// Make newform 315.2.i.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_315_i();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_315_i_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_315_2_i_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_315_2_i_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 poly := [729, -243, 162, -27, -36, 60, -38, 20, -4, -1, 2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7533, 15228, -12906, 45, -2973, 281, 865, -389, 301, -152, 10, -25], [-4860, -2349, 351, 1305, 822, -311, 110, -49, -46, -61, -25, 13], [-1458, 2997, -2673, 1629, -702, 79, 68, -181, 110, -49, 17, -5], [-6318, 4455, -4833, 2115, -672, -293, 272, -211, 116, -61, 11, -5], [-243, -3969, 5157, -4356, 2085, -946, 13, 316, -191, 157, -77, 8], [-2511, 2349, -1575, 774, -199, 8, 141, -102, 51, -21, 7, -2], [-7047, 7857, -3051, 2520, -1419, -284, 353, -292, 197, -61, 29, -14], [243, -162, 27, 36, -60, 38, -20, 4, 1, -2, 1, -1], [1215, -1863, 1269, -936, 483, -134, -37, 56, -67, 35, -13, 4], [-12150, 7371, -5805, 2223, -3048, 1411, -424, -331, 248, -217, 107, -65]]; Rf_basisdens := [1, 1, 17496, 8748, 2916, 2916, 4374, 972, 4374, 243, 972, 8748]; 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_315_i();" function MakeCharacter_315_i() N := 315; order := 3; char_gens := [281, 127, 136]; v := [2, 3, 3]; // 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_315_i_Hecke();" function MakeCharacter_315_i_Hecke(Kf) N := 315; order := 3; char_gens := [281, 127, 136]; char_values := [[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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 := [[1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, -1, 2, 1, 0, -2, 0, -1, 0, 0, 0, -1], [1, -1, -2, -1, -3, 1, 0, 1, 0, 0, -1, 2], [0, -1, 0, -2, 0, 1, 0, -1, 0, -1, -1, -1], [2, -2, 0, -2, 0, 0, -1, -1, -1, -2, -1, 0], [-1, 0, 0, 0, 1, 0, -1, 0, 2, 1, 1, 0], [-2, 0, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0], [0, 1, 2, 1, 1, -1, 1, -1, -2, -1, 0, -2], [2, 4, 0, 0, 0, 1, 0, 2, 0, 4, 2, -1], [0, -1, 0, 1, -2, 1, -1, 1, 2, 0, -1, 2], [-1, 0, -2, -1, 2, 2, 2, 2, -1, 2, -2, 1], [2, 0, -2, -1, -3, 2, 0, -1, 0, -1, 1, 1], [2, -4, 0, -2, 0, -1, -1, -1, -1, -4, -1, 1], [0, 0, -2, -3, 4, -1, 1, 0, -2, 0, 0, -2], [-2, 2, 0, 0, 0, 0, -2, 2, 1, 0, 0, 0], [2, 2, 0, 0, 1, 0, 2, -2, -4, -1, 1, 0], [-5, 2, 0, 2, 0, 1, 0, 