// Make newform 7350.2.a.ba in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_ba();". // 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_7350_2_a_ba();". // 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_7350_a();" function MakeCharacter_7350_a() N := 7350; order := 1; char_gens := [4901, 1177, 2551]; 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_7350_a_Hecke(Kf) return MakeCharacter_7350_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 := [[-1], [1], [0], [0], [-1], [7], [-4], [-1], [-1], [-8], [-6], [3], [-9], [4], [-3], [1], [-12], [4], [-12], [-14], [-14], [4], [12], [2], [-16], [0], [16], [18], [-10], [6], [-5], [13], [2], [4], [-4], [-2], [15], [-8], [-5], [-21], [13], [12], [-10], [-26], [3], [12], [-15], [-4], [-20], [22], [-26], [-6], [-7], [3], [-8], [16], [0], [-16], [2], [3], [-2], [9], [8], [-16], [-24], [-10], [-9], [0], [34], [28], [-8], [36], [19], [-26], [1], [13], [-14], [18], [-17], [-10], [-11], [-14], [-12], [40], [-16], [36], [-25], [-10], [28], [-33], [12], [-26], [8], [-12], [24], [28], [-30], [21], [-14], [2], [-36], [-45], [14], [-37], [-8], [-14], [42], [12], [6], [-22], [25], [15], [-8], [7], [14], [-23], [26], [-15], [-29], [-12], [-8], [12], [1], [-12], [-12], [6], [4], [-26], [17], [37], [41], [9], [-26], [2], [17], [29], [43], [-22], [-6], [53], [-37], [34], [-48], [-10], [-12], [-44], [1], [-22], [20], [-29], [-23], [25], [58], [20], [-46], [-58], [-44], [-31], [16], [30], [-46], [-44], [20], [-43], [-18], [-33], [10], [10], [-29], [39], [-20], [-22], [-2], [44], [16], [-30], [37], [30], [-15], [22], [-3], [9], [47], [-2], [39], [-12], [-23], [24], [-31], [-30], [2], [-2], [15], [54], [32], [36], [5], [-26], [-38], [12], [0], [2], [-22], [35], [0], [17], [56], [24], [-5], [37], [52], [-66], [-40], [10], [16], [41], [-13], [33], [-35], [51], [-4], [6], [-51], [59], [8], [-6], [-6], [36], [-53], [68], [-19], [23], [22], [19], [-30], [-64], [-18], [45], [-11], [-2], [60], [-67], [-24], [38], [-54], [6], [-51], [41], [-28], [-29], [50], [-17], [19], [57], [-2], [1], [-38], [-26], [23], [-52], [59], [66], [-14], [14], [-24], [5], [-20], [3], [22], [3], [-40], [-22], [14], [32], [58], [43], [20], [44], [-47], [11], [55], [44], [9], [-72], [50], [4], [-56], [-25], [-20], [31], [40], [10], [36], [53], [-9], [-48], [-18], [70], [71], [-46], [4], [-73], [-18], [-2], [36], [14], [60], [-78], [-18], [-39], [21], [-38], [66], [-38], [73], [67], [-63], [-12], [56], [66], [20], [-22], [-86], [49], [-56], [-51], [-11], [46], [0], [62], [46], [74], [-26], [84], [23], [50], [10], [72], [83], [-19], [30], [2], [14], [-38], [-18], [-20], [0], [-66], [24], [-67], [69], [-22], [-6], [-13], [-18], [22], [60], [-24], [36], [-92], [-35], [-42], [-48], [63], [-34], [6], [-1], [49], [27], [15], [-43], [-12], [-30], [-6], [-34], [39], [72], [-36], [62], [-10], [10], [-20], [37], [64], [77], [39], [60], [49], [16], [-8], [-50], [-54], [-60], [-4], [-62], [-38], [44], [-33], [85], [-30], [28], [-6], [-5], [68], [-22], [-60], [71], [-2], [32], [25], [-68], [-18], [-72], [-85], [-28], [-82], [52], [-35], [28], [63], [26], [-37], [48], [56], [-56], [-31], [-28], [-63], [-31], [94], [62], [61], [-21], [59], [-94], [-90], [-62], [-30], [-102], [-20], [18], [67], [-40], [-42], [-19], [24], [4], [-87], [-11], [-46], [74], [58], [84], [-79], [8], [-60], [-82], [93], [3], [-8], [11], [-68], [0], [22], [2], [-54], [-38], [-10], [-18], [32], [85], [-18], [44], [-83]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7350_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_ba();". // 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_7350_2_a_ba(:prec:=1) chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_ba();". // 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_7350_2_a_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![1, 1]>,<13,R![-7, 1]>,<17,R![4, 1]>,<19,R![1, 1]>,<23,R![1, 1]>,<31,R![6, 1]>],Snew); return Vf; end function;