
// Make newform 819.2.do.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_do();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_819_do_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_do_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_do_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 := [4, -2, -1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, -1, 1, 1], [2, 1, 1, -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_do();"
function MakeCharacter_819_do()
    N := 819;
    order := 6;
    char_gens := [92, 703, 379];
    v := [6, 4, 5];
    // 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_do_Hecke();"
function MakeCharacter_819_do_Hecke(Kf)
    N := 819;
    order := 6;
    char_gens := [92, 703, 379];
    char_values := [[1, 0, 0, 0], [-1, 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, 0, 1], [0, 0, 0, 0], [1, 0, 1, 0], [-1, 2, 2, -2], [2, 0, -4, 0], [-3, 0, 4, 0], [1, 0, -1, 0], [-2, 4, 4, -4], [-2, 4, -1, -2], [-7, 0, 7, 0], [-4, 0, 1, 2], [-2, 0, -1, 4], [1, -4, -3, 0], [-2, 4, 3, -2], [-1, -2, -3, 0], [-4, 8, 9, -4], [3, 2, 5, 0], [2, 4, 0, 4], [-2, 8, 4, -8], [-4, 0, 5, -6], [10, 0, -5, 0], [7, -2, -9, 4], [-2, 0, 4, 0], [18, 0, -9, 0], [6, 0, -1, -4], [-6, 4, 0, 4], [-9, 2, 11, -4], [2, -4, -9, 2], [10, 0, -1, -8], [4, -8, -3, 4], [-1, -2, -1, 4], [7, -2, -9, 4], [5, 0, 5, 0], [2, -4, -1, 2], [8, -8, -16, 8], [-4, 0, -3, 10], [-4, 8, 5, -4], [2, 0, -4, 0], [-1, 6, 5, 0], [-10, 8, 0, 8], [-8, -4, 0, -4], [22, 0, 0, 0], [0, -8, 0, -8], [-8, 8, 16, -8], [-7, 12, 5, 0], [3, -2, -5, 4], [-1, -2, -1, 4], [-1, 2, 1, 0], [11, -18, -7, 0], [5, -4, 1, 0], [9, 4, -5, -8], [-14, 12, 28, -12], [5, -12, -7, 0], [-2, 4, -13, -2], [-8, 16, 21, -8], [12, 4, 0, 4], [5, 4, -1, -8], [8, 0, -11, 14], [13, -8, -21, 16], [-8, 8, 16, -8], [-16, 8, 0, 8], [2, 0, -9, 16], [2, 8, -4, -8], [-2, 4, -1, -2], [25, -4, -29, 8], [1, 8, 9, 0], [-10, -4, 20, 4], [2, -12, 0, -12], [-9, -6, 3, 12], [5, -4, 1, 0], [4, -12, -8, 12], [7, -14, -7, 0], [8, -4, 0, -4], [-26, 0, 0, 0], [7, -14, -7, 0], [-10, 16, 20, -16], [8, -16, -3, 8], [-4, 20, 8, -20], [-34, 0, 15, 4], [17, -4, 13, 0], [-5, 6, 11, -12], [-12, 28, 24, -28], [-10, 4, 20, -4], [8, -16, -15, 8], [2, -4, 3, 2], [11, -6, -17, 12], [2, 0, 3, -8], [14, 0, -13, 12], [6, 0, 7, -20], [2, 16, -4, -16], [-1, -10, -9, 20], [-14, 24, 28, -24], [-9, -10, -19, 0], [-13, 6, 19, -12], [15, 2, 17, 0], [6, -12, -21, 6], [14, 0, -1, -12], [8, -16, -27, 8], [6, -12, -13, 6], [-7, -8, -15, 0], [-8, 0, 0, 0], [-4, -8, 8, 8], [11, -2, -13, 4], [-8, 16, 25, -8], [-6, 12, -25, -6], [-10, 0, 3, 4], [11, -6, 5, 0], [-7, 12, 5, 0], [-6, 12, 31, -6], [-15, -4, 11, 8], [-8, 0, 0, 0], [-4, -8, 8, 8], [6, 0, 3, -12], [-20, 0, 17, -14], [-32, 0, 13, 6], [17, -12, -29, 24], [24, 0, -7, -10], [24, -8, 0, -8], [12, -24, -19, 12], [14, -28, -33, 14], [12, -12, -24, 12], [13, 8, -5, -16], [41, -4, -45, 8], [-21, 6, -15, 0], [-24, 0, 17, -10], [-34, 0, 0, 0], [12, 4, -24, -4], [20, -8, 0, -8], [28, 0, 0, 0], [13, -16, -3, 0], [-6, -4, 12, 4], [-5, -2, -7, 0], [10, -20, -25, 10], [29, -12, -41, 24], [16, -12, -32, 12], [-3, -4, -7, 0], [-11, 12, 1, 0], [-17, 2, -15, 0], [4, -8, 9, 4], [26, -4, 0, -4], [10, -8, -20, 8], [-26, 0, 7, 12], [-25, -2, 23, 4], [6, 12, -12, -12], [-18, 16, 0, 16], [40, 0, -23, 6], [-28, 8, 56, -8], [8, -16, -19, 8], [19, 10, -9, -20], [-9, 10, 1, 0], [16, -20, -32, 20], [-15, 12, 27, -24], [36, -8, 0, -8], [2, -4, -45, 2], [16, 4, 0, 4], [-20, 16, 0, 16], [-12, -4, 0, -4], [-16, 32, 32, -32], [22, 12, 0, 12], [-14, 0, -1, 16], [-4, 0, 9, -14], [8, -16, 17, 8], [-2, 4, 4, -4], [-36, 8, 0, 8], [-8, 8, 16, -8], [24, 0, -11, -2], [4, -16, 0, -16], [25, 8, -17, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_819_do_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_do_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_do_d(:prec:=4)
    chi := MakeCharacter_819_do();
    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_do_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_do_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_do();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 2, 3, 1]>],Snew);
    return Vf;
end function;