
// Make newform 27.11.b.a in Magma, downloaded from the LMFDB on 30 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_27_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_27_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_27_11_b_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_27_11_b_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_27_b();"
function MakeCharacter_27_b()
    N := 27;
    order := 2;
    char_gens := [2];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_27_b_Hecke();"
function MakeCharacter_27_b_Hecke( : Kf := false)
    N := 27;
    order := 2;
    char_gens := [2];
    char_values := [[-1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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 := 11;
    raw_aps := [[0], [0], [0], [10907], [0], [-141961], [0], [4926251], [0], [0], [49326674], [-40895593], [0], [-282780982], [0], [0], [0], [-1354266001], [2698325411], [0], [1957684943], [-6059886949], [0], [0], [-15296411593], [0], [-18204368557], [0], [-17666350774], [0], [5265671186], [0], [0], [-62895680149], [0], [130666943699], [-178850287318], [-105959093461], [0], [0], [0], [339126572927], [0], [507232096943], [0], [405580685123], [60966462227], [-1010498535982], [0], [352910968826], [0], [0], [-1435380557473], [0], [0], [0], [0], [637452153299], [2921663134682], [0], [-3630460512982], [0], [-158857746118], [0], [581573030039], [0], [-7648110706501], [-1767220095289], [0], [1808583796151], [0], [0], [-6642793932589], [13833160690343], [8155864563923], [0], [0], [13794714220682], [0], [17621772820223], [0], [-2677485403801], [0], [-26991055164382], [-16663434550174], [0], [0], [28318715514386], [0], [-34014372075157], [0], [0], [-52923773296789], [0], [8038324413626], [0], [0], [0], [-73876473062557], [-83551859950201], [-95711494352293], [0], [0], [0], [-86731544799301], [86190254975207], [0], [0], [0], [-120560347522126], [81073139290907], [-172792838875057], [0], [171040147121051], [132320963167427], [0], [207428230817114], [0], [0], [0], [-251892336918001], [23564090590343], [0], [0], [-13604125758598], [0], [200395587836951], [0], [403308247742882], [-295451944755286], [435558582967898], [0], [268707010901099], [-369690952380793], [0], [87234906817823], [0], [-548998989200293], [0], [0], [-183163954607926], [0], [-485461686300061], [0], [-270118849141249], [0], [-894131124858361], [0], [577614412001651], [0], [662353879836407], [0], [1024668932414843], [0], [1224820205109107], [0], [833033518120226], [0], [953610622344407], [0], [0], [0], [-1031426297018293], [0], [0], [0], [-1533686014589101], [38245097892986]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_27_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_27_11_b_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_27_11_b_a(:prec:=1)
    chi := MakeCharacter_27_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 11));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_27_11_b_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_27_11_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_27_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,11,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;