488, -3071364, 20727744, -17291492, 5063202, -1016073, 1148906, -866962, 255324, -34905], [-2193056, 10569008, 2417208, -6494364, -1847616, 13863204, -4046658, 407037, -227874, 665338, -242212, 43053], [17709568, -25667824, 15774688, -1580084, 17934344, -26209616, 7769956, -1258503, 1078106, -1257142, 412146, -65619], [6409568, -16808976, -251880, -1136412, 6245376, -21848556, 6443046, -1079355, 532062, -1055638, 372468, -65883], [-33288832, 39098384, -21185216, -17165628, -54894792, 52377004, -14110440, 1600117, -2969054, 2628966, -805130, 112417], [16768, -54704, 36720, -228, 20464, -75424, 23804, -3655, 1778, -3622, 1290, -227], [-13456256, -25274576, 31536720, -2505732, -30221320, -54686860, 19696548, -2913901, -669562, -2530334, 1049078, -213873]];
    Rf_basisdens := [1, 1, 6321728, 6321728, 6321728, 6321728, 6321728, 6321728, 6321728, 6321728, 8768, 6321728];
    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_n();"
function MakeCharacter_210_n()
    N := 210;
    order := 6;
    char_gens := [71, 127, 31];
    v := [6, 3, 2];
    // 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_n_Hecke();"
function MakeCharacter_210_n_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], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, -1, 0, 0, -1, 1, -1, 0, -1, 0], [-1, 1, 0, 0, 0, 1, 0, 0, 0, 0, -1, -1], [-1, -2, -2, 1, 2, -1, 1, 1, -2, -1, 1, -1], [-2, 0, 0, -1, -1, 1, 0, 1, 1, 1, -2, -1], [2, -2, -2, -2, 2, 0, 0, -2, 0, -2, 2, 0], [1, 0, -1, 1, 1, 1, 1, 0, 1, 0, 0, 1], [-1, -1, -3, -1, 0, 0, -1, 1, -1, 1, -1, 0], [-6, 0, 0, -1, 1, -1, 0, 1, -2, -1, 0, -1], [-1, 1, 2, -1, -1, 0, 0, -1, 1, 1, -1, 0], [1, 1, 3, 1, 0, 0, 1, -1, 1, -1, 1, 0], [1, -1, -1, 2, -2, 1, -1, -1, 2, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [3, 3, -1, 3, 0, 0, 3, -3, 3, -3, 3, 0], [0, 0, -3, 2, 0, -2, -2, 2, 3, 2, 0, 2], [-7, 1, 4, -2, -1, -1, 1, -2, 1, 2, -5, -1], [0, 2, -2, 0, 2, 2, 0, 2, 0, -2, 2, 2], [4, -4, 2, -2, 4, -2, -2, -2, -6, -2, 4, 2], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 0, 0], [-1, 0, 1, -1, -1, -1, -1, 0, -1, 0, -9, -1], [-6, 2, -2, -1, -1, 3, -2, 3, 4, 3, -6, -3], [-4, 2, 2, -4, -2, -2, -4, 2, -4, -2, -2, -2], [-2, -1, 1, -2, -2, 1, 1, 1, 8, 1, -2, -1], [8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0], [2, 0, 