
// Make newform 2277.2.a.l in Magma, downloaded from the LMFDB on 01 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2277_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_2277_2_a_l();".
// 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_2277_2_a_l();".
// 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, 4, 0, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [3, 0, -6, -1, 1], [-1, -4, 4, 2, -1], [-3, -5, 5, 2, -1]];
    Rf_basisdens := [1, 1, 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_2277_a();"
function MakeCharacter_2277_a()
    N := 2277;
    order := 1;
    char_gens := [254, 1036, 1684];
    v := [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_2277_a_Hecke( : Kf := Kf)
    return MakeCharacter_2277_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 := 2;
    raw_aps := [[1, -1, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -1, 0, 1], [-1, 0, 0, 1, 1], [1, 0, 0, 0, 0], [-3, -1, -1, 0, 0], [2, -2, 1, 0, 1], [-1, -1, 1, -1, 1], [1, 0, 0, 0, 0], [2, 0, 0, 3, -1], [-1, 2, -1, 2, -1], [1, 1, -2, -1, 0], [1, 2, 3, -4, 1], [-1, -1, 2, -3, 0], [6, 1, -1, 1, 1], [-1, 4, 1, 1, 0], [3, 1, 5, -3, 1], [3, 3, 0, -1, 0], [0, -2, -2, 4, -2], [-3, 3, -4, 0, -1], [-6, 2, 3, 1, 0], [-2, -1, -2, -2, 3], [9, 3, 3, -3, -1], [-2, -1, -2, 4, -3], [3, -2, -2, -3, 1], [-2, -3, -3, 6, 0], [7, 6, 3, -3, -4], [4, 2, -5, 6, -1], [1, 4, 2, 1, 1], [5, -4, -5, -1, -2], [-4, 3, -2, 3, 4], [-1, -2, -5, 0, -5], [-3, 7, 6, -1, -4], [-2, -2, 4, -5, -1], [-1, 6, 9, -3, 0], [0, -4, 1, 3, 4], [-1, -1, -1, 5, -1], [-3, 7, 8, 0, -1], [-1, 5, -3, -3, 1], [1, 0, 6, 7, -1], [2, -1, 3, -5, 9], [8, 1, 2, -2, 1], [-9, -7, 0, -7, 4], [-2, -1, 2, -1, -6], [1, 3, -5, 4, -4], [0, 2, -1, 4, -7], [-5, -1, -8, 5, 0], [1, -4, 2, 1, -3], [20, -5, 0, 0, -3], [0, -2, -9, -4, 3], [-15, 8, -1, -4, 3], [0, -4, -1, -3, 0], [8, 2, 1, 4, -9], [2, -2, -3, -4, 1], [6, -5, -4, -5, 2], [1, 2, 0, -1, 3], [4, 3, 14, -5, 4], [-5, 6, 3, -1, 0], [-2, -1, 5, -1, 3], [2, -10, 6, -4, 2], [-2, 7, 1, 0, -4], [4, -5, 3, -2, -2], [0, 4, 3, -4, -3], [9, 1, -1, 0, -6], [-1, -2, 5, -5, 8], [-5, 6, 3, -3, 2], [0, -3, 2, -5, 8], [1, -5, -3, -5, -7], [4, -3, -6, 10, -9], [-8, 4, 9, -5, 2], [-3, -4, -9, 6, -1], [-14, -5, -6, 2, 1], [1, 2, -4, -1, -1], [5, 7, 5, 7, -7], [-6, -3, -5, 4, 0], [12, 7, 3, -6, -4], [-2, 6, 18, -10, 0], [5, 1, 12, -12, 7], [-6, 1, 3, 4, 6], [-4, 0, -1, -5, 0], [-1, 0, 3, -5, -8], [6, -8, 2, 8, -4], [-3, -1, 8, -1, 4], [-12, -1, 8, -2, 3], [-11, 1, -13, 8, -8], [16, 2, 5, 5, -6], [-12, -8, -9, 2, 7], [-1, 7, 7, -1, -13], [18, 1, -1, 5, -3], [12, -1, 4, -14, -5], [14, 2, -1, -4, -3], [13, -6, -3, 3, 