Properties

Label 98.10.a.k.1.4
Level $98$
Weight $10$
Character 98.1
Self dual yes
Analytic conductor $50.474$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.4735119441\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{4817})\)
Defining polynomial: \( x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-35.6165\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} +191.777 q^{3} +256.000 q^{4} +33.5898 q^{5} -3068.43 q^{6} -4096.00 q^{8} +17095.5 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +191.777 q^{3} +256.000 q^{4} +33.5898 q^{5} -3068.43 q^{6} -4096.00 q^{8} +17095.5 q^{9} -537.437 q^{10} -51540.4 q^{11} +49094.9 q^{12} +113303. q^{13} +6441.76 q^{15} +65536.0 q^{16} -461667. q^{17} -273528. q^{18} -836657. q^{19} +8598.99 q^{20} +824646. q^{22} +299404. q^{23} -785519. q^{24} -1.95200e6 q^{25} -1.81284e6 q^{26} -496229. q^{27} +2.58319e6 q^{29} -103068. q^{30} -6.84324e6 q^{31} -1.04858e6 q^{32} -9.88427e6 q^{33} +7.38666e6 q^{34} +4.37644e6 q^{36} +1.39547e7 q^{37} +1.33865e7 q^{38} +2.17288e7 q^{39} -137584. q^{40} +2.22245e7 q^{41} -3.24518e7 q^{43} -1.31943e7 q^{44} +574234. q^{45} -4.79046e6 q^{46} -3.18053e6 q^{47} +1.25683e7 q^{48} +3.12319e7 q^{50} -8.85371e7 q^{51} +2.90055e7 q^{52} +8.82912e7 q^{53} +7.93966e6 q^{54} -1.73123e6 q^{55} -1.60452e8 q^{57} -4.13311e7 q^{58} -1.39912e7 q^{59} +1.64909e6 q^{60} -1.13931e8 q^{61} +1.09492e8 q^{62} +1.67772e7 q^{64} +3.80581e6 q^{65} +1.58148e8 q^{66} -3.05493e8 q^{67} -1.18187e8 q^{68} +5.74188e7 q^{69} -2.86664e8 q^{71} -7.00231e7 q^{72} +4.78856e7 q^{73} -2.23275e8 q^{74} -3.74348e8 q^{75} -2.14184e8 q^{76} -3.47661e8 q^{78} -1.07582e8 q^{79} +2.20134e6 q^{80} -4.31656e8 q^{81} -3.55593e8 q^{82} +5.67595e7 q^{83} -1.55073e7 q^{85} +5.19228e8 q^{86} +4.95398e8 q^{87} +2.11109e8 q^{88} -8.48669e8 q^{89} -9.18774e6 q^{90} +7.66474e7 q^{92} -1.31238e9 q^{93} +5.08885e7 q^{94} -2.81032e7 q^{95} -2.01093e8 q^{96} +8.99384e8 q^{97} -8.81107e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9} + 9548 q^{11} - 417312 q^{15} + 262144 q^{16} - 254592 q^{18} - 152768 q^{22} - 2836736 q^{23} + 966144 q^{25} + 16669972 q^{29} + 6676992 q^{30} - 4194304 q^{32} + 4073472 q^{36} + 3226372 q^{37} + 69110568 q^{39} - 76480204 q^{43} + 2444288 q^{44} + 45387776 q^{46} - 15458304 q^{50} - 97261596 q^{51} + 13708112 q^{53} - 371557692 q^{57} - 266719552 q^{58} - 106831872 q^{60} + 67108864 q^{64} - 515731188 q^{65} - 814420776 q^{67} - 1324668744 q^{71} - 65175552 q^{72} - 51621952 q^{74} - 1105769088 q^{78} - 668411176 q^{79} - 1111307688 q^{81} - 1659264964 q^{85} + 1223683264 q^{86} - 39108608 q^{88} - 726204416 q^{92} - 3706577496 q^{93} + 977188688 q^{95} - 2791581660 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) 191.777 1.36695 0.683473 0.729976i \(-0.260469\pi\)
0.683473 + 0.729976i \(0.260469\pi\)
\(4\) 256.000 0.500000
\(5\) 33.5898 0.0240349 0.0120175 0.999928i \(-0.496175\pi\)
0.0120175 + 0.999928i \(0.496175\pi\)
\(6\) −3068.43 −0.966576
\(7\) 0 0
\(8\) −4096.00 −0.353553
\(9\) 17095.5 0.868540
\(10\) −537.437 −0.0169953
\(11\) −51540.4 −1.06140 −0.530702 0.847559i \(-0.678072\pi\)
−0.530702 + 0.847559i \(0.678072\pi\)
\(12\) 49094.9 0.683473
\(13\) 113303. 1.10026 0.550129 0.835080i \(-0.314579\pi\)
0.550129 + 0.835080i \(0.314579\pi\)
\(14\) 0 0
\(15\) 6441.76 0.0328544
\(16\) 65536.0 0.250000
\(17\) −461667. −1.34063 −0.670314 0.742078i \(-0.733841\pi\)
−0.670314 + 0.742078i \(0.733841\pi\)
\(18\) −273528. −0.614150
\(19\) −836657. −1.47284 −0.736421 0.676523i \(-0.763486\pi\)
−0.736421 + 0.676523i \(0.763486\pi\)
\(20\) 8598.99 0.0120175
\(21\) 0 0
\(22\) 824646. 0.750526
\(23\) 299404. 0.223091 0.111546 0.993759i \(-0.464420\pi\)
0.111546 + 0.993759i \(0.464420\pi\)
\(24\) −785519. −0.483288
\(25\) −1.95200e6 −0.999422
\(26\) −1.81284e6 −0.778000
\(27\) −496229. −0.179699
\(28\) 0 0
\(29\) 2.58319e6 0.678213 0.339106 0.940748i \(-0.389875\pi\)
0.339106 + 0.940748i \(0.389875\pi\)
\(30\) −103068. −0.0232316
\(31\) −6.84324e6 −1.33087 −0.665433 0.746458i \(-0.731753\pi\)
−0.665433 + 0.746458i \(0.731753\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) −9.88427e6 −1.45088
\(34\) 7.38666e6 0.947967
\(35\) 0 0
\(36\) 4.37644e6 0.434270
\(37\) 1.39547e7 1.22409 0.612043 0.790825i \(-0.290348\pi\)
0.612043 + 0.790825i \(0.290348\pi\)
\(38\) 1.33865e7 1.04146
\(39\) 2.17288e7 1.50399
\(40\) −137584. −0.00849763
\(41\) 2.22245e7 1.22830 0.614151 0.789188i \(-0.289498\pi\)
0.614151 + 0.789188i \(0.289498\pi\)
\(42\) 0 0
\(43\) −3.24518e7 −1.44754 −0.723770 0.690042i \(-0.757592\pi\)
−0.723770 + 0.690042i \(0.757592\pi\)
\(44\) −1.31943e7 −0.530702
\(45\) 574234. 0.0208753
\(46\) −4.79046e6 −0.157749
\(47\) −3.18053e6 −0.0950734 −0.0475367 0.998869i \(-0.515137\pi\)
−0.0475367 + 0.998869i \(0.515137\pi\)
\(48\) 1.25683e7 0.341736
\(49\) 0 0
\(50\) 3.12319e7 0.706698
\(51\) −8.85371e7 −1.83257
\(52\) 2.90055e7 0.550129
\(53\) 8.82912e7 1.53701 0.768504 0.639845i \(-0.221001\pi\)
0.768504 + 0.639845i \(0.221001\pi\)
\(54\) 7.93966e6 0.127066
\(55\) −1.73123e6 −0.0255107
\(56\) 0 0
\(57\) −1.60452e8 −2.01330
\(58\) −4.13311e7 −0.479569
\(59\) −1.39912e7 −0.150321 −0.0751606 0.997171i \(-0.523947\pi\)
−0.0751606 + 0.997171i \(0.523947\pi\)
