
// Make newform 9576.2.a.cl in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9576_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_9576_2_a_cl();".
// 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_9576_2_a_cl();".
// 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, -1, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [2, -4, -2, 1], [-2, -12, -2, 2]];
    Rf_basisdens := [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_9576_a();"
function MakeCharacter_9576_a()
    N := 9576;
    order := 1;
    char_gens := [7183, 4789, 5321, 4105, 1009];
    v := [1, 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_9576_a_Hecke(Kf)
    return MakeCharacter_9576_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], [1, 0, 0, 0], [0, -1, 0, -1], [-2, 1, 0, 0], [1, -1, 1, 0], [-1, 0, 0, 0], [0, 0, -2, 1], [-1, 0, -1, -1], [2, 0, 2, -1], [2, -1, 2, 0], [2, 0, 0, 0], [2, 1, 0, 1], [1, 1, -3, 1], [-1, 0, -1, -1], [0, 0, 0, 0], [-6, 0, 0, -2], [2, 1, -2, -1], [-1, 0, 1, 1], [0, 0, -2, -2], [8, 0, 2, -1], [-3, 2, -1, -1], [2, -2, 0, -2], [2, 1, 0, 3], [7, 0, 1, 2], [12, 1, -2, 4], [-3, 2, -1, -1], [4, 1, 4, -2], [3, 0, -1, 3], [6, 2, -4, 1], [-1, 0, -3, -1], [-6, 0, 0, -2], [2, 0, -2, -2], [0, 3, 4, 2], [18, 0, 0, 1], [6, -4, 0, 2], [6, 3, 4, 1], [2, 2, -6, 0], [2, 2, 0, -2], [-1, -2, 1, 3], [-2, 0, 6, 1], [-8, -2, 2, -1], [10, -4, 0, -2], [2, 1, -2, 2], [10, -6, 2, -2], [-4, 3, 0, -3], [10, 0, -2, -1], [-10, 0, 6, 2], [-8, -4, 2, -6], [-12, 2, 2, 0], [6, -2, 4, -1], [-6, 5, -4, 3], [13, 2, -5, 3], [12, 0, -2, 6], [-2, -2, 0, 3], [12, 0, 2, 0], [-4, -2, 0, -2], [-2, 1, 6, 1], [-3, 4, -3, 1], [10, 0, -2, -2], [-10, 0, 0, -2], [6, -5, 4, -1], [-5, -3, -1, -1], [2, 4, 4, 4], [-9, -2, -1, 1], [6, 3, -6, 5], [-16, 2, 2, -4], [10, -5, 2, -3], [-2, 0, -8, -2], [-5, 1, 3, -6], [8, -4, 2, -3], [10, 2, 2, 0], [2, 1, 2, -2], [2, -3, 2, -5], [6, -6, 6, 0], [-4, 3, -4, 0], [-2, 0, 8, 0], [7, 2, 3, 1], [-2, 4, 6, 6], [15, -4, 5, -3], [-16, 3, 0, 2], [-3, 6, -1, -3], [6, -1, 8, -3], [-2, -2, 10, -3], [2, 1, 2, 3], [-5, -4, 3, -3], [2, 5, 2, 1], [3, -2, 1, -2], [8, 0, 8, 0], [19, 2, 1, 1], [-1, -3, 11, -3], [-4, 2, -2, 1], [2, 5, 6, -3], [4, -6, 4, 0], [27, -3, -1, -1], [20, -4, 2, 0], [0, -2, -6, 4], [2, 3, 0, 3], [-6, 0, -8, 4], [-12, -5, -8, -1], [-12, 1, -4, 0], [-14, -4, 6, -6], [-7, -6, -3, 1], [-4, 4, -4, -2], [10, -2, -4, 6], [13, -2, -1, 5], [25, -1, 5, -2], [-9, 6, -3, -1], [10, 3, -12, 3], [4, -1, -2, 2], [2, 3, -2, -7], [-2, 8, 4, 4], [18, -6, -2, 0], [-26, -1, 8, 3], [-7, 0, -11, 3], [14, 4, -6, 6], [1, 1, 9, 3], [-18, 3, -6, 