2], [10, -2, 2, 3, -5], [3, -8, -2, -2, 12], [-7, -3, 2, -2, 5], [9, 6, 15, -9, 6], [8, -9, -9, 3, 3], [6, 1, 2, 8, -9], [-6, -2, -12, 4, -2], [-18, 3, 0, -1, -4], [-6, 8, 12, -5, 9], [-3, 13, 8, 3, -2], [8, -1, -5, 6, 4], [-8, -4, -15, 0, -5], [14, -1, 5, 6, 0], [7, -4, -4, -2, 14], [14, -2, -4, 3, 5], [-3, -2, -4, -1, -13], [-1, -12, -5, 7, 6], [-12, 0, -11, 7, -6], [19, 6, 5, 3, -2], [3, 3, 12, 7, 0], [4, -10, 0, -2, -2], [20, -3, -5, -6, -4], [-2, -15, -2, 4, 3], [7, -4, -7, -5, -4], [11, -12, -3, 3, -2], [-1, -12, 0, 4, 6], [-3, 0, -11, 0, -1], [9, 6, 10, -1, -11], [1, -4, -2, 3, 9], [7, 8, -2, -8, -4], [6, -3, -20, 14, -1], [7, -1, -2, 0, 5], [-6, -7, 2, 0, -3], [0, -11, -2, 0, 3], [3, -6, -6, 5, 3], [3, 8, 0, 7, -7], [2, -3, -6, -4, 9], [6, -17, -3, -2, 0], [26, -1, -5, 1, -3], [-2, 17, 13, -2, 0], [17, 13, 3, 11, -3], [-3, 8, 18, -11, 1], [10, 4, 2, 3, -11], [1, -10, -17, 9, 6], [0, -13, -18, 6, -7], [-1, 0, -6, -1, 1], [3, -10, 6, -7, 1], [-7, -3, -13, -7, 7], [-13, 15, -4, 8, 1], [5, 0, -5, 11, -14], [10, -4, -10, -7, 3], [7, 14, 11, 11, -2], [7, 11, 2, -5, 0], [16, 4, 1, 2, -3], [-7, -8, 5, 11, 0], [10, 6, 5, -3, 10], [-7, -9, -5, 10, -2], [25, -3, 3, -10, 0], [5, -14, -8, -1, 7], [15, -3, -13, 15, -9], [11, -11, -10, 1, -4], [9, -2, 5, -6, 15], [-12, -3, -6, 16, -5], [-7, 12, 18, 7, 5], [27, -4, -7, 5, 2], [-3, 8, 10, 0, -14], [-40, -7, 0, -10, 3], [-4, 9, 6, -16, 9], [-1, 4, 22, 0, 0], [6, 18, 21, -16, 5], [21, -17, -18, 4, -1], [13, 19, 15, 1, -7], [0, 11, 1, -2, -6], [11, -23, -10, 7, 4], [5, 5, -4, -9, 4], [3, 8, -3, -15, -2], [6, -4, 10, -4, -8], [4, -17, -4, -7, -4], [-10, 23, -4, 4, -7], [-12, 8, -14, 11, -1], [2, -1, -14, 6, -1], [-17, 0, -3, -25, 14], [6, -3, 5, -7, -1], [-7, -3, -15, 21, -13], [10, 9, 8, 5, -6], [15, 2, 12, 14, -4], [12, 7, 17, -16, 0], [-4, -13, 15, -8, 2], [-6, 10, -5, 15, -2], [0, -9, -23, 14, 8], [28, -19, -27, 5, 1], [22, -11, 16, -6, 5], [11, -3, -9, 9, -9], [-20, 7, 10, -4, -9], [-8, 4, -15, -6, 9], [-13, -11, 1, -13, 9], [-7, 10, 10, -8, -6], [-6, 0, -18, 6, 0], [31, 3, -1, 10, -8], [18, -3, -4, 8, -13], [-11, -1, -16, 11, -14], [28, 14, 3, -6, -3], [-8, 1, -12, -18, 3], [-19, -20, -8, -11, 7], [-20, 0, 16, 4, -2], [-10, 15, -3, 4, 8], [-3, 14, -2, -3, -1], [-5, 26, 16, -10, -4], [-22, -17, -8, -8, 11], [-9, -15, -8, -3, 10], [-11, -25, -6, 6, 13], [-4, -5, -1, 22, -10], [12, 4, 13, 16, -13], [-3, 20, 13, 2, 1], [22, 26, 7, 6, -15], [5, -2, -12, -2, 0], [3, -1, -6, 0, -19], [-21, 10, 13, 0, -5], [-17, -3, -14, -10, 1], [-3, -3, -6, 2, -1], [-2, -7, -2, -8, 7], [21, -9, 0, 13, -4], [7, -3, 5, 13, -3], [15, 4, 17, 3, -6], [-6, 8, -3, -3, -2], [-2, -3, -13, -11, 