
// Make newform 5929.2.a.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5929_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_5929_2_a_m();".
// 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_5929_2_a_m();".
// 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, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_5929_a();"
function MakeCharacter_5929_a()
    N := 5929;
    order := 1;
    char_gens := [1816, 2059];
    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_5929_a_Hecke(Kf)
    return MakeCharacter_5929_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, -1], [-1, 1], [2, 0], [0, 0], [0, 0], [1, 1], [-1, -1], [2, 2], [-2, 2], [-4, -2], [5, -1], [-4, -2], [-9, -1], [-8, 0], [-5, 1], [4, -2], [-1, 1], [-5, -1], [10, 2], [-6, -2], [-3, 1], [0, 4], [2, -6], [-2, 0], [-4, -6], [-7, 1], [-1, -3], [4, 0], [0, 2], [2, 0], [-12, 4], [4, 8], [12, 2], [-6, 2], [-14, 0], [0, -4], [-2, -4], [10, -6], [-4, 4], [-15, 1], [0, -4], [12, 6], [-12, 4], [6, -8], [2, 0], [-9, 5], [-4, 8], [-1, 5], [-12, 8], [0, 2], [4, -6], [-8, 8], [7, -9], [-11, 3], [6, 0], [0, 0], [0, -6], [-6, -2], [2, 8], [8, -2], [12, 8], [-5, -9], [0, -4], [15, -3], [-6, -4], [14, 0], [4, -8], [16, -2], [-12, -4], [-5, -1], [-2, -8], [16, 4], [3, 9], [-6, 0], [6, 14], [-11, 7], [-2, 8], [18, 8], [14, 4], [-11, 9], [-9, -7], [0, 10], [-12, 0], [4, 2], [6, 2], [-16, -4], [-24, -2], [2, 12], [-1, -5], [-10, 2], [5, -13], [18, -2], [-14, -10], [0, 0], [6, -2], [10, 6], [-36, -2], [-8, -10], [44, 0], [6, -12], [-28, 0], [30, 4], [26, -6], [-12, 2], [24, -4], [24, -2], [-7, -9], [-39, -3], [-10, -6], [-17, -9], [4, 4], [16, -14], [20, -6], [9, -9], [-24, 0], [-20, 2], [9, -9], [-7, -13], [34, -4], [0, 8], [14, 12], [8, 6], [-3, -3], [12, -8], [15, -15], [20, -2], [-6, -4], [9, 19], [13, -17], [11, -1], [-16, -4], [16, -8], [34, -2], [-2, 8], [-7, -11], [25, -3], [16, 10], [-30, -6], [6, 4], [-30, 4], [-8, -12], [-18, -12], [-32, 8], [4, 4], [-10, -12], [-15, -13], [-15, -7], [13, 1], [-7, 15], [-12, 4], [-28, 6], [-16, 6], [0, -4], [18, -10], [18, 2], [38, 2], [-24, -8], [12, 18], [5, -7], [-17, 11], [-32, 12], [-14, -4], [-4, 8], [9, -9], [14, 4], [-33, 5], [14, 18], [17, -7], [30, 8], [-20, -2], [-4, -16], [10, -8], [0, -4], [52, 2], [3, 1], [-36, -6], [4, 4], [12, -18], [-2, -6], [-15, 1], [-28, 4], [2, -14], [30, 8], [-13, 3], [4, -4], [16, 2], [-14, 16], [-39, 7], [8, 6], [-12, -4], [-40, 6], [4, 0], [42, 2], [-18, -12], [32, -8], [-34, 0], [-12, 2], [-36, -10], [-7, 13], [-30, 14], [-38, 0], [-32, 16], [-24, 18], [25, 13], [41, -1], [2, 16], [-1, 13], [16, -12], [-10, -16], [-1, 17], [32, -2], [4, 6], [24, 0], [-1, 1], [2, -18], [-18, 8], [4, -8], [-19, -7], [-22, -10], [10, 4], [-12, -10], [-46, -10], [36, 10], [24, 16], [32, 4], [-36, -10], [16, -10], [8, 16], [-22, 14], [28, -4], [-12, 6], [48, -12], [44, -4], [26, 12], [1, 7], [10, -10], [4, -14], [14, 12], [-2, -10], [17, 19], [-10, -2], [-18, -2], [-31, -5], [32, 2], [-21, 19], [36, -4], [1, 3], [-21, -11], [12, 8], [-52, -4], [18, 16], [-3, 9], [42, 6], [10, -12], [-37, -1], [-56, 4], [-4, -26], [46, 6], [30, -4], [15, 7], [-4, 16], [-32, 12], [-21, 23], [69, -3], [-22, 12], [1, 7], [-12, -26], [-24, -18], [-32, 8], [36, -10], [-2, -20], [-18, -26], [-8, 6], [0, 16], [45, 9], [7, 5], [-12, 8], [-26, -4], [10, -12], [-46, -6], [8, -8], [38, -14], [18, 14], [-11, 5], [-26, -18], [46, -14], [20, -22], [38, -4], [59, 1], [-13, -5], [-24, -6], [1, 23], [8, 30], [-10, 6], [-34, -4], [31, -21], [1, -13], [-12, 14], [6, -18], [14, 14], [25, -27], [28, 22], [52, -16], [-2, -26], [18, 2], [60, 2], [40, 12], [-6, -24], [-4, 4], [6, -20], [52, -12], [-26, -16], [12, 10], [-8, 28], [4, 0], [-51, -15], [-35, -5], [-4, 28], [-32, 6], [16, 6], [20, 8], [-10, 4], [-73, 3], [-12, 12], [-2, 16], [-10, 16], [-18, 30], [-31, -17], [-16, 20], [28, -18], [-42, -20], [12, 30], [-72, 8], [-6, -6], [12, -4], [-3, -21], [12, 30], [-49, 15], [22, 24], [-42, 2], [54, 16], [12, 10], [9, 1], [-72, 0], [34, 12], [-24, -4], [-20, -2], [-2, 14], [-28, -4], [-4, -2], [6, -18], [40, 6], [-58, -8], [14, 10], [2, -24], [7, 15], [71, -3], [-20, 0], [26, 20], [12, 4], [-18, -4], [31, -25], [-28, -12], [12, -4], [-95, -1], [-4, -30], [-63, 1], [56, 8], [-6, -28], [14, -2], [27, -27], [-4, 28], [-2, -44], [24, 32], [16, -14], [9, -1], [-76, 0], [71, -9], [23, 7], [15, 7], [10, -8], [-32, -2], [60, -8], [14, -8], [-6, -38], [11, 17], [56, 0], [42, 8], [36, -8], [-61, 1], [20, 2], [-32, 2], [26, 26], [-79, 7], [-12, -20], [24, 14], [-78, 6], [66, 12], [-42, -10], [-4, 22], [-53, -1], [42, -8], [0, 16], [-86, 4], [-39, 1], [-20, -28], [-66, 0], [-2, 16], [-41, 9], [-13, 13], [-31, 21], [40, 26], [16, 12], [-16, 0], [12, 18], [58, 0], [56, -8], [37, -17], [30, 24], [-52, 12], [-86, 4], [-21, 23], [-18, 26], [-20, -16], [-18, 16], [-24, 18], [10, 34], [-20, 26], [63, 1], [30, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5929_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5929_2_a_m();".
// 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_5929_2_a_m(:prec:=2)
    chi := MakeCharacter_5929_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(2999) - 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_5929_2_a_m();".
// 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_5929_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5929_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, 0, 1]>,<3,R![-4, 2, 1]>,<5,R![-2, 1]>],Snew);
    return Vf;
end function;