
// Make newform 4864.2.a.v in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4864_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_4864_2_a_v();".
// 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_4864_2_a_v();".
// 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 := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 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_4864_a();"
function MakeCharacter_4864_a()
    N := 4864;
    order := 1;
    char_gens := [3839, 2053, 4353];
    v := [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_4864_a_Hecke(Kf)
    return MakeCharacter_4864_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], [1, 0], [-2, 0], [-1, 1], [-4, 0], [1, -1], [-1, 2], [1, 0], [3, -1], [1, 1], [2, 2], [-2, 0], [0, 2], [-2, -2], [0, -2], [9, -1], [-3, 2], [-2, 2], [1, 2], [14, 0], [-7, 0], [-2, 0], [-4, -2], [6, 2], [6, -2], [4, 4], [4, 2], [7, -2], [3, 3], [6, -2], [6, 2], [14, 2], [-5, -4], [-2, 4], [4, -2], [16, 0], [-10, 0], [10, -4], [12, 2], [-10, 0], [12, -4], [-10, 0], [7, 1], [-4, -4], [0, 2], [-17, -1], [-11, 4], [20, -2], [5, 2], [-10, -2], [-2, 4], [-1, -5], [-4, 6], [6, -2], [6, 2], [-4, -2], [6, 4], [15, 3], [-18, 2], [-4, 0], [-12, 2], [-9, -5], [-12, -4], [3, 3], [-3, 4], [11, -1], [11, 6], [-14, -2], [-14, 4], [14, 0], [17, 2], [21, -1], [20, 2], [-1, 3], [-7, -4], [-4, 6], [-12, 4], [-2, 2], [-12, 0], [6, 0], [-10, 6], [-15, -1], [6, -2], [-4, 6], [-4, 0], [-4, 4], [20, 0], [-31, 0], [0, -4], [-4, -6], [-18, -6], [4, 6], [0, -10], [28, 4], [-2, 6], [-5, 5], [-18, 0], [20, 2], [31, 0], [-8, -6], [-20, -4], [-22, 0], [4, -8], [-12, -6], [-8, 6], [-9, -6], [0, 0], [30, 4], [-4, -4], [14, -4], [26, -4], [0, -10], [-14, 0], [14, -4], [4, -10], [-12, -6], [-26, 2], [-17, -1], [14, 6], [11, 4], [27, -1], [16, 2], [3, 3], [4, -4], [6, 4], [-20, -8], [-10, -6], [3, -3], [-17, -5], [30, -4], [4, 4], [-10, 2], [40, 2], [22, 6], [-21, 8], [-47, 0], [51, -1], [1, 6], [1, 9], [25, 2], [15, 4], [10, -10], [-27, -1], [13, 8], [-5, -5], [22, 0], [20, 2], [36, -4], [-28, -2], [6, 6], [7, -7], [-22, 0], [22, -8], [6, -2], [27, 4], [-6, -4], [5, 3], [-19, -8], [-23, 2], [-3, 11], [10, -2], [-28, -6], [4, -6], [12, -4], [6, 0], [-14, -10], [2, -12], [22, -6], [-18, -8], [-18, -6], [-9, -4], [17, 9], [20, 6], [-1, 2], [-20, -2], [-42, 4], [34, -4], [12, 0], [-10, -6], [28, -2], [-51, 1], [-13, -4], [2, -8], [-30, -2], [-13, -5], [36, -2], [3, 11], [20, 12], [30, -4], [23, 5], [16, -8], [-12, 6], [-20, 4], [-21, -11], [18, 4], [-6, 0], [9, 16], [-20, 14], [43, -4], [-27, -7], [-38, 0], [30, 8], [-11, -9], [-20, 8], [-24, 2], [48, 2], [-9, 17], [9, -6], [57, -2], [13, 10], [-23, -10], [14, -8], [16, -6], [-29, -6], [42, -4], [-16, 0], [-9, 11], [12, -16], [-10, 16], [-12, 2], [38, 8], [8, 14], [-6, -12], [19, -11], [-33, 0], [-40, -6], [-6, -8], [0, 4], [-42, 8], [-6, -16], [-26, 2], [61, 2], [-24, -6], [10, -10], [18, -10], [-7, -9], [22, 4], [12, 8], [-60, -2], [40, -8], [-20, 8], [18, 4], [-71, -1], [-19, -11], [64, -4], [-4, 2], [36, 2], [36, -8], [-8, 8], [-41, 9], [20, 14], [11, -2], [35, 1], [36, -6], [20, -4], [-40, -10], [-46, 0], [-4, 8], [-30, 4], [22, -8], [14, -4], [-12, 12], [28, 10], [-43, -9], [-18, 0], [-7, 0], [59, 1], [-21, 0], [55, 2], [-12, -10], [-23, 11], [27, 8], [-15, -18], [45, -3], [2, -2], [1, -7], [24, 4], [-35, 3], [58, -6], [-30, -8], [-30, 0], [-9, -9], [11, -11], [-11, -15], [6, 12], [44, 10], [29, -6], [10, 8], [25, 5], [44, 10], [-30, -12], [-64, 4], [28, 4], [48, 8], [-46, 4], [66, 6], [6, -4], [-6, -16], [59, 4], [-38, 14], [-10, 12], [-45, 7], [-37, 7], [21, -6], [40, 4], [38, 4], [-3, -18], [-1, -19], [52, 10], [-26, -4], [52, 6], [-30, 10], [22, 4], [-20, 12], [12, 10], [-48, -8], [-18, 8], [48, 12], [-13, -10], [42, 12], [-13, -2], [11, 10], [-3, 19], [32, 4], [11, -12], [50, 4], [1, 12], [19, -6], [48, -10], [12, 20], [-38, -8], [27, -17], [-9, 9], [-54, -10], [72, 0], [-18, 8], [-37, -7], [6, -10], [-29, -16], [53, 5], [51, 9], [-17, 2], [-3, 7], [18, -14], [-39, -9], [32, 4], [-18, 18], [15, 6], [-8, 6], [-48, -14], [-4, 16], [-8, -18], [-32, 16], [14, -12], [2, 0], [36, -2], [-1, 7], [42, -12], [3, -4], [20, 18], [48, 0], [-6, 4], [24, -12], [-15, -10], [46, 6], [16, -18], [78, 2], [-26, 14], [4, -6], [-16, -12], [7, -2], [7, 1], [-3, -1], [-4, -10], [52, 0], [3, 6], [32, -18], [-42, -12], [21, -4], [40, -16], [30, -8], [-6, -8], [85, -3], [-46, 8], [-59, -6], [-20, -12], [-23, -21], [-6, 24], [24, 2], [76, -4], [-8, -8], [15, 17], [-18, -20], [-10, -20], [84, -2], [64, -6], [-64, -4], [-14, 4], [-76, 0], [44, -2], [74, -4], [-6, 0], [6, 10], [-12, 0], [62, -8], [11, -5], [52, 12], [-36, 0], [27, -18], [-6, 12], [26, 4], [14, 2], [45, -8], [-34, 0], [-3, -2], [22, 16], [0, 2], [38, -6], [27, 2], [-48, 6], [-26, 20], [43, -13], [-27, 11], [51, -10], [10, -18], [-5, -3], [-19, 0], [-9, 10], [-52, 10], [27, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4864_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4864_2_a_v();".
// 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_4864_2_a_v(:prec:=2)
    chi := MakeCharacter_4864_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_4864_2_a_v();".
// 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_4864_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![2, 1]>,<7,R![-11, 2, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;