
// Make newform 6018.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6018_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_6018_2_a_f();".
// 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_6018_2_a_f();".
// 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_6018_a();"
function MakeCharacter_6018_a()
    N := 6018;
    order := 1;
    char_gens := [4013, 1771, 1123];
    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_6018_a_Hecke(Kf)
    return MakeCharacter_6018_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], [-1], [0], [-2], [-2], [-2], [1], [4], [4], [4], [0], [-2], [-2], [8], [-8], [-6], [-1], [6], [-4], [2], [8], [-14], [-12], [-10], [8], [-6], [0], [4], [-10], [-20], [-12], [-2], [-10], [-4], [22], [0], [-14], [-20], [10], [-2], [0], [0], [-8], [22], [8], [-10], [14], [-24], [18], [-22], [-28], [20], [10], [-12], [2], [24], [-2], [0], [-8], [-10], [18], [-10], [-4], [18], [-16], [-24], [20], [20], [-26], [30], [-6], [24], [36], [-14], [-20], [-8], [-26], [-22], [-20], [-30], [-10], [-6], [-8], [-34], [26], [4], [34], [-22], [-38], [-40], [8], [18], [-18], [36], [8], [4], [30], [30], [-20], [-34], [-32], [-22], [0], [14], [6], [-2], [12], [26], [24], [-20], [38], [34], [-10], [4], [-20], [42], [40], [12], [16], [0], [-22], [-44], [-32], [-26], [10], [6], [-36], [-16], [44], [34], [-36], [-6], [16], [22], [38], [-18], [6], [40], [-26], [48], [38], [14], [16], [12], [46], [-32], [-52], [0], [40], [8], [-52], [52], [36], [0], [-48], [6], [-8], [-52], [-58], [30], [-28], [38], [0], [-36], [30], [60], [20], [8], [-16], [6], [36], [-10], [42], [22], [24], [-42], [12], [18], [4], [-26], [-64], [0], [38], [22], [24], [10], [22], [-36], [-16], [12], [-12], [-14], [-24], [6], [-24], [32], [18], [10], [12], [16], [68], [58], [62], [50], [-20], [18], [-56], [-36], [32], [-24], [-28], [54], [-46], [-20], [-26], [-20], [28], [-10], [-32], [-62], [-14], [-36], [54], [30], [-40], [-22], [-62], [60], [16], [32], [-30], [-42], [48], [-6], [-32], [12], [56], [22], [-30], [-36], [-16], [-8], [-20], [54], [38], [-54], [40], [-6], [-52], [4], [22], [28], [48], [-2], [-34], [12], [-56], [54], [-6], [10], [12], [36], [36], [-68], [-6], [52], [42], [-30], [-64], [-18], [-50], [-68], [46], [72], [-14], [-8], [-4], [-30], [-18], [20], [52], [44], [12], [-70], [-86], [-32], [-12], [-60], [8], [18], [2], [-52], [6], [66], [-8], [70], [0], [10], [54], [8], [-6], [82], [-52], [-26], [-84], [-76], [4], [14], [-72], [-42], [54], [-24], [-6], [68], [-88], [-58], [-36], [-58], [14], [-78], [16], [-48], [-10], [-84], [74], [-50], [-10], [-4], [0], [-22], [62], [-54], [28], [-36], [40], [44], [50], [44], [20], [52], [-22], [38], [-30], [-48], [34], [66], [-46], [12], [-24], [18], [52], [2], [-42], [52], [-58], [6], [72], [12], [-78], [-28], [-2], [-78], [0], [30], [-36], [86], [-18], [20], [-4], [52], [74], [4], [-78], [32], [40], [6], [-76], [74], [-34], [-18], [-28], [46], [-46], [-88], [40], [84], [-4], [-10], [-32], [32], [-32], [-66], [0], [-88], [40], [-44], [-16], [-46], [4], [82], [-56], [86], [4], [98], [78], [8], [18], [74], [-4], [-16], [34], [-90], [-100], [44], [-82], [0], [-12], [-40], [-84], [0], [66], [-56], [2], [-68], [-2], [70], [-38], [42], [12], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6018_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6018_2_a_f();".
// 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_6018_2_a_f(:prec:=1)
    chi := MakeCharacter_6018_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_6018_2_a_f();".
// 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_6018_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;