// Make newform 8400.2.a.q in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8400_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_8400_2_a_q();". // 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_8400_2_a_q();". // 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_8400_a();" function MakeCharacter_8400_a() N := 8400; order := 1; char_gens := [3151, 2101, 2801, 5377, 3601]; 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_8400_a_Hecke(Kf) return MakeCharacter_8400_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], [0], [1], [-4], [2], [6], [-4], [8], [-2], [0], [2], [10], [4], [0], [-14], [-12], [-2], [-4], [0], [-2], [8], [-4], [-6], [6], [6], [-8], [20], [14], [14], [-16], [-4], [22], [20], [6], [-16], [18], [-4], [-8], [2], [-12], [-10], [-8], [-18], [18], [8], [4], [-16], [28], [-26], [6], [24], [2], [-12], [6], [-8], [-18], [32], [2], [26], [20], [26], [28], [8], [-2], [10], [-20], [-34], [-12], [30], [6], [0], [16], [-14], [12], [16], [22], [34], [18], [-6], [12], [-10], [24], [-26], [-8], [20], [-30], [6], [14], [-16], [-4], [0], [-24], [28], [36], [8], [30], [-38], [36], [-2], [28], [10], [-36], [-6], [12], [6], [36], [6], [-16], [10], [32], [2], [38], [-12], [16], [2], [-20], [24], [-6], [20], [-42], [-34], [42], [-44], [12], [-2], [-10], [-16], [-8], [34], [-44], [-24], [40], [-46], [42], [-14], [26], [28], [-14], [-22], [4], [-42], [56], [52], [-34], [8], [26], [46], [4], [16], [10], [-14], [-4], [24], [-12], [56], [-16], [50], [14], [-18], [12], [-42], [-40], [-60], [-18], [40], [-24], [-38], [18], [-6], [-4], [-34], [0], [-26], [-32], [-38], [12], [54], [56], [14], [-32], [-4], [34], [-50], [0], [-10], [42], [-4], [-22], [0], [-10], [-28], [-44], [62], [12], [14], [-46], [-38], [-58], [40], [46], [0], [-22], [-14], [20], [-62], [0], [-36], [10], [4], [-18], [22], [8], [-12], [24], [-38], [-48], [66], [56], [-54], [22], [24], [-62], [-32], [44], [-74], [-18], [-72], [40], [12], [-54], [44], [8], [-54], [-28], [-48], [50], [50], [-52], [56], [12], [20], [-40], [30], [-26], [-24], [32], [44], [-20], [0], [58], [50], [-8], [74], [-14], [36], [54], [52], [26], [-34], [16], [-68], [-26], [2], [-58], [-20], [-50], [10], [20], [-30], [-18], [-52], [62], [56], [-58], [56], [-60], [-18], [74], [-36], [0], [-32], [-72], [-58], [-12], [-24], [62], [-30], [8], [-78], [-50], [-52], [-10], [-44], [42], [30], [-32], [-38], [4], [-36], [62], [-6], [-24], [-68], [28], [-34], [4], [-50], [80], [2], [64], [-42], [-30], [-36], [56], [-38], [-36], [40], [22], [-78], [-36], [6], [-34], [-64], [-26], [-62], [-44], [-44], [-32], [82], [94], [42], [-64], [-52], [-68], [36], [-2], [-42], [58], [-64], [50], [6], [6], [48], [-6], [-28], [38], [-60], [-96], [10], [28], [-58], [-18], [16], [38], [78], [80], [-76], [30], [40], [-30], [26], [32], [-68], [92], [38], [98], [72], [-70], [-44], [-44], [0], [6], [-40], [-38], [12], [-24], [38], [-30], [94], [30], [-58], [72], [-34], [28], [-8], [-24], [-54], [20], [0], [-62], [90], [60], [28], [48], [-26], [80], [42], [-20], [54], [62], [46], [-16], [62], [-90], [24], [42], [82], [-52], [28], [-26], [-30], [-28], [-12], [70], [-50], [-8], [-40], [54], [-56], [78], [58], [0], [84], [14], [-78], [60], [-6], [20], [-40], [-70], [20], [44], [96], [-62], [-62], [-54], [102], [52], [-24], [36], [-46], [-58], [72], [-46], [30], [-60], [0], [-14], [46], [60], [40], [-36], [90], [-18], [-10], [30], [44], [-86], [-90], [92], [-80], [-76], [-26], [-60], [62], [64], [36], [66], [44], [-112], [-68], [96], [-30], [76], [2], [30], [32], [16], [-94], [14], [-22], [-106], [6], [-24], [4], [94], [-52], [-28], [16], [98], [72], [10], [2], [-108], [-42], [36], [18], [-88], [-28], [46], [64], [-58], [56], [-54], [-74], [-72], [48], [-46], [-28], [36], [88], [14], [-38], [-68], [62], [-42], [30], [-32], [-96], [-46], [-52], [-78], [24], [58], [-4], [6], [-102], [36], [-18], [-80], [-42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8400_2_a_q();". // 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_8400_2_a_q(:prec:=1) chi := MakeCharacter_8400_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(3833) - 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_8400_2_a_q();". // 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_8400_2_a_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![4, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<19,R![4, 1]>,<23,R![-8, 1]>],Snew); return Vf; end function;