4], [9, 0, -5, 7], [-18, 1, 4, 0], [6, -2, 4, -4], [0, 4, 2, 4], [-13, 4, 5, 5], [16, -6, 8, -2], [22, -9, 2, -2], [-6, -1, -6, -3], [13, 5, 5, 1], [-16, 6, -4, 0], [4, 0, 2, 0], [2, 1, 0, 3], [25, 2, -9, -1], [14, -2, 12, -5], [-10, -4, -8, -2], [-7, -7, -7, -6], [-6, 8, 4, 0], [-2, 4, 8, 2], [-10, 7, -4, -3], [6, 0, 8, 0], [14, 0, -8, -2], [-12, -4, 4, 2], [20, 1, 4, 2], [-2, 3, 4, 1], [-1, 0, 5, 1], [6, -8, 2, -3], [16, -2, 4, -6], [34, -10, 4, -4], [6, -4, 0, -4], [-8, -2, 0, -2], [7, 2, 1, 5], [-2, 3, -10, 4], [-9, 1, -1, -6], [-8, -2, 8, 4], [10, 2, 10, -2], [-12, 5, 0, 1], [-15, 8, -1, 7], [32, 0, 0, -4], [25, -1, -7, 0], [-18, 4, 4, -8], [-4, 2, 6, 6], [-16, 5, 0, -1], [-9, 0, -5, -1], [-6, 5, 0, 5], [16, 6, 0, 10], [3, 6, -1, 9], [-8, 2, -12, 4], [-8, 10, -10, 3], [-34, 0, 0, 2], [-16, -2, -2, 2], [-5, -2, 9, 2], [5, -4, 3, -1], [6, 7, -16, 4], [-2, 8, -4, 1], [14, 5, 6, -5], [8, -9, 2, -2], [-9, 7, -9, 8], [6, 7, -16, 9], [-2, -3, -2, -10], [6, 6, -18, 1], [-18, -4, 4, 6], [-8, -4, 4, -10], [-8, 8, 0, -4], [8, 1, 8, 6], [2, 0, 12, 0], [-8, 0, -14, -1], [15, 6, -3, 6], [4, 1, -8, 2], [2, 1, -8, -5], [-4, -2, -10, 2], [-5, -1, -9, 7], [10, 6, -14, -2], [0, -7, 16, -1], [-10, -3, 10, -9], [0, 10, -2, 10], [-12, 1, 4, -3], [-24, 14, -2, 8], [-22, -3, 2, -1], [-18, 4, 0, -6], [-13, -3, 11, -6], [11, 1, -1, 1], [-1, 4, -1, 3], [16, -9, 2, 2], [-6, -4, -14, 7], [-6, -2, -6, 2], [19, 0, 13, -7], [21, 0, -5, 0], [2, -2, -6, -2], [-17, -6, -3, -1], [-28, -4, -6, -2], [18, 3, 8, 1], [2, 5, 10, -3], [1, 0, 15, -4], [-14, -5, 0, 1], [-2, -6, -2, -4], [0, 6, -12, 10], [22, -8, 2, -6], [10, -3, 12, -1], [-32, -4, -6, 2], [1, -4, 23, -5], [-28, -5, -4, 4], [-8, -1, 12, -8], [-24, 1, 6, 2], [3, -10, 3, 3], [32, -8, 8, -4], [8, 2, 4, 2], [-30, -2, 0, 6], [8, -2, -2, -4], [5, -12, 15, -7], [-18, 4, -2, -7], [-6, -7, 6, -1], [-2, 0, -4, -12], [-14, -3, 4, -5], [-24, -6, 2, -11], [-27, 6, 1, 5], [-8, 6, -12, 6], [-11, 9, 1, -1], [24, 4, -2, 6], [-2, 7, -18, 6], [-14, -3, 10, -11], [-18, 6, 6, 0], [-39, 10, -1, -1], [12, -14, 4, -6], [-26, 8, 2, 6], [6, -1, -6, 6], [-28, 2, -6, 0], [41, -3, 1, -3], [14, 2, 6, -2], [-14, -6, 2, 0], [-12, -5, -12, 1], [-4, 4, -2, -3], [-26, -6, 0, -14], [-9, 11, -1, 0], [46, 0, 0, 1], [2, 2, 2, -6], [-21, -8, 9, -12], [16, -7, 12, 1], [30, 0, -4, 14], [18, -5, -4, 3], [28, 2, 2, -2], [-34, -4, 4, 0], [24, -14, 10, -9], [-11, 4, 7, 9], [-28, 12, 2, 4], [14, 7, -20, 0], [-7, -3, 1, -2], [-18, -3, 12, -1], [13, 12, -7, 13], [7, -1, -1, 12], [-12, -1, -16, 5], [0, 3, 8, 0], [6, -10, 18, -9], [-12, -1, -4, -3], [8, 