// Make newform 1800.4.a.z in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1800_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_1800_4_a_z();". // 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_1800_4_a_z();". // 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_1800_a();" function MakeCharacter_1800_a() N := 1800; order := 1; char_gens := [1351, 901, 1001, 577]; 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_1800_a_Hecke(Kf) return MakeCharacter_1800_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 := 4; raw_aps := [[0], [0], [0], [12], [-64], [-58], [32], [-136], [-128], [144], [20], [18], [288], [200], [384], [496], [128], [-458], [496], [-512], [602], [1108], [704], [960], [-206], [-432], [68], [-384], [-518], [-960], [-796], [-512], [1824], [-2160], [688], [-844], [-118], [-3576], [384], [2448], [4224], [-510], [384], [3454], [-3216], [1708], [2320], [116], [1344], [4594], [5056], [3712], [-978], [-1856], [7808], [1024], [-1328], [-5812], [-8386], [640], [-4832], [6384], [3312], [-9984], [-2586], [-2832], [-5920], [4674], [-9024], [-4362], [8768], [6144], [-4564], [8770], [-1096], [10368], [3248], [6106], [-7008], [1590], [192], [9074], [5248], [8222], [16236], [14528], [-6304], [-1958], [-4048], [-16988], [6720], [-9728], [8444], [15360], [6624], [-6912], [19920], [-3680], [11720], [11754], [18904], [3088], [-21440], [-22624], [-6000], [-19922], [-3584], [1984], [-14976], [25738], [8548], [-8558], [-10368], [-13088], [-4412], [30176], [-21288], [17024], [2256], [-23808], [-26242], [24590], [-2864], [-7616], [2168], [18000], [3506], [-15616], [15036], [19126], [-17392], [-32384], [-27708], [37246], [4192], [26882], [-17232], [-31816], [8272], [9184], [-19832], [-15216], [39772], [-18304], [4906], [-15360], [24802], [15072], [1800], [7552], [-20838], [-47744], [-28280], [7424], [5912], [26240], [35620], [3232], [11478], [39984], [24192], [-39456], [41668], [51648], [55776], [-36096], [42532], [-29806], [-9250], [47440], [13056], [45750], [39808], [-14326], [40772], [12544], [21056], [-42576], [8660], [9158], [-12692], [-40640], [2382], [18816], [-4992], [-336], [15254], [53280], [4246], [-63744], [-25918], [8832], [-21440], [-44944], [-28416], [12032], [26338], [9206], [17792], [-47616], [-8336], [-13204], [52894], [28366], [-58560], [-51600], [34316], [-20928], [38464], [-32888], [-18930], [76848], [15860], [58496], [93952], [-60166], [90412], [35168], [-80256], [79248], [-9262], [-8820], [-15072], [-85340], [34752], [-96610], [20928], [51456], [-3996], [-50176], [-48762], [-62648], [84476], [-59008], [33048], [5120], [-57922], [2512], [87232], [-24448], [94272], [39488], [3740], [89350], [-85248], [11776], [101084], [81792], [60640], [26624], [-4362], [-59712], [92288], [77686], [-100944], [127872], [-57022], [-7984], [112176], [-60778], [106964], [11136], [107026], [-73878], [67552], [-1984], [97840], [1216], [91288], [36560], [-95834], [-139952], [-68998], [64516], [66770], [50068], [74688], [-81706], [5146], [55424], [-87296], [6988], [-130688], [97934], [-107912], [-131584], [-145166], [78480], [133556], [16192], [111536], [-133248], [29248], [-21888], [86010], [48944], [52604], [-3024], [62016], [-49448], [-128522], [-54640], [43076], [-38144], [-112104], [150754], [-74688], [-6406], [-102656], [-76334], [-107904], [-12336], [12640], [38312], [147712], [80918], [-56832], [-7680], [158290], [-58176], [-10784], [20294], [-111344], [166604], [-90656], [38622], [-121512], [71496], [173056], [-66288], [93850], [111120], [143148], [108672], [-72816], [-158784], [84502], [-207936], [-162106], [2340], [-37342], [46176], [58704], [-127716], [-138704], [92864], [-9390], [-93760], [40704], [-198352], [-38712], [75610], [204944], [-101964], [169982], [-118528], [-206464], [-41984], [97312], [132864], [-86030], [133440], [-74112], [-73664], [-55216], [8454], [136944], [-135124], [-178550], [150592], [-137928], [104960], [-216688], [118180], [-52394], [-95040], [-117888], [141662], [-192864], [-239654], [-43536], [-239584], [100716], [65664], [64456], [-1152], [-215580], [194398], [-61744], [168576], [133438], [32848], [-103680], [49896], [-64128], [11210], [-74100], [161952], [-192040], [78704], [187190], [-151968], [123332], [25184], [-212080], [-158828], [-32026], [-102240], [129832], [-91968], [115198], [-56816], [-195520], [-204392], [2682], [-131056], [221568], [252172], [74592], [26624], [22576], [130930], [41984], [-137024], [248438], [51440], [104512], [204480], [-64248], [-234112]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1800_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1800_4_a_z();". // 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_1800_4_a_z(:prec:=1) chi := MakeCharacter_1800_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_1800_4_a_z();". // 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_1800_4_a_z( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![-12, 1]>,<11,R![64, 1]>,<17,R![-32, 1]>],Snew); return Vf; end function;