
// Make newform 1911.2.a.w in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1911_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_1911_2_a_w();".
// 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_1911_2_a_w();".
// 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, 7, 10, -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], [-3, 0, 1, 0, 0], [2, -4, -1, 1, 0], [6, 3, -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_1911_a();"
function MakeCharacter_1911_a()
    N := 1911;
    order := 1;
    char_gens := [638, 1522, 1471];
    v := [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_1911_a_Hecke(Kf)
    return MakeCharacter_1911_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, 1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, -1], [0, 0, 0, 0, 0], [1, 1, -1, 1, 0], [1, 0, 0, 0, 0], [-2, 0, 0, 2, 0], [2, -1, -1, -1, -1], [0, 1, 1, -1, -1], [4, -3, 1, 1, 1], [0, -1, 1, -1, 1], [0, 2, 0, -2, 0], [3, 1, -3, 1, 0], [0, -1, -1, -1, 1], [-1, -2, 4, 0, 1], [2, 3, -1, -1, 1], [-1, -3, -1, -1, 2], [-2, -4, 0, 0, 0], [2, 0, 2, 0, 2], [-1, -3, -1, 1, -2], [4, 1, -1, 1, 1], [0, -1, -1, -1, 3], [-3, 0, 2, -2, 1], [-9, 4, 0, 0, 1], [6, -3, 5, 1, -1], [-4, -2, 2, 0, 0], [2, 0, 2, 0, 0], [0, 0, -4, 2, 0], [8, -4, 0, 0, 2], [0, 3, -5, -1, -1], [-4, 0, 0, -4, -2], [-6, 4, -4, 0, -2], [5, -3, 5, -1, -2], [-2, 2, -4, 2, 2], [-3, -5, 3, -3, 4], [4, 2, 0, 2, 2], [0, 2, 2, 2, -2], [6, -4, 4, 4, -2], [-7, -2, -4, 0, 3], [-8, 0, 0, -2, -2], [6, -1, -5, -3, -1], [4, -6, 2, -2, 2], [6, -6, 2, 0, -4], [0, -2, 0, 2, 0], [5, -1, -1, -3, -4], [-4, 6, 2, 2, -2], [4, -5, -1, -1, -1], [2, 5, 1, -3, 1], [5, -5, 1, 5, 2], [8, 0, 2, -4, 2], [8, 1, 1, 1, 1], [-3, -3, 3, 1, 4], [0, 1, -5, 1, -3], [-2, -4, 4, 4, 2], [-8, 10, -2, 0, 4], [8, -5, 3, 1, -3], [8, -6, 6, 0, -4], [-2, 8, -10, 0, -2], [-6, -1, 3, -5, -3], [9, -3, 1, -1, 2], [-4, 4, 4, 0, 4], [13, -4, 0, 0, 3], [0, 7, -1, 3, 5], [-6, 6, 2, -2, 4], [-6, 0, 0, -4, 4], [5, 3, 3, 1, 0], [0, 6, -6, -2, 2], [2, 5, -1, 1, -3], [-2, -2, 6, 4, 0], [8, -1, -5, 3, -1], [9, -13, -1, 3, 0], [5, -3, 3, -3, 4], [-10, 4, -2, 0, 8], [10, -6, 0, 2, 0], [0, 8, -2, 0, -4], [-11, 7, 1, 5, 2], [10, 8, 0, 0, -4], [-2, -1, -3, -1, 1], [9, -1, -1, 5, 4], [-6, 5, -3, -7, 3], [-10, -8, 4, -4, 2], [2, -6, 0, 2, 2], [-3, 9, -5, -7, 0], [-10, 12, -8, 0, 4], [10, 10, 0, -2, 2], [-4, 5, -3, 3, -5], [5, 3, 7, -3, -4], [4, 10, -2, 6, 0], [11, 1, -7, -3, 0], [-14, 0, -6, 0, 6], [-2, -6, 6, -6, 0], [13, 6, 0, -4, -1], [8, -8, 6, 0, 0], [-16, -4, 0, 6, 0], [-4, 