// Make newform 810.3.j.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_810_j();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_810_j_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_810_3_j_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_810_3_j_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 := [961, 0, -434, 0, 165, 0, -14, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [721, 0, 0, 0, 0, 0, 1, 0], [0, 1711, 0, 0, 0, 0, 0, 1], [0, 721, 0, 0, 0, 0, 0, 1], [6076, 0, -2310, 0, 165, 0, -14, 0], [29078, 0, -11055, 0, 1155, 0, -67, 0], [0, 29078, 0, -11055, 0, 1155, 0, -67], [0, 12493, 0, -24915, 0, 2145, 0, -182]]; Rf_basisdens := [1, 495, 495, 495, 5115, 15345, 15345, 15345]; 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_810_j();" function MakeCharacter_810_j() N := 810; order := 6; char_gens := [731, 487]; v := [1, 3]; // 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_810_j_Hecke();" function MakeCharacter_810_j_Hecke(Kf) N := 810; order := 6; char_gens := [731, 487]; char_values := [[0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 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, 1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 3, 1, 0, 0], [0, 0, -2, 0, 0, 0, -1, -2], [0, 0, 3, 0, 0, 0, -2, 3], [0, 0, -3, 0, 0, 0, 0, 0], [3, -7, 0, 0, 0, 0, 0, 0], [-3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 15, 14, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0], [0, 0, 0, 0, 17, 15, 0, 0], [0, 0, 0, 14, 0, 0, -14, 2], [0, 0, -6, -1, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 7, -4], [-60, 14, 0, 0, 60, -14, 0, 0], [-51, 9, 0, 0, 0, 0, 0, 0], [0, 0, 9, -4, 0, 0, 0, 0], [49, -24, 0, 0, -49, 24, 0, 0], [0, 0, -26, 8, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, -6, 21], [0, 0, 0, 16, 0, 0, -16, 7], [-45, 63, 0, 0, 45, -63, 0, 0], [27, -58, 0, 0, -27, 58, 0, 0], [0, 0, 0, 29, 0, 0, -29, 0], [0, 0, -22, 0, 0, 0, 10, -22], [0, 0, -6, 0, 0, 0, 8, -6], [0, 0, 2, 28, 0, 0, 0, 0], [36, 30, 0, 0, 0, 0, 0, 0], [-19, -90, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12, 30, 0, 0], [0, 0, 0, -34, 0, 0, 34, 14], [0, 0, 12, 18, 0, 0, 0, 0], [27, 135, 0, 0, -27, -135, 0, 0], [0, 0, 0, 0, -38, 102, 0, 0], [0, 0, -12, 35, 0, 0, 0, 0], [20, -66, 0, 0, -20, 66, 0, 0], [0, 0, 19, -16, 0, 0, 0, 0], [0, 0, 0, -28, 0, 0, 28, -10], [0, 0, 0, 0, 21, -60, 0, 0], [69, -29, 0, 0, -69, 29, 0, 0], [0, 0, 0, -20, 0, 0, 20, -12], [113, -42, 0, 0, 0, 0, 0, 0], [0, 0, -12, 0, 0, 0, -66, -12], [0, 0, -17, -34, 0, 0, 0, 0], [-279, 1, 0, 0, 0, 0, 0, 0], [114, -30, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -171, -105, 0, 0], [0, 0, 2, 0, 0, 0, -74, 2], [-87, 232, 0, 0, 87, -232, 0, 0], [0, 0, 0, 0, 59, -114, 0, 0], [-126, -82, 0, 0, 0, 0, 0, 0], [0, 0, -48, -44, 0, 0, 0, 0], [191, 90, 0, 0, -191, -90, 0, 0], [0, 0, 0, 36, 0, 0, -36, 30], [0, 0, 0, 0, 87, -213, 0, 0], [60, -118, 0, 0, -60, 118, 0, 0], [0, 0, 0, 18, 0, 0, -18, 72], [115, -147, 0, 0, 0, 0, 0, 0], [0, 0, 49, 0, 0, 0, -22, 49], [0, 0, 0, 0, 0, 0, 27, 0], [0, 0, 86, -53, 0, 0, 0, 0], [0, 0, 0, 0, -45, -129, 0, 0], [0, 0, 0, 0, 0, 0, 0, 60], [0, 0, -21, -114, 0, 0, 0, 0], [0, 0, 20, 0, 0, 0, 112, 20], [57, 211, 0, 0, -57, -211, 0, 0], [266, 126, 0, 0, -266, -126, 0, 0], [0, 0, 95, -56, 0, 0, 0, 0], [0, 0, 0, 0, -150, -240, 0, 0], [37, -84, 0, 0, -37, 84, 0, 0], [6, -182, 0, 0, -6, 182, 0, 0], [0, 0, 0, 18, 0, 0, -18, -84], [0, 0, 16, 0, 0, 0, 107, 16], [0, 0, -12, -18, 0, 0, 0, 0], [91, 165, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 