
// Make newform 2960.2.a.bb in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2960_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_2960_2_a_bb();".
// 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_2960_2_a_bb();".
// 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 := [8, 15, -2, -8, 0, 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, 1, 0, 0], [-1, -4, 0, 1, 0], [7, 4, -6, -1, 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_2960_a();"
function MakeCharacter_2960_a()
    N := 2960;
    order := 1;
    char_gens := [2591, 741, 1777, 2481];
    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_2960_a_Hecke(Kf)
    return MakeCharacter_2960_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, 0, 0, 0, 0], [1, -1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 1, 0, -1], [1, 0, 0, 1, 1], [-1, 0, -1, 1, -1], [-1, -2, 3, 1, 1], [2, 1, 0, 1, 1], [2, 0, -2, -2, 0], [2, 0, -2, -2, 0], [2, 2, -1, 1, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 1, 1], [4, -2, 2, 0, 0], [3, 2, -1, 0, -1], [-1, 2, 0, -5, -1], [2, 0, -1, -1, 2], [0, 2, 2, -2, 0], [6, 3, -2, -1, -3], [3, 0, -2, -3, 3], [-3, 2, -2, 3, 1], [0, 0, 1, -3, 0], [3, -3, -4, 2, 0], [2, 2, -4, 0, 2], [-2, 4, -4, -4, -2], [-4, 2, 3, 2, 2], [-2, -4, 4, 4, 0], [6, -4, 3, 5, 4], [2, 4, -2, 0, 0], [-1, -4, 3, 5, -1], [3, 3, -2, -6, -2], [0, -3, 4, 5, -1], [-8, 4, -4, 0, 2], [2, 8, -6, -2, 2], [6, 4, -1, 0, 2], [-2, 2, -6, 0, 0], [-3, 2, 2, -1, -3], [8, -2, 0, -2, -4], [6, 4, -6, -4, 0], [1, -4, 2, 3, 3], [4, 3, 8, -3, 1], [2, 8, -3, 0, 2], [4, 4, -1, 3, -4], [6, -4, 4, 0, -4], [-5, -2, 0, -3, -1], [-6, -5, 2, 3, -5], [3, -2, 6, -3, -3], [-7, 1, 2, 10, 2], [4, 6, -2, 2, 0], [9, 4, 4, 1, -7], [4, 6, -2, 0, -4], [-2, -3, 2, -3, -3], [0, 4, -4, 0, 2], [-10, 1, 8, -1, 1], [-7, 0, 3, 7, -3], [-5, -1, -2, -4, 2], [-1, 0, 1, -5, -1], [-9, 10, 4, 1, 1], [11, -2, 3, -3, 1], [-10, 8, 0, -6, 0], [4, 2, -2, 2, 8], [6, 2, -6, 0, 2], [3, 0, 1, -4, 1], [8, 0, -3, -3, 4], [4, 0, 0, -2, 4], [-16, -6, -2, 4, -4], [-6, -1, 0, 5, -1], [9, 4, -12, -1, -1], [2, -2, -8, -2, -6], [11, 4, 5, 3, -5], [-4, -8, 4, 2, 2], [11, -8, -4, -1, -1], [-20, 1, 8, 3, 1], [-1, 4, -8, -3, 1], [3, -6, 12, 3, 1], [-10, 0, -4, 2, -4], [-10, 4, -2, 4, 4], [-3, -12, 2, 1, -1], [-6, -4, 0, 0, 4], [12, 4, -10, -8, -6], [9, 4, 4, -3, -11], [14, 4, 2, -4, -4], [12, -2, 3, 7, 6], [-3, -4, 2, 3, 3], [-6, -5, 6, -3, 3], [1, -8, -1, -4, -5], [0, -2, -4, -4, 4], [-8, -4, 0, 6, 8], [-12, -6, 2, 0, 4], [8, 4, -4, 0, 6], [6, -8, 0, 8, 0], [10, -7, 6, 1, -3], [0, -12, 8, 2, 2], [-6, 