// Make newform 3525.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_l();". // 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_3525_2_a_l();". // 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_3525_a();" function MakeCharacter_3525_a() N := 3525; order := 1; char_gens := [2351, 1552, 2026]; 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_3525_a_Hecke(Kf) return MakeCharacter_3525_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], [5], [6], [-3], [3], [-1], [5], [-7], [0], [0], [-5], [6], [-1], [-5], [-9], [-7], [8], [-3], [10], [-10], [8], [14], [-12], [-4], [11], [12], [4], [-2], [-12], [8], [16], [1], [2], [-23], [-4], [4], [-15], [13], [-20], [20], [-24], [-22], [19], [17], [20], [-8], [25], [-14], [-14], [-16], [1], [27], [12], [24], [-6], [-24], [-16], [19], [-28], [14], [-13], [-4], [-1], [12], [-8], [-36], [-12], [28], [-18], [-4], [-26], [26], [10], [-6], [-19], [32], [-22], [-34], [4], [-36], [5], [19], [-30], [-33], [17], [-20], [-18], [-24], [-13], [35], [-37], [13], [19], [-13], [11], [0], [1], [-5], [-10], [-12], [28], [-6], [-4], [3], [-28], [36], [-38], [-19], [-22], [-12], [-34], [6], [-8], [30], [-19], [-12], [-1], [-20], [7], [5], [-6], [-6], [1], [33], [-6], [-3], [-14], [36], [-34], [11], [-19], [6], [34], [22], [-6], [32], [12], [-30], [42], [39], [-52], [18], [50], [-10], [-16], [-28], [35], [6], [-35], [39], [-9], [32], [57], [-48], [-5], [-6], [14], [0], [46], [-18], [28], [20], [-17], [-15], [42], [-19], [30], [18], [9], [23], [54], [-11], [-7], [-31], [-40], [-4], [-4], [-40], [0], [-34], [48], [27], [-9], [-30], [-42], [-53], [-7], [-46], [-14], [43], [33], [-34], [-2], [38], [-54], [13], [-3], [64], [-48], [-2], [-61], [59], [-53], [30], [19], [4], [-38], [-36], [16], [-38], [40], [-57], [21], [28], [56], [-47], [44], [36], [14], [38], [30], [-52], [-18], [10], [-46], [34], [-48], [24], [-5], [14], [46], [-12], [-22], [68], [-62], [69], [-41], [53], [-7], [14], [-6], [-8], [-6], [56], [-41], [30], [-14], [-26], [-5], [-8], [80], [-24], [24], [53], [12], [-2], [-26], [-76], [41], [-5], [73], [-72], [18], [-58], [-36], [3], [-20], [38], [49], [67], [34], [8], [77], [64], [-66], [-65], [-44], [45], [30], [29], [-40], [-34], [-43], [-40], [19], [66], [23], [60], [-14], [12], [44], [-11], [64], [-25], [12], [22], [-40], [36], [13], [-24], [-62], [-12], [-12], [83], [-8], [-31], [19], [-4], [-64], [64], [15], [64], [27], [59], [18], [12], [-55], [-42], [10], [55], [66], [-29], [36], [6], [40], [-6], [-4], [-22], [-13], [63], [-55], [-48], [-22], [-9], [-38], [3], [53], [8], [-14], [-90], [-44], [12], [18], [-45], [-32], [14], [-15], [69], [-24], [-16], [-10], [62], [64], [68], [-56], [-41], [-6], [-87], [33], [-41], [14], [36], [52], [10], [28], [-57], [36], [4], [64], [-20], [74], [-31], [65], [54], [-87], [52], [-78], [-58], [-18], [-69], [-16], [-7], [8], [-49], [-31], [-23], [1], [92], [36], [-7], [78], [-35], [-7], [-56], [-25], [36], [60], [69], [12], [7], [70], [-5], [-26], [-30], [-82], [6], [-46], [46], [-43], [67], [0], [56], [-8], [-66], [-81], [-18], [-30], [82], [-18], [-76], [-11], [-64], [77], [32], [20], [-80], [11], [36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3525_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_l();". // 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_3525_2_a_l(:prec:=1) chi := MakeCharacter_3525_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_3525_2_a_l();". // 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_3525_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3525_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, 1]>,<7,R![-5, 1]>,<11,R![-6, 1]>],Snew); return Vf; end function;