// Make newform 7728.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7728_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_7728_2_a_w();". // 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_7728_2_a_w();". // 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_7728_a();" function MakeCharacter_7728_a() N := 7728; order := 1; char_gens := [4831, 5797, 5153, 6625, 6721]; v := [1, 1, 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_7728_a_Hecke(Kf) return MakeCharacter_7728_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 := [[0, 0], [-1, 0], [-2, -1], [-1, 0], [1, 0], [-3, 5], [-5, 0], [-3, 4], [-1, 0], [-3, -2], [-1, 6], [-5, 4], [-9, 2], [5, -7], [6, -6], [2, -1], [2, 1], [7, -9], [-4, 7], [-7, 7], [-7, 4], [1, 6], [7, -10], [-15, -1], [-9, -2], [-4, -1], [4, -4], [13, 1], [10, -17], [8, -3], [-3, -1], [-7, 10], [-11, 4], [-12, 7], [-10, 18], [-2, -2], [-8, 14], [6, 5], [-13, 10], [-7, 10], [14, -15], [3, -2], [6, -14], [12, 6], [21, 1], [14, -11], [-15, -2], [4, -9], [12, 3], [10, -11], [4, 9], [-24, 7], [-3, -14], [-20, 10], [-10, 6], [11, 10], [15, 3], [13, -4], [23, 5], [-8, 10], [-24, 13], [-4, 4], [13, -26], [7, 5], [-20, 12], [28, -1], [14, 12], [21, -21], [-11, 28], [24, -13], [7, -14], [2, 9], [3, 15], [7, 16], [-8, 16], [13, 2], [-9, 18], [14, -10], [9, -6], [9, 10], [8, -3], [17, 13], [3, -13], [4, -6], [19, 6], [-4, 8], [-24, 3], [18, -15], [-28, 15], [-29, 4], [1, -22], [-1, 16], [9, -32], [3, -5], [0, 13], [8, 7], [10, -2], [-14, 28], [2, 4], [-2, -2], [-8, 25], [-8, -10], [-23, -9], [-16, -16], [-5, -10], [14, -12], [41, 1], [20, -34], [-13, 11], [-38, 5], [21, -41], [-5, 26], [-9, -13], [27, -29], [-7, 4], [-15, -1], [19, -5], [-24, 5], [20, 9], [1, 0], [-29, -6], [17, 8], [-15, 9], [3, 8], [23, -23], [-34, 5], [-23, 31], [-31, 32], [-21, 36], [12, -6], [17, 2], [-2, -3], [34, -7], [-3, 12], [-20, 8], [-36, 14], [-11, 6], [6, -19], [-2, 26], [49, -1], [15, 20], [30, -14], [-4, 11], [-36, -11], [-1, -2], [-17, 15], [4, -30], [0, 14], [2, -16], [-8, 38], [-12, -28], [-40, 18], [-15, -7], [39, -15], [2, 29], [6, -30], [-19, 0], [18, 13], [-13, 0], [28, -24], [30, -20], [-33, 35], [34, 8], [15, -19], [29, 19], [1, -6], [35, -15], [-21, 2], [6, 21], [-5, 27], [-3, 31], [-50, 8], [35, -9], [-15, 0], [-12, 32], [21, -10], [35, -3], [-20, 15], [18, -9], [-2, -23], [28, -15], [-13, 31], [-40, 21], [41, 13], [-9, -20], [35, -31], [6, -16], [33, 10], [-40, 21], [-11, 14], [-12, 3], [-36, 39], [-16, -18], [-47, -2], [-21, -17], [-3, 3], [-25, 49], [8, -18], [-2, -32], [-12, 45], [18, 24], [-4, 8], [24, 13], [-1, 27], [-1, -24], [30, 20], [47, -26], [-24, -9], [0, -23], [29, 20], [-34, 2], [2, 3], [11, 26], [44, -36], [29, -33], [-14, 37], [37, -14], [-41, 25], [-5, -2], [31, -44], [-3, -43], [-26, -28], [-5, 39], [56, -20], [-49, 0], [2, 29], [-29, 46], [-36, 16], [7, 