// Make newform 1008.2.r.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1008_r();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_r_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_1008_2_r_k();". // 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_1008_2_r_k();". // 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 := [3, -12, 19, -15, 10, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [6, 0, 1, 5, -1, 1], [0, -3, 2, -5, 1, -1], [6, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2]]; Rf_basisdens := [1, 1, 3, 3, 3, 3]; 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_1008_r();" function MakeCharacter_1008_r() N := 1008; order := 3; char_gens := [127, 757, 785, 577]; v := [3, 3, 1, 3]; // 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_1008_r_Hecke();" function MakeCharacter_1008_r_Hecke(Kf) N := 1008; order := 3; char_gens := [127, 757, 785, 577]; char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 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 := 2; raw_aps := [[0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1], [2, 0, 1, 0, 2, -1], [0, 0, 0, 0, 1, 0], [-1, 3, 0, -1, 2, -1], [-1, 0, 0, 0, -1, 0], [-5, 1, 0, -2, 1, 0], [0, 1, 0, -2, 1, 0], [1, 1, 3, 1, 1, -3], [0, -1, 0, 0, 0, 1], [-2, -2, -3, -2, -2, 3], [-3, 3, 3, -3, 3, 0], [7, 0, -1, 0, 7, 1], [1, -5, 0, 1, -1, 3], [3, -6, 0, 3, 0, 0], [-7, 2, 3, -1, 2, 0], [-4, -1, -3, -1, -4, 3], [2, -1, 0, 2, -3, -3], [2, 7, 6, 7, 2, -6], [-4, 2, 3, -1, 2, 0], [4, -5, -6, 4, -5, 0], [1, 4, 0, 1, -4, -6], [-1, 2, 0, -1, 5, 0], [2, -1, 5, 7, -1, 0], [-2, 7, 0, -2, 4, -3], [3, -2, 0, 3, -6, -4], [-11, -1, -3, -1, -11, 3], [-5, 4, 4, -4, 4, 0], [-6, -3, -6, 0, -3, 0], [2, 6, 10, 6, 2, -10], [-4, -1, -3, -1, -1, 0], [8, 1, 5, 1, 8, -5], [-1, -1, 0, -1, -7, 3], [1, 1, 0, 1, 1, 0], [-3, -6, -3, -6, -3, 3], [4, 1, 0, 4, -4, -9], [-3, 1, 0, 1, -3, 0], [1, -1, 6, 8, -1, 0], [-1, 4, 8, 4, -1, -8], [1, -7, 0, 1, 1, 5], [-4, 5, 2, -8, 5, 0], [9, 5, 3, -7, 5, 0], [3, -5, 0, 3, -9, -1], [10, 9, 9, 9, 10, -9], [4, -2, 5, 9, -2, 0], [9, -6, 3, 15, -6, 0], [1, 0, 6, 0, 1, -6], [-6, 12, 0, -6, 11, 0], [0, 3, 0, 0, 6, -3], [-9, -10, -6, -10, -9, 6], [-9, 3, 9, 3, 3, 0], [11, 2, 0, 2, 11, 0], [-10, 17, 0, -10, 7, 3], [5, -7, -2, 12, -7, 0], [8, -2, -4, -2, 8, 4], [-3, -2, 0, -3, 0, 8], [-10, -4, -3, 5, -4, 0], [14, -3, -9, -3, -3, 0], [3, -6, 0, 3, -5, 0], [9, -8, 0, 9, 3, -10], [6, -4, 3, -4, 6, -3], [12, 11, 11, 11, 12, -11], [7, 5, 0, -10, 5, 0], [-3, 6, 9, 6, -3, -9], [4, -11, 0, 4, -12, 3], [-3, -3, 0, -3, -12, 9], [-1, -4, 0, -1, 9, 6], [12, -5, 3, -5, 12, -3], [15, -5, -2, -5, 15, 2], [0, -6, 0, 0, 10, 6], [0, 5, 0, 0, -21, -5], [-7, -7, -4, 10, -7, 0], [-11, 13, 0, -11, -5, 9], [-12, 2, 3, 2, -12, -3], [2, 2, 9, 5, 2, 0], [-2, 4, 3, 4, -2, -3], [-3, 0, 0, -3, 18, 6], [-7, 5, 15, 5, 5, 0], [13, 4, 3, 4, 13, -3], [16, -6, 0, -6, 16, 0], [15, 3, -3, 3, 15, 3], [-11, 10, 0, -11, -6, 12], [-30, 0, -3, -3, 0, 0], [-19, -2, -12, -8, -2, 0], [2, -7, 0, 2, 8, 3], [-2, -8, 0, -2, -8, 12], [3, -3, 