// Make newform 882.4.a.bg in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_882_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_882_4_a_bg();". // 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_882_4_a_bg();". // 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 poly := [-2, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_882_a();" function MakeCharacter_882_a() N := 882; order := 1; char_gens := [785, 199]; v := [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_882_a_Hecke(Kf) return MakeCharacter_882_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 := [[2, 0], [0, 0], [0, 14], [0, 0], [14, 0], [0, 36], [0, -1], [0, -1], [-140, 0], [286, 0], [0, -66], [-38, 0], [0, 89], [-34, 0], [0, -370], [74, 0], [0, -307], [0, 10], [684, 0], [-588, 0], [0, -191], [1220, 0], [0, -299], [0, -437], [0, 1049], [0, -798], [0, -614], [1684, 0], [-818, 0], [540, 0], [1720, 0], [0, 1227], [-828, 0], [0, -301], [-2050, 0], [-472, 0], [0, -1564], [3286, 0], [0, 1054], [0, 1464], [-540, 0], [0, 2676], [-1028, 0], [4592, 0], [-794, 0], [0, 1758], [-2748, 0], [0, -2424], [0, -3741], [0, -1944], [-72, 0], [-4308, 0], [0, -1089], [0, -659], [0, -663], [-7140, 0], [0, -3260], [0, -1672], [4006, 0], [5984, 0], [0, -3485], [0, -1394], [0, -3371], [0, 4792], [0, -4377], [-9826, 0], [-5738, 0], [-2254, 0], [1986, 0], [0, 4788], [0, -4945], [-5944, 0], [0, -596], [-5726, 0], [10330, 0], [0, -710], [-5210, 0], [0, -52], [498, 0], [0, 2373], [0, -10285], [10854, 0], [5364, 0], [0, -4587], [0, 9852], [-5996, 0], [-2622, 0], [11208, 0], [0, -6920], [3952, 0], [0, -12379], [0, -1618], [972, 0], [7404, 0], [-12244, 0], [0, 1708], [0, 4036], [0, -1], [0, -8667], [2050, 0], [14554, 0], [-6954, 0], [0, 1157], [7142, 0], [-20606, 0], [0, -6225], [0, -4599], [0, 16363], [11296, 0], [0, 6171], [0, 13956], [16962, 0], [19034, 0], [0, -13207], [-14716, 0], [-4730, 0], [0, 13475], [0, -6606], [-3774, 0], [21150, 0], [0, -7338], [-1164, 0], [0, -19200], [16596, 0], [0, 7989], [2754, 0], [29434, 0], [0, 12494], [0, 20114], [0, 15798], [20670, 0], [25400, 0], [29180, 0], [-26206, 0], [0, -4853], [0, -6405], [0, 94], [0, -6173], [0, -5316], [-3776, 0], [0, -25617], [16410, 0], [22072, 0], [11628, 0], [0, -21854], [0, -12646], [0, -14676], [0, -31763], [0, 49], [5452, 0], [31106, 0], [0, -4203], [34796, 0], [0, -7046], [-29756, 0], [-21440, 0], [-8288, 0], [0, 32231], [0, 8249], [0, -10], [-14034, 0], [42698, 0], [-48492, 0], [0, -37243], [55380, 0], [0, 35734], [-39712, 0], [0, 1546], [8464, 0], [0, 26428], [-8796, 0], [0, 21736], [-26796, 0], [12088, 0], [0, 37930], [0, -36439], [5364, 0], [5118, 0], [0, 26688], [0, 33302], [12868, 0], [0, 49711], [54942, 0], [0, 49291], [-32224, 0], [0, -19508], [-47778, 0], [0, 12303], [-53044, 0], [0, -35306], [0, -7785], [5436, 0], [-24292, 0], [0, -21880], [-65388, 0], [0, -8929], [49938, 0], [-26870, 0], [0, 39573], [0, -16000], [44274, 0], [0, -48930], [0, -34042], [0, -17533], [0, -24037], [0, -10238], [0, 56074], [78834, 0], [-68588, 0], [0, 3495], [-2966, 0], [0, -39282], [-49496, 0], [0, -27643], [0, 57376], [0, 4085], [2144, 0], [0, 24979], [4248, 0], [-27438, 0], [-64906, 0], [0, -12408], [-48462, 