// Make newform 7448.2.a.n in Magma, downloaded from the LMFDB on 28 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_n();". // 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_n();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; 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], [3], [0], [-1], [0], [-6], [-1], [-3], [6], [2], [0], [6], [9], [-3], [-2], [-12], [11], [-8], [6], [-15], [-12], [7], [-8], [-6], [9], [-14], [-4], [4], [-18], [-16], [-8], [-3], [-17], [-3], [6], [-7], [-19], [-14], [8], [-10], [22], [5], [-20], [3], [5], [-20], [-2], [12], [14], [14], [-24], [4], [3], [-22], [-24], [24], [15], [17], [6], [1], [14], [22], [0], [-13], [-2], [-22], [6], [-5], [22], [-30], [-5], [-8], [32], [-28], [-8], [18], [34], [0], [2], [-5], [2], [-6], [-24], [20], [28], [36], [-25], [1], [13], [33], [3], [26], [-33], [-13], [1], [6], [-12], [18], [11], [-2], [-15], [24], [-18], [23], [7], [12], [3], [14], [-16], [32], [-6], [45], [-1], [-23], [44], [-32], [25], [14], [50], [32], [28], [-48], [-42], [-4], [-19], [13], [32], [-33], [26], [0], [-32], [-40], [-53], [25], [19], [-36], [52], [44], [-29], [-30], [25], [-45], [16], [-24], [4], [-9], [-24], [13], [18], [-32], [-54], [-44], [-58], [26], [50], [-37], [-25], [-43], [-10], [12], [56], [-40], [-8], [12], [6], [52], [42], [-8], [29], [-6], [46], [-36], [47], [38], [-17], [13], [41], [2], [-1], [43], [-46], [-2], [-22], [-32], [10], [-30], [-28], [32], [43], [40], [0], [14], [2], [-32], [6], [-7], [-1], [19], [27], [-40], [26], [38], [40], [-24], [-39], [9], [-54], [-26], [22], [27], [-38], [7], [46], [66], [-4], [-57], [38], [18], [39], [-62], [-32], [-62], [-8], [-24], [11], [38], [-50], [-14], [65], [63], [50], [64], [48], [44], [56], [-51], [50], [-75], [30], [-56], [-41], [-11], [20], [-48], [-31], [-44], [58], [-4], [37], [15], [-50], [-72], [-68], [-5], [45], [41], [42], [-12], [-10], [24], [-26], [-14], [74], [67], [-66], [14], [43], [8], [19], [72], [18], [17], [-65], [-14], [-52], [39], [-66], [-74], [37], [-32], [-64], [33], [-76], [-12], [73], [-34], [-36], [-63], [24], [-3], [-3], [64], [-78], [4], [-62], [-2], [42], [24], [-59], [-67], [-14], [41], [-54], [-79], [72], [76], [-46], [-16], [-57], [85], [-21], [32], [-14], [-8], [46], [-45], [76], [-62], [69], [-2], [-42], [-42], [30], [-39], [50], [2], [52], [24], [-26], [-73], [-12], [-32], [91], [20], [-9], [46], [-60], [-69], [-7], [32], [-43], [4], [-14], [-56], [-6], [-10], [82], [46], [-11], [-10], [-2], [51], [-24], [-58], [-26], [9], [-45], [-19], [-26], [-18], [-38], [22], [84], [5], [6], [-42], [-28], [-22], [-36], [-10], [-9], [-6], [8], [-34], [-98], [23], [-69], [51], [-90], [-30], [71], [24], [59], [26], [-44], [65], [33], [-56], [70], [-48], [22], [-87], [-8], [-62], [-66], [64], [96], [-24], [-17], [10], [3], [-36], [64], [-24], [-27], [-98], [-20], [64], [84], [-4], [-93], [-72], [52], [13], [65], [-54], [80], [41], [-74], [24], [-92], [-57], [54], [-26], [-10], [-6], [71], [13], [53]]; 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_n();". // 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_n(:prec:=1) 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_n();". // 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_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7448_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![0, 1]>,<5,R![-3, 1]>,<11,R![1, 1]>,<13,R![0, 1]>,<17,R![6, 1]>],Snew); return Vf; end function;