
// Make newform 819.2.s.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_819_s_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_819_2_s_d();".
// 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_819_2_s_d();".
// 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 := [1, 1, 8, 1, 47, 17, 55, -11, 33, -2, 7, -1, 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], [-2119374, 5624108, -14058754, 8363982, -31119861, 16008568, -16878368, 2684563, -3846008, 682506, -478424, -29696], [-2229034, 21668998, -14803002, 8704710, -34289666, 16328192, -18206928, 1631143, -4188508, 495146, -498559, -73788], [-6198231, -133032, -450388, -1984215, -2183844, -4702740, 2259680, -3276280, 406680, -747485, 153752, -109660], [572347, 646135, 2423530, 22831649, 12171095, 21902645, -1556185, 14090715, -131145, 2921453, -329655, 439315], [-6707921, 12588213, -44235433, 27901335, -63065800, 58565408, -46353032, 24340355, -9944902, 5617249, -1732988, 566698], [3392561, -2155246, 22259302, -14729214, 12172746, -33035270, 20386532, -19018596, 4066147, -4336084, 1011662, -572347], [6345807, -4177460, 41650219, -27474213, 26529336, -61367800, 38513384, -34760912, 7725614, -7924683, 1869572, -1035034], [566698, 7274619, -8054629, 44802131, -1266529, 72699666, -27397018, 40119354, -5639321, 8811506, -1650363, 1166290], [-1035561, -15484451, 33752077, -96392052, 11027874, -158252610, 66981583, -90433689, 13501144, -19974443, 3882058, -2686072], [1, 8, 1, 47, 17, 55, -11, 33, -2, 7, -1, 1]];
    Rf_basisdens := [1, 1, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 1];
    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_819_s();"
function MakeCharacter_819_s()
    N := 819;
    order := 3;
    char_gens := [92, 703, 379];
    v := [3, 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_819_s_Hecke();"
function MakeCharacter_819_s_Hecke(Kf)
    N := 819;
    order := 3;
    char_gens := [92, 703, 379];
    char_values := [[1, 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, -1, 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, -1, 0, -1, 0, 0, 0, 0, 0, 1], [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, 1, 0, 0, -1, 0, 1, 1, -1, 0, 0], [0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0], [0, -1, 0, -1, 1, 0, -1, 0, 0, 0, -1, 0], [1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 1], [-1, -2, 0, 0, -2, -2, 0, 1, -2, -1, 1, 2], [-1, 0, 3, 1, -1, -2, 1, 0, 0, -3, 1, -1], [-1, 2, 