
// Make newform 1617.2.a.ba in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1617_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_1617_2_a_ba();".
// 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_1617_2_a_ba();".
// 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 := [-10, 11, 14, -8, -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], [-4, 0, 1, 0, 0], [0, -6, 0, 1, 0], [5, 0, -7, 0, 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_1617_a();"
function MakeCharacter_1617_a()
    N := 1617;
    order := 1;
    char_gens := [1079, 199, 442];
    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_1617_a_Hecke(Kf)
    return MakeCharacter_1617_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, 0, 0, 0, 0], [2, -1, 1, -1, -1], [0, 1, 0, 0, 0], [0, 0, 1, -1, -1], [3, 0, 1, 0, 1], [0, 0, 0, -1, 2], [1, 0, -1, 0, 0], [4, -2, 0, 0, -1], [-4, 1, 0, -1, -2], [2, -1, -1, -2, -1], [-2, 4, 1, 0, 0], [3, -2, 0, -2, -1], [3, 0, 1, 0, 1], [0, 3, 0, 3, 0], [3, -2, 1, -2, 0], [4, 2, -3, 0, 0], [2, 3, -3, 1, -1], [2, 1, -3, -1, -3], [-2, -2, -2, 0, 2], [0, -2, 0, -2, 2], [-4, -2, 1, 4, 2], [-6, 0, -2, -1, 0], [2, -4, -1, -2, 3], [4, -1, 0, -1, 2], [0, -1, -1, -1, -3], [1, 4, -2, 2, 1], [4, 0, 3, 3, -1], [6, -1, 2, 1, 0], [8, -6, 0, 2, 2], [2, -1, -1, 0, 3], [-2, 1, 2, -2, -2], [-6, 5, 2, 0, 0], [5, 0, -1, -6, -3], [5, -4, -2, 0, 3], [0, -2, 2, 0, -2], [-4, -1, 0, 1, -4], [4, -6, 3, 2, 0], [0, -2, 3, -2, 1], [4, -6, 2, 2, -2], [4, -4, 1, -4, 1], [-6, -1, 0, 2, 0], [2, 0, -2, 0, 2], [-2, -1, -4, 3, 0], [-1, 2, -4, 0, -3], [-6, 4, 2, 0, 4], [6, 2, -1, 2, -5], [-6, 4, 0, -3, -2], [6, -1, 4, -3, -4], [6, -8, 0, 0, 0], [4, -6, 3, -2, -2], [10, 4, -2, 0, -2], [18, -2, -2, -2, 0], [-15, 4, 2, 4, 1], [0, 7, 0, -1, -2], [0, 6, -5, 0, -1], [0, 0, 4, 1, 4], [0, -7, -1, 3, 1], [4, -2, 4, -1, -8], [4, -1, -3, 1, -1], [-5, 4, 3, 4, -3], [18, -4, -2, -2, 1], [-5, 0, -2, 6, 3], [-6, 2, -1, 2, 9], [0, 2, 3, 2, -1], [-4, 1, 4, -3, -4], [4, 5, -4, -3, 0], [9, -6, 6, 4, 1], [16, -1, 2, -1, -2], [5, -2, 5, 2, -2], [-20, 3, 1, 1, 7], [5, 8, -1, -2, -2], [-3, -2, 5, 2, 1], [-4, 2, -4, 2, 6], [2, -2, -2, 6, 5], [-11, 0, -4, -4, 7], [4, 3, 1, -1, -3], [4, -2, 3, -4, 8], [-3, -4, 0, -6, 4], [20, -3, -4, -3, -4], [-6, 8, -4, 0, 1], [14, 7, 0, 4, 0], [-1, 0, 5, 0, 7], [0, -4, 0, 0, 0], [-8, -3, -1, 5, -3], [-18, 1, -2, 2, 0], [8, 4, 3, -2, 3], [7, 4, -9, 2, 1], [-2, -4, -8, 0, 4], [7, -6, 7, 0, 8], [-12, -1, 0, 5, 0], [5, 0, 1, 0, 0], [10, 1, -2, 1, 8], [-10, -12, 6, -2, 0], [-4, -4, -6, -2, 