// Make newform 9360.2.a.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9360_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_9360_2_a_r();". // 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_9360_2_a_r();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_9360_a();" function MakeCharacter_9360_a() N := 9360; order := 1; char_gens := [8191, 2341, 2081, 5617, 5761]; 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_9360_a_Hecke(Kf) return MakeCharacter_9360_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], [1], [-3], [1], [3], [4], [-9], [6], [-2], [-1], [3], [-2], [-6], [-9], [-12], [5], [4], [9], [14], [7], [0], [-15], [5], [12], [-14], [3], [-10], [6], [-20], [12], [12], [1], [-15], [16], [8], [-11], [-6], [-6], [-18], [-7], [-18], [-13], [-24], [16], [4], [-8], [12], [-4], [-3], [-15], [-10], [-12], [6], [-24], [18], [-20], [26], [-18], [16], [30], [1], [-30], [-28], [6], [16], [-22], [-33], [20], [-18], [12], [-8], [-22], [-2], [24], [6], [-13], [-30], [32], [18], [-28], [0], [-28], [-11], [15], [-21], [17], [-21], [1], [15], [-9], [43], [18], [22], [-24], [-15], [-18], [4], [-28], [-38], [-18], [15], [6], [19], [41], [-12], [-42], [-6], [-37], [16], [-31], [-42], [-8], [34], [0], [-23], [-39], [-18], [30], [14], [-34], [3], [24], [28], [-24], [44], [-24], [-2], [-49], [-38], [-24], [-11], [14], [6], [14], [48], [52], [-39], [48], [22], [15], [4], [-18], [-22], [-51], [17], [33], [13], [-18], [-46], [54], [10], [9], [-50], [-48], [-29], [21], [-4], [33], [42], [-9], [-56], [0], [54], [-36], [-23], [-46], [20], [-57], [36], [32], [42], [-31], [49], [18], [-14], [-3], [16], [38], [37], [12], [20], [0], [18], [-30], [20], [19], [20], [-9], [20], [42], [43], [39], [0], [-9], [50], [-10], [12], [-57], [-42], [19], [-67], [-43], [-9], [21], [64], [12], [33], [-23], [38], [54], [-44], [60], [0], [14], [-2], [-54], [-48], [-36], [-13], [70], [18], [-56], [63], [17], [69], [18], [-62], [-3], [-16], [25], [-32], [0], [-44], [42], [14], [-24], [48], [-24], [24], [-11], [-62], [38], [54], [-48], [16], [57], [-26], [48], [-58], [18], [0], [-19], [69], [51], [-22], [-35], [42], [-43], [52], [-27], [-64], [38], [12], [-53], [21], [69], [-77], [-18], [50], [-41], [53], [40], [-52], [31], [0], [62], [-40], [24], [63], [-50], [-48], [-10], [-68], [6], [38], [66], [34], [-60], [60], [-12], [78], [36], [-4], [-18], [16], [69], [48], [25], [-4], [-48], [43], [-39], [4], [23], [33], [-37], [72], [-28], [63], [-42], [-36], [16], [78], [53], [-9], [-15], [-34], [-60], [-80], [50], [-60], [-35], [-24], [23], [4], [37], [-57], [-6], [-64], [-15], [28], [84], [4], [54], [44], [-30], [14], [34], [-79], [-39], [21], [-32], [-18], [-6], [5], [1], [48], [-39], [-20], [-37], [39], [40], [59], [-42], [-51], [12], [-78], [6], [-34], [54], [60], [48], [-32], [26], [36], [-23], [-7], [90], [100], [-78], [-24], [25], [-40], [-9], [36], [29], [-66], [80], [-18], [18], [28], [-18], [-86], [-54], [-80], [-28], [88], [-93], [80], [24], [-27], [10], [-39], [-52], [70], [-60], [85], [30], [86], [9], [-17], [48], [51], [-11], [-46], [78], [31], [-57], [-52], [33], [-45], [73], [92], [-84], [51], [82], [12], [-39], [66], [59], [12], [54], [-70], [24], [39], [75], [-26], [24], [32], [57], [-5], [-78], [17], [24], [-52], [-52], [-56], [-2], [-54], [45], [56], [-72], [-46], [-42], [34], [30], [-46], [38], [-35], [84], [66], [18], [23], [-78], [-10], [60], [-16], [84], [55], [40], [-12], [5], [13], [-61], [61], [-24], [-96], [4], [16], [12], [93], [-22], [-102], [-85], [90], [-56], [57], [48], [-16], [12], [-52], [114], [4], [3], [20], [-57], [-62], [-89], [29], [24], [-106], [-66], [66], [-22], [37], [-72], [28], [-11], [-93], [-44], [6], [-65], [-22], [78], [-111], [-47], [-34], [-110], [24], [-33], [59], [-66], [52], [-46], [-30], [-25], [54], [52], [26], [-32], [-18], [45], [77], [-60], [14], [-3], [-36], [-108], [-8], [36], [-20], [30], [2], [102], [-118], [21], [-34], [-5], [-48], [-21], [28], [12], [-30], [58], [-74], [-54], [-35], [27], [-30], [-26], [39], [99], [39], [71], [-32]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9360_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9360_2_a_r();". // 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_9360_2_a_r(:prec:=1) chi := MakeCharacter_9360_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(4027) - 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_9360_2_a_r();". // 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_9360_2_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9360_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-1, 1]>,<11,R![3, 1]>,<17,R![-3, 1]>,<19,R![-4, 1]>,<31,R![2, 1]>],Snew); return Vf; end function;