// Make newform 7350.2.a.du in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_du();". // 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_7350_2_a_du();". // 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 := [8, 8, -7, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [-4, -1, 1, 0], [-2, 8, 1, -1]]; Rf_basisdens := [1, 2, 2, 2]; 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_7350_a();" function MakeCharacter_7350_a() N := 7350; order := 1; char_gens := [4901, 1177, 2551]; 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_7350_a_Hecke(Kf) return MakeCharacter_7350_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, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 1, 0], [0, 1, -2, 0], [1, -1, 1, -1], [3, -1, 2, 1], [1, 1, -2, -1], [-2, 1, 1, 2], [4, -1, 1, 0], [-2, -1, -3, 0], [3, 0, 2, 1], [1, 2, 1, -1], [3, 2, -1, 1], [5, 1, -2, -1], [5, -1, -4, -1], [3, 0, 0, -3], [-1, 0, 1, 1], [-5, -1, -2, -3], [-3, 2, 2, -1], [-2, 2, 4, 2], [2, 0, -2, 2], [11, 0, 4, 1], [5, -2, -4, -1], [6, 3, -4, 2], [2, 0, 2, -2], [2, 0, -6, 2], [3, -1, -6, 1], [5, -1, 8, 3], [-8, -2, 6, 0], [3, 1, 0, 1], [5, -1, 6, 1], [1, 3, 0, 3], [-1, 0, 7, -1], [1, -1, -4, -1], [-3, -4, 4, 3], [-5, -2, -5, -3], [7, 4, -3, -3], [-2, -1, 2, 2], [-6, 1, -15, -2], [3, -2, -6, -3], [-3, -1, 4, 3], [0, -2, 2, 0], [7, 3, 2, -3], [14, -1, 3, -2], [0, -2, 6, 4], [0, -2, -2, 0], [6, 2, 0, -2], [15, 4, -6, 1], [4, 2, -8, 2], [1, 3, 8, -1], [0, 2, 1, 0], [-2, 2, 8, -2], [-3, -1, -9, -5], [3, -1, -6, -3], [7, -6, 0, 1], [8, -1, 9, 4], [-5, 0, 1, -3], [-17, -1, 6, 1], [-1, -5, 6, -3], [-16, -3, 4, 0], [0, 0, -8, 4], [-6, 2, -8, -2], [-3, 2, 6, 7], [5, -1, 0, -1], [-8, 2, -10, -4], [-19, 3, 2, 1], [0, 0, -8, 4], [-3, -2, -10, 3], [-9, -3, -1, 1], [3, 3, -2, -3], [2, 6, -12, -2], [-17, -2, -1, 1], [-12, 8, 4, 0], [5, 4, -13, -1], [8, -3, -13, 0], [-17, 2, 12, 1], [-1, 1, 4, 1], [4, 4, 7, 4], [-4, 0, 16, 0], [2, 6, 2, 0], [-19, 1, -2, 3], [-7, 6, -8, -5], [16, -3, -1, -4], [6, -2, 0, -2], [4, -4, 6, 2], [-12, 6, -10, -4], [-11, 2, 8, 3], [-8, -4, 20, 0], [10, -4, -6, 2], [6, 6, -16, 2], [-14, 0, -14, -2], [-2, -3, 9, 2], [2, -2, 12, 2], [-3, 0, 15, -1], [14, 1, 4, 2], [7, -4, -8, -3], [1, 1, -2, -5], [8, 4, -2, 2], [-23, -8, 11, -1], [1, -1, 4, -5], [-6, 0, 10, 2], [15, -3, 8, 1], [2, 6, -8, 2], [-11, 0, -18, -1], [-20, 