Properties

Label 7098.2.a.bt.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.56155 q^{10} -2.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} -1.56155 q^{15} +1.00000 q^{16} +0.123106 q^{17} +1.00000 q^{18} -2.56155 q^{19} +1.56155 q^{20} -1.00000 q^{21} -2.56155 q^{22} -1.12311 q^{23} -1.00000 q^{24} -2.56155 q^{25} -1.00000 q^{27} +1.00000 q^{28} -6.12311 q^{29} -1.56155 q^{30} +1.00000 q^{32} +2.56155 q^{33} +0.123106 q^{34} +1.56155 q^{35} +1.00000 q^{36} -2.43845 q^{37} -2.56155 q^{38} +1.56155 q^{40} -11.2462 q^{41} -1.00000 q^{42} -8.00000 q^{43} -2.56155 q^{44} +1.56155 q^{45} -1.12311 q^{46} -0.315342 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.56155 q^{50} -0.123106 q^{51} +7.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} +2.56155 q^{57} -6.12311 q^{58} +5.12311 q^{59} -1.56155 q^{60} -11.2462 q^{61} +1.00000 q^{63} +1.00000 q^{64} +2.56155 q^{66} +9.12311 q^{67} +0.123106 q^{68} +1.12311 q^{69} +1.56155 q^{70} -11.3693 q^{71} +1.00000 q^{72} -2.43845 q^{73} -2.43845 q^{74} +2.56155 q^{75} -2.56155 q^{76} -2.56155 q^{77} -6.56155 q^{79} +1.56155 q^{80} +1.00000 q^{81} -11.2462 q^{82} -5.12311 q^{83} -1.00000 q^{84} +0.192236 q^{85} -8.00000 q^{86} +6.12311 q^{87} -2.56155 q^{88} +3.43845 q^{89} +1.56155 q^{90} -1.12311 q^{92} -0.315342 q^{94} -4.00000 q^{95} -1.00000 q^{96} -0.876894 q^{97} +1.00000 q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} - q^{10} - q^{11} - 2q^{12} + 2q^{14} + q^{15} + 2q^{16} - 8q^{17} + 2q^{18} - q^{19} - q^{20} - 2q^{21} - q^{22} + 6q^{23} - 2q^{24} - q^{25} - 2q^{27} + 2q^{28} - 4q^{29} + q^{30} + 2q^{32} + q^{33} - 8q^{34} - q^{35} + 2q^{36} - 9q^{37} - q^{38} - q^{40} - 6q^{41} - 2q^{42} - 16q^{43} - q^{44} - q^{45} + 6q^{46} - 13q^{47} - 2q^{48} + 2q^{49} - q^{50} + 8q^{51} + 14q^{53} - 2q^{54} - 8q^{55} + 2q^{56} + q^{57} - 4q^{58} + 2q^{59} + q^{60} - 6q^{61} + 2q^{63} + 2q^{64} + q^{66} + 10q^{67} - 8q^{68} - 6q^{69} - q^{70} + 2q^{71} + 2q^{72} - 9q^{73} - 9q^{74} + q^{75} - q^{76} - q^{77} - 9q^{79} - q^{80} + 2q^{81} - 6q^{82} - 2q^{83} - 2q^{84} + 21q^{85} - 16q^{86} + 4q^{87} - q^{88} + 11q^{89} - q^{90} + 6q^{92} - 13q^{94} - 8q^{95} - 2q^{96} - 10q^{97} + 2q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.56155 0.493806
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −1.56155 −0.403191
\(16\) 1.00000 0.250000
\(17\) 0.123106 0.0298575 0.0149287 0.999889i \(-0.495248\pi\)
0.0149287 + 0.999889i \(0.495248\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.56155 −0.587661 −0.293830 0.955858i \(-0.594930\pi\)
−0.293830 + 0.955858i \(0.594930\pi\)
\(20\) 1.56155 0.349174
\(21\) −1.00000 −0.218218
\(22\) −2.56155 −0.546125
\(23\) −1.12311 −0.234184 −0.117092 0.993121i \(-0.537357\pi\)
−0.117092 + 0.993121i \(0.537357\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.12311 −1.13703 −0.568516 0.822672i \(-0.692482\pi\)
−0.568516 + 0.822672i \(0.692482\pi\)
\(30\) −1.56155 −0.285099
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.56155 0.445909
\(34\) 0.123106 0.0211124
\(35\) 1.56155 0.263951
\(36\) 1.00000 0.166667
\(37\) −2.43845 −0.400878 −0.200439 0.979706i \(-0.564237\pi\)
−0.200439 + 0.979706i \(0.564237\pi\)
\(38\) −2.56155 −0.415539
\(39\) 0 0
\(40\) 1.56155 0.246903
\(41\) −11.2462 −1.75636 −0.878182 0.478327i \(-0.841243\pi\)
−0.878182 + 0.478327i \(0.841243\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −2.56155 −0.386169
\(45\) 1.56155 0.232783
\(46\) −1.12311 −0.165593
\(47\) −0.315342 −0.0459973 −0.0229986 0.999735i \(-0.507321\pi\)
−0.0229986 + 0.999735i \(0.507321\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −2.56155 −0.362258
\(51\) −0.123106 −0.0172382
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 1.00000 0.133631
\(57\) 2.56155 0.339286
\(58\) −6.12311 −0.804003
\(59\) 5.12311 0.666972 0.333486 0.942755i \(-0.391775\pi\)
0.333486 + 0.942755i \(0.391775\pi\)
\(60\) −1.56155 −0.201596
\(61\) −11.2462 −1.43993 −0.719965 0.694010i \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.56155 0.315305
\(67\) 9.12311 1.11456 0.557282 0.830323i \(-0.311844\pi\)
0.557282 + 0.830323i \(0.311844\pi\)
\(68\) 0.123106 0.0149287
\(69\) 1.12311 0.135206
\(70\) 1.56155 0.186641
\(71\) −11.3693 −1.34929 −0.674645 0.738142i \(-0.735703\pi\)
−0.674645 + 0.738142i \(0.735703\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.43845 −0.285399 −0.142699 0.989766i \(-0.545578\pi\)
−0.142699 + 0.989766i \(0.545578\pi\)
\(74\) −2.43845 −0.283464
\(75\) 2.56155 0.295783
\(76\) −2.56155 −0.293830
\(77\) −2.56155 −0.291916
\(78\) 0 0
\(79\) −6.56155 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(80\) 1.56155 0.174587
\(81\) 1.00000 0.111111
\(82\) −11.2462 −1.24194
\(83\) −5.12311 −0.562334 −0.281167 0.959659i \(-0.590721\pi\)
