
// Make newform 6027.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6027_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_6027_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_6027_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_6027_a();"
function MakeCharacter_6027_a()
    N := 6027;
    order := 1;
    char_gens := [4019, 493, 2794];
    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_6027_a_Hecke(Kf)
    return MakeCharacter_6027_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], [-3], [0], [-6], [7], [6], [6], [5], [3], [0], [1], [-1], [6], [5], [-9], [8], [2], [9], [-10], [-4], [9], [2], [-16], [-13], [6], [7], [-11], [2], [-4], [-20], [6], [19], [13], [6], [8], [2], [-16], [23], [-15], [6], [6], [-18], [2], [2], [6], [-21], [-11], [7], [5], [-6], [24], [18], [10], [-16], [-14], [9], [-31], [-3], [3], [-20], [6], [-9], [-21], [9], [27], [-19], [11], [-18], [12], [30], [-33], [-25], [25], [22], [15], [-14], [-2], [-20], [32], [22], [-2], [-24], [14], [4], [-24], [0], [20], [-15], [1], [-6], [-24], [32], [0], [24], [21], [0], [42], [20], [-9], [9], [18], [-19], [-20], [-4], [-14], [-28], [36], [3], [-39], [-24], [19], [4], [35], [20], [18], [-40], [-10], [-18], [-38], [28], [34], [-22], [8], [20], [40], [26], [-12], [20], [28], [-34], [0], [-37], [-30], [47], [-32], [16], [39], [-14], [-13], [41], [18], [24], [-32], [6], [-27], [-30], [-21], [11], [11], [26], [41], [-13], [-33], [26], [-24], [-47], [-36], [-22], [47], [59], [-34], [-5], [-27], [-10], [-4], [-61], [-9], [21], [24], [48], [40], [-10], [13], [48], [48], [-17], [-16], [-46], [-1], [-48], [10], [-22], [31], [-51], [54], [18], [-19], [-40], [-8], [-46], [0], [42], [14], [-17], [-58], [30], [-30], [36], [36], [46], [13], [14], [-31], [-44], [18], [-7], [44], [-18], [52], [-2], [28], [-26], [2], [15], [-20], [41], [18], [18], [60], [38], [-27], [17], [-3], [0], [50], [49], [-47], [32], [13], [31], [-54], [44], [-30], [-44], [32], [-53], [13], [1], [-48], [-70], [26], [-68], [51], [29], [-20], [-53], [-5], [-10], [49], [-19], [13], [39], [-38], [38], [-1], [-40], [-22], [68], [59], [22], [-62], [-35], [-41], [-33], [-47], [-66], [-33], [8], [-45], [5], [-72], [20], [4], [-58], [64], [2], [-40], [20], [57], [24], [-9], [0], [-10], [46], [26], [62], [-44], [4], [-4], [-37], [48], [66], [10], [50], [46], [70], [-69], [17], [62], [-36], [7], [60], [70], [80], [-14], [33], [-30], [-50], [64], [-27], [15], [68], [-72], [-1], [84], [32], [-19], [-1], [26], [23], [27], [14], [64], [42], [61], [59], [-24], [6], [66], [-40], [0], [-37], [67], [30], [-75], [-74], [61], [-5], [-56], [56], [-12], [0], [-8], [16], [-57], [64], [38], [-3], [34], [58], [-49], [53], [64], [22], [-2], [-20], [8], [-36], [27], [1], [-28], [-54], [39], [48], [-16], [58], [10], [-55], [42], [76], [-8], [90], [49], [-22], [38], [65], [38], [30], [69], [-18], [86], [70], [42], [50], [22], [-53], [-85], [-54], [-57], [14], [-54], [26], [23], [75], [-52], [-46], [-63], [-48], [-22], [44], [-69], [60], [-12], [-55], [48], [-78], [30], [16], [-94], [10], [-84], [72], [77], [-74], [-41], [51], [-62], [90], [-52], [-76], [-10], [-27], [95], [56], [97], [-12], [81], [-94], [29]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6027_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_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_6027_2_a_b(:prec:=1)
    chi := MakeCharacter_6027_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_6027_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_6027_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![3, 1]>,<13,R![-7, 1]>],Snew);
    return Vf;
end function;