
// Make newform 1911.2.a.u in Magma, downloaded from the LMFDB on 25 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_u();".
// 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_u();".
// 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 := [-2, 2, 7, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 2, -5, 0, 1], [-3, 0, 6, 0, -1], [3, 2, -6, 0, 1], [-2, -6, 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, 0, 0, 1, 0], [1, 0, 0, 0, 0], [1, 0, -1, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, -1, 1], [1, 0, 0, 0, 0], [3, 1, 0, 0, 0], [1, 0, 1, 1, 0], [0, 1, 0, 0, 2], [-2, 0, -1, 2, 0], [1, 1, 1, -2, -1], [4, 2, -2, 0, 1], [1, -1, 1, 2, -1], [2, 1, 0, -2, -2], [-1, -1, 1, 1, 3], [0, 0, -3, -2, 3], [4, 3, -2, -3, 3], [5, 1, 0, 0, 0], [4, -2, 0, -1, -1], [-6, 1, 2, -1, -1], [3, -1, 1, 2, 1], [1, 1, 3, -2, 1], [1, -4, -1, 2, -1], [5, -1, 1, 0, 0], [2, 2, 2, -4, -3], [-3, -6, -1, 2, 2], [2, 1, 0, 4, 5], [-4, -5, 2, 4, -4], [-3, -3, -1, -2, -2], [-7, 0, 0, -2, 1], [-3, -5, 1, 2, 2], [4, 2, 2, -2, -5], [2, 5, 2, -2, 0], [1, -1, -3, -6, 3], [6, 2, -2, 2, 0], [3, 5, 1, -3, -3], [4, 7, -1, -2, 1], [12, -1, 2, 5, 1], [-5, -4, 1, 3, -2], [1, -4, -2, 2, 0], [-7, 2, 1, 6, -3], [2, -3, -1, 0, -4], [3, -6, -1, -2, 0], [-1, -1, -1, 4, 4], [3, -1, 1, -6, -5], [-11, -4, 3, 0, -1], [6, 3, -2, 2, 5], [6, 0, 0, -3, -4], [-2, 4, 4, -6, -2], [4, 1, -4, -2, 1], [-1, -1, 2, 2, 2], [-3, 1, 3, -1, -6], [13, -2, -3, -2, -2], [9, -4, 1, 0, 0], [-3, 6, -1, -8, 1], [-14, 0, 4, 0, -3], [8, 0, 3, 8, 1], [8, 3, 2, -1, -6], [-5, 0, 8, 2, -5], [-6, -5, 4, 0, -2], [-11, 1, -3, -6, -4], [-13, -1, 1, 6, 1], [-3, -3, 5, 3, -6], [-3, -4, 3, -6, -5], [-8, -1, -4, 4, 4], [-8, 0, -2, -6, -9], [15, 1, 3, 0, 2], [4, 5, 1, 2, -1], [-7, -1, 1, 0, 4], [7, 7, -3, 0, 6], [11, 3, -5, 0, 7], [-5, 1, 3, 0, -6], [-1, -6, -1, 8, -4], [3, 2, -2, -2, 8], [10, 4, 0, -4, 7], [4, -3, -4, 6, 2], [-16, -1, 1, -10, -6], [-9, -2, -5, 0, 6], [-6, 3, 0, 8, 3], [2, 3, 2, 10, 0], [8, 1, 2, 4, 3], [6, -3, -4, -8, 3], [-9, -1, -1, -2, -5], [11, 0, 4, -2, 1], [-5, -6, 5, 4, -5], [-11, 10, -1, -10, 1], [-11, -4, 1, 0, 1], [-5, -4, 3, -4, -10], [-17, -2, 3, -8, -5], [0, 2, 6, 8, -1], [1, -6, 3, 10, -4], [-13, 5, 1, -1, 6], [7, -2, 1, 5, 4], [7, 8, -1, 2, -3], [-12, -2, -2, 2, -6], [-6, 2, -2, 6, -3], [12, -3, 2, 6, 3], [8, -4, -6, -6, -5], [2, 0, 2, 6, -4], [-6, -1, -2, 0, 