Properties

Label 177.8.a.b.1.8
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-4.01497\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.01497 q^{2} +27.0000 q^{3} -91.8201 q^{4} -385.807 q^{5} -162.404 q^{6} +847.649 q^{7} +1322.21 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-6.01497 q^{2} +27.0000 q^{3} -91.8201 q^{4} -385.807 q^{5} -162.404 q^{6} +847.649 q^{7} +1322.21 q^{8} +729.000 q^{9} +2320.62 q^{10} -5526.47 q^{11} -2479.14 q^{12} +2987.05 q^{13} -5098.58 q^{14} -10416.8 q^{15} +3799.90 q^{16} -6651.02 q^{17} -4384.92 q^{18} +47421.5 q^{19} +35424.9 q^{20} +22886.5 q^{21} +33241.6 q^{22} +16902.5 q^{23} +35699.7 q^{24} +70722.2 q^{25} -17967.0 q^{26} +19683.0 q^{27} -77831.2 q^{28} +2638.69 q^{29} +62656.7 q^{30} +138735. q^{31} -192099. q^{32} -149215. q^{33} +40005.7 q^{34} -327029. q^{35} -66936.9 q^{36} -91219.9 q^{37} -285239. q^{38} +80650.4 q^{39} -510119. q^{40} +549406. q^{41} -137662. q^{42} -240843. q^{43} +507441. q^{44} -281253. q^{45} -101668. q^{46} -1.17880e6 q^{47} +102597. q^{48} -105035. q^{49} -425392. q^{50} -179578. q^{51} -274271. q^{52} -100058. q^{53} -118393. q^{54} +2.13215e6 q^{55} +1.12077e6 q^{56} +1.28038e6 q^{57} -15871.6 q^{58} -205379. q^{59} +956471. q^{60} +71464.3 q^{61} -834486. q^{62} +617936. q^{63} +669085. q^{64} -1.15243e6 q^{65} +897523. q^{66} -912586. q^{67} +610697. q^{68} +456367. q^{69} +1.96707e6 q^{70} -274897. q^{71} +963893. q^{72} -5.34005e6 q^{73} +548685. q^{74} +1.90950e6 q^{75} -4.35425e6 q^{76} -4.68451e6 q^{77} -485110. q^{78} +1.12488e6 q^{79} -1.46603e6 q^{80} +531441. q^{81} -3.30466e6 q^{82} -4.95003e6 q^{83} -2.10144e6 q^{84} +2.56601e6 q^{85} +1.44866e6 q^{86} +71244.6 q^{87} -7.30717e6 q^{88} -8.64528e6 q^{89} +1.69173e6 q^{90} +2.53197e6 q^{91} -1.55199e6 q^{92} +3.74584e6 q^{93} +7.09048e6 q^{94} -1.82956e7 q^{95} -5.18669e6 q^{96} -2.69412e6 q^{97} +631782. q^{98} -4.02880e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.01497 −0.531654 −0.265827 0.964021i \(-0.585645\pi\)
−0.265827 + 0.964021i \(0.585645\pi\)
\(3\) 27.0000 0.577350
\(4\) −91.8201 −0.717345
\(5\) −385.807 −1.38031 −0.690153 0.723664i \(-0.742457\pi\)
−0.690153 + 0.723664i \(0.742457\pi\)
\(6\) −162.404 −0.306950
\(7\) 847.649 0.934055 0.467028 0.884243i \(-0.345325\pi\)
0.467028 + 0.884243i \(0.345325\pi\)
\(8\) 1322.21 0.913032
\(9\) 729.000 0.333333
\(10\) 2320.62 0.733844
\(11\) −5526.47 −1.25191 −0.625956 0.779859i \(-0.715291\pi\)
−0.625956 + 0.779859i \(0.715291\pi\)
\(12\) −2479.14 −0.414159
\(13\) 2987.05 0.377087 0.188543 0.982065i \(-0.439623\pi\)
0.188543 + 0.982065i \(0.439623\pi\)
\(14\) −5098.58 −0.496594
\(15\) −10416.8 −0.796920
\(16\) 3799.90 0.231928
\(17\) −6651.02 −0.328335 −0.164167 0.986432i \(-0.552494\pi\)
−0.164167 + 0.986432i \(0.552494\pi\)
\(18\) −4384.92 −0.177218
\(19\) 47421.5 1.58613 0.793064 0.609139i \(-0.208485\pi\)
0.793064 + 0.609139i \(0.208485\pi\)
\(20\) 35424.9 0.990155
\(21\) 22886.5 0.539277
\(22\) 33241.6 0.665583
\(23\) 16902.5 0.289670 0.144835 0.989456i \(-0.453735\pi\)
0.144835 + 0.989456i \(0.453735\pi\)
\(24\) 35699.7 0.527139
\(25\) 70722.2 0.905244
\(26\) −17967.0 −0.200479
\(27\) 19683.0 0.192450
\(28\) −77831.2 −0.670040
\(29\) 2638.69 0.0200907 0.0100454 0.999950i \(-0.496802\pi\)
0.0100454 + 0.999950i \(0.496802\pi\)
\(30\) 62656.7 0.423685
\(31\) 138735. 0.836410 0.418205 0.908353i \(-0.362659\pi\)
0.418205 + 0.908353i \(0.362659\pi\)
\(32\) −192099. −1.03634
\(33\) −149215. −0.722791
\(34\) 40005.7 0.174560
\(35\) −327029. −1.28928
\(36\) −66936.9 −0.239115
\(37\) −91219.9 −0.296063 −0.148031 0.988983i \(-0.547294\pi\)
−0.148031 + 0.988983i \(0.547294\pi\)
\(38\) −285239. −0.843270
\(39\) 80650.4 0.217711
\(40\) −510119. −1.26026
\(41\) 549406. 1.24494 0.622472 0.782642i \(-0.286129\pi\)
0.622472 + 0.782642i \(0.286129\pi\)
\(42\) −137662. −0.286709
\(43\) −240843. −0.461949 −0.230974 0.972960i \(-0.574191\pi\)
−0.230974 + 0.972960i \(0.574191\pi\)
\(44\) 507441. 0.898052
\(45\) −281253. −0.460102
\(46\) −101668. −0.154004
\(47\) −1.17880e6 −1.65615 −0.828074 0.560618i \(-0.810563\pi\)
−0.828074 + 0.560618i \(0.810563\pi\)
\(48\) 102597. 0.133904
\(49\) −105035. −0.127540
\(50\) −425392. −0.481276
\(51\) −179578. −0.189564
\(52\) −274271. −0.270501
\(53\) −100058. −0.0923184 −0.0461592 0.998934i \(-0.514698\pi\)
−0.0461592 + 0.998934i \(0.514698\pi\)
\(54\) −118393. −0.102317
\(55\) 2.13215e6 1.72802
\(56\) 1.12077e6 0.852823
\(57\) 1.28038e6 0.915751
\(58\) −15871.6 −0.0106813
\(59\) −205379. −0.130189
\(60\) 956471. 0.571666
\(61\) 71464.3 0.0403120 0.0201560 0.999797i \(-0.493584\pi\)
0.0201560 + 0.999797i \(0.493584\pi\)
\(62\) −834486. −0.444680
\(63\) 617936. 0.311352
\(64\) 669085. 0.319045
\(65\) −1.15243e6 −0.520495
\(66\) 897523. 0.384275
\(67\) −912586. −0.370691 −0.185345 0.982673i \(-0.559340\pi\)
−0.185345 + 0.982673i \(0.559340\pi\)
\(68\) 610697. 0.235529
\(69\) 456367. 0.167241
\(70\) 1.96707e6 0.685451
\(71\) −274897. −0.0911520 −0.0455760 0.998961i \(-0.514512\pi\)
−0.0455760 + 0.998961i \(0.514512\pi\)
\(72\) 963893. 0.304344
\(73\) −5.34005e6 −1.60663 −0.803315 0.595555i \(-0.796932\pi\)
−0.803315 + 0.595555i \(0.796932\pi\)
\(74\) 548685. 0.157403
\(75\) 1.90950e6 0.522643
\(76\) −4.35425e6 −1.13780
\(77\) −4.68451e6 −1.16935
