// Make newform 7448.2.a.bd in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7448_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_7448_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_7448_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 := [-2, 0, 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_7448_a();" function MakeCharacter_7448_a() N := 7448; order := 1; char_gens := [1863, 3725, 3041, 3137]; v := [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_7448_a_Hecke(Kf) return MakeCharacter_7448_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], [2, 1], [-1, 0], [0, 0], [1, -3], [-2, -1], [-4, -2], [-1, 0], [3, 1], [-8, -1], [6, -1], [-8, 1], [0, -4], [1, -3], [3, -3], [-4, -1], [2, -2], [-5, 0], [6, -2], [-12, 3], [-7, 0], [-6, 7], [5, 7], [-2, 11], [0, -4], [-7, 6], [8, -4], [10, 5], [-8, -8], [4, 5], [-4, -11], [-4, 4], [-5, 4], [-3, -11], [-1, -4], [0, 4], [-13, 8], [7, 5], [20, -1], [-4, -4], [-2, 6], [-2, -9], [5, 3], [-2, 3], [-9, 6], [-5, 5], [-2, 10], [-12, -3], [0, 1], [16, 0], [-12, 4], [-10, 4], [-4, 9], [-1, -15], [14, 0], [-24, 2], [-16, 0], [15, -9], [-11, -12], [-8, -7], [-1, 11], [4, 4], [-4, 16], [8, 0], [-13, 10], [-6, -3], [-10, -6], [-2, 5], [11, 1], [0, 4], [24, 0], [7, 21], [-6, 16], [-12, 4], [14, -1], [-8, 3], [-4, -8], [-10, 0], [14, -16], [-18, -8], [-1, -21], [-26, 1], [-4, -7], [12, 0], [18, 6], [-28, 8], [8, 16], [3, 2], [-9, 14], [-15, -9], [5, -9], [-17, 5], [-26, 10], [11, -21], [-5, 3], [9, 9], [-2, 0], [-12, -7], [-4, 8], [-15, -10], [-6, -11], [27, 6], [16, 19], [6, -8], [-13, -21], [31, -2], [-34, 0], [-7, 8], [0, 15], [16, -23], [6, 7], [-24, -6], [-1, 20], [19, -7], [1, -13], [22, -8], [36, -6], [35, -5], [-16, -10], [6, 13], [-18, 19], [-18, -16], [0, 8], [-10, -10], [6, -16], [-1, -14], [13, 28], [-8, 8], [-1, 3], [-40, -8], [22, 16], [34, -9], [14, -6], [-43, 8], [1, -10], [-29, 18], [-8, 5], [16, -3], [4, -35], [-11, -12], [-10, 30], [25, -12], [-3, 9], [-12, 8], [28, 8], [22, 6], [-27, -18], [24, -12], [11, -27], [2, -19], [10, 19], [-4, 8], [-28, 2], [0, -20], [-34, -11], [-10, -11], [7, -1], [-13, -16], [39, -6], [26, 1], [-26, 0], [-4, -39], [0, 12], [22, 19], [6, 8], [-16, -24], [-18, -18], [-20, 28], [6, -36], [3, -38], [-20, 11], [-12, 17], [12, 22], [-3, -28], [22, -9], [21, 2], [45, -3], [9, 12], [-20, -19], [7, 16], [-19, 7], [-6, 30], [-8, 13], [-8, -3], [36, -6], [10, 32], [10, 0], [18, -6], [-8, 23], [-13, -15], [-38, 4], [4, 8], [0, -8], [-8, -33], [-12, -24], [8, 25], [9, -8], [11, -24], [-23, -8], [-39, -9], [54, 8], [-40, -12], [28, -4], [-18, -28], [20, -24], [-41, 6], [-3, 21], [32, 5], [20, -30], [40, 17], [-19, -26], [8, 30], [15, 15], [14, -14], [-28, 3], [-20, -3], [-9, 1], [16, -17], [44, 4], [25, -12], [16, -4], [-24, 16], [24, -23], [28, -14], [24, 1], [11, 12], [6, 27], [2, -42], [20, -9], [23, 11], [9, -16], [-24, -33], [12, 43], [-12, -35], [-24, -22], [4, -6], [33, 6], [-4, 16], [-17, -11], [42, 5], [-32, 7], [-1, -35], [-3, -7], [-24, 3], [-6, 32], [29, 15], [32, -8], [0, -1], [28, -21], [23, -19], [-65, 0], [4, 30], [0, -16], [-6, -9], [-3, 10], [-11, 33], [17, 14], [-26, 26], [-6, 25], [0, 48], [2, -19], [0, -29], [64, 12], [4, 39], [-9, 2], [16, 12], [-10, -16], [23, 8], [-6, 6], [-39, 12], [-20, 12], [-36, 20], [31, -26], [17, 11], [32, -9], [-16, 12], [49, -5], [-36, 28], [-20, -4], [9, -1], [18, -30], [40, 28], [21, 43], [-2, 20], [52, -18], [-25, -5], [-8, 8], [-26, -12], [15, 3], [-24, 0], [-1, -4], [11, -15], [8, 20], [-56, -4], [-8, 24], [8, -28], [32, -15], [-24, 12], [-14, 6], [-55, 21], [-69, -2], [-2, 56], [-11, -9], [26, 2], [21, -23], [20, 48], [-34, 22], [-18, -32], [-24, -40], [-69, -4], [5, 1], [-23, 8], [24, -23], [12, -21], [-2, -8], [50, -16], [29, -39], [-8, -25], [12, 36], [67, -16], [-24, -23], [-12, -26], [-34, -31], [-4, -12], [57, -2], [-10, 12], [-22, -47], [24, -43], [26, 27], [-42, 24], [13, -22], [-24, -4], [76, 10], [3, 27], [4, -24], [11, -23], [-6, -8], [-8, -52], [-49, 24], [15, 11], [-38, -36], [11, 32], [-58, 0], [-22, -14], [-24, 16], [56, 12], [8, 10], [-14, -26], [54, 27], [5, -12], [30, 3], [-24, -23], [-27, 48], [36, 12], [-14, -25], [6, 11], [9, -25], [-51, -7], [33, 2], [12, -16], [32, 6], [4, -6], [-50, -17], [12, -48], [35, 5], [-46, -32], [-6, 40], [-36, -1], [66, -12], [26, -12], [-44, 36], [-33, -25], [34, -17], [48, 4], [-84, -4], [-30, 23], [19, 11], [-29, -36], [5, 22], [34, 8], [-80, -5], [89, 6], [-32, -8], [61, -14], [-6, -2], [64, 16], [-19, -17], [-11, 12], [68, 2], [-22, -13], [12, -4], [10, 15], [-43, -1], [-32, 6], [14, 23], [-24, -27], [-22, -2], [-18, -1], [56, 16], [11, 6], [-24, -20], [45, 28], [4, 51], [0, 3], [-46, 20], [63, -19], [40, 2], [42, -12], [-22, 46], [-36, 18], [-2, -51], [57, -32], [-42, -19], [-20, 8], [-91, -6], [35, 26], [10, -15], [70, -6], [-7, -6], [-18, -22], [-18, -45], [18, 24], [79, 1], [-32, -37], [-38, -39], [-10, 55], [-78, 2], [31, 50], [-45, -9], [17, 21]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7448_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7448_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_7448_2_a_bd(:prec:=2) chi := MakeCharacter_7448_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_7448_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_7448_2_a_bd( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7448_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![2, -4, 1]>,<5,R![1, 1]>,<11,R![-17, -2, 1]>,<13,R![2, 4, 1]>,<17,R![8, 8, 1]>],Snew); return Vf; end function;