Properties

Label 7098.2.a.bt.1.1
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.56155 q^{10} +1.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} +2.56155 q^{15} +1.00000 q^{16} -8.12311 q^{17} +1.00000 q^{18} +1.56155 q^{19} -2.56155 q^{20} -1.00000 q^{21} +1.56155 q^{22} +7.12311 q^{23} -1.00000 q^{24} +1.56155 q^{25} -1.00000 q^{27} +1.00000 q^{28} +2.12311 q^{29} +2.56155 q^{30} +1.00000 q^{32} -1.56155 q^{33} -8.12311 q^{34} -2.56155 q^{35} +1.00000 q^{36} -6.56155 q^{37} +1.56155 q^{38} -2.56155 q^{40} +5.24621 q^{41} -1.00000 q^{42} -8.00000 q^{43} +1.56155 q^{44} -2.56155 q^{45} +7.12311 q^{46} -12.6847 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.56155 q^{50} +8.12311 q^{51} +7.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -1.56155 q^{57} +2.12311 q^{58} -3.12311 q^{59} +2.56155 q^{60} +5.24621 q^{61} +1.00000 q^{63} +1.00000 q^{64} -1.56155 q^{66} +0.876894 q^{67} -8.12311 q^{68} -7.12311 q^{69} -2.56155 q^{70} +13.3693 q^{71} +1.00000 q^{72} -6.56155 q^{73} -6.56155 q^{74} -1.56155 q^{75} +1.56155 q^{76} +1.56155 q^{77} -2.43845 q^{79} -2.56155 q^{80} +1.00000 q^{81} +5.24621 q^{82} +3.12311 q^{83} -1.00000 q^{84} +20.8078 q^{85} -8.00000 q^{86} -2.12311 q^{87} +1.56155 q^{88} +7.56155 q^{89} -2.56155 q^{90} +7.12311 q^{92} -12.6847 q^{94} -4.00000 q^{95} -1.00000 q^{96} -9.12311 q^{97} +1.00000 q^{98} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} - q^{10} - q^{11} - 2q^{12} + 2q^{14} + q^{15} + 2q^{16} - 8q^{17} + 2q^{18} - q^{19} - q^{20} - 2q^{21} - q^{22} + 6q^{23} - 2q^{24} - q^{25} - 2q^{27} + 2q^{28} - 4q^{29} + q^{30} + 2q^{32} + q^{33} - 8q^{34} - q^{35} + 2q^{36} - 9q^{37} - q^{38} - q^{40} - 6q^{41} - 2q^{42} - 16q^{43} - q^{44} - q^{45} + 6q^{46} - 13q^{47} - 2q^{48} + 2q^{49} - q^{50} + 8q^{51} + 14q^{53} - 2q^{54} - 8q^{55} + 2q^{56} + q^{57} - 4q^{58} + 2q^{59} + q^{60} - 6q^{61} + 2q^{63} + 2q^{64} + q^{66} + 10q^{67} - 8q^{68} - 6q^{69} - q^{70} + 2q^{71} + 2q^{72} - 9q^{73} - 9q^{74} + q^{75} - q^{76} - q^{77} - 9q^{79} - q^{80} + 2q^{81} - 6q^{82} - 2q^{83} - 2q^{84} + 21q^{85} - 16q^{86} + 4q^{87} - q^{88} + 11q^{89} - q^{90} + 6q^{92} - 13q^{94} - 8q^{95} - 2q^{96} - 10q^{97} + 2q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) −2.56155 −1.14556 −0.572781 0.819709i \(-0.694135\pi\)
−0.572781 + 0.819709i \(0.694135\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.56155 −0.810034
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 2.56155 0.661390
\(16\) 1.00000 0.250000
\(17\) −8.12311 −1.97014 −0.985071 0.172147i \(-0.944930\pi\)
−0.985071 + 0.172147i \(0.944930\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.56155 0.358245 0.179122 0.983827i \(-0.442674\pi\)
0.179122 + 0.983827i \(0.442674\pi\)
\(20\) −2.56155 −0.572781
\(21\) −1.00000 −0.218218
\(22\) 1.56155 0.332924
\(23\) 7.12311 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.12311 0.394251 0.197125 0.980378i \(-0.436839\pi\)
0.197125 + 0.980378i \(0.436839\pi\)
\(30\) 2.56155 0.467673
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.56155 −0.271831
\(34\) −8.12311 −1.39310
\(35\) −2.56155 −0.432981
\(36\) 1.00000 0.166667
\(37\) −6.56155 −1.07871 −0.539356 0.842078i \(-0.681332\pi\)
−0.539356 + 0.842078i \(0.681332\pi\)
\(38\) 1.56155 0.253317
\(39\) 0 0
\(40\) −2.56155 −0.405017
\(41\) 5.24621 0.819321 0.409660 0.912238i \(-0.365647\pi\)
0.409660 + 0.912238i \(0.365647\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.56155 0.235413
\(45\) −2.56155 −0.381854
\(46\) 7.12311 1.05024
\(47\) −12.6847 −1.85025 −0.925124 0.379666i \(-0.876039\pi\)
−0.925124 + 0.379666i \(0.876039\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.56155 0.220837
\(51\) 8.12311 1.13746
\(52\) 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
\(57\) −1.56155 −0.206833
\(58\) 2.12311 0.278777
\(59\) −3.12311 −0.406594 −0.203297 0.979117i \(-0.565166\pi\)
−0.203297 + 0.979117i \(0.565166\pi\)
\(60\) 2.56155 0.330695
\(61\) 5.24621 0.671709 0.335854 0.941914i \(-0.390975\pi\)
0.335854 + 0.941914i \(0.390975\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.56155 −0.192214
\(67\) 0.876894 0.107130 0.0535648 0.998564i \(-0.482942\pi\)
0.0535648 + 0.998564i \(0.482942\pi\)
\(68\) −8.12311 −0.985071
\(69\) −7.12311 −0.857521
\(70\) −2.56155 −0.306164
\(71\) 13.3693 1.58665 0.793323 0.608801i \(-0.208349\pi\)
0.793323 + 0.608801i \(0.208349\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.56155 −0.767972 −0.383986 0.923339i \(-0.625449\pi\)
−0.383986 + 0.923339i \(0.625449\pi\)
\(74\) −6.56155 −0.762765
\(75\) −1.56155 −0.180313
\(76\) 1.56155 0.179122
\(77\) 1.56155 0.177955
\(78\) 0 0
\(79\) −2.43845 −0.274347 −0.137173 0.990547i \(-0.543802\pi\)
−0.137173 + 0.990547i \(0.543802\pi\)
\(80\) −2.56155 −0.286390
\(81\) 1.00000 0.111111