\(78\) −485110. −0.115747
\(79\) 1.12488e6 0.256692 0.128346 0.991729i \(-0.459033\pi\)
0.128346 + 0.991729i \(0.459033\pi\)
\(80\) −1.46603e6 −0.320131
\(81\) 531441. 0.111111
\(82\) −3.30466e6 −0.661878
\(83\) −4.95003e6 −0.950243 −0.475122 0.879920i \(-0.657596\pi\)
−0.475122 + 0.879920i \(0.657596\pi\)
\(84\) −2.10144e6 −0.386848
\(85\) 2.56601e6 0.453202
\(86\) 1.44866e6 0.245597
\(87\) 71244.6 0.0115994
\(88\) −7.30717e6 −1.14304
\(89\) −8.64528e6 −1.29991 −0.649956 0.759972i \(-0.725213\pi\)
−0.649956 + 0.759972i \(0.725213\pi\)
\(90\) 1.69173e6 0.244615
\(91\) 2.53197e6 0.352220
\(92\) −1.55199e6 −0.207793
\(93\) 3.74584e6 0.482902
\(94\) 7.09048e6 0.880497
\(95\) −1.82956e7 −2.18934
\(96\) −5.18669e6 −0.598330
\(97\) −2.69412e6 −0.299720 −0.149860 0.988707i \(-0.547882\pi\)
−0.149860 + 0.988707i \(0.547882\pi\)
\(98\) 631782. 0.0678073
\(99\) −4.02880e6 −0.417304
\(100\) −6.49372e6 −0.649372
\(101\) 1.51552e7 1.46365 0.731826 0.681492i \(-0.238668\pi\)
0.731826 + 0.681492i \(0.238668\pi\)
\(102\) 1.08015e6 0.100782
\(103\) −1.40028e7 −1.26266 −0.631328 0.775516i \(-0.717490\pi\)
−0.631328 + 0.775516i \(0.717490\pi\)
\(104\) 3.94952e6 0.344292
\(105\) −8.82978e6 −0.744367
\(106\) 601849. 0.0490814
\(107\) −8.87562e6 −0.700415 −0.350208 0.936672i \(-0.613889\pi\)
−0.350208 + 0.936672i \(0.613889\pi\)
\(108\) −1.80730e6 −0.138053
\(109\) −2.49654e7 −1.84648 −0.923242 0.384219i \(-0.874471\pi\)
−0.923242 + 0.384219i \(0.874471\pi\)
\(110\) −1.28248e7 −0.918708
\(111\) −2.46294e6 −0.170932
\(112\) 3.22098e6 0.216633
\(113\) 1.44492e7 0.942038 0.471019 0.882123i \(-0.343886\pi\)
0.471019 + 0.882123i \(0.343886\pi\)
\(114\) −7.70146e6 −0.486862
\(115\) −6.52110e6 −0.399833
\(116\) −242285. −0.0144120
\(117\) 2.17756e6 0.125696
\(118\) 1.23535e6 0.0692154
\(119\) −5.63773e6 −0.306683
\(120\) −1.37732e7 −0.727614
\(121\) 1.10547e7 0.567283
\(122\) −429856. −0.0214320
\(123\) 1.48339e7 0.718768
\(124\) −1.27386e7 −0.599994
\(125\) 2.85605e6 0.130792
\(126\) −3.71687e6 −0.165531
\(127\) 1.71357e7 0.742315 0.371157 0.928570i \(-0.378961\pi\)
0.371157 + 0.928570i \(0.378961\pi\)
\(128\) 2.05642e7 0.866716
\(129\) −6.50275e6 −0.266706
\(130\) 6.93181e6 0.276723
\(131\) −1.82887e7 −0.710777 −0.355388 0.934719i \(-0.615651\pi\)
−0.355388 + 0.934719i \(0.615651\pi\)
\(132\) 1.37009e7 0.518491
\(133\) 4.01968e7 1.48153
\(134\) 5.48918e6 0.197079
\(135\) −7.59384e6 −0.265640
\(136\) −8.79406e6 −0.299780
\(137\) −4.79157e7 −1.59205 −0.796023 0.605266i \(-0.793067\pi\)
−0.796023 + 0.605266i \(0.793067\pi\)
\(138\) −2.74504e6 −0.0889143
\(139\) 2.76336e7 0.872740 0.436370 0.899767i \(-0.356264\pi\)
0.436370 + 0.899767i \(0.356264\pi\)
\(140\) 3.00278e7 0.924860
\(141\) −3.18277e7 −0.956178
\(142\) 1.65350e6 0.0484613
\(143\) −1.65079e7 −0.472079
\(144\) 2.77013e6 0.0773092
\(145\) −1.01803e6 −0.0277313
\(146\) 3.21203e7 0.854170
\(147\) −2.83594e6 −0.0736355
\(148\) 8.37582e6 0.212379
\(149\) 6.17248e6 0.152865 0.0764325 0.997075i \(-0.475647\pi\)
0.0764325 + 0.997075i \(0.475647\pi\)
\(150\) −1.14856e7 −0.277865
\(151\) −2.68093e7 −0.633673 −0.316837 0.948480i \(-0.602621\pi\)
−0.316837 + 0.948480i \(0.602621\pi\)
\(152\) 6.27013e7 1.44819
\(153\) −4.84859e6 −0.109445
\(154\) 2.81772e7 0.621692
\(155\) −5.35249e7 −1.15450
\(156\) −7.40533e6 −0.156174
\(157\) 1.87147e7 0.385952 0.192976 0.981203i \(-0.438186\pi\)
0.192976 + 0.981203i \(0.438186\pi\)
\(158\) −6.76614e6 −0.136471
\(159\) −2.70158e6 −0.0533000
\(160\) 7.41134e7 1.43046
\(161\) 1.43274e7 0.270568
\(162\) −3.19660e6 −0.0590726
\(163\) 9.00787e7 1.62917 0.814583 0.580047i \(-0.196966\pi\)
0.814583 + 0.580047i \(0.196966\pi\)
\(164\) −5.04465e7 −0.893053
\(165\) 5.75681e7 0.997673
\(166\) 2.97743e7 0.505200
\(167\) −6.92715e7 −1.15093 −0.575463 0.817828i \(-0.695178\pi\)
−0.575463 + 0.817828i \(0.695178\pi\)
\(168\) 3.02608e7 0.492377
\(169\) −5.38260e7 −0.857806
\(170\) −1.54345e7 −0.240947
\(171\) 3.45703e7 0.528709
\(172\) 2.21142e7 0.331376
\(173\) −1.01646e8 −1.49255 −0.746277 0.665635i \(-0.768161\pi\)
−0.746277 + 0.665635i \(0.768161\pi\)
\(174\) −428534. −0.00616685
\(175\) 5.99476e7 0.845548
\(176\) −2.10001e7 −0.290353
\(177\) −5.54523e6 −0.0751646
\(178\) 5.20011e7 0.691103
\(179\) 1.59338e7 0.207650 0.103825 0.994596i \(-0.466892\pi\)
0.103825 + 0.994596i \(0.466892\pi\)
\(180\) 2.58247e7 0.330052
\(181\) 8.29983e7 1.04038 0.520192 0.854049i \(-0.325860\pi\)
0.520192 + 0.854049i \(0.325860\pi\)
\(182\) −1.52297e7 −0.187259
\(183\) 1.92953e6 0.0232742
\(184\) 2.23487e7 0.264478
\(185\) 3.51933e7 0.408657
\(186\) −2.25311e7 −0.256736
\(187\) 3.67567e7 0.411046
\(188\) 1.08238e8 1.18803
\(189\) 1.66843e7 0.179759
\(190\) 1.10047e8 1.16397
\(191\) −2.94448e7 −0.305768 −0.152884 0.988244i \(-0.548856\pi\)
−0.152884 + 0.988244i \(0.548856\pi\)
\(192\) 1.80653e7 0.184201
\(193\) 8.61854e7 0.862945 0.431473 0.902126i \(-0.357994\pi\)
0.431473 + 0.902126i \(0.357994\pi\)
\(194\) 1.62051e7 0.159347
\(195\) −3.11155e7 −0.300508
\(196\) 9.64432e6 0.0914904
\(197\) −1.74170e8 −1.62308 −0.811541 0.584296i \(-0.801371\pi\)
−0.811541 + 0.584296i \(0.801371\pi\)
\(198\) 2.42331e7 0.221861
\(199\) 9.07182e7 0.816035 0.408017 0.912974i \(-0.366220\pi\)
0.408017 + 0.912974i \(0.366220\pi\)
\(200\) 9.35098e7 0.826517
\(201\) −2.46398e7 −0.214018
