// Make newform 1040.2.q.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1040_q();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_q_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_1040_2_q_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_1040_2_q_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; 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_1040_q();" function MakeCharacter_1040_q() N := 1040; order := 3; char_gens := [911, 261, 417, 561]; v := [3, 3, 3, 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_1040_q_Hecke();" function MakeCharacter_1040_q_Hecke(Kf) N := 1040; order := 3; char_gens := [911, 261, 417, 561]; 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], [1, -1], [1, 0], [0, -1], [3, -3], [3, -4], [0, 3], [0, -7], [-3, 3], [-3, 3], [4, 0], [7, -7], [9, -9], [0, 11], [0, 0], [-6, 0], [0, -3], [0, -11], [-7, 7], [0, -3], [2, 0], [-8, 0], [12, 0], [-15, 15], [0, 7], [9, -9], [-8, 0], [9, -9], [2, 0], [0, -9], [-19, 19], [12, 0], [0, 15], [0, 5], [0, 21], [-8, 0], [-10, 0], [0, -1], [-3, 3], [0, 3], [-21, 21], [2, 0], [0, -3], [-5, 5], [-21, 21], [0, 17], [11, -11], [-19, 19], [0, 27], [-22, 0], [18, 0], [0, 0], [0, 1], [0, 21], [-9, 9], [-3, 3], [0, -27], [23, -23], [0, 19], [6, 0], [5, -5], [0, 27], [-20, 0], [-24, 0], [-22, 0], [-18, 0], [0, -19], [-34, 0], [0, -33], [1, -1], [-9, 9], [24, 0], [5, -5], [0, 31], [-1, 1], [0, -9], [18, 0], [0, -5], [33, -33], [0, 25], [-9, 9], [2, 0], [-9, 9], [0, -29], [11, -11], [0, 0], [0, -3], [-5, 5], [0, -27], [4, 0], [36, 0], [39, -39], [0, 11], [-9, 9], [-32, 0], [0, 15], [9, -9], [30, 0], [29, -29], [-22, 0], [-8, 0], [39, -39], [0, 39], [-27, 27], [40, 0], [2, 0], [33, -33], [-6, 0], [-24, 0], [-35, 35], [0, -13], [43, -43], [0, -33], [-44, 0], [0, 17], [0, -27], [0, 11], [45, -45], [39, -39], [0, -39], [1, -1], [-17, 17], [-42, 0], [0, 51], [23, -23], [18, 0], [0, 37], [0, 9], [52, 0], [-34, 0], [47, -47], [21, -21], [-13, 13], [-29, 29], [0, -3], [13, -13], [0, 27], [0, -37], [0, 51], [-15, 15], [-32, 0], [-15, 15], [0, -13], [12, 0], [-11, 11], [0, 21], [-22, 0], [-30, 0], [-44, 0], [-24, 0], [0, -41], [-27, 27], [4, 0], [9, -9], [-19, 19], [12, 0], [0, 29], [0, -51], [2, 0], [42, 0], [21, -21], [0, 51], [40, 0], [0, 21], [-9, 9], [36, 0], [-61, 61], [0, 31], [2, 0], [6, 0], [-36, 0], [0, 13], [0, -3], [0, 7], [40, 0], [0, -15], [-1, 1], [-30, 0], [-19, 19], [25, -25], [-8, 0], [-36, 0], [14, 0], [6, 0], [21, -21], [0, -63], [-10, 0], [-44, 0], [37, -37], [0, -15], [0, 43], [0, 3], [-20, 0], [-27, 27], [0, -33], [15, -15], [-10, 0], [0, -29], [-18, 0], [-24, 0], [0, 9], [0, -7], [19, -19], [-10, 0], [-21, 21], [3, -3], [40, 0], [0, -9], [9, -9], [0, -31], [19, -19], [42, 0], [41, -41], [0, -57], [0, -3], [-10, 0], [-32, 0], [0, -27], [-15, 15], [-18, 0], [-59, 59], [40, 0], [-18, 0], [0, -73], [-63, 63], [50, 0], [51, -51], [0, 45], [0, -49], [12, 0], [55, -55], [35, -35], [-49, 49], [-18, 0], [52, 0], [72, 0], [0, 1], [27, -27], [0, 9], [-33, 33], [57, -57], [-25, 25], [0, 59], [73, -73], [0, 39], [0, 0], [0, 47], [15, -15], [0, -19], [-51, 51], [7, -7], [9, -9], [-48, 0], [49, -49], [-18, 0], [0, -39], [0, -47], [-19, 19], [-6, 0], [0, 43], [64, 0], [33, -33], [-22, 0], [-29, 29], [0, 27], [0, -67], [0, 9], [30, 0], [0, -13], [0, 51], [14, 0], [-20, 0], [-53, 53], [0, -31], [0, 31], [-79, 79], [0, 27], [26, 0], [0, 13], [0, 45], [-15, 15], [-13, 13], [0, 0], [13, -13], [4, 0], [-48, 0], [26, 0], [-18, 0], [0, -55], [0, 57], [-87, 87], [0, 63], [39, -39], [0, -15], [0, -17], [-54, 0], [-8, 0], [63, -63], [-9, 9], [53, -53], [0, -17], [6, 0], [11, -11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1040_q_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_q_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_1040_2_q_j(:prec:=2) chi := MakeCharacter_1040_q(); 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_1040_2_q_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_1040_2_q_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1040_q(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, -1, 1]>,<7,R![1, 1, 1]>,<11,R![9, -3, 1]>,<19,R![49, 7, 1]>],Snew); return Vf; end function;