
// Make newform 2160.4.a.f in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2160_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_2160_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_2160_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_2160_a();"
function MakeCharacter_2160_a()
    N := 2160;
    order := 1;
    char_gens := [271, 1621, 2081, 1297];
    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_2160_a_Hecke( : Kf := Kf)
    return MakeCharacter_2160_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], [6], [-47], [-5], [131], [56], [3], [157], [-225], [-70], [-140], [-397], [-347], [-4], [748], [-338], [-492], [32], [970], [1257], [-102], [1488], [974], [1335], [-686], [-1098], [-700], [1055], [1646], [-1833], [-1098], [1042], [2941], [-511], [-571], [-713], [1596], [-4134], [1828], [-520], [4826], [1670], [1380], [-4357], [4162], [5956], [4940], [4344], [5202], [1546], [-3659], [1221], [6255], [836], [2231], [4832], [6450], [1050], [1584], [-6594], [4343], [-2124], [-7516], [6880], [4986], [904], [8860], [-4454], [8781], [2928], [-9102], [-8183], [-6136], [5643], [-8991], [-12449], [-8076], [-2833], [-4777], [-6464], [10680], [11566], [1448], [-2376], [14894], [16204], [5082], [10326], [4184], [-15576], [-10220], [-2692], [-5764], [-2437], [5849], [17032], [-4147], [-3942], [13751], [-7944], [-6702], [2760], [-8930], [6944], [-4206], [6571], [-9490], [11861], [518], [15163], [-19011], [7906], [3416], [-4830], [-12549], [8164], [-23768], [21472], [12982], [-6006], [-1164], [26496], [-17110], [-30251], [18820], [31890], [11452], [7094], [3200], [-20831], [15605], [21349], [-3702], [-1393], [6906], [-30493], [-33488], [15304], [40122], [-25098], [43492], [11206], [-23964], [-34606], [-18477], [41342], [-21898], [-18487], [7593], [3038], [16732], [-8031], [38487], [-5120], [-28075], [12856], [-1374], [-8543], [13506], [-21775], [-3854], [12933], [-17521], [-12573], [-8945], [-58179], [-19639], [-22610], [22875], [39310], [-54222], [43380], [3575], [21240], [16128], [-29322], [1492], [-15554], [56944], [-10720], [-24899], [48655], [-38736], [11258], [34791], [40271], [3950], [-6458], [7402], [-23858], [39444], [-41113], [2196], [54238], [44206], [43623], [-35118], [1736], [-43125], [-48348], [-26694], [-52274], [1119], [-5040], [43579], [24778], [-30678], [44332], [-84472], [-54605], [35904], [-88814], [-29226], [40343], [-42034], [57582], [66269], [-80898], [25804], [27091], [12480], [58520], [55814], [-83506], [30819], [36662], [-14116], [51225], [78450], [-4798], [-25211], [-103510], [-11948], [-33684], [-56589], [61396], [83952], [-97884], [-32772], [-88260], [60458], [-7986], [57486], [-55652], [44794], [79015], [-37802], [4759], [-36983], [61364], [124632], [-75654], [-37178], [-83559], [-119684], [-123035], [-38454], [-130354], [26636], [10742], [-15730], [-59445], [24273], [-32080], [5546], [49576], [67051], [41960], [32564], [25889], [-1222], [-137109], [42090], [-91256], [-18912], [-74364], [23349], [20403], [-44035], [118487], [124475], [-31632], [-60597], [131142], [-52172], [89286], [-99425], [136722], [52027], [104168], [60453], [-23628], [-74014], [59526], [77824], [-53682], [-102723], [106108], [10632], [88280], [-56077], [77610], [-62578], [144278], [-145566], [145790], [-49638], [34953], [-54213], [-92826], [-133758], [84019], [-146941], [70495], [-69681], [-118464], [120432], [83346], [98464], [-123376], [-62779], [596], [156591], [-100415], [7780], [163148], [169539], [134312], [-56478], [-117958], [-195563], [-80446], [-20528], [-159166], [62060], [61882], [-76525], [-4134], [98785], [53479], [79242], [64191], [-138796], [134139], [18446], [105332], [-95436], [-218966], [105456], [223572], [47425], [-199354], [-174936], [-10827], [186572], [-47853], [-158427], [-10001], [174706], [81740], [-28788], [-104717], [-159545], [-197372], [43968], [190590], [18179], [-116061], [25506], [150757], [-34038], [79176], [114697], [27439], [-137280], [167348], [8262], [-211886], [113989], [255765], [52664], [96349], [104898], [191713], [23920], [234374], [-89517], [-156867], [27789], [133022], [34564], [228415], [210640], [95684], [65464], [-185616], [130682], [158403], [604], [-223647], [-174076], [-88907], [-231374], [-110992], [-122911], [-130934], [52908], [-77007], [111702], [53850], [-194858], [-184458], [-78046], [-265654], [-227926], [164708], [162834], [-250633], [-19403], [45214], [193263], [137389], [-215782], [213626], [300404], [-218964], [277006], [295402]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2160_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2160_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_2160_4_a_f(:prec:=1)
    chi := MakeCharacter_2160_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_2160_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_2160_4_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![-6, 1]>,<11,R![47, 1]>],Snew);
    return Vf;
end function;