\(82\) 5.24621 0.579347
\(83\) 3.12311 0.342805 0.171403 0.985201i \(-0.445170\pi\)
0.171403 + 0.985201i \(0.445170\pi\)
\(84\) −1.00000 −0.109109
\(85\) 20.8078 2.25692
\(86\) −8.00000 −0.862662
\(87\) −2.12311 −0.227621
\(88\) 1.56155 0.166462
\(89\) 7.56155 0.801523 0.400761 0.916182i \(-0.368746\pi\)
0.400761 + 0.916182i \(0.368746\pi\)
\(90\) −2.56155 −0.270011
\(91\) 0 0
\(92\) 7.12311 0.742635
\(93\) 0 0
\(94\) −12.6847 −1.30832
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −9.12311 −0.926311 −0.463156 0.886277i \(-0.653283\pi\)
−0.463156 + 0.886277i \(0.653283\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.56155 0.156942
\(100\) 1.56155 0.156155
\(101\) 14.8078 1.47343 0.736714 0.676205i \(-0.236377\pi\)
0.736714 + 0.676205i \(0.236377\pi\)
\(102\) 8.12311 0.804307
\(103\) −10.2462 −1.00959 −0.504795 0.863239i \(-0.668432\pi\)
−0.504795 + 0.863239i \(0.668432\pi\)
\(104\) 0 0
\(105\) 2.56155 0.249982
\(106\) 7.00000 0.679900
\(107\) −14.9309 −1.44342 −0.721711 0.692195i \(-0.756644\pi\)
−0.721711 + 0.692195i \(0.756644\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.24621 0.406713 0.203357 0.979105i \(-0.434815\pi\)
0.203357 + 0.979105i \(0.434815\pi\)
\(110\) −4.00000 −0.381385
\(111\) 6.56155 0.622795
\(112\) 1.00000 0.0944911
\(113\) −2.56155 −0.240971 −0.120485 0.992715i \(-0.538445\pi\)
−0.120485 + 0.992715i \(0.538445\pi\)
\(114\) −1.56155 −0.146253
\(115\) −18.2462 −1.70147
\(116\) 2.12311 0.197125
\(117\) 0 0
\(118\) −3.12311 −0.287505
\(119\) −8.12311 −0.744644
\(120\) 2.56155 0.233837
\(121\) −8.56155 −0.778323
\(122\) 5.24621 0.474970
\(123\) −5.24621 −0.473035
\(124\) 0 0
\(125\) 8.80776 0.787790
\(126\) 1.00000 0.0890871
\(127\) −20.4924 −1.81841 −0.909204 0.416350i \(-0.863309\pi\)
−0.909204 + 0.416350i \(0.863309\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −2.24621 −0.196252 −0.0981262 0.995174i \(-0.531285\pi\)
−0.0981262 + 0.995174i \(0.531285\pi\)
\(132\) −1.56155 −0.135916
\(133\) 1.56155 0.135404
\(134\) 0.876894 0.0757521
\(135\) 2.56155 0.220463
\(136\) −8.12311 −0.696551
\(137\) 16.5616 1.41495 0.707474 0.706739i \(-0.249834\pi\)
0.707474 + 0.706739i \(0.249834\pi\)
\(138\) −7.12311 −0.606359
\(139\) −9.56155 −0.811000 −0.405500 0.914095i \(-0.632903\pi\)
−0.405500 + 0.914095i \(0.632903\pi\)
\(140\) −2.56155 −0.216491
\(141\) 12.6847 1.06824
\(142\) 13.3693 1.12193
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.43845 −0.451638
\(146\) −6.56155 −0.543038
\(147\) −1.00000 −0.0824786
\(148\) −6.56155 −0.539356
\(149\) −11.4384 −0.937074 −0.468537 0.883444i \(-0.655219\pi\)
−0.468537 + 0.883444i \(0.655219\pi\)
\(150\) −1.56155 −0.127500
\(151\) 6.93087 0.564026 0.282013 0.959411i \(-0.408998\pi\)
0.282013 + 0.959411i \(0.408998\pi\)
\(152\) 1.56155 0.126659
\(153\) −8.12311 −0.656714
\(154\) 1.56155 0.125834
\(155\) 0 0
\(156\) 0 0
\(157\) −13.6847 −1.09215 −0.546077 0.837735i \(-0.683880\pi\)
−0.546077 + 0.837735i \(0.683880\pi\)
\(158\) −2.43845 −0.193992
\(159\) −7.00000 −0.555136
\(160\) −2.56155 −0.202509
\(161\) 7.12311 0.561379
\(162\) 1.00000 0.0785674
\(163\) −8.49242 −0.665178 −0.332589 0.943072i \(-0.607922\pi\)
−0.332589 + 0.943072i \(0.607922\pi\)
\(164\) 5.24621 0.409660
\(165\) 4.00000 0.311400
\(166\) 3.12311 0.242400
\(167\) −20.4924 −1.58575 −0.792876 0.609383i \(-0.791417\pi\)
−0.792876 + 0.609383i \(0.791417\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 20.8078 1.59588
\(171\) 1.56155 0.119415
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −2.12311 −0.160952
\(175\) 1.56155 0.118042
\(176\) 1.56155 0.117706
\(177\) 3.12311 0.234747
\(178\) 7.56155 0.566762
\(179\) −18.2462 −1.36379 −0.681893 0.731452i \(-0.738843\pi\)
−0.681893 + 0.731452i \(0.738843\pi\)
\(180\) −2.56155 −0.190927
\(181\) −3.24621 −0.241289 −0.120644 0.992696i \(-0.538496\pi\)
−0.120644 + 0.992696i \(0.538496\pi\)
\(182\) 0 0
\(183\) −5.24621 −0.387811
\(184\) 7.12311 0.525122
\(185\) 16.8078 1.23573
\(186\) 0 0
\(187\) −12.6847 −0.927594
\(188\) −12.6847 −0.925124
\(189\) −1.00000 −0.0727393
\(190\) −4.00000 −0.290191
\(191\) 3.12311 0.225980 0.112990 0.993596i \(-0.463957\pi\)
0.112990 + 0.993596i \(0.463957\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −8.12311 −0.584714 −0.292357 0.956309i \(-0.594440\pi\)
−0.292357 + 0.956309i \(0.594440\pi\)
\(194\) −9.12311 −0.655001
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 8.43845 0.601214 0.300607 0.953748i \(-0.402811\pi\)
0.300607 + 0.953748i \(0.402811\pi\)
\(198\) 1.56155 0.110975
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.56155 0.110418
\(201\) −0.876894 −0.0618514
\(202\) 14.8078 1.04187
\(203\) 2.12311 0.149013
\(204\) 8.12311 0.568731
\(205\) −13.4384 −0.938582
\(206\) −10.2462 −0.713887
\(207\) 7.12311 0.495090
\(208\) 0 0
