
// Make newform 5200.2.a.bz in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5200_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_5200_2_a_bz();".
// 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_5200_2_a_bz();".
// 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;
    Rf_num := [[1, 0], [-1, 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_5200_a();"
function MakeCharacter_5200_a()
    N := 5200;
    order := 1;
    char_gens := [1951, 1301, 4577, 1601];
    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_5200_a_Hecke( : Kf := Kf)
    return MakeCharacter_5200_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]];
    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, 1], [0, 0], [0, 0], [-3, 1], [1, 0], [-2, 0], [-3, 1], [-1, -1], [4, 2], [-1, 3], [4, 2], [-2, -4], [-1, -1], [4, 4], [2, 4], [-1, 3], [0, -2], [6, 2], [3, -1], [12, -2], [-2, -2], [4, 0], [10, 0], [10, 0], [14, 0], [-13, 3], [-5, -5], [-10, 4], [-2, -8], [-3, -3], [-8, -4], [2, 8], [-18, 2], [-6, 0], [5, -7], [2, 4], [-6, -2], [8, 0], [2, 4], [-8, -4], [-4, 2], [0, 8], [-6, 0], [-2, -8], [-12, -4], [0, -4], [8, -8], [14, 2], [10, 0], [2, 4], [13, -7], [2, -8], [2, -2], [-14, 4], [5, -3], [6, 0], [-3, -7], [2, -4], [10, 0], [15, 7], [0, -10], [-12, 0], [2, -6], [-6, 12], [-8, -10], [3, -1], [30, 0], [23, -1], [6, 4], [8, 2], [-5, -1], [-13, -5], [-6, 4], [-19, 1], [0, -8], [14, 8], [20, -6], [10, -8], [6, 12], [14, -6], [-14, 8], [-9, -13], [2, -12], [0, 0], [-7, -7], [-18, 4], [10, -8], [-22, 8], [-6, -6], [5, -3], [11, -9], [-2, 14], [30, 2], [-31, -3], [21, -3], [-18, 4], [-20, 6], [5, -11], [-26, 4], [11, 11], [-12, -6], [-21, 3], [28, 6], [18, -2], [28, 6], [-18, 2], [18, -8], [2, 2], [2, -8], [-5, -13], [6, 0], [18, 0], [-13, 7], [5, 17], [-14, 0], [24, -4], [17, 1], [-14, 12], [2, 14], [18, 0], [-26, -8], [-14, 12], [-22, -10], [-5, -17], [-34, -8], [2, -8], [20, 12], [-19, -3], [-2, 0], [3, -17], [24, -8], [-42, -2], [2, 12], [-30, 0], [6, 4], [-12, 18], [-4, -8], [22, 0], [0, 2], [-37, -1], [14, -4], [-1, -9], [0, 12], [24, 6], [35, 7], [-8, -2], [-30, 4], [-34, 10], [28, -4], [-18, 16], [-32, 10], [-21, 3], [7, -1], [-35, -3], [24, 0], [34, 2], [-10, -4], [18, 12], [18, 0], [-48, 4], [34, -12], [2, 26], [0, -12], [20, -2], [30, -10], [12, -4], [26, -12], [46, -4], [-14, -12], [53, 1], [14, 12], [30, 6], [-24, 10], [-16, -8], [2, -8], [9, 13], [22, 4], [-45, -5], [-10, 16], [-12, 4], [-30, -2], [6, -16], [-14, 16], [-30, 2], [16, -10], [2, -12], [20, -8], [-10, -4], [-15, 21], [22, -16], [-44, 8], [20, 0], [-22, 8], [-3, 5], [-30, 4], [6, -12], [-26, -8], [-16, -6], [-19, 21], [14, -20], [-4, -4], [-34, -8], [32, -6], [-5, -9], [-14, -4], [57, 5], [13, -11], [-2, -4], [-42, -6], [10, -4], [-8, -2], [57, 