// Make newform 9450.2.a.eu in Magma, downloaded from the LMFDB on 28 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_eu();". // 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_eu();". // 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_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], [-2, 2], [4, 0], [4, -1], [-1, -2], [2, 2], [-2, -1], [-2, -3], [-3, -4], [8, 2], [8, -2], [0, 8], [2, 4], [0, 8], [2, -6], [-6, -1], [-10, 4], [2, -4], [6, -2], [-1, 0], [0, -4], [6, -4], [7, -2], [-4, 0], [11, 2], [8, 2], [-2, 2], [14, -4], [10, -2], [4, 2], [6, -4], [0, -10], [-6, -2], [-6, -2], [6, -4], [6, -7], [-8, -2], [-10, -8], [-14, -3], [-4, 8], [16, -2], [6, 11], [0, 6], [7, -6], [-6, 12], [4, -4], [16, 2], [8, -2], [0, 6], [8, 8], [12, -9], [21, 2], [-1, 10], [0, -6], [10, 11], [0, -6], [-16, -4], [12, 4], [17, -4], [6, 6], [14, -10], [22, 4], [8, 12], [-20, 1], [-15, 4], [18, 8], [0, -5], [-32, -2], [13, -6], [2, -3], [-14, 0], [18, 4], [0, 4], [16, 6], [-18, -6], [-20, 10], [20, -2], [1, -2], [2, -13], [-14, 14], [-28, -6], [3, 8], [-10, 0], [8, 13], [-5, -6], [-14, -8], [4, 14], [0, -12], [16, -6], [-12, 17], [-14, -4], [-2, -6], [14, 16], [21, -12], [25, 4], [-1, 2], [-20, 0], [-16, -6], [14, -6], [-6, 16], [-28, 2], [-16, 10], [-2, -4], [0, -25], [30, -7], [-10, -6], [8, -12], [-30, -11], [2, 4], [-20, -5], [6, 8], [10, 4], [-23, -4], [24, 2], [6, 8], [0, -13], [-2, -14], [-4, -8], [-14, 19], [7, -8], [-8, -14], [2, -6], [-9, -2], [-34, 8], [28, -12], [14, -8], [-44, -2], [33, 6], [-2, 8], [0, -8], [22, -16], [-2, 30], [6, 21], [0, 8], [-2, -13], [-26, 2], [-36, -1], [-36, -10], [6, 28], [-7, 8], [40, -2], [14, 2], [8, 28], [-4, 1], [-12, -7], [28, -12], [-20, -12], [-39, 8], [-44, 4], [6, 18], [38, 4], [8, -14], [-28, -8], [-30, -8], [-8, -20], [17, 22], [-9, -24], [-6, -2], [-22, 8], [18, 10], [0, -24], [28, 6], [-26, 6], [4, -30], [-11, -28], [-38, -5], [0, -22], [12, -22], [-32, 6], [4, -23], [-38, 9], [27, 4], [-40, 0], [-26, 8], [15, 6], [-16, 20], [22, -4], [-16, 10], [-50, -3], [0, -4], [-37, 6], [51, 6], [0, 16], [-20, 16], [13, -24], [-24, -4], [-10, 22], [-24, -12], [-20, 18], [54, -4], [17, 0], [16, 25], [-19, 8], [24, 0], [20, 11], [2, 20], [-2, 6], [-14, -17], [-30, -4], [12, 0], [-66, 2], [-4, -32], [-24, -2], [13, 12], [-12, 30], [-28, 6], [52, 11], [-23, -18], [-36, 6], [-36, 2], [4, 12], [-4, -4], [-6, -22], [27, 0], [33, -18], [4, -36], [22, 8], [-28, -22], [18, 18], [-48, -10], [26, 24], [-3, 26], [-48, 3], [-22, 9], [43, 6], [-60, 5], [-10, 19], [0, -17], [26, -22], [18, -20], [25, 16], [6, -34], [22, -4], [6, 10], [36, -2], [-18, -30], [3, 24], [4, -25], [5, -30], [33, 6], [-24, 11], [54, 4], [-23, 6], [42, -18], [18, 26], [-39, 10], [-4, 28], [42, -16], [-32, 0], [26, 0], [-18, -3], [-20, -31], [23, 10], [-19, 24], [-36, -20], [14, 0], [16, -30], [-33, -20], [-26, -10], [-10, -28], [-32, 11], [4, 1], [-6, 18], [-38, 0], [-6, -20], [-30, -26], [-56, 10], [0, 8], [-14, 6], [-10, 26], [-6, 28], [-21, -30], [-39, -12], [22, 0], [38, -24], [42, 10], [-36, -10], [16, 26], [22, 0], [32, 14], [17, -36], [-66, 5], [-4, -11], [-6, 16], [26, -17], [-39, 0], [6, 4], [-60, -2], [-54, 2], [-40, -6], [14, 21], [19, -6], [70, 9], [-8, 32], [6, 2], [-19, 0], [20, 6], [-22, 14], [-24, -14], [-45, -4], [-32, -14], [4, -29], [1, 16], [20, -10], [6, 23], [34, 