// Make newform 2850.2.a.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2850_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_2850_2_a_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_2850_2_a_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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2850_a();" function MakeCharacter_2850_a() N := 2850; order := 1; char_gens := [1901, 1027, 1351]; 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_2850_a_Hecke(Kf) return MakeCharacter_2850_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 := [[-1], [-1], [0], [4], [-1], [0], [8], [1], [3], [-1], [1], [2], [-10], [8], [0], [-3], [4], [5], [5], [-6], [13], [-5], [-11], [3], [-10], [-12], [-1], [-14], [6], [1], [17], [-15], [-4], [6], [0], [0], [-2], [-10], [22], [11], [20], [12], [13], [14], [0], [26], [25], [-29], [30], [13], [4], [24], [8], [16], [27], [19], [-26], [-14], [-17], [-15], [-32], [-1], [5], [-16], [21], [-9], [-1], [16], [-12], [-25], [0], [16], [22], [-32], [-32], [18], [-22], [-35], [-5], [-18], [-12], [22], [-14], [22], [7], [-3], [-23], [22], [-12], [-4], [3], [21], [32], [4], [-4], [4], [25], [-1], [-4], [27], [-21], [-36], [-22], [6], [6], [21], [33], [20], [-24], [-38], [-27], [-38], [-6], [-20], [44], [6], [-2], [-43], [24], [0], [-8], [-16], [-3], [6], [10], [-8], [-35], [29], [18], [-23], [4], [24], [40], [-23], [-26], [-45], [-54], [-1], [46], [-16], [-49], [-10], [26], [28], [26], [-48], [-46], [18], [34], [-28], [-42], [-8], [26], [48], [48], [18], [-2], [-12], [-42], [-19], [52], [15], [-38], [-8], [-2], [-30], [-25], [13], [-30], [46], [44], [-40], [-20], [-21], [17], [-30], [28], [10], [16], [-47], [18], [-18], [18], [23], [-39], [-36], [44], [-17], [52], [7], [-64], [-61], [-9], [-9], [68], [22], [-39], [29], [58], [33], [-45], [8], [-20], [-28], [69], [52], [22], [-60], [2], [28], [29], [26], [-48], [42], [20], [-26], [-42], [-45], [18], [18], [-18], [69], [-42], [-26], [2], [22], [-49], [-42], [0], [-39], [-26], [29], [-75], [-5], [-34], [-24], [-58], [-18], [-12], [-42], [-52], [-16], [14], [42], [46], [-25], [44], [-50], [-49], [73], [17], [66], [-57], [-58], [-52], [-55], [-58], [-13], [78], [38], [-3], [20], [47], [58], [46], [-13], [10], [-32], [-13], [-64], [40], [-23], [65], [-10], [-36], [64], [-19], [-12], [20], [-5], [-36], [28], [-48], [62], [56], [-36], [31], [-61], [-64], [17], [-54], [1], [-54], [70], [-16], [78], [-4], [76], [-62], [-26], [39], [-11], [-10], [-16], [-14], [-44], [36], [-32], [31], [-25], [-67], [70], [-5], [-69], [16], [84], [-9], [-18], [-82], [-2], [48], [-2], [-34], [80], [-24], [42], [68], [92], [44], [58], [-65], [-79], [-36], [25], [-46], [-33], [62], [29], [1], [-68], [-4], [-2], [15], [-25], [-7], [54], [-53], [-37], [24], [48], [-45], [32], [88], [-37], [8], [57], [-19], [33], [-58], [-44], [-67], [0], [-78], [-50], [24], [64], [78], [64], [48], [45], [-71], [56], [-9], [-4], [74], [-50], [-83], [-47], [52], [14], [-18], [-30], [-40], [-96], [-35], [-12], [-28], [-94], [-24], [-46], [68], [57], [-9], [-10], [26], [-66], [64], [-27], [73], [48], [34], [26], [-4], [-57], [61], [-34], [-97], [-71], [-47], [-19], [-64], [54], [-72], [32], [-43], [106], [16], [-53], [-84], [78], [-38], [58], [-25], [-14], [-68], [42], [-64], [-12], [72], [-101]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2850_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_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_2850_2_a_f(:prec:=1) chi := MakeCharacter_2850_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_2850_2_a_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_2850_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2850_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 1]>,<11,R![1, 1]>,<13,R![0, 1]>,<23,R![-3, 1]>],Snew); return Vf; end function;