// Make newform 3025.2.a.bd in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3025_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_3025_2_a_bd();". // 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_3025_2_a_bd();". // 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, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -1, 1, 0], [1, -2, -1, 1]]; Rf_basisdens := [1, 1, 1, 1]; 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_3025_a();" function MakeCharacter_3025_a() N := 3025; order := 1; char_gens := [727, 2301]; v := [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_3025_a_Hecke(Kf) return MakeCharacter_3025_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], [1, 0, -1, -1], [0, 0, 0, 0], [0, 1, 2, -1], [0, 0, 0, 0], [0, -3, 0, 2], [-1, -1, -1, 3], [-5, -2, 1, 0], [-2, 1, 1, 0], [-5, 2, 1, 2], [0, -5, 0, 0], [-3, -3, -3, 7], [-1, 1, -3, -1], [6, -1, -2, 0], [-3, 1, 3, 0], [6, -1, -5, -1], [2, -1, -4, 5], [-2, 2, -2, -1], [5, 3, 4, -6], [3, -3, 0, -2], [1, 3, 2, 0], [-9, -4, -3, 6], [-1, -1, 3, -6], [1, 0, -6, 0], [-5, 0, 2, -8], [-2, 3, 7, -1], [1, 0, -4, -6], [-2, 8, 3, -7], [-5, 1, 4, -6], [-1, 1, -1, 2], [5, 2, -3, -3], [-6, 1, 6, -2], [8, -9, -4, 3], [-8, 5, 5, -7], [-1, -6, -5, 7], [-12, 2, 9, 5], [-6, -3, -6, 10], [-4, -2, 9, 4], [-7, -5, -6, 7], [-11, -1, -2, 3], [-5, 12, 2, 0], [0, 3, -8, 4], [-1, 5, 6, 0], [7, 5, -9, -2], [2, 5, -1, -7], [-4, -2, 3, -2], [-17, -1, -2, 12], [6, 2, -2, -10], [10, 2, -7, -7], [1, 1, 13, 2], [-12, 13, 6, 3], [4, 7, -7, -16], [-2, 10, 0, -9], [9, 2, -1, -9], [-7, 1, 3, 14], [-1, -7, 0, -4], [-4, 5, 6, 4], [-7, 7, 0, -5], [6, 2, 0, -1], [6, 8, -3, -16], [17, -9, -10, 3], [-10, 1, -2, -7], [0, 5, 7, 7], [-9, -13, -7, 16], [6, -8, 5, 2], [-6, 13, 7, -4], [-2, 20, 4, -12], [11, 6, -6, 5], [5, -3, 0, 1], [-10, -7, 0, 16], [4, 17, -2, -7], [5, -6, -8, 0], [3, 2, 11, -8], [1, -6, -5, -11], [2, -2, 4, 9], [16, 5, 5, -7], [11, 22, 6, -18], [11, 2, 0, 3], [-15, -11, -3, 13], [9, 8, 2, -3], [-11, -9, 9, 5], [5, 15, 0, -17], [-2, -11, -13, 15], [-8, -18, 7, 17], [0, 0, -5, 8], [3, 17, -2, 3], [16, 0, -13, -7], [9, 8, -6, -5], [8, -5, -3, 2], [-19, 2, 9, 17], [-18, -4, 6, -9], [8, -13, -1, 2], [12, -12, 3, 1], [-15, -5, -4, 0], [10, -12, -11, -5], [-38, 11, 5, 0], [3, 1, 6, -9], [-5, -13, 0, -3], [-12, 2, 5, 12], [4, 2, 4, -1], [2, -2, 1, 14], [-5, -20, 5, 22], [-9, -2, 9, -2], [-3, -5, 13, 20], [-6, -1, 0, -16], [-26, 13, 5, 8], [11, 10, -15, -16], [-36, 2, 1, 6], [-5, 3, 3, -12], [24, -9, 0, -16], [-2, -11, 13, -6], [-11, 15, 11, -4], [0, -5, -7, 15], [21, -10, -17, -5], [-15, -16, -12, 20], [-21, -12, 6, 12], [-9, 8, 17, 4], [-1, -4, -2, 29], [-9, 25, -3, -6], [-10, 7, 2, -9], [-18, -14, 9, 8], [4, 20, -13, -7], [19, -4, -22, 6], [15, -11, 1, 8], [10, -2, -5, 15], [-21, 23, 9, -1], [-15, 10, 7, 2], [-19, 9, 15, -6], [-19, 0, 18, 6], [-18, 4, 14, -11], [-4, 0, -17, -8], [2, 4, -6, -30], [-5, -4, 12, 13], [24, 0, -6, -6], [-7, 7, 9, -16], [5, -18, 10, 20], [-9, -19, 12, 7], [24, -6, -8, -11], [17, 