// Make newform 525.4.d.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_525_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_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_525_4_d_j();". // 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_525_4_d_j();". // 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 := [100, 0, 21, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 11, 0, 1], [11, 0, 1, 0]]; Rf_basisdens := [1, 1, 10, 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_525_d();" function MakeCharacter_525_d() N := 525; order := 2; char_gens := [176, 127, 451]; 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_525_d_Hecke();" function MakeCharacter_525_d_Hecke(Kf) N := 525; order := 2; char_gens := [176, 127, 451]; char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 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 := 4; raw_aps := [[0, 1, -1, 0], [0, 0, 3, 0], [0, 0, 0, 0], [0, 0, -7, 0], [30, 0, 0, 2], [0, 10, 2, 0], [0, -12, -26, 0], [62, 0, 0, -2], [0, 48, 32, 0], [-182, 0, 0, 12], [14, 0, 0, 38], [0, 80, 134, 0], [-46, 0, 0, 108], [0, -100, -248, 0], [0, 140, 164, 0], [0, -58, 462, 0], [-296, 0, 0, 76], [-482, 0, 0, 84], [0, -228, 64, 0], [198, 0, 0, -86], [0, 38, 182, 0], [-912, 0, 0, -8], [0, 224, -228, 0], [-566, 0, 0, 336], [0, 278, 474, 0], [950, 0, 0, -348], [0, -192, -48, 0], [0, 340, -356, 0], [-898, 0, 0, -252], [0, 254, -718, 0], [0, -520, -816, 0], [380, 0, 0, 72], [0, 134, 534, 0], [870, 0, 0, 390], [454, 0, 0, -344], [668, 0, 0, -36], [0, -614, 94, 0], [0, 316, -1968, 0], [0, 300, 964, 0], [0, -436, -3038, 0], [306, 0, 0, 574], [-3382, 0, 0, 184], [2434, 0, 0, 318], [0, 48, -4598, 0], [0, -370, 1434, 0], [-2174, 0, 0, 714], [1100, 0, 0, 712], [0, 4, 832, 0], [0, 416, -2116, 0], [2162, 0, 0, 1148], [0, -354, -1318, 0], [-694, 0, 0, -586], [-1026, 0, 0, -1172], [-1552, 0, 0, 1924], [0, 152, -2506, 0], [0, -436, -6288, 0], [1354, 0, 0, 1396], [5482, 0, 0, -830], [0, -368, 3214, 0], [1678, 0, 0, 1124], [0, -1176, -468, 0], [0, 1220, 482, 0], [0, 28, 4724, 0], [4104, 0, 0, -952], [0, -1398, -2338, 0], [0, 1298, 794, 0], [3588, 0, 0, -2776], [0, 976, -8506, 0], [0, -1296, 1564, 0], [5914, 0, 0, 1036], [0, 1148, -7678, 0], [-1026, 0, 0, -2654], [0, -2092, -5816, 0], [0, -1548, 4722, 0], [-7704, 0, 0, -236], [0, 228, 8012, 0], [390, 0, 0, -2600], [0, -3210, -2946, 0], [5738, 0, 0, 1464], [8198, 0, 0, -208], [-2212, 0, 0, -2008], [13294, 0, 0, -912], [-7122, 0, 0, -2486], [0, 418, -6298, 0], [-7982, 0, 0, -2798], [0, 2256, -7708, 0], [-890, 0, 0, 480], [0, -2060, -1826, 0], [2698, 0, 0, -2656], [0, 3224, 5752, 0], [0, 3520, 6884, 0], [-808, 0, 0, 3728], [0, 3272, 4024, 0], [-6090, 0, 0, 322], [-15120, 0, 0, 1300], [0, -3084, -6788, 0], [578, 0, 0, -3988], [7890, 0, 0, -4148], [0, -2108, -10948, 0], [5102, 0, 0, -1360], [0, -700, 11104, 0], [0, -1382, -12966, 0], [0, 3080, 6452, 0], [2302, 0, 0, -2152], [5624, 0, 0, -12], [0, -1130, 5354, 0], [0, 528, 15044, 0], [0, -2788, -6038, 0], [-13874, 0, 0, -3766], [-1282, 0, 0, 5924], [0, 1740, -8016, 0], [0, -4544, 1162, 0], [0, 1834, -12826, 0], [14110, 0, 0, 1350], [3280, 0, 0, -7848], [-1466, 0, 0, -132], [0, -7732, 1492, 0], [0, -5548, -16276, 0], [0, 2658, -738, 0], [-450, 0, 0, 1090], [-16522, 0, 0, 6164], [0, -1716, -5598, 0], [0, 3556, -2506, 0], [0, -3704, -25268, 0], [-8610, 0, 0, 3662], [6598, 0, 0, -5916], [27794, 0, 0, 136], [-14812, 0, 0, 1692], [0, -4164, -3096, 0], [0, -3726, -12558, 0], [2560, 0, 0, -4700], [0, 7596, -7448, 0], [15088, 0, 0, 304], [0, -5596, -4786, 0], [-14194, 0, 0, -4304], [6866, 0, 0, 7884], [0, 3092, -17998, 0], [0, 8060, 3444, 0], [0, -5588, -16946, 0], [-26422, 0, 0, 6212], [-7710, 0, 0, 8322], [-23082, 0, 0, 6364], [0, -160, 24592, 0], [0, -4900, 20804, 0], [9278, 0, 0, 352], [-11540, 0, 0, -1020], [0, 4322, 6922, 0], [0, -840, -39506, 0], [-18830, 0, 0, -3670], [0, -3928, -20688, 0], [0, 888, 17574, 0], [-370, 0, 0, 4792], [0, -1420, -10528, 0], [0, 5676, 18404, 0], [0, 8060, 11224, 0], [37866, 0, 0, -1194], [-40080, 0, 0, -440], [-31494, 0, 0, -4736], [0, 4210, -5926, 0], [-4758, 0, 0, 2400], [0, -8416, 5404, 0], [0, 14598, 17402, 0], [0, -3472, 22664, 0], [-17800, 0, 0, 8092], [0, 13794, 734, 0], [0, -4500, 34532, 0], [-23864, 0, 0, 11576], [0, 15318, 15094, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_525_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_525_4_d_j();". // 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_525_4_d_j(:prec:=4) chi := MakeCharacter_525_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(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_525_4_d_j();". // 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_525_4_d_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_525_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![64, 0, 25, 0, 1]>,<11,R![920, -62, 1]>],Snew); return Vf; end function;