\(209\) 2.43845 0.168671
\(210\) 2.56155 0.176764
\(211\) −3.12311 −0.215003 −0.107502 0.994205i \(-0.534285\pi\)
−0.107502 + 0.994205i \(0.534285\pi\)
\(212\) 7.00000 0.480762
\(213\) −13.3693 −0.916050
\(214\) −14.9309 −1.02065
\(215\) 20.4924 1.39757
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.24621 0.287590
\(219\) 6.56155 0.443389
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 6.56155 0.440383
\(223\) 26.2462 1.75758 0.878788 0.477212i \(-0.158353\pi\)
0.878788 + 0.477212i \(0.158353\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.56155 0.104104
\(226\) −2.56155 −0.170392
\(227\) −19.1231 −1.26925 −0.634623 0.772822i \(-0.718844\pi\)
−0.634623 + 0.772822i \(0.718844\pi\)
\(228\) −1.56155 −0.103416
\(229\) −20.0540 −1.32520 −0.662602 0.748972i \(-0.730548\pi\)
−0.662602 + 0.748972i \(0.730548\pi\)
\(230\) −18.2462 −1.20312
\(231\) −1.56155 −0.102743
\(232\) 2.12311 0.139389
\(233\) −23.3693 −1.53097 −0.765487 0.643451i \(-0.777502\pi\)
−0.765487 + 0.643451i \(0.777502\pi\)
\(234\) 0 0
\(235\) 32.4924 2.11957
\(236\) −3.12311 −0.203297
\(237\) 2.43845 0.158394
\(238\) −8.12311 −0.526543
\(239\) −5.36932 −0.347312 −0.173656 0.984806i \(-0.555558\pi\)
−0.173656 + 0.984806i \(0.555558\pi\)
\(240\) 2.56155 0.165348
\(241\) −20.8078 −1.34035 −0.670173 0.742205i \(-0.733780\pi\)
−0.670173 + 0.742205i \(0.733780\pi\)
\(242\) −8.56155 −0.550357
\(243\) −1.00000 −0.0641500
\(244\) 5.24621 0.335854
\(245\) −2.56155 −0.163652
\(246\) −5.24621 −0.334486
\(247\) 0 0
\(248\) 0 0
\(249\) −3.12311 −0.197919
\(250\) 8.80776 0.557052
\(251\) 27.6155 1.74308 0.871538 0.490327i \(-0.163123\pi\)
0.871538 + 0.490327i \(0.163123\pi\)
\(252\) 1.00000 0.0629941
\(253\) 11.1231 0.699304
\(254\) −20.4924 −1.28581
\(255\) −20.8078 −1.30303
\(256\) 1.00000 0.0625000
\(257\) 0.369317 0.0230374 0.0115187 0.999934i \(-0.496333\pi\)
0.0115187 + 0.999934i \(0.496333\pi\)
\(258\) 8.00000 0.498058
\(259\) −6.56155 −0.407715
\(260\) 0 0
\(261\) 2.12311 0.131417
\(262\) −2.24621 −0.138771
\(263\) −3.12311 −0.192579 −0.0962895 0.995353i \(-0.530697\pi\)
−0.0962895 + 0.995353i \(0.530697\pi\)
\(264\) −1.56155 −0.0961069
\(265\) −17.9309 −1.10148
\(266\) 1.56155 0.0957449
\(267\) −7.56155 −0.462760
\(268\) 0.876894 0.0535648
\(269\) 18.8769 1.15094 0.575472 0.817821i \(-0.304818\pi\)
0.575472 + 0.817821i \(0.304818\pi\)
\(270\) 2.56155 0.155891
\(271\) 26.7386 1.62426 0.812128 0.583479i \(-0.198309\pi\)
0.812128 + 0.583479i \(0.198309\pi\)
\(272\) −8.12311 −0.492536
\(273\) 0 0
\(274\) 16.5616 1.00052
\(275\) 2.43845 0.147044
\(276\) −7.12311 −0.428761
\(277\) −15.4384 −0.927606 −0.463803 0.885938i \(-0.653516\pi\)
−0.463803 + 0.885938i \(0.653516\pi\)
\(278\) −9.56155 −0.573464
\(279\) 0 0
\(280\) −2.56155 −0.153082
\(281\) −15.9309 −0.950356 −0.475178 0.879890i \(-0.657616\pi\)
−0.475178 + 0.879890i \(0.657616\pi\)
\(282\) 12.6847 0.755360
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 13.3693 0.793323
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 5.24621 0.309674
\(288\) 1.00000 0.0589256
\(289\) 48.9848 2.88146
\(290\) −5.43845 −0.319357
\(291\) 9.12311 0.534806
\(292\) −6.56155 −0.383986
\(293\) 21.9309 1.28122 0.640608 0.767868i \(-0.278683\pi\)
0.640608 + 0.767868i \(0.278683\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −6.56155 −0.381383
\(297\) −1.56155 −0.0906105
\(298\) −11.4384 −0.662611
\(299\) 0 0
\(300\) −1.56155 −0.0901563
\(301\) −8.00000 −0.461112
\(302\) 6.93087 0.398827
\(303\) −14.8078 −0.850684
\(304\) 1.56155 0.0895612
\(305\) −13.4384 −0.769483
\(306\) −8.12311 −0.464367
\(307\) 18.9309 1.08044 0.540221 0.841523i \(-0.318341\pi\)
0.540221 + 0.841523i \(0.318341\pi\)
\(308\) 1.56155 0.0889777
\(309\) 10.2462 0.582887
\(310\) 0 0
\(311\) −6.93087 −0.393014 −0.196507 0.980502i \(-0.562960\pi\)
−0.196507 + 0.980502i \(0.562960\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −13.6847 −0.772270
\(315\) −2.56155 −0.144327
\(316\) −2.43845 −0.137173
\(317\) 16.5616 0.930189 0.465095 0.885261i \(-0.346020\pi\)
0.465095 + 0.885261i \(0.346020\pi\)
\(318\) −7.00000 −0.392541
\(319\) 3.31534 0.185623
\(320\) −2.56155 −0.143195
\(321\) 14.9309 0.833360
\(322\) 7.12311 0.396955
\(323\) −12.6847 −0.705793
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.49242 −0.470352
\(327\) −4.24621 −0.234816
\(328\) 5.24621 0.289674
\(329\) −12.6847 −0.699328
\(330\) 4.00000 0.220193
\(331\) −5.36932 −0.295124 −0.147562 0.989053i \(-0.547143\pi\)
−0.147562 + 0.989053i \(0.547143\pi\)
\(332\) 3.12311 0.171403
\(333\) −6.56155 −0.359571
\(334\) −20.4924 −1.12130
\(335\) −2.24621 −0.122724
\(336\) −1.00000 −0.0545545
\(337\) 19.4924 1.06182 0.530910 0.847428i \(-0.321850\pi\)
0.530910 + 0.847428i \(0.321850\pi\)
\(338\) 0 0
\(339\) 2.56155 0.139124
\(340\) 20.8078 1.12846