−0.281167 + 0.959659i \(0.590721\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0.192236 0.0208509
\(86\) −8.00000 −0.862662
\(87\) 6.12311 0.656466
\(88\) −2.56155 −0.273062
\(89\) 3.43845 0.364475 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(90\) 1.56155 0.164602
\(91\) 0 0
\(92\) −1.12311 −0.117092
\(93\) 0 0
\(94\) −0.315342 −0.0325250
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −0.876894 −0.0890351 −0.0445176 0.999009i \(-0.514175\pi\)
−0.0445176 + 0.999009i \(0.514175\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.56155 −0.257446
\(100\) −2.56155 −0.256155
\(101\) −5.80776 −0.577894 −0.288947 0.957345i \(-0.593305\pi\)
−0.288947 + 0.957345i \(0.593305\pi\)
\(102\) −0.123106 −0.0121893
\(103\) 6.24621 0.615457 0.307729 0.951474i \(-0.400431\pi\)
0.307729 + 0.951474i \(0.400431\pi\)
\(104\) 0 0
\(105\) −1.56155 −0.152392
\(106\) 7.00000 0.679900
\(107\) 13.9309 1.34675 0.673374 0.739302i \(-0.264844\pi\)
0.673374 + 0.739302i \(0.264844\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.2462 −1.17297 −0.586487 0.809959i \(-0.699490\pi\)
−0.586487 + 0.809959i \(0.699490\pi\)
\(110\) −4.00000 −0.381385
\(111\) 2.43845 0.231447
\(112\) 1.00000 0.0944911
\(113\) 1.56155 0.146899 0.0734493 0.997299i \(-0.476599\pi\)
0.0734493 + 0.997299i \(0.476599\pi\)
\(114\) 2.56155 0.239911
\(115\) −1.75379 −0.163542
\(116\) −6.12311 −0.568516
\(117\) 0 0
\(118\) 5.12311 0.471620
\(119\) 0.123106 0.0112851
\(120\) −1.56155 −0.142550
\(121\) −4.43845 −0.403495
\(122\) −11.2462 −1.01818
\(123\) 11.2462 1.01404
\(124\) 0 0
\(125\) −11.8078 −1.05612
\(126\) 1.00000 0.0890871
\(127\) 12.4924 1.10852 0.554262 0.832343i \(-0.313001\pi\)
0.554262 + 0.832343i \(0.313001\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 14.2462 1.24470 0.622349 0.782740i \(-0.286179\pi\)
0.622349 + 0.782740i \(0.286179\pi\)
\(132\) 2.56155 0.222955
\(133\) −2.56155 −0.222115
\(134\) 9.12311 0.788116
\(135\) −1.56155 −0.134397
\(136\) 0.123106 0.0105562
\(137\) 12.4384 1.06269 0.531344 0.847156i \(-0.321687\pi\)
0.531344 + 0.847156i \(0.321687\pi\)
\(138\) 1.12311 0.0956051
\(139\) −5.43845 −0.461283 −0.230642 0.973039i \(-0.574082\pi\)
−0.230642 + 0.973039i \(0.574082\pi\)
\(140\) 1.56155 0.131975
\(141\) 0.315342 0.0265566
\(142\) −11.3693 −0.954092
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.56155 −0.794044
\(146\) −2.43845 −0.201807
\(147\) −1.00000 −0.0824786
\(148\) −2.43845 −0.200439
\(149\) −15.5616 −1.27485 −0.637426 0.770512i \(-0.720001\pi\)
−0.637426 + 0.770512i \(0.720001\pi\)
\(150\) 2.56155 0.209150
\(151\) −21.9309 −1.78471 −0.892354 0.451335i \(-0.850948\pi\)
−0.892354 + 0.451335i \(0.850948\pi\)
\(152\) −2.56155 −0.207769
\(153\) 0.123106 0.00995250
\(154\) −2.56155 −0.206416
\(155\) 0 0
\(156\) 0 0
\(157\) −1.31534 −0.104976 −0.0524878 0.998622i \(-0.516715\pi\)
−0.0524878 + 0.998622i \(0.516715\pi\)
\(158\) −6.56155 −0.522009
\(159\) −7.00000 −0.555136
\(160\) 1.56155 0.123452
\(161\) −1.12311 −0.0885131
\(162\) 1.00000 0.0785674
\(163\) 24.4924 1.91839 0.959197 0.282738i \(-0.0912426\pi\)
0.959197 + 0.282738i \(0.0912426\pi\)
\(164\) −11.2462 −0.878182
\(165\) 4.00000 0.311400
\(166\) −5.12311 −0.397630
\(167\) 12.4924 0.966693 0.483346 0.875429i \(-0.339421\pi\)
0.483346 + 0.875429i \(0.339421\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 0.192236 0.0147438
\(171\) −2.56155 −0.195887
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.12311 0.464191
\(175\) −2.56155 −0.193635
\(176\) −2.56155 −0.193084
\(177\) −5.12311 −0.385076
\(178\) 3.43845 0.257723
\(179\) −1.75379 −0.131084 −0.0655422 0.997850i \(-0.520878\pi\)
−0.0655422 + 0.997850i \(0.520878\pi\)
\(180\) 1.56155 0.116391
\(181\) 13.2462 0.984583 0.492292 0.870430i \(-0.336159\pi\)
0.492292 + 0.870430i \(0.336159\pi\)
\(182\) 0 0
\(183\) 11.2462 0.831344
\(184\) −1.12311 −0.0827964
\(185\) −3.80776 −0.279952
\(186\) 0 0
\(187\) −0.315342 −0.0230601
\(188\) −0.315342 −0.0229986
\(189\) −1.00000 −0.0727393
\(190\) −4.00000 −0.290191
\(191\) −5.12311 −0.370695 −0.185347 0.982673i \(-0.559341\pi\)
−0.185347 + 0.982673i \(0.559341\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0.123106 0.00886134 0.00443067 0.999990i \(-0.498590\pi\)
0.00443067 + 0.999990i \(0.498590\pi\)
\(194\) −0.876894 −0.0629573
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.5616 0.894974 0.447487 0.894291i \(-0.352319\pi\)
0.447487 + 0.894291i \(0.352319\pi\)
\(198\) −2.56155 −0.182042
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −2.56155 −0.181129
\(201\) −9.12311 −0.643494
\(202\) −5.80776 −0.408633
\(203\) −6.12311 −0.429758
\(204\) −0.123106 −0.00861912
\(205\) −17.5616 −1.22655
\(206\) 6.24621 0.435194
\(207\) −1.12311 −0.0780612
\(208\) 0 0
\(209\) 6.56155 0.453872
\(210\) −1.56155 −0.107757
\(211\) 5.12311 0.352689 0.176345 0.984328i \(-0.443573\pi\)
