// Make newform 9408.2.a.cv in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_2_a_cv();". // 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_9408_2_a_cv();". // 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_9408_a();" function MakeCharacter_9408_a() N := 9408; order := 1; char_gens := [1471, 6469, 3137, 4609]; 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_9408_a_Hecke(Kf) return MakeCharacter_9408_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], [2], [0], [-2], [-1], [0], [1], [0], [-4], [-9], [-3], [-10], [5], [6], [-12], [-12], [-10], [-5], [6], [-3], [1], [6], [16], [-6], [-2], [7], [-8], [-9], [10], [15], [-14], [-12], [-3], [12], [16], [14], [4], [14], [-8], [2], [-13], [-10], [11], [-16], [0], [4], [-16], [18], [19], [6], [-6], [14], [-8], [26], [-4], [-6], [-16], [-13], [-4], [-11], [-8], [-17], [6], [-1], [-24], [-25], [13], [32], [14], [34], [-20], [9], [-23], [3], [12], [6], [9], [-36], [5], [30], [7], [18], [31], [0], [12], [-18], [-11], [-20], [17], [6], [28], [-31], [-28], [37], [42], [-2], [12], [31], [19], [28], [2], [-26], [-26], [-19], [-17], [16], [-6], [-12], [-9], [-23], [-34], [-6], [-29], [-8], [0], [-19], [-2], [-18], [36], [41], [-41], [-12], [-12], [-37], [0], [-30], [18], [13], [15], [-15], [-42], [-13], [22], [-48], [-49], [34], [40], [8], [30], [32], [-2], [0], [-30], [-41], [44], [-35], [-32], [-40], [54], [38], [24], [-13], [34], [-37], [24], [-23], [14], [15], [4], [-10], [44], [-19], [36], [-18], [-36], [-17], [-19], [61], [-32], [48], [18], [-60], [5], [19], [-16], [-53], [-10], [-24], [-31], [19], [0], [-41], [-28], [54], [52], [-46], [-17], [-23], [38], [49], [-6], [-40], [36], [52], [-24], [35], [18], [-30], [-12], [-14], [-23], [-54], [-14], [38], [22], [-24], [-42], [-36], [52], [39], [24], [-4], [4], [-44], [7], [40], [-20], [-28], [30], [-5], [-61], [50], [64], [50], [-26], [-8], [46], [-47], [-6], [11], [4], [49], [46], [-63], [30], [-35], [10], [-60], [22], [-18], [-39], [-45], [50], [72], [20], [5], [50], [-8], [-24], [-77], [-24], [-10], [37], [-64], [56], [7], [-57], [78], [14], [-59], [56], [43], [61], [14], [-55], [6], [58], [-44], [26], [-19], [-4], [37], [-47], [10], [76], [-38], [81], [43], [8], [-56], [-63], [-2], [-55], [20], [-6], [25], [-6], [-53], [18], [-4], [-20], [54], [60], [29], [-46], [-25], [28], [-34], [-1], [-18], [6], [-28], [28], [-3], [-75], [-6], [63], [-16], [-19], [-36], [70], [90], [-48], [-8], [-7], [-6], [-48], [73], [-54], [-19], [-41], [42], [1], [-42], [-47], [7], [53], [52], [-30], [23], [66], [-17], [66], [-5], [58], [70], [-36], [-19], [59], [17], [-16], [-48], [-37], [-14], [-26], [-25], [4], [66], [70], [76], [-54], [62], [0], [65], [2], [-30], [-60], [32], [-6], [6], [-26], [-30], [2], [41], [70], [-8], [-1], [78], [84], [21], [6], [-54], [-95], [-49], [-56], [8], [-34], [36], [57], [16], [60], [56], [-4], [60], [28], [73], [-46], [11], [42], [25], [-54], [36], [-73], [22], [58], [44], [-78], [15], [-102], [-14], [-52], [-16], [-6], [-66], [-49], [18], [20], [101], [-36], [-46], [-44], [-16], [-44], [-15], [-90], [-66], [79], [12], [96], [24], [41], [88], [2], [-10], [-18], [58], [-72], [-12], [72], [81], [-108], [-21], [-34], [-35], [72], [1], [25], [-39], [40], [-44], [-32], [-3], [58], [-102], [44], [44], [42], [-26], [59], [-63], [58], [-30], [24], [57], [28], [-37], [38], [86], [-48], [-20], [-7], [52], [-59], [48], [42], [21], [-84], [-14], [-35], [31], [-32], [-32], [21], [0], [-46], [-62], [-11], [-64], [-40], [-79], [60], [-39], [100], [-76], [-42], [-89], [-48], [68], [40], [38], [8], [-45], [-78], [-100], [-71], [107], [-48], [-15], [-3], [60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9408_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9408_2_a_cv();". // 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_9408_2_a_cv(:prec:=1) chi := MakeCharacter_9408_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(3581) - 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_9408_2_a_cv();". // 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_9408_2_a_cv( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9408_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<11,R![2, 1]>,<13,R![1, 1]>,<17,R![0, 1]>,<19,R![-1, 1]>,<31,R![9, 1]>],Snew); return Vf; end function;