
// Make newform 4598.2.a.br in Magma, downloaded from the LMFDB on 23 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_br();".
// 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_br();".
// 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, 7, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [1, -8, 0, 1], [5, -8, -1, 1]];
    Rf_basisdens := [1, 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_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], [0, 1, 0, 0], [1, 0, 1, 1], [0, 0, 0, -1], [0, 0, 0, 0], [2, 0, -1, -1], [1, -1, 1, 1], [1, 0, 0, 0], [-1, 1, 0, -1], [0, -3, 1, 2], [7, -1, -1, 0], [2, -2, 2, 0], [1, -1, 1, -2], [-1, 2, 1, -1], [6, 0, 2, 2], [1, -3, -1, 1], [-1, 3, -1, -1], [-2, 0, 0, -2], [4, 0, -2, -1], [-5, 0, 1, -1], [-5, -1, -1, -1], [-4, 2, 2, -2], [7, -2, -1, 3], [4, 2, -2, 2], [4, -4, 2, 4], [-4, -2, 0, 4], [1, 2, -3, -3], [3, 3, 4, 3], [7, 1, 4, 3], [6, 4, 0, -2], [-4, 2, 2, 2], [-3, 1, -3, 4], [-4, -2, -3, 1], [3, -4, -3, -5], [10, 6, -2, 2], [4, 6, -4, 2], [7, -4, -1, -1], [6, -4, -2, 0], [-2, 2, -4, -2], [5, -4, -1, 5], [3, -1, 1, 0], [-4, -2, 0, -6], [5, -1, 0, -7], [5, 2, 1, 5], [0, 0, 4, 4], [3, -3, -2, 1], [9, -1, 0, -5], [-2, -2, 2, 2], [7, -3, -6, -1], [1, 3, 5, 4], [-8, 0, 0, 4], [-6, 3, -4, 2], [-1, -2, 3, 3], [-2, 4, -6, -4], [6, 6, -4, -6], [-9, 4, -9, -1], [-12, 0, 0, 4], [-8, 4, -4, -3], [8, -2, 6, -2], [15, 2, 3, -1], [1, 5, -1, 4], [10, 0, 5, 5], [-8, -2, 4, 2], [13, -9, -2, 3], [8, -9, -3, -2], [-1, -5, -7, -5], [-12, 1, 0, -4], [7, -11, 7, 8], [-6, -10, 0, -2], [-4, 4, -6, -6], [1, -5, 2, -7], [-2, 6, 2, 1], [-2, -2, -2, 4], [14, -3, 3, -4], [0, -4, 2, -3], [7, -1, -3, 8], [-3, -4, -1, 5], [-15, -5, -5, -6], [16, -4, 4, 6], [9, -12, 5, 9], [0, 8, -4, 2], [3, -3, 5, -1], [6, -12, 6, 4], [22, 2, 6, 4], [-2, 2, -8, -2], [4, -2, 6, 0], [6, 0, -2, -10], [-5, -1, 7, 1], [8, -2, 0, -2], [2, 6, -2, -8], [8, 4, 2, 8], [1, -7, -9, -4], [9, -2, -3, 5], [-3, 1, -9, -8], [10, 8, -4, 2], [-16, 9, 2, -8], [10, 8, -2, 2], [-18, 6, -2, 0], [15, 7, -2, 5], [22, -2, 6, 0], [-2, 4, 6, -2], [-2, -4, -8, -8], [-18, 4, -2, -2], [-17, 5, 1, 0], [-11, -3, 3, -4], [16, 6, -5, -5], [-10, 0, 6, -2], [2, -10, -6, -8], [7, 2, -5, -5], [-19, 11, 1, -4], [-8, 2, 0, -8], [8, 14, -4, 2], [3, 1, 3, -6], [-10, 6, 0, 0], [10, 4, -2, -6], [-4, 2, 4, 6], [10, -4, -2, -8], [5, -11, 0, 3], [-11, 7, 5, 0], [7, -5, -6, 1], [9, -7, 3, 3], [5, -1, 3, 12], [12, 1, -3, -6], [-2, 2, 6, -10], [-4, -2, -8, -4], [-14, 10, -10, -8], [7, 1, 1, -2], [-15, 17, -2, -7], [3, 3, 8, 5], [2, -6, 4, 2], [-11, 14, 9, 5], [32, -6, 2, 6], [-2, 14, -6, -10], [17, -4, -3, -11], [-15, -7, -3, -3], [-9, 5, -1, 3], [-25, 7, 1, -3], [5, -7, 8, 5], [-15, 11, -13, -11], [-5, -1, 3, 17], [-5, -9, -6, -11], [4, 6, -6, -10], [9, -7, 2, 