// Make newform 1155.2.a.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1155_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_1155_2_a_r();". // 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_1155_2_a_r();". // 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 := [-3, 0, 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_1155_a();" function MakeCharacter_1155_a() N := 1155; order := 1; char_gens := [386, 232, 661, 211]; 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_1155_a_Hecke(Kf) return MakeCharacter_1155_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, 1], [-1, 0], [-1, 0], [1, 0], [-1, 0], [1, 3], [-3, 2], [-5, -2], [5, -2], [4, -1], [-7, -1], [1, 3], [1, 5], [0, -1], [3, 1], [1, -2], [12, -1], [1, -6], [-2, -4], [-1, -5], [-2, -6], [-5, -5], [-1, -2], [6, -5], [-4, -1], [10, -4], [-16, -1], [7, 1], [-7, -1], [-3, -8], [-10, 7], [11, -3], [-10, -2], [8, -2], [12, -2], [2, 8], [-2, 7], [7, -7], [0, 6], [-22, 2], [-1, 11], [14, 4], [-8, -4], [-4, 2], [2, -2], [-8, 4], [8, 2], [6, 3], [5, 6], [23, -3], [-5, 3], [6, -7], [20, -4], [0, 4], [19, 3], [-6, -10], [-8, -9], [-11, 4], [-16, -4], [24, 4], [5, 3], [9, 0], [0, 6], [-10, -10], [0, -1], [12, -4], [-9, -10], [-2, 5], [2, 2], [-15, 0], [12, -2], [-12, 13], [22, 3], [24, -3], [7, -8], [31, -1], [7, 13], [24, -8], [-21, 7], [-16, -4], [28, -1], [-3, -16], [-14, 6], [-8, -10], [1, 16], [-20, 2], [-3, -11], [6, 5], [12, -10], [16, 2], [-10, 10], [-5, 1], [4, 2], [12, 5], [19, -6], [-23, 0], [-12, 3], [10, 11], [-10, 10], [-15, 11], [8, -9], [12, 2], [-18, 4], [4, 5], [5, -7], [-14, -4], [-3, 9], [16, 0], [4, -14], [-3, -12], [-8, -18], [14, -12], [-12, 20], [-18, 4], [11, 16], [6, 20], [-12, -3], [6, -22], [-33, -2], [-18, 7], [-1, 1], [-2, 9], [-15, 14], [-40, 4], [-26, 0], [28, -3], [33, -2], [24, -5], [10, -19], [20, -18], [12, 2], [-12, -18], [-1, -10], [12, 2], [-18, -4], [-3, 12], [12, -14], [-8, 20], [20, 18], [12, 2], [-16, -12], [16, 15], [-11, -17], [10, -2], [-4, 18], [24, 5], [23, 7], [22, -16], [-33, 9], [-29, -2], [-18, -21], [-16, -19], [-6, 18], [-11, 6], [27, 5], [42, 6], [2, 28], [12, 16], [1, -15], [-13, -23], [15, 16], [2, -4], [8, -13], [34, -5], [1, -6], [28, 10], [11, 24], [-2, 16], [-15, -3], [13, 21], [12, -7], [19, 1], [-18, -15], [24, -9], [-4, -2], [24, -14], [45, 1], [4, -28], [0, 26], [20, 12], [-16, 16], [2, -10], [40, -6], [3, -16], [27, -6], [-10, -15], [20, -1], [-8, 23], [-17, 1], [25, 13], [6, -8], [10, 0], [0, -2], [30, 19], [-21, 7], [-52, -8], [-4, 24], [13, -9], [-19, -24], [-36, -14], [-36, 17], [-17, -6], [-2, -5], [-6, -6], [-4, -27], [-56, -4], [25, -1], [-42, -12], [4, 3], [-37, -11], [-28, -23], [22, -3], [-2, -18], [13, -11], [16, 8], [40, -18], [4, -31], [-32, -6], [68, 2], [-11, 6], [-32, 8], [-10, -12], [15, -1], [-35, 19], [33, 8], [39, -5], [15, 1], [25, -11], [-18, 2], [16, -1], [-3, -21], [-50, 12], [-11, 9], [24, -15], [6, 7], [13, -26], [-23, -7], [-42, -10], [48, 6], [-6, 22], [42, -4], [-35, -2], [24, -22], [53, 0], [-9, -26], [-39, 15], [28, -4], [24, -15], [-59, -3], [-22, 6], [-2, 16], [-9, -1], [-8, 36], [29, -1], [-3, -2], [18, 3], [45, -14], [-11, -31], [29, -25], [16, 30], [-16, -19], [56, -4], [-5, 16], [5, 17], [-5, -5], [-15, 13], [-12, 16], [24, 15], [-18, 27], [5, -23], [-2, -36], [-18, 4], [18, -9], [-39, 7], [-25, -23], [8, -30], [-5, -14], [52, 8], [25, -1], [-28, 12], [-9, 24], [24, 10], [0, -36], [-63, 6], [5, 3], [-19, -5], [-27, 29], [-4, 24], [10, 16], [21, -7], [0, 24], [43, 7], [-19, 25], [12, -8], [26, -12], [-8, 4], [38, 18], [39, 19], [-64, -4], [40, 8], [16, -30], [15, -5], [1, -31], [-7, 26], [-11, 26], [-17, -21], [5, 4], [33, -11], [-3, 1], [26, -21], [-30, -8], [21, 33], [32, -25], [33, 9], [-53, -19], [-35, 22], [-6, -2], [54, -1], [-44, 10], [-16, -35], [16, 8], [9, 5], [-30, 0], [-11, 23], [-6, -42], [-6, 30], [62, -4], [-8, -37], [-21, 25], [-11, 1], [-37, 9], [34, -14], [11, -16], [-4, 0], [16, 38], [-24, -4], [-9, -8], [28, 20], [59, 7], [-1, -17], [19, 7], [-25, 28], [-15, -15], [-26, 28], [-13, 2], [-46, -17], [21, 17], [33, 5], [44, -12], [-1, -23], [-19, -16], [-74, -2], [-49, 5], [-33, 13], [-20, 24], [27, -14], [12, -34], [3, 17], [-36, 22], [1, -26], [-62, 19], [-6, 12], [16, 22], [44, 19], [-34, -9], [46, -4], [-13, 10], [-26, -31], [1, 23], [42, -10], [-38, -34], [2, 42], [-24, -27], [-29, -24], [-11, 31], [8, -18], [-32, -9], [28, -8], [21, -1], [-33, -15], [24, 33], [27, -28], [44, -8], [9, 43], [-43, -16], [11, -23], [7, -14], [-44, 25], [-26, -2], [-18, 38], [7, -50], [76, 2], [-13, 31], [-20, 5], [-14, 28], [4, 15], [-36, 26], [0, 22], [7, -22], [82, -5], [5, -2], [-3, 4], [28, 37], [-31, 21], [63, 9], [-33, -20], [-25, -17], [22, 8], [6, 11], [42, -19], [4, 24], [-1, -51], [34, 12], [39, 19], [-32, 26], [-2, 51], [-10, 27], [46, 27], [-5, -9], [-60, 22], [-59, 1], [64, -10], [-15, -6], [9, -11], [-32, 11], [3, 43], [-53, 2], [6, 7], [-34, 12], [73, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1155_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1155_2_a_r();". // 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_1155_2_a_r(:prec:=2) chi := MakeCharacter_1155_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_1155_2_a_r();". // 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_1155_2_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1155_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-2, -2, 1]>,<13,R![-26, -2, 1]>,<17,R![-3, 6, 1]>],Snew); return Vf; end function;