Properties

Label 98.10.a.k.1.3
Level $98$
Weight $10$
Character 98.1
Self dual yes
Analytic conductor $50.474$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 1.3
Root \(36.6165\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} +102.682 q^{3} +256.000 q^{4} -2094.80 q^{5} -1642.91 q^{6} -4096.00 q^{8} -9139.47 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +102.682 q^{3} +256.000 q^{4} -2094.80 q^{5} -1642.91 q^{6} -4096.00 q^{8} -9139.47 q^{9} +33516.8 q^{10} +56314.4 q^{11} +26286.5 q^{12} +124915. q^{13} -215098. q^{15} +65536.0 q^{16} +388641. q^{17} +146232. q^{18} -246657. q^{19} -536269. q^{20} -901030. q^{22} -1.71777e6 q^{23} -420584. q^{24} +2.43507e6 q^{25} -1.99863e6 q^{26} -2.95954e6 q^{27} +5.75179e6 q^{29} +3.44156e6 q^{30} -5.26785e6 q^{31} -1.04858e6 q^{32} +5.78246e6 q^{33} -6.21825e6 q^{34} -2.33970e6 q^{36} -1.23415e7 q^{37} +3.94651e6 q^{38} +1.28264e7 q^{39} +8.58031e6 q^{40} -5.54285e6 q^{41} -5.78833e6 q^{43} +1.44165e7 q^{44} +1.91454e7 q^{45} +2.74843e7 q^{46} +5.11932e7 q^{47} +6.72935e6 q^{48} -3.89611e7 q^{50} +3.99063e7 q^{51} +3.19782e7 q^{52} -8.14372e7 q^{53} +4.73526e7 q^{54} -1.17967e8 q^{55} -2.53272e7 q^{57} -9.20287e7 q^{58} -7.85852e7 q^{59} -5.50650e7 q^{60} +1.38027e7 q^{61} +8.42856e7 q^{62} +1.67772e7 q^{64} -2.61671e8 q^{65} -9.25193e7 q^{66} -1.01717e8 q^{67} +9.94920e7 q^{68} -1.76384e8 q^{69} -3.75670e8 q^{71} +3.74353e7 q^{72} -4.45945e8 q^{73} +1.97464e8 q^{74} +2.50037e8 q^{75} -6.31442e7 q^{76} -2.05223e8 q^{78} -2.26624e8 q^{79} -1.37285e8 q^{80} -1.23998e8 q^{81} +8.86857e7 q^{82} +5.26823e8 q^{83} -8.14125e8 q^{85} +9.26132e7 q^{86} +5.90604e8 q^{87} -2.30664e8 q^{88} +5.19381e8 q^{89} -3.06326e8 q^{90} -4.39750e8 q^{92} -5.40912e8 q^{93} -8.19091e8 q^{94} +5.16697e8 q^{95} -1.07670e8 q^{96} -8.75899e8 q^{97} -5.14684e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9}+O(q^{10}) \) 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
\(3\) 102.682 0.731893 0.365946 0.930636i \(-0.380745\pi\)
0.365946 + 0.930636i \(0.380745\pi\)
\(4\) 256.000 0.500000
\(5\) −2094.80 −1.49892 −0.749459 0.662051i \(-0.769686\pi\)
−0.749459 + 0.662051i \(0.769686\pi\)
\(6\) −1642.91 −0.517526
\(7\) 0 0
\(8\) −4096.00 −0.353553
\(9\) −9139.47 −0.464333
\(10\) 33516.8 1.05990
\(11\) 56314.4 1.15972 0.579859 0.814717i \(-0.303108\pi\)
0.579859 + 0.814717i \(0.303108\pi\)
\(12\) 26286.5 0.365946
\(13\) 124915. 1.21302 0.606510 0.795076i \(-0.292569\pi\)
0.606510 + 0.795076i \(0.292569\pi\)
\(14\) 0 0
\(15\) −215098. −1.09705
\(16\) 65536.0 0.250000
\(17\) 388641. 1.12857 0.564285 0.825580i \(-0.309152\pi\)
0.564285 + 0.825580i \(0.309152\pi\)
\(18\) 146232. 0.328333
\(19\) −246657. −0.434212 −0.217106 0.976148i \(-0.569662\pi\)
−0.217106 + 0.976148i \(0.569662\pi\)
\(20\) −536269. −0.749459
\(21\) 0 0
\(22\) −901030. −0.820044
\(23\) −1.71777e6 −1.27994 −0.639971 0.768399i \(-0.721054\pi\)
−0.639971 + 0.768399i \(0.721054\pi\)
\(24\) −420584. −0.258763
\(25\) 2.43507e6 1.24676
\(26\) −1.99863e6 −0.857735
\(27\) −2.95954e6 −1.07173
\(28\) 0 0
\(29\) 5.75179e6 1.51012 0.755061 0.655654i \(-0.227607\pi\)
0.755061 + 0.655654i \(0.227607\pi\)
\(30\) 3.44156e6 0.775729
\(31\) −5.26785e6 −1.02449 −0.512243 0.858841i \(-0.671185\pi\)
−0.512243 + 0.858841i \(0.671185\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) 5.78246e6 0.848789
\(34\) −6.21825e6 −0.798019
\(35\) 0 0
\(36\) −2.33970e6 −0.232167
\(37\) −1.23415e7 −1.08258 −0.541290 0.840836i \(-0.682064\pi\)
−0.541290 + 0.840836i \(0.682064\pi\)
\(38\) 3.94651e6 0.307035
\(39\) 1.28264e7 0.887801
\(40\) 8.58031e6 0.529948
\(41\) −5.54285e6 −0.306342 −0.153171 0.988200i \(-0.548948\pi\)
−0.153171 + 0.988200i \(0.548948\pi\)
\(42\) 0 0
\(43\) −5.78833e6 −0.258193 −0.129097 0.991632i \(-0.541208\pi\)
−0.129097 + 0.991632i \(0.541208\pi\)
\(44\) 1.44165e7 0.579859
\(45\) 1.91454e7 0.695997
\(46\) 2.74843e7 0.905055
\(47\) 5.11932e7 1.53028 0.765142 0.643861i \(-0.222669\pi\)
0.765142 + 0.643861i \(0.222669\pi\)
\(48\) 6.72935e6 0.182973
\(49\) 0 0
\(50\) −3.89611e7 −0.881589
\(51\) 3.99063e7 0.825991
\(52\) 3.19782e7 0.606510
\(53\) −8.14372e7 −1.41769 −0.708845 0.705364i \(-0.750783\pi\)
−0.708845 + 0.705364i \(0.750783\pi\)
\(54\) 4.73526e7 0.757831
\(55\) −1.17967e8 −1.73832
\(56\) 0 0
\(57\) −2.53272e7 −0.317797
\(58\) −9.20287e7 −1.06782
\(59\) −7.85852e7 −0.844319 −0.422160 0.906522i \(-0.638728\pi\)
−0.422160 + 0.906522i \(0.638728\pi\)
\(60\) −5.50650e7 −0.548523
\(61\) 1.38027e7 0.127638 0.0638188 0.997962i \(-0.479672\pi\)
0.0638188 + 0.997962i \(0.479672\pi\)
\(62\) 8.42856e7 0.724421
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) −2.61671e8 −1.81822
\(66\) −9.25193e7 −0.600184
\(67\) −1.01717e8 −0.616678 −0.308339 0.951276i \(-0.599773\pi\)
−0.308339 + 0.951276i \(0.599773\pi\)
\(68\) 9.94920e7 0.564285
\(69\) −1.76384e8 −0.936780
\(70\) 0 0
\(71\) −3.75670e8 −1.75446 −0.877232 0.480067i \(-0.840612\pi\)
−0.877232 + 0.480067i \(0.840612\pi\)
\(72\) 3.74353e7 0.164167
\(73\) −4.45945e8 −1.83793 −0.918964 0.394341i \(-0.870973\pi\)
−0.918964 + 0.394341i \(0.870973\pi\)
\(74\) 1.97464e8 0.765499
