// Make newform 2475.2.a.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2475_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_2475_2_a_s();". // 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_2475_2_a_s();". // 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_2475_a();" function MakeCharacter_2475_a() N := 2475; order := 1; char_gens := [551, 2377, 2026]; 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_2475_a_Hecke(Kf) return MakeCharacter_2475_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, 1], [0, 0], [0, 0], [-2, 3], [-1, 0], [-3, -2], [1, -1], [3, -6], [4, -5], [3, -1], [-3, 0], [-7, -2], [3, 0], [6, 0], [-1, 8], [-2, 7], [-7, 4], [-8, 5], [-8, 0], [8, -10], [-12, 1], [1, 3], [-15, 3], [15, -5], [0, -1], [-2, 5], [-7, 1], [-1, -2], [5, -5], [11, -14], [-1, 1], [-2, -5], [-9, 9], [-13, 6], [-4, 8], [-3, 0], [-14, 12], [2, 3], [-15, 6], [1, -4], [12, 1], [2, -5], [13, 5], [0, -8], [-12, -5], [5, -5], [17, 0], [9, -16], [-3, -13], [-9, 13], [-12, 17], [-8, 1], [-3, 15], [3, 15], [2, -8], [-21, 0], [-19, -7], [-3, 10], [5, 4], [-7, 20], [9, -1], [2, -16], [7, -25], [3, 10], [0, 2], [15, 1], [-3, -10], [18, -12], [6, 9], [-17, 24], [-11, 20], [-2, 4], [-12, 3], [-19, 10], [-21, 2], [18, -8], [-12, 4], [-23, 5], [-2, -20], [-9, 18], [-5, -10], [12, -15], [-17, 10], [-7, 16], [-25, 15], [-14, -14], [4, 7], [-26, 21], [-7, 10], [21, -20], [31, 4], [-9, -12], [11, 12], [23, -10], [12, 11], [16, -14], [-18, 1], [18, -10], [-10, -8], [-18, -15], [26, -3], [-13, 2], [8, -13], [-28, 1], [12, 15], [-1, -9], [17, -13], [9, 0], [11, -7], [2, -15], [-10, 4], [-21, 9], [-22, -10], [3, -16], [12, 15], [3, 0], [-30, 12], [35, -14], [-23, 19], [15, -15], [-33, -10], [17, -12], [6, 14], [-11, -20], [22, 5], [-32, 20], [4, -28], [-7, 14], [17, 15], [-7, -24], [-34, 3], [30, -7], [-13, 15], [15, 4], [-12, 0], [-5, 0], [16, 11], [-31, 6], [21, -11], [36, -12], [17, 0], [28, -20], [-8, -12], [-10, 26], [27, -39], [-18, 41], [-45, -3], [-37, -10], [0, 10], [35, -22], [-24, 12], [-2, -5], [9, 14], [-23, 32], [24, -24], [18, 0], [4, 12], [-19, -22], [0, 24], [23, -40], [-7, -20], [2, 4], [-13, -15], [-32, 5], [15, 26], [-8, 4], [-3, 15], [-37, 3], [-27, 14], [-6, 30], [-42, -1], [22, 5], [43, 5], [36, -5], [-11, -18], [-2, 4], [-8, 0], [13, 20], [-12, 6], [36, -12], [10, 14], [-12, -5], [13, 6], [-6, -32], [1, 16], [-21, 12], [-16, -14], [-24, 15], [7, -4], [8, 20], [27, -27], [-44, 16], [17, 0], [3, -5], [-22, 25], [-35, -2], [-18, 30], [-21, -6], [2, -23], [33, -33], [-58, 11], [-38, 15], [-31, -9], [39, -33], [26, 8], [-22, 5], [-32, 44], [27, 24], [-26, 52], [-8, 35], [-3, -30], [-37, 10], [-30, 37], [30, -54], [35, -10], [32, -50], [8, -12], [13, -25], [24, -27], [28, -43], [-33, 35], [-29, 8], [40, 20], [-7, -9], [-1, 38], [25, 30], [56, -19], [-46, 22], [24, -24], [23, -25], [-44, 15], [-33, -24], [-33, -20], [-17, 10], [27, -7], [-15, -4], [32, 6], [-4, 41], [-33, 11], [-42, 35], [15, 18], [-8, -30], [4, -6], [51, 8], [-12, 22], [-8, -19], [-6, -29], [-12, 15], [49, -33], [-6, -10], [-30, -26], [-57, 20], [28, 10], [3, -36], [22, -31], [24, -58], [7, -30], [-33, 40], [30, 1], [-22, 28], [32, -32], [-24, -6], [-7, -36], [-9, -25], [29, -2], [44, -38], [23, 4], [48, -25], [3, 26], [-25, -22], [62, -30], [45, -6], [18, -54], [35, -30], [-21, 21], [1, -40], [-24, -16], [-33, -9], [2, -45], [-7, 20], [-39, -19], [32, 0], [-42, 15], [-43, 30], [56, -33], [-32, 20], [-1, 9], [12, -33], [-24, 33], [27, 1], [-17, -40], [-3, -8], [24, -40], [3, 15], [-11, -6], [4, -23], [-3, 0], [-23, -26], [50, 20], [23, -12], [-33, 28], [6, -36], [48, -11], [69, 0], [7, 45], [3, 38], [-6, 38], [24, -33], [23, -1], [-17, 66], [-35, 53], [15, 35], [-32, 55], [33, -4], [29, 38], [20, -30], [-6, -8], [-12, 10], [-34, 30], [66, -27], [42, -5], [6, -53], [63, 0], [27, -42], [71, -19], [-38, 60], [5, -15], [74, 9], [22, -38], [-42, 52], [52, -30], [-33, 12], [-51, 27], [39, -30], [-48, 50], [-42, 40], [-22, -6], [-28, 34], [32, -65], [52, -15], [0, 42], [55, 16], [-35, -20], [12, -15], [-61, 20], [-47, 29], [-3, -15], [53, 18], [33, -55], [-51, 8], [-28, 55], [-8, 15], [33, 30], [73, -9], [-44, -12], [15, 18], [-10, 20], [-12, -25], [29, -52], [16, -4], [24, -69], [-57, -20], [-49, 34], [-9, -7], [46, -3], [47, 18], [10, 46], [15, 2], [-28, 40], [83, -25], [-4, 48], [-2, 47], [77, 11], [47, -65], [39, -24], [56, -7], [-42, 35], [-46, 4], [-16, -53], [-38, 45], [-82, 10], [7, -26], [28, 18], [61, 19], [-87, 24], [-21, 15], [-18, -25], [48, -72], [21, -60], [34, -32], [4, -13], [68, -28], [3, 24], [42, -30], [-37, -10], [-37, -24], [-19, 3], [20, -10], [-53, 20], [48, 5], [40, -60], [26, 11], [34, 21], [81, -36], [51, -12], [47, 0], [-24, 62], [13, 10], [-8, 48], [62, -14], [-45, 57], [43, -50], [15, -2], [-28, -35], [6, -8], [-47, 10], [-73, -4], [-31, 21], [-33, -18], [79, 5], [21, 38], [-75, 24], [-27, 50], [-61, 32], [-40, 42], [0, -24], [-18, 4], [-25, 65], [-18, 60], [23, -71]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2475_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2475_2_a_s();". // 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_2475_2_a_s(:prec:=2) chi := MakeCharacter_2475_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_2475_2_a_s();". // 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_2475_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2475_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -1, 1]>,<7,R![-11, 1, 1]>,<29,R![5, -5, 1]>],Snew); return Vf; end function;