6, -6, -2, 2], [-10, 4, -4, -4, -2], [1, -4, 4, -4, -5], [-16, 0, 4, 2, 6], [-2, 10, 0, 2, -6], [10, 12, -6, 4, 0], [-8, 17, -3, -3, -1], [-31, 1, 5, 3, 2], [-20, 2, -2, -2, 2], [16, 5, -7, 1, -3], [8, -1, 7, -9, 3], [-14, -6, 12, -2, 2], [-13, 6, -4, 0, 1], [-9, 10, -6, 2, -5], [-4, -11, 13, 3, 3], [2, -2, 10, -6, -6], [-2, -10, 0, 2, -6], [-6, 4, -8, 4, -8], [7, 1, 5, 3, 8], [-6, -4, 2, 4, 2], [-6, -16, 8, 0, -2], [26, -3, -3, -3, -5], [16, 4, -4, -4, 0], [4, 0, -8, -4, -4], [16, -2, 2, -6, 0], [-14, 13, 1, -1, 1], [4, 3, 3, 7, -1], [10, -1, 7, 3, -3], [-6, -10, 2, 4, -6], [7, -7, 11, 1, 6], [14, -9, 5, 3, 3], [14, -3, 5, -3, -1], [28, 2, -2, -2, -4], [22, -2, -2, 2, -4], [16, 4, -4, 4, -4], [24, -7, -5, 1, -3], [14, -6, 0, -6, -4], [-23, -3, -1, 5, 0], [-8, -9, 7, -1, 5], [6, 17, -9, 5, 1], [-21, 6, 6, 2, 7], [-4, 9, 5, -3, -3], [21, -5, -13, -5, 0], [2, -3, 7, -7, -3], [-22, -10, 6, 0, -2], [16, -1, 3, -5, 7], [-4, -6, 0, 6, -6], [25, -7, 1, 3, -6], [-8, -12, 0, 4, 4], [11, 3, 1, 7, 8], [-10, 6, -2, -2, -6], [-13, 9, -5, 3, -2], [2, 13, -1, -3, 7], [12, 10, -6, 4, 4], [12, -8, -4, -4, 8], [-17, 1, 3, -7, -6], [-6, 8, 0, 0, -4], [-8, 16, -4, -6, -6], [12, -4, -4, 4, -6], [-4, 18, -10, -6, -2], [4, -1, -1, -5, -1], [10, -15, -3, 3, 1], [0, -4, -12, 4, 6], [-7, -12, 0, -4, 3], [18, 2, -6, -2, -2], [1, 10, 2, 2, 1], [-11, 3, 9, -5, 2], [-4, 3, 3, 7, 3], [-10, 12, 0, 4, -6], [12, -2, -6, 2, -10], [17, -5, -5, 1, 0], [5, -18, 0, 8, -1], [4, 4, -12, 0, -8], [-12, -4, 0, -4, 4], [26, 6, 0, -2, 2], [-4, 8, -16, 6, -2], [-5, 5, -1, 9, -2], [18, -2, -4, 10, 6], [-16, 3, -13, 1, 5], [14, -12, -2, -4, -8], [2, -2, 12, -6, -6], [14, -8, 4, -2, 4], [-10, -4, 2, -4, -6], [-11, 5, 1, 3, -6], [2, 8, 2, 8, 4], [-6, 2, -6, -2, -14], [-34, 6, 4, -2, 0], [-4, -6, -6, -6, 6], [-6, 18, -4, 2, 0], [19, 2, -14, 2, -1], [9, 7, -15, -5, -2], [6, 0, 12, -2, -4], [-30, 9, 5, 5, -5], [-4, 7, 5, -1, -7], [-6, 8, 6, 0, -4], [7, -8, -2, -10, -9], [-4, -20, 16, 0, 8], [-13, -3, 3, -7, 6], [4, 11, -9, 11, -7], [13, -8, 0, 4, -9], [-6, -3, 1, -1, 5], [8, -8, 12, 2, 10], [12, -6, 0, 14, 8], [10, 1, 7, -3, 13], [-11, -2, 6, 6, 9], [-12, 0, 0, 0, 12], [-3, 7, 3, -11, -4], [18, -20, 6, 0, 0], [4, 13, -5, 5, -3], [-40, 2, -6, -2, 6], [-3, -4, 10, -2, 5], [18, -10, 2, 4, 2], [2, -13, 3, -5, -1], [-8, 1, 1, -1, 11], [11, -3, 1, 7, -4], [-36, 4, 0, 8, 4], [2, -9, 1, 3, 3], [14, -2, 18, -8, 2], [-12, -12, 4, 12, -2], [-17, 1, -9, -5, -2], [39, 6, 4, 4, -7], [2, 5, -5, -3, 3], [-20, 3, -5, 7, -1], [30, 6, -10, 8, 2], [9, -1, 