
// Make newform 1850.2.a.be in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1850_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_1850_2_a_be();".
// 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_1850_2_a_be();".
// 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, 13, 0, -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], [8, 5, -8, -1, 1], [13, -8, -16, 1, 2]];
    Rf_basisdens := [1, 1, 1, 3, 3];
    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_1850_a();"
function MakeCharacter_1850_a()
    N := 1850;
    order := 1;
    char_gens := [1777, 1001];
    v := [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_1850_a_Hecke(Kf)
    return MakeCharacter_1850_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 := [[1, 0, 0, 0, 0], [0, -1, 0, 0, 0], [0, 0, 0, 0, 0], [0, -1, -1, -1, 1], [0, -1, 0, 1, -1], [-1, -1, 0, -1, -1], [2, 1, -1, -1, 1], [0, 0, 0, 2, 0], [1, -1, 0, 1, 1], [2, 2, 1, 1, -1], [5, 0, 1, 1, 2], [1, 0, 0, 0, 0], [-3, 2, -1, 2, -2], [-4, -1, -1, 1, 1], [2, 0, 2, 2, -2], [-2, 1, -1, -1, -3], [4, 0, -2, -2, 0], [-3, 2, -1, 1, -2], [3, -1, 0, -2, -1], [2, 0, 0, 0, 4], [1, 1, 0, -4, 3], [2, 1, 0, 4, -2], [2, 4, 0, 0, -2], [4, -2, 2, -2, 0], [2, 1, 3, -1, -1], [4, 2, 2, 0, 2], [-8, 0, -1, 2, 0], [-4, 5, 2, 2, -2], [-2, 3, -3, -1, -1], [6, 1, -1, -1, 3], [2, 2, -2, -6, 2], [-4, 2, 0, 0, 2], [-3, -3, 0, 0, -1], [7, -4, 1, -2, -2], [-10, 2, 0, 6, -6], [8, -4, -2, 0, 2], [0, 5, -1, 5, -1], [-2, 3, 1, -7, 3], [-4, 2, 1, 0, 2], [-4, -1, 1, 3, -5], [-2, -4, -2, 2, 4], [2, -6, -2, 0, -2], [-5, 4, -1, -1, -2], [-12, 2, -2, -2, 2], [4, -2, 2, 0, -2], [2, -4, 0, -4, 2], [4, -3, 2, -1, 1], [-2, -5, 3, 3, 1], [-10, -1, -1, 3, -3], [0, 2, 4, 2, -4], [-2, -1, 2, 4, -8], [-7, 2, 3, 3, -4], [14, 2, -2, -4, 0], [-2, 4, 4, 6, -2], [0, 4, 0, -2, 2], [10, -3, 5, -3, 1], [-2, 2, 6, 8, -6], [12, -4, 0, 0, 2], [-8, 0, -1, 0, -4], [-16, 6, 2, -2, 2], [6, 6, 0, 0, -2], [6, -7, -3, 1, 3], [0, 3, 0, 0, 8], [0, 8, -1, -3, -1], [0, 2, 4, 4, -2], [4, 3, 1, -3, -5], [-10, -8, 4, 6, 0], [19, -3, 2, -6, 3], [-8, -2, -4, -6, 4], [2, 2, 0, -4, 0], [2, -3, -3, -3, 1], [-10, -6, -2, -6, 4], [6, -7, 1, 3, 3], [10, 4, -6, -6, 0], [-11, 3, 6, 1, 1], [8, 0, 0, 0, -4], [6, 2, -1, -13, 5], [26, 2, 0, 0, 0], [-4, -2, -8, -8, 6], [0, -4, -2, 12, -4], [11, -3, -2, -5, 7], [0, -3, -2, 0, 6], [-8, 1, 3, 1, -1], [-2, 9, 0, 2, 2], [4, -4, 3, 1, 7], [-3, -1, -2, 0, 7], [2, 8, 0, -4, -2], [6, 1, 1, -1, -1], [-6, -3, 3, 13, -7], [5, -5, 0, 1, -7], [-8, -1, 1, 1, -11], [-18, -7, 