// Make newform 1200.3.c.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_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_3_c_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_1200_3_c_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, 3, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 4, 0, 2], [0, 8, 0, 2], [3, 0, 2, 0]]; Rf_basisdens := [1, 1, 1, 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_1200_c();" function MakeCharacter_1200_c() N := 1200; order := 2; char_gens := [751, 901, 401, 577]; v := [2, 2, 1, 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_c_Hecke();" function MakeCharacter_1200_c_Hecke(Kf) N := 1200; order := 2; char_gens := [751, 901, 401, 577]; char_values := [[1, 0, 0, 0], [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 := 3; raw_aps := [[0, 0, 0, 0], [0, 1, 0, 1], [0, 0, 0, 0], [0, -3, 0, 0], [0, 0, -1, 0], [0, 8, 0, 0], [0, 0, 0, 2], [-2, 0, 0, 0], [0, 0, 0, 6], [0, 0, 7, 0], [18, 0, 0, 0], [0, 8, 0, 0], [0, 0, 14, 0], [0, -8, 0, 0], [0, 0, 0, 22], [0, 0, 0, -2], [0, 0, 1, 0], [82, 0, 0, 0], [0, 12, 0, 0], [0, 0, 28, 0], [0, -37, 0, 0], [138, 0, 0, 0], [0, 0, 0, -42], [0, 0, -24, 0], [0, 83, 0, 0], [0, 0, 15, 0], [0, -13, 0, 0], [0, 0, 0, -90], [-38, 0, 0, 0], [0, 0, 0, -14], [0, -13, 0, 0], [0, 0, -3, 0], [0, 0, 0, -54], [-82, 0, 0, 0], [0, 0, 25, 0], [158, 0, 0, 0], [0, -82, 0, 0], [0, -118, 0, 0], [0, 0, 0, -42], [0, 0, 0, -6], [0, 0, 43, 0], [2, 0, 0, 0], [0, 0, 46, 0], [0, -107, 0, 0], [0, 0, 0, -42], [-242, 0, 0, 0], [-2, 0, 0, 0], [0, -43, 0, 0], [0, 0, 0, 26], [282, 0, 0, 0], [0, 0, 0, 162], [0, 0, -56, 0], [262, 0, 0, 0], [0, 0, -105, 0], [0, 0, 0, 90], [0, 0, 0, -26], [0, 0, 83, 0], [-82, 0, 0, 0], [0, -12, 0, 0], [0, 0, -42, 0], [0, 72, 0, 0], [0, 0, 0, -210], [0, 92, 0, 0], [0, 0, -36, 0], [0, -197, 0, 0], [0, 0, 0, 202], [198, 0, 0, 0], [0, -197, 0, 0], [0, 0, 0, 82], [362, 0, 0, 0], [0, 0, 0, 138], [0, 0, 66, 0], [0, -93, 0, 0], [0, -22, 0, 0], [-362, 0, 0, 0], [0, 0, 0, -162], [0, 0, -99, 0], [0, -62, 0, 0], [0, 0, -60, 0], [-458, 0, 0, 0], [0, 0, -133, 0], [562, 0, 0, 0], [0, 0, -78, 0], [0, 113, 0, 0], [-2, 0, 0, 0], [0, 0, 0, 90], [0, 0, -70, 0], [0, -167, 0, 0], [0, 0, 21, 0], [0, -183, 0, 0], [0, 0, 0, -202], [0, 0, -132, 0], [0, -443, 0, 0], [0, 0, 91, 0], [-2, 0, 0, 0], [0, 0, 0, -98], [0, 0, 179, 0], [0, 0, -118, 0], [0, -188, 0, 0], [-198, 0, 0, 0], [0, 512, 0, 0], [0, 0, 0, 30], [0, 0, 0, 114], [0, 0, -192, 0], [-962, 0, 0, 0], [0, 443, 0, 0], [0, 0, 0, 294], [0, 0, 0, 50], [0, 0, 50, 0], [2, 0, 0, 0], [0, -253, 0, 0], [0, 278, 0, 0], [0, 0, 0, 42], [-802, 0, 0, 0], [698, 0, 0, 0], [0, 0, 204, 0], [0, -78, 0, 0], [0, 0, 0, -338], [0, 0, 0, 218], [0, 0, 91, 0], [682, 0, 0, 0], [0, -447, 0, 0], [0, 0, 0, -246], [0, 0, 0, 198], [758, 0, 0, 0], [0, 0, -175, 0], [2, 0, 0, 0], [0, 0, 192, 0], [0, 337, 0, 0], [0, 328, 0, 0], [598, 0, 0, 0], [0, 0, 0, 350], [338, 0, 0, 0], [0, 328, 0, 0], [0, 0, 66, 0], [82, 0, 0, 0], [0, 0, 0, 474], [0, -268, 0, 0], [0, 0, 0, -182], [0, 0, 244, 0], [558, 0, 0, 0], [0, 0, 87, 0], [0, 107, 0, 0], [0, 0, 0, -14], [-318, 0, 0, 0], [0, 0, 