
// Make newform 9200.2.a.ct in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9200_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_9200_2_a_ct();".
// 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_9200_2_a_ct();".
// 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 := [12, 18, -3, -9, 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], [-4, 0, 1, 0, 0], [-2, -5, 0, 1, 0], [9, 4, -7, -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_9200_a();"
function MakeCharacter_9200_a()
    N := 9200;
    order := 1;
    char_gens := [1151, 6901, 2577, 1201];
    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_9200_a_Hecke(Kf)
    return MakeCharacter_9200_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, -1, 0, 0, 0], [0, 0, 0, 0, 0], [-1, 0, 0, -1, 0], [0, 0, 0, 0, 1], [1, 0, 1, -1, 0], [2, -2, 2, 0, -2], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [2, 3, -1, 0, 1], [3, 1, -1, -1, 1], [1, 0, -2, -1, 1], [-1, 1, 0, 1, 2], [-2, -2, 0, 0, 1], [-3, -2, 3, 1, -1], [3, -2, 2, 1, -3], [0, 0, 0, -1, -2], [1, 0, -2, -1, -3], [-2, 2, 0, -2, -2], [0, -4, 3, 2, -2], [-1, 3, -5, -1, 2], [-1, 0, -2, -1, -2], [2, -4, 4, 0, -3], [2, 4, -2, 0, 0], [1, 0, 0, -1, 3], [1, -6, 4, 0, -3], [-1, -2, -2, 1, 2], [-6, 4, 0, 2, 2], [9, 2, 2, -1, 1], [5, 0, -4, 1, -1], [-6, -1, 1, 0, 2], [3, -1, -2, 1, 1], [-2, 0, 0, -2, 4], [0, 4, -3, 2, 0], [6, 4, -2, 0, 0], [6, 0, -1, 4, 4], [-1, 2, -6, 1, -3], [0, 3, -3, -2, 6], [-7, 2, -6, 2, 1], [8, 2, -2, 3, 3], [0, 5, -1, 2, -2], [-2, -2, -4, -2, 2], [3, 6, -8, -1, 0], [5, -5, 4, -3, -5], [2, -1, -6, -2, -1], [3, 0, 6, 1, 2], [-1, -2, -2, 0, -5], [2, -4, 4, 1, -2], [-10, 0, 0, 0, 2], [10, -6, 8, 2, -4], [12, -6, 7, 4, -3], [6, 1, 0, -2, 0], [-7, -2, -4, -1, 3], [-2, -4, 0, -2, -6], [3, 9, -4, 3, 5], [0, 0, -2, 4, 0], [3, 0, -3, 5, 2], [-2, 4, -4, -3, 6], [4, -1, 6, 4, 1], [1, -8, 0, -1, -5], [0, 2, 2, 0, 4], [14, 2, 2, 0, 2], [-8, 0, 0, 0, -8], [4, 3, 9, -2, 0], [-1, 0, 4, -5, -1], [7, -4, 2, -3, 0], [1, 1, 2, -1, 3], [7, -8, 6, 3, -3], [-1, 6, 2, 0, -3], [11, -4, 1, 1, 6], [15, 1, -4, 1, -2], [-7, 6, -6, -5, 4], [-1, 4, -2, 1, 4], [-16, 6, -10, 2, 8], [0, 4, -4, 0, 0], [1, 14, -8, -3, 6], [9, -10, 4, 3, -1], [-1, 3, 1, 1, 1], [7, -8, 6, -1, -5], [3, -1, -4, -3, 0], [12, 0, 4, 0, 3], [16, -2, 0, -2, 2], [-4, 2, 0, -2, 0], [3, 2, -2, -1, 7], [7, 1, 2, 5, -5], [-4, 4, 1, 2, 2], [-7, 2, -8, -4, -3], [-15, 8, 0, -5, -1], [7, -8, 5, 5, -6], [11, -6, 2, -2, -7], [-8, 6, 6, 2, -3], [7, -4, -6, -3, -4], [-3, 2, 1, -3, 3], [19, -6, 