0.176345 + 0.984328i \(0.443573\pi\)
\(212\) 7.00000 0.480762
\(213\) 11.3693 0.779013
\(214\) 13.9309 0.952295
\(215\) −12.4924 −0.851976
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −12.2462 −0.829418
\(219\) 2.43845 0.164775
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 2.43845 0.163658
\(223\) 9.75379 0.653162 0.326581 0.945169i \(-0.394103\pi\)
0.326581 + 0.945169i \(0.394103\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.56155 −0.170770
\(226\) 1.56155 0.103873
\(227\) −10.8769 −0.721925 −0.360962 0.932580i \(-0.617552\pi\)
−0.360962 + 0.932580i \(0.617552\pi\)
\(228\) 2.56155 0.169643
\(229\) 17.0540 1.12696 0.563479 0.826130i \(-0.309462\pi\)
0.563479 + 0.826130i \(0.309462\pi\)
\(230\) −1.75379 −0.115641
\(231\) 2.56155 0.168538
\(232\) −6.12311 −0.402002
\(233\) 1.36932 0.0897069 0.0448535 0.998994i \(-0.485718\pi\)
0.0448535 + 0.998994i \(0.485718\pi\)
\(234\) 0 0
\(235\) −0.492423 −0.0321221
\(236\) 5.12311 0.333486
\(237\) 6.56155 0.426219
\(238\) 0.123106 0.00797975
\(239\) 19.3693 1.25290 0.626448 0.779463i \(-0.284508\pi\)
0.626448 + 0.779463i \(0.284508\pi\)
\(240\) −1.56155 −0.100798
\(241\) −0.192236 −0.0123830 −0.00619150 0.999981i \(-0.501971\pi\)
−0.00619150 + 0.999981i \(0.501971\pi\)
\(242\) −4.43845 −0.285314
\(243\) −1.00000 −0.0641500
\(244\) −11.2462 −0.719965
\(245\) 1.56155 0.0997639
\(246\) 11.2462 0.717032
\(247\) 0 0
\(248\) 0 0
\(249\) 5.12311 0.324664
\(250\) −11.8078 −0.746789
\(251\) −13.6155 −0.859405 −0.429702 0.902971i \(-0.641381\pi\)
−0.429702 + 0.902971i \(0.641381\pi\)
\(252\) 1.00000 0.0629941
\(253\) 2.87689 0.180869
\(254\) 12.4924 0.783844
\(255\) −0.192236 −0.0120383
\(256\) 1.00000 0.0625000
\(257\) −24.3693 −1.52012 −0.760058 0.649855i \(-0.774830\pi\)
−0.760058 + 0.649855i \(0.774830\pi\)
\(258\) 8.00000 0.498058
\(259\) −2.43845 −0.151518
\(260\) 0 0
\(261\) −6.12311 −0.379011
\(262\) 14.2462 0.880134
\(263\) 5.12311 0.315904 0.157952 0.987447i \(-0.449511\pi\)
0.157952 + 0.987447i \(0.449511\pi\)
\(264\) 2.56155 0.157653
\(265\) 10.9309 0.671478
\(266\) −2.56155 −0.157059
\(267\) −3.43845 −0.210430
\(268\) 9.12311 0.557282
\(269\) 27.1231 1.65372 0.826862 0.562404i \(-0.190123\pi\)
0.826862 + 0.562404i \(0.190123\pi\)
\(270\) −1.56155 −0.0950331
\(271\) −22.7386 −1.38127 −0.690637 0.723202i \(-0.742670\pi\)
−0.690637 + 0.723202i \(0.742670\pi\)
\(272\) 0.123106 0.00746437
\(273\) 0 0
\(274\) 12.4384 0.751434
\(275\) 6.56155 0.395677
\(276\) 1.12311 0.0676030
\(277\) −19.5616 −1.17534 −0.587670 0.809101i \(-0.699955\pi\)
−0.587670 + 0.809101i \(0.699955\pi\)
\(278\) −5.43845 −0.326176
\(279\) 0 0
\(280\) 1.56155 0.0933206
\(281\) 12.9309 0.771391 0.385696 0.922626i \(-0.373962\pi\)
0.385696 + 0.922626i \(0.373962\pi\)
\(282\) 0.315342 0.0187783
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −11.3693 −0.674645
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −11.2462 −0.663843
\(288\) 1.00000 0.0589256
\(289\) −16.9848 −0.999109
\(290\) −9.56155 −0.561474
\(291\) 0.876894 0.0514045
\(292\) −2.43845 −0.142699
\(293\) −6.93087 −0.404906 −0.202453 0.979292i \(-0.564891\pi\)
−0.202453 + 0.979292i \(0.564891\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −2.43845 −0.141732
\(297\) 2.56155 0.148636
\(298\) −15.5616 −0.901457
\(299\) 0 0
\(300\) 2.56155 0.147891
\(301\) −8.00000 −0.461112
\(302\) −21.9309 −1.26198
\(303\) 5.80776 0.333647
\(304\) −2.56155 −0.146915
\(305\) −17.5616 −1.00557
\(306\) 0.123106 0.00703748
\(307\) −9.93087 −0.566785 −0.283392 0.959004i \(-0.591460\pi\)
−0.283392 + 0.959004i \(0.591460\pi\)
\(308\) −2.56155 −0.145958
\(309\) −6.24621 −0.355335
\(310\) 0 0
\(311\) 21.9309 1.24359 0.621793 0.783182i \(-0.286405\pi\)
0.621793 + 0.783182i \(0.286405\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −1.31534 −0.0742290
\(315\) 1.56155 0.0879835
\(316\) −6.56155 −0.369116
\(317\) 12.4384 0.698613 0.349306 0.937009i \(-0.386417\pi\)
0.349306 + 0.937009i \(0.386417\pi\)
\(318\) −7.00000 −0.392541
\(319\) 15.6847 0.878172
\(320\) 1.56155 0.0872935
\(321\) −13.9309 −0.777545
\(322\) −1.12311 −0.0625882
\(323\) −0.315342 −0.0175461
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 24.4924 1.35651
\(327\) 12.2462 0.677217
\(328\) −11.2462 −0.620968
\(329\) −0.315342 −0.0173853
\(330\) 4.00000 0.220193
\(331\) 19.3693 1.06463 0.532317 0.846545i \(-0.321321\pi\)
0.532317 + 0.846545i \(0.321321\pi\)
\(332\) −5.12311 −0.281167
\(333\) −2.43845 −0.133626
\(334\) 12.4924 0.683555
\(335\) 14.2462 0.778354
\(336\) −1.00000 −0.0545545
\(337\) −13.4924 −0.734979 −0.367490 0.930028i \(-0.619783\pi\)
−0.367490 + 0.930028i \(0.619783\pi\)
\(338\) 0 0
\(339\) −1.56155 −0.0848119
\(340\) 0.192236 0.0104255
\(341\) 0 0
\(342\) −2.56155 −0.138513
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 1.75379 0.0944208
