// Make newform 3675.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3675_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_3675_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_3675_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_3675_a();" function MakeCharacter_3675_a() N := 3675; order := 1; char_gens := [1226, 1177, 2551]; 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_3675_a_Hecke(Kf) return MakeCharacter_3675_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], [0], [0], [-1], [6], [-5], [-6], [-6], [-5], [7], [-12], [1], [6], [0], [6], [-2], [7], [12], [11], [-13], [-12], [-6], [-10], [0], [-1], [-18], [-7], [-18], [-11], [6], [-6], [-5], [-24], [-16], [-10], [4], [12], [0], [18], [7], [-6], [-5], [-6], [16], [-4], [-4], [12], [-29], [-6], [-6], [-2], [18], [-12], [12], [18], [16], [-23], [-24], [11], [12], [-7], [-12], [17], [-6], [23], [13], [24], [-26], [0], [-6], [-19], [-17], [-25], [0], [12], [29], [24], [-5], [30], [-7], [6], [29], [28], [36], [30], [-11], [6], [7], [30], [0], [7], [-12], [17], [-6], [18], [24], [-43], [-19], [-8], [-18], [-18], [-24], [-7], [-7], [-18], [36], [12], [1], [-19], [-38], [42], [-35], [-40], [-30], [-25], [-24], [24], [36], [13], [31], [-18], [-18], [-11], [6], [-10], [-6], [-7], [-13], [-7], [24], [-19], [-50], [42], [13], [-24], [-28], [18], [18], [28], [-24], [40], [6], [19], [30], [17], [42], [4], [6], [-2], [-12], [31], [12], [31], [-18], [47], [12], [29], [30], [-48], [24], [55], [24], [12], [18], [-43], [-31], [-7], [6], [-12], [-50], [-30], [13], [-53], [-60], [-1], [30], [-16], [1], [1], [-30], [-23], [24], [54], [-48], [46], [53], [-31], [-24], [-13], [-36], [8], [-12], [-66], [54], [11], [-26], [18], [54], [-18], [13], [50], [-26], [-36], [-42], [16], [30], [-12], [-8], [7], [-18], [40], [0], [66], [25], [-8], [12], [60], [-42], [-31], [-17], [-12], [16], [42], [26], [-24], [-30], [-19], [12], [25], [52], [23], [36], [35], [0], [-65], [12], [0], [24], [24], [-29], [-49], [-46], [24], [-60], [29], [-24], [20], [-36], [-11], [-60], [-42], [67], [12], [6], [29], [53], [-60], [38], [-65], [12], [19], [-7], [-78], [-17], [-48], [66], [-68], [12], [-11], [-20], [35], [11], [38], [-40], [12], [-1], [71], [-12], [12], [23], [-18], [49], [-16], [-60], [-59], [-60], [43], [-12], [-72], [12], [-48], [-18], [-29], [-54], [55], [54], [24], [5], [-58], [-54], [32], [12], [-31], [25], [30], [-5], [48], [-53], [78], [24], [-12], [76], [84], [-29], [-12], [6], [59], [-84], [-17], [-77], [18], [-17], [42], [-29], [-61], [-61], [48], [-54], [77], [-60], [-89], [48], [11], [6], [50], [18], [-77], [-73], [-89], [6], [24], [5], [72], [-66], [67], [-20], [-36], [-90], [-92], [70], [54], [8], [-25], [-6], [-42], [36], [0], [-24], [-62], [60], [-30], [90], [35], [22], [-54], [73], [-34], [-12], [43], [-60], [-60], [61], [25], [-30], [12], [-70], [-18], [-37], [96], [54], [-56], [48], [40], [12], [47], [86], [19], [-72], [-31], [60], [6], [17], [-24], [10], [4], [-90], [-13], [54], [22], [-18], [16], [-84], [-48], [-77], [-26], [36], [11], [24], [-22], [48], [42], [-76], [85], [72], [24], [-97], [-12], [24], [24], [-7], [-6], [30], [2], [-60], [54], [84], [-20], [-42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3675_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3675_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_3675_2_a_i(:prec:=1) chi := MakeCharacter_3675_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_3675_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_3675_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3675_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<11,R![0, 1]>,<13,R![1, 1]>],Snew); return Vf; end function;