
// Make newform 177.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_177_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_177_2_a_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_177_2_a_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
        poly := [-1, -1, 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_177_a();"
function MakeCharacter_177_a()
    N := 177;
    order := 1;
    char_gens := [119, 61];
    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_177_a_Hecke(Kf)
    return MakeCharacter_177_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], [1, 0], [-3, 0], [-4, 1], [-3, 4], [3, -6], [-1, -1], [-1, -3], [-3, 1], [-6, 1], [-4, 1], [-3, 5], [2, -5], [3, 6], [0, -3], [-1, 4], [1, 0], [4, 5], [-1, -2], [1, 2], [0, 5], [-3, 0], [-6, 3], [-2, 5], [-3, -4], [-10, 2], [-16, -2], [11, -13], [5, -3], [9, -2], [-9, 4], [10, -2], [-11, 4], [-9, -6], [-1, -13], [4, 3], [-5, 8], [-8, -1], [-16, 7], [16, -1], [3, -10], [6, -9], [-9, -6], [-20, 12], [-6, 10], [14, -17], [3, 1], [4, -12], [-11, 1], [-6, 21], [2, 2], [-15, 6], [0, -4], [7, -8], [-12, 16], [-2, -1], [21, -10], [-15, 10], [5, 0], [12, -18], [7, 3], [-9, 19], [-12, 0], [-3, 21], [-13, -9], [3, 0], [18, -2], [-21, -9], [16, -19], [23, 0], [9, -15], [-4, -16], [-15, 12], [-1, -9], [-13, 12], [25, 8], [14, -16], [-27, 12], [-21, 9], [6, 12], [2, 8], [3, -12], [4, -17], [15, -3], [3, 3], [-15, 9], [1, -26], [6, -3], [7, 7], [3, -13], [12, 8], [-13, 17], [-3, 9], [-22, 29], [5, 0], [-28, -5], [-6, -23], [2, 23], [-29, 20], [-7, 6], [11, -21], [-14, -8], [-31, 4], [-26, 23], [19, -24], [-9, 20], [27, -26], [22, 7], [8, -10], [-25, 13], [-39, 14], [36, -6], [9, -35], [-17, 26], [15, -12], [17, -22], [9, 1], [8, -4], [22, 7], [1, 5], [24, -9], [-10, 0], [-12, -10], [13, 10], [22, 14], [-26, 37], [-6, -6], [21, -9], [-14, 12], [11, -8], [23, -3], [3, 11], [10, -30], [-7, 25], [-23, 16], [-1, 15], [10, -14], [6, 6], [10, -26], [7, 19], [24, -30], [-32, 35], [11, -26], [-3, 0], [38, -15], [-15, 0], [39, 1], [9, 14], [-26, -4], [13, -13], [-10, 12], [-6, -14], [-40, 16], [-47, -2], [-27, 12], [36, -13], [42, -6], [-29, 20], [-11, 8], [-7, 2], [34, -25], [-16, -4], [9, 21], [-1, -29], [-21, 26], [-15, -24], [15, -3], [-16, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_177_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_177_2_a_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_177_2_a_a(:prec:=2)
    chi := MakeCharacter_177_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(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_177_2_a_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_177_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_177_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 1]>],Snew);
    return Vf;
end function;