
// Make newform 504.2.bk.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_504_bk();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_504_bk_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_504_2_bk_a();".
// 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_504_2_bk_a();".
// 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, -12, 96, -340, 624, -654, 431, -189, 48, 1, 0, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2179472, -3761958, 9818739, -15993902, 12502836, -6798430, 2511547, -142271, -97831, -86008, 45996, -3057], [-210768, 523802, -1625148, 1993822, -1557319, 558099, -118672, -69565, 21324, 17481, -5021, -1900], [1209724, -34487760, 63031010, -48405922, 29365260, -18686279, 5335597, -3089444, 1037153, 165954, 12805, -89065], [-1691468, 4532706, -733302, -3514016, 2030017, 58962, -420189, 761375, -163381, -100969, 7066, 17437], [3141260, 1803174, 50546510, -140407242, 167857153, -112773492, 52077825, -12628835, -885875, 6163, 871028, -246051], [7497160, -31852978, 45859296, -57471147, 64482219, -44577244, 21292555, -7300724, 229501, 291913, 319768, -146932], [-27316, 142154, -208368, -212358, 648191, -580379, 323728, -113135, -1316, 6307, 4785, -2236], [2200946, -16682018, 43381212, -27347851, -20414474, 35178775, -23124265, 10909881, -525407, -940304, -272019, 228352], [5254820, -15651176, -11011742, 127509630, -194515274, 151374901, -75835685, 23253862, 354939, -834872, -1135999, 475721], [-105412, 102322, 1295210, -5666228, 9211989, -7455618, 3777819, -1269817, 15779, 59599, 54578, -26339], [1072580, -8500670, 34473298, -80268948, 97308215, -69226294, 32474753, -9329027, -108091, 212817, 499534, -191973]];
    Rf_basisdens := [1, 1375087, 289492, 5500348, 785764, 5500348, 2750174, 41356, 2750174, 5500348, 289492, 785764];
    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_504_bk();"
function MakeCharacter_504_bk()
    N := 504;
    order := 6;
    char_gens := [127, 253, 281, 73];
    v := [3, 3, 6, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_504_bk_Hecke();"
function MakeCharacter_504_bk_Hecke(Kf)
    N := 504;
    order := 6;
    char_gens := [127, 253, 281, 73];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, -1, -1, -1, -1, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, -1, 0, 0, -1, 1], [0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 0, 0], [-1, 0, 0, -1, -2, -2, 0, 1, 1, 0, 2, -1], [0, 1, 0, -1, -1, 0, 0, 1, 1, -1, 0, -1], [-1, -1, 0, -2, -2, 0, 0, 1, 2, 1, 0, 0], [1, -2, 2, 1, -1, 1, -1, -1, 0, -2, 2, 0], [-1, 