
// Make newform 2960.2.a.ba in Magma, downloaded from the LMFDB on 25 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_ba();".
// 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_ba();".
// 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, 6, 6, -6, -2, 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], [0, 2, -4, -2, 1], [6, 4, -6, -2, 1], [12, -4, -10, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2, 2];
    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], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, -1], [-1, -1, 0, -2, 1], [1, -1, 1, -1, 0], [-1, -1, -1, -1, 0], [-3, 1, -2, 1, -1], [0, 0, -2, -2, 0], [0, 2, 0, 2, -2], [0, -3, 0, -2, 0], [-1, 0, 0, 0, 0], [-1, -3, 0, 0, 1], [-4, 2, 0, -2, 2], [-1, 4, 2, 0, 1], [-3, -1, 0, 0, 1], [-4, 3, 4, 2, 0], [2, 2, 0, -2, 0], [-1, 1, 4, 3, 1], [-3, -1, 4, 6, -1], [-3, 3, 0, 2, 1], [-8, 1, 2, -2, 2], [0, 2, -2, 1, 2], [0, -4, 2, 0, -4], [-4, 4, 2, 0, 6], [2, 0, -1, -1, -3], [0, 2, 2, 2, -4], [8, -5, -2, -2, 0], [-6, 4, -2, 2, 2], [-3, 7, 5, 3, 0], [-4, -6, -2, -3, 2], [-9, 1, -4, 3, -5], [-6, -4, -2, 0, 2], [2, -6, -2, -2, 0], [-10, 2, 1, -1, -1], [-4, 4, -2, 4, -2], [-3, -5, 4, 2, 1], [0, -4, 0, 2, -2], [4, 2, -4, 8, -2], [-5, 3, 2, -6, 1], [9, -3, -2, -1, -1], [-2, -2, 7, -3, -3], [-4, 3, -2, -2, -2], [10, -8, -8, -4, -4], [3, -3, -2, 4, -3], [-7, -1, 0, 3, 5], [-5, 3, -2, 2, 3], [0, 2, 6, -1, 6], [8, -4, 2, -4, -4], [-5, -3, -8, -4, 1], [0, -4, 6, 6, 0], [-1, 1, -4, 1, -1], [-2, 8, -2, 8, -2], [-3, -5, 0, 3, 1], [-1, 1, -7, 3, -6], [0, -4, 4, -1, 0], [-1, 1, 5, -7, 6], [-9, -1, 0, 4, 1], [11, -5, -1, -1, -4], [8, -2, -2, -4, 0], [0, 4, 6, 0, -4], [4, 0, -8, -6, 2], [11, -4, -6, -4, -3], [4, 1, -2, -2, -2], [-6, 6, -6, 4, -4], [-8, 8, -6, 6, -4], [11, 1, -4, 1, -5], [-13, -1, 4, -4, -1], [-6, 4, -6, 8, -2], [-9, 9, 5, -3, 2], [-4, -2, -4, 4, -6], [-11, -3, 4, -6, 3], [-11, -3, -6, 5, -7], [5, 9, 4, 8, -7], [11, -7, -10, -6, -5], [4, 0, 0, -4, 2], [6, 0, 6, 10, -2], [-3, 5, 0, 10, 1], [-10, 4, 4, -4, 0], [2, -4, -16, -2, 0], [-1, -5, 4, 2, 5], [6, -8, 2, -2, 2], [-2, -5, 4, 2, -2], [15, -1, -6, -10, 1], [-11, 9, 6, 5, 3], [13, -4, -6, -4, -1], [0, -4, -8, -8, 6], [14, -6, 6, -12, 0], [-4, -8, 2, -6, -4], [-10, 2, 0, -4, -4], [4, -2, -2, 2, 0], [-9, 5, 12, -3, -1], [-2, -4, -6, 6, 6], [4, 2, 6, -4, -2], [-8, 5, 6, -2, -6], [4, -4, 0, 0, 2], [19, -7, -2, 6, 1], [4, 0, 3, -15, 7], [4, -2, -4, -10, 6], [2, -4, -4, 