2, 0, 2, 2, -1], [1, 4, 0, 2, 0, 1, -1, 1, -1, 4, 1, -1], [6, 1, -2, 0, -2, 0, 4, 3, -2, 2, -2, 0], [1, 0, 2, 0, 1, 0, 2, -1, -1, -1, 1, 0], [-1, 4, 0, 0, 0, 0, 0, 3, 0, 4, 3, 0], [-7, 0, 2, 0, 5, 0, -2, -2, 1, -2, 2, 0], [2, -1, 0, 1, -1, -2, 0, 0, 0, 1, -1, -1], [-1, 0, 0, 0, -4, 0, -1, 0, 2, -2, -2, 0], [3, 0, 0, -2, 0, -1, -2, -1, -2, 0, -1, 1], [-5, 0, 0, -2, 0, 1, 0, 2, 0, 0, 2, -1], [-1, 2, -2, -3, -3, -1, 0, -2, 0, 1, 3, -2], [5, -1, 0, 4, 0, 2, 1, 0, 1, -1, 0, -2], [-1, 7, 6, 5, 4, -1, 0, -7, 0, -4, 3, -2], [-8, 3, -2, -3, -1, 6, -6, 1, 3, -2, 2, 3], [-3, -2, 6, 3, 5, -3, 0, 2, 0, -1, -3, -6], [1, -7, -2, -2, -2, 0, 1, 7, -2, 5, -2, 0], [-1, -1, -6, -1, 2, 2, 2, 4, -1, 5, -5, 1], [0, 4, -4, -4, -10, 0, 0, -4, 0, -3, 1, 0], [-9, -1, 0, 2, 0, -3, 1, 0, 1, -1, 0, 3], [-4, 4, 0, -2, -8, -2, -2, -4, 4, -1, 3, -4], [0, 1, 2, 1, -1, -2, -2, -4, 1, -5, 5, -1], [-5, -1, 0, -2, 0, 2, -1, -3, -1, -1, -3, -2], [12, -3, 0, 4, 0, -4, -1, -1, -1, -3, -1, 4], [-2, 2, 0, 0, 4, 0, 2, 2, -1, 0, 0, 0], [-2, 3, 6, 6, 6, 0, -2, -3, 4, -4, -1, 0], [7, 1, 0, 10, 0, -3, 2, 1, 2, 1, 1, 3], [0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2], [-3, -5, -2, -3, -2, -1, -2, 5, 4, 3, -2, -2], [-5, -1, -4, 1, 4, -2, -2, 4, 1, 5, -5, -1], [-9, 2, 10, 2, 9, -4, -2, -3, 1, -5, 5, -2], [2, 1, -2, 1, -6, 3, -1, -1, 2, -4, -3, 6], [6, 1, 0, 4, 0, -2, 0, 6, 0, 1, 6, 2], [3, -4, -10, -8, -13, 2, 1, 4, -2, -1, -5, 4], [-4, 5, -4, -1, 1, 2, -2, 4, 1, -1, 1, 1], [7, 1, 0, 4, 0, -6, 2, 1, 2, 1, 1, 6], [-3, 5, 2, 0, 3, -2, -1, -5, 2, 0, 5, -4], [-5, -6, 2, -1, -2, 2, -6, -1, 3, 5, -5, 1], [8, -4, 0, 6, 0, -2, 2, -2, 2, -4, -2, 2], [1, 1, 0, -6, 0, 4, -3, 0, -3, 1, 0, -4], [-6, 1, -2, -3, 1, 6, -2, 3, 1, 2, -2, 3], [-4, 5, 2, -4, -2, 8, -2, 6, 1, 1, -1, 4], [1, 0, 0, -1, 2, -1, 2, 0, -4, -5, -5, -2], [-2, 6, 4, 4, 0, 0, -2, -6, 4, 2, 8, 0], [7, -8, 0, -2, 0, 0, 2, -6, 2, -8, -6, 0], [2, -3, -8, -8, 0, 0, 2, 3, -4, 2, -1, 0], [-11, -3, 6, 0, 5, 0, -6, -3, 3, 0, 0, 0], [1, 2, -8, -1, 0, 2, 2, 3, -1, 1, -1, 1], [1, 2, 4, 0, -5, 0, -4, -4, 2, -6, 6, 0], [1, -1, -4, -2, -17, 2, -1, 1, 2, 3, 2, 4], [0, 4, 0, 1, -8, 1, -1, -4, 2, -4, 0, 2], [10, 3, -8, 1, -1, -2, 8, 5, -4, 2, -2, -1], [-11, -3, 2, 0, 15, 0, 4, -5, -2, -2, 2, 0], [6, -6, 0, -2, 0, 2, 1, -7, 1, -6, -7, -2], [0, 5, 8, -2, 0, 4, 2, 0, -1, -5, 5, 2], [-4, -2, 12, 11, -4, -1, -3, 2, 6, 8, 6, -2], [-10, 