
// Make newform 2898.2.a.be in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2898_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_2898_2_a_be();".
// 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_2898_2_a_be();".
// 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 := [2, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 1, 1], [-3, -1, 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_2898_a();"
function MakeCharacter_2898_a()
    N := 2898;
    order := 1;
    char_gens := [1289, 829, 1891];
    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_2898_a_Hecke(Kf)
    return MakeCharacter_2898_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 := [[-1, 0, 0], [0, 0, 0], [-1, 0, 1], [-1, 0, 0], [2, -1, 1], [0, 1, -1], [-3, 1, 0], [0, -1, -1], [1, 0, 0], [2, 1, -3], [-1, 1, 0], [-4, 2, 0], [-2, -2, -2], [0, -2, 2], [-1, 1, 0], [0, 2, 0], [-3, 0, -3], [-1, -2, 3], [2, -1, 1], [10, -2, 0], [-4, 4, -2], [-8, -1, -5], [-10, 1, -1], [1, -1, 2], [7, -1, 4], [-8, -1, 1], [-8, 0, 0], [4, -4, 4], [-2, 2, 2], [-8, 0, 2], [-10, 0, -2], [-13, 2, -3], [0, -4, -2], [3, 2, -3], [2, 0, 0], [6, 0, 6], [9, 0, -1], [-14, -2, -4], [9, -1, 0], [-18, 1, 1], [-4, -2, 6], [3, 2, 3], [0, 4, 4], [14, 0, -4], [-10, 0, -4], [-4, 0, 4], [12, -2, -2], [-7, 3, -4], [4, -1, -5], [3, -2, -1], [-8, -3, 3], [-2, 4, -6], [-5, -5, 4], [0, 1, -7], [-2, -2, 6], [-16, -1, -5], [-6, -3, 9], [-17, 1, 0], [-2, -1, -5], [-4, -2, 4], [-4, -5, -1], [-1, 0, 1], [-9, 0, -1], [-1, 1, 8], [-21, -1, 0], [-6, 5, -7], [-10, 8, -6], [-6, 2, 2], [10, 2, 8], [2, -5, 11], [4, 6, -4], [0, 3, -1], [-14, 4, -2], [6, 4, -4], [-14, 1, 3], [-2, -6, 8], [18, 2, -6], [-16, -3, -5], [14, -6, 2], [8, 2, 0], [4, -5, 7], [-4, -2, -4], [8, 0, -8], [7, -5, 0], [-15, -1, -8], [-8, -2, -6], [8, 3, -7], [16, -2, 4], [-6, 3, -9], [4, 4, 8], [-8, -1, -1], [4, -2, 10], [12, 4, 8], [8, 4, 8], [4, 0, 0], [-4, -2, -6], [-6, 3, 7], [13, 1, 8], [4, -1, -5], [6, -5, 7], [2, 6, 4], [6, 8, 4], [8, -7, 1], [6, -8, 8], [20, -2, 6], [0, 0, 6], [1, 2, 11], [-6, -2, 2], [12, 2, 14], [-4, -4, 6], [17, -11, -2], [-2, 6, -2], [-10, 4, -12], [-6, -7, -5], [-16, -5, 7], [30, 4, 4], [36, 1, -3], [-15, -1, -8], [18, 6, -2], [0, 6, 10], [3, -2, -1], [12, -1, 5], [-31, 4, -1], [8, 2, -2], [11, 8, -5], [4, 2, -4], [16, 4, -2], [9, 7, 0], [12, -2, 10], [13, 4, -9], [-6, 4, -14], [-8, -7, 5], [0, 1, -3], [22, -4, 4], [-20, 10, 0], [3, 7, -8], [7, -2, -1], [22, 3, 1], [21, -2, -3], [10, 0, 4], [17, 2, 3], [-6, -1, -13], [2, -12, 6], [0, -8, 12], [10, 3, 11], [-26, 0, 6], [0, 11, -7], [-6, 0, -12], [-33, -2, -11], [-2, 2, 0], [10, 3, 3], [-21, 9, -2], [4, -6, 2], [-21, -1, 10], [16, 4, 16], [0, 1, 13], [-20, 5, 5], [14, 0, 0], [25, -1, 6], [5, -4, 3], [6, -4, -2], [-30, -2, 2], [24, -2, 6], [-10, -3, -13], [4, 6, -4], [26, 0, 10], [-6, -10, 16], [10, -5, -5], [-8, 10, 0], [-12, -7, -1], [8, -4, 8], [6, -7, 5], [8, 12, -20], [36, -6, 4], [-15, 3, 4], [-11, -9, -2], [-24, -4, -8], [10, 2, -6], [-16, 0, 8], [-27, 16, -5], [18, 0, -6], [16, -15, 1], [18, 0, -4], [-4, -10, 12], [-32, 0, -16], [-21, -2, -13], [-30, -3, -3], [-10, -7, 15], [20, -7, -9], [-15, 13, -10], [-26, -8, 12], [-28, -4, -4], [12, 4, 20], [4, 3, -7], [-10, -5, 9], [3, 9, 14], [-34, 10, -2], [-14, -4, -8], [5, 3, -14], [5, 11, 8], [-12, 2, 12], [15, 9, -8], [2, -7, 9], [15, -7, -2], [-20, 5, -7], [14, -13, 11], [-18, 4, 10], [30, 6, -8], [8, -7, -1], [-47, 4, -3], [-8, 5, -1], [20, 5, 11], [-32, -1, -5], [-4, 5, -7], [-27, -3, 2], [-1, -13, -8], [-12, -6, 6], [16, -4, -10], [-12, 