1], [-23, -1, -4, -9], [7, 3, -3, -9], [9, -6, 3, 13], [-4, -12, 4, 2], [-5, -7, -13, -10], [-16, -4, -2, 14], [-1, 7, -1, 8], [0, 7, -5, 6], [11, 5, -1, 10], [20, 4, -6, 4], [-22, -10, 4, 0], [23, 7, -5, -17], [-22, -14, 4, -2], [-14, -9, 8, -10], [-12, -10, 7, -1], [-5, -15, -1, -1], [-1, 9, 0, -13], [-2, -4, 4, -6], [0, -6, 8, 10], [-2, 14, 2, -2], [17, -12, 5, 1], [20, -14, -6, -2], [1, 5, -7, 8], [-5, 11, -13, -2], [-26, 8, -8, -8], [13, -5, 5, 8], [-11, -9, 9, 16], [-5, -7, 6, 3], [5, 1, 3, 9], [-9, 8, 11, 3], [7, 3, -1, 11], [2, -10, 2, 14], [-1, -4, 13, -1], [-19, -7, 9, 2], [-8, -6, -8, -6], [38, 6, 2, 6], [40, 0, 0, -4], [1, -9, -6, -1], [23, 5, 2, 11], [8, 6, -8, 6], [-3, -10, -5, 1], [-13, 15, 6, -5], [16, -2, 8, 14], [-23, 3, -8, -7], [-23, -4, -7, -3], [-29, -4, 7, -1], [12, -1, -6, 0], [-22, 4, -14, -2], [9, -3, -1, 10], [9, -8, 13, 1], [1, 1, 7, -5], [19, -17, 3, 10], [-20, 16, 2, 0], [-25, 25, -1, -13], [-1, 1, 9, -4], [-9, -5, -5, -5], [-9, -1, 1, 7], [-1, -23, 3, 6], [0, 6, -2, 2], [17, 11, -1, 1], [3, 8, 9, 7], [20, -12, 2, 10], [18, -8, -8, -4], [3, -3, 12, 15], [13, -9, 2, -7], [-21, -11, -1, -9], [22, 5, 2, 14], [11, -7, -13, -7], [-3, -2, 7, 9], [44, -2, 12, 6], [-26, -6, 0, 17], [-30, 4, 6, 8], [-10, -16, 6, 2], [-13, -13, 1, -5], [-4, 18, 8, 2], [-35, -7, 5, 0], [21, 7, 1, -2], [-9, 17, -5, -2], [24, -6, -8, -14], [28, -14, -10, -8], [-7, 5, 12, -9], [17, -5, -2, -15], [-54, -2, 2, -2], [10, -4, -16, -2], [3, 14, 11, 19], [-14, 10, -2, 10], [-1, -2, 1, -13], [33, 0, 1, 1], [-7, -7, -10, -9], [-10, 6, -10, -8], [5, 15, -11, 0], [64, 4, -2, 0], [-30, 18, 10, -1], [-9, -15, -5, -14], [-6, 0, -6, -12], [-32, 10, 0, -10], [21, -7, 7, 4], [-5, -8, -9, -5], [-13, 11, -5, -6], [7, 9, 4, 7], [-25, -1, -11, -5], [4, -6, 4, -14], [-31, -11, 3, -4], [38, -2, -6, -12], [31, 0, 11, -1], [-18, 2, -2, -4], [11, -1, -3, -5], [-14, -4, 2, 2], [-11, -5, 5, 15], [11, -5, -2, 3], [-2, 4, 6, 4], [-8, 6, -4, -12], [-31, 16, 9, 5], [9, 2, 3, 1], [-4, -14, 0, 14], [2, 10, 2, 4], [-6, 18, -6, -4], [-34, -12, 2, 4], [-16, 18, -8, -6], [-10, 6, 18, -8], [32, -3, 5, -2], [13, -20, -5, 9], [-29, 11, 11, -9], [-23, 1, -5, -7], [-18, -5, -1, 4], [-1, -13, 4, 15], [-14, 2, -6, -8], [17, 3, -1, -7], [8, -1, -4, 4], [22, 8, 1, -1], [20, 9, 6, 0], [7, 6, -11, 7], [-27, -1, -10, -1], [-34, -2, -6, 4], [30, 1, -5, 4], [-27, 0, -3, 9], [17, -14, -5, 5], [10, 10, -2, 2], [3, -21, 6, 3], [0, 23, -14, -4], [14, 3, 1, 20], [-25, -14, 5, 7], [6, 24, -2, 2], [34, 4, 7, -11], [-13, 1, 11, 6], [9, 13, -12, -9], [-33, 10, 11, 3], [-37, 13, -9, -8], [-12, -6, -2, -12], [65, -5, -1, 0], [-4, 2, -12, -2], [23, -12, 1, -5], [-18, 8, -8, 4], [27, -1, -9, -18], [16, -22, -6, 2], [11, -9, -4, 3], [25, -14, 11, 9], [14, 20, -10, 6], [24, 2, 15, 