// Make newform 8034.2.a.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_f();". // 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_8034_2_a_f();". // 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_8034_a();" function MakeCharacter_8034_a() N := 8034; order := 1; char_gens := [5357, 1237, 5773]; 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_8034_a_Hecke(Kf) return MakeCharacter_8034_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], [4], [-5], [1], [1], [2], [0], [1], [0], [0], [-5], [9], [3], [-3], [-6], [0], [13], [-4], [8], [-10], [-8], [0], [4], [-8], [19], [1], [12], [-1], [1], [7], [4], [15], [-16], [11], [-12], [8], [5], [18], [13], [17], [-2], [16], [5], [-8], [3], [1], [13], [12], [-8], [9], [-20], [7], [28], [-14], [-14], [30], [18], [8], [18], [-13], [-30], [-16], [-15], [-19], [-22], [10], [13], [-3], [-14], [-30], [-22], [18], [-35], [-2], [21], [-3], [-2], [15], [28], [9], [-24], [10], [8], [1], [28], [2], [-3], [9], [-20], [15], [25], [-14], [24], [-14], [-9], [-13], [-3], [-26], [-10], [22], [30], [-24], [3], [0], [23], [28], [-10], [26], [-22], [34], [16], [28], [20], [-49], [12], [-33], [-11], [-5], [-48], [30], [41], [46], [0], [-44], [30], [-16], [38], [16], [-7], [-25], [0], [-30], [-9], [-18], [-31], [-33], [-13], [-42], [22], [-22], [-37], [-3], [39], [-28], [-30], [0], [-28], [19], [3], [51], [50], [-10], [-37], [2], [42], [27], [-29], [16], [-38], [5], [30], [-20], [0], [-1], [-14], [-52], [-22], [16], [-61], [-10], [-5], [-24], [-19], [-20], [-12], [32], [20], [-44], [24], [-8], [-5], [1], [36], [-64], [-36], [-60], [-55], [38], [-44], [-14], [-8], [20], [-50], [-32], [36], [8], [-34], [-44], [-42], [-58], [10], [48], [38], [52], [48], [6], [-54], [-4], [46], [27], [-27], [39], [12], [70], [38], [38], [3], [28], [57], [-68], [39], [-48], [-72], [0], [50], [1], [9], [-48], [-10], [-8], [-26], [48], [-19], [-54], [-39], [67], [2], [-12], [46], [14], [-12], [-43], [-50], [54], [-36], [-54], [-34], [-23], [55], [-64], [-39], [8], [-74], [-46], [25], [25], [12], [-42], [42], [-16], [39], [-39], [34], [15], [-79], [14], [4], [14], [-30], [-33], [-10], [36], [-50], [-33], [7], [-24], [-14], [-38], [77], [29], [-16], [8], [6], [-21], [8], [-58], [-27], [32], [10], [-70], [-70], [45], [44], [31], [63], [-28], [28], [36], [-36], [-73], [-36], [-20], [-27], [-35], [-38], [-58], [-39], [-54], [82], [22], [2], [74], [84], [-13], [17], [9], [0], [-14], [17], [61], [15], [-68], [48], [-26], [8], [41], [0], [-26], [26], [-88], [42], [62], [-60], [68], [72], [22], [46], [-38], [25], [-10], [-6], [-24], [24], [-74], [0], [-52], [51], [35], [-54], [59], [-74], [-28], [26], [50], [93], [-14], [60], [-19], [1], [62], [-12], [32], [-5], [-88], [1], [54], [-64], [29], [-8], [52], [32], [-39], [-42], [62], [-20], [-83], [72], [-12], [46], [-10], [9], [-20], [3], [-84], [-9], [64], [22], [-74], [-60], [-79], [30], [-36], [92], [66], [-58], [63], [22], [-84], [51], [62], [-47], [98], [70], [-55], [-56], [82], [62], [-32], [-6], [94], [-52], [-54], [7], [-42], [-25], [16], [5], [-47], [-9], [-36], [-8], [73], [37], [-103], [-74], [27], [-54], [-74], [38]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8034_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_f();". // 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_8034_2_a_f(:prec:=1) chi := MakeCharacter_8034_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_8034_2_a_f();". // 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_8034_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8034_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<7,R![5, 1]>],Snew); return Vf; end function;