// Make newform 294.3.h.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_294_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_294_h_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_294_3_h_g();". // 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_294_3_h_g();". // 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 := [81, 0, -72, 0, 55, 0, -8, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-148, 0, 0, 0, 0, 0, -1, 0], [576, 0, -440, 0, 55, 0, -8, 0], [0, 533, 0, 0, 0, 0, 0, 1], [0, -203, 0, 0, 0, 0, 0, -1], [1656, 0, -1265, 0, 220, 0, -23, 0], [0, 81, 0, -341, 0, 55, 0, -8], [0, -5688, 0, 4345, 0, -605, 0, 79]]; Rf_basisdens := [1, 55, 495, 165, 165, 495, 297, 1485]; 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_294_h();" function MakeCharacter_294_h() N := 294; order := 6; char_gens := [197, 199]; v := [3, 4]; // 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_294_h_Hecke();" function MakeCharacter_294_h_Hecke(Kf) N := 294; order := 6; char_gens := [197, 199]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 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, 1, 0, -1, 0], [0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, -2, 0, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 5, 0, 0, 0], [-10, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, -2, 0, 0, 0], [-16, 0, 16, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, -10], [0, 0, 0, 2, 0, 0, 20, 2], [0, 0, -32, 0, 0, 10, 0, 0], [-20, 0, 20, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 30, -9], [20, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, -6, 0], [0, 0, 0, 0, 36, 0, 0, 0], [0, 0, 0, 8, 20, 0, 0, 0], [-14, 20, 14, 0, 0, 20, 0, 0], [0, 0, -60, 0, 0, 4, 0, 0], [0, 0, 0, -14, 0, 0, 25, -14], [0, 0, 30, 0, 0, -16, 0, 0], [32, -20, -32, 0, 0, -20, 0, 0], [0, 0, 0, 20, 0, 0, 50, 20], [0, 0, 0, 0, -30, 0, 30, 3], [90, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 11, -70, 0, 0, 0], [-40, -30, 40, 0, 0, -30, 0, 0], [0, 0, 0, 0, -3, 0, 3, 30], [0, 0, 18, 0, 0, 40, 0, 0], [0, 0, 0, 20, 0, 0, -4, 20], [-80, 20, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -50, 0, 50, 20], [0, 0, 0, 10, -44, 0, 0, 0], [-120, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 60, 0, -60, -36], [0, 0, 0, 0, 0, -80, 0, 0], [0, 0, 50, 0, 0, -60, 0, 0], [-140, -48, 140, 0, 0, -48, 0, 0], [0, 0, 0, -20, 0, 0, 58, -20], [0, 0, 0, 0, 70, 0, -70, -25], [0, 0, 0, 6, 15, 0, 0, 0], [134, -20, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -105, 0, 105, -54], [0, 0, -80, 0, 0, -72, 0, 0], [0, 0, 0, -20, 0, 0, -8, -20], [0, 0, -88, 0, 0, 0, 0, 0], [-220, 60, 0, 0, 0, 0, 0, 0], [40, -72, 0, 0, 0, 0, 0, 0], [0, 0, 0, -60, 24, 0, 0, 0], [-90, -20, 90, 0, 0, -20, 0, 0], [0, 0, 0, 0, -32, 0, 32, -10], [0, 0, 0, -6, 0, 0, -135, -6], [0, 0, -130, 0, 0, 0, 0, 0], [0, 0, 0, -36, 0, 0, -180, -36], [0, 0, 0, 0, 198, 0, -198, -15], [0, 0, 0, -30, -231, 0, 0, 0], [0, 0, 0, 53, -130, 0, 0, 0], [88, -10, -88, 0, 0, -10, 0, 0], [0, 0, 40, 0, 0, 72, 0, 0], [0, 0, 0, -50, 0, 0, -80, -50], [0, 0, 120, 0, 0, 8, 0, 0], [0, 0, 0, 45, 0, 0, -114, 45], [0, -40, 0, 0, 0, 0, 0, 0], [0, 0, 0, -16, 230, 0, 0, 0], [90, 64, -90, 0, 0, 64, 0, 0], [0, 0, 0, 0, 132, 0, -132, 60], [172, -120, -172, 0, 0, -120, 0, 0], [-50, -208, 0, 0, 0, 0, 0, 0], [0, 0, 0, 30, -57, 0, 0, 0], [222, 140, 0, 0, 0, 0, 0, 0], [0, 0, 0, 105, 162, 0, 0, 0], [0, 0, 0, 0, -295, 0, 295, 22], [0, 0, 200, 0, 0, 48, 0, 0], [390, 32, -390, 0, 0, 32, 0, 0], [-12, -80, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 4, -140], [0, 0, 0, -26, 100, 0, 0, 