// Make newform 1350.2.a.v in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1350_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_1350_2_a_v();". // 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_1350_2_a_v();". // 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_1350_a();" function MakeCharacter_1350_a() N := 1350; order := 1; char_gens := [1001, 1027]; v := [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_1350_a_Hecke(Kf) return MakeCharacter_1350_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], [0], [0], [4], [5], [-3], [1], [-6], [1], [9], [-5], [-2], [2], [1], [-13], [0], [4], [8], [-4], [6], [-2], [9], [4], [14], [-10], [-17], [-10], [6], [-8], [-3], [4], [3], [-18], [-8], [13], [13], [-13], [-19], [12], [-16], [-20], [-22], [6], [4], [-22], [3], [0], [22], [-2], [2], [14], [-16], [-23], [-15], [21], [-8], [-9], [16], [-26], [18], [20], [-6], [21], [-18], [-14], [4], [10], [16], [-6], [16], [11], [-6], [22], [-21], [-28], [-19], [5], [13], [16], [11], [23], [-26], [8], [22], [0], [-30], [0], [16], [6], [-6], [6], [2], [-12], [-24], [-12], [9], [21], [22], [23], [10], [25], [42], [-4], [20], [-32], [-32], [36], [21], [42], [-19], [-10], [-23], [-29], [26], [-4], [20], [21], [0], [-26], [-20], [44], [48], [4], [-44], [22], [-31], [-24], [20], [0], [-46], [10], [-9], [-7], [43], [-12], [3], [-18], [13], [-14], [-12], [-26], [46], [24], [22], [0], [34], [-19], [-6], [2], [39], [-1], [18], [36], [51], [-3], [36], [33], [6], [-40], [-31], [48], [-61], [2], [9], [-39], [29], [-17], [15], [-19], [6], [27], [48], [62], [56], [-41], [12], [62], [-6], [-2], [-28], [4], [4], [-13], [57], [-16], [-54], [-59], [-19], [14], [-38], [6], [-20], [52], [-45], [-60], [-30], [-54], [-51], [42], [-16], [-57], [48], [-18], [22], [-1], [-6], [15], [46], [16], [28], [20], [-9], [-58], [-18], [-26], [19], [26], [-50], [-37], [64], [-36], [-41], [18], [-10], [48], [-12], [49], [-38], [-46], [53], [-46], [64], [5], [6], [-20], [-8], [-45], [50], [36], [-54], [-30], [8], [70], [46], [34], [6], [-18], [63], [62], [73], [-57], [-64], [8], [10], [14], [45], [-68], [59], [-52], [8], [36], [-14], [-56], [-55], [3], [32], [-18], [68], [61], [-6], [20], [-49], [44], [-29], [40], [-52], [44], [-40], [-7], [15], [51], [-1], [-43], [-44], [29], [80], [6], [44], [-37], [2], [23], [54], [27], [-24], [46], [-42], [-48], [60], [17], [72], [68], [0], [-25], [78], [-52], [70], [80], [-22], [76], [39], [-51], [42], [36], [-39], [37], [47], [43], [-44], [-50], [66], [38], [-4], [57], [40], [69], [29], [-24], [36], [-59], [22], [-44], [22], [-19], [14], [66], [42], [10], [-30], [31], [-74], [-21], [-67], [-70], [-61], [-20], [-33], [30], [-76], [4], [-12], [-70], [54], [45], [42], [38], [3], [-48], [27], [55], [13], [-62], [-60], [-24], [35], [-63], [-2], [-66], [-54], [-45], [-17], [-70], [-53], [-6], [-12], [-45], [35], [82], [90], [-84], [68], [33], [-25], [-100], [47], [-30], [63], [48], [38], [-39], [-5], [53], [-60], [-60], [-43], [10], [-14], [80], [10], [-40], [-9], [-8], [7], [8], [63], [54], [44], [39], [42], [-76], [-39], [86], [90], [-6], [76], [46], [-94], [-88], [-6], [-74], [81], [-39], [-46], [-47], [45], [-34], [-42], [-22], [-74], [-90], [-64]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1350_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1350_2_a_v();". // 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_1350_2_a_v(:prec:=1) chi := MakeCharacter_1350_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_1350_2_a_v();". // 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_1350_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 1]>,<11,R![-5, 1]>,<13,R![3, 1]>,<17,R![-1, 1]>],Snew); return Vf; end function;