
// Make newform 1600.4.a.bl in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_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_1600_4_a_bl();".
// 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_1600_4_a_bl();".
// 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_1600_a();"
function MakeCharacter_1600_a()
    N := 1600;
    order := 1;
    char_gens := [1151, 901, 577];
    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_1600_a_Hecke(Kf)
    return MakeCharacter_1600_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+1)]];
    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], [4], [0], [16], [60], [86], [-18], [-44], [-48], [186], [176], [254], [186], [-100], [-168], [-498], [252], [58], [-1036], [168], [-506], [272], [948], [-1014], [766], [1314], [448], [1548], [-278], [558], [-344], [-780], [-666], [-884], [114], [-40], [-154], [2180], [-3696], [1302], [4308], [-1550], [48], [-1058], [-3714], [-1768], [4036], [-680], [2388], [3874], [-3162], [5424], [-3886], [5100], [-2178], [6144], [-822], [8480], [-1138], [5706], [-3028], [3390], [-4156], [6552], [1366], [2598], [3292], [-6194], [-10020], [3130], [-4194], [-4104], [-7496], [-5842], [412], [-2568], [-13086], [10454], [-10830], [-8566], [-13884], [-4286], [6336], [8974], [-2968], [-12372], [11394], [358], [7530], [13768], [13380], [-6336], [5008], [-12900], [8116], [4944], [5466], [10074], [-13828], [15226], [-13228], [-8490], [-10284], [1770], [-6068], [-21506], [12108], [-15474], [-2520], [-12790], [-11576], [20126], [27942], [22540], [-5128], [-12798], [-21148], [-16464], [-24234], [22836], [-26318], [-28802], [2526], [-23076], [-7868], [-21510], [-30014], [-816], [9952], [-33946], [-23420], [14592], [9056], [-17554], [-36438], [-9022], [1470], [5252], [12294], [15546], [-19364], [7314], [-11984], [13500], [44602], [35448], [12590], [-24906], [-23204], [-19848], [27542], [-20718], [25172], [12864], [-23092], [-14208], [-26584], [162], [29734], [-17142], [26436], [-27882], [-12656], [-2916], [6894], [45264], [52016], [-13858], [4370], [-36786], [57756], [-6374], [-22008], [-40826], [-37264], [26058], [-12308], [27426], [4000], [41194], [-69176], [24372], [-34306], [42630], [-50904], [-67950], [47078], [32132], [-11782], [-20208], [-71234], [22812], [70180], [-12390], [-59532], [-39642], [47954], [51878], [3390], [-25440], [-55254], [-59488], [6446], [3746], [46380], [40518], [9728], [43620], [60186], [-76196], [-23762], [-82062], [60688], [-51972], [67608], [-51910], [-14120], [2418], [93504], [-68346], [-77246], [-78712], [20802], [72232], [27972], [27058], [-16170], [64800], [-19856], [79164], [-66058], [-56972], [-2512], [-74406], [85916], [94200], [14258], [-77394], [-46020], [-20760], [65172], [27244], [55840], [-15830], [-86130], [-57144], [111880], [33972], [-105044], [9576], [-18970], [58818], [-35088], [23066], [27702], [37332], [-121262], [-105892], [894], [3670], [-30824], [104244], [-96446], [-42010], [73086], [78244], [55626], [89322], [-14164], [-74274], [102058], [-4156], [62422], [15776], [-123026], [31504], [68316], [51610], [-23782], [-60732], [-99144], [91496], [14976], [109378], [-30772], [83904], [-11858], [131310], [-81880], [104802], [79722], [75684], [-29514], [63612], [104438], [-68262], [-60112], [-1554], [82140], [33428], [-139610], [164982], [33488], [-72684], [6028], [49726], [113772], [120010], [128616], [129566], [154248], [20370], [65250], [-108316], [-17520], [-110854], [58164], [-169968], [37822], [-73038], [-141788], [-22154], [48090], [-114728], [173190], [43346], [-113852], [-88180], [140808], [154398], [-21590], [16134], [-23200], [97284], [104956], [159708], [17050], [38430], [59066], [11896], [-208594], [-150954], [204066], [1640], [-30426], [-71388], [76930], [-105652], [-80736], [-150834], [-15500], [-98138], [-80502], [-226136], [71506], [-171306], [127488], [-41028], [40206], [-43632], [20798], [-169254], [10008], [-59556], [-96460], [181798], [-220170], [-126848], [-144310], [115092], [46348], [-71064], [242130], [-136456], [91430], [-168108], [231792], [-46562], [-89646], [-89834], [-233382], [-90906], [-208928], [166494], [-50204], [52512], [-3328], [161390], [-38356], [112296], [159170], [167646], [176844], [186500], [-83304], [-842], [68672], [9786], [74908], [-181806], [-51590], [238110], [-185576], [-110346], [-158910], [152840], [17270], [-79662], [127124], [-2460], [-102002], [2766], [-220884], [245908], [-95258], [120906], [-29376], [-68624], [132270], [81072], [-88230], [-103906], [168792], [204540], [83878], [-75594], [79956], [-95094], [65068], [-135576]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_4_a_bl();".
// 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_1600_4_a_bl(:prec:=1)
    chi := MakeCharacter_1600_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_1600_4_a_bl();".
// 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_1600_4_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-4, 1]>,<7,R![-16, 1]>,<11,R![-60, 1]>,<13,R![-86, 1]>],Snew);
    return Vf;
end function;