// Make newform 9450.2.a.bq in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9450_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_9450_2_a_bq();". // 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_9450_2_a_bq();". // 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_9450_a();" function MakeCharacter_9450_a() N := 9450; order := 1; char_gens := [9101, 6427, 6751]; 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_9450_a_Hecke(Kf) return MakeCharacter_9450_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 := [[-1], [0], [0], [1], [1], [-3], [4], [-6], [-3], [-6], [2], [7], [2], [2], [7], [2], [13], [-11], [0], [-13], [-2], [-8], [4], [4], [-17], [0], [2], [-3], [-6], [-2], [2], [-7], [18], [-20], [6], [-10], [2], [-14], [3], [-8], [9], [19], [12], [-14], [16], [2], [-20], [14], [-29], [-5], [24], [-11], [1], [-15], [-8], [-11], [0], [-24], [30], [-22], [-14], [-16], [14], [3], [-14], [-22], [22], [14], [-4], [-26], [-6], [17], [-4], [22], [-30], [-9], [2], [23], [-32], [-27], [36], [11], [9], [1], [16], [-1], [18], [-13], [-20], [-14], [19], [-24], [28], [12], [-24], [12], [-14], [-28], [-16], [3], [0], [8], [-11], [18], [-30], [17], [-44], [0], [12], [-9], [-22], [-21], [22], [-10], [38], [44], [-10], [-1], [-8], [-44], [-35], [-7], [-30], [-11], [10], [-32], [-41], [21], [36], [-31], [40], [-15], [32], [-31], [-30], [-14], [-42], [-16], [-42], [30], [20], [30], [26], [7], [35], [-31], [19], [-12], [-42], [-40], [19], [20], [-36], [-35], [-2], [45], [-10], [34], [13], [32], [-32], [-12], [8], [45], [36], [-41], [-44], [-17], [-23], [-16], [-40], [2], [45], [39], [-4], [58], [-32], [-30], [34], [25], [-26], [-21], [37], [22], [-12], [-42], [-58], [4], [-23], [-39], [-21], [-31], [40], [-22], [15], [36], [-41], [53], [-6], [40], [10], [50], [-3], [33], [-45], [58], [44], [-51], [30], [-16], [66], [24], [26], [-59], [-45], [14], [14], [-54], [-1], [18], [-13], [-4], [18], [66], [-41], [-7], [-10], [11], [-40], [8], [-67], [-12], [28], [72], [14], [9], [61], [64], [-76], [-11], [-76], [-10], [-66], [13], [30], [-9], [-56], [-45], [-16], [-29], [41], [0], [27], [45], [18], [55], [2], [-70], [54], [-3], [-54], [-28], [70], [7], [-6], [26], [-60], [50], [50], [-42], [-15], [-78], [-77], [-48], [-65], [70], [64], [62], [75], [19], [-13], [50], [-33], [9], [28], [-3], [-25], [54], [-10], [54], [-18], [-68], [56], [-7], [78], [36], [-2], [38], [60], [22], [-39], [76], [-44], [-39], [56], [-33], [-17], [-30], [-39], [10], [24], [36], [22], [16], [48], [53], [-60], [75], [66], [-76], [16], [17], [-28], [48], [48], [-7], [56], [-78], [-19], [24], [71], [54], [-8], [-33], [14], [24], [46], [42], [85], [-68], [59], [38], [90], [62], [24], [28], [18], [8], [-27], [12], [-58], [3], [-72], [-44], [-66], [-54], [-21], [4], [78], [81], [-77], [38], [-5], [43], [44], [86], [98], [-64], [55], [-12], [16], [-67], [-30], [-10], [14], [20], [15], [10], [-30], [54], [46], [-76], [-72], [66], [54], [48], [-12], [49], [82], [-88], [26], [8], [-80], [-78], [-8], [-58], [76], [-50], [-36], [54], [-27], [66], [38], [96], [10], [-38], [1], [0], [-24], [21], [-71], [24], [39], [40], [-2], [-62], [-48], [-66], [-72], [88], [16], [-14], [-57], [8], [18], [98], [4], [48], [-68], [-109], [-83], [-69], [-84], [57], [74], [-6], [-7], [-35], [102], [-10], [99], [90], [-31], [107], [-67], [-66], [20], [92], [-50], [-85], [-88], [63], [21], [36], [-5], [66], [-22], [-31], [-54], [-14], [100], [28], [-45], [-51], [32], [43], [-56], [63], [-68], [-18], [88], [20], [-65], [18], [100], [25], [0], [102], [53], [-24], [-5], [18], [-29], [-36], [-70], [-39], [-5], [84], [-32], [-50], [-53], [-63], [-42], [-78], [59], [62], [-6], [-60], [-14], [80], [44], [100], [100], [34], [-105], [32], [88], [40], [-11], [-40], [-27], [31], [13], [-54], [-54], [61], [-48], [-19], [-107], [38], [-34], [12], [98], [21], [-41], [-65], [86], [-52], [-48], [-74], [-6], [-32], [60], [40], [-107], [56], [43], [-108], [-74], [-12], [28], [102], [-116], [9], [-76], [54], [4], [-47], [78], [-108], [38], [76], [-13], [-118], [-33], [-125], [38], [-108], [74], [-67], [78], [-93], [53], [49], [-66], [46], [-71], [47], [90], [-68], [105], [22], [-90], [-21], [-127], [57], [-90], [110], [68], [-104], [-10], [-34], [22], [39], [-6], [-69], [7], [-36], [-70], [102]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9450_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9450_2_a_bq();". // 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_9450_2_a_bq(:prec:=1) chi := MakeCharacter_9450_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(4297) - 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_9450_2_a_bq();". // 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_9450_2_a_bq( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-1, 1]>,<13,R![3, 1]>,<17,R![-4, 1]>,<19,R![6, 1]>],Snew); return Vf; end function;