// Make newform 2800.2.a.bb in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2800_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_2800_2_a_bb();". // 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_2800_2_a_bb();". // 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_2800_a();" function MakeCharacter_2800_a() N := 2800; order := 1; char_gens := [351, 2101, 2577, 801]; v := [1, 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_2800_a_Hecke(Kf) return MakeCharacter_2800_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], [2], [0], [-1], [-1], [-4], [0], [-6], [3], [-3], [0], [-9], [2], [9], [-6], [-6], [-8], [-10], [-1], [7], [2], [9], [12], [-4], [-16], [-14], [-6], [20], [-11], [9], [-1], [12], [22], [6], [-17], [-17], [14], [12], [-12], [2], [-16], [22], [-20], [1], [-3], [18], [-4], [26], [-22], [-26], [19], [16], [18], [-20], [0], [-15], [4], [-4], [22], [-7], [14], [-8], [-12], [-4], [22], [33], [17], [-34], [17], [2], [0], [3], [10], [-15], [5], [-24], [-35], [-34], [-39], [14], [-12], [23], [4], [0], [4], [-36], [-9], [-11], [12], [-16], [-18], [-12], [17], [39], [4], [26], [-36], [30], [-20], [3], [11], [33], [-2], [37], [-15], [-2], [-22], [-44], [-15], [8], [20], [-5], [23], [16], [45], [49], [8], [42], [-6], [-44], [14], [-14], [-18], [-23], [-26], [-18], [22], [-26], [8], [4], [43], [0], [-4], [-17], [32], [38], [-40], [38], [28], [43], [-14], [2], [23], [-7], [-20], [-34], [-28], [38], [-26], [-7], [46], [-40], [-25], [-2], [12], [-23], [29], [-42], [-10], [-56], [20], [-11], [40], [26], [3], [-24], [-23], [-38], [19], [36], [45], [10], [9], [30], [26], [-36], [33], [-18], [24], [44], [-23], [-48], [3], [6], [11], [-10], [-11], [-34], [-47], [-30], [18], [-9], [-29], [-26], [-43], [-18], [5], [-1], [-26], [-40], [18], [-26], [-52], [-2], [-54], [-6], [-38], [40], [-65], [42], [-13], [-28], [-39], [-48], [-30], [48], [-47], [10], [-1], [-45], [61], [20], [-45], [-21], [-32], [30], [-34], [9], [-56], [-65], [51], [-60], [-60], [27], [-64], [-28], [50], [-35], [-25], [48], [-33], [-4], [-4], [-21], [-72], [56], [62], [-18], [40], [24], [38], [-50], [40], [-52], [18], [-60], [-35], [-20], [-54], [6], [36], [57], [12], [8], [42], [-32], [-5], [-50], [9], [31], [36], [-52], [-52], [41], [-52], [-44], [33], [-30], [-49], [8], [-70], [80], [68], [-76], [30], [-57], [-47], [9], [-60], [-30], [-21], [28], [-81], [46], [-23], [72], [-22], [68], [6], [-42], [14], [-14], [61], [25], [57], [2], [23], [36], [69], [-58], [54], [6], [29], [-28], [51], [-56], [26], [-32], [70], [21], [16], [22], [26], [23], [13], [-52], [45], [-50], [-85], [-49], [34], [-77], [-74], [-4], [36], [54], [85], [-16], [-60], [10], [-58], [75], [-2], [-75], [15], [3], [38], [-19], [62], [42], [14], [17], [10], [38], [19], [-62], [36], [-6], [34], [-76], [29], [72], [24], [-73], [-66], [46], [-14], [-79], [75], [-13], [-80], [-9], [54], [92], [49], [24], [71], [72], [42], [-18], [30], [45], [7], [13], [62], [28], [7], [-20], [61], [-34], [-55], [16], [-9], [94], [-5], [-17], [52], [-32], [76], [5], [36], [-55], [-13], [-76], [6], [-52], [17], [54], [-46], [104], [20], [-73], [84], [45], [59], [-30], [-24], [30], [4], [48], [-106], [18], [-32], [12], [36], [-46], [-91], [-42], [-28], [42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2800_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2800_2_a_bb();". // 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_2800_2_a_bb(:prec:=1) chi := MakeCharacter_2800_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_2800_2_a_bb();". // 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_2800_2_a_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 1]>,<11,R![1, 1]>,<13,R![4, 1]>],Snew); return Vf; end function;