\(341\) 0 0
\(342\) 1.56155 0.0844391
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 18.2462 0.982343
\(346\) −18.0000 −0.967686
\(347\) −22.0540 −1.18392 −0.591960 0.805968i \(-0.701646\pi\)
−0.591960 + 0.805968i \(0.701646\pi\)
\(348\) −2.12311 −0.113810
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 1.56155 0.0834685
\(351\) 0 0
\(352\) 1.56155 0.0832310
\(353\) −13.1922 −0.702152 −0.351076 0.936347i \(-0.614184\pi\)
−0.351076 + 0.936347i \(0.614184\pi\)
\(354\) 3.12311 0.165991
\(355\) −34.2462 −1.81760
\(356\) 7.56155 0.400761
\(357\) 8.12311 0.429920
\(358\) −18.2462 −0.964342
\(359\) −18.2462 −0.962998 −0.481499 0.876447i \(-0.659908\pi\)
−0.481499 + 0.876447i \(0.659908\pi\)
\(360\) −2.56155 −0.135006
\(361\) −16.5616 −0.871661
\(362\) −3.24621 −0.170617
\(363\) 8.56155 0.449365
\(364\) 0 0
\(365\) 16.8078 0.879759
\(366\) −5.24621 −0.274224
\(367\) −27.6155 −1.44152 −0.720759 0.693185i \(-0.756207\pi\)
−0.720759 + 0.693185i \(0.756207\pi\)
\(368\) 7.12311 0.371318
\(369\) 5.24621 0.273107
\(370\) 16.8078 0.873794
\(371\) 7.00000 0.363422
\(372\) 0 0
\(373\) −31.9309 −1.65332 −0.826659 0.562703i \(-0.809761\pi\)
−0.826659 + 0.562703i \(0.809761\pi\)
\(374\) −12.6847 −0.655908
\(375\) −8.80776 −0.454831
\(376\) −12.6847 −0.654161
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −3.61553 −0.185717 −0.0928586 0.995679i \(-0.529600\pi\)
−0.0928586 + 0.995679i \(0.529600\pi\)
\(380\) −4.00000 −0.205196
\(381\) 20.4924 1.04986
\(382\) 3.12311 0.159792
\(383\) 7.80776 0.398958 0.199479 0.979902i \(-0.436075\pi\)
0.199479 + 0.979902i \(0.436075\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) −8.12311 −0.413455
\(387\) −8.00000 −0.406663
\(388\) −9.12311 −0.463156
\(389\) −17.6847 −0.896648 −0.448324 0.893871i \(-0.647979\pi\)
−0.448324 + 0.893871i \(0.647979\pi\)
\(390\) 0 0
\(391\) −57.8617 −2.92619
\(392\) 1.00000 0.0505076
\(393\) 2.24621 0.113306
\(394\) 8.43845 0.425123
\(395\) 6.24621 0.314281
\(396\) 1.56155 0.0784710
\(397\) −0.930870 −0.0467190 −0.0233595 0.999727i \(-0.507436\pi\)
−0.0233595 + 0.999727i \(0.507436\pi\)
\(398\) −16.0000 −0.802008
\(399\) −1.56155 −0.0781754
\(400\) 1.56155 0.0780776
\(401\) 18.8078 0.939215 0.469607 0.882875i \(-0.344395\pi\)
0.469607 + 0.882875i \(0.344395\pi\)
\(402\) −0.876894 −0.0437355
\(403\) 0 0
\(404\) 14.8078 0.736714
\(405\) −2.56155 −0.127285
\(406\) 2.12311 0.105368
\(407\) −10.2462 −0.507886
\(408\) 8.12311 0.402154
\(409\) −7.43845 −0.367808 −0.183904 0.982944i \(-0.558873\pi\)
−0.183904 + 0.982944i \(0.558873\pi\)
\(410\) −13.4384 −0.663678
\(411\) −16.5616 −0.816921
\(412\) −10.2462 −0.504795
\(413\) −3.12311 −0.153678
\(414\) 7.12311 0.350082
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 9.56155 0.468231
\(418\) 2.43845 0.119268
\(419\) −1.36932 −0.0668955 −0.0334478 0.999440i \(-0.510649\pi\)
−0.0334478 + 0.999440i \(0.510649\pi\)
\(420\) 2.56155 0.124991
\(421\) 31.3002 1.52548 0.762739 0.646707i \(-0.223854\pi\)
0.762739 + 0.646707i \(0.223854\pi\)
\(422\) −3.12311 −0.152030
\(423\) −12.6847 −0.616749
\(424\) 7.00000 0.339950
\(425\) −12.6847 −0.615296
\(426\) −13.3693 −0.647746
\(427\) 5.24621 0.253882
\(428\) −14.9309 −0.721711
\(429\) 0 0
\(430\) 20.4924 0.988232
\(431\) −5.36932 −0.258631 −0.129315 0.991604i \(-0.541278\pi\)
−0.129315 + 0.991604i \(0.541278\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −33.6847 −1.61878 −0.809391 0.587271i \(-0.800202\pi\)
−0.809391 + 0.587271i \(0.800202\pi\)
\(434\) 0 0
\(435\) 5.43845 0.260754
\(436\) 4.24621 0.203357
\(437\) 11.1231 0.532090
\(438\) 6.56155 0.313523
\(439\) −9.75379 −0.465523 −0.232761 0.972534i \(-0.574776\pi\)
−0.232761 + 0.972534i \(0.574776\pi\)
\(440\) −4.00000 −0.190693
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 10.0540 0.477679 0.238839 0.971059i \(-0.423233\pi\)
0.238839 + 0.971059i \(0.423233\pi\)
\(444\) 6.56155 0.311398
\(445\) −19.3693 −0.918194
\(446\) 26.2462 1.24279
\(447\) 11.4384 0.541020
\(448\) 1.00000 0.0472456
\(449\) −31.3693 −1.48041 −0.740205 0.672381i \(-0.765271\pi\)
−0.740205 + 0.672381i \(0.765271\pi\)
\(450\) 1.56155 0.0736123
\(451\) 8.19224 0.385757
\(452\) −2.56155 −0.120485
\(453\) −6.93087 −0.325641
\(454\) −19.1231 −0.897492
\(455\) 0 0
\(456\) −1.56155 −0.0731264
\(457\) 0.0691303 0.00323378 0.00161689 0.999999i \(-0.499485\pi\)
0.00161689 + 0.999999i \(0.499485\pi\)
\(458\) −20.0540 −0.937061
\(459\) 8.12311 0.379154
\(460\) −18.2462 −0.850734
\(461\) −31.0540 −1.44633 −0.723164 0.690676i \(-0.757313\pi\)
−0.723164 + 0.690676i \(0.757313\pi\)
\(462\) −1.56155 −0.0726500
\(463\) 24.6847 1.14719 0.573597 0.819138i \(-0.305548\pi\)
0.573597 + 0.819138i \(0.305548\pi\)
\(464\) 2.12311 0.0985627
\(465\) 0 0
\(466\) −23.3693 −1.08256
\(467\) 28.9848 1.34126 0.670629 0.741793i \(-0.266024\pi\)