\(60\) 1.64909e6 0.0164272
\(61\) −1.13931e8 −1.05355 −0.526777 0.850004i \(-0.676600\pi\)
−0.526777 + 0.850004i \(0.676600\pi\)
\(62\) 1.09492e8 0.941064
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) 3.80581e6 0.0264446
\(66\) 1.58148e8 1.02593
\(67\) −3.05493e8 −1.85210 −0.926050 0.377400i \(-0.876818\pi\)
−0.926050 + 0.377400i \(0.876818\pi\)
\(68\) −1.18187e8 −0.670314
\(69\) 5.74188e7 0.304953
\(70\) 0 0
\(71\) −2.86664e8 −1.33878 −0.669392 0.742909i \(-0.733445\pi\)
−0.669392 + 0.742909i \(0.733445\pi\)
\(72\) −7.00231e7 −0.307075
\(73\) 4.78856e7 0.197357 0.0986785 0.995119i \(-0.468538\pi\)
0.0986785 + 0.995119i \(0.468538\pi\)
\(74\) −2.23275e8 −0.865559
\(75\) −3.74348e8 −1.36616
\(76\) −2.14184e8 −0.736421
\(77\) 0 0
\(78\) −3.47661e8 −1.06348
\(79\) −1.07582e8 −0.310753 −0.155377 0.987855i \(-0.549659\pi\)
−0.155377 + 0.987855i \(0.549659\pi\)
\(80\) 2.20134e6 0.00600873
\(81\) −4.31656e8 −1.11418
\(82\) −3.55593e8 −0.868541
\(83\) 5.67595e7 0.131277 0.0656383 0.997843i \(-0.479092\pi\)
0.0656383 + 0.997843i \(0.479092\pi\)
\(84\) 0 0
\(85\) −1.55073e7 −0.0322219
\(86\) 5.19228e8 1.02356
\(87\) 4.95398e8 0.927080
\(88\) 2.11109e8 0.375263
\(89\) −8.48669e8 −1.43378 −0.716891 0.697185i \(-0.754436\pi\)
−0.716891 + 0.697185i \(0.754436\pi\)
\(90\) −9.18774e6 −0.0147611
\(91\) 0 0
\(92\) 7.66474e7 0.111546
\(93\) −1.31238e9 −1.81922
\(94\) 5.08885e7 0.0672270
\(95\) −2.81032e7 −0.0353996
\(96\) −2.01093e8 −0.241644
\(97\) 8.99384e8 1.03151 0.515754 0.856737i \(-0.327512\pi\)
0.515754 + 0.856737i \(0.327512\pi\)
\(98\) 0 0
\(99\) −8.81107e8 −0.921871
\(100\) −4.99711e8 −0.499711
\(101\) 6.32202e7 0.0604519 0.0302259 0.999543i \(-0.490377\pi\)
0.0302259 + 0.999543i \(0.490377\pi\)
\(102\) 1.41659e9 1.29582
\(103\) 1.21580e9 1.06437 0.532185 0.846628i \(-0.321371\pi\)
0.532185 + 0.846628i \(0.321371\pi\)
\(104\) −4.64087e8 −0.389000
\(105\) 0 0
\(106\) −1.41266e9 −1.08683
\(107\) −1.28242e9 −0.945811 −0.472906 0.881113i \(-0.656795\pi\)
−0.472906 + 0.881113i \(0.656795\pi\)
\(108\) −1.27035e8 −0.0898494
\(109\) −5.06567e8 −0.343730 −0.171865 0.985120i \(-0.554979\pi\)
−0.171865 + 0.985120i \(0.554979\pi\)
\(110\) 2.76997e7 0.0180388
\(111\) 2.67619e9 1.67326
\(112\) 0 0
\(113\) 1.15512e9 0.666462 0.333231 0.942845i \(-0.391861\pi\)
0.333231 + 0.942845i \(0.391861\pi\)
\(114\) 2.56723e9 1.42361
\(115\) 1.00569e7 0.00536197
\(116\) 6.61298e8 0.339106
\(117\) 1.93696e9 0.955618
\(118\) 2.23859e8 0.106293
\(119\) 0 0
\(120\) −2.63854e7 −0.0116158
\(121\) 2.98463e8 0.126578
\(122\) 1.82289e9 0.744975
\(123\) 4.26216e9 1.67902
\(124\) −1.75187e9 −0.665433
\(125\) −1.31172e8 −0.0480559
\(126\) 0 0
\(127\) 3.41280e9 1.16411 0.582055 0.813150i \(-0.302249\pi\)
0.582055 + 0.813150i \(0.302249\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −6.22351e9 −1.97871
\(130\) −6.08930e7 −0.0186992
\(131\) 7.17987e8 0.213008 0.106504 0.994312i \(-0.466034\pi\)
0.106504 + 0.994312i \(0.466034\pi\)
\(132\) −2.53037e9 −0.725440
\(133\) 0 0
\(134\) 4.88789e9 1.30963
\(135\) −1.66682e7 −0.00431904
\(136\) 1.89099e9 0.473984
\(137\) −7.31370e9 −1.77376 −0.886880 0.462000i \(-0.847132\pi\)
−0.886880 + 0.462000i \(0.847132\pi\)
\(138\) −9.18701e8 −0.215634
\(139\) −4.88666e9 −1.11031 −0.555157 0.831746i \(-0.687342\pi\)
−0.555157 + 0.831746i \(0.687342\pi\)
\(140\) 0 0
\(141\) −6.09953e8 −0.129960
\(142\) 4.58662e9 0.946663
\(143\) −5.83966e9 −1.16782
\(144\) 1.12037e9 0.217135
\(145\) 8.67690e7 0.0163008
\(146\) −7.66170e8 −0.139552
\(147\) 0 0
\(148\) 3.57240e9 0.612043
\(149\) 6.90289e9 1.14734 0.573671 0.819086i \(-0.305519\pi\)
0.573671 + 0.819086i \(0.305519\pi\)
\(150\) 5.98957e9 0.966018
\(151\) −5.65547e9 −0.885263 −0.442632 0.896704i \(-0.645955\pi\)
−0.442632 + 0.896704i \(0.645955\pi\)
\(152\) 3.42695e9 0.520728
\(153\) −7.89241e9 −1.16439
\(154\) 0 0
\(155\) −2.29863e8 −0.0319872
\(156\) 5.56258e9 0.751996
\(157\) −6.59975e9 −0.866920 −0.433460 0.901173i \(-0.642707\pi\)
−0.433460 + 0.901173i \(0.642707\pi\)
\(158\) 1.72130e9 0.219736
\(159\) 1.69322e10 2.10101
\(160\) −3.52215e7 −0.00424881
\(161\) 0 0
\(162\) 6.90649e9 0.787843
\(163\) −7.98797e9 −0.886323 −0.443162 0.896442i \(-0.646143\pi\)
−0.443162 + 0.896442i \(0.646143\pi\)
\(164\) 5.68948e9 0.614151
\(165\) −3.32011e8 −0.0348718
\(166\) −9.08152e8 −0.0928266
\(167\) 4.34905e9 0.432683 0.216342 0.976318i \(-0.430588\pi\)
0.216342 + 0.976318i \(0.430588\pi\)
\(168\) 0 0
\(169\) 2.23297e9 0.210568
\(170\) 2.48117e8 0.0227843
\(171\) −1.43030e10 −1.27922
\(172\) −8.30765e9 −0.723770
\(173\) 1.38491e10 1.17548 0.587739 0.809051i \(-0.300018\pi\)
0.587739 + 0.809051i \(0.300018\pi\)
\(174\) −7.92636e9 −0.655545
\(175\) 0 0
\(176\) −3.37775e9 −0.265351
\(177\) −2.68319e9 −0.205481
\(178\) 1.35787e10 1.01384
\(179\) 1.99542e10 1.45277 0.726384 0.687289i \(-0.241199\pi\)
0.726384 + 0.687289i \(0.241199\pi\)
\(180\) 1.47004e8 0.0104376
\(181\) −3.33705e9 −0.231105 −0.115552 0.993301i \(-0.536864\pi\)
−0.115552 + 0.993301i \(0.536864\pi\)
\(182\) 0 0
\(183\) −2.18493e10 −1.44015
\(184\) −1.22636e9 −0.0788746
\(185\) 4.68735e8 0.0294208
\(186\) 2.09980e10 1.28638
\(187\) 2.37945e10 1.42295
\(188\) −8.14216e8 −0.0475367
\(189\) 0 0
\(190\) 4.49650e8 0.0250313
