// Make newform 1280.4.d.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1280_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_d_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_1280_4_d_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_1280_4_d_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, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_1280_d();" function MakeCharacter_1280_d() N := 1280; order := 2; char_gens := [511, 261, 257]; v := [2, 1, 2]; // 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_1280_d_Hecke();" function MakeCharacter_1280_d_Hecke(Kf) N := 1280; order := 2; char_gens := [511, 261, 257]; char_values := [[1, 0], [-1, 0], [1, 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 := 4; raw_aps := [[0, 0], [0, 2], [0, -5], [-6, 0], [0, 60], [0, -50], [-30, 0], [0, -40], [-178, 0], [0, -166], [20, 0], [0, 10], [250, 0], [0, 142], [214, 0], [0, 490], [0, -800], [0, -250], [0, 774], [-100, 0], [230, 0], [-1320, 0], [0, -982], [-874, 0], [-310, 0], [0, -1498], [-1402, 0], [0, 1194], [0, -650], [-1510, 0], [1246, 0], [0, -2660], [-2770, 0], [0, -560], [0, -2350], [-580, 0], [0, 1310], [0, -1862], [-726, 0], [0, -3250], [0, 1120], [0, -2842], [3180, 0], [-4670, 0], [0, -2990], [4240, 0], [0, 4060], [-5622, 0], [0, -1554], [0, 1134], [1710, 0], [4440, 0], [-850, 0], [0, -660], [-7590, 0], [-762, 0], [0, 150], [-6580, 0], [0, 4530], [-6950, 0], [0, -3882], [0, 1370], [0, -4106], [-2220, 0], [9430, 0], [0, 6470], [0, 900], [530, 0], [0, -414], [0, -8614], [-2270, 0], [-8080, 0], [2374, 0], [0, 1810], [0, 8120], [-11782, 0], [0, -4350], [0, 7470], [11698, 0], [3650, 0], [0, 1120], [0, 4850], [-12580, 0], [13130, 0], [-8560, 0], [0, -4258], [2550, 0], [6710, 0], [0, 14482], [162, 0], [0, 15974], [-10760, 0], [9266, 0], [0, 2860], [0, -7160], [1398, 0], [0, -7446], [16438, 0], [0, -7322], [0, -10878], [0, -16114], [0, -3690], [0, 2562], [6050, 0], [0, 8260], [-16870, 0], [0, -966], [26290, 0], [11640, 0], [25450, 0], [16694, 0], [0, 15890], [1230, 0], [0, -10840], [14060, 0], [-17650, 0], [0, -27358], [6786, 0], [0, 9030], [0, 15600], [0, 16850], [-7990, 0], [0, 18690], [0, 19182], [0, 23380], [0, -11850], [0, 25646], [-30280, 0], [-17446, 0], [0, 16750], [0, 36560], [30142, 0], [11860, 0], [0, 37010], [11718, 0], [4706, 0], [0, -28670], [0, -20434], [0, -3930], [4854, 0], [0, -13140], [0, 22050], [-14578, 0], [0, -37054], [0, 6150], [8200, 0], [0, -42990], [-32130, 0], [0, 15440], [46938, 0], [0, 31230], [25550, 0], [0, -4318], [-1766, 0], [0, 41906], [25140, 0], [-32920, 0], [10150, 0], [-28530, 0], [0, -9678], [0, -36986], [3350, 0], [-43774, 0], [0, 8740], [-48310, 0], [-2282, 0], [-31580, 0], [0, -2790], [23954, 0], [0, -20830], [0, 1720], [0, 24338], [-58140, 0], [37590, 0], [33280, 0], [-9786, 0], [0, -21980], [0, -21018], [30862, 0], [0, -32650], [27734, 0], [0, -14580], [0, 24010], [8390, 0], [-50502, 0], [0, -49746], [0, 9830], [0, 29122], [42050, 0], [-31180, 0], [-5990, 0], [0, -39138], [0, 58980], [0, -22650], [0, -16034], [32190, 0], [22350, 0], [0, 7590], [-67230, 0], [35478, 0], [0, -44134], [-27540, 0], [0, -4790], [32914, 0], [0, 34560], [0, -64290], [-44720, 0], [0, -49518], [-36874, 0], [0, -18700], [6130, 0], [0, -33950], [-49002, 0], [0, 50794], [-34280, 0], [7850, 0], [20446, 0], [1218, 0], [-24326, 0], [0, -27730], [0, -62750], [-9880, 0], [32946, 0], [-11278, 0], [0, 33246], [0, -16750], [-14890, 0], [-19760, 0], [-76414, 0], [0, 63780], [0, -71250], [0, 47480], [70300, 0], [-26550, 0], [0, 87718], [-46834, 0], [52950, 0], [0, -32790], [0, -16760], [-58380, 0], [0, 6778], [0, -107700], [49742, 0], [0, 8314], [33170, 0], [106080, 0], [-45874, 0], [0, 82020], [0, -71520], [71682, 0], [0, 64350], [402, 0], [-11694, 0], [-18750, 0], [0, 87710], [0, 67560], [0, 34438], [0, -2254], [0, -24030], [46550, 0], [23738, 0], [0, 58526], [0, 12494], [0, 75990], [-20190, 0], [0, 126920], [0, 38346], [-93962, 0], [0, -8578], [0, 85530], [0, 93378], [0, -68986], [-22130, 0], [78720, 0], [17850, 0], [-56018, 0], [0, -63134], [0, 29226], [106650, 0], [0, -74100], [-104822, 0], [-114460, 0], [-135214, 0], [0, 104118], [0, 56874], [42580, 0], [-46270, 0], [0, 99890], [-118840, 0], [92950, 0], [0, -42650], [0, -68994], [87270, 0], [0, -107140], [0, 84830], [0, 14950], [49300, 0], [0, -156470], [0, 13320], [0, 44934], [72110, 0], [0, -34530], [58280, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1280_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1280_4_d_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_1280_4_d_f(:prec:=2) chi := MakeCharacter_1280_d(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_1280_4_d_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_1280_4_d_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1280_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![4, 0, 1]>,<7,R![6, 1]>,<11,R![3600, 0, 1]>],Snew); return Vf; end function;