// Make newform 8470.2.a.cf in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8470_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_8470_2_a_cf();". // 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_8470_2_a_cf();". // 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, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_8470_a();" function MakeCharacter_8470_a() N := 8470; order := 1; char_gens := [6777, 6051, 7141]; 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_8470_a_Hecke(Kf) return MakeCharacter_8470_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, 0], [1, 1], [-1, 0], [1, 0], [0, 0], [2, 0], [0, -1], [2, 3], [6, 0], [0, 2], [2, -2], [0, -4], [1, -7], [-3, 3], [6, 2], [-2, 2], [-6, -1], [-10, 4], [-2, 5], [0, -6], [9, 3], [-2, -4], [-1, 7], [5, 5], [5, -9], [-4, 8], [-4, -8], [17, -1], [6, -4], [2, -7], [8, -12], [10, -11], [7, -5], [12, 4], [8, 10], [12, -6], [10, -6], [-2, 3], [0, 4], [12, 0], [-8, 1], [10, -4], [-8, 4], [-2, 0], [6, 0], [0, -12], [-13, 15], [16, -6], [0, -15], [-18, 18], [19, -5], [0, 0], [18, -15], [-12, 24], [15, 9], [4, 10], [4, -10], [10, -12], [16, -6], [-3, -13], [-12, 0], [10, 4], [-13, -9], [0, -18], [4, -3], [-24, 2], [-15, 13], [-14, -1], [-6, 7], [12, -4], [12, -25], [-4, 14], [12, 10], [-18, -4], [-23, -3], [-24, 0], [6, -24], [-4, -8], [1, -17], [2, 16], [1, 7], [26, -20], [28, 2], [-24, 25], [-6, -14], [7, 13], [-16, 25], [-7, -15], [-6, 8], [12, -12], [8, -16], [-12, -16], [12, -24], [-5, -17], [-33, 11], [12, -18], [-32, 4], [18, 11], [-28, 21], [-24, 24], [-26, 5], [22, 10], [31, -17], [0, -5], [0, 0], [-12, 23], [-28, 19], [4, 5], [-22, -2], [-21, -5], [6, -22], [-18, -12], [2, 13], [30, -7], [-14, -6], [29, 7], [19, -21], [-16, 10], [-12, 24], [-4, 5], [-20, -16], [-34, -3], [22, -22], [-44, 8], [26, -3], [4, -10], [-12, -8], [0, 20], [0, 24], [-10, -2], [-36, 7], [-38, 16], [-10, -14], [-6, 16], [-14, -11], [18, -16], [-2, -14], [-12, 39], [26, -20], [24, -45], [17, -29], [44, -8], [36, -24], [-10, -19], [18, 10], [-2, 4], [-14, 6], [15, -15], [-22, 9], [10, -2], [18, 0], [9, -3], [17, -9], [28, -20], [-2, 15], [-16, -22], [30, -24], [-34, 19], [-20, 33], [18, 10], [-50, 5], [-11, 25], [-28, 34], [12, -24], [-2, 28], [8, -14], [-2, 14], [6, -18], [35, 1], [22, -40], [53, -11], [-24, 14], [-6, -10], [-26, 15], [-48, -6], [-6, 0], [9, -11], [-24, 42], [-24, 48], [48, -10], [-52, 6], [44, -20], [-12, 28], [-30, 36], [12, 18], [-42, 14], [-14, -22], [23, 7], [-48, 19], [-22, 40], [32, -41], [34, -11], [16, -7], [-2, -14], [44, -4], [-2, 28], [25, 15], [-50, 24], [-2, -28], [28, -58], [22, 16], [-24, 18], [2, 42], [7, 35], [43, -11], [-8, 32], [-36, 2], [-10, -31], [-22, 4], [53, -5], [-22, 45], [-30, 18], [-30, -18], [7, -41], [-24, 4], [39, 3], [0, -14], [9, 9], [-22, 4], [-12, 10], [34, -40], [28, 18], [18, 20], [-18, 34], [-8, 40], [-6, 0], [24, -21], [26, -40], [-10, -14], [43, 