\(346\) −18.0000 −0.967686
\(347\) 15.0540 0.808140 0.404070 0.914728i \(-0.367595\pi\)
0.404070 + 0.914728i \(0.367595\pi\)
\(348\) 6.12311 0.328233
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) −2.56155 −0.136921
\(351\) 0 0
\(352\) −2.56155 −0.136531
\(353\) −33.8078 −1.79941 −0.899703 0.436503i \(-0.856217\pi\)
−0.899703 + 0.436503i \(0.856217\pi\)
\(354\) −5.12311 −0.272290
\(355\) −17.7538 −0.942273
\(356\) 3.43845 0.182237
\(357\) −0.123106 −0.00651544
\(358\) −1.75379 −0.0926906
\(359\) −1.75379 −0.0925614 −0.0462807 0.998928i \(-0.514737\pi\)
−0.0462807 + 0.998928i \(0.514737\pi\)
\(360\) 1.56155 0.0823011
\(361\) −12.4384 −0.654655
\(362\) 13.2462 0.696205
\(363\) 4.43845 0.232958
\(364\) 0 0
\(365\) −3.80776 −0.199307
\(366\) 11.2462 0.587849
\(367\) 13.6155 0.710725 0.355362 0.934729i \(-0.384357\pi\)
0.355362 + 0.934729i \(0.384357\pi\)
\(368\) −1.12311 −0.0585459
\(369\) −11.2462 −0.585454
\(370\) −3.80776 −0.197956
\(371\) 7.00000 0.363422
\(372\) 0 0
\(373\) −3.06913 −0.158914 −0.0794568 0.996838i \(-0.525319\pi\)
−0.0794568 + 0.996838i \(0.525319\pi\)
\(374\) −0.315342 −0.0163059
\(375\) 11.8078 0.609750
\(376\) −0.315342 −0.0162625
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 37.6155 1.93218 0.966090 0.258206i \(-0.0831312\pi\)
0.966090 + 0.258206i \(0.0831312\pi\)
\(380\) −4.00000 −0.205196
\(381\) −12.4924 −0.640006
\(382\) −5.12311 −0.262121
\(383\) −12.8078 −0.654446 −0.327223 0.944947i \(-0.606113\pi\)
−0.327223 + 0.944947i \(0.606113\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) 0.123106 0.00626591
\(387\) −8.00000 −0.406663
\(388\) −0.876894 −0.0445176
\(389\) −5.31534 −0.269499 −0.134749 0.990880i \(-0.543023\pi\)
−0.134749 + 0.990880i \(0.543023\pi\)
\(390\) 0 0
\(391\) −0.138261 −0.00699214
\(392\) 1.00000 0.0505076
\(393\) −14.2462 −0.718626
\(394\) 12.5616 0.632842
\(395\) −10.2462 −0.515543
\(396\) −2.56155 −0.128723
\(397\) 27.9309 1.40181 0.700905 0.713255i \(-0.252780\pi\)
0.700905 + 0.713255i \(0.252780\pi\)
\(398\) −16.0000 −0.802008
\(399\) 2.56155 0.128238
\(400\) −2.56155 −0.128078
\(401\) −1.80776 −0.0902754 −0.0451377 0.998981i \(-0.514373\pi\)
−0.0451377 + 0.998981i \(0.514373\pi\)
\(402\) −9.12311 −0.455019
\(403\) 0 0
\(404\) −5.80776 −0.288947
\(405\) 1.56155 0.0775942
\(406\) −6.12311 −0.303885
\(407\) 6.24621 0.309613
\(408\) −0.123106 −0.00609464
\(409\) −11.5616 −0.571682 −0.285841 0.958277i \(-0.592273\pi\)
−0.285841 + 0.958277i \(0.592273\pi\)
\(410\) −17.5616 −0.867303
\(411\) −12.4384 −0.613543
\(412\) 6.24621 0.307729
\(413\) 5.12311 0.252092
\(414\) −1.12311 −0.0551976
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 5.43845 0.266322
\(418\) 6.56155 0.320936
\(419\) 23.3693 1.14167 0.570833 0.821066i \(-0.306620\pi\)
0.570833 + 0.821066i \(0.306620\pi\)
\(420\) −1.56155 −0.0761960
\(421\) −22.3002 −1.08684 −0.543422 0.839459i \(-0.682872\pi\)
−0.543422 + 0.839459i \(0.682872\pi\)
\(422\) 5.12311 0.249389
\(423\) −0.315342 −0.0153324
\(424\) 7.00000 0.339950
\(425\) −0.315342 −0.0152963
\(426\) 11.3693 0.550845
\(427\) −11.2462 −0.544242
\(428\) 13.9309 0.673374
\(429\) 0 0
\(430\) −12.4924 −0.602438
\(431\) 19.3693 0.932987 0.466494 0.884525i \(-0.345517\pi\)
0.466494 + 0.884525i \(0.345517\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −21.3153 −1.02435 −0.512175 0.858881i \(-0.671160\pi\)
−0.512175 + 0.858881i \(0.671160\pi\)
\(434\) 0 0
\(435\) 9.56155 0.458441
\(436\) −12.2462 −0.586487
\(437\) 2.87689 0.137621
\(438\) 2.43845 0.116514
\(439\) −26.2462 −1.25266 −0.626332 0.779557i \(-0.715444\pi\)
−0.626332 + 0.779557i \(0.715444\pi\)
\(440\) −4.00000 −0.190693
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −27.0540 −1.28537 −0.642687 0.766129i \(-0.722180\pi\)
−0.642687 + 0.766129i \(0.722180\pi\)
\(444\) 2.43845 0.115724
\(445\) 5.36932 0.254530
\(446\) 9.75379 0.461855
\(447\) 15.5616 0.736036
\(448\) 1.00000 0.0472456
\(449\) −6.63068 −0.312921 −0.156461 0.987684i \(-0.550008\pi\)
−0.156461 + 0.987684i \(0.550008\pi\)
\(450\) −2.56155 −0.120753
\(451\) 28.8078 1.35650
\(452\) 1.56155 0.0734493
\(453\) 21.9309 1.03040
\(454\) −10.8769 −0.510478
\(455\) 0 0
\(456\) 2.56155 0.119956
\(457\) 28.9309 1.35333 0.676665 0.736291i \(-0.263425\pi\)
0.676665 + 0.736291i \(0.263425\pi\)
\(458\) 17.0540 0.796880
\(459\) −0.123106 −0.00574608
\(460\) −1.75379 −0.0817708
\(461\) 6.05398 0.281962 0.140981 0.990012i \(-0.454974\pi\)
0.140981 + 0.990012i \(0.454974\pi\)
\(462\) 2.56155 0.119174
\(463\) 12.3153 0.572342 0.286171 0.958178i \(-0.407617\pi\)
0.286171 + 0.958178i \(0.407617\pi\)
\(464\) −6.12311 −0.284258
\(465\) 0 0
\(466\) 1.36932 0.0634324
\(467\) −36.9848 −1.71145 −0.855727 0.517427i \(-0.826890\pi\)
−0.855727 + 0.517427i \(0.826890\pi\)
\(468\) 0 0