6, -5, -7], [-3, 2, 10, -8], [-4, 17, -3, 5], [-30, -7, 0, 31], [4, -24, 4, -1], [26, 1, -3, -35], [-7, -5, 5, 5], [12, -20, -3, 8], [-11, -10, -17, 12], [29, -20, -7, -7], [7, -3, -11, -2], [-5, -2, -8, 13], [13, 14, 4, -14], [14, 10, 17, -15], [0, -13, 8, -19], [11, -16, -7, -17], [-2, -23, 2, 8], [8, 8, -2, -20], [24, -8, -6, -22], [-8, 14, -11, 13], [12, -7, -13, 12], [-38, -8, 5, 4], [-9, 11, 6, -19], [-17, -10, 9, 13], [12, 11, -4, -7], [-17, -7, 2, 3], [6, -15, -17, -1], [-23, -8, 14, -17], [15, 7, 1, 0], [-8, -19, -2, 17], [2, 16, 25, -24], [32, 2, 3, -42], [-1, 20, -14, -25], [-33, -8, 3, 24], [-12, -10, 1, 4], [8, -24, -1, -1], [18, 1, -4, -36], [49, -3, -2, -17], [-8, 0, -17, 6], [-2, -6, 1, 29], [9, 20, -8, -16], [15, -23, -17, 11], [22, 5, 9, -6], [-9, 2, 1, -21], [18, 34, 5, -25], [23, 30, 12, -32], [-33, -16, -8, 41], [-14, -6, -2, -22], [5, 9, 20, -11], [-4, -14, -16, 8], [14, -17, -7, -1], [-9, 5, 3, 13], [11, 1, -14, 5], [-5, 6, 5, 2], [3, 16, -5, -14], [2, 16, -22, -19], [-50, 25, 18, 20], [-32, -20, 1, 27], [19, -11, -27, 2], [-7, 7, 18, 4], [2, 28, -10, -9], [-8, 15, -21, -20], [-6, -10, -5, -10], [26, 20, -20, 14], [-11, 7, 8, 17], [-17, -14, -9, 42], [-8, -15, 22, 3], [-26, 3, 20, 28], [29, 1, 7, 7], [-21, -25, -4, 30], [-31, 28, 6, 11], [28, 9, -8, 2], [14, 1, -5, -19], [16, 5, -13, 3], [-38, 16, 3, 18], [-36, 22, 10, 0], [-11, -12, -8, 13], [-20, 22, -3, 5], [-36, 9, 33, -12], [11, 1, -30, -27], [16, -15, -12, -6], [10, -13, 17, 17], [16, -37, 5, 17], [-7, 9, 4, -32], [-25, 3, 12, 45], [29, -6, -13, -2], [-18, -15, 8, -18], [31, 12, -7, -27], [14, -21, -2, 22], [-1, -8, -9, 1], [-44, -6, -11, 14], [-15, 5, 28, -6], [-4, -8, 14, 20], [-16, -9, -11, -12], [5, -14, 0, -23], [-29, -6, 0, 25], [-12, -30, -2, 10], [-37, 24, 4, 10], [6, -5, -7, 17], [-25, 4, 29, 10], [7, 21, 26, -23], [-44, 5, 16, -5], [-9, -16, 24, -1], [-11, -24, 14, 17], [-1, 24, -13, -35], [-7, 11, 16, -15], [-15, 3, 12, -2], [-31, -3, 28, -7], [44, 4, -3, 17], [-23, 22, 11, -2], [15, -16, -2, -18], [-34, -14, 2, 22], [23, -26, 4, 20], [15, 28, 24, -35], [-20, 18, 10, -27], [-13, 4, 3, 7], [21, -10, -5, 14], [-15, 16, -10, 5], [32, -15, -18, 17], [-7, 2, -21, 6], [-25, -33, 13, -1], [-18, 12, -6, 30], [18, 4, 18, 5], [-23, 33, 11, 9], [34, 1, -14, -33], [-16, 13, 24, -1], [15, -18, -18, -14], [-41, 5, 7, 6], [-1, 5, -4, 22], [2, 21, 15, -30], [60, -36, -26, -18], [24, -4, 5, -2], [39, 2, -10, -26], [-20, 10, -14, 7], [9, 13, 23, -9], [-19, 6, -13, 38], [21, -28, -40, 9], [-2, 24, 6, -10], [20, -21, 5, -22], [-9, -3, -22, -15], [3, 24, 10, -11], [-4, 17, 32, -14], [-11, -2, -4, -18], [-25, -15, 3, 23], [-47, -7, 29, -9], [8, 28, -7, 7], [22, 33, 21, -37], [22, -27, 15, 24], [15, 2, 2, 2], [12, 11, 21, -4], [18, 14, -10, -5], [20, -5, 5, -25], [2, 3, 35, -7], [36, -24, 3, 10], [30, -12, -29, 11], [13, 10, 20, -8], [24, 14, -35, -5], [7, 14, -9, -56], [-4, 2, -3, -20], [-9, -4, -1, 0], [30, -52, -21, 17], [-21, -1, 26, -20], [7, 12, -9, -40], [-40, -39, -13, 