
// Make newform 819.2.o.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_819_o_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_o_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_o_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, 4, 15, 6, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-20, 4, 15, -25, 5, -1], [216, 20, 75, -46, 25, -5], [5, 315, 95, 105, -21, 20], [128, 480, 299, 160, -32, 38]];
    Rf_basisdens := [1, 1, 79, 79, 79, 79];
    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_o();"
function MakeCharacter_819_o()
    N := 819;
    order := 3;
    char_gens := [92, 703, 379];
    v := [3, 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_819_o_Hecke();"
function MakeCharacter_819_o_Hecke(Kf)
    N := 819;
    order := 3;
    char_gens := [92, 703, 379];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 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 := [[1, 0, -1, -1, 1, 1], [0, 0, 0, 0, 0, 0], [-1, 0, 1, 2, 0, 0], [1, 0, 0, 0, 1, 0], [0, 1, 0, 0, 3, 0], [-1, -1, 0, 1, -1, -2], [1, 0, 0, 0, 1, -1], [3, 2, -2, 0, 3, 0], [1, -1, -1, -1, -3, 1], [3, 2, -3, -3, 4, 3], [-4, 0, 1, 4, 0, 0], [2, 1, -2, -2, 1, 2], [-4, 0, 4, 4, -2, -4], [7, 1, -1, 0, 7, 1], [6, 0, -3, 2, 0, 0], [-3, 0, -2, -2, 0, 0], [-1, 1, -1, 0, -1, -1], [-5, 0, 0, 0, -5, -2], [3, -2, -3, -3, 2, 3], [4, 5, -5, 0, 4, -1], [-1, 0, -2, 3, 0, 0], [-4, 0, 9, 2, 0, 0], [1, 0, 1, -6, 0, 0], [-6, -4, 6, 6, 3, -6], [-8, 1, -1, 0, -8, 2], [1, -1, -1, -1, -7, 1], [-9, 0, 6, 8, 0, 0], [3, 1, -3, -3, 8, 3], [-5, 0, -3, -3, 0, 0], [-6, -7, 7, 0, -6, -2], [4, 3, -4, -4, 1, 4], [-17, 0, -1, 2, 0, 0], [-1, -8, 8, 0, -1, 0], [-4, 0, 0, 0, -4, -1], [-7, -1, 1, 0, -7, -2], [4, 0, -8, -6, 0, 0], [12, 0, -1, 2, 0, 0], [11, 0, 0, 0, 11, -4], [2, -8, -2, -2, -5, 2], [15, 1, -1, 0, 15, 5], [3, -8, -3, -3, 0, 3], [0, 0, 2, -3, 0, 0], [-1, 1, -1, 0, -1, 1], [5, -3, -5, -5, 13, 5], [-11, -13, 11, 11, -6, -11], [-6, -11, 11, 0, -6, -3], [-3, -9, 3, 3, -13, -3], [-10, -4, 10, 10, -13, -10], [-2, 11, -11, 0, -2, 8], [-6, 0, 2, -1, 0, 0], [13, 0, 2, 0, 0, 0], [8, 0, -2, -2, 0, 0], [-2, -7, 7, 0, -2, -4], [-4, 1, -1, 0, -4, -6], [-3, -9, 3, 3, -8, -3], [-7, -11, 7, 7, -10, -7], [-20, 1, -1, 0, -20, -6], [3, -10, -3, -3, -9, 3], [8, 6, -6, 0, 8, -2], [-12, 0, 15, 2, 0, 0], [-5, -10, 5, 5, 2, -5], [-11, -1, 1, 0, -11, -8], [-1, 0, -7, -11, 0, 0], [14, 0, -7, -5, 0, 0], [-1, 0, 1, -8, 0, 0], [0, 0, 1, -7, 0, 0], [6, 8, -8, 0, 6, 7], [2, 0, -1, 2, 0, 0], [-10, 10, -10, 0, -10, 11], [-1, 6, 1, 1, -12, -1], [6, 1, -6, -6, 4, 6], [-14, 0, 8, 8, 0, 0], [11, 13, -11, -11, 5, 11], [5, 3, -3, 0, 5, 3], [-2, -4, 2, 2, -24, -2], [-8, 4, -4, 0, -8, 0], [-6, 0, 