// Make newform 7942.2.a.v in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7942_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_7942_2_a_v();". // 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_7942_2_a_v();". // 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 := [-6, 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_7942_a();" function MakeCharacter_7942_a() N := 7942; order := 1; char_gens := [5777, 6139]; 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_7942_a_Hecke(Kf) return MakeCharacter_7942_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], [0, 0], [0, 1], [-1, 0], [-1, -2], [-2, 1], [0, 0], [-4, -2], [3, 2], [2, 0], [6, -1], [-10, -1], [7, -1], [-9, -1], [2, 4], [-2, -2], [3, -2], [12, 1], [5, -1], [-12, 1], [2, -1], [-3, 1], [-9, 0], [-1, -2], [5, 0], [-1, 3], [3, -3], [11, -2], [1, -6], [-2, -6], [-3, -3], [-3, 2], [-4, 0], [4, -6], [-6, 1], [-6, 0], [-2, -2], [4, 7], [0, -4], [0, 4], [-8, -5], [-15, -3], [-16, 3], [11, 2], [13, -3], [20, 0], [-11, -5], [14, 4], [-8, -3], [-8, 1], [12, -4], [10, 0], [12, -5], [-22, 0], [-8, -4], [18, -1], [10, -6], [-7, 10], [0, 7], [8, 2], [-3, 2], [-7, 5], [-5, -5], [-5, 6], [-2, 9], [-2, -3], [22, 4], [-5, 1], [31, 0], [3, -10], [10, -6], [-23, 1], [-8, 4], [4, 3], [-12, 6], [4, 5], [-18, 3], [24, -6], [4, 11], [-10, 3], [-28, 3], [-6, -9], [17, -6], [-8, -8], [-14, 3], [20, 4], [4, 11], [-3, 4], [23, -7], [-14, -3], [-26, 7], [-15, -3], [-4, 4], [14, 6], [-6, 2], [4, -8], [-27, 6], [-21, 3], [28, -4], [23, 3], [-3, -8], [-4, -2], [-14, 8], [7, 11], [-28, 4], [-20, 3], [-4, -12], [27, -3], [24, 8], [-2, 7], [-17, 4], [-9, 12], [-10, 6], [23, 9], [-19, 10], [4, 3], [-27, 5], [-4, -3], [5, 11], [-8, -7], [-6, 15], [-1, -6], [-18, 7], [2, 17], [-27, 8], [-14, -12], [-23, -7], [-17, -11], [36, 2], [-5, -15], [-20, 7], [-1, 19], [26, 4], [-8, -16], [-10, 12], [-6, -7], [13, -5], [-20, -9], [18, 7], [-5, -1], [12, -14], [-7, 1], [3, 13], [-14, -7], [33, 5], [12, 6], [-6, -7], [-6, -2], [-19, -3], [18, -8], [8, -2], [-10, 1], [6, -3], [34, -2], [0, 10], [-18, 3], [-13, 10], [-4, 4], [-36, 4], [-14, 14], [40, 0], [-28, 4], [-12, 13], [-57, 0], [9, -15], [-1, -1], [10, 16], [-12, 8], [-24, 3], [-7, -11], [-24, -8], [-18, 10], [-28, 4], [16, 0], [-43, -8], [39, -1], [20, 7], [-32, -9], [5, 0], [43, -7], [-43, -5], [38, -4], [-16, -5], [9, -15], [-6, 17], [17, -2], [18, -13], [-6, 0], [-42, 5], [-4, 4], [-29, 3], [-24, -13], [28, -1], [-30, -12], [-27, -4], [-18, 20], [-12, -8], [-6, -8], [-24, -12], [-25, -6], [16, 17], [12, 0], [50, 4], [12, -8], [12, 0], [-35, 11], [-4, -20], [26, -12], [-10, -10], [-28, 0], [10, -7], [-11, -13], [20, -11], [-4, -12], [-7, -10], [34, -14], [-6, -19], [0, 2], [22, 7], [-15, -10], [26, -10], [33, -8], [8, 16], [33, -5], [52, -4], [10, 24], [24, -8], [-18, 14], [-25, 17], [14, -3], [29, -1], [4, -25], [-10, -9], [14, 10], [-32, 12], [13, -8], [1, 6], [-40, 0], [29, 13], [42, 14], [-31, 9], [24, -4], [10, 5], [48, 0], [34, -11], [-45, 9], [-34, 10], [-4, -2], [12, 1], [23, -10], [0, 5], [45, -5], [-47, 2], [2, 0], [34, -16], [16, -4], [8, -6], [30, -13], [12, -13], [-58, -2], [-11, -9], [-15, -10], [20, -8], [23, 6], [34, 2], [-14, 4], [31, 2], [5, 15], [23, 18], [-50, -4], [10, 10], [-17, 6], [-12, -16], [10, 11], [-33, 17], [8, 16], [-28, 6], [-28, -4], [48, 6], [48, 11], [11, 1], [-42, -14], [-1, 10], [-29, -15], [27, 13], [29, 12], [18, 0], [8, -8], [8, 9], [-18, -9], [36, 0], [28, 5], [9, 21], [-49, 10], [-32, 20], [-17, -27], [10, 27], [41, -11], [27, 13], [2, -24], [-11, -30], [46, 6], [-26, -10], [-2, -2], [-21, 28], [-30, 3], [30, -20], [16, -6], [-52, -6], [-16, 11], [34, -16], [-30, 20], [52, 9], [-26, 2], [22, 13], [-42, 6], [16, 0], [55, -14], [8, 16], [35, 7], [-24, 6], [-23, -2], [-24, 12], [0, 28], [-48, 2], [10, 29], [-32, 21], [-18, -15], [-25, -6], [43, -10], [-22, 15], [8, 24], [-3, -5], [3, 5], [-12, -3], [-24, -21], [34, -3], [2, 16], [6, -2], [12, 12], [-51, 8], [71, -8], [37, 17], [-10, 33], [3, 7], [-2, 28], [10, -26], [-76, 5], [18, 1], [28, 0], [-14, 0], [20, 13], [28, -9], [32, 0], [42, -12], [32, 22], [-69, 5], [-36, -4], [65, -1], [11, -14], [-2, -19], [39, 3], [-55, 5], [-62, -6], [-27, -28], [-17, 14], [38, 13], [6, 27], [10, 30], [34, -22], [-78, 2], [33, -2], [68, 13], [-28, -20], [-28, 27], [-10, -18], [16, 0], [0, -4], [-62, 0], [-36, 0], [-63, 8], [-26, -26], [44, -13], [17, -3], [39, 21], [17, -5], [54, -4], [-12, -32], [-28, -18], [12, -20], [-38, 16], [56, 3], [-12, -11], [-61, 5], [-34, 15], [-60, 8], [3, 20], [30, -6], [-5, 6], [-41, 10], [-17, -22], [12, 21], [-13, 6], [-45, 2], [-8, 25], [-6, -32], [-18, -21], [-54, 5], [-40, 24], [40, 4], [57, -16], [0, -32], [-13, -1], [-60, -7], [-60, -12], [-50, 10], [-23, -5], [-15, -28], [-42, -15], [-74, 7], [-11, -22], [-15, 25], [-29, -5], [37, 24], [-48, 11], [42, -2], [-4, -3], [-54, 3], [-12, -4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7942_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7942_2_a_v();". // 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_7942_2_a_v(:prec:=2) chi := MakeCharacter_7942_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_7942_2_a_v();". // 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_7942_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7942_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-6, 0, 1]>,<5,R![0, 1]>,<13,R![-23, 2, 1]>],Snew); return Vf; end function;