0.670629 + 0.741793i \(0.266024\pi\)
\(468\) 0 0
\(469\) 0.876894 0.0404912
\(470\) 32.4924 1.49876
\(471\) 13.6847 0.630556
\(472\) −3.12311 −0.143753
\(473\) −12.4924 −0.574402
\(474\) 2.43845 0.112002
\(475\) 2.43845 0.111884
\(476\) −8.12311 −0.372322
\(477\) 7.00000 0.320508
\(478\) −5.36932 −0.245587
\(479\) 26.9309 1.23050 0.615251 0.788331i \(-0.289054\pi\)
0.615251 + 0.788331i \(0.289054\pi\)
\(480\) 2.56155 0.116918
\(481\) 0 0
\(482\) −20.8078 −0.947768
\(483\) −7.12311 −0.324113
\(484\) −8.56155 −0.389161
\(485\) 23.3693 1.06115
\(486\) −1.00000 −0.0453609
\(487\) 11.3153 0.512747 0.256374 0.966578i \(-0.417472\pi\)
0.256374 + 0.966578i \(0.417472\pi\)
\(488\) 5.24621 0.237485
\(489\) 8.49242 0.384041
\(490\) −2.56155 −0.115719
\(491\) 16.4924 0.744293 0.372146 0.928174i \(-0.378622\pi\)
0.372146 + 0.928174i \(0.378622\pi\)
\(492\) −5.24621 −0.236517
\(493\) −17.2462 −0.776730
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 13.3693 0.599696
\(498\) −3.12311 −0.139950
\(499\) 13.7538 0.615704 0.307852 0.951434i \(-0.400390\pi\)
0.307852 + 0.951434i \(0.400390\pi\)
\(500\) 8.80776 0.393895
\(501\) 20.4924 0.915534
\(502\) 27.6155 1.23254
\(503\) 4.49242 0.200307 0.100154 0.994972i \(-0.468067\pi\)
0.100154 + 0.994972i \(0.468067\pi\)
\(504\) 1.00000 0.0445435
\(505\) −37.9309 −1.68790
\(506\) 11.1231 0.494482
\(507\) 0 0
\(508\) −20.4924 −0.909204
\(509\) −21.1922 −0.939329 −0.469665 0.882845i \(-0.655625\pi\)
−0.469665 + 0.882845i \(0.655625\pi\)
\(510\) −20.8078 −0.921383
\(511\) −6.56155 −0.290266
\(512\) 1.00000 0.0441942
\(513\) −1.56155 −0.0689442
\(514\) 0.369317 0.0162899
\(515\) 26.2462 1.15655
\(516\) 8.00000 0.352180
\(517\) −19.8078 −0.871144
\(518\) −6.56155 −0.288298
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 34.1231 1.49496 0.747480 0.664284i \(-0.231263\pi\)
0.747480 + 0.664284i \(0.231263\pi\)
\(522\) 2.12311 0.0929258
\(523\) 3.80776 0.166502 0.0832509 0.996529i \(-0.473470\pi\)
0.0832509 + 0.996529i \(0.473470\pi\)
\(524\) −2.24621 −0.0981262
\(525\) −1.56155 −0.0681518
\(526\) −3.12311 −0.136174
\(527\) 0 0
\(528\) −1.56155 −0.0679579
\(529\) 27.7386 1.20603
\(530\) −17.9309 −0.778867
\(531\) −3.12311 −0.135531
\(532\) 1.56155 0.0677019
\(533\) 0 0
\(534\) −7.56155 −0.327220
\(535\) 38.2462 1.65353
\(536\) 0.876894 0.0378761
\(537\) 18.2462 0.787382
\(538\) 18.8769 0.813841
\(539\) 1.56155 0.0672608
\(540\) 2.56155 0.110232
\(541\) −26.1771 −1.12544 −0.562720 0.826647i \(-0.690245\pi\)
−0.562720 + 0.826647i \(0.690245\pi\)
\(542\) 26.7386 1.14852
\(543\) 3.24621 0.139308
\(544\) −8.12311 −0.348275
\(545\) −10.8769 −0.465915
\(546\) 0 0
\(547\) 7.61553 0.325616 0.162808 0.986658i \(-0.447945\pi\)
0.162808 + 0.986658i \(0.447945\pi\)
\(548\) 16.5616 0.707474
\(549\) 5.24621 0.223903
\(550\) 2.43845 0.103976
\(551\) 3.31534 0.141238
\(552\) −7.12311 −0.303180
\(553\) −2.43845 −0.103693
\(554\) −15.4384 −0.655917
\(555\) −16.8078 −0.713450
\(556\) −9.56155 −0.405500
\(557\) 3.87689 0.164269 0.0821346 0.996621i \(-0.473826\pi\)
0.0821346 + 0.996621i \(0.473826\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.56155 −0.108245
\(561\) 12.6847 0.535547
\(562\) −15.9309 −0.672003
\(563\) −22.2462 −0.937566 −0.468783 0.883313i \(-0.655307\pi\)
−0.468783 + 0.883313i \(0.655307\pi\)
\(564\) 12.6847 0.534120
\(565\) 6.56155 0.276047
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) 13.3693 0.560964
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 4.00000 0.167542
\(571\) −12.8769 −0.538881 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(572\) 0 0
\(573\) −3.12311 −0.130470
\(574\) 5.24621 0.218973
\(575\) 11.1231 0.463866
\(576\) 1.00000 0.0416667
\(577\) −31.4384 −1.30880 −0.654400 0.756149i \(-0.727079\pi\)
−0.654400 + 0.756149i \(0.727079\pi\)
\(578\) 48.9848 2.03750
\(579\) 8.12311 0.337585
\(580\) −5.43845 −0.225819
\(581\) 3.12311 0.129568
\(582\) 9.12311 0.378165
\(583\) 10.9309 0.452710
\(584\) −6.56155 −0.271519
\(585\) 0 0
\(586\) 21.9309 0.905956
\(587\) −14.2462 −0.588004 −0.294002 0.955805i \(-0.594987\pi\)
−0.294002 + 0.955805i \(0.594987\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) −8.43845 −0.347111
\(592\) −6.56155 −0.269678
\(593\) 31.9848 1.31346 0.656730 0.754126i \(-0.271939\pi\)
0.656730 + 0.754126i \(0.271939\pi\)
\(594\) −1.56155 −0.0640713
\(595\) 20.8078 0.853035
\(596\) −11.4384 −0.468537
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −33.8617 −1.38355 −0.691777 0.722112i \(-0.743172\pi\)
−0.691777 + 0.722112i \(0.743172\pi\)
\(600\) −1.56155 −0.0637501
\(601\) −25.3002 −1.03202 −0.516008 0.856584i \(-0.672583\pi\)
−0.516008 + 0.856584i \(0.672583\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0.876894 0.0357099
\(604\) 6.93087 0.282013
