// Make newform 10000.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_10000_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_10000_2_a_i();". // 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_10000_2_a_i();". // 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 := [-1, -1, 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_10000_a();" function MakeCharacter_10000_a() N := 10000; order := 1; char_gens := [8751, 2501, 9377]; 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_10000_a_Hecke(Kf) return MakeCharacter_10000_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], [1, -2], [0, 0], [1, 1], [2, -2], [-5, 1], [-2, -2], [-6, -1], [-1, 4], [6, -1], [-5, 6], [5, 2], [-4, 6], [6, -1], [5, -3], [5, -8], [-4, 1], [3, -10], [4, 6], [7, 1], [-11, 0], [-6, 7], [-11, 4], [4, 0], [-9, 15], [-5, -8], [-8, 3], [7, 6], [-10, 8], [-4, 9], [14, -4], [-7, 5], [-5, 0], [-13, 2], [1, -12], [-2, 1], [-1, -2], [-1, 10], [2, -13], [-13, 4], [-13, 6], [-18, 6], [10, -6], [-2, 6], [18, -2], [-11, 1], [-7, -8], [-4, 6], [10, -22], [4, 10], [-6, 0], [19, 2], [5, 4], [21, -8], [2, 1], [-9, -3], [-24, 6], [0, -8], [-5, -10], [14, -3], [17, -15], [6, -4], [14, -2], [13, -5], [4, -6], [11, 2], [10, -26], [1, 5], [2, 3], [-4, 18], [15, 9], [-1, -15], [-5, 15], [-8, -5], [5, 0], [17, -2], [7, -12], [-12, -5], [-1, -22], [-27, 4], [3, -18], [-8, 8], [-13, 6], [22, -15], [1, -11], [-31, 14], [-32, 5], [-2, 4], [-31, 6], [-3, -8], [17, -15], [-24, 1], [-17, -6], [-7, 2], [-15, -9], [10, -16], [23, 5], [-11, 12], [-20, 1], [-14, 14], [-23, 20], [24, -10], [23, -13], [11, 4], [-13, 28], [10, -16], [-25, -4], [-5, -9], [-1, 10], [-37, 9], [-23, 15], [-11, 13], [5, -22], [31, 6], [31, -24], [23, -25], [32, -8], [0, -14], [-3, -17], [-26, 16], [-20, -9], [4, 18], [3, 3], [-11, -6], [42, 3], [-14, 10], [-1, -22], [-7, 28], [0, 7], [-6, -3], [0, -7], [12, -36], [35, -2], [-15, 0], [-30, 3], [46, -4], [-38, 7], [40, -6], [15, -6], [25, -11], [13, 20], [-9, -3], [22, -22], [28, 3], [-19, 9], [-8, 11], [14, -12], [8, -3], [5, -22], [14, -17], [-12, -14], [-24, -12], [24, 4], [-28, 16], [17, 10], [5, -20], [-10, 9], [53, -9], [-6, -9], [-35, -8], [0, -34], [5, 23], [-6, 4], [-42, 9], [9, 6], [-14, 11], [35, 10], [45, -4], [5, -12], [-9, -31], [-27, -14], [-4, -1], [2, -27], [7, -2], [1, 20], [7, 14], [-40, 2], [-25, 6], [-17, 12], [1, 35], [13, 10], [-16, 28], [54, -3], [-30, -4], [-7, -18], [-7, -6], [-13, 0], [3, -29], [-18, -22], [-6, -17], [-13, -5], [-20, 8], [47, -15], [14, 24], [-27, -25], [21, -35], [36, 8], [0, 19], [22, -20], [29, -12], [-10, 4], [11, -36], [36, -48], [-28, 1], [-32, -10], [42, 5], [13, -9], [13, -43], [13, -22], [56, -12], [-50, 3], [-13, 36], [1, -26], [5, -5], [21, -31], [-43, 24], [-13, 23], [11, 11], [-10, 2], [33, -24], [-37, 10], [12, -32], [-31, 50], [14, 9], [-36, 40], [32, -25], [15, 6], [-21, 0], [53, -13], [21, -57], [-36, 23], [-12, 52], [-52, 38], [-8, -18], [-44, 44], [-38, 34], [-34, -7], [-14, -15], [-13, -9], [41, -18], [-2, -3], [-38, 42], [-2, 41], [-23, 31], [-21, -3], [53, -3], [37, -38], [-13, 44], [11, -52], [-22, 58], [-23, 0], [-9, 44], [-22, -5], [-32, 39], [16, 6], [11, -4], [63, -11], [26, 22], [-42, 26], [-3, -26], [-10, -30], [-58, 15], [27, 3], [3, 6], [20, -50], [-42, 12], [-19, -25], [-30, 48], [49, -16], [-47, 2], [-29, 6], [5, -13], [-66, 21], [39, -32], [-19, 43], [21, -62], [43, -14], [14, -26], [27, -24], [6, -5], [-35, 31], [45, -39], [39, -21], [65, -10], [21, 21], [-54, 48], [-21, -4], [-1, -6], [-19, 56], [32, 5], [22, -24], [50, -42], [31, -30], [-67, 32], [-29, -7], [-43, 14], [86, 1], [-4, 31], [-46, 7], [-27, -12], [17, -37], [-15, 34], [-33, -28], [-33, 54], [-20, -11], [11, 2], [-16, 13], [7, -29], [-19, -30], [-66, 33], [-40, 68], [36, -46], [-56, 2], [10, -46], [13, 3], [43, 14], [13, -44], [-25, 27], [-33, -16], [5, -12], [15, -3], [-5, 27], [15, 30], [-1, -12], [59, -18], [-2, -14], [4, 25], [11, 28], [-42, 31], [30, -4], [22, -1], [-35, 71], [-3, 16], [48, -52], [-9, 55], [50, 9], [27, -40], [36, -27], [-15, 34], [4, 16], [-15, 50], [77, -6], [-42, 32], [-36, 15], [5, -9], [32, 14], [-44, -4], [-59, -10], [44, 20], [-18, 43], [26, 33], [-31, 6], [-17, -6], [16, 36], [28, -49], [-31, 7], [53, -17], [9, -37], [-7, 32], [23, -21], [-67, 9], [-24, -12], [42, 1], [-14, -30], [66, 5], [42, 25], [9, -35], [-2, -41], [61, 9], [-32, 10], [-47, 77], [31, -50], [1, 39], [-21, -7], [-44, 4], [6, 28], [18, 34], [-67, 6], [-39, -30], [-68, 18], [32, -80], [17, 9], [-60, 18], [-46, 7], [34, -26], [64, -27], [66, -33], [27, 32], [-51, 51], [-10, -14], [-25, -1], [34, -33], [-59, 41], [47, -17], [-13, 56], [67, -52], [16, 10], [6, -38], [21, 32], [50, -17], [-47, 74], [-38, -38], [-3, 57], [-11, -43], [28, -51], [25, -62], [-20, -49], [-32, 8], [76, 6], [1, -4], [-63, 28], [33, -13], [85, 9], [-4, -32], [12, 55], [-10, 46], [-16, -1], [-20, -47], [-35, 30], [19, -26], [15, -25], [-60, -11], [-62, -7], [-65, 9], [-10, 12], [-61, 11], [21, 53], [-45, 22], [-58, 42], [92, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_10000_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_10000_2_a_i();". // 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_10000_2_a_i(:prec:=2) chi := MakeCharacter_10000_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_10000_2_a_i();". // 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_10000_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_10000_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-5, 0, 1]>,<7,R![1, -3, 1]>,<11,R![-4, -2, 1]>],Snew); return Vf; end function;