\(469\) 9.12311 0.421266
\(470\) −0.492423 −0.0227138
\(471\) 1.31534 0.0606077
\(472\) 5.12311 0.235810
\(473\) 20.4924 0.942243
\(474\) 6.56155 0.301382
\(475\) 6.56155 0.301065
\(476\) 0.123106 0.00564254
\(477\) 7.00000 0.320508
\(478\) 19.3693 0.885932
\(479\) −1.93087 −0.0882237 −0.0441118 0.999027i \(-0.514046\pi\)
−0.0441118 + 0.999027i \(0.514046\pi\)
\(480\) −1.56155 −0.0712748
\(481\) 0 0
\(482\) −0.192236 −0.00875611
\(483\) 1.12311 0.0511031
\(484\) −4.43845 −0.201748
\(485\) −1.36932 −0.0621775
\(486\) −1.00000 −0.0453609
\(487\) 23.6847 1.07325 0.536627 0.843819i \(-0.319698\pi\)
0.536627 + 0.843819i \(0.319698\pi\)
\(488\) −11.2462 −0.509092
\(489\) −24.4924 −1.10759
\(490\) 1.56155 0.0705438
\(491\) −16.4924 −0.744293 −0.372146 0.928174i \(-0.621378\pi\)
−0.372146 + 0.928174i \(0.621378\pi\)
\(492\) 11.2462 0.507018
\(493\) −0.753789 −0.0339489
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −11.3693 −0.509984
\(498\) 5.12311 0.229572
\(499\) 30.2462 1.35401 0.677003 0.735980i \(-0.263278\pi\)
0.677003 + 0.735980i \(0.263278\pi\)
\(500\) −11.8078 −0.528059
\(501\) −12.4924 −0.558120
\(502\) −13.6155 −0.607691
\(503\) −28.4924 −1.27041 −0.635207 0.772342i \(-0.719085\pi\)
−0.635207 + 0.772342i \(0.719085\pi\)
\(504\) 1.00000 0.0445435
\(505\) −9.06913 −0.403571
\(506\) 2.87689 0.127894
\(507\) 0 0
\(508\) 12.4924 0.554262
\(509\) −41.8078 −1.85310 −0.926548 0.376176i \(-0.877239\pi\)
−0.926548 + 0.376176i \(0.877239\pi\)
\(510\) −0.192236 −0.00851235
\(511\) −2.43845 −0.107871
\(512\) 1.00000 0.0441942
\(513\) 2.56155 0.113095
\(514\) −24.3693 −1.07488
\(515\) 9.75379 0.429803
\(516\) 8.00000 0.352180
\(517\) 0.807764 0.0355254
\(518\) −2.43845 −0.107139
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 25.8769 1.13369 0.566844 0.823825i \(-0.308164\pi\)
0.566844 + 0.823825i \(0.308164\pi\)
\(522\) −6.12311 −0.268001
\(523\) −16.8078 −0.734952 −0.367476 0.930033i \(-0.619778\pi\)
−0.367476 + 0.930033i \(0.619778\pi\)
\(524\) 14.2462 0.622349
\(525\) 2.56155 0.111795
\(526\) 5.12311 0.223378
\(527\) 0 0
\(528\) 2.56155 0.111477
\(529\) −21.7386 −0.945158
\(530\) 10.9309 0.474807
\(531\) 5.12311 0.222324
\(532\) −2.56155 −0.111057
\(533\) 0 0
\(534\) −3.43845 −0.148796
\(535\) 21.7538 0.940498
\(536\) 9.12311 0.394058
\(537\) 1.75379 0.0756816
\(538\) 27.1231 1.16936
\(539\) −2.56155 −0.110334
\(540\) −1.56155 −0.0671985
\(541\) 19.1771 0.824487 0.412244 0.911074i \(-0.364745\pi\)
0.412244 + 0.911074i \(0.364745\pi\)
\(542\) −22.7386 −0.976708
\(543\) −13.2462 −0.568449
\(544\) 0.123106 0.00527811
\(545\) −19.1231 −0.819144
\(546\) 0 0
\(547\) −33.6155 −1.43730 −0.718648 0.695374i \(-0.755239\pi\)
−0.718648 + 0.695374i \(0.755239\pi\)
\(548\) 12.4384 0.531344
\(549\) −11.2462 −0.479977
\(550\) 6.56155 0.279786
\(551\) 15.6847 0.668189
\(552\) 1.12311 0.0478025
\(553\) −6.56155 −0.279026
\(554\) −19.5616 −0.831091
\(555\) 3.80776 0.161631
\(556\) −5.43845 −0.230642
\(557\) 12.1231 0.513672 0.256836 0.966455i \(-0.417320\pi\)
0.256836 + 0.966455i \(0.417320\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.56155 0.0659877
\(561\) 0.315342 0.0133137
\(562\) 12.9309 0.545456
\(563\) −5.75379 −0.242493 −0.121247 0.992622i \(-0.538689\pi\)
−0.121247 + 0.992622i \(0.538689\pi\)
\(564\) 0.315342 0.0132783
\(565\) 2.43845 0.102586
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) −11.3693 −0.477046
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 4.00000 0.167542
\(571\) −21.1231 −0.883974 −0.441987 0.897021i \(-0.645726\pi\)
−0.441987 + 0.897021i \(0.645726\pi\)
\(572\) 0 0
\(573\) 5.12311 0.214021
\(574\) −11.2462 −0.469408
\(575\) 2.87689 0.119975
\(576\) 1.00000 0.0416667
\(577\) −35.5616 −1.48045 −0.740223 0.672361i \(-0.765280\pi\)
−0.740223 + 0.672361i \(0.765280\pi\)
\(578\) −16.9848 −0.706476
\(579\) −0.123106 −0.00511610
\(580\) −9.56155 −0.397022
\(581\) −5.12311 −0.212542
\(582\) 0.876894 0.0363484
\(583\) −17.9309 −0.742621
\(584\) −2.43845 −0.100904
\(585\) 0 0
\(586\) −6.93087 −0.286312
\(587\) 2.24621 0.0927111 0.0463555 0.998925i \(-0.485239\pi\)
0.0463555 + 0.998925i \(0.485239\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) −12.5616 −0.516713
\(592\) −2.43845 −0.100220
\(593\) −33.9848 −1.39559 −0.697795 0.716297i \(-0.745835\pi\)
−0.697795 + 0.716297i \(0.745835\pi\)
\(594\) 2.56155 0.105102
\(595\) 0.192236 0.00788091
\(596\) −15.5616 −0.637426
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 23.8617 0.974964 0.487482 0.873133i \(-0.337915\pi\)
0.487482 + 0.873133i \(0.337915\pi\)
\(600\) 2.56155 0.104575
\(601\) 28.3002 1.15439 0.577194 0.816607i \(-0.304148\pi\)
0.577194 + 0.816607i \(0.304148\pi\)
\(602\) −8.00000 −0.326056
\(603\) 9.12311 0.371522
\(604\) −21.9309 −0.892354
\(605\) −6.93087 −0.281780
\(606\) 5.80776 0.235924