7], [-9, -5, 3, 10, -6], [-4, -7, 0, 2, 3], [-11, 2, 9, 0, -5], [-14, -9, 5, 2, 3], [5, -2, 1, 2, -6], [7, 6, -3, 8, 1], [6, -2, -4, -9, 3], [9, -6, 3, -4, 2], [-19, -1, 1, -14, -5], [1, -5, -6, 4, -3], [2, 3, 2, -16, -4], [10, -3, 0, 12, 4], [3, 9, -5, -4, 5], [-1, 7, 7, 2, -1], [-6, -5, -2, 6, -8], [-14, 2, -4, -10, -1], [-5, -1, -3, 1, 2], [-21, -3, -3, 10, -3], [-7, -5, 1, 4, 1], [2, -8, 0, -10, -4], [8, 2, 2, -10, -8], [4, 11, 0, 2, -5], [-6, -2, -3, 2, -7], [2, -2, -4, 4, 1], [-7, -9, -7, 5, 0], [2, 0, 0, -4, 12], [-11, -7, -5, 6, -1], [10, 2, 6, -12, -3], [-13, 12, 7, 2, 2], [2, 3, 4, -4, -9], [17, 2, -1, 0, 2], [1, -9, -1, 7, -2], [-10, 5, 8, 6, -5], [1, 3, -6, -2, -5], [4, 10, 8, -14, -3], [14, 1, -4, -2, 6], [-1, -3, -1, 0, 3], [11, -4, -5, 9, 1], [-2, 5, 6, -4, 4], [2, 4, 5, 18, -2], [3, 5, -5, -12, 10], [0, -8, 2, 4, 9], [-22, 6, 0, 4, 9], [-12, -1, -4, 13, -2], [8, 0, -1, -4, 3], [1, 3, -5, 1, -11], [7, -8, -9, 8, 2], [9, -8, -4, 12, 3], [-1, -7, -3, 14, 6], [-14, -9, -2, 0, -5], [5, -2, -1, -14, -5], [16, -5, 4, 8, 6], [3, 3, -1, 0, -2], [17, 4, -3, 4, 0], [22, -1, -14, 12, 4], [1, -8, -7, -4, 0], [25, -2, 5, 8, 2], [24, 3, 4, -6, 3], [8, -3, -11, 8, 7], [3, 4, 11, -6, 0], [13, -2, -3, -9, 4], [-17, -5, 2, 4, 1], [-23, -15, 3, 7, -1], [-2, -1, 2, 4, 7], [-11, -2, -1, 6, 2], [-20, 6, 2, -3, 2], [6, -8, -10, 0, 6], [-2, -9, 5, -4, 3], [-5, -4, 7, 8, 1], [22, 3, 1, -6, 11], [-23, 3, -7, -1, 7], [14, 4, 0, -6, 0], [-12, -7, 0, 8, -2], [6, 0, 4, 10, 10], [4, -2, -2, -12, 7], [-8, -1, 7, 6, -9], [15, 4, 1, -1, -9], [26, 8, -4, -4, 5], [-14, 1, -2, -4, 1], [-13, 1, 2, 10, -2], [2, -5, -6, -17, -1], [8, -7, 0, 6, 9], [11, 16, 2, -6, 3], [-6, 4, 8, 14, 6], [-8, -2, 6, 9, -3], [-17, 5, 7, 10, 0], [10, 0, -1, -8, 14], [-7, -11, -3, 0, 0], [11, 10, -7, -8, 6], [7, 10, 1, -11, -3], [9, -3, 2, 10, 8], [-9, 4, -5, -15, 6], [0, -6, 0, -14, -12], [-10, -8, -4, 8, 8], [-6, 0, 6, 6, -12], [6, -5, 3, -4, -12], [-11, -8, 3, -10, -1], [-21, 4, 5, 16, -3], [-24, -5, 2, 8, -13], [-17, -9, 3, -8, 1], [27, 3, 5, 6, -4], [-15, 0, 5, -6, -12], [-11, 13, -7, -8, 0], [17, 5, -7, 12, 9], [-8, -8, -2, 10, 4], [5, 15, -3, -14, 4], [20, -2, 0, 17, 0], [9, 5, -5, -12, 0], [8, -4, 6, 2, 1], [-14, 0, 2, -26, -6], [-3, -1, -2, 16, -2], [-1, 2, -10, -6, 2], [-26, -8, 14, 2, -5], [7, -7, 7, 11, -13], [-20, -18, 6, 10, -14], [26, 1, -4, -6, -2], [9, 11, 5, -16, -12], [20, 