// Make newform 4800.2.a.ct in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4800_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_4800_2_a_ct();". // 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_4800_2_a_ct();". // 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_4800_a();" function MakeCharacter_4800_a() N := 4800; order := 1; char_gens := [4351, 901, 1601, 577]; v := [1, 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_4800_a_Hecke(Kf) return MakeCharacter_4800_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], [1], [0], [5], [6], [-3], [2], [-1], [2], [-6], [3], [-6], [4], [11], [10], [-8], [6], [-3], [-1], [-12], [-10], [-8], [-6], [-16], [7], [8], [-4], [-8], [7], [-12], [-8], [-16], [-6], [12], [6], [9], [7], [-7], [-16], [-6], [-2], [19], [-10], [-3], [-2], [21], [15], [-3], [-12], [3], [-20], [24], [-7], [20], [12], [12], [30], [8], [1], [26], [13], [-2], [13], [-14], [29], [-16], [20], [23], [-18], [-10], [-18], [-4], [-13], [-25], [5], [-20], [12], [11], [0], [-11], [-32], [-30], [22], [-19], [-5], [0], [-4], [-22], [16], [8], [-14], [-6], [3], [-8], [-17], [-36], [-6], [-14], [21], [-17], [16], [-14], [-18], [10], [-1], [-43], [24], [-4], [44], [-13], [0], [2], [32], [19], [31], [36], [-12], [-12], [-6], [28], [-10], [34], [-18], [-24], [-16], [14], [35], [-42], [13], [-34], [44], [12], [24], [-3], [-52], [27], [28], [-3], [28], [-24], [-9], [10], [49], [-32], [-50], [26], [55], [52], [-48], [8], [-17], [-40], [19], [-42], [-44], [-14], [1], [-30], [-23], [-46], [-2], [-24], [-56], [-46], [-10], [36], [-25], [-54], [29], [-30], [14], [25], [50], [35], [5], [-24], [63], [30], [21], [1], [-7], [-8], [61], [6], [18], [-62], [5], [41], [-35], [-30], [23], [12], [-55], [-42], [26], [-62], [46], [-26], [-18], [-46], [52], [68], [-38], [-46], [-20], [54], [-19], [-62], [-12], [41], [13], [-2], [-7], [-52], [38], [-2], [5], [-2], [-4], [66], [55], [-55], [18], [-32], [-20], [-55], [-6], [-42], [-37], [-42], [41], [52], [-55], [70], [60], [-24], [10], [0], [50], [40], [-54], [-47], [-21], [-11], [-22], [-30], [-1], [30], [33], [-26], [22], [30], [68], [5], [46], [-12], [47], [-44], [-48], [-62], [17], [-10], [-22], [-49], [-74], [-16], [22], [16], [-44], [-22], [25], [-48], [-58], [-65], [-9], [81], [-78], [33], [15], [24], [60], [16], [62], [-67], [13], [6], [15], [-52], [-56], [12], [-34], [-12], [12], [32], [-77], [4], [-5], [36], [-18], [-4], [74], [68], [-65], [30], [36], [67], [44], [-81], [16], [38], [-2], [60], [44], [55], [-38], [-47], [14], [-12], [-33], [-78], [40], [-3], [48], [69], [38], [18], [23], [41], [-42], [20], [47], [-58], [79], [-34], [-73], [-4], [-3], [-24], [13], [72], [43], [36], [2], [-43], [-14], [-10], [7], [-47], [40], [84], [-41], [38], [-42], [-89], [25], [-94], [-36], [-74], [-46], [26], [47], [-18], [-42], [40], [28], [37], [72], [28], [47], [12], [-71], [-10], [92], [-64], [-83], [60], [-66], [9], [36], [91], [42], [-10], [48], [60], [-20], [34], [59], [-42], [-44], [-98], [93], [28], [-88], [-65], [22], [63], [19], [-16], [-47], [44], [-82], [-72], [-45], [88], [-96], [64], [-83], [36], [25], [-12], [9], [-36], [-74], [-1], [67], [68], [30], [67], [-60], [0], [-80], [43], [54], [-14], [50], [60], [52], [24], [17], [-102]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4800_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4800_2_a_ct();". // 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_4800_2_a_ct(:prec:=1) chi := MakeCharacter_4800_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_4800_2_a_ct();". // 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_4800_2_a_ct( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-5, 1]>,<11,R![-6, 1]>,<13,R![3, 1]>,<19,R![1, 1]>,<23,R![-2, 1]>],Snew); return Vf; end function;