// Make newform 7605.2.a.co in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7605_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_7605_2_a_co();". // 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_7605_2_a_co();". // 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, 6, 10, -6, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-3, 0, 1, 0, 0], [2, -4, -1, 1, 0], [5, 3, -6, -1, 1]]; Rf_basisdens := [1, 1, 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_7605_a();" function MakeCharacter_7605_a() N := 7605; order := 1; char_gens := [6761, 1522, 6931]; 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_7605_a_Hecke(Kf) return MakeCharacter_7605_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, 1, 0, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 1], [1, 1, 0, 0, 1], [0, 0, 0, 0, 0], [0, -1, 2, 0, 0], [-1, 0, 1, 0, 0], [-1, 0, -1, -1, 0], [3, 0, 0, -1, 1], [2, -1, 0, -1, 1], [0, -1, -1, -1, -1], [1, 0, 2, -1, -1], [2, 3, -1, 0, 0], [3, -1, -1, -1, -2], [-4, 0, -2, 0, 2], [3, 0, -2, 1, -1], [2, 1, 0, 1, -1], [2, -2, 0, 1, -1], [2, 1, -2, -1, 0], [-2, 5, 1, -1, 0], [-2, -4, 1, 2, 0], [3, 0, 1, -3, 0], [2, -3, 0, -1, 0], [-2, 1, -1, -2, -2], [-5, -5, 2, 0, 1], [6, 1, 3, -2, 0], [5, -1, 1, 1, -2], [3, -5, -1, 1, -3], [0, 0, 0, 4, 2], [-4, 7, -1, 1, 2], [-7, 6, -2, -3, -1], [9, -3, -3, 1, 2], [-5, -3, 1, -3, -1], [7, -3, 4, 4, -3], [-3, 9, -4, -3, 1], [4, -4, -2, -1, 3], [10, 0, 4, 0, -1], [4, -6, 2, 2, 2], [-8, 1, 4, 2, 2], [-4, 3, 2, 5, 0], [-7, 5, -6, -1, 1], [4, -5, -2, -1, -4], [-2, -2, 2, 4, 1], [3, 0, -1, -1, 4], [-7, 4, 0, -2, -4], [-1, -2, -2, -2, 2], [0, -7, 5, -1, -1], [9, 1, -3, -1, -2], [-5, 7, 0, 1, 1], [11, -2, -1, 1, -2], [7, -1, 0, 2, -1], [-3, -1, -4, 5, -1], [-11, 9, -2, -2, 3], [1, 5, 3, -5, 0], [4, 3, -6, 2, -4], [2, -1, 4, -3, 2], [-1, -6, 6, 0, 0], [2, 3, -1, -3, -3], [1, 4, -6, 1, -3], [14, 3, 3, -3, 0], [9, -3, 3, 1, 2], [0, 4, 0, 1, -3], [14, -3, -2, 3, 2], [-2, -8, 6, 1, 1], [-5, 6, -3, 3, 4], [-3, -5, 1, 1, -3], [-6, 1, -3, 4, 0], [8, 6, -4, -2, -2], [-2, -2, -5, -2, -2], [1, 6, -7, -1, -4], [7, -6, -2, 3, 1], [10, -3, -7, 0, -4], [-2, 1, -1, 1, 4], [-8, 4, 5, 4, -2], [7, 1, 1, -1, 2], [13, 3, -2, 0, 1], [-16, 4, -2, -5, -1], [-6, 12, -2, -4, -2], [6, -5, -2, 1, 3], [-6, -5, -6, 3, 0], [12, 8, -1, -4, 2], [1, 11, -6, 4, 3], [2, 0, 8, 0, 3], [4, -9, -2, 3, -1], [6, -7, 4, -4, -2], [10, -4, -2, 0, -2], [14, 5, -9, 2, -2], [10, 3, -6, -5, 0], [6, 0, -2, 3, -5], [9, 3, 1, -3, 0], [12, -3, -4, 5, 4], [2, -10, 6, -1, 4], [11, -14, 8, 