Properties

Label 4598.2.a.br.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.33452.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 7 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.79836\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.79836 q^{3} +1.00000 q^{4} +0.116104 q^{5} +2.79836 q^{6} +2.35735 q^{7} -1.00000 q^{8} +4.83081 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.79836 q^{3} +1.00000 q^{4} +0.116104 q^{5} +2.79836 q^{6} +2.35735 q^{7} -1.00000 q^{8} +4.83081 q^{9} -0.116104 q^{10} -2.79836 q^{12} +2.88390 q^{13} -2.35735 q^{14} -0.324902 q^{15} +1.00000 q^{16} +2.91446 q^{17} -4.83081 q^{18} +1.00000 q^{19} +0.116104 q^{20} -6.59672 q^{21} -1.44101 q^{23} +2.79836 q^{24} -4.98652 q^{25} -2.88390 q^{26} -5.12326 q^{27} +2.35735 q^{28} +5.15383 q^{29} +0.324902 q^{30} +8.32490 q^{31} -1.00000 q^{32} -2.91446 q^{34} +0.273699 q^{35} +4.83081 q^{36} +10.5436 q^{37} -1.00000 q^{38} -8.07017 q^{39} -0.116104 q^{40} +9.98652 q^{41} +6.59672 q^{42} -2.76591 q^{43} +0.560878 q^{45} +1.44101 q^{46} +4.23221 q^{47} -2.79836 q^{48} -1.44289 q^{49} +4.98652 q^{50} -8.15571 q^{51} +2.88390 q^{52} +5.56427 q^{53} +5.12326 q^{54} -2.35735 q^{56} -2.79836 q^{57} -5.15383 q^{58} -8.51118 q^{59} -0.324902 q^{60} +2.71470 q^{61} -8.32490 q^{62} +11.3879 q^{63} +1.00000 q^{64} +0.334833 q^{65} +3.41044 q^{67} +2.91446 q^{68} +4.03245 q^{69} -0.273699 q^{70} -1.16919 q^{71} -4.83081 q^{72} -1.31775 q^{73} -10.5436 q^{74} +13.9541 q^{75} +1.00000 q^{76} +8.07017 q^{78} -1.93510 q^{79} +0.116104 q^{80} -0.155711 q^{81} -9.98652 q^{82} +4.05120 q^{83} -6.59672 q^{84} +0.338382 q^{85} +2.76591 q^{86} -14.4223 q^{87} -9.25833 q^{89} -0.560878 q^{90} +6.79836 q^{91} -1.44101 q^{92} -23.2961 q^{93} -4.23221 q^{94} +0.116104 q^{95} +2.79836 q^{96} +8.71094 q^{97} +1.44289 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9} - 3 q^{10} + q^{12} + 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} + 2 q^{17} - 5 q^{18} + 4 q^{19} + 3 q^{20} - 2 q^{21} - 2 q^{23} - q^{24} + 15 q^{25} - 9 q^{26} - 2 q^{27} + q^{28} - 5 q^{29} - 5 q^{30} + 27 q^{31} - 4 q^{32} - 2 q^{34} - 12 q^{35} + 5 q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} - 3 q^{40} + 5 q^{41} + 2 q^{42} - q^{43} + 11 q^{45} + 2 q^{46} + 22 q^{47} + q^{48} - 7 q^{49} - 15 q^{50} - 12 q^{51} + 9 q^{52} + 2 q^{54} - q^{56} + q^{57} + 5 q^{58} + 5 q^{60} - 6 q^{61} - 27 q^{62} + 30 q^{63} + 4 q^{64} - 26 q^{65} + 17 q^{67} + 2 q^{68} + 14 q^{69} + 12 q^{70} - 19 q^{71} - 5 q^{72} - 20 q^{73} - 6 q^{74} + 23 q^{75} + 4 q^{76} + 2 q^{78} - 12 q^{79} + 3 q^{80} + 20 q^{81} - 5 q^{82} + 23 q^{83} - 2 q^{84} + 30 q^{85} + q^{86} - 45 q^{87} + 16 q^{89} - 11 q^{90} + 15 q^{91} - 2 q^{92} - 12 q^{93} - 22 q^{94} + 3 q^{95} - q^{96} + 8 q^{97} + 7 q^{98} + O(q^{100}) \)

