
// Make newform 1155.2.a.w in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1155_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_1155_2_a_w();".
// 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_1155_2_a_w();".
// 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 := [-4, 8, 3, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [6, 3, -8, -1, 1], [8, -5, -9, 0, 1], [9, -9, -10, 1, 1], [8, -4, -15, -1, 2]];
    Rf_basisdens := [1, 2, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1155_a();"
function MakeCharacter_1155_a()
    N := 1155;
    order := 1;
    char_gens := [386, 232, 661, 211];
    v := [1, 1, 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_1155_a_Hecke(Kf)
    return MakeCharacter_1155_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 := [[0, -1, 0, 0, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [2, 1, 0, 0, 1], [0, 0, 1, -1, 0], [1, 0, -1, 0, 0], [-1, -2, -1, 0, 0], [2, 1, 1, 1, 1], [2, 1, 0, 0, -1], [1, 1, 0, -1, 1], [1, 3, 0, 1, 1], [-1, -1, -3, 0, -1], [-1, -1, -2, 1, -1], [-2, 0, -1, -1, 0], [0, 3, -1, 1, 1], [3, 0, 1, 0, -2], [3, -2, 0, 1, 0], [-1, 3, 0, -1, -1], [1, 4, -2, 1, 0], [8, -1, 0, 0, -1], [-5, -2, 1, 0, -2], [-3, -3, -3, 0, 1], [1, 1, 5, 0, 1], [2, 4, 2, 0, -2], [1, 1, -1, 0, 1], [2, 3, 2, 0, -1], [1, -3, 0, -1, -3], [0, 2, 1, -1, 2], [3, 3, -1, 2, 3], [1, 3, 2, -1, -1], [-4, 2, 2, 0, 2], [-3, -4, 2, -1, 0], [5, 4, 0, -1, 0], [5, -2, -2, -1, -2], [-7, -3, -1, -2, -3], [6, 1, 4, 0, -1], [-1, 0, 2, -3, 0], [6, 2, -2, 2, 2], [3, -1, 0, 3, -1], [0, 4, -4, 2, 0], [2, 8, 0, 2, 0], [-2, 2, 2, 0, 0], [3, -4, 4, -1, -2], [3, 2, 2, -1, 0], [1, -4, -6, -1, 0], [5, -3, 1, 2, 1], [-5, -2, 3, -2, 2], [5, -7, 4, -1, 1], [-2, -3, -4, 2, -1], [9, -3, 3, 0, 1], [-3, 6, 0, 1, 4], [-19, -2, -2, 1, 0], [-5, -3, -2, 3, 1], [-1, 0, 0, -1, 0], [-9, -3, 3, 0, -3], [-1, -6, -5, 2, -2], [-2, -4, 0, 2, 2], [5, 6, -2, 1, 2], [1, 3, -6, -1, -1], [2, -6, 3, -1, 2], [4, -2, 6, 2, 2], [-17, 0, 2, -1, -2], [-5, -3, 1, -2, -3], [-4, 12, 2, 0, 0], [6, -2, 5, -1, 2], [-7, -7, 1, 2, -1], [-7, 0, -8, 1, -4], [13, -6, -1, 0, 0], [-12, -2, 4, 2, -2], [1, -7, -3, -2, 1], [-5, 5, 5, 0, -3], [-11, -3, -1, 0, 3], [0, 0, -1, -1, 0], [3, 1, 4, -1, 5], [1, 1, -4, -1, -3], [0, -8, 2, -4, -4], [7, 3, 2, -1, -1], [-1, -2, 4, -1, -4], [-6, -1, 3, 3, 1], [14, -8, -5, -1, 0], [14, 6, 0, 2, 2], [2, -6, 4, -4, -2], [9, -2, 3, 4, 2], [-7, 0, 0, -1, -2], [-1, -7, -6, -1, -3], [-9, -3, 1, 0, -5], [16, 6, 2, 2, -4], [-9, 0, 0, 1, 2], [0, 2, 0, 0, -2], [-13, 