// Make newform 900.3.c.j 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_j();". // 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_j();". // 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, -6], [0, -6], [-18, 0], [10, 0], [0, 8], [0, 4], [36, 0], [0, -4], [-54, 0], [-18, 0], [0, -12], [0, 0], [-26, 0], [0, 18], [-74, 0], [0, -24], [0, 60], [-36, 0], [0, 52], [0, -52], [18, 0], [72, 0], [-36, 0], [0, -6], [0, -108], [-26, 0], [-10, 0], [0, -126], [0, 78], [110, 0], [0, -108], [-288, 0], [0, -108], [-234, 0], [0, -72], [0, 76], [146, 0], [0, 42], [262, 0], [0, -108], [-180, 0], [-154, 0], [0, -108], [0, -140], [0, -54], [0, -124], [338, 0], [-182, 0], [0, 204], [-106, 0], [0, -186], [14, 0], [0, 108], [108, 0], [0, 188], [-270, 0], [234, 0], [0, 48], [-58, 0], [0, 156], [0, -156], [-468, 0], [250, 0], [0, -216], [468, 0], [0, -324], [-434, 0], [-158, 0], [0, -264], [0, -126], [270, 0], [0, -188], [0, -32], [288, 0], [306, 0], [450, 0], [-50, 0], [0, -426], [-286, 0], [0, 72], [36, 0], [0, 452], [0, -124], [-54, 0], [-288, 0], [288, 0], [0, -234], [0, 332], [0, 84], [0, 150], [0, -42], [0, -256], [0, 64], [252, 0], [-54, 0], [0, 360], [-650, 0], [0, 396], [574, 0], [0, 324], [-198, 0], [0, 104], [504, 0], [0, -236], [998, 0], [0, 312], [-614, 0], [0, -378], [-414, 0], [58, 0], [0, 108], [0, -476], [-810, 0], [0, 240], [0, 568], [-950, 0], [0, 654], [-242, 0], [-324, 0], [-806, 0], [0, 332], [0, 448], [-756, 0], [-310, 0], [0, -48], [0, 630], [-1206, 0], [0, 280], [0, -648], [0, -140], [-846, 0], [1458, 0], [1282, 0], [-422, 0], [0, 696], [-94, 0], [270, 0], [0, -108], [-1188, 0], [0, 222], [0, -260], [718, 0], [0, -528], [666, 0], [-182, 0], [0, -572], [0, 676], [774, 0], [1602, 0], [0, -240], [0, -212], [0, -828], [0, -624], [0, -108], [-54, 0], [-936, 0], [0, 0], [0, 596], [1550, 0], [0, -702], [0, 1062], [-206, 0], [0, -416], [0, -764], [198, 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_j();". // 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_j(: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_j();". // 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_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_900_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![108, 0, 1]>,<13,R![18, 1]>,<17,R![-10, 1]>],Snew); return Vf; end function;