2], [4, 3, 3, 1, 3], [-12, -1, 5, 5, 9], [-6, 2, -2, 1, -2], [6, -5, 6, -4, -4], [-4, 1, 4, 3, -4], [-4, 7, 2, -2, 4], [-6, 2, 1, 3, -3], [4, 2, 5, -2, 7], [18, -2, 4, -4, 2], [-12, 2, -6, -1, 2], [6, 8, -2, 6, 4], [10, -2, 3, 2, -1], [-2, 7, -11, -3, -3], [2, 2, 2, 4, 0], [11, 6, -4, 0, -7], [-17, 4, -3, 4, 4], [5, -12, -8, 2, 3], [-13, -8, 0, 2, 5], [-8, 2, -1, 6, 1], [-4, 0, -6, 0, -6], [-2, -2, -2, 8, 6], [-4, -12, 8, 2, 6], [1, -8, 0, 4, 2], [22, 1, -5, -3, -3], [-4, -7, 4, 0, 10], [4, 8, -5, -2, -4], [10, -8, -1, -4, -7], [-8, -5, 4, -4, -4], [-2, 2, 1, 0, 4], [8, -10, -3, 0, -2], [0, -14, 3, 4, -1], [14, 3, 1, 1, 3], [2, 7, 1, -3, -5], [10, 10, -4, 0, -2], [6, -4, -5, -6, -5], [7, 0, 1, 2, 7], [-10, -5, 2, 9, 6], [2, 6, 3, -2, -5], [10, 2, 0, -2, -8], [2, 1, 4, -2, 12], [-4, -4, 4, -2, 0], [-30, 5, 2, 3, 2], [16, 5, 4, -3, 2], [-10, 5, -4, -1, -4], [21, -6, 0, 2, -1], [0, 0, 4, 0, -2], [4, -2, -2, 2, 9], [8, -4, -6, -2, -2], [-2, -12, 5, -4, 7], [4, 0, 4, 7, 0], [-10, -12, 10, 0, 2], [18, 8, -4, 6, 2], [2, -15, 2, -1, -2], [-20, 16, 2, 4, -2], [14, 10, 1, 4, -3], [-28, 3, 6, -1, -4], [-3, -4, -15, 0, 2], [2, -6, 7, 0, 2], [-2, 4, 5, 1, 11], [5, 4, -4, -14, -5], [-26, 0, -3, -4, 7], [-6, 0, -14, -1, 2], [4, -4, 3, 6, 6], [-24, 11, -2, 5, 0], [14, -12, 1, -5, 11], [-4, 0, 4, -6, -12], [-3, 4, -6, 6, 7], [9, -6, -9, 0, -3], [-30, -4, -7, -4, 3], [14, 0, -1, -4, -13], [-14, 5, 5, -1, 3], [18, 2, 2, -4, -2], [6, 2, 6, -4, -4], [6, 2, -4, 10, 8], [26, -5, -2, 7, 8], [12, -8, 7, -4, -5], [-8, 8, -3, 2, -15], [-19, 8, 2, 6, 11], [4, -1, -1, -6, -5], [21, -10, 6, -8, 7], [-4, 9, -8, -3, -6], [8, 2, 3, 2, 5], [27, 6, 5, 0, -2], [4, -13, 2, 7, 2], [-17, 2, -4, 2, -10], [-24, 12, -4, 3, 12], [0, 16, 3, 10, -4], [-12, 0, 4, 4, 4], [-10, 12, -4, 0, -4], [-10, -4, -3, 2, -3], [-20, 12, 7, 2, -5], [16, -15, 6, 1, 0], [19, -4, -6, -10, -6], [0, -1, -2, 1, -4], [-2, 2, 6, -6, -6], [10, 2, -8, 2, 14], [-12, 6, 0, 4, -8], [-1, -2, 6, -2, 13], [-6, 7, -1, -7, 1], [-1, 8, 11, 2, -1], [16, 9, -6, -9, -8], [10, 8, 2, 2, -6], [-24, -3, 4, -1, -4], [-14, -13, 9, 0, -3], [2, -14, 0, 4, -8], [6, -14, 4, -4, -4], [46, 2, 1, -4, 0], [19, 24, -6, -2, 1], [-11, 12, 0, 0, -5], [20, -4, 4, -10, -6], [-20, 13, 10, -1, -8], [-7, -4, 8, -10, -13], [-18, -12, 5, -4, -1], [-10, -22, 10, -2, 8], [28, 12, 6, -4, 0], [10, -2, -2, 4, -4], [-20, 6, -6, 10, -10], [6, -6, -6, -8, 5], [10, 1, -2, 5, -4], [-6, 1, -8, 3, -8], [-4, 0, 0, 6, 8], [-8, 10, 4, -4, -2], [-6, 14, -5, 10, -7], [-4, -3, 3, 6, -7], [-18, -10, 