11, -1, -9], [16, 5, -4, -4, 6], [-11, -6, -2, 1, 0], [5, -7, 8, -3, -1], [15, 8, -4, 1, -1], [-2, 2, 4, -6, -1], [10, 9, -2, 2, 1], [-1, 18, -7, -1, 1], [3, 12, 4, -1, -5], [2, -2, -4, 8, -3], [-2, 6, 6, -4, 2], [2, 0, -2, 0, -2], [-1, -11, 0, 1, 2], [-3, 10, -9, -3, 1], [16, 6, -4, 0, 1], [11, -10, 6, 4, 3], [-11, -5, 1, -5, 3], [11, -10, -2, -3, -7], [-23, -2, 2, -1, 1], [-16, 2, -4, 0, 4], [4, 0, 2, 2, 0], [3, 4, 8, -3, -4], [-14, -4, 6, 4, -2], [2, 12, -10, 4, 1], [2, 9, -5, -4, 4], [4, 11, 3, 2, -1], [8, 2, 10, 0, -9], [13, -2, -8, 3, -1], [-5, -1, 0, 1, -4], [1, -8, 8, -1, 1], [-3, 1, -1, 1, -5], [15, -6, 2, 10, -3], [16, -4, 0, 2, 0], [12, 12, -2, -6, 6], [23, -2, 6, -3, 3], [2, 4, 4, 8, 2], [2, 16, -8, -2, 0], [3, -5, -7, -9, 5], [5, -10, 4, -1, -6], [-15, -6, -6, 5, -2], [-1, 0, -4, -9, -1], [7, 17, -7, 3, 4], [-3, 8, 4, 3, -7], [-9, 2, 4, -7, -1], [0, -6, 2, -6, -3], [15, -12, 0, 3, -1], [-3, -4, 0, -4, -8], [14, 8, 3, 0, 6], [4, -2, -2, 3, -9], [-3, 1, 2, -1, -15], [16, 4, 10, 0, -3], [2, 2, -6, 7, 3], [3, -14, 6, 3, -8], [-7, 0, -2, 7, -8], [-21, 3, 1, 5, 1], [10, 3, 5, 0, -6], [2, -3, 12, 2, 12], [-9, 10, 0, 6, 7], [1, -10, -4, 1, 9], [23, -14, 18, 4, -11], [-1, 5, 6, 1, -13], [-2, -20, 14, 6, -9], [-23, 8, -2, 5, 4], [0, 2, 2, -2, -12], [-11, 11, 3, -3, -4], [-7, 8, -2, 7, 7], [-1, 4, -12, -5, 1], [-15, 11, 1, 5, 7], [11, 14, 2, -3, 7], [36, 4, 3, -2, 4], [-32, 4, -4, -6, -1], [-5, 28, -16, 1, 13], [-19, 10, -8, -3, 0], [-2, -8, 4, -9, -6], [-10, -14, -2, 1, 1], [-4, 2, 8, 10, -4], [12, 0, 7, -2, -2], [-6, 2, -18, 0, -3], [13, 0, -7, -9, 8], [-17, -6, -2, 1, -4], [-6, -14, 4, -2, 0], [-12, -9, -8, 8, -2], [-14, -4, 0, -2, 6], [21, 3, -3, 5, 7], [12, 3, -6, -4, 9], [9, 0, 2, 3, 14], [14, 16, -4, 2, 2], [11, -1, -10, 5, -3], [-18, 8, 4, 8, 1], [-4, -21, 14, 0, -3], [-17, 8, -8, 0, 4], [-15, 8, 2, 5, 8], [-21, 16, -12, -7, 7], [5, -6, 8, 6, 9], [-12, -8, -4, 2, 1], [-2, -6, 3, -6, 7], [21, 6, 2, -1, 1], [-1, -3, -3, 3, 0], [8, -16, 20, 9, -10], [-10, -16, 14, -4, -3], [13, 2, -4, -4, -3], [-10, -2, -6, -2, 9], [8, 10, -10, 8, -6], [5, -20, 4, -3, -7], [-10, 10, -4, 8, 0], [23, 0, -2, -3, 7], [-24, 5, -8, 4, 14], [-11, 4, 2, 1, 3], [-1, -2, 2, -6, 3], [-2, -1, -14, 2, 3], [-7, -20, 10, -3, -11], [10, -2, 16, 6, -9], [32, -7, 13, 6, -7], [14, 16, 2, -2, 6], [7, 18, -5, -1, 7], [-24, -14, 1, -4, -1], [11, 3, 5, -5, 3], [11, 17, -11, -1, 0], [5, -16, 11, 7, 2], [-7, -4, 2, -3, -4], [-14, 2, -18, 0, 9], [0, 24, -4, 5, 8], [26, 10, 0, -2, 16], [28, -5, 0, 4, 8], [25, -9, -7, -5, -1], [3, -6, 