\(202\) −9.11583e7 −0.778155
\(203\) 2.23668e6 0.0187658
\(204\) 1.64888e7 0.135983
\(205\) −2.11965e8 −1.71840
\(206\) 8.42266e7 0.671296
\(207\) 1.23219e7 0.0965566
\(208\) 1.13505e7 0.0874568
\(209\) −2.62074e8 −1.98569
\(210\) 5.31109e7 0.395746
\(211\) −1.22589e8 −0.898387 −0.449194 0.893434i \(-0.648289\pi\)
−0.449194 + 0.893434i \(0.648289\pi\)
\(212\) 9.18737e6 0.0662241
\(213\) −7.42223e6 −0.0526266
\(214\) 5.33866e7 0.372378
\(215\) 9.29188e7 0.637631
\(216\) 2.60251e7 0.175713
\(217\) 1.17598e8 0.781254
\(218\) 1.50166e8 0.981690
\(219\) −1.44181e8 −0.927588
\(220\) −1.95775e8 −1.23959
\(221\) −1.98669e7 −0.123811
\(222\) 1.48145e7 0.0908765
\(223\) −5.48773e7 −0.331380 −0.165690 0.986178i \(-0.552985\pi\)
−0.165690 + 0.986178i \(0.552985\pi\)
\(224\) −1.62833e8 −0.967997
\(225\) 5.15565e7 0.301748
\(226\) −8.69114e7 −0.500838
\(227\) 1.64956e8 0.936003 0.468001 0.883728i \(-0.344974\pi\)
0.468001 + 0.883728i \(0.344974\pi\)
\(228\) −1.17565e8 −0.656909
\(229\) 1.18120e8 0.649980 0.324990 0.945717i \(-0.394639\pi\)
0.324990 + 0.945717i \(0.394639\pi\)
\(230\) 3.92243e7 0.212573
\(231\) −1.26482e8 −0.675127
\(232\) 3.48891e6 0.0183435
\(233\) −1.12335e8 −0.581795 −0.290898 0.956754i \(-0.593954\pi\)
−0.290898 + 0.956754i \(0.593954\pi\)
\(234\) −1.30980e7 −0.0668265
\(235\) 4.54791e8 2.28599
\(236\) 1.88579e7 0.0933903
\(237\) 3.03718e7 0.148201
\(238\) 3.39108e7 0.163049
\(239\) 2.96755e8 1.40607 0.703033 0.711158i \(-0.251829\pi\)
0.703033 + 0.711158i \(0.251829\pi\)
\(240\) −3.95828e7 −0.184828
\(241\) 1.37243e7 0.0631583 0.0315791 0.999501i \(-0.489946\pi\)
0.0315791 + 0.999501i \(0.489946\pi\)
\(242\) −6.64939e7 −0.301598
\(243\) 1.43489e7 0.0641500
\(244\) −6.56186e6 −0.0289176
\(245\) 4.05232e7 0.176045
\(246\) −8.92258e7 −0.382136
\(247\) 1.41651e8 0.598107
\(248\) 1.83437e8 0.763670
\(249\) −1.33651e8 −0.548623
\(250\) −1.71791e7 −0.0695359
\(251\) −2.78781e7 −0.111277 −0.0556385 0.998451i \(-0.517719\pi\)
−0.0556385 + 0.998451i \(0.517719\pi\)
\(252\) −5.67389e7 −0.223347
\(253\) −9.34112e7 −0.362641
\(254\) −1.03071e8 −0.394654
\(255\) 6.92823e7 0.261657
\(256\) −2.09336e8 −0.779837
\(257\) 1.86791e8 0.686419 0.343209 0.939259i \(-0.388486\pi\)
0.343209 + 0.939259i \(0.388486\pi\)
\(258\) 3.91139e7 0.141795
\(259\) −7.73224e7 −0.276539
\(260\) 1.05816e8 0.373374
\(261\) 1.92360e6 0.00669690
\(262\) 1.10006e8 0.377887
\(263\) 2.51415e8 0.852210 0.426105 0.904674i \(-0.359885\pi\)
0.426105 + 0.904674i \(0.359885\pi\)
\(264\) −1.97294e8 −0.659932
\(265\) 3.86033e7 0.127428
\(266\) −2.41783e8 −0.787661
\(267\) −2.33422e8 −0.750504
\(268\) 8.37937e7 0.265913
\(269\) −4.66642e8 −1.46168 −0.730838 0.682551i \(-0.760870\pi\)
−0.730838 + 0.682551i \(0.760870\pi\)
\(270\) 4.56768e7 0.141228
\(271\) 4.71085e8 1.43783 0.718914 0.695099i \(-0.244640\pi\)
0.718914 + 0.695099i \(0.244640\pi\)
\(272\) −2.52732e7 −0.0761500
\(273\) 6.83632e7 0.203354
\(274\) 2.88212e8 0.846417
\(275\) −3.90844e8 −1.13329
\(276\) −4.19037e7 −0.119969
\(277\) −4.94424e8 −1.39772 −0.698860 0.715259i \(-0.746309\pi\)
−0.698860 + 0.715259i \(0.746309\pi\)
\(278\) −1.66215e8 −0.463995
\(279\) 1.01138e8 0.278803
\(280\) −4.32402e8 −1.17716
\(281\) −1.88387e8 −0.506499 −0.253249 0.967401i \(-0.581499\pi\)
−0.253249 + 0.967401i \(0.581499\pi\)
\(282\) 1.91443e8 0.508355
\(283\) 3.45526e8 0.906208 0.453104 0.891458i \(-0.350317\pi\)
0.453104 + 0.891458i \(0.350317\pi\)
\(284\) 2.52411e7 0.0653874
\(285\) −4.93981e8 −1.26402
\(286\) 9.92943e7 0.250982
\(287\) 4.65703e8 1.16285
\(288\) −1.40040e8 −0.345446
\(289\) −3.66103e8 −0.892196
\(290\) 6.12340e6 0.0147435
\(291\) −7.27413e7 −0.173044
\(292\) 4.90324e8 1.15251
\(293\) 4.35989e8 1.01260 0.506301 0.862357i \(-0.331012\pi\)
0.506301 + 0.862357i \(0.331012\pi\)
\(294\) 1.70581e7 0.0391485
\(295\) 7.92367e7 0.179701
\(296\) −1.20612e8 −0.270315
\(297\) −1.08778e8 −0.240930
\(298\) −3.71273e7 −0.0812712
\(299\) 5.04886e7 0.109231
\(300\) −1.75330e8 −0.374915
\(301\) −2.04150e8 −0.431486
\(302\) 1.61257e8 0.336895
\(303\) 4.09191e8 0.845040
\(304\) 1.80197e8 0.367867
\(305\) −2.75714e7 −0.0556429
\(306\) 2.91642e7 0.0581868
\(307\) 9.90146e8 1.95306 0.976529 0.215387i \(-0.0691013\pi\)
0.976529 + 0.215387i \(0.0691013\pi\)
\(308\) 4.30132e8 0.838830
\(309\) −3.78076e8 −0.728995
\(310\) 3.21951e8 0.613795
\(311\) −2.78364e8 −0.524750 −0.262375 0.964966i \(-0.584506\pi\)
−0.262375 + 0.964966i \(0.584506\pi\)
\(312\) 1.06637e8 0.198777
\(313\) −7.58142e8 −1.39748 −0.698740 0.715376i \(-0.746256\pi\)
−0.698740 + 0.715376i \(0.746256\pi\)
\(314\) −1.12568e8 −0.205193
\(315\) −2.38404e8 −0.429761
\(316\) −1.03287e8 −0.184137
\(317\) −3.93211e8 −0.693296 −0.346648 0.937995i \(-0.612680\pi\)
−0.346648 + 0.937995i \(0.612680\pi\)
\(318\) 1.62499e7 0.0283371
\(319\) −1.45826e7 −0.0251518
\(320\) −2.58138e8 −0.440379
\(321\) −2.39642e8 −0.404385
\(322\) −8.61788e7 −0.143848
\(323\) −3.15402e8 −0.520781
\(324\) −4.87970e7 −0.0797049
\(325\) 2.11251e8 0.341355
\(326\) −5.41821e8 −0.866152
\(327\) −6.74066e8 −1.06607
\(328\) 7.26431e8 1.13667
\(329\) −9.99212e8 −1.54693
\(330\) −3.46271e8 −0.530417
\(331\) −9.50914e7 −0.144126 −0.0720632 0.997400i \(-0.522958\pi\)
−0.0720632 + 0.997400i \(0.522958\pi\)
\(332\) 4.54512e8 0.681652
\(333\) −6.64993e7 −0.0986876
\(334\) 4.16666e8 0.611894
