// Make newform 3150.2.a.w in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3150_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_3150_2_a_w();". // 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_3150_2_a_w();". // 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_3150_a();" function MakeCharacter_3150_a() N := 3150; order := 1; char_gens := [2801, 127, 451]; v := [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_3150_a_Hecke(Kf) return MakeCharacter_3150_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 := [[1], [0], [0], [-1], [-4], [2], [2], [-4], [-8], [-6], [-8], [2], [-2], [12], [-8], [6], [-4], [-2], [-12], [-8], [14], [0], [12], [-2], [-10], [-6], [-8], [-20], [-2], [-14], [0], [-12], [10], [20], [18], [8], [-14], [-12], [-16], [14], [20], [-10], [16], [14], [6], [-16], [20], [0], [12], [-26], [10], [16], [18], [-4], [2], [24], [-30], [-24], [-14], [-10], [-20], [6], [4], [-8], [-34], [14], [28], [-2], [12], [-34], [18], [8], [32], [-14], [28], [24], [18], [2], [-18], [10], [-12], [-26], [0], [6], [32], [-4], [14], [38], [34], [16], [-20], [0], [24], [-36], [4], [16], [-14], [30], [28], [14], [-28], [-2], [-36], [6], [-20], [-26], [-12], [18], [-8], [-22], [0], [34], [10], [-44], [24], [30], [20], [-32], [-34], [-28], [-10], [-2], [6], [-4], [-20], [-6], [-26], [0], [24], [-30], [36], [24], [-32], [34], [-2], [-14], [-26], [4], [30], [38], [36], [-46], [8], [12], [-34], [24], [26], [-22], [-28], [0], [10], [-42], [-28], [16], [12], [-48], [-8], [-10], [-18], [18], [-12], [-6], [8], [-36], [18], [32], [-32], [-38], [50], [-42], [-4], [62], [-56], [-26], [-40], [14], [-52], [18], [-8], [-18], [-64], [-28], [50], [-22], [16], [-38], [26], [4], [-22], [16], [-26], [44], [4], [50], [-12], [-54], [-46], [-22], [-30], [-48], [-54], [24], [10], [-30], [28], [-2], [-24], [-60], [-10], [-12], [14], [42], [-40], [-12], [-24], [58], [-32], [22], [8], [-50], [-58], [48], [-18], [-16], [44], [22], [26], [0], [-40], [12], [26], [44], [-48], [54], [76], [56], [34], [-10], [44], [40], [36], [4], [-72], [-34], [2], [-40], [-32], [-12], [-68], [48], [-22], [-42], [-24], [26], [-18], [-12], [38], [-36], [6], [-2], [-16], [-44], [-26], [2], [-62], [12], [-54], [30], [-20], [54], [-50], [20], [78], [0], [-42], [-8], [76], [14], [10], [-28], [24], [-40], [-16], [-26], [28], [0], [-2], [-58], [-80], [6], [-22], [-68], [26], [12], [-54], [-46], [-40], [-42], [-4], [-28], [-2], [-2], [-64], [-76], [44], [-2], [-36], [-18], [24], [34], [-24], [-62], [30], [36], [72], [42], [36], [-64], [-58], [30], [28], [6], [-30], [-64], [74], [-14], [52], [-36], [-48], [6], [-18], [-18], [56], [28], [-4], [-60], [-2], [18], [-22], [80], [-46], [58], [10], [-8], [46], [68], [6], [-52], [64], [22], [-20], [38], [-38], [32], [6], [74], [48], [-68], [-78], [-8], [-30], [-2], [-48], [-20], [-12], [-58], [14], [24], [58], [20], [-12], [-64], [-62], [-16], [10], [68], [80], [70], [-10], [78], [-46], [-54], [-8], [98], [44], [-16], [-64], [-22], [-4], [-72], [34], [-58], [-68], [20], [56], [38], [24], [46], [92], [18], [94], [2], [32], [90], [-54], [64], [-6], [-34], [-60], [-28], [6], [22], [12], [-44], [6], [2], [-48], [56], [18], [64], [26], [-6], [48], [-36], [-50], [-18], [20], [-42], [52], [72]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3150_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_a_w();". // 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_3150_2_a_w(:prec:=1) chi := MakeCharacter_3150_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_3150_2_a_w();". // 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_3150_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3150_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<29,R![6, 1]>],Snew); return Vf; end function;