Properties

Label 7098.2.a.ci.1.3
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.69202 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.69202 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.69202 q^{10} +3.15883 q^{11} -1.00000 q^{12} -1.00000 q^{14} -1.69202 q^{15} +1.00000 q^{16} +7.74094 q^{17} +1.00000 q^{18} +3.15883 q^{19} +1.69202 q^{20} +1.00000 q^{21} +3.15883 q^{22} +7.89977 q^{23} -1.00000 q^{24} -2.13706 q^{25} -1.00000 q^{27} -1.00000 q^{28} -4.96077 q^{29} -1.69202 q^{30} +4.82908 q^{31} +1.00000 q^{32} -3.15883 q^{33} +7.74094 q^{34} -1.69202 q^{35} +1.00000 q^{36} -0.185981 q^{37} +3.15883 q^{38} +1.69202 q^{40} +0.978230 q^{41} +1.00000 q^{42} -8.19806 q^{43} +3.15883 q^{44} +1.69202 q^{45} +7.89977 q^{46} +2.78017 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.13706 q^{50} -7.74094 q^{51} -0.652793 q^{53} -1.00000 q^{54} +5.34481 q^{55} -1.00000 q^{56} -3.15883 q^{57} -4.96077 q^{58} -4.80194 q^{59} -1.69202 q^{60} +3.21983 q^{61} +4.82908 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.15883 q^{66} -4.52111 q^{67} +7.74094 q^{68} -7.89977 q^{69} -1.69202 q^{70} -1.20775 q^{71} +1.00000 q^{72} -14.1564 q^{73} -0.185981 q^{74} +2.13706 q^{75} +3.15883 q^{76} -3.15883 q^{77} -7.33944 q^{79} +1.69202 q^{80} +1.00000 q^{81} +0.978230 q^{82} -8.32304 q^{83} +1.00000 q^{84} +13.0978 q^{85} -8.19806 q^{86} +4.96077 q^{87} +3.15883 q^{88} +13.5526 q^{89} +1.69202 q^{90} +7.89977 q^{92} -4.82908 q^{93} +2.78017 q^{94} +5.34481 q^{95} -1.00000 q^{96} -7.83340 q^{97} +1.00000 q^{98} +3.15883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + q^{11} - 3q^{12} - 3q^{14} + 3q^{16} + 9q^{17} + 3q^{18} + q^{19} + 3q^{21} + q^{22} + q^{23} - 3q^{24} - q^{25} - 3q^{27} - 3q^{28} - 2q^{29} + 4q^{31} + 3q^{32} - q^{33} + 9q^{34} + 3q^{36} + 14q^{37} + q^{38} + 6q^{41} + 3q^{42} - 29q^{43} + q^{44} + q^{46} + 7q^{47} - 3q^{48} + 3q^{49} - q^{50} - 9q^{51} + 16q^{53} - 3q^{54} - 7q^{55} - 3q^{56} - q^{57} - 2q^{58} - 10q^{59} + 11q^{61} + 4q^{62} - 3q^{63} + 3q^{64} - q^{66} + 2q^{67} + 9q^{68} - q^{69} + 14q^{71} + 3q^{72} + 7q^{73} + 14q^{74} + q^{75} + q^{76} - q^{77} - 2q^{79} + 3q^{81} + 6q^{82} - 5q^{83} + 3q^{84} + 21q^{85} - 29q^{86} + 2q^{87} + q^{88} + q^{92} - 4q^{93} + 7q^{94} - 7q^{95} - 3q^{96} + 6q^{97} + 3q^{98} + q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.69202 0.756695 0.378348 0.925664i \(-0.376492\pi\)
0.378348 + 0.925664i \(0.376492\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.69202 0.535064
\(11\) 3.15883 0.952424 0.476212 0.879330i \(-0.342009\pi\)
0.476212 + 0.879330i \(0.342009\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −1.69202 −0.436878
\(16\) 1.00000 0.250000
\(17\) 7.74094 1.87745 0.938727 0.344662i \(-0.112007\pi\)
0.938727 + 0.344662i \(0.112007\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.15883 0.724686 0.362343 0.932045i \(-0.381977\pi\)
0.362343 + 0.932045i \(0.381977\pi\)
\(20\) 1.69202 0.378348
\(21\) 1.00000 0.218218
\(22\) 3.15883 0.673466
\(23\) 7.89977 1.64722 0.823608 0.567159i \(-0.191958\pi\)
0.823608 + 0.567159i \(0.191958\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.13706 −0.427413
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −4.96077 −0.921192 −0.460596 0.887610i \(-0.652364\pi\)
−0.460596 + 0.887610i \(0.652364\pi\)
\(30\) −1.69202 −0.308919
\(31\) 4.82908 0.867329 0.433665 0.901074i \(-0.357220\pi\)
0.433665 + 0.901074i \(0.357220\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.15883 −0.549882
\(34\) 7.74094 1.32756
\(35\) −1.69202 −0.286004
\(36\) 1.00000 0.166667
\(37\) −0.185981 −0.0305750 −0.0152875 0.999883i \(-0.504866\pi\)
−0.0152875 + 0.999883i \(0.504866\pi\)
\(38\) 3.15883 0.512430
\(39\) 0 0
\(40\) 1.69202 0.267532
\(41\) 0.978230 0.152774 0.0763869 0.997078i \(-0.475662\pi\)
0.0763869 + 0.997078i \(0.475662\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.19806 −1.25019 −0.625096 0.780548i \(-0.714940\pi\)
−0.625096 + 0.780548i \(0.714940\pi\)
\(44\) 3.15883 0.476212
\(45\) 1.69202 0.252232
\(46\) 7.89977 1.16476
\(47\) 2.78017 0.405529 0.202765 0.979228i \(-0.435007\pi\)
0.202765 + 0.979228i \(0.435007\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −2.13706 −0.302226
\(51\) −7.74094 −1.08395
\(52\) 0 0
\(53\) −0.652793 −0.0896680 −0.0448340 0.998994i \(-0.514276\pi\)
−0.0448340 + 0.998994i \(0.514276\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.34481 0.720695
\(56\) −1.00000 −0.133631
\(57\) −3.15883 −0.418398
\(58\) −4.96077 −0.651381
\(59\) −4.80194 −0.625159 −0.312580 0.949892i \(-0.601193\pi\)
−0.312580 + 0.949892i \(0.601193\pi\)
\(60\) −1.69202 −0.218439
\(61\) 3.21983 0.412257 0.206129 0.978525i \(-0.433913\pi\)
0.206129 + 0.978525i \(0.433913\pi\)
\(62\) 4.82908 0.613294
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.15883 −0.388826
\(67\) −4.52111 −0.552341 −0.276171 0.961109i \(-0.589065\pi\)
−0.276171 + 0.961109i \(0.589065\pi\)
\(68\) 7.74094 0.938727
\(69\) −7.89977 −0.951021