\(191\) −2.45891e10 −1.33688 −0.668441 0.743765i \(-0.733038\pi\)
−0.668441 + 0.743765i \(0.733038\pi\)
\(192\) 3.21749e9 0.170868
\(193\) −1.80119e10 −0.934438 −0.467219 0.884142i \(-0.654744\pi\)
−0.467219 + 0.884142i \(0.654744\pi\)
\(194\) −1.43901e10 −0.729386
\(195\) 7.29868e8 0.0361483
\(196\) 0 0
\(197\) 4.02716e10 1.90503 0.952514 0.304496i \(-0.0984880\pi\)
0.952514 + 0.304496i \(0.0984880\pi\)
\(198\) 1.40977e10 0.651862
\(199\) 3.76852e10 1.70346 0.851730 0.523981i \(-0.175554\pi\)
0.851730 + 0.523981i \(0.175554\pi\)
\(200\) 7.99538e9 0.353349
\(201\) −5.85866e10 −2.53172
\(202\) −1.01152e9 −0.0427459
\(203\) 0 0
\(204\) −2.26655e10 −0.916283
\(205\) 7.46518e8 0.0295221
\(206\) −1.94527e10 −0.752624
\(207\) 5.11845e9 0.193763
\(208\) 7.42540e9 0.275065
\(209\) 4.31216e10 1.56328
\(210\) 0 0
\(211\) −1.57268e10 −0.546223 −0.273111 0.961982i \(-0.588053\pi\)
−0.273111 + 0.961982i \(0.588053\pi\)
\(212\) 2.26026e10 0.768504
\(213\) −5.49756e10 −1.83004
\(214\) 2.05188e10 0.668790
\(215\) −1.09005e9 −0.0347915
\(216\) 2.03255e9 0.0635331
\(217\) 0 0
\(218\) 8.10508e9 0.243054
\(219\) 9.18337e9 0.269776
\(220\) −4.43195e8 −0.0127554
\(221\) −5.23080e10 −1.47504
\(222\) −4.28190e10 −1.18317
\(223\) 6.29667e10 1.70506 0.852528 0.522681i \(-0.175068\pi\)
0.852528 + 0.522681i \(0.175068\pi\)
\(224\) 0 0
\(225\) −3.33703e10 −0.868038
\(226\) −1.84820e10 −0.471260
\(227\) 5.48011e10 1.36985 0.684925 0.728614i \(-0.259835\pi\)
0.684925 + 0.728614i \(0.259835\pi\)
\(228\) −4.10756e10 −1.00665
\(229\) −6.14118e10 −1.47568 −0.737840 0.674976i \(-0.764154\pi\)
−0.737840 + 0.674976i \(0.764154\pi\)
\(230\) −1.60911e8 −0.00379149
\(231\) 0 0
\(232\) −1.05808e10 −0.239784
\(233\) −8.42910e9 −0.187361 −0.0936806 0.995602i \(-0.529863\pi\)
−0.0936806 + 0.995602i \(0.529863\pi\)
\(234\) −3.09914e10 −0.675724
\(235\) −1.06833e8 −0.00228508
\(236\) −3.58174e9 −0.0751606
\(237\) −2.06317e10 −0.424783
\(238\) 0 0
\(239\) 3.13951e10 0.622402 0.311201 0.950344i \(-0.399269\pi\)
0.311201 + 0.950344i \(0.399269\pi\)
\(240\) 4.22167e8 0.00821360
\(241\) 1.45866e10 0.278533 0.139267 0.990255i \(-0.455525\pi\)
0.139267 + 0.990255i \(0.455525\pi\)
\(242\) −4.77541e9 −0.0895039
\(243\) −7.30144e10 −1.34332
\(244\) −2.91663e10 −0.526777
\(245\) 0 0
\(246\) −6.81945e10 −1.18725
\(247\) −9.47954e10 −1.62051
\(248\) 2.80299e10 0.470532
\(249\) 1.08852e10 0.179448
\(250\) 2.09876e9 0.0339807
\(251\) −3.56389e10 −0.566752 −0.283376 0.959009i \(-0.591454\pi\)
−0.283376 + 0.959009i \(0.591454\pi\)
\(252\) 0 0
\(253\) −1.54314e10 −0.236790
\(254\) −5.46048e10 −0.823150
\(255\) −2.97394e9 −0.0440455
\(256\) 4.29497e9 0.0625000
\(257\) 6.94569e10 0.993154 0.496577 0.867993i \(-0.334590\pi\)
0.496577 + 0.867993i \(0.334590\pi\)
\(258\) 9.95761e10 1.39916
\(259\) 0 0
\(260\) 9.74288e8 0.0132223
\(261\) 4.41609e10 0.589055
\(262\) −1.14878e10 −0.150619
\(263\) 7.23020e10 0.931857 0.465929 0.884822i \(-0.345720\pi\)
0.465929 + 0.884822i \(0.345720\pi\)
\(264\) 4.04860e10 0.512964
\(265\) 2.96569e9 0.0369419
\(266\) 0 0
\(267\) −1.62755e11 −1.95990
\(268\) −7.82062e10 −0.926050
\(269\) 9.03066e10 1.05156 0.525780 0.850621i \(-0.323774\pi\)
0.525780 + 0.850621i \(0.323774\pi\)
\(270\) 2.66692e8 0.00305403
\(271\) 4.02229e10 0.453014 0.226507 0.974010i \(-0.427269\pi\)
0.226507 + 0.974010i \(0.427269\pi\)
\(272\) −3.02558e10 −0.335157
\(273\) 0 0
\(274\) 1.17019e11 1.25424
\(275\) 1.00607e11 1.06079
\(276\) 1.46992e10 0.152477
\(277\) −2.06717e10 −0.210968 −0.105484 0.994421i \(-0.533639\pi\)
−0.105484 + 0.994421i \(0.533639\pi\)
\(278\) 7.81866e10 0.785110
\(279\) −1.16988e11 −1.15591
\(280\) 0 0
\(281\) −3.46833e10 −0.331850 −0.165925 0.986138i \(-0.553061\pi\)
−0.165925 + 0.986138i \(0.553061\pi\)
\(282\) 9.75925e9 0.0918957
\(283\) 1.04639e11 0.969741 0.484870 0.874586i \(-0.338867\pi\)
0.484870 + 0.874586i \(0.338867\pi\)
\(284\) −7.33860e10 −0.669392
\(285\) −5.38954e9 −0.0483894
\(286\) 9.34345e10 0.825772
\(287\) 0 0
\(288\) −1.79259e10 −0.153538
\(289\) 9.45481e10 0.797283
\(290\) −1.38830e9 −0.0115264
\(291\) 1.72481e11 1.41001
\(292\) 1.22587e10 0.0986785
\(293\) −1.60760e11 −1.27431 −0.637153 0.770737i \(-0.719888\pi\)
−0.637153 + 0.770737i \(0.719888\pi\)
\(294\) 0 0
\(295\) −4.69961e8 −0.00361296
\(296\) −5.71583e10 −0.432780
\(297\) 2.55758e10 0.190733
\(298\) −1.10446e11 −0.811293
\(299\) 3.39232e10 0.245458
\(300\) −9.58332e10 −0.683078
\(301\) 0 0
\(302\) 9.04875e10 0.625976
\(303\) 1.21242e10 0.0826344
\(304\) −5.48312e10 −0.368211
\(305\) −3.82691e9 −0.0253221
\(306\) 1.26279e11 0.823347
\(307\) 1.45170e11 0.932725 0.466362 0.884594i \(-0.345564\pi\)
0.466362 + 0.884594i \(0.345564\pi\)
\(308\) 0 0
\(309\) 2.33162e11 1.45494
\(310\) 3.67781e9 0.0226184
\(311\) −2.63090e11 −1.59471 −0.797356 0.603510i \(-0.793768\pi\)
−0.797356 + 0.603510i \(0.793768\pi\)
\(312\) −8.90013e10 −0.531742
\(313\) −8.38203e10 −0.493628 −0.246814 0.969063i \(-0.579384\pi\)
−0.246814 + 0.969063i \(0.579384\pi\)
\(314\) 1.05596e11 0.613005
\(315\) 0 0
\(316\) −2.75409e10 −0.155377
\(317\) 2.70716e11 1.50573 0.752865 0.658175i \(-0.228671\pi\)
0.752865 + 0.658175i \(0.228671\pi\)
\(318\) −2.70916e11 −1.48564
\(319\) −1.33139e11 −0.719858
\(320\) 5.63544e8 0.00300436
