
// Make newform 6018.2.a.c 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_c();".
// 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_c();".
// 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], [4], [-4], [4], [-1], [4], [8], [8], [-6], [4], [-10], [8], [-4], [0], [1], [12], [-4], [10], [2], [4], [-12], [10], [-10], [6], [6], [-16], [8], [-10], [12], [20], [22], [-4], [-22], [18], [-20], [4], [-10], [-22], [-12], [-6], [20], [10], [-8], [-16], [20], [-24], [-12], [8], [10], [-14], [-2], [-12], [10], [6], [-10], [-24], [-2], [10], [24], [28], [-4], [30], [14], [24], [-4], [2], [-4], [0], [-18], [30], [30], [22], [-20], [38], [-4], [8], [2], [-18], [28], [-12], [8], [14], [-16], [20], [14], [-10], [-28], [14], [28], [30], [0], [0], [-4], [32], [-18], [-18], [4], [32], [28], [-20], [-12], [-26], [24], [-2], [-36], [34], [18], [10], [8], [16], [-14], [28], [16], [30], [4], [-30], [8], [36], [14], [46], [-52], [-4], [-44], [-18], [-6], [40], [20], [-14], [-24], [-6], [-50], [-2], [-50], [30], [30], [-20], [26], [-30], [-28], [-2], [-2], [12], [-50], [32], [-34], [42], [28], [4], [-58], [-46], [36], [36], [36], [-30], [-14], [58], [-46], [-30], [-8], [-2], [-54], [0], [30], [0], [38], [-10], [-34], [-6], [24], [-28], [-30], [10], [24], [-6], [36], [30], [4], [22], [-28], [-48], [-40], [26], [-54], [-46], [-8], [60], [-34], [-36], [42], [-4], [-48], [-36], [-24], [58], [-30], [28], [18], [68], [40], [64], [-58], [50], [56], [-18], [-38], [-36], [10], [0], [38], [36], [8], [-40], [-46], [-2], [28], [-50], [20], [8], [-20], [30], [30], [-24], [-20], [56], [-46], [36], [-26], [-8], [-60], [-36], [-18], [30], [16], [42], [-34], [-22], [-12], [54], [-32], [-20], [4], [-54], [-14], [-42], [22], [12], [20], [-34], [-2], [14], [36], [-26], [4], [-72], [-38], [0], [-6], [-74], [-66], [-36], [-18], [70], [-42], [52], [-72], [-18], [-28], [-72], [76], [-68], [-14], [-60], [-14], [-38], [4], [-66], [-54], [64], [-46], [50], [12], [-28], [-32], [-40], [-42], [0], [-10], [18], [-2], [-56], [42], [84], [-74], [-70], [0], [20], [-36], [68], [18], [-46], [26], [20], [-12], [-34], [32], [-40], [24], [-60], [-6], [60], [-18], [-4], [-38], [-46], [84], [70], [62], [-30], [-68], [-18], [40], [36], [-2], [-50], [80], [-4], [-48], [-26], [32], [6], [-50], [36], [52], [16], [26], [-14], [2], [28], [14], [58], [18], [66], [-10], [12], [-4], [-60], [-24], [-78], [64], [-46], [-36], [22], [50], [-42], [36], [24], [-18], [70], [28], [42], [90], [60], [-12], [2], [-18], [-76], [-10], [32], [-16], [20], [-84], [26], [-50], [60], [-20], [2], [-62], [-90], [4], [74], [-52], [82], [-20], [72], [-28], [-34], [100], [-32], [-14], [66], [-12], [92], [-40], [-2], [44], [-26], [28], [8], [-26], [82], [-32], [-50], [-84], [50], [-72], [-50], [-28], [-8], [-2], [60], [-20], [44], [38], [0], [84], [-22], [30], [-54], [66], [106], [22], [-16], [34], [68], [-76], [-30], [-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_c();".
// 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_c(: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_c();".
// 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_c( : 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![-4, 1]>],Snew);
    return Vf;
end function;