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
\(3\) −2.79836 −1.61563 −0.807816 0.589434i \(-0.799351\pi\)
−0.807816 + 0.589434i \(0.799351\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.116104 0.0519234 0.0259617 0.999663i \(-0.491735\pi\)
0.0259617 + 0.999663i \(0.491735\pi\)
\(6\) 2.79836 1.14242
\(7\) 2.35735 0.890995 0.445498 0.895283i \(-0.353027\pi\)
0.445498 + 0.895283i \(0.353027\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.83081 1.61027
\(10\) −0.116104 −0.0367154
\(11\) 0 0
\(12\) −2.79836 −0.807816
\(13\) 2.88390 0.799849 0.399924 0.916548i \(-0.369036\pi\)
0.399924 + 0.916548i \(0.369036\pi\)
\(14\) −2.35735 −0.630029
\(15\) −0.324902 −0.0838892
\(16\) 1.00000 0.250000
\(17\) 2.91446 0.706861 0.353431 0.935461i \(-0.385015\pi\)
0.353431 + 0.935461i \(0.385015\pi\)
\(18\) −4.83081 −1.13863
\(19\) 1.00000 0.229416
\(20\) 0.116104 0.0259617
\(21\) −6.59672 −1.43952
\(22\) 0 0
\(23\) −1.44101 −0.300471 −0.150235 0.988650i \(-0.548003\pi\)
−0.150235 + 0.988650i \(0.548003\pi\)
\(24\) 2.79836 0.571212
\(25\) −4.98652 −0.997304
\(26\) −2.88390 −0.565578
\(27\) −5.12326 −0.985972
\(28\) 2.35735 0.445498
\(29\) 5.15383 0.957042 0.478521 0.878076i \(-0.341173\pi\)
0.478521 + 0.878076i \(0.341173\pi\)
\(30\) 0.324902 0.0593186
\(31\) 8.32490 1.49520 0.747598 0.664151i \(-0.231207\pi\)
0.747598 + 0.664151i \(0.231207\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.91446 −0.499826
\(35\) 0.273699 0.0462636
\(36\) 4.83081 0.805135
\(37\) 10.5436 1.73336 0.866681 0.498863i \(-0.166249\pi\)
0.866681 + 0.498863i \(0.166249\pi\)
\(38\) −1.00000 −0.162221
\(39\) −8.07017 −1.29226
\(40\) −0.116104 −0.0183577
\(41\) 9.98652 1.55963 0.779816 0.626009i \(-0.215313\pi\)
0.779816 + 0.626009i \(0.215313\pi\)
\(42\) 6.59672 1.01790
\(43\) −2.76591 −0.421797 −0.210898 0.977508i \(-0.567639\pi\)
−0.210898 + 0.977508i \(0.567639\pi\)
\(44\) 0 0
\(45\) 0.560878 0.0836107
\(46\) 1.44101 0.212465
\(47\) 4.23221 0.617331 0.308666 0.951171i \(-0.400118\pi\)
0.308666 + 0.951171i \(0.400118\pi\)
\(48\) −2.79836 −0.403908
\(49\) −1.44289 −0.206127
\(50\) 4.98652 0.705200
\(51\) −8.15571 −1.14203
\(52\) 2.88390 0.399924
\(53\) 5.56427 0.764311 0.382155 0.924098i \(-0.375182\pi\)
0.382155 + 0.924098i \(0.375182\pi\)
\(54\) 5.12326 0.697187
\(55\) 0 0
\(56\) −2.35735 −0.315014
\(57\) −2.79836 −0.370652
\(58\) −5.15383 −0.676731
\(59\) −8.51118 −1.10806 −0.554031 0.832496i \(-0.686911\pi\)
−0.554031 + 0.832496i \(0.686911\pi\)
\(60\) −0.324902 −0.0419446
\(61\) 2.71470 0.347582 0.173791 0.984783i \(-0.444398\pi\)
0.173791 + 0.984783i \(0.444398\pi\)
\(62\) −8.32490 −1.05726
\(63\) 11.3879 1.43474
\(64\) 1.00000 0.125000
\(65\) 0.334833 0.0415309
\(66\) 0 0