14, -3, 0], [36, -12, 11, 2, -6], [12, -3, 1, -2, 9], [7, -2, -2, -1, 0], [-6, 0, 3, 0, -16], [-7, 10, -10, -3, -4], [-8, 5, -17, 6, -4], [4, -9, -2, -12, 1], [16, 0, 0, 2, 22], [1, -3, 6, 11, -7], [26, 14, -2, 4, 2], [-19, -10, -6, 6, 3], [-36, -8, 0, 2, -6], [-26, 26, -18, -8, 6], [23, 4, 6, 3, 2], [-15, -8, 0, -4, -8], [-20, 10, -16, -2, 12], [-7, 14, -6, -7, -8], [-17, 14, -6, 1, -1], [9, -14, -2, 7, 1], [3, -5, 10, 1, -1], [-1, 5, -6, 1, -7], [-12, -2, -18, 4, -1], [18, -31, 8, -2, -10], [6, -12, 13, 0, -16], [-29, 1, -9, 1, 13], [35, 2, 8, 0, 3], [-5, 30, 1, -1, -1], [-10, 19, -16, 2, 4], [8, 6, 2, 0, 0], [-20, 10, -2, -10, -1], [-9, 6, 6, -1, 6], [16, -8, -4, 10, -2], [14, 2, -2, -4, -12], [-5, -34, 18, -3, -12], [-2, 2, 6, -2, -14], [-16, 6, -2, 5, 9], [21, 2, -1, 5, -9], [28, 0, -6, 12, 2], [26, -20, 2, -4, -9], [28, 5, 9, -6, 9], [12, 24, -3, 2, 2], [-10, -6, -14, 6, -6], [-4, -24, 12, -4, -9], [9, -22, 8, -6, -3], [-30, 0, 8, 4, -4], [18, -32, 4, 4, -4], [10, -12, 12, -6, -15], [-14, -8, 2, -2, 2], [-27, -6, -8, 7, 7], [-4, -4, -8, 10, -9], [18, -17, -2, -10, -5], [-18, 11, 3, 4, 9], [-2, 6, -8, -14, 14], [-8, 2, 12, 8, -12], [-19, 10, 2, 9, 4], [-14, -6, -7, -2, -7], [-3, 7, 9, 13, -11], [38, -8, 8, 5, 0], [40, 2, 6, 7, -9], [8, -6, 2, 4, -4], [-12, -10, 10, 10, -12], [-22, 7, -7, 0, -2], [-6, -28, 14, -2, -10], [-27, 16, -6, -7, 4], [12, -18, 8, 6, 2], [7, 2, 2, -4, -7], [26, -4, -16, -6, -4], [-27, -2, 4, 3, 1], [17, -16, 4, 9, 3], [-11, -10, -10, -4, -1], [-9, -12, -4, 2, -2], [15, 20, -2, 7, -3], [-18, 10, -12, -4, 15], [-11, -8, 8, -14, -6], [10, 6, -14, -14, 10], [11, 21, -9, -7, 13], [-2, 4, 0, 4, -10], [35, 0, -4, 5, 2], [-8, -5, 7, 6, 11], [-10, 17, -17, 0, 12], [-16, -9, 1, 2, 18], [-13, -12, 12, 5, -17], [28, 8, -4, -2, -12], [17, -24, 14, -5, -12], [16, -12, -5, 2, 14], [-18, 16, -24, 4, 10], [41, -17, 16, 15, -6], [-11, -10, 2, -5, -17], [29, 2, 20, -5, -13], [-38, 8, -2, 8, 2], [-19, 8, -14, 3, 16], [-21, -14, 15, -5, 1], [-47, 14, 0, -1, -3], [-2, -6, 4, 4, -8], [12, 0, 15, -10, 8], [-4, -12, -18, 4, -8], [0, 2, -14, 10, 10], [1, 7, -15, -3, 5], [-15, -18, 5, -3, -15], [8, 8, 10, 4, 18], [-19, 7, -20, 5, 7], [8, 2, 8, 8, -4], [23, 8, 0, -1, 9], [-8, 15, -10, -8, 1], [13, 6, -2, 0, 5], [10, 18, -14, 6, 16], [-31, 22, -2, 3, 11], [28, -12, 18, -12, -12], [-13, -3, 11, 7, 5], [-7, 4, 0, -5, -14], [-5, -8, 12, -5, -3], [8, 16, -29, -6, 2], [3, 16, 9, 1, 6], [4, 4, 1, -2, 14], [-9, 17, -25, -5, 9], [44, -16, 8, 4, 0], [7, 26, -3, -1, -3], [12, 