// Make newform 400.4.c.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_400_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_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_400_4_c_i();". // 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_400_4_c_i();". // 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; Rf_num := [[1, 0], [0, 2]]; Rf_basisdens := [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_400_c();" function MakeCharacter_400_c() N := 400; order := 2; char_gens := [351, 101, 177]; v := [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_400_c_Hecke();" function MakeCharacter_400_c_Hecke(Kf) N := 400; order := 2; char_gens := [351, 101, 177]; 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, 0], [0, 12], [44, 0], [0, 11], [0, -25], [44, 0], [0, 28], [-198, 0], [160, 0], [0, 81], [-198, 0], [0, -26], [0, 264], [0, -121], [-668, 0], [550, 0], [0, 94], [-728, 0], [0, 77], [-656, 0], [0, -118], [-714, 0], [0, 239], [1566, 0], [0, 484], [0, -390], [1994, 0], [0, -471], [0, 704], [2692, 0], [0, -813], [-684, 0], [-302, 0], [-1352, 0], [0, -1571], [0, -1518], [0, -132], [0, -1413], [3084, 0], [-2418, 0], [960, 0], [0, 1441], [0, -543], [88, 0], [3476, 0], [0, -464], [0, 78], [1634, 0], [0, -451], [1616, 0], [4818, 0], [2140, 0], [0, -385], [0, 3700], [2794, 0], [-8624, 0], [0, 937], [3338, 0], [0, -3586], [0, 2607], [0, 198], [4056, 0], [0, 1077], [0, 3693], [1132, 0], [0, 1671], [0, 1122], [6522, 0], [0, -5615], [1848, 0], [0, 3560], [0, 3175], [-7900, 0], [0, -5184], [-8830, 0], [0, -4939], [-13134, 0], [-906, 0], [-5412, 0], [-4642, 0], [-656, 0], [0, 4745], [5544, 0], [0, -3826], [446, 0], [0, -781], [10582, 0], [0, 5384], [0, -4938], [-352, 0], [0, -7588], [8844, 0], [19404, 0], [0, -8244], [12954, 0], [10970, 0], [0, 8470], [198, 0], [0, -7634], [0, -10427], [0, 9658], [-7018, 0], [-24420, 0], [0, -11617], [0, -5302], [0, -6919], [-3960, 0], [-5942, 0], [0, -1520], [0, -1265], [0, 9603], [10996, 0], [6680, 0], [6274, 0], [0, -4542], [0, -11828], [0, -3381], [15276, 0], [11054, 0], [0, -10639], [0, -4463], [0, -4058], [11764, 0], [-4698, 0], [-24638, 0], [16624, 0], [0, 15108], [0, -1661], [-14692, 0], [0, -14300], [29616, 0], [0, -1447], [14762, 0], [7678, 0], [0, 13695], [0, 9878], [0, -19427], [14278, 0], [716, 0], [-23538, 0], [0, 3308], [0, 13618], [-12070, 0], [-42024, 0], [0, 1207], [0, 18843], [40644, 0], [0, 9328], [0, 6501], [49490, 0], [0, -550], [0, -7052], [0, -6358], [39632, 0], [5704, 0], [-8162, 0], [0, 27555], [16374, 0], [0, 4230], [0, -10251], [0, -18260], [-20244, 0], [0, -25017], [0, -18564], [-27808, 0], [0, 14257]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_400_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_400_4_c_i();". // 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_400_4_c_i(:prec:=2) chi := MakeCharacter_400_c(); 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_400_4_c_i();". // 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_400_4_c_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_400_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![16, 0, 1]>,<7,R![576, 0, 1]>,<11,R![-44, 1]>],Snew); return Vf; end function;