\(67\) 3.41044 0.416651 0.208326 0.978060i \(-0.433199\pi\)
0.208326 + 0.978060i \(0.433199\pi\)
\(68\) 2.91446 0.353431
\(69\) 4.03245 0.485450
\(70\) −0.273699 −0.0327133
\(71\) −1.16919 −0.138757 −0.0693787 0.997590i \(-0.522102\pi\)
−0.0693787 + 0.997590i \(0.522102\pi\)
\(72\) −4.83081 −0.569316
\(73\) −1.31775 −0.154231 −0.0771153 0.997022i \(-0.524571\pi\)
−0.0771153 + 0.997022i \(0.524571\pi\)
\(74\) −10.5436 −1.22567
\(75\) 13.9541 1.61128
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 8.07017 0.913767
\(79\) −1.93510 −0.217716 −0.108858 0.994057i \(-0.534719\pi\)
−0.108858 + 0.994057i \(0.534719\pi\)
\(80\) 0.116104 0.0129809
\(81\) −0.155711 −0.0173012
\(82\) −9.98652 −1.10283
\(83\) 4.05120 0.444677 0.222339 0.974970i \(-0.428631\pi\)
0.222339 + 0.974970i \(0.428631\pi\)
\(84\) −6.59672 −0.719761
\(85\) 0.338382 0.0367027
\(86\) 2.76591 0.298255
\(87\) −14.4223 −1.54623
\(88\) 0 0
\(89\) −9.25833 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(90\) −0.560878 −0.0591217
\(91\) 6.79836 0.712662
\(92\) −1.44101 −0.150235
\(93\) −23.2961 −2.41569
\(94\) −4.23221 −0.436519
\(95\) 0.116104 0.0119121
\(96\) 2.79836 0.285606
\(97\) 8.71094 0.884462 0.442231 0.896901i \(-0.354187\pi\)
0.442231 + 0.896901i \(0.354187\pi\)
\(98\) 1.44289 0.145754
\(99\) 0 0
\(100\) −4.98652 −0.498652
\(101\) −7.83269 −0.779382 −0.389691 0.920946i \(-0.627418\pi\)
−0.389691 + 0.920946i \(0.627418\pi\)
\(102\) 8.15571 0.807536
\(103\) −1.94503 −0.191649 −0.0958247 0.995398i \(-0.530549\pi\)
−0.0958247 + 0.995398i \(0.530549\pi\)
\(104\) −2.88390 −0.282789
\(105\) −0.765907 −0.0747449
\(106\) −5.56427 −0.540449
\(107\) −6.57331 −0.635465 −0.317733 0.948180i \(-0.602921\pi\)
−0.317733 + 0.948180i \(0.602921\pi\)
\(108\) −5.12326 −0.492986
\(109\) 3.02341 0.289590 0.144795 0.989462i \(-0.453748\pi\)
0.144795 + 0.989462i \(0.453748\pi\)
\(110\) 0 0
\(111\) −29.5049 −2.80048
\(112\) 2.35735 0.222749
\(113\) −0.478728 −0.0450350 −0.0225175 0.999746i \(-0.507168\pi\)
−0.0225175 + 0.999746i \(0.507168\pi\)
\(114\) 2.79836 0.262090
\(115\) −0.167307 −0.0156015
\(116\) 5.15383 0.478521
\(117\) 13.9315 1.28797
\(118\) 8.51118 0.783518
\(119\) 6.87042 0.629810
\(120\) 0.324902 0.0296593
\(121\) 0 0
\(122\) −2.71470 −0.245778
\(123\) −27.9459 −2.51979
\(124\) 8.32490 0.747598
\(125\) −1.15948 −0.103707
\(126\) −11.3879 −1.01452
\(127\) −11.3645 −1.00844 −0.504219 0.863576i \(-0.668219\pi\)
−0.504219 + 0.863576i \(0.668219\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.74000 0.681469
\(130\) −0.334833 −0.0293668
\(131\) −19.6481 −1.71667 −0.858333 0.513093i \(-0.828500\pi\)
−0.858333 + 0.513093i \(0.828500\pi\)
\(132\) 0 0
\(133\) 2.35735 0.204408
\(134\) −3.41044 −0.294617
\(135\) −0.594833 −0.0511951
