
// Make newform 4598.2.a.bs in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4598_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_4598_2_a_bs();".
// 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_4598_2_a_bs();".
// 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 := [24, 6, -12, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [2, -8, -2, 1], [14, -4, -4, 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_4598_a();"
function MakeCharacter_4598_a()
    N := 4598;
    order := 1;
    char_gens := [3269, 3631];
    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_4598_a_Hecke(Kf)
    return MakeCharacter_4598_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, 0, 0, 0], [1, -1, 0, 0], [0, -1, 0, 0], [-1, 0, 0, -1], [0, 0, 0, 0], [0, 0, 1, 0], [2, 0, 0, 1], [-1, 0, 0, 0], [-1, 0, 1, 0], [3, 1, 1, -1], [2, -2, 0, 0], [3, 1, 0, 2], [-2, -1, 1, 1], [-4, 2, 2, 0], [-3, 0, -1, 1], [-3, 1, 1, -1], [1, -1, -1, 0], [-2, 1, 2, -2], [2, 2, -1, -2], [-2, -3, 1, 1], [2, 0, 0, -1], [-6, 1, -1, -1], [6, -1, 0, 0], [-2, 1, 1, 1], [-10, 0, 0, 0], [0, -3, -2, 2], [12, 1, 1, 1], [5, -3, 2, -1], [-1, -1, -1, 0], [4, -1, 1, 1], [8, -2, 0, 2], [-4, 0, 2, 2], [10, -2, -2, 1], [4, -4, 2, -2], [8, -2, 4, 0], [4, 1, -1, 1], [12, -1, -2, 4], [-8, 2, 2, -4], [4, -2, 0, 2], [-1, -3, 1, 3], [-9, 5, -1, 1], [10, 2, 0, 4], [5, -2, -2, 3], [-8, 0, 0, 0], [2, 4, 2, -4], [-4, 6, 0, -3], [3, -3, 2, 2], [12, 3, -1, -3], [-2, 2, -3, 2], [-2, 4, -2, 0], [11, 0, 3, 1], [-19, 2, -2, 0], [2, -7, 3, -1], [-2, 2, 2, 0], [-4, 2, 2, -4], [-14, 0, -3, -1], [-13, -5, 1, 3], [4, -2, 2, 1], [14, -1, 2, 0], [0, 1, -1, 1], [4, 2, -2, 2], [-11, -1, 0, 2], [-1, -1, -1, -3], [8, -8, 2, -1], [5, 2, 4, -1], [4, -4, -1, 6], [-1, 3, 2, 1], [-8, -1, -5, -1], [-4, -1, -4, -4], [4, 8, -2, -4], [-9, 2, -3, 3], [-16, 2, 2, -1], [15, 0, -1, -3], [-7, -1, -7, 3], [-3, -3, 0, 5], [-4, 8, 2, -4], [0, 3, 0, 0], [10, -4, 2, 2], [-2, 1, -1, -3], [-4, 4, 0, -4], [0, 5, -4, -2], [-1, 5, -2, -1], [-10, -1, -3, 1], [8, 2, 0, 0], [8, 4, 2, 0], [0, 0, -4, -2], [-14, 2, 2, 0], [1, 0, -3, 0], [-4, 5, -2, 0], [2, -2, 3, -3], [0, 4, 6, -6], [7, 2, -2, -2], [4, -6, -4, 2], [-6, -8, 2, -2], [-4, -3, -2, 2], [-5, -2, 2, -3], [-9, -5, -2, -4], [-4, 0, 4, -6], [-7, -3, -2, 7], [10, 6, 2, -2], [-11, 3, 2, -8], [-6, 6, 4, -10], [-5, 5, 4, -2], [8, 4, 2, -4], [16, -1, -4, -2], [6, -4, 2, -3], [-18, 3, -2, 2], [3, 6, -1, 1], [-16, -4, 2, -4], [-26, 6, -2, 2], [-20, -3, 1, 5], [30, -5, -2, 0], [7, -4, 3, 5], [2, 4, 6, -6], [-3, 6, 2, -6], [-30, 5, 3, -3], [6, 0, 2, -6], [-1, -2, 6, -8], [-4, 0, -2, -6], [-2, -2, 4, -5], [-1, -3, 6, 0], [-6, -1, -7, 5], [-9, 7, -6, 3], [4, -2, 2, 6], [-10, 7, -4, 4], [0, 6, -4, 4], [4, -4, -4, 2], [2, 4, -3, -8], [-15, 2, -1, 3], [2, -2, -6, 2], [-14, -9, 2, 4], [4, -10, 6, -4], [-10, -2, -8, 8], [16, 1, 0, -2], [23, 