3, -2, 1, -1], [-5, -4, -4, -3, 2], [-3, -7, 1, 2, 1], [8, 2, -7, 1, 2], [-21, -4, -3, 0, -4], [15, -7, 3, 0, 1], [-15, 1, -3, 0, 1], [-20, 2, -4, 0, -2], [15, 1, 0, 1, 5], [19, 3, 9, -4, -1], [9, 0, -2, -1, 2], [-7, -6, -6, -1, 6], [8, -9, -1, -3, -5], [12, -3, 4, 0, 1], [-4, 0, -8, 2, 4], [5, 11, 4, 1, 3], [0, 4, -2, -4, 0], [-14, -2, -2, 0, 6], [-5, -2, -5, -2, 4], [14, -2, 2, 4, 2], [10, 12, -2, 4, 2], [-6, 0, 2, -2, 0], [-25, -2, 0, -3, -4], [18, 4, -1, 5, -4], [14, 0, 0, -4, 0], [-25, 7, -3, 0, -1], [-14, -6, -4, 2, -2], [6, 4, -7, 1, 4], [-5, -1, -11, 0, -5], [-9, -7, 0, 1, -3], [-13, 1, -3, 0, -1], [16, 4, -7, -1, 0], [-3, 2, 2, 1, -4], [-5, -6, 8, -3, 4], [10, -17, -1, -1, -1], [8, -6, 1, -1, 2], [-12, -9, 3, -3, 5], [1, -5, 3, -4, -5], [3, 12, 0, 1, 0], [1, 0, -2, -5, -4], [-15, 4, -2, -1, -4], [-10, -10, 1, -1, 2], [14, -2, 0, 0, 6], [6, -8, -6, 0, 2], [9, -14, -1, -4, -4], [0, -2, 4, 2, -2], [2, 4, 8, -2, 4], [16, -2, -4, 2, -2], [11, 12, 4, 1, -4], [7, 6, 0, -1, 2], [20, 1, 9, -1, 5], [-4, 15, 8, -2, -3], [9, 0, 4, 5, 4], [8, 10, -8, 2, 2], [-12, 5, 5, -5, -1], [8, -7, 6, 0, 5], [-30, -8, 4, -4, 4], [-12, -5, 8, 2, -3], [-7, 6, 9, -4, 8], [5, 11, 3, 0, 5], [19, 3, 9, -2, 3], [1, 4, 2, -3, 2], [-3, 2, -9, 0, -2], [0, 1, 0, -2, 7], [-6, 10, 8, -2, 6], [-21, 2, 6, 1, -2], [-12, -12, 2, 4, 0], [-6, -13, 6, -2, -1], [7, 3, -2, -3, 5], [-3, 8, -1, -2, -2], [-25, -10, 2, -3, 0], [-7, -5, -7, -2, -1], [-32, -1, -11, 3, 1], [2, -4, -9, -1, -4], [-2, 2, -2, 0, -2], [12, 12, -1, -1, -4], [-3, 6, 4, 3, -2], [27, 11, -2, 3, -5], [5, -13, -2, 1, -1], [-1, 1, 5, 0, 1], [-13, -11, -8, 5, -7], [15, -11, 11, -2, 1], [-5, -1, -3, -4, -7], [-13, 0, 4, 5, 2], [-2, 6, 4, 0, 2], [2, 13, 4, 2, 5], [6, -8, -14, 2, -8], [-14, 2, 2, 0, -2], [7, 2, -4, 3, 8], [-5, 14, 10, -5, 4], [-14, -10, 0, -2, -10], [-2, 10, 4, 4, 2], [-14, 6, 5, -7, 2], [-17, -2, -15, -2, -4], [-11, 7, -1, 2, 7], [-3, 7, -7, 2, 5], [1, 1, -1, 0, -3], [-11, 7, -4, 3, 7], [-13, -9, -8, 7, -1], [-20, 0, 0, 2, 4], [-17, 2, -6, 1, 6], [-10, -6, -6, 0, -2], [-9, -3, 1, 2, -7], [8, 13, 0, -4, 1], [-6, 4, 2, -2, 4], [0, 0, 10, 0, -8], [-11, -3, 12, -1, -3], [4, -2, -13, 5, -10], [-3, 0, 6, 3, 4], [-8, -5, -1, -7, -1], [25, 12, -1, 0, 8], [27, 5, 11, -4, -3], [-11, 4, 4, 1, -2], [-34, -3, 1, 1, 3], [10, 8, 2, 2, 8], [-36, 9, 0, -2, -5], [-3, 2, -6, 3, 2], [8, 1, 5, 3, -3], [18, 15, -4, -4, 1], [-17, -3, 15, 2, -1], [13, 11, 3, -2, 3], [-17, 0, -2, 7, 2], [11, -5, 0, -1, -1], [-46, -8, 4, -2, 0], [-8, -6, -8, -2, 2], [9, -7, 11, -4, 