
// Make newform 592.2.a.i in Magma, downloaded from the LMFDB on 02 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_592_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_592_2_a_i();".
// 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_592_2_a_i();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 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_592_a();"
function MakeCharacter_592_a()
    N := 592;
    order := 1;
    char_gens := [223, 149, 113];
    v := [1, 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_592_a_Hecke(Kf)
    return MakeCharacter_592_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 := [[0, 0, 0], [-1, 0, -1], [0, 0, 1], [-2, -1, 1], [0, 3, 0], [1, -3, 0], [-2, 2, -2], [-2, -2, 2], [-3, 1, 0], [-3, 1, 0], [-6, 0, -1], [-1, 0, 0], [-5, -2, 1], [0, -2, -4], [-4, -3, -1], [-2, 5, -3], [2, 2, 4], [4, 2, -3], [0, 6, -5], [2, -3, 1], [-2, -3, 0], [1, -1, 2], [4, -5, 3], [4, 0, -4], [2, 2, 6], [-2, -5, 3], [-6, -2, -3], [5, -5, 8], [14, -2, -2], [-2, 4, 4], [-2, 9, 1], [2, 6, 0], [2, -6, 5], [3, -3, -4], [-2, -1, 1], [-4, -8, 2], [2, -5, 3], [-10, 0, -4], [-2, 4, 7], [-6, 1, 1], [-8, -2, -8], [4, 7, -1], [-2, 8, -5], [-4, 6, -2], [-10, -5, 3], [0, 2, 2], [-16, -1, 2], [-4, 5, -5], [4, 0, -2], [-16, -3, -5], [-3, 9, 2], [2, -2, 1], [4, -6, 6], [8, 2, 2], [2, 6, 6], [0, -11, -1], [12, -6, 10], [8, -5, -1], [-8, -6, 3], [4, -8, 0], [-12, 8, -2], [-24, -2, -4], [13, 0, 5], [-23, -1, -2], [-14, 4, -6], [8, 10, -6], [4, 6, -2], [-18, -3, 0], [28, -2, 0], [16, -2, 0], [8, 0, -2], [-14, 5, 5], [2, 6, -8], [-4, 11, -7], [25, 4, 1], [0, 12, -4], [21, -1, 2], [0, 9, -1], [6, 4, -6], [16, -12, -2], [-5, -6, 1], [-1, 17, -2], [0, -6, -10], [-3, 10, 7], [-11, 3, -6], [-4, -3, 8], [-2, 2, -10], [18, -4, 2], [-2, 4, 12], [13, -11, 0], [-4, 14, 2], [-5, -5, 8], [12, 0, -4], [3, -3, -8], [-2, 6, 0], [-10, 4, 1], [6, -7, 11], [22, 3, -5], [18, -12, 2], [4, 2, -3], [20, 0, 6], [-8, -4, -9], [2, -20, 12], [20, 0, 8], [-29, 2, 1], [6, 2, -4], [-14, 10, -6], [-18, -7, 8], [2, 13, -11], [-22, 2, 1], [-17, 1, 2], [-8, -11, -9], [-25, -8, -1], [-12, -5, -8], [25, 3, -2], [7, 2, 5], [2, 4, -2], [-9, -3, -2], [-11, 3, 2], [1, 2, 11], [-5, -15, -6], [-25, -2, -11], [-10, 13, -21], [-8, 2, -10], [-8, 12, -4], [16, -2, 7], [-16, -2, 1], [-12, 1, -15], [-14, -12, 1], [-26, -1, 7], [4, -13, -4], [-20, 7, 1], [-2, 1, 5], [8, -20, 13], [-7, -14, 3], [-10, 8, 0], [24, 1, 13], [4, -7, 17], [-37, -3, 6], [-6, 12, -10], [-24, 7, -12], [-6, 17, -15], [16, 8, 0], [14, -4, 10], [-1, 1, -2], [-24, 8, 10], [-16, 12, 5], [2, -18, 14], [10, 26, -18], [4, 12, -8], [6, -4, -12], [6, 2, -13], [-8, 12, -8], [-18, -15, 5], [-10, -10, 6], [4, -10, -2], [4, -10, 14], [10, -2, -5], [23, -4, -1], [8, -18, 8], [-12, 8, -8], [-2, 23, -18], [14, 10, 1], [-12, -10, -11], [-18, -6, -2], [-20, 15, -5], [-3, 21, 8], [2, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_592_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_592_2_a_i();".
// 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_592_2_a_i(:prec:=3)
    chi := MakeCharacter_592_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_592_2_a_i();".
// 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_592_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_592_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-7, -4, 2, 1]>,<5,R![2, -5, 1, 1]>],Snew);
    return Vf;
end function;