// Make newform 8025.2.a.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8025_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_8025_2_a_r();". // 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_8025_2_a_r();". // 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_8025_a();" function MakeCharacter_8025_a() N := 8025; order := 1; char_gens := [5351, 1927, 751]; 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_8025_a_Hecke(Kf) return MakeCharacter_8025_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], [-1, 0], [0, 0], [2, -2], [-2, 0], [1, 0], [1, 4], [1, -2], [4, 0], [-2, 2], [-2, 0], [-5, 8], [-4, -2], [2, -4], [2, -6], [10, -4], [-4, 8], [-5, 8], [-8, 4], [1, -6], [-6, 6], [-8, 0], [-4, -6], [-4, 12], [-10, 0], [-2, -8], [-6, 12], [-1, 0], [14, -6], [9, -16], [14, -6], [11, -10], [-12, 6], [-2, 2], [-10, -6], [-11, -2], [6, -6], [20, -8], [5, -14], [-1, -12], [5, 2], [14, -8], [-5, -10], [-14, 12], [-14, 6], [-13, -2], [-18, 10], [19, -10], [-18, -6], [-2, 4], [-4, 14], [-6, -4], [13, 4], [-10, -6], [22, 4], [12, 12], [17, 0], [-7, -10], [-14, -6], [10, 4], [-7, -10], [4, 12], [-2, -8], [13, -18], [-21, 0], [-10, 12], [-17, -6], [-3, -8], [15, 6], [18, -22], [-26, 12], [-17, 30], [-18, 10], [-1, -4], [0, 2], [24, -6], [-14, 0], [2, 0], [18, -8], [8, 10], [-3, -14], [-17, 20], [-8, 22], [16, -14], [-8, 20], [4, 0], [-25, 12], [-29, 0], [-12, 6], [-16, 12], [-24, 2], [-13, 10], [6, 18], [-21, 14], [14, 0], [0, -8], [18, -38], [-37, 8], [-16, 24], [-4, 2], [35, -6], [5, -16], [-5, 18], [10, -8], [7, -22], [-3, 20], [-6, 28], [25, -28], [-34, -4], [-22, -16], [10, -20], [-8, -12], [5, -4], [-24, 6], [12, -10], [-5, -8], [-9, -10], [4, 16], [-4, -8], [4, -8], [14, 16], [24, -8], [-18, 40], [-2, -18], [-3, -14], [7, -24], [-4, 10], [17, -10], [-27, 18], [32, -30], [-6, 22], [-10, 4], [-22, 34], [18, 16], [-8, 28], [-22, 4], [2, 16], [-24, 40], [30, -30], [13, 16], [-12, 24], [31, 4], [-12, 8], [17, -10], [10, 0], [8, -26], [26, 14], [-20, 38], [-37, 6], [25, -26], [-28, 40], [-8, -24], [41, -10], [-43, 14], [-20, -14], [-12, 28], [-6, 14], [-29, 20], [-7, -20], [16, -14], [36, -12], [-26, 4], [39, -26], [-1, 22], [-6, 28], [43, -6], [-30, -6], [-7, 8], [-12, -18], [-35, 0], [22, -12], [18, -16], [-5, -18], [14, -36], [-57, 6], [8, 12], [-16, -20], [9, 0], [-51, 2], [38, 4], [18, 10], [-5, 26], [-18, -8], [-2, 4], [4, -6], [36, 8], [-2, 8], [-44, 12], [-14, -10], [24, 16], [-9, -8], [28, -44], [28, 8], [-28, 26], [-8, -30], [-40, -2], [-42, 4], [-10, -16], [58, 0], [-20, 0], [48, -8], [30, -6], [-4, -22], [60, -2], [8, -24], [34, -2], [8, -16], [-20, 52], [-26, 28], [8, 26], [31, -44], [-2, -40], [-29, 50], [-24, 36], [16, 20], [-25, 36], [-24, 30], [-27, 24], [26, -22], [54, -24], [-14, -4], [48, -8], [47, -4], [24, -12], [-40, 40], [14, 34], [20, -12], [22, 2], [16, -8], [37, -30], [-59, 20], [12, 12], [4, -34], [2, -22], [-35, -14], [-9, -26], [-20, -10], [50, 4], [-16, 24], [-32, -6], [-18, 20], [35, -46], [-8, 4], [16, 0], [-35, 28], [-40, 10], [21, -50], [-23, -26], [-8, 0], [-28, 2], [1, 4], [37, -16], [25, 26], [-29, 0], [41, -24], [-8, 30], [-26, -20], [52, -32], [6, -28], [10, -24], [6, 12], [-24, 34], [14, 40], [2, -20], [-16, -10], [4, -4], [-10, -12], [26, 18], [-35, 38], [-62, 0], [2, 12], [-23, 2], [-10, 32], [-8, -16], [40, -8], [-30, 4], [20, 2], [28, -26], [37, 4], [60, 4], [-12, 50], [-27, 30], [31, 10], [-7, -4], [25, 10], [2, -38], [40, -38], [55, -36], [-22, -4], [35, 28], [-47, 4], [20, -16], [-75, 8], [-15, -10], [-58, 32], [-16, -10], [-43, 22], [14, 26], [8, 38], [-37, 6], [10, 22], [-13, 32], [-24, -28], [-65, 18], [0, 32], [-8, 10], [-14, 42], [70, -8], [-39, -22], [8, 40], [38, -44], [36, 16], [48, -54], [-4, -28], [-24, 32], [10, -40], [41, 26], [44, -16], [-18, -22], [45, 0], [-30, 8], [40, -72], [-34, 60], [5, -54], [-14, 36], [18, -36], [-40, 20], [10, 8], [21, 6], [-38, 12], [13, 12], [37, -40], [-1, 10], [40, -8], [45, -26], [23, -38], [-72, 10], [-30, 12], [3, -16], [29, -62], [-12, 6], [-7, -36], [-34, 12], [-41, -26], [-6, -2], [-12, -4], [16, -16], [19, -70], [-5, -30], [14, -24], [-62, 40], [19, -24], [6, -44], [0, -56], [-45, 28], [-74, -6], [-11, -30], [0, -32], [-43, -8], [56, -4], [10, 8], [-32, -20], [-27, 62], [-50, 6], [-14, 14], [7, -36], [-12, 52], [3, -38], [-14, -34], [-12, 40], [-38, 52], [-84, 12], [14, -48], [33, 26], [56, -34], [4, -46], [58, -10], [-6, -16], [-44, 28], [-10, 32], [-50, 14], [-10, -32], [-29, 22], [12, -12], [12, 24], [52, 24], [50, -46], [-42, 4], [-52, 32], [-4, 28], [-33, 40], [26, -24], [-43, 74], [-42, 70], [-36, 44], [-14, 42], [1, -38], [-41, 76], [29, -2], [74, -24], [-24, 22], [13, 24], [-48, 24], [30, 0], [38, -8], [48, -8], [78, -20], [-36, 50], [54, -2], [6, 34], [-2, 12], [-89, 4], [-82, 14], [20, -4], [-3, 24], [52, 22], [55, -50], [16, -68], [-2, 36], [18, -54], [23, 0], [-4, 24], [-24, -24], [31, -38], [-3, 16], [81, -16], [9, 6], [19, -20], [-60, 30], [18, -46]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8025_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8025_2_a_r();". // 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_8025_2_a_r(:prec:=2) chi := MakeCharacter_8025_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_8025_2_a_r();". // 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_8025_2_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8025_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -1, 1]>,<7,R![-4, -2, 1]>,<11,R![2, 1]>,<13,R![-1, 1]>],Snew); return Vf; end function;