0, 4, 6], [-4, 4, 0, -4, -4], [-1, -3, -2, -1, 3], [12, 4, 6, 6, -6], [12, 10, 1, -4, -6], [-2, -8, 6, 4, 0], [-12, 3, -3, 7, -9], [4, 2, 4, 12, 0], [9, -5, 0, -8, 7], [-2, 1, 5, 9, -5], [0, 6, -3, 2, -6], [-6, 1, 3, 3, 5], [0, -4, 4, -8, 2], [7, 2, 7, 8, -8], [10, 6, 0, 2, -4], [-10, 5, -1, -5, -1], [9, 11, 2, 2, -7], [-2, 4, 0, -4, 10], [16, -5, 4, -1, -1], [11, 11, -2, -3, 1], [-10, 3, -1, 7, 3], [10, -13, 2, -8, 4], [8, 7, -6, -11, 7], [8, 6, 5, -1, -7], [-1, -12, 3, 0, -4], [2, -5, -5, 1, -5], [5, -3, 2, -5, 7], [-5, 13, -2, 9, -3], [-5, 5, -10, -1, -5], [26, 0, 3, -5, 1], [-10, -11, 0, -6, -6], [-14, 2, 0, 2, 0], [-16, 1, 3, -5, 3], [10, -1, -3, -5, 11], [-9, 1, 0, -8, -3], [-1, 4, -5, 1, -4], [20, 2, 2, 2, 4], [-8, 6, -1, 4, -10], [20, 3, 1, -5, 1], [12, 3, 0, 5, -1], [38, 5, -3, -3, 1], [10, 20, -4, 4, 2], [16, -4, 5, -4, -4], [9, -2, 7, -2, -10], [14, -4, 4, 4, -6], [-20, -11, 3, 7, 5], [-6, -6, -6, -6, 0], [3, 7, -4, -9, 11], [20, -6, 0, 4, -8], [-2, -2, -3, 0, -6], [2, 6, 2, -2, 0], [4, 4, -4, 8, -4], [-22, 5, 5, -5, 5], [10, 12, -3, -7, 11], [-10, 0, -6, 10, -2], [12, 2, -1, -14, 14], [20, -3, 3, -7, 1], [26, -4, 0, 0, -6], [-4, -7, -1, 7, -3], [6, 1, 11, 7, 1], [8, 5, -6, -11, 9], [4, -5, 5, 9, -9], [0, -3, -5, -7, 7], [-6, 0, -4, 8, 6], [6, -4, 8, 8, 2], [22, -4, 8, -8, -2], [-20, -5, 0, -9, 3], [-2, -7, 0, -10, 2], [18, -8, -10, -6, 4], [0, -15, 7, -1, 3], [7, -9, 4, -4, -3], [12, -6, -1, -4, 6], [4, -19, 2, -5, 1], [-22, 9, 3, 7, -11], [-14, -5, 1, -5, -1], [4, 8, 3, -11, 3], [-14, -2, -6, 2, -10], [-4, 3, 2, 3, 15], [4, -6, 1, -2, -6], [-6, -6, -8, 10, 2], [6, 0, 5, 1, 3], [-15, -10, -1, -1, 0], [6, 1, 0, -2, 10], [20, 6, 0, -8, 4], [-4, 0, 8, 4, 8], [16, 2, -8, -2, 2], [24, -4, 8, -8, -4], [24, -11, -3, 7, 1], [-26, -8, -2, 0, 0], [12, 0, 4, 0, 12], [10, 14, -2, -14, 4], [4, -12, 3, 8, -4], [20, 13, -1, 5, 1], [-22, -7, -3, -3, 3], [0, 0, 2, 6, 0], [-14, -13, -3, 5, 7], [-8, -3, 5, -7, -3], [-10, -6, 2, 12, -12], [6, 6, -2, -4, -8], [0, 4, -4, 8, -2], [-6, 2, 2, 2, -12], [6, 10, -12, -4, -4], [-4, 4, -6, -2, -10], [-5, -1, 8, 22, -13], [8, -14, 2, -6, -2], [-8, 2, -12, 0, 4], [-10, 0, 8, 4, -4], [-10, -1, -10, -8, 12], [0, 4, -1, 6, 4], [-3, -5, 0, 4, 7], [12, 10, 2, -16, -4], [36, 5, -3, 1, -1], [-1, -2, 1, 2, -10], [19, -6, -5, -8, 8], [-8, 2, 3, -14, 10], [-2, -2, -4, 10, 4], [-19, 1, 0, 14, 1], [10, 0, 4, 2, 2], [22, -5, -3, -1, 11], [2, 3, 5, 1, 7], [-6, 15, 1, 3, 15], [-9, -11, 4, -5, 3], [11, -17, -4, -6, -1], [-3, -4, 1, 9, 10], [-12, 1, -2, 