Properties

Label 289.10.a.i.1.8
Level $289$
Weight $10$
Character 289.1
Self dual yes
Analytic conductor $148.845$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 289.1

$q$-expansion

\(f(q)\) \(=\) \(q-35.0759 q^{2} +118.571 q^{3} +718.321 q^{4} +1635.44 q^{5} -4158.98 q^{6} -10785.0 q^{7} -7236.92 q^{8} -5623.96 q^{9} +O(q^{10})\) \(q-35.0759 q^{2} +118.571 q^{3} +718.321 q^{4} +1635.44 q^{5} -4158.98 q^{6} -10785.0 q^{7} -7236.92 q^{8} -5623.96 q^{9} -57364.5 q^{10} +21197.3 q^{11} +85172.0 q^{12} -71034.1 q^{13} +378293. q^{14} +193915. q^{15} -113939. q^{16} +197266. q^{18} -239108. q^{19} +1.17477e6 q^{20} -1.27878e6 q^{21} -743516. q^{22} -2.36865e6 q^{23} -858087. q^{24} +721532. q^{25} +2.49159e6 q^{26} -3.00067e6 q^{27} -7.74707e6 q^{28} -1.37916e6 q^{29} -6.80176e6 q^{30} -4.72814e6 q^{31} +7.70182e6 q^{32} +2.51339e6 q^{33} -1.76381e7 q^{35} -4.03981e6 q^{36} +1.04815e7 q^{37} +8.38694e6 q^{38} -8.42257e6 q^{39} -1.18355e7 q^{40} -7.22217e6 q^{41} +4.48545e7 q^{42} -1.01797e7 q^{43} +1.52265e7 q^{44} -9.19763e6 q^{45} +8.30825e7 q^{46} +4.00577e7 q^{47} -1.35098e7 q^{48} +7.59618e7 q^{49} -2.53084e7 q^{50} -5.10253e7 q^{52} +2.21414e7 q^{53} +1.05251e8 q^{54} +3.46669e7 q^{55} +7.80499e7 q^{56} -2.83512e7 q^{57} +4.83753e7 q^{58} -9.89019e7 q^{59} +1.39293e8 q^{60} +2.14505e8 q^{61} +1.65844e8 q^{62} +6.06542e7 q^{63} -2.11812e8 q^{64} -1.16172e8 q^{65} -8.81594e7 q^{66} +1.65755e8 q^{67} -2.80852e8 q^{69} +6.18674e8 q^{70} -2.59100e7 q^{71} +4.07001e7 q^{72} -3.65612e8 q^{73} -3.67649e8 q^{74} +8.55527e7 q^{75} -1.71756e8 q^{76} -2.28612e8 q^{77} +2.95430e8 q^{78} +5.47273e8 q^{79} -1.86340e8 q^{80} -2.45095e8 q^{81} +2.53324e8 q^{82} +5.77055e8 q^{83} -9.18576e8 q^{84} +3.57064e8 q^{86} -1.63528e8 q^{87} -1.53403e8 q^{88} +3.72848e7 q^{89} +3.22616e8 q^{90} +7.66100e8 q^{91} -1.70145e9 q^{92} -5.60619e8 q^{93} -1.40506e9 q^{94} -3.91046e8 q^{95} +9.13211e8 q^{96} +8.89204e8 q^{97} -2.66443e9 q^{98} -1.19213e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + O(q^{10}) \) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + 156200q^{13} + 1207872q^{15} + 3407880q^{16} + 2193336q^{18} + 1185568q^{19} + 5198336q^{21} + 13827692q^{25} + 3618944q^{26} - 9167544q^{30} + 61884888q^{32} + 1635208q^{33} + 46992776q^{35} + 156027320q^{36} + 84813952q^{38} - 4635776q^{42} + 125448912q^{43} + 164193176q^{47} + 270850284q^{49} - 226223888q^{50} + 103553016q^{52} + 426167208q^{53} + 677761520q^{55} + 375214512q^{59} + 336918024q^{60} + 190014416q^{64} + 1377178928q^{66} + 311910088q^{67} + 533688136q^{69} + 1477690280q^{70} + 2757942680q^{72} + 4047975520q^{76} + 3440336432q^{77} + 3266558756q^{81} + 2072890608q^{83} + 2630025952q^{84} + 1538547296q^{86} - 1010436256q^{87} + 1873849184q^{89} - 1998451624q^{93} - 6880776704q^{94} - 4667454128q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −35.0759 −1.55015 −0.775076 0.631868i \(-0.782288\pi\)
−0.775076 + 0.631868i \(0.782288\pi\)
\(3\) 118.571 0.845147 0.422573 0.906329i \(-0.361127\pi\)
0.422573 + 0.906329i \(0.361127\pi\)
\(4\) 718.321 1.40297
\(5\) 1635.44 1.17022 0.585112 0.810952i \(-0.301051\pi\)
0.585112 + 0.810952i \(0.301051\pi\)
\(6\) −4158.98 −1.31011
\(7\) −10785.0 −1.69776 −0.848882 0.528582i \(-0.822724\pi\)
−0.848882 + 0.528582i \(0.822724\pi\)
\(8\) −7236.92 −0.624667
\(9\) −5623.96 −0.285727
\(10\) −57364.5 −1.81403
\(11\) 21197.3 0.436530 0.218265 0.975890i \(-0.429960\pi\)
0.218265 + 0.975890i \(0.429960\pi\)
\(12\) 85172.0 1.18572
\(13\) −71034.1 −0.689798 −0.344899 0.938640i \(-0.612087\pi\)
−0.344899 + 0.938640i \(0.612087\pi\)
\(14\) 378293. 2.63179
\(15\) 193915. 0.989011
\(16\) −113939. −0.434642
\(17\) 0 0
\(18\) 197266. 0.442920
\(19\) −239108. −0.420923 −0.210462 0.977602i \(-0.567497\pi\)
−0.210462 + 0.977602i \(0.567497\pi\)
\(20\) 1.17477e6 1.64179
\(21\) −1.27878e6 −1.43486
\(22\) −743516. −0.676688
\(23\) −2.36865e6 −1.76492 −0.882460 0.470388i \(-0.844114\pi\)
−0.882460 + 0.470388i \(0.844114\pi\)
\(24\) −858087. −0.527936
\(25\) 721532. 0.369424
\(26\) 2.49159e6 1.06929
\(27\) −3.00067e6 −1.08663
\(28\) −7.74707e6 −2.38192
\(29\) −1.37916e6 −0.362096 −0.181048 0.983474i \(-0.557949\pi\)
−0.181048 + 0.983474i \(0.557949\pi\)
\(30\) −6.80176e6 −1.53312
\(31\) −4.72814e6 −0.919523 −0.459761 0.888042i \(-0.652065\pi\)
−0.459761 + 0.888042i \(0.652065\pi\)
\(32\) 7.70182e6 1.29843
\(33\) 2.51339e6 0.368932
\(34\) 0 0
\(35\) −1.76381e7 −1.98676
\(36\) −4.03981e6 −0.400866
\(37\) 1.04815e7 0.919424 0.459712 0.888068i \(-0.347953\pi\)
0.459712 + 0.888068i \(0.347953\pi\)
\(38\) 8.38694e6 0.652495
\(39\) −8.42257e6 −0.582981
\(40\) −1.18355e7 −0.731001
\(41\) −7.22217e6 −0.399154 −0.199577 0.979882i \(-0.563957\pi\)
−0.199577 + 0.979882i \(0.563957\pi\)
\(42\) 4.48545e7 2.22425
\(43\) −1.01797e7 −0.454076 −0.227038 0.973886i \(-0.572904\pi\)
−0.227038 + 0.973886i \(0.572904\pi\)
\(44\) 1.52265e7 0.612439
\(45\) −9.19763e6 −0.334364
\(46\) 8.30825e7 2.73589
\(47\) 4.00577e7 1.19742 0.598709 0.800966i \(-0.295680\pi\)
0.598709 + 0.800966i \(0.295680\pi\)
\(48\) −1.35098e7 −0.367337
\(49\) 7.59618e7 1.88240
\(50\) −2.53084e7 −0.572664
\(51\) 0 0
\(52\) −5.10253e7 −0.967767
\(53\) 2.21414e7 0.385447 0.192723 0.981253i \(-0.438268\pi\)
0.192723 + 0.981253i \(0.438268\pi\)
\(54\) 1.05251e8 1.68444
\(55\) 3.46669e7 0.510838
\(56\) 7.80499e7 1.06054
\(57\) −2.83512e7 −0.355742