0, -2, -1, 0, 0, 0, 1, -1, 0, 0, -1], [2, -2, 3, -2, 0, 0, 1, -2, -2, -2, 1, 1], [-2, 0, -4, 1, 2, 1, -1, 2, 0, 0, -4, 0], [-1, -2, 0, -1, 0, 0, 0, 1, 1, 2, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, -1, 0, -1, -1, 0, 0, 1, -2, 2], [-2, -3, -1, 3, -2, -2, -1, 2, 5, -3, 1, -1], [-4, -2, 0, 2, 1, 0, 0, -3, -2, 2, 0, 1], [-4, 0, -2, 3, -4, -3, -3, 4, 6, 0, 2, -2], [0, 2, 0, 2, -1, 0, 0, -5, -2, -2, 0, 1], [-4, 2, -2, 0, -4, -2, -4, 4, 2, 2, 2, -2], [2, -2, 0, 2, 0, 0, 0, 1, -2, 2, 0, 0], [1, 0, 2, 3, -1, 3, -3, -1, 0, 0, 2, 0], [-4, 4, 0, 2, 0, 0, 0, -6, -2, -4, 0, -2], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-1, 2, 0, 1, 0, 0, 0, 5, -1, -2, 0, -7], [4, -2, 7, -2, -1, 0, 1, -4, -2, -2, 3, 2], [-5, 6, 0, -7, -5, -5, -5, 5, -2, 6, 0, 0], [1, 3, 0, 0, -2, 0, 0, 5, 0, -3, 0, -4], [-6, 4, -3, -4, -7, -6, -3, 6, 2, 4, 5, -4], [3, 0, 0, 1, 10, 0, 0, -5, -1, 0, 0, 5], [6, -2, 6, 2, 4, 2, 4, -6, 0, -2, -2, 4], [2, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0], [0, -1, 0, -1, -3, 0, 0, -3, 1, 1, 0, 3], [-4, 4, 0, 2, 0, 0, 0, -6, -2, -4, 0, -2], [-3, 3, -6, -1, 3, -1, 1, 3, 0, 3, -6, 0], [6, -4, 3, 4, 4, 6, 3, -6, -2, -4, 1, 1], [2, 1, 1, 1, 2, 0, 3, -2, 1, 1, -1, 1], [6, -2, 0, 0, 3, 0, 0, 0, 0, 2, 0, 6], [1, 0, 0, 1, 2, 2, 0, -1, -1, 0, -2, 1], [-8, 2, -6, 1, -8, -5, -5, 8, 6, 2, 6, -6], [0, -1, 0, -1, -4, 0, 0, 11, 1, 1, 0, 4], [-6, 10, -6, -8, 0, -6, 0, 6, -2, 10, -6, 0], [-4, 5, -8, 3, 4, 3, -3, 4, 0, 5, -8, 0], [0, 1, 0, 1, -3, 0, 0, -1, -1, -1, 0, 3], [7, -4, 2, -1, 8, 4, 8, -7, -5, -4, -4, 3], [6, -2, 3, -2, 6, 0, 9, -6, -2, -2, -3, 3], [-2, 0, 0, 4, -4, 0, 0, 2, -4, 0, 0, -2], [-4, -8, -4, 2, 0, -4, 0, 4, 6, -8, -4, 0], [18, -1, 0, 1, 12, 0, 0, -3, -1, 1, 0, 12], [3, -1, 4, 1, 3, 1, 1, -3, 0, -1, -4, 4], [8, 3, 0, 0, 1, 0, 0, 6, 0, -3, 0, 2], [11, -4, 10, 3, 8, 4, 8, -11, -1, -4, -4, 7], [-2, 2, 0, -2, 2, 0, 0, -3, 2, -2, 0, 2], [9, 2, 0, 1, 0, 0, 0, 19, -1, -2, 0, -1], [2, 5, 0, 10, 5, 0, 0, -2, -10, -5, 0, 0], [-2, -5, -1, 5, -7, -2, -1, 2, 7, -5, 11, -6], [6, 3, 11, 3, -2, 0, 1, -6, 3, 3, 5, 3], [-9, 6, -8, -7, -9, -5, -5, 9, -2, 6, 8, -8], [2, -11, 1, 11, 0, 2, 1, -2, 9, -11, 3, -1], [-3, 0, 0, -1, -10, 0, 0, 5, 1, 0, 0, -5], [26, -3, 0, 3, 10, 0, 0, 3, -3, 3, 0, 10], [-4, 2, -10, 0, 12, 2, 0, 4, -2, 2, -22, 6], [6, 12, 0, 6, 0, 0, 0, 16, -6, -12, 0, -4], [4, -7, 3, -7, 3, 0, 5, -4, -7, -7, -1, 2], [7, 0, 0, 0, -2, 0, 0, -7, 0, 0, 0, 0], [6, 2, 12, -1, -6, -1, 1, -6, 0, 2, 12, 0], [-6, 12, 0, 0, -5, 0, 0, 4, 0, -12, 0, -10], [-5, 0, 0, -5, -6, 0, 0, 3, 5, 0, 0, -3], [0, 