// Make newform 1520.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1520_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_1520_2_a_g();". // 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_1520_2_a_g();". // 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_1520_a();" function MakeCharacter_1520_a() N := 1520; order := 1; char_gens := [191, 1141, 1217, 401]; 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_1520_a_Hecke(Kf) return MakeCharacter_1520_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], [1], [-1], [1], [0], [-3], [-7], [1], [5], [-5], [-10], [2], [2], [-6], [0], [9], [7], [-4], [-7], [0], [-9], [10], [2], [-10], [-18], [-4], [0], [-9], [13], [8], [6], [20], [-3], [-12], [4], [6], [18], [18], [-14], [6], [-24], [-10], [7], [-2], [-10], [-17], [-5], [22], [25], [-18], [26], [27], [-28], [-4], [-6], [8], [-10], [-5], [28], [-26], [26], [31], [-4], [-33], [-29], [3], [17], [-14], [18], [10], [27], [-27], [-8], [3], [7], [0], [24], [-20], [18], [26], [26], [31], [-12], [12], [-10], [-24], [14], [-41], [-10], [-16], [8], [24], [2], [-14], [-42], [17], [-42], [0], [19], [-28], [20], [8], [44], [-8], [-10], [-11], [-30], [-34], [-30], [-10], [-22], [-16], [-2], [8], [0], [-12], [2], [9], [-12], [-15], [17], [-28], [19], [-36], [-10], [24], [-34], [-29], [-11], [24], [-50], [-34], [-46], [22], [-45], [49], [15], [-11], [5], [-39], [-11], [-56], [31], [9], [-51], [48], [-46], [24], [-56], [-24], [-5], [2], [-26], [-52], [-5], [32], [-11], [21], [-13], [-37], [28], [44], [-12], [-36], [-18], [18], [46], [14], [-48], [-62], [-15], [27], [24], [-23], [38], [30], [10], [-18], [-54], [-34], [-25], [-7], [10], [16], [-3], [-26], [-41], [-12], [36], [63], [8], [0], [52], [25], [24], [38], [25], [-44], [-7], [27], [-2], [-16], [41], [-2], [-30], [-36], [-69], [65], [49], [27], [-33], [-12], [-16], [-29], [46], [-12], [65], [38], [34], [6], [62], [26], [36], [65], [23], [-10], [-34], [-40], [-28], [22], [-32], [35], [44], [42], [-46], [3], [-14], [-22], [-42], [-66], [12], [8], [11], [11], [18], [-72], [72], [32], [40], [-11], [-56], [15], [73], [20], [-22], [-20], [10], [68], [6], [58], [-58], [-36], [-30], [-47], [14], [31], [-59], [-29], [31], [-24], [55], [-13], [-21], [5], [12], [-49], [48], [51], [-60], [-60], [2], [39], [21], [-65], [52], [36], [-25], [10], [61], [-2], [-80], [8], [12], [-4], [46], [-46], [30], [22], [5], [12], [34], [23], [51], [-71], [66], [-26], [-47], [37], [-72], [12], [8], [0], [12], [76], [76], [22], [-30], [24], [37], [-18], [-55], [-43], [39], [-32], [-43], [0], [7], [-45], [22], [54], [24], [75], [-29], [-30], [-50], [62], [41], [-72], [39], [-73], [27], [33], [-13], [-30], [17], [-24], [10], [31], [60], [-40], [-4], [16], [-44], [-52], [10], [-40], [-33], [8], [-79], [-38], [54], [42], [-28], [-57], [-56], [-78], [-22], [12], [30], [34], [37], [73], [55], [-32], [76], [21], [-88], [58], [9], [-58], [-18], [-50], [43], [18], [27], [90], [21], [-46], [-58], [12], [-2], [-25], [-66], [-28], [26], [-6], [66], [-88], [-52], [-30], [34], [-66], [46], [-18], [-18], [-53], [32], [-40], [-65], [78], [26], [-26], [37], [8], [-57], [-2], [52], [-102], [29], [72], [34], [63], [-13], [25], [44], [27], [81], [-41], [40], [-81]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1520_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1520_2_a_g();". // 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_1520_2_a_g(:prec:=1) chi := MakeCharacter_1520_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_1520_2_a_g();". // 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_1520_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1520_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<7,R![-1, 1]>],Snew); return Vf; end function;