
// Make newform 8954.2.a.y in Magma, downloaded from the LMFDB on 07 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8954_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_8954_2_a_y();".
// 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_8954_2_a_y();".
// 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, 12, 3, -7, -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, -1, 1, 0, 0], [1, -4, -1, 1, 0], [3, 6, -4, -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_8954_a();"
function MakeCharacter_8954_a()
    N := 8954;
    order := 1;
    char_gens := [1333, 3147];
    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_8954_a_Hecke( : Kf := Kf)
    return MakeCharacter_8954_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, 0, 0, 0, 0], [0, 0, -1, 0, 0], [1, 1, 0, 0, 0], [-2, 0, 0, -1, 0], [0, 0, 0, 0, 0], [-1, 2, 1, 1, 1], [-1, -1, 0, 1, 0], [0, -2, 1, -1, 0], [3, 0, 1, 0, -1], [-3, 0, -2, 0, -1], [1, 0, 0, 2, -1], [1, 0, 0, 0, 0], [-1, 1, 2, 2, 2], [1, -1, 0, 0, -2], [-2, -1, 1, -3, 1], [-1, -1, -2, 0, -1], [1, -2, 1, -1, 2], [0, 0, 0, 1, 2], [4, 1, 1, 0, -3], [1, 2, -1, 1, -3], [-3, -5, -4, 1, 2], [-3, -1, -2, -5, -1], [0, -2, 1, -1, -2], [4, 2, 3, 0, 2], [0, -3, -1, -1, -1], [-4, 6, 3, 5, -1], [3, 4, 5, 1, -2], [2, -4, 3, -3, -3], [6, 1, 1, 0, -3], [-3, 1, 5, 1, 3], [1, 1, 0, 4, 5], [0, 4, 1, 0, 1], [-1, 0, 1, -6, 1], [-5, -4, -4, -5, -1], [-2, -2, 2, -4, 1], [-5, 1, 5, 0, 0], [4, -3, 2, -2, 3], [-9, -2, -4, 1, 1], [-8, -3, -5, 3, 3], [-3, -7, 0, 1, 1], [-7, -3, -1, -1, -6], [3, -7, 0, 4, -1], [5, -7, 1, -4, -4], [1, 7, 0, 3, -4], [7, -9, -4, 0, 2], [-5, -1, -1, 3, 7], [-7, -3, -3, -7, -2], [6, -2, 0, 7, 0], [-2, 4, -1, -1, 1], [-5, 2, -2, -1, 2], [-8, 3, 5, 3, 2], [-13, 6, 1, 4, 1], [5, -8, -4, -4, 1], [-5, -2, -7, -3, 1], [6, 2, 2, 1, 2], [9, 2, 0, -3, 1], [6, 0, 3, 1, 6], [-1, 10, 7, 1, -1], [-10, 1, 4, 0, 2], [1, -2, -7, -1, 1], [-16, 6, 1, -2, 0], [4, -2, -8, 1, -4], [8, -5, -4, 5, -2], [2, 6, 7, -2, -2], [5, -5, 3, -10, -1], [13, 1, 4, -1, -4], [-2, 4, 6, 3, -5], [2, 2, 3, 5, -4], [-13, 2, -3, -3, 0], [-9, 2, 4, -1, -1], [-6, 8, 6, 0, 4], [-4, -9, 1, -2, 1], [-6, -7, -7, -1, -3], [9, 8, 2, -5, 1], [-3, -3, -8, -3, 3], [7, -5, -3, -4, 6], [3, 1, 5, -2, 3], [0, -5, 3, 1, 3], [6, -7, 2, -1, -1], [-14, 9, 2, 2, 1], [-13, -7, 3, 0, 0], [3, 4, -2, 12, 5], [7, -10, -5, -2, -5], [12, -5, 2, 4, -2], [3, 4, -9, 0, -2], [-1, -5, -5, -3, -3], [17, -11, -4, -5, -4], [2, 6, -9, 2, 4], [-15, 4, -9, -5, 2], [-5, -9, 4, -9, -2], [-1, 1, -2, 5, 6], [-10, -9, -7, 1, 3], [14, 0, 1, -5, 2], [2, 13, 8, -3, -7], [2, 