// Make newform 9450.2.a.ev in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9450_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_9450_2_a_ev();". // 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_9450_2_a_ev();". // 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_9450_a();" function MakeCharacter_9450_a() N := 9450; order := 1; char_gens := [9101, 6427, 6751]; 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_9450_a_Hecke(Kf) return MakeCharacter_9450_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, 0], [0, 0], [1, 0], [0, 1], [1, -1], [1, 0], [-2, -1], [-1, 1], [5, -1], [1, -1], [4, -3], [0, 2], [-1, 2], [2, 4], [-3, -1], [5, -2], [6, 2], [5, 0], [-3, -1], [2, 4], [-4, -1], [-6, -2], [11, 0], [12, -1], [0, 5], [13, 1], [-4, 3], [2, -7], [-4, -1], [4, 2], [-7, -4], [10, -2], [10, 5], [3, -1], [0, 5], [3, 1], [9, -2], [6, -1], [6, -7], [4, -2], [5, 1], [-8, -6], [17, -4], [-8, -2], [-3, 5], [-15, 0], [4, 0], [3, -6], [-8, 2], [4, -7], [-4, 10], [18, -3], [8, 8], [-12, 0], [-3, 11], [8, -2], [3, -3], [16, -4], [-6, -3], [20, -1], [6, 10], [-22, -1], [-6, -3], [2, 11], [-14, 4], [-13, 0], [13, 4], [6, -8], [-3, -7], [-9, 10], [19, 1], [7, -9], [20, 3], [10, 0], [12, -2], [-24, -2], [-8, -6], [6, -6], [10, -8], [-19, -8], [10, 4], [2, -4], [4, 8], [-1, 3], [30, -1], [32, 0], [5, -2], [10, 0], [16, -5], [22, -4], [-36, 0], [10, 12], [-24, 8], [-2, 2], [-14, 3], [-4, 7], [3, 0], [16, -8], [-22, 5], [-14, 6], [11, 9], [5, -6], [0, 13], [5, -10], [-22, 1], [-11, -10], [12, 2], [-3, -1], [-28, 6], [-1, -9], [-36, -5], [-34, -2], [34, 4], [0, 2], [-8, 1], [28, -7], [-6, 8], [3, 5], [-36, 5], [10, 10], [23, 10], [4, 4], [-8, 1], [16, -14], [-10, -12], [6, 7], [2, -7], [-11, 3], [35, 1], [34, 4], [-33, -3], [-44, -1], [-24, -8], [-5, -10], [16, -15], [-24, -6], [18, 14], [22, -10], [16, -4], [-6, -16], [-51, -1], [24, 13], [2, 14], [-2, -2], [-40, -1], [-1, -5], [5, -2], [-6, 11], [7, -3], [28, -6], [-21, 14], [7, -2], [0, 0], [-28, 4], [-16, 4], [2, -8], [-12, -8], [-8, -11], [-10, -8], [6, 9], [-12, -14], [20, 2], [-15, 10], [6, 12], [-20, -6], [-2, 12], [13, 15], [1, 14], [4, -2], [-54, -4], [-3, 5], [-11, 9], [-5, -14], [4, -12], [-15, -2], [-19, -12], [7, 17], [24, 0], [5, 5], [-14, 17], [-61, -2], [-2, 21], [45, 2], [5, 3], [-6, -18], [6, 19], [-8, -17], [-29, -6], [-12, -1], [-28, -2], [-30, -6], [-65, 0], [-36, 2], [-10, -15], [-5, 16], [41, 2], [-18, -6], [-1, 8], [32, 2], [19, 9], [-11, -11], [0, -22], [54, 6], [-9, -4], [-34, -11], [3, -11], [-32, -3], [-36, -6], [2, 18], [13, 2], [28, -3], [-30, -6], [9, -10], [-14, 6], [32, 8], [0, -19], [-24, 12], [-3, -9], [-35, 3], [-32, 11], [-16, 8], [0, 3], [-54, 5], [-4, 0], [-48, -1], [27, 12], [-4, 12], [-5, -15], [-6, -13], [-16, 16], [-12, 10], [20, 17], [24, 1], [-20, 16], [-24, -10], [-14, 5], [15, 19], [6, -5], [36, -15], [-6, 3], [10, 20], [-19, -3], [-6, -9], [57, 2], [-30, 16], [7, -27], [-6, -6], [-33, 8], [9, -5], [-10, -11], [51, -4], [55, -3], [-22, 3], [-54, -6], [-26, 14], [2, 25], [-16, 9], [-18, 1], [-28, 15], [-44, 15], [0, -23], [40, -10], [-10, 8], [-23, -4], [16, 10], [42, 12], [-8, -18], [9, -26], [54, -8], [61, 1], [30, 8], [-56, 6], [-28, -10], [-6, 19], [-27, 3], [42, -9], [10, 18], [-40, 0], [-15, 4], [-12, -25], [34, -16], [-56, 6], [-25, 11], [-16, 21], [48, -10], [-16, -26], [-15, -11], [23, -17], [45, -6], [-6, 16], [-39, -2], [12, 8], [-43, 0], [-40, -3], [10, 6], [52, -6], [-20, -7], [-21, 14], [-70, 3], [26, 0], [5, -19], [42, -14], [-54, 2], [-65, 