
// Make newform 90.6.a.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_90_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_90_6_a_f();".
// 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_90_6_a_f();".
// 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_90_a();"
function MakeCharacter_90_a()
    N := 90;
    order := 1;
    char_gens := [11, 37];
    v := [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_90_a_Hecke(Kf)
    return MakeCharacter_90_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 := 6;
    raw_aps := [[4], [0], [25], [-22], [768], [-46], [-378], [1100], [1986], [5610], [-3988], [-142], [-1542], [-5026], [-24738], [14166], [-28380], [5522], [-24742], [-42372], [-52126], [-39640], [59826], [-57690], [-144382], [141258], [139814], [-86418], [218450], [28806], [-216502], [244608], [239502], [30860], [100950], [12452], [-6022], [-500866], [-555258], [-417354], [52380], [546662], [452028], [485594], [-1010178], [-807640], [149552], [-443506], [-420018], [1058750], [1273446], [370680], [-561298], [-577152], [651462], [-917574], [735390], [-1121308], [-1660342], [-1452102], [309014], [1593006], [1247258], [665988], [-591286], [516342], [-3295768], [1910978], [-2420058], [2507270], [413166], [-1737720], [1160978], [343754], [573140], [2880546], [3085590], [885458], [3753438], [-1946530], [2991660], [3966602], [5171148], [-4534846], [-1082200], [1080786], [-2617830], [1590458], [-4254702], [3266054], [601542], [4579320], [7052258], [2623488], [-3612340], [-9156294], [-7261590], [-5810202], [-8170666], [-817378], [-3507502], [-9614898], [-2019414], [-1378590], [8542952], [-2314582], [-928338], [909486], [8511360], [6124982], [-4516462], [9639794], [9926502], [7633220], [18031412], [-9301902], [-13833226], [14899662], [19330566], [4061100], [-13515238], [14351954], [-1895298], [-2916414], [14427752], [15867918], [-301810], [-21267720], [15500858], [-12184966], [-29028220], [-16114494], [-2924308], [26032538], [-16326282], [25813250], [19059246], [-17341102], [25816902], [-8884890], [-24639568], [-11376762], [-13078306], [35718822], [16188050], [25549680], [-22295326], [-19687218], [6775820], [26380386], [29516138], [14856498], [-14534026], [17202822], [-34443382], [983748], [30885680], [32087370], [15251978], [-3481662], [25400982], [49735206], [30517298], [-31985352], [-2907858], [-34962054], [3004652], [32078858]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_90_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_90_6_a_f();".
// 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_90_6_a_f(:prec:=1)
    chi := MakeCharacter_90_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_90_6_a_f();".
// 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_90_6_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_90_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![22, 1]>,<11,R![-768, 1]>],Snew);
    return Vf;
end function;