0, -4, -2], [27, 10, -8, 0, -7], [4, 5, -12, 3, 0], [-16, -6, -4, 6, -6], [-7, -6, -6, 0, -7], [21, 2, -1, -12, -7], [-16, -12, 9, -7, -3], [-32, -18, 6, -4, 2], [-10, -4, -2, 12, 3], [10, 4, -6, -3, -2], [16, -9, 5, 12, 3], [-42, 8, 2, 3, 0], [-12, -10, -11, -2, 5], [2, -1, 10, 1, 4], [-1, 6, -6, 12, 2], [18, -9, 2, 1, -14], [-42, 12, -2, -4, -8], [8, -8, -6, -10, -2], [17, 0, -13, 4, 5], [18, -1, 3, -11, 7], [-8, 2, 9, -4, 5], [-15, 10, -9, 0, 13], [-2, -5, 12, 0, 10], [-18, 11, 0, -3, -4], [11, -8, 13, 2, 0], [-18, 4, 3, -6, -4], [-24, 3, 0, 0, -18], [-8, -3, -4, -1, -4], [-24, -16, 1, 4, -1], [36, -6, 0, 3, -8], [8, 4, 5, 0, -6], [16, 2, -5, -4, 9], [-38, 9, 2, 7, 2], [-14, 6, -10, 4, 10], [0, -14, -5, -4, 5], [-8, 2, -1, 5, -7], [4, 12, -4, -4, -6], [22, 3, 8, -5, -2], [-14, -11, 17, -1, 9], [-22, 0, -6, 12, 8], [38, -11, 1, -9, -5], [-34, 0, -5, 6, -1], [1, 6, -14, -8, -7], [7, -8, -5, 6, -2], [1, -2, 4, 6, 5], [-11, 2, -8, 14, 5], [2, -14, -6, -9, 10], [-20, -7, 4, -9, -6], [14, 6, -2, 2, 5], [16, -12, 6, 8, 0], [-12, 2, -1, 4, 17], [-4, -11, -1, 4, -13], [16, 2, 4, 0, -14], [-25, 2, -2, 2, 13], [10, -6, -1, -2, -4], [0, -10, 7, -6, 13], [10, -18, -5, 2, -1], [-20, 9, -12, -7, 6], [-8, 5, 6, 3, 8], [11, 6, 1, 4, -8], [24, 20, -2, 2, -4], [4, 4, 7, -4, 17], [-6, 7, 8, 4, 4], [-17, 12, -5, 12, -11], [22, -2, -1, -8, -9], [-2, -14, 0, 5, 4], [-21, -4, -13, 0, -2], [14, -7, 12, 2, -4], [24, 18, -8, 4, -10], [9, -14, -7, 6, 5], [-10, 17, -6, 7, -4], [-18, 1, -4, 5, -12], [-14, 16, -3, 4, -9], [-26, 13, -8, 7, 22], [-31, 16, -1, 10, -12], [22, 6, -10, 2, -4], [4, 16, 6, 4, -6], [-4, 12, 3, 7, 1], [16, -19, -14, -1, 8], [10, 0, 8, 13, 2], [30, -12, 8, -15, -10], [-9, 18, -1, -6, -13], [21, 4, 9, 2, -8], [-6, -4, 7, -6, -5], [12, 2, -2, -10, -2], [21, -22, 0, -12, -4], [19, -4, 21, 4, 1], [-24, -1, -7, -3, 11], [-8, 9, -18, 7, 8], [0, -4, 6, -2, -22], [-36, -3, 4, 10, 4], [-17, 2, -8, 4, 17], [48, -13, -2, 3, 2], [6, -7, -11, -11, -5], [41, 0, 5, 0, 9], [-4, 1, 6, -5, 0], [21, -16, 6, 4, -16], [20, 11, -14, -2, -6], [-12, -9, -1, 4, 9], [11, 8, -10, 8, 8], [-22, 3, -6, 8, -4], [23, 12, -19, 2, -4], [-4, 18, -12, -5, 2], [0, 26, 3, 6, -3], [1, -18, 19, 0, -2], [-8, 0, -5, -8, -3], [-6, 12, -18, -4, -10], [-24, -5, 0, -7, -4], [-12, 3, -11, -5, -3], [6, 8, 8, -6, -16], [-8, -1, -3, 0, 13], [-12, 0, -2, -4, -8], [-22, 3, -5, 5, 13], [-20, 4, -10, 6, 2], [10, 6, -11, 6, 12], [32, -21, 4, 0, -8], [-11, 6, -4, -6, -12], [0, -5, -7, -8, 9], [-12, 20, 1, 6, -9], [8, 6, -10, -8, 0], [-8, 