10, -3, -12, 2], [9, 1, 9, 13, -1], [9, -9, 1, 22, 4], [-47, -5, 6, 4, -4], [-9, 12, 1, 0, 16], [-5, -1, 9, 8, 2], [18, -8, 0, -2, 6], [-19, 6, 3, -4, 5], [-17, -7, -2, -6, -1], [12, 18, 6, -12, 0], [-8, 2, -12, -6, -3], [-27, 11, -1, -2, 9], [0, -2, -2, 15, 13], [-23, -11, 12, 16, -19], [19, -1, -5, 10, -6], [0, -13, 2, 11, -8], [-9, 8, -4, -8, 10], [-1, -1, 1, -2, -1], [-40, 1, -4, -8, -9], [-14, -8, 8, -4, -4], [11, 0, 11, 0, 12], [-6, -2, 10, -8, 7], [14, -6, -4, -12, -6], [-27, -4, 1, -5, 7], [-21, -12, 3, -2, 4], [-6, -2, 6, -4, -2], [33, 5, 1, -12, -1], [-4, 4, -4, -2, 7], [23, 3, 1, 16, 0], [-27, 6, 5, -5, -9], [-27, 4, 1, -19, -2], [-11, -6, 1, -5, 0], [13, -3, -17, 6, 8], [48, -5, -4, 0, -5], [28, 0, 0, -6, 3], [-11, 15, 3, 13, 12], [50, 3, 1, -6, 5], [2, 8, -3, -8, -11], [-28, -3, 0, 6, 2], [4, -11, 1, -8, 6], [-9, 4, -7, 1, -8], [0, 0, -11, -14, 8], [-10, -18, 2, 12, -13], [17, 5, -11, -14, 12], [-27, -5, 5, -14, -4], [-23, -2, -1, 8, -7], [-16, 2, -2, -10, -15], [-16, -4, 4, 8, 13], [-29, -8, -7, 14, -6], [43, 16, 3, -14, 0], [4, -6, 0, -18, 0], [-4, 0, -10, -6, 2], [-42, -8, -3, 12, -10], [-4, 0, -2, 6, -11], [7, -6, 1, -11, 7], [-14, 3, 0, 10, -1], [-5, 4, -11, -4, -11], [-7, 0, -9, -10, 6], [-13, 11, -3, -1, 10], [-11, -24, -9, 21, 0], [-18, -4, -10, 6, 15], [4, 18, -4, 2, 18], [-38, 18, 8, 4, 5], [-5, 6, -9, -16, 6], [-12, -11, 4, -10, -6], [-37, 3, 7, -6, -4], [-25, 2, -3, 14, 8], [-17, 6, 15, -5, 2], [-17, 5, 9, -8, 7], [2, -8, -5, 4, -5], [37, 14, -9, 12, 3], [-5, -5, 5, -13, 1], [31, 8, -14, -4, 15], [-9, -15, -6, 4, 1], [9, 0, 11, -6, -9], [29, -2, 1, 14, 9], [2, 2, -6, 7, -6], [-7, -7, 9, 6, 8], [16, 7, -5, -4, 3], [-3, 12, 13, -6, -4], [11, 5, -6, -4, -6], [-11, 1, -11, -22, 6], [-10, 0, -8, 0, -9], [0, -8, 1, -22, -6], [-19, 4, -3, 12, -5], [-7, 11, 13, 4, -2], [-28, 0, -4, -8, 6], [-1, -2, -3, 6, 15], [-16, -2, 6, 10, 6], [-8, -11, 4, -10, -17], [35, -7, 10, 8, -2], [26, -20, -6, 3, 0], [-7, -2, -18, -12, 7], [-3, 6, 11, -10, 4], [-19, -2, -1, -4, -23], [6, 9, 2, 2, -15], [33, 22, 1, -14, -3], [-21, 5, -3, 11, -9], [4, 0, -10, -18, -3], [12, 9, 8, -18, -18], [25, -7, -9, 7, 12], [-21, -21, 7, 2, 8], [20, 9, -5, -12, -9], [0, -5, -16, 4, 15], [-4, -5, 8, -2, -12], [11, 4, -6, -12, 5], [-1, 11, -9, -5, 4], [48, 7, 2, -2, -6], [22, 6, 5, -28, 3], [20, -3, 8, 0, 19], [6, 9, 8, -29, -8], [-32, 7, 6, 