
// Make newform 368.4.a.e in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_368_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_368_4_a_e();".
// 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_368_4_a_e();".
// 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_368_a();"
function MakeCharacter_368_a()
    N := 368;
    order := 1;
    char_gens := [47, 277, 97];
    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_368_a_Hecke(Kf)
    return MakeCharacter_368_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], [9], [-20], [-2], [52], [43], [-50], [74], [23], [-7], [273], [-4], [123], [152], [-75], [86], [444], [262], [-764], [21], [681], [-426], [-902], [-1272], [-342], [-1426], [1190], [1210], [-1680], [1030], [2279], [-987], [-1644], [-2189], [-946], [365], [-108], [1415], [-1756], [2358], [-1073], [2868], [-332], [-2143], [-2739], [-752], [1016], [1120], [-2706], [6140], [6567], [729], [-2912], [-398], [8131], [1978], [-8459], [7240], [-1319], [1770], [-4144], [-6812], [5692], [-5267], [6340], [-8794], [-8225], [-2576], [-596], [-9271], [8141], [2130], [2574], [-4504], [-2740], [-6948], [-1404], [-8641], [-1140], [12529], [-3252], [2206], [-14316], [7828], [16039], [-11747], [-2890], [-13126], [14481], [-5272], [13466], [4526], [8795], [1275], [9533], [13398], [-8031], [-21184], [21706], [5781], [-7809], [-20240], [-7612], [19484], [-6614], [-639], [829], [-20610], [17240], [-8459], [-17840], [2534], [-5610], [11948], [7840], [-2320], [-1864], [-11939], [10503], [-10950], [-3210], [-13517], [-7494], [-17865], [-22364], [7842], [11234], [17568], [35664], [-27914], [-39529], [-10062], [-25644], [37368], [105], [-15464], [35168], [21216], [9506], [-20550], [5161], [7866], [22317], [26196], [5886], [-32394], [-31286], [-2913], [-15451], [20627], [-6966], [-37590], [-27876], [-9471], [-28366], [7210], [-17198], [-51033], [33328], [-20166], [28629], [38146], [-44621], [-5950], [40836], [-26874], [-21472], [6286], [41998], [28739], [-55936], [-27347], [32734], [-39656], [-28085], [-18522], [48955], [-13053], [-4898], [-46858], [55075], [53548], [-59087], [31746], [-41936], [29490], [-20662], [-28902], [-7195], [-43840], [1367], [-4136], [-704], [4090], [52604], [47904], [-42748], [10172], [33452], [-5557], [74580], [3720], [-13843], [71108], [65424], [39457], [30880], [-28411], [6219], [86347], [29607], [-29189], [-90700], [77770], [-38632], [53880], [18173], [4485], [70776], [-83277], [25515], [24140], [-83415], [41442], [33033], [18723], [71792], [-5625], [24526], [7372], [47114], [68470], [69644], [-58414], [-108124], [9498], [-40052], [-18812], [-101623], [4171], [-87184], [-44915], [-87151], [-6185], [97086], [102417], [-23327], [37514], [-298], [7440], [-52506], [-108980], [-43554], [49534], [102090], [94415], [-4720], [12500], [-123319], [69821], [85840], [66900], [59906], [-96586], [28206], [-113562], [21500], [22364], [69218], [79685], [-115787], [-94970], [-89616], [-59274], [99249], [-9639], [50328], [5954], [70632], [53580], [136215], [-107492], [10008], [-152102], [89916], [20028], [39698], [28646], [-47708], [55794], [137642], [30942], [-11694], [125478], [677], [75774], [-130630], [31371], [99507], [-48213], [-39366], [-11478], [96914], [105799], [72952], [151819], [-84], [44312], [169464], [37234], [118313], [-86738], [109626], [50155], [-117434], [-106162], [20183], [-10051], [-9728], [-85241], [-37132], [46414], [-177473], [-63412], [-16824], [11628], [-143812], [-41953], [-159760], [52790], [-41335], [137475], [202231], [61239], [134014], [86863], [5912], [49058], [132539], [-82400], [85987], [-39440], [-4198], [-57358], [-46400], [-39968], [97475], [-130332], [-163422], [-49420], [-44427], [-32879], [-170178], [196058], [-163234], [23874], [56738], [131244], [-150239], [-48016], [170152], [96949], [-132317], [120660], [-56809], [-122139], [-4405], [-78672], [114434], [-7201], [-15756], [57745], [-29910], [27572], [-77037], [120575], [95712], [216578], [259020], [161079], [-100512], [-43998], [123404], [-29817], [-187598], [137011], [-132889], [122006], [5704], [124306], [-122674], [-30562], [235005], [-48316], [8406], [208492], [-198584], [127362], [174274], [-66566], [223303], [-135239], [-263178], [-93154], [-85901], [-182019], [-179710], [-133751], [-15266], [292068], [168967], [24915], [-21576], [75946], [67782], [78213], [93120], [45400], [-32148], [-157568], [60406], [4176], [-118224], [-110244], [-108299], [60431], [-26922], [60026], [-7515], [98881]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_368_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_368_4_a_e();".
// 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_368_4_a_e(:prec:=1)
    chi := MakeCharacter_368_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_368_4_a_e();".
// 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_368_4_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-9, 1]>],Snew);
    return Vf;
end function;