10, 2, -2, -2, 2, 0, 2, 0, 2, 2], [-3, -1, 4, -3, 2, 2, -3, 1, -3, 1, -3, -2], [2, -4, 0, 0, 4, -4, 4, 0, -4, 0, 10, -4], [4, 0, 0, 2, 2, -2, 0, -2, -2, -2, 4, 2], [-8, 3, -3, -1, -1, 4, -3, 4, -9, 4, -8, -4], [0, -1, 1, 0, -1, -1, 0, -1, 0, 1, -3, -1], [-6, 6, 0, 4, -6, 2, 2, 4, 6, 4, -6, -2], [-12, -2, -2, 0, 0, -2, -2, 2, -4, -2, 0, -2], [-2, 0, 2, -2, -2, -2, -2, 0, -2, 0, -16, -2], [-7, 3, 6, -3, -3, 0, 0, -3, 3, 3, -7, 0], [-1, 1, -9, 2, -1, -1, -1, 2, 10, 2, -1, 1], [0, 2, -10, 0, 2, 2, 0, -2, 0, -2, 0, -2], [-2, 2, -2, 1, 1, 1, -2, 1, -5, 1, -2, -1], [-3, -1, -1, -3, 2, 2, -3, 1, -3, 1, -3, -2], [10, -3, -6, 3, 3, 0, 0, 3, -3, -3, 10, 0], [2, 6, 6, 0, 0, 6, 6, -6, 12, 6, 0, 6], [0, -4, 4, 0, -4, -4, 0, -4, 0, 4, 0, -4], [1, -1, -10, -1, 1, 0, 0, -1, 9, -1, 1, 0], [6, -2, 2, 1, 1, -3, 2, -3, -5, -3, 6, 3], [-12, 2, 4, -2, -2, 0, 0, -2, 2, 2, -12, 0], [-7, -1, -1, 0, 0, -1, -1, 1, -2, -1, 0, -1], [10, -1, 1, 4, 4, -5, 1, -5, -4, -5, 10, 5], [1, -1, -12, -2, 1, 1, 1, -2, 11, -2, 1, -1], [4, -4, 0, 4, 0, 0, 4, -4, 4, 4, 16, 0], [0, 2, 2, 0, 2, 2, 0, -2, 0, -2, 0, -2], [-12, 2, 2, -2, 2, 0, 2, 0, 0, 0, 0, 0], [-1, -2, -4, 2, 2, 0, 0, 2, -2, -2, -1, 0], [-11, -5, -5, -5, 5, -10, -5, 10, -20, -10, 0, -10], [-8, -4, -2, -8, 4, 4, -8, 4, -8, 4, -8, -4], [-6, 6, -8, 2, -6, 4, 4, 2, 14, 2, -6, -4], [-7, -1, -4, 2, 1, 1, -1, 2, -1, -2, -9, 1], [5, -4, -1, 5, 1, 1, 5, -4, 5, 4, -5, 1], [-6, 6, 14, 4, -6, 2, 2, 4, -8, 4, -6, -2], [7, 1, 1, 0, 0, 1, 1, -1, 2, 1, 0, 1], [8, -8, -2, -4, 8, -4, -4, -4, -6, -4, 8, 4], [4, -2, 2, 0, 0, -2, 2, -2, 7, -2, 4, 2], [4, 2, -2, 4, 4, -2, -2, -2, 10, -2, 4, 2], [12, 2, 4, -2, -2, 0, 0, -2, 2, 2, 12, 0], [-1, 0, 17, -1, 1, 1, -1, 0, -1, 0, -1, -1], [3, 3, -8, 3, 0, 0, 3, -3, 3, -3, 3, 0], [-7, 2, 5, -7, -5, -5, -7, 2, -7, -2, 4, -5], [-10, 1, -1, -4, -4, 5, -1, 5, -2, 5, -10, -5], [4, -4, -8, -4, 4, 0, 0, -4, 4, -4, 4, 0], [-20, 2, 2, -4, 4, -2, 2, 2, -4, -2, 0, -2], [4, -4, 0, -4, 4, 0, 0, -4, -4, -4, 4, 0], [-2, 2, 0, -2, 0, 0, -2, 2, -2, -2, -6, 0], [2, -2, 13, 1, 2, -3, -3, 1, -15, 1, 2, 3], [10, 2, 10, 10, -8, -8, 10, -2, 10, -2, 10, 8], [7, -3, -3, 6, -6, 3, -3, -3, 6, 3, 0, 3], [1, -1, 11, 1, -2, -2, 1, 1, 1, 1, 1, 2], [16, 2, -4, 2, -2, 4, -4, 2, 2, -2, 8, 4], [-2, 2, -22, -2, 4, 4, -2, -2, -2, -2, -2, -4], [5, 