// Make newform 2800.2.a.bg in Magma, downloaded from the LMFDB on 28 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_bg();". // 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_bg();". // 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], [3], [0], [1], [5], [-6], [-1], [3], [0], [-6], [4], [8], [11], [8], [-2], [4], [-4], [-2], [-9], [10], [-7], [2], [-11], [-11], [-10], [0], [-4], [-3], [-18], [-1], [14], [-8], [-3], [11], [12], [-8], [4], [19], [-12], [2], [-3], [10], [6], [-19], [-22], [10], [-1], [-22], [28], [-14], [6], [-4], [-5], [-27], [2], [10], [18], [6], [-30], [14], [13], [-14], [-13], [-6], [-10], [4], [17], [-29], [-19], [-8], [-18], [-26], [8], [-4], [-9], [6], [8], [10], [37], [-21], [39], [20], [36], [-1], [-28], [-37], [33], [25], [-38], [-8], [4], [6], [34], [12], [-36], [30], [-14], [11], [-13], [-42], [-27], [4], [-20], [21], [-12], [13], [13], [-39], [24], [21], [-28], [18], [14], [4], [16], [2], [-16], [8], [-28], [-1], [-50], [34], [-48], [13], [-49], [-32], [4], [48], [-6], [40], [-4], [24], [-50], [-2], [27], [19], [36], [-52], [42], [6], [28], [-24], [-10], [-41], [-4], [-2], [-34], [3], [51], [-4], [32], [26], [-15], [34], [4], [12], [34], [46], [-7], [56], [4], [9], [2], [51], [21], [4], [-4], [-58], [1], [-14], [19], [26], [-30], [39], [4], [-35], [-49], [60], [-26], [58], [28], [-45], [62], [-19], [-42], [-24], [34], [19], [25], [42], [33], [35], [-25], [-40], [-27], [-6], [19], [-26], [21], [-14], [10], [-2], [-30], [-39], [12], [-4], [10], [0], [30], [-32], [-6], [-48], [8], [7], [54], [-13], [32], [27], [56], [60], [-52], [-50], [33], [36], [-5], [-64], [-47], [46], [56], [43], [-50], [57], [-22], [-21], [44], [60], [-53], [-10], [7], [20], [53], [13], [42], [-14], [33], [-18], [34], [-36], [-5], [10], [-2], [-39], [-42], [21], [12], [35], [44], [-55], [60], [-21], [18], [23], [22], [44], [-33], [55], [-6], [-46], [41], [-24], [-36], [51], [41], [18], [-70], [64], [37], [-10], [6], [-37], [-34], [40], [-48], [-42], [25], [12], [-2], [42], [-44], [78], [-78], [-68], [31], [25], [18], [-36], [14], [40], [-60], [13], [-69], [38], [26], [-3], [-65], [-15], [28], [72], [-60], [2], [-66], [60], [37], [-8], [30], [-10], [75], [-18], [9], [35], [-40], [25], [8], [-24], [66], [-34], [-17], [-36], [92], [-36], [-10], [18], [68], [-21], [-47], [-12], [22], [42], [-47], [10], [52], [-23], [-72], [64], [-90], [92], [0], [-25], [-66], [54], [-59], [-30], [54], [82], [-36], [-3], [-54], [-37], [-78], [18], [8], [-55], [82], [80], [77], [55], [-44], [-16], [-47], [-36], [-61], [50], [30], [-50], [-68], [43], [18], [65], [-57], [42], [16], [49], [36], [27], [36], [-66], [12], [68], [-20], [-22], [-73], [-32], [-85], [-15], [26], [-11], [-28], [-17], [69], [-24], [30], [33], [62], [63], [-80], [-22], [68], [30], [-84], [-39], [29], [-12], [75], [56], [-13], [-98], [54], [-6], [51], [10], [-32], [14], [-36], [49], [-33], [-44], [4], [-9], [47], [-56]]; 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_bg();". // 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_bg(: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_bg();". // 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_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 1]>,<11,R![-5, 1]>,<13,R![6, 1]>],Snew); return Vf; end function;