// Make newform 5445.2.a.be in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5445_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_5445_2_a_be();". // 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_5445_2_a_be();". // 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, 1, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -1, 1, 0], [1, -2, -1, 1]]; 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_5445_a();" function MakeCharacter_5445_a() N := 5445; order := 1; char_gens := [3026, 4357, 3511]; v := [1, 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; function MakeCharacter_5445_a_Hecke(Kf) return MakeCharacter_5445_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 := [[-2, 1, 1, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 2, -1, -1], [0, 0, 0, 0], [1, -1, 1, -1], [-5, 0, 0, 0], [-1, -3, 2, 3], [0, 1, 1, 1], [1, -1, -4, 0], [0, 3, 0, -2], [0, 1, 1, -5], [-6, -4, -1, 5], [-3, 0, -1, 6], [6, 4, -1, -3], [-1, -6, -3, 5], [5, -1, -1, -6], [3, -3, -6, -2], [-5, -5, 5, 1], [4, -3, 2, 4], [2, -3, 5, 4], [2, 2, 2, -7], [-10, -5, -2, 8], [-2, -2, 0, -3], [-1, 4, 3, -3], [3, -5, -1, 0], [-7, 5, 0, 5], [-1, -7, -6, 5], [-4, 6, 7, -2], [6, -5, -4, -6], [-8, -4, 3, 4], [-4, 11, 6, -5], [-9, 3, 1, -1], [3, 4, -3, -6], [0, 3, -1, -4], [1, 0, -4, -3], [-3, -11, -6, 10], [-2, 8, 8, -1], [-8, 6, 6, -8], [-6, -7, 1, 6], [3, 3, 0, -5], [6, -3, 6, 5], [-7, -11, -6, 12], [9, -2, -10, 6], [-17, 4, 3, -2], [-4, 9, 0, -7], [-14, 6, 3, 11], [-13, 6, 4, 11], [-7, -1, -5, -2], [-10, -10, 3, 2], [-2, 5, -5, 5], [-4, 6, -3, -3], [1, 4, -4, -2], [9, 6, -6, 6], [-11, 7, 6, -3], [7, 4, -9, -10], [-2, 0, -7, -6], [11, 1, 1, 0], [-8, -7, 4, 0], [-5, 8, 8, -2], [5, -3, 3, -10], [8, -11, 0, -3], [9, -2, -9, -5], [-17, 2, 15, -1], [-8, 10, 6, -7], [-6, 4, -6, 2], [-6, -10, -3, 7], [5, 11, -6, 7], [-18, -5, 1, 11], [18, -9, -16, -2], [12, -4, 6, 5], [8, -12, -11, 1], [2, 2, -11, 1], [-2, -5, -4, -2], [1, -2, -11, 2], [-21, 12, 11, 5], [-1, 6, 13, 5], [5, -9, -13, 14], [-5, 11, 19, -9], [18, 4, 0, -20], [8, -4, -9, 17], [-11, 8, 21, 1], [-4, 2, 16, -15], [-10, 9, -4, 0], [-13, 5, -3, 3], [-8, -8, -6, 7], [-4, -7, -6, 7], [-4, 21, 16, -12], [9, 16, 6, -21], [6, 14, 8, -23], [-12, 17, 5, 4], [-13, -1, 8, 27], [4, -8, 0, 13], [-5, 2, -5, 12], [29, -11, -14, -10], [-7, 6, 0, 6], [-5, -3, 7, 15], [8, 11, 17, -18], [-21, -3, 4, -2], [0, -7, -17, 1], [-15, 4, 18, -11], [-4, -4, 5, 7], [-13, 0, -9, -5], [3, -3, 14, -7], [-7, 7, -10, -19], [7, 3, -2, 3], [-10, 6, 13, 3], [-11, -12, 8, -1], [3, 10, -8, -15], [-4, -3, -11, 22], [-33, 5, 7, 8], [-4, 5, 1, 7], [-4, 23, -8, -18], [13, 28, 6, -26], [-13, 17, 5, 4], [2, -4, -7, 2], [1, -6, -6, -1], [-16, 1, -2, 1], [-7, -4, 10, 9], [19, -4, 6, -16], [-13, 0, 0, 14], [-4, 23, -4, -9], [-11, -22, 3, 4], [-17, 14, 6, 5], [-28, 7, 23, -3], [25, 13, -7, -4], [8, -2, -6, 25], [-18, 22, 3, -14], [-10, 10, -10, -4], [23, -7, 6, 8], [-7, -4, 15, -11], [-4, 11, 16, -3], [3, 19, -1, -12], [16, -18, -28, 8], [-16, -5, -8, 2], [21, -6, 0, 10], [6, -5, 8, 11], [-16, -1, -4, 