1, -7], [16, 6, 13, 8, -10], [16, -2, -2, 8, -2], [0, 16, -1, -2, -4], [14, -1, -11, -1, -5], [-8, -18, 2, -4, 6], [0, 2, 2, 2, -4], [0, -7, -9, -15, 5], [7, 9, 8, -1, -5], [-43, -1, -2, 0, 3], [-36, -7, 6, 0, 2], [22, 7, 0, -22, 6], [8, 14, -4, 0, 0], [14, -5, -5, 5, 3], [18, 12, -4, -12, 6], [12, -3, 1, -9, 11], [26, -2, 7, 12, -6], [-4, 4, 0, -10, 14], [-26, -4, -5, -4, 12], [19, -3, 2, 4, -7], [-2, -15, 5, 7, -15], [3, 5, 0, 1, 5], [-1, 11, 0, 11, -5], [-16, 0, 8, 4, -8], [9, 3, 8, -6, 5], [-24, -3, -5, 7, 5], [6, 16, 14, 14, -20], [-4, -14, 8, -8, 4], [23, 6, 5, -17, -2], [10, 9, -4, -10, -10], [-4, -8, -5, -3, 7], [-13, -3, -8, 7, -5], [-24, 4, 8, 4, -2], [-15, -20, 7, 6, -14], [-24, -4, -4, 0, 16], [4, -6, -1, 2, 14], [6, 4, -3, -16, 4], [0, 7, 9, -1, -3], [24, -4, -8, -10, 14], [5, -3, 0, 25, -11], [-3, 12, 3, -2, -6], [26, -2, 4, -4, 2], [10, 2, 10, 2, -4], [-4, 9, 9, 7, -17], [-14, -11, 5, -11, 1], [-5, -1, 6, 5, 5], [10, 2, 10, -2, 10], [-24, -12, -4, -4, 2], [-20, -7, 5, -5, 7], [-10, -14, 0, 4, 0], [-5, 14, -1, 4, 20], [-16, 2, -14, -6, 2], [-18, 4, -4, 2, 18], [7, -11, -2, -4, 9], [-35, -3, -4, 5, -3], [21, -5, 8, 4, 7], [0, 3, 9, 5, 11], [-54, -2, 2, 2, -8], [16, 8, -11, -13, 17], [-18, 3, 18, 12, -4], [8, 9, -5, -13, 7], [-8, -13, 4, 8, 0], [-17, -7, -12, -12, 13], [-23, -8, -7, 0, 0], [10, 2, -4, 10, -6], [20, -6, -6, -4, 4], [8, 13, -5, -11, 19], [-4, -9, -5, -11, 19], [-4, 10, 2, -24, 6], [-12, 12, -4, 6, -12], [-18, 16, -2, 16, -6], [2, -3, 11, 13, -21], [2, 5, -1, -7, 13], [7, -11, 0, -22, 11], [10, -2, 4, -2, -4], [22, 5, 3, -11, 1], [-16, -12, -4, -2, 4], [-12, -4, -12, -4, 20], [-54, 5, -1, 5, -11], [14, -9, 13, 7, -11], [-22, 14, 2, -4, -8], [-4, 16, 4, 14, -8], [6, -16, -2, -20, 8], [26, -12, -10, -8, 10], [-44, 9, -10, 6, 6], [46, -7, -5, -11, 19], [-10, -8, -10, -18, 4], [-32, -4, 2, 6, -18], [14, -12, 8, -8, 2], [-2, 2, -10, 0, 4], [-22, 11, 1, 3, -11], [-42, -15, -3, -15, 5], [-4, 17, -2, 0, 6], [6, -2, 6, 16, 2], [-61, 1, 2, 1, 5], [-14, 2, -8, 4, -6], [2, 16, 2, -6, -16], [-20, -7, -2, 1, 1], [-1, 11, 0, 6, 3], [-1, -1, -6, -11, 5], [10, -4, -10, -12, -6], [-22, 3, 1, 15, -13], [22, -5, -4, -8, -6], [26, 21, -4, 2, 2], [12, -12, 4, -8, 0], [-18, 0, -6, -2, -12], [23, -11, 10, 6, -9], [-29, -18, 13, 19, -10], [-8, -1, -15, -5, -13], [-39, 11, 10, -6, 5], [4, -10, -6, -8, -2], [-21, -2, 3, -8, -8], [-18, 11, -5, 7, 5], [-4, 2, -7, 20, -2], [-14, 11, -11, 5, 15], [14, -4, -10, -4, 14], [-4, 8, -7, -28, 12], [29, 12, -11, -3, 6], [-10, -8, 0, 22, -6], [-12, -8, 