\(321\) −2.45940e11 −1.29287
\(322\) 0 0
\(323\) 3.86257e11 1.97453
\(324\) −1.10504e11 −0.557089
\(325\) −2.21166e11 −1.09962
\(326\) 1.27807e11 0.626725
\(327\) −9.71480e10 −0.469861
\(328\) −9.10317e10 −0.434271
\(329\) 0 0
\(330\) 5.31217e9 0.0246581
\(331\) 1.93741e11 0.887146 0.443573 0.896238i \(-0.353711\pi\)
0.443573 + 0.896238i \(0.353711\pi\)
\(332\) 1.45304e10 0.0656383
\(333\) 2.38562e11 1.06317
\(334\) −6.95847e10 −0.305953
\(335\) −1.02615e10 −0.0445151
\(336\) 0 0
\(337\) 2.23909e11 0.945665 0.472832 0.881152i \(-0.343232\pi\)
0.472832 + 0.881152i \(0.343232\pi\)
\(338\) −3.57275e10 −0.148894
\(339\) 2.21526e11 0.911017
\(340\) −3.96987e9 −0.0161109
\(341\) 3.52703e11 1.41259
\(342\) 2.28849e11 0.904547
\(343\) 0 0
\(344\) 1.32922e11 0.511782
\(345\) 1.92869e9 0.00732953
\(346\) −2.21586e11 −0.831188
\(347\) −2.57729e11 −0.954289 −0.477145 0.878825i \(-0.658328\pi\)
−0.477145 + 0.878825i \(0.658328\pi\)
\(348\) 1.26822e11 0.463540
\(349\) 1.38744e11 0.500609 0.250304 0.968167i \(-0.419469\pi\)
0.250304 + 0.968167i \(0.419469\pi\)
\(350\) 0 0
\(351\) −5.62240e10 −0.197715
\(352\) 5.40440e10 0.187631
\(353\) 1.86980e11 0.640929 0.320464 0.947261i \(-0.396161\pi\)
0.320464 + 0.947261i \(0.396161\pi\)
\(354\) 4.29310e10 0.145297
\(355\) −9.62899e9 −0.0321776
\(356\) −2.17259e11 −0.716891
\(357\) 0 0
\(358\) −3.19268e11 −1.02726
\(359\) 2.07563e11 0.659516 0.329758 0.944066i \(-0.393033\pi\)
0.329758 + 0.944066i \(0.393033\pi\)
\(360\) −2.35206e9 −0.00738053
\(361\) 3.77307e11 1.16926
\(362\) 5.33928e10 0.163416
\(363\) 5.72384e10 0.173025
\(364\) 0 0
\(365\) 1.60847e9 0.00474346
\(366\) 3.49589e11 1.01834
\(367\) 1.79331e11 0.516009 0.258004 0.966144i \(-0.416935\pi\)
0.258004 + 0.966144i \(0.416935\pi\)
\(368\) 1.96217e10 0.0557728
\(369\) 3.79939e11 1.06683
\(370\) −7.49976e9 −0.0208036
\(371\) 0 0
\(372\) −3.35968e11 −0.909610
\(373\) −5.44836e11 −1.45739 −0.728696 0.684837i \(-0.759873\pi\)
−0.728696 + 0.684837i \(0.759873\pi\)
\(374\) −3.80711e11 −1.00618
\(375\) −2.51559e10 −0.0656899
\(376\) 1.30274e10 0.0336135
\(377\) 2.92682e11 0.746209
\(378\) 0 0
\(379\) −1.24662e11 −0.310353 −0.155177 0.987887i \(-0.549595\pi\)
−0.155177 + 0.987887i \(0.549595\pi\)
\(380\) −7.19441e9 −0.0176998
\(381\) 6.54497e11 1.59127
\(382\) 3.93426e11 0.945319
\(383\) −4.08245e11 −0.969453 −0.484726 0.874666i \(-0.661081\pi\)
−0.484726 + 0.874666i \(0.661081\pi\)
\(384\) −5.14798e10 −0.120822
\(385\) 0 0
\(386\) 2.88190e11 0.660748
\(387\) −5.54778e11 −1.25725
\(388\) 2.30242e11 0.515754
\(389\) 5.13640e11 1.13733 0.568663 0.822570i \(-0.307461\pi\)
0.568663 + 0.822570i \(0.307461\pi\)
\(390\) −1.16779e10 −0.0255607
\(391\) −1.38225e11 −0.299082
\(392\) 0 0
\(393\) 1.37694e11 0.291170
\(394\) −6.44346e11 −1.34706
\(395\) −3.61364e9 −0.00746893
\(396\) −2.25563e11 −0.460936
\(397\) 3.60814e11 0.728998 0.364499 0.931204i \(-0.381240\pi\)
0.364499 + 0.931204i \(0.381240\pi\)
\(398\) −6.02963e11 −1.20453
\(399\) 0 0
\(400\) −1.27926e11 −0.249856
\(401\) 2.38594e11 0.460797 0.230399 0.973096i \(-0.425997\pi\)
0.230399 + 0.973096i \(0.425997\pi\)
\(402\) 9.37385e11 1.79020
\(403\) −7.75356e11 −1.46430
\(404\) 1.61844e10 0.0302259
\(405\) −1.44992e10 −0.0267792
\(406\) 0 0
\(407\) −7.19229e11 −1.29925
\(408\) 3.62648e11 0.647910
\(409\) −7.74690e11 −1.36890 −0.684452 0.729058i \(-0.739959\pi\)
−0.684452 + 0.729058i \(0.739959\pi\)
\(410\) −1.19443e10 −0.0208753
\(411\) −1.40260e12 −2.42463
\(412\) 3.11244e11 0.532185
\(413\) 0 0
\(414\) −8.18952e10 −0.137011
\(415\) 1.90654e9 0.00315522
\(416\) −1.18806e11 −0.194500
\(417\) −9.37150e11 −1.51774
\(418\) −6.89946e11 −1.10541
\(419\) −5.87231e11 −0.930777 −0.465389 0.885106i \(-0.654085\pi\)
−0.465389 + 0.885106i \(0.654085\pi\)
\(420\) 0 0
\(421\) −1.12541e12 −1.74599 −0.872993 0.487733i \(-0.837824\pi\)
−0.872993 + 0.487733i \(0.837824\pi\)
\(422\) 2.51629e11 0.386238
\(423\) −5.43726e10 −0.0825750
\(424\) −3.61641e11 −0.543414
\(425\) 9.01172e11 1.33985
\(426\) 8.79610e11 1.29404
\(427\) 0 0
\(428\) −3.28300e11 −0.472906
\(429\) −1.11991e12 −1.59634
\(430\) 1.74408e10 0.0246013
\(431\) −6.77668e11 −0.945952 −0.472976 0.881075i \(-0.656820\pi\)
−0.472976 + 0.881075i \(0.656820\pi\)
\(432\) −3.25209e10 −0.0449247
\(433\) 5.66584e11 0.774585 0.387292 0.921957i \(-0.373410\pi\)
0.387292 + 0.921957i \(0.373410\pi\)
\(434\) 0 0
\(435\) 1.66403e10 0.0222823
\(436\) −1.29681e11 −0.171865
\(437\) −2.50498e11 −0.328578
\(438\) −1.46934e11 −0.190761
\(439\) −7.41236e11 −0.952503 −0.476252 0.879309i \(-0.658005\pi\)
−0.476252 + 0.879309i \(0.658005\pi\)
\(440\) 7.09113e9 0.00901941
\(441\) 0 0
\(442\) 8.36928e11 1.04301
\(443\) 9.25207e10 0.114136 0.0570679 0.998370i \(-0.481825\pi\)
0.0570679 + 0.998370i \(0.481825\pi\)
\(444\) 6.85104e11 0.836629
\(445\) −2.85066e10 −0.0344608
\(446\) −1.00747e12 −1.20566
\(447\) 1.32382e12 1.56835
\(448\) 0 0
\(449\) −1.26884e11 −0.147333 −0.0736664 0.997283i \(-0.523470\pi\)
−0.0736664 + 0.997283i \(0.523470\pi\)
\(450\) 5.33925e11 0.613796
\(451\) −1.14546e12 −1.30372
\(452\) 2.95711e11 0.333231
\(453\) −1.08459e12 −1.21011
\(454\) −8.76817e11 −0.968630
\(455\) 0 0
\(456\) 6.57210e11 0.711807
\(457\) 1.02815e12 1.10264 0.551321 0.834293i \(-0.314124\pi\)