9, -5, -10], [17, -5, -1, -7, 0], [-2, -9, 3, -13, -3], [-38, 8, 2, 0, 2], [-5, 11, 3, -11, -2], [-8, 18, 4, -6, 6], [-14, 20, 8, -4, -6], [6, -7, 5, -11, -1], [-30, -12, 4, -6, 0], [0, 7, -1, -7, 1], [0, 8, -16, -8, -8], [-19, -1, 1, -5, -2], [-26, 16, -8, -8, 0], [-6, 6, -8, 10, 2], [4, 2, 10, -6, 6], [2, -8, -4, -12, 8], [14, -10, 8, -6, -6], [1, 4, 10, -6, 5], [0, 4, -2, 16, 10], [-3, 3, -13, 5, -12], [-26, 13, 1, 3, 5], [-8, 10, 6, 6, 10], [15, -17, -7, 7, 8], [4, 12, -4, 4, -4], [4, 22, -2, 2, -2], [0, -2, 4, -6, 0], [1, 2, 6, 10, 1], [-6, -24, 4, 0, 2], [30, -5, -3, -9, -5], [-11, 6, -8, -4, -5], [18, 2, -10, 2, -12], [0, -1, -1, 1, -3], [-4, -18, 18, 2, 8], [-1, 13, -3, -15, 0], [-39, 5, 15, 5, 8], [-6, 4, 4, 4, 12], [4, -2, 2, -16, 0], [-17, 11, -11, 7, -4], [-22, 7, 5, 3, -5], [-6, 2, 2, -2, -12], [-38, -4, -4, -2, 4], [-4, -1, 15, 3, -1], [-8, -1, 7, -1, -1], [18, -11, -7, -5, -3], [-26, 17, -7, -3, -1], [16, 6, -2, -2, 6], [-15, 19, -13, -13, 4], [4, 10, -2, 10, -2], [5, -13, -5, 1, 0], [23, -4, -12, -12, -3], [-26, -10, 8, -2, -4], [8, 2, 6, 14, -4], [26, -20, 0, 8, 4], [-12, 12, 4, -4, -12], [40, -4, -2, -8, -6], [-20, -3, -7, 5, -5], [-2, -24, 16, 0, 0], [6, -4, 10, -4, 2], [-11, -3, -9, -11, 8], [16, 0, -6, -4, -6], [12, -8, -10, -4, -2], [-6, -16, -8, 12, 2], [-20, 4, -12, 8, 0], [-10, -26, 4, -2, 0], [-32, -20, 8, -8, 8], [-18, -1, -15, 7, -3], [26, 8, -2, 8, -6], [22, -33, -5, 1, -1], [14, -9, 21, -5, -1], [-3, 5, -11, 7, 10], [-12, -14, -8, 14, -2], [0, 16, -4, -2, -6], [-22, -1, -9, -1, 13], [-12, 0, 0, -4, 0], [-7, 1, -19, 7, 2], [5, 14, -4, 4, -5], [-22, -25, 11, -5, 5], [22, -10, 10, 4, 14], [-46, 10, -4, -10, -10], [-38, 6, 10, 8, 2], [-12, -18, 14, -10, -6], [14, -3, -9, 9, 5], [16, 6, -10, -2, -10], [3, -13, 11, 13, 2], [-28, 15, 7, 3, 9], [6, 3, -1, -19, -1], [24, -23, 21, 13, -1], [-4, 20, -8, 8, 2], [6, 14, -2, 4, 8], [8, -24, 4, 12, 0], [17, -19, 7, 5, 8], [14, -13, 17, -1, -5], [-2, 2, -6, -10, 0], [17, -5, -13, 13, 4], [-4, -18, 10, -6, 20], [24, -15, 17, 5, 1], [3, 15, -11, -9, -8], [10, 6, 6, -2, 10], [-5, -8, 10, 2, -1], [-7, -17, 5, -5, -10], [-14, 15, 11, 3, -11], [10, 5, 1, -19, 7], [-2, -20, 10, -8, -8], [28, 2, -8, 6, -4], [14, -6, -22, 4, 2], [-20, -2, -16, 6, 2], [13, 9, -7, -5, -10], [-6, -2, 6, -10, 2], [26, -12, 0, -12, -10], [20, 3, -17, -1, -7], [-18, 6, -2, 16, 8], [-24, -14, 10, -10, 8], [16, -10, 18, 2, 0], [-44, 5, 1, -7, 1], [12, 26, -18, -6, -6], [-47, 4, 6, -2, 1], [-2, 18, -22, 2, 0], [-23, 2, -4, 12, -13], [2, 8, 12, 0, -8], [7, 9, -7, 1, 4], [-42, 