\(58\) 4.83753e7 0.561304
\(59\) −9.89019e7 −1.06260 −0.531301 0.847183i \(-0.678297\pi\)
−0.531301 + 0.847183i \(0.678297\pi\)
\(60\) 1.39293e8 1.38755
\(61\) 2.14505e8 1.98360 0.991798 0.127812i \(-0.0407955\pi\)
0.991798 + 0.127812i \(0.0407955\pi\)
\(62\) 1.65844e8 1.42540
\(63\) 6.06542e7 0.485097
\(64\) −2.11812e8 −1.57812
\(65\) −1.16172e8 −0.807218
\(66\) −8.81594e7 −0.571901
\(67\) 1.65755e8 1.00492 0.502458 0.864602i \(-0.332429\pi\)
0.502458 + 0.864602i \(0.332429\pi\)
\(68\) 0 0
\(69\) −2.80852e8 −1.49162
\(70\) 6.18674e8 3.07979
\(71\) −2.59100e7 −0.121005 −0.0605027 0.998168i \(-0.519270\pi\)
−0.0605027 + 0.998168i \(0.519270\pi\)
\(72\) 4.07001e7 0.178484
\(73\) −3.65612e8 −1.50684 −0.753421 0.657539i \(-0.771598\pi\)
−0.753421 + 0.657539i \(0.771598\pi\)
\(74\) −3.67649e8 −1.42525
\(75\) 8.55527e7 0.312218
\(76\) −1.71756e8 −0.590544
\(77\) −2.28612e8 −0.741125
\(78\) 2.95430e8 0.903709
\(79\) 5.47273e8 1.58082 0.790409 0.612579i \(-0.209868\pi\)
0.790409 + 0.612579i \(0.209868\pi\)
\(80\) −1.86340e8 −0.508629
\(81\) −2.45095e8 −0.632634
\(82\) 2.53324e8 0.618749
\(83\) 5.77055e8 1.33465 0.667323 0.744768i \(-0.267440\pi\)
0.667323 + 0.744768i \(0.267440\pi\)
\(84\) −9.18576e8 −2.01307
\(85\) 0 0
\(86\) 3.57064e8 0.703887
\(87\) −1.63528e8 −0.306024
\(88\) −1.53403e8 −0.272686
\(89\) 3.72848e7 0.0629908 0.0314954 0.999504i \(-0.489973\pi\)
0.0314954 + 0.999504i \(0.489973\pi\)
\(90\) 3.22616e8 0.518315
\(91\) 7.66100e8 1.17111
\(92\) −1.70145e9 −2.47613
\(93\) −5.60619e8 −0.777132
\(94\) −1.40506e9 −1.85618
\(95\) −3.91046e8 −0.492575
\(96\) 9.13211e8 1.09736
\(97\) 8.89204e8 1.01983 0.509916 0.860224i \(-0.329676\pi\)
0.509916 + 0.860224i \(0.329676\pi\)
\(98\) −2.66443e9 −2.91801
\(99\) −1.19213e8 −0.124728
\(100\) 5.18292e8 0.518292
\(101\) −1.51590e9 −1.44952 −0.724760 0.689002i \(-0.758049\pi\)
−0.724760 + 0.689002i \(0.758049\pi\)
\(102\) 0 0
\(103\) 4.65778e8 0.407766 0.203883 0.978995i \(-0.434644\pi\)
0.203883 + 0.978995i \(0.434644\pi\)
\(104\) 5.14068e8 0.430894
\(105\) −2.09137e9 −1.67911
\(106\) −7.76632e8 −0.597501
\(107\) 1.88637e9 1.39124 0.695618 0.718412i \(-0.255131\pi\)
0.695618 + 0.718412i \(0.255131\pi\)
\(108\) −2.15544e9 −1.52451
\(109\) 5.88893e8 0.399593 0.199796 0.979837i \(-0.435972\pi\)
0.199796 + 0.979837i \(0.435972\pi\)
\(110\) −1.21598e9 −0.791877
\(111\) 1.24280e9 0.777049
\(112\) 1.22883e9 0.737920
\(113\) −1.46525e9 −0.845393 −0.422696 0.906271i \(-0.638916\pi\)
−0.422696 + 0.906271i \(0.638916\pi\)
\(114\) 9.94447e8 0.551454
\(115\) −3.87377e9 −2.06535
\(116\) −9.90680e8 −0.508010
\(117\) 3.99493e8 0.197094
\(118\) 3.46908e9 1.64720
\(119\) 0 0
\(120\) −1.40335e9 −0.617803
\(121\) −1.90862e9 −0.809441
\(122\) −7.52397e9 −3.07488
\(123\) −8.56338e8 −0.337343
\(124\) −3.39632e9 −1.29006
\(125\) −2.01419e9 −0.737915
\(126\) −2.12750e9 −0.751973
\(127\) −2.87764e9 −0.981566 −0.490783 0.871282i \(-0.663289\pi\)
−0.490783 + 0.871282i \(0.663289\pi\)
\(128\) 3.48616e9 1.14790
\(129\) −1.20702e9 −0.383761
\(130\) 4.07484e9 1.25131
\(131\) 5.71635e9 1.69589 0.847946 0.530083i \(-0.177839\pi\)
0.847946 + 0.530083i \(0.177839\pi\)
\(132\) 1.80542e9 0.517601
\(133\) 2.57877e9 0.714629
\(134\) −5.81401e9 −1.55777
\(135\) −4.90740e9 −1.27160
\(136\) 0 0
\(137\) −2.50282e9 −0.606997 −0.303499 0.952832i \(-0.598155\pi\)
−0.303499 + 0.952832i \(0.598155\pi\)
\(138\) 9.85116e9 2.31223
\(139\) 8.09336e8 0.183892 0.0919459 0.995764i \(-0.470691\pi\)
0.0919459 + 0.995764i \(0.470691\pi\)
\(140\) −1.26699e10 −2.78737
\(141\) 4.74968e9 1.01199
\(142\) 9.08817e8 0.187577
\(143\) −1.50573e9 −0.301118
\(144\) 6.40787e8 0.124189
\(145\) −2.25553e9 −0.423733
\(146\) 1.28242e10 2.33583
\(147\) 9.00685e9 1.59091
\(148\) 7.52909e9 1.28993
\(149\) 4.26492e9 0.708880 0.354440 0.935079i \(-0.384672\pi\)
0.354440 + 0.935079i \(0.384672\pi\)
\(150\) −3.00084e9 −0.483985
\(151\) 8.41177e9 1.31671 0.658357 0.752706i \(-0.271252\pi\)
0.658357 + 0.752706i \(0.271252\pi\)
\(152\) 1.73041e9 0.262937
\(153\) 0 0
\(154\) 8.01880e9 1.14886
\(155\) −7.73258e9 −1.07605
\(156\) −6.05012e9 −0.817905
\(157\) 1.40154e9 0.184101 0.0920504 0.995754i \(-0.470658\pi\)
0.0920504 + 0.995754i \(0.470658\pi\)
\(158\) −1.91961e10 −2.45051
\(159\) 2.62533e9 0.325759
\(160\) 1.25958e10 1.51945
\(161\) 2.55458e10 2.99642
\(162\) 8.59695e9 0.980678
\(163\) 9.27080e9 1.02866 0.514331 0.857592i \(-0.328040\pi\)
0.514331 + 0.857592i \(0.328040\pi\)
\(164\) −5.18784e9 −0.560001
\(165\) 4.11049e9 0.431733
\(166\) −2.02408e10 −2.06890
\(167\) −1.35795e10 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(168\) 9.25444e9 0.896310
\(169\) −5.55865e9 −0.524179
\(170\) 0 0
\(171\) 1.34473e9 0.120269
\(172\) −7.31232e9 −0.637055
\(173\) −6.63039e9 −0.562771 −0.281386 0.959595i \(-0.590794\pi\)
−0.281386 + 0.959595i \(0.590794\pi\)
\(174\) 5.73590e9 0.474384
\(175\) −7.78170e9 −0.627196
\(176\) −2.41520e9 −0.189734
\(177\) −1.17269e10 −0.898055
\(178\) −1.30780e9 −0.0976453
\(179\) −4.34584e9 −0.316399 −0.158200 0.987407i \(-0.550569\pi\)
−0.158200 + 0.987407i \(0.550569\pi\)
\(180\) −6.60686e9 −0.469104
\(181\) −1.32517e10 −0.917737 −0.458869 0.888504i \(-0.651745\pi\)
−0.458869 + 0.888504i \(0.651745\pi\)
\(182\) −2.68717e10 −1.81541
\(183\) 2.54340e10 1.67643
\(184\) 1.71417e10 1.10249
\(185\) 1.71419e10 1.07593
\(186\) 1.96642e10 1.20467
\(187\) 0 0