\(607\) −48.9848 −1.98823 −0.994117 0.108314i \(-0.965455\pi\)
−0.994117 + 0.108314i \(0.965455\pi\)
\(608\) −2.56155 −0.103885
\(609\) 6.12311 0.248121
\(610\) −17.5616 −0.711046
\(611\) 0 0
\(612\) 0.123106 0.00497625
\(613\) 21.5616 0.870863 0.435431 0.900222i \(-0.356596\pi\)
0.435431 + 0.900222i \(0.356596\pi\)
\(614\) −9.93087 −0.400777
\(615\) 17.5616 0.708150
\(616\) −2.56155 −0.103208
\(617\) 23.1771 0.933074 0.466537 0.884502i \(-0.345501\pi\)
0.466537 + 0.884502i \(0.345501\pi\)
\(618\) −6.24621 −0.251259
\(619\) 2.06913 0.0831654 0.0415827 0.999135i \(-0.486760\pi\)
0.0415827 + 0.999135i \(0.486760\pi\)
\(620\) 0 0
\(621\) 1.12311 0.0450687
\(622\) 21.9309 0.879348
\(623\) 3.43845 0.137758
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) −6.00000 −0.239808
\(627\) −6.56155 −0.262043
\(628\) −1.31534 −0.0524878
\(629\) −0.300187 −0.0119692
\(630\) 1.56155 0.0622138
\(631\) −10.0691 −0.400846 −0.200423 0.979709i \(-0.564232\pi\)
−0.200423 + 0.979709i \(0.564232\pi\)
\(632\) −6.56155 −0.261005
\(633\) −5.12311 −0.203625
\(634\) 12.4384 0.493994
\(635\) 19.5076 0.774135
\(636\) −7.00000 −0.277568
\(637\) 0 0
\(638\) 15.6847 0.620962
\(639\) −11.3693 −0.449763
\(640\) 1.56155 0.0617258
\(641\) 8.43845 0.333299 0.166649 0.986016i \(-0.446705\pi\)
0.166649 + 0.986016i \(0.446705\pi\)
\(642\) −13.9309 −0.549808
\(643\) 47.6847 1.88050 0.940250 0.340486i \(-0.110592\pi\)
0.940250 + 0.340486i \(0.110592\pi\)
\(644\) −1.12311 −0.0442566
\(645\) 12.4924 0.491889
\(646\) −0.315342 −0.0124069
\(647\) −8.94602 −0.351705 −0.175852 0.984417i \(-0.556268\pi\)
−0.175852 + 0.984417i \(0.556268\pi\)
\(648\) 1.00000 0.0392837
\(649\) −13.1231 −0.515127
\(650\) 0 0
\(651\) 0 0
\(652\) 24.4924 0.959197
\(653\) 20.5616 0.804636 0.402318 0.915500i \(-0.368205\pi\)
0.402318 + 0.915500i \(0.368205\pi\)
\(654\) 12.2462 0.478865
\(655\) 22.2462 0.869231
\(656\) −11.2462 −0.439091
\(657\) −2.43845 −0.0951329
\(658\) −0.315342 −0.0122933
\(659\) 2.06913 0.0806019 0.0403009 0.999188i \(-0.487168\pi\)
0.0403009 + 0.999188i \(0.487168\pi\)
\(660\) 4.00000 0.155700
\(661\) −20.0540 −0.780009 −0.390005 0.920813i \(-0.627527\pi\)
−0.390005 + 0.920813i \(0.627527\pi\)
\(662\) 19.3693 0.752810
\(663\) 0 0
\(664\) −5.12311 −0.198815
\(665\) −4.00000 −0.155113
\(666\) −2.43845 −0.0944879
\(667\) 6.87689 0.266274
\(668\) 12.4924 0.483346
\(669\) −9.75379 −0.377103
\(670\) 14.2462 0.550379
\(671\) 28.8078 1.11211
\(672\) −1.00000 −0.0385758
\(673\) −2.75379 −0.106151 −0.0530754 0.998591i \(-0.516902\pi\)
−0.0530754 + 0.998591i \(0.516902\pi\)
\(674\) −13.4924 −0.519709
\(675\) 2.56155 0.0985942
\(676\) 0 0
\(677\) −24.7386 −0.950783 −0.475391 0.879774i \(-0.657694\pi\)
−0.475391 + 0.879774i \(0.657694\pi\)
\(678\) −1.56155 −0.0599711
\(679\) −0.876894 −0.0336521
\(680\) 0.192236 0.00737191
\(681\) 10.8769 0.416803
\(682\) 0 0
\(683\) 40.4924 1.54940 0.774700 0.632329i \(-0.217901\pi\)
0.774700 + 0.632329i \(0.217901\pi\)
\(684\) −2.56155 −0.0979434
\(685\) 19.4233 0.742126
\(686\) 1.00000 0.0381802
\(687\) −17.0540 −0.650650
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 1.75379 0.0667656
\(691\) −32.9848 −1.25480 −0.627401 0.778696i \(-0.715881\pi\)
−0.627401 + 0.778696i \(0.715881\pi\)
\(692\) −18.0000 −0.684257
\(693\) −2.56155 −0.0973053
\(694\) 15.0540 0.571441
\(695\) −8.49242 −0.322136
\(696\) 6.12311 0.232096
\(697\) −1.38447 −0.0524406
\(698\) −18.0000 −0.681310
\(699\) −1.36932 −0.0517923
\(700\) −2.56155 −0.0968176
\(701\) −39.3002 −1.48435 −0.742174 0.670207i \(-0.766205\pi\)
−0.742174 + 0.670207i \(0.766205\pi\)
\(702\) 0 0
\(703\) 6.24621 0.235580
\(704\) −2.56155 −0.0965422
\(705\) 0.492423 0.0185457
\(706\) −33.8078 −1.27237
\(707\) −5.80776 −0.218423
\(708\) −5.12311 −0.192538
\(709\) −38.3002 −1.43839 −0.719197 0.694806i \(-0.755490\pi\)
−0.719197 + 0.694806i \(0.755490\pi\)
\(710\) −17.7538 −0.666288
\(711\) −6.56155 −0.246077
\(712\) 3.43845 0.128861
\(713\) 0 0
\(714\) −0.123106 −0.00460711
\(715\) 0 0
\(716\) −1.75379 −0.0655422
\(717\) −19.3693 −0.723360
\(718\) −1.75379 −0.0654508
\(719\) 0.807764 0.0301245 0.0150623 0.999887i \(-0.495205\pi\)
0.0150623 + 0.999887i \(0.495205\pi\)
\(720\) 1.56155 0.0581956
\(721\) 6.24621 0.232621
\(722\) −12.4384 −0.462911
\(723\) 0.192236 0.00714933
\(724\) 13.2462 0.492292
\(725\) 15.6847 0.582514
\(726\) 4.43845 0.164726
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.80776 −0.140932
\(731\) −0.984845 −0.0364258
\(732\) 11.2462 0.415672
\(733\) −28.8617 −1.06603 −0.533016 0.846105i \(-0.678942\pi\)
−0.533016 + 0.846105i \(0.678942\pi\)
\(734\) 13.6155 0.502558
\(735\) −1.56155 −0.0575987
\(736\) −1.12311 −0.0413982
\(737\) −23.3693 −0.860820
\(738\) −11.2462 −0.413979
\(739\) 15.3693 0.565369 0.282685 0.959213i \(-0.408775\pi\)
0.282685 + 0.959213i \(0.408775\pi\)
