// Make newform 7360.2.a.bh in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7360_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_7360_2_a_bh();". // 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_7360_2_a_bh();". // 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_7360_a();" function MakeCharacter_7360_a() N := 7360; order := 1; char_gens := [1151, 5061, 4417, 6721]; 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_7360_a_Hecke(Kf) return MakeCharacter_7360_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, -1], [-1, 0], [1, -1], [-2, 3], [-1, 5], [-2, 5], [3, -3], [1, 0], [6, 2], [1, 5], [0, -4], [-8, 7], [0, 0], [6, -6], [6, -4], [8, -6], [-2, 7], [-8, -4], [4, -5], [0, 2], [8, -4], [-6, 8], [-4, -4], [14, -1], [10, -6], [2, 9], [-2, -10], [-3, 9], [14, -8], [0, 10], [10, -8], [1, -7], [-16, 2], [3, 3], [12, -9], [-6, 6], [-3, 1], [-8, 0], [16, -9], [14, -10], [17, -3], [-4, -6], [4, -2], [8, 9], [2, 0], [-14, 0], [-8, -8], [22, -2], [-10, 0], [16, -6], [12, -20], [0, 0], [-11, 3], [-24, 10], [-10, 19], [-2, 10], [-19, -7], [-10, -12], [26, -2], [14, 8], [-6, 16], [-12, 25], [4, 0], [-5, 11], [3, 5], [-16, -2], [25, -7], [8, -1], [2, 0], [24, 0], [-6, 20], [0, 4], [-4, -10], [-18, 25], [8, -16], [-10, -7], [4, -7], [8, -6], [7, -23], [-8, -4], [17, -19], [26, -14], [-19, 17], [-15, 1], [-17, -5], [13, -9], [14, -32], [36, 2], [2, 0], [-12, 18], [-6, -16], [-36, 10], [10, -10], [20, -20], [25, 7], [-14, -14], [-22, 4], [26, 0], [16, -24], [19, -29], [-16, 24], [-12, 16], [-2, 0], [18, -21], [-10, 4], [-12, -1], [36, 2], [-20, 1], [-9, -15], [20, 4], [26, -28], [26, 5], [-15, 11], [0, 26], [-32, 20], [18, -6], [22, -36], [-13, 13], [-12, 36], [-30, 7], [2, -8], [38, 8], [17, 9], [26, 2], [23, -41], [18, 5], [11, 15], [46, -11], [10, 12], [-12, 16], [-23, 35], [8, -34], [-8, 16], [-13, 37], [-26, 12], [-16, -8], [32, -18], [-4, -6], [-16, -19], [-38, 10], [46, -8], [6, 12], [-12, -6], [-6, -8], [-22, 34], [-7, 11], [-26, 12], [-18, 26], [36, -2], [12, -19], [28, -12], [-1, -9], [-44, 24], [22, -18], [-6, 26], [40, 0], [-10, 4], [-19, -11], [50, -17], [11, 27], [-14, 11], [32, -30], [-10, -15], [-9, -9], [10, -15], [0, 17], [14, -44], [-28, 34], [20, -11], [28, -20], [6, -26], [2, 10], [-2, -5], [32, -1], [26, -30], [-34, 4], [-36, 24], [-10, -23], [-49, -5], [16, -18], [-14, 33], [-33, 29], [-10, 2], [38, -3], [11, -41], [15, -11], [-16, -24], [-40, 23], [-52, 15], [40, -2], [-8, 8], [60, 0], [-12, 32], [-22, 0], [17, -11], [18, 10], [-2, -28], [30, 0], [26, -14], [-22, 39], [25, 13], [-24, 21], [-16, -8], [24, -24], [2, -12], [46, -16], [8, -15], [-43, -5], [26, 6], [-46, 6], [-26, 44], [-43, 15], [-30, -6], [-28, -4], [-14, 26], [52, -22], [26, 4], [-17, -3], [10, -16], [30, -34], [-18, 8], [-44, 