
// Make newform 6018.2.a.l in Magma, downloaded from the LMFDB on 19 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_l();".
// 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_l();".
// 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], [2], [-4], [4], [-2], [1], [-4], [4], [10], [4], [2], [-6], [12], [-8], [6], [-1], [10], [-4], [-12], [10], [12], [12], [10], [10], [-18], [8], [-12], [2], [10], [-16], [-4], [-6], [4], [6], [-16], [-10], [4], [12], [2], [20], [-6], [24], [-6], [26], [-20], [-20], [8], [-12], [-2], [18], [-16], [10], [12], [-30], [-24], [-30], [-16], [18], [-6], [20], [-18], [4], [12], [2], [18], [12], [34], [12], [30], [-30], [0], [4], [-18], [-36], [-32], [30], [-38], [2], [-6], [-12], [-10], [12], [-30], [-20], [-36], [-30], [10], [14], [16], [-12], [-36], [4], [-20], [-20], [-12], [-34], [-6], [36], [10], [-36], [30], [28], [42], [-4], [34], [-20], [34], [16], [42], [-4], [-2], [18], [-20], [0], [2], [-44], [-24], [-22], [-36], [-34], [-6], [42], [-36], [28], [22], [34], [4], [-8], [6], [-4], [-20], [44], [-10], [42], [34], [38], [44], [6], [10], [-20], [-46], [44], [36], [-18], [-20], [-30], [-22], [28], [-16], [-38], [-54], [-20], [28], [52], [-36], [16], [18], [26], [-22], [-52], [58], [8], [-12], [-30], [-52], [-20], [-38], [10], [34], [20], [-2], [-60], [58], [-32], [18], [-20], [34], [-32], [-18], [8], [-36], [2], [-6], [16], [-50], [-6], [-12], [-30], [-52], [34], [-20], [-20], [-42], [-28], [-30], [34], [-38], [-6], [-8], [-30], [-20], [-42], [-62], [20], [30], [16], [36], [-14], [60], [-22], [62], [28], [-20], [-44], [-62], [-8], [66], [28], [-2], [-50], [-36], [34], [-4], [36], [-10], [34], [60], [-64], [-52], [30], [4], [-16], [-22], [20], [8], [26], [2], [-60], [0], [-60], [-52], [24], [-34], [42], [60], [68], [20], [-4], [-24], [-2], [-22], [0], [42], [-74], [-12], [-22], [76], [2], [-6], [20], [12], [-38], [-22], [-14], [12], [-50], [-6], [60], [-34], [34], [52], [26], [-40], [2], [-8], [52], [-58], [58], [-52], [80], [-4], [28], [-42], [-44], [16], [10], [42], [0], [-14], [66], [-12], [26], [-28], [2], [66], [-8], [-26], [76], [36], [74], [-66], [-20], [-28], [60], [74], [36], [58], [-24], [14], [20], [-62], [66], [-68], [-16], [26], [4], [52], [50], [-62], [-60], [2], [-58], [-56], [-6], [50], [-20], [-44], [68], [34], [-78], [26], [-4], [28], [28], [20], [-42], [-6], [-38], [-8], [14], [42], [34], [-24], [38], [36], [2], [20], [-36], [-62], [20], [2], [86], [-28], [-26], [90], [48], [-36], [18], [24], [-14], [66], [48], [-68], [76], [-38], [-78], [-56], [2], [92], [-92], [28], [86], [-48], [-70], [20], [-12], [-30], [82], [58], [10], [10], [44], [42], [20], [-36], [16], [38], [92], [-96], [34], [34], [4], [-44], [-32], [-14], [-16], [42], [36], [-10], [-62], [18], [-56], [-54], [78], [60], [-10], [2], [36], [-28], [50], [14], [-12], [-68], [26], [18], [60], [28], [-94], [-24], [6], [2], [-84], [12], [-62], [38], [-52], [82], [-12], [-108]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![4, 1]>],Snew);
    return Vf;
end function;