15], [27, 4], [61, -4], [-43, 21], [-12, 7], [48, -32], [-25, -3], [23, -11], [2, 26], [9, 24], [11, -4], [-19, -6], [37, -4], [20, 11], [12, 18], [29, -14], [-35, -2], [-13, -19], [16, 12], [-36, 64], [23, -36], [-3, -2], [55, 4], [70, 0], [35, 16], [49, 8], [-63, 8], [-40, 42], [-63, 12], [47, -40], [45, 19], [37, -30], [13, -17], [-18, -12], [-40, 3], [-13, 32], [-11, 31], [-69, 2], [-6, -36], [-63, 6], [-25, 6], [-4, -3], [39, 6], [-17, 24], [-15, 1], [14, 26], [7, 34], [22, -16], [21, -12], [45, 14], [-29, -16], [-31, -10], [-58, 14], [-21, -27], [-23, 10], [-35, 6], [-6, 25], [-51, 0], [15, -52], [-34, -8], [19, -15], [-56, 12], [-7, -7], [-49, 15], [-41, -13], [-3, 52], [-24, 12], [25, -10], [44, 12], [23, -5], [-51, 12], [17, 38], [-41, 33], [-42, -4], [7, 46], [-4, 5], [-4, -12], [11, -26], [20, 39], [21, -52], [-8, 34], [-35, 31], [3, 0], [47, -32], [-16, 17], [-11, 0], [-1, 18], [-8, 16], [-5, 12], [65, 4], [-5, -52], [-18, -30], [14, 20], [-11, -33], [-18, -5], [-33, 0], [-35, 22], [-22, 51], [5, -22], [2, -33], [-66, 6], [-44, 46], [16, -56], [21, -32], [-63, 21], [52, -51], [-10, -38], [-14, -48], [-52, 46], [-2, 22], [44, -11], [36, -5], [13, -4], [-54, 4], [21, 17], [59, -20], [18, -44], [57, 1], [4, -35], [6, -25], [82, -11], [61, 4], [-21, -19], [-27, 40], [-63, -8], [-46, 81], [-23, 46], [44, -16], [32, -58], [41, -20], [-13, 18], [21, 4], [45, 10], [23, -14], [27, 40], [-76, 0], [-42, 26], [24, 12], [29, 3], [6, 34], [33, -69], [-31, 27], [-39, -18], [-28, -32], [-20, -30], [20, 24], [37, 32], [34, 35], [28, -51], [6, -54], [58, -24], [71, -10], [-59, 32], [-29, -37], [-26, 54], [11, -25], [18, 23], [-23, -17], [25, -25], [-26, 8], [29, -32], [-48, -10], [55, 18], [-14, -29], [62, -36], [46, 14], [-47, 49], [-18, -29], [-47, 12], [11, -8], [-15, 35], [55, -32], [37, 10], [27, -25], [-16, 10], [-29, 41], [-29, 25], [-3, -15], [-12, 52], [-40, 50], [-52, 40], [-18, 43], [41, 21], [6, 22], [-28, 8], [-63, 7], [-59, 25], [38, -20], [27, -2], [19, -6], [39, 6], [6, -34], [17, 4], [53, -66], [-32, 43], [-1, 45], [-37, -1], [-14, 14], [67, 13], [-32, 42], [20, -49], [-61, 45], [-1, -12], [-20, 36], [24, 23], [-31, 65], [28, 38], [14, -4], [-18, 43], [48, -66], [-17, 50]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7728_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7728_2_a_w();". // 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_7728_2_a_w(:prec:=2) chi := MakeCharacter_7728_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(3067) - 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_7728_2_a_w();". // 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_7728_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7728_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![5, 5, 1]>,<11,R![-1, 1]>,<13,R![-31, 1, 1]>,<17,R![5, 1]>],Snew); return Vf; end function;