
// Make newform 1008.4.a.f in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_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_1008_4_a_f();".
// 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_1008_4_a_f();".
// 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_1008_a();"
function MakeCharacter_1008_a()
    N := 1008;
    order := 1;
    char_gens := [127, 757, 785, 577];
    v := [1, 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_1008_a_Hecke( : Kf := Kf)
    return MakeCharacter_1008_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], [-6], [-7], [-12], [-82], [30], [-68], [216], [-246], [112], [110], [246], [172], [192], [-558], [540], [110], [-140], [-840], [-550], [208], [516], [1398], [1586], [1242], [-680], [996], [1382], [750], [-176], [-1548], [-378], [2500], [-846], [2536], [-1186], [-2108], [-1944], [1362], [1596], [-1690], [3552], [-2686], [1410], [2968], [1348], [-3872], [5364], [-874], [-378], [1920], [4322], [5292], [5118], [3768], [-3918], [-4880], [-3538], [5430], [6436], [-1350], [-3332], [-4728], [5114], [-7206], [-6260], [-5326], [36], [3134], [-1218], [-10008], [1072], [-274], [-7652], [2160], [1074], [6926], [-1938], [-9574], [-5052], [3422], [2208], [-6814], [-12584], [6996], [-9474], [5786], [-3438], [-9392], [-4956], [-20592], [13432], [-14172], [5956], [16968], [-5214], [1398], [18580], [-18970], [16036], [-8310], [7092], [7158], [-6500], [21794], [9756], [-5586], [-24], [4298], [-8480], [-1906], [-7482], [7348], [-4520], [19806], [5020], [28392], [17562], [4716], [-22762], [4802], [-21558], [3780], [5500], [-10230], [10190], [9408], [33064], [-6322], [20740], [32040], [12832], [-19906], [-10842], [28274], [32346], [-30116], [6594], [43014], [14164], [-34830], [-31016], [9876], [-3154], [36936], [9638], [-10266], [4084], [-192], [19910], [-14802], [32548], [1464], [49564], [8448], [-14600], [21102], [-20806], [-24510], [-44148], [-27114], [10264], [51468], [23790], [26424], [-39488], [30854], [-19246], [-43782], [62676], [42302], [-19272], [-52198], [38080], [62742], [-22964], [25650], [48904], [-21250], [37168], [-1500], [11630], [5766], [-5568], [42042], [-1114], [62476], [39194], [-16944], [38498], [-15516], [-34940], [13218], [-17556], [7782], [12434], [25238], [35838], [-83496], [43338], [53632], [27542], [-2590], [-15924], [3474], [-48080], [-61956], [8742], [53764], [-26926], [61770], [27880], [17820], [42168], [14954], [-47504], [-68706], [50664], [75498], [-80146], [-63272], [-39810], [-55712], [66756], [63614], [-78282], [42000], [45160], [-64812], [-10618], [-103412], [83008], [-104730], [34612], [61392], [-83614], [-70350], [14004], [-1800], [-28884], [-114812], [32392], [7094], [-90354], [-62376], [5536], [12948], [38284], [-88656], [112646], [60702], [49176], [-85414], [-19614], [60828], [-122242], [60916], [36762], [-63430], [-23792], [-3588], [-48442], [34382], [107406], [-80420], [16362], [108822], [-49556], [55314], [11918], [104884], [-18454], [-39056], [13970], [65176], [-15324], [88358], [150362], [-31164], [88032], [328], [-18504], [12902], [118132], [-112320], [-106894], [101586], [37480], [-65154], [-72726], [-140700], [122454], [31692], [-39850], [-110334], [-27584], [45786], [80364], [-63044], [30314], [100026], [-2864], [61548], [144124], [-90238], [120084], [48734], [159384], [134078], [-32976], [81474], [-66690], [-74972], [80376], [127610], [95604], [-100800], [58562], [16206], [-36692], [-72022], [98610], [60256], [70086], [-46366], [-63164], [3652], [71664], [-32622], [34934], [-60150], [-5168], [-98940], [-66116], [53724], [-172618], [-74178], [-176806], [-153632], [-24562], [-146442], [52458], [-148376], [164010], [-22212], [98678], [-45236], [-227376], [-125046], [122236], [-61990], [123018], [22336], [112622], [168918], [227856], [-64596], [113646], [28536], [105374], [-142938], [-86112], [64020], [-40436], [44282], [-131886], [-65960], [-138166], [-220116], [-26588], [-124368], [-38766], [130360], [157382], [-89148], [-82992], [187490], [40734], [91322], [169122], [212262], [113848], [14046], [-22916], [195192], [19888], [13142], [5548], [-106416], [-56062], [-113286], [240900], [10348], [-14760], [-39382], [-160976], [-229866], [-189284], [14274], [74414], [257022], [49552], [118758], [-109398], [-120680], [-187594], [200814], [-260852], [178404], [-31246], [-51390], [-63036], [99700], [51194], [-66894], [11328], [87448], [98046], [109032], [-117174], [-236602], [212400], [-281076], [-72790], [-134718], [27852], [-242826], [-156332], [227352]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_4_a_f();".
// 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_1008_4_a_f(:prec:=1)
    chi := MakeCharacter_1008_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_1008_4_a_f();".
// 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_1008_4_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![6, 1]>,<11,R![12, 1]>],Snew);
    return Vf;
end function;