\(70\) −1.69202 −0.202235
\(71\) −1.20775 −0.143334 −0.0716668 0.997429i \(-0.522832\pi\)
−0.0716668 + 0.997429i \(0.522832\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.1564 −1.65689 −0.828443 0.560073i \(-0.810773\pi\)
−0.828443 + 0.560073i \(0.810773\pi\)
\(74\) −0.185981 −0.0216198
\(75\) 2.13706 0.246767
\(76\) 3.15883 0.362343
\(77\) −3.15883 −0.359982
\(78\) 0 0
\(79\) −7.33944 −0.825751 −0.412876 0.910787i \(-0.635476\pi\)
−0.412876 + 0.910787i \(0.635476\pi\)
\(80\) 1.69202 0.189174
\(81\) 1.00000 0.111111
\(82\) 0.978230 0.108027
\(83\) −8.32304 −0.913573 −0.456786 0.889576i \(-0.651000\pi\)
−0.456786 + 0.889576i \(0.651000\pi\)
\(84\) 1.00000 0.109109
\(85\) 13.0978 1.42066
\(86\) −8.19806 −0.884020
\(87\) 4.96077 0.531851
\(88\) 3.15883 0.336733
\(89\) 13.5526 1.43657 0.718285 0.695749i \(-0.244928\pi\)
0.718285 + 0.695749i \(0.244928\pi\)
\(90\) 1.69202 0.178355
\(91\) 0 0
\(92\) 7.89977 0.823608
\(93\) −4.82908 −0.500753
\(94\) 2.78017 0.286752
\(95\) 5.34481 0.548366
\(96\) −1.00000 −0.102062
\(97\) −7.83340 −0.795361 −0.397680 0.917524i \(-0.630185\pi\)
−0.397680 + 0.917524i \(0.630185\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.15883 0.317475
\(100\) −2.13706 −0.213706
\(101\) 15.2228 1.51473 0.757363 0.652994i \(-0.226487\pi\)
0.757363 + 0.652994i \(0.226487\pi\)
\(102\) −7.74094 −0.766467
\(103\) −11.6528 −1.14818 −0.574092 0.818791i \(-0.694645\pi\)
−0.574092 + 0.818791i \(0.694645\pi\)
\(104\) 0 0
\(105\) 1.69202 0.165124
\(106\) −0.652793 −0.0634048
\(107\) 14.5797 1.40947 0.704737 0.709469i \(-0.251065\pi\)
0.704737 + 0.709469i \(0.251065\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.29590 0.603038 0.301519 0.953460i \(-0.402506\pi\)
0.301519 + 0.953460i \(0.402506\pi\)
\(110\) 5.34481 0.509608
\(111\) 0.185981 0.0176525
\(112\) −1.00000 −0.0944911
\(113\) 15.1860 1.42858 0.714288 0.699851i \(-0.246750\pi\)
0.714288 + 0.699851i \(0.246750\pi\)
\(114\) −3.15883 −0.295852
\(115\) 13.3666 1.24644
\(116\) −4.96077 −0.460596
\(117\) 0 0
\(118\) −4.80194 −0.442054
\(119\) −7.74094 −0.709611
\(120\) −1.69202 −0.154460
\(121\) −1.02177 −0.0928882
\(122\) 3.21983 0.291510
\(123\) −0.978230 −0.0882040
\(124\) 4.82908 0.433665
\(125\) −12.0761 −1.08012
\(126\) −1.00000 −0.0890871
\(127\) 11.8629 1.05267 0.526333 0.850279i \(-0.323567\pi\)
0.526333 + 0.850279i \(0.323567\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.19806 0.721799
\(130\) 0 0
\(131\) −1.56465 −0.136704 −0.0683519 0.997661i \(-0.521774\pi\)
−0.0683519 + 0.997661i \(0.521774\pi\)
\(132\) −3.15883 −0.274941
\(133\) −3.15883 −0.273906
\(134\) −4.52111 −0.390564
\(135\) −1.69202 −0.145626
\(136\) 7.74094 0.663780
\(137\) 20.5700 1.75742 0.878708 0.477360i \(-0.158406\pi\)
0.878708 + 0.477360i \(0.158406\pi\)
\(138\) −7.89977 −0.672473
\(139\) −22.5308 −1.91104 −0.955519 0.294931i \(-0.904703\pi\)
−0.955519 + 0.294931i \(0.904703\pi\)
\(140\) −1.69202 −0.143002
\(141\) −2.78017 −0.234132
\(142\) −1.20775 −0.101352
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.39373 −0.697061
\(146\) −14.1564 −1.17160
\(147\) −1.00000 −0.0824786
\(148\) −0.185981 −0.0152875
\(149\) −1.60925 −0.131835 −0.0659175 0.997825i \(-0.520997\pi\)
−0.0659175 + 0.997825i \(0.520997\pi\)
\(150\) 2.13706 0.174490
\(151\) 13.1709 1.07183 0.535917 0.844271i \(-0.319966\pi\)
0.535917 + 0.844271i \(0.319966\pi\)
\(152\) 3.15883 0.256215
\(153\) 7.74094 0.625818
\(154\) −3.15883 −0.254546
\(155\) 8.17092 0.656304
\(156\) 0 0
\(157\) −2.61596 −0.208776 −0.104388 0.994537i \(-0.533288\pi\)
−0.104388 + 0.994537i \(0.533288\pi\)
\(158\) −7.33944 −0.583894
\(159\) 0.652793 0.0517698
\(160\) 1.69202 0.133766
\(161\) −7.89977 −0.622589
\(162\) 1.00000 0.0785674
\(163\) 0.853248 0.0668315 0.0334158 0.999442i \(-0.489361\pi\)
0.0334158 + 0.999442i \(0.489361\pi\)
\(164\) 0.978230 0.0763869
\(165\) −5.34481 −0.416093
\(166\) −8.32304 −0.645993
\(167\) 14.8944 1.15256 0.576281 0.817251i \(-0.304503\pi\)
0.576281 + 0.817251i \(0.304503\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 13.0978 1.00456
\(171\) 3.15883 0.241562
\(172\) −8.19806 −0.625096
\(173\) −7.28621 −0.553960 −0.276980 0.960876i \(-0.589334\pi\)
−0.276980 + 0.960876i \(0.589334\pi\)
\(174\) 4.96077 0.376075
\(175\) 2.13706 0.161547
\(176\) 3.15883 0.238106
\(177\) 4.80194 0.360936
\(178\) 13.5526 1.01581
\(179\) −4.10992 −0.307190 −0.153595 0.988134i \(-0.549085\pi\)
−0.153595 + 0.988134i \(0.549085\pi\)
\(180\) 1.69202 0.126116
\(181\) −8.63640 −0.641939 −0.320969 0.947090i \(-0.604009\pi\)
−0.320969 + 0.947090i \(0.604009\pi\)
\(182\) 0 0
\(183\) −3.21983 −0.238017
\(184\) 7.89977 0.582379
\(185\) −0.314683 −0.0231360
\(186\) −4.82908 −0.354086
\(187\) 24.4523 1.78813
\(188\) 2.78017 0.202765
\(189\) 1.00000 0.0727393
\(190\) 5.34481 0.387754
\(191\) 26.9922 1.95309 0.976545 0.215315i \(-0.0690779\pi\)
0.976545 + 0.215315i \(0.0690779\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 16.2174 1.16736 0.583678 0.811985i \(-0.301613\pi\)