2, -7, 5, 7, 7, 5, 2, 5, -2, 4, 7], [1, -1, 2, -1, 1, -2, 2, -1, -1, 1, 5, -2], [-9, -7, -7, 0, 0, -7, -7, 7, -14, -7, 0, -7], [8, 4, 4, 6, -6, 10, 4, -10, 20, 10, 0, 10], [-16, 2, 12, -6, -2, -4, 4, -6, 2, 6, -8, -4], [8, 0, 0, 4, 4, -4, 0, -4, -12, -4, 8, 4], [3, 0, -3, 3, 3, 3, 3, 0, 3, 0, 21, 3], [5, -1, 10, 5, -6, -6, 5, 1, 5, 1, 5, 6], [1, 1, 1, 4, -4, 5, 1, -5, 10, 5, 0, 5], [-1, 0, 1, -1, 1, 1, -1, 0, -1, 0, -1, -1], [-4, 8, 8, 0, 0, 8, 8, -8, 16, 8, 0, 8], [18, -4, 4, 5, 5, -9, 4, -9, -3, -9, 18, 9], [2, 6, 6, 2, 4, 4, 2, -6, 2, -6, 2, -4], [6, 6, -4, 2, -6, 8, -8, 2, 6, -2, -10, 8], [-1, 1, 20, -3, -1, 4, 4, -3, -19, -3, -1, -4], [2, 2, 2, -3, 3, -1, 2, 1, -2, -1, 0, -1], [2, 0, -2, 2, 2, 2, 2, 0, 2, 0, -6, 2], [-8, 2, -2, -2, -2, 4, -2, 4, 4, 4, -8, -4], [-1, -5, 6, -1, -6, -6, -1, -5, -1, 5, -15, -6], [14, 7, 2, -1, -7, 6, -6, -1, 7, 1, 2, 6], [-8, -6, 22, -8, 2, 2, -8, 6, -8, 6, -8, -2], [10, -6, -4, 10, 4, 4, 10, -6, 10, 6, 22, 4], [12, -10, 10, -4, -4, -6, 10, -6, 2, -6, 12, 6], [8, -8, 5, -6, 8, -2, -2, -6, -13, -6, 8, 2], [-7, 7, -8, 6, -7, 1, 1, 6, 15, 6, -7, -1], [11, -4, -7, 11, 7, 7, 11, -4, 11, 4, 16, 7], [-2, -2, 0, 0, 2, -2, 2, 0, -2, 0, 2, -2], [9, -9, 8, -7, 9, -2, -2, -7, -17, -7, 9, 2], [2, -2, 2, -1, -1, -1, 2, -1, -28, -1, 2, 1], [2, 4, -4, 2, 2, 2, 2, -4, 2, -4, 2, -2], [-6, -2, 8, -4, 2, -6, 6, -4, -2, 4, 6, -6], [12, -5, -5, -2, 2, -7, -5, 7, -14, -7, 0, -7], [-6, 1, -26, -6, 7, 7, -6, -1, -6, -1, -6, -7], [3, -3, -29, 0, 3, -3, -3, 0, 26, 0, 3, 3], [0, 2, -2, 2, 2, 0, -2, 0, 26, 0, 0, 0], [-6, -13, -18, 9, 13, -4, 4, 9, -13, -9, 2, -4], [4, -7, -7, -4, 4, -11, -7, 11, -22, -11, 0, -11], [-6, 1, 6, -3, -1, -2, 2, -3, 1, 3, -2, -2], [-24, 2, -2, -10, -10, 12, -2, 12, 6, 12, -24, -12], [-5, 5, 13, 0, -5, 5, 5, 0, -8, 0, -5, -5], [-10, -4, 15, -10, 6, 6, -10, 4, -10, 4, -10, -6], [-16, 0, 0, 2, -2, 2, 0, -2, 4, 2, 0, 2], [6, -10, -16, 8, 10, -2, 2, 8, -10, -8, 10, -2], [2, -7, 7, -6, -6, -1, 7, -1, -4, -1, 2, 1], [2, 0, 25, 2, -2, -2, 2, 0, 2, 0, 2, 2], [15, -15, -2, -6, 15, -9, -9, -6, -13, -6, 15, 9], [8, -8, 0, 8, 0, 0, 8, -8, 8, 8, 4, 0], [0, -2, -2, -1, 1, -3, -2, 3, -6, -3, 0, -3], [-14, 4, 10, -14, -10, -10, -14, 4, -14, -4, -18, -10], [-8, 10, -2, -8, 2, 2, -8, 10, -8, -10, -26, 2], [-28, 5, -5, -9, -9, 14, -5, 