
// Make newform 10000.2.a.bb in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_10000_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_10000_2_a_bb();".
// 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_10000_2_a_bb();".
// 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 := [5, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -3, 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_10000_a();"
function MakeCharacter_10000_a()
    N := 10000;
    order := 1;
    char_gens := [8751, 2501, 9377];
    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_10000_a_Hecke(Kf)
    return MakeCharacter_10000_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, -1, 0, -1], [1, -1, 1, 2], [-4, 0, -1, -1], [-2, 0, 0, 1], [-1, 0, -2, -1], [0, -1, -2, 1], [-1, -1, 3, 0], [-1, 0, 2, -2], [-5, -1, -1, -1], [-5, -2, -4, 4], [2, 2, -3, 3], [4, 3, 4, -3], [0, -2, 2, 0], [0, 0, 0, -4], [-2, -1, -3, -3], [0, -2, 1, -1], [-5, 2, -6, 0], [-6, 0, 0, 4], [-4, 4, 2, 2], [6, 0, 5, 5], [-8, 0, -6, 0], [-6, 1, 2, 2], [-1, -2, 4, -1], [0, 0, -2, 5], [-1, -5, -1, 3], [3, -1, 1, -4], [2, 7, 6, 1], [-4, -6, -2, -4], [-1, 5, -3, 2], [-4, 7, -4, -2], [-6, -1, -7, -1], [-1, 3, 3, 6], [-4, -2, 6, -4], [1, -5, 6, -6], [7, 5, 7, -5], [-6, -5, -6, -2], [0, -2, 2, 6], [9, -1, 13, 0], [-8, 6, -10, -6], [-3, -8, -2, 0], [-6, 0, 0, 0], [4, 0, -8, 2], [1, -8, -3, 2], [0, 11, -1, -3], [4, -1, 6, 5], [-6, 2, -1, 3], [-4, 5, 7, 9], [10, 2, 2, 0], [-16, 4, 3, 2], [-9, -8, 3, -6], [-1, 9, -3, 8], [0, 3, -6, 2], [2, -5, 12, 2], [-10, 3, -5, -1], [8, 8, 5, -2], [8, 0, 10, -6], [-3, 4, 0, -2], [0, -4, 8, -6], [5, -3, 2, -6], [-2, -6, -8, -2], [-5, 0, -1, -2], [8, 2, -2, -5], [21, -3, 6, -4], [-1, -8, -18, -7], [4, 2, -8, 4], [-11, 3, -21, 3], [-11, 9, -7, -8], [12, 0, 6, 14], [9, 2, 13, 12], [14, -1, 4, -3], [6, -6, -11, -7], [5, 5, 5, -12], [-8, 12, 2, 3], [-7, 2, 16, 3], [6, 0, 21, 3], [6, 4, 8, 10], [-19, 0, 2, 2], [-4, -6, 2, -6], [7, 6, 10, 7], [-12, 10, -10, 0], [-24, 1, -6, -7], [-1, 0, 8, -2], [12, 8, -8, 2], [-7, -6, -14, -6], [2, -1, -2, -2], [-6, 4, 4, 8], [2, 5, -8, -4], [12, 0, 15, -7], [2, -6, -1, -2], [6, 1, -22, 1], [-16, 6, -18, -10], [-15, -6, 0, 11], [-4, -15, 0, 4], [-2, 1, -19, -7], [-14, -4, 3, -8], [4, 0, -14, -6], [-3, -15, -5, 4], [5, -11, 6, 12], [-8, 0, -7, 13], [-17, 5, -1, -1], [-15, 6, -10, -4], [-10, 1, -11, 17], [-2, 11, 10, -7], [18, 16, 12, -10], [0, 10, 12, 4], [-22, 4, -4, -10], [18, 14, 2, -6], [2, 7, -10, -2], [-31, 3, -10, -2], [0, 2, -16, -6], [19, 3, 13, -4], [22, -2, 28, -6], [4, 10, 4, 0], [-12, 12, -1, 3], [-22, -3, -8, -5], [0, -4, 7, -9], [9, -15, 3, -8], [-17, 6, -8, -11], [-4, 14, 4, -12], [-19, 3, -34, -2], [6, 4, -15, 5], [-8, 3, -7, 1], [8, -2, 12, 2], [7, 11, 9, -12], [-23, 6, -1, 2], [32, 6, 10, -12], [-12, -8, -12, 6], [24, -8, 18, 16], [-8, -3, 12, -6], [14, -8, 7, -14], [-16, -8, -13, 5], [7, -6, 15, -6], [18, -4, -4, 8], [7, -5, 21, 5], [-34, -2, -22, 