0.583678 + 0.811985i \(0.301613\pi\)
\(194\) −7.83340 −0.562405
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −21.6625 −1.54339 −0.771694 0.635994i \(-0.780590\pi\)
−0.771694 + 0.635994i \(0.780590\pi\)
\(198\) 3.15883 0.224489
\(199\) −8.81700 −0.625021 −0.312510 0.949914i \(-0.601170\pi\)
−0.312510 + 0.949914i \(0.601170\pi\)
\(200\) −2.13706 −0.151113
\(201\) 4.52111 0.318894
\(202\) 15.2228 1.07107
\(203\) 4.96077 0.348178
\(204\) −7.74094 −0.541974
\(205\) 1.65519 0.115603
\(206\) −11.6528 −0.811889
\(207\) 7.89977 0.549072
\(208\) 0 0
\(209\) 9.97823 0.690209
\(210\) 1.69202 0.116761
\(211\) 0.753020 0.0518401 0.0259200 0.999664i \(-0.491748\pi\)
0.0259200 + 0.999664i \(0.491748\pi\)
\(212\) −0.652793 −0.0448340
\(213\) 1.20775 0.0827537
\(214\) 14.5797 0.996649
\(215\) −13.8713 −0.946015
\(216\) −1.00000 −0.0680414
\(217\) −4.82908 −0.327820
\(218\) 6.29590 0.426412
\(219\) 14.1564 0.956604
\(220\) 5.34481 0.360347
\(221\) 0 0
\(222\) 0.185981 0.0124822
\(223\) −2.14377 −0.143557 −0.0717787 0.997421i \(-0.522868\pi\)
−0.0717787 + 0.997421i \(0.522868\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.13706 −0.142471
\(226\) 15.1860 1.01016
\(227\) 21.3327 1.41590 0.707952 0.706261i \(-0.249619\pi\)
0.707952 + 0.706261i \(0.249619\pi\)
\(228\) −3.15883 −0.209199
\(229\) −15.3502 −1.01437 −0.507185 0.861837i \(-0.669314\pi\)
−0.507185 + 0.861837i \(0.669314\pi\)
\(230\) 13.3666 0.881366
\(231\) 3.15883 0.207836
\(232\) −4.96077 −0.325691
\(233\) 23.1468 1.51639 0.758197 0.652025i \(-0.226080\pi\)
0.758197 + 0.652025i \(0.226080\pi\)
\(234\) 0 0
\(235\) 4.70410 0.306862
\(236\) −4.80194 −0.312580
\(237\) 7.33944 0.476748
\(238\) −7.74094 −0.501771
\(239\) −17.8538 −1.15487 −0.577434 0.816437i \(-0.695946\pi\)
−0.577434 + 0.816437i \(0.695946\pi\)
\(240\) −1.69202 −0.109220
\(241\) 16.4179 1.05757 0.528785 0.848756i \(-0.322648\pi\)
0.528785 + 0.848756i \(0.322648\pi\)
\(242\) −1.02177 −0.0656819
\(243\) −1.00000 −0.0641500
\(244\) 3.21983 0.206129
\(245\) 1.69202 0.108099
\(246\) −0.978230 −0.0623696
\(247\) 0 0
\(248\) 4.82908 0.306647
\(249\) 8.32304 0.527451
\(250\) −12.0761 −0.763757
\(251\) −23.9124 −1.50934 −0.754670 0.656104i \(-0.772203\pi\)
−0.754670 + 0.656104i \(0.772203\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 24.9541 1.56885
\(254\) 11.8629 0.744347
\(255\) −13.0978 −0.820218
\(256\) 1.00000 0.0625000
\(257\) 8.77479 0.547356 0.273678 0.961821i \(-0.411760\pi\)
0.273678 + 0.961821i \(0.411760\pi\)
\(258\) 8.19806 0.510389
\(259\) 0.185981 0.0115563
\(260\) 0 0
\(261\) −4.96077 −0.307064
\(262\) −1.56465 −0.0966642
\(263\) 6.78448 0.418349 0.209174 0.977878i \(-0.432922\pi\)
0.209174 + 0.977878i \(0.432922\pi\)
\(264\) −3.15883 −0.194413
\(265\) −1.10454 −0.0678513
\(266\) −3.15883 −0.193681
\(267\) −13.5526 −0.829404
\(268\) −4.52111 −0.276171
\(269\) −5.13706 −0.313212 −0.156606 0.987661i \(-0.550055\pi\)
−0.156606 + 0.987661i \(0.550055\pi\)
\(270\) −1.69202 −0.102973
\(271\) −25.7995 −1.56721 −0.783605 0.621259i \(-0.786622\pi\)
−0.783605 + 0.621259i \(0.786622\pi\)
\(272\) 7.74094 0.469363
\(273\) 0 0
\(274\) 20.5700 1.24268
\(275\) −6.75063 −0.407078
\(276\) −7.89977 −0.475510
\(277\) 19.9976 1.20154 0.600770 0.799422i \(-0.294861\pi\)
0.600770 + 0.799422i \(0.294861\pi\)
\(278\) −22.5308 −1.35131
\(279\) 4.82908 0.289110
\(280\) −1.69202 −0.101118
\(281\) 18.5851 1.10869 0.554347 0.832286i \(-0.312968\pi\)
0.554347 + 0.832286i \(0.312968\pi\)
\(282\) −2.78017 −0.165557
\(283\) −4.35152 −0.258671 −0.129335 0.991601i \(-0.541284\pi\)
−0.129335 + 0.991601i \(0.541284\pi\)
\(284\) −1.20775 −0.0716668
\(285\) −5.34481 −0.316599
\(286\) 0 0
\(287\) −0.978230 −0.0577431
\(288\) 1.00000 0.0589256
\(289\) 42.9221 2.52483
\(290\) −8.39373 −0.492897
\(291\) 7.83340 0.459202
\(292\) −14.1564 −0.828443
\(293\) 7.14138 0.417204 0.208602 0.978001i \(-0.433109\pi\)
0.208602 + 0.978001i \(0.433109\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −8.12498 −0.473055
\(296\) −0.185981 −0.0108099
\(297\) −3.15883 −0.183294
\(298\) −1.60925 −0.0932215
\(299\) 0 0
\(300\) 2.13706 0.123383
\(301\) 8.19806 0.472528
\(302\) 13.1709 0.757901
\(303\) −15.2228 −0.874528
\(304\) 3.15883 0.181172
\(305\) 5.44803 0.311953
\(306\) 7.74094 0.442520
\(307\) −3.13467 −0.178905 −0.0894525 0.995991i \(-0.528512\pi\)
−0.0894525 + 0.995991i \(0.528512\pi\)
\(308\) −3.15883 −0.179991
\(309\) 11.6528 0.662904
\(310\) 8.17092 0.464077
\(311\) −32.7875 −1.85921 −0.929603 0.368562i \(-0.879850\pi\)
−0.929603 + 0.368562i \(0.879850\pi\)
\(312\) 0 0
\(313\) 11.5985 0.655586 0.327793 0.944750i \(-0.393695\pi\)
0.327793 + 0.944750i \(0.393695\pi\)
\(314\) −2.61596 −0.147627
\(315\) −1.69202 −0.0953346
\(316\) −7.33944 −0.412876
\(317\) −3.71810 −0.208830 −0.104415 0.994534i \(-0.533297\pi\)
−0.104415 + 0.994534i \(0.533297\pi\)
\(318\) 0.652793 0.0366068
\(319\) −15.6703 −0.877366
\(320\) 1.69202 0.0945869
\(321\) −14.5797 −0.813760
\(322\) −7.89977 −0.440237
