
// Make newform 2898.2.a.bg in Magma, downloaded from the LMFDB on 18 April 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_bg();".
// 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_bg();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -2, 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], [0, 0, -1], [-1, 0, 0], [0, 2, 0], [2, -1, 0], [0, 1, -1], [2, 1, 1], [-1, 0, 0], [4, -2, -1], [2, -1, 1], [2, -2, -1], [4, -2, 1], [2, 2, 3], [4, 1, 2], [0, -4, -2], [0, 0, 0], [2, 2, -2], [0, 4, 0], [-4, -2, -4], [6, 2, 2], [2, -2, 0], [2, 1, 3], [0, 5, 5], [2, 1, 0], [4, 5, 3], [10, 0, -3], [-8, 2, 0], [0, -2, 1], [0, 0, 1], [6, 0, 1], [-8, -4, -2], [8, 2, 3], [6, -8, -1], [-2, 0, 6], [-6, 0, -3], [2, 0, -4], [6, 2, 6], [6, -1, 1], [-4, -5, 1], [-2, -2, -1], [-6, -2, 2], [-10, -2, 4], [-2, -6, -7], [-4, 0, -5], [-6, 0, -5], [-2, 2, -4], [10, 3, 5], [-8, -1, 0], [14, 2, 6], [-10, 0, 0], [-4, -2, -4], [2, 5, 2], [4, -11, -6], [2, -2, 2], [-6, 2, 3], [-12, 3, -3], [10, -7, 1], [2, -8, 6], [0, -2, 1], [10, -13, -3], [-6, 6, 2], [-10, 12, 5], [-10, -5, -3], [12, 1, -1], [-8, 8, -1], [-16, 4, 2], [2, 10, -2], [-12, 8, -1], [-8, -7, -9], [0, -6, 7], [-8, 0, 1], [2, -4, -1], [8, -4, -6], [-2, -4, -5], [-12, 0, -2], [12, -4, 4], [12, 3, 1], [-10, 0, 4], [2, 4, -4], [-2, 7, 1], [4, 2, 9], [-6, -6, -1], [-2, 11, 4], [-2, 7, -9], [8, 4, 5], [2, 12, -2], [-10, -4, 8], [24, -3, -3], [18, 2, -3], [-12, -1, 2], [4, -14, -2], [-22, 2, -3], [2, -2, 12], [-2, -6, 12], [-8, 4, 2], [8, -3, 5], [12, -11, 3], [-10, -5, 3], [-26, -4, -4], [0, 0, -2], [14, 8, 12], [-24, -1, 0], [-8, -14, -15], [-16, 8, 4], [-18, 6, 0], [24, 2, 2], [-20, 4, 3], [16, -4, 8], [14, -14, 2], [6, -1, -1], [-28, -6, -5], [-2, 4, 8], [-26, 7, 3], [-6, 2, -10], [32, -2, 1], [6, -1, 7], [-10, 11, 9], [-4, -6, -1], [8, 0, 8], [-34, 2, 0], [14, -2, 13], [2, -14, -6], [-6, 2, 0], [18, 4, 9], [-6, 8, 8], [-4, -4, 14], [0, 7, -4], [4, 2, -10], [10, 8, -2], [-14, 2, -10], [-30, 2, -4], [-24, -4, -8], [14, -20, 0], [2, 10, -10], [22, -1, 2], [0, -2, -3], [14, -3, -5], [24, -2, -7], [22, -12, -4], [-10, -14, -7], [-2, 14, 2], [-10, 12, -3], [0, -14, -4], [-12, 3, 5], [-4, 0, -12], [14, 13, 4], [24, 6, -3], [10, -16, -1], [-4, -10, -2], [10, 6, 0], [8, -9, 5], [6, -6, -4], [14, -13, -3], [26, 8, -1], [20, 4, 1], [6, 10, 10], [16, 6, -3], [-2, -17, 2], [-8, 8, -5], [8, 8, -3], [10, -14, 6], [24, 8, -8], [2, -15, -5], [12, -18, -5], [-48, 0, -4], [-8, -14, -18], [-4, 5, 11], [-26, -4, 4], [16, -7, -1], [-16, -10, 8], [-10, 13, -2], [-36, 0, -1], [-14, 6, -2], [-14, -9, -23], [-8, 3, 11], [28, 4, 0], [26, -4, 4], [-2, 18, 5], [46, -4, 2], [0, -6, -18], [4, 3, -10], [26, -6, 8], [-8, -10, -5], [6, 6, 13], [16, 10, 21], [6, 12, 6], [2, -11, -7], [2, -14, -7], [-8, 7, 0], [10, -4, -14], [32, -4, 5], [-28, 2, 6], [-12, 9, -11], [4, -8, 2], [4, -3, 7], [18, 2, 14], [-22, 16, 0], [24, 3, -7], [-16, 5, -4], [12, 8, 10], [10, 15, -11], [26, -5, -4], [4, -11, 3], [-24, 17, 8], [20, -25, -13], [10, 14, 5], [-46, -6, 8], [32, 8, 8], [4, 2, 10], [-24, -14, -17], [32, -1, 11], [-4, 4, -18], [26, -7, 11], [-56, -3, -8], [12, 7, -15], [24, -12, -12], [-12, 8, 11], [4, 4, -9], [0, -6, -9], [-6, 10, -14], [-24, 16, -4], [32, -16, 0], [-10, -2, 14], [8, -16, -14], [22, 10, 0], [-24, 15, 