// Make newform 7350.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_e();". // 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_7350_2_a_e();". // 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_7350_a();" function MakeCharacter_7350_a() N := 7350; order := 1; char_gens := [4901, 1177, 2551]; 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_7350_a_Hecke(Kf) return MakeCharacter_7350_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], [0], [-4], [4], [3], [6], [7], [4], [-5], [2], [7], [2], [1], [-2], [-14], [12], [12], [-9], [6], [-17], [-4], [-7], [-7], [-18], [-7], [14], [6], [-3], [16], [-8], [15], [2], [-6], [24], [-4], [-8], [0], [6], [2], [12], [15], [25], [20], [15], [-22], [9], [-22], [8], [-6], [3], [-2], [4], [-2], [-27], [-10], [5], [-16], [17], [-8], [-12], [20], [21], [-27], [18], [-34], [3], [32], [-18], [-15], [4], [-32], [12], [20], [-31], [10], [-22], [34], [29], [-26], [16], [-39], [37], [17], [-26], [-13], [18], [36], [-7], [0], [1], [5], [38], [28], [36], [-16], [-9], [44], [32], [22], [30], [-2], [-45], [40], [-2], [42], [13], [-31], [46], [29], [4], [1], [-16], [7], [-9], [14], [-32], [16], [0], [30], [-11], [32], [22], [-24], [12], [14], [-7], [-49], [-10], [14], [1], [-16], [-16], [-21], [10], [-18], [22], [-12], [54], [0], [6], [-24], [20], [-38], [-25], [-2], [-26], [24], [-11], [38], [-35], [6], [56], [-28], [19], [-39], [-30], [-6], [42], [-32], [-26], [-29], [-24], [3], [28], [7], [-16], [29], [-48], [42], [34], [-16], [43], [29], [17], [-4], [-2], [-16], [-20], [-12], [-58], [6], [26], [45], [18], [-38], [30], [25], [12], [6], [12], [-34], [-32], [44], [26], [2], [-50], [-22], [40], [-16], [-55], [-4], [-65], [-26], [24], [-35], [-52], [-15], [8], [-47], [10], [-19], [66], [-25], [25], [31], [-18], [48], [-14], [8], [-19], [19], [5], [-42], [46], [29], [29], [-37], [-40], [-4], [28], [-5], [25], [20], [43], [-22], [72], [-4], [-43], [-54], [26], [-32], [8], [-3], [-27], [-17], [8], [-6], [-15], [-8], [-6], [-3], [46], [-60], [74], [20], [28], [2], [41], [36], [42], [2], [24], [79], [40], [-30], [10], [-34], [-68], [-8], [18], [-49], [9], [18], [-69], [-42], [56], [-49], [66], [-11], [-67], [63], [-80], [-46], [-15], [-7], [0], [16], [49], [-20], [-42], [57], [-46], [34], [-48], [28], [66], [6], [-18], [-35], [-30], [-19], [-66], [-50], [29], [-2], [-40], [-24], [-22], [51], [44], [-18], [-20], [-45], [-22], [82], [-45], [7], [-7], [72], [2], [66], [16], [-9], [2], [0], [-56], [87], [54], [78], [10], [-57], [6], [6], [32], [-38], [9], [22], [-5], [14], [51], [-6], [-80], [38], [-18], [46], [8], [0], [-28], [-50], [-43], [48], [-73], [60], [-87], [-21], [-6], [-57], [-28], [92], [50], [39], [-16], [-6], [46], [28], [9], [90], [-18], [-48], [15], [60], [41], [32], [84], [-8], [-81], [-89], [41], [-48], [6], [-24], [-78], [-86], [15], [-67], [18], [56], [7], [-41], [-66], [-72], [-38], [-61], [85], [-8], [-90], [36], [6], [94], [6], [9], [54], [24], [-63], [70], [-69], [-60], [24], [18], [-4], [24], [-10], [-57], [-14], [-56], [-15], [-53], [-8], [-8], [-40], [-36], [18], [-11], [106], [50], [-102], [28], [-39], [-83], [-12], [-56], [93], [46], [-45], [-33], [70], [-82], [23], [18], [45], [90], [-59], [79], [41], [86], [57], [-14], [12], [88], [59], [-78], [83], [63], [50], [24], [-22], [-20], [-35], [-54], [-17], [-34], [-90], [-14], [3], [3], [2], [-1], [-68], [-60], [34], [21], [-101]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7350_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_e();". // 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_7350_2_a_e(:prec:=1) chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_e();". // 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_7350_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![4, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>,<19,R![-6, 1]>,<23,R![-7, 1]>,<31,R![5, 1]>],Snew); return Vf; end function;