\(323\) 24.4523 1.36056
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0.853248 0.0472570
\(327\) −6.29590 −0.348164
\(328\) 0.978230 0.0540137
\(329\) −2.78017 −0.153276
\(330\) −5.34481 −0.294222
\(331\) 15.8532 0.871373 0.435687 0.900098i \(-0.356506\pi\)
0.435687 + 0.900098i \(0.356506\pi\)
\(332\) −8.32304 −0.456786
\(333\) −0.185981 −0.0101917
\(334\) 14.8944 0.814985
\(335\) −7.64981 −0.417954
\(336\) 1.00000 0.0545545
\(337\) −26.2597 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(338\) 0 0
\(339\) −15.1860 −0.824789
\(340\) 13.0978 0.710330
\(341\) 15.2543 0.826065
\(342\) 3.15883 0.170810
\(343\) −1.00000 −0.0539949
\(344\) −8.19806 −0.442010
\(345\) −13.3666 −0.719633
\(346\) −7.28621 −0.391709
\(347\) −16.3099 −0.875561 −0.437781 0.899082i \(-0.644235\pi\)
−0.437781 + 0.899082i \(0.644235\pi\)
\(348\) 4.96077 0.265925
\(349\) 34.3937 1.84105 0.920527 0.390679i \(-0.127760\pi\)
0.920527 + 0.390679i \(0.127760\pi\)
\(350\) 2.13706 0.114231
\(351\) 0 0
\(352\) 3.15883 0.168366
\(353\) 16.3284 0.869074 0.434537 0.900654i \(-0.356912\pi\)
0.434537 + 0.900654i \(0.356912\pi\)
\(354\) 4.80194 0.255220
\(355\) −2.04354 −0.108460
\(356\) 13.5526 0.718285
\(357\) 7.74094 0.409694
\(358\) −4.10992 −0.217216
\(359\) −21.0248 −1.10964 −0.554822 0.831969i \(-0.687214\pi\)
−0.554822 + 0.831969i \(0.687214\pi\)
\(360\) 1.69202 0.0891774
\(361\) −9.02177 −0.474830
\(362\) −8.63640 −0.453919
\(363\) 1.02177 0.0536290
\(364\) 0 0
\(365\) −23.9530 −1.25376
\(366\) −3.21983 −0.168303
\(367\) −23.0858 −1.20507 −0.602533 0.798094i \(-0.705842\pi\)
−0.602533 + 0.798094i \(0.705842\pi\)
\(368\) 7.89977 0.411804
\(369\) 0.978230 0.0509246
\(370\) −0.314683 −0.0163596
\(371\) 0.652793 0.0338913
\(372\) −4.82908 −0.250376
\(373\) −16.6189 −0.860496 −0.430248 0.902711i \(-0.641574\pi\)
−0.430248 + 0.902711i \(0.641574\pi\)
\(374\) 24.4523 1.26440
\(375\) 12.0761 0.623605
\(376\) 2.78017 0.143376
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −16.4131 −0.843085 −0.421542 0.906809i \(-0.638511\pi\)
−0.421542 + 0.906809i \(0.638511\pi\)
\(380\) 5.34481 0.274183
\(381\) −11.8629 −0.607757
\(382\) 26.9922 1.38104
\(383\) 0.241603 0.0123453 0.00617266 0.999981i \(-0.498035\pi\)
0.00617266 + 0.999981i \(0.498035\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −5.34481 −0.272397
\(386\) 16.2174 0.825446
\(387\) −8.19806 −0.416731
\(388\) −7.83340 −0.397680
\(389\) −28.5042 −1.44522 −0.722611 0.691255i \(-0.757058\pi\)
−0.722611 + 0.691255i \(0.757058\pi\)
\(390\) 0 0
\(391\) 61.1517 3.09257
\(392\) 1.00000 0.0505076
\(393\) 1.56465 0.0789260
\(394\) −21.6625 −1.09134
\(395\) −12.4185 −0.624842
\(396\) 3.15883 0.158737
\(397\) 13.8009 0.692646 0.346323 0.938115i \(-0.387430\pi\)
0.346323 + 0.938115i \(0.387430\pi\)
\(398\) −8.81700 −0.441956
\(399\) 3.15883 0.158139
\(400\) −2.13706 −0.106853
\(401\) −20.3134 −1.01440 −0.507200 0.861828i \(-0.669320\pi\)
−0.507200 + 0.861828i \(0.669320\pi\)
\(402\) 4.52111 0.225492
\(403\) 0 0
\(404\) 15.2228 0.757363
\(405\) 1.69202 0.0840772
\(406\) 4.96077 0.246199
\(407\) −0.587482 −0.0291204
\(408\) −7.74094 −0.383234
\(409\) −21.0683 −1.04176 −0.520880 0.853630i \(-0.674396\pi\)
−0.520880 + 0.853630i \(0.674396\pi\)
\(410\) 1.65519 0.0817438
\(411\) −20.5700 −1.01464
\(412\) −11.6528 −0.574092
\(413\) 4.80194 0.236288
\(414\) 7.89977 0.388253
\(415\) −14.0828 −0.691296
\(416\) 0 0
\(417\) 22.5308 1.10334
\(418\) 9.97823 0.488051
\(419\) −0.486663 −0.0237751 −0.0118875 0.999929i \(-0.503784\pi\)
−0.0118875 + 0.999929i \(0.503784\pi\)
\(420\) 1.69202 0.0825622
\(421\) 26.4644 1.28980 0.644898 0.764268i \(-0.276900\pi\)
0.644898 + 0.764268i \(0.276900\pi\)
\(422\) 0.753020 0.0366565
\(423\) 2.78017 0.135176
\(424\) −0.652793 −0.0317024
\(425\) −16.5429 −0.802447
\(426\) 1.20775 0.0585157
\(427\) −3.21983 −0.155819
\(428\) 14.5797 0.704737
\(429\) 0 0
\(430\) −13.8713 −0.668933
\(431\) 17.2476 0.830786 0.415393 0.909642i \(-0.363644\pi\)
0.415393 + 0.909642i \(0.363644\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.78687 0.133929 0.0669643 0.997755i \(-0.478669\pi\)
0.0669643 + 0.997755i \(0.478669\pi\)
\(434\) −4.82908 −0.231803
\(435\) 8.39373 0.402449
\(436\) 6.29590 0.301519
\(437\) 24.9541 1.19371
\(438\) 14.1564 0.676421
\(439\) −21.4808 −1.02522 −0.512612 0.858621i \(-0.671322\pi\)
−0.512612 + 0.858621i \(0.671322\pi\)
\(440\) 5.34481 0.254804
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 9.85623 0.468284 0.234142 0.972202i \(-0.424772\pi\)
0.234142 + 0.972202i \(0.424772\pi\)
\(444\) 0.185981 0.00882625
\(445\) 22.9312 1.08704
\(446\) −2.14377 −0.101510
\(447\) 1.60925 0.0761150
\(448\) −1.00000 −0.0472456
\(449\) 10.8358 0.511373 0.255686 0.966760i \(-0.417699\pi\)
0.255686 + 0.966760i \(0.417699\pi\)
\(450\) −2.13706 −0.100742
\(451\) 3.09006 0.145505
\(452\) 15.1860 0.714288
\(453\) −13.1709 −0.618824
\(454\) 21.3327 1.00119
\(455\) 0 0
\(456\) −3.15883 −0.147926
\(457\) −15.5386 −0.726863 −0.363432 0.931621i \(-0.618395\pi\)