2, -4], [-23, -5, -4, 0, -4], [3, 2, -3, -20, -6], [11, -15, -5, 22, 2], [-18, -5, 2, 4, -7], [-18, -11, 2, -1, -10], [36, 14, -12, 7, 7], [-22, -7, -12, 13, -14], [21, -10, 1, 24, 8], [29, 14, 9, -28, -6], [23, 11, -1, 16, 4], [-33, 1, 5, 10, 5], [-23, 25, -3, -16, 1], [-14, 22, 1, -16, 5], [-1, 6, 13, -10, -8], [2, 19, 4, -4, 6], [9, -3, -3, -8, 6], [10, 3, -8, -6, -19], [19, -24, 4, 32, 3], [9, 10, -1, -3, 2], [15, -7, -7, -3, 1], [11, 0, -11, 6, 20], [11, -9, -21, 2, 5], [33, -6, -15, -14, 8], [19, 8, -7, 16, 4], [9, 6, 13, -14, -2], [-5, -13, 2, 6, -1], [-37, -1, 11, -10, -10], [-11, 0, 9, 11, 8], [42, -8, -8, 3, 5], [-13, 3, 6, 4, 17], [-15, 19, 3, -11, -3], [12, 1, 4, 8, -1], [0, -6, -5, 0, 1], [-13, -6, -9, -6, 18], [13, 11, 1, -3, -14], [-11, -8, -5, 26, 4], [1, -5, -8, 8, 13], [-33, 3, -5, -30, -3], [-24, -10, 12, 17, -22], [-24, 2, 3, -18, -19], [1, 6, 5, 12, 15], [20, 5, 6, 14, -3], [-37, -1, -3, 5, -10], [8, 14, 2, -30, 3], [-8, 26, 4, 2, 7], [-21, 25, -1, -14, 8], [11, 0, -3, -7, 13], [5, -9, -13, -6, -6], [-8, -18, 12, 16, -14], [24, -8, -18, 18, 7], [18, 11, -11, -6, -3], [38, 0, -10, 4, -5], [-61, -3, 5, 10, -9], [-10, -2, -8, -3, -10], [36, -5, -4, -10, -16], [34, 6, 0, -1, 5], [-10, 4, -4, -12, 19], [-6, -9, 4, -3, -7], [-32, -10, -9, -2, 0], [36, -13, 8, 23, 7], [34, 7, -2, 14, 5], [15, -5, -15, 18, 10], [47, 18, -13, 4, 25], [-14, 0, 8, 8, -7], [3, 3, -17, -16, 12], [-28, -9, 14, 5, -17], [2, -3, 13, 2, -12], [-6, -5, 6, -5, -3], [17, 13, 2, 4, 24], [8, 6, -12, 2, -1], [26, 12, 10, -24, 5], [-1, -7, 9, 10, 15], [25, 1, -15, -14, 0], [-21, -17, -1, 11, 10], [14, -2, -10, -10, -5], [12, 5, -2, 28, -2], [13, -3, -1, 4, 7], [10, 4, 0, 16, 20], [-29, -12, -5, 16, -23], [-27, 13, -7, -3, -1], [9, 12, 9, 23, -11], [26, -5, 5, 6, 6], [-9, 1, 7, -10, -8], [-19, -10, -5, 22, 10], [-57, -19, 17, -16, -5], [40, 10, -1, -16, 9], [-13, 7, 4, -16, -11], [34, 9, -2, 4, 1], [32, -4, 10, -4, 24], [18, 15, 2, -30, 4], [-2, -9, -12, 2, -7], [4, -19, -8, 8, 12], [-18, -11, 0, 4, 16], [8, -11, 4, 28, 11], [-39, 19, -7, -20, 3], [6, 5, 6, 8, -19], [-27, 9, 15, -16, -9], [-8, -1, 12, 0, -5], [-8, -6, -18, 8, 14], [62, -4, 0, 6, -2], [27, 7, -5, 0, 15]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1911_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 13, -1, -8, 0, 1]>,<5,R![-144, 48, 47, -16, -3, 1]>],Snew);
    return Vf;
end function;