1], [-16, 12], [5, -15], [2, 8], [-29, 3], [10, 8], [14, 6], [30, 0], [8, -18], [-17, 11], [-6, 16], [-16, 0], [25, -7], [32, -10], [-10, -8], [-18, -2], [-39, -7], [-59, 1], [-30, 12], [-14, 6], [27, -1], [-30, 8], [-27, -3], [22, 14], [-6, -8], [32, -10], [32, 12], [-44, 12], [4, 8], [-10, -6], [-39, 9], [26, -8], [-30, 0], [-20, 12], [-16, -16], [-7, -19], [-71, -3], [43, 3], [-4, 2], [26, -24], [64, 0], [-6, 16], [30, 0], [31, -5], [-32, -10], [26, 6], [-18, -8], [-4, -18], [-33, 7], [-37, -13], [-6, 8], [30, -8], [0, -22], [-18, 2], [-10, -20], [-10, -4], [-34, -14], [54, 0], [46, -8], [18, -2], [-14, -16], [24, 8], [10, -20], [0, 32], [14, -22], [-34, -20], [46, 12], [22, -6], [73, 1], [9, -19], [55, 7], [6, -4], [-12, 8], [2, -22], [-6, 4], [38, 0], [-33, 3], [-4, 30], [40, 6], [39, -1], [42, 16], [49, -3], [-2, 32], [-10, -8], [44, -12], [2, -4], [-10, 2], [-20, -16], [-26, 16], [-2, 32], [28, 12], [21, 21], [4, 16], [-12, 14], [45, -19], [6, -32], [63, -5], [-2, 16], [-39, -15], [-50, -12], [20, -10], [37, -19], [-12, 12], [-24, -22], [65, -11], [27, 15], [-36, 6], [34, 0], [-26, -6], [-28, -10], [-16, -2], [-18, -18], [-12, -10], [4, 22], [31, -21], [4, 24], [33, -23], [-46, 12], [-22, -8], [38, 0], [-2, -10], [10, -10], [67, -1], [18, 6], [-2, -12], [-30, -8], [-70, 0], [-35, -3], [-10, 16], [-66, -8], [42, 8], [2, 18], [14, 0], [-44, -16], [-26, -16], [28, 16], [-7, 5], [18, -28], [31, 19], [58, 0], [2, -8], [-61, 3], [-10, 16], [34, -12], [19, 15], [-9, 19], [30, 24], [-52, -4], [8, -2], [8, -6], [63, -1], [9, -11], [-27, 29], [46, -8], [-66, 0], [34, -14], [-62, 8], [-36, 8], [-8, 36], [62, 14], [52, -6], [36, 12], [-26, -8], [29, 25], [-38, -14], [-14, 32], [24, -14], [-10, 8], [42, -16], [-4, -2], [-72, 0], [-36, -26], [23, -5], [78, 6], [15, 11], [82, -4], [24, 12], [15, -1], [26, 24], [44, -6], [7, -5], [71, -1], [73, -11], [-54, -12], [5, 9], [12, 6], [44, -8], [18, -32], [-54, 16], [30, 16], [60, -4], [48, -14], [-38, -8], [12, -12], [-16, -2], [50, 0], [-6, -2], [-5, 7], [-50, -8], [-34, -16], [29, -3], [30, 2], [22, 16], [-34, 0], [-11, 9], [3, -37], [8, 18], [27, 11], [40, 6], [38, 0], [-54, 2], [-22, 22], [58, 0], [48, -18], [-21, 19], [42, 8], [-15, -27], [-10, -42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5200_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5200_2_a_bz();".
// 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_5200_2_a_bz(:prec:=2)
    chi := MakeCharacter_5200_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_5200_2_a_bz();".
// 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_5200_2_a_bz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -2, 1]>,<7,R![0, 1]>,<11,R![4, 6, 1]>],Snew);
    return Vf;
end function;