14, 0], [0, -342, 0, 0], [0, 0, 0, -670], [-842, 0, 0, 0], [0, 0, 0, 454], [0, 78, 0, 0], [0, 0, 28, 0], [0, 482, 0, 0], [0, 0, 0, -426], [0, 642, 0, 0], [0, 0, 14, 0], [418, 0, 0, 0], [0, 0, -38, 0], [0, -267, 0, 0], [0, 0, 29, 0], [0, 0, 0, -678], [0, 0, 0, -182], [0, 337, 0, 0], [0, 0, -297, 0], [0, 0, 0, -166], [0, 0, 0, 198], [-962, 0, 0, 0], [0, -12, 0, 0], [1342, 0, 0, 0], [0, 0, 0, -454], [0, 0, -363, 0], [-278, 0, 0, 0], [0, 0, 312, 0], [0, 243, 0, 0], [1018, 0, 0, 0], [0, 0, 0, 0], [-1962, 0, 0, 0], [0, 0, 351, 0], [0, -253, 0, 0], [842, 0, 0, 0], [0, -953, 0, 0], [0, 0, 301, 0], [0, -622, 0, 0], [0, 0, 0, -302], [0, 0, 0, 702], [0, 0, -211, 0], [0, 618, 0, 0], [0, 562, 0, 0], [562, 0, 0, 0], [0, 0, -190, 0], [0, 713, 0, 0], [0, 0, 0, -406], [1878, 0, 0, 0], [0, 0, -347, 0], [0, 0, 0, 54], [0, 0, 0, 530], [262, 0, 0, 0], [0, 468, 0, 0], [0, 0, 0, 882], [0, 0, 0, -554], [0, 0, -393, 0], [-702, 0, 0, 0], [0, -972, 0, 0], [-1858, 0, 0, 0], [0, 0, 21, 0], [0, 0, 0, -306], [-982, 0, 0, 0], [0, 0, 0, -862], [0, 0, 336, 0], [-962, 0, 0, 0], [0, 323, 0, 0], [0, 0, -385, 0], [0, -813, 0, 0], [0, 0, 0, 610], [0, 0, 582, 0], [782, 0, 0, 0], [0, -793, 0, 0], [0, 0, 36, 0], [0, 0, 0, 638], [0, 0, 0, 294], [-1598, 0, 0, 0], [1598, 0, 0, 0], [0, 0, -66, 0], [0, 757, 0, 0], [0, 0, 0, -154], [-2438, 0, 0, 0], [0, 0, 0, -678], [0, 0, -496, 0], [0, -463, 0, 0], [0, 0, -465, 0], [0, -492, 0, 0], [-2762, 0, 0, 0], [-1682, 0, 0, 0], [0, 0, -62, 0], [0, 1112, 0, 0], [0, 0, 0, 894], [2382, 0, 0, 0], [0, 0, 0, -370], [0, 0, -315, 0], [0, 0, 504, 0], [0, 0, 0, 786], [-1482, 0, 0, 0], [0, -593, 0, 0], [-2238, 0, 0, 0], [0, 0, 0, -462], [0, 0, -74, 0], [0, -293, 0, 0], [0, 0, 293, 0], [1718, 0, 0, 0], [0, 0, 0, -1162], [0, -302, 0, 0], [0, 0, 220, 0], [0, 0, 0, -450], [1062, 0, 0, 0], [0, 0, 0, 1286], [0, 0, 87, 0], [682, 0, 0, 0], [0, -338, 0, 0], [0, 0, 0, 966], [0, 953, 0, 0], [0, -383, 0, 0], [0, 0, 0, -742], [802, 0, 0, 0], [0, -972, 0, 0], [0, 0, 0, -1062], [838, 0, 0, 0], [0, 0, -301, 0], [0, 0, 182, 0], [0, -988, 0, 0], [0, 0, 0, 922], [2042, 0, 0, 0], [0, 992, 0, 0], [0, -317, 0, 0], [138, 0, 0, 0], [0, -57, 0, 0], [0, 407, 0, 0], [0, 0, 0, 834], [-318, 0, 0, 0], [1902, 0, 0, 0], [0, 0, -171, 0], [0, 0, 0, -194], [3518, 0, 0, 0], [0, 0, 0, -1498], [282, 0, 0, 0], [0, -918, 0, 0], [0, 0, 438, 0], [0, 1013, 0, 0], [0, 0, 0, -966], [878, 0, 0, 0], [0, 0, -294, 0], [0, 0, -55, 0], [0, 0, 0, -730], [0, 0, 0, 666], [0, 0, 147, 0], [0, 988, 0, 0], [0, 0, -225, 0], [898, 0, 0, 0], [0, 0, 0, -966], [0, 0, 633, 0], [0, 1492, 0, 0], [0, -1317, 0, 0], [0, 0, 0, -22], [1438, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1200_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_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_1200_3_c_f(:prec:=4) chi := MakeCharacter_1200_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_3_c_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_1200_3_c_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1200_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![36, 0, 1]>,<11,R![20, 0, 1]>],Snew); return Vf; end function;