
// Make newform 2023.4.a.a in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2023_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_2023_4_a_a();".
// 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_2023_4_a_a();".
// 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_2023_a();"
function MakeCharacter_2023_a()
    N := 2023;
    order := 1;
    char_gens := [290, 1737];
    v := [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_2023_a_Hecke( : Kf := Kf)
    return MakeCharacter_2023_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 := [[-1], [2], [-16], [7], [8], [28], [0], [-110], [-48], [110], [-12], [246], [-182], [128], [324], [-162], [810], [488], [244], [768], [702], [-440], [-1302], [730], [-294], [-688], [1388], [-244], [-90], [-1318], [-1776], [1118], [2274], [210], [-2010], [1112], [124], [-2008], [-2884], [-2228], [-820], [-3892], [-5048], [2962], [-3334], [-1860], [4268], [-5432], [2046], [-2980], [-4458], [4440], [-3302], [1582], [2354], [-3872], [-180], [2032], [5426], [842], [3782], [-4312], [2674], [3768], [-2438], [3186], [8672], [-814], [-9344], [-5180], [12178], [440], [9816], [-442], [3960], [6708], [-13350], [1356], [-6222], [5150], [-2310], [1262], [4488], [17038], [-16200], [-8772], [-2130], [10534], [-9268], [-9392], [-10806], [-4940], [5216], [4412], [-19060], [-12768], [-5500], [7338], [-17582], [1618], [-16144], [4654], [10078], [-5930], [19048], [-14366], [-3626], [-1062], [-10200], [25158], [-25664], [19018], [-17334], [-18730], [-6928], [-16302], [-4718], [-21436], [-4458], [-26640], [7432], [-58], [21516], [-18108], [10078], [18762], [-6810], [-4860], [-13636], [2088], [-5160], [28152], [16808], [21674], [7422], [13790], [-6232], [1766], [1204], [7050], [-23282], [-10142], [9192], [46716], [11240], [-700], [37492], [-28894], [-2770], [17688], [33566], [16758], [11468], [50356], [8716], [-7632], [-23080], [-45110], [16674], [-43832], [736], [38138], [26224], [18762], [38394], [-5388], [-25472], [17096], [8930], [-53468], [-33300], [50652], [7008], [-63862], [-8500], [39550], [-15892], [-30442], [-29232], [-44840], [-6616], [-65982], [-338], [-40486], [53808], [-16060], [39866], [-6302], [-55810], [13148], [-7618], [-73068], [-47188], [2172], [-22924], [74622], [-5102], [48462], [-19194], [2088], [-230], [-5012], [-5976], [74170], [57890], [19304], [-64860], [-41112], [-48750], [29062], [-8574], [-49728], [42152], [-28166], [12280], [60338], [21104], [-72578], [-56104], [32538], [66002], [73640], [-65710], [-56888], [20594], [6050], [-52138], [46600], [12224], [12888], [-21302], [45530], [13232], [-107318], [58338], [86784], [26570], [12202], [-55260], [-7168], [-21888], [90162], [94848], [-17690], [-92778], [-40420], [69776], [-68402], [-31760], [-84432], [-78706], [55998], [23344], [-92890], [-34232], [-49540], [-52562], [-67254], [84336], [92654], [-72008], [-25576], [85360], [-120008], [-34214], [24950], [47070], [-77798], [87932], [-99422], [-22612], [-128416], [-21422], [-5440], [73234], [-122512], [-8576], [128190], [108142], [56282], [74408], [48168], [74116], [-89768], [-51074], [75312], [-82018], [34746], [53660], [17010], [-122422], [75686], [-63162], [5838], [143162], [79340], [33952], [49028], [-22430], [63854], [130398], [46794], [40160], [-92248], [42288], [-24474], [-4656], [42000], [-119120], [-2402], [-23928], [113590], [84918], [12648], [-115456], [-173350], [-11970], [-167712], [-82138], [-86910], [-122582], [-154594], [145992], [91688], [59802], [-120818], [142640], [189502], [83056], [-176178], [-72862], [-177834], [-162260], [-79242], [22268], [-14434], [-85510], [-72018], [41398], [82084], [-82202], [-150106], [-116340], [13792], [-109222], [-197920], [-75892], [-202616], [-172872], [129736], [40462], [-53514], [-188858], [198732], [-41210], [-119182], [154380], [184998], [13966], [-123072], [33066], [155678], [-95376], [-85700], [42194], [132838], [11116], [147328], [-53622], [126972], [174690], [-164688], [-115050], [-241768], [-4654], [-118690], [55928], [-63142], [-103290], [22974], [-41732], [159798], [-60664], [-49874], [101010], [-18968], [-190408], [19024], [-6908], [-129276], [-148990], [-196128], [-155100], [164054], [-114808], [134702], [-209540], [-54810], [245628], [12522], [146200], [155378], [-67376], [-8094], [-230820], [-199852], [-57786], [-94498], [275798], [53730], [-17978], [33074], [-209032], [119708], [-2406], [295188], [-125880], [-1244], [-225414], [-89472], [159690], [189656], [-128144], [269990], [-129878], [145584], [-254848], [-197850], [122022], [-205140]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2023_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2023_4_a_a();".
// 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_2023_4_a_a(:prec:=1)
    chi := MakeCharacter_2023_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_2023_4_a_a();".
// 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_2023_4_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2023_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 1]>,<3,R![-2, 1]>],Snew);
    return Vf;
end function;