
// Make newform 5929.2.a.i in Magma, downloaded from the LMFDB on 24 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_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_5929_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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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 := [[-1, -1], [1, -1], [-1, 0], [0, 0], [0, 0], [0, -2], [0, 5], [3, 2], [2, -5], [-4, 1], [3, -2], [4, -4], [5, -10], [-7, 9], [6, -1], [-3, -1], [-8, 5], [7, -1], [-4, 7], [-13, 5], [13, -2], [7, 1], [1, 6], [-5, 3], [-7, 0], [8, -11], [-7, 3], [-1, -6], [-4, -4], [2, 3], [-2, -8], [-14, 8], [-11, 7], [-7, 8], [-7, 3], [5, 8], [-10, 16], [-5, 6], [-4, 4], [16, 1], [-5, -2], [0, -12], [17, 2], [-11, 7], [-8, -6], [7, -2], [15, -1], [6, 13], [8, -13], [-8, -2], [-22, 12], [-14, 5], [-7, 15], [-23, 0], [3, -9], [-6, 18], [-15, 3], [-2, -11], [15, -13], [-19, 10], [-7, 8], [11, 0], [17, -10], [-9, 0], [-14, 7], [-10, -9], [0, -10], [21, -8], [11, 14], [-26, 11], [-21, 5], [-15, 1], [-18, 24], [-21, -9], [16, -26], [21, -22], [-28, 10], [7, 10], [-20, -4], [14, -6], [-20, 16], [2, 2], [-12, 3], [17, 13], [-4, -1], [0, -5], [8, -7], [-2, -3], [8, -17], [21, -2], [-6, -17], [-23, -2], [-19, 22], [24, 3], [-7, 5], [20, -24], [-1, -19], [11, -2], [-11, 9], [-3, -6], [-11, -15], [-11, -2], [19, -21], [20, -28], [17, -28], [-1, 14], [-1, 0], [0, 18], [-16, 0], [-10, 6], [-27, -11], [9, 16], [2, -12], [36, -6], [3, -24], [23, -14], [-9, -12], [0, 10], [17, -2], [29, -4], [29, -9], [-14, -12], [27, -15], [11, 17], [0, 30], [8, -17], [1, 13], [-7, -16], [-3, -25], [-48, 8], [-12, -1], [-18, -5], [24, -28], [-16, -2], [2, 28], [-43, 9], [-39, 2], [-15, 9], [-31, 2], [22, -17], [-16, -24], [27, -48], [-22, 16], [-6, 22], [35, -21], [-17, 22], [38, 5], [12, -5], [-16, 9], [-43, 8], [13, 0], [-16, -3], [10, -35], [-43, 23], [-17, -2], [-39, 10], [-5, 16], [-24, 0], [27, -40], [-11, 26], [10, 6], [-5, 2], [12, -25], [6, -36], [-17, -10], [13, 1], [-10, -15], [33, -37], [23, -34], [3, -22], [-27, 32], [30, -1], [2, 26], [1, 20], [11, -37], [-6, -3], [15, 28], [27, -45], [-22, -21], [-26, -5], [1, -34], [-28, 46], [16, 13], [43, -24], [35, -32], [35, 16], [0, 1], [35, 2], [13, -42], [-28, 23], [-40, -3], [-19, -3], [-2, -17], [-21, 18], [-34, 25], [2, -27], [-28, 32], [-11, 4], [41, 7], [23, -28], [-1, 24], [45, -35], [-21, 12], [9, -23], [-33, 4], [1, 11], [43, -4], [23, 18], [-54, 11], [48, -23], [-34, 7], [-14, 39], [12, 30], [-5, -16], [31, -15], [-26, -23], [12, -26], [42, -16], [6, 19], [-33, 44], [41, -25], [33, 2], [-25, 32], [-1, 11], [14, 22], [29, -43], [26, 1], [16, 2], [0, -4], [4, 11], [6, -36], [0, 42], [-12, 35], [-41, 6], [-42, -4], [-44, 35], [39, -28], [37, -48], [-19, -2], [-11, 8], [7, -31], [13, -8], [13, -34], [33, -8], [-17, 46], [-16, 20], [13, 9], [-1, -13], [38, -19], [45, -13], [-14, 16], [-3, -6], [-57, 12], [43, -36], [-30, 2], [39, -2], [31, -46], [11, 33], [55, 7], [25, 4], [-31, -20], [-40, 27], [-13, 17], [-15, 18], [-10, 48], [24, -16], [20, -40], [45, -6], [-2, 2], [5, -13], [-48, 38], [17, -37], [-26, 24], [-5, -14], [21, 12], [29, -50], [-14, 7], [16, -7], [-7, 33], [-17, 30], [11, 7], [13, 32], [17, 20], [-65, -4], [-37, 0], [-39, 14], [12, -23], [27, -10], [-32, -22], [62, -14], [-63, 3], [-46, -17], [-17, 18], [-29, 39], [-39, 65], [-31, -18], [-28, 19], [19, -44], [12, -39], [-6, 5], [11, 6], [30, 31], [19, -34], [-24, 23], [13, 22], [40, -8], [25, -12], [30, 30], [-32, -1], [16, -29], [29, 30], [2, 37], [-59, 31], [-21, -10], [29, -54], [-42, -3], [3, 32], [41, -50], [19, -60], [-25, 36], [-15, 14], [-5, -7], [28, -18], [-30, -30], [-57, -11], [43, -16], [-18, -21], [82, -10], [-39, -25], [11, 30], [19, 25], [-27, 54], [-31, -1], [30, 36], [-40, 29], [21, -3], [6, -36], [-26, 36], [46, -60], [29, -16], [-43, 35], [23, 8], [-23, -20], [0, -39], [52, 0], [-7, -28], [18, 44], [-4, -29], [57, -13], [25, 26], [33, -42], [25, 21], [-11, 38], [38, -38], [-56, 9], [-1, 54], [44, -18], [-7, 21], [-3, 2], [-64, -10], [50, 26], [46, -10], [-1, 6], [-73, 5], [-22, 13], [68, -9], [-65, 10], [-57, -13], [38, 17], [-35, -19], [57, -21], [8, 37], [21, 46], [-43, 5], [66, 5], [31, 28], [13, -15], [-17, 42], [35, -83], [-11, 4], [12, -1], [-21, -16], [-65, 10], [-1, -8], [70, 7], [-84, -2], [5, -51], [-14, -14], [7, 34], [-56, -4], [-45, 55], [8, 28], [-63, -20], [-29, -34], [-49, 32], [8, -10], [51, -39], [-75, 18], [40, -8], [5, -32], [-14, -16], [-67, 17], [34, -74], [-6, 50], [-17, -35], [-59, 41], [-39, 48], [27, -32], [52, 18], [-59, -4], [-77, -9], [43, -28], [-11, 60], [-53, 36], [66, -30], [-47, 84], [-16, 22], [-78, 20], [54, 18], [-59, 6], [-15, 38], [-58, 62], [-38, 52], [-17, 38], [17, -34], [42, -12], [-27, 26], [-80, -1], [32, -49], [51, -30]];
    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_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_5929_2_a_i(: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_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_5929_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5929_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 1]>,<3,R![-1, -1, 1]>,<5,R![1, 1]>],Snew);
    return Vf;
end function;