// Make newform 9576.2.a.bc in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9576_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_9576_2_a_bc();". // 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_9576_2_a_bc();". // 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_9576_a();" function MakeCharacter_9576_a() N := 9576; order := 1; char_gens := [7183, 4789, 5321, 4105, 1009]; v := [1, 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_9576_a_Hecke(Kf) return MakeCharacter_9576_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], [2, 2], [-4, 4], [-1, 0], [-4, 6], [-4, 3], [-2, 4], [8, -5], [-2, 5], [2, 3], [7, 3], [5, -1], [4, -5], [-3, -3], [-2, -8], [1, -9], [12, -2], [-2, -3], [-4, 0], [7, -5], [11, -9], [-2, 8], [4, 6], [4, -10], [-5, 9], [-4, -8], [7, -3], [6, 10], [-4, -9], [4, -4], [-4, 8], [-20, 0], [17, 1], [-13, 9], [2, 0], [16, -10], [8, -18], [-4, -2], [8, -6], [4, 2], [-18, 0], [-20, 13], [12, -16], [-14, 10], [0, 8], [2, -7], [15, -3], [18, -6], [-10, 9], [-26, -2], [14, -7], [22, 2], [10, 12], [12, 3], [-28, 6], [-8, 16], [12, -4], [-2, -2], [12, -5], [15, -25], [-8, 0], [19, -7], [18, -28], [-6, -2], [-4, 4], [10, -16], [-8, 20], [16, -2], [17, -7], [-12, 13], [6, 18], [16, -26], [2, -20], [9, 17], [16, -16], [-1, -5], [-22, -4], [-14, -4], [15, 5], [6, -15], [22, -28], [-18, 15], [-10, -10], [5, 15], [-5, -13], [2, 2], [-12, 10], [-16, 33], [22, -3], [-20, 16], [10, 13], [12, -3], [-8, 2], [22, -28], [8, 16], [4, 6], [-16, -6], [-18, 10], [-3, -11], [-4, -6], [-3, 23], [-6, -8], [8, -18], [-16, 18], [25, 11], [-18, 16], [6, 2], [4, 8], [-40, 5], [28, -6], [8, 20], [0, 14], [44, -4], [-4, 15], [10, 14], [-4, 8], [22, -24], [-22, -2], [-6, 0], [2, -16], [-16, -14], [-36, -2], [2, 10], [0, -16], [1, 9], [-42, 9], [-18, -9], [14, -11], [22, -5], [28, -4], [-2, 8], [8, 14], [28, 6], [-40, 7], [-40, -6], [12, -13], [35, -25], [-14, -12], [-10, -12], [24, -26], [-12, -12], [34, -8], [10, -5], [-50, 0], [28, -22], [18, -15], [-49, 9], [18, -22], [-41, 13], [-2, 12], [-26, 34], [10, 21], [4, 22], [6, -2], [0, -16], [10, 30], [31, -27], [30, 4], [4, -12], [19, -9], [-40, 26], [44, -8], [35, -7], [26, -25], [14, 18], [-38, 4], [-36, -2], [-32, -10], [-28, 22], [-32, 25], [-2, -4], [42, 12], [2, 29], [-16, -2], [6, -4], [-2, 5], [44, -34], [-9, -21], [11, -19], [-9, -9], [24, -30], [-5, 7], [-12, -19], [-32, 17], [-40, -10], [1, 9], [22, 12], [-54, -3], [38, -12], [-22, -10], [37, -45], [26, -35], [19, 21], [-2, -26], [-24, 40], [28, -23], [42, -28], [32, -38], [6, 22], [-37, 39], [0, 12], [-15, -15], [-11, -11], [18, 20], [-30, 12], [-5, -25], [28, -25], [47, -21], [-26, 34], [0, -9], [10, 34], [42, -33], [-4, 14], [14, 23], [50, 11], [-40, 32], [15, 17], [-10, 10], [8, -16], [40, -12], [-19, 37], [-18, -24], [14, -15], [-24, 8], [52, -12], [2, -41], [22, -30], [-15, 41], [-7, -17], [-18, 30], [-24, 10], [8, -39], [20, -44], [34, -4], [-66, 5], [32, -50], [2, 18], [28, 12], [16, -40], [-37, 45], [10, -4], [-31, -23], [-12, 25], [-56, 20], [18, 26], [16, 20], [-52, 32], [30, -20], [-34, 38], [22, 8], [26, -24], [14, -37], [24, -24], [-21, 19], [-40, 44], [42, 12], [30, 7], [28, -38], [53, -31], [4, 10], [-30, 6], [43, -29], [50, -4], [-8, 46], [-8, 8], [18, 22], [-38, -24], [0, 0], [-64, 6], [6, -14], [6, -23], [28, -23], [-40, 