\(136\) −2.91446 −0.249913
\(137\) −5.18101 −0.442643 −0.221322 0.975201i \(-0.571037\pi\)
−0.221322 + 0.975201i \(0.571037\pi\)
\(138\) −4.03245 −0.343265
\(139\) 21.5598 1.82868 0.914340 0.404947i \(-0.132710\pi\)
0.914340 + 0.404947i \(0.132710\pi\)
\(140\) 0.273699 0.0231318
\(141\) −11.8432 −0.997380
\(142\) 1.16919 0.0981163
\(143\) 0 0
\(144\) 4.83081 0.402567
\(145\) 0.598382 0.0496929
\(146\) 1.31775 0.109057
\(147\) 4.03772 0.333026
\(148\) 10.5436 0.866681
\(149\) −14.4518 −1.18393 −0.591967 0.805962i \(-0.701649\pi\)
−0.591967 + 0.805962i \(0.701649\pi\)
\(150\) −13.9541 −1.13934
\(151\) −23.3987 −1.90416 −0.952079 0.305853i \(-0.901058\pi\)
−0.952079 + 0.305853i \(0.901058\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 14.0792 1.13824
\(154\) 0 0
\(155\) 0.966557 0.0776357
\(156\) −8.07017 −0.646131
\(157\) 19.0773 1.52254 0.761268 0.648437i \(-0.224577\pi\)
0.761268 + 0.648437i \(0.224577\pi\)
\(158\) 1.93510 0.153948
\(159\) −15.5708 −1.23485
\(160\) −0.116104 −0.00917885
\(161\) −3.39696 −0.267718
\(162\) 0.155711 0.0122338
\(163\) 14.2465 1.11587 0.557937 0.829883i \(-0.311593\pi\)
0.557937 + 0.829883i \(0.311593\pi\)
\(164\) 9.98652 0.779816
\(165\) 0 0
\(166\) −4.05120 −0.314434
\(167\) −8.77584 −0.679095 −0.339547 0.940589i \(-0.610274\pi\)
−0.339547 + 0.940589i \(0.610274\pi\)
\(168\) 6.59672 0.508948
\(169\) −4.68315 −0.360242
\(170\) −0.338382 −0.0259527
\(171\) 4.83081 0.369421
\(172\) −2.76591 −0.210898
\(173\) 2.93321 0.223008 0.111504 0.993764i \(-0.464433\pi\)
0.111504 + 0.993764i \(0.464433\pi\)
\(174\) 14.4223 1.09335
\(175\) −11.7550 −0.888593
\(176\) 0 0
\(177\) 23.8173 1.79022
\(178\) 9.25833 0.693942
\(179\) 7.27182 0.543521 0.271760 0.962365i \(-0.412394\pi\)
0.271760 + 0.962365i \(0.412394\pi\)
\(180\) 0.560878 0.0418054
\(181\) 15.7408 1.17001 0.585003 0.811031i \(-0.301093\pi\)
0.585003 + 0.811031i \(0.301093\pi\)
\(182\) −6.79836 −0.503928
\(183\) −7.59672 −0.561565
\(184\) 1.44101 0.106232
\(185\) 1.22416 0.0900021
\(186\) 23.2961 1.70815
\(187\) 0 0
\(188\) 4.23221 0.308666
\(189\) −12.0773 −0.878496
\(190\) −0.116104 −0.00842309
\(191\) 24.2998 1.75827 0.879137 0.476569i \(-0.158120\pi\)
0.879137 + 0.476569i \(0.158120\pi\)
\(192\) −2.79836 −0.201954
\(193\) −10.9100 −0.785321 −0.392660 0.919684i \(-0.628445\pi\)
−0.392660 + 0.919684i \(0.628445\pi\)
\(194\) −8.71094 −0.625409
\(195\) −0.936982 −0.0670987
\(196\) −1.44289 −0.103064
\(197\) −3.53558 −0.251900 −0.125950 0.992037i \(-0.540198\pi\)
−0.125950 + 0.992037i \(0.540198\pi\)
\(198\) 0 0
\(199\) 6.09081 0.431766 0.215883 0.976419i \(-0.430737\pi\)
0.215883 + 0.976419i \(0.430737\pi\)
\(200\) 4.98652 0.352600
\(201\) −9.54363 −0.673156
\(202\) 7.83269 0.551106
\(203\) 12.1494 0.852720
\(204\) −8.15571 −0.571014