6, -4], [6, -14, -2, -14, 2], [5, 1, -1, -9, 10], [8, -2, 12, -6, -2], [-2, 6, 10, -6, 6], [5, 5, 4, 8, 3], [7, -3, 5, -13, 6], [14, 10, -2, 2, 2], [3, 1, 0, 4, 7], [-5, -3, -8, 4, 1], [-9, -5, 7, -3, 4], [-10, -4, 8, 6, -4], [15, -5, -6, 0, -7], [5, 5, -6, -4, -1], [-5, 7, 6, -2, 7], [-5, -9, 6, 3, -3], [2, -8, 3, -13, 1], [0, 0, -4, -6, 6], [-2, 2, 18, 10, 6], [0, 6, -4, 8, -2], [17, -9, -4, 10, -5], [-4, 6, -4, 0, 6], [-3, -9, 10, -2, 3], [-5, -3, 0, 10, -5], [26, -4, -2, -12, -2], [2, 0, 10, 2, 4], [-8, -4, -6, 8, -12], [6, -10, 6, -14, 2], [19, -3, -2, -6, -1], [-20, 6, 6, -12, 8], [-3, 7, 2, 10, -1], [13, 7, 10, 12, -3], [-7, 18, -2, 16, -1], [5, -1, -10, -10, 5], [10, -4, 0, -4, 0], [19, 5, 2, -6, 9], [-16, 4, -8, 10, 2], [11, 1, 10, 0, 1], [19, -10, -2, -12, 1], [3, 3, 11, 7, 8], [16, 0, 0, 2, 6], [1, 3, 6, 16, -7], [-5, 9, 14, 14, 1], [-4, 3, 4, 14, -10], [14, 10, 10, 8, -2], [-12, -6, -8, 0, -2], [-16, 8, 4, -4, 4], [-10, -6, -2, -4, -4], [16, 12, -2, 8, -2], [4, -17, -6, -10, 2], [7, -13, -10, -1, -5], [-4, 0, 2, 6, 0], [-23, 17, 1, 7, 0], [16, -24, -10, -4, -8], [-7, -4, -8, 0, 5], [-18, -8, -2, 2, 6], [14, 5, 4, 6, -10], [-10, 5, 8, -6, -2], [-8, 14, 6, 10, -2], [-15, 11, 2, 8, 3], [1, 1, -19, 11, 4], [-8, 8, 2, 0, 16], [13, 1, -2, 16, -13], [-16, 10, -2, 10, 0], [14, -4, 0, -6, -10], [4, 12, -10, 0, -2], [18, 2, 4, -9, -2], [15, 13, 0, -3, 7], [-22, -8, 0, 4, 4], [-18, 6, 3, -7, -3], [12, 0, 2, -16, 10], [-14, 15, -2, 18, 0], [2, 6, -4, 20, -8], [-17, -1, 6, 7, 1], [-5, -7, 12, 12, -9], [-17, -1, -4, 0, -3], [14, -18, -16, -10, -4], [-16, -13, -20, -10, 0], [7, -1, 11, -3, 0], [-29, 0, 2, 12, 1], [-12, -6, -10, -2, 14], [-26, 8, -4, 2, 8], [-27, -1, -16, 3, -7], [1, -11, 13, -3, 0], [-28, 12, 10, -8, 2], [-4, 17, -2, 2, 8], [1, -1, -8, -28, 7], [-25, 11, -12, 14, -5], [2, -12, -10, -4, 2], [-2, 8, 6, -4, 2], [13, 5, 10, 0, -7], [23, -5, -7, 5, -10], [17, 1, 6, -3, 13], [-18, 1, 2, -18, 12], [-5, -15, -10, -10, 9], [15, -8, 0, 0, -1], [23, -5, -14, 10, 1], [-16, 8, 8, -8, 2], [9, 9, -11, -11, -8], [1, 1, -4, 2, -15], [26, 12, -4, 16, -4], [18, 12, -4, -6, 4], [-9, 13, -2, -4, -9], [6, -4, 10, 16, -2], [-14, 8, -5, -9, -3], [-40, 10, -4, 12, 0], [33, -5, 3, -3, 6], [14, -6, 6, -10, 8], [-16, 19, -8, 6, -2], [12, 8, 18, 16, -4], [9, 11, -6, 4, 1], [-17, 9, -5, 9, 0], [8, -4, 4, -2, -6], [-10, 18, 2, 10, 10], [-25, 15, 18, 11, 3], [-15, 1, -18, -7, -1], [-22, 8, -1, -9, 21], [-50, -2, 0, 0, 4], [-10, -10, 