\(188\) 2.87743e10 1.67994
\(189\) 3.23621e10 1.84484
\(190\) 1.37163e10 0.763566
\(191\) 4.77283e8 0.0259493 0.0129746 0.999916i \(-0.495870\pi\)
0.0129746 + 0.999916i \(0.495870\pi\)
\(192\) −2.51147e10 −1.33374
\(193\) 1.38605e10 0.719070 0.359535 0.933132i \(-0.382935\pi\)
0.359535 + 0.933132i \(0.382935\pi\)
\(194\) −3.11897e10 −1.58090
\(195\) −1.37746e10 −0.682218
\(196\) 5.45650e10 2.64096
\(197\) −2.83003e10 −1.33873 −0.669364 0.742934i \(-0.733433\pi\)
−0.669364 + 0.742934i \(0.733433\pi\)
\(198\) 4.18151e9 0.193348
\(199\) 2.49935e10 1.12977 0.564884 0.825171i \(-0.308921\pi\)
0.564884 + 0.825171i \(0.308921\pi\)
\(200\) −5.22167e9 −0.230767
\(201\) 1.96537e10 0.849302
\(202\) 5.31716e10 2.24698
\(203\) 1.48742e10 0.614753
\(204\) 0 0
\(205\) −1.18114e10 −0.467099
\(206\) −1.63376e10 −0.632100
\(207\) 1.33212e10 0.504284
\(208\) 8.09355e9 0.299815
\(209\) −5.06845e9 −0.183746
\(210\) 7.33567e10 2.60287
\(211\) −1.80312e10 −0.626260 −0.313130 0.949710i \(-0.601377\pi\)
−0.313130 + 0.949710i \(0.601377\pi\)
\(212\) 1.59047e10 0.540771
\(213\) −3.07217e9 −0.102267
\(214\) −6.61663e10 −2.15663
\(215\) −1.66483e10 −0.531370
\(216\) 2.17156e10 0.678781
\(217\) 5.09928e10 1.56113
\(218\) −2.06560e10 −0.619429
\(219\) −4.33509e10 −1.27350
\(220\) 2.49020e10 0.716691
\(221\) 0 0
\(222\) −4.35924e10 −1.20454
\(223\) −3.30245e10 −0.894262 −0.447131 0.894468i \(-0.647554\pi\)
−0.447131 + 0.894468i \(0.647554\pi\)
\(224\) −8.30638e10 −2.20443
\(225\) −4.05787e9 −0.105554
\(226\) 5.13950e10 1.31049
\(227\) 3.66514e10 0.916166 0.458083 0.888909i \(-0.348536\pi\)
0.458083 + 0.888909i \(0.348536\pi\)
\(228\) −2.03653e10 −0.499096
\(229\) 5.79474e9 0.139243 0.0696217 0.997573i \(-0.477821\pi\)
0.0696217 + 0.997573i \(0.477821\pi\)
\(230\) 1.35876e11 3.20161
\(231\) −2.71068e10 −0.626360
\(232\) 9.98087e9 0.226189
\(233\) −7.30737e7 −0.00162428 −0.000812138 1.00000i \(-0.500259\pi\)
−0.000812138 1.00000i \(0.500259\pi\)
\(234\) −1.40126e10 −0.305525
\(235\) 6.55120e10 1.40125
\(236\) −7.10434e10 −1.49080
\(237\) 6.48906e10 1.33602
\(238\) 0 0
\(239\) 6.99325e10 1.38640 0.693200 0.720745i \(-0.256200\pi\)
0.693200 + 0.720745i \(0.256200\pi\)
\(240\) −2.20945e10 −0.429866
\(241\) 5.86797e10 1.12050 0.560250 0.828324i \(-0.310705\pi\)
0.560250 + 0.828324i \(0.310705\pi\)
\(242\) 6.69467e10 1.25476
\(243\) 3.00010e10 0.551960
\(244\) 1.54084e11 2.78293
\(245\) 1.24231e11 2.20283
\(246\) 3.00369e10 0.522934
\(247\) 1.69848e10 0.290352
\(248\) 3.42171e10 0.574396
\(249\) 6.84219e10 1.12797
\(250\) 7.06497e10 1.14388
\(251\) −3.69311e10 −0.587300 −0.293650 0.955913i \(-0.594870\pi\)
−0.293650 + 0.955913i \(0.594870\pi\)
\(252\) 4.35692e10 0.680577
\(253\) −5.02090e10 −0.770440
\(254\) 1.00936e11 1.52158
\(255\) 0 0
\(256\) −1.38329e10 −0.201295
\(257\) 3.07821e10 0.440149 0.220074 0.975483i \(-0.429370\pi\)
0.220074 + 0.975483i \(0.429370\pi\)
\(258\) 4.23373e10 0.594888
\(259\) −1.13043e11 −1.56097
\(260\) −8.34488e10 −1.13250
\(261\) 7.75634e9 0.103460
\(262\) −2.00506e11 −2.62889
\(263\) 2.21843e10 0.285920 0.142960 0.989728i \(-0.454338\pi\)
0.142960 + 0.989728i \(0.454338\pi\)
\(264\) −1.81892e10 −0.230460
\(265\) 3.62110e10 0.451059
\(266\) −9.04528e10 −1.10778
\(267\) 4.42089e9 0.0532365
\(268\) 1.19065e11 1.40987
\(269\) 5.70776e10 0.664631 0.332316 0.943168i \(-0.392170\pi\)
0.332316 + 0.943168i \(0.392170\pi\)
\(270\) 1.72132e11 1.97117
\(271\) −3.32663e10 −0.374664 −0.187332 0.982297i \(-0.559984\pi\)
−0.187332 + 0.982297i \(0.559984\pi\)
\(272\) 0 0
\(273\) 9.08371e10 0.989764
\(274\) 8.77887e10 0.940938
\(275\) 1.52946e10 0.161265
\(276\) −2.01742e11 −2.09270
\(277\) −3.42210e10 −0.349248 −0.174624 0.984635i \(-0.555871\pi\)
−0.174624 + 0.984635i \(0.555871\pi\)
\(278\) −2.83882e10 −0.285060
\(279\) 2.65908e10 0.262732
\(280\) 1.27646e11 1.24107
\(281\) 6.69886e10 0.640947 0.320474 0.947257i \(-0.396158\pi\)
0.320474 + 0.947257i \(0.396158\pi\)
\(282\) −1.66599e11 −1.56875
\(283\) −2.10268e10 −0.194865 −0.0974327 0.995242i \(-0.531063\pi\)
−0.0974327 + 0.995242i \(0.531063\pi\)
\(284\) −1.86117e10 −0.169767
\(285\) −4.63667e10 −0.416298
\(286\) 5.28150e10 0.466778
\(287\) 7.78908e10 0.677669
\(288\) −4.33147e10 −0.370996
\(289\) 0 0
\(290\) 7.91148e10 0.656851
\(291\) 1.05434e11 0.861908
\(292\) −2.62627e11 −2.11406
\(293\) −3.54172e10 −0.280743 −0.140372 0.990099i \(-0.544830\pi\)
−0.140372 + 0.990099i \(0.544830\pi\)
\(294\) −3.15924e11 −2.46615
\(295\) −1.61748e11 −1.24348
\(296\) −7.58538e10 −0.574334
\(297\) −6.36062e10 −0.474346
\(298\) −1.49596e11 −1.09887
\(299\) 1.68255e11 1.21744
\(300\) 6.14543e10 0.438033
\(301\) 1.09788e11 0.770914
\(302\) −2.95051e11 −2.04111
\(303\) −1.79741e11 −1.22506
\(304\) 2.72437e10 0.182951
\(305\) 3.50810e11 2.32125
\(306\) 0 0
\(307\) −9.48586e9 −0.0609472 −0.0304736 0.999536i \(-0.509702\pi\)
−0.0304736 + 0.999536i \(0.509702\pi\)
\(308\) −1.64217e11 −1.03978
\(309\) 5.52277e10 0.344622
\(310\) 2.71227e11 1.66804
\(311\) 1.56456e11 0.948354 0.474177 0.880429i \(-0.342746\pi\)
0.474177 + 0.880429i \(0.342746\pi\)
\(312\) 6.09535e10 0.364169
\(313\) 7.93382e10 0.467232 0.233616 0.972329i \(-0.424944\pi\)
0.233616 + 0.972329i \(0.424944\pi\)
\(314\) −4.91602e10 −0.285384
\(315\) 9.91961e10 0.567672
\(316\) 3.93118e11 2.21784
\(317\) −1.26108e11 −0.701417 −0.350709 0.936485i \(-0.614059\pi\)
−0.350709 + 0.936485i \(0.614059\pi\)