12, 20, 2], [-21, 1, -2, 16, -3], [16, 14, 6, 10, -8], [24, 2, 6, 26, 2], [-6, -2, 6, 10, 4], [8, -10, -4, 6, 8], [38, 10, 0, -2, -4], [14, 5, -4, -14, -10], [4, -7, -2, 8, -14], [3, -26, 9, -7, -4], [1, -21, -6, 7, -9], [32, 4, 4, -14, -12], [16, -14, -10, -10, 20], [6, 20, 0, 4, -2], [0, 14, 7, 13, -9], [-8, -3, 5, 23, -13], [9, 4, 3, -14, 10], [-27, -5, 2, -2, -15], [-6, 11, -3, -1, 11], [-12, 10, 14, -6, 6], [14, 16, -8, 8, -2], [-22, 1, -8, 2, 10], [-4, -10, 1, -5, -3], [-38, -8, -8, -12, -2], [-19, 1, -4, -8, 3], [28, 4, 7, -6, -8], [33, -17, -6, 3, 3], [14, -20, -7, 0, -8], [4, 6, -7, 0, 2], [-28, -8, 6, 14, 2], [16, -7, 12, 8, 8], [-14, -9, 3, 9, 9], [-9, 9, 2, -11, 21], [12, -12, -1, 10, 4], [-22, 2, 6, 12, -8], [-20, -8, 0, 8, -14], [40, -4, 6, -2, -6], [28, -5, -11, -3, 7], [-6, 12, -2, 0, 24], [8, -2, 3, -27, 3], [-24, 5, 3, 7, -17], [-24, 0, 8, 4, -16], [12, 2, 18, 8, -10], [42, -11, -4, -18, 30], [-22, 16, -2, 20, -10], [-2, 29, -4, 6, -14], [6, -8, 1, 13, -9], [-22, 22, 4, 2, -16], [8, -6, -3, -8, -2], [-35, -3, 0, 8, -9], [-16, 28, 0, 2, 14], [40, -15, -5, -9, 7], [-18, 10, 12, 10, 10], [24, -24, -13, 0, 0], [-18, 15, -1, 15, -11], [56, -2, -1, 0, 10], [-8, -11, -5, -11, -3], [-8, -6, 9, -10, -6], [-10, 20, 6, -2, 8], [20, 17, 1, -17, 3], [18, 8, 2, 4, 2], [-24, -2, -6, -2, 10], [-7, -1, -8, 18, -3], [-4, 7, -2, -1, 7], [4, -19, -14, 3, -3], [2, 2, 20, 2, -2], [-8, -8, -8, -12, -20], [-34, 25, 9, -15, 7], [11, -3, 0, -1, 11], [-4, -2, -6, -8, 2], [32, -3, -10, -8, -14], [22, -18, 10, 0, 6], [-12, 10, 1, 6, 6], [-32, -5, -2, 3, -9], [-12, 15, -14, -2, 22], [-31, 6, -11, 4, -12], [-32, -16, 4, 12, 12], [24, 3, -3, -3, -19], [8, -9, -5, -11, 25], [16, 8, 4, 22, -6], [-24, -23, 3, 15, 7], [14, 28, -8, -12, -4], [40, 4, 6, 0, 18], [-20, -9, -3, -27, 3], [28, -6, 2, -10, -10], [19, -11, 12, -1, 7], [-31, 7, -8, 0, -13], [16, -14, -9, -6, -14], [22, -22, -10, -8, 6], [50, 5, 7, -11, -19], [-8, -30, 6, 10, -2], [0, 19, 13, 11, 7], [-9, 31, 8, -2, -13], [-28, -1, -8, -5, -1], [-42, -10, 11, 16, -10], [0, -23, 3, -7, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1850_2_a_be();".
// 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_1850_2_a_be(:prec:=5)
    chi := MakeCharacter_1850_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_1850_2_a_be();".
// 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_1850_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 13, 0, -9, 0, 1]>,<7,R![400, 240, -44, -32, 1, 1]>],Snew);
    return Vf;
end function;