3], [6, -3, -6, 9], [-5, -13, 2, 11], [-2, 3, -15, 14], [13, 13, 19, -17], [12, -17, -25, 8], [8, 13, 2, -11], [2, -9, 6, -24], [25, 4, -10, -14], [-4, -7, -3, 10], [-12, -6, -15, 5], [28, 21, 3, -12], [-2, 11, -11, 15], [9, 13, 0, -12], [0, -17, -8, 12], [-10, -34, 12, 7], [7, 29, -3, -14], [-9, 6, 0, 24], [7, -4, 16, -7], [-19, 15, 6, 13], [-8, -7, -2, 25], [25, -5, -9, 14], [17, -18, -4, -4], [15, 12, 15, -12], [11, -15, -23, 9], [-3, 9, 1, -23], [-20, 22, 2, 8], [-20, 6, 11, -16], [-20, 10, -5, -13], [-30, -11, -4, 21], [18, 0, 6, -6], [-12, 3, 26, -23], [-42, 12, 7, 5], [12, -1, 5, 25], [-7, -8, 9, 17], [20, -1, 4, -11], [13, -16, -2, 1], [2, -2, 3, -15], [16, 4, -30, -13], [13, 14, 14, -12], [-33, 8, 20, 15], [-1, 21, -3, -19], [-4, 4, 19, -21], [20, 4, -7, -1], [15, -8, 10, 13], [-5, -17, 3, 23], [-21, -6, 2, -8], [-2, -11, 9, -20], [22, -24, 1, -13], [25, -16, -3, 12], [9, 9, 13, -15], [4, 16, -2, -6], [-21, -7, 11, 13], [22, 13, -2, -34], [-31, -3, -8, -2], [-51, 19, 23, 1], [12, -5, -10, -20], [-2, 6, 1, 8], [4, -2, 19, 6], [25, -12, -11, -11], [37, 5, -8, -11], [-3, 30, -7, -27], [1, -5, -27, -1], [18, 9, 15, -42], [-12, -12, 7, 21], [40, -11, -18, -12], [16, -15, 4, 29], [17, 16, 7, -46], [27, 1, -5, -49], [27, -24, -11, 0], [-30, 0, 8, -5], [21, 8, -1, -23], [1, -18, -27, 19], [36, -11, -6, -7], [10, 6, -25, 18], [-36, -9, 2, 14], [-5, -25, 11, 25], [16, 17, 21, -34], [-44, -3, 9, 11], [-34, -7, 7, 19], [-35, -4, 2, 44], [8, -11, -8, -7], [8, 21, -18, -8], [10, 21, -5, -21], [-13, 36, -8, -24], [-1, -26, 5, 24], [-20, 9, -1, -10], [10, 2, 6, 9], [-9, 1, -24, -15], [-11, -1, 5, 6], [22, 1, -17, -33], [-12, -24, -5, 11], [-31, 4, 6, 47], [-7, -15, 6, 16], [-10, 27, -1, -26], [-28, 3, 33, -8], [-38, 0, 16, 2], [8, 19, -14, -15], [22, 8, -8, -1], [-4, 14, -20, -34], [12, 4, -25, -4], [16, -20, -1, 7], [-20, 3, 12, 16], [27, -31, -2, 8], [-28, -12, -5, 60], [11, -13, 9, -29], [19, -8, -6, -7], [11, 14, 4, -45], [30, 13, -21, -8], [-2, 20, -11, -40], [-17, -18, 1, 18], [-28, -22, 21, -17], [13, 5, -16, 13], [-18, 6, 8, 11], [27, 2, 5, -56], [-4, 13, 0, -19], [-48, 6, 3, 21], [7, -21, -9, 20], [-38, 43, 16, -6], [-27, -15, -10, 31], [-45, -17, -2, 28], [35, -10, -1, -4], [33, 12, -5, -17], [0, 2, -5, -26], [43, 20, 12, -30], [11, -8, 6, 13], [-8, -11, -15, 21], [15, -13, -36, 8], [35, 1, 2, -9], [6, -44, 4, 14], [26, 7, -11, 28], [9, 16, -4, -30], [4, 22, -12, -6], [-18, -15, -28, 11], [-7, -33, 6, 13], [-37, -12, 1, 36], [-9, -37, -9, 19], [8, -32, -25, 14], [-8, -24, 7, 43], [11, -34, 1, 32], [-29, 21, 25, 19], [-15, -6, 20, -23], [19, -1, 4, 12], [11, -18, 4, 13], [18, 8, -14, 32], [25, -12, 4, -19], [47, -6, -5, -45], [-23, -2, 26, -16], [53, -30, -32, 3], [20, 31, 2, -7], [-38, 14, 12, 23], [-29, 16, 35, 22], [-29, 1, 14, 14], [32, 6, -13, -33], [31, -17, -12, -6], [-23, -25, -1, 15], [-6, 33, 12, 3], [16, 5, -42, -7], [13, -6, 3, 15], [-10, -34, 15, 30], [15, 7, -32, -1], [31, -10, -16, -19], [6, 5, 7, -47], [8, -1, 13, 27], [-20, 8, -30, -17], [-36, 0, 12, -8], [30, 1, 17, 