
// Make newform 2898.2.a.w in Magma, downloaded from the LMFDB on 19 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_w();".
// 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_w();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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], [-1, 1], [1, 0], [0, -2], [2, 2], [-1, -1], [-2, 0], [-1, 0], [-8, 0], [-5, -3], [4, 2], [2, 0], [0, 0], [3, -3], [0, 2], [3, -5], [3, -3], [-8, -2], [2, 6], [-4, 2], [-6, -4], [-4, -6], [-9, 3], [-7, 1], [-2, -2], [16, 0], [0, 4], [-2, 4], [-16, 2], [2, 10], [5, 1], [0, 10], [-7, -3], [-2, 12], [-14, 2], [-3, 3], [2, -6], [-3, 11], [0, 4], [0, -8], [-5, -11], [-16, 0], [-2, -4], [-2, 4], [-4, 12], [0, -16], [-23, -1], [-6, -4], [-5, -3], [-6, 6], [-6, -6], [9, -7], [-6, -8], [10, -8], [2, -4], [12, 8], [-9, 1], [-8, 0], [16, -2], [18, 4], [-25, 1], [-11, -11], [23, 1], [-7, -7], [16, -8], [10, 6], [-18, 8], [-30, 2], [-8, 12], [-16, 6], [-6, -4], [-2, 14], [2, -4], [-4, -2], [22, 2], [-14, -4], [-14, 10], [2, -16], [4, -2], [18, 4], [36, 2], [0, 16], [17, 9], [17, -1], [-28, 4], [6, -10], [4, 6], [-8, 16], [4, -4], [-10, 8], [-16, -4], [-12, -12], [8, 12], [12, 8], [-8, 20], [40, 0], [3, -5], [-6, 4], [-8, 0], [34, -6], [18, -8], [-6, -16], [6, -8], [20, 4], [0, -10], [-21, -1], [6, -12], [0, -12], [-4, 2], [-15, 3], [-18, -12], [6, -16], [8, 14], [10, 12], [-18, -16], [22, 12], [13, -13], [-2, 16], [8, -16], [-1, 17], [-14, -10], [41, 3], [-4, 4], [-7, 1], [28, 14], [-4, -2], [21, 3], [-40, -4], [5, -5], [-2, 10], [6, -4], [14, -20], [26, 12], [24, -10], [9, -7], [-1, 5], [-8, -14], [-3, 27], [10, -12], [-17, 19], [28, 16], [22, -2], [4, 8], [-8, 4], [2, 10], [6, 2], [-46, 4], [-11, -7], [-18, 2], [44, -4], [45, 5], [-40, 0], [-9, 13], [28, 0], [14, -4], [-22, -8], [-2, -8], [23, 15], [9, -21], [-30, -10], [38, -4], [-4, 8], [-4, -22], [24, -2], [14, 6], [-30, -2], [24, 12], [-4, -18], [-2, -22], [28, 16], [-8, -8], [-24, 8], [-8, -6], [-19, -5], [11, 23], [24, -12], [-50, 0], [-32, -8], [29, 3], [22, -2], [-6, -16], [2, 28], [24, 22], [0, -8], [-25, -3], [-16, -20], [20, -14], [-2, -2], [13, -9], [-18, 12], [-20, 0], [0, 12], [34, 10], [-4, -14], [-31, 17], [26, 20], [-10, -16], [7, 3], [-3, 11], [-20, 22], [-45, 13], [44, 12], [-3, 9], [-34, 12], [4, -12], [-38, 18], [-18, -2], [-46, -2], [-5, -13], [-30, -6], [6, 2], [-14, -4], [14, -4], [-19, -17], [-3, 21], [16, 28], [0, 26], [20, -4], [18, -12], [2, -12], [-20, -20], [-46, 0], [-16, -24], [-13, 19], [32, -20], [-19, -3], [14, -26], [2, -34], [34, -2], [2, 0], [-14, 0], [10, -24], [-22, 6], [18, -16], [-30, 10], [65, 1], [0, -18], [-44, -8], [-3, 7], [16, 0], [55, -5], [13, 15], [-14, 4], [-6, -16], [-61, 5], [7, 9], [48, -6], [28, -20], [-16, 32], [4, -2], [37, -11], [-38, -16], [67, -5], [-42, 2], [-12, -4], [38, 0], [-2, 4], [-16, 12], [28, -18], [6, 16], [0, 8], [20, 0], [-29, -19], [-6, 4], [46, 8], [-26, -24], [13, -27], [-16, 26], [-18, 0], [16, -24], [28, -14], [49, -11], [-6, 12], [12, -10], [-31, 11], [-10, -14], [-8, -12], [-28, -14], [6, -28], [-13, -3], [0, -32], [-38, 10], [13, -9], [-17, 19], [22, 18], [10, 32], [26, -24], [59, 1], [32, 6], [-42, 20], [26, 4], [-54, -2], [28, 18], [-30, 12], [-1, -15], [-40, 24], [-10, 10], [-45, -9], [35, -21], [-25, -21], [54, -12], [-44, -4], [42, 22], [12, -4], [-18, 24], [14, 30], [63, -15], [-36, -4], [-34, -4], [-41, 13], [-6, 8], [32, -26], [-66, 10], [-38, -8], [-1, -9], [-25, 19], [2, -26], [-53, 11], [-30, 24], [52, 2], [-12, 10], [10, -14], [-28, -12], [-24, -14], [55, -21], [-12, 28], [25, -31], [-6, -48], [-12, 18], [-16, 32], [-6, 24], [51, -11], [-39, 1], [-20, -10], [1, 5], [4, 22], [3, -33], [50, 20], [-14, 6], [-20, 8], [-78, 4], [-18, -30], [24, 0], [34, -28], [22, -38], [50, -14], [-26, 18], [0, 4], [-75, -1], [-7, -27], [-26, -32], [-40, 24], [62, 2], [-34, 8], [-52, 2], [6, 6], [10, 40], [-50, -6], [48, -16], [-4, 22], [12, -2], [-12, 12], [16, -2], [31, -21], [22, -24], [30, -22], [30, 20], [4, 42], [-24, 0], [-39, 21], [16, 28], [-30, 20], [-38, 10], [-78, -8], [75, 3], [-20, 4], [9, -11], [-13, -13], [-50, 12], [67, 1], [24, 30], [50, 26], [30, -8], [6, 28], [11, 29], [34, -26], [-16, -24], [-12, 2], [-18, 6], [18, 20], [28, 12], [-20, -36], [17, -19], [62, -4], [-40, -14], [50, -6], [9, -11], [36, 24], [-16, 4], [-14, -38], [54, -6], [30, -24], [33, 21], [22, -14], [7, 37], [26, 0], [-2, -12], [-20, 38], [57, -23], [62, 0], [-54, -24], [-60, 20], [36, 0], [-36, 2], [60, 8], [-4, -32], [29, -13], [7, 31], [-36, -16], [24, 6], [-37, 25], [60, 8], [-15, -31], [-40, 26], [30, -14], [-24, -6], [-30, 16], [-51, -15], [21, -25]];
    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_w();".
// 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_w(:prec:=2)
    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_w();".
// 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_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2898_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 2, 1]>,<11,R![-12, 0, 1]>,<13,R![-8, -4, 1]>],Snew);
    return Vf;
end function;