// Make newform 9072.2.a.y in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9072_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_9072_2_a_y();". // 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_9072_2_a_y();". // 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_9072_a();" function MakeCharacter_9072_a() N := 9072; order := 1; char_gens := [1135, 6805, 3809, 2593]; 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_9072_a_Hecke(Kf) return MakeCharacter_9072_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], [-2, 1], [1, 0], [1, 3], [-3, -2], [-7, 0], [1, -1], [1, -3], [-5, 2], [-3, -3], [2, 5], [-6, 2], [-2, 2], [3, 1], [6, 2], [-1, 3], [-3, 4], [5, 1], [10, -2], [10, -1], [-3, 7], [1, 9], [-3, -4], [4, 4], [-2, -4], [-2, 6], [0, 0], [-2, 3], [4, 1], [-12, 0], [14, -2], [10, -1], [5, -1], [-9, 0], [11, -3], [-1, 0], [10, 2], [-7, 3], [-8, 1], [3, -3], [-6, -6], [3, -5], [5, 8], [5, -12], [7, 7], [19, -1], [22, 2], [-12, -4], [1, -2], [8, -3], [9, -11], [-16, 1], [-3, -11], [-3, -2], [-15, 5], [10, -9], [-3, 13], [2, -12], [8, 3], [2, -10], [-12, 5], [8, 8], [5, 3], [10, 9], [21, 6], [-2, -6], [-6, -6], [18, 2], [-2, 2], [18, 0], [-4, -4], [1, -7], [-20, 8], [5, 13], [-8, 16], [10, 2], [-21, 0], [12, -13], [26, -5], [-6, -2], [-12, 7], [-18, 2], [14, 1], [-6, -10], [-17, 1], [-2, -8], [7, -8], [-14, 12], [17, -9], [-1, -9], [1, -13], [13, -9], [30, -2], [-25, 3], [21, 11], [-18, -8], [30, 0], [-2, -18], [12, -5], [21, 1], [-17, -4], [-8, 16], [-18, -3], [21, 1], [16, 5], [13, 1], [-9, 18], [-3, -7], [14, 3], [9, 9], [6, 12], [14, -15], [-4, 16], [-5, 5], [-10, -17], [25, -9], [6, -6], [6, 14], [-26, -6], [1, -8], [-1, -10], [-36, 0], [12, -8], [28, 0], [-17, -6], [-16, 7], [-6, -2], [-34, -10], [30, 10], [15, -1], [-30, -6], [36, -8], [0, 12], [-37, 0], [-20, 9], [-8, 7], [18, 6], [0, 17], [-20, -7], [-13, -3], [-19, -4], [20, -12], [-6, -10], [-14, 0], [-2, 2], [-22, -16], [3, 8], [-14, -6], [5, -7], [-14, -13], [-6, -14], [1, 5], [-27, -17], [36, 0], [18, 14], [-1, -15], [3, 28], [14, -23], [22, 7], [-22, -14], [26, 9], [-5, 5], [21, -7], [-30, -16], [24, 20], [19, 5], [17, 14], [11, -8], [6, -8], [-10, -10], [-5, -18], [21, 21], [27, 12], [-18, -10], [-2, 6], [-36, 8], [5, 2], [-2, 2], [-19, 12], [-18, -10], [-20, 20], [-20, -11], [10, 22], [-28, 0], [60, -1], [18, -7], [25, 3], [30, 6], [33, 15], [4, 15], [-6, -6], [-1, 15], [56, 1], [48, 0], [26, -2], [29, 2], [22, -4], [-27, 18], [-27, -1], [15, 14], [45, -3], [-10, 10], [36, 17], [-15, -19], [30, 20], [-41, 7], [21, 19], [-18, 16], [28, 16], [-34, 0], [4, 11], [-6, 14], [32, 4], [-12, 8], [26, -12], [14, -10], [-67, 2], [34, -6], [-43, -6], [-8, -3], [-49, 3], [-42, -9], [-22, -2], [-6, 34], [-22, -9], [38, 10], [56, -4], [-12, 16], [57, -9], [-36, 7], [14, 6], [-19, -25], [-24, 19], [-11, 35], [12, -28], [16, -13], [18, 18], [-27, -21], [11, 21], [1, -25], [-35, 11], [-20, 12], [-36, 8], [38, -22], [-21, 17], [-30, 14], [22, 6], [-14, 30], [-21, 