// Make newform 1725.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1725_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_1725_2_a_k();". // 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_1725_2_a_k();". // 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_1725_a();" function MakeCharacter_1725_a() N := 1725; order := 1; char_gens := [1151, 277, 1201]; 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_1725_a_Hecke(Kf) return MakeCharacter_1725_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], [3], [-4], [0], [3], [-8], [-1], [9], [-5], [9], [7], [-4], [2], [-13], [-3], [-14], [-13], [-13], [4], [0], [1], [-8], [-10], [9], [-8], [-11], [-16], [-1], [18], [-4], [-2], [-3], [-14], [-8], [3], [2], [22], [24], [0], [0], [-2], [-4], [-2], [-18], [7], [-10], [4], [4], [8], [13], [20], [18], [-18], [23], [-17], [-23], [-18], [24], [5], [-19], [0], [-8], [-25], [-30], [7], [-10], [16], [-1], [6], [-2], [-31], [6], [-8], [23], [-30], [-8], [30], [25], [0], [4], [20], [37], [-24], [-6], [-11], [35], [26], [20], [13], [6], [-2], [3], [21], [-21], [30], [-12], [20], [-10], [-20], [-21], [11], [-6], [-22], [20], [-14], [-18], [-4], [-41], [10], [-14], [33], [-4], [38], [-20], [-13], [-12], [-16], [12], [-14], [34], [45], [-26], [8], [-6], [22], [-7], [37], [-51], [27], [48], [-16], [-45], [-41], [6], [46], [27], [-27], [-39], [-25], [-10], [16], [-37], [-27], [-6], [46], [18], [43], [-54], [22], [30], [34], [46], [-53], [-40], [24], [57], [-10], [10], [8], [22], [-10], [-12], [25], [-15], [-33], [-8], [-28], [10], [-24], [-3], [22], [-7], [-32], [-54], [-36], [49], [-15], [-16], [-16], [22], [50], [48], [-41], [-24], [4], [-3], [-15], [-33], [-4], [-54], [-52], [-49], [48], [-39], [-40], [-14], [10], [46], [-2], [0], [4], [-44], [-46], [-58], [70], [-64], [47], [-8], [-54], [42], [-32], [36], [-11], [-10], [50], [-9], [-73], [-56], [5], [50], [41], [-19], [-12], [-5], [6], [-59], [28], [55], [12], [48], [-6], [-11], [-1], [12], [10], [54], [21], [-65], [-3], [63], [74], [-50], [-24], [-8], [22], [52], [50], [-51], [50], [2], [7], [-62], [16], [33], [14], [64], [-60], [74], [-23], [-69], [-21], [51], [24], [62], [42], [-50], [61], [16], [53], [31], [-35], [-34], [2], [-12], [-70], [-47], [22], [-18], [26], [-50], [60], [40], [-8], [19], [-23], [-42], [24], [54], [-10], [39], [44], [-2], [-6], [12], [-20], [-68], [51], [54], [15], [51], [-16], [26], [16], [-46], [-60], [50], [42], [-2], [12], [-44], [10], [40], [-33], [90], [49], [45], [-35], [-54], [10], [15], [2], [22], [23], [-2], [-38], [92], [24], [69], [15], [56], [77], [-40], [0], [-28], [-58], [-46], [-22], [35], [10], [-90], [30], [-66], [-47], [-73], [13], [-72], [-66], [-3], [37], [38], [-7], [17], [-74], [-46], [-12], [-18], [76], [66], [13], [-13], [62], [36], [-22], [-26], [14], [-57], [-12], [-65], [-53], [-58], [-12], [28], [-68], [-67], [-18], [41], [-24], [-98], [18], [86], [32], [-16], [40], [66], [67], [-30], [-61], [-11], [-28], [58], [3], [-86], [40], [17], [10], [-96], [8], [87], [-49], [14], [17], [-27], [57], [-71], [35], [30], [32], [-60], [-34], [-64], [36], [36], [-13], [50], [60], [-54], [87], [-72], [96], [-6], [24], [84], [94], [-24], [64], [-31], [-52], [87]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1725_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1725_2_a_k();". // 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_1725_2_a_k(:prec:=1) chi := MakeCharacter_1725_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_1725_2_a_k();". // 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_1725_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1725_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<7,R![-3, 1]>,<11,R![4, 1]>],Snew); return Vf; end function;