4, -10, -2], [20, 4, -16, 14], [10, 11, -4, 7], [10, 4, 14, 2], [-14, 15, -2, 3], [18, -3, 2, 4], [26, -12, 8, 0], [-31, -2, -13, 5], [-46, 0, -2, -8], [2, -1, 0, -7], [-5, 5, -9, -5], [-22, -2, 10, -1], [-16, -6, 20, -6], [2, -6, 12, -11], [22, 4, 0, 12], [-11, -2, -19, 5], [6, 0, 2, 0], [12, -6, -10, 0], [4, 1, -4, 12], [-45, 0, -3, -1], [-23, -8, 9, 1], [-20, 2, 4, 6], [-18, -3, -2, -4], [13, -6, -5, -2], [2, 8, -6, 9], [-15, 6, 11, -8], [-38, 4, -10, 6], [18, 2, -6, -8], [-10, 6, 12, -6], [3, 4, -5, 7], [-24, 0, 12, 6], [-29, 10, -19, 9], [10, 11, 4, 7], [4, 10, -2, 2], [51, 2, -7, -1], [6, 3, 4, 16], [-2, 10, -28, 11], [-6, -3, 14, -1], [23, -3, -1, -7], [32, -5, 8, 6], [-3, 4, 9, 3], [-10, -5, 6, 3], [44, -4, 2, -5], [10, -5, -12, -5], [-15, 0, -1, 5], [9, -2, -13, 9], [32, -4, -10, -2], [-22, -6, 4, 10], [-4, 6, -4, 12], [-38, -1, 6, 3], [-12, 0, 6, 6], [-10, -2, -8, -3], [-20, 4, 2, 4], [10, -3, -20, -5], [-20, 7, 4, 7], [8, -6, 28, -10], [-55, -2, 11, 5], [26, 5, 10, 6], [14, -10, -8, 6], [-51, 4, -3, -5], [22, -2, -10, -2], [13, 6, -5, 13], [4, -2, 0, 2], [-13, 10, -15, 9], [-2, -13, -2, -4], [-38, 2, -4, -6], [8, -6, 14, 0], [40, 0, -8, 6], [-10, -3, 6, -2], [-14, 2, -4, 4], [-6, -10, 4, -4], [-16, 2, 18, 11], [-29, 4, 7, 1], [-9, -14, 17, -17], [-36, -2, 14, -4], [16, 9, 0, 9], [4, 10, 8, 2], [-9, 12, -15, -2], [-12, 10, 4, 0], [-60, 10, -6, 2], [-46, -5, 10, -4], [44, 1, 2, 2], [2, -5, -2, -4], [-8, 0, -10, -10], [29, 1, -15, 17], [5, -4, -9, 1], [28, 8, 2, -8], [3, -12, 21, -3], [-34, 6, -16, 6], [17, 7, -3, 14], [-13, -12, 13, 1], [23, 2, -19, 15], [-18, -10, 18, -16], [18, -8, -4, 0], [-7, -10, -1, 6], [10, 2, -16, 1], [2, -4, -16, -2], [-28, -3, 8, -1], [-30, 3, -4, 5], [22, 6, -8, 5], [3, 8, -9, -11], [-4, 2, -8, 10], [-14, 4, 0, -6], [-8, 8, -4, 0], [-8, 14, -6, -5], [-42, -6, 4, -10], [41, 9, 17, 2], [-34, -5, 4, 9], [48, -16, 10, -8], [-46, 4, 12, 6], [14, -9, 8, -1], [18, -4, 4, -4], [14, 13, -4, 19], [2, 8, 6, 0], [-30, 11, -4, -5], [-4, 14, 2, 2], [-30, -5, -4, 3], [-26, -10, 14, -12], [-36, -4, 18, -14], [-6, -4, -16, 10], [12, 11, -16, 5], [32, -6, -4, -2], [9, -2, -9, -9], [0, -12, 6, 8], [-12, -5, -6, -4], [-46, 6, 0, -6], [-30, 13, -6, 11], [-4, -11, 8, -10], [22, -1, 4, 4], [-2, 16, -8, 10], [12, -6, -14, -9], [16, 6, -18, -4], [28, 6, -2, -6], [-14, -10, 2, 6], [-2, -2, -12, 2], [-15, -14, 5, -15], [28, -14, 0, -6], [-29, 4, -19, 3], [6, 5, 4, 5], [-36, -9, -20, 0], [23, -8, 9, 3], [14, 7, 12, 9], [14, -8, 8, 8], [7, -2, 9, -12], [43, 4, -7, 11], [-2, 0, 10, -15], [27, -1, 31, -10], [8, 2, 4, -4], [41, 0, -15, 7], [14, -12, 