
// Make newform 6018.2.a.a 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_a();".
// 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_a();".
// 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], [-4], [-2], [-2], [2], [1], [-4], [-8], [0], [-4], [-2], [-10], [4], [-8], [-2], [-1], [-10], [8], [6], [-16], [10], [-12], [-14], [-16], [-10], [-8], [-4], [-10], [0], [12], [6], [6], [-12], [-22], [8], [-2], [-4], [-2], [6], [-16], [16], [0], [-2], [-12], [22], [-14], [8], [2], [-10], [0], [24], [10], [-4], [-6], [4], [6], [16], [8], [14], [6], [18], [20], [-18], [0], [4], [-12], [-4], [-10], [-14], [-10], [-4], [8], [-6], [-4], [36], [18], [-22], [-24], [10], [-34], [-26], [-20], [-2], [10], [28], [-14], [2], [-34], [16], [-24], [22], [14], [-12], [-40], [-24], [34], [22], [-4], [-2], [16], [14], [-40], [-38], [2], [-2], [4], [18], [12], [28], [46], [-26], [-10], [-28], [-28], [42], [16], [8], [28], [40], [-14], [-12], [-28], [38], [-34], [-30], [44], [12], [52], [-14], [-24], [14], [-12], [6], [-18], [-10], [26], [24], [-14], [-12], [-14], [30], [-20], [20], [-42], [28], [-44], [28], [-4], [24], [-44], [48], [-20], [-36], [-32], [42], [-40], [-16], [14], [-58], [-52], [-50], [-16], [-36], [18], [-24], [-16], [16], [-56], [-18], [-28], [2], [-2], [38], [-32], [46], [44], [42], [52], [-34], [48], [8], [-2], [-50], [44], [-42], [38], [32], [24], [-64], [-20], [2], [20], [-38], [8], [36], [26], [-14], [-8], [24], [-16], [-14], [54], [10], [-44], [-42], [-8], [28], [68], [-52], [52], [2], [-46], [52], [10], [-36], [-44], [62], [12], [-26], [-58], [0], [58], [-66], [-48], [-26], [34], [0], [-8], [-48], [14], [18], [0], [-34], [12], [-72], [-16], [-50], [34], [-16], [32], [-20], [-4], [70], [-42], [6], [-28], [-30], [-76], [-72], [-34], [24], [0], [-2], [50], [20], [-40], [-62], [2], [-22], [64], [-60], [68], [20], [-14], [-32], [-74], [6], [-8], [66], [-50], [-20], [-74], [-8], [66], [16], [28], [42], [54], [4], [-40], [-64], [40], [62], [-2], [-24], [-4], [48], [56], [18], [58], [-20], [6], [-6], [-64], [-66], [-56], [-26], [-26], [-44], [58], [86], [56], [78], [-20], [-52], [4], [-26], [-56], [10], [-38], [-52], [-14], [-24], [-28], [-10], [-60], [-38], [-2], [-6], [88], [-32], [42], [-12], [-54], [14], [-22], [4], [60], [66], [46], [-86], [-72], [60], [-80], [-28], [74], [48], [-68], [4], [50], [-30], [90], [40], [-2], [-30], [34], [12], [76], [90], [40], [-54], [-22], [8], [30], [74], [80], [-36], [-6], [16], [70], [-78], [12], [54], [52], [-34], [78], [-4], [-84], [84], [86], [24], [54], [-48], [-24], [94], [-48], [-6], [-94], [-34], [-72], [-30], [18], [-12], [-64], [-48], [28], [86], [-72], [-72], [-40], [30], [96], [44], [0], [-4], [24], [26], [-68], [-18], [16], [66], [4], [98], [18], [-92], [-82], [-14], [40], [48], [-22], [2], [-28], [20], [-98], [-52], [40], [84], [-48], [-52], [-34], [-48], [62], [60], [30], [90], [-46], [106], [84], [-42]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;