// Make newform 900.3.c.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_900_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_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_900_3_c_e();". // 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_900_3_c_e();". // 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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [-1, 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_900_c();" function MakeCharacter_900_c() N := 900; order := 2; char_gens := [451, 101, 577]; v := [1, 2, 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_900_c_Hecke();" function MakeCharacter_900_c_Hecke(Kf) N := 900; order := 2; char_gens := [451, 101, 577]; char_values := [[-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 := 3; raw_aps := [[-1, 1], [0, 0], [0, 0], [0, -4], [0, 4], [-2, 0], [10, 0], [0, -12], [0, 16], [26, 0], [0, -4], [-26, 0], [-58, 0], [0, -28], [0, -40], [-74, 0], [0, -52], [26, 0], [0, 4], [0, 0], [46, 0], [0, -68], [0, -28], [-82, 0], [-2, 0], [74, 0], [0, -44], [0, -12], [-46, 0], [-110, 0], [0, -84], [0, 68], [10, 0], [0, -28], [2, 0], [0, 52], [214, 0], [0, 12], [0, -56], [334, 0], [0, -108], [2, 0], [0, -128], [-290, 0], [-26, 0], [0, -228], [0, 140], [0, -196], [0, 164], [-142, 0], [82, 0], [0, 224], [-46, 0], [0, 84], [-254, 0], [0, -88], [-262, 0], [0, -12], [-290, 0], [-226, 0], [0, 172], [-362, 0], [0, 84], [0, -136], [478, 0], [-170, 0], [0, -236], [-338, 0], [0, -116], [506, 0], [178, 0], [0, 96], [0, 116], [310, 0], [0, 252], [0, 352], [578, 0], [-26, 0], [-250, 0], [290, 0], [0, -196], [674, 0], [0, 312], [334, 0], [0, -68], [0, 44], [-394, 0], [478, 0], [-142, 0], [0, 364], [0, -12], [0, 424], [0, 60], [0, -532], [0, 44], [0, 336], [842, 0], [326, 0], [0, 180], [530, 0], [0, -196], [766, 0], [0, -284], [422, 0], [0, 164], [46, 0], [0, -364], [82, 0], [0, 32], [-334, 0], [0, -212], [214, 0], [-1118, 0], [0, 388], [0, 84], [-10, 0], [0, 700], [0, 192], [670, 0], [0, -476], [-1222, 0], [334, 0], [1006, 0], [0, 108], [0, -572], [1034, 0], [530, 0], [0, -408], [0, 140], [-194, 0], [0, 780], [0, -392], [0, 380], [1006, 0], [758, 0], [2, 0], [262, 0], [0, -836], [-866, 0], [-10, 0], [0, 252], [-838, 0], [0, 508], [0, 420], [1298, 0], [0, 112], [-506, 0], [-998, 0], [0, -292], [0, -96], [646, 0], [-898, 0], [0, -420], [0, -488], [0, 788], [0, -224], [0, -348], [-1594, 0], [-674, 0], [-430, 0], [0, 44], [730, 0], [0, 532], [0, 852], [346, 0], [0, -424], [0, -564], [-458, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_900_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_c_e();". // 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_900_3_c_e(:prec:=2) chi := MakeCharacter_900_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_900_3_c_e();". // 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_900_3_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_900_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![48, 0, 1]>,<13,R![2, 1]>,<17,R![-10, 1]>],Snew); return Vf; end function;