// Make newform 8280.2.a.bm in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8280_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_8280_2_a_bm();". // 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_8280_2_a_bm();". // 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 := [2, -4, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]]; Rf_basisdens := [1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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_8280_a();" function MakeCharacter_8280_a() N := 8280; order := 1; char_gens := [2071, 4141, 4601, 1657, 3961]; 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_8280_a_Hecke(Kf) return MakeCharacter_8280_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], [0, 0, 0], [1, 0, 0], [-2, 0, 1], [-1, 1, 1], [-1, 3, -1], [1, 3, 0], [-1, 1, -1], [-1, 0, 0], [1, 3, -2], [-2, 0, 3], [-2, -2, 3], [5, -1, 2], [-2, 2, -4], [7, -3, 3], [7, -3, 0], [-5, 3, -2], [-5, -1, 1], [-4, -2, 1], [3, -5, 6], [-1, 3, 3], [0, 0, 0], [1, 1, -4], [4, -2, 0], [2, -4, 2], [9, -1, 0], [0, -8, 2], [9, -3, -2], [5, -7, -3], [7, -7, 0], [-9, 5, -3], [-4, 4, -4], [14, 0, -2], [6, -4, 7], [-1, -1, 3], [2, 2, -4], [-6, 2, -1], [8, 0, -2], [5, -1, -3], [2, 8, -2], [-6, 10, -8], [-12, 6, -4], [-11, 7, -7], [8, -10, 0], [4, -6, -2], [-6, 10, -4], [2, 12, -7], [8, -8, -6], [-4, 0, 0], [4, 6, -8], [2, 4, -6], [-15, 1, 2], [-1, -5, -1], [4, -8, 2], [15, -5, 11], [-1, 7, -6], [5, -5, -4], [8, 6, -3], [10, 0, 0], [7, 3, 1], [12, 6, 1], [-7, 11, -2], [9, 7, -3], [-8, 0, 4], [-14, 6, -3], [11, 11, -1], [12, -2, -7], [-2, 4, 0], [-26, -2, -2], [4, 4, 1], [7, -5, 11], [-5, 9, -1], [-10, 0, 1], [-12, 10, -8], [12, -8, 8], [1, 1, -6], [0, -10, -2], [14, -8, 2], [-12, -6, 6], [-4, 4, 11], [-7, 7, -1], [1, -3, 9], [-14, -2, 6], [-18, 10, 5], [12, 0, -2], [9, -1, -7], [-3, -9, -6], [14, 2, 1], [12, -6, -2], [-9, 9, -9], [-11, 9, 10], [-19, -1, -7], [-5, -15, 1], [-3, -7, -4], [16, -2, -9], [29, -3, -2], [-22, 8, 6], [21, 5, 3], [-16, 4, -4], [6, -12, -2], [28, 0, -4], [19, -7, 4], [11, -9, -2], [0, -18, 6], [1, -21, 3], [18, -4, -6], [4, -4, -10], [9, 9, 5], [-18, -14, 6], [0, -8, 3], [5, -1, 15], [2, 0, 12], [1, 7, 2], [-20, 0, 4], [-3, 3, 15], [1, 9, -1], [2, 0, -9], [9, -9, 7], [-25, 11, -3], [-1, 1, -7], [0, -18, 14], [23, -9, -7], [-27, 3, 8], [23, 1, -1], [6, 2, -4], [-7, -3, -3], [7, 3, 5], [-7, 5, -8], [-8, 10, 9], [0, 12, -3], [0, 2, 5], [4, -12, 0], [3, -15, 11], [4, -16, -7], [-3, 3, 0], [39, 7, -3], [16, -10, 6], [-2, 12, -1], [13, -21, -4], [19, 21, -6], [18, 4, 1], [22, -4, -10], [12, 16, 0], [21, -3, 0], [-6, -10, 7], [-12, -8, 2], [12, 14, -10], [38, 4, -4], [10, -12, -3], [14, 22, -8], [16, 2, 6], [11, -5, -5], [-33, -3, 