\(335\) 3.52082e8 0.511667
\(336\) 8.69665e7 0.125073
\(337\) −6.91531e8 −0.984254 −0.492127 0.870524i \(-0.663780\pi\)
−0.492127 + 0.870524i \(0.663780\pi\)
\(338\) 3.23762e8 0.456055
\(339\) 3.90128e8 0.543886
\(340\) −2.35611e8 −0.325102
\(341\) −7.66714e8 −1.04711
\(342\) −2.07939e8 −0.281090
\(343\) −7.87108e8 −1.05319
\(344\) −3.18445e8 −0.421774
\(345\) −1.76070e8 −0.230844
\(346\) 6.11400e8 0.793522
\(347\) −5.15844e8 −0.662774 −0.331387 0.943495i \(-0.607516\pi\)
−0.331387 + 0.943495i \(0.607516\pi\)
\(348\) −6.54169e6 −0.00832075
\(349\) −8.89896e8 −1.12060 −0.560300 0.828290i \(-0.689314\pi\)
−0.560300 + 0.828290i \(0.689314\pi\)
\(350\) −3.60583e8 −0.449539
\(351\) 5.87941e7 0.0725703
\(352\) 1.06163e9 1.29740
\(353\) −3.85325e8 −0.466246 −0.233123 0.972447i \(-0.574895\pi\)
−0.233123 + 0.972447i \(0.574895\pi\)
\(354\) 3.33544e7 0.0399615
\(355\) 1.06057e8 0.125818
\(356\) 7.93810e8 0.932485
\(357\) −1.52219e8 −0.177063
\(358\) −9.58411e7 −0.110398
\(359\) −1.51447e9 −1.72755 −0.863777 0.503875i \(-0.831907\pi\)
−0.863777 + 0.503875i \(0.831907\pi\)
\(360\) −3.71877e8 −0.420088
\(361\) 1.35493e9 1.51580
\(362\) −4.99232e8 −0.553124
\(363\) 2.98478e8 0.327521
\(364\) −2.32486e8 −0.252663
\(365\) 2.06023e9 2.21764
\(366\) −1.16061e7 −0.0123738
\(367\) −8.13769e8 −0.859350 −0.429675 0.902984i \(-0.641372\pi\)
−0.429675 + 0.902984i \(0.641372\pi\)
\(368\) 6.42279e7 0.0671825
\(369\) 4.00517e8 0.414981
\(370\) −2.11687e8 −0.217264
\(371\) −8.48144e7 −0.0862305
\(372\) −3.43943e8 −0.346407
\(373\) −1.20212e9 −1.19940 −0.599702 0.800223i \(-0.704714\pi\)
−0.599702 + 0.800223i \(0.704714\pi\)
\(374\) −2.21090e8 −0.218534
\(375\) 7.71133e7 0.0755127
\(376\) −1.55863e9 −1.51212
\(377\) 7.88190e6 0.00757594
\(378\) −1.00355e8 −0.0955695
\(379\) −4.34931e8 −0.410377 −0.205189 0.978722i \(-0.565781\pi\)
−0.205189 + 0.978722i \(0.565781\pi\)
\(380\) 1.67990e9 1.57051
\(381\) 4.62663e8 0.428576
\(382\) 1.77110e8 0.162563
\(383\) 3.58726e7 0.0326263 0.0163131 0.999867i \(-0.494807\pi\)
0.0163131 + 0.999867i \(0.494807\pi\)
\(384\) 5.55233e8 0.500399
\(385\) 1.80732e9 1.61407
\(386\) −5.18403e8 −0.458788
\(387\) −1.75574e8 −0.153983
\(388\) 2.47374e8 0.215003
\(389\) −1.13638e9 −0.978815 −0.489407 0.872055i \(-0.662787\pi\)
−0.489407 + 0.872055i \(0.662787\pi\)
\(390\) 1.87159e8 0.159766
\(391\) −1.12419e8 −0.0951087
\(392\) −1.38878e8 −0.116448
\(393\) −4.93795e8 −0.410367
\(394\) 1.04762e9 0.862917
\(395\) −4.33988e8 −0.354314
\(396\) 3.69925e8 0.299351
\(397\) −5.01657e8 −0.402383 −0.201192 0.979552i \(-0.564481\pi\)
−0.201192 + 0.979552i \(0.564481\pi\)
\(398\) −5.45667e8 −0.433848
\(399\) 1.08531e9 0.855362
\(400\) 2.68738e8 0.209951
\(401\) 6.17044e8 0.477871 0.238936 0.971035i \(-0.423202\pi\)
0.238936 + 0.971035i \(0.423202\pi\)
\(402\) 1.48208e8 0.113784
\(403\) 4.14408e8 0.315399
\(404\) −1.39155e9 −1.04994
\(405\) −2.05034e8 −0.153367
\(406\) −1.34536e7 −0.00997692
\(407\) 5.04124e8 0.370644
\(408\) −2.37440e8 −0.173078
\(409\) 1.67124e9 1.20783 0.603915 0.797048i \(-0.293606\pi\)
0.603915 + 0.797048i \(0.293606\pi\)
\(410\) 1.27496e9 0.913595
\(411\) −1.29372e9 −0.919168
\(412\) 1.28574e9 0.905760
\(413\) −1.74089e8 −0.121604
\(414\) −7.41160e7 −0.0513347
\(415\) 1.90976e9 1.31163
\(416\) −5.73811e8 −0.390789
\(417\) 7.46106e8 0.503877
\(418\) 1.57637e9 1.05570
\(419\) −6.96670e7 −0.0462677 −0.0231339 0.999732i \(-0.507364\pi\)
−0.0231339 + 0.999732i \(0.507364\pi\)
\(420\) 8.10751e8 0.533968
\(421\) −2.02237e9 −1.32091 −0.660457 0.750864i \(-0.729637\pi\)
−0.660457 + 0.750864i \(0.729637\pi\)
\(422\) 7.37371e8 0.477631
\(423\) −8.59349e8 −0.552050
\(424\) −1.32298e8 −0.0842896
\(425\) −4.70375e8 −0.297223
\(426\) 4.46445e7 0.0279791
\(427\) 6.05766e7 0.0376537
\(428\) 8.14960e8 0.502439
\(429\) −4.45712e8 −0.272555
\(430\) −5.58904e8 −0.338999
\(431\) −2.20133e9 −1.32439 −0.662193 0.749333i \(-0.730374\pi\)
−0.662193 + 0.749333i \(0.730374\pi\)
\(432\) 7.47935e7 0.0446345
\(433\) −1.11885e9 −0.662316 −0.331158 0.943575i \(-0.607439\pi\)
−0.331158 + 0.943575i \(0.607439\pi\)
\(434\) −7.07351e8 −0.415356
\(435\) −2.74867e7 −0.0160107
\(436\) 2.29232e9 1.32457
\(437\) 8.01542e8 0.459453
\(438\) 8.67248e8 0.493155
\(439\) 3.23226e8 0.182340 0.0911698 0.995835i \(-0.470939\pi\)
0.0911698 + 0.995835i \(0.470939\pi\)
\(440\) 2.81916e9 1.57774
\(441\) −7.65705e7 −0.0425134
\(442\) 1.19499e8 0.0658244
\(443\) −2.28214e9 −1.24718 −0.623591 0.781751i \(-0.714327\pi\)
−0.623591 + 0.781751i \(0.714327\pi\)
\(444\) 2.26147e8 0.122617
\(445\) 3.33541e9 1.79428
\(446\) 3.30086e8 0.176179
\(447\) 1.66657e8 0.0882566
\(448\) 5.67149e8 0.298005
\(449\) 2.89839e9 1.51111 0.755553 0.655087i \(-0.227368\pi\)
0.755553 + 0.655087i \(0.227368\pi\)
\(450\) −3.10111e8 −0.160425
\(451\) −3.03627e9 −1.55856
\(452\) −1.32672e9 −0.675766
\(453\) −7.23850e8 −0.365852
\(454\) −9.92205e8 −0.497629
\(455\) −9.76852e8 −0.486171
\(456\) 1.69294e9 0.836110
\(457\) −1.61157e9 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(458\) −7.10490e8 −0.345564
\(459\) −1.30912e8 −0.0631881
\(460\) 5.98768e8 0.286818
\(461\) 4.98049e8 0.236766 0.118383 0.992968i \(-0.462229\pi\)
0.118383 + 0.992968i \(0.462229\pi\)
\(462\) 7.60784e8 0.358934
\(463\) −1.13218e9 −0.530131 −0.265065 0.964230i \(-0.585394\pi\)
−0.265065 + 0.964230i \(0.585394\pi\)