\(605\) 21.9309 0.891617
\(606\) −14.8078 −0.601524
\(607\) 16.9848 0.689394 0.344697 0.938714i \(-0.387982\pi\)
0.344697 + 0.938714i \(0.387982\pi\)
\(608\) 1.56155 0.0633293
\(609\) −2.12311 −0.0860326
\(610\) −13.4384 −0.544107
\(611\) 0 0
\(612\) −8.12311 −0.328357
\(613\) 17.4384 0.704332 0.352166 0.935938i \(-0.385445\pi\)
0.352166 + 0.935938i \(0.385445\pi\)
\(614\) 18.9309 0.763988
\(615\) 13.4384 0.541890
\(616\) 1.56155 0.0629168
\(617\) −22.1771 −0.892816 −0.446408 0.894830i \(-0.647297\pi\)
−0.446408 + 0.894830i \(0.647297\pi\)
\(618\) 10.2462 0.412163
\(619\) 30.9309 1.24322 0.621608 0.783328i \(-0.286480\pi\)
0.621608 + 0.783328i \(0.286480\pi\)
\(620\) 0 0
\(621\) −7.12311 −0.285840
\(622\) −6.93087 −0.277903
\(623\) 7.56155 0.302947
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) −6.00000 −0.239808
\(627\) −2.43845 −0.0973822
\(628\) −13.6847 −0.546077
\(629\) 53.3002 2.12522
\(630\) −2.56155 −0.102055
\(631\) −38.9309 −1.54981 −0.774907 0.632076i \(-0.782203\pi\)
−0.774907 + 0.632076i \(0.782203\pi\)
\(632\) −2.43845 −0.0969962
\(633\) 3.12311 0.124132
\(634\) 16.5616 0.657743
\(635\) 52.4924 2.08310
\(636\) −7.00000 −0.277568
\(637\) 0 0
\(638\) 3.31534 0.131256
\(639\) 13.3693 0.528882
\(640\) −2.56155 −0.101254
\(641\) 12.5616 0.496152 0.248076 0.968741i \(-0.420202\pi\)
0.248076 + 0.968741i \(0.420202\pi\)
\(642\) 14.9309 0.589274
\(643\) 35.3153 1.39270 0.696351 0.717702i \(-0.254806\pi\)
0.696351 + 0.717702i \(0.254806\pi\)
\(644\) 7.12311 0.280690
\(645\) −20.4924 −0.806888
\(646\) −12.6847 −0.499071
\(647\) −46.0540 −1.81057 −0.905284 0.424806i \(-0.860342\pi\)
−0.905284 + 0.424806i \(0.860342\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.87689 −0.191435
\(650\) 0 0
\(651\) 0 0
\(652\) −8.49242 −0.332589
\(653\) 16.4384 0.643286 0.321643 0.946861i \(-0.395765\pi\)
0.321643 + 0.946861i \(0.395765\pi\)
\(654\) −4.24621 −0.166040
\(655\) 5.75379 0.224819
\(656\) 5.24621 0.204830
\(657\) −6.56155 −0.255991
\(658\) −12.6847 −0.494499
\(659\) 30.9309 1.20490 0.602448 0.798158i \(-0.294192\pi\)
0.602448 + 0.798158i \(0.294192\pi\)
\(660\) 4.00000 0.155700
\(661\) 17.0540 0.663323 0.331661 0.943398i \(-0.392391\pi\)
0.331661 + 0.943398i \(0.392391\pi\)
\(662\) −5.36932 −0.208684
\(663\) 0 0
\(664\) 3.12311 0.121200
\(665\) −4.00000 −0.155113
\(666\) −6.56155 −0.254255
\(667\) 15.1231 0.585569
\(668\) −20.4924 −0.792876
\(669\) −26.2462 −1.01474
\(670\) −2.24621 −0.0867787
\(671\) 8.19224 0.316258
\(672\) −1.00000 −0.0385758
\(673\) −19.2462 −0.741887 −0.370943 0.928655i \(-0.620966\pi\)
−0.370943 + 0.928655i \(0.620966\pi\)
\(674\) 19.4924 0.750820
\(675\) −1.56155 −0.0601042
\(676\) 0 0
\(677\) 24.7386 0.950783 0.475391 0.879774i \(-0.342306\pi\)
0.475391 + 0.879774i \(0.342306\pi\)
\(678\) 2.56155 0.0983758
\(679\) −9.12311 −0.350113
\(680\) 20.8078 0.797941
\(681\) 19.1231 0.732799
\(682\) 0 0
\(683\) 7.50758 0.287269 0.143635 0.989631i \(-0.454121\pi\)
0.143635 + 0.989631i \(0.454121\pi\)
\(684\) 1.56155 0.0597075
\(685\) −42.4233 −1.62091
\(686\) 1.00000 0.0381802
\(687\) 20.0540 0.765107
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 18.2462 0.694621
\(691\) 32.9848 1.25480 0.627401 0.778696i \(-0.284119\pi\)
0.627401 + 0.778696i \(0.284119\pi\)
\(692\) −18.0000 −0.684257
\(693\) 1.56155 0.0593185
\(694\) −22.0540 −0.837157
\(695\) 24.4924 0.929051
\(696\) −2.12311 −0.0804761
\(697\) −42.6155 −1.61418
\(698\) −18.0000 −0.681310
\(699\) 23.3693 0.883909
\(700\) 1.56155 0.0590211
\(701\) 14.3002 0.540111 0.270055 0.962845i \(-0.412958\pi\)
0.270055 + 0.962845i \(0.412958\pi\)
\(702\) 0 0
\(703\) −10.2462 −0.386443
\(704\) 1.56155 0.0588532
\(705\) −32.4924 −1.22374
\(706\) −13.1922 −0.496496
\(707\) 14.8078 0.556903
\(708\) 3.12311 0.117373
\(709\) 15.3002 0.574611 0.287305 0.957839i \(-0.407241\pi\)
0.287305 + 0.957839i \(0.407241\pi\)
\(710\) −34.2462 −1.28524
\(711\) −2.43845 −0.0914489
\(712\) 7.56155 0.283381
\(713\) 0 0
\(714\) 8.12311 0.304000
\(715\) 0 0
\(716\) −18.2462 −0.681893
\(717\) 5.36932 0.200521
\(718\) −18.2462 −0.680943
\(719\) −19.8078 −0.738705 −0.369352 0.929289i \(-0.620420\pi\)
−0.369352 + 0.929289i \(0.620420\pi\)
\(720\) −2.56155 −0.0954634
\(721\) −10.2462 −0.381589
\(722\) −16.5616 −0.616357
\(723\) 20.8078 0.773849
\(724\) −3.24621 −0.120644
\(725\) 3.31534 0.123129
\(726\) 8.56155 0.317749
\(727\) 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
\(730\) 16.8078 0.622083
\(731\) 64.9848 2.40355
\(732\) −5.24621 −0.193906
\(733\) 28.8617 1.06603 0.533016 0.846105i \(-0.321058\pi\)
0.533016 + 0.846105i \(0.321058\pi\)
\(734\) −27.6155 −1.01931
\(735\) 2.56155 0.0944843
\(736\) 7.12311 0.262561
\(737\) 1.36932 0.0504394
\(738\) 5.24621 0.193116
\(739\) −9.36932 −0.344656 −0.172328 0.985040i \(-0.555129\pi\)
