
// Make newform 2000.2.a.r in Magma, downloaded from the LMFDB on 17 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2000_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_2000_2_a_r();".
// 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_2000_2_a_r();".
// 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 := [11, 0, -7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [0, -4, 0, 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_2000_a();"
function MakeCharacter_2000_a()
    N := 2000;
    order := 1;
    char_gens := [751, 501, 1377];
    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_2000_a_Hecke(Kf)
    return MakeCharacter_2000_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, 1, 0, 0], [0, 0, 0, 0], [2, 0, 1, -1], [0, 0, 0, 2], [0, 2, 2, 0], [-2, 0, -2, -2], [0, -2, 0, -2], [2, 0, -3, 1], [4, -2, 3, 2], [4, 0, 0, -2], [-2, 2, -4, 2], [-2, -2, -5, -2], [3, 1, 0, 0], [3, -1, -2, -2], [-8, 0, -4, 0], [0, 0, 4, 2], [1, -4, -1, 0], [-4, 2, 0, -2], [4, 2, -4, 2], [4, -2, 10, 0], [12, 2, 0, 0], [0, -2, -1, 3], [5, -4, 5, 0], [0, -2, 2, -4], [3, 2, -1, 4], [-4, -6, 4, -2], [5, -5, 0, 0], [2, 0, 1, 0], [-8, -2, -2, 4], [7, -1, 2, 2], [0, 2, 0, 0], [2, -8, 2, 2], [4, 2, 4, -4], [-1, 4, 5, 0], [16, -2, 8, 4], [2, 0, -2, -2], [4, 2, 5, 3], [-15, -1, -10, 2], [-2, 0, -2, -6], [4, -4, 4, -2], [8, -2, 9, -6], [-4, -2, 0, -2], [6, 2, -8, 2], [2, -4, 14, -2], [-4, -6, -8, 2], [-8, 6, 4, 0], [-9, 1, -10, 2], [4, 6, -11, 5], [1, -2, -11, -4], [-6, 4, -2, 10], [8, -2, 0, 0], [1, 2, 7, 8], [-8, 4, -8, 0], [-12, 0, -4, 0], [7, -3, 6, 2], [-10, 0, -8, 0], [16, 0, 0, 0], [-14, 0, -10, -2], [-7, -6, 3, 0], [-16, 2, -4, -2], [-8, -2, -14, 0], [0, 2, -5, -7], [8, -4, 12, -6], [0, 4, 4, 0], [2, -6, 8, -2], [12, -2, 20, -4], [6, -10, -8, -2], [8, 2, -5, 1], [1, -2, 5, 4], [-16, 2, -10, -8], [-4, 8, 0, 2], [6, 0, 7, -5], [-2, 2, 8, 2], [-4, -8, -8, 4], [9, 9, -2, 6], [-6, 4, -3, 4], [-4, -6, -6, 4], [3, 0, 7, -8], [-9, -2, -7, -8], [-12, 0, 4, -4], [-6, 4, -1, -4], [0, 2, 16, 0], [6, 10, 4, -2], [16, -6, 0, 2], [8, -10, 3, -5], [-14, 8, 8, 8], [-4, 4, 8, 4], [-1, -2, -1, -4], [6, 4, 1, 1], [0, -6, -5, -11], [-4, -2, 0, 12], [5, -1, 18, 6], [-12, -6, -12, -2], [-12, -6, 4, 2], [14, 0, -1, -3], [-2, 8, 4, 0], [0, 12, -15, 4], [-17, -7, -16, 8], [16, 6, 1, 2], [8, 2, 7, 1], [-2, -4, 2, 10], [0, -6, -4, 6], [10, 6, 5, -2], [4, 12, 12, 0], [12, 4, -12, 8], [4, -6, -12, 2], [6, -6, 4, -6], [-12, -2, -8, 0], [2, 6, -9, 14], [-20, -2, -8, 14], [-16, -4, -4, 4], [-26, 0, -6, -2], [-12, -2, -4, -6], [12, 12, 0, -2], [-10, 10, 3, -6], [0, 6, -5, -9], [12, 6, 20, 2], [-26, -2, 0, 6], [-8, 2, -4, 10], [10, 8, 23, 0], [4, 6, 10, 8], [8, -12, -4, -8], [1, -3, -4, -4], [20, 2, 0, -4], [14, 8, 0, 0], [-7, 10, -19, -4], [-24, -10, -24, 8], [17, 3, 22, 2], [4, -10, -26, -4], [12, 14, -8, 4], [0, -10, 4, 6], [8, -10, -4, -12], [12, 6, 26, 4], [11, 8, -9, 