\(464\) 1.00268e7 0.00465959
\(465\) −1.44517e9 −0.666552
\(466\) 6.75693e8 0.309313
\(467\) 3.04377e9 1.38294 0.691470 0.722405i \(-0.256963\pi\)
0.691470 + 0.722405i \(0.256963\pi\)
\(468\) −1.99944e8 −0.0901670
\(469\) −7.73552e8 −0.346246
\(470\) −2.73556e9 −1.21536
\(471\) 5.05296e8 0.222829
\(472\) −2.71555e8 −0.118867
\(473\) 1.33101e9 0.578319
\(474\) −1.82686e8 −0.0787917
\(475\) 3.35376e9 1.43583
\(476\) 5.17657e8 0.219997
\(477\) −7.29426e7 −0.0307728
\(478\) −1.78497e9 −0.747539
\(479\) 1.85843e9 0.772630 0.386315 0.922367i \(-0.373748\pi\)
0.386315 + 0.922367i \(0.373748\pi\)
\(480\) 2.00106e9 0.825878
\(481\) −2.72479e8 −0.111641
\(482\) −8.25512e7 −0.0335783
\(483\) 3.86839e8 0.156212
\(484\) −1.01505e9 −0.406937
\(485\) 1.03941e9 0.413705
\(486\) −8.63083e7 −0.0341056
\(487\) −3.95871e8 −0.155311 −0.0776555 0.996980i \(-0.524743\pi\)
−0.0776555 + 0.996980i \(0.524743\pi\)
\(488\) 9.44909e7 0.0368062
\(489\) 2.43212e9 0.940600
\(490\) −2.43746e8 −0.0935948
\(491\) 8.60610e8 0.328111 0.164056 0.986451i \(-0.447542\pi\)
0.164056 + 0.986451i \(0.447542\pi\)
\(492\) −1.36205e9 −0.515605
\(493\) −1.75500e7 −0.00659648
\(494\) −8.52025e8 −0.317986
\(495\) 1.55434e9 0.576007
\(496\) 5.27179e8 0.193987
\(497\) −2.33016e8 −0.0851410
\(498\) 8.03906e8 0.291677
\(499\) 2.92320e9 1.05319 0.526595 0.850116i \(-0.323469\pi\)
0.526595 + 0.850116i \(0.323469\pi\)
\(500\) −2.62243e8 −0.0938228
\(501\) −1.87033e9 −0.664487
\(502\) 1.67686e8 0.0591609
\(503\) 2.74156e9 0.960529 0.480264 0.877124i \(-0.340541\pi\)
0.480264 + 0.877124i \(0.340541\pi\)
\(504\) 8.17042e8 0.284274
\(505\) −5.84700e9 −2.02029
\(506\) 5.61866e8 0.192799
\(507\) −1.45330e9 −0.495254
\(508\) −1.57340e9 −0.532495
\(509\) 3.66404e9 1.23154 0.615769 0.787927i \(-0.288846\pi\)
0.615769 + 0.787927i \(0.288846\pi\)
\(510\) −4.16731e8 −0.139111
\(511\) −4.52649e9 −1.50068
\(512\) −1.37307e9 −0.452113
\(513\) 9.33398e8 0.305250
\(514\) −1.12354e9 −0.364937
\(515\) 5.40239e9 1.74285
\(516\) 5.97083e8 0.191320
\(517\) 6.51463e9 2.07335
\(518\) 4.65092e8 0.147023
\(519\) −2.74445e9 −0.861727
\(520\) −1.52375e9 −0.475229
\(521\) −4.43075e9 −1.37260 −0.686302 0.727317i \(-0.740767\pi\)
−0.686302 + 0.727317i \(0.740767\pi\)
\(522\) −1.15704e7 −0.00356043
\(523\) −5.16773e9 −1.57959 −0.789794 0.613372i \(-0.789813\pi\)
−0.789794 + 0.613372i \(0.789813\pi\)
\(524\) 1.67927e9 0.509872
\(525\) 1.61858e9 0.488178
\(526\) −1.51226e9 −0.453080
\(527\) −9.22728e8 −0.274623
\(528\) −5.67002e8 −0.167635
\(529\) −3.11913e9 −0.916091
\(530\) −2.32198e8 −0.0677473
\(531\) −1.49721e8 −0.0433963
\(532\) −3.69087e9 −1.06277
\(533\) 1.64110e9 0.469451
\(534\) 1.40403e9 0.399008
\(535\) 3.42428e9 0.966787
\(536\) −1.20663e9 −0.338453
\(537\) 4.30211e8 0.119887
\(538\) 2.80684e9 0.777105
\(539\) 5.80473e8 0.159669
\(540\) 6.97267e8 0.190555
\(541\) 5.16094e9 1.40132 0.700662 0.713493i \(-0.252888\pi\)
0.700662 + 0.713493i \(0.252888\pi\)
\(542\) −2.83356e9 −0.764426
\(543\) 2.24095e9 0.600666
\(544\) 1.27766e9 0.340266
\(545\) 9.63183e9 2.54871
\(546\) −4.11203e8 −0.108114
\(547\) −1.77725e9 −0.464295 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(548\) 4.39962e9 1.14205
\(549\) 5.20974e7 0.0134373
\(550\) 2.35092e9 0.602515
\(551\) 1.25131e8 0.0318664
\(552\) 6.03414e8 0.152696
\(553\) 9.53505e8 0.239765
\(554\) 2.97394e9 0.743102
\(555\) 9.50219e8 0.235938
\(556\) −2.53732e9 −0.626055
\(557\) −2.65766e9 −0.651637 −0.325819 0.945432i \(-0.605640\pi\)
−0.325819 + 0.945432i \(0.605640\pi\)
\(558\) −6.08340e8 −0.148227
\(559\) −7.19409e8 −0.174195
\(560\) −1.24268e9 −0.299020
\(561\) 9.92430e8 0.237318
\(562\) 1.13314e9 0.269282
\(563\) −2.85664e9 −0.674646 −0.337323 0.941389i \(-0.609522\pi\)
−0.337323 + 0.941389i \(0.609522\pi\)
\(564\) 2.92242e9 0.685909
\(565\) −5.57459e9 −1.30030
\(566\) −2.07833e9 −0.481789
\(567\) 4.50475e8 0.103784
\(568\) −3.63472e8 −0.0832247
\(569\) −1.18310e9 −0.269233 −0.134617 0.990898i \(-0.542980\pi\)
−0.134617 + 0.990898i \(0.542980\pi\)
\(570\) 2.97128e9 0.672019
\(571\) 7.04152e8 0.158285 0.0791426 0.996863i \(-0.474782\pi\)
0.0791426 + 0.996863i \(0.474782\pi\)
\(572\) 1.51575e9 0.338643
\(573\) −7.95010e8 −0.176535
\(574\) −2.80119e9 −0.618231
\(575\) 1.19538e9 0.262222
\(576\) 4.87763e8 0.106348
\(577\) −5.26118e9 −1.14017 −0.570083 0.821587i \(-0.693089\pi\)
−0.570083 + 0.821587i \(0.693089\pi\)
\(578\) 2.20210e9 0.474339
\(579\) 2.32701e9 0.498222
\(580\) 9.34752e7 0.0198929
\(581\) −4.19589e9 −0.887580
\(582\) 4.37537e8 0.0919992
\(583\) 5.52970e8 0.115574
\(584\) −7.06068e9 −1.46690
\(585\) −8.40119e8 −0.173498
\(586\) −2.62246e9 −0.538354
\(587\) 7.97467e9 1.62734 0.813672 0.581324i \(-0.197465\pi\)
0.813672 + 0.581324i \(0.197465\pi\)
\(588\) 2.60397e8 0.0528220
\(589\) 6.57902e9 1.32665
\(590\) −4.76607e8 −0.0955384
\(591\) −4.70258e9 −0.937087
\(592\) −3.46627e8 −0.0686651
\(593\) −5.84550e9 −1.15115 −0.575573 0.817750i \(-0.695221\pi\)
−0.575573 + 0.817750i \(0.695221\pi\)
\(594\) 6.54294e8 0.128092
\(595\) 2.17508e9 0.423316
\(596\) −5.66758e8 −0.109657
\(597\) 2.44939e9 0.471138
\(598\) −3.03688e8 −0.0580729
\(599\) −2.93990e9 −0.558906 −0.279453 0.960159i \(-0.590153\pi\)
−0.279453 + 0.960159i \(0.590153\pi\)
\(600\) 2.52476e9 0.477190
\(601\) −6.68201e9 −1.25559 −0.627793 0.778380i \(-0.716042\pi\)
