
// Make newform 5200.2.a.ci in Magma, downloaded from the LMFDB on 04 May 2024.
// 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_ci();".
// 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_ci();".
// 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, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, 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_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)
    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+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], [1, 0, 1], [0, 0, 0], [-1, -1, 0], [0, -1, 1], [-1, 0, 0], [0, 0, 2], [-2, 1, 1], [-3, 0, 1], [3, -1, 0], [4, -1, 1], [1, -1, 2], [0, 0, 2], [-1, 0, -1], [-3, 1, 0], [6, 2, -2], [6, -1, 3], [5, 1, 0], [5, -1, 2], [0, -1, 1], [-5, -3, 2], [-2, 0, 2], [-3, 1, -4], [0, -2, 4], [-8, 2, 0], [6, 0, 0], [-1, 2, 1], [-3, -2, -1], [2, 0, 4], [6, 2, 2], [5, 2, -1], [-6, 2, -4], [0, -2, -4], [-14, 0, 2], [0, 2, 4], [4, 1, -5], [-2, 0, 4], [-1, -3, 6], [-9, -1, 0], [-12, 0, -6], [6, 2, -4], [-1, -1, -4], [-12, 0, 0], [4, -2, 0], [6, 0, 4], [10, 2, -4], [-2, -2, 4], [-1, -1, -4], [3, -3, -2], [8, 0, 6], [6, 0, 0], [6, 3, -1], [-4, 0, 6], [-12, -2, -2], [12, 2, 0], [3, 4, 3], [-6, -4, 4], [16, 1, -1], [4, 4, -2], [12, 0, -2], [11, 0, -5], [3, -3, -6], [-1, 3, 4], [6, 0, 6], [-20, -4, 2], [15, -3, -2], [4, -3, -1], [-2, 2, 2], [-21, 0, -5], [-10, -2, 2], [9, 3, 6], [-18, 1, -3], [-7, 4, 1], [-2, 4, 0], [-2, 5, -7], [-9, -1, -4], [6, -4, -8], [-23, 3, -2], [0, -6, 0], [-16, 0, -6], [12, -2, 2], [-10, 4, -12], [-12, -1, 1], [22, 0, 8], [-8, 0, -8], [-3, -4, 1], [-6, -6, 6], [-8, -2, 4], [6, -6, -2], [-25, 1, 2], [-9, 2, 5], [-6, -3, 5], [-1, 5, 2], [12, 2, -6], [-2, -3, -3], [21, 6, -5], [-18, 2, -2], [3, 3, -8], [-7, -4, 5], [-16, 4, -6], [5, -2, -1], [-3, -5, 10], [-21, 4, -1], [3, -5, 4], [10, 0, 6], [-5, -3, -6], [27, -3, 14], [0, 6, -4], [-6, -8, -2], [-10, 0, 0], [-13, -2, 1], [-2, -4, -8], [0, 2, 0], [-2, 7, 3], [16, 5, -1], [6, 4, -4], [5, 5, 8], [-15, 6, 7], [12, -2, 0], [0, 6, 6], [2, 4, 0], [-20, 0, -10], [18, 2, 2], [-15, -1, -6], [-32, -3, 3], [18, -4, -4], [8, 4, 2], [6, 6, 4], [-1, -4, 7], [10, 0, 0], [10, -1, 11], [-21, -5, 4], [-8, -6, -6], [4, -8, 10], [18, 4, -4], [14, 8, -4], [9, 3, -2], [35, -1, 4], [-18, 0, -4], [9, -7, -4], [-8, -3, -5], [-24, 2, 4], [-25, 6, -3], [-3, 5, 4], [17, 1, -4], [6, 9, 1], [-29, 1, -2], [-12, 2, -8], [4, -2, 2], [15, 3, 8], [10, 0, 0], [-3, 9, 4], [11, 4, -9], [-15, 4, 9], [11, 6, -3], [-30, 2, -8], [-14, 0, 6], [18, 10, -2], [-26, 0, 0], [-36, 0, -6], [-3, -3, 4], [12, -2, 4], [-1, 1, -2], [-18, -2, 4], [39, 1, 2], [15, 5, -2], [-8, 0, 12], [-2, 10, -2], [-10, -4, 0], [6, -6, 6], [30, -3, 1], [-16, 8, -10], [-36, -2, 2], [25, -5, 14], [-32, 0, 0], [6, -4, -8], [-20, 3, -19], [-18, -2, -6], [-31, 0, -3], [2, 0, 8], [-31, 5, 8], [18, -8, 14], [-8, -4, 10], [-24, -2, -4], [-3, -9, 6], [-3, -7, 12], [-2, -12, 0], [41, -3, 12], [-28, 4, -6], [12, 5, -1], [-14, -2, -6], [21, -7, 4], [-38, -2, 0], [0, -6, 12], [9, -4, 5], [6, 8, -16], [-4, -4, -14], [-14, -4, -12], [-15, -1, -6], [-3, -10, -5], [-6, 6, 6], [-14, -2, -4], [-38, -8, 8], [-7, 1, -4], [30, 3, 3], [12, 4, -2], [-50, -1, 3], [33, 0, 9], [6, 8, -4], [16, -2, 10], [-50, 4, 4], [-15, -3, 0], [29, -2, -5], [39, 7, -8], [18, 3, 7], [2, -2, -14], [-19, 0, 1], [6, -8, 