14, 11, 14, -28, -14], [5, 7, 3, 5, 2, 2, 5, -7, 5, -7, 5, -2], [-2, -1, 10, -5, 1, -6, 6, -5, -1, 5, 10, -6], [2, -2, 2, -1, -1, -1, 2, -1, -31, -1, 2, 1], [5, -8, 3, 5, -3, -3, 5, -8, 5, 8, -1, -3], [0, 4, -4, 4, 4, 0, -4, 0, 14, 0, 0, 0], [5, 0, -5, 5, 5, 5, 5, 0, 5, 0, 24, 5], [-7, 2, 2, 4, -4, 6, 2, -6, 12, 6, 0, 6], [-3, 3, 3, 6, -3, -3, -3, 6, 0, 6, -3, 3], [-4, 4, 22, 4, -4, 0, 0, 4, -18, 4, -4, 0], [-2, -8, 8, -9, -9, 1, 8, 1, -12, 1, -2, -1], [-16, 1, 6, -3, -1, -2, 2, -3, 1, 3, -12, -2], [25, -3, -3, 10, -10, 7, -3, -7, 14, 7, 0, 7], [-5, -1, 6, -5, -6, -6, -5, -1, -5, 1, 1, -6], [-10, 10, -16, 2, -10, 8, 8, 2, 26, 2, -10, -8], [-14, 6, -6, -1, -1, 7, -6, 7, 0, 7, -14, -7], [4, 2, 0, 0, -2, 2, -2, 0, 2, 0, 0, 2], [-6, 4, 4, 2, -2, 6, 4, -6, 12, 6, 0, 6], [6, -4, 4, -1, -1, -3, 4, -3, -13, -3, 6, 3], [2, -2, -16, 6, 2, -8, -8, 6, 14, 6, 2, 8], [2, -6, 4, 2, -4, -4, 2, -6, 2, 6, -2, -4], [-3, -5, 11, -3, -2, -2, -3, 5, -3, 5, -3, 2], [-5, -3, -25, -5, 2, 2, -5, 3, -5, 3, -5, -2], [-33, -3, -3, -4, 4, -7, -3, 7, -14, -7, 0, -7], [-8, 8, -8, 4, 4, 4, -8, 4, 4, 4, -8, -4], [4, -6, -6, 4, -10, -10, 4, 6, 4, 6, 4, 10], [-12, 12, 12, 6, -12, 6, 6, 6, 0, 6, -12, -6], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, -4, 0, 4, 4, 0, 4, 0, -4, 24, 4], [9, 2, -11, 9, 11, 11, 9, 2, 9, -2, -6, 11], [6, -3, 3, 0, 0, -3, 3, -3, 34, -3, 6, 3], [-3, -7, -16, 8, 7, 1, -1, 8, -7, -8, -5, 1], [-12, -12, 12, -12, 0, 0, -12, 12, -12, 12, -12, 0], [20, -2, 2, 8, 8, -10, 2, -10, 16, -10, 20, 10], [-2, 5, -5, 4, 4, 1, -5, 1, -4, 1, -2, -1], [0, 7, -7, 0, 7, 7, 0, 7, 0, -7, 21, 7], [-8, 8, -2, 6, -8, 2, 2, 6, 10, 6, -8, -2], [1, -1, -3, -2, 1, 1, 1, -2, 2, -2, 1, -1], [5, -7, -14, 7, 7, 0, 0, 7, -7, -7, 5, 0], [-12, 12, -6, 10, -12, 2, 2, 10, 18, 10, -12, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_210_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_210_2_n_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_n_b(:prec:=12)
    chi := MakeCharacter_210_n();
    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_n_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_n_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_210_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2401, 1323, 876, 17, 36, 3, 1]>],Snew);
    return Vf;
end function;