−0.627793 + 0.778380i \(0.716042\pi\)
\(602\) 1.22796e9 0.229401
\(603\) −6.65275e8 −0.123564
\(604\) 2.46163e9 0.454562
\(605\) −4.26500e9 −0.783024
\(606\) −2.46127e9 −0.449268
\(607\) 4.48841e8 0.0814577 0.0407289 0.999170i \(-0.487032\pi\)
0.0407289 + 0.999170i \(0.487032\pi\)
\(608\) −9.10965e9 −1.64376
\(609\) 6.03904e7 0.0108345
\(610\) 1.65841e8 0.0295827
\(611\) −3.52115e9 −0.624511
\(612\) 4.45198e8 0.0785097
\(613\) 7.78075e9 1.36430 0.682150 0.731212i \(-0.261045\pi\)
0.682150 + 0.731212i \(0.261045\pi\)
\(614\) −5.95570e9 −1.03835
\(615\) −5.72305e9 −0.992120
\(616\) −6.19391e9 −1.06766
\(617\) −6.97925e9 −1.19622 −0.598109 0.801414i \(-0.704081\pi\)
−0.598109 + 0.801414i \(0.704081\pi\)
\(618\) 2.27412e9 0.387573
\(619\) −3.49030e9 −0.591488 −0.295744 0.955267i \(-0.595568\pi\)
−0.295744 + 0.955267i \(0.595568\pi\)
\(620\) 4.91466e9 0.828176
\(621\) 3.32692e8 0.0557470
\(622\) 1.67435e9 0.278985
\(623\) −7.32816e9 −1.21419
\(624\) 3.06464e8 0.0504932
\(625\) −6.62706e9 −1.08578
\(626\) 4.56021e9 0.742975
\(627\) −7.07600e9 −1.14644
\(628\) −1.71838e9 −0.276861
\(629\) 6.06705e8 0.0972077
\(630\) 1.43399e9 0.228484
\(631\) −7.42461e9 −1.17644 −0.588222 0.808700i \(-0.700172\pi\)
−0.588222 + 0.808700i \(0.700172\pi\)
\(632\) 1.48733e9 0.234368
\(633\) −3.30991e9 −0.518684
\(634\) 2.36516e9 0.368593
\(635\) −6.61107e9 −1.02462
\(636\) 2.48059e8 0.0382345
\(637\) −3.13745e8 −0.0480938
\(638\) 8.77142e7 0.0133720
\(639\) −2.00400e8 −0.0303840
\(640\) −7.93382e9 −1.19633
\(641\) −2.52065e9 −0.378015 −0.189007 0.981976i \(-0.560527\pi\)
−0.189007 + 0.981976i \(0.560527\pi\)
\(642\) 1.44144e9 0.214993
\(643\) −7.35494e9 −1.09104 −0.545520 0.838098i \(-0.683668\pi\)
−0.545520 + 0.838098i \(0.683668\pi\)
\(644\) −1.31554e9 −0.194090
\(645\) 2.50881e9 0.368136
\(646\) 1.89713e9 0.276875
\(647\) −9.35379e8 −0.135776 −0.0678879 0.997693i \(-0.521626\pi\)
−0.0678879 + 0.997693i \(0.521626\pi\)
\(648\) 7.02678e8 0.101448
\(649\) 1.13502e9 0.162985
\(650\) −1.27067e9 −0.181483
\(651\) 3.17515e9 0.451057
\(652\) −8.27103e9 −1.16867
\(653\) 1.22407e10 1.72032 0.860161 0.510022i \(-0.170363\pi\)
0.860161 + 0.510022i \(0.170363\pi\)
\(654\) 4.05449e9 0.566779
\(655\) 7.05591e9 0.981089
\(656\) 2.08769e9 0.288737
\(657\) −3.89290e9 −0.535543
\(658\) 6.01023e9 0.822433
\(659\) 5.21215e9 0.709444 0.354722 0.934972i \(-0.384575\pi\)
0.354722 + 0.934972i \(0.384575\pi\)
\(660\) −5.28591e9 −0.715676
\(661\) −1.11350e10 −1.49964 −0.749820 0.661642i \(-0.769860\pi\)
−0.749820 + 0.661642i \(0.769860\pi\)
\(662\) 5.71972e8 0.0766253
\(663\) −5.36407e8 −0.0714821
\(664\) −6.54499e9 −0.867603
\(665\) −1.55082e10 −2.04497
\(666\) 3.99992e8 0.0524676
\(667\) 4.46004e7 0.00581967
\(668\) 6.36052e9 0.825610
\(669\) −1.48169e9 −0.191322
\(670\) −2.11776e9 −0.272029
\(671\) −3.94945e8 −0.0504671
\(672\) −4.39649e9 −0.558873
\(673\) 9.15611e9 1.15787 0.578933 0.815375i \(-0.303469\pi\)
0.578933 + 0.815375i \(0.303469\pi\)
\(674\) 4.15954e9 0.523282
\(675\) 1.39203e9 0.174214
\(676\) 4.94231e9 0.615342
\(677\) 1.14733e10 1.42111 0.710557 0.703639i \(-0.248443\pi\)
0.710557 + 0.703639i \(0.248443\pi\)
\(678\) −2.34661e9 −0.289159
\(679\) −2.28367e9 −0.279955
\(680\) 3.39281e9 0.413788
\(681\) 4.45381e9 0.540401
\(682\) 4.61176e9 0.556701
\(683\) −5.63555e9 −0.676806 −0.338403 0.941001i \(-0.609887\pi\)
−0.338403 + 0.941001i \(0.609887\pi\)
\(684\) −3.17425e9 −0.379267
\(685\) 1.84862e10 2.19751
\(686\) 4.73443e9 0.559930
\(687\) 3.18924e9 0.375266
\(688\) −9.15179e8 −0.107139
\(689\) −2.98880e8 −0.0348120
\(690\) 1.05906e9 0.122729
\(691\) −6.22407e9 −0.717632 −0.358816 0.933408i \(-0.616819\pi\)
−0.358816 + 0.933408i \(0.616819\pi\)
\(692\) 9.33317e9 1.07068
\(693\) −3.41501e9 −0.389785
\(694\) 3.10279e9 0.352366
\(695\) −1.06612e10 −1.20465
\(696\) 9.42005e7 0.0105906
\(697\) −3.65411e9 −0.408758
\(698\) 5.35270e9 0.595771
\(699\) −3.03305e9 −0.335900
\(700\) −5.50439e9 −0.606550
\(701\) 8.57820e9 0.940553 0.470276 0.882519i \(-0.344154\pi\)
0.470276 + 0.882519i \(0.344154\pi\)
\(702\) −3.53645e8 −0.0385823
\(703\) −4.32579e9 −0.469593
\(704\) −3.69768e9 −0.399416
\(705\) 1.22794e10 1.31982
\(706\) 2.31772e9 0.247881
\(707\) 1.28463e10 1.36713
\(708\) 5.09164e8 0.0539189
\(709\) 2.72444e9 0.287088 0.143544 0.989644i \(-0.454150\pi\)
0.143544 + 0.989644i \(0.454150\pi\)
\(710\) −6.37932e8 −0.0668914
\(711\) 8.20039e8 0.0855640
\(712\) −1.14309e10 −1.18686
\(713\) 2.34496e9 0.242283
\(714\) 9.15591e8 0.0941364
\(715\) 6.36885e9 0.651613
\(716\) −1.46304e9 −0.148957
\(717\) 8.01239e9 0.811792
\(718\) 9.10952e9 0.918460
\(719\) −2.39574e9 −0.240375 −0.120188 0.992751i \(-0.538350\pi\)
−0.120188 + 0.992751i \(0.538350\pi\)
\(720\) −1.06874e9 −0.106710
\(721\) −1.18695e10 −1.17939
\(722\) −8.14988e9 −0.805881
\(723\) 3.70556e8 0.0364644
\(724\) −7.62091e9 −0.746314
\(725\) 1.86614e8 0.0181870
\(726\) −1.79534e9 −0.174128
\(727\) 1.12427e9 0.108517 0.0542586 0.998527i \(-0.482720\pi\)
0.0542586 + 0.998527i \(0.482720\pi\)
\(728\) 3.34780e9 0.321588
\(729\) 3.87420e8 0.0370370
\(730\) −1.23922e10 −1.17902
\(731\) 1.60185e9 0.151674
\(732\) −1.77170e8 −0.0166956
\(733\) 1.04255e10 0.977759 0.488880 0.872351i \(-0.337406\pi\)
0.488880 + 0.872351i \(0.337406\pi\)
\(734\) 4.89480e9 0.456876