0], [-19684, 0], [0, 7241], [-51578, 0], [0, 15883], [91772, 0], [0, -43480], [-20610, 0], [66854, 0], [0, 53799], [33520, 0], [100110, 0], [0, 34903], [0, -25704], [0, -35055], [35878, 0], [64132, 0], [0, 19964], [-62586, 0], [0, 29973], [0, -36012], [47238, 0], [0, 72275], [0, 54566], [0, 69892], [0, 70969], [-6922, 0], [61432, 0], [91754, 0], [0, -10593], [-32924, 0], [0, -64403], [0, 7222], [-94996, 0], [-8258, 0], [0, 73391], [0, 34034], [0, 53203], [6388, 0], [-95542, 0], [0, 49542], [0, -15022], [0, -1743], [0, 68829], [126030, 0], [0, -117], [-30244, 0], [130322, 0], [0, 6744], [55762, 0], [0, 77865], [42904, 0], [0, 57265], [0, 12076], [83730, 0], [-48718, 0], [31292, 0], [0, -48265], [0, 12212], [43912, 0], [0, 66286], [0, 29034], [0, 479], [-57216, 0], [86878, 0], [-113114, 0], [0, -106598], [0, -47431], [115754, 0], [0, 20447], [50362, 0], [0, -5261], [132950, 0], [0, -23828], [0, -1816], [0, 28328], [0, -4129], [0, -25183], [0, 35805], [-67086, 0], [173256, 0], [4122, 0], [-38146, 0], [-53004, 0], [51234, 0], [0, -70918], [-67620, 0], [-52130, 0], [0, 48424], [-59058, 0], [-2492, 0], [-9602, 0], [-35532, 0], [0, -95725], [0, 57509], [36488, 0], [0, 93181], [-189096, 0], [0, 47855], [34372, 0], [0, 89638], [-21252, 0], [-37512, 0], [0, -37181], [-158950, 0], [0, 50997], [-195176, 0], [-29990, 0], [118602, 0], [-128110, 0], [0, -131138], [0, 41459], [177860, 0], [0, 55931], [-151170, 0], [0, -122015], [0, 37179], [0, -80622], [62210, 0], [-13118, 0], [0, -153212], [177036, 0], [-105542, 0], [-117794, 0], [0, -139152], [151402, 0], [0, -101296], [0, 67146], [0, -122997], [15638, 0], [77234, 0], [0, 24598], [-156982, 0], [0, -107911], [0, 48226], [0, 110551], [-107682, 0], [-44764, 0], [-138398, 0], [0, 88199], [-110352, 0], [119244, 0], [0, 98271], [190954, 0], [0, -160160], [-18908, 0], [-9496, 0], [29172, 0], [0, 121803], [36512, 0], [144030, 0], [0, 67212], [-132966, 0], [0, 86673], [-127668, 0], [0, 109893], [0, -8631], [0, 46917], [0, 157060], [49568, 0], [-134232, 0], [146662, 0], [0, 128513], [0, -44244], [77508, 0], [0, 137742], [112852, 0], [0, 113922], [82130, 0], [0, -173022], [137044, 0], [0, -73761], [59480, 0], [-113412, 0], [0, -55934], [0, 20503], [47988, 0], [145574, 0], [0, 179598], [-166542, 0], [91904, 0], [0, -14597], [0, -780], [0, 56122], [-175802, 0], [238544, 0], [0, 85461], [0, 98709], [0, -128085], [70434, 0], [138558, 0], [245620, 0], [148000, 0], [0, 151416], [-146700, 0], [0, -101810], [0, -52511], [0, 83804], [-187350, 0], [0, -20174], [-80376, 0], [0, 96563], [0, 32145], [0, 104470], [107262, 0], [-175688, 0], [0, 136861], [0, -199926]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_882_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_882_4_a_bg();". // 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_882_4_a_bg(:prec:=2) chi := MakeCharacter_882_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_882_4_a_bg();". // 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_882_4_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_882_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![-392, 0, 1]>,<11,R![-14, 1]>,<13,R![-2592, 0, 1]>],Snew); return Vf; end function;