\(318\) −9.20859e10 −0.504976
\(319\) −2.92345e10 −0.158066
\(320\) −3.46405e11 −1.84675
\(321\) 2.23669e11 1.17580
\(322\) −8.96041e11 −4.64490
\(323\) 0 0
\(324\) −1.76057e11 −0.887567
\(325\) −5.12534e10 −0.254828
\(326\) −3.25182e11 −1.59458
\(327\) 6.98256e10 0.337714
\(328\) 5.22662e10 0.249338
\(329\) −4.32021e11 −2.03294
\(330\) −1.44179e11 −0.669252
\(331\) 1.07985e11 0.494467 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(332\) 4.14511e11 1.87247
\(333\) −5.89475e10 −0.262704
\(334\) 4.76315e11 2.09428
\(335\) 2.71082e11 1.17598
\(336\) 1.45703e11 0.623651
\(337\) −2.39230e11 −1.01037 −0.505186 0.863011i \(-0.668576\pi\)
−0.505186 + 0.863011i \(0.668576\pi\)
\(338\) 1.94975e11 0.812557
\(339\) −1.73736e11 −0.714481
\(340\) 0 0
\(341\) −1.00224e11 −0.401399
\(342\) −4.71678e10 −0.186435
\(343\) −3.84033e11 −1.49811
\(344\) 7.36699e10 0.283646
\(345\) −4.59317e11 −1.74553
\(346\) 2.32567e11 0.872381
\(347\) 1.33301e11 0.493573 0.246786 0.969070i \(-0.420625\pi\)
0.246786 + 0.969070i \(0.420625\pi\)
\(348\) −1.17466e11 −0.429343
\(349\) −1.00639e11 −0.363120 −0.181560 0.983380i \(-0.558115\pi\)
−0.181560 + 0.983380i \(0.558115\pi\)
\(350\) 2.72950e11 0.972249
\(351\) 2.13150e11 0.749554
\(352\) 1.63258e11 0.566803
\(353\) 9.24641e10 0.316947 0.158474 0.987363i \(-0.449343\pi\)
0.158474 + 0.987363i \(0.449343\pi\)
\(354\) 4.11331e11 1.39212
\(355\) −4.23742e10 −0.141603
\(356\) 2.67825e10 0.0883743
\(357\) 0 0
\(358\) 1.52435e11 0.490467
\(359\) 4.04248e11 1.28447 0.642234 0.766509i \(-0.278008\pi\)
0.642234 + 0.766509i \(0.278008\pi\)
\(360\) 6.65625e10 0.208866
\(361\) −2.65515e11 −0.822824
\(362\) 4.64816e11 1.42263
\(363\) −2.26307e11 −0.684097
\(364\) 5.50306e11 1.64304
\(365\) −5.97936e11 −1.76334
\(366\) −8.92123e11 −2.59872
\(367\) −5.43157e11 −1.56289 −0.781444 0.623975i \(-0.785517\pi\)
−0.781444 + 0.623975i \(0.785517\pi\)
\(368\) 2.69881e11 0.767109
\(369\) 4.06172e10 0.114049
\(370\) −6.01267e11 −1.66786
\(371\) −2.38795e11 −0.654398
\(372\) −4.02705e11 −1.09029
\(373\) 7.02135e10 0.187815 0.0939077 0.995581i \(-0.470064\pi\)
0.0939077 + 0.995581i \(0.470064\pi\)
\(374\) 0 0
\(375\) −2.38825e11 −0.623646
\(376\) −2.89895e11 −0.747988
\(377\) 9.79674e10 0.249773
\(378\) −1.13513e12 −2.85978
\(379\) 3.37065e11 0.839146 0.419573 0.907722i \(-0.362180\pi\)
0.419573 + 0.907722i \(0.362180\pi\)
\(380\) −2.80897e11 −0.691068
\(381\) −3.41204e11 −0.829568
\(382\) −1.67411e10 −0.0402253
\(383\) −2.69746e11 −0.640562 −0.320281 0.947323i \(-0.603777\pi\)
−0.320281 + 0.947323i \(0.603777\pi\)
\(384\) 4.13357e11 0.970142
\(385\) −3.73881e11 −0.867283
\(386\) −4.86170e11 −1.11467
\(387\) 5.72504e10 0.129742
\(388\) 6.38735e11 1.43080
\(389\) −6.98737e11 −1.54718 −0.773590 0.633687i \(-0.781541\pi\)
−0.773590 + 0.633687i \(0.781541\pi\)
\(390\) 4.83157e11 1.05754
\(391\) 0 0
\(392\) −5.49729e11 −1.17588
\(393\) 6.77792e11 1.43328
\(394\) 9.92658e11 2.07523
\(395\) 8.95031e11 1.84991
\(396\) −8.56332e10 −0.174990
\(397\) −5.38044e11 −1.08708 −0.543538 0.839384i \(-0.682916\pi\)
−0.543538 + 0.839384i \(0.682916\pi\)
\(398\) −8.76672e11 −1.75131
\(399\) 3.05767e11 0.603966
\(400\) −8.22106e10 −0.160568
\(401\) 9.77776e10 0.188838 0.0944190 0.995533i \(-0.469901\pi\)
0.0944190 + 0.995533i \(0.469901\pi\)
\(402\) −6.89372e11 −1.31655
\(403\) 3.35859e11 0.634285
\(404\) −1.08890e12 −2.03363
\(405\) −4.00838e11 −0.740323
\(406\) −5.21726e11 −0.952961
\(407\) 2.22180e11 0.401356
\(408\) 0 0
\(409\) 8.67529e11 1.53295 0.766477 0.642271i \(-0.222008\pi\)
0.766477 + 0.642271i \(0.222008\pi\)
\(410\) 4.14296e11 0.724075
\(411\) −2.96761e11 −0.513002
\(412\) 3.34578e11 0.572085
\(413\) 1.06665e12 1.80405
\(414\) −4.67252e11 −0.781718
\(415\) 9.43738e11 1.56183
\(416\) −5.47092e11 −0.895654
\(417\) 9.59637e10 0.155416
\(418\) 1.77781e11 0.284834
\(419\) −5.57776e11 −0.884091 −0.442045 0.896993i \(-0.645747\pi\)
−0.442045 + 0.896993i \(0.645747\pi\)
\(420\) −1.50227e12 −2.35574
\(421\) 2.55514e11 0.396411 0.198206 0.980160i \(-0.436489\pi\)
0.198206 + 0.980160i \(0.436489\pi\)
\(422\) 6.32462e11 0.970798
\(423\) −2.25283e11 −0.342134
\(424\) −1.60236e11 −0.240776
\(425\) 0 0
\(426\) 1.07759e11 0.158530
\(427\) −2.31343e12 −3.36768
\(428\) 1.35502e12 1.95186
\(429\) −1.78536e11 −0.254489
\(430\) 5.83956e11 0.823705
\(431\) 8.68561e11 1.21242 0.606210 0.795305i \(-0.292689\pi\)
0.606210 + 0.795305i \(0.292689\pi\)
\(432\) 3.41893e11 0.472294
\(433\) 4.20026e11 0.574223 0.287112 0.957897i \(-0.407305\pi\)
0.287112 + 0.957897i \(0.407305\pi\)
\(434\) −1.78862e12 −2.41999
\(435\) −2.67440e11 −0.358117
\(436\) 4.23015e11 0.560617
\(437\) 5.66362e11 0.742896
\(438\) 1.52057e12 1.97412
\(439\) 2.86798e11 0.368541 0.184270 0.982876i \(-0.441008\pi\)
0.184270 + 0.982876i \(0.441008\pi\)
\(440\) −2.50882e11 −0.319104
\(441\) −4.27206e11 −0.537853
\(442\) 0 0
\(443\) 2.71905e11 0.335428 0.167714 0.985836i \(-0.446361\pi\)
0.167714 + 0.985836i \(0.446361\pi\)
\(444\) 8.92731e11 1.09018
\(445\) 6.09770e10 0.0737133
\(446\) 1.15837e12 1.38624
\(447\) 5.05695e11 0.599107
\(448\) 2.28438e12 2.67928
\(449\) −2.15739e10 −0.0250507 −0.0125254 0.999922i \(-0.503987\pi\)
−0.0125254 + 0.999922i \(0.503987\pi\)
\(450\) 1.42333e11 0.163625
\(451\) −1.53091e11 −0.174243
\(452\) −1.05252e12 −1.18606
\(453\) 9.97391e11 1.11282
\(454\) −1.28558e12 −1.42020
\(455\) 1.25291e12 1.37047