15], [10, 27], [15, 10], [0, 20], [-2, -27], [-10, -29], [19, 1], [8, -9], [-9, 21], [-26, 13], [-22, 16], [35, 1], [-31, -9], [7, -20], [-50, 16], [6, -2], [-7, 3], [-21, -14], [-5, -8], [26, 2], [-3, -6], [-21, 6], [43, 11], [30, 8], [22, 26], [22, 1], [-24, -32], [43, -15], [-24, -17], [-5, 16], [18, -22], [21, -21], [15, 17], [18, -6], [5, 14], [-51, -13], [10, 30], [-69, -4], [38, -14], [49, -11], [21, 28], [6, -18], [-9, 23], [24, -33], [-43, -15], [2, -31], [6, -19], [31, -9], [18, 1], [12, 0], [-6, 26], [-20, 31], [-6, 0], [26, 6], [-30, 22], [25, -15], [2, -6], [56, -4], [-34, 8], [-59, -3], [26, 3], [6, 18], [34, 26], [46, 1], [43, 23], [-16, -12], [2, 4], [39, 11], [8, 20], [2, 17], [-18, -28], [-29, -23], [22, -10], [12, -19], [-25, 13], [22, -12], [-22, 9], [16, -16], [-64, -8], [-5, -25], [-10, -2], [-34, 8], [17, 20], [15, -17], [-68, -4], [-64, -8], [11, -5], [36, -29], [1, 30], [-10, -44], [-13, -21], [44, -3], [18, 24], [0, -27], [42, 14], [27, -34], [-45, 9], [5, -28], [34, -10], [18, -2], [6, 9], [-8, -16], [47, -2], [-27, 10], [-24, -4], [-8, 37], [59, 14], [-4, -4], [-48, -4], [-22, 4], [-20, 4], [40, 27], [-67, 14], [-22, 26], [-68, 12], [77, 3], [41, 18], [-6, -40], [-3, 11], [10, -10], [-44, -8], [8, 40], [-36, 24], [49, 20], [-40, 28], [28, 33], [57, -3], [-14, 30], [28, -19], [26, -8], [-40, -1], [0, -25], [52, -17], [38, 6], [-18, 9], [53, -1], [-22, 2], [-23, 35], [-5, -2], [-37, 3], [-11, -29], [35, 6], [-48, 29], [7, -35], [34, -26], [41, 9], [45, 24], [23, 31], [-7, 26], [-30, -14], [-7, 14], [-43, -26], [60, 20], [86, -10], [-17, -50], [42, -23], [19, -37], [34, 39], [-48, -13], [7, -35], [4, -4], [-50, 25], [-19, -18], [13, 27], [45, 29], [-1, -26], [-28, -43], [-23, -19], [-37, 11], [-15, -20], [71, -19], [2, 8], [-22, -6], [-50, 18], [39, -17], [38, -29], [-38, 19], [-7, -37], [18, -9], [78, -6], [43, 21], [34, 5], [-42, -18], [25, 41], [29, 39], [-2, -18], [-41, -24], [10, 14], [14, -29], [-79, 5], [-93, 7], [11, -45], [-24, -40], [-32, -1], [-24, -16], [-16, 43], [-30, -1], [-10, 14], [-43, 3], [-80, 1], [10, 26], [-20, -20], [-15, -17], [-51, -25], [21, 22], [58, 8], [27, 12], [40, -28], [15, -45], [27, -30], [-6, -8], [-64, 16], [49, -29], [-29, 9], [16, -35], [1, -33], [35, -8], [11, -55], [44, -16], [-18, 40], [-28, 8], [1, -21], [-72, -24], [-10, -42], [-19, 42], [-36, 8], [33, -40], [-42, 10], [49, 21], [15, -7], [-37, -34], [-42, 16], [39, -2], [102, 1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9072_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9072_2_a_y();". // 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_9072_2_a_y(:prec:=2) chi := MakeCharacter_9072_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_9072_2_a_y();". // 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_9072_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9072_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 4, 1]>,<11,R![-26, -2, 1]>,<13,R![-3, 6, 1]>],Snew); return Vf; end function;