
// Make newform 576.4.a.u in Magma, downloaded from the LMFDB on 01 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_576_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_576_4_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_576_4_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_576_a();"
function MakeCharacter_576_a()
    N := 576;
    order := 1;
    char_gens := [127, 325, 65];
    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_576_a_Hecke( : Kf := Kf)
    return MakeCharacter_576_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]];
    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], [14], [-24], [-28], [74], [-82], [-92], [-8], [-138], [80], [-30], [-282], [-4], [-240], [-130], [596], [218], [436], [-856], [-998], [-32], [-1508], [246], [866], [270], [-1496], [-1692], [-406], [-786], [1744], [652], [-1530], [-516], [1342], [-424], [-262], [2292], [1896], [-2874], [-1188], [3474], [-192], [4802], [1518], [5128], [-1084], [688], [4812], [-2494], [-698], [6320], [-6510], [628], [4862], [-5816], [3526], [-256], [-142], [-8842], [7180], [7374], [-1500], [7608], [-4758], [4374], [7804], [5106], [-4716], [-7302], [4382], [-7224], [1408], [1714], [-884], [-10368], [398], [5098], [-10002], [-9270], [-6516], [2626], [4304], [11794], [-5544], [-3788], [13342], [-4390], [5798], [-14656], [8412], [-14848], [18568], [-14364], [-21660], [17112], [11478], [-13114], [4508], [-22950], [6580], [7046], [8252], [6838], [-23316], [-10558], [1028], [-1202], [3576], [8650], [12656], [3298], [-5370], [16220], [-20360], [-14498], [-21612], [-12184], [-28122], [-5700], [29458], [19810], [-10450], [23300], [14212], [-15978], [8866], [-7760], [13080], [-16934], [7060], [12520], [-9792], [-13166], [23222], [-39934], [-17106], [9956], [-9130], [-11482], [-4612], [-35010], [13688], [11668], [29306], [2664], [-26030], [-44202], [32748], [-45344], [8778], [4142], [-22076], [40376], [26396], [-24368], [-5096], [18494], [-33222], [27846], [41052], [-5706], [-39352], [-33180], [4014], [-20328], [11728], [-50974], [44690], [-59746], [3540], [20058], [-35320], [37978], [-58912], [28790], [7948], [-12690], [232], [5578], [-57616], [-212], [65442], [5862], [-1968], [-50370], [218], [9684], [13114], [20352], [-17150], [-48636], [-26620], [-5674], [31596], [-38074], [22546], [9914], [-36034], [40392], [56614], [29120], [-39374], [45730], [-58428], [11478], [7376], [-52884], [26726], [-66148], [-35022], [15870], [-392], [-13580], [-42424], [22202], [-2176], [-530], [-19944], [42582], [-70526], [792], [49566], [-3712], [23004], [34066], [36534], [40896], [-35544], [26660], [40362], [-51004], [-1456], [-89114], [-53412], [1904], [-28366], [34814], [-19244], [91336], [72060], [-95060], [67432], [-85846], [-2610], [-61000], [32464], [-47316], [-102820], [21872], [109434], [89214], [121256], [91546], [13446], [-69252], [-29774], [11276], [122190], [-31574], [26320], [54540], [41762], [-19590], [105406], [-127532], [-76154], [-124330], [16780], [39502], [32042], [64132], [-78134], [27856], [20434], [-53992], [28596], [115354], [-120614], [-44356], [10048], [-98840], [6232], [-65342], [-11492], [-59120], [46130], [-129474], [536], [107294], [73670], [154908], [-34538], [121092], [-15702], [-96106], [-31312], [160638], [-44172], [17044], [-13414], [19942], [-41824], [-8644], [145452], [124514], [67332], [-57686], [-40936], [54274], [2448], [23454], [-99586], [-460], [-154712], [104762], [64668], [-154464], [-151102], [-56370], [-109756], [35146], [-73418], [-98288], [175654], [21138], [11412], [-162708], [182176], [103662], [3626], [-67338], [-188208], [-132852], [-152388], [-144300], [88250], [-36066], [114554], [-159264], [-157582], [-182922], [-38034], [150984], [-143434], [93260], [96642], [132956], [-24816], [51646], [40980], [-60838], [226502], [-71808], [114514], [188726], [-62816], [-193052], [77326], [153720], [-213950], [31590], [43536], [-135020], [-47788], [-94278], [-44250], [166888], [-111542], [57676], [19068], [-80272], [89150], [-82952], [-111494], [-4260], [45216], [610], [-207474], [163370], [156534], [-44602], [265368], [-83490], [184820], [-173400], [91968], [-106670], [-154516], [189600], [-190718], [22862], [222468], [41892], [270744], [32842], [-116688], [76134], [-186308], [158366], [-25382], [-6306], [189568], [238934], [-118258], [18376], [66794], [190158], [-195452], [-80532], [162866], [-43906], [95188], [44276], [-133062], [-3034], [-204096], [-161560], [-199122], [30584], [185302], [279586], [177968], [-170060], [-10022], [-698], [86300], [11958], [185004], [-87656]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_576_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_576_4_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_576_4_a_u(:prec:=1)
    chi := MakeCharacter_576_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_576_4_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_576_4_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-14, 1]>,<7,R![24, 1]>,<11,R![28, 1]>],Snew);
    return Vf;
end function;