0.551321 + 0.834293i \(0.314124\pi\)
\(458\) 9.82588e11 1.04346
\(459\) 2.29092e11 0.240909
\(460\) 2.57457e9 0.00268099
\(461\) −8.10233e11 −0.835518 −0.417759 0.908558i \(-0.637184\pi\)
−0.417759 + 0.908558i \(0.637184\pi\)
\(462\) 0 0
\(463\) −1.44793e12 −1.46431 −0.732155 0.681139i \(-0.761485\pi\)
−0.732155 + 0.681139i \(0.761485\pi\)
\(464\) 1.69292e11 0.169553
\(465\) −4.40825e10 −0.0437248
\(466\) 1.34866e11 0.132484
\(467\) −4.47034e11 −0.434926 −0.217463 0.976069i \(-0.569778\pi\)
−0.217463 + 0.976069i \(0.569778\pi\)
\(468\) 4.95862e11 0.477809
\(469\) 0 0
\(470\) 1.70933e9 0.00161580
\(471\) −1.26568e12 −1.18503
\(472\) 5.73079e10 0.0531466
\(473\) 1.67258e12 1.53642
\(474\) 3.30107e11 0.300367
\(475\) 1.63315e12 1.47199
\(476\) 0 0
\(477\) 1.50938e12 1.33495
\(478\) −5.02321e11 −0.440105
\(479\) −8.69551e10 −0.0754719 −0.0377360 0.999288i \(-0.512015\pi\)
−0.0377360 + 0.999288i \(0.512015\pi\)
\(480\) −6.75467e9 −0.00580790
\(481\) 1.58110e12 1.34681
\(482\) −2.33386e11 −0.196953
\(483\) 0 0
\(484\) 7.64066e10 0.0632888
\(485\) 3.02101e10 0.0247922
\(486\) 1.16823e12 0.949872
\(487\) −4.81935e8 −0.000388247 0 −0.000194124 1.00000i \(-0.500062\pi\)
−0.000194124 1.00000i \(0.500062\pi\)
\(488\) 4.66660e11 0.372487
\(489\) −1.53191e12 −1.21156
\(490\) 0 0
\(491\) −5.88640e11 −0.457070 −0.228535 0.973536i \(-0.573394\pi\)
−0.228535 + 0.973536i \(0.573394\pi\)
\(492\) 1.09111e12 0.839511
\(493\) −1.19257e12 −0.909231
\(494\) 1.51673e12 1.14587
\(495\) −2.95962e10 −0.0221571
\(496\) −4.48478e11 −0.332716
\(497\) 0 0
\(498\) −1.74163e11 −0.126889
\(499\) −2.47946e12 −1.79021 −0.895107 0.445852i \(-0.852901\pi\)
−0.895107 + 0.445852i \(0.852901\pi\)
\(500\) −3.35801e10 −0.0240280
\(501\) 8.34048e11 0.591454
\(502\) 5.70223e11 0.400754
\(503\) 1.35604e12 0.944530 0.472265 0.881457i \(-0.343437\pi\)
0.472265 + 0.881457i \(0.343437\pi\)
\(504\) 0 0
\(505\) 2.12356e9 0.00145296
\(506\) 2.46902e11 0.167436
\(507\) 4.28232e11 0.287835
\(508\) 8.73676e11 0.582055
\(509\) 1.68824e12 1.11482 0.557409 0.830238i \(-0.311795\pi\)
0.557409 + 0.830238i \(0.311795\pi\)
\(510\) 4.75831e10 0.0311449
\(511\) 0 0
\(512\) −6.87195e10 −0.0441942
\(513\) 4.15173e11 0.264668
\(514\) −1.11131e12 −0.702266
\(515\) 4.08383e10 0.0255821
\(516\) −1.59322e12 −0.989354
\(517\) 1.63926e11 0.100911
\(518\) 0 0
\(519\) 2.65594e12 1.60681
\(520\) −1.55886e10 −0.00934958
\(521\) 9.42135e11 0.560200 0.280100 0.959971i \(-0.409632\pi\)
0.280100 + 0.959971i \(0.409632\pi\)
\(522\) −7.06575e11 −0.416525
\(523\) 2.44405e12 1.42841 0.714206 0.699936i \(-0.246788\pi\)
0.714206 + 0.699936i \(0.246788\pi\)
\(524\) 1.83805e11 0.106504
\(525\) 0 0
\(526\) −1.15683e12 −0.658923
\(527\) 3.15929e12 1.78419
\(528\) −6.47775e11 −0.362720
\(529\) −1.71151e12 −0.950230
\(530\) −4.74510e10 −0.0261218
\(531\) −2.39186e11 −0.130560
\(532\) 0 0
\(533\) 2.51810e12 1.35145
\(534\) 2.60408e12 1.38586
\(535\) −4.30764e10 −0.0227325
\(536\) 1.25130e12 0.654816
\(537\) 3.82677e12 1.98586
\(538\) −1.44490e12 −0.743565
\(539\) 0 0
\(540\) −4.26707e9 −0.00215952
\(541\) −6.58195e11 −0.330344 −0.165172 0.986265i \(-0.552818\pi\)
−0.165172 + 0.986265i \(0.552818\pi\)
\(542\) −6.43566e11 −0.320329
\(543\) −6.39970e11 −0.315908
\(544\) 4.84092e11 0.236992
\(545\) −1.70155e10 −0.00826153
\(546\) 0 0
\(547\) −2.50775e12 −1.19768 −0.598840 0.800869i \(-0.704372\pi\)
−0.598840 + 0.800869i \(0.704372\pi\)
\(548\) −1.87231e12 −0.886880
\(549\) −1.94770e12 −0.915053
\(550\) −1.60971e12 −0.750092
\(551\) −2.16125e12 −0.998901
\(552\) −2.35187e11 −0.107817
\(553\) 0 0
\(554\) 3.30747e11 0.149177
\(555\) 8.98926e10 0.0402166
\(556\) −1.25099e12 −0.555157
\(557\) −2.60505e12 −1.14675 −0.573375 0.819293i \(-0.694366\pi\)
−0.573375 + 0.819293i \(0.694366\pi\)
\(558\) 1.87181e12 0.817351
\(559\) −3.67687e12 −1.59267
\(560\) 0 0
\(561\) 4.56324e12 1.94509
\(562\) 5.54933e11 0.234654
\(563\) 2.59402e11 0.108814 0.0544071 0.998519i \(-0.482673\pi\)
0.0544071 + 0.998519i \(0.482673\pi\)
\(564\) −1.56148e11 −0.0649801
\(565\) 3.88004e10 0.0160183
\(566\) −1.67423e12 −0.685710
\(567\) 0 0
\(568\) 1.17418e12 0.473332
\(569\) 2.64332e12 1.05717 0.528585 0.848880i \(-0.322723\pi\)
0.528585 + 0.848880i \(0.322723\pi\)
\(570\) 8.62327e10 0.0342165
\(571\) −2.06794e12 −0.814098 −0.407049 0.913406i \(-0.633442\pi\)
−0.407049 + 0.913406i \(0.633442\pi\)
\(572\) −1.49495e12 −0.583909
\(573\) −4.71564e12 −1.82745
\(574\) 0 0
\(575\) −5.84435e11 −0.222962
\(576\) 2.86814e11 0.108567
\(577\) −2.56500e12 −0.963377 −0.481688 0.876343i \(-0.659976\pi\)
−0.481688 + 0.876343i \(0.659976\pi\)
\(578\) −1.51277e12 −0.563764
\(579\) −3.45426e12 −1.27733
\(580\) 2.22129e10 0.00815039
\(581\) 0 0
\(582\) −2.75970e12 −0.997030
\(583\) −4.55056e12 −1.63139
\(584\) −1.96140e11 −0.0697762
\(585\) 6.50621e10 0.0229682
\(586\) 2.57216e12 0.901071
\(587\) 1.59591e12 0.554800 0.277400 0.960755i \(-0.410527\pi\)
0.277400 + 0.960755i \(0.410527\pi\)
\(588\) 0 0
\(589\) 5.72544e12 1.96015
\(590\) 7.51938e9 0.00255475
\(591\) 7.72318e12 2.60407
\(592\) 9.14533e11 0.306021
\(593\) −1.97116e12 −0.654600 −0.327300 0.944920i \(-0.606139\pi\)
−0.327300 + 0.944920i \(0.606139\pi\)
\(594\) −4.09213e11 −0.134869
\(595\) 0 0
\(596\) 1.76714e12 0.573671