0, 0, 0, -3, 0, 0, -23, 0, 0, 0, 3], [7, 8, 4, 1, 6, 10, 0, -7, -9, 8, -2, 3], [-4, -3, 0, 3, 1, 0, 0, -3, -3, 3, 0, 1], [0, -1, 0, -1, 0, -1, 1, 0, 0, -1, 0, 0], [-4, 1, 1, 1, -7, 0, -9, 4, 1, 1, 5, -2], [2, 2, 4, 1, -2, 1, -1, -2, 0, 2, 4, 0], [3, 0, 0, 9, 2, 0, 0, -1, -9, 0, 0, 1], [8, -1, 2, -5, 8, 7, 7, -8, -12, -1, -2, 2], [22, -12, 11, 12, 21, 22, 11, -22, -10, -12, -9, 10], [-1, -1, -4, 1, -1, 1, 1, 1, 0, -1, 4, -4], [-6, -1, 0, 0, 3, 0, 0, -12, 0, 1, 0, 6], [-12, -3, 0, 3, 7, 0, 0, -13, -3, 3, 0, 7], [-14, 4, 0, 2, 0, 0, 0, -20, -2, -4, 0, -8], [1, 4, 14, -13, -8, -4, -8, -1, -9, 4, 4, 5], [-14, 4, -7, -4, -10, -14, -7, 14, 10, 4, -1, -3], [-1, -14, 0, -7, 0, 0, 0, -11, 7, 14, 0, 9], [-2, -1, -6, 1, -2, 1, 1, 2, 0, -1, 6, -6], [11, -1, 0, 0, 4, 0, 0, 3, 0, 1, 0, 8], [-1, 0, 0, -5, -6, 0, 0, 3, 5, 0, 0, -3], [-15, 6, 0, 0, 0, 0, 0, -15, 0, -6, 0, 0], [-1, 8, 4, -7, -10, -6, 0, 1, -1, 8, 14, -5], [-4, -2, -14, 14, 4, 2, 4, 4, 12, -2, -2, -6], [4, 0, 0, 0, 13, 0, 0, -4, 0, 0, 0, 0], [-12, 5, -5, 5, -13, 0, -19, 12, 5, 5, 7, -6], [-6, 1, -3, -1, -9, -6, -3, 6, 5, 1, 9, -6], [-15, 0, 0, -1, -10, 0, 0, 5, 1, 0, 0, -5], [2, -2, 0, 0, -5, 0, 0, 12, 0, 2, 0, -10], [4, -8, -4, 10, 16, 12, 0, -4, -2, -8, -20, 8], [-11, 3, -4, 3, -11, -9, -9, 11, 12, 3, 4, -4], [-14, -6, 0, 6, 6, 0, 0, -13, -6, 6, 0, 6], [7, -1, 0, -2, 10, 0, 0, -7, 2, 1, 0, 0], [-1, 9, -2, -3, 1, -3, 3, 1, 0, 9, -2, 0], [9, 0, 0, 11, 22, 0, 0, -11, -11, 0, 0, 11], [-10, 0, -5, 0, -17, -10, -5, 10, 10, 0, 19, -12], [16, -2, 0, 2, -9, 0, 0, 17, -2, 2, 0, -9], [-25, 2, 0, 0, -8, 0, 0, -9, 0, -2, 0, -16], [0, -5, 0, -5, -2, 0, 0, 17, 5, 5, 0, 2], [10, -4, 0, 4, -4, 0, 0, 9, -4, 4, 0, -4], [14, 4, 0, 2, 0, 0, 0, 28, -2, -4, 0, 0], [-8, -3, 0, -6, -13, 0, 0, 8, 6, 3, 0, 0], [6, -5, 3, 5, -1, 6, 3, -6, -1, -5, 11, -4], [-11, 6, 0, 3, 0, 0, 0, -5, -3, -6, 0, -17], [16, -3, 14, -3, 16, 9, 9, -16, -12, -3, -14, 14], [10, 2, 5, -2, 11, 10, 5, -10, -12, 2, -7, 6], [-5, 0, 0, -13, 2, 0, 0, -1, 13, 0, 0, 1], [20, 8, 0, -8, -9, 0, 0, 19, 8, -8, 0, -9], [-18, 10, -6, 2, -20, -10, -20, 18, 12, 10, 10, -8], [0, 6, 0, 6, -2, 0, 0, -3, -6, -6, 0, 2], [6, -4, 0, -2, 0, 0, 0, 0, 2, 4, 0, 12], [8, 3, 11, 3, 1, 0, 5, -8, 3, 3, 3, 4], [-4, 16, -8, 3, 4, 3, -3, 4, 0, 16, -8, 0], [20, 0, 0, 6, 16, 0, 0, -8, -6, 0, 0, 8], [-10, 3, -15, 3, 0, 0, -5, 10, 3, 3, -5, -5], [11, 0, 0, -13, 2, 0, 0, -1, 13, 0, 0, 1], [15, -13, 8, 15, 15, 11, 11, -15, 4, -13, -8, 8], [0, -11, 0, -11, 0, 0, 0, -21, 11, 11, 0, 0], [-10, -8, 0, -16, -3, 0, 0, 10, 16, 8, 0, 0], [-20, 