2], [-1, 17, 12, 0], [9, 2, -2, 0], [-6, -16, 2, -8], [-1, 8, 14, 13], [24, 2, 26, 6], [-7, 7, -8, 26], [-18, -12, 14, 0], [1, 4, 12, 12], [7, 7, -19, 11], [8, 18, 10, -7], [35, 2, 10, 7], [-2, 14, -26, 2], [11, -7, 11, 1], [6, 8, -2, -4], [-5, 13, -2, 6], [-18, 1, 10, 7], [1, 3, -2, -6], [-15, 8, 14, -4], [13, 4, -4, 16], [-7, 2, -24, 4], [-20, -7, 4, -14], [-5, 6, -14, 19], [15, 5, 21, -5], [8, -9, -12, -12], [0, -10, -14, 10], [-32, 4, -20, -12], [2, -15, 18, -1], [14, -13, 16, 3], [14, -4, 17, 2], [4, 6, -3, 25], [19, -6, 23, -10], [-21, -7, -15, -13], [-8, 9, -11, 11], [26, 1, 3, -5], [-21, -8, 2, 4], [16, -18, 14, 12], [-19, 2, -33, 2], [-15, 18, -5, 0], [13, -13, -5, -12], [-5, -5, -29, 2], [-6, 15, 6, -3], [-3, -3, -21, 2], [6, -8, 8, 10], [10, -13, 9, 5], [-8, -18, 6, 8], [18, -5, 10, -19], [4, 14, 16, -8], [13, 3, 11, 14], [-4, -10, -24, 8], [2, -10, 7, 3], [1, 18, 7, 0], [-11, 8, 17, 6], [2, -9, 6, -1], [14, -14, 16, -4], [28, -6, 0, -6], [7, 11, 15, -8], [6, 10, 3, -19], [0, 8, -18, 17], [21, -10, 8, 6], [21, -15, 34, 6], [-8, -10, 18, -12], [-36, 7, -4, -8], [17, -6, 14, -13], [-32, 8, 0, 12], [8, 0, 15, -7], [0, -2, -25, 12], [-40, 1, -5, -11], [15, -7, 19, -1], [10, 16, 20, 0], [-31, -7, -4, 8], [5, -16, -5, -8], [4, -18, -8, 10], [2, -20, -12, 5], [16, -19, 13, 1], [14, 1, -4, -9], [23, 5, -13, 9], [-26, -16, -2, -12], [21, -4, 8, 4], [0, 34, -4, -9], [-7, -20, 12, 6], [-8, 2, 0, -11], [9, -1, -5, -1], [1, 10, -12, 13], [-21, 6, -22, 10], [4, -14, -22, -18], [0, 20, -12, -1], [4, -10, -11, 21], [35, 6, 10, 3], [-14, 17, 14, 4], [-28, -4, -11, 6], [-12, -4, 2, -16], [-18, -23, 3, 5], [0, 16, -3, -1], [18, -6, -4, -6], [10, -10, 14, 2], [-1, -8, -6, 10], [32, 2, 22, -24], [8, -16, -28, -15], [-3, -14, -11, 16], [6, -16, -1, 11], [5, 19, -5, 10], [-9, -18, 11, -26], [41, -3, -10, -10], [36, 14, -14, 4], [28, -13, 4, 11], [5, -7, 25, 8], [-28, 4, -14, 5], [-15, -8, -10, 0], [-28, -20, 0, 6], [36, 3, -19, 7], [26, -3, -13, 7], [32, 6, 2, 3], [-4, -10, -14, -8], [-5, -6, -4, -2], [-16, 0, -36, 4], [28, -12, 6, 16], [-6, -14, 15, 13], [-5, 13, -25, -6], [-52, -3, -13, 5], [8, -8, -18, 2], [-13, -5, -27, -21], [-24, 4, -34, -21], [-24, 1, -20, -15], [-6, -10, 19, -23], [-29, -2, -18, -15], [0, 26, 27, 3], [-26, -8, 2, 7], [40, 9, 5, -19], [-4, 18, -8, -2], [32, -12, -15, -8], [-11, -1, -4, -8], [-3, 25, -24, 2], [12, -12, 0, 4], [34, -4, 34, 14], [4, -5, 40, 0], [-7, -16, -29, -12], [18, 4, 10, -10], [-34, -11, -10, 11], [6, 20, 8, 12], [-25, -11, -5, 0], [-24, -6, -7, 14], [6, 16, -34, 2], [0, 28, 8, 2], [47, -10, 18, 4], [18, -12, -28, 3], [-22, 26, 2, -2], [38, 16, 42, -10], [28, -8, 30, 4], [30, -4, 52, -7], [39, 11, 12, 4], [23, -18, 26, 14], [-4, 22, 22, -8], [54, 12, 34, -22], [-1, -25, -6, -8], [-12, -20, -22, -2], [33, 13, 25, -26], [20, -16, 17, 15], [-28, 0, -26, 18], [9, -2, 12, 26], [-4, 12, -11, 9], [-25, -5, -25, 12], [-2, -12, -23, -25], [20, -6, 2, -3], [10, 