5], [-2, 6, -4], [12, -10, 13], [-12, 4, 16], [8, 0, -8], [-9, -19, 8], [-14, -4, -2], [-15, 13, -19], [26, 10, -10], [-14, -16, 0], [-33, -3, -7], [18, -10, -10], [3, 5, 4], [21, 9, 4], [-2, 4, -11], [4, -6, 0], [14, -8, -14], [3, 27, -9], [-44, -4, 4], [14, 14, -7], [16, -4, 6], [8, -16, -1], [-40, -8, 0], [23, -9, 15], [32, -8, 18], [-9, -3, 8], [-28, 2, -5], [-13, 19, -1], [-7, -9, -11], [15, -11, 7], [-4, -10, 2], [41, 5, -5], [-1, -9, 2], [39, -5, -7], [-9, -29, 11], [-24, -2, -1], [6, -2, 15], [-9, 7, 6], [-13, -9, 7], [-16, 8, 10], [1, -1, 1], [-5, -31, 12], [6, 2, 20], [29, -9, -4], [3, -1, -1], [-6, 12, -6], [-10, 4, 4], [52, -8, 6], [-13, 11, -5], [6, -10, 4], [-7, 1, -11], [21, -11, 9], [-31, 11, -15], [40, -2, -6], [-57, 1, -3], [-3, -25, 3], [21, -9, 2], [-2, 2, -12], [32, -14, 14], [-20, 18, 6], [-16, 0, -18], [-14, -26, 16], [1, 13, 0], [7, 7, 17], [-6, 6, -24], [25, 3, -6], [55, -1, 6], [-7, -15, 7], [34, -18, -17], [-22, -6, 4], [11, -7, -16], [0, -26, 19], [-7, 19, -17], [-28, 4, 1], [0, 10, 6], [-11, -11, -10], [0, 12, -4], [-7, 21, -24], [3, -9, -1], [-3, -21, 9], [-38, -6, 0], [-7, 3, -2], [-10, 12, -9], [-2, 22, 8], [-31, 21, -9], [-22, -12, 8], [25, 13, -4], [39, -5, 8], [31, -5, 6], [10, 32, -13], [6, -22, 4], [30, -4, 6], [-6, -12, -6], [-2, 10, 14], [15, -19, 21], [5, -37, 7], [-34, -6, 8], [7, -17, 6], [38, -16, -14], [10, 20, -4], [-19, -11, -10], [20, -18, -14], [-19, -35, 11], [23, -17, 22], [-25, -5, 5], [12, -32, -4], [13, -19, 7], [10, 32, -4], [10, 20, -15], [-19, -3, 8], [-54, 10, -19], [-12, 8, -15], [20, 2, 16], [-31, -5, -1], [-23, 5, 5], [14, -16, -20], [36, 6, -3], [-34, -12, 2], [-28, 4, -11], [-18, -4, 19], [-38, 10, 7], [-24, 12, -10], [49, -11, 3], [17, -1, -19], [-19, -21, 13], [-48, 8, -11], [-15, 21, -13], [-17, -11, -9], [-4, -16, 6], [-25, 9, 17], [-10, -14, -2], [22, 12, -14], [19, -19, 7], [37, -11, 2], [24, 16, -1], [14, 8, 8], [40, -12, -14], [16, 14, -2], [-36, 2, -14], [35, 27, -12], [24, -18, 12], [-9, 13, 15], [5, -7, -17], [18, 16, 2], [-27, 7, -11], [-5, 23, -3], [15, -13, 16], [32, 24, -6], [8, -4, -7], [-39, 15, 16], [14, -6, -14], [-5, 13, -1], [-32, 12, -20], [31, 7, 1], [-23, 11, -9], [-26, -8, 16], [-5, -7, -13], [1, -3, 3], [-20, 28, -16], [15, -13, 21], [-29, -5, 15], [8, -4, 20], [23, 11, 0], [-17, 7, -11], [-15, 5, -32], [1, 9, -10], [-42, 18, -27], [-22, 8, -6], [-21, 9, -15], [-30, 22, 13], [-2, -32, 20], [-8, 40, -10], [11, -27, -4], [-7, -3, 23], [3, 9, 17], [25, -17, 13], [52, 12, 0], [-3, 23, 10], [24, -16, 9], [19, -17, 1], [2, -32, 29], [-11, 39, 3], [32, -4, 0], [0, -20, 4], [56, -10, -10], [-32, 10, -10], [-22, -28, 14], [14, -20, -9], [-32, 18, -14], [-34, 