// Make newform 441.6.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_441_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_441_6_a_l();". // 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_441_6_a_l();". // 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 := [-14, -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_441_a();" function MakeCharacter_441_a() N := 441; order := 1; char_gens := [344, 199]; v := [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_441_a_Hecke(Kf) return MakeCharacter_441_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 := 6; raw_aps := [[-4, -1], [0, 0], [-14, 10], [0, 0], [-260, 124], [238, -126], [938, -76], [1624, 18], [-760, -568], [-3222, -252], [280, -540], [2846, 540], [-2478, -1092], [884, -4788], [3976, 3748], [-4838, 208], [-20944, -2050], [29974, 4806], [13364, -1944], [-50808, 4200], [-11354, 5256], [18176, 14904], [50904, 15750], [53242, -22208], [-5978, -8820], [-91630, 27434], [-30464, 10476], [76436, -21904], [-100330, 3348], [-35226, -23940], [-105040, 8064], [-273224, 23278], [-217946, 51448], [-168728, -46242], [2474, -33496], [38984, 2592], [-257474, 81234], [-457684, -5508], [120624, -48468], [31234, -20906], [-96020, 71440], [165382, -103950], [229232, 50504], [-49510, -158436], [6522, 83832], [217168, -118476], [515300, 54936], [-812336, 65016], [-463624, 12398], [186886, -284742], [-384682, 117872], [207408, 10920], [-395402, -237924], [-1615992, -13902], [763042, -8432], [-164984, 91432], [1915298, -25738], [-169904, 440640], [-210874, 571104], [-168954, -351456], [962416, -132462], [1047522, -462462], [-294056, -359982], [23576, -542800], [947422, 579492], [-2999518, 13208], [-824428, 418860], [-1308214, 337932], [462076, -191228], [-1520498, 388458], [2943458, 125840], [-1377688, -87760], [661024, 259272], [-2821066, -313704], [2299796, -393372], [-3722600, -91004], [-2009246, -610892], [-2912546, -505494], [-165610, 385076], [3702286, 194508], [-92400, 47586], [2764022, -976752], [2327632, -796280], [-2218202, 591444], [-2333240, -782064], [-637228, -800584], [1543134, 1268400], [-1542334, 780732], [-2402190, 67662], [-1879120, 734832], [-996296, -606554], [412328, -661828], [-432760, -1113768], [2113284, -1755096], [-2196316, -257256], [-2435832, -1921752], [-7195118, 1041526], [-3041710, 1874300], [-6175568, 879498], [7100510, -1550520], [-1785796, 2204244], [1072210, 799048], [-7662928, 1286090], [5789486, 1296644], [-4339852, 817812], [4014430, -415584], [4086600, 1878954], [373954, 1784296], [-12553976, -476240], [-5229434, 3251808], [2752288, 2468736], [-4030018, 1492668], [-9635586, 84], [-4784864, 2256498], [-10271224, 859320], [-6019322, -1972700], [9083200, 1494486], [17863664, -280468], [6647882, 1566620], [-8726412, 2978556], [1201102, -2807442], [-1517710, 2907576], [-10551926, -707522], [-5738972, 4306888], [-1746248, -1256958], [13224762, -750708], [-1749730, -1659636], [-6002696, 2943580], [-18698960, -33012], [-12918122, 3672918], [-24555796, -79812], [-683832, 1063104], [-10469392, 4022712], [17946350, 896364], [-9496718, -3792044], [-2259530, 3671892], [7082306, 5491298], [15421672, -3121686], [1781010, -1816794], [2094910, -5095148], [1412656, -2319030], [5845178, -1048720], [-6722344, 1254528], [7251396, -4052496], [10499062, -1136682], [-10312512, -309372], [-10630634, -4184838], [15262898, -5899564], [-2753576, 4137102], [-31247248, -1695016], [22767686, -3126276], [1762530, -154056], [14697524, -1551312], [-6522992, -1980668], [10979564, -1603152], [-2718144, -8821008], [11778680, 2356992], [6726874, 925732], [28148806, 2751336], [7422842, -1426774], [23082356, 1064300], [-10610490, 9684528], [2057288, 203112], [26226592, -5484434], [28487038, -8415080], [-20543152, 9011780], [25179680, 2830032], [-45264002, -813618], [13904522, -4628484], [-3057110, 8686870], [24657268, 7846984], [1673182, -10494414], [-20851576, 6592424], [9499274, -5354928], [-7363160, 3969972], [27229482, -3272472], [-27457540, -6052032], [-17656766, 1118500], [38383912, -9619848], [33690118, 1283958], [-11673280, 11080728], [11075232, -12086214], [-17072914, -8800092], [31990378, 5138416], [-18948224, 1955056], [-7850374, -15023938], [-5002738, 12219552], [8613640, 964242], [-4052422, 296136], [-42069160, -8500876], [-23606114, 12562848], [-27487644, 3967320], [-20579884, -14586480], [38476354, 1300078], [21638140, -2745248], [24511130, 12932984], [-7584982, -4663188], [-19721866, 4941756], [48801522, 4955160], [-17988824, 7227328], [13669858, -14790512], [-86096360, 3105900], [-28404530, -8218962], [39279982, -4835160], [-22428336, -515802], [-49722358, 3419018], [10448872, 2973996], [-13946188, 4500404], [-24430674, -12701892], [-16960328, 4150278], [66822410, -11170116], [6842178, -2343222], [-69540280, 159264], [55348048, -2309906], [-38204152, 15375200], [42554974, 18535500], [-15631216, -22132080], [34738298, 5001788], [8944904, 24977840], [29230458, 16471644], [-5047570, 10894500], [-15421784, -637056], [-42457762, -14629840], [13941680, 23975064], [-113077272, -90930], [86633822, -7977564], [68562970, -1085648], [33121696, -3866984], [24828328, 2222280], [1489148, -23235892], [59202326, -4271580], [45770368, -15671538], [60779840, 1483776], [-4216010, 