3, -1], [7, 3, 2, -3, 5], [20, 8, -2, -6, 2], [5, -7, 2, -2, 1], [-14, 15, -6, -3, 4], [4, -4, -2, -4, -6], [-9, 0, 0, 0, -2], [16, -11, 1, 4, -2], [3, -14, 1, 3, -6], [-16, 6, -4, -2, -2], [-12, 7, -2, -3, 0], [-5, -2, -5, 4, -2], [-2, -8, 6, -2, -2], [5, -11, 7, 9, -4], [-6, 8, -2, 2, -2], [-16, -3, 10, 3, 2], [1, -12, 5, 0, -4], [18, -4, -10, -1, -4], [-10, 19, 1, -2, 2], [15, -6, 7, 1, 2], [-9, -7, 11, -1, 5], [0, 10, -1, 4, 0], [1, 15, -4, -2, -3], [-4, -13, 3, 5, -2], [-11, 10, -9, -3, -8], [-6, 3, -4, -4, 4], [10, 10, -2, 2, -4], [13, -3, -3, -3, -1], [16, -14, 4, 4, 1], [7, -12, -3, 3, 4], [6, -1, -6, 0, 0], [-4, 0, -1, -6, -6], [24, -8, 2, 4, 2], [5, -3, -1, 3, 3], [-5, -12, 6, 1, -3], [14, -18, 4, 3, 3], [-2, 1, -1, 4, -2], [10, -13, 1, 5, 3], [24, 9, 2, -4, 4], [-1, -7, -6, -1, 1], [-6, -6, 0, 3, 4], [-13, 11, 4, 0, -1], [-31, 2, 3, -4, -2], [11, -4, -3, 3, -4], [-16, 9, 9, 3, 0], [3, -9, 13, 1, 2], [2, -6, 10, 2, 0], [5, 0, -2, 0, -6], [16, 6, 0, -10, 2], [18, -5, 3, 1, 3], [9, -3, -1, -3, 0], [0, -4, -1, 2, -10], [11, 3, -4, -2, 1], [4, -13, 7, -1, -6], [11, 3, 3, 3, -10], [-22, -8, 5, -8, -4], [0, 12, 0, 6, -6], [-4, -18, 4, 0, -6], [-10, -12, -6, 8, -6], [2, 6, -6, 3, -9], [-16, 7, -2, -6, -4], [-4, -16, 6, 5, -2], [0, 12, -6, 0, -6], [-19, 1, 0, -5, 5], [8, -8, 6, -4, 0], [10, -1, -5, 5, -3], [13, -6, 12, 3, -5], [26, -13, 8, 4, -2], [13, -2, 1, 5, -10], [-18, 12, -4, 0, 10], [9, -7, 2, 6, -9], [12, 0, 2, 2, -2], [5, 6, 1, 3, 8], [-11, -3, 10, 3, -1], [30, 0, -6, -6, 1], [-14, 6, 3, 6, 8], [16, 9, -4, 0, -2], [9, -6, 2, -9, -3], [5, 12, -3, -6, 0], [-11, 9, 14, -2, -3], [22, 1, -1, -5, -2], [7, -6, 11, 0, 2], [0, 19, -6, 5, 4], [-3, -1, 5, -1, -1], [-20, 8, -8, -10, -4], [14, 3, -5, -6, 6], [1, -5, -3, -11, 7], [0, -3, -7, 9, -5], [-12, -9, 8, -9, 0], [-16, -5, 9, -2, 2], [3, 9, 5, -3, 12], [-4, -3, 2, -4, 0], [15, -15, -8, -2, -1], [-8, 2, 4, -4, -9], [-22, 9, -3, -9, -3], [30, 6, -1, -6, 4], [-17, 6, 2, 1, 3], [-4, 4, -4, -2, -13], [13, 9, 1, 7, 0], [-4, 6, 3, 0, 12], [-13, 12, -10, -1, -9], [9, 3, -5, 7, -4], [16, -12, 4, 2, -4], [-2, -9, 6, 3, -7], [-4, -1, -9, -4, 4], [-22, -3, 12, 0, 2], [0, 3, -12, 6, 6], [-28, 3, 8, 1, 10], [-2, 3, -10, 3, -1], [42, 0, -4, -3, 1], [-10, -4, -3, 8, -2], [10, -24, 14, 8, -4], [-15, -9, 3, -1, -2], [-1, -19, 3, 5, -5], [15, -22, -3, 3, 0], [17, -11, -4, 6, -1], [-16, 1, 15, 1, -1], [-18, 10, 4, -2, 3], [-4, 6, 8, -6, 2], [-10, -8, 8, 1, -1], [8, -3, 4, 10, -2], [-1, -9, 10, -2, 7], [-9, 12, 5, 0, -2], [34, 7, -9, 1, -2], [-4, -6, 2, -8, 