\(456\) 2.05176e11 0.222220
\(457\) 1.57168e12 1.68555 0.842775 0.538266i \(-0.180921\pi\)
0.842775 + 0.538266i \(0.180921\pi\)
\(458\) −2.03256e11 −0.215848
\(459\) 0 0
\(460\) −2.78261e12 −2.89763
\(461\) 9.53263e11 0.983012 0.491506 0.870874i \(-0.336447\pi\)
0.491506 + 0.870874i \(0.336447\pi\)
\(462\) 9.50795e11 0.970953
\(463\) 1.32544e12 1.34044 0.670219 0.742164i \(-0.266200\pi\)
0.670219 + 0.742164i \(0.266200\pi\)
\(464\) 1.57140e11 0.157382
\(465\) −9.16858e11 −0.909418
\(466\) 2.56313e9 0.00251787
\(467\) −2.18931e11 −0.213001 −0.106500 0.994313i \(-0.533965\pi\)
−0.106500 + 0.994313i \(0.533965\pi\)
\(468\) 2.86964e11 0.276517
\(469\) −1.78766e12 −1.70611
\(470\) −2.29789e12 −2.17215
\(471\) 1.66181e11 0.155592
\(472\) 7.15745e11 0.663773
\(473\) −2.15783e11 −0.198218
\(474\) −2.27610e12 −2.07104
\(475\) −1.72524e11 −0.155499
\(476\) 0 0
\(477\) −1.24523e11 −0.110132
\(478\) −2.45295e12 −2.14913
\(479\) 1.42034e12 1.23277 0.616387 0.787443i \(-0.288596\pi\)
0.616387 + 0.787443i \(0.288596\pi\)
\(480\) 1.49350e12 1.28416
\(481\) −7.44544e11 −0.634217
\(482\) −2.05825e12 −1.73694
\(483\) 3.02898e12 2.53241
\(484\) −1.37100e12 −1.13562
\(485\) 1.45424e12 1.19343
\(486\) −1.05231e12 −0.855621
\(487\) 7.40911e11 0.596878 0.298439 0.954429i \(-0.403534\pi\)
0.298439 + 0.954429i \(0.403534\pi\)
\(488\) −1.55236e12 −1.23909
\(489\) 1.09925e12 0.869371
\(490\) −4.35751e12 −3.41473
\(491\) 2.27702e12 1.76807 0.884036 0.467418i \(-0.154816\pi\)
0.884036 + 0.467418i \(0.154816\pi\)
\(492\) −6.15126e11 −0.473283
\(493\) 0 0
\(494\) −5.95759e11 −0.450090
\(495\) −1.94965e11 −0.145960
\(496\) 5.38719e11 0.399664
\(497\) 2.79438e11 0.205439
\(498\) −2.39996e12 −1.74853
\(499\) −1.41291e11 −0.102014 −0.0510071 0.998698i \(-0.516243\pi\)
−0.0510071 + 0.998698i \(0.516243\pi\)
\(500\) −1.44684e12 −1.03527
\(501\) −1.61014e12 −1.14181
\(502\) 1.29539e12 0.910405
\(503\) 1.69321e12 1.17938 0.589692 0.807628i \(-0.299249\pi\)
0.589692 + 0.807628i \(0.299249\pi\)
\(504\) −4.38949e11 −0.303024
\(505\) −2.47916e12 −1.69626
\(506\) 1.76113e12 1.19430
\(507\) −6.59094e11 −0.443008
\(508\) −2.06707e12 −1.37711
\(509\) 1.99900e12 1.32003 0.660015 0.751253i \(-0.270550\pi\)
0.660015 + 0.751253i \(0.270550\pi\)
\(510\) 0 0
\(511\) 3.94311e12 2.55826
\(512\) −1.29971e12 −0.835859
\(513\) 7.17484e11 0.457387
\(514\) −1.07971e12 −0.682297
\(515\) 7.61751e11 0.477178
\(516\) −8.67028e11 −0.538405
\(517\) 8.49117e11 0.522709
\(518\) 3.96508e12 2.41973
\(519\) −7.86171e11 −0.475624
\(520\) 8.40726e11 0.504243
\(521\) −2.07948e12 −1.23648 −0.618238 0.785991i \(-0.712153\pi\)
−0.618238 + 0.785991i \(0.712153\pi\)
\(522\) −2.72061e11 −0.160379
\(523\) 9.71740e11 0.567927 0.283963 0.958835i \(-0.408351\pi\)
0.283963 + 0.958835i \(0.408351\pi\)
\(524\) 4.10618e12 2.37929
\(525\) −9.22682e11 −0.530073
\(526\) −7.78134e11 −0.443219
\(527\) 0 0
\(528\) −2.86372e11 −0.160353
\(529\) 3.80933e12 2.11494
\(530\) −1.27013e12 −0.699210
\(531\) 5.56220e11 0.303614
\(532\) 1.85239e12 1.00260
\(533\) 5.13020e11 0.275335
\(534\) −1.55067e11 −0.0825246
\(535\) 3.08505e12 1.62806
\(536\) −1.19956e12 −0.627738
\(537\) −5.15290e11 −0.267404
\(538\) −2.00205e12 −1.03028
\(539\) 1.61019e12 0.821726
\(540\) −3.52509e12 −1.78402
\(541\) 2.59266e12 1.30124 0.650620 0.759403i \(-0.274509\pi\)
0.650620 + 0.759403i \(0.274509\pi\)
\(542\) 1.16685e12 0.580787
\(543\) −1.57127e12 −0.775623
\(544\) 0 0
\(545\) 9.63098e11 0.467613
\(546\) −3.18620e12 −1.53428
\(547\) 5.58809e11 0.266883 0.133441 0.991057i \(-0.457397\pi\)
0.133441 + 0.991057i \(0.457397\pi\)
\(548\) −1.79783e12 −0.851600
\(549\) −1.20637e12 −0.566766
\(550\) −5.36471e11 −0.249985
\(551\) 3.29768e11 0.152415
\(552\) 2.03251e12 0.931764
\(553\) −5.90232e12 −2.68386
\(554\) 1.20033e12 0.541387
\(555\) 2.03252e12 0.909321
\(556\) 5.81364e11 0.257995
\(557\) 3.73545e12 1.64435 0.822175 0.569235i \(-0.192761\pi\)
0.822175 + 0.569235i \(0.192761\pi\)
\(558\) −9.32699e11 −0.407275
\(559\) 7.23108e11 0.313221
\(560\) 2.00967e12 0.863532
\(561\) 0 0
\(562\) −2.34969e12 −0.993566
\(563\) 4.79014e11 0.200937 0.100469 0.994940i \(-0.467966\pi\)
0.100469 + 0.994940i \(0.467966\pi\)
\(564\) 3.41180e12 1.41980
\(565\) −2.39632e12 −0.989299
\(566\) 7.37535e11 0.302071
\(567\) 2.64334e12 1.07406
\(568\) 1.87508e11 0.0755881
\(569\) −4.70752e12 −1.88273 −0.941363 0.337395i \(-0.890454\pi\)
−0.941363 + 0.337395i \(0.890454\pi\)
\(570\) 1.62636e12 0.645325
\(571\) 4.64774e11 0.182970 0.0914848 0.995806i \(-0.470839\pi\)
0.0914848 + 0.995806i \(0.470839\pi\)
\(572\) −1.08160e12 −0.422459
\(573\) 5.65918e10 0.0219310
\(574\) −2.73209e12 −1.05049
\(575\) −1.70905e12 −0.652004
\(576\) 1.19122e12 0.450911
\(577\) 1.23582e11 0.0464158 0.0232079 0.999731i \(-0.492612\pi\)
0.0232079 + 0.999731i \(0.492612\pi\)
\(578\) 0 0
\(579\) 1.64345e12 0.607719
\(580\) −1.62020e12 −0.594486
\(581\) −6.22352e12 −2.26591
\(582\) −3.69819e12 −1.33609
\(583\) 4.69340e11 0.168259
\(584\) 2.64590e12 0.941274
\(585\) 6.53346e11 0.230644
\(586\) 1.24229e12 0.435195
\(587\) −2.85606e12 −0.992878 −0.496439 0.868072i \(-0.665359\pi\)
−0.496439 + 0.868072i \(0.665359\pi\)
\(588\) 6.46981e12 2.23200
\(589\) 1.13054e12 0.387049
\(590\) 5.67346e12 1.92759
\(591\) −3.35559e12 −1.13142
\(592\) −1.19425e12 −0.399621
\(593\) −6.66611e11 −0.221374 −0.110687 0.993855i \(-0.535305\pi\)
−0.110687 + 0.993855i \(0.535305\pi\)