4, 0, 0, -1], [8, 4, 0, -5, -5], [-3, 1, -4, 0, 3], [0, 4, -3, 9, -1], [0, 4, 2, -9, -4], [0, -11, -6, -4, -5], [9, 2, 1, 0, -1], [8, 5, -2, 8, -1], [-1, 7, -2, -1, 6], [11, -12, -8, -5, -5], [-7, 5, 1, -10, -2], [2, 2, -6, 0, -2], [0, -8, -14, -6, 4], [8, -3, -2, 8, -5], [-3, -4, -6, -10, -8], [-17, 18, 4, 6, 2], [-3, -4, -10, 0, -2], [-2, -2, -4, -4, -9], [-14, 0, -1, 8, 2], [8, 6, -9, 4, 0], [-2, 6, 2, 13, 15], [-2, 1, -1, -1, 1], [-8, -10, -4, -1, -2], [12, 7, 3, 7, -7], [-5, -2, -7, -8, -11], [-8, 12, 5, 5, -1], [-4, 12, -5, -1, -1], [-18, -4, -8, 2, -4], [11, 6, 0, 12, 8], [-20, 10, 7, 5, 2], [-21, -7, -11, 1, 4], [8, -7, 7, -5, -1], [-2, -14, -16, 2, -1], [-25, -6, 8, 0, 0], [4, -13, -4, -3, 10], [-1, -2, -1, -8, -2], [-10, 18, 2, -8, -7], [-16, -7, 0, -4, 0], [-10, 18, -5, -4, 2], [0, -16, 8, 3, -3], [4, -2, -13, -4, -5], [-25, 16, 4, 13, 5], [-12, 5, 4, 7, 0], [-15, -8, -5, 6, -3], [-7, -1, -11, -8, -4], [-14, 0, -14, -1, 3], [-1, -13, -8, -9, 4], [3, 2, 6, -11, 4], [-11, 6, 6, -2, -7], [18, 9, -3, 21, -3], [-19, -6, -5, 4, -1], [-10, 4, 4, 8, -12], [8, -7, 2, 1, -3], [8, 12, 7, -1, -2], [4, 5, 1, 14, 2], [7, -1, -7, 11, 3], [5, -11, 7, -9, -1], [-1, 0, -1, -13, 3], [25, 6, 14, -3, -9], [7, -21, -6, 1, 10], [-31, 7, -2, -2, 1], [-6, -7, -17, -5, 6], [-21, -15, -2, -5, 10], [1, -6, -13, -5, 2], [3, -6, 3, 2, 14], [21, 4, -4, -1, -9], [4, 19, 1, -1, -10], [4, 16, 5, 12, 1], [4, -5, -3, -6, 2], [-10, 6, -5, 8, 7], [23, 9, 15, -7, -7], [-18, 2, -2, -3, 6], [11, -22, -13, -10, 8], [3, -11, -14, -10, 3], [15, 8, 16, 4, -10], [-3, -5, -2, -13, -8], [7, 2, 17, 6, -9], [-29, 9, 1, -8, 1], [-10, -1, -7, 0, -8], [-2, 15, -5, -5, -7], [-27, -2, 13, -3, 8], [-2, -20, -5, -4, 8], [-6, 0, -6, -9, -1], [-17, 8, -4, 0, 0], [6, 10, 12, 10, 4], [-15, 9, 1, 11, 4], [-7, 3, -9, -3, 0], [2, 3, 2, 5, -3], [-28, 5, -5, 10, 6], [-2, -16, -7, 6, 3], [-4, 4, -5, -3, 3], [-1, -7, 0, 1, 12], [-10, -4, 2, -8, -6], [-8, 3, 16, -1, -2], [-24, 0, -4, 3, 6], [16, -11, 9, 4, 3], [-5, 8, 10, 11, 8], [-23, 1, 11, 5, 1], [-20, -16, -12, 7, 0], [1, -4, 13, 2, -8], [27, -3, 7, 10, 2], [-5, 29, -5, 0, 1], [-11, 0, -4, 4, 1], [9, -1, -4, -2, 9], [-11, 2, -2, 9, 10], [23, -14, -11, -9, 0], [0, -6, 4, 8, -10], [12, 11, 3, 13, -9], [-12, 6, 2, 10, 5], [-6, -5, -12, -12, 4], [-1, -11, -5, -12, -6], [-7, 16, 8, 5, 2], [-1, -10, 4, 4, 15], [21, 4, 3, 10, 8], [12, -25, -11, -5, 3], [-15, 20, 15, 9, 5], [-11, -2, -24, 0, -1], [18, -4, 16, 8, -3], [-3, -14, -3, -10, -6], [8, 3, -21, -2, -4], [21, 3, 11, 1, -5], [8, 7, -3, 1, 5], [1, -2, 7, -12, -2], [16, 9, 14, 3, 14], [-5, 