
// Make newform 819.2.j.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_819_j_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_j_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_819_2_j_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 := [4, 16, 64, 28, 58, 18, 41, 7, 8, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [30518, 2376, 9416, 84549, 6908, 880, -5913, -220, 176, -364], [-22604, -98928, -392048, -39747, -287624, -36640, -87336, 9160, -7328, -983], [-348592, -132894, -526654, 25194, -386377, -49220, -109053, 12305, -9844, -1159], [-180842, -41094, -162854, -129531, -119477, -15220, -3978, 3805, -3044, 916], [-3988, -20316, -46906, 39066, -152578, 31195, -34617, 26720, -9541, 2249], [-2692, -957744, -258154, -871206, -272902, -637445, -107253, -121720, 14531, -15259], [532, 1362924, 255184, 981276, 92092, 719570, 96813, 145570, -21776, 18139], [-136, 50058, 11648, 53832, 2993, 39580, 3726, 9041, -1552, 1058]];
    Rf_basisdens := [1, 1, 118350, 118350, 118350, 59175, 78900, 236700, 118350, 4734];
    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_j();"
function MakeCharacter_819_j()
    N := 819;
    order := 3;
    char_gens := [92, 703, 379];
    v := [3, 2, 3];
    // 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_j_Hecke();"
function MakeCharacter_819_j_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, -1, 0, 0], [1, 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, -1, 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, 1, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0, -1], [2, 0, 0, 1, 0, 0, -1, -2, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, -2, 0, -1, -1, 0, 1, 1, 1], [0, -2, 0, 0, 0, 0, 1, -2, 1, 1], [0, 0, 0, 0, 0, 0, 0, 2, 0, 1], [1, 0, 2, 0, 1, 0, 0, 0, 0, 0], [2, 0, 0, 0, -1, -2, 0, -2, 2, 1], [0, -2, 0, 0, 0, 0, -2, 0, 0, -1], [-4, 0, -2, 2, -3, 1, 0, 0, 0, 0], [0, 0, 2, 2, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -4], [3, -4, -4, -2, -2, 0, 2, -3, 0, 2], [2, 0, 0, 1, 0, 0, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, -4, 3, 0, 2], [-4, 2, 2, 3, 1, 3, -3, 4, -3, -1], [-2, 0, 4, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 1, 2, 0, -1, 0], [0, -2, 0, 0, 0, 0, 2, 0, -1, -2], [-2, 0, 2, 0, 1, 3, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, 0, -2, 3], [-2, 0, -2, -2, 1, -1, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -3, 0, 2, 3, -2], [0, 8, 0, 0, 0, 0, 2, 0, -2, -3], [0, 2, 0, 0, 0, 0, 0, 12, -1, -2], [0, -6, -6, 2, -1, 0, -2, 0, 0, 1], [-1, 0, -2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, -4, -3, -2], [6, 4, 4, -2, 6, -1, 2, -6, 1, -6], [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, -2, -4, 1, 2], [10, 2, 2, -3, 5, -1, 3, -10, 1, -5], [7, -2, -2, 2, -1, -1, -2, -7, 