\(594\) 2.23105e12 0.735308
\(595\) 0 0
\(596\) 3.06358e12 0.994538
\(597\) 2.96351e12 0.954819
\(598\) −5.90169e12 −1.88721
\(599\) −2.92151e11 −0.0927230 −0.0463615 0.998925i \(-0.514763\pi\)
−0.0463615 + 0.998925i \(0.514763\pi\)
\(600\) −6.19138e11 −0.195032
\(601\) −3.82913e12 −1.19720 −0.598598 0.801050i \(-0.704275\pi\)
−0.598598 + 0.801050i \(0.704275\pi\)
\(602\) −3.85092e12 −1.19503
\(603\) −9.32199e11 −0.287131
\(604\) 6.04236e12 1.84731
\(605\) −3.12143e12 −0.947228
\(606\) 6.30460e12 1.89902
\(607\) −1.66205e12 −0.496928 −0.248464 0.968641i \(-0.579926\pi\)
−0.248464 + 0.968641i \(0.579926\pi\)
\(608\) −1.84157e12 −0.546539
\(609\) 1.76364e12 0.519557
\(610\) −1.23050e13 −3.59829
\(611\) −2.84547e12 −0.825977
\(612\) 0 0
\(613\) −3.34407e12 −0.956540 −0.478270 0.878213i \(-0.658736\pi\)
−0.478270 + 0.878213i \(0.658736\pi\)
\(614\) 3.32725e11 0.0944774
\(615\) −1.40049e12 −0.394767
\(616\) 1.65445e12 0.462957
\(617\) −3.02182e12 −0.839431 −0.419716 0.907656i \(-0.637870\pi\)
−0.419716 + 0.907656i \(0.637870\pi\)
\(618\) −1.93716e12 −0.534217
\(619\) −4.41312e12 −1.20820 −0.604099 0.796909i \(-0.706467\pi\)
−0.604099 + 0.796909i \(0.706467\pi\)
\(620\) −5.55448e12 −1.50966
\(621\) 7.10752e12 1.91781
\(622\) −5.48784e12 −1.47009
\(623\) −4.02115e11 −0.106944
\(624\) 9.59659e11 0.253388
\(625\) −4.70333e12 −1.23295
\(626\) −2.78286e12 −0.724281
\(627\) −6.00971e11 −0.155292
\(628\) 1.00675e12 0.258288
\(629\) 0 0
\(630\) −3.47940e12 −0.879977
\(631\) 2.58843e12 0.649986 0.324993 0.945716i \(-0.394638\pi\)
0.324993 + 0.945716i \(0.394638\pi\)
\(632\) −3.96057e12 −0.987486
\(633\) −2.13798e12 −0.529281
\(634\) 4.42336e12 1.08730
\(635\) −4.70620e12 −1.14865
\(636\) 1.88583e12 0.457031
\(637\) −5.39588e12 −1.29848
\(638\) 1.02543e12 0.245026
\(639\) 1.45717e11 0.0345745
\(640\) 5.70141e12 1.34330
\(641\) 2.23675e12 0.523307 0.261653 0.965162i \(-0.415732\pi\)
0.261653 + 0.965162i \(0.415732\pi\)
\(642\) −7.84539e12 −1.82267
\(643\) 4.25453e12 0.981527 0.490764 0.871293i \(-0.336718\pi\)
0.490764 + 0.871293i \(0.336718\pi\)
\(644\) 1.83501e13 4.20389
\(645\) −1.97401e12 −0.449086
\(646\) 0 0
\(647\) 3.30319e11 0.0741079 0.0370539 0.999313i \(-0.488203\pi\)
0.0370539 + 0.999313i \(0.488203\pi\)
\(648\) 1.77373e12 0.395186
\(649\) −2.09646e12 −0.463858
\(650\) 1.79776e12 0.395023
\(651\) 6.04626e12 1.31939
\(652\) 6.65941e12 1.44318
\(653\) 1.35293e12 0.291184 0.145592 0.989345i \(-0.453491\pi\)
0.145592 + 0.989345i \(0.453491\pi\)
\(654\) −2.44920e12 −0.523509
\(655\) 9.34874e12 1.98457
\(656\) 8.22885e11 0.173489
\(657\) 2.05619e12 0.430545
\(658\) 1.51535e13 3.15136
\(659\) 3.08677e12 0.637558 0.318779 0.947829i \(-0.396727\pi\)
0.318779 + 0.947829i \(0.396727\pi\)
\(660\) 2.95265e12 0.605709
\(661\) 6.15803e12 1.25469 0.627343 0.778743i \(-0.284142\pi\)
0.627343 + 0.778743i \(0.284142\pi\)
\(662\) −3.78768e12 −0.766500
\(663\) 0 0
\(664\) −4.17610e12 −0.833710
\(665\) 4.21742e12 0.836276
\(666\) 2.06764e12 0.407231
\(667\) 3.26674e12 0.639070
\(668\) −9.75448e12 −1.89544
\(669\) −3.91575e12 −0.755783
\(670\) −9.50845e12 −1.82294
\(671\) 4.54694e12 0.865900
\(672\) −9.84894e12 −1.86306
\(673\) −7.92971e12 −1.49001 −0.745005 0.667058i \(-0.767553\pi\)
−0.745005 + 0.667058i \(0.767553\pi\)
\(674\) 8.39121e12 1.56623
\(675\) −2.16508e12 −0.401427
\(676\) −3.99290e12 −0.735408
\(677\) −2.46341e12 −0.450701 −0.225350 0.974278i \(-0.572353\pi\)
−0.225350 + 0.974278i \(0.572353\pi\)
\(678\) 6.09395e12 1.10755
\(679\) −9.59003e12 −1.73143
\(680\) 0 0
\(681\) 4.34579e12 0.774295
\(682\) 3.51545e12 0.622230
\(683\) −5.68009e12 −0.998763 −0.499381 0.866382i \(-0.666439\pi\)
−0.499381 + 0.866382i \(0.666439\pi\)
\(684\) 9.65951e11 0.168734
\(685\) −4.09320e12 −0.710323
\(686\) 1.34703e13 2.32230
\(687\) 6.87088e11 0.117681
\(688\) 1.15987e12 0.197361
\(689\) −1.57280e12 −0.265880
\(690\) 1.61110e13 2.70583
\(691\) −5.44139e12 −0.907943 −0.453971 0.891016i \(-0.649993\pi\)
−0.453971 + 0.891016i \(0.649993\pi\)
\(692\) −4.76275e12 −0.789552
\(693\) 1.28571e12 0.211759
\(694\) −4.67566e12 −0.765112
\(695\) 1.32362e12 0.215195
\(696\) 1.18344e12 0.191163
\(697\) 0 0
\(698\) 3.53000e12 0.562892
\(699\) −8.66441e9 −0.00137275
\(700\) −5.58976e12 −0.879938
\(701\) 7.42727e11 0.116171 0.0580856 0.998312i \(-0.481500\pi\)
0.0580856 + 0.998312i \(0.481500\pi\)
\(702\) −7.47643e12 −1.16192
\(703\) −2.50621e12 −0.387007
\(704\) −4.48984e12 −0.688897
\(705\) 7.76781e12 1.18426
\(706\) −3.24326e12 −0.491316
\(707\) 1.63489e13 2.46094
\(708\) −8.42367e12 −1.25995
\(709\) −5.72142e12 −0.850347 −0.425173 0.905112i \(-0.639787\pi\)
−0.425173 + 0.905112i \(0.639787\pi\)
\(710\) 1.48631e12 0.219507
\(711\) −3.07784e12 −0.451682
\(712\) −2.69827e11 −0.0393483
\(713\) 1.11993e13 1.62288
\(714\) 0 0
\(715\) −2.46253e12 −0.352375
\(716\) −3.12171e12 −0.443899
\(717\) 8.29196e12 1.17171
\(718\) −1.41794e13 −1.99112
\(719\) 4.81513e12 0.671936 0.335968 0.941873i \(-0.390937\pi\)
0.335968 + 0.941873i \(0.390937\pi\)
\(720\) 1.04797e12 0.145329
\(721\) −5.02340e12 −0.692291
\(722\) 9.31319e12 1.27550
\(723\) 6.95771e12 0.946986
\(724\) −9.51899e12 −1.28756
\(725\) −9.95108e11 −0.133767
\(726\) 7.93792e12 1.06045
\(727\) −1.09017e13 −1.44740 −0.723700 0.690114i \(-0.757560\pi\)
−0.723700 + 0.690114i \(0.757560\pi\)
\(728\) −5.54420e12 −0.731557
\(729\) 8.38145e12 1.09912