20], [-27, 3, -10], [-6, 0, 8], [-7, 13, 8], [10, 5, 1], [-12, -6, 4], [17, -7, 8], [-21, -2, -3], [-31, 1, 4], [12, 2, -8], [12, 10, 2], [-13, 0, -9], [-36, 3, 5], [4, 10, -4], [-56, -2, -2], [4, 7, -11], [18, 4, 8], [29, 4, 9], [-9, -3, 2], [8, -4, 22], [-33, -11, 6], [-18, 2, -4], [0, 8, -12], [27, 7, 8], [-38, 4, -14], [17, 2, 11], [20, -4, -10], [12, -2, 8], [0, -4, -12], [-43, -3, 4], [36, -1, -11], [10, 1, -7], [3, 2, -11], [49, 3, 6], [-42, 4, 8], [-9, 7, 0], [2, 4, -8], [36, -2, 8], [42, 11, -5], [5, -3, 0], [5, -5, -2], [-12, 2, 4], [19, 17, -10], [-19, -4, 5], [27, 8, -5], [-10, -6, -2], [-2, -6, 14], [-27, 3, -6], [10, 0, -6], [60, 0, 2], [-6, 0, 0], [-13, 5, -22], [-6, -10, -2], [38, 8, 0], [-13, 1, 2], [4, -2, 16], [-14, 6, 0], [-8, -4, -2], [-37, 3, 12], [15, 9, -18], [-4, 0, 18], [-34, -6, 2], [-6, -12, 2], [-15, 2, -13], [-20, 5, -13], [27, -10, 1], [-16, 0, 6], [11, -1, -4], [0, -2, 6], [-14, 0, -12], [-30, 4, -12], [22, 5, 1], [51, -5, -4], [21, -7, -8], [21, -10, 7], [-12, 2, 4], [-12, -9, 5], [4, -10, 16], [6, 4, 8], [-2, -6, -12], [-18, -8, 4], [18, 8, 10], [-19, 9, -8], [22, -4, -12], [-18, 8, 12], [10, -6, 12], [-3, 4, 9], [16, 8, 0], [7, 5, 6], [3, -2, 5], [26, -4, -16], [6, 5, -11], [28, -2, -8], [-21, 4, 11], [24, 8, -10], [-9, 7, 8], [23, -2, 9], [-21, -1, -20], [17, 9, 4], [-42, -3, 5], [12, -1, 13], [-29, 9, -2], [18, -8, -4], [16, -10, 18], [19, -11, -14], [9, 5, -8], [-61, 1, -10], [-21, 5, -6], [-13, -9, 16], [-14, -13, 3], [29, -7, -4], [21, 18, -17], [36, 10, -4], [14, -12, 12], [-12, -10, 16], [-2, -8, 10], [-9, -11, 14], [-20, -3, 19], [-15, 7, -10], [-16, 2, -20], [-12, 6, 8], [-16, -16, 2], [-55, -6, 3], [-14, -12, -12], [6, -2, 14], [12, -10, 28], [22, -8, 14], [42, -4, 4], [-6, 2, -16], [-34, 0, -4], [5, -7, 8], [12, -11, -1], [-12, -10, -8], [-8, 13, -13], [64, -6, 0], [-12, 14, -12], [-1, -6, 25], [-22, -4, 8], [42, -2, 2], [-6, -11, 17], [48, 1, 15], [-36, 6, -12], [33, -3, -12], [55, 1, 10], [9, -3, 0], [21, -4, -11], [18, 1, 17], [29, -8, 5], [-8, -8, -6], [-30, -4, -4], [23, 1, 2], [-10, -4, -4], [18, -2, -16], [-8, -8, -4], [33, -9, 2], [-33, 3, -12], [-26, 2, -4], [-8, 12, -6], [-30, 1, -19], [6, -16, 14], [16, -10, 20], [-15, 9, 0], [34, 2, -2], [-42, -6, -10], [51, -7, 22], [11, -5, 16], [-21, 1, -10], [10, -1, 27], [39, -3, 6], [4, 15, -15], [-44, 0, 2], [-19, 13, -24], [27, -6, -3], [-22, -6, 6], [-15, 7, 18], [6, 3, -9], [41, 14, -13], [-48, 5, 3], [4, 2, 0], [-26, -9, 3], [-45, -1, -12], [16, -8, -8], [0, 2, -4], [8, 2, -4], [24, 0, 2], [29, -15, -16], [-3, -13, -6], [6, 0, -8], [-8, -8, 4], [43, 13, -6], [-48, 6, 12], [-25, 5, 14], [-6, 15, 7], [-50, -8, 8], [-18, 6, -14], [-21, -6, -7], [-44, 10, -14], [4, 2, -24], [66, 4, 8], [78, -5, -1], [35, 2, -3], [-15, -5, 6], [-9, 6, -3], [-39, 1, 4], [22, 4, 8], [39, 9, 6], [-24, 6, -18], [4, 10, 4], [-21, -7, 22], [3, 16, -13], [60, -4, 2], [-8, -13, 1], [42, -4, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5200_2_a_ci();".
// 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_ci(:prec:=3)
    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_ci();".
// 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_ci( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![12, -8, -2, 1]>,<7,R![-24, -20, 2, 1]>,<11,R![-36, -24, 0, 1]>],Snew);
    return Vf;
end function;