14, 3], [6, 12, 4, -3], [0, -2, 24, -14], [30, -8, -10, 10], [-6, 2, 0, 2], [-41, -10, -19, -3], [-52, 8, -2, 6], [-22, 2, -14, -4], [45, 11, -3, 13], [62, 4, -2, -2], [8, 5, -8, 13], [-14, 13, 4, 11], [20, 4, 4, -10], [-30, -6, -4, -14], [35, 11, -9, 6], [-22, 21, -6, 9], [10, -9, -6, -14], [-14, -11, 10, 5], [-32, 0, 8, -8], [9, 12, -5, 15], [10, 14, -12, 2], [-6, 7, -14, 6], [-55, -2, 7, -13], [-40, 4, -6, -2], [-7, 6, 17, -5], [-2, -12, 14, 2], [-44, -6, -8, -4], [14, 0, 10, -6], [-34, -5, -20, 4], [-4, -15, 32, -7], [52, 4, -12, 12], [62, -9, -2, 5], [9, 13, 9, 2], [-50, -4, 12, -10], [-5, 6, -1, 11], [-24, 9, 0, -8], [-16, 4, 16, -8], [-28, 10, -14, -6], [65, 6, -11, 9], [-4, -5, 20, -1], [22, -18, 16, -11], [11, 6, 17, 11], [14, 9, 22, 2], [-16, -10, 16, 2], [-2, 16, -20, 10], [0, -2, -10, -1], [-29, 0, -15, -7], [66, 2, 4, 8], [-2, 2, -6, -4], [44, 2, 14, -5], [-33, -12, 1, -5], [-46, 4, -10, 4], [10, 17, -2, 7], [25, -20, 7, -15], [26, 8, 6, 3], [46, -17, 18, -8], [34, -4, -2, -4], [-53, -1, -13, 1], [3, 6, 19, 9], [-10, 11, 16, 1], [26, 12, 20, -8], [6, 12, -2, -10], [50, 4, 4, 0], [-20, -8, 4, 12], [-14, 5, 2, -11], [-2, -13, 14, -17], [-46, 0, 10, -10], [-20, 8, 20, 6], [26, 0, 0, -1], [-22, 2, -2, -7], [-12, 0, 12, 8], [14, -14, 8, 10], [-60, -4, 2, -16], [28, 13, -4, -13], [-34, -14, 4, -20], [-4, -2, -24, -6], [36, -15, 12, 6], [-34, -10, -2, 0], [80, 3, -4, -5], [4, 5, -8, 16], [12, 18, 0, 14], [-11, 8, 9, -1], [56, 4, 16, 0], [46, -7, -2, -6], [49, 1, -7, 10], [-11, 14, -5, 11], [58, -10, -6, -9], [-40, -5, -4, 10], [-64, -4, 16, -2], [-1, 2, 9, 13], [-23, -7, 5, -13], [46, -6, 4, 2], [-69, -2, -1, -9], [-44, 26, -8, 2], [-30, -7, 26, -1], [-24, -2, 2, 8], [22, -6, -8, -6], [7, 0, 1, 9], [-22, 4, -26, 3], [10, -1, -26, 3], [-6, -5, 6, 3], [53, -8, -7, 3], [16, -8, -18, -3], [-14, 10, 12, 18], [-33, 18, -3, -7], [82, 0, -2, -2], [1, 10, -33, 2], [-5, 0, -23, -3], [-20, 0, -2, 4], [16, -4, 12, -2], [29, 14, -3, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9576_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_cl();".
// 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_9576_2_a_cl(:prec:=4)
    chi := MakeCharacter_9576_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(3833) - 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_9576_2_a_cl();".
// 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_9576_2_a_cl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 20, -6, -4, 1]>,<11,R![192, 0, -36, 0, 1]>,<13,R![-96, -96, -12, 6, 1]>,<17,R![-32, 64, -30, -2, 1]>],Snew);
    return Vf;
end function;