
// Make newform 2898.2.a.t 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_t();".
// 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_t();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [3], [1], [-6], [5], [6], [2], [1], [-9], [-4], [-1], [9], [11], [3], [-6], [12], [2], [-4], [-6], [2], [-16], [-12], [12], [5], [-6], [11], [-6], [11], [-15], [-19], [-6], [-15], [11], [24], [-19], [14], [14], [-24], [-18], [-3], [2], [0], [11], [-3], [-7], [8], [8], [21], [-10], [18], [-18], [11], [21], [-18], [3], [-30], [14], [-4], [3], [-28], [-6], [17], [0], [-10], [-3], [26], [-22], [-27], [-10], [-21], [-3], [17], [14], [-19], [6], [-24], [26], [6], [-22], [0], [-37], [-33], [-37], [-10], [-9], [12], [2], [12], [23], [-21], [0], [-25], [12], [8], [-6], [0], [30], [-22], [38], [-22], [30], [21], [-9], [-4], [-4], [-24], [21], [-36], [-22], [-22], [5], [-18], [26], [-34], [3], [26], [12], [-21], [-36], [32], [-37], [6], [12], [-31], [-42], [-10], [-21], [8], [20], [-10], [0], [-4], [-10], [6], [5], [27], [-16], [15], [-6], [11], [-6], [29], [42], [38], [-12], [-37], [33], [23], [-24], [-52], [30], [-16], [-24], [41], [45], [26], [57], [53], [-33], [27], [-30], [-16], [36], [57], [-12], [-4], [2], [26], [-30], [54], [-31], [39], [-46], [56], [60], [-16], [-54], [29], [-4], [44], [-21], [-4], [27], [-3], [63], [44], [26], [-25], [-33], [-52], [-9], [-40], [-36], [6], [-6], [-58], [62], [18], [-15], [6], [56], [5], [38], [3], [24], [29], [36], [0], [-4], [23], [0], [-22], [-12], [45], [2], [32], [-63], [-45], [-9], [-10], [-52], [-36], [-34], [-30], [32], [60], [18], [71], [-24], [50], [38], [-22], [30], [2], [48], [-49], [6], [51], [0], [6], [-16], [50], [56], [-9], [45], [-34], [-39], [59], [15], [5], [-6], [63], [23], [36], [-15], [-37], [68], [12], [-70], [-10], [42], [-67], [-34], [-18], [62], [24], [72], [-13], [-66], [-31], [-52], [-19], [-10], [68], [-28], [45], [-22], [38], [45], [-9], [8], [6], [-46], [-25], [48], [26], [-6], [8], [57], [12], [-33], [-24], [15], [-16], [30], [17], [12], [84], [47], [11], [18], [20], [-69], [-37], [5], [21], [-22], [-63], [-34], [-39], [-48], [42], [-28], [0], [2], [30], [-24], [5], [-30], [44], [32], [-24], [29], [-63], [14], [-4], [59], [57], [84], [50], [51], [-82], [54], [71], [54], [83], [84], [-16], [23], [-10], [-81], [18], [-10], [-24], [9], [53], [26], [66], [57], [20], [-43], [69], [-64], [-49], [-93], [30], [96], [66], [-18], [-46], [-15], [-30], [-6], [47], [-85], [-12], [-76], [2], [-60], [-67], [-12], [-18], [-40], [50], [-6], [21], [-67], [54], [62], [42], [54], [-88], [24], [26], [57], [-49], [-91], [-43], [-30], [38], [-48], [9], [71], [12], [-40], [-13], [-54], [-73], [-33], [41], [-102], [-76], [24], [-72], [2], [-67], [72], [-16], [-12], [98], [-57], [-66], [-52], [62], [84], [-18], [-10], [-18], [-30], [102], [56], [-18], [-60], [-70], [42], [-36], [-60], [65], [-45]];
    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_t();".
// 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_t(:prec:=1)
    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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2898_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<11,R![6, 1]>,<13,R![-5, 1]>],Snew);
    return Vf;
end function;