// Make newform 3072.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3072_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_3072_2_a_l();". // 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_3072_2_a_l();". // 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, -4, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, -3, 0, 1], [0, 5, 0, -1], [-4, 0, 2, 0]]; Rf_basisdens := [1, 1, 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_3072_a();" function MakeCharacter_3072_a() N := 3072; order := 1; char_gens := [2047, 2053, 1025]; v := [1, 1, 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; function MakeCharacter_3072_a_Hecke(Kf) return MakeCharacter_3072_a(); 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 := 2; raw_aps := [[0, 0, 0, 0], [-1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 1, 0], [-2, 0, 0, 1], [0, 3, 0, 0], [0, 0, 0, -1], [-4, 0, 0, -1], [0, 2, 0, 0], [0, 1, 3, 0], [0, 0, 3, 0], [0, -3, 2, 0], [8, 0, 0, -1], [0, 0, 0, 1], [0, -2, 0, 0], [0, -5, 1, 0], [0, 0, 0, -4], [0, 3, -2, 0], [-8, 0, 0, -2], [0, 8, -2, 0], [4, 0, 0, 0], [0, 0, -1, 0], [-2, 0, 0, -1], [2, 0, 0, -2], [8, 0, 0, -2], [0, -5, -3, 0], [0, -4, 1, 0], [4, 0, 0, 0], [0, 1, 0, 0], [18, 0, 0, 0], [0, 12, -1, 0], [8, 0, 0, 2], [4, 0, 0, 3], [0, 0, 0, -2], [0, -9, 3, 0], [0, -4, 5, 0], [0, 5, -2, 0], [-8, 0, 0, 1], [0, -2, -2, 0], [0, -7, -3, 0], [12, 0, 0, 2], [0, 5, 0, 0], [0, -2, -2, 0], [6, 0, 0, 0], [0, -1, -3, 0], [0, 8, 3, 0], [0, 0, 0, -2], [0, -12, 1, 0], [6, 0, 0, -5], [0, -5, -4, 0], [6, 0, 0, 2], [0, 2, 2, 0], [-6, 0, 0, -6], [10, 0, 0, -1], [-2, 0, 0, -6], [0, 16, 0, 0], [0, 9, -1, 0], [0, -4, 1, 0], [0, -5, 8, 0], [-6, 0, 0, 2], [4, 0, 0, -4], [0, -1, 5, 0], [-16, 0, 0, -2], [0, 16, 4, 0], [-10, 0, 0, 4], [0, 3, 1, 0], [-8, 0, 0, 2], [-4, 0, 0, 4], [-6, 0, 0, 1], [0, 9, 6, 0], [6, 0, 0, -2], [0, -4, 10, 0], [0, 4, -9, 0], [0, 1, -6, 0], [4, 0, 0, -5], [0, 10, 6, 0], [0, 5, 1, 0], [0, 1, -10, 0], [-8, 0, 0, -3], [-16, 0, 0, 2], [10, 0, 0, -3], [0, 7, 4, 0], [0, 10, -4, 0], [-16, 0, 0, -4], [0, -8, 7, 0], [18, 0, 0, 5], [4, 0, 0, -3], [0, 0, 0, -6], [0, 3, 11, 0], [0, -12, -5, 0], [6, 0, 0, 7], [0, -24, 2, 0], [0, -4, 7, 0], [8, 0, 0, -4], [-4, 0, 0, 0], [0, 20, -2, 0], [0, 7, -7, 0], [-4, 0, 0, -5], [-28, 0, 0, -1], [0, -13, -4, 0], [8, 0, 0, -9], [0, -25, 3, 0], [2, 0, 0, -1], [0, 0, 0, -1], [4, 0, 0, 8], [-8, 0, 0, -4], [36, 0, 0, 2], [10, 0, 0, 8], [0, -2, 4, 0], [-8, 0, 0, 8], [0, 0, -13, 0], [0, 13, -10, 0], [-26, 0, 0, -4], [12, 0, 0, 0], [0, -8, -5, 0], [-36, 0, 0, 3], [32, 0, 0, -1], [0, 22, 4, 0], [0, -9, -5, 0], [24, 0, 0, 0], [0, 15, -6, 0], [8, 0, 0, 0], [0, 11, -5, 0], [6, 0, 0, -9], [-16, 0, 0, -3], [0, 5, -9, 0], [0, -7, -16, 0], [0, -12, 