\(735\) 1.09413e9 0.101639
\(736\) −3.24696e9 −0.300196
\(737\) 5.04338e9 0.464072
\(738\) −2.40910e9 −0.220626
\(739\) −1.61439e8 −0.0147147 −0.00735737 0.999973i \(-0.502342\pi\)
−0.00735737 + 0.999973i \(0.502342\pi\)
\(740\) −3.23145e9 −0.293148
\(741\) 3.82457e9 0.345317
\(742\) 5.10156e8 0.0458447
\(743\) 1.59100e10 1.42302 0.711508 0.702678i \(-0.248013\pi\)
0.711508 + 0.702678i \(0.248013\pi\)
\(744\) 4.95279e9 0.440905
\(745\) −2.38139e9 −0.211000
\(746\) 7.23070e9 0.637668
\(747\) −3.60857e9 −0.316748
\(748\) −3.37500e9 −0.294862
\(749\) −7.52341e9 −0.654226
\(750\) −4.63834e8 −0.0401466
\(751\) 9.63052e9 0.829679 0.414839 0.909895i \(-0.363838\pi\)
0.414839 + 0.909895i \(0.363838\pi\)
\(752\) −4.47934e9 −0.384107
\(753\) −7.52710e8 −0.0642459
\(754\) −4.74094e7 −0.00402777
\(755\) 1.03432e10 0.874663
\(756\) −1.53195e9 −0.128949
\(757\) −1.23984e10 −1.03880 −0.519398 0.854532i \(-0.673844\pi\)
−0.519398 + 0.854532i \(0.673844\pi\)
\(758\) 2.61610e9 0.218178
\(759\) −2.52210e9 −0.209371
\(760\) −2.41906e10 −1.99894
\(761\) −3.84482e9 −0.316249 −0.158125 0.987419i \(-0.550545\pi\)
−0.158125 + 0.987419i \(0.550545\pi\)
\(762\) −2.78291e9 −0.227854
\(763\) −2.11619e10 −1.72472
\(764\) 2.70363e9 0.219341
\(765\) 1.87062e9 0.151067
\(766\) −2.15773e8 −0.0173459
\(767\) −6.13478e8 −0.0490925
\(768\) −5.65207e9 −0.450239
\(769\) 1.85468e10 1.47071 0.735354 0.677683i \(-0.237016\pi\)
0.735354 + 0.677683i \(0.237016\pi\)
\(770\) −1.08710e10 −0.858125
\(771\) 5.04335e9 0.396304
\(772\) −7.91355e9 −0.619029
\(773\) 9.18489e9 0.715230 0.357615 0.933869i \(-0.383590\pi\)
0.357615 + 0.933869i \(0.383590\pi\)
\(774\) 1.05607e9 0.0818656
\(775\) 9.81163e9 0.757156
\(776\) −3.56220e9 −0.273654
\(777\) −2.08771e9 −0.159660
\(778\) 6.83530e9 0.520390
\(779\) 2.60537e10 1.97464
\(780\) 2.85703e9 0.215568
\(781\) 1.51921e9 0.114114
\(782\) 6.76196e8 0.0505649
\(783\) 5.19373e7 0.00386646
\(784\) −3.99123e8 −0.0295801
\(785\) −7.22025e9 −0.532732
\(786\) 2.97016e9 0.218173
\(787\) −1.00821e10 −0.737291 −0.368646 0.929570i \(-0.620178\pi\)
−0.368646 + 0.929570i \(0.620178\pi\)
\(788\) 1.59923e10 1.16431
\(789\) 6.78821e9 0.492024
\(790\) 2.61043e9 0.188372
\(791\) 1.22478e10 0.879916
\(792\) −5.32693e9 −0.381012
\(793\) 2.13467e8 0.0152011
\(794\) 3.01745e9 0.213928
\(795\) 1.04229e9 0.0735703
\(796\) −8.32975e9 −0.585378
\(797\) −1.03385e10 −0.723361 −0.361681 0.932302i \(-0.617797\pi\)
−0.361681 + 0.932302i \(0.617797\pi\)
\(798\) −6.52813e9 −0.454756
\(799\) 7.84025e9 0.543771
\(800\) −1.35857e10 −0.938139
\(801\) −6.30241e9 −0.433304
\(802\) −3.71150e9 −0.254062
\(803\) 2.95117e10 2.01136
\(804\) 2.26243e9 0.153525
\(805\) −5.52760e9 −0.373466
\(806\) −2.49265e9 −0.167683
\(807\) −1.25993e10 −0.843899
\(808\) 2.00384e10 1.33636
\(809\) 2.64371e10 1.75547 0.877735 0.479146i \(-0.159053\pi\)
0.877735 + 0.479146i \(0.159053\pi\)
\(810\) 1.23327e9 0.0815383
\(811\) 2.22378e10 1.46392 0.731961 0.681346i \(-0.238605\pi\)
0.731961 + 0.681346i \(0.238605\pi\)
\(812\) −2.05372e8 −0.0134616
\(813\) 1.27193e10 0.830130
\(814\) −3.03229e9 −0.197054
\(815\) −3.47530e10 −2.24875
\(816\) −6.82377e8 −0.0439652
\(817\) −1.14211e10 −0.732710
\(818\) −1.00524e10 −0.642147
\(819\) 1.84581e9 0.117407
\(820\) 1.94626e10 1.23269
\(821\) 1.19117e10 0.751229 0.375615 0.926776i \(-0.377432\pi\)
0.375615 + 0.926776i \(0.377432\pi\)
\(822\) 7.78171e9 0.488679
\(823\) 1.73151e9 0.108274 0.0541372 0.998534i \(-0.482759\pi\)
0.0541372 + 0.998534i \(0.482759\pi\)
\(824\) −1.85147e10 −1.15285
\(825\) −1.05528e10 −0.654303
\(826\) 1.04714e9 0.0646510
\(827\) 2.70946e10 1.66576 0.832882 0.553451i \(-0.186689\pi\)
0.832882 + 0.553451i \(0.186689\pi\)
\(828\) −1.13140e9 −0.0692644
\(829\) 1.95311e10 1.19065 0.595326 0.803484i \(-0.297023\pi\)
0.595326 + 0.803484i \(0.297023\pi\)
\(830\) −1.14871e10 −0.697331
\(831\) −1.33494e10 −0.806974
\(832\) 1.99859e9 0.120307
\(833\) 6.98589e8 0.0418759
\(834\) −4.48781e9 −0.267888
\(835\) 2.67255e10 1.58863
\(836\) 2.40637e10 1.42442
\(837\) 2.73072e9 0.160967
\(838\) 4.19045e8 0.0245984
\(839\) −2.20343e10 −1.28805 −0.644024 0.765005i \(-0.722737\pi\)
−0.644024 + 0.765005i \(0.722737\pi\)
\(840\) −1.16748e10 −0.679632
\(841\) −1.72429e10 −0.999596
\(842\) 1.21645e10 0.702268
\(843\) −5.08644e9 −0.292427
\(844\) 1.12562e10 0.644453
\(845\) 2.07665e10 1.18403
\(846\) 5.16896e9 0.293499
\(847\) 9.37053e9 0.529873
\(848\) −3.80212e8 −0.0214112
\(849\) 9.32919e9 0.523199
\(850\) 2.82929e9 0.158020
\(851\) −1.54184e9 −0.0857604
\(852\) 6.81510e8 0.0377514
\(853\) 2.56385e10 1.41440 0.707199 0.707014i \(-0.249958\pi\)
0.707199 + 0.707014i \(0.249958\pi\)
\(854\) −3.64366e8 −0.0200187
\(855\) −1.33375e10 −0.729780
\(856\) −1.17354e10 −0.639502
\(857\) −1.90552e10 −1.03414 −0.517072 0.855942i \(-0.672978\pi\)
−0.517072 + 0.855942i \(0.672978\pi\)
\(858\) 2.68095e9 0.144905
\(859\) 1.94648e10 1.04779 0.523896 0.851782i \(-0.324478\pi\)
0.523896 + 0.851782i \(0.324478\pi\)
\(860\) −8.53182e9 −0.457401
\(861\) 1.25740e10 0.671369
\(862\) 1.32409e10 0.704115
\(863\) −6.07650e9 −0.321822 −0.160911 0.986969i \(-0.551443\pi\)
−0.160911 + 0.986969i \(0.551443\pi\)
\(864\) −3.78109e9 −0.199443
\(865\) 3.92159e10 2.06018
\(866\) 6.72987e9 0.352123
\(867\) −9.88477e9 −0.515110
\(868\) −1.07979e10 −0.560428
