
// Make newform 6018.2.a.b 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_b();".
// 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_b();".
// 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], [2], [4], [-4], [-1], [4], [-2], [6], [-4], [-10], [6], [4], [0], [-8], [-1], [-10], [4], [12], [-2], [-2], [12], [10], [-6], [-14], [-6], [-12], [-10], [-6], [4], [20], [-18], [20], [2], [10], [16], [0], [12], [4], [12], [-8], [0], [-26], [-22], [6], [4], [-12], [4], [-20], [26], [-18], [14], [-24], [2], [-6], [-16], [-24], [-32], [-30], [4], [16], [4], [-4], [-6], [34], [28], [-18], [16], [16], [-34], [-34], [-8], [2], [-20], [10], [-24], [-22], [-2], [-10], [28], [-20], [-26], [2], [30], [-20], [-30], [-2], [-12], [-14], [20], [0], [-30], [-12], [-32], [-2], [30], [-30], [12], [34], [-32], [-12], [20], [34], [20], [-10], [-28], [42], [-6], [26], [2], [20], [-34], [-44], [12], [-18], [-12], [-34], [-18], [36], [-14], [10], [-34], [-24], [-36], [-22], [4], [-30], [36], [6], [-32], [24], [12], [-2], [30], [-10], [14], [12], [-42], [-26], [28], [-52], [16], [-28], [-10], [-22], [16], [54], [-32], [52], [48], [42], [-52], [30], [-28], [-28], [26], [-58], [-38], [24], [-36], [30], [-2], [8], [34], [-46], [-28], [28], [-34], [-24], [8], [36], [36], [2], [-16], [-54], [-16], [-12], [24], [38], [8], [28], [2], [-22], [30], [46], [-38], [40], [34], [2], [46], [64], [0], [24], [52], [42], [50], [-2], [-30], [-20], [10], [-54], [-18], [-30], [32], [-34], [-50], [60], [50], [-12], [-50], [24], [-10], [60], [-60], [22], [-40], [-2], [22], [24], [-24], [-40], [-2], [-38], [-36], [-8], [-34], [26], [22], [28], [44], [-36], [-46], [-18], [72], [-6], [-66], [24], [48], [30], [-36], [56], [-36], [2], [14], [-20], [56], [-20], [52], [78], [14], [34], [-8], [-22], [56], [12], [72], [76], [-40], [70], [32], [44], [-68], [-40], [-14], [-4], [56], [-18], [-44], [-60], [-14], [20], [-38], [8], [-42], [-50], [-76], [34], [82], [24], [54], [44], [14], [28], [28], [64], [-62], [26], [22], [-22], [-32], [-52], [-46], [-64], [-68], [-20], [-20], [12], [-72], [72], [-58], [-46], [72], [36], [-8], [14], [-72], [-6], [-24], [16], [0], [-78], [42], [28], [58], [-6], [44], [-20], [2], [-22], [-52], [10], [-20], [8], [-14], [70], [20], [56], [-6], [-20], [-10], [36], [-20], [-4], [68], [28], [62], [-38], [-10], [32], [-6], [-42], [32], [-50], [74], [-12], [-26], [20], [38], [-8], [60], [-58], [-48], [-48], [34], [-6], [36], [36], [-50], [-86], [-18], [-54], [54], [-36], [-28], [-62], [-16], [-84], [-78], [32], [28], [-54], [44], [-62], [48], [-64], [-10], [26], [-6], [6], [-38], [-78], [-22], [-38], [20], [-78], [-28], [70], [56], [88], [22], [-48], [-60], [24], [12], [-26], [-36], [30], [-76], [88], [68], [-66], [4], [82], [0], [12], [68], [6], [-24], [-12], [-14], [-8], [36], [-44], [-22], [106], [-6], [-84], [-18], [22], [-34], [-20], [16], [-28], [-74], [-32], [40], [-6], [28], [12]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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![-2, 1]>],Snew);
    return Vf;
end function;