// Make newform 320.3.h.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_320_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_320_h_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_320_3_h_f();". // 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_320_3_h_f();". // 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, 14, 0, 9, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [23, 0, 20, 0, 2, 0], [0, 76, 0, 40, 0, 4], [7, -92, -10, -70, -2, -8], [21, -92, 30, -70, 4, -8], [0, 224, 0, 140, 0, 16]]; Rf_basisdens := [1, 5, 5, 5, 5, 5]; 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_320_h();" function MakeCharacter_320_h() N := 320; order := 2; char_gens := [191, 261, 257]; v := [1, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_320_h_Hecke();" function MakeCharacter_320_h_Hecke(Kf) N := 320; order := 2; char_gens := [191, 261, 257]; char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 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 := 3; raw_aps := [[0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [2, 0, 0, 1, -1, 0], [-2, -2, 1, 2, 2, 0], [0, 0, 0, 0, 0, 1], [1, 1, -4, -1, -1, -1], [0, 0, -1, 0, 0, 2], [-12, -2, 0, 3, -3, 0], [-10, -8, 0, 0, 0, 0], [-2, -2, 4, 2, 2, 2], [1, 1, 8, -1, -1, -2], [-13, -5, 0, -1, 1, 0], [13, 1, 0, 6, -6, 0], [42, -8, 0, 1, -1, 0], [8, 8, -8, -8, -8, 3], [0, 0, -15, 0, 0, -2], [17, 1, 0, -7, 7, 0], [-47, 13, 0, 2, -2, 0], [6, 6, 2, -6, -6, -2], [-1, -1, 4, 1, 1, 3], [6, 6, -24, -6, -6, 2], [33, -3, 0, -14, 14, 0], [10, -8, 0, -16, 16, 0], [5, 5, -12, -5, -5, -1], [-58, -8, 0, 16, -16, 0], [20, -2, 0, 3, -3, 0], [-61, -21, 0, -20, 20, 0], [-39, -7, 0, -7, 7, 0], [12, 12, 0, -12, -12, -8], [-20, -26, 0, 3, -3, 0], [-12, -12, 3, 12, 12, -10], [2, 2, -8, -2, -2, -4], [6, 6, -31, -6, -6, 8], [-7, -23, 0, 17, -17, 0], [-12, -12, 26, 12, 12, 8], [3, 3, -32, -3, -3, -6], [-141, -21, 0, 20, -20, 0], [-16, 34, 0, -1, 1, 0], [-13, -13, -8, 13, 13, -10], [14, 14, -29, -14, -14, 12], [-10, 72, 0, -16, 16, 0], [20, 20, -8, -20, -20, 12], [-13, -13, -20, 13, 13, 1], [6, 6, -32, -6, -6, 1], [-34, -34, -14, 34, 34, 6], [-12, -12, 39, 12, 12, 14], [72, -54, 0, 31, -31, 0], [123, 35, 0, -4, 4, 0], [54, -24, 0, 0, 0, 0], [31, 31, -28, -31, -31, -5], [6, 6, 0, -6, -6, 2], [139, -29, 0, 7, -7, 0], [-6, -6, -27, 6, 6, -4], [-28, -28, 56, 28, 28, -12], [-18, 20, 0, -3, 3, 0], [-111, 65, 0, -7, 7, 0], [-6, -6, 32, 6, 6, 14], [-35, -35, -24, 35, 35, -2], [-213, -13, 0, -33, 33, 0], [-389, -13, 0, -12, 12, 0], [19, 19, -32, -19, -19, 20], [253, 1, 0, -2, 2, 0], [-12, -12, -46, 12, 12, -24], [10, 10, -72, -10, -10, 8], [-14, -14, 0, 14, 14, -5], [-44, -44, 13, 44, 44, -14], [-26, -26, 16, 26, 26, -26], [165, 41, 0, 14, -14, 0], [118, 56, 0, 48, -48, 0], [-18, -18, 24, 18, 18, 34], [-20, -20, 54, 20, 20, -24], [-108, -34, 0, -21, 21, 0], [-31, -31, 72, 31, 31, 18], [34, 34, 21, -34, -34, 4], [-116, -122, 0, -45, 45, 0], [281, 9, 0, -47, 47, 0], [54, 54, -64, -54, -54, -21], [266, -24, 0, 16, -16, 0], [179, 91, 