\(869\) −6.21663e9 −0.321356
\(870\) 1.65332e8 0.00851214
\(871\) −2.72594e9 −0.139783
\(872\) −3.30095e10 −1.68590
\(873\) −1.96401e9 −0.0999067
\(874\) −4.82126e9 −0.244270
\(875\) 2.42092e9 0.122167
\(876\) 1.32388e10 0.665400
\(877\) −1.82857e10 −0.915404 −0.457702 0.889106i \(-0.651327\pi\)
−0.457702 + 0.889106i \(0.651327\pi\)
\(878\) −1.94420e9 −0.0969415
\(879\) 1.17717e10 0.584626
\(880\) 8.10198e9 0.400776
\(881\) 1.41923e10 0.699259 0.349629 0.936888i \(-0.386308\pi\)
0.349629 + 0.936888i \(0.386308\pi\)
\(882\) 4.60569e8 0.0226024
\(883\) 1.61031e10 0.787131 0.393566 0.919297i \(-0.371241\pi\)
0.393566 + 0.919297i \(0.371241\pi\)
\(884\) 1.82418e9 0.0888149
\(885\) 2.13939e9 0.103750
\(886\) 1.37270e10 0.663068
\(887\) −1.49039e9 −0.0717077 −0.0358539 0.999357i \(-0.511415\pi\)
−0.0358539 + 0.999357i \(0.511415\pi\)
\(888\) −3.25653e9 −0.156066
\(889\) 1.45250e10 0.693363
\(890\) −2.00624e10 −0.953933
\(891\) −2.93699e9 −0.139101
\(892\) 5.03884e9 0.237713
\(893\) −5.59007e10 −2.62686
\(894\) −1.00244e9 −0.0469220
\(895\) −6.14736e9 −0.286621
\(896\) 1.74312e10 0.809561
\(897\) 1.36319e9 0.0630643
\(898\) −1.74338e10 −0.803385
\(899\) 3.66078e8 0.0168041
\(900\) −4.73392e9 −0.216457
\(901\) 6.65490e8 0.0303113
\(902\) 1.82631e10 0.828613
\(903\) −5.51205e9 −0.249118
\(904\) 1.91049e10 0.860111
\(905\) −3.20213e10 −1.43605
\(906\) 4.35394e9 0.194506
\(907\) −1.44651e10 −0.643717 −0.321858 0.946788i \(-0.604307\pi\)
−0.321858 + 0.946788i \(0.604307\pi\)
\(908\) −1.51463e10 −0.671436
\(909\) 1.10482e10 0.487884
\(910\) 5.87574e9 0.258475
\(911\) −2.79187e10 −1.22343 −0.611717 0.791077i \(-0.709521\pi\)
−0.611717 + 0.791077i \(0.709521\pi\)
\(912\) 4.86533e9 0.212388
\(913\) 2.73562e10 1.18962
\(914\) 9.69353e9 0.419923
\(915\) −7.44429e8 −0.0321254
\(916\) −1.08458e10 −0.466259
\(917\) −1.55024e10 −0.663905
\(918\) 7.87432e8 0.0335942
\(919\) 1.89312e10 0.804590 0.402295 0.915510i \(-0.368213\pi\)
0.402295 + 0.915510i \(0.368213\pi\)
\(920\) −8.62228e9 −0.365061
\(921\) 2.67339e10 1.12760
\(922\) −2.99575e9 −0.125877
\(923\) −8.21133e8 −0.0343722
\(924\) 1.16136e10 0.484299
\(925\) −6.45127e9 −0.268009
\(926\) 6.81005e9 0.281846
\(927\) −1.02081e10 −0.420886
\(928\) −5.06891e8 −0.0208208
\(929\) −3.58709e10 −1.46787 −0.733934 0.679221i \(-0.762318\pi\)
−0.733934 + 0.679221i \(0.762318\pi\)
\(930\) 8.69267e9 0.354375
\(931\) −4.98092e9 −0.202295
\(932\) 1.03146e10 0.417348
\(933\) −7.51584e9 −0.302964
\(934\) −1.83082e10 −0.735245
\(935\) −1.41810e10 −0.567369
\(936\) 2.87920e9 0.114764
\(937\) 1.18752e10 0.471577 0.235789 0.971804i \(-0.424233\pi\)
0.235789 + 0.971804i \(0.424233\pi\)
\(938\) 4.65289e9 0.184083
\(939\) −2.04698e10 −0.806835
\(940\) −4.17590e10 −1.63984
\(941\) −2.99086e10 −1.17013 −0.585063 0.810988i \(-0.698930\pi\)
−0.585063 + 0.810988i \(0.698930\pi\)
\(942\) −3.03934e9 −0.118468
\(943\) 9.28632e9 0.360623
\(944\) −7.80420e8 −0.0301944
\(945\) −6.43691e9 −0.248122
\(946\) −8.00599e9 −0.307465
\(947\) −1.97326e10 −0.755021 −0.377510 0.926005i \(-0.623220\pi\)
−0.377510 + 0.926005i \(0.623220\pi\)
\(948\) −2.78874e9 −0.106311
\(949\) −1.59510e10 −0.605838
\(950\) −2.01728e10 −0.763366
\(951\) −1.06167e10 −0.400275
\(952\) −7.45427e9 −0.280011
\(953\) 3.05905e10 1.14488 0.572442 0.819945i \(-0.305996\pi\)
0.572442 + 0.819945i \(0.305996\pi\)
\(954\) 4.38748e8 0.0163605
\(955\) 1.13600e10 0.422053
\(956\) −2.72481e10 −1.00863
\(957\) −3.93731e8 −0.0145214
\(958\) −1.11784e10 −0.410772
\(959\) −4.06157e10 −1.48706
\(960\) −6.96972e9 −0.254253
\(961\) −8.26528e9 −0.300418
\(962\) 1.63895e9 0.0593545
\(963\) −6.47033e9 −0.233472
\(964\) −1.26017e9 −0.0453062
\(965\) −3.32510e10 −1.19113
\(966\) −2.32683e9 −0.0830509
\(967\) 1.80021e10 0.640221 0.320111 0.947380i \(-0.396280\pi\)
0.320111 + 0.947380i \(0.396280\pi\)
\(968\) 1.46167e10 0.517947
\(969\) −8.51584e9 −0.300673
\(970\) −6.25203e9 −0.219948
\(971\) −6.56410e9 −0.230095 −0.115048 0.993360i \(-0.536702\pi\)
−0.115048 + 0.993360i \(0.536702\pi\)
\(972\) −1.31752e9 −0.0460177
\(973\) 2.34235e10 0.815188
\(974\) 2.38115e9 0.0825716
\(975\) 5.70378e9 0.197082
\(976\) 2.71557e8 0.00934947
\(977\) 1.67236e10 0.573718 0.286859 0.957973i \(-0.407389\pi\)
0.286859 + 0.957973i \(0.407389\pi\)
\(978\) −1.46292e10 −0.500073
\(979\) 4.77779e10 1.62737
\(980\) −3.72085e9 −0.126285
\(981\) −1.81998e10 −0.615495
\(982\) −5.17655e9 −0.174442
\(983\) −6.60882e9 −0.221915 −0.110957 0.993825i \(-0.535392\pi\)
−0.110957 + 0.993825i \(0.535392\pi\)
\(984\) 1.96136e10 0.656259
\(985\) 6.71959e10 2.24035
\(986\) 1.05563e8 0.00350704
\(987\) −2.69787e10 −0.893123
\(988\) −1.30064e10 −0.429049
\(989\) −4.07084e9 −0.133813
\(990\) −9.34931e9 −0.306236
\(991\) −2.45768e10 −0.802173 −0.401086 0.916040i \(-0.631367\pi\)
−0.401086 + 0.916040i \(0.631367\pi\)
\(992\) −2.66509e10 −0.866803
\(993\) −2.56747e9 −0.0832114
\(994\) 1.40159e9 0.0452655
\(995\) −3.49997e10 −1.12638
\(996\) 1.22718e10 0.393552
\(997\) 6.18690e10 1.97715 0.988575 0.150727i \(-0.0481614\pi\)
0.988575 + 0.150727i \(0.0481614\pi\)
\(998\) −1.75830e10 −0.559932
\(999\) −1.79548e9 −0.0569773
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.b.1.8 17
3.2 odd 2 531.8.a.d.1.10 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.8 17 1.1 even 1 trivial
531.8.a.d.1.10 17 3.2 odd 2