
// Make newform 567.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_567_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_567_2_a_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_567_2_a_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, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 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_567_a();"
function MakeCharacter_567_a()
    N := 567;
    order := 1;
    char_gens := [407, 325];
    v := [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;

function MakeCharacter_567_a_Hecke(Kf)
    return MakeCharacter_567_a();
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], [0, 0, 0], [-1, -1, 1], [1, 0, 0], [-2, 1, -1], [-1, 2, -4], [-2, 0, 1], [-1, -2, 1], [-4, -1, -2], [-3, -5, 4], [-1, 3, 3], [-1, 3, 3], [0, -1, -1], [-1, -1, -1], [-1, -2, -3], [-2, 3, -2], [1, 5, 0], [2, 3, 0], [-4, 0, 3], [-3, -3, 6], [-7, 4, 1], [-7, 0, 3], [6, 4, 1], [-4, 3, -7], [-1, -7, 8], [7, -2, -1], [2, -3, 0], [-1, 0, -4], [5, 0, -3], [-12, 2, -4], [-4, -13, 5], [5, -7, 10], [-5, 7, -7], [5, -10, 5], [2, -7, 7], [-1, 9, -12], [-1, 10, -5], [2, 0, -3], [3, 14, -4], [-13, 7, 3], [7, 0, 1], [-7, -3, -3], [-1, 1, -9], [5, -3, 0], [-15, -3, 6], [2, -3, 0], [-1, 2, 2], [5, 6, -6], [7, 1, 2], [5, -12, 0], [-7, -9, 5], [-3, 8, 2], [-10, -3, 0], [6, -6, -9], [-16, -4, -2], [1, -14, 17], [7, 9, -2], [8, -3, 3], [2, 6, 3], [-18, -8, 7], [5, -15, 3], [-3, -7, 2], [2, 8, -7], [16, -1, -3], [-1, -17, 10], [8, -11, 3], [14, 4, 1], [5, 3, 12], [-5, 8, -15], [8, -10, 8], [-8, -5, 2], [2, 6, 5], [5, -5, -5], [5, 1, -5], [-1, 5, -10], [11, -1, -11], [7, 4, -13], [-19, -3, -3], [-4, 11, 7], [-4, 12, -18], [12, -13, 8], [-13, 6, -15], [-5, 15, 4], [17, -10, 14], [11, -11, -8], [2, 4, 6], [0, -6, 3], [-19, 12, -15], [-6, -2, 7], [-4, -3, 6], [23, -3, 11], [25, 4, -7], [-7, 1, 16], [9, -13, -7], [2, 6, -18], [12, -9, -9], [8, 11, -2], [13, 6, -2], [11, 15, -12], [11, 10, -2], [8, -1, 20], [1, -21, 4], [10, -13, 21], [17, 10, -6], [-13, 13, -14], [-16, -9, 0], [-18, 7, 13], [-19, -6, -7], [-14, 14, -12], [-22, 5, -1], [17, 22, -8], [-1, 6, -24], [-24, 5, 11], [-22, 15, 0], [2, -1, 17], [-14, 13, -10], [-19, -13, 2], [7, -18, 19], [8, 2, -23], [3, 28, -5], [5, -5, 22], [14, 9, -3], [-1, 1, -24], [-32, 0, -2], [-4, -9, 15], [-6, 12, -6], [2, -6, 12], [11, 21, -16], [-13, 17, -19], [11, 6, 12], [-31, 3, 0], [6, -10, 23], [11, 16, -11], [14, 8, -13], [-10, 2, 16], [20, 9, -12], [1, -12, 16], [-19, -15, 9], [-24, 2, -19], [-8, -12, 7], [-4, 9, -24], [5, 22, 3], [32, 0, -12], [-3, -3, -18], [5, 10, 1], [-9, -8, 10], [11, 0, -15], [4, -17, 14], [-31, -12, 0], [2, -12, 11], [2, 0, -9], [-33, -21, 15], [5, 3, -21], [-4, 17, -26], [17, -18, -6], [-9, -11, 22], [8, 3, 9], [-25, 19, -12], [2, -16, 17], [31, -1, -15], [8, -17, 12], [-18, 24, -12], [-22, -28, 11], [26, 18, -19], [-17, -16, 24], [-32, 13, -4], [35, 0, 15], [2, -12, 24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_567_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_567_2_a_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_567_2_a_c(:prec:=3)
    chi := MakeCharacter_567_a();
    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_567_2_a_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_567_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_567_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 0, 3, 1]>],Snew);
    return Vf;
end function;