\(597\) 7.22716e12 2.32854
\(598\) −5.42771e11 −0.173565
\(599\) 4.97079e12 1.57763 0.788814 0.614632i \(-0.210696\pi\)
0.788814 + 0.614632i \(0.210696\pi\)
\(600\) 1.53333e12 0.483009
\(601\) 4.73669e12 1.48095 0.740475 0.672084i \(-0.234601\pi\)
0.740475 + 0.672084i \(0.234601\pi\)
\(602\) 0 0
\(603\) −5.22255e12 −1.60862
\(604\) −1.44780e12 −0.442632
\(605\) 1.00253e10 0.00304228
\(606\) −1.93987e11 −0.0584314
\(607\) 3.26903e12 0.977395 0.488697 0.872453i \(-0.337472\pi\)
0.488697 + 0.872453i \(0.337472\pi\)
\(608\) 8.77298e11 0.260364
\(609\) 0 0
\(610\) 6.12306e10 0.0179054
\(611\) −3.60362e11 −0.104605
\(612\) −2.02046e12 −0.582194
\(613\) −6.53577e11 −0.186950 −0.0934748 0.995622i \(-0.529797\pi\)
−0.0934748 + 0.995622i \(0.529797\pi\)
\(614\) −2.32272e12 −0.659536
\(615\) 1.43165e11 0.0403552
\(616\) 0 0
\(617\) −6.63823e12 −1.84404 −0.922018 0.387148i \(-0.873460\pi\)
−0.922018 + 0.387148i \(0.873460\pi\)
\(618\) −3.73059e12 −1.02880
\(619\) 7.25374e10 0.0198589 0.00992943 0.999951i \(-0.496839\pi\)
0.00992943 + 0.999951i \(0.496839\pi\)
\(620\) −5.88450e10 −0.0159936
\(621\) −1.48573e11 −0.0400892
\(622\) 4.20943e12 1.12763
\(623\) 0 0
\(624\) 1.42402e12 0.375998
\(625\) 3.80809e12 0.998267
\(626\) 1.34113e12 0.349048
\(627\) 8.26974e12 2.13692
\(628\) −1.68954e12 −0.433460
\(629\) −6.44240e12 −1.64104
\(630\) 0 0
\(631\) −2.35325e12 −0.590930 −0.295465 0.955354i \(-0.595475\pi\)
−0.295465 + 0.955354i \(0.595475\pi\)
\(632\) 4.40654e11 0.109868
\(633\) −3.01604e12 −0.746657
\(634\) −4.33145e12 −1.06471
\(635\) 1.14635e11 0.0279793
\(636\) 4.33465e12 1.05050
\(637\) 0 0
\(638\) 2.13022e12 0.509016
\(639\) −4.90066e12 −1.16279
\(640\) −9.01670e9 −0.00212441
\(641\) −7.52887e11 −0.176144 −0.0880722 0.996114i \(-0.528071\pi\)
−0.0880722 + 0.996114i \(0.528071\pi\)
\(642\) 3.93503e12 0.914199
\(643\) 4.82355e12 1.11280 0.556400 0.830915i \(-0.312182\pi\)
0.556400 + 0.830915i \(0.312182\pi\)
\(644\) 0 0
\(645\) −2.09047e11 −0.0475581
\(646\) −6.18010e12 −1.39621
\(647\) 8.19172e12 1.83783 0.918916 0.394454i \(-0.129066\pi\)
0.918916 + 0.394454i \(0.129066\pi\)
\(648\) 1.76806e12 0.393922
\(649\) 7.21111e11 0.159552
\(650\) 3.53866e12 0.777551
\(651\) 0 0
\(652\) −2.04492e12 −0.443162
\(653\) −5.81217e12 −1.25092 −0.625460 0.780257i \(-0.715088\pi\)
−0.625460 + 0.780257i \(0.715088\pi\)
\(654\) 1.55437e12 0.332242
\(655\) 2.41171e10 0.00511963
\(656\) 1.45651e12 0.307076
\(657\) 8.18628e11 0.171412
\(658\) 0 0
\(659\) −3.45418e12 −0.713446 −0.356723 0.934210i \(-0.616106\pi\)
−0.356723 + 0.934210i \(0.616106\pi\)
\(660\) −8.49947e10 −0.0174359
\(661\) −5.45581e12 −1.11161 −0.555805 0.831313i \(-0.687590\pi\)
−0.555805 + 0.831313i \(0.687590\pi\)
\(662\) −3.09985e12 −0.627307
\(663\) −1.00315e13 −2.01629
\(664\) −2.32487e11 −0.0464133
\(665\) 0 0
\(666\) −3.81699e12 −0.751773
\(667\) 7.73418e11 0.151303
\(668\) 1.11336e12 0.216342
\(669\) 1.20756e13 2.33072
\(670\) 1.64183e11 0.0314769
\(671\) 5.87203e12 1.11825
\(672\) 0 0
\(673\) 4.93782e12 0.927828 0.463914 0.885880i \(-0.346445\pi\)
0.463914 + 0.885880i \(0.346445\pi\)
\(674\) −3.58255e12 −0.668686
\(675\) 9.68637e11 0.179595
\(676\) 5.71639e11 0.105284
\(677\) −9.07898e12 −1.66107 −0.830535 0.556967i \(-0.811965\pi\)
−0.830535 + 0.556967i \(0.811965\pi\)
\(678\) −3.54442e12 −0.644186
\(679\) 0 0
\(680\) 6.35179e10 0.0113922
\(681\) 1.05096e13 1.87251
\(682\) −5.64325e12 −0.998848
\(683\) −5.09917e12 −0.896615 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(684\) −3.66158e12 −0.639611
\(685\) −2.45666e11 −0.0426322
\(686\) 0 0
\(687\) −1.17774e13 −2.01717
\(688\) −2.12676e12 −0.361885
\(689\) 1.00036e13 1.69111
\(690\) −3.08590e10 −0.00518276
\(691\) −6.84110e12 −1.14150 −0.570748 0.821125i \(-0.693347\pi\)
−0.570748 + 0.821125i \(0.693347\pi\)
\(692\) 3.54537e12 0.587739
\(693\) 0 0
\(694\) 4.12366e12 0.674785
\(695\) −1.64142e11 −0.0266863
\(696\) −2.02915e12 −0.327772
\(697\) −1.02603e13 −1.64670
\(698\) −2.21990e12 −0.353984
\(699\) −1.61651e12 −0.256112
\(700\) 0 0
\(701\) 9.89108e12 1.54708 0.773539 0.633748i \(-0.218484\pi\)
0.773539 + 0.633748i \(0.218484\pi\)
\(702\) 8.99584e11 0.139806
\(703\) −1.16753e13 −1.80289
\(704\) −8.64704e11 −0.132675
\(705\) −2.04882e10 −0.00312358
\(706\) −2.99169e12 −0.453205
\(707\) 0 0
\(708\) −6.86896e11 −0.102740
\(709\) −1.14431e13 −1.70074 −0.850368 0.526189i \(-0.823620\pi\)
−0.850368 + 0.526189i \(0.823620\pi\)
\(710\) 1.54064e11 0.0227530
\(711\) −1.83916e12 −0.269902
\(712\) 3.47615e12 0.506919
\(713\) −2.04889e12 −0.296904
\(714\) 0 0
\(715\) −1.96153e11 −0.0280684
\(716\) 5.10828e12 0.726384
\(717\) 6.02086e12 0.850790
\(718\) −3.32101e12 −0.466348
\(719\) −7.20450e12 −1.00537 −0.502683 0.864471i \(-0.667654\pi\)
−0.502683 + 0.864471i \(0.667654\pi\)
\(720\) 3.76330e10 0.00521882
\(721\) 0 0
\(722\) −6.03692e12 −0.826795
\(723\) 2.79738e12 0.380740
\(724\) −8.54285e11 −0.115552
\(725\) −5.04239e12 −0.677821
\(726\) −9.15815e11 −0.122347
\(727\) 7.27861e12 0.966371 0.483185 0.875518i \(-0.339480\pi\)
0.483185 + 0.875518i \(0.339480\pi\)
\(728\) 0 0
\(729\) −5.50621e12 −0.722070
\(730\) −2.57355e10 −0.00335413
\(731\) 1.49819e13 1.94061
\(732\) −5.59342e12 −0.720075