−0.363432 + 0.931621i \(0.618395\pi\)
\(458\) −15.3502 −0.717267
\(459\) −7.74094 −0.361316
\(460\) 13.3666 0.623220
\(461\) −5.55389 −0.258671 −0.129335 0.991601i \(-0.541284\pi\)
−0.129335 + 0.991601i \(0.541284\pi\)
\(462\) 3.15883 0.146962
\(463\) 8.29291 0.385404 0.192702 0.981257i \(-0.438275\pi\)
0.192702 + 0.981257i \(0.438275\pi\)
\(464\) −4.96077 −0.230298
\(465\) −8.17092 −0.378917
\(466\) 23.1468 1.07225
\(467\) −22.7071 −1.05076 −0.525379 0.850868i \(-0.676077\pi\)
−0.525379 + 0.850868i \(0.676077\pi\)
\(468\) 0 0
\(469\) 4.52111 0.208765
\(470\) 4.70410 0.216984
\(471\) 2.61596 0.120537
\(472\) −4.80194 −0.221027
\(473\) −25.8963 −1.19071
\(474\) 7.33944 0.337112
\(475\) −6.75063 −0.309740
\(476\) −7.74094 −0.354805
\(477\) −0.652793 −0.0298893
\(478\) −17.8538 −0.816616
\(479\) 22.9245 1.04745 0.523724 0.851888i \(-0.324542\pi\)
0.523724 + 0.851888i \(0.324542\pi\)
\(480\) −1.69202 −0.0772299
\(481\) 0 0
\(482\) 16.4179 0.747815
\(483\) 7.89977 0.359452
\(484\) −1.02177 −0.0464441
\(485\) −13.2543 −0.601846
\(486\) −1.00000 −0.0453609
\(487\) 7.97525 0.361393 0.180696 0.983539i \(-0.442165\pi\)
0.180696 + 0.983539i \(0.442165\pi\)
\(488\) 3.21983 0.145755
\(489\) −0.853248 −0.0385852
\(490\) 1.69202 0.0764377
\(491\) −25.0116 −1.12876 −0.564379 0.825516i \(-0.690884\pi\)
−0.564379 + 0.825516i \(0.690884\pi\)
\(492\) −0.978230 −0.0441020
\(493\) −38.4010 −1.72950
\(494\) 0 0
\(495\) 5.34481 0.240232
\(496\) 4.82908 0.216832
\(497\) 1.20775 0.0541750
\(498\) 8.32304 0.372965
\(499\) 11.7168 0.524515 0.262257 0.964998i \(-0.415533\pi\)
0.262257 + 0.964998i \(0.415533\pi\)
\(500\) −12.0761 −0.540058
\(501\) −14.8944 −0.665433
\(502\) −23.9124 −1.06726
\(503\) 6.62133 0.295231 0.147615 0.989045i \(-0.452840\pi\)
0.147615 + 0.989045i \(0.452840\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 25.7573 1.14619
\(506\) 24.9541 1.10934
\(507\) 0 0
\(508\) 11.8629 0.526333
\(509\) 33.6728 1.49252 0.746259 0.665655i \(-0.231848\pi\)
0.746259 + 0.665655i \(0.231848\pi\)
\(510\) −13.0978 −0.579982
\(511\) 14.1564 0.626244
\(512\) 1.00000 0.0441942
\(513\) −3.15883 −0.139466
\(514\) 8.77479 0.387039
\(515\) −19.7168 −0.868825
\(516\) 8.19806 0.360900
\(517\) 8.78209 0.386236
\(518\) 0.185981 0.00817152
\(519\) 7.28621 0.319829
\(520\) 0 0
\(521\) −6.73125 −0.294901 −0.147451 0.989069i \(-0.547107\pi\)
−0.147451 + 0.989069i \(0.547107\pi\)
\(522\) −4.96077 −0.217127
\(523\) 4.01938 0.175755 0.0878776 0.996131i \(-0.471992\pi\)
0.0878776 + 0.996131i \(0.471992\pi\)
\(524\) −1.56465 −0.0683519
\(525\) −2.13706 −0.0932691
\(526\) 6.78448 0.295817
\(527\) 37.3817 1.62837
\(528\) −3.15883 −0.137471
\(529\) 39.4064 1.71332
\(530\) −1.10454 −0.0479781
\(531\) −4.80194 −0.208386
\(532\) −3.15883 −0.136953
\(533\) 0 0
\(534\) −13.5526 −0.586477
\(535\) 24.6692 1.06654
\(536\) −4.52111 −0.195282
\(537\) 4.10992 0.177356
\(538\) −5.13706 −0.221475
\(539\) 3.15883 0.136061
\(540\) −1.69202 −0.0728130
\(541\) 44.8286 1.92733 0.963666 0.267109i \(-0.0860685\pi\)
0.963666 + 0.267109i \(0.0860685\pi\)
\(542\) −25.7995 −1.10819
\(543\) 8.63640 0.370623
\(544\) 7.74094 0.331890
\(545\) 10.6528 0.456316
\(546\) 0 0
\(547\) −30.5603 −1.30667 −0.653333 0.757071i \(-0.726630\pi\)
−0.653333 + 0.757071i \(0.726630\pi\)
\(548\) 20.5700 0.878708
\(549\) 3.21983 0.137419
\(550\) −6.75063 −0.287848
\(551\) −15.6703 −0.667575
\(552\) −7.89977 −0.336237
\(553\) 7.33944 0.312105
\(554\) 19.9976 0.849617
\(555\) 0.314683 0.0133576
\(556\) −22.5308 −0.955519
\(557\) 44.0277 1.86552 0.932758 0.360504i \(-0.117395\pi\)
0.932758 + 0.360504i \(0.117395\pi\)
\(558\) 4.82908 0.204431
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) −24.4523 −1.03238
\(562\) 18.5851 0.783965
\(563\) −30.7138 −1.29443 −0.647216 0.762307i \(-0.724067\pi\)
−0.647216 + 0.762307i \(0.724067\pi\)
\(564\) −2.78017 −0.117066
\(565\) 25.6950 1.08100
\(566\) −4.35152 −0.182908
\(567\) −1.00000 −0.0419961
\(568\) −1.20775 −0.0506761
\(569\) 35.8963 1.50485 0.752426 0.658677i \(-0.228884\pi\)
0.752426 + 0.658677i \(0.228884\pi\)
\(570\) −5.34481 −0.223870
\(571\) −30.2218 −1.26474 −0.632370 0.774666i \(-0.717918\pi\)
−0.632370 + 0.774666i \(0.717918\pi\)
\(572\) 0 0
\(573\) −26.9922 −1.12762
\(574\) −0.978230 −0.0408305
\(575\) −16.8823 −0.704041
\(576\) 1.00000 0.0416667
\(577\) 10.6799 0.444612 0.222306 0.974977i \(-0.428642\pi\)
0.222306 + 0.974977i \(0.428642\pi\)
\(578\) 42.9221 1.78533
\(579\) −16.2174 −0.673974
\(580\) −8.39373 −0.348531
\(581\) 8.32304 0.345298
\(582\) 7.83340 0.324705
\(583\) −2.06206 −0.0854020
\(584\) −14.1564 −0.585798
\(585\) 0 0
\(586\) 7.14138 0.295007
\(587\) 5.42865 0.224064 0.112032 0.993705i \(-0.464264\pi\)
0.112032 + 0.993705i \(0.464264\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 15.2543 0.628541
\(590\) −8.12498 −0.334500
\(591\) 21.6625 0.891075
\(592\) −0.185981 −0.00764376
\(593\) 12.3037 0.505251 0.252626 0.967564i \(-0.418706\pi\)
0.252626 + 0.967564i \(0.418706\pi\)