18, -11, -7], [17, 22, -11, 16], [-36, -4, 11, -3], [3, -13, 15, 14], [-14, 8, 6, 5], [50, -14, 26, 26], [32, 10, -6, -14], [10, -8, -5, 6], [3, 31, 3, -5], [-4, 18, 0, 13], [-20, -2, -20, 6], [-28, -10, 5, -2], [-25, -9, -37, -1], [40, -13, 6, -1], [-18, 10, 4, 40], [-28, -3, -11, 9], [7, -6, 33, 2], [-38, -10, -34, 26], [-8, -10, -11, 20], [-4, -13, 1, 5], [-6, -21, -8, 24], [-67, 10, -8, -3], [-46, -14, -14, -1], [-54, 21, -26, -23], [6, -4, -2, -14], [12, -17, -31, -17], [-3, 17, 7, -21], [26, -1, 38, 13], [64, 4, 0, -2], [33, 1, -3, 2], [22, 8, 23, -17], [-19, 6, -8, 32], [26, -4, -10, 6], [20, 4, 4, 34], [-13, 27, -25, 1], [-12, 14, -4, -8], [18, 14, -2, -11], [13, -16, -8, -28], [-12, 10, 32, -6], [-2, -10, -12, 10], [-8, -32, 28, -15], [-15, -31, 5, -10], [-27, 12, -50, 6], [-23, -7, 1, 7], [11, 18, 22, -11], [-28, 10, 10, -2], [16, -22, 3, -13], [-5, 28, -1, -4], [-2, -34, 0, -2], [-5, 1, -21, -26], [10, -1, 34, 20], [49, 9, 59, -12], [-2, 8, 14, 9], [19, -25, 3, -4], [-46, -18, -40, 12], [-6, -24, 23, -2], [48, 6, 10, -8], [2, -19, 8, -3], [2, 27, -28, 8], [44, -4, -3, 3], [24, -6, 29, -28], [62, -2, 20, -6], [10, -33, -5, 9], [5, 9, -4, -8], [6, -37, -6, -5], [-14, -14, -14, 2], [0, 11, 18, -7], [24, 2, 4, 0], [21, 4, 6, 35], [-13, 27, 9, 10], [24, 23, 16, -28], [-68, -8, -26, 20], [-29, 2, 1, -24], [3, 19, 5, -7], [-4, 27, 7, -5], [-1, 20, -18, 10], [-32, -7, -32, 8], [0, 2, -20, -26], [-20, 7, -36, -3], [-40, -36, -4, 10], [-2, -4, -22, -2], [-44, 6, -32, -22], [-26, -7, 2, -1], [11, 30, 12, -5], [24, 3, 6, 4], [12, 8, 33, -20], [30, 20, -11, -11], [-2, -18, -1, 3], [4, -30, -36, -7], [21, 4, -3, 8], [11, 7, -26, -4], [40, -26, 10, 0], [-33, 9, -20, -6], [-42, 30, -20, -28], [6, 6, 24, 25], [46, 8, -3, 16], [-59, 12, 7, 2], [-24, 5, -29, 5], [-29, -27, -27, 4], [-16, 7, -27, 5], [-6, -34, 20, -6], [-12, -2, -8, 47], [-8, -3, -2, 33], [-21, -15, 13, -34], [5, -28, -36, -18], [24, 42, 7, -11], [-31, 34, 4, 2], [-7, -7, 44, 6], [28, 2, 26, 26], [52, -13, 26, 33], [52, -2, 3, -3], [-39, -19, -15, 33], [19, 12, 12, 17], [20, 1, -26, 6], [19, -6, 14, 7], [-5, 10, -40, 0], [4, 8, -16, 0], [8, 19, -10, 15], [8, -5, 34, -29], [-2, -8, -24, 28], [-1, 11, 12, 44], [48, -4, 2, -23], [48, -25, -9, 5], [-14, 13, 4, -17], [58, -6, 5, -13], [-5, -11, -17, -41], [21, 20, -3, -22], [-20, -32, -6, 14], [-28, -12, -51, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_10000_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_10000_2_a_bb();".
// 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_10000_2_a_bb(:prec:=4)
    chi := MakeCharacter_10000_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_10000_2_a_bb();".
// 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_10000_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_10000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 6, 1, -4, 1]>,<7,R![-19, 8, 14, -8, 1]>,<11,R![-59, 82, -26, -2, 1]>],Snew);
    return Vf;
end function;