4], [28, -3, -6, -10, 2], [-34, -3, 8, 0, 2], [-18, 0, -7, 6, -14], [-4, -3, 2, -3, 3], [-9, -15, -2, 0, 3], [12, -2, 6, 7, 3], [33, 0, -9, -1, -2], [-1, -5, -3, 13, 5], [-27, 14, 7, 1, 8], [20, -16, 10, 0, -10], [2, 1, 7, -5, -4], [3, -2, 8, -5, -1], [2, 5, -3, 11, -8], [-12, 11, -4, 5, 5], [32, -8, 6, 4, -10], [34, -1, 0, -1, -4], [-24, -7, -1, 5, 2], [-14, -2, 0, 4, 0], [0, -7, 2, 5, 5], [-3, 11, 7, 1, 2], [-8, -19, -2, 9, -2], [-31, 7, -4, -6, -1], [22, 0, 4, 8, 8], [-11, 9, -6, -15, -1], [4, -4, 10, 8, 6], [-18, 9, 8, -9, -5], [2, 6, -2, -8, -2], [-12, -4, 8, 8, -2], [-30, -1, 5, -4, -4], [-6, 4, -2, 0, 12], [-38, 11, -14, -1, 9], [10, 8, -10, -8, 0], [-30, 4, -4, -2, 9], [20, 9, 0, 5, 0], [-12, -3, 0, 12, -6], [2, 12, -7, -12, 6], [11, -9, -3, 3, 8], [1, -10, 22, 3, 1], [-29, 12, 1, 0, 2], [26, -20, -2, 7, 8], [9, -7, 5, -7, 10], [-2, 18, -14, 3, 2], [-8, 0, -8, 6, 8], [-38, 5, 2, 4, 6], [19, 3, -5, 3, 5], [14, -1, 3, 5, -8], [-30, -3, 10, -2, -4], [-31, 15, -3, -3, 3], [7, -8, -4, -1, -13], [-37, -6, 6, 3, 5], [8, 9, -11, 15, 3], [5, 30, -5, 1, 10], [-7, -5, 15, 1, 5], [-12, -15, -1, 0, 4], [26, 7, -1, 7, -7], [3, 10, 9, -2, 0], [6, -12, -10, 6, -5], [-20, 16, -14, -8, 10], [-10, -12, 0, -2, 4], [-36, 9, -13, -3, -11], [11, 4, -2, -8, 2], [-23, -1, -8, 6, 1], [-18, 27, -12, -10, 10], [-14, 4, -14, -2, -8], [17, -8, -9, -5, -8], [13, -20, 4, 4, 10], [-4, 15, -7, -15, 3], [-14, -6, 10, 18, -2], [-18, 18, -2, -9, 7], [13, 11, -23, -7, -4], [19, -12, 4, 6, 2], [25, -2, -10, -13, -1], [-12, 9, 2, 11, -2], [20, 5, 6, 0, 8], [-4, 6, 4, -4, -6], [-14, -2, 4, 2, 2], [-10, 5, -3, -13, 10], [18, -15, 14, 3, 12], [26, 1, 7, -5, 5], [-19, -18, 3, -1, 0], [-34, 0, -4, -4, -6], [8, 18, 12, -6, -3], [12, 15, -13, -12, -2], [-21, -5, 17, 7, 2], [6, 8, -4, 2, -10], [-17, -21, 21, -3, 4], [-10, 1, 0, -11, -1], [20, -18, -2, 0, -15], [-7, 10, 3, -5, 4], [-12, 1, -1, 1, 9], [18, 13, 10, -15, 0], [-12, 12, -16, -6, -2], [43, -17, 7, -1, -10], [6, -18, -4, 8, -2], [37, -20, 6, 1, -3], [0, 23, -7, -7, 2], [-7, -3, 7, -11, 4], [-15, 0, 2, 0, 10], [8, 9, -6, 7, -14], [7, 9, -6, 0, 1], [-32, -4, -8, -1, 3], [-10, -1, 2, 7, -14], [16, -8, -1, 4, -14], [-8, 0, -14, 6, -13], [-13, 14, -2, 3, -7], [2, -5, -3, -8, -4], [-41, -8, 1, -5, -6], [-8, 14, 1, -4, 0], [-7, 1, 13, -11, -7], [26, 12, 6, -6, 0], [-23, -5, -7, 3, -2], [31, -3, 9, 11, -16], [-15, 14, -1, 14, 14], [-25, 11, 3, -1, 0], [3, -2, 16, 4, -6], [18, 3, -8, 10, 8], [0, 27, -2, -3, 1], [18, -6, 12, 0, 0], [16, -5, 2, 7, -5], [-38, -12, 4, 8, -4], [-42, 25, -19, -5, 9], [6, 