0, 4, 0, 8], [6, -4, -6, -12, -4], [30, 6, -18, -2, -2], [32, 2, 2, -10, 8], [7, 13, -7, 13, 4], [8, -16, 0, 4, 16], [-11, -1, -13, 1, 4], [-22, 18, 2, 4, 0], [-14, -6, -6, 10, -2], [12, -24, 10, -8, -12], [-31, -6, -12, 16, -1], [-22, -14, 2, 4, -2], [-20, 29, -11, -15, -5], [-18, -12, -2, -4, 4], [3, -29, -1, -3, 6], [-28, 4, -4, 0, 8], [34, -22, 0, 2, 0], [24, -16, 12, 2, -6], [-43, 20, -2, -6, -7], [-51, 16, -14, -6, 9], [0, 6, -6, 14, 0], [25, 3, 9, 3, -6], [6, 10, 2, 10, 6], [10, 0, -12, 2, 8], [4, -23, -7, 11, -1], [25, -13, -15, 3, 6], [46, -20, -10, -4, 4], [-14, 17, -3, 5, -9], [33, 0, 12, 0, -1], [50, 8, -8, 0, -6], [38, -3, 13, 1, -9], [-8, -12, 0, 14, 0], [14, 16, -6, 4, -12], [-25, 11, 1, 15, 0], [-6, -13, 3, 19, -7], [-20, -30, 10, -6, 10], [-34, 1, 1, 5, 3], [-13, 25, -21, -5, 2], [2, 22, -26, -4, 0], [-18, -14, 4, -2, 6], [-10, 18, 2, 12, -14], [-20, 2, -14, -2, -10], [-3, 4, 8, 0, 3], [-23, -11, 1, -13, 10], [0, -10, 10, -18, 6], [-5, -9, 23, 1, 10], [8, -7, -5, 9, 7], [11, -26, 20, 8, -3], [20, -16, 8, 0, -8], [18, -18, 10, 14, -10], [-2, 20, 0, 4, -6], [-6, -24, 16, -12, -2], [22, -16, -6, 8, 0], [-7, -2, -6, -2, -3], [19, 11, -7, -9, 8], [-24, -4, 4, -16, 8], [-3, 23, -7, -9, -6], [34, 17, -5, 5, -7], [48, -8, 0, -8, 0], [30, -2, 2, -12, -2], [-4, 21, 9, -3, -3], [-9, -29, 11, -3, 14], [22, 17, -9, -7, 7], [-18, 3, 11, 7, -7], [26, -4, 14, 4, -2], [-23, -18, 30, 6, 5], [-9, 10, 6, 22, -1], [-8, 6, 22, 2, -6], [-50, 18, 4, 2, 6], [1, -3, -7, -9, -2], [10, -9, 1, -5, 7], [9, -36, 10, -2, -7], [4, -16, 12, 0, -8], [10, 9, 13, 1, 3], [6, -31, 5, -1, -3], [-24, 24, 4, -4, -20], [14, 17, -11, 5, -5], [-38, 8, -16, -2, 20], [5, 3, -11, 7, 6], [0, -20, 12, 12, 12], [-21, 5, 13, 9, 8], [-10, 14, -2, 2, 12], [32, -7, 9, -7, -15], [20, -10, 10, 6, 24], [-9, 15, -15, 3, -4], [-14, 8, -28, -4, 6], [-28, -10, -2, 22, -8], [27, -6, 18, 2, -9], [24, 21, -19, -9, -1], [35, 13, -7, -13, -4], [-4, -3, -1, 5, -13], [-20, 22, 14, -10, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1911_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1911_2_a_w();".
// 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_1911_2_a_w(:prec:=5)
    chi := MakeCharacter_1911_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_1911_2_a_w();".
// 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_1911_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1911_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-8, 7, 10, -6, -2, 1]>,<5,R![-4, 20, 40, -16, -3, 1]>],Snew);
    return Vf;
end function;