// Make newform 1200.4.f.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1200_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_f_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_1200_4_f_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_1200_4_f_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 := [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_1200_f();" function MakeCharacter_1200_f() N := 1200; order := 2; char_gens := [751, 901, 401, 577]; v := [2, 2, 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_1200_f_Hecke();" function MakeCharacter_1200_f_Hecke(Kf) N := 1200; order := 2; char_gens := [751, 901, 401, 577]; char_values := [[1, 0], [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, -3], [0, 0], [0, 16], [-12, 0], [0, -38], [0, -126], [20, 0], [0, 168], [-30, 0], [88, 0], [0, 254], [42, 0], [0, -52], [0, 96], [0, -198], [-660, 0], [-538, 0], [0, -884], [-792, 0], [0, -218], [-520, 0], [0, -492], [-810, 0], [0, 1154], [-618, 0], [0, 128], [0, 1476], [-1190, 0], [0, 462], [0, 2536], [-2292, 0], [0, -726], [380, 0], [-1590, 0], [-2432, 0], [0, 614], [0, -1852], [0, 2136], [0, -1758], [-540, 0], [1982, 0], [2688, 0], [0, 2302], [0, 4374], [-1600, 0], [-3332, 0], [0, 2648], [0, -2244], [5650, 0], [0, -4698], [-1200, 0], [-718, 0], [-6012, 0], [0, -2046], [0, -6072], [6930, 0], [-1352, 0], [0, -1186], [2442, 0], [0, 2828], [0, -4758], [0, 8476], [-4632, 0], [0, 4822], [0, -3426], [2788, 0], [0, 434], [0, -6684], [-2630, 0], [0, 7422], [-10440, 0], [0, -10424], [0, -3278], [6140, 0], [0, -3072], [-6150, 0], [0, -106], [-1758, 0], [3670, 0], [-9660, 0], [8462, 0], [-9792, 0], [0, 7342], [10640, 0], [0, -17412], [1710, 0], [0, -646], [-6018, 0], [0, -6712], [0, -5364], [9840, 0], [0, -1424], [4548, 0], [6500, 0], [0, 12168], [21090, 0], [-5238, 0], [0, 8588], [3062, 0], [0, 8476], [0, -12546], [0, -12], [-19290, 0], [12148, 0], [0, -10366], [0, -7644], [0, -8658], [25800, 0], [16202, 0], [0, 24136], [0, 4642], [0, -6726], [-21220, 0], [-29792, 0], [-10158, 0], [0, 29828], [0, -1944], [0, -26718], [4260, 0], [22862, 0], [0, 32542], [0, 14214], [0, -7092], [13228, 0], [28062, 0], [27250, 0], [-14400, 0], [0, -17984], [0, -16598], [1460, 0], [0, -30072], [18088, 0], [0, 24734], [-22278, 0], [-16130, 0], [0, -29718], [0, -9524], [0, -33906], [630, 0], [20788, 0], [-43098, 0], [0, -14272], [0, -13644], [2410, 0], [23160, 0], [0, -32078], [0, -14406], [30620, 0], [0, 17568], [0, -21706], [-14958, 0], [0, -32812], [0, 38856], [0, 28276], [-8112, 0], [-26080, 0], [-49170, 0], [0, 48314], [34782, 0], [0, 25116], [0, 15462], [0, 736], [29268, 0], [0, 16674], [0, -31272], [15928, 0], [0, 42014], [-12530, 0], [0, 19722], [-18420, 0], [-53818, 0], [-13992, 0], [0, 28582], [-19240, 0], [8310, 0], [-39692, 0], [26982, 0], [0, -17152], [31210, 0], [0, -61544], [-24132, 0], [0, -9038], [0, 28314], [0, 32208], [44970, 0], [0, 51734], [0, -47212], [21670, 0], [-29952, 0], [0, -58178], [0, -23172], [-55172, 0], [9102, 0], [0, -1044], [0, 59142], [-5038, 0], [0, -3638], [0, 1794], [0, 74088], [-14910, 0], [38968, 0], [0, -38626], [37150, 0], [37260, 0], [0, 11934], [21080, 0], [0, 15108], [59910, 0], [-56972, 0], [0, 6194], [-87978, 0], [0, -17872], [0, -64524], [92760, 0], [61562, 0], [0, 8296], [49602, 0], [0, 75816], [0, 15762], [-60658, 0], [-29680, 0], [-46770, 0], [0, -47032], [0, -59604], [-85070, 0], [0, -59178], [-12240, 0], [0, -62624], [95748, 0], [0, 14362], [10100, 0], [28168, 0], [68442, 0], [0, -46132], [0, 40416], [37870, 0], [0, 75402], [60, 0], [12168, 0], [0, 64308], [89908, 0], [0, 62768], [44170, 0], [0, 69582], [-13560, 0], [0, -84584], [-8052, 0], [-39940, 0], [0, -6192], [0, 63254], [23202, 0], [0, 69096], [-58970, 0], [0, 55122], [71460, 0], [542, 0], [0, -94124], [0, 46134], [0, -119446], [0, -80872], [0, -62004], [-11630, 0], [0, -86438], [0, -122286], [-15820, 0], [-132030, 0], [106122, 0], [0, -120292], [0, -27078], [53462, 0], [0, -14564], [0, -54218], [-11320, 0], [0, 42194], [0, 29648], [0, 24996], [33370, 0], [-20518, 0], [-83172, 0], [0, 82608], [-65792, 0], [0, 70296], [13502, 0], [0, 6676], [97488, 0], [0, -30098], [0, -23466], [-125440, 0], [19710, 0], [-57858, 0], [0, 107676], [0, -114378], [70788, 0], [0, 25162], [-66030, 0], [2248, 0], [0, -45318], [-106020, 0], [0, 40156], [0, -49178], [0, 81054], [78680, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1200_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1200_4_f_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_1200_4_f_j(:prec:=2) chi := MakeCharacter_1200_f(); 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_1200_4_f_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_1200_4_f_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1200_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![256, 0, 1]>,<11,R![12, 1]>],Snew); return Vf; end function;