\(75\) 2.50037e8 0.912491
\(76\) −6.31442e7 −0.217106
\(77\) 0 0
\(78\) −2.05223e8 −0.627770
\(79\) −2.26624e8 −0.654612 −0.327306 0.944918i \(-0.606141\pi\)
−0.327306 + 0.944918i \(0.606141\pi\)
\(80\) −1.37285e8 −0.374730
\(81\) −1.23998e8 −0.320061
\(82\) 8.86857e7 0.216616
\(83\) 5.26823e8 1.21847 0.609233 0.792992i \(-0.291478\pi\)
0.609233 + 0.792992i \(0.291478\pi\)
\(84\) 0 0
\(85\) −8.14125e8 −1.69163
\(86\) 9.26132e7 0.182570
\(87\) 5.90604e8 1.10525
\(88\) −2.30664e8 −0.410022
\(89\) 5.19381e8 0.877468 0.438734 0.898617i \(-0.355427\pi\)
0.438734 + 0.898617i \(0.355427\pi\)
\(90\) −3.06326e8 −0.492145
\(91\) 0 0
\(92\) −4.39750e8 −0.639971
\(93\) −5.40912e8 −0.749814
\(94\) −8.19091e8 −1.08207
\(95\) 5.16697e8 0.650849
\(96\) −1.07670e8 −0.129382
\(97\) −8.75899e8 −1.00457 −0.502286 0.864701i \(-0.667508\pi\)
−0.502286 + 0.864701i \(0.667508\pi\)
\(98\) 0 0
\(99\) −5.14684e8 −0.538495
\(100\) 6.23378e8 0.623378
\(101\) −8.73621e6 −0.00835366 −0.00417683 0.999991i \(-0.501330\pi\)
−0.00417683 + 0.999991i \(0.501330\pi\)
\(102\) −6.38501e8 −0.584064
\(103\) 9.59029e8 0.839584 0.419792 0.907620i \(-0.362103\pi\)
0.419792 + 0.907620i \(0.362103\pi\)
\(104\) −5.11650e8 −0.428868
\(105\) 0 0
\(106\) 1.30299e9 1.00246
\(107\) −1.48471e9 −1.09500 −0.547499 0.836806i \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(108\) −7.57642e8 −0.535867
\(109\) −1.53530e9 −1.04178 −0.520888 0.853625i \(-0.674399\pi\)
−0.520888 + 0.853625i \(0.674399\pi\)
\(110\) 1.88748e9 1.22918
\(111\) −1.26724e9 −0.792332
\(112\) 0 0
\(113\) 3.29034e9 1.89840 0.949199 0.314677i \(-0.101896\pi\)
0.949199 + 0.314677i \(0.101896\pi\)
\(114\) 4.05235e8 0.224716
\(115\) 3.59839e9 1.91853
\(116\) 1.47246e9 0.755061
\(117\) −1.14165e9 −0.563246
\(118\) 1.25736e9 0.597024
\(119\) 0 0
\(120\) 8.81040e8 0.387865
\(121\) 8.13362e8 0.344945
\(122\) −2.20843e8 −0.0902534
\(123\) −5.69150e8 −0.224209
\(124\) −1.34857e9 −0.512243
\(125\) −1.00958e9 −0.369866
\(126\) 0 0
\(127\) −5.41672e9 −1.84765 −0.923826 0.382814i \(-0.874955\pi\)
−0.923826 + 0.382814i \(0.874955\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −5.94355e8 −0.188970
\(130\) 4.18674e9 1.28567
\(131\) −3.38792e9 −1.00511 −0.502553 0.864546i \(-0.667606\pi\)
−0.502553 + 0.864546i \(0.667606\pi\)
\(132\) 1.48031e9 0.424394
\(133\) 0 0
\(134\) 1.62748e9 0.436057
\(135\) 6.19965e9 1.60644
\(136\) −1.59187e9 −0.399009
\(137\) 4.74729e9 1.15134 0.575669 0.817683i \(-0.304742\pi\)
0.575669 + 0.817683i \(0.304742\pi\)
\(138\) 2.82214e9 0.662403
\(139\) 2.17838e9 0.494958 0.247479 0.968893i \(-0.420398\pi\)
0.247479 + 0.968893i \(0.420398\pi\)
\(140\) 0 0
\(141\) 5.25661e9 1.12000
\(142\) 6.01073e9 1.24059
\(143\) 7.03449e9 1.40676
\(144\) −5.98964e8 −0.116083
\(145\) −1.20489e10 −2.26355
\(146\) 7.13512e9 1.29961
\(147\) 0 0
\(148\) −3.15942e9 −0.541290
\(149\) −1.18058e10 −1.96226 −0.981129 0.193354i \(-0.938064\pi\)
−0.981129 + 0.193354i \(0.938064\pi\)
\(150\) −4.00059e9 −0.645228
\(151\) −6.31601e9 −0.988660 −0.494330 0.869274i \(-0.664586\pi\)
−0.494330 + 0.869274i \(0.664586\pi\)
\(152\) 1.01031e9 0.153517
\(153\) −3.55197e9 −0.524032
\(154\) 0 0
\(155\) 1.10351e10 1.53562
\(156\) 3.28357e9 0.443900
\(157\) −1.03026e9 −0.135331 −0.0676655 0.997708i \(-0.521555\pi\)
−0.0676655 + 0.997708i \(0.521555\pi\)
\(158\) 3.62598e9 0.462881
\(159\) −8.36211e9 −1.03760
\(160\) 2.19656e9 0.264974
\(161\) 0 0
\(162\) 1.98397e9 0.226318
\(163\) −3.34930e9 −0.371629 −0.185814 0.982585i \(-0.559492\pi\)
−0.185814 + 0.982585i \(0.559492\pi\)
\(164\) −1.41897e9 −0.153171
\(165\) −1.21131e10 −1.27226
\(166\) −8.42917e9 −0.861585
\(167\) 1.63487e10 1.62652 0.813261 0.581899i \(-0.197690\pi\)
0.813261 + 0.581899i \(0.197690\pi\)
\(168\) 0 0
\(169\) 4.99917e9 0.471420
\(170\) 1.30260e10 1.19616
\(171\) 2.25431e9 0.201619
\(172\) −1.48181e9 −0.129097
\(173\) −7.47099e9 −0.634119 −0.317059 0.948406i \(-0.602695\pi\)
−0.317059 + 0.948406i \(0.602695\pi\)
\(174\) −9.44966e9 −0.781528
\(175\) 0 0
\(176\) 3.69062e9 0.289929
\(177\) −8.06926e9 −0.617951
\(178\) −8.31010e9 −0.620464
\(179\) −6.23008e9 −0.453581 −0.226790 0.973944i \(-0.572823\pi\)
−0.226790 + 0.973944i \(0.572823\pi\)
\(180\) 4.90122e9 0.347999
\(181\) 5.46976e9 0.378804 0.189402 0.981900i \(-0.439345\pi\)
0.189402 + 0.981900i \(0.439345\pi\)
\(182\) 0 0
\(183\) 1.41728e9 0.0934170
\(184\) 7.03599e9 0.452528
\(185\) 2.58530e10 1.62270
\(186\) 8.65459e9 0.530198
\(187\) 2.18861e10 1.30882
\(188\) 1.31055e10 0.765142
\(189\) 0 0
\(190\) −8.26716e9 −0.460220
\(191\) 6.93485e9 0.377039 0.188520 0.982069i \(-0.439631\pi\)
0.188520 + 0.982069i \(0.439631\pi\)
\(192\) 1.72271e9 0.0914866
\(193\) 4.17331e9 0.216508 0.108254 0.994123i \(-0.465474\pi\)
0.108254 + 0.994123i \(0.465474\pi\)
\(194\) 1.40144e10 0.710340
\(195\) −2.68689e10 −1.33074
\(196\) 0 0
\(197\) −3.62569e10 −1.71511 −0.857556 0.514391i \(-0.828018\pi\)
−0.857556 + 0.514391i \(0.828018\pi\)
\(198\) 8.23494e9 0.380774
\(199\) 1.79583e10 0.811759 0.405880 0.913927i \(-0.366965\pi\)
0.405880 + 0.913927i \(0.366965\pi\)
\(200\) −9.97404e9 −0.440795
\(201\) −1.04445e10 −0.451342