\(205\) 1.15948 0.0809815
\(206\) 1.94503 0.135517
\(207\) −6.96122 −0.483839
\(208\) 2.88390 0.199962
\(209\) 0 0
\(210\) 0.765907 0.0528526
\(211\) 23.5851 1.62367 0.811833 0.583889i \(-0.198470\pi\)
0.811833 + 0.583889i \(0.198470\pi\)
\(212\) 5.56427 0.382155
\(213\) 3.27182 0.224181
\(214\) 6.57331 0.449342
\(215\) −0.321134 −0.0219011
\(216\) 5.12326 0.348594
\(217\) 19.6247 1.33221
\(218\) −3.02341 −0.204771
\(219\) 3.68753 0.249180
\(220\) 0 0
\(221\) 8.40501 0.565382
\(222\) 29.5049 1.98024
\(223\) 1.82893 0.122474 0.0612369 0.998123i \(-0.480495\pi\)
0.0612369 + 0.998123i \(0.480495\pi\)
\(224\) −2.35735 −0.157507
\(225\) −24.0889 −1.60593
\(226\) 0.478728 0.0318445
\(227\) 8.91169 0.591489 0.295745 0.955267i \(-0.404432\pi\)
0.295745 + 0.955267i \(0.404432\pi\)
\(228\) −2.79836 −0.185326
\(229\) −9.45720 −0.624949 −0.312475 0.949926i \(-0.601158\pi\)
−0.312475 + 0.949926i \(0.601158\pi\)
\(230\) 0.167307 0.0110319
\(231\) 0 0
\(232\) −5.15383 −0.338365
\(233\) −17.4294 −1.14184 −0.570919 0.821006i \(-0.693413\pi\)
−0.570919 + 0.821006i \(0.693413\pi\)
\(234\) −13.9315 −0.910734
\(235\) 0.491378 0.0320540
\(236\) −8.51118 −0.554031
\(237\) 5.41510 0.351748
\(238\) −6.87042 −0.445343
\(239\) −25.0036 −1.61735 −0.808674 0.588256i \(-0.799815\pi\)
−0.808674 + 0.588256i \(0.799815\pi\)
\(240\) −0.324902 −0.0209723
\(241\) 1.94503 0.125290 0.0626452 0.998036i \(-0.480046\pi\)
0.0626452 + 0.998036i \(0.480046\pi\)
\(242\) 0 0
\(243\) 15.8055 1.01392
\(244\) 2.71470 0.173791
\(245\) −0.167526 −0.0107028
\(246\) 27.9459 1.78176
\(247\) 2.88390 0.183498
\(248\) −8.32490 −0.528632
\(249\) −11.3367 −0.718435
\(250\) 1.15948 0.0733318
\(251\) −12.6048 −0.795606 −0.397803 0.917471i \(-0.630227\pi\)
−0.397803 + 0.917471i \(0.630227\pi\)
\(252\) 11.3879 0.717371
\(253\) 0 0
\(254\) 11.3645 0.713073
\(255\) −0.946913 −0.0592980
\(256\) 1.00000 0.0625000
\(257\) −2.53986 −0.158432 −0.0792161 0.996857i \(-0.525242\pi\)
−0.0792161 + 0.996857i \(0.525242\pi\)
\(258\) −7.74000 −0.481871
\(259\) 24.8551 1.54442
\(260\) 0.334833 0.0207654
\(261\) 24.8972 1.54110
\(262\) 19.6481 1.21387
\(263\) −31.0972 −1.91753 −0.958767 0.284192i \(-0.908275\pi\)
−0.958767 + 0.284192i \(0.908275\pi\)
\(264\) 0 0
\(265\) 0.646035 0.0396856
\(266\) −2.35735 −0.144539
\(267\) 25.9081 1.58555
\(268\) 3.41044 0.208326
\(269\) −21.4294 −1.30657 −0.653287 0.757110i \(-0.726611\pi\)
−0.653287 + 0.757110i \(0.726611\pi\)
\(270\) 0.594833 0.0362004
\(271\) −18.0152 −1.09435 −0.547173 0.837020i \(-0.684296\pi\)
−0.547173 + 0.837020i \(0.684296\pi\)
\(272\) 2.91446 0.176715
\(273\) −19.0242 −1.15140
\(274\) 5.18101 0.312996
\(275\) 0 0
\(276\) 4.03245 0.242725
\(277\) 27.1522 1.63142 0.815708 0.578464i \(-0.196348\pi\)
0.815708 + 0.578464i \(0.196348\pi\)
