// Make newform 3528.2.a.ba in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3528_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_3528_2_a_ba();". // 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_3528_2_a_ba();". // 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_3528_a();" function MakeCharacter_3528_a() N := 3528; order := 1; char_gens := [2647, 1765, 785, 1081]; 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_3528_a_Hecke(Kf) return MakeCharacter_3528_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], [0], [4], [0], [0], [3], [-4], [-7], [-4], [8], [5], [3], [8], [11], [4], [-4], [12], [2], [-3], [-12], [-1], [1], [12], [8], [2], [0], [-3], [12], [-11], [12], [-1], [-8], [-16], [-15], [16], [-20], [18], [-8], [12], [4], [20], [-5], [4], [-9], [-12], [-20], [-16], [12], [-8], [-5], [-28], [8], [2], [20], [12], [-24], [0], [-20], [13], [-12], [1], [-24], [11], [-24], [13], [-24], [-7], [-19], [12], [34], [12], [24], [-17], [-1], [21], [12], [4], [-7], [36], [15], [16], [1], [12], [5], [12], [-4], [-20], [1], [0], [5], [-12], [-4], [-35], [-24], [-29], [-40], [-12], [-24], [-29], [-23], [-8], [-8], [20], [-24], [-25], [-27], [-16], [24], [16], [41], [35], [-26], [0], [-9], [-4], [28], [-7], [48], [-16], [-4], [-35], [-29], [0], [36], [-17], [-20], [-38], [0], [-5], [35], [19], [8], [11], [10], [12], [-31], [8], [-40], [12], [-12], [-24], [36], [-4], [-36], [19], [0], [17], [-48], [8], [8], [14], [-28], [-47], [-24], [53], [4], [-11], [-28], [45], [-36], [-8], [-20], [57], [20], [48], [12], [3], [29], [21], [8], [-32], [-30], [48], [-11], [21], [40], [-31], [-12], [-4], [-43], [-5], [-8], [61], [-28], [48], [24], [2], [43], [45], [24], [-41], [60], [32], [-60], [-20], [0], [-29], [-46], [-16], [-4], [-40], [51], [-26], [2], [24], [48], [28], [-24], [-16], [8], [-9], [-48], [-44], [36], [-44], [57], [-36], [-32], [12], [-20], [-39], [-19], [-28], [-12], [56], [-10], [24], [32], [71], [36], [-23], [-56], [13], [-56], [-19], [60], [43], [24], [-8], [-12], [-28], [-47], [25], [46], [0], [-28], [47], [-20], [0], [-16], [-43], [28], [64], [-45], [8], [-24], [-7], [7], [36], [6], [-43], [72], [-69], [-7], [72], [-59], [-32], [-4], [24], [-48], [5], [16], [19], [-15], [-30], [76], [-52], [-65], [-73], [-36], [48], [65], [48], [-3], [-8], [24], [81], [28], [-39], [-36], [-72], [32], [44], [-64], [35], [32], [-63], [52], [-12], [-17], [-74], [0], [60], [-72], [3], [-7], [12], [71], [88], [-1], [-44], [20], [-8], [40], [-16], [59], [-4], [-44], [59], [72], [-35], [-85], [-48], [73], [-16], [-29], [-23], [29], [-72], [12], [-3], [-12], [-55], [-40], [65], [52], [-50], [-64], [-65], [41], [3], [48], [-36], [35], [-4], [0], [23], [64], [-88], [64], [40], [70], [48], [-84], [3], [60], [84], [-68], [-12], [-96], [-10], [12], [0], [-24], [45], [26], [-72], [-33], [-86], [44], [-11], [36], [16], [11], [93], [92], [-60], [-50], [88], [11], [-76], [-88], [-20], [-36], [-56], [-24], [-55], [22], [89], [-52], [89], [-64], [72], [63], [64], [-10], [44], [-64], [-83], [-60], [66], [-100], [-60], [-48], [40], [1], [50], [88], [9], [-96], [78], [-96], [32], [40], [-87], [-48], [-56], [-31], [-48], [-52], [-12], [1], [-48], [36], [54], [12], [52], [72], [-56], [104]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3528_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_ba();". // 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_3528_2_a_ba(:prec:=1) chi := MakeCharacter_3528_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_3528_2_a_ba();". // 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_3528_2_a_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3528_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<11,R![0, 1]>,<13,R![-3, 1]>,<23,R![4, 1]>],Snew); return Vf; end function;