9], [18, -4, 8, -6, -6], [16, -9, -26, 13, -6], [33, 16, 2, 18, -6], [-1, -11, -1, 9, 1], [16, -12, 23, -3, 4], [25, 2, 8, 11, -17], [5, -21, -29, 8, 6], [10, -5, 4, 9, -18], [-12, 2, 7, 4, 5], [34, -16, 1, 9, -8], [29, 6, 4, 11, -11], [-28, -12, -18, -4, -6], [-6, 17, 9, 2, 4], [18, -10, -8, -8, -16], [15, -4, -9, -1, 10], [21, 9, 17, 9, 5], [-22, -6, -5, 6, 1], [44, -20, 3, 2, -5], [15, 16, -4, 19, -1], [-40, -7, -16, 14, -7], [0, -1, 6, -5, -6], [-10, 11, 4, 3, 16], [22, -12, -6, 14, -12], [5, -5, 24, -10, 7], [-7, -16, -2, -4, -8], [3, 14, 8, -8, -2], [31, -5, 9, -1, -3], [-13, 8, 7, -4, 21], [4, 14, 11, 15, 14], [25, -11, -6, 21, -12], [-7, -10, 14, -25, 9], [-5, -19, 24, -19, 0], [-28, 9, -7, 4, 10], [-13, -16, -23, 15, -2], [-25, -16, -5, 13, 14], [11, -13, -13, -17, -9], [7, -10, 4, -19, 5], [-10, -1, 3, -1, 7], [15, -13, -4, 5, -4], [-2, 11, -1, 4, 0], [-8, 0, 1, -19, 18], [0, 27, -6, 1, -12], [-17, 16, 6, -9, 15], [-7, -19, -8, -1, 24], [1, -17, -28, 21, 0], [-2, 15, 14, -18, 9], [-20, 1, 8, 20, -5], [-7, 10, 11, 11, -4], [-20, 1, -21, -2, -16], [-29, 11, 0, 5, 4], [4, -18, -3, -14, -7], [5, 10, 24, 14, 2], [-4, 7, 28, 9, 0], [-7, 3, -9, 5, -7], [13, -4, 10, 21, -13], [-8, -4, 10, 10, -8], [-30, 8, 24, -15, 1], [23, 17, 5, 20, -20], [2, -24, -34, 2, -12], [-8, -20, -9, 10, 3], [-10, 8, 5, 6, -5], [18, 6, -24, 10, -2], [29, -7, -8, 18, -15], [3, 3, 0, 5, -20], [-12, -3, -16, -2, -13], [-33, -19, 3, -5, 9], [-7, 12, 4, -7, -9], [-4, 2, 20, 0, -22], [17, 11, 3, -23, 3], [-60, -3, -6, -4, -3], [-22, -25, -17, 0, 2], [-36, 13, 15, 4, 8], [-3, 26, 15, -9, -4], [-26, 12, 7, 0, 5], [25, 7, -7, -7, 7], [15, -8, 26, -19, 13], [36, -17, 9, -11, -15], [-10, 17, 0, 10, 7], [-20, -9, 13, -16, 14], [-28, -34, -9, -1, 12], [-10, -15, -6, 19, -10], [8, 8, -22, 17, -7], [24, -18, -13, 14, 1], [-6, -6, -21, 0, -11], [-13, -14, -18, -3, 7], [-6, 19, 20, -1, -12], [-18, 16, -13, -6, 3], [37, -5, -16, 0, 5], [30, -13, 5, -12, -8], [19, 4, 22, -13, -1], [30, 0, 7, 6, -7], [-10, -10, -20, 16, -2], [-9, 33, 15, -2, 2], [-1, 14, -1, -11, 6], [-28, -4, 2, 18, 10], [-27, 11, -14, 4, -15], [-4, 21, 18, -12, 15], [17, -8, -2, -9, 1], [-38, -2, 4, 9, 3], [-22, 21, 21, -3, -3], [14, -25, -40, 4, 5], [-43, 18, 16, -12, 8], [1, 29, 13, -5, -15], [-23, 1, -13, 17, 5], [-12, 26, 11, -1, -6], [3, -7, 13, 13, -3], [21, 16, 22, -27, 1], [-8, 8, -18, 12, 2], [67, -10, -10, 7, -1], [14, -17, 0, 5, 10], [-24, -9, -9, 2, -14], [-7, -23, -21, -3, 3], [-46, -3, -14, 19, -12], [1, 7, -13, 6, -20], [-47, -1, -6, 12, 7], [27, 16, 8, 6, 10], [-30, 2, 5, 4, 19], [0, 18, 9, -13, 0], [9, -12, 18, -15, 21], [-14, -6, -3, -12, -7], [-45, 