
// Make newform 720.4.a.bc in Magma, downloaded from the LMFDB on 25 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_720_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_720_4_a_bc();".
// 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_720_4_a_bc();".
// 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_720_a();"
function MakeCharacter_720_a()
    N := 720;
    order := 1;
    char_gens := [271, 181, 641, 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_720_a_Hecke( : Kf := Kf)
    return MakeCharacter_720_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], [5], [30], [50], [-20], [10], [44], [120], [50], [-108], [-40], [-400], [-280], [-280], [610], [50], [-518], [180], [700], [-410], [516], [660], [1500], [-1630], [450], [-770], [660], [1754], [310], [1070], [1950], [-1050], [-1676], [-2050], [-448], [-100], [1900], [1920], [-2550], [3650], [-4342], [-3500], [3350], [-90], [-3664], [268], [3670], [3760], [-1434], [3450], [-4900], [4822], [-4650], [-5130], [-1280], [-3350], [-5512], [4920], [-4500], [6900], [-1530], [-3040], [-5700], [3110], [950], [-2292], [-7730], [-1120], [1186], [-3630], [-1800], [-8490], [100], [8084], [-9480], [10950], [13840], [-9300], [-2854], [1150], [-11162], [-1200], [1510], [-424], [12360], [1300], [-7190], [150], [-2670], [1180], [-14100], [9850], [-2450], [17036], [-20600], [-5750], [15500], [13940], [-20478], [-12040], [-23550], [-6120], [11700], [8188], [11690], [21060], [22910], [-1400], [-11002], [-4630], [24040], [1890], [-19244], [-15892], [12600], [-7260], [7400], [4790], [-1450], [11818], [5550], [-12930], [32580], [-10228], [-8350], [-14954], [29400], [16330], [30800], [9524], [-28600], [8252], [-24920], [-27900], [-11506], [12510], [1100], [4490], [-28600], [-10068], [14250], [-6830], [-8920], [-3534], [-8000], [5160], [-7670], [-25804], [400], [-35100], [18700], [2980], [-35880], [-45240], [33200], [-35356], [25700], [52890], [-38050], [-29640], [-15170], [5470], [15150], [-31190], [-7560], [-32672], [-4740], [38054], [-47230], [-15150], [-338], [40200], [-15570], [49124], [-21300], [-8148], [-17850], [15010], [94], [30970], [-30650], [-44360], [-29050], [16320], [-36050], [660], [-49060], [-22234], [-72100], [-50930], [28760], [-11988], [-4250], [34980], [56990], [-3998], [-11700], [43470], [48640], [-83850], [-24108], [53220], [15886], [-75150], [-14370], [6916], [13640], [45900], [86572], [53150], [44750], [35430], [1640], [54300], [20738], [58790], [96300], [-14600], [-88110], [-28658], [-96536], [24600], [-105070], [49540], [-34066], [-50370], [-39800], [78410], [-16950], [-21060], [56396], [22888], [-31100], [96940], [53040], [-98526], [94430], [18450], [63900], [9840], [16692], [-44350], [-44286], [70530], [117500], [25510], [72650], [9916], [25400], [95200], [-41600], [103320], [81954], [29930], [40950], [-37862], [22780], [107730], [-88330], [-3730], [-127040], [114206], [-77040], [111270], [96836], [110350], [-10600], [-102860], [-72290], [-115522], [12080], [-123010], [-39396], [13710], [-33770], [-35760], [29226], [74202], [-89850], [-50600], [-39208], [-112760], [-35082], [-82560], [32700], [-138710], [-85170], [-3584], [21600], [-155050], [34780], [55890], [6450], [72440], [-60450], [-37052], [25170], [51450], [-53840], [-1170], [-143210], [-72136], [74700], [-172132], [-130270], [113700], [130334], [99900], [-7260], [-50160], [43650], [131100], [75160], [-20600], [-88874], [-55350], [3300], [-4770], [112200], [-49892], [109910], [-81350], [-102670], [-104550], [44182], [9316], [74960], [166320], [-89870], [-7962], [49090], [55924], [-167300], [156748], [141460], [-43594], [-71830], [-12742], [170670], [147820], [-132070], [21450], [-111432], [73690], [67050], [4022], [-207280], [87300], [-75790], [80412], [118410], [143450], [147830], [-17574], [-103710], [-47600], [173250], [122430], [135680], [19660], [85600], [-169920], [158450], [-138620], [110770], [201450], [1350], [-188362], [41850], [90324], [-143080], [126450], [-165048], [-81920], [137450], [150600], [-88150], [131700], [-136430], [-113050], [176190], [74150], [-39430], [-47004], [-161920], [52588], [-27880], [174120], [188400], [38574], [32170], [67850], [-139580], [40400], [163490], [-235544], [250200], [51908], [-91050], [68686], [203070], [-14650], [-123610], [-221350], [14828], [-45880], [-33700], [120780], [-174450], [-26310], [63870], [239180], [195348], [144110], [-163550], [224200], [83870], [192630], [134320], [143150], [5740], [-96720], [-28450], [28430], [18190], [20260], [-8400], [-315788], [266000]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_720_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_720_4_a_bc();".
// 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_720_4_a_bc(:prec:=1)
    chi := MakeCharacter_720_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_720_4_a_bc();".
// 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_720_4_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![-30, 1]>,<11,R![-50, 1]>],Snew);
    return Vf;
end function;