
// Make newform 6018.2.a.d in Magma, downloaded from the LMFDB on 18 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_d();".
// 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_d();".
// 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], [5], [-6], [2], [-1], [-1], [-9], [6], [2], [-10], [-3], [-10], [-6], [-3], [1], [2], [-4], [-12], [-7], [8], [-9], [3], [-7], [0], [-1], [-15], [20], [0], [-10], [12], [-18], [-22], [24], [-19], [5], [2], [12], [6], [9], [2], [12], [2], [18], [-7], [-16], [26], [18], [-13], [6], [0], [26], [12], [-12], [21], [3], [32], [-1], [0], [11], [6], [-7], [-24], [-34], [12], [-25], [-13], [-6], [17], [-3], [-3], [-28], [14], [2], [-9], [-18], [-16], [-24], [14], [36], [35], [-27], [-31], [20], [-15], [15], [8], [-21], [20], [-12], [18], [-13], [24], [-4], [9], [-6], [-39], [20], [20], [-22], [-30], [-21], [-9], [-13], [-34], [-15], [12], [-33], [-19], [-25], [11], [-27], [26], [32], [-3], [-28], [21], [6], [27], [-4], [-43], [18], [-18], [-43], [24], [26], [0], [38], [-46], [-28], [36], [8], [2], [42], [-4], [-36], [8], [18], [24], [11], [21], [2], [15], [-52], [-36], [-22], [-48], [-40], [30], [-13], [-48], [-43], [-51], [-52], [42], [56], [-18], [8], [3], [3], [-42], [23], [-18], [-18], [-57], [-34], [29], [-34], [54], [-12], [35], [24], [-4], [-4], [-21], [32], [-39], [20], [62], [-28], [-45], [32], [12], [-9], [12], [26], [-10], [41], [9], [38], [48], [50], [-27], [24], [0], [-58], [26], [42], [-12], [24], [8], [26], [20], [-6], [48], [-64], [-21], [30], [-10], [62], [21], [-19], [-18], [-30], [-19], [-40], [-48], [24], [21], [17], [-58], [-9], [-28], [9], [53], [66], [57], [-13], [-39], [-13], [11], [11], [39], [56], [9], [50], [15], [54], [15], [39], [-70], [-4], [-34], [-69], [30], [-28], [6], [-13], [-63], [50], [48], [-72], [-40], [-39], [-30], [-13], [68], [51], [29], [26], [6], [5], [71], [-21], [44], [-18], [51], [50], [-9], [14], [-28], [-46], [-22], [23], [-16], [-72], [-22], [62], [36], [-81], [50], [48], [-58], [32], [60], [-61], [60], [-49], [6], [18], [-3], [6], [-84], [-10], [-39], [8], [-21], [-42], [38], [-31], [-24], [50], [-12], [-58], [38], [-6], [62], [72], [41], [-36], [-48], [-30], [14], [-24], [-40], [-15], [-12], [26], [66], [50], [-22], [-33], [26], [-54], [-70], [20], [26], [12], [-39], [32], [6], [-64], [-36], [-52], [63], [44], [66], [-49], [-70], [-28], [63], [-45], [-64], [12], [30], [38], [-4], [-27], [-69], [-4], [2], [66], [-16], [-76], [-54], [18], [-60], [45], [72], [26], [-33], [45], [66], [47], [44], [-21], [56], [-31], [42], [-64], [-45], [75], [-31], [-49], [66], [-84], [2], [66], [2], [-18], [-57], [-97], [-12], [20], [-96], [-16], [92], [-52], [0], [-19], [-57], [-9], [8], [48], [11], [68], [-6], [-10], [33], [-55], [27], [-40], [-87], [81], [-4], [35], [78], [-58], [-27], [62], [54], [-63], [56], [-79], [-84], [-96], [56], [96], [24], [6], [17], [-42], [-72], [80], [21], [6], [-21], [8], [-60]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![-5, 1]>],Snew);
    return Vf;
end function;