// Make newform 1350.4.c.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1350_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1350_c_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_1350_4_c_g();". // 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_1350_4_c_g();". // 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; 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_1350_c();" function MakeCharacter_1350_c() N := 1350; order := 2; char_gens := [1001, 1027]; v := [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_1350_c_Hecke();" function MakeCharacter_1350_c_Hecke(Kf) N := 1350; order := 2; char_gens := [1001, 1027]; char_values := [[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, -2], [0, 0], [0, 0], [0, 8], [-18, 0], [0, -8], [0, -15], [-23, 0], [0, 63], [156, 0], [-85, 0], [0, 74], [-246, 0], [0, 190], [0, -288], [0, -177], [792, 0], [-907, 0], [0, -322], [270, 0], [0, -254], [1123, 0], [0, -771], [-198, 0], [0, -1192], [1692, 0], [0, -1748], [0, 948], [-593, 0], [0, -1062], [0, 326], [990, 0], [0, 147], [-1604, 0], [1218, 0], [-2248, 0], [0, -2998], [0, -3470], [0, -387], [0, -855], [-264, 0], [-2551, 0], [2238, 0], [0, -2180], [0, -2577], [-1412, 0], [-307, 0], [0, -5234], [0, -1509], [-1211, 0], [0, 6246], [4650, 0], [-3145, 0], [-1020, 0], [0, -6741], [0, 2340], [-6198, 0], [875, 0], [0, 5486], [3204, 0], [0, -7322], [0, 1353], [0, 1658], [1044, 0], [0, -2588], [0, 1449], [4880, 0], [0, -7744], [0, -804], [2815, 0], [0, -3738], [-11022, 0], [0, 7544], [0, 5404], [2335, 0], [0, -6633], [-7566, 0], [0, -7420], [8502, 0], [1903, 0], [13482, 0], [-1537, 0], [10368, 0], [0, 13168], [-7319, 0], [0, -4119], [-5388, 0], [0, -2752], [4314, 0], [0, 5794], [0, -6309], [14826, 0], [0, 6758], [14574, 0], [-12611, 0], [0, 15639], [15420, 0], [-10494, 0], [0, 10708], [23030, 0], [0, -3814], [0, -22266], [0, -23844], [7488, 0], [5111, 0], [0, 6986], [0, -20571], [0, 23241], [20208, 0], [-9055, 0], [0, 15554], [0, 5632], [0, 9141], [13372, 0], [11165, 0], [912, 0], [0, 27952], [0, 6285], [0, -16497], [14844, 0], [31934, 0], [0, 24352], [0, 10374], [0, -7347], [-5371, 0], [-7086, 0], [-17186, 0], [-23814, 0], [0, -22732], [0, -4664], [-5501, 0], [0, -27096], [-5659, 0], [0, 37694], [6588, 0], [19, 0], [0, 33639], [0, 23474], [0, -7917], [-41202, 0], [35492, 0], [7146, 0], [0, -8882], [0, 21705], [-29018, 0], [31164, 0], [0, -49160], [0, 2349], [28195, 0], [0, -23997], [0, 46286], [-39636, 0], [0, 16744], [0, -1251], [0, -36988], [-16404, 0], [664, 0], [-39642, 0], [0, -36028], [-23058, 0], [0, -19953], [0, 25638], [0, -27034], [-14802, 0], [0, 9186], [0, 31647], [-48823, 0], [0, -13066], [24361, 0], [0, 23013], [31260, 0], [-8953, 0], [-2760, 0], [0, -55628], [-11321, 0], [17064, 0], [-59629, 0], [-55644, 0], [0, 44794], [-38114, 0], [0, 2030], [-47892, 0], [0, -29162], [0, -32118], [0, 22689], [-42180, 0], [0, -17044], [0, -70844], [57379, 0], [18636, 0], [0, 13954], [0, -62709], [-28783, 0], [29208, 0], [0, 14643], [0, 31929], [-50407, 0], [0, -53876], [0, 32046], [0, -71448], [-50706, 0], [-12400, 0], [0, 11774], [-22979, 0], [65556, 0], [0, 9174], [48280, 0], [0, -70743], [-86340, 0], [-11020, 0], [0, 7016], [-44142, 0], [0, -33680], [0, 34788], [64356, 0], [-72655, 0], [0, -29746], [-52968, 0], [0, 62661], [0, 43923], [30434, 0], [-9299, 0], [-46164, 0], [0, -19370], [0, -34041], [68293, 0], [0, -60087], [74466, 0], [0, -20074], [-9012, 0], [0, 1918], [-32339, 0], [48320, 0], [-94338, 0], [0, 36340], [0, 36015], [-9461, 0], [0, -58239], [-29886, 0], [28146, 0], [0, 56700], [22871, 0], [0, -37478], [-84161, 0], [0, 106422], [33198, 0], [0, 115484], [-79008, 0], [-96476, 0], [0, -74823], [0, 7958], [13902, 0], [0, -94245], [85147, 0], [0, 1875], [85632, 0], [110693, 0], [0, 49376], [0, -93147], [0, -46960], [0, -44402], [0, 56973], [-88823, 0], [0, -1772], [0, -78021], [-19028, 0], [3192, 0], [-76434, 0], [0, 84022], [0, 64938], [-11773, 0], [0, 84998], [0, 134206], [-52571, 0], [0, -29002], [0, 89152], [0, 44253], [40783, 0], [-34663, 0], [129258, 0], [0, 62640], [95807, 0], [0, -113535], [6311, 0], [0, -94084], [59988, 0], [0, 60418], [0, 85566], [93751, 0], [22440, 0], [-32568, 0], [0, -100995], [0, 11850], [122166, 0], [0, 31930], [-139104, 0], [-25213, 0], [0, 165102], [73872, 0], [0, 80012], [0, 112246], [0, -13251], [2800, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1350_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1350_4_c_g();". // 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_1350_4_c_g(:prec:=2) chi := MakeCharacter_1350_c(); 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_1350_4_c_g();". // 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_1350_4_c_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1350_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![64, 0, 1]>,<11,R![18, 1]>],Snew); return Vf; end function;