
// Make newform 1950.4.a.l in Magma, downloaded from the LMFDB on 12 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_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_1950_4_a_l();".
// 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_1950_4_a_l();".
// 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_1950_a();"
function MakeCharacter_1950_a()
    N := 1950;
    order := 1;
    char_gens := [1301, 1327, 301];
    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_1950_a_Hecke( : Kf := Kf)
    return MakeCharacter_1950_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 := [[2], [-3], [0], [8], [40], [-13], [-130], [-20], [0], [-18], [-184], [74], [-362], [-76], [452], [-382], [464], [358], [700], [-748], [-1058], [-976], [1008], [-386], [614], [518], [-112], [372], [934], [-1914], [-1296], [-892], [-2326], [1932], [882], [-1776], [2410], [-3212], [-1668], [-3598], [1068], [-4786], [-1312], [350], [342], [-3368], [-2004], [5608], [1928], [-3938], [-2562], [7164], [-6182], [-1396], [-6906], [6848], [-6034], [4832], [4082], [3350], [-7796], [-3922], [-5956], [2352], [-8442], [5550], [140], [6174], [2988], [-162], [10754], [3588], [-11272], [10914], [8100], [-6180], [-7522], [-6078], [1830], [12434], [-14188], [8638], [4292], [5982], [256], [-12556], [5574], [-1266], [7554], [6752], [-7924], [-11084], [-4432], [-1140], [1764], [-16976], [9474], [14114], [-20284], [-14362], [20956], [4134], [16228], [2514], [-11612], [-6354], [13240], [1146], [10464], [6650], [6664], [-2134], [714], [29228], [-13536], [17218], [-15044], [-25176], [16034], [25356], [18310], [-24802], [22706], [14792], [-1148], [14870], [-6354], [9288], [21544], [-19990], [532], [25452], [6440], [786], [-1498], [14738], [3822], [11900], [21274], [-27566], [-11244], [13554], [-14384], [2488], [-20858], [23116], [-934], [-12642], [-22796], [76], [46130], [6682], [-47404], [-33672], [14540], [-7840], [47720], [7502], [-22058], [23338], [30488], [-9522], [7616], [51316], [48666], [-17388], [11496], [-48862], [-30326], [-3334], [-960], [16062], [28112], [31758], [-46520], [-54150], [-52228], [9346], [-36336], [3718], [-27760], [-15284], [-9182], [8706], [-21132], [47582], [22970], [10292], [12130], [6732], [36398], [40872], [30308], [4858], [18724], [13038], [-41502], [-63654], [70650], [-51064], [61122], [12888], [-70702], [15410], [60752], [27866], [-65992], [11132], [70590], [-7324], [-34882], [28686], [79496], [-11400], [40108], [25722], [-20160], [40434], [-26084], [53318], [-76194], [100744], [45542], [72328], [30876], [-17570], [56230], [-64344], [83000], [3744], [11082], [-11532], [-70000], [4890], [-73564], [-64044], [-88598], [174], [-1804], [-73992], [89408], [-38252], [-32424], [49246], [-99670], [-98352], [-62496], [-34296], [-19020], [-3672], [-21494], [-87266], [-68876], [-23830], [120834], [-84240], [-72594], [-47780], [92274], [-100490], [-86272], [38388], [14638], [67930], [-17190], [25788], [-26862], [76182], [-17084], [43626], [-117530], [17340], [12782], [61944], [-25426], [37032], [-11472], [-6818], [72794], [-28956], [-91920], [82280], [32440], [-67514], [-66332], [-90336], [-37314], [-99698], [100304], [-73110], [89846], [-56924], [64066], [14544], [164122], [-59610], [-151736], [66066], [60756], [63724], [172870], [-128754], [-86728], [-4236], [133388], [108790], [48540], [-31210], [140748], [-96254], [-162088], [-93542], [-164214], [-20780], [143868], [11194], [-155832], [-4332], [-153722], [-94702], [-191756], [37446], [25014], [185848], [26322], [164482], [-130868], [-16508], [40256], [-45318], [69126], [49186], [-33824], [-164432], [48532], [824], [58094], [-78382], [200482], [-48664], [47162], [-101010], [140754], [105680], [186702], [-85412], [-31906], [50252], [113324], [78530], [76692], [-22930], [120330], [-27344], [-71010], [-168978], [-115812], [77328], [-45834], [-121980], [-61078], [102298], [156648], [-94624], [182684], [235910], [18366], [-129968], [-89654], [126164], [42060], [177700], [-105490], [98456], [-175238], [-147840], [99840], [-16786], [-68766], [-39466], [185898], [-106406], [18104], [45706], [-161988], [-152684], [165936], [264306], [204492], [138136], [-26390], [22014], [219176], [-88636], [121508], [-17018], [-42400], [149578], [163700], [-20198], [-155154], [256030], [67680], [-145838], [-247014], [-68328], [-85734], [12870], [96172], [271568], [-58770], [-196926], [105356], [52788], [108438], [203270], [222140], [173544], [286474], [77184], [-196786], [184090], [81140], [-190820], [131382], [-73162], [-259492], [49182], [-88540], [-123000]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_4_a_l();".
// 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_1950_4_a_l(:prec:=1)
    chi := MakeCharacter_1950_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_1950_4_a_l();".
// 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_1950_4_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![-8, 1]>,<11,R![-40, 1]>],Snew);
    return Vf;
end function;