\(202\) 1.39779e8 0.00590693
\(203\) 0 0
\(204\) 1.02160e10 0.412996
\(205\) 1.16112e10 0.459181
\(206\) −1.53445e10 −0.593675
\(207\) 1.56995e10 0.594320
\(208\) 8.18641e9 0.303255
\(209\) −1.38903e10 −0.503564
\(210\) 0 0
\(211\) 6.33838e9 0.220144 0.110072 0.993924i \(-0.464892\pi\)
0.110072 + 0.993924i \(0.464892\pi\)
\(212\) −2.08479e10 −0.708845
\(213\) −3.85745e10 −1.28408
\(214\) 2.37553e10 0.774281
\(215\) 1.21254e10 0.387011
\(216\) 1.21223e10 0.378915
\(217\) 0 0
\(218\) 2.45648e10 0.736647
\(219\) −4.57904e10 −1.34517
\(220\) −3.01997e10 −0.869161
\(221\) 4.85469e10 1.36898
\(222\) 2.02759e10 0.560263
\(223\) −4.34746e10 −1.17724 −0.588619 0.808411i \(-0.700328\pi\)
−0.588619 + 0.808411i \(0.700328\pi\)
\(224\) 0 0
\(225\) −2.22552e10 −0.578910
\(226\) −5.26454e10 −1.34237
\(227\) −1.89487e10 −0.473657 −0.236828 0.971552i \(-0.576108\pi\)
−0.236828 + 0.971552i \(0.576108\pi\)
\(228\) −6.48375e9 −0.158898
\(229\) 3.72378e10 0.894796 0.447398 0.894335i \(-0.352351\pi\)
0.447398 + 0.894335i \(0.352351\pi\)
\(230\) −5.75743e10 −1.35660
\(231\) 0 0
\(232\) −2.35593e10 −0.533909
\(233\) 1.83837e10 0.408632 0.204316 0.978905i \(-0.434503\pi\)
0.204316 + 0.978905i \(0.434503\pi\)
\(234\) 1.82665e10 0.398275
\(235\) −1.07240e11 −2.29377
\(236\) −2.01178e10 −0.422160
\(237\) −2.32701e10 −0.479106
\(238\) 0 0
\(239\) −3.65476e10 −0.724550 −0.362275 0.932071i \(-0.618000\pi\)
−0.362275 + 0.932071i \(0.618000\pi\)
\(240\) −1.40966e10 −0.274262
\(241\) 2.85905e10 0.545941 0.272970 0.962022i \(-0.411994\pi\)
0.272970 + 0.962022i \(0.411994\pi\)
\(242\) −1.30138e10 −0.243913
\(243\) 4.55203e10 0.837484
\(244\) 3.53348e9 0.0638188
\(245\) 0 0
\(246\) 9.10639e9 0.158540
\(247\) −3.08111e10 −0.526709
\(248\) 2.15771e10 0.362210
\(249\) 5.40951e10 0.891786
\(250\) 1.61532e10 0.261535
\(251\) −1.00172e10 −0.159300 −0.0796499 0.996823i \(-0.525380\pi\)
−0.0796499 + 0.996823i \(0.525380\pi\)
\(252\) 0 0
\(253\) −9.67353e10 −1.48437
\(254\) 8.66676e10 1.30649
\(255\) −8.35957e10 −1.23809
\(256\) 4.29497e9 0.0625000
\(257\) −1.69681e10 −0.242625 −0.121312 0.992614i \(-0.538710\pi\)
−0.121312 + 0.992614i \(0.538710\pi\)
\(258\) 9.50968e9 0.133622
\(259\) 0 0
\(260\) −6.69879e10 −0.909109
\(261\) −5.25683e10 −0.701200
\(262\) 5.42067e10 0.710717
\(263\) 7.07192e9 0.0911458 0.0455729 0.998961i \(-0.485489\pi\)
0.0455729 + 0.998961i \(0.485489\pi\)
\(264\) −2.36849e10 −0.300092
\(265\) 1.70595e11 2.12500
\(266\) 0 0
\(267\) 5.33310e10 0.642212
\(268\) −2.60397e10 −0.308339
\(269\) −3.69954e8 −0.00430787 −0.00215393 0.999998i \(-0.500686\pi\)
−0.00215393 + 0.999998i \(0.500686\pi\)
\(270\) −9.91944e10 −1.13593
\(271\) 1.24288e11 1.39981 0.699904 0.714237i \(-0.253226\pi\)
0.699904 + 0.714237i \(0.253226\pi\)
\(272\) 2.54700e10 0.282142
\(273\) 0 0
\(274\) −7.59566e10 −0.814119
\(275\) 1.37129e11 1.44588
\(276\) −4.51542e10 −0.468390
\(277\) 1.69613e10 0.173101 0.0865506 0.996247i \(-0.472416\pi\)
0.0865506 + 0.996247i \(0.472416\pi\)
\(278\) −3.48542e10 −0.349988
\(279\) 4.81454e10 0.475703
\(280\) 0 0
\(281\) 1.16441e11 1.11411 0.557053 0.830477i \(-0.311932\pi\)
0.557053 + 0.830477i \(0.311932\pi\)
\(282\) −8.41057e10 −0.791962
\(283\) −2.65533e10 −0.246082 −0.123041 0.992402i \(-0.539265\pi\)
−0.123041 + 0.992402i \(0.539265\pi\)
\(284\) −9.61716e10 −0.877232
\(285\) 5.30554e10 0.476351
\(286\) −1.12552e11 −0.994731
\(287\) 0 0
\(288\) 9.58343e9 0.0820833
\(289\) 3.24537e10 0.273668
\(290\) 1.92782e11 1.60057
\(291\) −8.99388e10 −0.735239
\(292\) −1.14162e11 −0.918964
\(293\) −9.43424e8 −0.00747829 −0.00373915 0.999993i \(-0.501190\pi\)
−0.00373915 + 0.999993i \(0.501190\pi\)
\(294\) 0 0
\(295\) 1.64620e11 1.26557
\(296\) 5.05507e10 0.382750
\(297\) −1.66665e11 −1.24291
\(298\) 1.88892e11 1.38753
\(299\) −2.14575e11 −1.55260
\(300\) 6.40095e10 0.456245
\(301\) 0 0
\(302\) 1.01056e11 0.699088
\(303\) −8.97049e8 −0.00611398
\(304\) −1.61649e10 −0.108553
\(305\) −2.89139e10 −0.191318
\(306\) 5.68315e10 0.370547
\(307\) 2.20668e10 0.141780 0.0708902 0.997484i \(-0.477416\pi\)
0.0708902 + 0.997484i \(0.477416\pi\)
\(308\) 0 0
\(309\) 9.84747e10 0.614485
\(310\) −1.76562e11 −1.08585
\(311\) −2.30243e11 −1.39561 −0.697805 0.716288i \(-0.745840\pi\)
−0.697805 + 0.716288i \(0.745840\pi\)
\(312\) −5.25371e10 −0.313885
\(313\) −1.57588e11 −0.928055 −0.464027 0.885821i \(-0.653596\pi\)
−0.464027 + 0.885821i \(0.653596\pi\)
\(314\) 1.64841e10 0.0956935
\(315\) 0 0
\(316\) −5.80158e10 −0.327306
\(317\) 2.41500e11 1.34323 0.671614 0.740901i \(-0.265601\pi\)
0.671614 + 0.740901i \(0.265601\pi\)
\(318\) 1.33794e11 0.733692
\(319\) 3.23909e11 1.75132
\(320\) −3.51449e10 −0.187365
\(321\) −1.52452e11 −0.801422
\(322\) 0 0
\(323\) −9.58610e10 −0.490039
\(324\) −3.17436e10 −0.160031
\(325\) 3.04176e11 1.51234
\(326\) 5.35887e10 0.262781
\(327\) −1.57647e11 −0.762468
\(328\) 2.27035e10 0.108308
\(329\) 0 0
\(330\) 1.93810e11 0.899627
\(331\) 1.43537e11 0.657259 0.328630 0.944459i \(-0.393413\pi\)
0.328630 + 0.944459i \(0.393413\pi\)
\(332\) 1.34867e11 0.609233
\(333\) 1.12795e11 0.502678
\(334\) −2.61580e11 −1.15012
