// Make newform 1440.2.x.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1440_x();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1440_x_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_1440_2_x_f();". // 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_1440_2_x_f();". // 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; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_1440_x();" function MakeCharacter_1440_x() N := 1440; order := 4; char_gens := [991, 901, 641, 577]; v := [2, 4, 4, 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_1440_x_Hecke();" function MakeCharacter_1440_x_Hecke(Kf) N := 1440; order := 4; char_gens := [991, 901, 641, 577]; char_values := [[-1, 0], [1, 0], [1, 0], [0, 1]]; 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], [-1, -2], [3, -3], [0, -2], [3, -3], [-1, -1], [-4, 0], [-1, -1], [0, 0], [0, 10], [-1, -1], [10, 0], [-5, -5], [-3, 3], [5, -5], [-12, 0], [2, 0], [1, -1], [0, -2], [1, -1], [-8, 0], [5, 5], [0, -16], [-3, -3], [-6, 0], [9, 9], [3, -3], [0, -4], [3, -3], [7, -7], [0, -10], [11, 11], [-12, 0], [0, 4], [0, -6], [-1, -1], [-1, -1], [1, -1], [5, -5], [-12, 0], [22, 0], [0, 14], [-15, 15], [13, 13], [16, 0], [0, -14], [1, 1], [-5, 5], [0, -8], [-21, 21], [16, 0], [-2, 0], [0, 6], [-5, -5], [11, 11], [0, -20], [0, -14], [11, 11], [-6, 0], [7, 7], [-11, 11], [17, -17], [0, -18], [9, -9], [13, 13], [0, 26], [-15, -15], [-9, 9], [0, -8], [15, -15], [32, 0], [15, -15], [-9, 9], [-20, 0], [-1, -1], [0, 4], [15, 15], [22, 0], [0, 20], [28, 0], [34, 0], [0, 30], [21, -21], [-16, 0], [25, 25], [0, 12], [9, 9], [2, 0], [-11, -11], [-13, 13], [-40, 0], [19, -19], [0, -10], [-28, 0], [-17, -17], [0, -24], [14, 0], [15, 15], [30, 0], [-11, 11], [-27, -27], [33, 33], [0, -12], [0, 34], [-19, -19], [23, -23], [7, -7], [0, 0], [-10, 0], [-5, 5], [15, -15], [-13, -13], [12, 0], [0, -14], [-30, 0], [27, 27], [29, -29], [1, -1], [-36, 0], [-30, 0], [-3, 3], [-3, -3], [-11, -11], [0, -14], [-34, 0], [0, 48], [16, 0], [3, -3], [27, -27], [28, 0], [-21, -21], [0, 2], [19, 19], [-18, 0], [0, -8], [17, -17], [-31, 31], [37, 37], [0, -16], [0, -54], [-10, 0], [-7, -7], [-25, 25], [0, 20], [-24, 0], [-9, 9], [35, 35], [28, 0], [-13, -13], [7, 7], [18, 0], [-33, -33], [5, -5], [-27, 27], [0, 6], [48, 0], [0, -28], [21, 21], [34, 0], [31, -31], [31, -31], [-1, 1], [0, -2], [-21, -21], [-5, -5], [0, 10], [-17, -17], [0, 8], [-31, 31], [36, 0], [-50, 0], [0, 6], [-15, 15], [40, 0], [0, 24], [0, -22], [-6, 0], [-23, -23], [0, -36], [-25, 25], [0, 30], [19, -19], [23, 23], [35, 35], [0, 48], [3, 3], [-21, -21], [0, 44], [0, -66], [-19, 19], [25, 25], [0, 10], [-42, 0], [-41, 41], [-37, 37], [14, 0], [-9, 9], [-9, -9], [15, 15], [0, 64], [0, 42], [-17, -17], [0, 56], [-36, 0], [49, 49], [-24, 0], [5, 5], [0, 24], [0, 42], [-15, -15], [-2, 0], [45, 45], [15, -15], [0, 0], [-26, 0], [-25, 25], [30, 0], [25, -25], [5, -5], [18, 0], [56, 0], [0, 48], [25, 25], [7, -7], [0, 68], [-1, 1], [32, 0], [-25, 25], [0, -42], [-37, 37], [28, 0], [0, -54], [-38, 0], [3, 3], [49, -49], [0, 20], [37, -37], [52, 0], [0, 22], [-51, -51], [0, 26], [-19, -19], [0, 40], [39, -39], [-16, 0], [39, -39], [0, -18], [-36, 0], [19, 19], [-1, -1], [-74, 0], [-11, 11], [0, -4], [-19, 19], [-60, 0], [-26, 0], [-15, 15], [-11, -11], [-39, -39], [5, 5], [11, -11], [0, 40], [15, -15], [-25, -25], [-12, 0], [0, 24], [-50, 0], [-1, -1], [-3, 3], [-2, 0], [-43, 43], [13, -13], [-32, 0], [13, 13], [21, 21], [-29, 29], [0, -16], [-58, 0], [0, 62], [3, 3], [0, 10], [-7, 7], [-10, 0], [33, -33], [0, 6], [17, -17], [-3, -3], [-8, 0], [0, 60], [-42, 0], [7, -7], [-1, 1], [0, -74], [-5, 5], [0, -36], [0, 26], [45, -45], [12, 0], [-3, 3], [-19, 19], [41, 41], [-24, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1440_x_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1440_2_x_f();". // 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_1440_2_x_f(:prec:=2) chi := MakeCharacter_1440_x(); 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_1440_2_x_f();". // 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_1440_2_x_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1440_x(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![18, -6, 1]>,<11,R![4, 0, 1]>,<17,R![2, 2, 1]>,<19,R![4, 1]>],Snew); return Vf; end function;