14, 8, -6], [20, 12, 0, 6, -6], [-2, 0, 6, -8, 10], [-4, 0, -17, 9, -5], [-25, -11, 4, -2, -3], [-6, 12, 11, -5, 13], [-18, 14, -4, -8, -4], [12, -10, 2, 5, -2], [-22, 6, -8, -2, -4], [-2, 6, 14, -16, 6], [-7, -15, -10, -8, 5], [-9, 6, -2, -4, -3], [-41, 5, -4, -3, -1], [-5, -9, -12, 12, -17], [17, 11, -8, -4, 3], [-11, 7, -8, -20, 11], [15, -3, 6, 22, -3], [-3, 0, -8, -4, -11], [14, -18, -8, -17, 2], [-4, -18, 2, 6, -14], [25, -1, -7, -17, 0], [7, 3, 10, 1, 1], [10, 11, 2, 6, 12], [-4, -22, 6, -12, -4], [-16, 31, 22, 14, 6], [-20, -14, 14, 3, -6], [6, -14, 0, 0, 12], [-2, 4, -4, 8, 4], [-22, -1, -12, -6, -2], [10, -6, 6, -4, -2], [-12, 1, 0, -22, 16], [3, 5, 12, -14, 13], [22, -12, -6, 2, -18], [-35, 17, 9, 9, 0], [-5, -11, -23, -3, -18], [27, -19, -8, -1, -15], [10, -18, 2, -14, 6], [-6, -8, -8, 0, 8], [1, 3, 2, 0, -3], [45, 3, -4, 4, 3], [19, 13, -12, -13, -3], [-19, 7, 8, 4, 21], [18, -16, -8, -4, -16], [-32, 8, 4, 4, 18], [-18, -16, -8, -6, -4], [-34, -8, -4, 0, 0], [-8, -6, 4, 6, 10], [-15, 5, 17, -7, -4], [25, 3, 6, -6, 13], [-24, -6, 11, -5, -5], [-14, 20, -6, -6, -6], [34, 2, 0, -7, -2], [33, -3, 1, -11, 4], [8, 10, 22, 4, 16], [4, 10, 4, 0, 2], [-3, -9, -7, -1, 4], [-2, -27, 4, -6, -6], [-16, 8, 10, 2, 4], [12, -1, 8, -6, -14], [1, 1, -6, 21, -11], [14, 18, 2, 10, -2], [11, 1, 0, 0, -3], [-34, 3, -8, 10, 14], [-17, 22, 4, -4, 7], [7, 5, 0, 25, -1], [-20, -3, -2, -10, 4], [23, -11, -2, -22, -3], [14, -2, 14, -20, 10], [9, 15, -4, -8, -5], [-13, -15, -5, -3, 12], [12, -16, 4, -14, -6], [26, 1, -10, -6, 4], [-30, 10, 12, 20, 4], [-26, 12, 10, -14, 10], [24, -2, 16, -2, 14], [5, -5, 6, -20, 11], [-17, -3, -10, 0, -17], [-21, 3, -12, -6, -1], [37, 7, -10, -2, 3], [5, -21, -10, -6, -3], [-14, 2, 6, 12, -18], [-14, -13, -4, -14, 2], [4, 6, 6, 11, 6], [18, -12, -20, -8, -12], [23, -3, 12, -2, 15], [-35, 31, 12, 12, 7], [-2, -2, -6, -10, -10], [-28, -11, -16, -10, 0], [-7, -3, -1, 23, -18], [-26, 22, 4, 4, 0], [4, 4, -2, -2, 12], [23, -9, 10, -10, -13], [11, 1, -1, -19, 0], [-27, 6, 28, 16, 1], [12, -18, 19, 3, 7], [-12, 10, -5, 11, 11], [-5, -2, 8, 0, 7], [-2, 16, 16, 10, 8], [2, 14, -4, 26, -12], [-5, -3, -26, -12, -13], [15, 11, 10, 11, -3], [1, 5, -10, -20, 7], [12, -8, -10, 10, 8], [-44, 21, 14, 10, 10], [-3, 5, -6, 6, -5], [-10, -2, -14, 16, 14], [-4, -25, -8, -22, 10], [23, 1, 4, -18, -1], [24, 2, 14, 0, 0], [50, -2, 12, -10, 14], [-8, 26, -8, -4, 2], [-16, 8, -10, 2, -4], [-44, 14, 6, 18, 