−0.172328 + 0.985040i \(0.555129\pi\)
\(740\) 16.8078 0.617866
\(741\) 0 0
\(742\) 7.00000 0.256978
\(743\) 28.4924 1.04529 0.522643 0.852552i \(-0.324946\pi\)
0.522643 + 0.852552i \(0.324946\pi\)
\(744\) 0 0
\(745\) 29.3002 1.07348
\(746\) −31.9309 −1.16907
\(747\) 3.12311 0.114268
\(748\) −12.6847 −0.463797
\(749\) −14.9309 −0.545562
\(750\) −8.80776 −0.321614
\(751\) −20.1922 −0.736825 −0.368413 0.929662i \(-0.620099\pi\)
−0.368413 + 0.929662i \(0.620099\pi\)
\(752\) −12.6847 −0.462562
\(753\) −27.6155 −1.00637
\(754\) 0 0
\(755\) −17.7538 −0.646127
\(756\) −1.00000 −0.0363696
\(757\) 3.26137 0.118536 0.0592682 0.998242i \(-0.481123\pi\)
0.0592682 + 0.998242i \(0.481123\pi\)
\(758\) −3.61553 −0.131322
\(759\) −11.1231 −0.403743
\(760\) −4.00000 −0.145095
\(761\) −14.9848 −0.543200 −0.271600 0.962410i \(-0.587553\pi\)
−0.271600 + 0.962410i \(0.587553\pi\)
\(762\) 20.4924 0.742362
\(763\) 4.24621 0.153723
\(764\) 3.12311 0.112990
\(765\) 20.8078 0.752306
\(766\) 7.80776 0.282106
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 13.1231 0.473231 0.236616 0.971603i \(-0.423962\pi\)
0.236616 + 0.971603i \(0.423962\pi\)
\(770\) −4.00000 −0.144150
\(771\) −0.369317 −0.0133006
\(772\) −8.12311 −0.292357
\(773\) −28.7386 −1.03366 −0.516828 0.856089i \(-0.672887\pi\)
−0.516828 + 0.856089i \(0.672887\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −9.12311 −0.327500
\(777\) 6.56155 0.235394
\(778\) −17.6847 −0.634026
\(779\) 8.19224 0.293517
\(780\) 0 0
\(781\) 20.8769 0.747034
\(782\) −57.8617 −2.06913
\(783\) −2.12311 −0.0758736
\(784\) 1.00000 0.0357143
\(785\) 35.0540 1.25113
\(786\) 2.24621 0.0801197
\(787\) 45.1771 1.61039 0.805195 0.593011i \(-0.202061\pi\)
0.805195 + 0.593011i \(0.202061\pi\)
\(788\) 8.43845 0.300607
\(789\) 3.12311 0.111186
\(790\) 6.24621 0.222230
\(791\) −2.56155 −0.0910783
\(792\) 1.56155 0.0554874
\(793\) 0 0
\(794\) −0.930870 −0.0330353
\(795\) 17.9309 0.635942
\(796\) −16.0000 −0.567105
\(797\) 17.5076 0.620150 0.310075 0.950712i \(-0.399646\pi\)
0.310075 + 0.950712i \(0.399646\pi\)
\(798\) −1.56155 −0.0552784
\(799\) 103.039 3.64525
\(800\) 1.56155 0.0552092
\(801\) 7.56155 0.267174
\(802\) 18.8078 0.664125
\(803\) −10.2462 −0.361581
\(804\) −0.876894 −0.0309257
\(805\) −18.2462 −0.643094
\(806\) 0 0
\(807\) −18.8769 −0.664498
\(808\) 14.8078 0.520935
\(809\) −19.9309 −0.700732 −0.350366 0.936613i \(-0.613943\pi\)
−0.350366 + 0.936613i \(0.613943\pi\)
\(810\) −2.56155 −0.0900038
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 2.12311 0.0745064
\(813\) −26.7386 −0.937765
\(814\) −10.2462 −0.359130
\(815\) 21.7538 0.762002
\(816\) 8.12311 0.284366
\(817\) −12.4924 −0.437055
\(818\) −7.43845 −0.260079
\(819\) 0 0
\(820\) −13.4384 −0.469291
\(821\) −26.3002 −0.917883 −0.458941 0.888467i \(-0.651771\pi\)
−0.458941 + 0.888467i \(0.651771\pi\)
\(822\) −16.5616 −0.577650
\(823\) −21.7538 −0.758289 −0.379145 0.925337i \(-0.623782\pi\)
−0.379145 + 0.925337i \(0.623782\pi\)
\(824\) −10.2462 −0.356944
\(825\) −2.43845 −0.0848958
\(826\) −3.12311 −0.108667
\(827\) 17.7538 0.617360 0.308680 0.951166i \(-0.400113\pi\)
0.308680 + 0.951166i \(0.400113\pi\)
\(828\) 7.12311 0.247545
\(829\) 24.3693 0.846381 0.423191 0.906041i \(-0.360910\pi\)
0.423191 + 0.906041i \(0.360910\pi\)
\(830\) −8.00000 −0.277684
\(831\) 15.4384 0.535554
\(832\) 0 0
\(833\) −8.12311 −0.281449
\(834\) 9.56155 0.331089
\(835\) 52.4924 1.81658
\(836\) 2.43845 0.0843355
\(837\) 0 0
\(838\) −1.36932 −0.0473023
\(839\) 1.75379 0.0605475 0.0302738 0.999542i \(-0.490362\pi\)
0.0302738 + 0.999542i \(0.490362\pi\)
\(840\) 2.56155 0.0883820
\(841\) −24.4924 −0.844566
\(842\) 31.3002 1.07868
\(843\) 15.9309 0.548688
\(844\) −3.12311 −0.107502
\(845\) 0 0
\(846\) −12.6847 −0.436108
\(847\) −8.56155 −0.294178
\(848\) 7.00000 0.240381
\(849\) 20.0000 0.686398
\(850\) −12.6847 −0.435080
\(851\) −46.7386 −1.60218
\(852\) −13.3693 −0.458025
\(853\) 5.63068 0.192791 0.0963955 0.995343i \(-0.469269\pi\)
0.0963955 + 0.995343i \(0.469269\pi\)
\(854\) 5.24621 0.179522
\(855\) −4.00000 −0.136797
\(856\) −14.9309 −0.510327
\(857\) 20.5616 0.702369 0.351185 0.936306i \(-0.385779\pi\)
0.351185 + 0.936306i \(0.385779\pi\)
\(858\) 0 0
\(859\) −10.9309 −0.372956 −0.186478 0.982459i \(-0.559707\pi\)
−0.186478 + 0.982459i \(0.559707\pi\)
\(860\) 20.4924 0.698786
\(861\) −5.24621 −0.178790
\(862\) −5.36932 −0.182880
\(863\) 32.8769 1.11914 0.559571 0.828782i \(-0.310966\pi\)
0.559571 + 0.828782i \(0.310966\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 46.1080 1.56772
\(866\) −33.6847 −1.14465
\(867\) −48.9848 −1.66361
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) 5.43845 0.184381
\(871\) 0 0
\(872\) 4.24621 0.143795
\(873\) −9.12311 −0.308770
\(874\) 11.1231 0.376245
\(875\) 8.80776 0.297757
