// Make newform 2700.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2700_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_2700_2_a_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_2700_2_a_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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2700_a();" function MakeCharacter_2700_a() N := 2700; order := 1; char_gens := [1351, 1001, 2377]; 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_2700_a_Hecke(Kf) return MakeCharacter_2700_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], [1], [-6], [1], [0], [-1], [6], [-6], [8], [7], [6], [4], [12], [-6], [0], [11], [7], [6], [-11], [-1], [6], [12], [13], [0], [13], [-18], [-10], [-18], [16], [-12], [-18], [5], [12], [11], [-2], [-11], [12], [12], [-6], [-7], [12], [19], [24], [5], [17], [16], [6], [-10], [24], [-30], [17], [-6], [0], [-12], [-24], [-25], [-2], [0], [28], [30], [16], [0], [-17], [12], [-19], [-23], [-6], [-1], [-36], [18], [-23], [7], [11], [24], [0], [-14], [6], [-31], [24], [-19], [18], [-26], [8], [-12], [-36], [10], [12], [13], [12], [18], [-23], [30], [32], [-6], [-30], [30], [-5], [-43], [-35], [12], [-12], [24], [-13], [31], [6], [6], [18], [14], [13], [-29], [6], [11], [-25], [0], [4], [-30], [-12], [-48], [-13], [-17], [12], [-18], [-28], [-18], [5], [-6], [-8], [22], [20], [-24], [-13], [-23], [-42], [-37], [6], [-17], [0], [48], [-4], [-24], [-17], [36], [-7], [-12], [-41], [42], [41], [-12], [-5], [18], [37], [48], [19], [-24], [-16], [-18], [7], [42], [24], [0], [31], [36], [-18], [-42], [5], [10], [5], [48], [-30], [-22], [24], [37], [17], [30], [20], [42], [-41], [50], [-53], [-36], [22], [-12], [36], [54], [1], [-35], [-7], [-48], [37], [-24], [65], [-12], [24], [-6], [23], [-35], [-30], [30], [12], [35], [-23], [-55], [6], [-54], [35], [-42], [-42], [23], [7], [0], [-5], [42], [18], [35], [52], [42], [-6], [-18], [-43], [-4], [-60], [16], [54], [11], [12], [-54], [37], [36], [-23], [-52], [-16], [-30], [1], [60], [-67], [12], [30], [12], [6], [71], [-53], [71], [36], [6], [-53], [-24], [44], [54], [-50], [-24], [12], [41], [18], [-36], [5], [-8], [42], [70], [13], [-42], [5], [79], [24], [-28], [30], [-30], [-68], [-48], [-49], [-47], [-26], [-25], [82], [49], [-78], [74], [14], [-54], [18], [-4], [-54], [-13], [-83], [24], [85], [0], [-55], [6], [-18], [-66], [42], [36], [10], [24], [5], [-48], [-6], [-17], [31], [-30], [44], [0], [-67], [-62], [24], [-19], [24], [7], [-84], [60], [-18], [61], [30], [26], [-60], [78], [-38], [78], [-73], [-53], [78], [-32], [-6], [-43], [-16], [-44], [30], [78], [-67], [-78], [35], [42], [-52], [-30], [-13], [90], [2], [-56], [67], [66], [-54], [71], [24], [36], [-70], [4], [-60], [-78], [-13], [61], [-12], [64], [47], [-54], [-78], [-30], [30], [54], [-77], [-72], [18], [-60], [-47], [-17], [48], [-47], [23], [-36], [-7], [36], [6], [-31], [-71], [6], [-12], [31], [-78], [-11], [-60], [30], [-29], [-42], [-49], [-12], [-64], [-23], [61], [48], [74], [-48], [42], [19], [18], [37], [-61], [-60], [20], [-102], [50], [6], [-8], [-42], [-84], [-52], [13], [42], [-47], [-42], [34], [-12], [60], [41], [13], [18], [72], [-77], [42], [-54], [66], [-26], [-48], [-96], [-74], [12], [18], [-90], [56], [-54]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2700_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2700_2_a_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_2700_2_a_k(:prec:=1) chi := MakeCharacter_2700_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_2700_2_a_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_2700_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2700_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-1, 1]>,<11,R![6, 1]>,<13,R![-1, 1]>,<17,R![0, 1]>],Snew); return Vf; end function;