0], [56, -17], [4, -4], [12, 10], [10, -4], [0, -20], [53, 5], [22, 26], [-8, -20], [14, -14], [33, -41], [28, 1], [25, -7], [66, -12], [44, -52], [-62, 40], [-28, 18], [-12, -1], [-22, -24], [2, -15], [-16, 14], [6, 13], [61, -15], [28, -24], [26, -38], [22, -4], [2, 44], [-44, 6], [34, -49], [-48, 16], [-18, 42], [12, 12], [20, -4], [20, -12], [28, -33], [22, -52], [-16, -12], [28, -8], [12, -2], [-34, 41], [46, -72], [-34, 29], [-62, -8], [-7, 27], [21, 9], [-4, 25], [38, 0], [-1, -15], [-6, 4], [54, -35], [-54, 10], [-12, 11], [41, 7], [-42, 42], [34, -12], [32, 17], [-25, 27], [40, -24], [-62, 17], [32, 8], [-29, 29], [23, -21], [54, -4], [-61, -1], [-47, 11], [67, 1], [34, -56], [-28, 56], [-68, 19], [22, -46], [-53, 39], [42, -20], [-6, 40], [62, -39], [-28, -7], [26, -16], [4, 24], [-2, -50], [20, -51], [25, -27], [-29, -9], [14, -28], [5, 3], [-19, 19], [-56, 42], [38, -4], [8, 20], [-10, 50], [10, -28], [-38, 24], [-62, -1], [10, -37], [-56, 20], [-26, 33], [-40, 67], [72, -6], [38, 28], [0, -40], [-80, 4], [-45, 47], [-38, 40], [-62, 42], [-66, 52], [-30, 1], [-10, 40], [23, -47], [-72, 14], [-8, 25], [47, -9], [-52, -22], [-6, 48], [54, 27], [-36, 72], [44, -14], [54, -28], [-8, -40], [14, 51], [64, -64], [-38, 67], [-8, 24], [28, -54], [-19, 25], [40, -40], [-36, 12], [-10, 36], [-25, 41], [29, 15], [-85, 19], [-48, -8], [-4, 19], [-60, 8], [-18, 4], [20, -64], [-20, 16], [79, -21], [38, -62], [-26, 36], [-5, -15], [-4, -13], [7, 55], [16, 28], [-4, -6], [34, 12], [-56, 42], [5, 11], [24, 16], [-22, -24], [-24, 12], [13, 1], [-46, -2], [56, 6], [-4, 35], [-10, 50], [28, 32], [-22, 59], [-78, 9], [80, -8], [-48, -2], [1, -51], [60, 8], [44, -29], [33, -57], [56, -50], [4, -33], [-52, 4], [2, -44], [-4, -15], [-52, 24], [-6, 52], [-10, -42], [0, 30], [-16, 46], [-18, -48], [-40, -8], [-62, -12], [-62, 22], [-14, -41], [16, 30], [-8, 31], [48, -15], [69, 15], [-68, 27], [10, -20], [-10, 4], [-66, 40], [-78, 3], [74, 0], [-87, 3], [-78, 32], [19, -71], [74, -63], [-58, 80], [95, 1], [-30, 22], [-20, -8], [10, 32], [50, -54], [-4, -17], [22, -14], [-2, 6], [-46, 12], [-24, 80], [60, -11], [32, -27], [46, -8], [-70, -17], [51, -65], [28, 2], [-45, 55], [-45, 33], [52, -24], [-83, 25], [65, -59], [-57, 57], [46, -38], [52, -26], [-10, 46], [32, -84], [27, -27], [98, -19], [88, 13], [42, 22], [-24, 53], [44, -20], [16, -24], [85, -43], [0, 2], [14, -32], [-26, 12], [54, -58], [38, -89], [-18, -40], [-59, -5], [104, -9], [56, 6], [45, -51], [62, -22], [-88, -17], [-26, -17], [65, -33], [52, -30], [-18, 81], [-33, -3], [40, 18], [32, 4], [-38, 36], [-34, -36], [54, -32], [-62, -30], [66, -42], [-38, -11], [14, -53], [-20, 52], [-8, 32], [2, -76], [-58, 60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9576_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_bc();". // 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_9576_2_a_bc(:prec:=2) chi := MakeCharacter_9576_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(3833) - 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_9576_2_a_bc();". // 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_9576_2_a_bc( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9576_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![5, 5, 1]>,<11,R![-5, -5, 1]>,<13,R![4, -6, 1]>,<17,R![-16, 4, 1]>],Snew); return Vf; end function;