// Make newform 2352.4.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2352_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_2352_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_2352_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_2352_a();" function MakeCharacter_2352_a() N := 2352; order := 1; char_gens := [1471, 1765, 785, 2257]; 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_2352_a_Hecke(Kf) return MakeCharacter_2352_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 := 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 := 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], [-3], [-6], [0], [30], [53], [-84], [97], [-84], [-180], [-179], [-145], [126], [325], [366], [-768], [264], [818], [523], [342], [-43], [1171], [810], [-600], [386], [618], [-1475], [-1884], [413], [-882], [-2483], [-2118], [3012], [37], [-1644], [-1088], [506], [-1844], [-162], [-2724], [1254], [-1807], [-714], [-3709], [-1044], [136], [-1484], [2032], [-6198], [-4591], [4530], [-1530], [5534], [468], [-2490], [-1572], [1806], [6112], [-4231], [-3816], [3997], [4608], [631], [-3894], [-2185], [3504], [-2945], [4277], [-7188], [-9406], [3390], [4812], [7099], [2963], [11899], [-2568], [-10146], [-6229], [-2472], [-7075], [4158], [-6595], [-1518], [8567], [-10640], [-7032], [-14814], [-11251], [-3852], [475], [-5934], [13368], [-6653], [-15444], [-683], [-9882], [4206], [9060], [15679], [-7711], [-4292], [-9858], [13890], [19038], [8053], [-17137], [-18144], [-24702], [2172], [4175], [-2261], [-16318], [-26550], [-19925], [6832], [10212], [-3779], [-16998], [-21750], [10944], [10955], [25103], [-5604], [-10968], [-8405], [468], [-25066], [-11082], [-13481], [24317], [18217], [-19782], [4921], [18098], [-24468], [21719], [-30306], [-27296], [-35100], [44394], [8584], [-9834], [-43856], [-13266], [17453], [35172], [3503], [22848], [13456], [-40710], [2906], [-19188], [17251], [2094], [40267], [-17604], [-3509], [-34638], [-17353], [-46920], [-18354], [35568], [27343], [-51024], [-2226], [-35304], [2341], [29015], [44765], [-40632], [51600], [-34522], [61692], [41285], [-10511], [-45576], [21691], [-58530], [-52208], [-41653], [34153], [8424], [-12907], [34392], [7986], [64500], [-37810], [52783], [11321], [-37878], [-62791], [28422], [26560], [20928], [20208], [-71688], [12035], [-10810], [76914], [63168], [48174], [44731], [55790], [-52534], [-26370], [20994], [59104], [52638], [-16020], [31492], [92735], [-23112], [40636], [20256], [-33576], [-23545], [-26480], [-55524], [31128], [-26514], [-85879], [9289], [41646], [-2360], [54522], [74906], [54768], [-10230], [22003], [-12738], [20729], [-9164], [-18053], [52566], [-53783], [-80610], [-32515], [7590], [-34164], [24750], [116418], [-21287], [29329], [-29386], [69624], [-84792], [104527], [98310], [54304], [-21528], [40553], [-74652], [60402], [33605], [-1704], [-37776], [-78859], [47791], [-81666], [67646], [13903], [-50832], [34577], [114623], [95910], [16849], [-14382], [92790], [-61652], [10446], [45623], [-28844], [3077], [-68165], [91730], [-134180], [65526], [-74845], [20915], [72924], [87816], [-34901], [-89910], [-43021], [44956], [-37290], [-115831], [-74286], [-55727], [81198], [-44292], [-9060], [11226], [110988], [-107209], [-31674], [-61307], [-150696], [1626], [19351], [110870], [-37806], [-29036], [18900], [133381], [-5347], [-53034], [79445], [129768], [-96649], [-152112], [46446], [-88206], [-122264], [69336], [59201], [-114786], [-112668], [-133543], [68562], [53917], [27047], [83358], [-32597], [-48234], [104129], [32671], [-145763], [-42204], [-121158], [72629], [-7086], [47149], [134346], [-130517], [2214], [-89902], [-24468], [79829], [138457], [168659], [7776], [56556], [115297], [-142278], [-56394], [-27379], [-93260], [175806], [-12318], [-125756], [-147502], [55818], [-152120], [88523], [-27090], [6450], [61896], [139548], [-26802], [-72622], [-21678], [200838], [226986], [53137], [-78274], [-40320], [229285], [-76258], [41952], [-71675], [70566], [122670], [188683], [-111619], [21180], [-82836], [180710], [28092], [187265], [-92460], [1332], [91072], [159936], [235684], [-115332], [-58445], [192446], [-126965], [-124542], [-178471], [47262], [-27348], [-51713], [221730], [189890], [-20420], [-34398], [23671], [-44022], [61094], [-185712], [-129296], [202746], [-20622], [65053], [-113194], [234828], [-212507], [203244], [257042], [175980], [-9648], [-201428], [245609], [271218], [12642], [87829], [-252324], [274164], [28932], [-267013], [-146376], [-3330], [-145474], [58614], [-92514], [21888], [92356], [277824]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2352_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_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_2352_4_a_f(:prec:=1) chi := MakeCharacter_2352_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_2352_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_2352_4_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2352_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![6, 1]>,<11,R![-30, 1]>],Snew); return Vf; end function;