\(730\) 2.09732e13 2.73345
\(731\) 0 0
\(732\) 1.82698e13 2.35198
\(733\) −5.31441e12 −0.679966 −0.339983 0.940432i \(-0.610421\pi\)
−0.339983 + 0.940432i \(0.610421\pi\)
\(734\) 1.90517e13 2.42272
\(735\) 1.47301e13 1.86172
\(736\) −1.82429e13 −2.29162
\(737\) 3.51356e12 0.438676
\(738\) −1.42468e12 −0.176793
\(739\) −9.75394e12 −1.20304 −0.601520 0.798857i \(-0.705438\pi\)
−0.601520 + 0.798857i \(0.705438\pi\)
\(740\) 1.23134e13 1.50950
\(741\) 2.01391e12 0.245390
\(742\) 8.37594e12 1.01442
\(743\) 3.69441e12 0.444729 0.222365 0.974964i \(-0.428622\pi\)
0.222365 + 0.974964i \(0.428622\pi\)
\(744\) 4.05716e12 0.485449
\(745\) 6.97501e12 0.829548
\(746\) −2.46281e12 −0.291142
\(747\) −3.24534e12 −0.381344
\(748\) 0 0
\(749\) −2.03445e13 −2.36199
\(750\) 8.37700e12 0.966747
\(751\) −9.80164e12 −1.12440 −0.562198 0.827003i \(-0.690044\pi\)
−0.562198 + 0.827003i \(0.690044\pi\)
\(752\) −4.56413e12 −0.520449
\(753\) −4.37895e12 −0.496355
\(754\) −3.43630e12 −0.387186
\(755\) 1.37569e13 1.54085
\(756\) 2.32464e13 2.58826
\(757\) 3.18101e11 0.0352073 0.0176037 0.999845i \(-0.494396\pi\)
0.0176037 + 0.999845i \(0.494396\pi\)
\(758\) −1.18229e13 −1.30080
\(759\) −5.95332e12 −0.651135
\(760\) 2.82997e12 0.307695
\(761\) 1.17912e13 1.27446 0.637230 0.770674i \(-0.280080\pi\)
0.637230 + 0.770674i \(0.280080\pi\)
\(762\) 1.19681e13 1.28596
\(763\) −6.35119e12 −0.678414
\(764\) 3.42842e11 0.0364061
\(765\) 0 0
\(766\) 9.46161e12 0.992969
\(767\) 7.02541e12 0.732981
\(768\) −1.64018e12 −0.170124
\(769\) 6.40708e11 0.0660681 0.0330341 0.999454i \(-0.489483\pi\)
0.0330341 + 0.999454i \(0.489483\pi\)
\(770\) 1.31142e13 1.34442
\(771\) 3.64986e12 0.371990
\(772\) 9.95629e12 1.00883
\(773\) 1.64262e13 1.65474 0.827371 0.561656i \(-0.189835\pi\)
0.827371 + 0.561656i \(0.189835\pi\)
\(774\) −2.00811e12 −0.201119
\(775\) −3.41150e12 −0.339694
\(776\) −6.43510e12 −0.637056
\(777\) −1.34036e13 −1.31925
\(778\) 2.45089e13 2.39836
\(779\) 1.72688e12 0.168013
\(780\) −9.89459e12 −0.957132
\(781\) −5.49223e11 −0.0528225
\(782\) 0 0
\(783\) 4.13840e12 0.393464
\(784\) −8.65500e12 −0.818172
\(785\) 2.29213e12 0.215439
\(786\) −2.37742e13 −2.22180
\(787\) 2.95065e12 0.274177 0.137089 0.990559i \(-0.456226\pi\)
0.137089 + 0.990559i \(0.456226\pi\)
\(788\) −2.03287e13 −1.87820
\(789\) 2.63041e12 0.241644
\(790\) −3.13940e13 −2.86764
\(791\) 1.58027e13 1.43528
\(792\) 8.62734e11 0.0779137
\(793\) −1.52372e13 −1.36828
\(794\) 1.88724e13 1.68513
\(795\) 4.29356e12 0.381211
\(796\) 1.79534e13 1.58503
\(797\) −1.14843e13 −1.00819 −0.504094 0.863649i \(-0.668173\pi\)
−0.504094 + 0.863649i \(0.668173\pi\)
\(798\) −1.07251e13 −0.936240
\(799\) 0 0
\(800\) 5.55711e12 0.479671
\(801\) −2.09688e11 −0.0179981
\(802\) −3.42964e12 −0.292728
\(803\) −7.75000e12 −0.657782
\(804\) 1.41177e13 1.19155
\(805\) 4.17785e13 3.50648
\(806\) −1.17806e13 −0.983238
\(807\) 6.76774e12 0.561711
\(808\) 1.09704e13 0.905467
\(809\) −9.69132e12 −0.795453 −0.397727 0.917504i \(-0.630201\pi\)
−0.397727 + 0.917504i \(0.630201\pi\)
\(810\) 1.40598e13 1.14761
\(811\) −1.92605e13 −1.56341 −0.781707 0.623646i \(-0.785651\pi\)
−0.781707 + 0.623646i \(0.785651\pi\)
\(812\) 1.06844e13 0.862482
\(813\) −3.94441e12 −0.316647
\(814\) −7.79317e12 −0.622163
\(815\) 1.51618e13 1.20377
\(816\) 0 0
\(817\) 2.43406e12 0.191131
\(818\) −3.04294e13 −2.37631
\(819\) −4.30851e12 −0.334619
\(820\) −8.48438e12 −0.655327
\(821\) −2.01071e13 −1.54456 −0.772279 0.635284i \(-0.780883\pi\)
−0.772279 + 0.635284i \(0.780883\pi\)
\(822\) 1.04092e13 0.795231
\(823\) 1.02420e13 0.778188 0.389094 0.921198i \(-0.372788\pi\)
0.389094 + 0.921198i \(0.372788\pi\)
\(824\) −3.37080e12 −0.254718
\(825\) 1.81349e12 0.136293
\(826\) −3.74139e13 −2.79655
\(827\) −8.99775e12 −0.668897 −0.334448 0.942414i \(-0.608550\pi\)
−0.334448 + 0.942414i \(0.608550\pi\)
\(828\) 9.56888e12 0.707497
\(829\) 8.74712e12 0.643235 0.321617 0.946870i \(-0.395774\pi\)
0.321617 + 0.946870i \(0.395774\pi\)
\(830\) −3.31025e13 −2.42108
\(831\) −4.05761e12 −0.295166
\(832\) 1.50459e13 1.08858
\(833\) 0 0
\(834\) −3.36602e12 −0.240918
\(835\) −2.22085e13 −1.58099
\(836\) −3.64078e12 −0.257790
\(837\) 1.41876e13 0.999179
\(838\) 1.95645e13 1.37047
\(839\) 1.38983e13 0.968352 0.484176 0.874971i \(-0.339120\pi\)
0.484176 + 0.874971i \(0.339120\pi\)
\(840\) 1.51351e13 1.04888
\(841\) −1.26051e13 −0.868887
\(842\) −8.96241e12 −0.614498
\(843\) 7.94289e12 0.541695
\(844\) −1.29522e13 −0.878624
\(845\) −9.09083e12 −0.613407
\(846\) 7.90201e12 0.530361
\(847\) 2.05844e13 1.37424
\(848\) −2.52277e12 −0.167532
\(849\) −2.49317e12 −0.164690
\(850\) 0 0
\(851\) −2.48270e13 −1.62271
\(852\) −2.20680e12 −0.143478
\(853\) −1.12894e13 −0.730127 −0.365064 0.930983i \(-0.618953\pi\)
−0.365064 + 0.930983i \(0.618953\pi\)
\(854\) 8.11457e13 5.22042
\(855\) 2.19923e12 0.140742
\(856\) −1.36515e13 −0.869059
\(857\) −1.71291e13 −1.08473 −0.542365 0.840143i \(-0.682471\pi\)
−0.542365 + 0.840143i \(0.682471\pi\)
\(858\) 6.26232e12 0.394496
\(859\) −4.07005e12 −0.255053 −0.127527 0.991835i \(-0.540704\pi\)
−0.127527 + 0.991835i \(0.540704\pi\)
\(860\) −1.19588e13 −0.745498
\(861\) 9.23557e12 0.572730
\(862\) −3.04656e13 −1.87943
\(863\) −2.43498e13 −1.49433 −0.747167 0.664637i \(-0.768586\pi\)
−0.747167 + 0.664637i \(0.768586\pi\)
\(864\) −2.31106e13 −1.41091
\(865\) −1.08436e13 −0.658568
\(866\) −1.47328e13 −0.890134
