// Make newform 1764.4.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1764_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_1764_4_a_e();". // 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_1764_4_a_e();". // 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_1764_a();" function MakeCharacter_1764_a() N := 1764; order := 1; char_gens := [883, 785, 1081]; 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_1764_a_Hecke(Kf) return MakeCharacter_1764_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 := 4; raw_aps := [[0], [0], [0], [0], [0], [-89], [0], [163], [0], [0], [19], [-433], [0], [449], [0], [0], [0], [-182], [1007], [0], [919], [503], [0], [0], [1330], [0], [19], [0], [-1567], [0], [-2647], [0], [0], [3043], [0], [1748], [3850], [-3400], [0], [0], [0], [-1241], [0], [5111], [0], [5236], [6032], [3220], [0], [6823], [0], [0], [7378], [0], [0], [0], [0], [-812], [-5167], [0], [-3869], [0], [6697], [0], [901], [0], [9899], [-7363], [0], [11914], [0], [0], [13753], [251], [14687], [0], [0], [-13067], [0], [-8297], [0], [-10459], [0], [14149], [-14924], [0], [0], [-19207], [0], [7811], [0], [0], [-14797], [0], [-6607], [0], [0], [0], [11881], [20789], [1640], [0], [0], [0], [629], [-27323], [0], [0], [0], [-11951], [-22283], [17390], [0], [37], [1892], [0], [32419], [0], [0], [0], [13249], [-33949], [0], [0], [-36251], [0], [36146], [0], [27253], [-24281], [-37423], [0], [41021], [-41470], [0], [-39023], [0], [-43400], [0], [0], [-39368], [0], [-12220], [0], [47143], [0], [-38411], [0], [-31304], [0], [50150], [0], [-31429], [0], [4607], [0], [-49573], [0], [-15227], [0], [0], [0], [53927], [0], [0], [0], [59669], [-33983], [8693], [0], [0], [40138], [0], [63629], [65683], [0], [2771], [0], [-67340], [69697], [54557], [0], [-63631], [0], [0], [0], [56810], [-32471], [52037], [0], [-52649], [0], [-37312], [0], [0], [0], [-68221], [-56950], [0], [0], [0], [82549], [83230], [49714], [0], [0], [-85484], [0], [0], [41272], [-1513], [0], [48620], [0], [0], [-11951], [15860], [0], [0], [0], [-73891], [-72557], [0], [-77380], [0], [53534], [0], [0], [-41543], [0], [99989], [69496], [90539], [0], [50311], [0], [76267], [0], [0], [0], [0], [115939], [-9161], [-49786], [0], [0], [-81863], [0], [-118816], [0], [8513], [0], [0], [-41813], [0], [0], [-7939], [103717], [0], [-4970], [-103069], [0], [-96857], [-37259], [0], [-75833], [0], [0], [142760], [0], [30529], [92720], [144811], [-13033], [71890], [-127820], [0], [144467], [129599], [0], [0], [155231], [0], [111169], [138040], [0], [-71119], [0], [150967], [0], [0], [0], [0], [0], [163799], [0], [-146249], [0], [0], [60517], [-75530], [0], [171236], [0], [-10711], [-145207], [0], [151867], [0], [-182629], [0], [0], [0], [-131560], [0], [-138077], [0], [0], [-181781], [0], [-168929], [-125677], [0], [-36721], [0], [19171], [203363], [147079], [0], [0], [62441], [0], [64783], [0], [109619], [0], [6806], [0], [-160361], [-217457], [-31249], [0], [0], [132749], [0], [0], [112051], [188480], [0], [0], [222712], [70310], [0], [-178220], [182447], [0], [0], [0], [0], [0], [7310], [0], [0], [0], [160543], [152570], [0], [147041], [-179062], [0], [-245537], [0], [0], [140149], [-93367], [0], [0], [-178850], [0], [209827], [0], [0], [-205900], [0], [271096], [0], [74159], [-219170], [-145909], [0], [47753], [0], [0], [115813], [0], [-254410], [-206444], [0], [-209161], [0], [112714], [0], [191900], [0], [0], [259471], [-41830], [0], [281251], [0], [-108290], [0], [0], [235352], [241793], [0], [0], [-249713], [0], [0], [0], [18037], [0], [0], [-277130], [0], [0], [0], [323512], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1764_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1764_4_a_e();". // 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_1764_4_a_e(:prec:=1) chi := MakeCharacter_1764_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_1764_4_a_e();". // 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_1764_4_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1764_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![89, 1]>],Snew); return Vf; end function;