\(876\) 6.56155 0.221694
\(877\) 41.5464 1.40292 0.701461 0.712708i \(-0.252531\pi\)
0.701461 + 0.712708i \(0.252531\pi\)
\(878\) −9.75379 −0.329174
\(879\) −21.9309 −0.739710
\(880\) −4.00000 −0.134840
\(881\) 57.0540 1.92220 0.961099 0.276205i \(-0.0890770\pi\)
0.961099 + 0.276205i \(0.0890770\pi\)
\(882\) 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
\(886\) 10.0540 0.337770
\(887\) −46.0540 −1.54634 −0.773171 0.634198i \(-0.781330\pi\)
−0.773171 + 0.634198i \(0.781330\pi\)
\(888\) 6.56155 0.220191
\(889\) −20.4924 −0.687294
\(890\) −19.3693 −0.649261
\(891\) 1.56155 0.0523140
\(892\) 26.2462 0.878788
\(893\) −19.8078 −0.662842
\(894\) 11.4384 0.382559
\(895\) 46.7386 1.56230
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −31.3693 −1.04681
\(899\) 0 0
\(900\) 1.56155 0.0520518
\(901\) −56.8617 −1.89434
\(902\) 8.19224 0.272772
\(903\) 8.00000 0.266223
\(904\) −2.56155 −0.0851960
\(905\) 8.31534 0.276411
\(906\) −6.93087 −0.230263
\(907\) 7.12311 0.236519 0.118259 0.992983i \(-0.462269\pi\)
0.118259 + 0.992983i \(0.462269\pi\)
\(908\) −19.1231 −0.634623
\(909\) 14.8078 0.491143
\(910\) 0 0
\(911\) −27.6155 −0.914943 −0.457472 0.889224i \(-0.651245\pi\)
−0.457472 + 0.889224i \(0.651245\pi\)
\(912\) −1.56155 −0.0517082
\(913\) 4.87689 0.161402
\(914\) 0.0691303 0.00228663
\(915\) 13.4384 0.444261
\(916\) −20.0540 −0.662602
\(917\) −2.24621 −0.0741764
\(918\) 8.12311 0.268102
\(919\) −17.0691 −0.563059 −0.281529 0.959553i \(-0.590842\pi\)
−0.281529 + 0.959553i \(0.590842\pi\)
\(920\) −18.2462 −0.601560
\(921\) −18.9309 −0.623793
\(922\) −31.0540 −1.02271
\(923\) 0 0
\(924\) −1.56155 −0.0513713
\(925\) −10.2462 −0.336893
\(926\) 24.6847 0.811188
\(927\) −10.2462 −0.336530
\(928\) 2.12311 0.0696944
\(929\) −19.2462 −0.631448 −0.315724 0.948851i \(-0.602247\pi\)
−0.315724 + 0.948851i \(0.602247\pi\)
\(930\) 0 0
\(931\) 1.56155 0.0511778
\(932\) −23.3693 −0.765487
\(933\) 6.93087 0.226906
\(934\) 28.9848 0.948413
\(935\) 32.4924 1.06262
\(936\) 0 0
\(937\) −38.5616 −1.25975 −0.629876 0.776696i \(-0.716894\pi\)
−0.629876 + 0.776696i \(0.716894\pi\)
\(938\) 0.876894 0.0286316
\(939\) 6.00000 0.195803
\(940\) 32.4924 1.05979
\(941\) −35.3693 −1.15301 −0.576503 0.817095i \(-0.695583\pi\)
−0.576503 + 0.817095i \(0.695583\pi\)
\(942\) 13.6847 0.445870
\(943\) 37.3693 1.21691
\(944\) −3.12311 −0.101648
\(945\) 2.56155 0.0833273
\(946\) −12.4924 −0.406164
\(947\) 54.4384 1.76901 0.884506 0.466529i \(-0.154496\pi\)
0.884506 + 0.466529i \(0.154496\pi\)
\(948\) 2.43845 0.0791971
\(949\) 0 0
\(950\) 2.43845 0.0791137
\(951\) −16.5616 −0.537045
\(952\) −8.12311 −0.263271
\(953\) 21.1231 0.684245 0.342122 0.939655i \(-0.388854\pi\)
0.342122 + 0.939655i \(0.388854\pi\)
\(954\) 7.00000 0.226633
\(955\) −8.00000 −0.258874
\(956\) −5.36932 −0.173656
\(957\) −3.31534 −0.107170
\(958\) 26.9309 0.870097
\(959\) 16.5616 0.534800
\(960\) 2.56155 0.0826738
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −14.9309 −0.481141
\(964\) −20.8078 −0.670173
\(965\) 20.8078 0.669826
\(966\) −7.12311 −0.229182
\(967\) 4.49242 0.144467 0.0722333 0.997388i \(-0.476987\pi\)
0.0722333 + 0.997388i \(0.476987\pi\)
\(968\) −8.56155 −0.275179
\(969\) 12.6847 0.407490
\(970\) 23.3693 0.750344
\(971\) 0.384472 0.0123383 0.00616914 0.999981i \(-0.498036\pi\)
0.00616914 + 0.999981i \(0.498036\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.56155 −0.306529
\(974\) 11.3153 0.362567
\(975\) 0 0
\(976\) 5.24621 0.167927
\(977\) 4.17708 0.133637 0.0668183 0.997765i \(-0.478715\pi\)
0.0668183 + 0.997765i \(0.478715\pi\)
\(978\) 8.49242 0.271558
\(979\) 11.8078 0.377378
\(980\) −2.56155 −0.0818258
\(981\) 4.24621 0.135571
\(982\) 16.4924 0.526294
\(983\) −30.7386 −0.980410 −0.490205 0.871607i \(-0.663078\pi\)
−0.490205 + 0.871607i \(0.663078\pi\)
\(984\) −5.24621 −0.167243
\(985\) −21.6155 −0.688728
\(986\) −17.2462 −0.549231
\(987\) 12.6847 0.403757
\(988\) 0 0
\(989\) −56.9848 −1.81201
\(990\) −4.00000 −0.127128
\(991\) 49.6695 1.57780 0.788902 0.614519i \(-0.210650\pi\)
0.788902 + 0.614519i \(0.210650\pi\)
\(992\) 0 0
\(993\) 5.36932 0.170390
\(994\) 13.3693 0.424049
\(995\) 40.9848 1.29931
\(996\) −3.12311 −0.0989594
\(997\) 23.9848 0.759608 0.379804 0.925067i \(-0.375991\pi\)
0.379804 + 0.925067i \(0.375991\pi\)
\(998\) 13.7538 0.435369
\(999\) 6.56155 0.207598
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.1.1 2
13.3 even 3 546.2.l.l.295.1 yes 4
13.9 even 3 546.2.l.l.211.1 4
13.12 even 2 7098.2.a.bi.1.2 2
39.29 odd 6 1638.2.r.y.1387.2 4
39.35 odd 6 1638.2.r.y.757.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.l.211.1 4 13.9 even 3
546.2.l.l.295.1 yes 4 13.3 even 3
1638.2.r.y.757.2 4 39.35 odd 6
1638.2.r.y.1387.2 4 39.29 odd 6
7098.2.a.bi.1.2 2 13.12 even 2
7098.2.a.bt.1.1 2 1.1 even 1 trivial