11], [0, -36], [-13, -27], [-2, -30], [30, -12], [1, 33], [18, 6], [67, -1], [-40, 2], [-25, 55], [46, -2], [-18, 23], [-21, 33], [-22, 12], [-10, -24], [20, -31], [-56, 16], [-8, 0], [15, -51], [-16, 12], [-42, 50], [-42, 26], [22, -4], [-6, 48], [-46, 21], [0, -24], [8, 11], [20, 6], [12, 5], [54, 0], [12, -21], [-6, -32], [-40, 64], [0, 42], [-42, -12], [2, 20], [51, -3], [10, -50], [6, -11], [-10, 43], [-16, -26], [-2, 6], [55, -7], [-48, 27], [-42, -18], [0, -19], [62, 6], [10, -55], [-4, -24], [56, -3], [12, -28], [-58, 38], [10, -6], [6, -34], [-36, 44], [20, -21], [-62, 28], [-10, -33], [38, -46], [-38, 24], [2, 32], [-34, 62], [41, -7], [-45, 35], [54, -9], [-36, -10], [-28, 8], [48, -42], [-46, 58], [-23, 43], [63, -9], [-1, 21], [22, -40], [-20, -36], [65, 7], [8, -21], [63, -9], [17, 11], [40, -42], [-40, 22], [42, -22], [-10, 2], [22, -38], [-24, 35], [50, -3], [0, -26], [47, 19], [-11, -37], [36, -22], [27, -3], [-15, -33], [-35, 15], [-61, 43], [-6, 28], [24, 18], [-30, 29], [-25, 15], [-48, 16], [24, -16], [74, -8], [-26, -20], [-24, 48], [50, -34], [24, -4], [-9, 39], [0, 3], [-40, -16], [-20, 0], [-72, 33], [-26, 69], [-40, 34], [-28, 54], [27, -53], [-14, 26], [-42, -12], [-28, -2], [-16, -8], [-24, -4], [16, 24], [-30, -24], [-30, -6], [34, -45], [-33, -5], [2, 46], [-18, 0], [-44, 30], [29, -1], [0, 44], [18, -9], [62, -7], [-26, 4], [50, -2], [-34, 52], [40, -64], [0, -33], [48, 0], [1, -7], [-10, -4], [-40, 10], [-77, 21], [37, -47], [16, 32], [-2, -38], [-46, 14], [40, -34], [-60, -22], [6, -31], [-54, 20], [-16, 45], [57, -11], [27, -3], [-2, 8], [-13, -17], [-12, -42], [38, -64], [25, -33], [26, -16], [56, -2], [26, -56], [-54, 9], [24, 24], [-48, -33], [-32, 46], [27, 23], [-36, -4], [18, -42], [-54, 36], [6, 42], [50, 11], [-3, -47], [36, 6], [16, -34], [-69, 23], [18, -6], [-68, 17], [-28, -16], [58, 12], [-72, 18], [8, -5], [48, 24], [54, 7], [-32, 5], [6, 42], [48, -24], [12, -39], [-45, -21], [42, -14], [46, -50], [-36, -24], [20, -7], [24, -42], [36, -24], [-30, 6], [8, 26], [-15, 27], [-27, -13], [-62, 16], [-15, 27], [12, -59], [69, -21], [54, -24], [54, 3], [12, 48], [28, 24], [4, -44], [-74, 54], [58, -1], [16, 33], [-6, -36], [53, -33], [-60, 40], [17, 41], [-21, -3], [-60, 34], [52, 28], [18, -9], [38, 16], [-64, 12], [-42, 64], [38, 15]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8470_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8470_2_a_cf();". // 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_8470_2_a_cf(:prec:=2) chi := MakeCharacter_8470_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(3169) - 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_8470_2_a_cf();". // 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_8470_2_a_cf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8470_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, -3, 1]>,<13,R![-2, 1]>,<17,R![-1, 1, 1]>,<19,R![1, -7, 1]>],Snew); return Vf; end function;