// Make newform 7920.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7920_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_7920_2_a_i();". // 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_7920_2_a_i();". // 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_7920_a();" function MakeCharacter_7920_a() N := 7920; order := 1; char_gens := [991, 5941, 3521, 6337, 6481]; v := [1, 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_7920_a_Hecke(Kf) return MakeCharacter_7920_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], [0], [-1], [0], [-1], [2], [-6], [4], [4], [-6], [8], [-2], [-2], [-4], [-12], [2], [4], [-10], [16], [8], [14], [-8], [-4], [-10], [10], [10], [4], [12], [-18], [6], [-16], [-12], [-18], [-12], [10], [-8], [-2], [-16], [-8], [6], [4], [-10], [8], [-26], [-2], [0], [-4], [4], [-20], [-10], [-6], [8], [10], [12], [-18], [24], [18], [0], [10], [-18], [-4], [-10], [-20], [-24], [-22], [18], [-4], [6], [-4], [-10], [-18], [-32], [-4], [18], [-20], [-12], [-6], [30], [-2], [-6], [-28], [6], [-24], [-22], [8], [8], [-2], [-26], [34], [36], [0], [24], [-28], [-28], [36], [-16], [-14], [6], [20], [-34], [-12], [-10], [36], [-26], [-36], [-22], [-24], [-22], [24], [2], [32], [34], [-2], [20], [-40], [-34], [16], [20], [10], [36], [-10], [-26], [38], [-16], [28], [-22], [-10], [0], [-52], [42], [-4], [40], [-16], [6], [-10], [-22], [-14], [-52], [2], [-18], [12], [18], [36], [12], [-2], [-32], [2], [18], [-28], [4], [-22], [-18], [16], [56], [40], [8], [-24], [14], [6], [2], [0], [42], [-8], [36], [-10], [-4], [-8], [-46], [26], [-30], [28], [-34], [40], [-10], [-32], [-10], [-36], [-6], [16], [-34], [36], [28], [22], [-6], [-36], [-38], [-6], [24], [58], [-8], [42], [-12], [44], [-30], [28], [38], [-22], [38], [2], [-8], [-38], [40], [62], [10], [12], [50], [40], [52], [30], [44], [-18], [58], [36], [8], [-48], [-6], [56], [-34], [36], [42], [30], [32], [-50], [-52], [68], [-58], [-34], [24], [-8], [-36], [14], [-20], [40], [-66], [16], [-72], [-14], [54], [20], [0], [64], [-76], [4], [-18], [-30], [-48], [28], [36], [52], [48], [2], [14], [-12], [58], [6], [36], [-74], [-36], [26], [-58], [-32], [12], [46], [-14], [62], [20], [66], [38], [-52], [-42], [14], [-16], [50], [-64], [14], [-52], [-40], [22], [18], [20], [-48], [-16], [-32], [22], [28], [24], [-6], [22], [40], [78], [18], [-40], [-6], [-84], [34], [50], [0], [2], [-4], [-68], [-74], [-42], [64], [-8], [60], [-30], [-40], [-50], [-8], [-46], [56], [42], [-50], [88], [24], [18], [84], [-80], [26], [-50], [-52], [2], [90], [36], [-6], [82], [-20], [-40], [56], [78], [22], [2], [-8], [-12], [-76], [16], [-66], [-46], [-54], [40], [86], [-58], [10], [0], [42], [-68], [54], [96], [72], [-46], [68], [10], [-30], [56], [62], [-30], [-24], [-20], [2], [28], [-70], [-42], [-92], [20], [-40], [2], [-42], [-24], [-30], [-28], [-44], [16], [-22], [-48], [-18], [36], [-32], [14], [30], [38], [-94], [-90], [8], [42], [52], [28], [-64], [54], [20], [-4], [-30], [18], [-60], [-96], [-72], [-10], [-48], [86], [84], [-30], [94], [6], [8], [6], [-54], [88], [-2], [78], [-40], [-52], [62], [-42], [8], [-76], [-34], [-46], [48], [28], [-66], [16], [34], [74], [28], [36], [26], [74], [48], [6], [-12], [-96], [42], [-100], [-20], [-68], [30], [-18], [-30], [-10], [56], [8], [-40], [30], [-10], [16], [-78], [-62], [-12], [-32], [-62], [-26], [28], [24], [-28], [6], [18], [-54], [-2], [92], [42], [46], [20], [40], [12], [-26], [36], [86], [-40], [-88], [-42], [68], [-56], [72], [80], [-86], [-100], [-30], [-78], [-24], [72], [90], [74], [-26]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7920_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_i();". // 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_7920_2_a_i(:prec:=1) chi := MakeCharacter_7920_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(3449) - 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_7920_2_a_i();". // 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_7920_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7920_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![-4, 1]>,<23,R![-4, 1]>,<29,R![6, 1]>],Snew); return Vf; end function;