2, -11, 5, 7], [-21, -7, -17, -21, -3], [-5, -19, -7, 2, -4], [-18, -14, 1, -19, -3], [7, 1, 17, -6, -2], [-23, 5, 14, 9, -2], [-8, 7, -7, -6, -2], [-12, -12, -1, -6, -6], [7, -9, -3, -11, 4], [37, 4, 1, 6, 11], [0, 16, 8, 24, -1], [0, 9, -4, -4, -11], [4, -11, -10, -5, 11], [10, 21, 3, 1, -12], [22, 21, 11, 0, -11], [-38, 6, -1, -8, -1], [7, 1, -14, 8, -9], [31, 3, 23, 4, -7], [-47, 1, 2, -3, -4], [-13, -2, -23, 9, 3], [-15, -5, -4, 5, 15], [15, -7, -3, 5, -5], [6, -6, -18, -1, 0], [23, 8, 5, 3, 0], [15, -24, -6, 5, 7], [-2, 8, 6, -2, 9], [27, 12, 12, -2, -12], [-7, -4, 18, -4, 0], [-3, 2, -16, 8, -6], [21, 0, 20, 2, -2], [19, 8, 10, 4, -3], [-2, 0, 1, -12, 9], [10, 14, 14, 9, 2], [-15, 31, 3, -8, -1], [-2, -19, -7, 2, 12], [-8, -4, 0, 9, -6], [24, 2, -9, 0, -8], [29, 4, -4, -4, 5], [3, 3, 9, 0, 4], [-10, 10, 12, 17, 21], [41, 3, -7, 5, -9], [9, 0, 2, -1, -12], [9, -5, -8, 12, -6], [10, -11, 5, 9, 7], [-14, -15, -9, 9, 3], [7, -5, 9, 0, -4], [0, 13, 27, 12, -1], [-11, 21, -5, 1, 4], [-7, -2, -5, -17, 9], [-4, 8, 31, 6, 11], [1, 0, 8, -3, 10], [28, 1, 15, 17, 0], [-5, -6, 5, -1, -9], [-51, 10, 4, 0, 2], [-29, -3, -12, 14, 2], [0, 10, 5, -15, 2], [-17, -18, 4, -12, -10], [31, 15, 11, -3, -15], [-2, 4, 17, -16, 4], [-36, -17, 6, -8, 4], [18, -16, -5, -4, -13], [36, 7, -3, -6, 17], [-1, 4, 15, 3, 13], [-18, -13, 6, 8, -4], [9, 6, -27, -5, -3], [-17, 6, 0, -6, -6], [-16, 0, 6, 9, -2], [10, -5, -16, -8, 7], [7, 7, 25, -7, 3], [-5, 16, 11, -2, 7], [-25, -1, -6, 2, 10], [13, 6, 1, 7, -3], [23, -9, 7, -9, -8], [-15, 20, -2, 5, 6], [10, 10, 21, 3, 18], [-13, -5, -2, 11, -3], [-22, 11, -13, -8, -3], [41, 17, -2, 0, -4], [14, 10, 2, 25, -6], [-1, 11, -5, 29, 2], [9, -25, 0, -12, -2], [15, 2, -3, 0, 0], [4, 3, 1, -8, -4], [13, 11, 8, 2, 6], [7, -2, 18, 15, 11], [-32, -12, 13, -22, 6], [-3, -8, -21, 3, -6], [2, 10, -9, -13, -10], [-19, 22, -2, -16, -6], [-17, -11, -7, 3, 2], [-24, 5, -6, 2, -2], [-13, -3, 9, -4, 10], [53, 4, 16, 15, 0], [18, -4, -8, 6, -9], [-35, -1, -22, 8, 3], [-41, -8, 1, -13, 0], [-32, -3, 5, 12, 2], [-29, -13, -22, -16, 5], [20, -4, -1, 13, 7], [-7, 12, -26, -4, 7], [42, -12, 4, 16, 9], [25, 5, -26, -3, -8], [-1, 15, -8, 11, 0], [1, -7, -14, -7, -4], [-29, -9, -20, -13, -6], [28, -3, 17, 12, -3], [-7, 4, 4, -15, -12], [14, -10, -6, -12, 4], [6, -5, -15, -16, 6], [18, 9, 7, -2, -12], [5, 11, 17, 13, -8], [22, 0, 11, 23, 17], [-4, 23, -7, 12, 0], [21, -5, -8, 10, 10], [35, 8, 14, 8, 6], [-25, -8, -5, -7, 0], [6, 5, -15, 22, -9], [4, -5, -7, -12, 1], [12, 13, -11, 5, -10], [9, 10, -7, -3, -18], [9, 10, -2, 10, 14], [6, -4, 19, 8, 