\(278\) −21.5598 −1.29307
\(279\) 40.2160 2.40767
\(280\) −0.273699 −0.0163566
\(281\) 16.1810 0.965278 0.482639 0.875819i \(-0.339678\pi\)
0.482639 + 0.875819i \(0.339678\pi\)
\(282\) 11.8432 0.705254
\(283\) −23.8947 −1.42039 −0.710195 0.704005i \(-0.751393\pi\)
−0.710195 + 0.704005i \(0.751393\pi\)
\(284\) −1.16919 −0.0693787
\(285\) −0.324902 −0.0192455
\(286\) 0 0
\(287\) 23.5417 1.38963
\(288\) −4.83081 −0.284658
\(289\) −8.50591 −0.500348
\(290\) −0.598382 −0.0351382
\(291\) −24.3763 −1.42897
\(292\) −1.31775 −0.0771153
\(293\) 5.58052 0.326018 0.163009 0.986625i \(-0.447880\pi\)
0.163009 + 0.986625i \(0.447880\pi\)
\(294\) −4.03772 −0.235485
\(295\) −0.988185 −0.0575343
\(296\) −10.5436 −0.612836
\(297\) 0 0
\(298\) 14.4518 0.837168
\(299\) −4.15571 −0.240331
\(300\) 13.9541 0.805639
\(301\) −6.52022 −0.375819
\(302\) 23.3987 1.34644
\(303\) 21.9187 1.25920
\(304\) 1.00000 0.0573539
\(305\) 0.315189 0.0180477
\(306\) −14.0792 −0.804855
\(307\) −1.22416 −0.0698666 −0.0349333 0.999390i \(-0.511122\pi\)
−0.0349333 + 0.999390i \(0.511122\pi\)
\(308\) 0 0
\(309\) 5.44289 0.309635
\(310\) −0.966557 −0.0548968
\(311\) 28.1663 1.59716 0.798581 0.601888i \(-0.205584\pi\)
0.798581 + 0.601888i \(0.205584\pi\)
\(312\) 8.07017 0.456884
\(313\) 33.4796 1.89238 0.946188 0.323617i \(-0.104899\pi\)
0.946188 + 0.323617i \(0.104899\pi\)
\(314\) −19.0773 −1.07660
\(315\) 1.32219 0.0744968
\(316\) −1.93510 −0.108858
\(317\) 14.4644 0.812399 0.406200 0.913784i \(-0.366854\pi\)
0.406200 + 0.913784i \(0.366854\pi\)
\(318\) 15.5708 0.873168
\(319\) 0 0
\(320\) 0.116104 0.00649043
\(321\) 18.3945 1.02668
\(322\) 3.39696 0.189305
\(323\) 2.91446 0.162165
\(324\) −0.155711 −0.00865059
\(325\) −14.3806 −0.797692
\(326\) −14.2465 −0.789042
\(327\) −8.46059 −0.467872
\(328\) −9.98652 −0.551413
\(329\) 9.97681 0.550039
\(330\) 0 0
\(331\) −5.36895 −0.295104 −0.147552 0.989054i \(-0.547139\pi\)
−0.147552 + 0.989054i \(0.547139\pi\)
\(332\) 4.05120 0.222339
\(333\) 50.9343 2.79118
\(334\) 8.77584 0.480193
\(335\) 0.395967 0.0216340
\(336\) −6.59672 −0.359880
\(337\) 29.2373 1.59266 0.796329 0.604864i \(-0.206772\pi\)
0.796329 + 0.604864i \(0.206772\pi\)
\(338\) 4.68315 0.254730
\(339\) 1.33965 0.0727600
\(340\) 0.338382 0.0183513
\(341\) 0 0
\(342\) −4.83081 −0.261220
\(343\) −19.9029 −1.07465
\(344\) 2.76591 0.149128
\(345\) 0.468185 0.0252062
\(346\) −2.93321 −0.157691
\(347\) 26.6983 1.43324 0.716620 0.697464i \(-0.245688\pi\)
0.716620 + 0.697464i \(0.245688\pi\)
\(348\) −14.4223 −0.773114
\(349\) −9.89006 −0.529403 −0.264701 0.964330i \(-0.585273\pi\)
−0.264701 + 0.964330i \(0.585273\pi\)
\(350\) 11.7550 0.628330
\(351\) −14.7749 −0.788628
\(352\) 0 0
\(353\) 34.4402 1.83307 0.916533 0.399960i \(-0.130976\pi\)