\(594\) −3.15883 −0.129609
\(595\) −13.0978 −0.536959
\(596\) −1.60925 −0.0659175
\(597\) 8.81700 0.360856
\(598\) 0 0
\(599\) −42.9487 −1.75484 −0.877418 0.479727i \(-0.840736\pi\)
−0.877418 + 0.479727i \(0.840736\pi\)
\(600\) 2.13706 0.0872452
\(601\) 6.95348 0.283638 0.141819 0.989893i \(-0.454705\pi\)
0.141819 + 0.989893i \(0.454705\pi\)
\(602\) 8.19806 0.334128
\(603\) −4.52111 −0.184114
\(604\) 13.1709 0.535917
\(605\) −1.72886 −0.0702880
\(606\) −15.2228 −0.618385
\(607\) 4.25129 0.172555 0.0862773 0.996271i \(-0.472503\pi\)
0.0862773 + 0.996271i \(0.472503\pi\)
\(608\) 3.15883 0.128108
\(609\) −4.96077 −0.201021
\(610\) 5.44803 0.220584
\(611\) 0 0
\(612\) 7.74094 0.312909
\(613\) −9.09113 −0.367187 −0.183594 0.983002i \(-0.558773\pi\)
−0.183594 + 0.983002i \(0.558773\pi\)
\(614\) −3.13467 −0.126505
\(615\) −1.65519 −0.0667435
\(616\) −3.15883 −0.127273
\(617\) 12.2067 0.491423 0.245711 0.969343i \(-0.420978\pi\)
0.245711 + 0.969343i \(0.420978\pi\)
\(618\) 11.6528 0.468744
\(619\) 49.1183 1.97423 0.987115 0.160012i \(-0.0511532\pi\)
0.987115 + 0.160012i \(0.0511532\pi\)
\(620\) 8.17092 0.328152
\(621\) −7.89977 −0.317007
\(622\) −32.7875 −1.31466
\(623\) −13.5526 −0.542972
\(624\) 0 0
\(625\) −9.74764 −0.389906
\(626\) 11.5985 0.463569
\(627\) −9.97823 −0.398492
\(628\) −2.61596 −0.104388
\(629\) −1.43967 −0.0574032
\(630\) −1.69202 −0.0674117
\(631\) −16.4620 −0.655343 −0.327671 0.944792i \(-0.606264\pi\)
−0.327671 + 0.944792i \(0.606264\pi\)
\(632\) −7.33944 −0.291947
\(633\) −0.753020 −0.0299299
\(634\) −3.71810 −0.147665
\(635\) 20.0723 0.796547
\(636\) 0.652793 0.0258849
\(637\) 0 0
\(638\) −15.6703 −0.620391
\(639\) −1.20775 −0.0477779
\(640\) 1.69202 0.0668830
\(641\) 43.0428 1.70009 0.850044 0.526711i \(-0.176575\pi\)
0.850044 + 0.526711i \(0.176575\pi\)
\(642\) −14.5797 −0.575415
\(643\) −20.3220 −0.801421 −0.400710 0.916205i \(-0.631237\pi\)
−0.400710 + 0.916205i \(0.631237\pi\)
\(644\) −7.89977 −0.311295
\(645\) 13.8713 0.546182
\(646\) 24.4523 0.962064
\(647\) 11.0049 0.432647 0.216324 0.976322i \(-0.430593\pi\)
0.216324 + 0.976322i \(0.430593\pi\)
\(648\) 1.00000 0.0392837
\(649\) −15.1685 −0.595417
\(650\) 0 0
\(651\) 4.82908 0.189267
\(652\) 0.853248 0.0334158
\(653\) −17.9355 −0.701872 −0.350936 0.936399i \(-0.614137\pi\)
−0.350936 + 0.936399i \(0.614137\pi\)
\(654\) −6.29590 −0.246189
\(655\) −2.64742 −0.103443
\(656\) 0.978230 0.0381935
\(657\) −14.1564 −0.552295
\(658\) −2.78017 −0.108382
\(659\) 29.1594 1.13589 0.567945 0.823067i \(-0.307739\pi\)
0.567945 + 0.823067i \(0.307739\pi\)
\(660\) −5.34481 −0.208047
\(661\) −34.5217 −1.34274 −0.671369 0.741123i \(-0.734293\pi\)
−0.671369 + 0.741123i \(0.734293\pi\)
\(662\) 15.8532 0.616154
\(663\) 0 0
\(664\) −8.32304 −0.322997
\(665\) −5.34481 −0.207263
\(666\) −0.185981 −0.00720660
\(667\) −39.1890 −1.51740
\(668\) 14.8944 0.576281
\(669\) 2.14377 0.0828829
\(670\) −7.64981 −0.295538
\(671\) 10.1709 0.392644
\(672\) 1.00000 0.0385758
\(673\) −9.46203 −0.364734 −0.182367 0.983231i \(-0.558376\pi\)
−0.182367 + 0.983231i \(0.558376\pi\)
\(674\) −26.2597 −1.01148
\(675\) 2.13706 0.0822556
\(676\) 0 0
\(677\) −17.1438 −0.658889 −0.329444 0.944175i \(-0.606861\pi\)
−0.329444 + 0.944175i \(0.606861\pi\)
\(678\) −15.1860 −0.583214
\(679\) 7.83340 0.300618
\(680\) 13.0978 0.502279
\(681\) −21.3327 −0.817472
\(682\) 15.2543 0.584116
\(683\) −21.0339 −0.804838 −0.402419 0.915456i \(-0.631830\pi\)
−0.402419 + 0.915456i \(0.631830\pi\)
\(684\) 3.15883 0.120781
\(685\) 34.8049 1.32983
\(686\) −1.00000 −0.0381802
\(687\) 15.3502 0.585646
\(688\) −8.19806 −0.312548
\(689\) 0 0
\(690\) −13.3666 −0.508857
\(691\) −9.23490 −0.351312 −0.175656 0.984452i \(-0.556205\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(692\) −7.28621 −0.276980
\(693\) −3.15883 −0.119994
\(694\) −16.3099 −0.619115
\(695\) −38.1226 −1.44607
\(696\) 4.96077 0.188038
\(697\) 7.57242 0.286826
\(698\) 34.3937 1.30182
\(699\) −23.1468 −0.875491
\(700\) 2.13706 0.0807734
\(701\) −17.2989 −0.653370 −0.326685 0.945133i \(-0.605932\pi\)
−0.326685 + 0.945133i \(0.605932\pi\)
\(702\) 0 0
\(703\) −0.587482 −0.0221573
\(704\) 3.15883 0.119053
\(705\) −4.70410 −0.177167
\(706\) 16.3284 0.614528
\(707\) −15.2228 −0.572513
\(708\) 4.80194 0.180468
\(709\) 19.2911 0.724493 0.362246 0.932082i \(-0.382010\pi\)
0.362246 + 0.932082i \(0.382010\pi\)
\(710\) −2.04354 −0.0766927
\(711\) −7.33944 −0.275250
\(712\) 13.5526 0.507904
\(713\) 38.1487 1.42868
\(714\) 7.74094 0.289697
\(715\) 0 0
\(716\) −4.10992 −0.153595
\(717\) 17.8538 0.666764
\(718\) −21.0248 −0.784637
\(719\) −3.61165 −0.134692 −0.0673458 0.997730i \(-0.521453\pi\)
−0.0673458 + 0.997730i \(0.521453\pi\)
\(720\) 1.69202 0.0630579
\(721\) 11.6528 0.433973
\(722\) −9.02177 −0.335756
\(723\) −16.4179 −0.610588
\(724\) −8.63640 −0.320969
\(725\) 10.6015 0.393729
\(726\) 1.02177 0.0379215
\(727\) −18.6614 −0.692114 −0.346057 0.938214i \(-0.612480\pi\)