0, 0, 2, 2, 0, 1, 2, 0, 1, 1], [3, 2, 0, 0, 0, 2, 0, -3, 0, 2, 1, 0], [2, 0, 3, 0, 0, 0, -1, 0, 0, -3, -1, 0], [2, 0, 0, 0, 0, 0, 0, -2, 0, -1, -2, -1], [0, 2, 0, -1, 0, 0, 2, -3, 0, 0, 0, 0], [0, 4, 0, -3, 0, 0, 2, -2, -2, 0, 0, 0], [0, -1, 0, 0, -2, -1, 0, 0, -2, -3, 0, 0], [1, 0, 1, -1, 2, 1, 2, 0, 0, -1, 2, 1], [2, 6, 0, 0, 0, 6, 0, -2, 0, 2, -2, -5], [0, -1, -2, -2, 0, 0, 3, -3, -2, 0, 0, 0], [0, 0, 1, -3, 0, 0, 2, -2, 2, 0, 0, 0], [-4, 5, 0, 0, 0, 5, 0, 4, 0, 0, 0, -1], [0, 1, -2, 3, 0, 0, 0, 2, 1, 0, 0, 0], [4, 0, 1, -1, 3, 0, -1, 0, 0, -1, -1, 1], [4, 0, 0, 5, 1, 8, -4, 0, 0, 0, -4, -5], [0, 2, 0, -4, 0, 0, 3, -8, -1, 0, 0, 0], [0, 3, 1, -4, 0, 0, 1, -7, -1, 0, 0, 0], [0, 6, 1, -1, 0, 0, -2, 4, 1, 0, 0, 0], [8, 0, -2, -1, -3, 1, 1, 0, 0, 2, 1, 1], [0, -5, 0, 0, -1, -5, 0, 0, -1, -1, -1, -4], [0, 8, 0, -5, 0, 0, 2, 1, 2, 0, 0, 0], [1, 8, 0, 0, 1, 8, 0, -1, 1, 2, 0, -3], [0, 2, 2, 1, 0, 0, -2, -4, 1, 0, 0, 0], [-5, 0, 1, -1, 4, -1, -1, 0, 0, -1, -1, 1], [0, -6, 0, 0, 0, 0, 0, 4, -3, 0, 0, 0], [4, -6, 0, 0, -5, -6, 0, -4, -5, -3, 2, 0], [2, -6, 0, 0, -2, -6, 0, -2, -2, 3, 0, 6], [0, -1, 0, 0, 5, -1, 0, 0, 5, -2, 2, -2], [3, 4, 0, 0, 2, 4, 0, -3, 2, 2, 3, -1], [2, 0, 0, 0, 2, 0, 0, -2, 2, 2, 4, 3], [-6, 4, 0, 0, 0, 4, 0, 6, 0, -1, 3, 2], [0, -7, -3, 8, 0, 0, 0, 3, -4, 0, 0, 0], [4, 0, 2, 1, -3, -3, 2, 0, 0, -2, 2, -1], [-2, 6, 0, 0, -1, 6, 0, 2, -1, 3, -1, -7], [0, 2, -1, 3, 0, 0, 3, 0, -4, 0, 0, 0], [-4, -4, 0, 0, 1, -4, 0, 4, 1, -5, -5, -1], [2, 0, 3, 3, -3, 6, -4, 0, 0, -3, -4, -3], [-6, -2, 0, 0, 4, -2, 0, 6, 4, 4, -1, 6], [-2, -6, 0, 0, 1, -6, 0, 2, 1, -5, 3, -1], [-5, 0, 4, -2, -3, -7, -7, 0, 0, -4, -7, 2], [0, -10, -3, 5, 0, 0, 1, -2, -4, 0, 0, 0], [0, -1, -1, -1, 0, 0, 2, 10, 4, 0, 0, 0], [-8, 0, -1, 7, -4, 0, -1, 0, 0, 1, -1, -7], [-9, 0, 1, -1, 0, -6, -2, 0, 0, -1, -2, 1], [0, -7, -8, 3, 0, 0, -1, 4, -3, 0, 0, 0], [10, 0, -1, 0, 2, 2, -7, 0, 0, 1, -7, 0], [0, 10, 8, -1, 0, 0, 2, 1, 5, 0, 0, 0], [13, 0, -1, -3, -2, 9, 2, 0, 0, 1, 2, 3], [5, 0, 4, -7, 5, -3, 0, 0, 0, -4, 0, 7], [0, 0, -3, -6, 7, 0, -2, 0, 0, 3, -2, 6], [5, 0, 1, 5, 6, 7, -2, 0, 0, -1, -2, -5], [0, 2, 6, -7, 0, 0, 1, 5, 1, 0, 0, 0], [0, -6, -3, 1, 0, 0, 0, -9, -4, 0, 0, 0], [-2, 0, -7, -8, 3, 0, 2, 0, 0, 7, 2, 8], [-6, 8, 0, 0, 9, 8, 0, 6, 9, 5, 3, -2], [0, -1, -6, -1, 0, 0, -1, 5, 1, 0, 0, 0], [0, 0, 2, -7, 0, 0, -2, -7, 8, 0, 0, 0], [-5, 4, 0, 0, -2, 4, 0, 5, -2, 0, 2, -8], [-3, 0, -5, 8, -3, 6, 0, 0, 0, 5, 0, -8], [-1, 0, 