// Make newform 7350.2.a.s in Magma, downloaded from the LMFDB on 28 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_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_7350_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_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], [6], [4], [3], [-4], [-3], [-6], [5], [-8], [-3], [-8], [-9], [-12], [6], [2], [-8], [-9], [-14], [-7], [6], [3], [-17], [12], [13], [-6], [-4], [-3], [16], [-18], [15], [2], [-6], [-16], [-14], [-8], [0], [6], [-18], [2], [15], [-5], [0], [5], [8], [-11], [18], [-22], [-6], [3], [-22], [24], [18], [3], [0], [-25], [-26], [27], [22], [18], [-20], [21], [-17], [-12], [26], [13], [-18], [-28], [-15], [-36], [28], [-8], [-10], [-21], [0], [-2], [-6], [29], [24], [-4], [21], [7], [-13], [24], [-3], [-2], [36], [13], [30], [21], [25], [-12], [38], [-24], [-6], [21], [34], [-28], [-8], [30], [-12], [-15], [20], [-2], [-18], [33], [9], [-34], [-11], [34], [21], [-46], [17], [-39], [34], [-12], [36], [-30], [-10], [19], [-18], [-18], [-34], [-48], [-46], [-27], [31], [-50], [-16], [-9], [-16], [4], [9], [50], [12], [-8], [18], [-6], [-40], [36], [-44], [-30], [2], [15], [28], [-6], [14], [-21], [-2], [15], [-14], [-24], [-38], [-21], [11], [-30], [34], [12], [48], [-6], [31], [6], [-57], [48], [-43], [4], [29], [12], [42], [-16], [24], [13], [59], [27], [26], [-42], [4], [-10], [28], [42], [-14], [-54], [-45], [18], [-38], [10], [5], [-48], [-14], [12], [26], [18], [54], [6], [-58], [10], [18], [0], [-36], [-25], [-14], [35], [54], [-36], [-25], [18], [15], [38], [-17], [-60], [1], [36], [-45], [-55], [31], [42], [48], [36], [-22], [11], [69], [25], [18], [56], [-51], [69], [-17], [0], [16], [38], [5], [15], [40], [33], [-22], [-18], [-24], [-3], [-24], [-34], [-32], [38], [-63], [-27], [43], [18], [44], [15], [52], [54], [27], [-34], [-60], [24], [20], [28], [-18], [-29], [16], [-18], [-58], [-56], [-21], [-10], [-30], [30], [-14], [42], [2], [-2], [1], [-1], [-62], [31], [78], [26], [11], [6], [-81], [23], [33], [-70], [4], [-15], [43], [-30], [-64], [-21], [0], [-12], [-3], [-36], [4], [-78], [8], [-24], [36], [52], [-5], [30], [11], [-66], [20], [-41], [-42], [-40], [-24], [-2], [81], [-66], [-78], [-50], [45], [38], [42], [-45], [37], [3], [2], [22], [6], [16], [51], [62], [20], [4], [-3], [-66], [-52], [-30], [53], [-24], [-34], [-18], [-58], [-51], [-58], [55], [-26], [-9], [-36], [-40], [-42], [72], [26], [-62], [0], [-48], [-70], [7], [-42], [67], [-10], [33], [-21], [-36], [3], [72], [82], [30], [9], [84], [4], [-14], [-42], [49], [50], [-18], [92], [45], [0], [-49], [-38], [24], [72], [49], [-39], [91], [-48], [6], [16], [-18], [-76], [-15], [23], [58], [-74], [-63], [-1], [24], [-72], [22], [39], [55], [-28], [-30], [-64], [36], [14], [66], [-71], [-6], [24], [-13], [10], [-99], [-80], [24], [-2], [-54], [84], [-10], [13], [6], [24], [25], [-33], [-48], [102], [-20], [-36], [-12], [79], [66], [0], [78], [8], [-39], [77], [-102], [-76], [3], [-14], [-15], [47], [-40], [-62], [-7], [-12], [15], [80], [21], [-1], [-39], [-104], [-33], [26], [-58], [-32], [39], [-108], [93], [-107], [-90], [-16], [18], [-50], [-75], [-4], [-67], [-54], [20], [-14], [13], [-7], [42], [-51], [-88], [40], [-96], [81], [-61]]; 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_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_7350_2_a_s(: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_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_7350_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-6, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>,<19,R![4, 1]>,<23,R![3, 1]>,<31,R![-5, 1]>],Snew); return Vf; end function;