// Make newform 8048.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8048_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_8048_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_8048_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_8048_a();" function MakeCharacter_8048_a() N := 8048; order := 1; char_gens := [1007, 6037, 2017]; 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_8048_a_Hecke(Kf) return MakeCharacter_8048_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], [-1], [5], [-5], [0], [6], [-1], [6], [-8], [-4], [0], [5], [9], [4], [12], [7], [9], [2], [14], [0], [1], [-8], [-2], [2], [-18], [18], [10], [5], [-12], [5], [-2], [-12], [0], [-10], [-12], [6], [20], [25], [18], [-8], [8], [-2], [13], [-24], [8], [19], [16], [-7], [-13], [16], [10], [-8], [-1], [15], [-6], [5], [6], [3], [-20], [31], [26], [-18], [4], [-27], [32], [-4], [18], [0], [28], [8], [-35], [27], [-35], [12], [10], [27], [-23], [-16], [18], [-6], [-16], [-18], [0], [-11], [12], [2], [-6], [33], [18], [26], [2], [36], [-6], [-1], [-25], [15], [-14], [-4], [44], [19], [-24], [-42], [-14], [44], [-28], [-36], [0], [-7], [27], [-24], [12], [36], [-7], [-31], [22], [0], [43], [33], [-41], [14], [2], [-6], [-16], [-33], [-50], [20], [-37], [-28], [35], [44], [-32], [8], [-1], [-4], [44], [-12], [-6], [-54], [-35], [-42], [14], [-20], [-10], [39], [31], [-15], [-16], [-32], [38], [-6], [-52], [-24], [2], [-52], [-12], [34], [32], [-45], [58], [-27], [28], [3], [30], [-42], [7], [-24], [30], [18], [37], [-32], [-24], [37], [-1], [-5], [-20], [14], [-20], [61], [-37], [15], [-26], [37], [-51], [-6], [18], [-15], [-24], [-9], [-49], [-2], [0], [-61], [30], [64], [24], [15], [6], [-2], [3], [-3], [-29], [69], [-36], [-66], [39], [24], [7], [-56], [43], [72], [63], [28], [-56], [-34], [-48], [50], [15], [-3], [-62], [-31], [-36], [-12], [-44], [-23], [-33], [27], [-7], [69], [-38], [-44], [55], [-10], [-26], [28], [41], [-18], [-20], [-21], [71], [-27], [14], [74], [-9], [-15], [-34], [-72], [40], [-36], [34], [-70], [-63], [33], [-66], [-8], [38], [-20], [23], [-42], [20], [12], [34], [-25], [7], [59], [29], [14], [-20], [-14], [-66], [12], [-9], [52], [-25], [-56], [-18], [16], [-25], [-54], [20], [-31], [-15], [-34], [8], [46], [-79], [58], [55], [63], [25], [7], [4], [-4], [34], [60], [12], [-42], [-22], [8], [-55], [38], [-6], [-40], [-58], [10], [-40], [-86], [23], [16], [18], [84], [-14], [-26], [-63], [55], [36], [69], [80], [-71], [-18], [-28], [13], [16], [18], [-88], [-4], [76], [-12], [-17], [-18], [-2], [0], [45], [44], [-44], [-66], [38], [58], [56], [55], [-22], [85], [-23], [4], [-84], [58], [-18], [-37], [-18], [62], [38], [-44], [14], [80], [-24], [53], [66], [-16], [2], [34], [-75], [-29], [-94], [-32], [-62], [9], [58], [75], [23], [-17], [33], [46], [5], [-50], [33], [-36], [37], [-66], [-80], [-66], [67], [-89], [82], [-44], [-48], [-91], [-17], [-101], [39], [58], [-54], [-3], [-21], [-32], [-11], [60], [50], [-1], [-74], [-27], [-18], [41], [-38], [8], [1], [-18], [21], [-69], [72], [-2], [1], [24], [75], [-86], [26], [24], [40], [58], [-49], [-51], [-2], [36], [60], [61], [10], [-39], [62], [8], [-72]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8048_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8048_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_8048_2_a_g(:prec:=1) chi := MakeCharacter_8048_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_8048_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_8048_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8048_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<5,R![0, 1]>,<7,R![1, 1]>,<13,R![5, 1]>],Snew); return Vf; end function;