0.916533 + 0.399960i \(0.130976\pi\)
\(354\) −23.8173 −1.26588
\(355\) −0.135748 −0.00720476
\(356\) −9.25833 −0.490691
\(357\) −19.2259 −1.01754
\(358\) −7.27182 −0.384327
\(359\) −18.2006 −0.960590 −0.480295 0.877107i \(-0.659470\pi\)
−0.480295 + 0.877107i \(0.659470\pi\)
\(360\) −0.560878 −0.0295609
\(361\) 1.00000 0.0526316
\(362\) −15.7408 −0.827320
\(363\) 0 0
\(364\) 6.79836 0.356331
\(365\) −0.152996 −0.00800818
\(366\) 7.59672 0.397087
\(367\) −8.77961 −0.458292 −0.229146 0.973392i \(-0.573593\pi\)
−0.229146 + 0.973392i \(0.573593\pi\)
\(368\) −1.44101 −0.0751176
\(369\) 48.2430 2.51143
\(370\) −1.22416 −0.0636411
\(371\) 13.1169 0.680997
\(372\) −23.2961 −1.20784
\(373\) 36.2449 1.87669 0.938344 0.345704i \(-0.112360\pi\)
0.938344 + 0.345704i \(0.112360\pi\)
\(374\) 0 0
\(375\) 3.24464 0.167552
\(376\) −4.23221 −0.218259
\(377\) 14.8631 0.765489
\(378\) 12.0773 0.621191
\(379\) 21.2124 1.08961 0.544804 0.838563i \(-0.316604\pi\)
0.544804 + 0.838563i \(0.316604\pi\)
\(380\) 0.116104 0.00595603
\(381\) 31.8020 1.62926
\(382\) −24.2998 −1.24329
\(383\) −13.4808 −0.688838 −0.344419 0.938816i \(-0.611924\pi\)
−0.344419 + 0.938816i \(0.611924\pi\)
\(384\) 2.79836 0.142803
\(385\) 0 0
\(386\) 10.9100 0.555305
\(387\) −13.3616 −0.679207
\(388\) 8.71094 0.442231
\(389\) −5.06679 −0.256896 −0.128448 0.991716i \(-0.541000\pi\)
−0.128448 + 0.991716i \(0.541000\pi\)
\(390\) 0.936982 0.0474459
\(391\) −4.19976 −0.212391
\(392\) 1.44289 0.0728769
\(393\) 54.9825 2.77350
\(394\) 3.53558 0.178120
\(395\) −0.224673 −0.0113045
\(396\) 0 0
\(397\) 5.76862 0.289519 0.144759 0.989467i \(-0.453759\pi\)
0.144759 + 0.989467i \(0.453759\pi\)
\(398\) −6.09081 −0.305305
\(399\) −6.59672 −0.330249
\(400\) −4.98652 −0.249326
\(401\) 18.9431 0.945976 0.472988 0.881069i \(-0.343175\pi\)
0.472988 + 0.881069i \(0.343175\pi\)
\(402\) 9.54363 0.475993
\(403\) 24.0081 1.19593
\(404\) −7.83269 −0.389691
\(405\) −0.0180787 −0.000898337 0
\(406\) −12.1494 −0.602964
\(407\) 0 0
\(408\) 8.15571 0.403768
\(409\) 28.7314 1.42068 0.710338 0.703860i \(-0.248542\pi\)
0.710338 + 0.703860i \(0.248542\pi\)
\(410\) −1.15948 −0.0572626
\(411\) 14.4983 0.715149
\(412\) −1.94503 −0.0958247
\(413\) −20.0638 −0.987277
\(414\) 6.96122 0.342126
\(415\) 0.470362 0.0230892
\(416\) −2.88390 −0.141395
\(417\) −60.3321 −2.95448
\(418\) 0 0
\(419\) −32.9954 −1.61193 −0.805965 0.591963i \(-0.798353\pi\)
−0.805965 + 0.591963i \(0.798353\pi\)
\(420\) −0.765907 −0.0373725
\(421\) 21.1197 1.02931 0.514656 0.857397i \(-0.327920\pi\)
0.514656 + 0.857397i \(0.327920\pi\)
\(422\) −23.5851 −1.14811
\(423\) 20.4450 0.994070
\(424\) −5.56427 −0.270225
\(425\) −14.5330 −0.704955
\(426\) −3.27182 −0.158520
\(427\) 6.39952 0.309694
\(428\) −6.57331 −0.317733
\(429\) 0 0
