// Make newform 625.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_625_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_a_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_625_2_a_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 := [1, 16, -27, -35, 26, 20, -8, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0], [1, -15, -1, 17, -2, -4, 1, 0], [0, 4, -15, -1, 17, -2, -4, 1], [-4, 36, -5, -26, 5, 4, -1, 0], [0, -8, 51, -4, -43, 7, 8, -2], [-6, 1, 57, -7, -43, 7, 8, -2]]; Rf_basisdens := [1, 1, 1, 3, 3, 3, 3, 3]; 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_625_a();" function MakeCharacter_625_a() N := 625; order := 1; char_gens := [2]; v := [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; function MakeCharacter_625_a_Hecke(Kf) return MakeCharacter_625_a(); 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 := 2; raw_aps := [[1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, -1, 0, 1], [2, 1, 0, 1, -2, 0, 2, -1], [3, 0, 0, 1, -1, 0, 2, -1], [4, 0, 1, 3, 0, -1, 1, 1], [-2, -2, -1, -2, 1, 0, -1, -2], [3, 1, 0, 0, 1, -1, 0, 1], [-2, 1, -1, -3, 1, 1, -2, -1], [1, 0, 2, 4, 0, -1, 1, 4], [-2, 1, -2, -2, -1, -1, -1, -2], [-4, 2, -1, -3, 1, -2, -4, 1], [-2, 0, 0, -1, 1, 2, -2, -1], [2, -1, 0, -1, 1, 1, -1, 2], [1, -3, -1, -2, 1, -1, -3, -3], [-3, 0, -1, -4, 1, -1, -2, -2], [-1, -1, -1, 0, 2, 3, 0, -1], [1, -2, -1, -1, -1, 0, -4, -1], [-1, 2, 2, 2, 1, -1, 0, 2], [2, 3, 1, 4, -3, 1, -1, 0], [0, -1, -1, 4, 0, 1, -1, -4], [9, -1, 2, 1, -1, -3, 2, -2], [3, 0, 4, 3, -1, -3, 1, 3], [-1, 0, 3, -4, -1, 2, 1, -1], [-4, 1, -1, 1, -2, 3, -1, 2], [2, 0, -2, -1, 0, 2, -2, 3], [2, 1, -1, 0, 0, 2, -1, 3], [0, 2, 2, 3, -3, 0, 6, 2], [0, -2, -2, 2, 3, 1, -3, 2], [-1, -3, 2, 1, 4, -1, -1, 0], [3, -2, -3, 6, -1, -1, 2, 1], [-9, 0, 0, -4, 1, 3, 2, 1], [-4, 2, -3, 0, 0, -2, -6, 2], [6, -6, 3, 0, 2, 1, 2, -3], [0, 2, 0, 3, 1, -5, 4, 3], [-5, 4, 1, 1, -6, 1, 3, 3], [-6, 5, -1, 1, -1, -2, 1, 6], [10, 2, 0, -3, -4, 2, 3, -6], [-2, -5, -5, -2, 3, 4, -4, -2], [-3, 2, 0, 4, -1, -2, 3, 10], [-4, -5, 0, -7, 5, 0, 0, -5], [4, -1, 2, 9, -3, -1, -1, 0], [-4, -6, 2, 2, 3, -4, 1, 2], [-6, 2, -2, -5, 5, -3, -1, 10], [0, 0, -3, 6, 0, -1, -4, 4], [-3, -4, 4, 3, 3, -2, 4, 2], [4, -4, 4, -5, 1, 2, -1, -7], [20, 2, 1, 11, -6, -3, 8, 2], [6, -2, 0, -2, -8, 3, 0, -6], [-5, 5, 1, 3, 2, 0, -5, 3], [2, -4, -1, -8, -3, 2, -2, -3], [-2, 5, 0, 2, 0, 0, 5, 10], [5, -3, -3, -1, 6, -1, 0, -3], [-1, -1, -3, -4, -1, -2, -11, -4], [9, 5, -1, 1, -6, 3, 1, 1], [-6, 7, 1, -5, -2, 4, -7, 3], [-4, -5, 2, -1, 5, 4, 1, 4], [-6, -4, -3, -11, 1, 4, -5, -3], [-10, 1, -2, -9, -2, 1, 6, 0], [-9, -1, -5, -14, 4, -2, -10, -3], [-7, 6, 0, -9, 4, 2, 2, 2], [14, -4, 2, 15, 0, -2, 7, 7], [-1, -6, 1, -4, 2, 2, 1, -12], [-7, -3, -5, -5, 0, 5, 0, -15], [-8, 5, -7, 3, 0, 3, 3, -3], [-4, 5, 1, 1, 0, -3, -2, 12], [-10, 4, -4, 4, 3, 5, -1, 1], [1, -3, -4, 5, 5, 3, -3, -1], [-12, -3, -1, -15, -4, 3, -9, -9], [6, -1, -1, 7, 2, -1, 10, -2], [-3, 5, -2, -2, -4, 2, -5, 1], [-10, -2, 0, -11, 9, -3, -8, 2], [6, 4, 2, 16, 0, -2, 4, 11], [4, 5, -1, -2, -2, 6, -4, 0], [7, -3, -1, -3, 3, -4, 1, -2], [-11, -6, -6, -6, 6, 7, -6, -11], [6, 6, 6, 8, -3, -7, 3, 1], [9, -2, 2, 8, -1, -6, -3, 11], [0, 2, 3, 6, 7, 2, -1, 9], [9, -8, -6, -1, 3, 5, -8, -1], [4, -3, 