\(335\) 2.13078e11 0.924350
\(336\) 0 0
\(337\) 1.35343e11 0.571612 0.285806 0.958288i \(-0.407739\pi\)
0.285806 + 0.958288i \(0.407739\pi\)
\(338\) −7.99867e10 −0.333344
\(339\) 3.37857e11 1.38942
\(340\) −2.08416e11 −0.845816
\(341\) −2.96656e11 −1.18811
\(342\) −3.60690e10 −0.142566
\(343\) 0 0
\(344\) 2.37090e10 0.0912851
\(345\) 3.69489e11 1.40416
\(346\) 1.19536e11 0.448390
\(347\) 3.87493e11 1.43477 0.717383 0.696679i \(-0.245340\pi\)
0.717383 + 0.696679i \(0.245340\pi\)
\(348\) 1.51195e11 0.552624
\(349\) −1.02561e11 −0.370056 −0.185028 0.982733i \(-0.559238\pi\)
−0.185028 + 0.982733i \(0.559238\pi\)
\(350\) 0 0
\(351\) −3.69690e11 −1.30004
\(352\) −5.90499e10 −0.205011
\(353\) −1.30016e10 −0.0445667 −0.0222834 0.999752i \(-0.507094\pi\)
−0.0222834 + 0.999752i \(0.507094\pi\)
\(354\) 1.29108e11 0.436957
\(355\) 7.86955e11 2.62980
\(356\) 1.32962e11 0.438734
\(357\) 0 0
\(358\) 9.96812e10 0.320730
\(359\) −8.55526e10 −0.271837 −0.135918 0.990720i \(-0.543398\pi\)
−0.135918 + 0.990720i \(0.543398\pi\)
\(360\) −7.84195e10 −0.246072
\(361\) −2.61848e11 −0.811460
\(362\) −8.75161e10 −0.267855
\(363\) 8.35174e10 0.252463
\(364\) 0 0
\(365\) 9.34166e11 2.75490
\(366\) −2.26765e10 −0.0660558
\(367\) −1.13117e11 −0.325484 −0.162742 0.986669i \(-0.552034\pi\)
−0.162742 + 0.986669i \(0.552034\pi\)
\(368\) −1.12576e11 −0.319985
\(369\) 5.06588e10 0.142245
\(370\) −4.13647e11 −1.14742
\(371\) 0 0
\(372\) −1.38473e11 −0.374907
\(373\) −1.94719e11 −0.520857 −0.260428 0.965493i \(-0.583864\pi\)
−0.260428 + 0.965493i \(0.583864\pi\)
\(374\) −3.50177e11 −0.925477
\(375\) −1.03665e11 −0.270702
\(376\) −2.09687e11 −0.541037
\(377\) 7.18483e11 1.83181
\(378\) 0 0
\(379\) −2.22727e11 −0.554493 −0.277247 0.960799i \(-0.589422\pi\)
−0.277247 + 0.960799i \(0.589422\pi\)
\(380\) 1.32275e11 0.325424
\(381\) −5.56198e11 −1.35228
\(382\) −1.10958e11 −0.266607
\(383\) −3.36594e11 −0.799305 −0.399652 0.916667i \(-0.630869\pi\)
−0.399652 + 0.916667i \(0.630869\pi\)
\(384\) −2.75634e10 −0.0646908
\(385\) 0 0
\(386\) −6.67730e10 −0.153094
\(387\) 5.29023e10 0.119888
\(388\) −2.24230e11 −0.502286
\(389\) −3.47900e11 −0.770338 −0.385169 0.922846i \(-0.625857\pi\)
−0.385169 + 0.922846i \(0.625857\pi\)
\(390\) 4.29902e11 0.940976
\(391\) −6.67596e11 −1.44450
\(392\) 0 0
\(393\) −3.47877e11 −0.735630
\(394\) 5.80110e11 1.21277
\(395\) 4.74732e11 0.981210
\(396\) −1.31759e11 −0.269248
\(397\) 2.75509e10 0.0556645 0.0278323 0.999613i \(-0.491140\pi\)
0.0278323 + 0.999613i \(0.491140\pi\)
\(398\) −2.87333e11 −0.574001
\(399\) 0 0
\(400\) 1.59585e11 0.311689
\(401\) −6.21170e11 −1.19967 −0.599834 0.800124i \(-0.704767\pi\)
−0.599834 + 0.800124i \(0.704767\pi\)
\(402\) 1.67112e11 0.319147
\(403\) −6.58032e11 −1.24272
\(404\) −2.23647e9 −0.00417683
\(405\) 2.59752e11 0.479746
\(406\) 0 0
\(407\) −6.95003e11 −1.25549
\(408\) −1.63456e11 −0.292032
\(409\) 1.46395e10 0.0258686 0.0129343 0.999916i \(-0.495883\pi\)
0.0129343 + 0.999916i \(0.495883\pi\)
\(410\) −1.85779e11 −0.324690
\(411\) 4.87459e11 0.842656
\(412\) 2.45511e11 0.419792
\(413\) 0 0
\(414\) −2.51192e11 −0.420247
\(415\) −1.10359e12 −1.82638
\(416\) −1.30983e11 −0.214434
\(417\) 2.23680e11 0.362256
\(418\) 2.22245e11 0.356073
\(419\) 1.21505e11 0.192589 0.0962946 0.995353i \(-0.469301\pi\)
0.0962946 + 0.995353i \(0.469301\pi\)
\(420\) 0 0
\(421\) −5.61016e11 −0.870373 −0.435187 0.900340i \(-0.643318\pi\)
−0.435187 + 0.900340i \(0.643318\pi\)
\(422\) −1.01414e11 −0.155666
\(423\) −4.67879e11 −0.710562
\(424\) 3.33567e11 0.501229
\(425\) 9.46367e11 1.40705
\(426\) 6.17192e11 0.907981
\(427\) 0 0
\(428\) −3.80085e11 −0.547499
\(429\) 7.22313e11 1.02960
\(430\) −1.94006e11 −0.273658
\(431\) 1.04432e12 1.45776 0.728878 0.684644i \(-0.240042\pi\)
0.728878 + 0.684644i \(0.240042\pi\)
\(432\) −1.93956e11 −0.267934
\(433\) −1.05446e12 −1.44156 −0.720781 0.693163i \(-0.756216\pi\)
−0.720781 + 0.693163i \(0.756216\pi\)
\(434\) 0 0
\(435\) −1.23720e12 −1.65668
\(436\) −3.93037e11 −0.520888
\(437\) 4.23700e11 0.555767
\(438\) 7.32646e11 0.951176
\(439\) −1.17037e12 −1.50395 −0.751973 0.659194i \(-0.770897\pi\)
−0.751973 + 0.659194i \(0.770897\pi\)
\(440\) 4.83195e11 0.614589
\(441\) 0 0
\(442\) −7.76751e11 −0.968014
\(443\) −7.10660e11 −0.876688 −0.438344 0.898807i \(-0.644435\pi\)
−0.438344 + 0.898807i \(0.644435\pi\)
\(444\) −3.24415e11 −0.396166
\(445\) −1.08800e12 −1.31525
\(446\) 6.95594e11 0.832433
\(447\) −1.21224e12 −1.43616
\(448\) 0 0
\(449\) 9.40485e11 1.09205 0.546026 0.837768i \(-0.316140\pi\)
0.546026 + 0.837768i \(0.316140\pi\)
\(450\) 3.56084e11 0.409351
\(451\) −3.12142e11 −0.355270
\(452\) 8.42326e11 0.949199
\(453\) −6.48539e11 −0.723593
\(454\) 3.03180e11 0.334926
\(455\) 0 0
\(456\) 1.03740e11 0.112358
\(457\) −1.15083e12 −1.23420 −0.617102 0.786883i \(-0.711693\pi\)
−0.617102 + 0.786883i \(0.711693\pi\)
\(458\) −5.95805e11 −0.632716
\(459\) −1.15020e12 −1.20953
\(460\) 9.21188e11 0.959264
\(461\) −2.66471e11 −0.274786 −0.137393 0.990517i \(-0.543872\pi\)
−0.137393 + 0.990517i \(0.543872\pi\)
\(462\) 0 0
\(463\) −4.48766e11 −0.453843 −0.226921 0.973913i \(-0.572866\pi\)
