// Make newform 3969.2.a.v in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3969_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_3969_2_a_v();". // 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_3969_2_a_v();". // 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 poly := [1, 0, -5, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -5, 0, 1]]; Rf_basisdens := [1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_3969_a();" function MakeCharacter_3969_a() N := 3969; order := 1; char_gens := [2108, 3727]; v := [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_3969_a_Hecke(Kf) return MakeCharacter_3969_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, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 0, -3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 2, 0, -2], [0, -4, 0, 4], [0, 0, 0, 0], [-1, 0, 4, 0], [0, 0, 0, 0], [-5, 0, 2, 0], [0, 0, 0, 0], [0, 7, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [-5, 0, -6, 0], [0, 9, 0, 7], [0, 0, 0, 0], [-1, 0, 6, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -9, 0, -11], [-18, 0, 0, 0], [0, -5, 0, 3], [-11, 0, -6, 0], [0, 0, 0, 0], [0, -9, 0, -1], [0, 0, 0, 0], [0, 9, 0, 13], [-13, 0, -2, 0], [0, 0, 0, 0], [7, 0, -6, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 10, 0, -10], [0, 0, 0, 0], [0, 9, 0, -1], [-18, 0, 0, 0], [0, 11, 0, 15], [0, 0, 0, 0], [-11, 0, -10, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -8, 0, 8], [0, -13, 0, -3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -17, 0, -15], [0, 0, 0, 0], [0, 0, 0, 0], [-1, 0, -12, 0], [0, -9, 0, -17], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -4, 0, 4], [-36, 0, 0, 0], [11, 0, -8, 0], [0, 9, 0, -5], [0, 0, 0, 0], [0, 0, 0, 0], [0, -11, 0, 3], [0, 0, 0, 0], [-5, 0, 12, 0], [1, 0, 14, 0], [0, 0, 0, 0], [0, 4, 0, -4], [0, 0, 0, 0], [0, 13, 0, 21], [0, 0, 0, 0], [0, 0, 0, 0], [19, 0, 12, 0], [0, -10, 0, 10], [0, 0, 0, 0], [0, 0, 0, 0], [0, -14, 0, 14], [0, 9, 0, -7], [5, 0, 16, 0], [0, 0, 0, 0], [23, 0, 6, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-19, 0, -14, 0], [0, 2, 0, -2], [-36, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [11, 0, -12, 0], [-19, 0, 6, 0], [0, -25, 0, -21], [0, 0, 0, 0], [0, -19, 0, -3], [4, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 9, 0, 23], [0, 0, 0, 0], [0, 0, 0, 0], [25, 0, 12, 0], [0, 16, 0, -16], [0, 0, 0, 0], [16, 0, 0, 0], [0, 27, 0, 19], [0, 0, 0, 0], [0, 0, 0, 0], [0, 27, 0, 23], [0, 27, 0, 17], [0, 0, 0, 0], [23, 0, 16, 0], [0, 0, 0, 0], [0, -27, 0, -25], [0, 0, 0, 0], [0, -9, 0, 11], [-13, 0, -20, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [29, 0, 6, 0], [0, -27, 0, -13], [-29, 0, -10, 0], [54, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -27, 0, -11], [0, 0, 0, 0], [0, -1, 0, -21], [32, 0, 0, 0], [0, -29, 0, -15], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 7, 0, -15], [-31, 0, -12, 0], [0, 0, 0, 0], [-17, 0, -22, 0], [0, 0, 0, 0], [31, 0, 2, 0], [0, 22, 0, -22], [-17, 0, 14, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 22, 0, -22], [0, -8, 0, 8], [-40, 0, 0, 0], [0, 0, 0, 0], [0, 16, 0, -16], [0, 0, 0, 0], [-1, 0, 22, 0], [0, 0, 0, 0], [11, 0, 24, 0], [0, 0, 0, 0], [0, 14, 0, -14], [0, 0, 0, 0], [0, -2, 0, 2], [17, 0, 24, 0], [0, 0, 0, 0], [0, 0, 0, 0], [44, 0, 0, 0], [0, -9, 0, -29], [0, 0, 0, 0], [0, 0, 0, 0], [-35, 0, -6, 0], [0, 0, 0, 0], [-23, 0, 12, 0], [0, 0, 0, 0], [0, 27, 0, 5], [0, 0, 0, 0], [-35, 0, -4, 0], [0, 0, 0, 0], [-23, 0, -24, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 35, 0, 33], [-36, 0, 0, 0], [0, 0, 0, 0], [0, -26, 0, 26], [0, 0, 0, 0], [-29, 0, 8, 0], [37, 0, 12, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -20, 0, 20], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 23, 0, -3], [0, -37, 0, -21], [0, 0, 0, 0], [29, 0, 24, 0], [0, 0, 0, 0], [-72, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-25, 