
// Make newform 4600.2.a.x in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4600_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_4600_2_a_x();".
// 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_4600_2_a_x();".
// 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 := [8, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-6, 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_4600_a();"
function MakeCharacter_4600_a()
    N := 4600;
    order := 1;
    char_gens := [1151, 2301, 2577, 1201];
    v := [1, 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_4600_a_Hecke(Kf)
    return MakeCharacter_4600_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], [0, -1, 0], [0, 0, 0], [-1, 1, 0], [2, 1, 0], [0, 0, 1], [-3, 0, -1], [-4, 0, -1], [-1, 0, 0], [5, -1, -1], [-3, 1, 0], [-3, 3, 1], [3, 0, -1], [-8, 0, 0], [0, 0, -2], [1, -3, -1], [-5, -1, -1], [4, -1, 2], [-3, 3, 1], [-7, -2, 1], [-6, 4, 2], [0, -4, 0], [5, 1, -1], [-4, 4, 0], [2, -3, 0], [9, -1, -1], [-4, 1, -2], [-3, 3, 3], [8, -2, 1], [1, -3, 3], [0, -2, 0], [-8, -2, 2], [4, -2, -3], [-3, -3, 1], [-14, 4, 1], [-4, -5, 0], [3, 1, -3], [-8, 2, -1], [8, 0, 0], [0, 3, 0], [-4, 0, 2], [2, 4, -1], [0, -2, 4], [-4, 2, 0], [-2, -1, -2], [-14, 0, 0], [3, -1, 1], [-6, -2, 2], [0, -4, 2], [0, 2, -2], [-8, -2, 0], [-1, 5, -1], [-2, -6, -2], [-20, 0, 1], [-12, 2, -4], [3, 0, -3], [-17, -3, 1], [-9, -1, -2], [6, 4, 0], [6, 2, 4], [-1, 3, 5], [-5, -5, -3], [-2, -1, 2], [0, -4, 4], [-3, -3, 4], [10, 2, -1], [3, 11, -1], [8, 2, 1], [8, -9, 0], [9, -7, -1], [8, 0, 0], [0, -2, -2], [-17, 1, 3], [10, -8, -2], [24, -3, -2], [-9, -11, 3], [2, 7, 0], [8, -7, 4], [6, 4, -2], [1, -1, 2], [-10, 6, -2], [-12, 10, 1], [4, 0, -6], [-9, -5, 0], [-2, 8, 5], [-2, 8, -5], [7, -3, 2], [-9, -9, 1], [-22, 0, -2], [-6, -4, 4], [1, -7, 1], [0, 2, 6], [8, 2, 4], [13, -1, -5], [9, 3, 1], [5, 3, -2], [-22, 10, 4], [22, -8, -4], [-24, 2, -2], [2, -10, 2], [-6, 8, 3], [-3, 7, 1], [-1, 1, -1], [6, 0, 0], [4, -5, -2], [0, 6, -2], [-10, -3, -6], [12, -10, 0], [8, -7, -4], [-21, 5, -4], [-32, 0, -4], [-2, -8, 4], [-21, -10, 1], [-2, 2, -1], [12, 6, 4], [30, -2, -2], [21, 3, -1], [6, -8, -4], [8, 4, -3], [-2, -6, 2], [-2, 1, -8], [0, -2, 2], [7, 3, 5], [10, 4, -3], [-16, 12, 2], [-2, 0, -7], [28, -3, -2], [-23, 1, 2], [11, 6, -3], [17, 1, 3], [-9, -7, 7], [18, -10, -1], [14, -8, -2], [-13, 1, -9], [9, -1, 6], [-40, 2, 2], [-14, -2, -6], [1, 1, 5], [-21, -9, 3], [-43, 4, 1], [-5, 13, 1], [10, 0, 0], [-4, -14, 2], [-5, 7, -5], [23, -5, -3], [-18, -2, 4], [12, -4, -5], [-30, 4, 0], [7, -3, -7], [22, 8, 2], [-2, -1, -6], [16, 0, 4], [-18, 4, 3], [-38, 2, -2], [33, 7, -1], [-8, 4, -10], [8, 0, 0], [-3, 1, -5], [24, 2, 5], [12, 3, -2], [-22, -8, 3], [10, -11, -4], [8, -10, -8], [-4, 9, -6], [-15, 1, -4], [17, 0, 7], [-13, 14, 7], [4, 6, -2], [28, -6, 0], [16, -3, -4], [8, -8, -4], [23, -7, -7], [2, 2, 0], [-23, -2, -5], [8, 3, 8], [4, 0, 2], [40, -2, -2], [-19, 11, -3], [-1, 6, -7], [-12, 6, -1], [22, 4, -2], [0, 9, -6], [24, 0, 1], [-22, 6, 0], [3, 6, -3], [-12, -18, 3], [0, 0, 9], [35, 1, -1], [-21, 0, 7], [-27, 6, 5], [46, 4, -2], [2, -10, 2], [-2, -10, 10], [-37, 5, 3], [-36, 6, 2], [-11, 13, 10], [-26, -2, -4], [6, 12, 0], [-10, 12, 4], [26, 10, -4], [24, -1, 2], [-14, -8, -7], [-10, 11, 6], [-14, 6, 10], [-4, -4, -4], [-18, -8, -4], [-12, 2, -2], [-18, 3, -2], [9, 15, 4], [16, -16, 2], [-18, -6, 0], [-10, 0, -4], [-46, -2, 