// Make newform 3465.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3465_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_3465_2_a_i();". // 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_3465_2_a_i();". // 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_3465_a();" function MakeCharacter_3465_a() N := 3465; order := 1; char_gens := [1541, 1387, 2971, 2521]; 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_3465_a_Hecke(Kf) return MakeCharacter_3465_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], [-1], [0], [-3], [-3], [1], [7], [6], [-8], [2], [-5], [6], [3], [5], [-7], [6], [10], [12], [-16], [3], [-3], [-1], [14], [-5], [-6], [-2], [19], [19], [-4], [2], [4], [6], [14], [-1], [12], [-12], [14], [18], [10], [16], [6], [0], [24], [-18], [1], [3], [-14], [16], [21], [10], [12], [-20], [-16], [1], [-1], [-10], [22], [-8], [-19], [26], [32], [21], [-2], [17], [-1], [8], [-19], [10], [21], [7], [11], [5], [-32], [-12], [-10], [10], [-6], [-33], [-23], [-36], [22], [1], [-32], [-26], [-37], [-6], [-18], [0], [0], [-22], [15], [-21], [13], [11], [27], [-20], [-14], [13], [-6], [4], [-13], [36], [18], [-36], [34], [-30], [15], [-38], [-38], [-2], [46], [29], [30], [31], [4], [-39], [15], [-18], [-43], [25], [4], [2], [-11], [-15], [-15], [-7], [-6], [22], [-18], [53], [-6], [30], [9], [-26], [-40], [30], [-10], [12], [-23], [-24], [36], [38], [17], [-10], [-50], [46], [-39], [45], [-15], [20], [-41], [-44], [4], [-14], [-6], [-38], [-40], [45], [-30], [-17], [-47], [-49], [6], [31], [-42], [32], [36], [-39], [-30], [15], [-11], [50], [42], [-20], [-24], [58], [-2], [32], [-32], [54], [39], [-39], [-47], [-11], [37], [-26], [42], [2], [-50], [20], [55], [0], [-44], [-24], [24], [-23], [-12], [23], [-39], [-55], [34], [41], [12], [62], [-70], [-13], [42], [43], [35], [52], [12], [-28], [-18], [-35], [-10], [8], [9], [-32], [8], [-14], [-8], [65], [-40], [-6], [6], [-24], [29], [-32], [-52], [20], [11], [-9], [63], [-8], [8], [50], [24], [-4], [1], [-16], [23], [-3], [-24], [-64], [47], [4], [48], [30], [-22], [76], [18], [39], [49], [-7], [42], [42], [38], [-25], [-72], [-49], [-2], [-48], [-16], [32], [31], [-9], [-28], [-10], [-62], [-49], [-74], [-68], [48], [13], [64], [10], [52], [47], [-64], [12], [25], [32], [-64], [38], [-52], [22], [66], [60], [36], [-36], [32], [-34], [60], [36], [84], [80], [-80], [-10], [-48], [40], [13], [53], [82], [9], [52], [2], [-41], [48], [-54], [-87], [-14], [30], [75], [4], [87], [38], [-9], [-72], [22], [-66], [20], [16], [-60], [-60], [29], [78], [-44], [-30], [-66], [-7], [36], [-30], [48], [15], [-6], [84], [18], [-94], [17], [78], [-8], [-7], [-31], [88], [14], [-56], [2], [-13], [4], [-12], [-76], [-40], [-67], [32], [-48], [6], [71], [-79], [-70], [12], [-87], [93], [-2], [-79], [5], [-6], [72], [-76], [-48], [-9], [31], [72], [12], [-29], [16], [44], [-52], [-51], [-63], [8], [6], [35], [14], [-29], [57], [-60], [-60], [-55], [38], [30], [69], [22], [65], [100], [54], [-67], [-53], [-89], [-81], [41], [68], [28], [-35], [-34], [-2], [-61], [-69], [-44], [-62], [-92], [-28], [102], [31], [55], [-53], [-56], [56], [-38], [-42], [-27], [42], [97], [-96], [-27], [9], [-46], [28]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3465_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3465_2_a_i();". // 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_3465_2_a_i(:prec:=1) chi := MakeCharacter_3465_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_3465_2_a_i();". // 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_3465_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3465_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<13,R![0, 1]>,<17,R![3, 1]>,<19,R![3, 1]>,<23,R![-1, 1]>],Snew); return Vf; end function;