2, -6, 0], [-5, -7, 7, -3], [1, -5, 4, 7], [8, 6, -1, 0], [12, -2, 26, 4], [-6, 1, -5, 8], [13, -9, -6, -3], [5, 1, -6, -1], [3, -7, 4, -3], [7, -7, 4, -7], [-17, 3, 6, 5], [-5, -4, -7, 1], [-6, 0, 4, 4], [0, -3, 3, 0], [-1, 2, 16, 1], [-19, 5, 4, -3], [-20, 3, -2, 4], [-4, 2, -2, 8], [5, 11, -4, -1], [2, 11, -13, -2], [-13, 1, -8, 9], [-8, -6, 2, 0], [-6, 0, -6, -10], [-19, 0, 4, -5], [-2, -2, 4, 6], [0, 12, 0, 0], [-7, 7, 4, -1], [-24, 3, -13, -6], [-19, 4, -4, -1], [-14, -4, -3, -2], [-10, -1, -4, -6], [-8, 0, -8, -4], [-32, 3, 6, 0], [16, 4, 10, -2], [-5, -11, 8, 1], [-3, 8, -7, -7], [-2, -6, -8, -6], [18, -2, 12, 10], [-1, -6, 12, 1], [2, 4, 10, -2], [-23, 2, -4, 7], [-15, 3, 7, -1], [-15, -1, -12, 3], [20, 4, 0, 4], [-2, -7, 7, -4], [15, 2, 2, 5], [7, 4, 5, 9], [-1, 2, -17, 5], [7, 4, -11, 9], [15, 5, -16, 1], [-21, -3, 0, -3], [-5, 6, 8, 1], [-11, -6, -8, 7], [2, -9, 0, 6], [16, 6, 2, -4], [8, -2, -6, -8], [-40, 0, -12, 0], [-2, -10, 0, -2], [-4, 6, 8, 6], [1, -2, 17, -5], [-3, 3, 0, -5], [-23, -2, -12, -1], [13, 9, -4, 9], [-16, -1, -10, 0], [-4, 7, 9, -4], [5, -2, -2, -5], [9, -13, -4, -1], [21, 7, 2, -11], [-12, -1, 17, 0], [-5, -6, -4, -7], [-6, -4, -2, 10], [-4, 5, 7, 4], [6, -8, -6, -6], [-35, -2, 0, 3], [-30, 8, -6, -2], [23, -3, 12, -3], [-10, -9, 9, 4], [-23, 3, -11, -1], [5, -7, -6, -5], [1, 0, -4, -9], [1, 0, 19, -1], [-11, -3, 2, 7], [-7, 7, -20, -9], [4, -2, 18, -4], [19, -2, 28, 1], [16, -2, 6, -12], [20, 4, -4, 8], [1, -2, 16, 7], [-14, -4, -18, -6], [-21, -3, 23, -3], [-2, -6, 0, 2], [18, -3, 19, 12], [9, 5, -9, 7], [17, 4, 7, 3], [17, 0, 17, 9], [2, 15, -5, 2], [-18, -11, 22, -6], [-30, 0, 5, 6], [2, 2, 16, 10], [-4, -5, -4, 4], [-16, 9, -13, -4], [-26, -2, 4, -2], [4, -2, 0, -10], [-13, 9, -8, 1], [-11, -11, -4, 5], [-13, -4, -16, -3], [4, 4, -12, -4], [28, -2, -6, -8], [-26, -4, -6, 2], [-10, -2, 3, 2], [18, -4, -22, -2], [-35, 4, -18, -1], [9, 1, 18, -1], [8, -2, 0, 6], [5, -5, -12, 7], [-10, 5, 9, -2], [4, 4, -34, 2], [-20, 10, -10, -4], [20, -6, 2, 0], [-33, -11, 8, 5], [15, 1, -7, 9], [11, 5, 12, -3], [36, 4, 16, -12], [10, 1, -3, 6], [6, -5, 13, 12], [-17, 3, -2, -3], [-12, 4, 4, 4], [0, 2, 16, -6], [8, -12, -8, 4], [1, -2, 25, 3], [-4, 0, -39, -4], [27, 7, -6, -7], [2, -5, -17, -2], [-28, -2, 2, -4], [6, 12, -10, 6], [7, -9, 18, -3], [-6, -6, -20, -10], [6, -2, -6, -16], [43, 