// Make newform 9075.2.a.s in Magma, downloaded from the LMFDB on 19 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_s();". // 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_s();". // 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 := [[2], [1], [0], [-3], [0], [1], [2], [5], [-6], [-10], [-3], [-2], [8], [1], [-2], [4], [-10], [-7], [3], [-8], [-14], [0], [6], [0], [-17], [-12], [4], [12], [-5], [4], [-8], [-12], [18], [-20], [-10], [-7], [13], [-11], [12], [6], [-10], [17], [22], [11], [-18], [-5], [13], [19], [-8], [-15], [-24], [-20], [23], [12], [-12], [16], [-10], [8], [-3], [18], [-9], [6], [7], [-18], [-11], [8], [12], [-23], [2], [-10], [-6], [0], [-27], [-29], [25], [-36], [0], [-7], [12], [-5], [-20], [22], [18], [29], [35], [24], [20], [22], [-12], [24], [38], [-30], [13], [8], [-5], [16], [-10], [22], [31], [3], [-8], [42], [6], [-30], [13], [13], [-12], [16], [-20], [13], [-8], [-14], [-12], [-25], [-23], [12], [-36], [28], [14], [40], [-38], [6], [42], [24], [-8], [-22], [25], [30], [43], [-34], [20], [-4], [-8], [23], [-12], [-35], [24], [7], [-52], [20], [-27], [18], [-41], [-28], [-30], [10], [51], [-28], [40], [-16], [27], [32], [-41], [-18], [-12], [-58], [-55], [-50], [-33], [-22], [18], [56], [32], [42], [-2], [-36], [17], [42], [-5], [14], [30], [-53], [18], [21], [-5], [0], [-27], [-18], [-19], [5], [-17], [8], [-1], [-38], [54], [-10], [-63], [29], [-5], [-2], [-1], [-64], [-3], [-58], [42], [-46], [-22], [-6], [42], [46], [0], [-12], [-42], [-10], [60], [-42], [35], [26], [-60], [-43], [-13], [-38], [39], [28], [30], [-58], [37], [-42], [8], [-6], [-37], [25], [-30], [-16], [-8], [-35], [54], [10], [-53], [-42], [-21], [20], [-47], [58], [4], [72], [-30], [-44], [-30], [12], [-66], [3], [29], [15], [6], [10], [73], [2], [-5], [-34], [22], [-2], [-12], [45], [-14], [60], [37], [-28], [-12], [62], [1], [22], [10], [-69], [-42], [40], [-70], [-8], [-4], [26], [-13], [8], [-26], [15], [57], [-41], [18], [-75], [33], [48], [16], [-8], [-38], [33], [-33], [-78], [-1], [-48], [-40], [0], [82], [-52], [-84], [-72], [71], [0], [77], [24], [30], [-28], [6], [-28], [-25], [54], [52], [43], [-12], [5], [60], [6], [-54], [60], [-12], [19], [22], [-25], [-30], [-12], [39], [-30], [-32], [13], [28], [-61], [66], [-38], [-85], [29], [-78], [56], [3], [-62], [-65], [46], [13], [-52], [-25], [-44], [-43], [-48], [-31], [8], [30], [-93], [-86], [-30], [77], [53], [-12], [68], [-27], [-42], [22], [-89], [45], [6], [40], [78], [42], [34], [-53], [58], [-42], [-80], [-92], [-91], [72], [-84], [-47], [12], [-75], [26], [-60], [-72], [23], [80], [-42], [-49], [60], [37], [22], [-6], [-8], [-8], [-60], [94], [-43], [58], [36], [78], [45], [64], [20], [-77], [2], [-79], [-5], [-60], [-83], [-12], [-10], [-36], [67], [68], [20], [8], [-17], [8], [69], [60], [-9], [12], [-26], [-27], [77], [92], [-10], [-67], [-72], [-44], [40], [7], [18], [70], [34], [-32], [-16], [-60], [77], [-50]]; 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_s();". // 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_s(: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_s();". // 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_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9075_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-2, 1]>,<7,R![3, 1]>,<13,R![-1, 1]>,<17,R![-2, 1]>,<19,R![-5, 1]>,<23,R![6, 1]>,<37,R![2, 1]>],Snew); return Vf; end function;