10, 14, 0, 0], [-1, 11, -11, 0, -1, 2], [-1, 6, 1, 1, 8, -1], [-14, -2, 2, 0, -14, -5], [8, 1, -8, -8, -9, 8], [28, 0, -3, -8, 0, 0], [-2, -15, 2, 2, -14, -2], [3, 4, -4, 0, 3, 14], [-3, 12, 3, 3, 10, -3], [3, 0, -6, -6, 0, 0], [-17, -11, 11, 0, -17, -10], [1, 7, -1, -1, 31, 1], [2, 13, -13, 0, 2, 10], [2, 0, -6, 1, 0, 0], [1, 0, 5, -9, 0, 0], [13, -1, -13, -13, 16, 13], [6, 2, -2, 0, 6, 1], [1, 8, -1, -1, -6, 1], [-12, 0, -5, -3, 0, 0], [-7, -8, 8, 0, -7, -6], [-4, 10, 4, 4, 0, -4], [25, 0, 5, 1, 0, 0], [6, -11, -6, -6, 0, 6], [-29, 0, 1, 11, 0, 0], [-14, 0, -15, -9, 0, 0], [-15, 6, 15, 15, -10, -15], [5, -8, 8, 0, 5, -1], [3, 5, -3, -3, 19, 3], [2, 0, -23, -21, 0, 0], [20, 0, 2, 3, 0, 0], [-12, -5, 12, 12, -10, -12], [-9, 0, 13, 11, 0, 0], [-33, 0, 0, 8, 0, 0], [0, 3, 0, 0, 13, 0], [1, 8, -8, 0, 1, 13], [2, -13, -2, -2, 3, 2], [11, -3, 3, 0, 11, -12], [16, 0, -10, -10, 0, 0], [-7, -4, 4, 0, -7, -19], [1, -13, 13, 0, 1, -15], [-19, -14, 14, 0, -19, 2], [5, 6, -5, -5, 8, 5], [-6, 8, 6, 6, 7, -6], [-6, 19, -19, 0, -6, 24], [-16, -3, 16, 16, 0, -16], [-7, -1, 7, 7, 11, -7], [13, 0, -13, -3, 0, 0], [23, 3, -3, 0, 23, 7], [-3, -13, 3, 3, 3, -3], [7, 0, -7, -19, 0, 0], [0, -9, 9, 0, 0, -1], [10, -9, 9, 0, 10, 1], [5, 0, -3, 9, 0, 0], [32, 0, -9, -11, 0, 0], [-7, -1, 7, 7, -11, -7], [-9, -12, 9, 9, 16, -9], [15, 18, -15, -15, 14, 15], [1, 5, -1, -1, -13, 1], [-26, 11, -11, 0, -26, 4], [4, 2, -4, -4, 30, 4], [-11, -4, 4, 0, -11, 1], [-4, -8, 8, 0, -4, 11], [-8, -17, 17, 0, -8, -17], [-5, -6, 5, 5, -39, -5], [-25, 0, -4, 7, 0, 0], [16, 14, -16, -16, -4, 16], [-7, 12, -12, 0, -7, -14], [24, 0, 1, -17, 0, 0], [12, -4, -12, -12, 12, 12], [-8, 8, -8, 0, -8, 1], [-5, 0, 28, 21, 0, 0], [-25, 0, -1, 9, 0, 0], [14, 0, -9, -12, 0, 0], [0, 0, 10, 15, 0, 0], [-20, -6, 6, 0, -20, -1], [-21, -4, 21, 21, -8, -21], [-14, 0, 13, 4, 0, 0], [12, 12, -12, -12, 6, 12], [-4, 7, 4, 4, 1, -4], [15, 0, -24, -22, 0, 0], [4, 7, -7, 0, 4, -2], [6, 7, -7, 0, 6, 23], [-4, 0, -6, 2, 0, 0], [-7, 0, -7, -6, 0, 0], [-6, -5, 6, 6, 16, -6], [31, 12, -12, 0, 31, 19], [1, 0, 2, 18, 0, 0], [-9, 18, -18, 0, -9, 16], [-1, 19, 1, 1, -1, -1], [-31, 0, 10, 24, 0, 0], [21, 22, -21, -21, 2, 21], [8, -8, 8, 0, 8, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_819_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_o_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_o_d(:prec:=6)
    chi := MakeCharacter_819_o();
    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_o_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_o_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![25, 15, 19, 4, 7, 2, 1]>,<11,R![25, 85, 249, 126, 47, 8, 1]>],Snew);
    return Vf;
end function;