// Make newform 1472.2.a.u in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1472_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_1472_2_a_u();". // 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_1472_2_a_u();". // 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 poly := [-4, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_1472_a();" function MakeCharacter_1472_a() N := 1472; order := 1; char_gens := [1151, 645, 833]; 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_1472_a_Hecke(Kf) return MakeCharacter_1472_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], [0, 1], [-2, 0], [0, 0], [0, -2], [-2, -1], [2, -2], [0, -2], [-1, 0], [-2, 1], [-4, -1], [-2, 4], [-2, 5], [8, 0], [4, 3], [-2, 0], [-4, 4], [-2, -4], [0, 2], [12, -1], [-10, 3], [0, -2], [-8, 4], [2, -6], [2, -6], [-6, 4], [-8, 4], [-8, 2], [-10, 2], [-6, -2], [-4, -3], [-8, -3], [10, 2], [8, 3], [14, -4], [-12, -1], [-18, 2], [-8, 7], [-16, 0], [10, 0], [8, 3], [-10, -4], [16, -6], [-2, -1], [6, -7], [8, -2], [4, 0], [-16, 8], [0, -4], [-18, -2], [-2, 9], [12, -9], [-6, -8], [0, -4], [-10, -3], [-16, 2], [-18, -3], [24, 0], [-10, 5], [2, -2], [0, 2], [6, -6], [12, -4], [4, -1], [-6, 6], [10, 4], [-8, 15], [-6, 10], [4, -8], [30, -3], [14, -1], [-8, 10], [-8, 4], [-2, -8], [0, -8], [16, 4], [14, -6], [-18, -1], [-22, 4], [6, -13], [-8, 8], [22, 2], [-8, -8], [-6, 12], [-12, 9], [24, -7], [2, -4], [10, -6], [-2, 5], [-8, 8], [0, 6], [0, 4], [20, -3], [0, -5], [0, 9], [-8, 6], [-18, 13], [-6, 10], [-16, 4], [-18, -7], [0, -17], [6, -2], [-16, 4], [2, -4], [32, -8], [-2, 7], [0, 9], [-14, -12], [8, 0], [-26, -5], [16, -12], [-18, 18], [-6, 16], [-8, -8], [24, -12], [2, -6], [8, 6], [4, -13], [-2, -9], [8, -4], [-10, 4], [-26, 9], [-2, 16], [-24, 3], [20, -16], [6, -14], [22, 6], [-16, 8], [-24, -6], [-34, -6], [-8, 3], [24, 4], [0, 16], [22, -8], [-26, 3], [-6, 0], [-18, 6], [16, 4], [-2, -4], [26, 0], [-16, -7], [-6, 12], [52, -1], [16, -8], [10, 4], [40, 4], [-6, 16], [14, 1], [-40, 9], [20, -11], [10, -16], [10, -2], [4, -16], [12, 3], [-48, 2], [-32, -2], [48, -4], [22, 5], [26, -4], [-34, -2], [0, 1], [34, 4], [12, 9], [-16, -6], [-6, -20], [-24, 18], [-32, 16], [-14, 0], [26, -16], [6, -1], [24, 6], [-2, 13], [8, -12], [-14, -2], [-20, -15], [18, 2], [16, -5], [-26, 7], [-16, 6], [-2, 14], [4, 13], [24, 4], [-26, -11], [-22, 16], [0, -18], [14, 2], [-14, -16], [56, 2], [-26, -5], [8, -24], [6, -15], [-36, 16], [24, -14], [18, -20], [32, -22], [-30, 2], [18, -12], [-2, -6], [26, 14], [28, -19], [22, -2], [-40, -8], [-26, 9], [-14, 12], [-56, -4], [14, 1], [-16, 6], [8, 13], [30, -3], [16, -9], [22, -23], [22, -13], [24, -24], [16, 2], [0, -16], [58, -2], [-4, -9], [30, -21], [-16, 30], [46, -5], [6, 19], [32, -10], [14, -5], [-16, 32], [16, -19], [54, -5], [-6, 16], [60, -3], [-32, -10], [20, 8], [-22, 20], [32, -16], [40, 2], [-6, -16], [-8, -20], [-40, 4], [2, 22], [38, 0], [32, -3], [-28, 17], [-16, -24], [40, -11], [4, -17], [30, -29], [-30, 20], [4, 7], [-28, 3], [-32, -14], [0, -30], [8, 2], [-26, 2], [18, 0], [32, -6], [50, 2], [-14, -4], [56, -5], [-26, 20], [-40, -10], [6, -11], [-10, -1], [-64, -4], [40, 4], [50, 4], [-26, -6], [-46, 4], [-24, 14], [-18, 4], [66, -6], [16, 14], [22, -15], [-18, 21], [-8, -2], [34, 14], [-32, 16], [-2, -17], [-4, 11], [-52, 16], [2, -12], [-6, -4], [-32, 24], [-28, 25], [-16, 18], [40, -10], [-26, -12], [36, 4], [-32, 24], [10, -16], [-58, -4], [32, 8], [34, 12], [-58, 16], [-24, 8], [-22, 8], [-16, -10], [-18, 1], [6, 24], [40, 8], [-26, 15], [0, 7], [32, -21], [42, -22], [-50, 8], [-72, 8], [-16, -1], [-32, 20], [6, 5], [4, 16], [6, 6], [-64, -6], [42, 4], [-4, -17], [-34, -12], [-46, 26], [32, -1], [8, -30], [-30, 14], [-56, 3], [52, 7], [-6, 6], [6, 29], [0, 24], [-6, -6], [-50, 7], [-48, 8], [34, -24], [-14, 14], [-40, 14], [40, -1], [-8, 6], [-50, 14], [-34, 29], [-2, -21], [44, 1], [16, -25], [0, 14], [-32, 7], [70, 4], [10, 6], [-18, -11], [-16, -12], [54, 3], [-22, 22], [-22, -16], [-8, -14], [-18, -24], [-52, 16], [-26, -9], [-44, 8], [64, -14], [30, 20], [0, -27], [-58, -5], [-38, -8], [-32, 30], [-18, -18], [-70, 16], [64, 0], [8, -22], [-34, -13], [-48, 4], [-58, 24], [22, -37], [12, -3], [-16, -14], [-32, 7], [-26, 15], [-18, 3], [24, -14], [82, -8], [-8, -17], [4, 8], [4, -25], [-18, 2], [-40, 8], [-18, -25], [-16, 15], [-48, 2], [-46, 4], [-46, -6], [62, -3], [14, 4], [-14, 4], [0, 20], [22, -3], [-40, -8], [-4, -15], [60, 1], [2, -8], [8, -2], [24, -28], [42, 22], [-14, 8], [0, -19], [36, -8], [-16, -20], [18, -24], [-16, 12], [42, -10], [8, 8], [10, 32], [62, -17], [-34, 3], [-24, -28], [-22, -2], [-74, -5], [-12, -5], [46, -20], [-50, 3], [48, 16], [-20, -8], [38, -27], [-10, -25], [40, -22], [16, 14], [-6, 24], [-50, 1], [8, 32], [-8, 4], [-30, -18], [-48, 4], [38, 6], [54, 2], [-32, 0], [12, 0], [-42, 3], [14, -21], [56, 18], [74, -4], [32, 17], [-4, 3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1472_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1472_2_a_u();". // 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_1472_2_a_u(:prec:=2) chi := MakeCharacter_1472_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_1472_2_a_u();". // 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_1472_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1472_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-4, -1, 1]>,<5,R![2, 1]>,<7,R![0, 1]>],Snew); return Vf; end function;