11], [29, -11, -7, -13, 0], [-6, -6, 11, -25, 10], [32, 14, 10, 10, -1], [-2, -19, -3, -21, -3], [8, 20, -2, -2, -10], [4, -3, 11, -20, -1], [22, 10, -2, -10, -21], [-30, -11, -5, 7, 11], [16, -5, -16, 13, -8], [13, 24, -25, 4, 2], [-17, -11, -3, 5, 8], [-3, -6, -13, -5, 2], [7, -13, -5, -14, -2], [9, 16, 0, 13, 1], [10, -9, -2, -1, 2], [24, -1, 1, -2, 6], [-15, 32, 3, -4, -5], [15, -17, -5, -10, -2], [34, -1, 14, 1, -9], [37, 8, -18, 6, -1], [-28, 9, 8, 10, 14], [-46, -2, 1, -9, -15], [8, 26, 9, 4, -3], [-1, 2, -3, 26, 0], [15, 10, 20, -2, -5], [0, 3, 0, 7, -11], [15, 10, 15, -16, -7], [-20, 2, -9, -4, -5], [35, 15, -4, 16, 5], [16, 22, -16, 1, 4], [-27, 6, 26, -11, 11], [31, 5, 1, 2, -2], [-35, -9, -19, -5, 2], [-26, 17, 10, 19, -6], [5, 28, -3, -4, -4], [3, -5, 18, 10, 17], [-27, 9, 18, 2, 0], [12, -14, 2, -14, -7], [-10, 1, 11, -17, -1], [-19, 14, 9, 8, 8], [-31, -12, -5, 10, 19], [59, 0, 10, 1, -5], [16, -26, -3, -7, 5], [-10, -8, -11, 15, -15], [-23, 4, -25, 8, 6], [-28, -2, -1, 7, 5], [40, 13, 14, -6, -1], [-29, -8, -18, -2, 7], [-2, 22, -3, -8, -4], [-3, -11, -9, -5, -13], [-9, 42, 13, -8, -3], [-10, -11, 7, -20, -9], [4, 0, 3, -16, -5], [4, 23, -14, -4, -10], [-17, -21, 9, -18, 1], [-11, -3, 32, 1, 7], [12, 0, 18, 7, -1], [-8, 6, 8, -23, -4], [5, 6, -20, -9, -1], [-3, 3, 13, 2, -10], [-28, 16, 16, 3, -4], [11, 21, 1, 13, 7], [-26, 3, 23, 5, 19], [-14, -15, -21, -4, 4], [-32, -21, -7, 9, 17], [-51, 2, 9, 13, -12], [-17, 0, -7, 21, 24], [1, -4, -2, 13, 25], [-58, 7, 12, 5, -1], [-22, 18, 2, 3, 0], [-36, 7, -4, 3, -1], [-2, -5, -30, -17, 15], [-41, -2, -1, -7, 7], [-37, 22, -14, 5, 5], [-19, 0, 15, -13, 7], [7, 23, -12, 9, 11], [-28, -3, -14, -3, -14], [22, 5, 6, 13, -9], [10, -39, -15, 13, 1], [-2, -7, -17, 15, -2], [22, -6, 20, -12, -10], [-8, 13, -10, 14, 3], [-27, -6, 20, 1, 4], [31, 12, 18, -3, 4], [53, -2, 28, 10, -5], [1, -27, 0, 14, 7], [-18, 12, 7, 18, 11], [1, 15, -18, 16, 4], [16, -27, 3, -28, -10], [25, 21, -7, 29, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8954_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8954_2_a_y();".
// 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_8954_2_a_y(:prec:=5)
    chi := MakeCharacter_8954_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_8954_2_a_y();".
// 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_8954_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8954_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 11, 3, -8, -1, 1]>,<5,R![-3, -6, 8, 7, -6, 1]>,<7,R![-37, -69, -7, 22, 9, 1]>,<17,R![9, -3, -44, -3, 7, 1]>],Snew);
    return Vf;
end function;