Properties

Label 98.10.a.k.1.2
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 1.2
Root \(33.7881\) 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 \(-0.619255\pi\)
−0.365946 + 0.930636i \(0.619255\pi\)
\(4\) 256.000 0.500000
\(5\) 2094.80 1.49892 0.749459 0.662051i \(-0.230314\pi\)
0.749459 + 0.662051i \(0.230314\pi\)
\(6\) 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 \(-0.707431\pi\)
−0.606510 + 0.795076i \(0.707431\pi\)
\(14\) 0 0
\(15\) −215098. −1.09705
\(16\) 65536.0 0.250000
\(17\) −388641. −1.12857 −0.564285 0.825580i \(-0.690848\pi\)
−0.564285 + 0.825580i \(0.690848\pi\)
\(18\) 146232. 0.328333
\(19\) 246657. 0.434212 0.217106 0.976148i \(-0.430338\pi\)
0.217106 + 0.976148i \(0.430338\pi\)
\(20\) 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 \(-0.328815\pi\)
0.512243 + 0.858841i \(0.328815\pi\)
\(32\) −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 \(-0.451052\pi\)
0.153171 + 0.988200i \(0.451052\pi\)
\(42\) 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 \(-0.777331\pi\)
−0.765142 + 0.643861i \(0.777331\pi\)
\(48\) −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 \(-0.361272\pi\)
0.422160 + 0.906522i \(0.361272\pi\)
\(60\) −5.50650e7 −0.548523
\(61\) −1.38027e7 −0.127638 −0.0638188 0.997962i \(-0.520328\pi\)
−0.0638188 + 0.997962i \(0.520328\pi\)
\(62\) −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 \(-0.129027\pi\)
0.918964 + 0.394341i \(0.129027\pi\)
\(74\) 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 \(-0.708522\pi\)
−0.609233 + 0.792992i \(0.708522\pi\)
\(84\) 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 \(-0.644573\pi\)
−0.438734 + 0.898617i \(0.644573\pi\)
\(90\) 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 \(-0.332492\pi\)
0.502286 + 0.864701i \(0.332492\pi\)
\(98\) 0 0
\(99\) −5.14684e8 −0.538495
\(100\) 6.23378e8 0.623378
\(101\) 8.73621e6 0.00835366 0.00417683 0.999991i \(-0.498670\pi\)
0.00417683 + 0.999991i \(0.498670\pi\)
\(102\) −6.38501e8 −0.584064
\(103\) −9.59029e8 −0.839584 −0.419792 0.907620i \(-0.637897\pi\)
−0.419792 + 0.907620i \(0.637897\pi\)
\(104\) 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 \(-0.332394\pi\)
0.502553 + 0.864546i \(0.332394\pi\)
\(132\) −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 \(-0.579602\pi\)
−0.247479 + 0.968893i \(0.579602\pi\)
\(140\) 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 \(-0.478445\pi\)
0.0676655 + 0.997708i \(0.478445\pi\)
\(158\) 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 \(-0.802310\pi\)
−0.813261 + 0.581899i \(0.802310\pi\)
\(168\) 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 \(-0.397305\pi\)
0.317059 + 0.948406i \(0.397305\pi\)
\(174\) 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 \(-0.560655\pi\)
−0.189402 + 0.981900i \(0.560655\pi\)
\(182\) 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 \(-0.633035\pi\)
−0.405880 + 0.913927i \(0.633035\pi\)
\(200\) −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 \(-0.299672\pi\)
0.588619 + 0.808411i \(0.299672\pi\)
\(224\) 0 0
\(225\) −2.22552e10 −0.578910
\(226\) −5.26454e10 −1.34237
\(227\) 1.89487e10 0.473657 0.236828 0.971552i \(-0.423892\pi\)
0.236828 + 0.971552i \(0.423892\pi\)
\(228\) −6.48375e9 −0.158898
\(229\) −3.72378e10 −0.894796 −0.447398 0.894335i \(-0.647649\pi\)
−0.447398 + 0.894335i \(0.647649\pi\)
\(230\) 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 \(-0.588006\pi\)
−0.272970 + 0.962022i \(0.588006\pi\)
\(242\) −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 \(-0.474620\pi\)
0.0796499 + 0.996823i \(0.474620\pi\)
\(252\) 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 \(-0.461290\pi\)
0.121312 + 0.992614i \(0.461290\pi\)
\(258\) −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 \(-0.499314\pi\)