1, -7, -3], [-24, 2, 6, 0], [14, 8, -10, 2], [23, 5, -8, -3], [-4, -6, 26, 8], [-20, -4, 0, 12], [-9, 4, 17, 9], [-23, 4, 14, -5], [4, 0, 4, 4], [-2, -2, -15, 2], [-16, 7, -4, 0], [26, -3, 5, -2], [15, -1, -2, -3], [31, -5, -6, 5], [-2, 5, -4, -14], [-5, 4, 12, 9], [-12, 8, -24, 4], [32, 4, -12, -4], [-13, -4, 16, -3], [-2, -7, 22, -6], [-5, 9, -5, 5], [-35, 3, 16, -1], [21, 6, -9, -3], [-11, 0, -20, -1], [-27, 12, -1, 3], [-36, -12, 18, -10], [-5, 2, -22, 5], [3, 14, -29, -3], [35, 4, -14, -7], [26, -10, -24, -2], [19, 0, 26, 9], [2, -10, 0, -2], [-36, -10, -2, 8], [-22, -8, 16, -8], [33, -3, 16, 1], [-57, 1, -4, 5], [7, 2, 31, 5], [-22, -10, 28, -2], [35, -6, -5, 1], [-27, -4, 24, -5], [15, 15, -6, 5], [13, 13, -14, -5], [-18, 8, 18, 2], [-17, -5, -14, -11], [8, -9, -7, -20], [-1, -10, 0, -11], [21, 2, -5, 13], [14, 6, -28, -2], [-34, 4, -24, -4], [-3, 11, 0, -1], [-5, -2, -29, -11], [-38, 3, -16, 6], [-36, -1, 5, 8], [-12, -1, 2, 12], [18, 2, 12, 10], [16, 12, -8, -4], [-25, -8, -12, -3], [42, -12, -12, -8], [-8, 8, -12, 0], [2, -12, -6, 2], [4, -4, 16, 0], [-19, -7, -16, 13], [22, -16, -6, -2], [13, 2, 28, 3], [33, 1, -6, -9], [-5, 6, 11, -3], [27, -10, -9, -7], [6, -11, 5, 2], [-7, -5, -24, -9], [-3, -8, 23, -5], [-16, -12, 12, -8], [2, 14, -3, 14], [-20, -8, 0, -16], [27, 7, -10, -11], [21, 10, -4, 7], [1, 21, -10, -1], [-5, -7, -36, 1], [-17, -5, -24, -1], [12, -3, -24, -4], [-14, 2, 12, 6], [8, 2, 38, -8], [-20, 0, -15, -4], [-46, -2, -24, 2], [12, -12, 12, 8], [-8, -8, 12, 0], [16, 16, -14, 2], [-30, -12, 12, 0], [-4, 6, 36, 2], [2, -9, -13, -14], [0, 0, -4, 12], [4, 2, -10, -8], [16, -4, 24, -12], [15, -21, 2, 5], [-3, 11, -29, -5], [54, -6, 27, 10], [12, 16, -20, 4], [-41, 2, 3, 1], [-15, 11, -2, 5], [34, 1, -20, -2], [-48, 8, -12, 0], [-12, -8, 14, -10], [2, 5, -7, 14], [3, -6, -2, -3], [1, 0, -13, -17], [4, 6, -26, 4], [-30, -3, 8, -2], [29, 7, 4, -9], [-14, 8, 14, -2], [-29, -6, 13, -9], [50, -6, 28, 6], [-20, -8, 14, 6], [-5, 1, 11, -3], [-22, 0, 2, 14], [-8, 12, -16, 4], [-13, -3, -12, -3], [-15, -3, -6, 7], [7, -14, 3, 9], [7, -2, 50, -3], [-11, -3, -26, 3], [28, -1, -31, -4], [36, 12, -8, 8], [-29, 5, -2, 7], [-6, 7, -34, -10], [22, -4, 34, 10], [-6, 0, -16, 0], [0, 13, 15, 16], [29, 3, 12, 7], [-31, 1, -6, 15], [9, 0, 5, -3], [32, -1, 13, 12], [9, -12, 