−0.226921 + 0.973913i \(0.572866\pi\)
\(464\) 3.76949e11 0.377531
\(465\) 1.13310e12 1.12391
\(466\) −2.94140e11 −0.288946
\(467\) 1.77815e12 1.72999 0.864993 0.501783i \(-0.167323\pi\)
0.864993 + 0.501783i \(0.167323\pi\)
\(468\) −2.92263e11 −0.281623
\(469\) 0 0
\(470\) 1.71583e12 1.62194
\(471\) −1.05789e11 −0.0990478
\(472\) 3.21885e11 0.298512
\(473\) −3.25966e11 −0.299431
\(474\) 3.72322e11 0.338779
\(475\) −6.00627e11 −0.541357
\(476\) 0 0
\(477\) 7.44293e11 0.658281
\(478\) 5.84762e11 0.512334
\(479\) −1.98906e12 −1.72639 −0.863194 0.504872i \(-0.831540\pi\)
−0.863194 + 0.504872i \(0.831540\pi\)
\(480\) 2.25546e11 0.193932
\(481\) −1.54163e12 −1.31319
\(482\) −4.57448e11 −0.386038
\(483\) 0 0
\(484\) 2.08221e11 0.172472
\(485\) 1.83484e12 1.50577
\(486\) −7.28324e11 −0.592191
\(487\) 6.88592e11 0.554730 0.277365 0.960765i \(-0.410539\pi\)
0.277365 + 0.960765i \(0.410539\pi\)
\(488\) −5.65357e10 −0.0451267
\(489\) −3.43911e11 −0.271992
\(490\) 0 0
\(491\) 1.86881e12 1.45111 0.725553 0.688166i \(-0.241584\pi\)
0.725553 + 0.688166i \(0.241584\pi\)
\(492\) −1.45702e11 −0.112105
\(493\) 2.23538e12 1.70428
\(494\) 4.92977e11 0.372439
\(495\) 1.07816e12 0.807161
\(496\) −3.45234e11 −0.256121
\(497\) 0 0
\(498\) −8.65521e11 −0.630588
\(499\) −8.77411e11 −0.633506 −0.316753 0.948508i \(-0.602593\pi\)
−0.316753 + 0.948508i \(0.602593\pi\)
\(500\) −2.58452e11 −0.184933
\(501\) 1.67872e12 1.19044
\(502\) 1.60275e11 0.112642
\(503\) −2.12177e11 −0.147789 −0.0738945 0.997266i \(-0.523543\pi\)
−0.0738945 + 0.997266i \(0.523543\pi\)
\(504\) 0 0
\(505\) 1.83006e10 0.0125215
\(506\) 1.54776e12 1.04961
\(507\) 5.13323e11 0.345029
\(508\) −1.38668e12 −0.923826
\(509\) −1.82871e12 −1.20758 −0.603790 0.797144i \(-0.706343\pi\)
−0.603790 + 0.797144i \(0.706343\pi\)
\(510\) 1.33753e12 0.875464
\(511\) 0 0
\(512\) −6.87195e10 −0.0441942
\(513\) 7.29991e11 0.465361
\(514\) 2.71490e11 0.171561
\(515\) −2.00897e12 −1.25847
\(516\) −1.52155e11 −0.0944849
\(517\) 2.88291e12 1.77470
\(518\) 0 0
\(519\) −7.67134e11 −0.464107
\(520\) 1.07181e12 0.642837
\(521\) 2.68357e12 1.59567 0.797834 0.602877i \(-0.205979\pi\)
0.797834 + 0.602877i \(0.205979\pi\)
\(522\) 8.41093e11 0.495823
\(523\) 9.07444e11 0.530349 0.265175 0.964200i \(-0.414570\pi\)
0.265175 + 0.964200i \(0.414570\pi\)
\(524\) −8.67307e11 −0.502553
\(525\) 0 0
\(526\) −1.13151e11 −0.0644498
\(527\) −2.04730e12 −1.15620
\(528\) 3.78959e11 0.212197
\(529\) 1.14959e12 0.638251
\(530\) −2.72952e12 −1.50260
\(531\) 7.18227e11 0.392046
\(532\) 0 0
\(533\) −6.92384e11 −0.371599
\(534\) −8.53295e11 −0.454113
\(535\) 3.11017e12 1.64131
\(536\) 4.16635e11 0.218029
\(537\) −6.39715e11 −0.331972
\(538\) 5.91926e9 0.00304612
\(539\) 0 0
\(540\) 1.58711e12 0.803221
\(541\) −4.83134e11 −0.242482 −0.121241 0.992623i \(-0.538687\pi\)
−0.121241 + 0.992623i \(0.538687\pi\)
\(542\) −1.98861e12 −0.989814
\(543\) 5.61644e11 0.277244
\(544\) −4.07519e11 −0.199505
\(545\) 3.21615e12 1.56154
\(546\) 0 0
\(547\) 2.24340e12 1.07143 0.535716 0.844398i \(-0.320042\pi\)
0.535716 + 0.844398i \(0.320042\pi\)
\(548\) 1.21531e12 0.575669
\(549\) −1.26149e11 −0.0592664
\(550\) −2.19407e12 −1.02239
\(551\) −1.41872e12 −0.655714
\(552\) 7.22468e11 0.331202
\(553\) 0 0
\(554\) −2.71381e11 −0.122401
\(555\) 2.65463e12 1.18764
\(556\) 5.57666e11 0.247479
\(557\) 7.24610e11 0.318974 0.159487 0.987200i \(-0.449016\pi\)
0.159487 + 0.987200i \(0.449016\pi\)
\(558\) −7.70326e11 −0.336373
\(559\) −7.23047e11 −0.313194
\(560\) 0 0
\(561\) 2.24730e12 0.957917
\(562\) −1.86305e12 −0.787792
\(563\) −4.21854e12 −1.76960 −0.884799 0.465973i \(-0.845704\pi\)
−0.884799 + 0.465973i \(0.845704\pi\)
\(564\) 1.34569e12 0.560002
\(565\) −6.89260e12 −2.84554
\(566\) 4.24852e11 0.174006
\(567\) 0 0
\(568\) 1.53875e12 0.620297
\(569\) 4.34809e12 1.73898 0.869488 0.493954i \(-0.164449\pi\)
0.869488 + 0.493954i \(0.164449\pi\)
\(570\) −8.48886e11 −0.336831
\(571\) −2.24603e12 −0.884206 −0.442103 0.896964i \(-0.645767\pi\)
−0.442103 + 0.896964i \(0.645767\pi\)
\(572\) 1.80083e12 0.703381
\(573\) 7.12082e11 0.275952
\(574\) 0 0
\(575\) −4.18289e12 −1.59577
\(576\) −1.53335e11 −0.0580417
\(577\) −2.64828e12 −0.994655 −0.497328 0.867563i \(-0.665685\pi\)
−0.497328 + 0.867563i \(0.665685\pi\)
\(578\) −5.19260e11 −0.193513
\(579\) 4.28523e11 0.158460
\(580\) −3.08451e12 −1.13177
\(581\) 0 0
\(582\) 1.43902e12 0.519893
\(583\) −4.58608e12 −1.64412
\(584\) 1.82659e12 0.649806
\(585\) 2.39154e12 0.844259
\(586\) 1.50948e10 0.00528795
\(587\) −2.72201e12 −0.946276 −0.473138 0.880988i \(-0.656879\pi\)
−0.473138 + 0.880988i \(0.656879\pi\)
\(588\) 0 0
\(589\) 1.29935e12 0.444845
\(590\) −2.63393e12 −0.894890
\(591\) −3.72292e12 −1.25528
\(592\) −8.08812e11 −0.270645
\(593\) 2.15619e11 0.0716046 0.0358023 0.999359i \(-0.488601\pi\)
0.0358023 + 0.999359i \(0.488601\pi\)
\(594\) 2.66663e12 0.878870
\(595\) 0 0
\(596\) −3.02228e12 −0.981129
\(597\) 1.84399e12 0.594121
\(598\) 3.43320e12 1.09785
\(599\) −1.11903e12 −0.355156 −0.177578 0.984107i \(-0.556826\pi\)
−0.177578 + 0.984107i \(0.556826\pi\)
\(600\) −1.02415e12 −0.322614