9, 13, 1, -2, 11, 1], [-7, -2, 2, -9, -1, 9, -8, -9], [-5, -2, 1, -6, 1, 0, -3, 7], [12, 6, -1, -3, 2, -3, 3, -3], [14, -8, -1, 3, 2, 0, 10, -3], [7, -2, -6, -1, -1, -7, -2, 0], [-8, 1, 9, -10, 5, -8, 4, 3], [7, 6, 4, 18, -7, -2, 4, 7], [-21, -4, -1, -12, 7, -2, -10, -1], [10, -5, 0, 8, 0, 2, 0, 3], [-15, -6, -2, -9, 5, 9, 1, -6], [0, -6, 1, -1, 3, 5, -6, 6], [6, -3, 3, 4, -4, 1, 8, -1], [-2, -3, -5, 3, 6, 0, 4, -7], [3, 1, 2, -4, 8, 0, -9, 8], [8, -11, -5, 0, 8, -4, -6, 6], [3, 8, 0, 5, -1, -5, 6, -3], [-9, -2, 2, -6, 7, 3, 10, 3], [5, 8, 2, 5, -6, -1, 12, -9], [-1, 2, 3, 10, -5, 2, 4, 0], [-6, -6, 2, 6, 4, 3, -1, -9], [-7, 3, 0, -18, 3, 0, -11, 8], [-11, -3, 4, -11, 3, -5, -3, 9], [-23, 4, -5, -26, 7, 10, -17, -4], [-12, 9, -2, 5, -5, -4, -7, 4], [6, 1, 0, 13, 4, -7, 6, 9], [4, -2, -1, -10, -10, 0, -1, -15], [4, 6, 8, 3, -2, 6, 16, 10], [1, 11, 6, 7, 3, -8, 0, 21], [3, -1, -7, 1, -6, 0, 10, -6], [-9, 0, -3, -4, 9, 4, 4, 0], [13, -2, 12, 24, -1, -4, 19, 8], [-17, 13, -5, 1, 1, 7, -3, 10], [28, 3, 1, 8, -6, -3, 4, -1], [-8, 11, -6, -2, -7, -2, -1, 2], [-4, 4, 2, 2, 3, -9, 6, 12], [37, 6, 10, 18, -13, 1, 22, -2], [-6, -4, -9, -4, 3, -2, 5, 0], [-3, -5, -3, -5, -4, 5, 2, -21], [-5, 3, -5, 19, -1, -5, 1, 7], [19, -14, 9, 15, -4, -3, 9, -2], [-5, 6, 1, -16, 6, 2, -11, -1], [-8, 0, 4, -22, 2, 6, 3, -13], [-3, 11, -1, 8, 3, -2, -1, 7], [-5, -3, 4, 5, 1, 8, 6, 11], [2, 5, 0, 9, 1, -9, -9, 5], [-15, 1, -2, 2, 5, 8, 2, -7], [2, -12, 4, -7, -3, -2, 5, -1], [4, 7, -3, -3, -7, -3, -6, 7], [-27, 0, -3, -13, 4, 3, -5, 4], [21, 1, 12, 16, -6, -4, 4, 3], [14, 8, 0, 2, -6, 5, 6, -13], [3, 2, -7, -3, 3, 0, 7, -2], [9, -16, -8, -1, 7, -6, -6, 0], [4, 2, 6, 10, -7, -6, -2, 3], [-15, 2, -7, -18, 4, -5, -8, 8], [2, 3, -1, -8, 0, 1, 3, -18], [7, -15, 0, 6, 9, 6, -7, -5], [-2, -1, 11, -2, -2, -7, 7, 8], [-15, 3, -6, -14, 4, -2, -7, 0], [1, 0, 3, -8, -5, -9, 4, -9], [1, 12, -8, 7, -11, 3, -4, 0], [6, -9, 0, 9, 6, 0, -12, -9], [9, 0, 4, 8, -2, -4, 6, 10], [-7, -5, -4, -1, 4, -9, -13, 7], [-3, 4, 8, 17, -8, -3, 4, 10], [-3, 3, 4, -11, -4, 7, -13, -1], [17, -10, 4, 1, 1, 4, 8, -2], [19, -3, -1, 5, -5, -2, -1, 2], [-6, -1, -8, -15, 10, 9, -8, -14], [-24, 0, -8, 4, 10, -1, -14, -1], [-15, 10, 1, -12, -3, 0, 0, 13], [23, -15, -2, 11, 11, -6, -10, 0], [-2, 13, 3, 9, -9, 0, -4, 11], [26, -8, 1, 3, -4, 2, -3, -15], [-4, 0, -1, -4, 7, -10, -7, 1], [13, 6, 11, 13, -4, -5, 19, 3], [-2, -3, -7, -10, 7, 9, -7, -4], [14, -6, -2, -8, 2, 1, -3, 7], [-6, 6, -7, -7, -8, 7, 15, -23], [9, 14, 11, 2, -13, 3, 17, 9], [17, -5, 3, -7, -5, -2, 3, -18], [10, 11, 0, -13, -7, 10, 2, -11], [7, -2, -1, 13, 7, -2, 10, 4], [10, -3, -9, 12, -2, 5, 3, 4], [6, -3, 5, 0, -4, 0, -6, -12], [1, -10, 1, -14, -1, 3, 14, -7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_625_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_a_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_625_2_a_g(:prec:=8) chi := MakeCharacter_625_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_625_2_a_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_625_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_625_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-9, 0, 69, -60, -29, 35, -1, -5, 1]>,<3,R![-29, 40, 71, -105, -19, 45, -4, -5, 1]>],Snew); return Vf; end function;