17], [51, 5], [-52, 8], [-6, -26], [23, -57], [-43, 5], [-17, -27], [-48, 12], [-15, -5], [14, -33], [-12, 8], [-8, 15], [14, -50], [-12, 12], [-26, -6], [-22, 6], [-2, -6], [-4, 5], [30, 2], [-26, -8], [-2, -4], [-20, 30], [22, -2], [3, -37], [6, -22], [31, 9], [-30, -19], [-18, 43], [20, -14], [16, 6], [59, 5], [4, -44], [6, -20], [-8, 26], [-46, 13], [12, 30], [27, -23], [-18, 56], [18, -1], [18, -20], [16, -20], [14, 2], [50, 16], [2, -13], [14, -13], [14, 20], [48, -14], [45, -13], [-38, 14], [-4, 4], [-1, 7], [-8, 24], [-26, 0], [-24, 34], [-4, 49], [-26, 10], [18, -16], [3, -11], [20, -44], [0, -48], [14, 13], [54, -28], [3, -25], [-40, -17], [42, -4], [-4, 54], [-8, 21], [32, -47], [-18, 24], [-64, 34], [-28, 18], [1, 31], [48, -17], [-14, -33], [-54, 34], [4, 39], [-2, -16], [-8, 11], [31, -29], [-14, -34], [36, 18], [48, -40], [-31, 63], [20, -34], [40, -11], [19, 35], [-18, 58], [16, -17], [-66, 10], [9, -11], [4, -14], [36, 31], [-49, 9], [-34, 14], [-38, 40], [-13, -5], [-42, 0], [-12, 35], [66, -27], [-54, 31], [10, -32], [26, -26], [-14, -19], [-16, 0], [-1, 41], [53, 5], [1, 43], [58, -18], [8, 42], [12, 15], [64, 16], [5, 29], [20, -40], [-19, 29], [-74, 23], [-14, 0], [20, -67], [30, -13], [-54, 12], [26, -15], [-8, -20], [-30, 30], [-6, -32], [-24, 48], [-26, 4], [-27, 33], [-2, 44], [30, -14], [-58, 10], [-2, 12], [44, -18], [-54, 43], [14, 48], [-46, -6], [16, -42], [15, 45], [10, -36], [46, -60], [48, -22], [50, -40], [44, 18], [-62, 33], [16, -27], [26, -4], [40, -19], [-8, -32], [-34, -10], [-38, 20], [-28, -34], [10, 12], [-14, -7], [72, -40], [69, -5], [-22, 26], [32, -42], [58, 7], [70, -14], [-4, -32], [-6, 17], [-54, 45], [36, -18], [-78, 24], [-4, -4], [0, -16], [-15, 9], [-17, 69], [-34, 36], [53, -37], [-38, 0], [-47, 67], [40, -36], [-24, -44], [94, 0], [45, -23], [-10, 28], [-4, 40], [-35, 27], [42, 8], [64, -38], [40, -42], [-4, 0], [-1, -1], [32, 4], [24, -23], [-6, 6], [63, -11], [-14, -10], [-8, 56], [16, -38], [-28, 73], [44, -40], [42, 19], [61, 9], [14, 40], [-57, 55], [-26, -14], [14, -27], [37, -45], [18, -39], [74, -22], [18, 18], [62, -30], [54, 10], [-38, 40], [20, -20], [-37, 1], [-12, -58], [16, 53]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7360_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7360_2_a_bh();". // 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_7360_2_a_bh(:prec:=2) chi := MakeCharacter_7360_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_7360_2_a_bh();". // 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_7360_2_a_bh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7360_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1, 1]>,<7,R![-1, -1, 1]>,<11,R![-11, 1, 1]>,<13,R![-29, -3, 1]>,<17,R![-31, -1, 1]>],Snew); return Vf; end function;