1, 1], [0, -2, 0, 0, 0, 0, -3, -4, 3, 1], [-6, 0, -8, -3, 1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, -1, -2, 2], [6, 0, 0, 0, -3, 2, 0, -6, -2, 3], [3, 0, 6, -2, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, 6, 1, 0], [-4, -2, -2, -2, -1, 4, 2, 4, -4, 1], [0, 0, -2, 0, -5, 1, 0, 0, 0, 0], [-12, -4, -4, -2, 5, 0, 2, 12, 0, -5], [0, 0, -2, -6, -3, -1, 0, 0, 0, 0], [2, 0, -10, 1, -8, 1, 0, 0, 0, 0], [-2, 4, 4, -4, 1, 2, 4, 2, -2, -1], [0, 2, 0, 0, 0, 0, 0, -8, -2, 7], [0, 4, 0, 0, 0, 0, -2, -1, -2, -4], [-8, 0, -2, 1, -4, -5, 0, 0, 0, 0], [2, -6, -6, 4, -5, 0, -4, -2, 0, 5], [14, 0, 2, -4, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 6, -1, 4], [8, 0, 0, 0, -5, -2, 0, -8, 2, 5], [-7, 6, 6, -2, 3, 3, 2, 7, -3, -3], [0, 2, 0, 0, 0, 0, -1, 4, -1, -1], [5, 8, 8, -4, 2, -2, 4, -5, 2, -2], [-2, 0, -2, -4, 7, 3, 0, 0, 0, 0], [10, -6, -6, 0, 0, 3, 0, -10, -3, 0], [8, 0, -6, -4, 5, -9, 0, 0, 0, 0], [0, 0, -6, -5, 4, -3, 0, 0, 0, 0], [0, 4, 4, -4, 5, 4, 4, 0, -4, -5], [0, 4, 0, 0, 0, 0, -2, 0, -3, -4], [0, 0, 0, 0, 0, 0, 6, -8, 5, 2], [0, 0, 0, 0, 0, 0, 4, 6, 4, -9], [-7, 0, -2, -2, -3, 6, 0, 0, 0, 0], [16, -4, -4, 0, -5, 2, 0, -16, -2, 5], [-2, 0, -2, 2, -3, -5, 0, 0, 0, 0], [8, 2, 2, 4, -1, -4, -4, -8, 4, 1], [0, 4, 0, 0, 0, 0, 8, -2, -4, -7], [-14, 2, 2, -4, -6, -3, 4, 14, 3, 6], [0, 2, 0, 0, 0, 0, -6, 7, -5, 7], [0, 0, 10, -2, 11, 1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, 0, -2, 3], [5, 6, 6, -4, -7, -1, 4, -5, 1, 7], [0, -4, 0, 0, 0, 0, 6, -4, 5, 0], [0, 6, 0, 0, 0, 0, 0, 4, -4, -1], [10, 4, 4, 6, -2, 1, -6, -10, -1, 2], [6, 0, 0, 4, -4, 2, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 2, -5, 0, -2, 12, 0, 5], [-7, 0, -4, 4, -2, 1, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 20, -3, 4], [0, -10, 0, 0, 0, 0, -2, 22, -1, 2], [8, 0, 4, -4, 6, 4, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -2, -9], [8, 0, 2, -2, 11, 3, 0, 0, 0, 0], [2, 0, 2, -4, -3, 3, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 8, -1, 0], [10, -8, -8, 7, -4, -6, -7, -10, 6, 4], [12, 12, 12, 5, 2, 6, -5, -12, -6, -2], [-10, 0, 4, 8, -6, 4, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 6, 4, 3], [-2, 0, 10, -2, -1, 7, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 16, 4, 1], [-8, 4, 4, 10, -4, 3, -10, 8, -3, 4], [0, 8, 0, 0, 0, 0, 4, -14, 6, -3], [0, -4, 0, 0, 0, 0, 0, 12, -3, 2], [-6, 0, -14, 4, -7, -5, 0, 0, 0, 0], [-2, 8, 8, 4, 0, 7, -4, 2, -7, 0], [-8, -10, -10, -4, 8, 3, 4, 8, -3, -8], [0, -8, 0, 0, 0, 0, -6, -17, 4, -2], [-2, -4, -4, -6, -1, -4, 6, 2, 4, 1], [-2, 2, 2, -6, 17, -4, 6, 2, 4, -17], [10, 0, -8, 5, 1, -6, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -12, 12, -1, -2], [6, -6, -6, -2, 2, -11, 2, -6, 11, -2], [13, 0, -6, 8, 5, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 8, 14, -7, 0], [8, -4, -4, 4, -6, -7, -4, -8, 7, 6], [-4, 0, -2, 4, 7, -7, 0, 0, 0, 0], [-16, 12, 12, -6, 5, 4, 6, 16, -4, -5], [-6, 0, -6, -4, 9, -1, 0, 0, 0, 0], [0, 0, 0, -6, 18, -7, 6, 0, 7, -18], [-10, 0, -8, 7, 3, 6, 0, 0, 0, 0], [0, 2, 2, -10, -6, -1, 10, 0, 1, 6], [0, 4, 0, 0, 0, 0, 6, 4, 1, 10], [4, 0, -16, -4, -4, -8, 0, 0, 0, 0], [12, -2, -2, 2, 9, -2, -2, -12, 2, -9], [-12, 0, -8, 10, -10, -2, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 8, 1, 4, -14], [28, 12, 12, 2, 3, 4, -2, -28, -4, -3], [0, -16, 0, 0, 0, 0, 5, 6, 4, 0], [-2, 0, 4, 4, -10, -2, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 2, 8, 7, 2], [0, 8, 0, 0, 0, 0, 0, -18, -4, 1], [6, 0, 14, 2, 5, 1, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, -2, -14, 4, 1], [-10, -4, -4, 8, -1, -2, -8, 10, 2, 1], [-6, 0, 10, 11, -4, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -6, 0, -7], [11, 0, 0, -2, 2, -7, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 4, 4, 3, -6], [-2, 0, -4, 4, 2, -6, 0, 0, 0, 0], [-4, 6, 6, 4, 3, 0, -4, 4, 0, -3], [-26, -10, -10, -1, 3, 3, 1, 26, -3, -3], [-8, 0, 8, 10, 6, 2, 0, 0, 0, 0], [-3, -2, -2, 4, 3, 5, -4, 3, -5, -3], [-26, 0, 6, 0, -5, 5, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, -2, -2, -7, 8], [12, 8, 8, -4, 1, -10, 4, -12, 10, -1], [-28, 0, 4, 1, -5, 2, 0, 0, 0, 0], [-1, 12, 12, -2, 10, 6, 2, 1, -6, -10], [-20, 0, -2, 7, 2, 7, 0, 0, 0, 0], [0, 0, -12, 2, -12, -4, 0, 0, 0, 0], [-3, -8, -8, -12, -8, -6, 12, 3, 6, 8], [0, -6, 0, 0, 0, 0, 4, 20, 3, -8], [0, -8, 0, 0, 0, 0, -16, 18, -2, 15], [0, -14, 0, 0, 0, 0, 2, -16, -2, 17], [0, 0, -12, -14, -4, -4, 0, 0, 0, 0], [16, 0, -2, 4, 7, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 16, -14, 4, -11], [-2, 2, 2, 0, -4, -13, 0, 2, 13, 4], [16, 0, 2, -2, 3, 5, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 8, -10, 4, -13], [0, 2, 0, 0, 0, 0, 2, 22, 4, 7], [-9, 0, 0, -12, 8, 1, 0, 0, 0, 0], [-12, 10, 10, 6, -5, 6, -6, 12, -6, 5], [0, 0, 0, 0, 0, 0, 7, 16, -2, -8], [17, 0, 4, -10, 12, 1, 0, 0, 0, 0], [8, 0, 0, 11, -5, -8, 0, 0, 0, 0], [0, 18, 0, 0, 0, 0, 2, -6, 5, -8], [-6, -8, -8, 0, 10, -3, 0, 6, 3, -10], [0, 6, 6, 1, 5, 7, -1, 0, -7, -5], [8, 4, 4, -4, 7, 0, 4, -8, 0, -7], [-3, -8, -8, -14, 4, 2, 14, 3, -2, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_819_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_j_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_819_2_j_h(:prec:=10)
    chi := MakeCharacter_819_j();
    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_j_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_819_2_j_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 36, 195, -210, 265, -116, 81, -30, 17, -4, 1]>],Snew);
    return Vf;
end function;