0, 6, -3, 0], [1, -8, 0, 1, -10, 6], [-1, 12, 0, -1, 23, -10], [17, 1, 3, 1, 17, -3], [1, 7, 3, -11, 7, 0], [-1, 4, 0, -1, -16, -2], [-6, 4, -3, -11, 4, 0], [2, -6, -11, -6, 2, 11], [-20, -12, -6, -12, -20, 6], [-12, 3, -9, -15, 3, 0], [-11, -9, -1, -9, -11, 1], [-17, -2, -12, -8, -2, 0], [-26, 2, 3, -1, 2, 0], [-13, 6, 6, -6, 6, 0], [-4, -1, 0, -4, 1, 9], [12, 12, 11, -13, 12, 0], [11, -14, -19, -14, 11, 19], [6, -10, 0, 6, -9, -2], [12, 1, 15, 1, 12, -15], [-5, 6, 3, -9, 6, 0], [-3, 1, 0, -3, -24, 5], [-14, 1, -12, -14, 1, 0], [16, -2, -6, -2, 16, 6], [-1, 11, 0, -1, 8, -9], [-5, 8, 12, 8, -5, -12], [25, 0, 6, 6, 0, 0], [7, -2, -3, -2, 7, 3], [-10, 5, 0, -10, -5, 15], [-18, -1, -15, -13, -1, 0], [-10, 17, 0, -10, -1, 3], [-6, 1, -3, 1, -6, 3], [18, -15, -4, 26, -15, 0], [8, -1, -12, -1, 8, 12], [-1, 8, 0, -1, 2, -6], [2, -9, -21, -9, 2, 21], [-3, -9, 0, -3, 4, 15], [2, 5, 0, 2, 8, -9], [-2, 10, 24, 4, 10, 0], [-7, 11, 0, -7, 36, 3], [20, -10, -24, -4, -10, 0], [-12, 6, 0, -12, 12, 18], [16, -2, -9, -5, -2, 0], [3, -9, 0, 3, -23, 3], [-9, 12, 12, 12, -9, -12], [8, -4, 15, 23, -4, 0], [-18, -9, 0, -9, -18, 0], [9, 5, -6, 5, 9, 6], [-23, -1, 6, 8, -1, 0], [4, 10, 6, 10, 4, -6], [-15, -7, -21, -7, -15, 21], [-1, -4, 10, 18, -4, 0], [2, -14, -21, -14, 2, 21], [-24, 7, -2, 7, -24, 2], [-3, 9, 18, 0, 9, 0], [7, 6, 15, 3, 6, 0], [1, 2, 0, 1, -26, -4], [-36, 0, 0, 0, -36, 0], [32, -10, 3, 23, -10, 0], [10, -1, -12, -10, -1, 0], [14, -30, 0, 14, -7, 2], [-9, 0, 0, -9, -10, 18], [0, -7, 0, 0, -33, 7], [1, 2, 18, 2, 1, -18], [9, 3, 18, 12, 3, 0], [20, 9, 6, 9, 20, -6], [1, 1, -9, -11, 1, 0], [29, -11, -3, 19, -11, 0], [-31, -9, -5, -9, -31, 5], [2, -10, 0, 2, -33, 6], [-10, 23, 0, -10, 2, -3], [12, 11, 21, -1, 11, 0], [14, -9, 0, 14, -10, -19], [9, 3, 0, -6, 3, 0], [22, 12, 17, 12, 22, -17], [-8, 21, 0, -8, 31, -5], [-30, 12, 4, -20, 12, 0], [-8, -21, -30, -21, -8, 30], [7, -5, -18, -8, -5, 0], [11, -16, -12, -16, 11, 12], [18, -15, 0, 18, -3, -21], [10, -3, -6, 0, -3, 0], [4, -20, 0, 4, -34, 12], [27, -1, 12, 14, -1, 0], [-19, 1, -10, 1, -19, 10], [10, -29, 0, 10, -14, 9], [7, -2, 3, -2, 7, -3], [-5, 4, -12, 4, -5, 12], [8, -4, 0, 8, 13, -12], [-21, -2, -12, -2, -21, 12], [9, 2, 23, 2, 9, -23], [3, -3, 0, 3, -19, -3], [-3, -9, 5, 23, -9, 0], [-25, 14, 3, -25, 14, 0], [9, 3, 0, 9, -11, -21], [-6, 3, 0, -6, -11, 9], [9, 0, 0, 9, -42, -18], [26, 10, 18, 10, 26, -18], [12, -9, -8, 10, -9, 0], [27, -3, 15, -3, 27, -15], [-1, -6, 0, -1, 2, 8], [-25, 1, 9, 7, 1, 0], [11, -16, 0, 11, -5, -6], [-51, 4, 3, 4, -51, -3], [22, 1, 10, 8, 1, 0], [-32, 18, -9, -45, 18, 0], [1, 7, 0, 1, 13, -9], [-14, 12, -3, -27, 12, 0], [-11, -5, 0, -11, 28, 27], [-13, -7, -5, 9, -7, 0], [1, -23, -33, -23, 1, 33], [-1, -17, -18, -17, -1, 18], [9, -12, 0, 9, -18, -6], [17, -34, 0, 17, -19, 0], [25, 16, 12, -20, 16, 0], [-3, 2, 8, 2, -3, -8], [-24, 24, 0, -24, 1, 24], [2, 17, 3, 17, 2, -3], [-9, -3, 0, -9, 16, 21], [2, 5, 17, 7, 5, 0], [-2, 13, 7, -19, 13, 0], [-40, -4, -6, 2, -4, 0], [27, -3, -3, -3, 27, 3], [-12, 