6, 14, 3, -8], [-14, 8, -9, 14, 15], [-31, 2, -8, 4, 9], [8, -13, 6, 17, 2], [19, -14, 2, 12, 10], [16, 2, -8, 2, 12], [-12, -6, 12, -2, 22], [9, 18, 4, 8, -7], [36, -11, 2, -4, -2], [1, -4, -3, 2, 17], [-9, -4, 0, 14, 1], [-10, 20, -2, -7, 6], [4, -7, -7, -3, -1], [24, -10, 2, -11, -12], [-4, 12, 4, -8, 0], [20, -10, 4, -2, -12], [30, -6, -2, 1, -4], [12, 8, -4, -2, 16], [-38, 7, 0, 1, -2], [-10, 11, 10, 4, 2], [-14, 16, -2, -2, 24], [-24, -3, 8, -7, -2], [-35, -2, 3, 4, 0], [23, 8, -7, -6, -9], [6, 8, 0, -7, -8], [14, -15, -11, 7, 7], [-32, 21, 2, -5, -4], [9, 16, -15, 6, -5], [-33, 16, 7, 18, 10], [0, 1, -4, -11, -14], [-14, -1, -8, 5, -14], [38, -13, -15, -3, -3], [29, 6, -8, -6, 4], [14, -6, 7, 2, -16], [50, 0, 2, 0, 6], [-32, -5, -2, -11, -8], [22, 3, 26, 1, -4], [-16, 16, -3, 16, 7], [9, -22, 12, -4, -3], [28, -6, 6, -4, 6], [12, 19, -6, -1, -14], [22, 7, -12, 2, 2], [6, -16, 2, -13, -6], [-20, 12, -10, 12, 10], [-11, 6, -9, -4, -6], [-18, -2, 15, -6, -4], [-18, -7, 7, 12, 9], [-20, 2, 3, -14, -4], [26, -12, 4, 2, -13], [-20, -10, 4, -6, -10], [-14, 24, 5, -6, 2], [17, -16, 15, 8, 2], [3, -16, 15, 2, -5], [-22, -1, -2, -1, 14], [24, 7, -8, -2, -6], [3, 12, -16, 0, 3], [-48, 4, -3, 8, -13], [-4, 17, -6, 8, -14], [2, -18, -6, -18, 0], [26, 20, -2, 10, -8], [14, 15, 0, -5, 12], [-3, -6, -12, 4, -5], [8, 14, -6, 19, 4], [14, 13, -15, 3, -9], [46, -8, -1, 0, 8], [-2, 9, -10, -6, -12], [-37, 14, -7, 8, -2], [-16, 10, -7, 16, 2], [16, 2, 0, 6, -6], [8, 19, 10, -8, -6], [-32, 2, -5, -2, 18], [-4, -13, 14, -8, -18], [-2, -22, -1, -10, 9], [18, -12, -16, -2, -8], [40, 10, -10, 8, 8], [12, -10, 3, -18, -7], [32, -6, 12, -20, 4], [26, -23, 6, 5, 2], [12, -16, 6, 10, -8], [-24, 15, 1, 7, -9], [30, -4, 7, 4, -12], [-12, 23, -10, -7, 0], [11, 24, -9, 10, -1], [-51, 10, -12, 6, 1], [-24, 16, -6, 6, -26], [-10, -4, 4, 13, 10], [-2, 10, -2, 4, -22], [-24, 20, 0, -2, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1617_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_2_a_ba();".
// 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_1617_2_a_ba(:prec:=5)
    chi := MakeCharacter_1617_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_1617_2_a_ba();".
// 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_1617_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1617_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-10, 11, 14, -8, -2, 1]>,<5,R![32, 20, -28, -8, 4, 1]>,<13,R![476, 524, 80, -38, -5, 1]>,<17,R![-10, 11, 14, -8, -2, 1]>],Snew);
    return Vf;
end function;