3, 0, -5, -2, 4, 0, 0, -3, 4, 0], [-12, 5, 0, 0, -1, 5, 0, 12, -1, -5, 4, 4], [0, -8, -6, -5, 0, 0, -7, 7, -9, 0, 0, 0], [0, -4, -1, -1, 0, 0, -6, 6, -4, 0, 0, 0], [0, -9, -1, 7, 0, 0, 3, 9, -3, 0, 0, 0], [6, -3, 0, 0, -1, -3, 0, -6, -1, 0, 9, 0], [-1, -4, 0, 0, -1, -4, 0, 1, -1, -4, -6, 0], [-5, 4, 0, 0, -3, 4, 0, 5, -3, -2, 8, 0], [12, 5, 0, 0, 4, 5, 0, -12, 4, 4, -6, 0], [17, 5, 0, 0, 2, 5, 0, -17, 2, 5, 1, 6], [-11, 0, 6, 0, 3, -5, -1, 0, 0, -6, -1, 0], [7, 0, 3, 2, -3, -5, -2, 0, 0, -3, -2, -2], [-13, 2, 0, 0, 2, 2, 0, 13, 2, 8, 1, 1], [1, 0, 9, 1, -2, -10, 8, 0, 0, -9, 8, -1], [0, -1, -3, 9, 0, 0, 4, -1, 2, 0, 0, 0], [0, 1, -1, -7, 0, 0, 14, -6, 1, 0, 0, 0], [0, 0, 1, 11, 0, 9, -8, 0, 0, -1, -8, -11], [0, 6, -5, 7, 0, 0, -1, 4, -2, 0, 0, 0], [0, 2, -6, 0, 0, 0, -2, 2, 5, 0, 0, 0], [8, 0, -2, 4, -3, 7, -3, 0, 0, 2, -3, -4], [0, -6, 5, 3, 0, 0, 1, -10, 4, 0, 0, 0], [5, 0, 5, 3, 5, 7, -5, 0, 0, -5, -5, -3], [0, 9, 8, -9, 0, 0, 1, -10, 7, 0, 0, 0], [0, -1, 2, 12, 0, 0, 2, 10, 0, 0, 0, 0], [-4, 0, 5, 12, -7, 1, 0, 0, 0, -5, 0, -12], [-13, -2, 0, 0, -5, -2, 0, 13, -5, 3, 10, 0], [0, -1, -1, 3, 0, 0, 2, 18, 11, 0, 0, 0], [0, 5, 6, 1, 0, 0, -11, -7, -2, 0, 0, 0], [-12, 0, 0, -11, 3, 1, 0, 0, 0, 0, 0, 11], [-5, -9, 0, 0, -10, -9, 0, 5, -10, -6, -1, -1], [-9, 0, 5, -10, 4, -10, 2, 0, 0, -5, 2, 10], [0, -12, -1, 4, 0, 0, -11, 0, -3, 0, 0, 0], [5, 0, 2, -1, -6, -10, -8, 0, 0, -2, -8, 1], [0, 6, -4, -2, 0, 0, -2, -7, -3, 0, 0, 0], [6, 0, -7, 4, 8, 14, 0, 0, 0, 7, 0, -4], [2, 0, 4, -1, -10, 5, -2, 0, 0, -4, -2, 1], [-3, 1, 0, 0, -3, 1, 0, 3, -3, 0, 1, -5], [2, -3, 0, 0, 1, -3, 0, -2, 1, -3, 3, 0], [-5, -3, 0, 0, -5, -3, 0, 5, -5, -4, 2, 6], [0, -18, -11, 0, 0, 0, 8, -5, -7, 0, 0, 0], [-5, 11, 0, 0, 5, 11, 0, 5, 5, 1, -10, -6], [4, 16, 0, 0, 11, 16, 0, -4, 11, 10, 0, -15], [-2, -4, 0, 0, 5, -4, 0, 2, 5, 12, -2, 5], [0, 4, -1, 2, 0, 0, -2, -6, 4, 0, 0, 0], [0, 13, 2, 4, 0, 0, 6, 6, 1, 0, 0, 0], [-3, 5, 0, 0, 3, 5, 0, 3, 3, -1, 4, -3], [0, 0, -12, -3, 0, 0, -5, 15, -8, 0, 0, 0], [-18, 0, -2, -4, 1, -17, 3, 0, 0, 2, 3, 4], [0, 10, -4, 2, 0, 0, -4, -10, 2, 0, 0, 0], [0, -1, 4, -3, 0, 0, 5, -20, -6, 0, 0, 0], [-1, 0, -6, -2, 6, -3, 4, 0, 0, 6, 4, 2], [0, -13, -2, 8, 0, 0, 0, 3, -1, 0, 0, 0], [0, 10, 10, 0, 0, 0, 9, -18, 3, 0, 0, 0], [2, -4, 0, 0, -4, -4, 0, -2, -4, 2, 3, 0], [0, -17, -7, 4, 0, 0, -8, -7, -10, 0, 0, 0], [-4, 0, -5, 8, -12, -2, -5, 0, 0, 5, -5, -8], [-20, 0, -4, 5, 5, 17, 1, 0, 0, 4, 1, -5], [22, 0, -2, 