// Make newform 550.6.b.f in Magma, downloaded from the LMFDB on 19 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_550_b();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_550_b_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_550_6_b_f();". // 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_550_6_b_f();". // 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_550_b();" function MakeCharacter_550_b() N := 550; order := 2; char_gens := [177, 101]; v := [1, 2]; // 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_550_b_Hecke();" function MakeCharacter_550_b_Hecke(Kf) N := 550; order := 2; char_gens := [177, 101]; 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 := 6; raw_aps := [[0, 4], [0, 1], [0, 0], [0, 166], [-121, 0], [0, 692], [0, 738], [-1424, 0], [0, -1779], [2064, 0], [6245, 0], [0, 14785], [5304, 0], [0, 17798], [0, 17184], [0, -30726], [34989, 0], [-45940, 0], [0, -25343], [13311, 0], [0, -53260], [-77234, 0], [0, 55014], [-125415, 0], [0, 88807], [1482, 0], [0, -117496], [0, 79362], [-87842, 0], [0, -47247], [0, 239416], [-98142, 0], [0, -400137], [-205766, 0], [-87726, 0], [-432778, 0], [0, 34075], [0, 45020], [0, -482556], [0, -766254], [-303399, 0], [-285181, 0], [767067, 0], [0, 411668], [0, 759258], [46600, 0], [-932428, 0], [0, 169745], [0, -198078], [849997, 0], [0, -401832], [-855174, 0], [1125464, 0], [-1197513, 0], [0, -37758], [0, -631254], [1080342, 0], [-816100, 0], [0, -1688198], [-879042, 0], [0, 1540268], [0, 720840], [0, 1039048], [-1251348, 0], [0, -1443361], [0, 2012079], [2017337, 0], [0, -264122], [0, 1710492], [-218822, 0], [0, 3681915], [-1885284, 0], [0, 3116659], [0, 1394414], [4260355, 0], [0, 201765], [-1948821, 0], [0, 1468258], [2246178, 0], [3614878, 0], [3812388, 0], [1973462, 0], [-2083590, 0], [0, -72691], [-594392, 0], [0, 4566507], [5443821, 0], [0, -6703124], [-1259940, 0], [0, -5023081], [0, 2356605], [6722580, 0], [0, -1960013], [-579624, 0], [-1369052, 0], [0, 1833426], [1712661, 0], [-789435, 0], [0, 627392], [3208952, 0], [0, -3426584], [0, -10519830], [0, 5472876], [11778672, 0], [-8356276, 0], [0, 13775839], [0, 12709308], [0, 1008252], [-10510032, 0], [-199390, 0], [0, -16190], [0, -11525344], [0, -16997394], [18487471, 0], [-4542811, 0], [18428631, 0], [0, 9666035], [0, 4514301], [0, -5372349], [-9879558, 0], [10805201, 0], [0, 11327486], [0, 12059490], [0, -5141664], [13124255, 0], [3659562, 0], [-10225211, 0], [-24168285, 0], [0, -16824641], [0, -5041684], [6263746, 0], [0, 3639756], [-18736957, 0], [0, -489242], [14696940, 0], [-24207236, 0], [0, 13526010], [0, -14209424], [0, 7933335], [10468512, 0], [11914742, 0], [-1861122, 0], [0, 23015285], [0, 16835148], [23529859, 0], [-29154921, 0], [0, -9490522], [0, 1815528], [10781209, 0], [0, -28335456], [0, 26891884], [-19213227, 0], [0, 11593100], [0, -13185678], [0, -2981948], [29657868, 0], [-31805678, 0], [23344446, 0], [0, -20737208], [26919282, 0], [0, -10189563], [0, 10392354], [0, 8188768], [-17327439, 0], [0, 438963], [0, -27912387], [-42684640, 0], [0, 22104418]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_550_b_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_550_6_b_f();". // 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_550_6_b_f(:prec:=2) chi := MakeCharacter_550_b(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 6)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_550_6_b_f();". // 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_550_6_b_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_550_b(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign))); Vf := Kernel([<3,R![1, 0, 1]>,<7,R![27556, 0, 1]>],Snew); return Vf; end function;