3], [8, -4, 6], [35, 9, -5], [-16, -20, 6], [14, -4, 2], [23, -13, -7], [21, -1, -4], [22, 4, -4], [31, 1, 1], [14, -16, 0], [1, 0, -7], [21, 3, 0], [-34, -2, -6], [21, 7, -5], [24, -2, -1], [-19, 5, -12], [2, 6, 5], [-3, -13, -7], [-8, -8, -1], [-42, 9, 4], [22, -2, 10], [9, 18, -3], [-47, 15, 7], [-36, 4, 0], [12, 4, 6], [10, 6, 0], [-37, 1, -3], [-47, 8, 1], [29, 7, -5], [3, -5, -3], [-14, -4, 8], [26, -4, -10], [0, 4, -2], [-18, 12, 3], [-4, 4, 10], [12, -8, -5], [6, -9, -8], [-15, -14, -5], [10, -4, -2], [-2, 0, 6], [-27, -5, 0], [2, -10, -2], [34, 0, -4], [7, -5, 3], [-32, 1, 2], [0, -14, 12], [12, 4, 1], [-2, 20, -4], [-13, 8, -7], [-9, -13, 1], [13, -13, 1], [-25, 5, 5], [24, 6, -6], [-28, 3, -2], [-36, -5, -6], [6, 4, 0], [-21, 3, -1], [10, 2, -3], [3, -7, 1], [29, 7, -3], [-5, 23, -2], [-22, 6, 10], [24, -6, -2], [14, 8, -2], [0, 13, 4], [7, 5, 9], [-8, -6, -2], [-14, 26, 5], [-2, 10, 10], [-4, 4, 4], [-12, -15, 2], [-4, -14, -6], [-12, -6, -3], [17, 14, 5], [-39, 5, 7], [-2, 12, 10], [24, 9, 0], [-42, -5, -2], [10, -16, -8], [23, 7, -5], [-28, -2, 8], [-66, -2, -1], [40, -5, 8], [4, 11, -10], [38, -10, 4], [-2, -15, 2], [13, 9, 7], [34, 1, 2], [33, -3, 4], [17, 11, 3], [28, 2, -8], [-36, -4, -4], [14, 14, -1], [-4, -6, 8], [26, -5, 10], [8, -10, 7], [-44, 12, 6], [12, -1, 12], [16, 16, -10], [0, 14, -11], [24, 6, -4], [-16, -5, -4], [25, -1, 0], [-8, 4, 2], [15, 11, 5], [9, 21, -6], [-7, -11, -13], [26, -3, -2], [20, 1, -2], [-21, 12, 1], [10, 4, 12], [6, 2, 16], [-39, 8, -1], [-24, -8, 8], [-14, -6, -7], [4, 2, 19], [34, -10, 7], [11, 17, -3], [-19, -11, 9], [-42, 1, 2], [-31, 3, 5], [-52, 2, 3], [-4, 0, 8], [26, -12, 9], [-30, 17, 0], [-22, 20, -4], [14, 11, 2], [17, -6, 11], [38, -16, -4], [-54, -9, 4], [-2, -2, 6], [-20, 20, 2], [23, -1, 17], [33, -13, -11], [-33, -13, 1], [0, 2, -9], [26, 16, -4], [7, 5, 5], [-15, -5, 3], [24, -10, -10], [17, 1, -7], [-73, 8, 5], [36, 22, -6], [-10, 14, -8], [-22, 4, 2], [44, 4, 3], [-42, 4, 0], [50, 4, 8], [37, -23, -7], [43, 11, 5], [-26, 4, -6], [-48, 25, 10], [6, 7, -2], [-22, 16, 12], [28, -11, -12], [25, 11, -3], [8, 12, -6], [1, -11, -1], [-49, 19, 7], [14, -12, -8], [-16, -11, -14], [20, 4, -12], [-4, 8, 7], [27, -11, -7], [-30, 4, -6], [-3, -6, -5], [-24, -8, 2], [2, -14, -2], [-10, -17, 8], [-74, 7, 0], [24, -2, 4], [-90, 0, 0], [-32, 0, 4], [-10, 10, 14], [1, 1, 12], [-50, -6, 7], [-15, -19, -1], [-25, -7, -4], [22, 4, -4], [24, -16, 7], [37, 11, -7], [-16, -4, -8], [38, 0, -8], [3, 7, 12], [-2, 20, 8], [0, -12, 12], [-2, -10, -13], [11, -17, -7], [9, -15, -11], [18, 24, -6], [-65, -7, -1], [-19, -11, 0], [17, -1, 5], [5, -20, -7], [15, -11, -15], [0, -8, 5], [56, 8, -2], [12, 4, 4], [-20, -6, -12], [14, 11, -14], [44, -8, 8], [16, -1, 2], [-75, 13, 6], [-62, 0, -8], [-54, 16, 3], [44, 4, 2], [13, -16, 15], [6, 6, 15], [-26, 23, 16], [22, 14, -4], [-58, 6, 0], [-24, -8, -2], [-12, -20, -6], [36, 6, 2], [-68, 20, 8], [-13, 1, 0], [-4, -14, 0], [13, 16, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4600_2_a_x();".
// 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_4600_2_a_x(:prec:=3)
    chi := MakeCharacter_4600_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_4600_2_a_x();".
// 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_4600_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, -9, 1, 1]>,<7,R![-1, -8, 2, 1]>,<11,R![14, 7, -7, 1]>],Snew);
    return Vf;
end function;