
// Make newform 27.9.b.a in Magma, downloaded from the LMFDB on 18 April 2024.
// 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_9_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_9_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)
    N := 27;
    order := 2;
    char_gens := [2];
    char_values := [[-1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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 := 9;
    raw_aps := [[0], [0], [0], [239], [0], [56447], [0], [100559], [0], [0], [-1809406], [-3468481], [0], [3492194], [0], [0], [0], [24133919], [-31874833], [0], [-55236481], [-56007121], [0], [0], [-94775521], [0], [168881039], [0], [203180834], [0], [-400561918], [0], [0], [-153241201], [0], [734407439], [36173474], [-96818353], [0], [0], [0], [-1093758433], [0], [146318399], [0], [-3130362673], [-3337587793], [-4727321086], [0], [5358773474], [0], [0], [-519543553], [0], [0], [0], [0], [-7483095601], [7428074594], [0], [-10369724446], [0], [-14125556446], [0], [7328506847], [0], [23429178959], [11864612447], [0], [13028709119], [0], [0], [-35481533233], [-35724839041], [37354760687], [0], [0], [-15761131486], [0], [42701110847], [0], [-47669963521], [0], [65168330114], [67549311554], [0], [0], [-27260779198], [0], [-18397201681], [0], [0], [11784067247], [0], [123330154274], [0], [0], [0], [-140080969681], [106700917439], [176439353999], [0], [0], [0], [170953033679], [-60697122721], [0], [0], [0], [-62260657726], [180170097839], [-236286690721], [0], [-275593854001], [-112983590833], [0], [188543652002], [0], [0], [0], [-297468341281], [409331240159], [0], [0], [155801183522], [0], [358629195839], [0], [121500363074], [-577329540958], [-435661607518], [0], [564936257519], [-176287050241], [0], [-507612834433], [0], [-22520421361], [0], [0], [88854548834], [0], [702936569807], [0], [530164547039], [0], [438996242687], [0], [-235827738001], [0], [902799884159], [0], [642290797199], [0], [-1136429694481], [0], [-544533789886], [0], [1198925101919], [0], [0], [0], [1513463580719], [0], [0], [0], [-805453099441], [581390427362]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_27_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_27_9_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_9_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, 9));
    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_9_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_9_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_27_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;