2, -8, -6, -4], [4, -6, -1, -3, -2], [-6, 0, 4, -2, 0], [-11, -2, 2, -7, 1], [-12, 0, -3, 2, -4], [-4, 6, 6, -8, 0], [4, 4, 14, -4, 2], [-4, 4, -10, -4, -10], [-15, -10, 3, -5, 7], [-12, -14, 2, 0, -8], [18, 6, -6, -2, 2], [1, 2, -8, -1, -1], [-15, 12, -5, -1, 1], [-4, 16, 2, -4, -6], [-7, -6, 0, -7, -11], [-1, 2, 2, 3, 5], [13, 2, -3, -11, -1], [-2, 10, -14, -2, 2], [-5, -4, 8, 11, 13], [-11, -8, -4, 1, -1], [3, 18, -14, -3, 5], [2, 5, 2, -5, -3], [-20, -10, 3, 2, 2], [4, 6, 4, 2, -4], [-12, -8, 10, 0, 2], [-8, -12, 0, 6, -6], [19, 8, -6, -3, 3], [12, 0, -12, -10, -10], [-13, -14, 6, 3, -5], [-5, 10, -18, -7, -1], [14, -2, -4, 2, 2], [-22, 0, 0, -4, 2], [-10, 10, -4, 4, -2], [-2, 10, -2, 6, 2], [-1, 14, 0, -5, -11], [4, 10, -12, -4, 0], [-5, 2, 6, 7, 7], [19, -8, 12, 5, -1], [-11, 6, 11, -2, 1], [-3, 10, -12, -7, 7], [10, -12, -4, 0, 4], [-7, -10, -2, -7, 9], [6, -6, -2, 16, -2], [-13, -6, 4, 1, -5], [15, 0, -9, 6, -3], [-13, 6, -5, 1, 1], [10, 6, 10, -8, -6], [3, 0, 0, -3, 3], [5, -2, 2, -7, 1], [0, 16, -9, -11, -2], [6, 14, 6, -4, 2], [-4, -4, -8, -2, 6], [0, -8, 12, 8, 8], [4, -4, 0, 2, -8], [-18, 8, 4, 4, 2], [0, -14, 17, 3, -6], [20, 11, 4, -3, 11], [20, -6, -6, -12, -8], [-9, -2, 15, 7, 9], [4, -6, -2, 10, 12], [-13, 4, -9, 8, 5], [12, 8, 4, 2, -10], [16, -2, -11, -7, 2], [0, -2, 9, 9, 6], [-4, -14, -10, 6, 2], [1, 2, 4, -17, 1], [-21, -6, 7, 3, -3], [4, 6, -18, -6, 4], [-19, -8, 12, 5, -1], [-14, -4, -4, 4, 4], [-16, 6, -6, -8, -6], [-18, 16, 4, 0, -2], [5, -9, 6, 14, 12], [-2, 9, -2, -7, -3], [26, -4, 12, -4, -8], [-14, 0, -3, -8, -14], [18, 8, 4, -4, -6], [2, -2, 3, -21, 0], [-2, 10, -8, 6, -2], [18, -11, -6, -5, -3], [5, 2, 4, -7, -3], [-1, 14, -20, -15, -3], [8, -22, 2, 14, 0], [2, -8, 7, 7, -2], [13, -14, 1, 13, 3], [7, 22, -17, -8, -5], [32, -6, 2, 6, 2], [-18, -2, 6, 2, -2], [-18, -7, 20, 11, 1], [-3, -6, 9, -9, -5], [18, 16, -4, 0, -6], [-10, -22, 23, 7, 2], [-21, -8, 4, 11, -5], [-5, 8, -6, -9, 9], [18, -10, -4, 6, 2], [4, -4, 12, 8, -6], [19, 0, 6, -1, 11], [-33, 6, -7, 1, 5], [-8, -1, 10, 11, -5], [14, -14, -11, -3, -4], [-3, 14, -6, -7, 9], [1, -6, -1, -12, 5], [9, 0, 14, 7, 3], [4, 10, -8, -4, -16], [1, 14, -15, -9, -5], [9, 8, -2, -15, -9], [16, 8, -10, -2, -6], [12, -8, -10, -8, 6], [-15, -12, 4, 7, 17], [16, 0, 0, 16, -6], [-16, -14, 7, -6, -10], [16, 8, -2, 2, -12], [-5, -16, -3, 13, 7], [22, -4, -2, -10, -2], [-4, -6, -15, 5, 0], [-6, -10, 4, 8, 18], [-25, -12, 16, 9, -1], [9, 4, 1, 11, -3], [14, -2, -6, 0, 14], [12, 0, 2, -8, -2], [0, 1, 6, -3, -3], [22, -5, -10, -15, -9], [36, -10, 7, 10, 2], [-28, 4, 