// Make newform 7623.2.a.u in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7623_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_7623_2_a_u();". // 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_7623_2_a_u();". // 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 := [-1, -1, 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_7623_a();" function MakeCharacter_7623_a() N := 7623; order := 1; char_gens := [848, 4357, 4600]; 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_7623_a_Hecke(Kf) return MakeCharacter_7623_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], [0, -1], [1, 0], [0, 0], [-1, 4], [3, 0], [4, 0], [4, 2], [-3, 0], [-3, 2], [-2, -4], [-5, 5], [-9, 0], [-2, -5], [4, -5], [-11, -1], [3, -6], [6, -6], [-9, -2], [1, -3], [2, 3], [10, -8], [-14, 7], [3, -12], [0, -2], [9, 2], [3, 8], [15, -9], [-5, 4], [-8, 9], [-10, 7], [4, 4], [-3, -9], [9, 7], [-9, -1], [-21, 3], [17, -3], [-7, -5], [-11, -1], [-19, 4], [11, -2], [-23, 4], [-2, -12], [-14, 10], [3, -9], [-9, 14], [1, -10], [-1, 1], [-14, 26], [15, -12], [2, -1], [9, -6], [-9, 12], [-6, 4], [15, 0], [-9, 18], [9, -16], [-6, -4], [16, -17], [-11, -6], [22, -14], [3, -6], [-3, -13], [10, -7], [-11, -2], [-17, 30], [3, -2], [-2, 16], [-12, 9], [-15, -6], [18, -21], [-15, -6], [-18, -10], [6, -11], [-18, 15], [-14, 26], [-6, 15], [21, -15], [8, 1], [2, 7], [3, 1], [-18, -1], [6, 8], [-9, 24], [-30, 14], [-7, -4], [-16, 3], [15, -12], [-13, 27], [-5, -14], [-16, 7], [3, -15], [18, 14], [-5, 0], [-27, 2], [-21, 10], [-7, 20], [27, -12], [-24, 25], [-13, 22], [8, -10], [-27, 28], [5, 4], [3, -16], [-20, -3], [-6, 0], [-6, 20], [-34, 11], [-12, 16], [31, 3], [-18, 21], [4, -11], [22, 6], [-1, -14], [28, -17], [-32, 0], [17, -8], [-25, -4], [-21, 18], [1, -12], [9, -12], [-31, 5], [-23, 14], [16, -7], [-5, 10], [-12, 10], [15, -5], [-17, 33], [-17, 26], [3, -9], [-12, 5], [12, 9], [14, -18], [-6, 27], [30, -4], [-6, -8], [24, -27], [-42, -3], [15, -19], [-26, -6], [30, -26], [8, 24], [23, -38], [22, -9], [15, 0], [31, -15], [-11, -8], [21, 12], [-1, 2], [31, -25], [41, -26], [36, -25], [-47, 10], [41, -27], [14, -23], [-28, 4], [-36, 0], [-27, 4], [38, -34], [3, -9], [-41, 1], [-10, -14], [21, 12], [-19, -22], [15, -38], [25, 11], [50, -8], [-8, 11], [-9, 18], [-9, -26], [-6, 39], [10, -20], [-43, -12], [-36, -3], [5, 25], [18, 25], [-13, -13], [13, 20], [39, 3], [32, 4], [5, 5], [-25, 6], [-42, 0], [40, -23], [30, 2], [3, -10], [54, -20], [-43, 24], [-9, 42], [0, -12], [3, 33], [24, -24], [3, 27], [51, -18], [25, -5], [-3, -6], [24, -51], [12, 23], [7, 11], [-31, 19], [-5, -22], [-8, 42], [-26, 56], [-39, 27], [-32, 16], [-4, 15], [-12, 4], [7, 25], [-15, 6], [-8, 18], [36, -32], [39, -27], [31, -46], [49, -10], [11, -27], [39, -2], [8, 14], [-49, 2], [21, -24], [-16, 16], [53, 6], [-7, 11], [24, -27], [-40, -7], [-9, 1], [-16, -7], [7, -7], [-31, 39], [-2, 10], [-7, -23], [-36, -15], [-8, -19], [-16, 14], [10, -22], [48, -21], [29, -36], [-40, -1], [-34, 44], [16, -28], [-15, -17], [34, -21], [-22, -11], [22, 9], [49, 8], [-16, 11], [0, 9], [-63, 12], [0, 22], [10, -8], [-30, -15], [-9, 48], [8, 8], [9, 30], [53, -4], [-15, 18], [25, 24], [27, 20], [-30, 16], [20, -29], [40, -9], [-26, 4], [33, 8], [54, 9], [32, -13], [52, -1], [-8, -2], [-42, 36], [-39, 1], [9, 36], [7, 17], [1, 12], [-35, -16], [30, -24], [38, -36], [-40, 7], [39, -42], [14, 22], [25, -13], [-8, -34], [21, -1], [-43, 10], [57, -38], [22, -42], [17, -9], [17, -19], [12, 0], [-7, -16], [-40, -9], [-21, -29], [-35, 10], [38, -22], [-5, 49], [27, -63], [-9, -30], [10, -11], [6, -18], [17, -4], [18, 1], [17, -60], [-36, 32], [-48, 37], [-55, 15], [-54, 2], [-66, -3], [-63, 30], [6, 33], [-34, 33], [-16, -8], [-15, -30], [-19, 2], [-6, 30], [27, -48], [9, 24], [3, -42], [-33, 37], [-15, 30], [-35, 5], [-65, 9], [-57, -11], [-29, 53], [-71, 30], [38, -59], [-6, 21], [34, 4], [-6, 33], [45, -30], [36, -55], [-35, 64], [-54, -8], [-17, 8], [0, -56], [0, 12], [69, 12], [-4, 36], [3, 24], [69, -12], [84, 0], [26, -45], [8, 28], [-63, 4], [-10, 11], [45, -66], [-49, -6], [-3, -11], [-24, 66], [31, -49], [-33, 30], [11, -2], [38, -77], [21, -52], [17, -47], [36, 21], [24, 30], [44, -75], [-14, -41], [47, 4], [15, -19], [-9, 39], [-45, 30], [7, 8], [-46, 19], [15, 34], [-42, 24], [-19, 48], [-24, 48], [-75, 22], [-27, 15], [7, 50], [-6, -48], [5, -29], [-40, 47], [53, -10], [18, -35], [-42, 18], [13, -16], [18, -20], [13, -42], [7, -59], [-15, 12], [-29, 4], [15, -37], [53, -24], [77, -13], [20, 25], [7, -61], [60, -8], [49, 18], [-73, 9], [28, -44], [12, -32], [63, 15], [43, -56], [-18, 25], [3, 6], [-82, 1], [8, -24], [-3, 0], [10, 36], [69, -12], [-79, 18], [-62, 20], [-6, -24], [2, 22], [-29, 5], [64, -25], [6, -23], [-28, 58], [7, -69], [-26, 43], [-28, -12], [-2, 25], [35, 20], [59, -20], [-17, 26], [-17, -38], [-9, 21], [25, -78], [28, -7], [-50, 73], [39, -62], [37, 27], [-21, 36], [-42, 51], [34, 20], [-87, 9], [-39, 24], [-52, 47], [61, 17], [69, -64]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7623_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7623_2_a_u();". // 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_7623_2_a_u(:prec:=2) chi := MakeCharacter_7623_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_7623_2_a_u();". // 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_7623_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7623_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 3, 1]>,<5,R![-1, 1, 1]>,<13,R![-19, -2, 1]>],Snew); return Vf; end function;