8, -8, 2, 14], [28, -20, 18, 8, -4], [17, -5, 3, 5, 12], [-34, 14, -16, -10, 14], [21, -1, -7, 15, 11], [-32, 4, -6, -10, 4], [15, 24, -10, 13, 0], [-16, 14, -12, 6, 20], [-12, 16, 4, 10, -8], [7, -18, -14, 9, -1], [22, 0, 25, -4, -12], [8, 0, -28, -3, -8], [5, 8, -2, 9, -2], [38, 10, -12, 8, -6], [-6, -11, 12, -6, 10], [-20, 4, -3, -2, 14], [-15, 8, 2, -13, -12], [4, 8, 2, 12, 12], [9, 6, 16, -9, 11], [20, 10, -8, 0, -17], [-3, -36, 4, 1, 0], [-24, -14, 12, 8, -11], [41, -5, 19, 5, -4], [-53, 6, -2, 4, 3], [-3, -26, 4, -1, 7], [-18, 18, 7, 10, 3], [20, -11, 0, -8, -16], [-26, -24, 4, 0, 1], [21, 6, -3, -7, 27], [-8, 24, -7, -2, 2], [7, 27, -19, -11, 21], [12, -2, 0, 6, -16], [36, 4, 0, 4, -8], [20, -5, -4, 8, 14], [16, -32, 16, -3, -20], [-17, 5, -14, -3, -7], [-30, -16, 6, -8, 12], [-7, 8, -24, 3, -12], [18, 11, -9, 0, -15], [8, -16, -3, 14, -6], [-45, 6, -4, 7, 6], [-6, 8, -30, -10, 4], [-9, 28, -2, 7, 13], [27, -3, -3, 15, 4], [-12, 10, -4, 2, 10], [11, -6, 4, 15, 3], [15, 18, -10, 10, 11], [13, -23, 16, 1, -9], [22, 16, -6, 12, 13], [-22, 15, -3, 4, 6], [26, -15, -1, 16, 0], [51, 2, 8, -2, -1], [12, 20, -30, 4, -4], [-16, 18, 2, 18, 8], [18, -30, 22, -4, -10], [46, -10, 22, -3, -15], [-21, -10, 13, -1, -9], [24, -8, 8, -12, -8], [36, 2, -2, -10, 8], [0, 18, -20, 12, 8], [-23, 4, -8, 11, 10], [9, -20, 14, 17, -13], [14, 22, 0, 6, -5], [24, 24, 0, -6, 14], [22, -13, 14, -6, -25], [40, 0, 1, -6, 14], [-5, 6, 18, 11, -8], [20, 8, 6, 2, -10], [6, 12, -15, 12, 4], [52, 3, -7, 10, 0], [26, -20, 22, -8, -14], [-3, 5, 8, -3, 15], [-26, -12, -4, -4, -9], [44, -24, 32, 9, -8], [-40, 0, -21, 2, 18], [-21, 4, 1, -11, -14], [-16, -18, 10, -16, -5], [76, 2, 0, 0, -3], [-17, 22, -8, 9, 13], [22, -5, 7, 12, -11], [-9, -18, -2, -11, -1], [50, -8, 16, 7, -16], [-6, 34, -24, -6, 10], [23, -14, 26, 5, -18], [6, 10, 2, -6, 14], [28, -12, 24, 12, -6], [-2, 4, -8, -9, 4], [-42, 12, -12, -9, -12], [8, 10, 9, 0, -17], [-1, 20, -7, 13, 26], [-22, -4, 4, 2, -5], [-9, -8, -8, 12, -8], [-32, 5, -4, -4, 2], [7, 9, -1, 3, 19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9200_2_a_ct();".
// 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_9200_2_a_ct(:prec:=5)
    chi := MakeCharacter_9200_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_9200_2_a_ct();".
// 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_9200_2_a_ct( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-12, 18, 3, -9, 0, 1]>,<7,R![200, 20, -76, -17, 4, 1]>,<11,R![-24, 48, 6, -15, 0, 1]>],Snew);
    return Vf;
end function;