// Make newform 7350.2.a.dh 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_dh();". // 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_dh();". // 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_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, 0], [-1, 0], [0, 0], [0, 0], [-2, 3], [0, 4], [4, -1], [4, 2], [-6, 2], [4, -1], [6, 3], [-4, 3], [-2, -6], [-2, -5], [2, -3], [-6, 4], [6, -2], [6, 4], [-2, -5], [4, -6], [-2, 0], [8, 0], [8, -2], [-2, 2], [2, -6], [-12, -4], [12, 2], [-4, -4], [2, -4], [-10, 0], [0, -2], [6, -2], [-8, -4], [8, -4], [4, 13], [6, 10], [-14, 4], [10, 5], [-2, 3], [18, 2], [22, 3], [-10, -6], [4, 0], [2, 2], [6, 14], [10, -3], [-12, -8], [4, -8], [-8, -12], [2, 10], [-2, -12], [-8, -8], [4, 11], [-4, 12], [-20, 1], [-2, 10], [-18, 4], [-2, -21], [0, 11], [2, -18], [18, -6], [10, 8], [4, -8], [8, 2], [22, 6], [14, 8], [-12, 4], [-2, -22], [28, -4], [2, -18], [-8, 17], [-20, -6], [-8, -2], [-8, 1], [-8, 6], [18, 1], [-20, -7], [18, -8], [-6, 0], [0, -3], [20, 12], [2, -4], [-12, 18], [2, -14], [-6, 21], [-8, -20], [-2, -6], [6, -6], [-2, 0], [20, 4], [4, 6], [0, 6], [-8, -2], [-10, -5], [4, 2], [-6, 1], [0, 12], [-2, -26], [22, 14], [-10, -14], [10, 13], [38, 0], [-12, -6], [26, -12], [16, -8], [-18, 10], [-8, 2], [-20, -15], [4, 0], [36, -3], [4, 0], [4, 5], [0, -24], [12, -22], [6, -10], [26, 0], [26, 2], [-18, 19], [-18, 14], [2, 13], [22, -12], [-2, 20], [-2, -30], [12, 14], [-20, 12], [4, -15], [26, -10], [16, -24], [-8, -16], [-34, 0], [-20, 22], [0, -20], [2, 14], [-16, -7], [10, 12], [8, -9], [-6, -12], [44, -4], [-14, 10], [26, 6], [-16, 4], [-20, 15], [-12, -12], [-44, 6], [-22, -8], [0, -6], [-18, -16], [12, 9], [-4, 28], [-40, -4], [0, -3], [-26, 0], [30, 3], [-6, -19], [-10, 19], [28, 12], [6, -26], [-50, -4], [42, -4], [0, 20], [12, -2], [-16, 8], [16, -4], [-12, 12], [18, 8], [-10, -5], [-30, 14], [-34, -16], [12, 20], [-50, 2], [-42, -5], [-22, -2], [12, -24], [-2, -2], [-2, 31], [14, 20], [-12, -18], [-28, -7], [16, 4], [30, 4], [-8, -22], [26, -2], [8, 11], [20, 31], [-14, 14], [6, -4], [-12, 1], [-18, -14], [22, -8], [-40, -4], [-34, 14], [24, -12], [8, 2], [-14, 4], [-8, 10], [4, -15], [12, -20], [20, -3], [20, -19], [18, 9], [-8, -17], [34, -11], [32, 20], [0, -13], [-12, 32], [-6, -16], [18, -3], [-12, 22], [-22, 16], [-20, 16], [14, 12], [2, -24], [44, 8], [-16, -10], [32, 6], [32, -23], [0, -2], [18, 6], [-30, 2], [-14, 34], [-42, -22], [6, 15], [-34, -22], [-4, 14], [-16, -22], [-54, -6], [-20, 35], [28, 0], [-8, -4], [14, 23], [12, -21], [-8, -14], [-32, -4], [-26, 24], [-28, -28], [14, 15], [-32, -17], [22, 6], [-6, -31], [-36, 10], [-20, 4], [-48, -10], [20, 8], [14, 22], [24, -17], [8, 0], [-8, 8], [-2, -10], [12, -4], [-24, -12], [24, 11], [-30, 