12, 12, -6], [24, -6, 12, -6, -8], [-28, -2, -8, 2, 8], [-12, -12, 16, 4, -6], [0, -4, 1, -6, -16], [5, 8, 6, -9, -3], [-4, 14, -21, -10, -2], [20, -8, 0, 0, 10], [-5, 5, 16, 2, 12], [-16, -10, 6, 2, 8], [16, -2, 4, 10, -8], [-25, 20, -6, -17, -9], [-21, -2, -11, 2, 3], [-6, -9, -12, 1, -1], [5, -8, 8, 11, -9], [5, -18, 14, 5, -1], [5, -14, -2, -3, -1], [5, 6, -14, -7, 1], [23, 6, -15, 4, 3], [9, 3, 10, 6, -16], [8, -14, -6, -10, 2], [3, -16, 7, -3, -13], [-16, 1, 4, 15, 7], [2, -8, -19, 11, -2], [-8, -2, -12, 20, 4], [8, -22, 5, -5, -2], [17, -15, 2, 2, -6], [22, -14, -8, 6, 2], [-14, -4, 0, -12, 0], [8, -6, 7, -5, 2], [-10, 6, 14, 0, -6], [26, -12, -3, -15, -6], [17, -12, 6, 19, 1], [-12, 0, -8, 10, 18], [13, 2, 21, -9, -5], [15, -4, 1, 11, -1], [10, -27, 8, 11, -1], [42, 2, 6, 2, 10], [10, -16, -8, 0, 0], [31, 16, -16, 5, -1], [27, -4, 4, 7, 11], [10, 7, -14, -21, -3], [3, 26, -16, -5, 7], [18, 0, -12, -12, -8], [10, 20, -12, 6, 4], [14, -10, -2, 6, -2], [2, -4, -8, -8, 12], [-26, -16, 2, 10, -12], [13, 10, 5, -9, -1], [-15, -10, 8, 9, 11], [-34, -6, 23, 4, 4], [-2, 8, 14, 8, -12], [27, 15, -4, -2, 2], [1, -14, 1, -1, -9], [2, 16, 4, -6, -4], [6, -12, -6, 4, 4], [11, -20, -1, 1, -5], [8, -14, -11, 5, 2], [12, -10, -2, 12, -4], [16, 16, -1, 1, -2], [-8, 11, -2, -17, -13], [-6, -2, -2, -22, 10], [13, -8, 0, -3, -3], [-20, -4, 17, 15, 12], [-19, -16, -1, -6, 15], [-6, 29, -18, -15, -3], [-10, -14, 15, -1, -2], [21, 14, -10, -11, -23], [-2, -14, 10, 16, -10], [25, -6, 14, -3, -13], [21, -16, -11, -1, -3], [2, -22, 6, -8, 2], [10, 0, 3, -23, -14], [-18, -18, 20, 2, 6], [14, 4, 2, 12, 4], [-28, -2, 2, 16, 0], [21, 6, 0, -1, 1], [21, -4, -12, -1, -3], [-5, 28, -10, -17, -3], [-5, 2, 18, 15, -17], [-23, -6, 20, -3, 11], [0, 14, -20, -18, 4], [-16, -32, 9, 3, 0], [33, -1, 0, -10, -8], [6, 4, 0, -12, -8], [-25, 6, -2, -15, -5], [17, 26, -18, -11, 3], [24, 12, -18, 12, -6], [18, -4, 5, 25, 2], [-11, 6, -13, -9, -1], [-32, -12, 8, 0, 6], [-28, -10, 10, -4, 12], [15, -2, 6, -7, 9], [5, 16, 5, 3, -23], [-17, 16, -3, 6, 1], [-6, 10, 15, -8, 0], [-6, -18, 3, 12, -12], [41, -10, -7, 6, 5], [-2, 18, -14, -10, 6], [-4, 6, -2, -10, -16], [-27, -12, 12, -17, 5], [46, 9, 6, -5, -11], [13, 16, -16, -7, 7], [-24, 2, 2, 0, 0], [-20, -10, 3, 13, 6], [-1, -12, 30, 9, -3], [44, -18, 8, 18, 8], [0, 4, -5, -7, 2], [-13, -14, 6, -7, 11], [14, -4, 28, 2, -12], [-4, -2, 8, -6, -10], [2, 4, 18, -4, 8], [38, 16, -22, -10, -8], [16, 20, -26, -14, -14], [15, -4, -4, -13, -1], [-18, -20, 8, -4, -6], [14, 9, 2, -23, 9], [-22, 22, -2, -30, -6], [-22, -3, 8, -3, 