15], [-74, 0], [36, -2], [24, -16], [66, -2], [33, -18], [32, 4], [-30, 18], [22, 17], [52, -4], [-12, -30], [-32, -14], [64, 9], [-7, 42], [-64, 6], [16, 8], [56, -16], [40, 14], [13, -32], [-72, 4], [-28, 10], [31, -30], [-36, -10], [-26, -15], [42, 26], [-38, 10], [-28, 8], [3, 26], [32, -15], [-8, 42], [-25, 18], [22, -15], [-2, 2], [-30, -12], [22, 6], [-30, -20], [-44, 2], [-28, 10], [-6, 22], [-6, -14], [-16, 0], [-12, 11], [16, 0], [32, 4], [57, 18], [41, 14], [-44, 27], [10, 32], [84, -4], [-6, 8], [-36, 12], [-12, -28], [-21, -32], [16, 8], [32, -25], [-56, -20], [-12, 1], [34, -29], [-22, 28], [21, -8], [-12, -11], [32, 33], [-12, 12], [-48, -2], [58, -15], [46, -1], [2, -8], [-12, -34], [0, 26], [-47, -24], [-26, 14], [57, -6], [28, -6], [-18, -46], [-66, 4], [20, 3], [-84, 6], [52, 18], [-28, 0], [-40, 12], [36, -20], [62, -12], [-18, -11], [16, 7], [16, 8], [-72, -10], [12, 15], [46, -15], [37, -32], [2, -16], [2, -38], [0, 18], [14, -30], [30, -20], [72, -13], [57, 10], [-58, -19], [-30, 29], [-42, 30], [35, -12], [0, 8], [12, 32], [-18, 36], [68, 16], [-48, 20], [-64, -4], [27, 22], [18, 15], [-28, -20], [-48, -16], [-30, 12], [0, -16], [68, -8], [-1, -30], [-22, 20], [0, -2], [-36, 20], [-32, 44], [-24, 4], [24, 21], [-38, 26], [-34, -12], [-44, -1], [-44, -20], [-78, 14], [-56, -2], [78, 8], [10, 8], [31, -14], [-44, -22], [-2, -27], [-2, -32], [-60, -18], [29, 34], [26, 3], [52, 30], [-36, 4], [21, -36], [-22, 16], [-4, -12], [-2, 12], [-16, 2], [-22, -46], [-40, -8], [-30, -16], [-16, -31], [-8, -16], [-23, 22], [12, 30], [-24, -12], [0, 10], [-66, -14], [-14, 10], [-64, 7], [46, 24], [41, 0], [-48, -8], [-46, 28], [14, 30], [66, -22], [28, 1], [-16, -24], [33, -8], [-60, -18], [43, -16], [60, 26], [24, 20], [-74, -22], [2, 21], [-14, -26], [12, -22], [22, 16], [-11, -8], [40, 28], [70, -8], [51, -22], [24, 0], [37, 2], [42, 26], [-48, -18], [18, 8], [-6, 40], [36, 30], [49, 28], [62, 23], [32, -30], [34, -28], [32, -20], [-38, -14], [-52, -10], [-14, -14], [0, 44], [-53, 2], [6, -62], [4, 0], [-30, 47], [0, 41], [29, -26], [-8, -30], [56, -8], [31, -24], [22, -38], [24, -14], [28, -30], [10, 18], [-18, -16], [40, 31], [82, 16], [28, -8], [-74, -9], [37, 10], [70, 1], [-26, -18], [52, 2], [0, -28], [14, 10], [12, -14], [-16, 12], [-6, -41], [11, -48], [-26, 16], [-36, 30], [54, 4], [40, 12], [-12, 45], [-24, -6], [92, -8], [-14, 18], [1, -16], [-35, 8], [-80, -6], [22, -30], [54, 16], [-70, 23], [84, -12], [-48, 20], [-84, -8], [-4, 22], [48, 32], [44, -4], [-6, 16], [34, 46], [-63, -16], [18, 52], [24, 18], [-20, -19], [-28, 22], [42, 0], [6, 8], [12, 17], [40, -13], [36, 48], [52, 6], [4, 28], [-52, -2], [74, -22], [74, 8], [39, 2], [30, -26], [44, 24], [8, 23], [16, -55], [-86, -15], [-36, -22], [8, -48], [10, 0], [0, 17], [32, 15], [-48, 15], [24, 20], [80, -10], [-43, -8], [54, 0], [-14, 8], [100, 4], [14, -18], [-2, -45], [-10, -26], [-4, -60], [-27, 24], [-8, -24]]; 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_eu();". // 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_eu(: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_eu();". // 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_eu( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-3, 0, 1]>,<13,R![-8, 4, 1]>,<17,R![-4, 1]>,<19,R![13, -8, 1]>],Snew); return Vf; end function;