12, -3, 19, -1, 0, 0, 2, -1, -12], [2, 14, 0, 0, 0, 14, 0, -2, 0, 6, -5, -22], [-20, -8, 0, 0, 2, -8, 0, 20, 2, 6, 11, 10], [-5, 0, -9, -9, 2, -6, 8, 0, 0, 9, 8, 9], [0, 7, -5, -9, 0, 0, -4, -6, -3, 0, 0, 0], [0, -5, -9, 19, 0, 0, -2, 2, 0, 0, 0, 0], [-3, 1, 0, 0, -9, 1, 0, 3, -9, 3, -3, -1], [-6, 0, 5, 8, -2, -7, 4, 0, 0, -5, 4, -8], [8, -6, 0, 0, -13, -6, 0, -8, -13, -8, 0, -4], [-4, 16, 0, 0, 7, 16, 0, 4, 7, 19, -2, 0], [2, 1, 0, 0, -4, 1, 0, -2, -4, 6, -3, 10], [-8, 0, 11, -3, 1, -6, 2, 0, 0, -11, 2, 3], [-10, 0, -5, -12, -5, -7, 2, 0, 0, 5, 2, 12], [0, 18, 8, -16, 0, 0, 11, -8, 6, 0, 0, 0], [0, 10, -2, -3, 0, 0, -2, -4, 0, 0, 0, 0], [5, 0, -7, 13, 7, 21, 7, 0, 0, 7, 7, -13], [14, 0, -1, 11, 4, 12, -17, 0, 0, 1, -17, -11], [-9, 0, -1, -5, -7, -3, -1, 0, 0, 1, -1, 5], [-3, 0, 6, -11, -1, -10, -3, 0, 0, -6, -3, 11], [0, 14, -1, -12, 0, 0, 12, -2, -2, 0, 0, 0], [0, -10, 3, 10, 0, 0, 4, 11, -2, 0, 0, 0], [2, 0, 6, 7, 5, -7, -11, 0, 0, -6, -11, -7], [-11, -7, 0, 0, -6, -7, 0, 11, -6, 2, 5, -1], [0, -1, 8, -12, 0, 0, -6, -5, -14, 0, 0, 0], [0, 10, -2, 8, 0, 0, 5, -17, -1, 0, 0, 0], [14, 17, 0, 0, 9, 17, 0, -14, 9, 6, -2, -4], [-7, 15, 0, 0, 9, 15, 0, 7, 9, 18, -2, 2], [-13, 0, -3, -5, -1, -4, 3, 0, 0, 3, 3, 5], [21, 0, -6, 10, -6, 10, 0, 0, 0, 6, 0, -10], [0, 10, 6, 7, 0, 0, -1, -8, -5, 0, 0, 0], [-13, 0, -7, -10, 2, 2, 16, 0, 0, 7, 16, 10], [-10, 11, 0, 0, 3, 11, 0, 10, 3, -6, -2, 5], [-4, -9, 0, 0, -7, -9, 0, 4, -7, -10, -5, 0], [15, 0, 7, 10, -5, 10, 0, 0, 0, -7, 0, -10], [16, 10, 0, 0, 5, 10, 0, -16, 5, 7, 2, -23], [3, 0, 0, 4, -6, -8, -3, 0, 0, 0, -3, -4], [0, 2, 9, -3, 0, 0, 1, -19, -7, 0, 0, 0], [0, 0, -10, 1, 5, 15, 0, 0, 0, 10, 0, -1], [-4, 12, 0, 0, 0, 12, 0, 4, 0, 10, 1, 11], [0, -10, -7, 5, 0, 0, 5, 16, -11, 0, 0, 0], [-5, 8, 0, 0, -4, 8, 0, 5, -4, -5, -10, -18], [0, 12, -9, -21, 0, 0, 14, -8, 3, 0, 0, 0], [12, 0, 6, 1, -9, 2, 4, 0, 0, -6, 4, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_819_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_s_d();".
// 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_819_2_s_d(:prec:=12)
    chi := MakeCharacter_819_s();
    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_819_2_s_d();".
// 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_819_2_s_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -14, 8, 11, -6, -2, 1]>,<11,R![6561, 16767, 36288, 29079, 23994, 2862, 6811, 1155, 664, 68, 37, 4, 1]>],Snew);
    return Vf;
end function;