30, 0, -12, 18, -6], [15, -4, -15, -4, 15, 15], [22, -11, -21, 1, -11, 0], [-29, -5, 6, -5, -29, -6], [-6, 30, 0, -6, 20, -18], [7, -13, 0, 7, 40, -1], [-34, 3, -2, 3, -34, 2], [0, -3, 0, 0, 22, 3], [-21, 1, -3, 1, -21, 3], [15, 5, 0, 15, -15, -35], [9, 9, 10, -8, 9, 0], [12, -21, -9, -21, 12, 9], [20, -11, -15, -11, 20, 15], [-10, -8, 12, -8, -10, -12], [20, 28, 32, 28, 20, -32], [-13, 1, -3, -5, 1, 0], [-16, 6, 10, 6, -16, -10], [-15, 30, 0, -15, 11, 0], [-11, 21, 0, -11, 34, 1], [-7, 14, -7, -35, 14, 0], [3, 21, 0, 3, 22, -27], [1, 19, 0, 1, 31, -21], [-12, -2, 9, -2, -12, -9], [-32, 8, -30, -46, 8, 0], [18, 4, 18, 4, 18, -18], [-15, -11, -26, -11, -15, 26], [-6, 12, 0, -6, -24, 0], [13, -41, 0, 13, -23, 15], [16, -16, -18, -16, 16, 18], [11, -4, -29, -21, -4, 0], [-30, 6, 3, 6, -30, -3], [48, -9, -4, 14, -9, 0], [-5, 1, 0, -5, 16, 9], [36, -12, -9, 15, -12, 0], [17, 14, 33, 14, 17, -33], [-12, 8, 9, -7, 8, 0], [19, 19, 27, 19, 19, -27], [-13, 24, 0, -13, 35, 2], [-2, -28, -21, 35, -28, 0], [17, -6, 19, -6, 17, -19], [1, -3, 12, -3, 1, -12], [3, -6, -6, 6, -6, 0], [-11, -18, 3, -18, -11, -3], [8, 2, 0, -4, 2, 0], [-21, -13, -1, -13, -21, 1], [3, -6, 0, 3, -11, 0], [9, -22, 0, 9, 24, 4], [-27, 15, 8, -22, 15, 0], [-32, -7, -24, -10, -7, 0], [-5, -2, 0, -5, 52, 12], [-1, -7, -27, -13, -7, 0], [29, -12, -6, 18, -12, 0], [18, -21, 0, 18, -11, -15], [27, -34, 0, 27, -9, -20], [13, -12, -3, -12, 13, 3], [35, -12, 0, 24, -12, 0], [16, 24, 8, 24, 16, -8], [-6, 0, 0, -6, -56, 12], [8, 11, 6, -16, 11, 0], [-5, -3, 0, -5, 7, 13], [-31, -2, -9, -2, -31, 9], [13, -1, 13, -1, 13, -13], [-18, -17, 0, -17, -18, 0], [-20, 9, 0, -18, 9, 0], [18, -6, 0, 18, -29, -30], [22, -1, 6, -1, 22, -6], [-6, -17, -15, -17, -6, 15], [-1, 10, 36, 16, 10, 0], [-31, 19, 17, 19, -31, -17], [9, 12, 0, 9, -36, -30], [8, -2, 24, 28, -2, 0], [-3, 11, 0, -3, 21, -5], [25, -10, -26, -10, 25, 26], [-25, 20, 3, 20, -25, -3], [-14, 9, 0, -14, 49, 19], [-22, 17, 0, -22, 18, 27], [48, -11, -12, -11, 48, 12], [-52, 5, 0, -10, 5, 0], [-14, 9, 18, 0, 9, 0], [-44, -5, 3, -5, -44, -3], [8, -31, 0, 8, -24, 15], [1, 1, 12, 10, 1, 0], [-2, 9, 0, -2, 25, -5], [10, -8, -20, -4, -8, 0], [3, -15, 12, -15, 3, -12], [5, 11, 18, 11, 5, -18], [18, -27, 0, 18, -5, -9], [-2, -28, -5, -28, -2, 5], [-16, 17, 0, -16, 10, 15], [0, -26, 0, 0, -36, 26], [49, 1, 11, 9, 1, 0], [12, -30, 0, 12, -35, 6], [31, 22, 30, 22, 31, -30], [29, -22, -21, 23, -22, 0], [-32, 20, 39, -1, 20, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1008_r_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_r_k();". // 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_1008_2_r_k(:prec:=6) chi := MakeCharacter_1008_r(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_1008_2_r_k();". // 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_1008_2_r_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1008_r(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![121, 22, 59, -32, 23, -5, 1]>,<11,R![2209, 893, 455, 56, 23, 2, 1]>],Snew); return Vf; end function;