0, -17, 17, 0], [26, 26, 103, -26, -26, -8], [-439, -71, 0, -15, 15, 0], [-6, -6, -104, 6, 6, 14], [3, 3, -92, -3, -3, 13], [-52, -52, 98, 52, 52, -16], [351, -17, 0, -8, 8, 0], [-293, -77, 0, -9, 9, 0], [80, 80, -16, -80, -80, -26], [470, -8, 0, 16, -16, 0], [306, 112, 0, 25, -25, 0], [-361, -25, 0, 56, -56, 0], [-40, -40, 88, 40, 40, -24], [62, -124, 0, 45, -45, 0], [16, 16, -131, -16, -16, 6], [-86, -86, 75, 86, 86, 0], [-252, -18, 0, 35, -35, 0], [-42, -136, 0, 0, 0, 0], [-350, 80, 0, 48, -48, 0], [-281, -89, 0, 48, -48, 0], [230, 40, 0, 80, -80, 0], [-225, -161, 0, -16, 16, 0], [-25, -25, 48, 25, 25, -22], [99, 59, 0, 4, -4, 0], [131, -117, 0, -65, 65, 0], [-26, -26, 61, 26, 26, 32], [-78, -78, -24, 78, 78, 22], [-523, 89, 0, -34, 34, 0], [12, 12, 152, -12, -12, 24], [-26, -26, 126, 26, 26, 54], [-229, 67, 0, -33, 33, 0], [-54, 184, 0, 17, -17, 0], [-14, -14, 24, 14, 14, -59], [13, 13, -44, -13, -13, 13], [-40, -40, 61, 40, 40, -50], [-52, -52, 18, 52, 52, -32], [347, -13, 0, 39, -39, 0], [53, 73, 0, -58, 58, 0], [-366, -208, 0, -63, 63, 0], [-78, -78, 56, 78, 78, -9], [26, 26, 51, -26, -26, -16], [-463, -207, 0, -31, 31, 0], [-87, -87, 4, 87, 87, -37], [72, 72, 96, -72, -72, -9], [-111, -99, 0, 106, -106, 0], [64, 64, -9, -64, -64, -14], [-119, -311, 0, -55, 55, 0], [118, -120, 0, -16, 16, 0], [-34, -34, 120, 34, 34, 42], [214, 12, 0, -75, 75, 0], [65, 65, 200, -65, -65, 4], [22, 22, 31, -22, -22, -20], [-342, 216, 0, -7, 7, 0], [76, 76, 60, -76, -76, -36], [35, 35, -216, -35, -35, -48], [-438, 40, 0, -112, 112, 0], [2, -16, 0, -48, 48, 0], [-32, -32, 40, 32, 32, 15], [569, 37, 0, -118, 118, 0], [48, 48, -216, -48, -48, 45], [418, 144, 0, -80, 80, 0], [98, 98, 49, -98, -98, -36], [-399, -47, 0, 177, -177, 0], [184, 58, 0, -41, 41, 0], [-343, -59, 0, -126, 126, 0], [769, 17, 0, -87, 87, 0], [100, 100, 14, -100, -100, -48], [-24, -24, 208, 24, 24, 51], [-108, -108, 144, 108, 108, -66], [72, 72, -183, -72, -72, 22], [-114, -268, 0, 133, -133, 0], [-27, -27, 48, 27, 27, 92], [547, -37, 0, 55, -55, 0], [-441, 183, 0, -56, 56, 0], [-502, -40, 0, 41, -41, 0], [1195, 67, 0, 68, -68, 0], [-58, -58, -8, 58, 58, -78], [-26, -26, -50, 26, 26, -10], [75, 323, 0, 7, -7, 0], [-45, -45, 92, 45, 45, -17], [390, -120, 0, -48, 48, 0], [401, -243, 0, -14, 14, 0], [-24, -24, -16, 24, 24, 18], [-88, -22, 0, -25, 25, 0], [-184, -184, 69, 184, 184, 30], [27, 27, 4, -27, -27, 13], [846, -44, 0, -3, 3, 0], [-200, -200, 56, 200, 200, -8], [72, 72, -280, -72, -72, -1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_320_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_320_3_h_f();". // 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_320_3_h_f(:prec:=6) chi := MakeCharacter_320_h(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_320_3_h_f();". // 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_320_3_h_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_320_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<3,R![8, -16, 2, 1]>],Snew); return Vf; end function;