48], [9, 20, -29, -5], [5, 16, 6, -25], [42, -37, -29, 6], [20, 19, 1, -38], [15, 3, -16, 0], [19, -25, 19, 20], [45, -18, -17, -2], [-10, 9, 23, 1], [-22, 31, 16, -28], [34, -1, -19, -19], [-7, -18, 24, 40], [21, 5, -15, 22], [-47, -19, 0, 33], [20, -20, 9, -15], [-25, -7, -13, 45], [-2, -17, -22, 21], [-10, 3, 32, -10], [19, 0, 4, 2], [19, -23, -6, 23], [-3, -20, 5, 28], [4, -33, -12, 1], [-33, 43, 20, -3], [19, -12, 15, -4], [46, -19, -27, 3], [0, 29, -1, -44], [43, -23, -15, 7], [-37, 8, 24, 8], [-4, -3, 3, -36], [-20, 4, 11, -13], [46, -30, -20, -20], [25, -9, 4, -38], [13, -43, 3, 13], [8, -15, 31, 4], [-9, -24, 0, 33], [7, 29, 18, -24], [15, 31, -9, -47], [-49, 15, 18, 34], [-2, -34, -26, 16], [21, 0, -15, -40], [20, -15, 13, 12], [8, 41, 10, -29], [-4, 11, 42, -28], [45, -25, -42, 9], [23, -16, -11, -23], [-31, -3, -6, -10], [-51, 1, -6, 9], [25, -36, -16, -7], [-13, -1, 21, -24], [-36, -7, -11, 24], [50, 27, -6, -5], [23, -16, -7, -14], [-24, -26, -17, 50], [45, 5, 4, 8], [7, -10, -30, -7], [-37, -2, -7, 64], [27, 3, -14, 39], [15, -22, -47, 7], [-2, -7, -5, 35], [-26, -2, 37, -15], [-39, -27, -16, 37], [-39, 9, 14, -14], [-28, 25, 20, 19], [34, -8, 3, 10], [-24, 30, -4, -18], [27, 19, 38, -12], [17, -31, 1, 24], [-54, -9, 28, -2], [-1, 7, 5, -29], [58, -17, -37, 4], [-29, -1, 16, 41], [-7, -16, 8, -13], [17, -19, 18, 31], [-25, 23, 28, 10], [-1, 16, 15, -2], [11, -31, 17, -4], [0, 33, 4, -34], [4, -11, 4, -2], [35, -13, -18, -31], [-15, 53, 18, -14], [-11, -28, -46, 22], [53, 3, -12, -40], [8, 26, 15, -4], [-35, -17, -16, 12], [14, -39, 11, 12], [-10, 19, 27, -35], [-39, 18, 1, -21], [34, 8, 30, -10], [-2, 22, 46, -30], [-4, -5, -36, 20], [-3, 16, -13, 33], [17, 1, 11, -37], [-4, 9, 19, -1], [20, 26, 11, -35], [7, -1, -36, 20], [-2, 6, 22, 6], [28, -38, -2, 24], [48, -37, -50, 5], [41, 45, 23, -69], [29, -4, -27, -4], [2, 49, 11, -22], [35, -19, 8, 16], [3, -1, -17, 20], [25, -33, -41, 2], [-60, 52, 22, 15], [15, 10, -33, -19], [17, 36, -13, -3], [33, -20, -56, 4], [36, -23, -16, -12], [25, 3, -16, -25], [-20, -34, 12, 25], [27, 24, 37, -33], [22, -45, -19, 12], [67, -30, -26, -8], [17, -9, -3, -56], [-4, 13, 11, 32], [-29, -17, -6, 8], [33, -2, 23, -28], [-22, 23, -13, 23], [-61, -6, 9, -8], [60, -31, -30, -4], [-48, -31, -25, 33], [-12, -22, 32, -28], [-8, 31, 50, -30], [-13, 31, 16, -23], [35, 13, -27, 16], [2, -1, 3, 37], [2, -11, -15, 51], [-33, -35, -3, 46]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3025_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3025_2_a_bd();". // 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_3025_2_a_bd(:prec:=4) chi := MakeCharacter_3025_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_3025_2_a_bd();". // 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_3025_2_a_bd( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3025_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 1, -3, -1, 1]>,<3,R![-1, -5, -6, 0, 1]>,<19,R![25, 275, 130, 20, 1]>],Snew); return Vf; end function;