4, -16], [6, 6, 18], [34, -4, 8], [-36, -4, 0], [-18, 2, 10], [-40, 8, 0], [-35, -4, -15], [-6, 5, -11], [-49, -3, -2], [-22, 0, 2], [10, 10, -4], [-10, 4, 14], [6, -6, 10], [-16, -3, -11], [4, -7, 9], [0, -3, 11], [-16, 5, -11], [-18, 8, -2], [-5, 3, 12], [-20, 16, 2], [4, 12, -4], [5, 1, -10], [8, -2, -6], [41, 6, 15], [21, 1, 6], [18, 0, -12], [6, -6, 10], [-17, -11, 12], [-33, 5, -12], [6, -5, 17], [20, -6, 18], [-8, 0, 8], [0, 0, 2], [-17, 13, -10], [36, 9, 1], [-3, 3, -18], [-4, 1, -9], [-24, -2, 2], [-6, 0, 12], [-20, -3, 9], [38, -3, 13], [16, -18, 4], [-12, -5, 3], [-56, 6, 2], [14, 7, -21], [7, -6, -9], [-6, -8, -12], [56, -1, 7], [6, -18, 18], [7, -11, 14], [-26, 17, -9], [18, -14, 2], [6, -3, 9], [-34, 3, -15], [-17, -11, 18], [-24, -9, -13], [-16, -14, 16], [-23, 1, 10], [-6, -12, -14], [-30, -3, -3], [12, 12, -10], [36, 1, 5], [-29, -3, -16], [32, 8, 0], [14, 12, -22], [5, 14, -7], [29, 10, -9], [-50, 6, 4], [10, 4, -4], [30, 0, -4], [-41, 7, -2], [12, 14, 12], [58, 4, 4], [4, 11, -9], [8, 13, -13], [-42, 17, -9], [18, -12, 8], [3, 4, -15], [-32, -8, 8], [20, -11, 11], [9, 6, 7], [-3, -8, -11], [-35, 9, -4], [-22, -8, -8], [4, -12, 24], [38, -8, -6], [-4, 2, 2], [-22, 2, 2], [-34, 2, -4], [-37, -6, 11], [12, -20, 0], [10, 14, 10], [-33, 7, -18], [30, 4, 16], [-20, -2, 12], [2, -12, -6], [-4, -11, 5], [5, -1, 2], [-19, -2, 11], [18, 2, -12], [25, -5, 26], [-42, -2, 14], [-10, 13, -13], [28, -4, -10], [36, -1, -11], [36, 8, -4], [0, -8, 10], [9, -3, -12], [28, -10, 14], [3, -4, 15], [-20, -3, 21], [24, 4, 2], [-42, -1, -5], [2, 16, 0], [15, 5, 20], [-9, 8, -17], [-6, -7, -13], [-5, -6, 5], [-8, -8, -14], [37, -5, 10], [54, -2, 2], [2, 6, 8], [6, 9, 21], [26, -8, 12], [-36, -3, -21], [8, 8, -8], [-22, 2, -22], [18, -4, 10], [24, -3, 11], [26, 0, 22], [4, 14, -22], [-15, 8, 3], [39, 2, 9], [-22, 6, -18], [50, -7, 5], [-26, -10, -4], [82, -4, 0], [0, -6, 4], [-46, 8, 2], [16, -19, 5], [12, 9, -21], [-16, 4, -4], [0, 20, -2], [36, 8, -2], [-28, -4, -16], [18, -13, -11], [49, 10, -13], [-46, 2, 2], [4, -11, 19], [-32, 9, -3], [-36, 8, -14], [12, 6, -18], [19, 10, -27], [-12, -6, 22], [-46, 2, -6], [30, -16, -2], [30, 6, -18], [1, 8, 17], [52, 0, 4], [-45, -3, -26], [-3, -11, 36], [-34, 0, -28], [-9, -2, -9], [-4, -18, 20], [14, -20, 10], [38, 12, 4], [-44, 1, -3], [-53, 21, -8], [10, 10, 4], [-2, 13, -7], [22, -1, -19], [66, -4, 6], [14, -18, -14], [30, -9, -9], [48, -10, 10], [11, 2, -3], [24, 13, -23], [-4, 6, -8], [-26, 4, -14], [-13, -5, -24], [-8, 8, 12], [38, -5, 11], [-24, -7, -1], [-28, 13, -25], [-36, -5, -5], [11, -1, 4], [-28, -15, 23], [-21, 15, -22], [26, -10, 2], [18, -10, -14], [-38, -9, 9], [-21, -4, -25], [-46, 2, -6], [-54, 4, 4], [20, 14, -2], [-32, -12, -24], [-20, -4, 6], [34, 13, 1], [16, -10, -2], [1, 15, -24], [25, -15, -4], [40, -14, -6], [-4, 0, -6], [-33, -4, -15], [64, -6, 2], [-41, 8, 7], [20, -10, 16], [-12, 13, -21], [38, 9, -1], [54, 6, -2], [15, -14, 25], [-15, -3, -34]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2898_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2898_2_a_be();".
// 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_2898_2_a_be(:prec:=3)
    chi := MakeCharacter_2898_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_2898_2_a_be();".
// 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_2898_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2898_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, -2, 4, 1]>,<11,R![16, -12, -4, 1]>,<13,R![16, -16, -2, 1]>],Snew);
    return Vf;
end function;