// Make newform 7605.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7605_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_7605_2_a_i();". // 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_7605_2_a_i();". // 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_7605_a();" function MakeCharacter_7605_a() N := 7605; order := 1; char_gens := [6761, 1522, 6931]; 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_7605_a_Hecke(Kf) return MakeCharacter_7605_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], [-1], [-3], [-3], [0], [3], [0], [-3], [6], [-6], [9], [-3], [10], [-12], [3], [12], [1], [0], [9], [-6], [1], [6], [15], [-9], [18], [4], [9], [-18], [-18], [2], [-12], [18], [-13], [3], [-6], [-4], [-3], [12], [-6], [-24], [25], [12], [-3], [6], [-16], [4], [-24], [0], [-6], [-15], [-27], [-30], [-6], [18], [-24], [24], [-6], [-8], [-6], [-22], [6], [-15], [-6], [-8], [6], [24], [14], [9], [18], [0], [-24], [26], [4], [-30], [-30], [30], [-3], [-18], [24], [18], [-18], [0], [38], [19], [21], [-15], [-33], [9], [-15], [-15], [3], [15], [12], [24], [-36], [15], [-12], [2], [-12], [26], [30], [9], [-30], [-31], [-21], [-18], [42], [-48], [-17], [-4], [-21], [-18], [-30], [-6], [-24], [-27], [39], [18], [6], [24], [-44], [39], [-30], [30], [12], [0], [30], [26], [-15], [-30], [36], [-5], [16], [-18], [-12], [-18], [-12], [-51], [-24], [42], [33], [-14], [12], [2], [9], [-9], [21], [-5], [30], [30], [0], [-16], [-33], [8], [-18], [-11], [27], [34], [-9], [-24], [27], [48], [18], [24], [-36], [-25], [8], [-42], [-33], [12], [12], [-18], [-33], [-5], [-12], [-12], [-45], [16], [-26], [33], [-6], [-14], [48], [-6], [24], [52], [3], [54], [51], [-16], [48], [-7], [21], [0], [-9], [-42], [-10], [30], [9], [-54], [-55], [39], [-55], [15], [-51], [-54], [-12], [-45], [23], [2], [-48], [-26], [-42], [-60], [-30], [-68], [0], [-42], [42], [59], [66], [-30], [-24], [33], [-71], [9], [-12], [-46], [-3], [40], [11], [18], [-30], [-46], [66], [-48], [-24], [-24], [-24], [-36], [-25], [-14], [36], [-6], [-72], [0], [-63], [-42], [60], [-54], [54], [-12], [-19], [-63], [63], [-38], [21], [42], [-33], [26], [15], [60], [56], [-66], [37], [27], [27], [15], [30], [14], [63], [51], [40], [40], [15], [-6], [60], [-24], [6], [9], [-24], [24], [0], [-36], [18], [34], [-48], [-24], [0], [30], [-60], [78], [-60], [62], [-30], [44], [-63], [30], [33], [-14], [-12], [-47], [-33], [-76], [-27], [39], [-49], [12], [-70], [81], [66], [6], [40], [-42], [25], [27], [-87], [30], [-6], [-44], [30], [12], [-3], [30], [-65], [-48], [-39], [33], [18], [18], [-3], [-44], [12], [-24], [6], [78], [42], [0], [70], [-45], [-51], [39], [-44], [-36], [-48], [-11], [45], [-48], [-87], [-84], [33], [-39], [80], [-85], [54], [-75], [0], [-18], [24], [90], [-84], [-48], [-60], [34], [8], [72], [-39], [25], [-60], [-20], [18], [-72], [-41], [38], [-21], [36], [39], [-72], [-44], [42], [-48], [0], [-36], [66], [36], [-42], [-40], [84], [-51], [24], [-66], [-87], [100], [21], [-68], [12], [-66], [-5], [18], [30], [-75], [51], [24], [-51], [43], [18], [30], [-81], [51], [-62], [-27], [69], [-53], [4], [-6], [-33], [28], [-48], [-21], [42], [-63], [-42], [-42], [6], [12], [-15], [57], [-42], [-12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7605_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7605_2_a_i();". // 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_7605_2_a_i(:prec:=1) chi := MakeCharacter_7605_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_7605_2_a_i();". // 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_7605_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7605_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<7,R![3, 1]>,<11,R![3, 1]>],Snew); return Vf; end function;