1], [6, 6, -6, 0, 0], [-3, 12, 0, 3, 6], [22, -16, 3, -5, -8], [-44, -8, -2, 4, 0], [-5, -10, -6, 5, 6], [-1, -9, -10, 3, -5], [-4, 7, 4, 2, 5], [-9, 8, 1, -4, 0], [15, 17, 2, -1, 9], [-15, -9, 4, -1, -13], [-9, -11, -8, -1, -7], [2, -6, -8, 2, -6], [-17, 3, -5, -6, -1], [9, 5, 0, -5, 1], [9, 14, 6, -5, 6], [14, -11, -4, -2, 1], [14, -21, -1, 3, 7], [-13, -3, -7, 4, 1], [-15, -6, -5, -4, 2], [-7, -15, -2, -7, -7], [-7, -4, 4, 1, -6], [-28, -6, 2, 6, -2], [-11, 4, -2, 9, 2], [-23, -2, 0, -5, 0], [-15, -24, 9, -2, -4], [-18, 18, 10, 0, 10], [18, -8, 3, -5, -8], [30, -8, 3, 3, 4], [1, 1, -2, 3, 1], [-2, 0, -8, -2, 8], [6, 21, 5, 1, -1], [2, -7, 4, 2, 1], [5, -12, 4, 1, 4], [-22, -12, 2, -8, 2], [25, 1, 0, 1, 7], [35, 6, 6, -1, 0], [13, 7, -12, -1, 3], [8, -16, -11, -5, -4], [15, -5, 1, -4, 3], [-20, 14, -7, 3, -2], [-15, -11, -8, 9, -3], [19, -3, 0, 1, 9], [21, 0, 4, 3, -4], [23, 1, 7, 2, -3], [23, 22, -4, 3, 2], [15, -2, -3, -8, -4], [10, -17, -2, -2, -1], [-37, 1, -6, -1, -11], [-14, -9, 10, -2, -3], [10, -24, -6, -2, 0], [5, 13, 7, -2, 5], [-27, 11, 7, 0, -5], [24, -1, -4, 4, 5], [-10, 0, 4, -4, 8], [5, -6, 4, -1, -4], [-27, -3, -17, 6, 5], [8, -1, -6, -6, -1], [8, -5, 8, -2, -1], [16, -2, -6, 2, 14], [11, -6, -3, 2, 8], [12, -4, -2, 0, 8], [-21, 5, -16, 5, 5], [-32, -8, 0, -8, -4], [17, 8, -5, 4, -8], [38, -10, -2, -4, -2], [41, 6, 4, 1, -2], [-15, 0, 13, -6, 6], [21, -9, 2, -1, -1], [29, -13, 10, -5, -5], [9, 7, -14, -3, -1], [-3, 18, 8, 1, 0], [-17, -14, 0, 5, -12], [5, -13, -12, 5, -3], [12, -8, -10, 4, 0], [-9, -23, 8, -5, -3], [-20, -21, 4, 0, -3], [-8, -16, 0, -8, -4], [12, 16, 2, 2, 6], [18, 4, 12, 6, 2], [1, 4, -2, 1, 2], [7, 9, 10, -9, 5], [-16, 0, -16, 0, -12], [8, 0, 12, 0, 0], [3, -4, 0, 1, 4], [40, 5, 12, -8, -1], [18, 13, -8, 6, -7], [37, 2, 15, -2, 0], [-6, 6, -13, 5, -2], [-33, 5, 2, -1, 1], [13, -16, 5, -4, -2], [47, 9, -2, -1, -3], [3, 9, -8, -1, 1], [-5, -1, 3, 2, -11], [-9, 18, 6, -3, 6], [-7, -23, 6, -7, -3], [0, -7, -3, 3, -3], [22, 5, 10, -2, 3], [-18, 3, 10, -4, -5], [25, 12, 1, -2, -2], [19, -4, -10, -1, -2], [27, -21, -3, 0, 3], [-2, -18, -4, 0, -6], [36, 19, -5, 5, -1], [19, -6, 0, -1, -6], [-23, -1, -6, 5, -1], [6, -12, 10, -4, 6], [14, -17, -16, 4, -15], [11, 20, 8, -9, -2], [24, -6, 10, 0, -6], [18, 28, 2, -4, 0], [-57, -3, -1, 2, -3], [22, 15, 2, 8, 11], [4, -25, -8, 0, 1], [-43, -13, 0, 3, 3], [-46, 6, 10, -2, 2], [9, 2, 1, -6, -2], [-5, -18, 4, 3, 2], [1, -8, -2, 7, 8], [26, -20, -12, 2, -4], [2, -12, -3, -3, -12], [6, 28, -2, 2, 0], [-7, 