\(733\) 1.36945e13 1.75218 0.876092 0.482144i \(-0.160142\pi\)
0.876092 + 0.482144i \(0.160142\pi\)
\(734\) −2.86929e12 −0.364873
\(735\) 0 0
\(736\) −3.13948e11 −0.0394373
\(737\) 1.57452e13 1.96583
\(738\) −6.07902e12 −0.754363
\(739\) −1.65987e12 −0.204726 −0.102363 0.994747i \(-0.532640\pi\)
−0.102363 + 0.994747i \(0.532640\pi\)
\(740\) 1.19996e11 0.0147104
\(741\) −1.81796e13 −2.21514
\(742\) 0 0
\(743\) −4.66431e11 −0.0561484 −0.0280742 0.999606i \(-0.508937\pi\)
−0.0280742 + 0.999606i \(0.508937\pi\)
\(744\) 5.37550e12 0.643191
\(745\) 2.31867e11 0.0275763
\(746\) 8.71738e12 1.03053
\(747\) 9.70331e11 0.114019
\(748\) 6.09138e12 0.711474
\(749\) 0 0
\(750\) 4.02494e11 0.0464497
\(751\) −5.33602e12 −0.612122 −0.306061 0.952012i \(-0.599011\pi\)
−0.306061 + 0.952012i \(0.599011\pi\)
\(752\) −2.08439e11 −0.0237684
\(753\) −6.83473e12 −0.774719
\(754\) −4.68292e12 −0.527650
\(755\) −1.89966e11 −0.0212772
\(756\) 0 0
\(757\) 8.40938e12 0.930748 0.465374 0.885114i \(-0.345920\pi\)
0.465374 + 0.885114i \(0.345920\pi\)
\(758\) 1.99458e12 0.219453
\(759\) −2.95939e12 −0.323678
\(760\) 1.15111e11 0.0125157
\(761\) 6.16386e12 0.666227 0.333113 0.942887i \(-0.391901\pi\)
0.333113 + 0.942887i \(0.391901\pi\)
\(762\) −1.04719e13 −1.12520
\(763\) 0 0
\(764\) −6.29482e12 −0.668441
\(765\) −2.65104e11 −0.0279860
\(766\) 6.53192e12 0.685507
\(767\) −1.58524e12 −0.165392
\(768\) 8.23677e11 0.0854341
\(769\) 9.39193e12 0.968470 0.484235 0.874938i \(-0.339098\pi\)
0.484235 + 0.874938i \(0.339098\pi\)
\(770\) 0 0
\(771\) 1.33203e13 1.35759
\(772\) −4.61104e12 −0.467219
\(773\) −4.22637e12 −0.425755 −0.212877 0.977079i \(-0.568283\pi\)
−0.212877 + 0.977079i \(0.568283\pi\)
\(774\) 8.87645e12 0.889007
\(775\) 1.33580e13 1.33010
\(776\) −3.68388e12 −0.364693
\(777\) 0 0
\(778\) −8.21823e12 −0.804212
\(779\) −1.85943e13 −1.80910
\(780\) 1.86846e11 0.0180742
\(781\) 1.47748e13 1.42099
\(782\) 2.21160e12 0.211483
\(783\) −1.28186e12 −0.121874
\(784\) 0 0
\(785\) −2.21684e11 −0.0208363
\(786\) −2.20310e12 −0.205889
\(787\) −5.89826e11 −0.0548072 −0.0274036 0.999624i \(-0.508724\pi\)
−0.0274036 + 0.999624i \(0.508724\pi\)
\(788\) 1.03095e13 0.952514
\(789\) 1.38659e13 1.27380
\(790\) 5.78183e10 0.00528133
\(791\) 0 0
\(792\) 3.60901e12 0.325931
\(793\) −1.29086e13 −1.15918
\(794\) −5.77303e12 −0.515480
\(795\) 5.68751e11 0.0504975
\(796\) 9.64741e12 0.851730
\(797\) −2.83029e12 −0.248467 −0.124233 0.992253i \(-0.539647\pi\)
−0.124233 + 0.992253i \(0.539647\pi\)
\(798\) 0 0
\(799\) 1.46834e12 0.127458
\(800\) 2.04682e12 0.176675
\(801\) −1.45084e13 −1.24530
\(802\) −3.81751e12 −0.325833
\(803\) −2.46804e12 −0.209475
\(804\) −1.49982e13 −1.26586
\(805\) 0 0
\(806\) 1.24057e13 1.03541
\(807\) 1.73187e13 1.43743
\(808\) −2.58950e11 −0.0213730
\(809\) 1.16033e13 0.952388 0.476194 0.879340i \(-0.342016\pi\)
0.476194 + 0.879340i \(0.342016\pi\)
\(810\) 2.31988e11 0.0189357
\(811\) −4.13817e12 −0.335903 −0.167952 0.985795i \(-0.553715\pi\)
−0.167952 + 0.985795i \(0.553715\pi\)
\(812\) 0 0
\(813\) 7.71383e12 0.619245
\(814\) 1.15077e13 0.918708
\(815\) −2.68314e11 −0.0213027
\(816\) −5.80237e12 −0.458141
\(817\) 2.71510e13 2.13200
\(818\) 1.23950e13 0.967961
\(819\) 0 0
\(820\) 1.91109e11 0.0147611
\(821\) −1.95703e11 −0.0150332 −0.00751662 0.999972i \(-0.502393\pi\)
−0.00751662 + 0.999972i \(0.502393\pi\)
\(822\) 2.24416e13 1.71447
\(823\) −8.51398e12 −0.646895 −0.323447 0.946246i \(-0.604842\pi\)
−0.323447 + 0.946246i \(0.604842\pi\)
\(824\) −4.97990e12 −0.376312
\(825\) 1.92941e13 1.45004
\(826\) 0 0
\(827\) −1.19931e12 −0.0891576 −0.0445788 0.999006i \(-0.514195\pi\)
−0.0445788 + 0.999006i \(0.514195\pi\)
\(828\) 1.31032e12 0.0968817
\(829\) −1.64084e13 −1.20662 −0.603311 0.797506i \(-0.706152\pi\)
−0.603311 + 0.797506i \(0.706152\pi\)
\(830\) −3.05047e10 −0.00223108
\(831\) −3.96436e12 −0.288382
\(832\) 1.90090e12 0.137532
\(833\) 0 0
\(834\) 1.49944e13 1.07320
\(835\) 1.46084e11 0.0103995
\(836\) 1.10391e13 0.781640
\(837\) 3.39581e12 0.239155
\(838\) 9.39570e12 0.658159
\(839\) −1.67670e13 −1.16822 −0.584111 0.811674i \(-0.698557\pi\)
−0.584111 + 0.811674i \(0.698557\pi\)
\(840\) 0 0
\(841\) −7.83425e12 −0.540027
\(842\) 1.80065e13 1.23460
\(843\) −6.65147e12 −0.453622
\(844\) −4.02607e12 −0.273111
\(845\) 7.50049e10 0.00506098
\(846\) 8.69962e11 0.0583894
\(847\) 0 0
\(848\) 5.78625e12 0.384252
\(849\) 2.00674e13 1.32558
\(850\) −1.44187e13 −0.947419
\(851\) 4.17808e12 0.273083
\(852\) −1.40738e13 −0.915022
\(853\) −6.93277e12 −0.448370 −0.224185 0.974547i \(-0.571972\pi\)
−0.224185 + 0.974547i \(0.571972\pi\)
\(854\) 0 0
\(855\) −4.80437e11 −0.0307460
\(856\) 5.25281e12 0.334395
\(857\) −2.60920e11 −0.0165232 −0.00826158 0.999966i \(-0.502630\pi\)
−0.00826158 + 0.999966i \(0.502630\pi\)
\(858\) 1.79186e13 1.12879
\(859\) −2.54206e13 −1.59300 −0.796502 0.604636i \(-0.793319\pi\)
−0.796502 + 0.604636i \(0.793319\pi\)
\(860\) −2.79053e11 −0.0173957
\(861\) 0 0
\(862\) 1.08427e13 0.668889
\(863\) 7.18619e12 0.441012 0.220506 0.975386i \(-0.429229\pi\)
0.220506 + 0.975386i \(0.429229\pi\)
\(864\) 5.20334e11 0.0317666
\(865\) 4.65189e11 0.0282525
\(866\) −9.06535e12 −0.547714
\(867\) 1.81322e13 1.08984
\(868\) 0 0
\(869\) 5.54479e12 0.329835
\(870\) −2.66245e11 −0.0157560