6, 0], [0, -4, -1, 0], [0, 5, 0, 0], [-36, 0, 0, 0], [0, 2, 10, 0], [0, 24, 5, 0], [0, -3, -12, 0], [-28, 0, 0, -7], [-22, 0, 0, 0], [0, 19, 9, 0], [20, 0, 0, 1], [0, -7, 13, 0], [-24, 0, 0, 5], [-44, 0, 0, -3], [0, -23, 3, 0], [0, -12, 11, 0], [-24, 0, 0, -6], [0, -15, 8, 0], [0, -8, -6, 0], [0, 9, 14, 0], [12, 0, 0, 9], [8, 0, 0, 7], [0, -2, -14, 0], [0, -1, -14, 0], [-2, 0, 0, -10], [16, 0, 0, -3], [0, -12, -8, 0], [32, 0, 0, 3], [0, 26, -6, 0], [0, 16, -9, 0], [-28, 0, 0, -5], [2, 0, 0, 0], [0, -17, -5, 0], [-36, 0, 0, 2], [36, 0, 0, 3], [0, -16, 5, 0], [2, 0, 0, -7], [-4, 0, 0, -9], [0, 4, -12, 0], [0, 0, 5, 0], [0, 9, 2, 0], [-16, 0, 0, 10], [0, -15, -7, 0], [-8, 0, 0, -4], [0, -25, -2, 0], [0, 2, -14, 0], [14, 0, 0, -4], [0, -12, -11, 0], [-6, 0, 0, 14], [44, 0, 0, 0], [0, -1, -17, 0], [0, -4, 11, 0], [0, 21, 6, 0], [0, 8, 7, 0], [-10, 0, 0, -5], [0, 13, 12, 0], [-6, 0, 0, -2], [0, 6, 10, 0], [0, -3, -7, 0], [0, 15, 12, 0], [4, 0, 0, -12], [-26, 0, 0, -6], [0, 8, 8, 0], [-2, 0, 0, 4], [-24, 0, 0, 8], [-4, 0, 0, -16], [0, -19, -15, 0], [16, 0, 0, 6], [38, 0, 0, 8], [32, 0, 0, 8], [0, 3, -16, 0], [42, 0, 0, -6], [0, 4, 16, 0], [0, 11, -17, 0], [0, 20, -15, 0], [0, -23, 4, 0], [12, 0, 0, 8], [6, 0, 0, 9], [0, 17, 11, 0], [0, -24, -7, 0], [2, 0, 0, 3], [2, 0, 0, -8], [-40, 0, 0, -1], [50, 0, 0, 4], [0, 19, -13, 0], [0, 12, 11, 0], [-8, 0, 0, 0], [0, 0, -2, 0], [-16, 0, 0, -4], [0, 8, 7, 0], [24, 0, 0, -7], [0, -6, 22, 0], [0, -41, -1, 0], [0, -39, -4, 0], [0, -12, -21, 0], [-20, 0, 0, -3], [0, -36, 3, 0], [-26, 0, 0, 3], [0, -35, 2, 0], [22, 0, 0, -12], [0, 12, -14, 0], [0, 36, -3, 0], [-16, 0, 0, 4], [0, -3, 20, 0], [-32, 0, 0, 7], [0, -4, -13, 0], [-22, 0, 0, 2], [56, 0, 0, -1], [0, 4, 2, 0], [-40, 0, 0, 2], [0, 7, -1, 0], [-42, 0, 0, 7], [0, -10, 0, 0], [26, 0, 0, -1], [-32, 0, 0, -2], [0, -24, -13, 0], [0, 19, 10, 0], [6, 0, 0, 8], [0, -20, -14, 0], [0, -12, -11, 0], [-36, 0, 0, 0], [8, 0, 0, -10], [0, 12, -4, 0], [0, -9, 0, 0], [50, 0, 0, 4], [0, -2, -8, 0], [42, 0, 0, 4], [0, 27, -3, 0], [-48, 0, 0, 6], [0, 3, 0, 0], [-20, 0, 0, -12], [0, -7, 11, 0], [24, 0, 0, -4], [0, -16, -15, 0], [8, 0, 0, -8], [0, 15, -4, 0], [0, 3, 2, 0], [-32, 0, 0, 7], [-16, 0, 0, -6], [0, 13, 5, 0], [58, 0, 0, 4], [4, 0, 0, -15], [0, 31, -1, 0], [0, -3, 2, 0], [-4, 0, 0, 0], [-10, 0, 0, -4], [0, 8, 5, 0], [30, 0, 0, -8], [0, 28, 3, 0], [-58, 0, 0, -3], [0, 41, 6, 0], [40, 0, 0, 12], [32, 0, 0, 14], [0, 44, -6, 0], [0, 8, -9, 0], [0, 6, 8, 0], [0, 13, 22, 0], [60, 0, 0, 1], [0, 44, 0, 0], [34, 0, 0, -6], [0, 5, 5, 0], [0, 8, 1, 0], [6, 0, 0, -14], [0, -29, 3, 0], [-4, 0, 0, 2], [6, 0, 0, -12], [12, 0, 0, 10], [0, -13, -6, 0], [0, -9, 11, 0], [0, 4, -25, 0], [0, -9, -19, 0], [-22, 0, 0, 19], [12, 