7, 0, -6, 0, 2, 2, 1, 2, 7, 1, -2], [9, -10, -12, -7, -8, 5, 4, 10, -8, -3, -13, 10], [4, -2, 8, 5, 3, -10, 2, -6, -1, -4, 4, -5], [12, 3, 0, 14, 0, -1, 6, 4, 6, 3, 4, 1], [2, -1, 6, 4, 7, -2, 4, 1, -8, 3, 2, -4], [-3, 5, 4, 4, -12, 0, -3, -5, 6, 0, 5, 0], [-4, 1, 2, 1, 2, -1, -3, -1, 6, 4, 5, -2], [-5, -3, 6, -1, 6, 2, 2, -5, -1, -2, 2, 1], [-7, 11, 0, 8, 0, 4, 4, 8, 4, 11, 8, -4], [-7, 2, 0, -6, 0, 4, -2, -1, -2, 2, -1, -4], [2, 1, -8, -5, -1, 10, 6, 1, -3, 0, 0, 5], [0, -1, -6, 1, 5, -2, 4, 6, -2, 7, -7, -1], [-4, -6, 0, 6, 0, -3, 0, -2, 0, -6, -2, 3], [5, 2, 4, 1, -2, -2, 2, -7, -1, -9, 9, -1], [-11, -5, 2, 2, 13, -4, 0, 1, 0, 6, -6, -2], [2, 1, 2, 6, 4, 4, -2, -1, 4, 2, 3, 8], [8, 0, 0, -2, 0, -1, -2, 3, -2, 0, 3, 1], [-3, -10, 4, 1, 4, -2, 0, -8, 0, 2, -2, -1], [-6, 5, 0, -2, 0, -1, -2, 8, -2, 5, 8, 1], [4, 1, -4, -5, -3, -1, 5, -1, -10, -4, -3, -2], [0, 0, 2, 0, 2, -2, 2, 0, -4, 5, 5, -4], [-10, 2, 0, -4, 0, 2, 2, 0, 2, 2, 0, -2], [-3, 4, 2, 0, 17, -2, -1, -4, 2, 1, 5, -4], [8, -1, 0, 8, 0, 1, -2, 2, -2, -1, 2, -1], [9, 0, 0, 0, 0, 1, 2, -2, 2, 0, -2, -1], [-8, 3, 0, -10, 0, 4, -2, 1, -2, 3, 1, -4], [1, -3, 14, 4, 3, -8, 0, -7, 0, -4, 4, -4], [-1, -4, 0, -8, 0, -1, 2, -1, 2, -4, -1, 1], [-3, 5, 4, 0, -12, -4, 1, -5, -2, -5, 0, -8], [8, -7, -2, -2, -10, 4, 0, -13, 0, -6, 6, 2], [4, -11, -12, -9, 1, 3, 1, 11, -2, -2, -13, 6], [5, 7, 0, 2, 0, 1, -1, 9, -1, 7, 9, -1], [-4, 11, -6, -3, 1, 6, 0, 5, 0, -6, 6, 3], [-7, 3, 0, -2, 0, -1, -5, 6, -5, 3, 6, 1], [-3, -2, 14, 12, 10, -2, -1, 2, 2, 3, 1, -4], [-6, 3, 4, -3, -7, 6, -10, -1, 5, -4, 4, 3], [-1, -2, 6, 2, 5, -4, 3, 2, -6, 5, 3, -8], [5, -8, 0, -12, 0, 1, 3, 2, 3, -8, 2, -1], [4, 0, -14, -13, -2, 1, 3, 0, -6, -12, -12, 2], [2, -8, -6, -6, -10, 12, -2, -1, 1, 7, -7, 6], [-9, -13, 0, 4, 0, -2, 2, -6, 2, -13, -6, 2], [12, -5, -6, -5, -11, 10, 6, -6, -3, -1, 1, 5], [11, -6, -10, -5, 1, 5, 6, 6, -12, -2, -8, 10], [9, -6, 0, -4, 0, -2, -1, -8, -1, -6, -8, 2], [-3, 7, 6, 2, 3, -4, 1, -7, -2, -5, 2, -8], [-18, 9, 2, 0, 12, 0, -6, 4, 3, -5, 5, 0], [-2, 0, -4, -3, -2, 1, -3, 0, 6, -2, -2, 2], [13, 4, -12, -3, -14, 6, 2, 7, -1, 3, -3, 3], [13, -1, 8, 7, 2, -14, 8, -2, -4, -1, 1, -7], [-3, -9, 0, -16, 0, 1, -7, -8, -7, -9, -8, -1], [-8, -4, -2, 0, 10, 0, 2, 11, -1, 15, -15, 0], [18, 6, 0, -4, 0, -1, 0, 5, 0, 6, 5, 1], [-17, 6, 4, -3, 18, 6, 4, 3, -2, -3, 3, 3], [5, -3, 0, -10, 