\(601\) 2.89926e12 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(602\) 0 0
\(603\) 9.29643e11 0.286344
\(604\) −1.61690e12 −0.494330
\(605\) −1.70383e12 −0.517044
\(606\) 1.43528e10 0.00432324
\(607\) 1.76725e12 0.528384 0.264192 0.964470i \(-0.414895\pi\)
0.264192 + 0.964470i \(0.414895\pi\)
\(608\) 2.58639e11 0.0767586
\(609\) 0 0
\(610\) 4.62622e11 0.135283
\(611\) 6.39478e12 1.85627
\(612\) −9.09305e11 −0.262016
\(613\) −7.94293e11 −0.227200 −0.113600 0.993527i \(-0.536238\pi\)
−0.113600 + 0.993527i \(0.536238\pi\)
\(614\) −3.53068e11 −0.100254
\(615\) 1.19226e12 0.336071
\(616\) 0 0
\(617\) 3.17799e12 0.882815 0.441407 0.897307i \(-0.354479\pi\)
0.441407 + 0.897307i \(0.354479\pi\)
\(618\) −1.57559e12 −0.434507
\(619\) 6.59526e12 1.80561 0.902804 0.430051i \(-0.141505\pi\)
0.902804 + 0.430051i \(0.141505\pi\)
\(620\) 2.82499e12 0.767810
\(621\) 5.08381e12 1.37176
\(622\) 3.68388e12 0.986845
\(623\) 0 0
\(624\) 8.40594e11 0.221950
\(625\) −2.64113e12 −0.692357
\(626\) 2.52141e12 0.656234
\(627\) −1.42628e12 −0.368555
\(628\) −2.63746e11 −0.0676655
\(629\) −4.79640e12 −1.22177
\(630\) 0 0
\(631\) −5.22143e12 −1.31117 −0.655583 0.755123i \(-0.727577\pi\)
−0.655583 + 0.755123i \(0.727577\pi\)
\(632\) 9.28252e11 0.231440
\(633\) 6.50836e11 0.161122
\(634\) −3.86399e12 −0.949806
\(635\) 1.13470e13 2.76948
\(636\) −2.14070e12 −0.518798
\(637\) 0 0
\(638\) −5.18254e12 −1.23837
\(639\) 3.43343e12 0.814656
\(640\) 5.62319e11 0.132487
\(641\) 3.69727e12 0.865008 0.432504 0.901632i \(-0.357630\pi\)
0.432504 + 0.901632i \(0.357630\pi\)
\(642\) 2.43923e12 0.566691
\(643\) −5.23082e12 −1.20676 −0.603379 0.797455i \(-0.706179\pi\)
−0.603379 + 0.797455i \(0.706179\pi\)
\(644\) 0 0
\(645\) 1.24506e12 0.283250
\(646\) 1.53378e12 0.346510
\(647\) −2.92473e12 −0.656170 −0.328085 0.944648i \(-0.606403\pi\)
−0.328085 + 0.944648i \(0.606403\pi\)
\(648\) 5.07897e11 0.113159
\(649\) −4.42548e12 −0.979172
\(650\) −4.86681e12 −1.06939
\(651\) 0 0
\(652\) −8.57420e11 −0.185814
\(653\) −4.32042e11 −0.0929859 −0.0464930 0.998919i \(-0.514805\pi\)
−0.0464930 + 0.998919i \(0.514805\pi\)
\(654\) 2.52236e12 0.539146
\(655\) 7.09701e12 1.50657
\(656\) −3.63257e11 −0.0765854
\(657\) 4.07570e12 0.853411
\(658\) 0 0
\(659\) −6.16777e12 −1.27393 −0.636963 0.770894i \(-0.719810\pi\)
−0.636963 + 0.770894i \(0.719810\pi\)
\(660\) −3.10095e12 −0.636132
\(661\) −2.25373e12 −0.459193 −0.229596 0.973286i \(-0.573741\pi\)
−0.229596 + 0.973286i \(0.573741\pi\)
\(662\) −2.29659e12 −0.464752
\(663\) 4.98488e12 1.00194
\(664\) −2.15787e12 −0.430793
\(665\) 0 0
\(666\) −1.80471e12 −0.355447
\(667\) −9.88027e12 −1.93287
\(668\) 4.18528e12 0.813261
\(669\) −4.46405e12 −0.861611
\(670\) −3.40925e12 −0.653614
\(671\) 7.77289e11 0.148024
\(672\) 0 0
\(673\) 4.91498e12 0.923536 0.461768 0.887001i \(-0.347215\pi\)
0.461768 + 0.887001i \(0.347215\pi\)
\(674\) −2.16549e12 −0.404191
\(675\) −7.20668e12 −1.33619
\(676\) 1.27979e12 0.235710
\(677\) 1.00923e12 0.184647 0.0923233 0.995729i \(-0.470571\pi\)
0.0923233 + 0.995729i \(0.470571\pi\)
\(678\) −5.40571e12 −0.982470
\(679\) 0 0
\(680\) 3.33466e12 0.598082
\(681\) −1.94569e12 −0.346666
\(682\) 4.74649e12 0.840124
\(683\) −2.33611e12 −0.410772 −0.205386 0.978681i \(-0.565845\pi\)
−0.205386 + 0.978681i \(0.565845\pi\)
\(684\) 5.77105e11 0.100810
\(685\) −9.94462e12 −1.72576
\(686\) 0 0
\(687\) 3.82364e12 0.654895
\(688\) −3.79344e11 −0.0645483
\(689\) −1.01727e13 −1.71969
\(690\) −5.91182e12 −0.992888
\(691\) 1.10788e13 1.84860 0.924300 0.381666i \(-0.124650\pi\)
0.924300 + 0.381666i \(0.124650\pi\)
\(692\) −1.91257e12 −0.317059
\(693\) 0 0
\(694\) −6.19989e12 −1.01453
\(695\) −4.56328e12 −0.741901
\(696\) −2.41911e12 −0.390764
\(697\) −2.15418e12 −0.345728
\(698\) 1.64098e12 0.261669
\(699\) 1.88767e12 0.299075
\(700\) 0 0
\(701\) 8.11682e11 0.126956 0.0634782 0.997983i \(-0.479781\pi\)
0.0634782 + 0.997983i \(0.479781\pi\)
\(702\) 5.91504e12 0.919265
\(703\) 3.04411e12 0.470069
\(704\) 9.44799e11 0.144965
\(705\) −1.10115e13 −1.67879
\(706\) 2.08026e11 0.0315134
\(707\) 0 0
\(708\) −2.06573e12 −0.308976
\(709\) 8.96980e12 1.33314 0.666568 0.745444i \(-0.267762\pi\)
0.666568 + 0.745444i \(0.267762\pi\)
\(710\) −1.25913e13 −1.85955
\(711\) 2.07122e12 0.303958
\(712\) −2.12739e12 −0.310232
\(713\) 9.04897e12 1.31128
\(714\) 0 0
\(715\) −1.47359e13 −2.10862
\(716\) −1.59490e12 −0.226790
\(717\) −3.75277e12 −0.530293
\(718\) 1.36884e12 0.192218
\(719\) 1.05576e13 1.47328 0.736638 0.676287i \(-0.236412\pi\)
0.736638 + 0.676287i \(0.236412\pi\)
\(720\) 1.25471e12 0.173999
\(721\) 0 0
\(722\) 4.18957e12 0.573789
\(723\) 2.93572e12 0.399570
\(724\) 1.40026e12 0.189402
\(725\) 1.40060e13 1.88275
\(726\) −1.33628e12 −0.178518
\(727\) 2.48043e12 0.329323 0.164661 0.986350i \(-0.447347\pi\)
0.164661 + 0.986350i \(0.447347\pi\)
\(728\) 0 0
\(729\) 7.11476e12 0.933010
\(730\) −1.49467e13 −1.94801
\(731\) −2.24958e12 −0.291389
\(732\) 3.62824e11 0.0467085
\(733\) −9.95078e12 −1.27318 −0.636589 0.771203i \(-0.719655\pi\)
−0.636589 + 0.771203i \(0.719655\pi\)
\(734\) 1.80987e12 0.230152
\(735\) 0 0
\(736\) 1.80121e12 0.226264
\(737\) −5.72815e12 −0.715173
