// Make newform 256.6.b.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_256_b();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_256_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_256_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_256_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; 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_256_b();" function MakeCharacter_256_b() N := 256; order := 2; char_gens := [255, 5]; 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_256_b_Hecke();" function MakeCharacter_256_b_Hecke(Kf) N := 256; order := 2; char_gens := [255, 5]; 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, 0], [0, 10], [0, 37], [24, 0], [0, -62], [0, 239], [-1198, 0], [0, 1522], [-184, 0], [0, -1641], [-5728, 0], [0, -5163], [8886, 0], [0, 4594], [23664, 0], [0, -5843], [0, -8438], [0, -9241], [0, -7766], [31960, 0], [4886, 0], [44560, 0], [0, 33682], [-71994, 0], [48866, 0], [0, -25803], [-180424, 0], [0, 32850], [0, -56353], [-23502, 0], [-94592, 0], [0, 35146], [-277290, 0], [0, 65154], [0, 200765], [75976, 0], [0, -197161], [0, -5862], [551928, 0], [0, 216447], [0, 279810], [0, -302355], [-409152, 0], [540866, 0], [0, 314949], [-283048, 0], [0, 71378], [889696, 0], [0, 571578], [0, 347893], [347126, 0], [-1642960, 0], [-1167438, 0], [0, 395306], [-129790, 0], [-70888, 0], [0, 895087], [-1773616, 0], [0, 137725], [-594170, 0], [0, -546214], [0, -166827], [0, 529986], [1336488, 0], [-1644186, 0], [0, -861849], [0, -1374814], [-3414894, 0], [0, -365382], [0, -1148745], [-1170718, 0], [-3886536, 0], [933040, 0], [0, 196109], [0, 2364650], [1897344, 0], [0, 1861477], [0, 1669039], [4274898, 0], [2573190, 0], [0, 2634138], [0, 486677], [3557360, 0], [-1954958, 0], [3296808, 0], [0, 2529098], [2127298, 0], [-289130, 0], [0, 1334351], [7586192, 0], [0, -709806], [-1884064, 0], [6013880, 0], [0, -2146158], [0, 672546], [-202008, 0], [0, 4891719], [10483030, 0], [0, -3105086], [0, 2540439], [0, 1673434], [0, 3501887], [0, -6490942], [-1899418, 0], [0, 831498], [8773442, 0], [0, -2591966], [8498578, 0], [-11247096, 0], [3464390, 0], [-999712, 0], [0, -4906699], [5347446, 0], [0, 3413842], [3609704, 0], [-13385342, 0], [0, -4955574], [17835928, 0], [0, -2161617], [0, 9892914], [0, -7888579], [6787618, 0], [0, 7497109], [0, -5779022], [0, -110078], [0, 2394663], [0, -2134459], [-16195952, 0], [-6534264, 0], [0, 6580855], [0, -7117382], [21583544, 0], [18659376, 0], [0, 1283405], [25958566, 0], [5532674, 0], [0, -4164699], [0, -6826126], [0, -4273129], [-7584842, 0], [0, -3092366], [0, 1390509], [-16389464, 0], [0, -11475574], [0, -1750681], [-5296680, 0], [0, 10147421], [4827846, 0], [0, 6510522], [-39238688, 0], [0, 6731103], [-917710, 0], [0, 12274402], [16146280, 0], [0, 10168066], [10772560, 0], [-41856568, 0], [29984546, 0], [-14240202, 0], [0, -20727297], [0, -7703934], [20632806, 0], [-11872424, 0], [0, -766110], [17432082, 0], [-2232696, 0], [22250080, 0], [0, -26633099]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_256_b_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_256_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_256_6_b_f(:prec:=2) chi := MakeCharacter_256_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_256_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_256_6_b_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_256_b(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign))); Vf := Kernel([<3,R![400, 0, 1]>,<7,R![-24, 1]>],Snew); return Vf; end function;