1, -18, 22, 7], [-6, -7, -11, 0, 0], [-21, 2, -4, 4, 18], [-14, 5, -2, 20, 15], [-7, 10, -17, -25, 8], [5, -8, -7, -13, -10], [-4, 8, -2, 4, 2], [-40, 1, -2, -10, 23], [-26, 0, -16, 15, -3], [3, -2, -9, 7, 4], [10, -34, -27, 4, 19], [-12, -17, -16, 6, 17], [-2, 16, -1, 1, -12], [-26, 0, 20, -3, -5], [-2, -12, -9, -12, -3], [1, 19, -6, 21, 10], [-17, -6, -8, -23, 11], [-6, 19, -12, 14, -1], [1, -5, 19, -23, 11], [12, 4, 12, -8, 6], [9, 12, 12, 10, 4], [15, 7, 28, -15, 12], [15, 27, 30, -23, 0], [-29, -9, 12, 2, -3], [14, -4, -23, 1, 16], [-22, 22, -3, -10, 7], [3, 14, -2, -8, 2], [30, -16, 10, 9, -11], [7, -25, -11, 4, 10], [-33, -12, 13, -5, 10], [-20, 19, -7, 12, -28], [10, 24, 3, 5, -2], [12, 5, 24, -14, 17], [4, -9, -21, 15, -21], [0, 32, 11, 2, -27], [-40, 25, -10, 6, 1], [-8, -8, 1, 9, 14], [37, 14, -7, -8, -5], [-3, -15, -9, -13, -5], [43, -7, -6, -13, 14], [44, 14, 17, -8, 9], [-30, 11, 1, -7, -15], [19, -1, -26, -15, 12], [20, 6, 31, -10, 3], [11, -17, 16, -7, 4], [3, 15, 34, -8, 5], [49, -8, 17, -5, 14], [8, -4, -6, -13, -11], [29, 31, 20, -16, -21], [-28, 5, 18, -36, 21], [0, 14, -5, 8, -15], [37, -23, 10, -7, 0], [-32, -4, -23, -2, -9], [-9, 4, -1, -11, 20], [17, 34, 17, 10, -19], [25, 15, 10, 3, -12], [-2, 4, -11, 4, 25], [-24, -3, -7, 6, 6], [35, -9, -3, -3, -1], [10, -26, -13, 20, -13], [3, -10, -4, -15, 3], [-2, 30, 1, 0, 5], [-15, -24, -3, -10, 11], [1, -11, 19, -4, 0], [-41, 23, 12, -1, -14], [-5, 16, -7, 13, -10], [19, 22, -3, 6, -23], [14, -32, -40, 17, 3], [5, 10, 8, 15, 5], [13, -1, -8, 0, -1], [-39, 8, 0, 19, 13], [1, -12, -7, -3, 2], [29, -28, -26, -2, -6], [-7, 29, 22, -2, -23], [33, -16, 13, -3, -6], [39, 6, 0, 9, 5], [2, 4, 12, 0, 10], [9, -22, 0, -10, 36], [19, -13, 10, -19, -18], [-11, 22, 45, -19, 4], [26, -16, 0, -18, -4], [13, -17, -2, 3, -4], [-48, -14, -4, -14, 8], [26, 5, -7, 24, -22], [10, 0, 4, -6, 10], [-12, 27, 51, -26, 0], [-60, -13, 1, 5, 7], [7, -8, -10, 30, -8], [52, 9, -2, -12, 1], [-7, -20, -20, -5, 21], [-11, -38, -29, -12, 1], [34, -11, -15, -15, 17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2277_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2277_2_a_l();".
// 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_2277_2_a_l(:prec:=5)
    chi := MakeCharacter_2277_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_2277_2_a_l();".
// 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_2277_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2277_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -13, 14, 0, -4, 1]>,<5,R![-16, -12, 43, -14, -3, 1]>,<17,R![-1492, -72, 257, -16, -9, 1]>],Snew);
    return Vf;
end function;