0, -1, -7, -10, -7, -3, -10, 1], [6, -5, -12, 2, -8, -4, -4, -5, 2, 0, 0, -2], [-5, 0, 6, 2, -13, -4, -1, 0, 2, 1, 1, -8], [4, -5, 0, 4, 0, 0, -4, 0, -4, -5, 0, 0], [-2, 5, 6, 9, 5, 3, -5, -5, 10, -5, 0, 6], [1, -7, 4, 5, 12, 1, 0, 7, 0, 6, -1, 2], [-7, 7, 0, 0, 0, -2, 3, 2, 3, 7, 2, 2], [5, -6, 2, 8, -2, 6, -1, 6, 2, 2, -4, 12], [6, -7, -18, -12, -6, 6, 0, 7, 0, -2, -9, 12], [8, 1, 0, 6, 0, 1, 4, 1, 4, 1, 1, -1], [5, -6, -2, 6, -11, 8, -3, 6, 6, 8, 2, 16], [2, -11, -8, -8, -6, 0, 2, 11, -4, 3, -8, 0], [-19, -6, 0, 0, 0, -4, 2, -6, 2, -6, -6, 4], [5, -7, 0, 14, 0, -4, -1, -4, -1, -7, -4, 4], [-1, -4, 4, 0, 3, 0, 2, -6, -1, -2, 2, 0], [0, 0, 8, 8, 7, 0, 0, 0, 0, 1, 1, 0], [-24, 5, 0, -2, 0, 6, 0, 2, 0, 5, 2, -6], [28, -3, 0, 4, 0, 0, 1, -9, 1, -3, -9, 0], [13, -7, -6, -2, -19, 4, -4, -3, 2, 4, -4, 2], [-14, 5, -2, -1, 17, 2, 4, 0, -2, -5, 5, 1], [-1, -3, 0, 3, 2, -6, -2, -1, 1, 2, -2, -3], [-2, -2, 4, 7, -5, 3, -5, 2, 10, 3, 1, 6], [-10, -15, 0, 2, 0, -3, -2, -8, -2, -15, -8, 3], [-4, 1, -6, -13, 5, -7, 3, -1, -6, -1, 0, -14], [-9, -10, 0, -10, 0, 1, -3, -12, -3, -10, -12, -1], [-4, 13, 0, 12, 0, 5, 2, 4, 2, 13, 4, -5], [2, 8, 6, 8, -7, 2, 0, -8, 0, -5, 3, 4], [-4, 1, -4, -5, -3, 10, -2, -2, 1, -3, 3, 5], [17, -2, 0, 2, -15, -4, 0, -4, 0, -2, 2, -2], [13, -1, 0, 22, 0, -6, 6, -3, 6, -1, -3, 6], [12, 4, -16, -6, -30, 12, -12, 9, 6, 5, -5, 6], [1, -9, 0, 2, 0, -5, 1, 1, 1, -9, 1, 5], [-4, -6, 6, 0, 3, -6, 2, 6, -4, 3, -3, -12], [9, 6, -14, -1, -14, 2, -4, 9, 2, 3, -3, 1], [-8, -2, 0, -8, 0, -2, 1, 3, 1, -2, 3, 2], [-1, 2, 10, 8, -1, -2, 1, -2, -2, 11, 13, -4], [-11, -2, 0, -10, 0, -2, 1, -8, 1, -2, -8, 2], [-8, 3, 18, 16, -7, -2, -6, -3, 12, 11, 14, -4], [-11, 0, 6, 2, 11, -4, -2, 5, 1, 5, -5, -2], [-6, 5, 0, 10, 0, 3, 3, -2, 3, 5, -2, -3], [-11, -2, 12, 11, 32, -22, 10, -10, -5, -8, 8, -11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_315_i_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_315_2_i_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_315_2_i_e(:prec:=12) chi := MakeCharacter_315_i(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_315_2_i_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_315_2_i_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_315_i(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![441, -756, 1401, -954, 1144, -702, 628, -255, 135, -33, 16, -3, 1]>],Snew); return Vf; end function;