18], [-24, -4], [12, 7], [10, -16], [14, -9], [18, -36], [2, 18], [-30, -4], [6, 30], [24, 4], [-20, -36], [2, 36], [16, -28], [56, 13], [56, -4], [-8, -19], [2, 18], [18, -11], [2, -50], [-38, -22], [-14, 5], [-30, 2], [40, -8], [2, -42], [8, -18], [0, 12], [-14, -34], [-12, 20], [-46, 6], [-38, -23], [8, -16], [34, 1], [50, -20], [34, -10], [-60, 14], [-18, -30], [46, 8], [14, -37], [-10, -12], [-16, -7], [20, -36], [-34, -4], [-54, -10], [52, -19], [-40, -4], [38, 27], [14, 42], [60, 8], [12, 52], [54, -16], [34, -18], [48, 20], [-32, 2], [-12, 36], [10, -44], [28, -24], [6, -16], [48, -14], [48, 13], [-30, -7], [-36, -29], [-42, -4], [-14, 21], [-16, 48], [24, -25], [20, -12], [4, 2], [-38, -6], [66, -18], [-12, 28], [-34, -8], [0, -4], [-32, 40], [-4, -28], [36, -9], [0, -12], [20, -12], [-24, -36], [66, 8], [42, -12], [-6, -34], [34, 13], [48, -8], [-40, -18], [24, 36], [2, 2], [-24, -27], [8, 29], [4, -22], [16, -23], [-36, 28], [-20, -24], [8, -44], [-78, 6], [-26, -9], [26, -10], [-14, -43], [4, -2], [-30, 24], [-52, -4], [2, -10], [-52, 15], [12, -32], [-58, 10], [40, -3], [4, 20], [4, -64], [58, 12], [-54, 6], [16, 25], [14, -28], [-46, -22], [-66, -13], [-44, -16], [78, -14], [30, -14], [32, 2], [-34, 20], [50, 29], [-24, 32], [2, -22], [16, -1], [38, -5], [0, 33], [34, 6], [8, 46], [-26, 10], [-2, -18], [-58, 20], [2, 24], [-2, 36], [28, 30], [42, 0], [-28, -24], [82, 5], [80, 4], [30, -16], [30, -1], [14, 15], [-4, 44], [66, -20], [-42, 19], [84, 8], [-16, 46], [-38, -8], [2, 17], [-86, -4], [0, 30], [8, 33], [2, -52], [-64, 16], [80, 6], [-32, 23], [-52, -16], [30, 15], [48, -15], [-82, 2], [22, 50], [-14, 54], [10, 24], [2, -20], [-44, -12], [52, 30], [10, 38], [44, 16], [68, -8], [8, 18], [0, -33], [54, -1], [-12, -15], [-20, 12], [-48, 4], [-68, 8], [-14, 18], [10, 4], [64, -26], [50, 14], [40, -2], [72, -18], [48, -35], [-2, -17], [-64, 6], [26, 21], [-6, 0], [58, -22], [46, 12], [-26, -40], [22, -5], [-30, -35], [32, -22], [-42, -14], [-26, -14], [24, 16], [36, 5], [56, 36], [38, -14], [26, -15], [48, -43], [-14, 56], [-10, -33], [-16, 0], [24, 52], [10, -42], [-50, -32], [-48, 39], [-38, 0], [-28, -32], [-40, -12], [-20, 4], [-12, 44], [-26, -6], [-14, -15], [-70, 8], [-22, -6], [-70, -18], [0, -48], [52, 32], [-14, 4], [12, -38], [0, 58], [-96, -6], [-24, 12], [-72, -4]]; 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_dh();". // 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_dh(:prec:=2) 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_dh();". // 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_dh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-14, 4, 1]>,<13,R![-32, 0, 1]>,<17,R![14, -8, 1]>,<19,R![8, -8, 1]>,<23,R![28, 12, 1]>,<31,R![18, -12, 1]>],Snew); return Vf; end function;