2, 1, 6], [-1, 2, 0, -4], [25, -5, 0, -1], [-9, 3, 1, -3], [24, -4, 1, 2], [-3, 2, -1, 1], [21, -11, -5, 3], [-14, 7, -2, -2], [21, -10, 2, 0], [-18, 10, 2, -5], [-4, 4, -1, -2], [16, -5, 1, 7], [8, 0, 0, -4], [12, -6, -2, 8], [22, -3, 8, -4], [14, -2, 6, -8], [4, -8, -4, 1], [13, -4, 2, -6], [30, 3, 0, -4], [-12, -1, 7, -1], [3, -3, 1, 4], [18, -3, 1, 5], [14, 6, -1, 6], [5, -4, 2, 2], [-15, -12, 3, 1], [4, -10, 3, 2], [-10, 10, 8, -10], [-32, 5, 3, -1], [-11, -6, 3, -3], [-9, -1, -7, 7], [-2, 8, -2, -2], [4, -9, 3, 5], [6, -1, 3, 5], [38, -1, 4, -2], [-34, -3, 1, -5], [-14, 4, 4, -2], [-18, -6, 5, 4], [21, -5, -1, 9], [-2, 2, 3, -7], [-9, 0, -5, -3], [-8, -8, 6, -2], [-31, -2, 3, 1], [-16, 1, 4, -8], [8, 8, -6, 4], [6, 9, 1, -9], [-18, -5, 0, 10], [5, 0, 3, 0], [-11, -11, 4, 1], [19, -13, 9, -5], [-14, 1, -7, -3], [15, -4, -2, 8], [-8, 11, 6, 2], [-13, 7, 9, 2], [-19, -9, 3, -3], [12, -7, 9, -5], [9, -12, 2, 4], [28, 1, 3, -5], [-38, 3, -2, -2], [-8, 0, -4, -4], [-24, 8, -7, 4], [16, -13, 2, 6], [-22, 4, 4, 0], [-6, 2, 0, 7], [-12, 7, -2, -2], [5, 0, 2, 2], [-26, 2, -1, 6], [-15, 13, 4, -10], [12, 4, 4, 0], [-4, 2, 7, -4], [12, 4, -4, -4], [14, 10, 4, 0], [22, -4, -6, 8], [-3, 2, -4, -3], [23, -15, -6, 10], [-17, 8, 0, -4], [1, 5, -1, -4], [3, -6, -10, 14], [16, 4, 6, -10], [-11, 4, -5, -3], [-1, 7, 1, -3], [-12, 12, 2, 4], [-4, 6, -4, -2], [17, 2, 8, 6], [14, 6, 2, -4], [-26, -7, 3, -1], [-16, 7, -10, 8], [1, 1, 1, -1], [12, 0, 2, -10], [10, -1, -5, 7], [-8, 10, 8, -7], [-27, 3, 2, -2], [30, -3, 0, 4], [16, -15, 5, 3], [2, 0, 2, 2], [-14, 10, 2, 4], [-22, 9, 4, -6], [14, -8, -8, -4], [-13, -7, 1, 4], [0, 15, -7, 5], [-8, -2, -6, -4], [8, 6, 6, 6], [7, -4, 3, 2], [8, 0, 9, 3], [10, -9, -6, -4], [6, 15, -6, 6], [-8, 11, 5, -9], [25, 11, -5, -11], [-38, 1, 2, -2], [17, -4, 13, -10], [-3, 3, 7, -11], [30, 1, 3, 3], [23, -2, 9, -5], [-13, -8, -1, 1], [-20, -8, -2, -2], [-3, 7, -2, -12], [35, 4, 1, -3], [-6, 16, 0, -2], [17, 10, -2, -6], [24, 12, -2, 11], [4, -11, -5, 3], [2, 10, 0, -2], [-6, 1, 8, -8], [26, -9, -4, -2], [-13, 1, 9, -1], [6, 2, 3, -9], [31, 4, 3, 9], [20, -16, -6, 8], [3, 11, 6, -12], [10, 10, 4, -10], [0, 0, -4, -1], [16, -8, 3, 5], [34, -2, -7, 8], [3, -3, 6, 0], [8, 6, -3, -2], [-7, -5, -4, 0], [32, -5, 0, 10], [5, 3, 5, -5], [-17, 17, -3, 4], [-35, -2, -2, -3], [-36, 18, 2, -1], [-34, -11, 5, 3], [35, -10, -7, 1], [16, 10, 0, -4], [-1, 7, 0, -10], [-16, -5, 5, 3], [26, -1, -6, 10], [-6, 12, 2, -14], [9, -6, 7, 7], [44, 6, -2, -3], [-13, -11, 4, 10], [-12, 6, 2, 2], [-23, 8, 10, -14], [-8, 10, -5, -6], [-9, 25, -4, -2], [-7, -2, -2, 8], [-24, 17, -1, 1], [-24, 11, 0, 0], [-10, 1, -2, -12], [-36, -3, 3, -9], [-39, 7, -8, 2], [35, 11, -1, -3], [16, -1, -2, -2], [-16, -9, -3, 15], [14, -6, 8, -4], [-3, -9, -14, 11], [26, 6, 0, 2], [-3, 10, 4, -12], [4, 0, 0, -7], [-25, 6, -7, -3], [-30, 