// Make newform 2450.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2450_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_2450_2_a_g();". // 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_2450_2_a_g();". // 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_2450_a();" function MakeCharacter_2450_a() N := 2450; order := 1; char_gens := [1177, 101]; v := [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_2450_a_Hecke(Kf) return MakeCharacter_2450_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], [0], [-3], [4], [3], [-5], [6], [0], [-2], [2], [3], [-4], [-12], [6], [0], [-2], [-13], [12], [-11], [-10], [9], [-15], [-2], [18], [4], [-3], [-10], [-9], [2], [-12], [-3], [-5], [0], [2], [-2], [11], [-12], [24], [-15], [-2], [-18], [-19], [-18], [-20], [-13], [4], [-12], [-20], [6], [0], [-17], [-27], [18], [6], [0], [-2], [32], [-18], [-11], [-6], [-17], [18], [-26], [12], [17], [-13], [-3], [10], [-6], [30], [28], [26], [-25], [-6], [30], [28], [-3], [-5], [15], [-28], [-18], [19], [40], [-9], [-15], [17], [-12], [-4], [-12], [-30], [2], [12], [20], [-36], [30], [3], [-41], [32], [17], [12], [24], [-15], [-28], [13], [33], [9], [30], [13], [-32], [-34], [-18], [-20], [2], [42], [4], [-12], [6], [15], [-32], [26], [48], [-39], [-17], [12], [20], [30], [-32], [4], [20], [6], [32], [32], [3], [-5], [24], [28], [-42], [-30], [28], [42], [-4], [-33], [-20], [0], [-26], [3], [-5], [-24], [-28], [18], [41], [-12], [-28], [12], [-10], [-30], [13], [-12], [-48], [-39], [32], [3], [57], [-6], [-58], [28], [5], [24], [15], [-2], [-18], [11], [40], [-15], [-13], [-18], [-56], [-50], [-28], [-57], [-34], [33], [36], [60], [2], [19], [-55], [48], [49], [51], [-13], [18], [-3], [54], [-13], [26], [-27], [-6], [-30], [28], [-32], [-35], [60], [-42], [-50], [36], [30], [-32], [2], [-42], [-64], [-57], [30], [-17], [2], [3], [-18], [66], [62], [-20], [-15], [-34], [-57], [-10], [9], [0], [28], [-33], [26], [25], [2], [-33], [64], [18], [-5], [-24], [-45], [-12], [-39], [-47], [-26], [20], [-51], [0], [58], [-12], [-25], [-24], [32], [-27], [-18], [-35], [24], [15], [32], [73], [78], [43], [-34], [27], [40], [4], [3], [55], [30], [18], [41], [-54], [28], [47], [-71], [-40], [-2], [4], [57], [-40], [62], [63], [-36], [-58], [-72], [28], [13], [72], [-34], [-48], [-80], [30], [42], [-12], [-69], [-27], [-4], [-30], [-2], [24], [-60], [73], [-41], [-18], [80], [21], [-13], [17], [12], [10], [-30], [-34], [-6], [-30], [-63], [-64], [12], [10], [75], [72], [49], [-45], [28], [77], [-12], [26], [-54], [-2], [35], [4], [42], [36], [-28], [42], [10], [-21], [17], [48], [-40], [-66], [-47], [-2], [56], [-63], [-30], [92], [6], [60], [28], [-13], [48], [18], [73], [2], [-18], [4], [50], [-51], [60], [63], [-18], [-84], [2], [93], [-18], [60], [-47], [11], [-72], [56], [-43], [12], [-5], [-84], [60], [-32], [-58], [-45], [42], [-11], [15], [-2], [-42], [81], [-88], [87], [40], [-66], [32], [-32], [56], [-42], [-85], [-36], [-45], [73], [42], [-19], [-20], [15], [-43], [-78], [40], [-99], [92], [-57], [0], [88], [2], [42], [4], [-75], [49], [-48], [-9], [-28], [-73], [-102], [30], [-2], [-57], [-6], [0], [-62], [-78], [-15], [-101], [48], [96], [75], [-17], [-30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2450_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2450_2_a_g();". // 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_2450_2_a_g(:prec:=1) chi := MakeCharacter_2450_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_2450_2_a_g();". // 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_2450_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<11,R![3, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>,<19,R![5, 1]>,<23,R![-6, 1]>,<37,R![-2, 1]>],Snew); return Vf; end function;