23, 6, -3, 3], [17, -1, 4, -7, -1], [17, 5, 10, 5, 9], [-6, 10, -1, -5, 6], [23, 19, -8, -5, 1], [-4, 8, 14, -2, -4], [30, -16, -7, 1, -8], [-17, -5, 1, 0, 7], [-31, -7, 2, 1, 1], [16, 9, 10, 4, -5], [10, 16, -8, 6, 12], [1, -5, 8, -1, 7], [-7, 2, 15, 0, -6], [8, -10, -6, 0, -10], [59, 9, 0, 5, -3], [1, -7, 2, -9, 1], [-8, 12, -2, 4, -8], [8, -8, -9, 5, 0], [-45, -20, 4, -7, 6], [-9, 7, -10, 1, -5], [-3, 8, 22, -1, 10], [-5, 2, 11, 8, 4], [3, 11, -1, 2, -11], [-26, -8, -2, 0, -10], [21, 0, 0, 3, -14], [17, 11, 7, 4, 3], [-27, 5, 9, -2, -7], [-42, 8, -4, -4, -4], [2, 12, -5, 7, -4], [-21, -7, -11, 0, 1], [-37, -11, -12, 9, 5], [0, -26, -16, 4, 6], [-21, -12, 6, -5, 6], [17, -12, -4, -3, -4], [42, 3, 3, 3, 11], [33, -22, 1, -6, -10], [33, -11, 0, -5, 1], [-35, 8, -4, -1, -6], [7, 15, -1, 2, 11], [7, 10, 10, -5, -2], [1, 9, 6, -5, -5], [-50, 3, 8, 0, 3], [-11, 15, 21, 4, 3], [-6, 10, -13, 7, 6], [-10, 24, 4, 2, 8], [-10, -15, 2, 0, 3], [33, 6, -1, 8, 6], [-7, -33, -2, -5, -1], [12, -18, 9, 1, -2], [-25, -13, 3, 0, -13], [-8, 2, 8, 0, -2], [10, 6, -12, -2, -2], [4, -34, 3, -7, -10], [16, 10, -4, 2, 2], [25, 15, -4, 5, -1], [27, 3, -19, -4, 3], [0, 4, -10, 10, 8], [-33, -1, -7, -4, 9], [-15, -12, -2, 3, 0], [-22, -14, -12, 8, 2], [32, 18, 1, 5, 14], [14, -11, -1, 3, -3], [31, 12, -3, -4, -2], [8, -26, 13, -5, -2], [1, 21, 3, 8, 1], [-7, -1, -26, 1, -5], [11, 3, -10, -3, 9], [-13, -2, 5, 4, 6], [-15, -11, 6, 1, 9], [39, -6, 4, -7, -2], [37, 17, 1, 10, 9], [-16, -7, 7, 1, 7], [9, 18, 10, 9, 10], [-26, -5, -2, -12, -7], [9, 2, 4, -13, 0], [12, 3, 16, 0, -5], [-22, -14, -6, 0, -4], [-1, -7, -15, 2, 5], [3, -13, -15, -12, -5], [9, -3, -3, -2, -11], [-29, 13, 2, -1, 1], [5, 16, -6, -1, -4], [-29, 5, 2, -1, 1], [21, 0, 4, 3, 16], [-25, -14, -1, 0, -12], [13, 15, 16, -3, -1], [15, 1, -11, 6, 1], [23, 9, 6, 11, 5], [47, -2, -7, 6, -8], [12, 1, -1, 1, 5], [7, 2, -8, 1, 12], [17, -15, -18, -1, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1155_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1155_2_a_w();".
// 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_1155_2_a_w(:prec:=5)
    chi := MakeCharacter_1155_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(2999) - 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_1155_2_a_w();".
// 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_1155_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-6, 16, 7, -9, -1, 1]>,<13,R![-784, -40, 164, -10, -8, 1]>,<17,R![-504, 380, 70, -63, 0, 1]>],Snew);
    return Vf;
end function;