// Make newform 2450.2.a.x in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2450_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_2450_2_a_x();". // 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_2450_2_a_x();". // 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_2450_a();" function MakeCharacter_2450_a() N := 2450; order := 1; char_gens := [1177, 101]; v := [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_2450_a_Hecke(Kf) return MakeCharacter_2450_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], [3], [-2], [-3], [7], [0], [-6], [4], [8], [9], [8], [6], [-12], [-12], [10], [-7], [6], [-5], [14], [9], [15], [10], [0], [-20], [3], [14], [-9], [2], [0], [21], [7], [12], [8], [-20], [5], [-12], [6], [3], [-2], [-6], [5], [-6], [-14], [17], [-14], [-12], [-26], [6], [-12], [25], [-15], [30], [6], [6], [-2], [2], [-18], [1], [6], [7], [18], [10], [-12], [-25], [-13], [-21], [-8], [-30], [-6], [-8], [-4], [17], [30], [-24], [-2], [-27], [25], [3], [20], [-36], [-11], [4], [21], [-15], [17], [-18], [8], [-36], [-18], [-34], [-12], [-28], [6], [-42], [-39], [7], [38], [35], [36], [-12], [-27], [-4], [7], [-39], [27], [24], [7], [-44], [2], [-18], [28], [32], [-30], [40], [24], [36], [9], [10], [2], [0], [3], [19], [0], [-28], [0], [34], [40], [20], [-24], [-46], [-34], [9], [-23], [12], [4], [30], [6], [-44], [24], [26], [9], [4], [6], [10], [-15], [31], [-12], [32], [6], [47], [-6], [-4], [36], [-34], [18], [-29], [-24], [12], [57], [-34], [-9], [-3], [-12], [20], [-62], [-7], [6], [-3], [-2], [30], [-1], [52], [63], [17], [12], [-50], [46], [-28], [-33], [14], [63], [-54], [24], [50], [55], [-31], [-30], [-29], [45], [-7], [-24], [27], [6], [35], [-10], [-57], [-6], [-54], [22], [22], [-53], [36], [36], [34], [-48], [30], [-8], [26], [0], [56], [27], [-66], [49], [-16], [-15], [-72], [-36], [-4], [-26], [-15], [-52], [63], [-16], [-21], [66], [-8], [-3], [14], [-11], [-58], [-21], [4], [-36], [1], [54], [9], [-12], [-45], [-47], [-14], [50], [27], [6], [-38], [-12], [29], [-42], [14], [27], [-6], [-41], [36], [-3], [-52], [37], [-60], [-47], [62], [33], [34], [52], [45], [19], [-54], [-18], [23], [24], [4], [-43], [19], [-34], [70], [16], [-45], [-10], [-58], [15], [-42], [-40], [48], [-14], [-35], [-12], [62], [-6], [4], [-30], [-30], [-12], [-33], [-51], [2], [-12], [-26], [-72], [-84], [-23], [-23], [-42], [-10], [-69], [-7], [41], [-12], [40], [12], [50], [6], [12], [-3], [-40], [-78], [-38], [39], [-78], [43], [75], [16], [-31], [-24], [-40], [-30], [-62], [-7], [52], [-12], [12], [-58], [-30], [-44], [-9], [47], [-84], [38], [54], [-5], [34], [-28], [9], [24], [80], [6], [84], [-32], [-31], [6], [18], [49], [2], [54], [-86], [-52], [63], [-30], [15], [18], [-66], [8], [-45], [-18], [48], [1], [71], [-36], [-64], [41], [36], [-65], [-66], [-18], [-26], [-52], [15], [-66], [-5], [-3], [-26], [0], [9], [92], [27], [76], [-42], [20], [-68], [-76], [-78], [-25], [-96], [-75], [13], [-90], [-67], [4], [-27], [-37], [72], [58], [-15], [2], [-75], [48], [34], [-28], [30], [52], [-75], [31], [-60], [93], [-88], [-85], [42], [-102], [82], [-63], [18], [48], [-38], [-12], [93], [61], [60], [108], [-33], [-53], [-24]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2450_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2450_2_a_x();". // 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_2450_2_a_x(:prec:=1) chi := MakeCharacter_2450_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_2450_2_a_x();". // 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_2450_2_a_x( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<11,R![-3, 1]>,<13,R![2, 1]>,<17,R![3, 1]>,<19,R![-7, 1]>,<23,R![0, 1]>,<37,R![-8, 1]>],Snew); return Vf; end function;