12, -20], [-24, -2, -22], [49, -1, -9], [-33, 1, -10], [27, 3, -6], [-18, -14, 3], [-3, 19, -3], [0, -4, 22], [-25, 29, -8], [-42, 24, 13], [-37, 11, 1], [-21, -27, 6], [24, -18, -9], [-29, 7, -27], [18, 8, 8], [-49, 9, -11], [-24, -4, 14], [-47, -3, -9], [-2, 38, -20], [-32, 12, 5], [17, -13, -10], [6, -18, 28], [50, -10, 10], [26, -2, 4], [20, -2, 18], [5, 25, 9], [42, -24, 11], [-34, -32, 10], [-21, 31, 4], [-40, 2, 9], [-12, -48, 22], [-4, -26, 8], [36, -4, 12], [-7, -7, 1], [1, -11, 8], [-38, 2, 8], [-2, 18, -11], [40, -22, 8], [40, -22, 38], [16, -2, -18], [-20, -2, -30], [56, -12, 24], [57, -11, 11], [-22, -6, -18], [8, 12, -6], [32, -6, -13], [-29, 11, -3], [40, 6, 19], [35, 19, -6], [31, -13, 35], [-48, 14, -22], [-17, -9, 14], [27, 33, -7], [-9, 9, 29], [-10, 18, -11], [-26, 6, 8], [8, 2, 20], [12, 8, -12], [1, -21, 18], [6, -38, 15], [-14, -16, 20], [-50, -12, 11], [55, 1, 0], [-9, 9, -2], [-22, 24, -7], [28, -12, -1], [16, -34, 2], [-2, 10, -24], [-20, 12, -24], [27, -5, -11], [9, 1, -5], [4, -24, 0], [-2, -10, -18], [26, -30, -13], [-30, 20, 28], [14, -6, -18], [-23, -29, -1], [31, -11, 6], [10, 6, -22], [-33, 27, -15], [-66, -8, 2], [43, -35, -19], [-4, -4, -28], [55, -17, 9], [-30, 32, -38], [6, -10, 18], [13, 3, -4], [32, -16, 2], [-39, 33, 6], [-46, 12, -2], [32, 36, -14], [48, -8, 18], [-31, 5, -4], [25, 13, 15], [-21, 19, -13], [-4, -4, -1], [-34, 8, 4], [16, -8, 18], [26, -22, 16], [8, 16, 2], [-15, -7, 19], [-14, 42, -31], [40, -16, -16], [6, -18, -13], [37, -27, -7], [20, 36, -22], [27, 17, -17], [-28, 0, 21], [6, -32, 12], [11, -31, 25], [29, -29, -5], [41, -21, 23], [16, -2, -10], [14, -8, 6], [25, 3, -22], [46, 2, 11], [-14, 6, -14], [-18, 18, -19], [73, 3, 4], [-22, 24, 7], [-59, -9, 1], [4, -28, 28], [-38, 30, 3], [26, 34, -6], [-49, 31, 1], [-23, 7, 19], [-27, -3, -8], [3, 7, -3], [20, -4, 10], [26, 34, -6], [47, -11, -15], [38, 14, 8], [-12, 10, 0], [-15, 33, 2], [32, -12, 33], [-43, 31, -18], [-8, 8, 16], [-19, 31, -31], [-28, -22, 6], [11, 19, -19], [6, -36, -6], [34, 22, -11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8280_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8280_2_a_bm();". // 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_8280_2_a_bm(:prec:=3) chi := MakeCharacter_8280_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(3457) - 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_8280_2_a_bm();". // 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_8280_2_a_bm( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8280_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 5, 6, 1]>,<11,R![-32, -14, 2, 1]>,<13,R![76, -34, 0, 1]>,<17,R![86, -27, -6, 1]>],Snew); return Vf; end function;