\(738\) −8.10540e11 −0.100582
\(739\) −3.52201e12 −0.434401 −0.217200 0.976127i \(-0.569692\pi\)
−0.217200 + 0.976127i \(0.569692\pi\)
\(740\) 6.61836e12 0.811349
\(741\) −3.16373e12 −0.385494
\(742\) 0 0
\(743\) −1.25259e13 −1.50786 −0.753930 0.656955i \(-0.771844\pi\)
−0.753930 + 0.656955i \(0.771844\pi\)
\(744\) 2.21558e12 0.265099
\(745\) 2.47307e13 2.94126
\(746\) 3.11550e12 0.368301
\(747\) −4.81488e12 −0.565774
\(748\) 5.60283e12 0.654411
\(749\) 0 0
\(750\) 1.65864e12 0.191415
\(751\) 6.91859e12 0.793666 0.396833 0.917891i \(-0.370109\pi\)
0.396833 + 0.917891i \(0.370109\pi\)
\(752\) 3.35500e12 0.382571
\(753\) −1.02858e12 −0.116590
\(754\) −1.14957e13 −1.29529
\(755\) 1.32308e13 1.48192
\(756\) 0 0
\(757\) −2.68669e12 −0.297362 −0.148681 0.988885i \(-0.547503\pi\)
−0.148681 + 0.988885i \(0.547503\pi\)
\(758\) 3.56363e12 0.392086
\(759\) −9.93294e12 −1.08640
\(760\) −2.11639e12 −0.230110
\(761\) 4.54213e12 0.490940 0.245470 0.969404i \(-0.421058\pi\)
0.245470 + 0.969404i \(0.421058\pi\)
\(762\) 8.89917e12 0.956208
\(763\) 0 0
\(764\) 1.77532e12 0.188520
\(765\) 7.44067e12 0.785481
\(766\) 5.38551e12 0.565194
\(767\) −9.81644e12 −1.02418
\(768\) 4.41014e11 0.0457433
\(769\) −2.34852e12 −0.242173 −0.121087 0.992642i \(-0.538638\pi\)
−0.121087 + 0.992642i \(0.538638\pi\)
\(770\) 0 0
\(771\) −1.74232e12 −0.177575
\(772\) 1.06837e12 0.108254
\(773\) 3.78193e11 0.0380983 0.0190492 0.999819i \(-0.493936\pi\)
0.0190492 + 0.999819i \(0.493936\pi\)
\(774\) −8.46436e11 −0.0847734
\(775\) −1.28276e13 −1.27728
\(776\) 3.58768e12 0.355170
\(777\) 0 0
\(778\) 5.56640e12 0.544711
\(779\) 1.36718e12 0.133017
\(780\) −6.87843e12 −0.665370
\(781\) −2.11556e13 −2.03468
\(782\) 1.06815e13 1.02142
\(783\) −1.70227e13 −1.61845
\(784\) 0 0
\(785\) 2.15818e12 0.202850
\(786\) 5.56603e12 0.520169
\(787\) 1.21085e13 1.12514 0.562568 0.826751i \(-0.309814\pi\)
0.562568 + 0.826751i \(0.309814\pi\)
\(788\) −9.28176e12 −0.857556
\(789\) 7.26157e11 0.0667089
\(790\) −7.59572e12 −0.693820
\(791\) 0 0
\(792\) 2.10814e12 0.190387
\(793\) 1.72416e12 0.154827
\(794\) −4.40815e11 −0.0393608
\(795\) 1.75170e13 1.55527
\(796\) 4.59733e12 0.405880
\(797\) 2.16394e13 1.89969 0.949846 0.312719i \(-0.101240\pi\)
0.949846 + 0.312719i \(0.101240\pi\)
\(798\) 0 0
\(799\) 1.98958e13 1.72703
\(800\) −2.55335e12 −0.220397
\(801\) −4.74687e12 −0.407438
\(802\) 9.93873e12 0.848293
\(803\) −2.51131e13 −2.13148
\(804\) −2.67380e12 −0.225671
\(805\) 0 0
\(806\) 1.05285e13 0.878738
\(807\) −3.79875e10 −0.00315289
\(808\) 3.57835e10 0.00295347
\(809\) −4.36869e12 −0.358578 −0.179289 0.983796i \(-0.557380\pi\)
−0.179289 + 0.983796i \(0.557380\pi\)
\(810\) −4.15603e12 −0.339231
\(811\) −1.39017e13 −1.12843 −0.564215 0.825628i \(-0.690821\pi\)
−0.564215 + 0.825628i \(0.690821\pi\)
\(812\) 0 0
\(813\) 1.27621e13 1.02451
\(814\) 1.11201e13 0.887763
\(815\) 7.01611e12 0.557041
\(816\) 2.61530e12 0.206498
\(817\) 1.42773e12 0.112111
\(818\) −2.34233e11 −0.0182919
\(819\) 0 0
\(820\) 2.97246e12 0.229591
\(821\) 8.03927e12 0.617551 0.308775 0.951135i \(-0.400081\pi\)
0.308775 + 0.951135i \(0.400081\pi\)
\(822\) −7.79935e12 −0.595848
\(823\) −1.95539e13 −1.48571 −0.742854 0.669454i \(-0.766528\pi\)
−0.742854 + 0.669454i \(0.766528\pi\)
\(824\) −3.92818e12 −0.296838
\(825\) 1.40807e13 1.05823
\(826\) 0 0
\(827\) 1.31390e13 0.976760 0.488380 0.872631i \(-0.337588\pi\)
0.488380 + 0.872631i \(0.337588\pi\)
\(828\) 4.01908e12 0.297160
\(829\) 2.18347e13 1.60565 0.802827 0.596211i \(-0.203328\pi\)
0.802827 + 0.596211i \(0.203328\pi\)
\(830\) 1.76574e13 1.29145
\(831\) 1.74161e12 0.126692
\(832\) 2.09572e12 0.151628
\(833\) 0 0
\(834\) −3.57888e12 −0.256154
\(835\) −3.42474e13 −2.43802
\(836\) −3.55593e12 −0.251782
\(837\) 1.55904e13 1.09798
\(838\) −1.94408e12 −0.136181
\(839\) −1.23225e13 −0.858561 −0.429281 0.903171i \(-0.641233\pi\)
−0.429281 + 0.903171i \(0.641233\pi\)
\(840\) 0 0
\(841\) 1.85760e13 1.28047
\(842\) 8.97625e12 0.615447
\(843\) 1.19563e13 0.815406
\(844\) 1.62263e12 0.110072
\(845\) −1.04723e13 −0.706620
\(846\) 7.48606e12 0.502443
\(847\) 0 0
\(848\) −5.33707e12 −0.354423
\(849\) −2.72654e12 −0.180105
\(850\) −1.51419e13 −0.994934
\(851\) 2.11999e13 1.38564
\(852\) −9.87506e12 −0.642040
\(853\) −9.07085e12 −0.586647 −0.293324 0.956013i \(-0.594761\pi\)
−0.293324 + 0.956013i \(0.594761\pi\)
\(854\) 0 0
\(855\) −4.72234e12 −0.302211
\(856\) 6.08136e12 0.387141
\(857\) −1.66440e12 −0.105401 −0.0527003 0.998610i \(-0.516783\pi\)
−0.0527003 + 0.998610i \(0.516783\pi\)
\(858\) −1.15570e13 −0.728036
\(859\) 1.40232e12 0.0878775 0.0439388 0.999034i \(-0.486009\pi\)
0.0439388 + 0.999034i \(0.486009\pi\)
\(860\) 3.10410e12 0.193505
\(861\) 0 0
\(862\) −1.67091e13 −1.03079
\(863\) −2.05982e12 −0.126410 −0.0632049 0.998001i \(-0.520132\pi\)
−0.0632049 + 0.998001i \(0.520132\pi\)
\(864\) 3.10330e12 0.189458
\(865\) 1.56502e13 0.950492
\(866\) 1.68713e13 1.01934
\(867\) 3.33240e12 0.200296
\(868\) 0 0
\(869\) −1.27622e13 −0.759165
\(870\) 1.97952e13 1.17145
\(871\) −1.27060e13 −0.748044
\(872\) 6.28859e12 0.368323
\(873\) 8.00526e12 0.466457
