
// Make newform 2898.2.a.s in Magma, downloaded from the LMFDB on 23 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_s();".
// 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_s();".
// 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], [4], [4], [8], [-2], [-1], [-2], [-6], [-10], [-6], [-8], [-6], [-2], [0], [10], [8], [12], [6], [0], [-2], [-12], [12], [-8], [-20], [4], [18], [14], [-20], [0], [18], [-4], [10], [-20], [-18], [4], [14], [0], [-20], [-6], [16], [-10], [-6], [20], [4], [-10], [-6], [2], [-6], [-12], [-8], [22], [14], [-8], [-16], [2], [22], [-6], [-2], [-22], [-28], [10], [-8], [10], [20], [-26], [28], [-16], [10], [24], [-12], [-10], [4], [-24], [6], [20], [18], [10], [6], [-14], [0], [4], [-10], [4], [34], [26], [24], [16], [-22], [-4], [16], [-36], [28], [-24], [-24], [-20], [-34], [6], [-12], [2], [38], [10], [-40], [30], [-44], [30], [32], [-2], [-22], [26], [-6], [10], [8], [-38], [-38], [46], [6], [-24], [18], [46], [18], [-36], [-4], [42], [-10], [-42], [-12], [-50], [-20], [-24], [24], [42], [-18], [16], [46], [18], [22], [18], [8], [-6], [-12], [12], [-12], [-24], [4], [-6], [4], [-4], [-6], [24], [36], [54], [4], [16], [-40], [-54], [-8], [-6], [12], [14], [-40], [14], [-30], [-24], [52], [28], [-38], [28], [-28], [40], [16], [46], [-34], [-60], [-52], [6], [-40], [-34], [-28], [-2], [-14], [42], [-32], [-18], [38], [-6], [38], [50], [50], [-20], [68], [-52], [32], [12], [-10], [-46], [-32], [-34], [30], [-30], [-28], [8], [-30], [-4], [32], [44], [6], [20], [-6], [-40], [-8], [-38], [26], [-24], [-32], [-18], [24], [30], [18], [-8], [-18], [24], [24], [-6], [-68], [60], [-28], [-44], [-70], [74], [-32], [10], [46], [12], [-8], [38], [20], [-14], [8], [-40], [-62], [22], [6], [-6], [2], [-10], [-32], [-48], [18], [-28], [-48], [40], [24], [68], [-22], [-62], [28], [50], [80], [0], [-28], [26], [18], [-54], [-14], [28], [-32], [-2], [40], [68], [36], [-48], [54], [42], [84], [14], [42], [-6], [-30], [-16], [72], [-38], [84], [60], [-46], [-38], [-58], [66], [-14], [42], [-66], [-18], [38], [66], [64], [-84], [-36], [72], [84], [78], [40], [-68], [32], [-38], [84], [-70], [-48], [22], [14], [38], [-30], [44], [-32], [-52], [68], [-4], [-44], [14], [22], [46], [32], [-56], [-42], [48], [16], [28], [-88], [-62], [38], [78], [2], [24], [8], [76], [14], [52], [26], [-16], [-2], [74], [-20], [-32], [62], [4], [-52], [-84], [-72], [-14], [-24], [-86], [-78], [-32], [38], [66], [-40], [-78], [26], [-40], [58], [-70], [40], [0], [88], [-42], [76], [0], [-46], [68], [60], [-8], [-54], [28], [-66], [-4], [0], [-64], [-84], [30], [54], [6], [28], [30], [26], [-14], [-20], [-76], [-88], [-56], [-10], [28], [36], [100], [24], [46], [80], [-12], [20], [-34], [-8], [26], [-48], [-20], [68], [-18], [62], [26], [-10], [60], [82], [6], [-24], [20], [-42], [-72], [-24], [62], [-12], [40], [-70], [78], [48], [24], [-38], [16], [36], [22], [108], [-14]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : 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![-4, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;