0.00215393 + 0.999998i \(0.499314\pi\)
\(270\) −9.91944e10 −1.13593
\(271\) −1.24288e11 −1.39981 −0.699904 0.714237i \(-0.746774\pi\)
−0.699904 + 0.714237i \(0.746774\pi\)
\(272\) −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 \(-0.460735\pi\)
0.123041 + 0.992402i \(0.460735\pi\)
\(284\) −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 \(-0.498810\pi\)
0.00373915 + 0.999993i \(0.498810\pi\)
\(294\) 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 \(-0.522584\pi\)
−0.0708902 + 0.997484i \(0.522584\pi\)
\(308\) 0 0
\(309\) 9.84747e10 0.614485
\(310\) −1.76562e11 −1.08585
\(311\) 2.30243e11 1.39561 0.697805 0.716288i \(-0.254160\pi\)
0.697805 + 0.716288i \(0.254160\pi\)
\(312\) −5.25371e10 −0.313885
\(313\) 1.57588e11 0.928055 0.464027 0.885821i \(-0.346404\pi\)
0.464027 + 0.885821i \(0.346404\pi\)
\(314\) −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 \(-0.440762\pi\)
0.185028 + 0.982733i \(0.440762\pi\)
\(350\) 0 0
\(351\) −3.69690e11 −1.30004
\(352\) −5.90499e10 −0.205011
\(353\) 1.30016e10 0.0445667 0.0222834 0.999752i \(-0.492906\pi\)
0.0222834 + 0.999752i \(0.492906\pi\)
\(354\) 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 \(-0.447966\pi\)
0.162742 + 0.986669i \(0.447966\pi\)
\(368\) −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 \(-0.369131\pi\)
0.399652 + 0.916667i \(0.369131\pi\)
\(384\) 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 \(-0.508860\pi\)
−0.0278323 + 0.999613i \(0.508860\pi\)
\(398\) 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 \(-0.504117\pi\)
−0.0129343 + 0.999916i \(0.504117\pi\)
\(410\) −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 \(-0.530699\pi\)
−0.0962946 + 0.995353i \(0.530699\pi\)
\(420\) 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 \(-0.243784\pi\)
0.720781 + 0.693163i \(0.243784\pi\)
\(434\) 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 \(-0.229103\pi\)
0.751973 + 0.659194i \(0.229103\pi\)
\(440\) −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 \(-0.456128\pi\)
0.137393 + 0.990517i \(0.456128\pi\)
\(462\) 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 \(-0.832677\pi\)
−0.864993 + 0.501783i \(0.832677\pi\)
\(468\) 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 \(-0.168460\pi\)
0.863194 + 0.504872i \(0.168460\pi\)
\(480\) 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 \(-0.476457\pi\)
0.0738945 + 0.997266i \(0.476457\pi\)
\(504\) 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 \(-0.293657\pi\)
0.603790 + 0.797144i \(0.293657\pi\)
\(510\) −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 \(-0.794021\pi\)
−0.797834 + 0.602877i \(0.794021\pi\)
\(522\) 8.41093e11 0.495823
\(523\) −9.07444e11 −0.530349 −0.265175 0.964200i \(-0.585430\pi\)
−0.265175 + 0.964200i \(0.585430\pi\)
\(524\) 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 \(-0.154296\pi\)
0.884799 + 0.465973i \(0.154296\pi\)
\(564\) 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 \(-0.334315\pi\)
0.497328 + 0.867563i \(0.334315\pi\)
\(578\) −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 \(-0.343121\pi\)
0.473138 + 0.880988i \(0.343121\pi\)
\(588\) 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 \(-0.511399\pi\)
−0.0358023 + 0.999359i \(0.511399\pi\)
\(594\) −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 \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(602\) 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 \(-0.585105\pi\)
−0.264192 + 0.964470i \(0.585105\pi\)
\(608\) −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 \(-0.858495\pi\)
−0.902804 + 0.430051i \(0.858495\pi\)
\(620\) 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 \(-0.293821\pi\)
0.603379 + 0.797455i \(0.293821\pi\)