33163408], [-107713424, 6953562], [3849104, 10666832], [-28418474, 5903244], [106106570, -18678472], [81399540, -14374248], [-110056968, -2759400], [72282580, -3737876], [85983520, 7888122], [866824, -23956920], [88037534, 6539184], [81474498, 8229816], [-54616352, 21162844], [128158912, 3436272], [131978168, 4340414], [109935092, 5343228], [49256016, -36824592], [63876542, -20910456], [35963746, -15527264], [19952008, 16100824], [3650038, 23267664], [52772090, -24378598], [-2990740, 37350368], [33886910, -36284652], [-13210232, -37024506], [80486994, 27367914], [94431694, -9346788], [10078496, -21271752], [-76069548, 10398276], [-27723122, 22539222], [15205078, 25910262], [137150426, 18762836], [-9063824, -21542706], [41530578, 30650424], [64509018, 3113376], [71353340, -8796816], [-182461070, -6260], [-75895778, 16793802], [-122579764, -9365004], [94840438, 2156328], [-148215952, 26503632], [129468430, 17998848], [-121394504, 21071880], [-118151044, -22502236], [-51328066, -24414696], [105089618, -35826300], [19772536, 51179446], [52005296, -8377816], [-15553432, 31933008], [-31596432, -21073164], [-67831778, -16977618], [49975408, -5606982], [-40086400, 19834352], [-98632078, 41857776], [140501898, -675864], [-11112080, 33301260], [-6531294, -29423856], [37437154, 41255968], [16933784, 12212054], [103374638, -14238988], [-85228752, -36502410], [-107000962, 19866672], [-33957742, 17140166], [-33732608, -26110080], [63811482, 54209022], [-151878104, 2109922], [104225632, 2583126], [-70320194, 33422652], [-13249366, -53023972], [-25129264, -10776960], [-228594396, 17151540], [-64101628, -5191668], [173540978, 15126552], [-110072852, 31800004], [134163190, -7852194], [14947208, -11486392], [143537846, 33211872], [125788768, 4023400], [-93255086, 81889204], [176872622, -23869000], [-22512916, 18405900], [-210876024, -11091864], [-148277738, 50045904], [-166442640, 9387210], [-48244064, -42208784], [264518926, -9822456], [113094798, 28993776], [-174228992, 12539358], [130386890, 59186520], [-49617750, 58191882], [-231221296, -19963944], [59042206, -47874332], [-199009370, 25994196], [-139895332, -6030828], [145425280, -49252374], [-9643120, 38143328], [199708626, 36995196], [-142618738, 27518184], [71460346, -52304660], [-171960152, 5608764], [-201931408, 27386066], [213832172, 37687680], [-11344032, -16966026], [-11002906, -58696092], [-164386334, -4744688], [-230531882, 30522240], [207271960, -7854948], [-204694762, 19263960], [76500366, -20721036], [196435050, 23878302], [230481176, 45582264], [-136632262, 102447260], [267683652, -72993060], [-289984394, -25301214], [193299236, -77114988], [-25425120, -85461096], [293407282, 37016962], [240520672, 3303990], [-29678326, 41761008], [-102326814, -35662032], [92320984, -84245652], [-144889930, -34682472], [115891146, -103123944], [-169984472, 35172028], [397892768, -24171202], [-24062770, 14095136], [-339496824, 26114424], [-114263098, -88844616], [-248774246, 8408536], [297384352, -19801616], [81237572, 46481168], [-50535128, 105190434], [-115670110, -73881828], [185895234, -76708170], [-340716760, -22666824], [-243801262, 66707676], [-58451492, 77663920], [80714536, -91548306], [-356667952, -21626704], [424301610, -33405288], [19490632, 118907424], [-15594946, 55568808], [331965704, -25932514], [-154067328, 50268456], [-91694162, -71611128], [-238135478, -28026044], [243198382, 2660364], [348423698, -37808266], [511136862, -51215052], [-244468936, 17329536], [193445566, -74525936], [-179757536, -15881418], [111760040, -11573680], [-32206432, 95009112], [247308670, -34919802], [-170235796, -3395160], [-53002152, 114481332], [422554106, 63595980], [243955474, 67250074], [327775636, 52361440], [-139737920, 118048158], [-79495336, -46496752], [-37169110, -124173432], [-73792712, -114226092], [213043026, 11804268], [11483852, -84639744], [-501324398, -56896940], [-5498570, -73896714], [228714398, 18523280], [-172104208, -78116688], [226398130, -174560684], [495051578, 28386758], [284324992, 57569292], [-660799138, 26671248], [-167903178, -136018428], [106968736, 150365214], [125032096, 58014106], [114214366, -11479464], [396650306, -85621684], [429635820, -26487636], [407599700, 91527192], [-67297270, 191911104], [75604690, 67411438], [-279342928, 54976088], [-362380256, 43551612], [329492394, 78736644], [-296336600, -104337272], [-403923206, 41672524], [316050910, -75391074], [-79449456, -164169264], [-657533352, -1945398], [-636370994, 53746308], [-532388374, 66127874], [-74051212, -122466772], [297732006, -1746864], [-169623608, -90217674], [-485184448, -59527348]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_441_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_441_6_a_l();". // 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_441_6_a_l(:prec:=2) chi := MakeCharacter_441_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 6)); 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_441_6_a_l();". // 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_441_6_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_441_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign))); Vf := Kernel([<2,R![6, 9, 1]>,<5,R![-1344, 18, 1]>,<13,R![-195608, -350, 1]>],Snew); return Vf; end function;