465, 76, 0, 0], [0, 0, 78, 0, 0, 0, 119, 78], [0, 0, 0, -106, 0, 0, 106, 23], [0, 0, 18, 106, 0, 0, 0, 0], [0, 0, 0, 0, 491, 30, 0, 0], [0, 0, 63, -50, 0, 0, 0, 0], [-391, -24, 0, 0, 391, 24, 0, 0], [0, 0, 0, -152, 0, 0, 152, 33], [0, 0, 0, -8, 0, 0, 8, 103], [-135, -189, 0, 0, 135, 189, 0, 0], [447, -6, 0, 0, -447, 6, 0, 0], [0, 0, 0, -33, 0, 0, 33, -132], [0, 0, 82, 0, 0, 0, 98, 82], [0, 0, -30, 0, 0, 0, 78, -30], [0, 0, -96, 132, 0, 0, 0, 0], [675, 48, 0, 0, 0, 0, 0, 0], [0, 0, 93, 0, 0, 0, -34, 93], [0, 0, 0, 51, 0, 0, -51, 30], [0, 0, -132, -18, 0, 0, 0, 0], [0, 0, 0, 0, 681, -57, 0, 0], [357, 54, 0, 0, 0, 0, 0, 0], [0, 0, -192, 54, 0, 0, 0, 0], [0, 0, 0, 17, 0, 0, -17, 6], [0, 0, 0, 137, 0, 0, -137, 86], [-80, 210, 0, 0, 0, 0, 0, 0], [0, 0, 10, 0, 0, 0, -67, 10], [-210, 66, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 384, 106, 0, 0], [0, 0, -12, 0, 0, 0, -19, -12], [0, 0, 0, 0, -127, 699, 0, 0], [0, 0, 0, 170, 0, 0, -170, 77], [-309, -162, 0, 0, 309, 162, 0, 0], [549, 237, 0, 0, 0, 0, 0, 0], [0, 0, 75, -162, 0, 0, 0, 0], [-437, 30, 0, 0, 437, -30, 0, 0], [0, 0, 16, 53, 0, 0, 0, 0], [0, 0, 0, -176, 0, 0, 176, -14], [0, 0, 0, 0, 315, -269, 0, 0], [266, -174, 0, 0, -266, 174, 0, 0], [-427, 153, 0, 0, 0, 0, 0, 0], [0, 0, -192, 0, 0, 0, -38, -192], [0, 0, 12, -45, 0, 0, 0, 0], [339, 252, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -855, -49, 0, 0], [0, 0, -27, 0, 0, 0, 94, -27], [0, 0, 0, 0, -668, -258, 0, 0], [0, 0, 33, 0, 0, 0, -84, 33], [444, 94, 0, 0, -444, -94, 0, 0], [147, 268, 0, 0, 0, 0, 0, 0], [65, -345, 0, 0, -65, 345, 0, 0], [0, 0, 0, 100, 0, 0, -100, -114], [-636, 378, 0, 0, 636, -378, 0, 0], [0, 0, 0, -28, 0, 0, 28, -18], [0, 0, 108, 0, 0, 0, -36, 108], [0, 0, 92, 334, 0, 0, 0, 0], [827, -51, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 498, 576, 0, 0], [0, 0, 0, 0, -473, -537, 0, 0], [0, 0, 0, -166, 0, 0, 166, 2], [0, 0, -156, 153, 0, 0, 0, 0], [0, 0, 0, 0, 687, -102, 0, 0], [459, 45, 0, 0, 0, 0, 0, 0], [0, 0, -82, -155, 0, 0, 0, 0], [0, 0, 0, 0, 1011, -281, 0, 0], [0, 0, 0, 15, 0, 0, -15, -90], [-692, 282, 0, 0, 0, 0, 0, 0], [0, 0, 192, 0, 0, 0, -100, 192], [0, 0, 114, -189, 0, 0, 0, 0], [615, -424, 0, 0, 0, 0, 0, 0], [180, -930, 0, 0, 0, 0, 0, 0], [0, 0, 72, 0, 0, 0, -68, 72], [0, 0, 40, 0, 0, 0, -88, 40], [945, -537, 0, 0, -945, 537, 0, 0], [0, 0, 0, 0, 653, 315, 0, 0], [-75, 30, 0, 0, 0, 0, 0, 0], [0, 0, -83, -226, 0, 0, 0, 0], [0, 0, 0, 117, 0, 0, -117, 342], [0, 0, 0, -123, 0, 0, 123, -90], [0, 0, 0, 0, 531, 798, 0, 0], [0, 0, -102, 0, 0, 0, 318, -102], [0, 0, -111, 0, 0, 0, -348, -111], [-762, -366, 0, 0, 0, 0, 0, 0], [0, 0, 120, 0, 0, 0, 129, 120], [0, 0, 0, -14, 0, 0, 14, -176], [0, 0, -180, -4, 0, 0, 0, 0], [399, -150, 0, 0, -399, 150, 0, 0], [294, -814, 0, 0, 0, 0, 0, 0], [0, 0, 274, -136, 0, 0, 0, 0], [0, 0, 0, -342, 0, 0, 342, 90], [0, 0, 0, 0, -678, 402, 0, 0], [243, 70, 0, 0, -243, -70, 0, 0], [433, -207, 0, 0, 0, 0, 0, 0], [0, 0, 159, 0, 0, 0, -120, 159]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_810_j_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_810_3_j_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_810_3_j_f(:prec:=8) chi := MakeCharacter_810_j(); 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_810_3_j_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_810_3_j_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_810_j(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![15376, 0, -23312, 0, 35220, 0, -188, 0, 1]>,<17,R![-89, -6, 1]>],Snew); return Vf; end function;