0], [310, -4, -310, 0, 0, -4, 0, 0], [0, 0, 0, 0, -180, 0, 180, 6], [0, 0, -106, 0, 0, 240, 0, 0], [0, 0, 0, 24, 0, 0, 0, 24], [-248, 200, 0, 0, 0, 0, 0, 0], [0, 0, 0, 42, 285, 0, 0, 0], [670, -48, 0, 0, 0, 0, 0, 0], [448, -40, -448, 0, 0, -40, 0, 0], [0, 0, 0, 0, 415, 0, -415, 50], [0, 0, 0, 8, 0, 0, 500, 8], [-750, 32, 750, 0, 0, 32, 0, 0], [0, 0, 0, -89, 0, 0, 10, -89], [360, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -440, 0, 440, 20], [0, 0, 0, 40, 10, 0, 0, 0], [0, 0, 40, 0, 0, -24, 0, 0], [0, 0, 0, -90, 0, 0, -165, -90], [276, 80, -276, 0, 0, 80, 0, 0], [0, 0, 0, -60, 0, 0, 228, -60], [0, 0, 0, 0, 90, 0, -90, -51], [0, 0, 0, -183, -30, 0, 0, 0], [-160, -18, 160, 0, 0, -18, 0, 0], [-496, -200, 496, 0, 0, -200, 0, 0], [-740, 68, 0, 0, 0, 0, 0, 0], [0, 0, 0, -20, 460, 0, 0, 0], [0, 0, 0, 160, -92, 0, 0, 0], [0, 0, 0, 0, 120, 0, -120, -90], [0, 0, 20, 0, 0, 20, 0, 0], [0, 0, 150, 0, 0, 152, 0, 0], [0, 0, 0, -100, 0, 0, 524, -100], [0, 0, 0, 0, -558, 0, 558, -45], [0, 0, 0, 78, -225, 0, 0, 0], [22, -360, 0, 0, 0, 0, 0, 0], [-440, 248, 440, 0, 0, 248, 0, 0], [0, 0, 290, 0, 0, 8, 0, 0], [0, 0, 0, -50, 0, 0, 508, -50], [0, 0, 240, 0, 0, -70, 0, 0], [-136, -120, 0, 0, 0, 0, 0, 0], [0, 0, 0, 50, -460, 0, 0, 0], [-440, -144, 0, 0, 0, 0, 0, 0], [0, 0, 0, -60, -546, 0, 0, 0], [0, 0, 0, 0, 144, 0, -144, 150], [0, 0, 0, -158, 0, 0, 325, -158], [0, 0, -198, 0, 0, 20, 0, 0], [260, -24, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -78, 0, 78, -15], [0, 0, 0, -30, 609, 0, 0, 0], [872, -150, -872, 0, 0, -150, 0, 0], [0, 0, 0, -274, 0, 0, -160, -274], [334, -120, -334, 0, 0, -120, 0, 0], [0, 0, 0, 0, -780, 0, 780, 60], [-120, 122, 0, 0, 0, 0, 0, 0], [190, -292, -190, 0, 0, -292, 0, 0], [0, 0, -484, 0, 0, -280, 0, 0], [0, 0, 0, 270, 0, 0, -309, 270], [632, -20, -632, 0, 0, -20, 0, 0], [130, 392, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 230, 0, -230, 301], [666, -160, 0, 0, 0, 0, 0, 0], [0, 0, 0, 305, 86, 0, 0, 0], [0, 0, 40, 0, 0, -494, 0, 0], [0, 0, 0, -285, 0, 0, -102, -285], [0, 0, 0, 136, -140, 0, 0, 0], [-384, 170, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 220, 0, -220, -16], [0, 0, -240, 0, 0, 76, 0, 0], [0, 0, 0, 30, 0, 0, 3, 30], [0, 0, -1146, 0, 0, -100, 0, 0], [0, 0, 0, -220, 0, 0, -370, -220], [490, 180, 0, 0, 0, 0, 0, 0], [0, 0, 0, 165, 258, 0, 0, 0], [992, 120, -992, 0, 0, 120, 0, 0], [0, 0, 0, 0, -523, 0, 523, 70], [-750, 296, 750, 0, 0, 296, 0, 0], [0, 0, 0, -309, 0, 0, 150, -309], [500, -168, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 462, 0, -462, 180], [0, 0, 500, 0, 0, 272, 0, 0], [0, 0, 0, -102, 0, 0, -315, -102], [24, 380, -24, 0, 0, 380, 0, 0], [0, 0, 0, 0, 630, 0, -630, -147], [-1010, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -13, 170, 0, 0, 0], [0, 0, 0, 0, 23, 0, -23, -230], [0, 0, 0, 160, 0, 0, 496, 160], [-280, -36, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -150, 0, 150, -336], [0, 0, 0, 150, -72, 0, 0, 0], [0, 0, 0, -240, 708, 0, 0, 0], [0, 0, -536, 0, 0, 300, 0, 0], [0, 0, 950, 0, 0, 108, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_294_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_294_3_h_g();". // 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_294_3_h_g(:prec:=8) chi := MakeCharacter_294_h(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_294_3_h_g();". // 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_294_3_h_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_294_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![1296, 0, -1584, 0, 1900, 0, -44, 0, 1]>,<13,R![-12, 20, 1]>],Snew); return Vf; end function;