\(867\) 0 0
\(868\) 3.66292e13 2.19023
\(869\) 1.16007e13 0.690075
\(870\) 9.38071e12 0.555136
\(871\) −1.17743e13 −0.693189
\(872\) −4.26177e12 −0.249612
\(873\) −5.00085e12 −0.291393
\(874\) −1.98657e13 −1.15160
\(875\) 2.17230e13 1.25281
\(876\) −3.11399e13 −1.78669
\(877\) 3.24681e13 1.85336 0.926679 0.375854i \(-0.122651\pi\)
0.926679 + 0.375854i \(0.122651\pi\)
\(878\) −1.00597e13 −0.571294
\(879\) −4.19944e12 −0.237269
\(880\) −3.94991e12 −0.222032
\(881\) 5.75612e12 0.321913 0.160956 0.986962i \(-0.448542\pi\)
0.160956 + 0.986962i \(0.448542\pi\)
\(882\) 1.49846e13 0.833754
\(883\) 1.16594e13 0.645438 0.322719 0.946495i \(-0.395403\pi\)
0.322719 + 0.946495i \(0.395403\pi\)
\(884\) 0 0
\(885\) −1.91786e13 −1.05093
\(886\) −9.53731e12 −0.519965
\(887\) 1.26043e13 0.683696 0.341848 0.939755i \(-0.388947\pi\)
0.341848 + 0.939755i \(0.388947\pi\)
\(888\) −8.99405e12 −0.485397
\(889\) 3.10352e13 1.66647
\(890\) −2.13883e12 −0.114267
\(891\) −5.19537e12 −0.276164
\(892\) −2.37222e13 −1.25462
\(893\) −9.57813e12 −0.504022
\(894\) −1.77377e13 −0.928708
\(895\) −7.10736e12 −0.370258
\(896\) −3.75981e13 −1.94886
\(897\) 1.99501e13 1.02891
\(898\) 7.56726e11 0.0388325
\(899\) 6.52086e12 0.332955
\(900\) −2.91485e12 −0.148090
\(901\) 0 0
\(902\) 5.36980e12 0.270103
\(903\) 1.30177e13 0.651535
\(904\) 1.06039e13 0.528089
\(905\) −2.16724e13 −1.07396
\(906\) −3.49844e13 −1.72503
\(907\) −2.63722e13 −1.29394 −0.646969 0.762516i \(-0.723964\pi\)
−0.646969 + 0.762516i \(0.723964\pi\)
\(908\) 2.63275e13 1.28536
\(909\) 8.52535e12 0.414166
\(910\) −4.39470e13 −2.12443
\(911\) 4.17031e12 0.200602 0.100301 0.994957i \(-0.468019\pi\)
0.100301 + 0.994957i \(0.468019\pi\)
\(912\) 3.23031e12 0.154621
\(913\) 1.22320e13 0.582613
\(914\) −5.51282e13 −2.61286
\(915\) 4.15958e13 1.96180
\(916\) 4.16249e12 0.195354
\(917\) −6.16506e13 −2.87922
\(918\) 0 0
\(919\) 3.08220e13 1.42541 0.712706 0.701462i \(-0.247469\pi\)
0.712706 + 0.701462i \(0.247469\pi\)
\(920\) 2.80342e13 1.29016
\(921\) −1.12475e12 −0.0515093
\(922\) −3.34366e13 −1.52382
\(923\) 1.84049e12 0.0834692
\(924\) −1.94714e13 −0.878765
\(925\) 7.56274e12 0.339658
\(926\) −4.64912e13 −2.07788
\(927\) −2.61951e12 −0.116510
\(928\) −1.06220e13 −0.470156
\(929\) 2.37926e12 0.104802 0.0524011 0.998626i \(-0.483313\pi\)
0.0524011 + 0.998626i \(0.483313\pi\)
\(930\) 3.21597e13 1.40974
\(931\) −1.81631e13 −0.792348
\(932\) −5.24904e10 −0.00227881
\(933\) 1.85511e13 0.801499
\(934\) 7.67920e12 0.330183
\(935\) 0 0
\(936\) −2.89110e12 −0.123118
\(937\) 3.37885e13 1.43199 0.715995 0.698105i \(-0.245973\pi\)
0.715995 + 0.698105i \(0.245973\pi\)
\(938\) 6.27039e13 2.64473
\(939\) 9.40719e12 0.394880
\(940\) 4.70586e13 1.96591
\(941\) −1.76401e13 −0.733411 −0.366705 0.930337i \(-0.619514\pi\)
−0.366705 + 0.930337i \(0.619514\pi\)
\(942\) −5.82897e12 −0.241192
\(943\) 1.71068e13 0.704474
\(944\) 1.12688e13 0.461852
\(945\) 5.29262e13 2.15887
\(946\) 7.56880e12 0.307268
\(947\) −3.73290e13 −1.50824 −0.754121 0.656736i \(-0.771937\pi\)
−0.754121 + 0.656736i \(0.771937\pi\)
\(948\) 4.66123e13 1.87440
\(949\) 2.59709e13 1.03942
\(950\) 6.05145e12 0.241048
\(951\) −1.49528e13 −0.592801
\(952\) 0 0
\(953\) −4.03536e13 −1.58476 −0.792381 0.610026i \(-0.791159\pi\)
−0.792381 + 0.610026i \(0.791159\pi\)
\(954\) 4.36774e12 0.170722
\(955\) 7.80566e11 0.0303665
\(956\) 5.02340e13 1.94508
\(957\) −3.46636e12 −0.133589
\(958\) −4.98199e13 −1.91099
\(959\) 2.69928e13 1.03054
\(960\) −4.10735e13 −1.56078
\(961\) −4.08433e12 −0.154478
\(962\) 2.61156e13 0.983133
\(963\) −1.06089e13 −0.397513
\(964\) 4.21509e13 1.57203
\(965\) 2.26680e13 0.841472
\(966\) −1.06244e14 −3.92562
\(967\) −4.76728e13 −1.75328 −0.876641 0.481145i \(-0.840221\pi\)
−0.876641 + 0.481145i \(0.840221\pi\)
\(968\) 1.38125e13 0.505632
\(969\) 0 0
\(970\) −5.10088e13 −1.85000
\(971\) 1.28032e13 0.462204 0.231102 0.972930i \(-0.425767\pi\)
0.231102 + 0.972930i \(0.425767\pi\)
\(972\) 2.15504e13 0.774384
\(973\) −8.72866e12 −0.312205
\(974\) −2.59882e13 −0.925252
\(975\) −6.07716e12 −0.215367
\(976\) −2.44405e13 −0.862155
\(977\) 4.43718e13 1.55805 0.779026 0.626992i \(-0.215714\pi\)
0.779026 + 0.626992i \(0.215714\pi\)
\(978\) −3.85571e13 −1.34766
\(979\) 7.90339e11 0.0274974
\(980\) 8.92376e13 3.09051
\(981\) −3.31191e12 −0.114174
\(982\) −7.98686e13 −2.74078
\(983\) −3.07326e13 −1.04981 −0.524903 0.851162i \(-0.675898\pi\)
−0.524903 + 0.851162i \(0.675898\pi\)
\(984\) 6.19725e12 0.210727
\(985\) −4.62833e13 −1.56661
\(986\) 0 0
\(987\) −5.12251e13 −1.71813
\(988\) 1.22006e13 0.407356
\(989\) 2.41122e13 0.801407
\(990\) 6.83859e12 0.226260
\(991\) 1.60170e13 0.527533 0.263766 0.964587i \(-0.415035\pi\)
0.263766 + 0.964587i \(0.415035\pi\)
\(992\) −3.64152e13 −1.19394
\(993\) 1.28039e13 0.417898
\(994\) −9.80156e12 −0.318461
\(995\) 4.08754e13 1.32208
\(996\) 4.91489e13 1.58251
\(997\) 1.27762e13 0.409518 0.204759 0.978812i \(-0.434359\pi\)
0.204759 + 0.978812i \(0.434359\pi\)
\(998\) 4.95590e12 0.158138
\(999\) −3.14515e13 −0.999072
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 289.10.a.i.1.8 52
17.3 odd 16 17.10.d.a.9.2 yes 52
17.6 odd 16 17.10.d.a.2.2 52
17.16 even 2 inner 289.10.a.i.1.7 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.d.a.2.2 52 17.6 odd 16
17.10.d.a.9.2 yes 52 17.3 odd 16
289.10.a.i.1.7 52 17.16 even 2 inner
289.10.a.i.1.8 52 1.1 even 1 trivial