−0.346057 + 0.938214i \(0.612480\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −23.9530 −0.886540
\(731\) −63.4607 −2.34718
\(732\) −3.21983 −0.119008
\(733\) −41.8872 −1.54714 −0.773570 0.633711i \(-0.781531\pi\)
−0.773570 + 0.633711i \(0.781531\pi\)
\(734\) −23.0858 −0.852111
\(735\) −1.69202 −0.0624112
\(736\) 7.89977 0.291189
\(737\) −14.2814 −0.526063
\(738\) 0.978230 0.0360091
\(739\) −46.7622 −1.72018 −0.860088 0.510145i \(-0.829592\pi\)
−0.860088 + 0.510145i \(0.829592\pi\)
\(740\) −0.314683 −0.0115680
\(741\) 0 0
\(742\) 0.652793 0.0239648
\(743\) −13.3260 −0.488885 −0.244442 0.969664i \(-0.578605\pi\)
−0.244442 + 0.969664i \(0.578605\pi\)
\(744\) −4.82908 −0.177043
\(745\) −2.72289 −0.0997589
\(746\) −16.6189 −0.608463
\(747\) −8.32304 −0.304524
\(748\) 24.4523 0.894066
\(749\) −14.5797 −0.532731
\(750\) 12.0761 0.440956
\(751\) 21.8888 0.798732 0.399366 0.916792i \(-0.369230\pi\)
0.399366 + 0.916792i \(0.369230\pi\)
\(752\) 2.78017 0.101382
\(753\) 23.9124 0.871418
\(754\) 0 0
\(755\) 22.2855 0.811051
\(756\) 1.00000 0.0363696
\(757\) 12.6974 0.461495 0.230747 0.973014i \(-0.425883\pi\)
0.230747 + 0.973014i \(0.425883\pi\)
\(758\) −16.4131 −0.596151
\(759\) −24.9541 −0.905775
\(760\) 5.34481 0.193877
\(761\) −53.5096 −1.93972 −0.969861 0.243659i \(-0.921652\pi\)
−0.969861 + 0.243659i \(0.921652\pi\)
\(762\) −11.8629 −0.429749
\(763\) −6.29590 −0.227927
\(764\) 26.9922 0.976545
\(765\) 13.0978 0.473553
\(766\) 0.241603 0.00872946
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 22.2892 0.803769 0.401884 0.915690i \(-0.368355\pi\)
0.401884 + 0.915690i \(0.368355\pi\)
\(770\) −5.34481 −0.192614
\(771\) −8.77479 −0.316016
\(772\) 16.2174 0.583678
\(773\) −32.3726 −1.16436 −0.582180 0.813060i \(-0.697800\pi\)
−0.582180 + 0.813060i \(0.697800\pi\)
\(774\) −8.19806 −0.294673
\(775\) −10.3201 −0.370708
\(776\) −7.83340 −0.281203
\(777\) −0.185981 −0.00667202
\(778\) −28.5042 −1.02193
\(779\) 3.09006 0.110713
\(780\) 0 0
\(781\) −3.81508 −0.136514
\(782\) 61.1517 2.18678
\(783\) 4.96077 0.177284
\(784\) 1.00000 0.0357143
\(785\) −4.42626 −0.157980
\(786\) 1.56465 0.0558091
\(787\) 31.0984 1.10854 0.554270 0.832337i \(-0.312998\pi\)
0.554270 + 0.832337i \(0.312998\pi\)
\(788\) −21.6625 −0.771694
\(789\) −6.78448 −0.241534
\(790\) −12.4185 −0.441830
\(791\) −15.1860 −0.539951
\(792\) 3.15883 0.112244
\(793\) 0 0
\(794\) 13.8009 0.489775
\(795\) 1.10454 0.0391740
\(796\) −8.81700 −0.312510
\(797\) −23.1997 −0.821776 −0.410888 0.911686i \(-0.634781\pi\)
−0.410888 + 0.911686i \(0.634781\pi\)
\(798\) 3.15883 0.111821
\(799\) 21.5211 0.761362
\(800\) −2.13706 −0.0755566
\(801\) 13.5526 0.478856
\(802\) −20.3134 −0.717290
\(803\) −44.7178 −1.57806
\(804\) 4.52111 0.159447
\(805\) −13.3666 −0.471110
\(806\) 0 0
\(807\) 5.13706 0.180833
\(808\) 15.2228 0.535537
\(809\) −5.47948 −0.192648 −0.0963242 0.995350i \(-0.530709\pi\)
−0.0963242 + 0.995350i \(0.530709\pi\)
\(810\) 1.69202 0.0594516
\(811\) 52.0592 1.82805 0.914023 0.405663i \(-0.132959\pi\)
0.914023 + 0.405663i \(0.132959\pi\)
\(812\) 4.96077 0.174089
\(813\) 25.7995 0.904830
\(814\) −0.587482 −0.0205912
\(815\) 1.44371 0.0505711
\(816\) −7.74094 −0.270987
\(817\) −25.8963 −0.905997
\(818\) −21.0683 −0.736636
\(819\) 0 0
\(820\) 1.65519 0.0578016
\(821\) 2.61655 0.0913182 0.0456591 0.998957i \(-0.485461\pi\)
0.0456591 + 0.998957i \(0.485461\pi\)
\(822\) −20.5700 −0.717462
\(823\) 50.4596 1.75891 0.879456 0.475980i \(-0.157907\pi\)
0.879456 + 0.475980i \(0.157907\pi\)
\(824\) −11.6528 −0.405944
\(825\) 6.75063 0.235027
\(826\) 4.80194 0.167081
\(827\) 2.91079 0.101218 0.0506090 0.998719i \(-0.483884\pi\)
0.0506090 + 0.998719i \(0.483884\pi\)
\(828\) 7.89977 0.274536
\(829\) −6.09724 −0.211766 −0.105883 0.994379i \(-0.533767\pi\)
−0.105883 + 0.994379i \(0.533767\pi\)
\(830\) −14.0828 −0.488820
\(831\) −19.9976 −0.693709
\(832\) 0 0
\(833\) 7.74094 0.268208
\(834\) 22.5308 0.780178
\(835\) 25.2016 0.872139
\(836\) 9.97823 0.345104
\(837\) −4.82908 −0.166918
\(838\) −0.486663 −0.0168115
\(839\) −4.22654 −0.145916 −0.0729581 0.997335i \(-0.523244\pi\)
−0.0729581 + 0.997335i \(0.523244\pi\)
\(840\) 1.69202 0.0583803
\(841\) −4.39075 −0.151405
\(842\) 26.4644 0.912024
\(843\) −18.5851 −0.640104
\(844\) 0.753020 0.0259200
\(845\) 0 0
\(846\) 2.78017 0.0955841
\(847\) 1.02177 0.0351084
\(848\) −0.652793 −0.0224170
\(849\) 4.35152 0.149344
\(850\) −16.5429 −0.567416
\(851\) −1.46921 −0.0503637
\(852\) 1.20775 0.0413769
\(853\) 42.0538 1.43990 0.719948 0.694028i \(-0.244166\pi\)
0.719948 + 0.694028i \(0.244166\pi\)
\(854\) −3.21983 −0.110180
\(855\) 5.34481 0.182789
\(856\) 14.5797 0.498324
\(857\) −40.6039 −1.38700 −0.693501 0.720456i \(-0.743933\pi\)
−0.693501 + 0.720456i \(0.743933\pi\)
\(858\) 0 0
\(859\) 4.53643 0.154781 0.0773906 0.997001i \(-0.475341\pi\)
0.0773906 + 0.997001i \(0.475341\pi\)
\(860\) −13.8713 −0.473007
\(861\) 0.978230 0.0333380
\(862\) 17.2476 0.587455
\(863\) 27.8592 0.948339 0.474169 0.880434i \(-0.342748\pi\)