\(740\) −3.80776 −0.139976
\(741\) 0 0
\(742\) 7.00000 0.256978
\(743\) −4.49242 −0.164811 −0.0824055 0.996599i \(-0.526260\pi\)
−0.0824055 + 0.996599i \(0.526260\pi\)
\(744\) 0 0
\(745\) −24.3002 −0.890290
\(746\) −3.06913 −0.112369
\(747\) −5.12311 −0.187445
\(748\) −0.315342 −0.0115300
\(749\) 13.9309 0.509023
\(750\) 11.8078 0.431159
\(751\) −40.8078 −1.48910 −0.744548 0.667569i \(-0.767335\pi\)
−0.744548 + 0.667569i \(0.767335\pi\)
\(752\) −0.315342 −0.0114993
\(753\) 13.6155 0.496177
\(754\) 0 0
\(755\) −34.2462 −1.24635
\(756\) −1.00000 −0.0363696
\(757\) 52.7386 1.91682 0.958409 0.285398i \(-0.0921257\pi\)
0.958409 + 0.285398i \(0.0921257\pi\)
\(758\) 37.6155 1.36626
\(759\) −2.87689 −0.104425
\(760\) −4.00000 −0.145095
\(761\) 50.9848 1.84820 0.924100 0.382152i \(-0.124817\pi\)
0.924100 + 0.382152i \(0.124817\pi\)
\(762\) −12.4924 −0.452553
\(763\) −12.2462 −0.443343
\(764\) −5.12311 −0.185347
\(765\) 0.192236 0.00695030
\(766\) −12.8078 −0.462763
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 4.87689 0.175865 0.0879327 0.996126i \(-0.471974\pi\)
0.0879327 + 0.996126i \(0.471974\pi\)
\(770\) −4.00000 −0.144150
\(771\) 24.3693 0.877640
\(772\) 0.123106 0.00443067
\(773\) 20.7386 0.745917 0.372958 0.927848i \(-0.378343\pi\)
0.372958 + 0.927848i \(0.378343\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −0.876894 −0.0314787
\(777\) 2.43845 0.0874788
\(778\) −5.31534 −0.190564
\(779\) 28.8078 1.03215
\(780\) 0 0
\(781\) 29.1231 1.04211
\(782\) −0.138261 −0.00494419
\(783\) 6.12311 0.218822
\(784\) 1.00000 0.0357143
\(785\) −2.05398 −0.0733095
\(786\) −14.2462 −0.508146
\(787\) −0.177081 −0.00631225 −0.00315613 0.999995i \(-0.501005\pi\)
−0.00315613 + 0.999995i \(0.501005\pi\)
\(788\) 12.5616 0.447487
\(789\) −5.12311 −0.182387
\(790\) −10.2462 −0.364544
\(791\) 1.56155 0.0555224
\(792\) −2.56155 −0.0910208
\(793\) 0 0
\(794\) 27.9309 0.991229
\(795\) −10.9309 −0.387678
\(796\) −16.0000 −0.567105
\(797\) 50.4924 1.78853 0.894267 0.447534i \(-0.147698\pi\)
0.894267 + 0.447534i \(0.147698\pi\)
\(798\) 2.56155 0.0906780
\(799\) −0.0388203 −0.00137336
\(800\) −2.56155 −0.0905646
\(801\) 3.43845 0.121492
\(802\) −1.80776 −0.0638344
\(803\) 6.24621 0.220424
\(804\) −9.12311 −0.321747
\(805\) −1.75379 −0.0618129
\(806\) 0 0
\(807\) −27.1231 −0.954779
\(808\) −5.80776 −0.204316
\(809\) 8.93087 0.313993 0.156996 0.987599i \(-0.449819\pi\)
0.156996 + 0.987599i \(0.449819\pi\)
\(810\) 1.56155 0.0548674
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) −6.12311 −0.214879
\(813\) 22.7386 0.797479
\(814\) 6.24621 0.218930
\(815\) 38.2462 1.33971
\(816\) −0.123106 −0.00430956
\(817\) 20.4924 0.716939
\(818\) −11.5616 −0.404240
\(819\) 0 0
\(820\) −17.5616 −0.613276
\(821\) 27.3002 0.952783 0.476392 0.879233i \(-0.341944\pi\)
0.476392 + 0.879233i \(0.341944\pi\)
\(822\) −12.4384 −0.433841
\(823\) −38.2462 −1.33318 −0.666590 0.745425i \(-0.732247\pi\)
−0.666590 + 0.745425i \(0.732247\pi\)
\(824\) 6.24621 0.217597
\(825\) −6.56155 −0.228444
\(826\) 5.12311 0.178256
\(827\) 34.2462 1.19086 0.595429 0.803408i \(-0.296982\pi\)
0.595429 + 0.803408i \(0.296982\pi\)
\(828\) −1.12311 −0.0390306
\(829\) −0.369317 −0.0128269 −0.00641345 0.999979i \(-0.502041\pi\)
−0.00641345 + 0.999979i \(0.502041\pi\)
\(830\) −8.00000 −0.277684
\(831\) 19.5616 0.678583
\(832\) 0 0
\(833\) 0.123106 0.00426536
\(834\) 5.43845 0.188318
\(835\) 19.5076 0.675088
\(836\) 6.56155 0.226936
\(837\) 0 0
\(838\) 23.3693 0.807280
\(839\) 18.2462 0.629929 0.314965 0.949103i \(-0.398007\pi\)
0.314965 + 0.949103i \(0.398007\pi\)
\(840\) −1.56155 −0.0538787
\(841\) 8.49242 0.292842
\(842\) −22.3002 −0.768515
\(843\) −12.9309 −0.445363
\(844\) 5.12311 0.176345
\(845\) 0 0
\(846\) −0.315342 −0.0108417
\(847\) −4.43845 −0.152507
\(848\) 7.00000 0.240381
\(849\) 20.0000 0.686398
\(850\) −0.315342 −0.0108161
\(851\) 2.73863 0.0938792
\(852\) 11.3693 0.389506
\(853\) 30.3693 1.03983 0.519913 0.854219i \(-0.325964\pi\)
0.519913 + 0.854219i \(0.325964\pi\)
\(854\) −11.2462 −0.384837
\(855\) −4.00000 −0.136797
\(856\) 13.9309 0.476147
\(857\) 16.4384 0.561527 0.280763 0.959777i \(-0.409412\pi\)
0.280763 + 0.959777i \(0.409412\pi\)
\(858\) 0 0
\(859\) 17.9309 0.611793 0.305897 0.952065i \(-0.401044\pi\)
0.305897 + 0.952065i \(0.401044\pi\)
\(860\) −12.4924 −0.425988
\(861\) 11.2462 0.383270
\(862\) 19.3693 0.659722
\(863\) 41.1231 1.39985 0.699923 0.714218i \(-0.253217\pi\)
0.699923 + 0.714218i \(0.253217\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.1080 −0.955699
\(866\) −21.3153 −0.724325
\(867\) 16.9848 0.576836
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 9.56155 0.324167
\(871\) 0 0
\(872\) −12.2462 −0.414709
\(873\) −0.876894 −0.0296784
\(874\) 2.87689 0.0973124
\(875\) −11.8078 −0.399175
\(876\) 2.43845 0.0823875