8], [-4, 4, -5, 4], [8, 4, -16, 0], [-9, 3, -4, -4], [8, -6, 10, 8], [9, -12, -7, 0], [-16, -4, 4, -6], [-19, 0, -17, 0], [4, 18, 16, 2], [-28, 2, -12, -6], [22, 4, 1, -4], [0, -6, 4, -8], [20, 8, 16, -4], [32, -10, 10, 8], [-8, -8, -24, 4], [-6, 8, 13, 5], [10, -6, 0, -14], [-35, -6, -5, 8], [8, -2, 27, -5], [-14, -12, -21, 3], [-12, 6, -19, 13], [16, 8, 0, 14], [16, -6, -4, 6], [11, 10, 13, 8], [-8, 4, -4, -4], [14, 8, -12, 8], [23, -13, 4, -4], [8, 8, -4, 8], [-7, -9, -18, -14], [4, 2, 12, 8], [22, 16, -2, 2], [-12, 2, -16, -6], [8, 0, 4, 8], [14, 8, -6, 6], [4, 4, -5, -12], [18, -10, 8, 6], [24, -6, 12, -6], [20, 10, -3, -2], [-32, 0, -24, 2], [4, 10, -6, 4], [20, -6, 28, -2], [12, 0, 31, 0], [0, 8, -4, 4], [12, -10, 9, -14], [33, 1, 26, -6], [22, 0, 9, -8], [23, 7, 2, -6], [28, 16, -4, -2], [-8, -10, 6, -8], [-14, -6, -36, -6], [-1, -3, -14, 14], [-4, -14, -13, -18], [-28, 6, 2, -4], [17, -3, -16, -8], [34, -8, 4, -8], [0, 10, 12, -12], [24, -4, 32, 12], [-9, 9, 0, 8], [0, -2, 32, 2], [-10, 0, -25, -8], [20, 6, 11, 17], [22, -2, 8, -2], [21, 6, 27, -8], [-32, 0, -16, 4], [12, 2, 2, 8], [-26, 0, -9, -3], [18, -8, 16, 8], [-8, 12, -28, 8], [36, 4, 16, 8], [-8, -4, -9, 4], [-4, 8, -32, -2], [26, 8, 10, 10], [-12, 8, 8, 10], [9, -11, -8, -16], [-7, 8, -11, 8], [16, -24, 0, 0], [-20, -10, 2, -16], [-26, 4, -9, -4], [1, -11, -6, -6], [16, -6, 45, 1], [-8, 28, -4, -6], [10, -8, 8, 0], [-23, 7, 22, 10], [4, -8, -11, 8], [30, -4, -3, 7], [-2, 4, 2, -18], [-2, -16, 12, 0], [0, 2, -16, 0], [8, 8, 7, 0], [38, -4, -7, -3], [-7, 11, 12, 20], [-17, 16, -11, -16], [6, -14, 24, -2], [-20, 8, -28, -4], [22, 0, -1, -13], [4, -16, 8, 8], [20, -8, -8, -20], [0, -2, -24, 0], [12, 22, -8, 6], [25, -2, -5, -8], [28, 10, -7, 11], [-4, -2, 24, -10], [14, 6, -7, 22], [-24, 4, -12, -8], [0, -12, 0, -14], [-16, -20, -12, 4], [-21, 9, 0, -8], [28, 2, 32, 6], [42, 8, 9, 9], [-11, -6, -11, 4], [-16, -6, 10, 0], [20, 12, 20, -8], [16, 2, 8, 14], [20, 20, 24, -8], [-20, -2, -40, 6], [5, -7, -2, -10], [14, 10, 32, 10], [-16, 4, 25, -4], [-6, 0, 21, -13], [14, -8, 0, 16], [44, 2, 14, 12], [-12, 8, -8, 0], [23, 2, 31, 4], [52, -2, 21, 9], [-24, 0, 12, 8], [-10, 2, 8, -6], [23, 13, -10, 10], [-8, 14, 4, -6], [12, 30, 11, -6], [-16, 14, -14, -16], [-14, 2, -40, 2], [16, -2, -20, -8], [21, -6, 21, 4], [-24, 8, -27, 0], [-48, -2, 1, -9], [-6, 6, 16, 2], [-5, -10, -5, -12], [-35, -5, -12, -4], [26, -12, 26, 6], [4, -14, 24, -6], [-2, 26, 8, -6], [0, -10, 4, 14], [0, 18, -27, -3], [20, 14, -1, -6], [-10, -16, -16, -8], [16, 6, -12, 10], [-10, -8, -17, 1], [-8, -14, -24, 10], [-3, -5, -18, -14], [0, -2, 21, 2], [-31, 11, -28, -12], [-4, -16, 4, 10], [14, -8, 22, -6], [6, -12, 14, 6], [-20, 26, 16, 6], [-30, 0, -32, -8], [22, -12, 35, 12], [-5, -17, 16, -16], [12, 18, -14, 16], [0, 14, -20, 6], [-16, -20, -4, 8], [47, -12, 33, 8], [-44, -4, -20, -2], [30, -4, 6, 10], [-4, 10, -4, -16], [0, 