\(871\) −3.46131e13 −2.03779
\(872\) 2.07490e12 0.121527
\(873\) 1.53754e13 0.895905
\(874\) 4.00797e12 0.232340
\(875\) 0 0
\(876\) 2.35094e12 0.134888
\(877\) −2.68806e13 −1.53441 −0.767205 0.641402i \(-0.778353\pi\)
−0.767205 + 0.641402i \(0.778353\pi\)
\(878\) 1.18598e13 0.673521
\(879\) −3.08301e13 −1.74191
\(880\) −1.13458e11 −0.00637769
\(881\) 5.33553e11 0.0298391 0.0149195 0.999889i \(-0.495251\pi\)
0.0149195 + 0.999889i \(0.495251\pi\)
\(882\) 0 0
\(883\) 6.42561e12 0.355706 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(884\) −1.33908e13 −0.737518
\(885\) −9.01278e10 −0.00493872
\(886\) −1.48033e12 −0.0807062
\(887\) −2.88437e12 −0.156457 −0.0782285 0.996935i \(-0.524926\pi\)
−0.0782285 + 0.996935i \(0.524926\pi\)
\(888\) −1.09617e13 −0.591586
\(889\) 0 0
\(890\) 4.56106e11 0.0243675
\(891\) 2.22477e13 1.18259
\(892\) 1.61195e13 0.852528
\(893\) 2.66101e12 0.140028
\(894\) −2.11811e13 −1.10899
\(895\) 6.70259e11 0.0349172
\(896\) 0 0
\(897\) 6.50570e12 0.335527
\(898\) 2.03015e12 0.104180
\(899\) −1.76774e13 −0.902610
\(900\) −8.54280e12 −0.434019
\(901\) −4.07611e13 −2.06056
\(902\) 1.83274e13 0.921873
\(903\) 0 0
\(904\) −4.73138e12 −0.235630
\(905\) −1.12091e11 −0.00555459
\(906\) 1.73534e13 0.855675
\(907\) −1.74881e13 −0.858044 −0.429022 0.903294i \(-0.641142\pi\)
−0.429022 + 0.903294i \(0.641142\pi\)
\(908\) 1.40291e13 0.684925
\(909\) 1.08078e12 0.0525049
\(910\) 0 0
\(911\) −1.68088e13 −0.808543 −0.404271 0.914639i \(-0.632475\pi\)
−0.404271 + 0.914639i \(0.632475\pi\)
\(912\) −1.05154e13 −0.503324
\(913\) −2.92541e12 −0.139337
\(914\) −1.64504e13 −0.779686
\(915\) −7.33914e11 −0.0346139
\(916\) −1.57214e13 −0.737840
\(917\) 0 0
\(918\) −3.66548e12 −0.170348
\(919\) −2.48947e13 −1.15130 −0.575648 0.817698i \(-0.695250\pi\)
−0.575648 + 0.817698i \(0.695250\pi\)
\(920\) −4.11931e10 −0.00189574
\(921\) 2.78403e13 1.27498
\(922\) 1.29637e13 0.590800
\(923\) −3.24798e13 −1.47301
\(924\) 0 0
\(925\) −2.72395e13 −1.22338
\(926\) 2.31669e13 1.03542
\(927\) 2.07846e13 0.924449
\(928\) −2.70868e12 −0.119892
\(929\) 3.84486e13 1.69360 0.846799 0.531914i \(-0.178527\pi\)
0.846799 + 0.531914i \(0.178527\pi\)
\(930\) 7.05320e11 0.0309181
\(931\) 0 0
\(932\) −2.15785e12 −0.0936806
\(933\) −5.04546e13 −2.17988
\(934\) 7.15255e12 0.307539
\(935\) 7.99252e11 0.0342004
\(936\) −7.93379e12 −0.337862
\(937\) −4.89574e11 −0.0207486 −0.0103743 0.999946i \(-0.503302\pi\)
−0.0103743 + 0.999946i \(0.503302\pi\)
\(938\) 0 0
\(939\) −1.60748e13 −0.674763
\(940\) −2.73493e10 −0.00114254
\(941\) 2.20721e13 0.917679 0.458840 0.888519i \(-0.348265\pi\)
0.458840 + 0.888519i \(0.348265\pi\)
\(942\) 2.02509e13 0.837944
\(943\) 6.65411e12 0.274023
\(944\) −9.16926e11 −0.0375803
\(945\) 0 0
\(946\) −2.67612e13 −1.08642
\(947\) 4.86143e11 0.0196422 0.00982108 0.999952i \(-0.496874\pi\)
0.00982108 + 0.999952i \(0.496874\pi\)
\(948\) −5.28171e12 −0.212391
\(949\) 5.42556e12 0.217144
\(950\) −2.61304e13 −1.04086
\(951\) 5.19171e13 2.05825
\(952\) 0 0
\(953\) −2.77682e13 −1.09051 −0.545254 0.838271i \(-0.683567\pi\)
−0.545254 + 0.838271i \(0.683567\pi\)
\(954\) −2.41501e13 −0.943954
\(955\) −8.25945e11 −0.0321319
\(956\) 8.03714e12 0.311201
\(957\) −2.55330e13 −0.984006
\(958\) 1.39128e12 0.0533667
\(959\) 0 0
\(960\) 1.08075e11 0.00410680
\(961\) 2.03903e13 0.771202
\(962\) −2.52976e13 −0.952339
\(963\) −2.19236e13 −0.821475
\(964\) 3.73417e12 0.139267
\(965\) −6.05015e11 −0.0224591
\(966\) 0 0
\(967\) −2.53282e12 −0.0931505 −0.0465752 0.998915i \(-0.514831\pi\)
−0.0465752 + 0.998915i \(0.514831\pi\)
\(968\) −1.22251e12 −0.0447519
\(969\) 7.40752e13 2.69908
\(970\) −4.83362e11 −0.0175307
\(971\) 8.52422e12 0.307729 0.153864 0.988092i \(-0.450828\pi\)
0.153864 + 0.988092i \(0.450828\pi\)
\(972\) −1.86917e13 −0.671661
\(973\) 0 0
\(974\) 7.71096e9 0.000274532 0
\(975\) −4.24146e13 −1.50312
\(976\) −7.46657e12 −0.263388
\(977\) −3.37176e13 −1.18394 −0.591972 0.805958i \(-0.701651\pi\)
−0.591972 + 0.805958i \(0.701651\pi\)
\(978\) 2.45106e13 0.856699
\(979\) 4.37407e13 1.52182
\(980\) 0 0
\(981\) −8.66001e12 −0.298544
\(982\) 9.41824e12 0.323197
\(983\) 3.27904e12 0.112010 0.0560049 0.998430i \(-0.482164\pi\)
0.0560049 + 0.998430i \(0.482164\pi\)
\(984\) −1.74578e13 −0.593624
\(985\) 1.35272e12 0.0457872
\(986\) 1.90812e13 0.642923
\(987\) 0 0
\(988\) −2.42676e13 −0.810253
\(989\) −9.71618e12 −0.322933
\(990\) 4.73540e11 0.0156674
\(991\) 2.29294e13 0.755200 0.377600 0.925969i \(-0.376749\pi\)
0.377600 + 0.925969i \(0.376749\pi\)
\(992\) 7.17566e12 0.235266
\(993\) 3.71550e13 1.21268
\(994\) 0 0
\(995\) 1.26584e12 0.0409425
\(996\) 2.78661e12 0.0897240
\(997\) 1.25509e13 0.402297 0.201148 0.979561i \(-0.435533\pi\)
0.201148 + 0.979561i \(0.435533\pi\)
\(998\) 3.96714e13 1.26587
\(999\) −6.92471e12 −0.219967
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 98.10.a.k.1.4 yes 4
7.2 even 3 98.10.c.m.67.1 8
7.3 odd 6 98.10.c.m.79.4 8
7.4 even 3 98.10.c.m.79.1 8
7.5 odd 6 98.10.c.m.67.4 8
7.6 odd 2 inner 98.10.a.k.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.10.a.k.1.1 4 7.6 odd 2 inner
98.10.a.k.1.4 yes 4 1.1 even 1 trivial
98.10.c.m.67.1 8 7.2 even 3
98.10.c.m.67.4 8 7.5 odd 6
98.10.c.m.79.1 8 7.4 even 3
98.10.c.m.79.4 8 7.3 odd 6