0, 0, 13], [26, 0, 0, 4], [0, 29, 5, 0], [0, -4, 5, 0], [28, 0, 0, -8], [0, 0, 0, -3], [2, 0, 0, 10], [-22, 0, 0, -13], [0, 33, 4, 0], [0, 8, 18, 0], [0, 3, -14, 0], [0, 2, -30, 0], [0, 15, 9, 0], [52, 0, 0, 11], [28, 0, 0, -13], [0, 8, 16, 0], [-18, 0, 0, -8], [-12, 0, 0, -6], [0, -16, 2, 0], [20, 0, 0, 16], [2, 0, 0, 6], [-52, 0, 0, 4], [-6, 0, 0, 4], [0, -31, -9, 0], [0, -32, 11, 0], [-20, 0, 0, 9], [-8, 0, 0, 2], [36, 0, 0, -1], [48, 0, 0, -11], [0, 24, 20, 0], [0, -5, 15, 0], [0, 23, 0, 0], [0, -1, -5, 0], [0, -16, 5, 0], [-28, 0, 0, 16], [64, 0, 0, 2], [-54, 0, 0, -1], [0, 3, -14, 0], [30, 0, 0, -16], [-12, 0, 0, -4], [0, 4, 23, 0], [0, 51, -2, 0], [-32, 0, 0, 1], [0, 1, 19, 0], [0, -20, 13, 0], [0, -5, 9, 0], [2, 0, 0, -21], [0, 37, 4, 0], [8, 0, 0, -6], [0, -20, -18, 0], [0, 37, -13, 0], [44, 0, 0, -3], [-24, 0, 0, 2], [0, -7, 7, 0], [0, 8, 9, 0], [0, -21, 6, 0], [22, 0, 0, -18], [0, 26, 6, 0], [12, 0, 0, 10], [36, 0, 0, 7], [0, -4, -8, 0], [0, -7, 6, 0], [34, 0, 0, -14], [0, -14, 6, 0], [42, 0, 0, -9], [-52, 0, 0, 12], [40, 0, 0, -12], [0, -13, -7, 0], [0, 40, -11, 0], [-8, 0, 0, 4], [42, 0, 0, -5], [-12, 0, 0, 11], [0, -16, 12, 0], [0, -1, -15, 0], [0, 12, 7, 0], [0, -3, -16, 0], [10, 0, 0, -1], [0, 2, -10, 0], [46, 0, 0, -4], [-72, 0, 0, -1], [-34, 0, 0, -18], [0, 1, 1, 0], [-36, 0, 0, 9], [0, 36, -3, 0], [26, 0, 0, 4], [0, 0, 0, -5], [0, 22, -14, 0], [0, -24, 21, 0], [0, 15, -12, 0], [-8, 0, 0, 6], [0, -50, 12, 0], [-34, 0, 0, 12], [0, 9, -3, 0], [32, 0, 0, -2], [52, 0, 0, -13], [0, -38, 2, 0], [0, 0, 0, 2], [0, -24, -23, 0], [-24, 0, 0, 1], [52, 0, 0, -5], [0, 15, 31, 0], [0, -1, 10, 0], [0, 0, 0, 19], [0, 16, 11, 0], [24, 0, 0, 15], [0, 9, -17, 0], [0, 60, 1, 0], [0, 3, 28, 0], [-48, 0, 0, 13], [-4, 0, 0, 11], [-6, 0, 0, 19], [24, 0, 0, 20], [0, 37, 17, 0], [18, 0, 0, 23], [-24, 0, 0, -15], [20, 0, 0, 8], [0, -25, 17, 0], [0, -2, -30, 0], [0, -64, -3, 0], [60, 0, 0, -9], [0, -24, 10, 0], [0, 1, 5, 0], [0, 35, -10, 0], [0, -4, -8, 0], [70, 0, 0, -5], [6, 0, 0, -12], [0, -9, -11, 0], [-6, 0, 0, 1], [-78, 0, 0, -2], [-32, 0, 0, 15], [0, -14, 24, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3072_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3072_2_a_l();". // 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_3072_2_a_l(:prec:=4) chi := MakeCharacter_3072_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 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_3072_2_a_l();". // 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_3072_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3072_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![16, 0, -16, 0, 1]>,<7,R![-6, 0, 1]>,<11,R![-8, 4, 1]>,<19,R![4, 8, 1]>],Snew); return Vf; end function;