// Make newform 900.4.d.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_900_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_d_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_4_d_i();". // 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_4_d_i();". // 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_900_d();" function MakeCharacter_900_d() N := 900; order := 2; char_gens := [451, 101, 577]; v := [2, 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_900_d_Hecke();" function MakeCharacter_900_d_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 := 4; raw_aps := [[0, 0], [0, 0], [0, 0], [0, 1], [30, 0], [0, 2], [0, 45], [28, 0], [0, -60], [-210, 0], [-4, 0], [0, 100], [240, 0], [0, 68], [0, -60], [0, 15], [450, 0], [-166, 0], [0, 454], [-1020, 0], [0, 125], [916, 0], [0, 570], [420, 0], [0, 769], [450, 0], [0, 575], [0, -810], [1702, 0], [0, -675], [0, 1225], [690, 0], [0, 1035], [1924, 0], [-2910, 0], [176, 0], [0, 1174], [0, 998], [0, 1560], [0, -885], [2130, 0], [-1654, 0], [1740, 0], [0, -43], [0, -1245], [832, 0], [2084, 0], [0, 587], [0, -1560], [58, 0], [0, 2955], [-3300, 0], [-2986, 0], [-6630, 0], [0, -765], [0, -1320], [7350, 0], [3512, 0], [0, -2684], [-3060, 0], [0, 2522], [0, 1005], [0, -1376], [9540, 0], [0, -4627], [0, 75], [1892, 0], [0, -3689], [0, 3360], [-5186, 0], [0, -1665], [-9000, 0], [0, -4379], [0, -2362], [-7292, 0], [0, -7260], [-7110, 0], [0, -5744], [780, 0], [-5402, 0], [2190, 0], [-7162, 0], [9360, 0], [0, -6403], [-11288, 0], [0, 4260], [1260, 0], [0, -6875], [-3210, 0], [0, 6425], [0, -4110], [7020, 0], [0, -4061], [-13470, 0], [-2468, 0], [0, 2220], [11190, 0], [-4020, 0], [0, 4538], [-7486, 0], [0, 3700], [0, 5745], [0, 9660], [-8340, 0], [21044, 0], [0, 709], [0, -11010], [0, -12615], [-8280, 0], [-18874, 0], [0, 5275], [0, -5500], [0, -5655], [17572, 0], [1604, 0], [31320, 0], [0, 15650], [0, 5460], [0, 1605], [11910, 0], [-3382, 0], [0, -7975], [0, 16095], [0, -11070], [-6172, 0], [19170, 0], [21898, 0], [-16680, 0], [0, 3259], [0, 11600], [16324, 0], [0, 60], [30548, 0], [0, 8476], [-20220, 0], [20722, 0], [0, -2175], [0, 20986], [0, -19755], [-16680, 0], [-15484, 0], [4170, 0], [0, 15113], [0, -7380], [9934, 0], [23520, 0], [0, -14908], [0, -17715], [36196, 0], [0, -240], [0, 14266], [-20340, 0], [0, 5378], [0, 300], [0, 12700], [-36240, 0], [-6572, 0], [-2340, 0], [0, 1261], [-52770, 0], [0, 14100], [0, 7785], [0, -4175], [43650, 0], [0, 9405], [0, 12660], [-6736, 0], [0, -10250]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_900_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_900_4_d_i();". // 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_4_d_i(:prec:=2) chi := MakeCharacter_900_d(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_4_d_i();". // 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_4_d_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_900_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![4, 0, 1]>,<11,R![-30, 1]>],Snew); return Vf; end function;