12], [-61, 6, 0, 1], [32, 4, -15, 6], [-10, -15, 23, 16], [5, 27, 20, -4], [-12, 11, -15, -3], [-31, -37, -10, 29], [5, 9, 23, -1], [5, -23, 6, 2], [33, -8, -2, -5], [-34, 5, 24, -6], [-19, -17, 16, 30], [55, -5, -21, 2], [21, 22, 24, -11], [8, 10, 3, -22], [8, 2, -21, 5], [14, 11, 13, -11], [-15, -22, 22, 1], [25, -32, -24, 14], [-20, -27, -28, 45], [-19, -39, 8, 45], [49, 3, 2, -28], [11, -5, -17, -31], [3, -33, -2, 16], [9, -2, 11, 9], [-16, -6, 23, 5], [13, 1, 25, -23], [-47, 11, 25, 30], [-4, -24, 7, 1], [-7, -10, -32, 14], [-1, -10, -33, 19], [9, -27, -6, 19], [37, -13, -50, 14], [2, -10, -36, 15], [-19, -16, -2, 18], [-11, 31, 16, -6], [-35, -13, -5, 0], [-48, -11, 5, 10], [-37, 21, -13, -19], [-6, -15, -7, 44], [62, 1, 11, -7], [-20, 40, -5, -35], [2, -12, 17, -12], [24, 25, 0, -19], [-12, 8, 21, 30], [-9, -5, -9, -35], [14, 4, 14, 22], [-6, 10, 6, 20], [28, 4, 8, -47], [-14, -42, 5, 55], [-45, 12, -9, -8], [-27, -15, 25, -13], [-29, -8, 9, -19], [-32, 19, 6, 38], [-12, -14, -37, 10], [-18, -24, -26, 48], [-4, -29, 0, 34], [22, -48, -21, 9], [27, 31, -11, -20], [23, -19, 1, -9], [-54, 5, 16, 33], [35, -18, -48, 16], [23, 46, -4, -59], [-33, -34, 6, 22], [-12, 10, 45, -3], [15, -29, 4, 21], [-29, -33, 2, 13], [43, 5, -12, 9], [-14, 28, 6, -32], [82, -10, -10, -8], [5, 33, 14, -24], [-5, -56, 16, 47], [14, -15, -2, 27], [19, 22, -11, 23], [-9, 25, 26, -17], [-4, -6, -2, -19], [-18, -14, -13, 36], [16, 29, 7, -34], [13, -13, -18, -19], [16, 29, -1, -23], [-12, 26, -20, -9], [-11, 37, -5, 7], [49, -8, 5, -9], [-6, 17, 31, -13], [-5, -15, -11, -19], [51, -3, -5, 5], [41, -7, -16, 10], [15, -31, 0, 1], [13, 34, 9, -58], [-55, 27, 12, 33], [26, 12, 29, -44], [-21, 14, 17, -12], [39, 6, -30, 10], [-10, -9, 21, -28], [0, 27, 0, -11], [-42, -1, 32, -12], [53, -25, -42, -6], [17, -36, -6, 26], [2, 20, -19, -32], [-49, -2, -6, 30], [12, 19, 3, -7], [1, -24, -7, 34], [44, 41, -2, -30], [-48, 41, 29, -3], [35, 2, -34, -16], [-9, 4, -6, -15], [-34, 3, -2, 32], [-42, 19, 7, 21], [-41, -20, 13, -11], [-15, -36, 2, 20], [44, -7, -18, -19], [77, -45, -43, -10], [-12, 4, -20, 41], [30, -4, -28, -1], [32, -3, -54, -18], [5, 15, 25, -29], [-3, -7, -4, 12], [-37, -12, -11, -9], [18, 7, 12, -15], [39, 1, 0, -9], [24, -4, -17, -14], [12, 6, -42, -1], [42, -41, -18, 10], [-31, -17, -7, 41], [14, -7, -21, 31], [28, -17, -49, 22], [29, 7, -5, 6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5445_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5445_2_a_be();". // 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_5445_2_a_be(:prec:=4) chi := MakeCharacter_5445_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(2999) - 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_5445_2_a_be();". // 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_5445_2_a_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5445_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-11, -10, 4, 5, 1]>,<7,R![41, -17, -16, 2, 1]>,<23,R![-1, 5, -1, -5, 1]>,<53,R![-1271, -777, -100, 6, 1]>],Snew); return Vf; end function;