// Make newform 1800.4.f.v in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1800_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_f_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_1800_4_f_v();". // 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_1800_4_f_v();". // 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 := [1, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [0, 2]]; Rf_basisdens := [1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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_1800_f();" function MakeCharacter_1800_f() N := 1800; order := 2; char_gens := [1351, 901, 1001, 577]; v := [2, 2, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_f_Hecke();" function MakeCharacter_1800_f_Hecke(Kf) N := 1800; order := 2; char_gens := [1351, 901, 1001, 577]; char_values := [[1, 0], [1, 0], [1, 0], [-1, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0], [0, 0], [0, 0], [0, 10], [56, 0], [0, 43], [0, 53], [-4, 0], [0, 68], [-206, 0], [-152, 0], [0, 141], [246, 0], [0, -206], [0, -20], [0, -63], [56, 0], [-2, 0], [0, -194], [672, 0], [0, -585], [-408, 0], [0, 334], [66, 0], [0, -463], [198, 0], [0, 766], [0, 222], [-62, 0], [0, 207], [0, -498], [264, 0], [0, -1139], [-1812, 0], [-1534, 0], [-3016, 0], [0, -907], [0, 922], [0, -1884], [0, 469], [3968, 0], [-3514, 0], [1480, 0], [0, 1387], [0, 1903], [856, 0], [3020, 0], [0, 842], [0, -1002], [5042, 0], [0, -1545], [2136, 0], [98, 0], [5040, 0], [0, 993], [0, 708], [-6670, 0], [48, 0], [0, 3469], [1694, 0], [0, 3182], [0, -1567], [0, -118], [-3776, 0], [0, 3959], [0, -2181], [7980, 0], [0, -4147], [0, 482], [-8670, 0], [0, -1157], [-1896, 0], [0, 742], [0, -6185], [-5620, 0], [0, 2940], [2082, 0], [0, -871], [3270, 0], [6134, 0], [-10392, 0], [-12690, 0], [7408, 0], [0, 2531], [7160, 0], [0, 8550], [8634, 0], [0, 1493], [2406, 0], [0, 7158], [0, 146], [14056, 0], [0, -5602], [-4608, 0], [-2468, 0], [0, 6096], [1714, 0], [18014, 0], [0, 8374], [-14018, 0], [0, -206], [0, -9109], [0, 11762], [23330, 0], [-13124, 0], [0, 5857], [0, 8814], [0, -1401], [-2664, 0], [23962, 0], [0, 5970], [0, -8397], [0, 10353], [-10724, 0], [-5744, 0], [-27906, 0], [0, -10278], [0, 5112], [0, -6491], [-1512, 0], [16710, 0], [0, -3981], [0, -6113], [0, -4374], [-7324, 0], [21934, 0], [10690, 0], [13792, 0], [0, -12002], [0, -4281], [13836, 0], [0, -11112], [11544, 0], [0, -1907], [25662, 0], [-30658, 0], [0, -15447], [0, 10798], [0, 4323], [24954, 0], [40004, 0], [-16570, 0], [0, 2194], [0, -7182], [21170, 0], [10664, 0], [0, 1595], [0, -10407], [18988, 0], [0, 5832], [0, -4123], [-22890, 0], [0, -16774], [0, 16132], [0, 25614], [2144, 0], [-33584, 0], [-3590, 0], [0, -10843], [5174, 0], [0, -17762], [0, -8061], [0, 9594], [38464, 0], [0, 21965], [0, 8664], [-18160, 0], [0, -4551], [-14306, 0], [0, -2855], [-56056, 0], [-12170, 0], [4360, 0], [0, -4225], [-31400, 0], [-31806, 0], [-60428, 0], [-21170, 0], [0, -1334], [62474, 0], [0, 4378], [8952, 0], [0, 12779], [0, 2509], [0, 11932], [33698, 0], [0, 18085], [0, 1154], [-64490, 0], [63080, 0], [0, -9157], [0, -18138], [380, 0], [49318, 0], [0, -23054], [0, 38227], [51666, 0], [0, -3789], [0, 5033], [0, 14776], [-77238, 0], [-63992, 0], [0, -14687], [59694, 0], [23160, 0], [0, 18659], [-20104, 0], [0, -14542], [-40398, 0], [-2356, 0], [0, -23707], [-22322, 0], [0, -15802], [0, -20022], [16672, 0], [-74262, 0], [0, 29390], [37702, 0], [0, -6216], [0, -31219], [-13330, 0], [84680, 0], [-92318, 0], [0, -12642], [0, 33690], [-28990, 0], [0, 3331], [-4768, 0], [0, -50762], [-105272, 0], [0, -27221], [-21852, 0], [-12400, 0], [68030, 0], [0, 13470], [0, 47304], [41070, 0], [0, 26265], [-94040, 0], [-38160, 0], [0, 33386], [52252, 0], [0, 20794], [-42094, 0], [0, 28267], [96800, 0], [0, 47414], [-80240, 0], [-23548, 0], [0, -53264], [0, 47193], [-21250, 0], [0, 24012], [-56810, 0], [0, 4381], [10968, 0], [-35242, 0], [0, -1206], [0, -41433], [0, 58953], [0, 55162], [0, 41510], [-31518, 0], [0, -27361], [0, -41311], [93836, 0], [56602, 0], [-28026, 0], [0, 22854], [0, -49807], [-29530, 0], [0, -4582], [0, 1407], [30040, 0], [0, 15661], [0, 8966], [0, -67478], [-111758, 0], [133370, 0], [-120216, 0], [0, -25200], [31656, 0], [0, 50888], [-63698, 0], [0, 28502], [127064, 0], [0, 14615], [0, -11337], [67704, 0], [-4062, 0], [38726, 0], [0, -60942], [0, 34863], [-125928, 0], [0, 79567], [-31926, 0], [66680, 0], [0, 69801], [-27848, 0], [0, -12606], [0, 26319], [0, -45477], [69280, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1800_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1800_4_f_v();". // 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_1800_4_f_v(:prec:=2) chi := MakeCharacter_1800_f(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_1800_4_f_v();". // 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_1800_4_f_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1800_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![400, 0, 1]>,<11,R![-56, 1]>],Snew); return Vf; end function;