// Make newform 9075.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9075_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_9075_2_a_j();". // 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_9075_2_a_j();". // 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_9075_a();" function MakeCharacter_9075_a() N := 9075; order := 1; char_gens := [3026, 727, 5326]; 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_9075_a_Hecke(Kf) return MakeCharacter_9075_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], [-1], [0], [-1], [0], [2], [6], [7], [6], [0], [-1], [7], [6], [8], [0], [6], [-12], [1], [7], [6], [-13], [-11], [0], [-18], [1], [12], [1], [12], [7], [12], [17], [-18], [12], [4], [-6], [16], [-17], [1], [-6], [-6], [-12], [17], [12], [5], [-24], [11], [-23], [-29], [-12], [14], [24], [12], [22], [6], [6], [-6], [-24], [-20], [11], [30], [-1], [18], [-13], [-30], [22], [6], [29], [-13], [-30], [-35], [6], [-6], [4], [-31], [8], [-6], [30], [7], [18], [-5], [0], [-10], [6], [-5], [19], [18], [-6], [-10], [-30], [-32], [42], [-24], [-8], [-42], [23], [-24], [24], [0], [-25], [-14], [8], [-42], [36], [0], [-23], [43], [-36], [-42], [42], [-11], [32], [5], [6], [20], [8], [30], [-29], [24], [42], [-6], [-1], [29], [48], [30], [-13], [6], [-34], [-24], [-8], [-22], [19], [-36], [-37], [25], [6], [25], [-12], [-28], [-18], [-30], [-11], [42], [13], [12], [-7], [0], [5], [54], [-49], [36], [5], [0], [25], [-6], [-35], [-12], [-11], [30], [-43], [-30], [12], [-6], [23], [18], [18], [30], [-40], [41], [-14], [0], [-42], [-1], [24], [38], [-16], [24], [37], [-6], [-28], [25], [-35], [54], [-50], [24], [36], [24], [11], [-17], [-26], [66], [61], [-36], [-25], [36], [18], [6], [-14], [-11], [-24], [42], [24], [-17], [13], [-29], [0], [66], [-7], [6], [-36], [11], [-43], [24], [-35], [0], [-30], [-10], [17], [-54], [-54], [-24], [-5], [40], [48], [-53], [-54], [22], [-24], [36], [-40], [30], [46], [19], [-53], [72], [25], [-12], [-31], [54], [6], [30], [24], [28], [-23], [11], [42], [-6], [-68], [-42], [40], [-6], [47], [-48], [-24], [5], [30], [0], [14], [-28], [18], [53], [-16], [6], [-53], [29], [36], [41], [42], [-48], [65], [-60], [26], [-65], [13], [-20], [-49], [-47], [-36], [-26], [-47], [-12], [-72], [-25], [-18], [-50], [-61], [-24], [61], [24], [-7], [-54], [12], [-36], [-54], [42], [-10], [30], [-55], [60], [-36], [-7], [41], [36], [49], [-6], [-76], [25], [-18], [5], [30], [-79], [-48], [72], [-72], [-8], [12], [7], [42], [-12], [-47], [60], [-11], [-29], [66], [61], [-42], [-31], [20], [-11], [30], [-30], [-5], [-84], [-59], [-18], [-20], [6], [47], [42], [-67], [71], [-38], [-12], [-42], [-37], [78], [54], [65], [-17], [66], [48], [-44], [25], [66], [-64], [61], [-24], [-12], [84], [-42], [90], [-19], [-12], [12], [-66], [-29], [-2], [-30], [56], [-47], [18], [47], [24], [36], [40], [-47], [48], [-12], [53], [42], [71], [18], [-36], [41], [-54], [-5], [66], [8], [94], [92], [-18], [53], [-18], [-48], [73], [-90], [62], [-29], [-96], [-103], [30], [94], [48], [104], [42], [-54], [-20], [73], [24], [-56], [-54], [26], [-48], [6], [31], [-82], [-36], [84], [73], [66], [96], [24], [-7], [42], [-12], [19], [84], [48], [-48], [-43], [6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9075_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9075_2_a_j();". // 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_9075_2_a_j(:prec:=1) chi := MakeCharacter_9075_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_9075_2_a_j();". // 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_9075_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9075_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<7,R![1, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<19,R![-7, 1]>,<23,R![-6, 1]>,<37,R![-7, 1]>],Snew); return Vf; end function;