\(874\) −6.77921e12 −0.392986
\(875\) 0 0
\(876\) −1.17223e13 −0.672583
\(877\) 1.74285e13 0.994859 0.497429 0.867504i \(-0.334277\pi\)
0.497429 + 0.867504i \(0.334277\pi\)
\(878\) 1.87259e13 1.06345
\(879\) −9.68723e10 −0.00547331
\(880\) −7.73112e12 −0.434580
\(881\) 1.33834e11 0.00748471 0.00374235 0.999993i \(-0.498809\pi\)
0.00374235 + 0.999993i \(0.498809\pi\)
\(882\) 0 0
\(883\) 2.44009e12 0.135077 0.0675386 0.997717i \(-0.478485\pi\)
0.0675386 + 0.997717i \(0.478485\pi\)
\(884\) 1.24280e13 0.684489
\(885\) 1.69035e13 0.926258
\(886\) 1.13706e13 0.619912
\(887\) 1.36032e13 0.737877 0.368939 0.929454i \(-0.379721\pi\)
0.368939 + 0.929454i \(0.379721\pi\)
\(888\) 5.19063e12 0.280132
\(889\) 0 0
\(890\) 1.74080e13 0.930024
\(891\) −6.98289e12 −0.371181
\(892\) −1.11295e13 −0.588619
\(893\) −1.26272e13 −0.664468
\(894\) 1.93958e13 1.01552
\(895\) 1.30508e13 0.679880
\(896\) 0 0
\(897\) −2.20329e13 −1.13633
\(898\) −1.50478e13 −0.772197
\(899\) −3.02996e13 −1.54710
\(900\) −5.69734e12 −0.289455
\(901\) −3.16498e13 −1.59996
\(902\) 4.99428e12 0.251214
\(903\) 0 0
\(904\) −1.34772e13 −0.671185
\(905\) −1.14581e13 −0.567796
\(906\) 1.03766e13 0.511657
\(907\) −6.73549e12 −0.330473 −0.165237 0.986254i \(-0.552839\pi\)
−0.165237 + 0.986254i \(0.552839\pi\)
\(908\) −4.85087e12 −0.236828
\(909\) 7.98443e10 0.00387888
\(910\) 0 0
\(911\) 6.74134e12 0.324275 0.162137 0.986768i \(-0.448161\pi\)
0.162137 + 0.986768i \(0.448161\pi\)
\(912\) −1.65984e12 −0.0794492
\(913\) 2.96677e13 1.41308
\(914\) 1.84132e13 0.872714
\(915\) −2.96892e12 −0.140024
\(916\) 9.53287e12 0.447398
\(917\) 0 0
\(918\) 1.84032e13 0.855264
\(919\) −2.79444e13 −1.29233 −0.646167 0.763196i \(-0.723629\pi\)
−0.646167 + 0.763196i \(0.723629\pi\)
\(920\) −1.47390e13 −0.678302
\(921\) 2.26585e12 0.103768
\(922\) 4.26353e12 0.194303
\(923\) −4.69267e13 −2.12820
\(924\) 0 0
\(925\) −3.00524e13 −1.34971
\(926\) 7.18025e12 0.320915
\(927\) −8.76501e12 −0.389847
\(928\) −6.03119e12 −0.266954
\(929\) 1.03095e13 0.454118 0.227059 0.973881i \(-0.427089\pi\)
0.227059 + 0.973881i \(0.427089\pi\)
\(930\) −1.81297e13 −0.794724
\(931\) 0 0
\(932\) 4.70624e12 0.204316
\(933\) −2.36417e13 −1.02144
\(934\) −2.84504e13 −1.22329
\(935\) −4.58470e13 −1.96182
\(936\) 4.67621e12 0.199138
\(937\) −2.54975e13 −1.08061 −0.540306 0.841468i \(-0.681692\pi\)
−0.540306 + 0.841468i \(0.681692\pi\)
\(938\) 0 0
\(939\) −1.61814e13 −0.679236
\(940\) −2.74533e13 −1.14689
\(941\) −1.65839e13 −0.689498 −0.344749 0.938695i \(-0.612036\pi\)
−0.344749 + 0.938695i \(0.612036\pi\)
\(942\) 1.69262e12 0.0700373
\(943\) 9.52136e12 0.392100
\(944\) −5.15016e12 −0.211080
\(945\) 0 0
\(946\) 5.21546e12 0.211730
\(947\) 2.26232e13 0.914067 0.457034 0.889449i \(-0.348912\pi\)
0.457034 + 0.889449i \(0.348912\pi\)
\(948\) −5.95716e12 −0.239553
\(949\) −5.57051e13 −2.22945
\(950\) 9.61003e12 0.382797
\(951\) 2.47976e13 0.983099
\(952\) 0 0
\(953\) −3.02695e11 −0.0118874 −0.00594370 0.999982i \(-0.501892\pi\)
−0.00594370 + 0.999982i \(0.501892\pi\)
\(954\) −1.19087e13 −0.465475
\(955\) −1.45271e13 −0.565151
\(956\) −9.35618e12 −0.362275
\(957\) 3.32595e13 1.28177
\(958\) 3.18250e13 1.22074
\(959\) 0 0
\(960\) −3.60874e12 −0.137131
\(961\) 1.31065e12 0.0495715
\(962\) 2.46661e13 0.928566
\(963\) 1.35694e13 0.508444
\(964\) 7.31918e12 0.272970
\(965\) −8.74227e12 −0.324527
\(966\) 0 0
\(967\) 3.03988e12 0.111799 0.0558994 0.998436i \(-0.482197\pi\)
0.0558994 + 0.998436i \(0.482197\pi\)
\(968\) −3.33153e12 −0.121956
\(969\) −9.84316e12 −0.358656
\(970\) −2.93574e13 −1.06474
\(971\) −2.15206e13 −0.776906 −0.388453 0.921468i \(-0.626990\pi\)
−0.388453 + 0.921468i \(0.626990\pi\)
\(972\) 1.16532e13 0.418742
\(973\) 0 0
\(974\) −1.10175e13 −0.392254
\(975\) 3.12333e13 1.10687
\(976\) 9.04572e11 0.0319094
\(977\) −1.56454e13 −0.549364 −0.274682 0.961535i \(-0.588573\pi\)
−0.274682 + 0.961535i \(0.588573\pi\)
\(978\) 5.50258e12 0.192328
\(979\) 2.92486e13 1.01762
\(980\) 0 0
\(981\) 1.40318e13 0.483731
\(982\) −2.99010e13 −1.02609
\(983\) 2.28763e13 0.781437 0.390719 0.920510i \(-0.372227\pi\)
0.390719 + 0.920510i \(0.372227\pi\)
\(984\) 2.33124e12 0.0792699
\(985\) 7.59510e13 2.57081
\(986\) −3.57661e13 −1.20511
\(987\) 0 0
\(988\) −7.88763e12 −0.263354
\(989\) 9.94303e12 0.330472
\(990\) −1.72506e13 −0.570749
\(991\) 4.47356e13 1.47340 0.736701 0.676218i \(-0.236382\pi\)
0.736701 + 0.676218i \(0.236382\pi\)
\(992\) 5.52374e12 0.181105
\(993\) 1.47386e13 0.481043
\(994\) 0 0
\(995\) −3.76192e13 −1.21676
\(996\) 1.38483e13 0.445893
\(997\) −3.24564e13 −1.04033 −0.520166 0.854065i \(-0.674130\pi\)
−0.520166 + 0.854065i \(0.674130\pi\)
\(998\) 1.40386e13 0.447956
\(999\) 3.65251e13 1.16024
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.1.3 yes 4
7.2 even 3 98.10.c.m.67.2 8
7.3 odd 6 98.10.c.m.79.3 8
7.4 even 3 98.10.c.m.79.2 8
7.5 odd 6 98.10.c.m.67.3 8
7.6 odd 2 inner 98.10.a.k.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.10.a.k.1.2 4 7.6 odd 2 inner
98.10.a.k.1.3 yes 4 1.1 even 1 trivial
98.10.c.m.67.2 8 7.2 even 3
98.10.c.m.67.3 8 7.5 odd 6
98.10.c.m.79.2 8 7.4 even 3
98.10.c.m.79.3 8 7.3 odd 6