
// Make newform 256.4.a.l in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_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_256_4_a_l();".
// 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_256_4_a_l();".
// 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 := [-7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_256_a();"
function MakeCharacter_256_a()
    N := 256;
    order := 1;
    char_gens := [255, 5];
    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_256_a_Hecke(Kf)
    return MakeCharacter_256_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 := 4;
    raw_aps := [[0, 0], [0, 1], [0, 2], [8, 0], [0, 3], [0, 10], [-14, 0], [0, -7], [152, 0], [0, -30], [224, 0], [0, -46], [70, 0], [0, 83], [336, 0], [0, -6], [0, -101], [0, 18], [0, 33], [72, 0], [294, 0], [-464, 0], [0, -103], [-266, 0], [994, 0], [0, -142], [-1176, 0], [0, 51], [0, -358], [-1710, 0], [-1664, 0], [0, -127], [1062, 0], [0, -509], [0, -150], [-744, 0], [0, 34], [0, -335], [1960, 0], [0, 378], [0, 617], [0, 442], [3904, 0], [3330, 0], [0, 226], [1736, 0], [0, -551], [1568, 0], [0, -239], [0, -974], [838, 0], [6288, 0], [-2926, 0], [0, -1029], [2562, 0], [5896, 0], [0, -1014], [-1680, 0], [0, 298], [2742, 0], [0, -565], [0, 1746], [0, 489], [2744, 0], [-2282, 0], [0, 1810], [0, 803], [6130, 0], [0, -469], [0, -62], [-10206, 0], [3176, 0], [-11760, 0], [0, 2074], [0, -581], [2688, 0], [0, -1982], [0, -1078], [12402, 0], [12278, 0], [0, -1551], [0, 274], [7632, 0], [3794, 0], [1848, 0], [0, 2331], [-3582, 0], [-2714, 0], [0, 1578], [2224, 0], [0, 1945], [17696, 0], [-1304, 0], [0, -3149], [0, 585], [19320, 0], [0, -846], [2982, 0], [0, -381], [0, 2914], [0, -2383], [0, -1350], [0, -1431], [-15594, 0], [0, 3163], [6594, 0], [0, 4387], [14322, 0], [16088, 0], [21238, 0], [-13664, 0], [0, -3854], [3782, 0], [0, -1101], [-2056, 0], [11842, 0], [0, -3071], [-19320, 0], [0, -438], [0, 5241], [0, -4246], [-10078, 0], [0, -3054], [0, 3091], [0, -5527], [0, 466], [0, 5986], [-9296, 0], [-21672, 0], [0, -1790], [0, 1297], [-17432, 0], [-11632, 0], [0, 3162], [-39466, 0], [35266, 0], [0, -3070], [0, 6585], [0, 482], [24390, 0], [0, 1811], [0, 1658], [21688, 0], [0, 3675], [0, -3694], [18760, 0], [0, -5430], [-8778, 0], [0, 1067], [-9312, 0], [0, 26], [-31150, 0], [0, 2377], [-37128, 0], [0, -6653], [-10512, 0], [46104, 0], [-5726, 0], [-1274, 0], [0, -4998], [0, 4521], [-38250, 0], [-4664, 0], [0, -5853], [-4814, 0], [12376, 0], [45344, 0], [0, 4914], [-37998, 0], [0, -550], [0, -1477], [0, 8498], [-20984, 0], [-12826, 0], [53872, 0], [-30282, 0], [0, 9739], [0, 1234], [60200, 0], [0, 11450], [-48704, 0], [0, 10977], [0, -4446], [46886, 0], [-9168, 0], [0, 6858], [0, -5438], [0, 3025], [-24698, 0], [-34944, 0], [3458, 0], [0, -9981], [0, -12487], [0, -11614], [0, -4175], [50694, 0], [27410, 0], [0, -2606], [-58814, 0], [-44856, 0], [0, 6186], [8624, 0], [0, -1142], [8610, 0], [0, 13939], [0, 3762], [8960, 0], [0, 2049], [8934, 0], [0, -9341], [11506, 0], [0, 8106], [-33896, 0], [0, -6837], [15400, 0], [-28602, 0], [-16816, 0], [36274, 