0.474169 + 0.880434i \(0.342748\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −12.3284 −0.419179
\(866\) 2.78687 0.0947018
\(867\) −42.9221 −1.45771
\(868\) −4.82908 −0.163910
\(869\) −23.1841 −0.786465
\(870\) 8.39373 0.284574
\(871\) 0 0
\(872\) 6.29590 0.213206
\(873\) −7.83340 −0.265120
\(874\) 24.9541 0.844084
\(875\) 12.0761 0.408245
\(876\) 14.1564 0.478302
\(877\) −45.6892 −1.54281 −0.771407 0.636343i \(-0.780446\pi\)
−0.771407 + 0.636343i \(0.780446\pi\)
\(878\) −21.4808 −0.724942
\(879\) −7.14138 −0.240873
\(880\) 5.34481 0.180174
\(881\) 17.2935 0.582633 0.291316 0.956627i \(-0.405907\pi\)
0.291316 + 0.956627i \(0.405907\pi\)
\(882\) 1.00000 0.0336718
\(883\) 41.8179 1.40728 0.703641 0.710555i \(-0.251556\pi\)
0.703641 + 0.710555i \(0.251556\pi\)
\(884\) 0 0
\(885\) 8.12498 0.273118
\(886\) 9.85623 0.331127
\(887\) −11.4196 −0.383431 −0.191715 0.981451i \(-0.561405\pi\)
−0.191715 + 0.981451i \(0.561405\pi\)
\(888\) 0.185981 0.00624110
\(889\) −11.8629 −0.397870
\(890\) 22.9312 0.768657
\(891\) 3.15883 0.105825
\(892\) −2.14377 −0.0717787
\(893\) 8.78209 0.293881
\(894\) 1.60925 0.0538214
\(895\) −6.95407 −0.232449
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.8358 0.361595
\(899\) −23.9560 −0.798977
\(900\) −2.13706 −0.0712354
\(901\) −5.05323 −0.168347
\(902\) 3.09006 0.102888
\(903\) −8.19806 −0.272814
\(904\) 15.1860 0.505078
\(905\) −14.6130 −0.485752
\(906\) −13.1709 −0.437574
\(907\) −31.6396 −1.05058 −0.525289 0.850924i \(-0.676043\pi\)
−0.525289 + 0.850924i \(0.676043\pi\)
\(908\) 21.3327 0.707952
\(909\) 15.2228 0.504909
\(910\) 0 0
\(911\) 14.5875 0.483305 0.241652 0.970363i \(-0.422311\pi\)
0.241652 + 0.970363i \(0.422311\pi\)
\(912\) −3.15883 −0.104599
\(913\) −26.2911 −0.870109
\(914\) −15.5386 −0.513970
\(915\) −5.44803 −0.180106
\(916\) −15.3502 −0.507185
\(917\) 1.56465 0.0516692
\(918\) −7.74094 −0.255489
\(919\) −47.3019 −1.56034 −0.780172 0.625565i \(-0.784869\pi\)
−0.780172 + 0.625565i \(0.784869\pi\)
\(920\) 13.3666 0.440683
\(921\) 3.13467 0.103291
\(922\) −5.55389 −0.182908
\(923\) 0 0
\(924\) 3.15883 0.103918
\(925\) 0.397452 0.0130682
\(926\) 8.29291 0.272522
\(927\) −11.6528 −0.382728
\(928\) −4.96077 −0.162845
\(929\) −40.5526 −1.33049 −0.665243 0.746627i \(-0.731672\pi\)
−0.665243 + 0.746627i \(0.731672\pi\)
\(930\) −8.17092 −0.267935
\(931\) 3.15883 0.103527
\(932\) 23.1468 0.758197
\(933\) 32.7875 1.07341
\(934\) −22.7071 −0.742999
\(935\) 41.3739 1.35307
\(936\) 0 0
\(937\) 18.7186 0.611509 0.305755 0.952110i \(-0.401091\pi\)
0.305755 + 0.952110i \(0.401091\pi\)
\(938\) 4.52111 0.147619
\(939\) −11.5985 −0.378503
\(940\) 4.70410 0.153431
\(941\) 27.2285 0.887622 0.443811 0.896120i \(-0.353626\pi\)
0.443811 + 0.896120i \(0.353626\pi\)
\(942\) 2.61596 0.0852325
\(943\) 7.72779 0.251652
\(944\) −4.80194 −0.156290
\(945\) 1.69202 0.0550415
\(946\) −25.8963 −0.841962
\(947\) 51.9963 1.68965 0.844826 0.535041i \(-0.179704\pi\)
0.844826 + 0.535041i \(0.179704\pi\)
\(948\) 7.33944 0.238374
\(949\) 0 0
\(950\) −6.75063 −0.219019
\(951\) 3.71810 0.120568
\(952\) −7.74094 −0.250885
\(953\) −37.5265 −1.21560 −0.607801 0.794089i \(-0.707948\pi\)
−0.607801 + 0.794089i \(0.707948\pi\)
\(954\) −0.652793 −0.0211349
\(955\) 45.6714 1.47789
\(956\) −17.8538 −0.577434
\(957\) 15.6703 0.506547
\(958\) 22.9245 0.740658
\(959\) −20.5700 −0.664241
\(960\) −1.69202 −0.0546098
\(961\) −7.67994 −0.247740
\(962\) 0 0
\(963\) 14.5797 0.469825
\(964\) 16.4179 0.528785
\(965\) 27.4403 0.883333
\(966\) 7.89977 0.254171
\(967\) 49.0786 1.57826 0.789130 0.614226i \(-0.210532\pi\)
0.789130 + 0.614226i \(0.210532\pi\)
\(968\) −1.02177 −0.0328409
\(969\) −24.4523 −0.785522
\(970\) −13.2543 −0.425569
\(971\) −37.6746 −1.20903 −0.604517 0.796592i \(-0.706634\pi\)
−0.604517 + 0.796592i \(0.706634\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 22.5308 0.722304
\(974\) 7.97525 0.255543
\(975\) 0 0
\(976\) 3.21983 0.103064
\(977\) 10.0852 0.322653 0.161326 0.986901i \(-0.448423\pi\)
0.161326 + 0.986901i \(0.448423\pi\)
\(978\) −0.853248 −0.0272839
\(979\) 42.8103 1.36822
\(980\) 1.69202 0.0540496
\(981\) 6.29590 0.201013
\(982\) −25.0116 −0.798152
\(983\) −24.1299 −0.769624 −0.384812 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(984\) −0.978230 −0.0311848
\(985\) −36.6534 −1.16787
\(986\) −38.4010 −1.22294
\(987\) 2.78017 0.0884937
\(988\) 0 0
\(989\) −64.7628 −2.05934
\(990\) 5.34481 0.169869
\(991\) −39.6553 −1.25969 −0.629846 0.776720i \(-0.716882\pi\)
−0.629846 + 0.776720i \(0.716882\pi\)
\(992\) 4.82908 0.153324
\(993\) −15.8532 −0.503088
\(994\) 1.20775 0.0383075
\(995\) −14.9186 −0.472950
\(996\) 8.32304 0.263726
\(997\) −29.5394 −0.935523 −0.467761 0.883855i \(-0.654939\pi\)
−0.467761 + 0.883855i \(0.654939\pi\)
\(998\) 11.7168 0.370888
\(999\) 0.185981 0.00588417
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 7098.2.a.ci.1.3 yes 3
13.12 even 2 7098.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cd.1.1 3 13.12 even 2
7098.2.a.ci.1.3 yes 3 1.1 even 1 trivial