
// Make newform 819.2.j.c in Magma, downloaded from the LMFDB on 25 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_c();".
// 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_c();".
// 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, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 1], [-1, -2, 2, -1]];
    Rf_basisdens := [1, 1, 2, 2];
    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], [-1, 0, 0, -1], [1, 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 := [[-1, -1, 0, -1], [0, 0, 0, 0], [-1, 2, 0, -1], [-1, 0, 0, 2], [0, 0, 0, 3], [-1, 0, 0, 0], [0, -4, -4, 5], [-3, 0, 0, -3], [-5, -2, 0, -5], [2, 0, 4, 0], [0, 0, 0, 5], [5, -6, 0, 5], [-2, 0, -4, 0], [-8, 0, 0, 0], [-1, -4, 0, -1], [0, 4, 4, 1], [0, 4, 4, -5], [-3, 0, 0, -3], [0, 0, 0, -3], [-4, 0, -8, 0], [0, -6, -6, 7], [-7, 6, 0, -7], [0, 0, 0, 0], [-1, 2, 0, -1], [-10, 0, -12, 0], [0, 0, 0, 9], [-7, 6, 0, -7], [-13, 2, 0, -13], [0, -6, -6, 7], [10, 0, 8, 0], [8, 0, 12, 0], [-5, -2, 0, -5], [0, 2, 2, 5], [-4, 0, 12, 0], [-9, 6, 0, -9], [0, 12, 12, 1], [0, 0, 0, 7], [-3, 12, 0, -3], [-16, 0, 4, 0], [3, -12, 0, 3], [0, 18, 18, -9], [-6, 0, 12, 0], [-5, 10, 0, -5], [0, 6, 6, -9], [22, 0, 8, 0], [0, -6, -6, -11], [4, 0, 0, 0], [-4, 0, 0, 0], [0, -8, -8, 7], [5, -18, 0, 5], [-1, 8, 0, -1], [0, 0, 12, 0], [0, 6, 6, -5], [-8, 0, -4, 0], [-13, 8, 0, -13], [0, -18, -18, 3], [0, -4, -4, -7], [1, 12, 0, 1], [0, -12, -12, 19], [-14, 0, 8, 0], [0, 18, 18, -3], [2, 0, -8, 0], [0, 0, -12, 0], [0, -2, -2, 31], [25, -12, 0, 25], [-5, -2, 0, -5], [21, -12, 0, 21], [18, 0, 0, 0], [0, -26, -26, 19], [-10, 0, 12, 0], [0, 6, 6, -27], [-1, 8, 0, -1], [0, -6, -6, -3], [21, -12, 0, 21], [4, 0, 12, 0], [15, 0, 0, 15], [0, -4, -4, 5], [-3, 18, 0, -3], [11, 2, 0, 11], [0, 6, 6, -1], [-20, 0, 8, 0], [-6, 0, -12, 0], [0, 20, 20, -1], [10, 0, -12, 0], [5, -18, 0, 5], [-1, 2, 0, -1], [2, 0, -20, 0], [-23, 18, 0, -23], [-2, 0, 20, 0], [-24, 0, 0, 0], [-13, 14, 0, -13], [0, -24, -24, 15], [0, -24, -24, 17], [32, 0, 4, 0], [-7, -12, 0, -7], [-8, 0, 8, 0], [-17, -2, 0, -17], [0, 8, 8, 5], [-23, -6, 0, -23], [-11, 6, 0, -11], [-24, 0, -12, 0], [0, 6, 6, 9], [0, -22, -22, 23], [19, 4, 0, 19], [0, -18, -18, -7], [0, 6, 6, -29], [32, 0, 16, 0], [31, -2, 0, 31], [0, 26, 26, -25], [-22, 0, 0, 0], [5, -18, 0, 5], [0, -18, -18, 11], [-2, 0, -4, 0], [0, 0, 0, 17], [-16, 0, -24, 0], [0, -8, -8, 1], [-4, 0, -24, 0], [0, -2, -2, -17], [11, -4, 0, 11], [-4, 0, -8, 0], [0, -6, -6, 3], [10, 0, 12, 0], [23, -16, 0, 23], [0, 4, 4, 7], [13, 24, 0, 13], [-10, 0, -20, 0], [21, 18, 0, 21], [15, 6, 0, 15], [0, 0, 24, 0], [-19, -6, 0, -19], [0, -24, -24, 21], [-20, 0, 8, 0], [-19, 18, 0, -19], [-14, 0, 24, 0], [15, -30, 0, 15], [46, 0, 0, 0], [0, 26, 26, 5], [0, 12, 12, 1], [-22, 0, -8, 0], [0, -12, -12, -3], [24, 0, 24, 0], [35, 2, 0, 35], [0, 18, 18, 5], [12, 0, 24, 0], [0, 0, 0, -25], [-20, 0, -16, 0], [6, 0, 36, 0], [0, 20, 20, -31], [-11, -6, 0, -11], [11, 8, 0, 11], [-3, -18, 0, -3], [2, 0, 28, 0], [20, 0, 0, 0], [-17, -2, 0, -17], [0, 6, 6, 9], [-24, 0, -36, 0], [9, -18, 0, 9], [-41, 10, 0, -41], [6, 0, -12, 0], [0, -2, -2, -47], [-29, 4, 0, -29], [-10, 0, -32, 0], [36, 0, 12, 0], [-21, 6, 0, -21], [0, -14, -14, 25], [0, -4, -4, -1], [0, 6, 6, 21], [0, 12, 12, 7]];
    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_c();".
// 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_c(:prec:=4)
    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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 8, 3, 1]>],Snew);
    return Vf;
end function;