// Make newform 3136.2.a.s in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3136_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_3136_2_a_s();". // 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_3136_2_a_s();". // 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_3136_a();" function MakeCharacter_3136_a() N := 3136; order := 1; char_gens := [1471, 197, 1473]; 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_3136_a_Hecke(Kf) return MakeCharacter_3136_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], [-3], [0], [-3], [-2], [3], [-1], [-3], [6], [7], [1], [6], [-4], [9], [-3], [9], [1], [-7], [0], [-1], [13], [12], [15], [-10], [-15], [-11], [15], [1], [6], [-8], [3], [-21], [20], [-3], [-17], [13], [11], [12], [9], [21], [10], [9], [11], [-18], [7], [-4], [-8], [-3], [-11], [-21], [12], [-1], [0], [3], [3], [-3], [-11], [13], [30], [29], [-6], [-28], [27], [23], [9], [-13], [-34], [9], [-26], [-21], [-15], [-5], [25], [8], [33], [-15], [37], [3], [11], [-12], [22], [15], [-10], [1], [-9], [-18], [23], [6], [16], [-21], [3], [19], [24], [11], [0], [-3], [39], [-1], [-35], [8], [33], [9], [-9], [29], [-1], [12], [-21], [27], [14], [-47], [25], [6], [-31], [16], [15], [20], [-21], [-39], [12], [-11], [14], [-27], [21], [-13], [18], [1], [21], [-32], [25], [-19], [48], [25], [-2], [3], [-34], [33], [-31], [-42], [-33], [-52], [-27], [-5], [-36], [1], [24], [22], [15], [23], [-51], [13], [18], [44], [-39], [41], [-48], [1], [-9], [26], [-27], [-45], [-18], [16], [51], [27], [-33], [19], [-59], [-10], [33], [-39], [-50], [-3], [23], [7], [6], [32], [-3], [16], [13], [13], [12], [-38], [15], [27], [21], [-47], [5], [11], [15], [-1], [-60], [47], [-27], [21], [-45], [11], [37], [30], [21], [9], [40], [-23], [35], [36], [-3], [61], [39], [-6], [41], [-37], [30], [-8], [27], [63], [-13], [31], [51], [57], [66], [61], [-20], [39], [25], [60], [-26], [-45], [-21], [-23], [-45], [61], [65], [16], [27], [-4], [3], [11], [21], [60], [36], [-39], [-1], [-29], [-47], [-30], [45], [-8], [45], [5], [36], [-2], [15], [-9], [14], [21], [-57], [37], [-67], [54], [-61], [-5], [-60], [25], [-26], [-45], [23], [-18], [-42], [-4], [-63], [61], [29], [59], [-47], [-58], [-11], [3], [13], [71], [63], [63], [-65], [24], [-50], [-49], [-15], [-1], [30], [67], [54], [57], [21], [-33], [24], [-50], [-63], [-23], [6], [51], [-40], [23], [-51], [-65], [60], [-25], [62], [33], [70], [-39], [-35], [45], [-75], [-45], [-31], [-72], [35], [12], [63], [14], [6], [-67], [11], [-90], [28], [87], [71], [-37], [-13], [-27], [54], [-23], [9], [40], [-39], [-19], [-36], [-2], [3], [-70], [37], [-23], [30], [-42], [-32], [-63], [-36], [37], [71], [-12], [57], [-85], [11], [-42], [19], [25], [90], [-3], [21], [3], [24], [46], [27], [87], [27], [-55], [35], [78], [-89], [-70], [33], [71], [-39], [42], [-65], [73], [-27], [0], [11], [87], [14], [-87], [6], [-56], [-57], [68], [-57], [-5], [85], [-37], [-96], [50], [-27], [-87], [-85], [9], [-49], [-29], [-42], [-52], [9], [-23], [51], [-95], [-33], [-39], [-35], [37], [30], [29], [99], [71], [9], [-36], [-73], [38], [-15], [-15], [-41], [-18], [105], [-51], [1], [-48], [-12], [38], [-27], [-81], [-18], [65], [-33]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3136_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3136_2_a_s();". // 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_3136_2_a_s(:prec:=1) chi := MakeCharacter_3136_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_3136_2_a_s();". // 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_3136_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3136_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<5,R![3, 1]>,<11,R![3, 1]>],Snew); return Vf; end function;