// Make newform 4650.2.a.bv in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4650_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_4650_2_a_bv();". // 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_4650_2_a_bv();". // 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_4650_a();" function MakeCharacter_4650_a() N := 4650; order := 1; char_gens := [3101, 2977, 1801]; 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_4650_a_Hecke(Kf) return MakeCharacter_4650_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], [3], [3], [2], [4], [-3], [-5], [4], [1], [0], [4], [-1], [-10], [-3], [6], [-2], [-2], [7], [-5], [-1], [-12], [1], [10], [-1], [-16], [-9], [20], [-9], [-8], [6], [10], [-14], [-11], [0], [5], [-14], [19], [22], [4], [5], [16], [6], [6], [7], [-19], [6], [27], [13], [-15], [12], [-18], [-12], [23], [-24], [2], [13], [-12], [16], [24], [-30], [-16], [0], [-30], [8], [12], [2], [32], [-24], [-18], [15], [-2], [9], [15], [-12], [-26], [-35], [25], [8], [12], [-22], [16], [1], [18], [-15], [2], [-10], [0], [-4], [-24], [3], [-32], [-37], [4], [-20], [0], [-18], [43], [22], [-22], [1], [4], [-23], [22], [-4], [-44], [-22], [-27], [42], [-29], [22], [27], [-38], [17], [26], [-19], [-3], [26], [-4], [-14], [-26], [-27], [19], [-31], [-21], [-25], [26], [-45], [18], [-20], [37], [-50], [-50], [45], [9], [31], [35], [-2], [17], [-27], [40], [18], [-20], [-35], [51], [41], [-42], [32], [-43], [-46], [26], [-11], [46], [52], [20], [-38], [-41], [52], [40], [-30], [52], [0], [-54], [-38], [56], [-25], [42], [2], [1], [-36], [-1], [-32], [24], [6], [51], [-20], [55], [-43], [27], [-24], [-23], [-2], [2], [-10], [-30], [13], [16], [48], [45], [7], [-51], [-8], [60], [-8], [-12], [46], [1], [33], [-6], [-13], [-7], [13], [-59], [6], [22], [-56], [-30], [-20], [-60], [-51], [36], [-16], [5], [-10], [-41], [-16], [46], [-8], [10], [27], [-4], [4], [19], [-53], [29], [38], [16], [54], [-22], [-10], [28], [6], [-3], [4], [-35], [-3], [-6], [-47], [72], [-48], [-10], [70], [-2], [-4], [45], [-14], [13], [-40], [56], [-7], [30], [-47], [-17], [12], [30], [2], [-41], [-34], [-72], [69], [70], [-11], [-59], [-14], [-49], [39], [-40], [-10], [28], [28], [-20], [67], [75], [70], [40], [-55], [34], [42], [15], [40], [-6], [58], [-34], [38], [66], [-29], [66], [-40], [66], [-35], [11], [-24], [-18], [-20], [-64], [-60], [26], [41], [81], [10], [59], [40], [51], [54], [88], [82], [58], [-66], [-30], [-16], [68], [34], [-9], [30], [-16], [-12], [48], [-60], [3], [20], [45], [-79], [55], [-57], [28], [-25], [62], [-60], [61], [22], [-38], [86], [-6], [5], [21], [-2], [-51], [25], [52], [-44], [62], [-12], [71], [-42], [-18], [72], [-59], [-78], [-48], [-40], [37], [87], [34], [-54], [-14], [-62], [3], [36], [-26], [55], [5], [19], [-6], [-18], [51], [14], [62], [-65], [-12], [61], [22], [-22], [18], [-58], [-19], [-69], [56], [36], [-57], [12], [-38], [-42], [28], [102], [-62], [-12], [0], [-101], [-57], [44], [65], [-32], [-78], [65], [-38], [63], [14], [82], [-28], [11], [57], [13], [46], [-60], [70], [-46], [30], [-11], [101], [-82], [36], [68], [-20], [64], [41], [-42], [-11], [3], [-6], [-42], [92], [17], [-32], [76], [79], [3], [45], [96], [66]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4650_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4650_2_a_bv();". // 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_4650_2_a_bv(:prec:=1) chi := MakeCharacter_4650_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_4650_2_a_bv();". // 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_4650_2_a_bv( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4650_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-3, 1]>,<11,R![-3, 1]>,<13,R![-2, 1]>,<19,R![3, 1]>],Snew); return Vf; end function;