// Make newform 2646.2.a.bk in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2646_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_2646_2_a_bk();". // 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_2646_2_a_bk();". // 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, 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_2646_a();" function MakeCharacter_2646_a() N := 2646; order := 1; char_gens := [785, 1081]; v := [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_2646_a_Hecke(Kf) return MakeCharacter_2646_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, 0], [-3, 1], [0, 0], [1, 0], [0, -3], [0, -2], [-3, 2], [-1, 3], [-4, -3], [-3, 3], [1, -3], [-9, -2], [2, 6], [-6, 4], [-8, 0], [-6, 4], [12, -2], [-6, -3], [-3, -3], [-6, -2], [2, 6], [-6, -7], [-9, -4], [0, 8], [6, -6], [3, -5], [2, 12], [-11, 3], [0, 3], [-8, 3], [-6, 6], [-6, -3], [-6, 2], [10, 3], [2, -9], [12, 5], [-2, 9], [-6, -13], [9, -3], [-6, -12], [6, -3], [5, 3], [-4, -6], [-10, -9], [9, 3], [0, -9], [3, -3], [0, 9], [12, 0], [-12, 3], [18, 0], [6, 6], [0, 8], [3, -6], [9, 9], [-3, -13], [18, 8], [7, 15], [-2, 6], [6, 10], [-6, -4], [3, -14], [0, 11], [-6, -3], [10, 0], [16, 6], [5, 6], [27, 6], [18, 3], [3, 4], [-8, -18], [-21, 7], [-11, 3], [-14, -12], [6, 17], [20, -12], [18, -1], [-6, -3], [-6, 0], [12, -17], [-3, 15], [-15, -3], [18, 9], [-12, -12], [5, -24], [-24, 9], [3, -12], [3, -7], [-26, 9], [18, 4], [-24, 4], [-30, 6], [-9, 6], [2, 12], [24, 11], [-6, 20], [-9, 10], [-9, 6], [-29, -3], [16, -6], [-6, 9], [18, 1], [-32, -3], [-20, -15], [-24, -3], [12, 3], [-3, -4], [-7, 9], [18, -9], [-30, 0], [-9, 9], [10, -12], [3, 14], [14, 0], [20, -15], [-3, 2], [30, -6], [6, -24], [-9, 24], [-12, 9], [24, 18], [9, 21], [3, -12], [12, -24], [24, -3], [-23, -3], [18, 4], [0, -10], [-6, 24], [36, 9], [19, -3], [14, 18], [-22, -18], [18, 8], [-6, -27], [-21, -13], [-42, -8], [21, 3], [14, 3], [-3, -30], [30, -12], [34, -3], [21, -24], [-30, 10], [-36, 13], [24, 0], [-3, 26], [-27, -2], [14, -12], [2, -12], [-21, -24], [20, -6], [-12, -25], [4, -9], [-4, -6], [4, 6], [12, -20], [-12, 8], [-3, -5], [-33, 12], [22, -3], [-6, -33], [-6, -35], [0, 12], [-18, 14], [-4, 3], [30, 7], [-35, 6], [-33, -13], [-12, 0], [30, 16], [-12, 18], [1, 18], [-15, -17], [-51, -2], [-24, -6], [18, -3], [-21, -21], [0, -28], [22, 3], [-12, -1], [-3, -15], [-24, 8], [-29, -21], [-21, -13], [-6, 30], [-18, 0], [-31, 0], [12, 8], [18, -27], [-18, -6], [-16, 21], [-30, -2], [39, 0], [-33, -8], [39, 0], [10, -30], [-3, -18], [42, 1], [12, -21], [3, 27], [-18, 8], [-36, 8], [-54, -11], [54, -4], [12, 22], [15, -6], [26, 6], [42, 6], [-47, -6], [-21, 9], [-14, -18], [6, -14], [-48, -4], [0, -4], [8, 3], [-27, -16], [39, 21], [-36, 12], [26, 12], [-48, 12], [-18, -3], [-10, 12], [36, 7], [19, -33], [-3, 20], [13, -9], [-57, 11], [39, 18], [23, 9], [-15, 0], [28, -15], [-36, -27], [15, -8], [60, -3], [-48, 3], [-16, 15], [2, 42], [12, -35], [-3, 0], [51, 6], [-33, -21], [21, 15], [9, 28], [-6, -38], [21, -29], [-6, -13], [-22, -15], [-7, -3], [20, -18], [24, 24], [8, 0], [-12, -5], [-9, 11], [-17, 30], [-23, -21], [-27, -16], [-36, -14], [-30, 22], [-26, 18], [13, -30], [-12, 33], [42, 14], [27, -24], [-15, 18], [0, 15], [-6, 36], [-52, 9], [-44, -3], [18, -6], [52, 0], [-30, 3], [-8, 21], [-54, 4], [3, -35], [9, -42], [20, 6], [20, -24], [18, 43], [-54, 10], [-50, -15], [54, -7], [30, 12], [-9, -34], [-3, -39], [-17, -6], [14, -42], [21, -35], [-21, 16], [-2, 45], [6, -8], [14, -15], [-12, 23], [-61, -9], [-63, -13], [57, -7], [-6, -2], [-42, 25], [21, 32], [36, 9], [36, 9], [18, -21], [-33, 18], [-40, 9], [21, 18], [3, -12], [-18, -10], [33, -9], [-11, -27], [-18, -7], [42, -24], [20, -18], [-50, 15], [-3, 3], [-12, -40], [54, 7], [25, 9], [12, 16], [6, 24], [-6, -40], [-5, 0], [57, -19], [78, 0], [-12, 18], [12, 4], [32, -3], [69, 12], [-64, 18], [14, 51], [27, 33], [-50, 21], [-45, -17], [-66, 1], [26, 15], [48, 22], [-33, -27], [-21, 8], [-66, -6], [-9, 33], [53, -27], [-22, -51], [30, 14], [12, 9], [2, -42], [2, -6], [30, -12], [22, -9], [-24, -23], [66, 6], [-63, -6], [-40, 0], [-56, 18], [33, 1], [17, 27], [-3, 10], [-18, 26], [0, 56], [30, 30], [88, 6], [-20, -6], [-9, 28], [12, 48], [-9, -24], [-18, 30], [25, -6], [15, -1], [46, 27], [17, 12], [69, -6], [-27, 0], [-50, 12], [-6, 42], [27, 21], [-15, 51], [-42, -6], [12, 24], [-84, 3], [63, 24], [-6, -57], [3, -49], [64, 3], [-36, 21], [38, 36], [-15, 2], [18, 6], [-30, -21], [60, -6], [76, -6], [24, 12], [-64, -18], [-9, 3], [-1, -36], [0, 36], [46, 0], [21, -6], [39, 19], [15, -16], [30, -6], [-8, 21], [-6, 42], [78, -3], [14, 27], [-15, 32], [-45, -33], [-9, 19], [-5, 45], [-14, 51], [-9, 18], [12, 18], [0, 6], [-54, -18], [12, 12], [28, -12], [46, 12], [-15, -63], [-57, -21], [-18, 28], [54, 8], [18, -17], [-72, 6], [12, -26], [-75, 15], [-6, -26], [48, 27], [60, 14], [-30, 18], [20, 30], [87, 4], [-24, 15]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2646_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_a_bk();". // 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_2646_2_a_bk(:prec:=2) chi := MakeCharacter_2646_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_2646_2_a_bk();". // 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_2646_2_a_bk( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2646_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![7, 6, 1]>,<11,R![-1, 1]>,<13,R![-18, 0, 1]>,<17,R![-8, 0, 1]>],Snew); return Vf; end function;