4], [-17, -14], [-8, 10], [35, 5], [12, -22], [-14, -9], [46, 15], [9, -5], [18, 20], [45, 4], [-15, 3], [-28, 4], [3, 18], [16, -4], [-68, -4], [36, 0], [34, 13], [-50, -12], [2, -21], [-4, 33], [-8, -10], [8, -25], [-17, 2], [-50, 13], [15, 1], [2, 16], [-32, -18], [48, 2], [-50, -12], [-18, 14], [31, -8], [-19, -10], [-12, 17], [-13, -6], [-50, 15], [32, 0], [-20, -9], [10, -7], [48, 18], [-32, 21], [33, 7], [-26, -13], [35, 7], [3, -20], [0, -2], [50, -3], [12, -25], [43, 0], [47, -5], [-51, 3], [-46, 5], [10, -24], [-14, -25], [-6, 8], [42, 20], [0, 36], [-36, -4], [-3, -20], [-29, 11], [-10, 8], [-14, -23], [-12, 22], [-14, 19], [-76, -2], [-3, -8], [-6, -23], [20, -6], [-57, -9], [21, -29], [43, 3], [-50, -4], [47, -8], [-27, 23], [48, 8], [16, -24], [70, -4], [30, 6], [-68, 6], [54, 11], [6, -23], [-28, -27], [-24, 9], [70, 12], [-7, 9], [-1, -18], [-18, 19], [-7, -18], [-6, -23], [70, 10], [-50, 8], [39, 11], [-3, 0], [-77, -3], [73, -2], [-19, 2], [-73, 1], [15, 27], [20, -24], [42, 4], [33, -16], [30, 25], [-28, 20], [30, -1], [36, 21], [-4, -15], [-63, 8], [18, -16], [45, -21], [-40, -11], [-39, 26], [-31, 26], [72, -5], [-21, -9], [-4, -38], [-3, 20], [-78, 0], [69, 5], [23, 7], [44, -18], [17, 20], [-48, -11], [-18, 5], [-8, -4], [0, 28], [30, -12], [58, -1], [50, 3], [-44, 17], [-5, 34], [-54, 8], [-5, 31], [-66, 8], [-17, -10], [-62, 13], [37, 20], [-47, 3], [52, 16], [-18, -3], [14, 10], [-13, -21], [-50, 12], [-12, -8], [24, 9], [-22, -5], [26, 6], [-41, -21], [68, -4], [-34, -1], [38, -26], [27, -22], [-1, 14], [49, 5], [28, 10], [-21, -12], [-3, -9], [42, -2], [45, 10], [-56, 1], [-46, 4], [-8, -27], [-58, -15], [21, 28], [-70, 16], [38, -24], [24, -19], [-14, -41], [44, -9], [-24, -29], [-30, 30], [-20, 2], [36, 11], [48, -22], [-2, 8], [68, 6], [34, -8], [-45, 11], [82, 7], [55, -2], [-14, 25], [58, -8], [-60, -10], [-6, -15], [22, -13], [30, -16], [-48, 24], [60, 5], [27, 25], [-12, -15], [19, -14], [-52, 20], [-4, -15], [-59, -1], [56, 22], [55, -5], [-7, 17], [-38, -8], [3, -13], [43, 23], [54, 19], [-50, 6], [30, -10], [36, -22], [68, -14], [76, 6], [-22, 4], [70, 15], [-41, -8], [72, -2], [28, 30], [-53, -27], [-36, 8], [19, -2], [34, 1], [31, -5], [7, -15], [43, -5], [4, 13], [43, 6], [58, -4], [-63, -7], [32, 28], [-75, 10], [-4, -30], [-42, 16], [-70, 12], [-32, 4], [-12, -36], [-6, 22], [4, -21], [-30, -15], [-63, -21], [-94, -8], [64, 6], [46, 8], [67, -4], [19, 6], [94, 12], [33, 25], [-40, 26], [9, 18], [18, 17], [18, 4], [62, -17], [-15, 8], [38, 20], [48, -20], [-2, -29], [12, -19], [-72, -5], [18, 30], [-50, 2], [41, 1], [18, 0], [27, -16], [-12, 29], [31, 8], [72, 0], [64, 21], [-71, -8], [47, -17], [14, -10], [-52, 2], [57, -19], [48, -14], [-52, 3], [-64, -22], [7, 2], [-68, -21], [42, -28], [8, -21], [-35, 12], [-72, -10], [11, -24], [60, 15], [3, 11], [-19, 5], [71, -14], [42, 6], [37, -1], [-2, -38], [47, 3], [37, -27], [86, 4], [-73, 6], [-12, 18], [34, -3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9450_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9450_2_a_ev();". // 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_9450_2_a_ev(:prec:=2) chi := MakeCharacter_9450_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(4297) - 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_9450_2_a_ev();". // 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_9450_2_a_ev( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-6, 0, 1]>,<13,R![-5, -2, 1]>,<17,R![-1, 1]>,<19,R![-2, 4, 1]>],Snew); return Vf; end function;