
// Make newform 81.4.a.c in Magma, downloaded from the LMFDB on 28 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_81_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_81_4_a_c();".
// 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_81_4_a_c();".
// 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
        poly := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_81_a();"
function MakeCharacter_81_a()
    N := 81;
    order := 1;
    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;

function MakeCharacter_81_a_Hecke(Kf)
    return MakeCharacter_81_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]];
    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 := 4;
    raw_aps := [[0, 1], [0, 0], [0, -7], [-22, 0], [0, 34], [-49, 0], [0, 21], [-70, 0], [0, 14], [0, -161], [-112, 0], [281, 0], [0, -28], [50, 0], [0, -140], [0, 216], [0, 56], [-679, 0], [-274, 0], [0, -258], [-511, 0], [-526, 0], [0, -224], [0, -21], [1778, 0], [0, -140], [896, 0], [0, -216], [-205, 0], [0, 209], [146, 0], [0, 1526], [0, 1135], [-3136, 0], [0, -181], [-2092, 0], [1337, 0], [-1744, 0], [0, -1582], [0, 1267], [0, -1428], [-70, 0], [0, 1198], [-121, 0], [0, 1707], [-4228, 0], [-202, 0], [-238, 0], [0, -3206], [2975, 0], [0, -105], [0, 3128], [-2443, 0], [0, 3066], [0, -4501], [0, -1436], [0, -63], [-5614, 0], [6026, 0], [0, 2689], [140, 0], [0, -1435], [-2968, 0], [0, -3052], [-5593, 0], [0, 4333], [7754, 0], [2570, 0], [0, -3226], [-6790, 0], [0, -4760], [0, -4278], [-9436, 0], [1742, 0], [8408, 0], [0, 1442], [0, -3080], [8813, 0], [0, 4703], [6125, 0], [0, 2996], [3695, 0], [0, -1512], [-15757, 0], [-868, 0], [0, -3332], [0, -10716], [-2659, 0], [0, -224], [-8968, 0], [0, -6132], [0, 8806], [-3688, 0], [0, -6118], [18086, 0], [0, 12180], [0, 1400], [0, -9156], [7238, 0], [-15541, 0], [12956, 0], [0, 10947], [0, 10724], [0, -1331], [968, 0], [18011, 0], [0, -13622], [0, 1575], [0, -9232], [-21217, 0], [2786, 0], [-9178, 0], [0, 9023], [13748, 0], [8948, 0], [0, -3317], [3920, 0], [0, -1638], [0, 11936], [0, -1910], [-32431, 0], [20399, 0], [0, -224], [0, 14820], [-5782, 0], [0, 7533], [17879, 0], [0, -6090], [-32746, 0], [27482, 0], [24716, 0], [0, -18388], [7358, 0], [-30886, 0], [0, 13727], [-7609, 0], [0, 5943], [-40054, 0], [0, 5831], [0, 5643], [35840, 0], [0, 4633], [14828, 0], [0, -22506], [27146, 0], [0, -13958], [19670, 0], [0, -77], [14756, 0], [0, -16290], [-31963, 0], [0, 18228], [34112, 0], [0, 21182], [36716, 0], [0, 31486], [-4318, 0], [0, 19285], [-13489, 0], [0, -11893], [0, 8974], [0, 1491], [-41494, 0], [0, 1638], [0, -27412], [0, 10276], [-21706, 0], [-25375, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_81_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_81_4_a_c();".
// 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_81_4_a_c(:prec:=2)
    chi := MakeCharacter_81_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_81_4_a_c();".
// 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_81_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_81_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-3, 0, 1]>],Snew);
    return Vf;
end function;