1], [-21, -5, -23, -21], [-8, 1, -12, 0], [36, 8, 4, 0], [-8, -20, 2, 10], [-10, -2, -16, -17], [15, 9, 1, -17], [18, -6, 2, -4], [48, -2, -10, -2], [-3, 5, 19, 4], [3, 2, -11, -13], [-29, -7, 3, 8], [9, -16, -3, 1], [-3, -3, -9, -16], [10, 12, 4, -8], [-6, -4, 0, 14], [10, 6, 12, 6], [-22, 15, 7, 8], [-26, -2, 2, -4], [-43, -3, -10, -1], [21, -9, 6, -7], [-53, -3, 0, -9], [10, 2, 2, -26], [3, 11, -5, -3], [-28, 16, -6, -4], [18, 10, 4, -9], [34, -7, -10, 6], [-22, 14, -10, -12], [14, 8, -2, -20], [28, -4, 14, 0], [-31, 15, 11, -9], [42, 13, 4, 2], [-26, -6, 0, 10], [-35, 8, -3, -3], [-12, 2, 12, -12], [13, 5, -13, -27], [15, 9, 3, 8], [19, -15, 18, 23], [-6, 23, 4, -10], [-17, 7, 17, 11], [24, -3, -5, -2], [6, 16, -5, 7], [-23, -2, -1, 21], [31, -15, -7, 1], [16, -20, -12, -8], [-13, 5, -9, -16], [-4, 24, -10, -25], [-18, 0, 14, 26], [8, 6, -12, 14], [26, -12, 14, 6], [6, 8, -18, 8], [-55, 19, 5, -6], [10, -10, -10, -6], [9, 3, -7, -6], [11, 2, 7, -5], [51, 1, 2, -7], [-11, -10, -21, -11], [-35, 15, -7, -1], [23, 19, 1, 0], [8, -8, -12, -10], [8, -6, -12, -22], [19, 8, -5, -9], [-17, -7, -14, -9], [-4, 14, -8, 16], [2, 8, 18, 0], [18, -4, -20, 16], [-20, 4, -8, -2], [31, 12, -7, -25], [34, 16, -10, -10], [-24, 8, -22, -5], [19, -1, -19, 3], [32, -11, 9, 14], [2, -6, 10, 22], [22, 24, 4, 8], [2, 10, 8, -7], [-31, -3, 7, 32], [-38, 14, 14, 16], [27, -9, 15, 19], [-24, -6, 8, -2], [-40, 14, 12, -10], [19, -11, -1, -10], [-2, -6, 22, 7], [30, 2, 18, -4], [-41, -11, 4, -3], [-38, 6, 6, -10], [-1, -11, -2, 1], [-35, 18, -21, -15], [-15, -11, 3, 8], [25, -8, 21, -3], [28, -28, 16, 14], [19, 13, -7, -21], [2, -28, 2, 0], [4, 20, -2, 0], [17, -7, 3, -4], [-21, 20, 13, -5], [30, -14, -18, 12], [8, 6, 6, -6], [17, -15, 3, 12], [36, -6, 4, 8], [19, 1, -17, -6], [-37, 12, -3, 3], [44, 0, 10, 16], [-16, -4, 8, 26], [-37, 4, 1, 11], [23, 3, 5, 11], [9, 7, 1, 6], [-20, 10, 0, -14], [-11, 13, 17, 33], [2, 2, 12, -8], [38, -4, -6, 0], [4, 10, 6, 10], [-9, -17, 15, -13], [-3, -6, 3, -11], [-5, -23, 11, 31], [-44, -2, 2, 14], [50, -14, -4, 10], [55, 11, -1, 4], [-30, 4, -1, -15], [14, -6, 12, 18], [20, -10, 8, 10], [-31, 27, 0, -7], [7, 3, -10, -17], [-7, 11, -10, -7], [-10, -8, 4, 4], [51, 15, -7, 3], [2, 26, -8, -17], [-13, 23, 7, -9], [-8, -16, 14, 2], [-30, 11, 4, 14]];
    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_br();".
// 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_br(: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_br();".
// 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_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4598_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 7, -8, -1, 1]>,<5,R![-4, 36, -13, -3, 1]>,<7,R![-7, 19, -10, -1, 1]>,<13,R![-13, 15, 14, -9, 1]>],Snew);
    return Vf;
end function;