0, 3, -6], [26, -15, -10, 10], [-6, 1, -13, 3], [-15, 15, -6, 6], [-11, -7, 6, -11], [-10, -4, -9, 13], [34, -9, -6, 0], [21, -6, -5, 5], [6, -9, 10, -4], [32, 1, -7, 5], [11, -5, 7, 5], [-37, 12, 4, -12], [20, 16, -6, 2], [2, 7, -4, -10], [-30, -2, 4, -4], [37, -4, -5, 9], [32, -20, -3, 7], [-3, 7, -4, 5], [55, 6, 0, 4], [-4, -8, 4, -9], [18, 10, 2, -6], [27, 4, -7, 19], [4, -9, -3, 13], [-3, -7, 11, 3], [-53, 3, 4, 3], [22, 10, 2, 0], [20, -1, -8, 6], [-2, 3, 4, -4], [-15, 5, -13, 10], [7, 6, -3, -3], [4, 0, 16, -2], [14, -11, -12, 14], [8, -8, -4, 20], [-5, 15, 3, -11], [30, 2, 0, -6], [-14, -8, -4, -13], [-19, 4, 10, -5], [23, -1, -2, 8], [25, -4, -7, 12], [37, -7, -7, 0], [18, 0, 10, 4], [-11, 15, -5, -11], [-35, -5, -8, -2], [-4, -14, 2, -8], [-2, 2, -9, 16], [-8, 11, 7, -5], [-40, -8, 4, 0], [-8, -12, -4, -8], [24, 5, 5, -9], [24, -1, -10, -2], [-14, 1, -7, 1], [-23, 11, 4, -10], [8, -4, 8, -10], [-21, -8, -3, 0], [-18, -1, -16, 16], [-17, 4, -1, 12], [-24, 15, 9, -9], [32, -5, 0, -4], [-3, -8, -3, 3], [22, -4, 14, 0], [-5, 21, 3, -1], [12, -15, -8, 16], [28, -20, 4, 0], [4, 8, 16, -8], [-8, -10, -4, 10], [22, 1, -1, -5], [-24, 6, -6, -6], [30, -10, 11, 0], [28, 8, 4, 7], [41, -3, 9, 4], [21, 6, 7, -3], [60, -7, -8, 4], [30, -12, 9, -6], [-34, -8, -5, -3], [-30, -6, 8, 8], [3, -4, -10, 10], [34, -3, -9, 9], [-21, -7, -3, -9], [-33, -10, -1, -11], [-20, 14, -8, 5], [35, -20, -7, 11], [-13, 19, -6, 2], [-14, 2, 12, -4], [-16, 0, 6, -15], [42, -23, -6, 2], [16, 1, -4, 16], [52, -7, -5, 13], [-24, 16, -4, 0], [-25, 7, 0, -2], [10, -2, -4, 12], [14, -13, -14, 8], [-16, 10, 12, -14], [32, -15, -11, 7], [-42, -7, -3, 1], [2, -23, 7, -9], [-8, 12, 7, 3], [50, -10, -2, 6], [-46, 10, 0, -4], [29, -2, -6, 10], [-4, -18, 4, 4], [-38, 21, 7, -11], [29, 9, 5, 11], [40, 6, 6, -3], [14, -2, 8, -8], [-36, 9, 11, -5], [19, 1, -2, -3], [40, -7, 18, -10], [-8, -6, 0, -12], [26, -8, -2, 0], [25, -5, 4, -11], [30, -3, -6, 10], [37, -2, 17, -5], [10, 1, -14, 4], [-36, -2, 2, -14], [-8, 23, -3, -11], [18, 2, 8, -5], [2, -19, 9, -5], [-21, 13, 16, -10], [-51, 7, 3, 1], [-2, -8, -5, 4], [-38, 4, -5, 10], [-8, 13, -1, 11], [13, 9, 3, -10], [21, 9, -5, -16], [-23, 6, 2, -3], [28, 9, 4, -6], [-35, -16, 3, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4598_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4598_2_a_bs();".
// 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_4598_2_a_bs(:prec:=4)
    chi := MakeCharacter_4598_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_4598_2_a_bs();".
// 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_4598_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4598_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![17, 20, -12, -2, 1]>,<5,R![24, -6, -12, 2, 1]>,<7,R![-48, -67, -21, 3, 1]>,<13,R![104, 20, -30, 1, 1]>],Snew);
    return Vf;
end function;