// Make newform 2205.4.a.q in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2205_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_2205_4_a_q();". // 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_2205_4_a_q();". // 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_2205_a();" function MakeCharacter_2205_a() N := 2205; order := 1; char_gens := [1226, 442, 1081]; 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_2205_a_Hecke(Kf) return MakeCharacter_2205_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 := [[4], [0], [-5], [0], [-32], [38], [26], [-100], [78], [50], [108], [266], [22], [442], [-514], [-2], [500], [518], [126], [-412], [878], [600], [282], [-150], [-386], [702], [598], [1194], [-550], [-1562], [1846], [-2208], [2334], [700], [-2050], [1852], [2494], [2762], [3126], [-78], [1300], [-1742], [-3772], [-358], [2214], [2600], [-1168], [6478], [646], [-3750], [-1482], [-1400], [-3022], [-1248], [2106], [3638], [-6550], [4388], [546], [6858], [-9282], [4842], [2594], [7332], [-1562], [-1426], [-4008], [8866], [1714], [-1150], [-4398], [-1800], [5874], [-2078], [7900], [-7518], [1950], [-13786], [-6402], [-11150], [-13700], [-5438], [-7692], [1118], [2600], [11958], [17050], [-9494], [-11418], [7962], [6526], [17400], [1166], [-7072], [100], [2602], [11150], [-3638], [2078], [5622], [16486], [-11706], [-25038], [-17550], [10712], [13654], [14166], [17842], [17600], [-27302], [3794], [-13238], [11574], [-8300], [-7508], [27378], [-1842], [-10114], [-10402], [-7100], [7118], [-31278], [-30054], [4518], [-29272], [5798], [8950], [7800], [8554], [-2882], [18700], [-12242], [-31148], [-7694], [-4518], [39550], [22122], [16634], [27586], [-3850], [-10032], [-20562], [10322], [-8846], [25350], [46000], [16998], [-26494], [21500], [-25762], [30546], [32942], [-27118], [-38634], [-1794], [-41732], [29200], [-48650], [11334], [-31178], [-4686], [598], [41726], [24312], [-40946], [42282], [1172], [31614], [2450], [50362], [12900], [-1462], [30108], [8082], [-5200], [29850], [-46248], [24018], [30238], [-35050], [27566], [-8528], [63442], [-67214], [40398], [51350], [36426], [27278], [6350], [-47548], [-2602], [-23162], [8712], [-358], [-77766], [-15158], [20302], [-25038], [22626], [-56678], [66450], [-79692], [75534], [3950], [-56700], [40146], [-24800], [-43682], [26550], [42328], [-6214], [57202], [-49398], [57206], [30000], [-6662], [-20954], [-13518], [64874], [13878], [-45758], [-20000], [-28050], [11722], [-29354], [-38350], [17682], [-44800], [-93886], [77448], [82602], [37100], [68012], [-71942], [34318], [-17034], [89850], [106958], [58500], [15732], [-84522], [-36192], [-47842], [-46050], [-51198], [33800], [-91526], [-57688], [114500], [34918], [60586], [-25198], [101394], [-64350], [-42638], [9900], [-23138], [1954], [-58734], [-73706], [-82238], [55274], [74650], [-19242], [39786], [5900], [-104750], [-28838], [-34678], [-58682], [102778], [14686], [-102602], [-78000], [139254], [43918], [-68766], [105150], [80502], [89232], [75922], [-140108], [-38114], [97218], [-49926], [120588], [-87878], [24054], [120400], [40950], [-127402], [113206], [151038], [30192], [59282], [83850], [33148], [23322], [152700], [-89966], [-96642], [66814], [-96200], [88398], [149632], [-24574], [-20046], [-67550], [-1800], [51402], [92562], [38250], [-151842], [59282], [47834], [74150], [-114300], [71868], [-70362], [-9750], [39208], [90866], [26522], [-36758], [-22002], [98018], [156500], [116198], [79794], [-165362], [85962], [102334], [-57800], [-161558], [101552], [28726], [-20950], [53722], [69658], [62634], [-156758], [34214], [-7950], [-78468], [67318], [81900], [43678], [41086], [113652], [210106], [-116112], [142746], [125558], [-53482], [-177450], [-35358], [69400], [142632], [-28626], [-102722], [-54734], [-59878], [199314], [103900], [42874], [-181478], [49146], [107802], [-34138], [80208], [-128700], [358], [-22050], [98148], [-187094], [-25900], [-75972], [-74642], [111750], [-18226], [32662], [-81282], [-146114], [-13906], [-161700], [-67438], [-212788], [-174746], [234282], [-62034], [206550], [261842], [-16100], [-79406], [136468], [-162238], [92200], [-40950], [49792], [-227922], [-25450], [-160602], [62926], [-249854], [-189950], [-66172], [-171214], [77198], [-201402], [-142500], [-212082], [-113466], [-6242], [-32448], [67106], [-53818], [245600], [-247366], [-170814], [-278798], [-276250], [53174], [-174846], [-250300], [157798], [-214894], [191438], [-750], [-51512], [260600]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2205_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2205_4_a_q();". // 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_2205_4_a_q(:prec:=1) chi := MakeCharacter_2205_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_2205_4_a_q();". // 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_2205_4_a_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2205_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-4, 1]>,<11,R![32, 1]>,<13,R![-38, 1]>,<17,R![-26, 1]>],Snew); return Vf; end function;