
// Make newform 1728.4.a.h in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_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_1728_4_a_h();".
// 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_1728_4_a_h();".
// 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_1728_a();"
function MakeCharacter_1728_a()
    N := 1728;
    order := 1;
    char_gens := [703, 325, 1217];
    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_1728_a_Hecke( : Kf := Kf)
    return MakeCharacter_1728_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], [-9], [1], [63], [28], [72], [98], [-126], [126], [259], [-386], [-450], [-34], [54], [693], [180], [280], [-586], [-504], [161], [-440], [999], [882], [-721], [441], [532], [819], [1294], [1134], [1807], [-2205], [1386], [476], [1575], [-449], [-1820], [-1828], [810], [1323], [-315], [2800], [3276], [3221], [-3339], [-3689], [-6022], [952], [5292], [-2198], [5166], [-3402], [1862], [-5472], [5292], [-1638], [1206], [-4319], [2248], [-2016], [-2338], [8514], [6104], [4338], [-8155], [4977], [5678], [2906], [6993], [-7910], [2466], [-7182], [11431], [6616], [-9820], [-1440], [-5985], [11284], [-7308], [6335], [6372], [-3320], [2898], [-4291], [8323], [-3780], [12474], [16679], [17271], [-17387], [3087], [5238], [4384], [-3843], [12566], [5238], [6309], [-5868], [6776], [9388], [15056], [-16569], [14553], [4284], [692], [4886], [-11025], [-14562], [-11466], [10955], [-1232], [-25622], [21042], [6524], [-20663], [-882], [-7252], [-21168], [-14301], [15057], [4690], [-5203], [-6174], [756], [17948], [-34083], [5284], [24516], [12481], [3094], [-376], [-29484], [-11681], [-890], [-32832], [3185], [21222], [12320], [11907], [-14994], [-38878], [-3906], [10207], [8064], [10486], [33696], [15778], [-26136], [23912], [-17892], [-44162], [-8820], [-4654], [-33084], [47258], [-36666], [2995], [-13284], [-2989], [-46485], [11529], [37044], [-12455], [-20169], [-39942], [-54234], [26137], [33460], [-49141], [34578], [11781], [-9758], [-63378], [-50086], [1351], [2718], [42842], [-4725], [-48671], [-2072], [-23057], [23427], [17890], [-9396], [60228], [-8685], [67258], [-31024], [10691], [-32958], [69650], [-14931], [-19270], [-1773], [-9324], [47700], [3239], [83098], [61740], [-59148], [-21357], [-40481], [-12824], [-77203], [-62739], [9270], [10108], [33327], [75726], [70784], [-46999], [-882], [-11411], [-43479], [85752], [-22015], [50560], [44982], [43218], [42210], [28594], [-65324], [56952], [14524], [35019], [-36296], [-77544], [52290], [-33992], [-39060], [-22454], [-69496], [70216], [-16380], [-84322], [19404], [45710], [91098], [-63315], [-33948], [66276], [-40474], [-93023], [86536], [-7056], [-92052], [7651], [-13221], [-30850], [35532], [-10676], [21168], [-25074], [88361], [-39447], [16947], [31564], [-82600], [19845], [-53431], [-9431], [34209], [44002], [-56168], [-61740], [-65212], [30870], [71442], [68564], [35154], [-8792], [33920], [-47635], [-122291], [-32767], [30079], [9135], [4192], [79031], [85500], [-20988], [104371], [17082], [-92330], [-64414], [125874], [77939], [-107289], [98560], [117504], [110754], [-61497], [-113652], [-52407], [-119846], [-82854], [343], [-118683], [-26325], [-79534], [-163702], [21735], [-24383], [-152649], [10898], [-20113], [-1260], [118090], [-57960], [-90872], [51156], [-113211], [5670], [16388], [59976], [32186], [-182565], [-146412], [-51982], [92610], [-85708], [77813], [-177399], [7261], [-17640], [-70273], [-15982], [-188440], [-204120], [-46557], [153694], [-87822], [-149849], [-50328], [13880], [58707], [138430], [58518], [18641], [205765], [-53738], [-129024], [27081], [-34424], [42021], [-116172], [-98714], [-15154], [84978], [-120834], [167552], [-18397], [125370], [-107345], [-86246], [-11898], [-145530], [-169083], [4662], [180810], [109936], [-167904], [-168588], [58149], [201194], [-4402], [-105642], [129907], [190961], [-120897], [-8092], [-116298], [-56259], [-142289], [211636], [211680], [-243810], [-169813], [-89082], [-58114], [5175], [31122], [-216872], [73080], [-72604], [210798], [-177140], [-23156], [193850], [15876], [-99583], [-122598], [274932], [-240814], [76356], [-182674], [202384], [156438], [-100294], [-31437], [-60788], [115794], [27448], [-82728], [141561], [-157493], [-94562], [-164556], [-178948], [-235935], [150815], [193851], [-76293], [-255268], [-111007], [-163611], [-56574], [40600], [50094], [-157878], [116550], [66094], [154476], [111267], [-1267], [-1395], [-185157], [207522], [111944], [202860]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_4_a_h();".
// 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_1728_4_a_h(:prec:=1)
    chi := MakeCharacter_1728_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_1728_4_a_h();".
// 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_1728_4_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![9, 1]>,<7,R![-1, 1]>,<11,R![-63, 1]>],Snew);
    return Vf;
end function;