18, 29, 21], [-40, -4, -28, 12], [20, -4, -4, -16], [12, 14, -20, 10], [-36, -14, 7, 3], [-32, 14, 12, 8], [-18, 10, 24, 6], [28, -2, 1, -15], [-39, -14, 1, -4], [-8, -8, -16, 6], [-18, 0, -50, -10], [13, 29, 6, -10], [-19, 2, -27, -4], [-38, 0, -20, 16], [-12, -6, 21, -21], [-41, 7, 14, 6], [-36, 4, -1, 4], [20, 22, 28, 4], [40, 4, -4, -14], [-56, 8, -8, -8], [7, -2, 9, -16], [-4, -6, 0, -12], [0, 0, 4, 12], [-31, -12, 7, 0], [-5, -7, -10, -30], [4, 30, -10, 12], [-7, 22, -13, -8], [-48, -2, -40, -10], [-49, 9, -8, 8], [-14, 4, 23, 7], [22, -10, 36, 6], [10, 4, 27, -12], [-6, -16, -42, 2], [24, -4, 52, 6], [-32, 6, -3, -1], [16, -2, 0, -14], [-3, -21, 12, 12], [7, 8, 37, 8], [30, -10, -4, -14], [-2, -8, -16, 16], [-2, -16, 3, 11], [6, 8, 10, 14], [-34, -12, 6, -18], [11, 16, -15, 8], [-16, 14, -36, 2], [26, -10, 40, 2], [-40, 2, -32, 0], [-6, -4, -9, -12], [39, -21, -12, -4], [36, -4, 36, 4], [-26, -20, -18, 22], [-16, -30, 24, -4], [-56, -8, -32, 20], [-18, 8, -21, 16], [-63, 1, -10, -10], [-11, 6, 13, 4], [46, -16, -26, -10], [28, 0, 8, 26], [12, 28, -8, 10], [6, -4, 22, 14], [4, 18, 4, 22], [-32, -18, -38, 0], [50, 8, 28, -16], [18, 24, -3, -1], [4, -4, -20, -14], [-20, 22, -3, 9], [-16, -6, 34, 0], [-40, 10, -2, 0], [-30, 16, 9, 1], [-34, 2, -41, 2], [-28, -14, -36, -2], [-48, -22, -20, 10], [10, -12, -5, -11], [-22, 16, 9, 16], [-24, 4, -8, 4], [-4, -14, 46, 0], [56, -4, 48, 8], [-8, -22, 28, -10], [-40, -14, -6, 4], [22, 2, 1, 18], [-46, 12, 2, 6], [39, 10, 11, 12], [30, 8, 14, -18], [-58, -4, -39, 11], [-54, 12, -46, -14], [-32, -2, -12, -10], [-58, -8, -21, -7], [20, -2, -4, -24], [-22, 4, -6, -30], [-43, 5, -20, 12], [-9, -13, -6, -30], [11, -26, 5, 0], [-14, 22, 16, -2], [44, -16, 28, 16], [15, -1, 32, 8], [68, -10, -4, -10], [38, 2, -16, 10], [28, 8, 28, -20], [10, -8, 20, 8], [-36, -16, 0, 4], [17, -12, 15, 0], [6, -12, 1, -20], [-10, -6, 8, -10], [-26, -20, -11, 15], [30, 14, 16, -6], [-50, -20, -11, 4], [24, -6, -4, -16], [-8, -6, 30, 8], [12, 0, 13, 8], [-12, 2, -19, -9], [48, 0, 8, 0], [-28, -8, -20, -12], [22, 2, -4, -22], [-31, -15, 16, -16], [-20, -4, -20, -8], [8, 6, 14, -4], [39, 2, -5, 12], [28, 4, 4, -12], [10, -24, 29, -17], [-10, 22, -4, 10], [18, -4, -1, -7], [11, 8, -27, -8], [-6, -18, -48, -6], [29, -17, 2, 6], [36, 10, 28, 0], [10, 18, -20, 14], [8, 4, -36, 12], [17, -19, 40, 8], [-18, 10, -15, -22], [-12, 12, -4, 0], [40, 8, 4, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2000_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2000_2_a_r();".
// 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_2000_2_a_r(:prec:=4)
    chi := MakeCharacter_2000_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_2000_2_a_r();".
// 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_2000_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 10, -1, -4, 1]>,<7,R![-31, 28, 3, -6, 1]>],Snew);
    return Vf;
end function;