7, -1], [33, -1, -20, 3], [27, 3, -2, 13], [3, -18, 20, 1], [39, -2, -12, -11], [23, -10, 16, 5], [25, 2, 4, -1], [-27, 3, -12, -5], [56, -2, 10, -8], [-3, -9, 40, -5], [63, 1, 0, -11], [1, 0, -45, -5], [-18, 2, -12, 10], [17, -8, 4, 7], [-9, 12, 13, 9], [-11, -14, 13, 7], [-16, 4, -4, -16], [-60, 1, 4, -4], [-8, -1, -19, 0], [8, -4, -24, 4], [27, 3, 22, 5], [3, 11, -12, -5], [-14, 19, -1, 2], [-3, 6, 2, 7], [-4, 8, 0, 8], [24, 13, -13, 0], [-17, -4, -32, 1], [80, 2, -2, 0], [-8, 6, 26, 0], [-13, -13, 29, 5], [8, 11, -36, -8], [6, -15, 17, 6], [57, -8, -13, -9], [0, -16, -10, -10], [37, 5, 14, 7], [-5, 13, -16, -15], [29, 4, -28, -9], [-10, 8, 12, 4], [24, -16, 16, 4], [-2, -12, 42, -2], [37, 7, -14, 5], [18, -9, 36, 14], [33, -13, 4, 7], [10, 14, 20, 6], [-27, 17, 9, 3], [17, 2, 25, 19], [-32, 1, -9, 0], [-14, -17, 6, 6], [20, 4, -8, -4], [-10, 6, 28, 14], [47, -2, -10, -3], [-16, -11, 0, 0], [-24, 20, 4, 12], [-55, 5, -2, 7], [41, 17, -14, -5], [10, -4, 34, -2], [48, -14, -1, 0], [28, 3, -23, 4], [32, 2, 30, -12], [29, 0, 11, -9], [-47, 6, -8, -9], [29, -4, -34, -1], [-23, -1, -44, -1], [5, 3, 44, -9], [3, -4, -11, -11], [4, -3, -9, 8], [26, 8, 22, -14], [27, -9, 34, 5], [37, 7, 24, -9], [41, -17, -16, -9], [-30, 20, 3, 6], [-2, -4, 14, 10], [-15, 1, -18, -13], [51, 10, -25, 9], [-6, 20, -33, -10], [-5, -2, -4, -11], [-3, -14, 37, -5], [24, -6, -6, 0], [38, 2, 24, 6], [13, -16, 30, 15], [-4, 0, 36, 4], [-33, -2, 25, 7], [7, -7, -4, -19], [-30, -22, 24, 2], [-38, 11, 2, -2], [-6, 0, -2, -2], [36, 0, 12, 8], [5, -5, 12, -13], [-4, 1, 19, -12], [-30, 6, -14, 0], [-9, -9, 38, 5], [11, -19, -14, -13], [-34, 10, 0, 2], [58, -14, 4, 2], [27, 9, -16, 5], [59, 3, -14, 1], [2, 8, 2, 14], [-4, -12, 20, -8], [72, -14, -18, -8], [25, -7, -16, -3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7350_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_du();". // 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_7350_2_a_du(:prec:=4) chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_du();". // 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_7350_2_a_du( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![32, -16, -18, 0, 1]>,<13,R![-16, 48, -26, 0, 1]>,<17,R![124, 8, -32, -4, 1]>,<19,R![-448, 224, 12, -12, 1]>,<23,R![128, 0, -36, -4, 1]>,<31,R![64, -128, 78, -16, 1]>],Snew); return Vf; end function;