// Make newform 7744.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7744_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_7744_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_7744_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_7744_a();" function MakeCharacter_7744_a() N := 7744; order := 1; char_gens := [5567, 4357, 6657]; 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_7744_a_Hecke(Kf) return MakeCharacter_7744_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], [-1], [-2], [0], [4], [2], [0], [1], [0], [-7], [-3], [8], [6], [-8], [6], [5], [12], [-7], [3], [-4], [-10], [6], [15], [-7], [2], [16], [-18], [10], [9], [8], [18], [-7], [-10], [-10], [2], [7], [4], [-12], [-6], [-15], [-7], [-17], [-4], [-2], [0], [-12], [-19], [-18], [-15], [-24], [-30], [8], [-23], [-2], [14], [-10], [-28], [-2], [18], [-4], [24], [-8], [-12], [-1], [-13], [7], [22], [-28], [30], [-21], [-20], [17], [-26], [-5], [1], [15], [2], [2], [30], [20], [-22], [-18], [-11], [40], [-11], [35], [12], [12], [11], [-27], [20], [-23], [8], [20], [-26], [-15], [-3], [16], [-8], [-8], [-2], [-4], [0], [28], [33], [28], [-44], [-40], [-2], [-22], [-16], [18], [-25], [-7], [-33], [29], [7], [41], [-10], [-37], [-14], [-42], [-16], [17], [2], [25], [-15], [-3], [-36], [-50], [4], [23], [22], [-12], [-20], [6], [32], [-53], [0], [38], [22], [-39], [52], [-25], [5], [14], [-8], [-15], [-24], [-12], [-43], [4], [-22], [-12], [-12], [10], [-30], [-8], [42], [-27], [-34], [-32], [47], [-27], [-39], [8], [38], [10], [-39], [10], [-22], [32], [16], [-5], [-55], [-2], [13], [44], [-20], [-8], [58], [51], [42], [51], [30], [48], [24], [-50], [2], [-31], [-34], [-3], [18], [12], [-21], [-2], [41], [42], [14], [60], [-18], [-18], [-40], [-25], [47], [15], [36], [0], [-8], [-48], [-27], [-39], [28], [-30], [47], [68], [-12], [72], [-39], [-68], [60], [-15], [-29], [12], [-70], [54], [0], [28], [-52], [71], [20], [22], [-32], [49], [58], [-15], [-36], [55], [-37], [-41], [-32], [36], [15], [56], [-60], [52], [-28], [30], [34], [-32], [-2], [-33], [-10], [-6], [20], [-22], [-78], [-33], [2], [4], [-48], [50], [-6], [-42], [40], [45], [-3], [46], [-6], [-17], [-57], [34], [-40], [-8], [-59], [-57], [10], [-52], [-12], [-56], [43], [-52], [62], [-28], [3], [-6], [18], [35], [-70], [-77], [-52], [36], [18], [54], [-40], [23], [-79], [-30], [22], [66], [-72], [-20], [4], [-13], [-17], [63], [-45], [-60], [84], [24], [-70], [18], [89], [48], [10], [35], [-38], [-86], [-20], [68], [73], [-58], [91], [26], [-13], [-45], [-1], [48], [4], [22], [-78], [-70], [56], [48], [93], [-25], [-4], [7], [38], [-29], [-57], [60], [13], [-59], [-10], [-67], [-37], [-48], [57], [28], [3], [18], [-36], [-50], [-54], [75], [-62], [22], [31], [-82], [-42], [-3], [50], [3], [-11], [48], [14], [-72], [57], [0], [34], [-20], [-98], [-13], [20], [-58], [-14], [30], [-18], [-22], [39], [38], [-38], [-40], [-39], [-72], [7], [16], [-23], [5], [41], [-55], [-17], [-87], [56], [-70], [30], [-68], [-58], [50], [49], [48], [-42], [-20], [42], [42], [-52], [44], [-25], [6], [-62], [4], [-2], [82], [63], [-40], [57], [38], [54], [25], [88], [72], [50], [-86], [-3], [-81], [-70], [-53], [-80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7744_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7744_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_7744_2_a_k(:prec:=1) chi := MakeCharacter_7744_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_7744_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_7744_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7744_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<5,R![1, 1]>,<7,R![2, 1]>,<13,R![-4, 1]>],Snew); return Vf; end function;