11], [-28, 18, 4], [30, 8, -3], [20, -20, 0], [10, -6, 10], [6, 15, -5], [10, 10, -10], [34, 0, 4], [30, -17, -9], [8, -10, 6], [-14, 21, 6], [-12, -8, -18], [18, 6, -11], [-34, -5, -3], [-8, -20, 2], [8, 0, 24], [42, -7, 1], [-6, 0, -18], [20, -12, -7], [-28, 7, 14], [6, -17, -1], [-20, -11, -18], [-10, -18, 3], [-24, 0, -5], [56, 6, 11], [12, 19, 21], [2, 6, -13], [42, -15, -6], [-32, -11, -19], [-2, 10, -9], [-24, 6, 9], [-14, -9, -3], [36, -21, -1], [38, 0, 4], [12, -28, -2], [12, 12, -14], [2, 17, 10], [-2, 6, -18], [-18, -8, -16], [-6, -23, -15], [-42, 12, 6], [0, 25, -3], [10, -14, -13], [-14, 4, 0], [30, -9, 0], [8, 18, 6], [10, -13, 10], [44, 12, 2], [30, 2, -4], [2, -13, 5], [14, 6, -1], [6, 0, 0], [-34, 36, 8], [-8, -17, -24], [0, -9, -8], [-40, -8, 8], [-40, 12, 10], [-2, 22, -2], [-22, 8, 7], [-44, 6, 2], [18, 8, 0], [22, 0, -20], [14, 17, -13], [-36, -10, 1], [10, 16, -14], [32, 5, 16], [8, 4, 0], [48, 5, 16], [2, 24, -10], [-6, 10, 2], [-6, 14, 9], [-28, -11, 9], [20, -6, -10], [10, 14, 5], [6, 15, 8], [48, 16, 22], [-8, -12, 12], [40, 4, 17], [22, 10, 31], [4, -2, 1], [-10, 26, 11], [10, -18, -22], [-18, 22, 17], [14, 4, -12], [0, -23, 8], [-16, 4, -8], [-10, -8, 8], [50, 6, 12], [-16, -20, -12], [16, 7, -11], [32, -14, -12], [-48, 14, 10], [-6, -1, 6], [16, -8, -18], [-38, -1, -3], [42, -14, -8], [-16, -1, -9], [-46, -2, -5], [-16, -8, 1], [-12, 9, -9], [-12, 4, -20], [-30, 4, -1], [-28, 12, -3], [-62, 4, -6], [-14, -8, -24], [64, 10, 11], [26, 13, 1], [-28, 32, 6], [-34, 10, -5], [0, 6, 4], [-22, 14, 11], [-32, 1, 1], [-46, 8, 6], [22, -6, -1], [-6, 12, -12], [52, -26, -7], [0, -9, 9], [12, 16, 6], [20, -28, -4], [14, -10, -21], [2, -27, -10], [-24, -12, -26], [-48, -16, -10], [-72, -4, 1], [-24, -8, -4], [-28, -6, -27], [32, 8, -1], [-36, 4, 16], [-14, -2, 23], [-28, 18, 11], [8, 20, -10], [30, -1, -7], [42, 12, 0], [8, 16, 18], [-2, 16, 0], [-16, -10, 21], [20, -16, -10], [32, -4, 6], [30, 2, -3], [-2, -26, -11], [4, 1, 1], [-8, 0, 0], [22, -8, 0], [6, -6, 12], [10, -28, -11], [-16, 12, -4], [-26, 8, -8], [4, -14, -18], [30, 0, 12], [-24, -10, -28], [-2, -18, 5], [22, -9, 10], [-12, 7, -7], [-2, -18, 2], [30, -8, -16], [-46, 6, -2], [-48, 2, 2], [24, 0, 20], [10, 5, 25], [-56, -3, -4], [-10, -16, -1], [18, -7, 18], [-6, 22, 1], [-24, -28, -26], [-18, 30, 10], [12, 27, 3], [-10, 2, 15], [14, 4, -25], [14, 10, 4], [-6, -18, -32], [38, 26, 35], [-44, 7, 5], [-62, 16, 5], [56, -14, -9], [-34, -17, -8], [-10, 4, 0], [-2, 2, 20], [0, -1, 15], [-4, 9, -11], [-34, 3, -13], [22, 10, 13], [-10, -16, -30], [-26, 9, -21], [28, 14, 26], [-82, 4, -4], [-16, -26, -29], [0, -16, -18], [-20, 6, -18], [-14, 10, 2], [-56, -5, -17], [-8, 20, 18], [-26, 3, 3], [-36, -3, -17], [-28, 16, -10], [10, -24, 12], [6, 16, -10], [12, 24, -2], [-32, -16, 0], [-6, 16, 12], [-12, 19, -7], [8, 6, -14], [-52, 12, -4], [34, 20, 17], [-8, 3, -8]];
    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_bg();".
// 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_bg(: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_bg();".
// 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_bg( : 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, -8, -1, 1]>,<11,R![-16, -24, -2, 1]>,<13,R![10, 2, -5, 1]>],Snew);
    return Vf;
end function;