0], [-60968, 0], [0, 4802], [0, -2702], [91896, 0], [-73918, 0], [18160, 0], [0, 2105], [0, -3734], [-86730, 0], [72032, 0], [-39256, 0], [0, 3603], [0, -12166], [0, 553], [9280, 0], [46758, 0], [0, -5341], [31920, 0], [1330, 0], [0, -18550], [0, 8491], [107240, 0], [0, -695], [0, 9659], [53256, 0], [0, -20982], [42546, 0], [51352, 0], [96992, 0], [0, 15153], [0, -9005], [27984, 0], [0, -1134], [-117054, 0], [53832, 0], [-14042, 0], [0, 19946], [0, 7193], [0, -3126], [0, 10027], [0, 3954], [58198, 0], [-23168, 0], [0, -18495], [0, -2622], [0, 19618], [-7518, 0], [0, 625], [0, -22790], [71638, 0], [0, 23515], [0, 10850], [0, 17194], [0, 25625], [-54922, 0], [-83424, 0], [-51758, 0], [-68488, 0], [0, 9275], [0, -8654], [-112922, 0], [0, -5255], [34272, 0], [-12248, 0], [110264, 0], [0, -5598], [0, 18531], [-61136, 0], [-77390, 0], [0, -12598], [-126504, 0], [138530, 0], [0, 18970], [0, -14903], [-116842, 0], [0, 21443], [0, 3018], [0, 5026], [-158368, 0], [0, -16326], [0, -17253], [0, -18719], [94822, 0], [0, -25942], [52400, 0], [0, 28633], [0, -27605], [180322, 0], [0, 19315], [0, -6822], [-17544, 0], [0, 21634], [68208, 0], [0, -726], [78626, 0], [0, 19505], [89896, 0], [51590, 0], [0, -18903], [149952, 0], [-27454, 0], [-95758, 0], [0, 10905], [-20554, 0], [0, 25666], [31904, 0], [86662, 0], [-140014, 0], [0, -3135], [0, -23541], [73824, 0], [0, 34706], [0, -12678], [0, -2990], [30016, 0], [0, -33567], [0, -26205], [0, -22165], [0, -6654], [135842, 0], [-73402, 0], [173712, 0], [0, -5542], [-85034, 0], [0, -21246], [-187640, 0], [0, -31262], [0, -2703], [0, 35282], [0, -7405], [156240, 0], [0, 10746], [0, -20255], [30182, 0], [0, -10518], [-184016, 0], [0, -21174], [-88074, 0], [-158816, 0], [0, -8205], [-101358, 0], [-113416, 0], [0, -15550], [-118426, 0], [165104, 0], [0, 11851], [0, -19535], [150278, 0], [0, 44538], [88520, 0], [98870, 0], [0, -19503], [0, 39539], [-140912, 0], [0, 36186], [26488, 0], [0, -5326], [0, -4039], [-102688, 0], [-231070, 0], [133714, 0], [146454, 0], [0, 9362], [207846, 0], [91352, 0], [80418, 0], [0, -19631], [-212632, 0], [-161520, 0], [0, -44134], [0, -2117], [97664, 0], [57986, 0], [0, -35646], [0, -12861], [0, -36487], [-122600, 0], [-119562, 0], [1120, 0], [-77882, 0], [0, 7699], [0, -38598], [0, 5714], [-223998, 0], [168880, 0], [246582, 0], [0, -1998], [180712, 0], [0, -53030], [-129390, 0], [0, 37065], [0, 30273], [221746, 0], [0, -214], [0, 20363], [0, 19953], [2182, 0], [0, -8646], [-138560, 0], [115528, 0], [90034, 0], [-94248, 0], [0, -7614], [0, -21262], [-59376, 0], [0, -46661], [23590, 0], [0, 3658], [0, -52039], [195382, 0], [0, -14069], [-130760, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_4_a_l();".
// 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_256_4_a_l(:prec:=2)
    chi := MakeCharacter_256_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_256_4_a_l();".
// 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_256_4_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-28, 0, 1]>,<5,R![-112, 0, 1]>,<7,R![-8, 1]>],Snew);
    return Vf;
end function;