
// Make newform 2898.2.a.g in Magma, downloaded from the LMFDB on 16 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_g();".
// 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_g();".
// 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], [2], [-1], [2], [-4], [6], [0], [-1], [2], [4], [0], [-6], [6], [0], [12], [10], [2], [-2], [-8], [2], [8], [16], [-6], [-2], [8], [0], [18], [-20], [6], [16], [-22], [2], [-14], [12], [0], [18], [-4], [16], [24], [-16], [-2], [16], [-18], [-18], [24], [4], [-16], [12], [6], [-10], [16], [14], [12], [6], [12], [-24], [20], [26], [22], [-12], [-14], [10], [-24], [-10], [-2], [16], [26], [-12], [24], [-10], [4], [0], [4], [2], [24], [28], [-20], [-6], [-34], [-4], [-20], [-28], [-14], [20], [16], [-30], [-18], [-8], [16], [40], [8], [8], [0], [-24], [32], [0], [-10], [16], [-6], [-40], [-32], [-24], [42], [-38], [-38], [-18], [6], [-40], [-46], [-40], [44], [-18], [-28], [-20], [-30], [24], [-12], [-26], [6], [-30], [-50], [22], [12], [46], [-20], [-24], [20], [-8], [38], [20], [-48], [0], [44], [18], [26], [-10], [20], [-14], [42], [-2], [-54], [8], [-14], [12], [0], [-44], [-38], [-42], [8], [-2], [-18], [12], [8], [-46], [-36], [-16], [34], [30], [10], [-60], [26], [24], [-36], [-30], [-24], [40], [-36], [30], [28], [10], [32], [-52], [-54], [0], [58], [-48], [-42], [-24], [14], [56], [12], [-50], [-2], [-20], [34], [-26], [36], [54], [32], [-46], [36], [10], [36], [-46], [6], [-46], [-64], [10], [8], [-64], [32], [-12], [38], [12], [-12], [-24], [8], [54], [-50], [10], [40], [32], [36], [28], [-26], [-8], [42], [-4], [6], [-62], [16], [22], [48], [-18], [2], [-54], [24], [-8], [-52], [22], [-8], [-52], [-6], [28], [8], [-2], [16], [76], [60], [-14], [74], [8], [-74], [78], [-4], [-52], [-8], [-34], [36], [8], [46], [4], [-6], [32], [40], [20], [-20], [4], [-26], [-32], [-50], [-68], [-38], [-14], [64], [-16], [-42], [58], [26], [-56], [78], [58], [-64], [58], [20], [0], [-34], [-46], [76], [-56], [36], [72], [2], [-14], [-16], [70], [-48], [16], [-34], [4], [8], [-26], [-44], [10], [70], [8], [4], [-70], [2], [10], [-28], [56], [-56], [-70], [42], [-28], [10], [0], [70], [24], [72], [-22], [16], [24], [-10], [26], [72], [50], [66], [64], [-74], [-88], [-56], [-6], [10], [-42], [14], [32], [-12], [18], [-18], [-40], [-10], [-38], [-58], [-48], [18], [-38], [0], [-74], [34], [-12], [-16], [-44], [-12], [-28], [-12], [-56], [18], [22], [6], [-34], [88], [-32], [-34], [-48], [88], [-42], [48], [-8], [22], [-8], [6], [-58], [26], [-52], [44], [-18], [0], [-58], [32], [0], [64], [46], [-46], [4], [34], [-14], [-18], [42], [2], [-56], [42], [-4], [-64], [-8], [20], [14], [64], [-30], [60], [-44], [-38], [36], [-90], [-8], [26], [-22], [-66], [80], [2], [-24], [-38], [-44], [-76], [-40], [-42], [24], [-78], [58], [46], [-98], [22], [-14], [-24], [40], [-28], [2], [-56], [60], [62], [72], [-42], [70], [32], [34], [6], [-30], [24]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : 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, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;