18, 2, 9, 7], [-8, -8, 14, -5, 4], [-29, -6, 7, 7, 6], [-31, 12, 12, -3, 5], [-9, 6, 12, 12, -8], [16, -1, 4, -12, -8], [-33, 13, -10, -8, -13], [20, 9, 4, 9, -9], [-4, -24, 12, 0, 0], [48, -3, -4, -1, -2], [2, 6, -6, -2, 4], [17, 5, -3, 17, 7], [-10, -13, 9, -1, -5], [-9, -3, 12, 14, 7], [10, -14, 12, 7, -5], [-5, -9, 2, -15, -1], [0, -24, 6, -12, -12], [-14, 21, 2, 7, 6], [29, -14, 16, -7, -15], [25, -20, -3, 1, -12], [-4, 12, 6, -6, 8], [-54, 1, 1, 1, -9], [-12, 6, -10, -10, -14], [11, 4, -5, 3, 8], [20, -11, 14, 9, 8], [-10, -8, 16, 4, 5], [-8, 7, 9, 13, 4], [24, 17, -6, -18, 6], [-8, -27, 1, 3, 2], [-13, -3, 20, 9, 9], [2, 9, -4, -5, 16], [3, 23, -25, -1, -7], [45, -12, 15, 3, -6], [-1, 17, -4, 8, 3], [5, 7, 8, -5, 17], [6, 10, -2, 13, -18], [53, -8, 6, 7, 13], [31, -15, 12, -8, -9], [36, 2, 2, -7, 11], [-14, -7, -8, -9, 10], [22, -23, 3, -2, -14], [-17, 3, -6, 6, 1], [-53, -9, -3, 13, -6], [8, -7, -3, 11, 7], [-4, 5, 12, 8, 0], [17, 16, -24, -8, -4], [22, 27, -30, -4, -4], [-15, 25, 2, 7, 3], [-20, 2, -8, 5, -9], [12, -5, -13, 7, 6], [50, -12, -2, 4, 10], [3, 13, -12, -11, -3], [-21, 21, -5, -7, 4], [35, -25, 0, 2, -9], [8, -14, 18, 13, -10], [35, -26, 4, -1, -9], [2, 9, 9, -3, -12], [-5, -6, 17, 18, -4], [10, -15, -6, 3, -2], [-13, 4, 9, 22, -4], [-20, -23, 8, -3, -2], [6, 8, -19, 8, 8], [-38, 11, -6, -2, -12], [-26, 6, -8, 3, 17], [-43, -3, 3, -9, 8], [1, 10, -8, 5, 11], [-3, 9, 5, 3, -11], [30, 13, -7, 4, -12], [17, -4, -8, -9, 11], [10, -28, 4, 11, -9], [21, 9, -10, -4, -11], [28, -10, -8, 8, -3], [16, 7, -4, -24, 4], [-27, 9, 7, 7, -4], [-16, 0, -6, -12, 18], [-22, -6, 12, 9, -3], [-25, -36, 0, 9, -7], [14, 0, 22, -8, 4], [60, -1, -1, 5, -4], [-4, 21, -2, -10, 6], [-42, 15, 20, 10, 2], [-30, 9, 8, -1, -2], [8, -1, 21, -1, 4], [16, 9, 16, -4, -10], [-8, 38, -8, 2, 8], [-30, 5, -7, -7, -10], [2, -15, 10, 4, -8], [-20, 15, -16, 6, 10], [35, -33, -2, -6, -13], [5, -7, 21, -1, 1], [33, 5, 8, 6, 9]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7605_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7605_2_a_co();". // 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_7605_2_a_co(:prec:=5) chi := MakeCharacter_7605_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_7605_2_a_co();". // 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_7605_2_a_co( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7605_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-6, 6, 10, -6, -2, 1]>,<7,R![-49, 47, 22, -22, -1, 1]>,<11,R![-24, 48, 64, 0, -8, 1]>],Snew); return Vf; end function;