0, -26, 0], [0, 0, 0, 0], [0, -25, 0, -39], [0, 13, 0, -15], [11, 0, 28, 0], [0, 0, 0, 0], [0, 16, 0, -16], [-35, 0, -22, 0], [0, 0, 0, 0], [54, 0, 0, 0], [0, 0, 0, 0], [0, 26, 0, -26], [0, 0, 0, 0], [0, 37, 0, 39], [-18, 0, 0, 0], [0, 0, 0, 0], [-72, 0, 0, 0], [0, -41, 0, -33], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 28, 0, -28], [0, -27, 0, -41], [0, 0, 0, 0], [0, -22, 0, 22], [0, 0, 0, 0], [0, 0, 0, 0], [37, 0, -4, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [13, 0, 30, 0], [0, 26, 0, -26], [29, 0, 28, 0], [0, 0, 0, 0], [0, -2, 0, 2], [0, 0, 0, 0], [0, 0, 0, 0], [0, -10, 0, 10], [-43, 0, -12, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [37, 0, -6, 0], [0, 9, 0, 35], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 43, 0, 39], [0, 0, 0, 0], [-68, 0, 0, 0], [0, 5, 0, 33], [0, 0, 0, 0], [19, 0, -22, 0], [0, 0, 0, 0], [-11, 0, 26, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -45, 0, -31], [-23, 0, 20, 0], [-19, 0, -32, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-31, 0, -30, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -43, 0, -21], [-18, 0, 0, 0], [0, 45, 0, 41], [0, 0, 0, 0], [0, 0, 0, 0], [0, 9, 0, 37], [0, 0, 0, 0], [0, 27, 0, -5], [0, 0, 0, 0], [35, 0, -12, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -28, 0, 28], [-47, 0, -6, 0], [0, -45, 0, -23], [-76, 0, 0, 0], [-1, 0, -32, 0], [0, -19, 0, 15], [0, 0, 0, 0], [0, -34, 0, 34], [90, 0, 0, 0], [0, 0, 0, 0], [0, 27, 0, 47], [0, 32, 0, -32], [43, 0, 26, 0], [0, -9, 0, 25], [0, 0, 0, 0], [0, 0, 0, 0], [0, -49, 0, -39], [0, 0, 0, 0], [0, -16, 0, 16], [0, 0, 0, 0], [90, 0, 0, 0], [0, 0, 0, 0], [-72, 0, 0, 0], [0, -32, 0, 32], [0, 0, 0, 0], [-43, 0, 6, 0], [0, 0, 0, 0], [0, 45, 0, 19], [0, 28, 0, -28], [49, 0, 20, 0], [0, -45, 0, -49], [0, 0, 0, 0], [0, 0, 0, 0], [11, 0, -30, 0], [0, 0, 0, 0], [2, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-10, 0, 0, 0], [0, -35, 0, -51], [0, 0, 0, 0], [-49, 0, -2, 0], [0, -45, 0, -17], [0, -26, 0, 26], [0, 0, 0, 0], [-36, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-25, 0, 24, 0], [0, 20, 0, -20], [0, 0, 0, 0], [22, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 9, 0, 41], [0, 37, 0, 3], [-26, 0, 0, 0], [0, 0, 0, 0], [0, 29, 0, 51], [0, 14, 0, -14], [0, 0, 0, 0], [90, 0, 0, 0], [0, 0, 0, 0], [88, 0, 0, 0], [54, 0, 0, 0], [0, 17, 0, -21], [0, 0, 0, 0], [0, 38, 0, -38], [0, -53, 0, -33], [0, 0, 0, 0], [-34, 0, 0, 0], [0, 0, 0, 0], [0, 27, 0, -11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 45, 0, 13], [-55, 0, -14, 0], [0, 55, 0, 39], [0, 0, 0, 0], [0, 0, 0, 0], [-31, 0, -38, 0], [0, 0, 0, 0], [92, 0, 0, 0], [0, 0, 0, 0], [53, 0, 24, 0], [0, 0, 0, 0], [0, -9, 0, -43], [0, 0, 0, 0], [0, 53, 0, 51], [47, 0, -8, 0], [0, 0, 0, 0], [0, 0, 0, 0], [49, 0, 30, 0], [0, -23, 0, -51], [0, 0, 0, 0], [0, 47, 0, 15], [49, 0, -6, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [46, 0, 0, 0], [0, -40, 0, 40], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 4, 0, -4], [0, -43, 0, -57], [-36, 0, 0, 0], [-31, 0, 24, 0], [0, 0, 0, 0], [0, 10, 0, -10], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 37, 0, 57], [0, 0, 0, 0], [0, 1, 0, 39], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -22, 0, 22], [0, -7, 0, 33], [0, 0, 0, 0], [0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3969_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3969_2_a_v();". // 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_3969_2_a_v(:prec:=4) chi := MakeCharacter_3969_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_3969_2_a_v();". // 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_3969_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3969_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 0, -5, 0, 1]>,<5,R![0, 1]>,<11,R![25, 0, -38, 0, 1]>,<13,R![0, 1]>],Snew); return Vf; end function;