9], [-18, -24, 31, 5, 0], [-17, -8, 24, 11, -1], [-11, -24, 6, 5, 15], [16, 8, 2, 4, 18], [-12, 0, -9, -7, 14], [-33, -2, -2, 13, 3], [1, 10, 2, -1, -15], [-8, -22, 16, 6, -4], [-6, 19, -2, -7, -9], [10, -4, 8, 24, 12], [-30, -6, -3, 1, -20], [-3, -18, -2, 5, 5], [-15, -4, -9, 16, 11], [32, -4, -23, -19, -8], [5, -4, 20, -9, 7], [-40, -30, 20, 2, -6], [-39, 8, 6, 17, 3], [-28, -12, 8, -6, 6], [26, -10, 22, 12, -10], [-25, 18, -21, -5, 9], [20, -38, -4, -2, -4], [-16, -20, -1, -1, -12], [26, 11, -20, -9, -5], [-39, -6, 16, 9, 9], [-12, 6, -12, -6, -16], [15, -8, -21, -3, 7], [27, -12, 0, -17, -13], [-30, -10, 22, 4, 2], [-22, 5, 12, -21, -3], [-23, -6, -21, 20, -3], [24, 20, -8, 10, -8], [45, 4, 1, -17, 5], [2, -10, 14, 20, 14], [10, 4, -10, -26, -4], [-8, 17, 4, -5, 11], [16, 4, -21, 1, -4], [0, -3, -20, -9, 13], [35, 8, 5, -9, -5], [16, -6, 5, 1, -18], [7, 4, 24, 1, -3], [-10, -4, 0, 12, 22], [3, 12, 24, 1, -5], [-26, -6, -2, 4, 10], [2, 4, 18, -2, -4], [-15, -6, -10, 13, -7], [32, -16, -2, 20, -2], [-33, 8, -31, 5, 7], [-8, 22, -16, -8, 4], [-24, 6, 0, 18, 8], [-4, 6, -21, -7, 10], [23, -21, 2, -2, -2], [13, -2, -8, 11, 3], [-16, 0, -4, -2, -8], [50, 0, -22, -14, -4], [4, -22, 16, 6, -8], [3, 6, -6, 11, -9], [16, 4, 28, -6, 4], [-2, 33, -24, -21, -3], [2, -10, 6, -8, 10], [13, -16, -12, -5, 1], [-36, 0, 0, 28, 6], [-48, 0, -1, 19, 8], [-17, -4, -8, -21, 11], [-29, 18, -8, -9, -7], [40, 2, -2, -18, -22], [-16, -14, 21, 22, 6], [34, 12, -20, 4, -12], [10, -8, 16, 18, -4], [6, 8, -20, -16, -8], [14, 4, -16, -12, 0], [-13, 16, -28, 1, 11], [6, -4, -12, -16, -4], [-4, 18, -6, -6, 6], [-28, 18, 5, 1, 8], [-23, -10, 10, 9, 11], [5, 12, 8, -5, -17], [5, 22, -12, -7, -15], [-28, -6, 16, 26, 12], [-24, -7, -4, 17, 11], [22, 16, -12, 4, -6], [-13, -10, 35, 3, 1], [38, 0, -22, -10, -6], [37, -9, -2, -10, 8], [20, 2, -12, 2, 12], [10, 0, 0, -6, 0], [-24, 6, -14, -6, -12], [-6, 32, -8, 8, 8], [34, 10, -15, -11, 2], [-77, -6, -6, 3, -7], [-8, 10, 26, -16, 0], [-53, 2, 8, 13, -9], [8, -15, -18, -7, -7], [-19, 22, 2, 9, -3], [-22, -8, -8, 16, 6], [10, -5, -10, -1, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_bb();".
// 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_2960_2_a_bb(:prec:=5)
    chi := MakeCharacter_2960_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_2960_2_a_bb();".
// 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_2960_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-14, -8, 16, 2, -5, 1]>,<7,R![62, -160, 100, -12, -5, 1]>,<13,R![-8, -196, -172, -28, 6, 1]>],Snew);
    return Vf;
end function;