12], [-15, -9, -4, -4, 7], [-12, 4, -10, -20, 14], [-9, 1, -12, -27, -1], [38, -20, -16, -10, -20], [-15, -15, -2, 1, -9], [56, -21, -2, -10, 0], [1, -1, 16, -8, -1], [-23, -11, 0, -22, 9], [-18, 0, -16, -10, 12], [-20, 19, 20, 14, 2], [19, -39, 8, -18, 7], [17, -25, -8, 2, -7], [2, -6, 10, 24, -10], [5, -11, -22, -7, -5], [-42, 8, 12, 0, 0], [6, 9, 18, -10, 8], [11, 29, 10, 18, 1], [5, -12, 2, 16, -21], [24, -3, -10, -18, 10], [7, -5, -4, -16, 7], [-14, 10, 40, -6, -2], [5, 1, 0, 26, -19], [16, -18, -20, -8, -10], [-6, -14, -2, -12, 14], [9, 9, -19, -17, 12], [10, -14, -10, -8, -18], [24, 7, 14, 6, -10], [-17, 15, -18, 35, -11], [-5, 1, 4, -16, 31], [-4, 12, 4, 2, -2], [-19, 27, 17, 11, 20], [25, -21, -12, 8, -17], [-38, 26, 14, 4, 18], [-39, 11, 0, -1, 9], [7, 8, -12, 20, 3], [-2, 10, -6, -20, 4], [-45, 21, -13, 17, -12], [-2, -2, -2, -4, -22], [4, -34, -4, -10, -10], [-17, 23, -10, 13, 9], [18, -3, -16, 2, -18], [21, -19, -22, -3, -3], [11, -11, 1, -3, 6], [-6, 21, -8, 30, -10], [-15, 29, -12, 36, -7], [-6, 12, 18, 10, -8], [-7, -13, -6, 8, -15], [16, 20, -8, 6, -10], [-24, -26, -8, -18, 14], [7, -7, 14, -6, -11], [-30, 12, -8, -2, 4], [-7, 3, 11, 17, 6], [-8, 6, 22, 8, 0], [10, -18, 10, -16, 6], [-16, -11, -22, -10, -10], [32, -6, 2, -11, -10], [1, -3, -12, -12, 19], [34, 2, 14, 16, 0], [-52, -12, -2, 2, 0], [24, -4, -16, -2, -6], [13, -9, 2, 22, -5], [-2, 2, 6, -8, 12], [-51, 19, 4, 19, 25], [42, -10, 2, 0, -2], [31, -39, -6, -28, -5], [-50, -8, -6, 4, -10], [0, 23, -2, 10, 14], [25, 7, -8, 0, 15], [7, 5, -2, 14, -1], [-28, 18, -6, 26, 10], [18, -8, 9, -31, -1], [-14, -12, 4, -20, -4], [-24, 4, 16, 24, 10], [34, 4, 12, 20, -8], [-18, 12, -12, -12, 8], [29, -33, -22, -8, -23], [22, 20, 24, 12, -8], [-48, 10, 10, 6, 14], [36, -9, -16, 2, 2], [13, -25, -12, -24, -5], [19, 3, 28, 8, -5], [15, -11, 0, -24, 7], [16, 16, -4, 18, 2], [-37, 29, 4, -1, 7], [-20, 32, 6, 0, 18], [-35, 9, 21, 7, 8], [50, -6, -2, -6, 12], [26, -10, -12, -13, 10], [-18, -2, 2, 8, 6], [-16, 0, -8, -36, 14], [26, 14, 16, 14, -16], [14, -20, -8, 8, -20], [4, 3, -6, 6, 16], [15, -3, 24, -12, 5], [-12, 20, 38, 6, 0], [31, -27, 14, -16, 5], [-37, -1, 14, 15, -9], [27, -27, -38, -2, -7], [-6, -8, 14, -22, 12], [9, -13, 16, -29, 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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 4, 4, -8, -1, 1]>,<7,R![2, 0, -24, 6, 7, 1]>,<13,R![-88, 76, 20, -20, -2, 1]>],Snew);
    return Vf;
end function;