// Make newform 1575.4.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1575_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_1575_4_a_e();". // 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_1575_4_a_e();". // 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_1575_a();" function MakeCharacter_1575_a() N := 1575; order := 1; char_gens := [1226, 127, 451]; 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_1575_a_Hecke(Kf) return MakeCharacter_1575_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 := [[-1], [0], [0], [7], [8], [-28], [54], [-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_1575_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1575_4_a_e();". // 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_1575_4_a_e(:prec:=1) chi := MakeCharacter_1575_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_1575_4_a_e();". // 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_1575_4_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1575_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![1, 1]>,<11,R![-8, 1]>,<13,R![28, 1]>],Snew); return Vf; end function;