
// Make newform 6018.2.a.h 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_h();".
// 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_h();".
// 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], [0], [0], [-1], [-4], [2], [-6], [0], [-2], [-10], [0], [8], [-8], [1], [-2], [-8], [0], [10], [10], [-12], [6], [-2], [-10], [10], [-4], [-18], [2], [4], [-8], [14], [-4], [6], [10], [4], [16], [-24], [-4], [4], [-4], [-8], [-18], [-18], [-22], [28], [-4], [0], [0], [18], [6], [22], [8], [10], [-6], [8], [8], [-28], [-6], [-4], [-16], [20], [24], [-10], [-2], [4], [-30], [-36], [-28], [-30], [6], [28], [-6], [-20], [-22], [0], [-6], [6], [14], [24], [32], [-38], [-14], [-30], [-12], [-6], [22], [-12], [18], [12], [-12], [-2], [-12], [16], [34], [42], [-6], [4], [-22], [0], [36], [12], [22], [-28], [22], [28], [2], [26], [30], [-26], [-16], [-42], [-44], [-12], [46], [4], [-2], [26], [-4], [10], [14], [2], [36], [28], [30], [0], [22], [-44], [22], [-12], [-12], [-40], [-42], [6], [22], [2], [20], [18], [30], [20], [12], [-12], [-12], [-10], [22], [-52], [-42], [20], [4], [-36], [-30], [-52], [26], [52], [-40], [50], [38], [58], [-8], [-52], [30], [54], [56], [-34], [-2], [0], [-48], [58], [24], [-48], [0], [56], [26], [-32], [26], [-16], [60], [16], [-18], [-32], [36], [-14], [-22], [30], [10], [-14], [20], [38], [46], [-30], [-12], [60], [48], [-28], [2], [-30], [22], [-6], [28], [-2], [-2], [46], [10], [-48], [-38], [54], [44], [-70], [-40], [-38], [48], [-6], [-60], [56], [-30], [16], [6], [2], [16], [4], [12], [66], [-74], [28], [44], [38], [22], [22], [-68], [-16], [4], [-54], [42], [36], [58], [-70], [8], [12], [-34], [-12], [-36], [28], [66], [-10], [-24], [-4], [64], [60], [46], [-10], [18], [-32], [42], [56], [-4], [-4], [-20], [0], [-42], [20], [-44], [-8], [-28], [42], [32], [32], [-22], [4], [4], [-54], [-52], [-22], [16], [-58], [-34], [52], [10], [2], [-80], [-66], [16], [-54], [24], [-60], [-32], [-50], [-26], [-66], [-14], [-32], [-36], [-54], [36], [16], [-44], [20], [-12], [36], [-44], [-58], [-18], [-28], [64], [-32], [-30], [-32], [-70], [-24], [4], [12], [-58], [-78], [-72], [18], [2], [20], [-24], [34], [-6], [-12], [70], [4], [16], [26], [-18], [4], [-40], [6], [4], [14], [-4], [-56], [4], [-28], [44], [6], [-54], [-70], [-88], [-62], [-54], [24], [62], [6], [0], [38], [-20], [-30], [-16], [-24], [22], [0], [-28], [-6], [86], [36], [-44], [-34], [-78], [-50], [42], [46], [-24], [-68], [18], [-24], [76], [38], [0], [44], [-90], [4], [-22], [20], [-12], [42], [10], [14], [38], [-10], [-74], [-74], [-38], [-52], [-26], [-28], [-10], [-16], [8], [-46], [-16], [84], [60], [4], [-6], [-52], [54], [44], [-16], [40], [-78], [12], [-54], [48], [32], [24], [-26], [76], [36], [-6], [48], [-12], [-28], [74], [14], [-90], [-32], [54], [-18], [-14], [-48], [-60], [36], [-2], [32], [-20], [-54], [76], [80]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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;