\(877\) −28.5464 −0.963943 −0.481972 0.876187i \(-0.660079\pi\)
−0.481972 + 0.876187i \(0.660079\pi\)
\(878\) −26.2462 −0.885767
\(879\) 6.93087 0.233772
\(880\) −4.00000 −0.134840
\(881\) 19.9460 0.671999 0.335999 0.941862i \(-0.390926\pi\)
0.335999 + 0.941862i \(0.390926\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) −27.0540 −0.908896
\(887\) −8.94602 −0.300378 −0.150189 0.988657i \(-0.547988\pi\)
−0.150189 + 0.988657i \(0.547988\pi\)
\(888\) 2.43845 0.0818289
\(889\) 12.4924 0.418982
\(890\) 5.36932 0.179980
\(891\) −2.56155 −0.0858152
\(892\) 9.75379 0.326581
\(893\) 0.807764 0.0270308
\(894\) 15.5616 0.520456
\(895\) −2.73863 −0.0915424
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −6.63068 −0.221269
\(899\) 0 0
\(900\) −2.56155 −0.0853851
\(901\) 0.861739 0.0287087
\(902\) 28.8078 0.959194
\(903\) 8.00000 0.266223
\(904\) 1.56155 0.0519365
\(905\) 20.6847 0.687581
\(906\) 21.9309 0.728604
\(907\) −1.12311 −0.0372921 −0.0186461 0.999826i \(-0.505936\pi\)
−0.0186461 + 0.999826i \(0.505936\pi\)
\(908\) −10.8769 −0.360962
\(909\) −5.80776 −0.192631
\(910\) 0 0
\(911\) 13.6155 0.451103 0.225551 0.974231i \(-0.427582\pi\)
0.225551 + 0.974231i \(0.427582\pi\)
\(912\) 2.56155 0.0848215
\(913\) 13.1231 0.434311
\(914\) 28.9309 0.956948
\(915\) 17.5616 0.580567
\(916\) 17.0540 0.563479
\(917\) 14.2462 0.470451
\(918\) −0.123106 −0.00406309
\(919\) −45.9309 −1.51512 −0.757560 0.652766i \(-0.773609\pi\)
−0.757560 + 0.652766i \(0.773609\pi\)
\(920\) −1.75379 −0.0578207
\(921\) 9.93087 0.327233
\(922\) 6.05398 0.199377
\(923\) 0 0
\(924\) 2.56155 0.0842689
\(925\) 6.24621 0.205374
\(926\) 12.3153 0.404707
\(927\) 6.24621 0.205152
\(928\) −6.12311 −0.201001
\(929\) −2.75379 −0.0903489 −0.0451744 0.998979i \(-0.514384\pi\)
−0.0451744 + 0.998979i \(0.514384\pi\)
\(930\) 0 0
\(931\) −2.56155 −0.0839515
\(932\) 1.36932 0.0448535
\(933\) −21.9309 −0.717984
\(934\) −36.9848 −1.21018
\(935\) −0.492423 −0.0161039
\(936\) 0 0
\(937\) −34.4384 −1.12506 −0.562528 0.826779i \(-0.690171\pi\)
−0.562528 + 0.826779i \(0.690171\pi\)
\(938\) 9.12311 0.297880
\(939\) 6.00000 0.195803
\(940\) −0.492423 −0.0160611
\(941\) −10.6307 −0.346550 −0.173275 0.984873i \(-0.555435\pi\)
−0.173275 + 0.984873i \(0.555435\pi\)
\(942\) 1.31534 0.0428561
\(943\) 12.6307 0.411312
\(944\) 5.12311 0.166743
\(945\) −1.56155 −0.0507973
\(946\) 20.4924 0.666266
\(947\) 58.5616 1.90299 0.951497 0.307657i \(-0.0995449\pi\)
0.951497 + 0.307657i \(0.0995449\pi\)
\(948\) 6.56155 0.213109
\(949\) 0 0
\(950\) 6.56155 0.212885
\(951\) −12.4384 −0.403344
\(952\) 0.123106 0.00398988
\(953\) 12.8769 0.417124 0.208562 0.978009i \(-0.433122\pi\)
0.208562 + 0.978009i \(0.433122\pi\)
\(954\) 7.00000 0.226633
\(955\) −8.00000 −0.258874
\(956\) 19.3693 0.626448
\(957\) −15.6847 −0.507013
\(958\) −1.93087 −0.0623836
\(959\) 12.4384 0.401658
\(960\) −1.56155 −0.0503989
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 13.9309 0.448916
\(964\) −0.192236 −0.00619150
\(965\) 0.192236 0.00618829
\(966\) 1.12311 0.0361353
\(967\) −28.4924 −0.916255 −0.458127 0.888887i \(-0.651480\pi\)
−0.458127 + 0.888887i \(0.651480\pi\)
\(968\) −4.43845 −0.142657
\(969\) 0.315342 0.0101302
\(970\) −1.36932 −0.0439661
\(971\) 41.6155 1.33551 0.667753 0.744383i \(-0.267256\pi\)
0.667753 + 0.744383i \(0.267256\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −5.43845 −0.174349
\(974\) 23.6847 0.758905
\(975\) 0 0
\(976\) −11.2462 −0.359982
\(977\) −41.1771 −1.31737 −0.658686 0.752418i \(-0.728887\pi\)
−0.658686 + 0.752418i \(0.728887\pi\)
\(978\) −24.4924 −0.783181
\(979\) −8.80776 −0.281497
\(980\) 1.56155 0.0498820
\(981\) −12.2462 −0.390991
\(982\) −16.4924 −0.526294
\(983\) 18.7386 0.597670 0.298835 0.954305i \(-0.403402\pi\)
0.298835 + 0.954305i \(0.403402\pi\)
\(984\) 11.2462 0.358516
\(985\) 19.6155 0.625003
\(986\) −0.753789 −0.0240055
\(987\) 0.315342 0.0100374
\(988\) 0 0
\(989\) 8.98485 0.285701
\(990\) −4.00000 −0.127128
\(991\) −28.6695 −0.910717 −0.455358 0.890308i \(-0.650489\pi\)
−0.455358 + 0.890308i \(0.650489\pi\)
\(992\) 0 0
\(993\) −19.3693 −0.614667
\(994\) −11.3693 −0.360613
\(995\) −24.9848 −0.792073
\(996\) 5.12311 0.162332
\(997\) −41.9848 −1.32967 −0.664837 0.746989i \(-0.731499\pi\)
−0.664837 + 0.746989i \(0.731499\pi\)
\(998\) 30.2462 0.957427
\(999\) 2.43845 0.0771491
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.bt.1.2 2
13.3 even 3 546.2.l.l.295.2 yes 4
13.9 even 3 546.2.l.l.211.2 4
13.12 even 2 7098.2.a.bi.1.1 2
39.29 odd 6 1638.2.r.y.1387.1 4
39.35 odd 6 1638.2.r.y.757.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.l.211.2 4 13.9 even 3
546.2.l.l.295.2 yes 4 13.3 even 3
1638.2.r.y.757.1 4 39.35 odd 6
1638.2.r.y.1387.1 4 39.29 odd 6
7098.2.a.bi.1.1 2 13.12 even 2
7098.2.a.bt.1.2 2 1.1 even 1 trivial