8, -16, -4, -16, -8, -16, 20, 4, 8, 8, -12], [-4, -6, -8, -7, 4, -7, 7, 4, 0, -6, -8, 0], [5, -6, 0, 7, 5, 5, 5, -5, 2, -6, 0, 0], [4, -8, -4, 10, 16, 12, 0, -4, -2, -8, -20, 8], [2, 0, 10, 3, 2, -3, -3, -2, 6, 0, -10, 10], [0, 7, 0, 7, -4, 0, 0, -19, -7, -7, 0, 4], [8, -2, 10, 4, 4, 2, 4, -8, 2, -2, -2, 6], [2, -3, 4, -7, -2, -7, 7, -2, 0, -3, 4, 0], [11, -8, -2, -5, 16, 8, 16, -11, -13, -8, -8, 3], [-18, -3, 0, -6, 5, 0, 0, 18, 6, 3, 0, 0], [11, -2, 0, -1, 0, 0, 0, 9, 1, 2, 0, 13], [16, -8, 11, -8, 13, 0, 21, -16, -8, -8, -5, 8], [-16, 8, 0, -8, 9, 0, 0, -17, 8, -8, 0, 9], [1, 8, 4, -5, -6, -2, 0, -1, -3, 8, 10, -3], [0, 12, 0, 12, 2, 0, 0, -3, -12, -12, 0, -2], [-6, 4, 0, 2, 0, 0, 0, 0, -2, -4, 0, -12], [1, -11, 2, -9, -1, -9, 9, -1, 0, -11, 2, 0], [10, -7, 5, 7, 5, 10, 5, -10, -3, -7, 5, 0], [-10, 0, 0, 10, 12, 0, 0, -6, -10, 0, 0, 6], [-4, 12, 1, 12, -7, 0, -9, 4, 12, 12, 5, -2], [-6, 0, 0, -6, -12, -12, 0, 6, 6, 0, 12, -6], [5, 0, 0, 5, 10, 10, 0, -5, -5, 0, -10, 5], [22, 6, 0, -6, -12, 0, 0, 23, 6, -6, 0, -12], [5, -1, 0, -2, -4, 0, 0, -5, 2, 1, 0, 0], [-3, 10, -6, -3, 3, -3, 3, 3, 0, 10, -6, 0], [9, -5, 18, -7, -9, -7, 7, -9, 0, -5, 18, 0], [4, -20, 0, -10, 0, 0, 0, 12, 10, 20, 0, -4], [-4, 0, 0, 2, -24, 0, 0, 12, -2, 0, 0, -12], [20, -3, 10, -9, 20, 15, 15, -20, -24, -3, -10, 10], [0, 10, 0, 10, -7, 0, 0, 7, -10, -10, 0, 7], [-12, 2, -18, 14, -4, -2, -4, 12, 16, 2, 2, -10], [-8, -3, -16, 1, 8, 1, -1, 8, 0, -3, -16, 0], [7, 2, 0, 4, -12, 0, 0, -7, -4, -2, 0, 0], [4, -8, 0, -4, 0, 0, 0, 24, 4, 8, 0, -16], [-14, 7, -21, 7, 0, 0, -7, 14, 7, 7, -7, -7], [0, 14, 0, 0, 7, 0, 0, -14, 0, -14, 0, 14], [25, 0, 0, 11, 22, 0, 0, -11, -11, 0, 0, 11], [0, 2, 6, 4, -4, -2, -4, 0, 6, 2, 2, 2], [0, 10, 0, 20, -3, 0, 0, 0, -20, -10, 0, 0], [0, 19, 0, 19, 11, 0, 0, 17, -19, -19, 0, -11], [-18, 10, -23, 10, -4, 0, -13, 18, 10, 10, -5, -9], [-6, 19, -3, -19, 3, -6, -3, 6, -13, 19, -15, 6], [-2, 3, -15, 3, 12, 0, 11, 2, 3, 3, -13, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_504_bk_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_504_2_bk_a();".
// 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_504_2_bk_a(:prec:=12)
    chi := MakeCharacter_504_bk();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_504_2_bk_a();".
// 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_504_2_bk_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_504_bk();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![441, 0, 1071, 0, 2286, 0, 723, 0, 174, 0, 15, 0, 1]>],Snew);
    return Vf;
end function;