\(644\) 0 0
\(645\) 1.24506e12 0.283250
\(646\) 1.53378e12 0.346510
\(647\) 2.92473e12 0.656170 0.328085 0.944648i \(-0.393597\pi\)
0.328085 + 0.944648i \(0.393597\pi\)
\(648\) 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 \(-0.426259\pi\)
0.229596 + 0.973286i \(0.426259\pi\)
\(662\) −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 \(-0.529429\pi\)
−0.0923233 + 0.995729i \(0.529429\pi\)
\(678\) 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 \(-0.875350\pi\)
−0.924300 + 0.381666i \(0.875350\pi\)
\(692\) 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 \(-0.763588\pi\)
−0.736638 + 0.676287i \(0.763588\pi\)
\(720\) −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 \(-0.552653\pi\)
−0.164661 + 0.986350i \(0.552653\pi\)
\(728\) 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 \(-0.280345\pi\)
0.636589 + 0.771203i \(0.280345\pi\)
\(734\) −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 \(-0.578942\pi\)
−0.245470 + 0.969404i \(0.578942\pi\)
\(762\) −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 \(-0.461362\pi\)
0.121087 + 0.992642i \(0.461362\pi\)
\(770\) 0 0
\(771\) −1.74232e12 −0.177575
\(772\) 1.06837e12 0.108254
\(773\) −3.78193e11 −0.0380983 −0.0190492 0.999819i \(-0.506064\pi\)
−0.0190492 + 0.999819i \(0.506064\pi\)
\(774\) −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 \(-0.690186\pi\)
−0.562568 + 0.826751i \(0.690186\pi\)
\(788\) −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 \(-0.898760\pi\)
−0.949846 + 0.312719i \(0.898760\pi\)
\(798\) 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 \(-0.309179\pi\)
0.564215 + 0.825628i \(0.309179\pi\)
\(812\) 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 \(-0.796672\pi\)
−0.802827 + 0.596211i \(0.796672\pi\)
\(830\) 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 \(-0.358767\pi\)
0.429281 + 0.903171i \(0.358767\pi\)
\(840\) 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 \(-0.405239\pi\)
0.293324 + 0.956013i \(0.405239\pi\)
\(854\) 0 0
\(855\) −4.72234e12 −0.302211
\(856\) 6.08136e12 0.387141
\(857\) 1.66440e12 0.105401 0.0527003 0.998610i \(-0.483217\pi\)
0.0527003 + 0.998610i \(0.483217\pi\)
\(858\) −1.15570e13 −0.728036
\(859\) −1.40232e12 −0.0878775 −0.0439388 0.999034i \(-0.513991\pi\)
−0.0439388 + 0.999034i \(0.513991\pi\)
\(860\) −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 \(-0.501191\pi\)
−0.00374235 + 0.999993i \(0.501191\pi\)
\(882\) 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 \(-0.620279\pi\)
−0.368939 + 0.929454i \(0.620279\pi\)
\(888\) −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 \(-0.572911\pi\)
−0.227059 + 0.973881i \(0.572911\pi\)
\(930\) 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 \(-0.318308\pi\)
0.540306 + 0.841468i \(0.318308\pi\)
\(938\) 0 0
\(939\) −1.61814e13 −0.679236
\(940\) −2.74533e13 −1.14689
\(941\) 1.65839e13 0.689498 0.344749 0.938695i \(-0.387964\pi\)
0.344749 + 0.938695i \(0.387964\pi\)
\(942\) 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 \(-0.373010\pi\)
0.388453 + 0.921468i \(0.373010\pi\)
\(972\) −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 \(-0.627773\pi\)
−0.390719 + 0.920510i \(0.627773\pi\)
\(984\) 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 \(-0.325870\pi\)
0.520166 + 0.854065i \(0.325870\pi\)
\(998\) 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.1.2 4
7.2 even 3 98.10.c.m.67.3 8
7.3 odd 6 98.10.c.m.79.2 8
7.4 even 3 98.10.c.m.79.3 8
7.5 odd 6 98.10.c.m.67.2 8
7.6 odd 2 inner 98.10.a.k.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.10.a.k.1.2 4 1.1 even 1 trivial
98.10.a.k.1.3 yes 4 7.6 odd 2 inner
98.10.c.m.67.2 8 7.5 odd 6
98.10.c.m.67.3 8 7.2 even 3
98.10.c.m.79.2 8 7.3 odd 6
98.10.c.m.79.3 8 7.4 even 3