\(430\) 0.321134 0.0154865
\(431\) 38.9916 1.87816 0.939080 0.343698i \(-0.111680\pi\)
0.939080 + 0.343698i \(0.111680\pi\)
\(432\) −5.12326 −0.246493
\(433\) 15.8146 0.760002 0.380001 0.924986i \(-0.375924\pi\)
0.380001 + 0.924986i \(0.375924\pi\)
\(434\) −19.6247 −0.942017
\(435\) −1.67449 −0.0802855
\(436\) 3.02341 0.144795
\(437\) −1.44101 −0.0689327
\(438\) −3.68753 −0.176197
\(439\) −14.6697 −0.700145 −0.350072 0.936723i \(-0.613843\pi\)
−0.350072 + 0.936723i \(0.613843\pi\)
\(440\) 0 0
\(441\) −6.97032 −0.331920
\(442\) −8.40501 −0.399785
\(443\) 18.4375 0.875990 0.437995 0.898977i \(-0.355689\pi\)
0.437995 + 0.898977i \(0.355689\pi\)
\(444\) −29.5049 −1.40024
\(445\) −1.07493 −0.0509567
\(446\) −1.82893 −0.0866021
\(447\) 40.4412 1.91280
\(448\) 2.35735 0.111374
\(449\) 26.6266 1.25659 0.628294 0.777976i \(-0.283754\pi\)
0.628294 + 0.777976i \(0.283754\pi\)
\(450\) 24.0889 1.13556
\(451\) 0 0
\(452\) −0.478728 −0.0225175
\(453\) 65.4779 3.07642
\(454\) −8.91169 −0.418246
\(455\) 0.789319 0.0370038
\(456\) 2.79836 0.131045
\(457\) 5.75520 0.269217 0.134609 0.990899i \(-0.457022\pi\)
0.134609 + 0.990899i \(0.457022\pi\)
\(458\) 9.45720 0.441906
\(459\) −14.9315 −0.696945
\(460\) −0.167307 −0.00780073
\(461\) 18.3114 0.852848 0.426424 0.904523i \(-0.359773\pi\)
0.426424 + 0.904523i \(0.359773\pi\)
\(462\) 0 0
\(463\) 1.12176 0.0521324 0.0260662 0.999660i \(-0.491702\pi\)
0.0260662 + 0.999660i \(0.491702\pi\)
\(464\) 5.15383 0.239260
\(465\) −2.70477 −0.125431
\(466\) 17.4294 0.807402
\(467\) −19.1053 −0.884090 −0.442045 0.896993i \(-0.645747\pi\)
−0.442045 + 0.896993i \(0.645747\pi\)
\(468\) 13.9315 0.643986
\(469\) 8.03961 0.371235
\(470\) −0.491378 −0.0226656
\(471\) −53.3852 −2.45986
\(472\) 8.51118 0.391759
\(473\) 0 0
\(474\) −5.41510 −0.248724
\(475\) −4.98652 −0.228797
\(476\) 6.87042 0.314905
\(477\) 26.8799 1.23075
\(478\) 25.0036 1.14364
\(479\) 16.7568 0.765638 0.382819 0.923823i \(-0.374953\pi\)
0.382819 + 0.923823i \(0.374953\pi\)
\(480\) 0.324902 0.0148297
\(481\) 30.4067 1.38643
\(482\) −1.94503 −0.0885937
\(483\) 9.50591 0.432534
\(484\) 0 0
\(485\) 1.01138 0.0459243
\(486\) −15.8055 −0.716953
\(487\) −1.61042 −0.0729749 −0.0364874 0.999334i \(-0.511617\pi\)
−0.0364874 + 0.999334i \(0.511617\pi\)
\(488\) −2.71470 −0.122889
\(489\) −39.8669 −1.80284
\(490\) 0.167526 0.00756804
\(491\) −0.200650 −0.00905520 −0.00452760 0.999990i \(-0.501441\pi\)
−0.00452760 + 0.999990i \(0.501441\pi\)
\(492\) −27.9459 −1.25990
\(493\) 15.0206 0.676495
\(494\) −2.88390 −0.129753
\(495\) 0 0
\(496\) 8.32490 0.373799
\(497\) −2.75619 −0.123632
\(498\) 11.3367 0.508010
\(499\) −22.9954 −1.02942 −0.514708 0.857366i \(-0.672100\pi\)
−0.514708 + 0.857366i \(0.672100\pi\)
\(500\) −1.15948 −0.0518534
\(501\) 24.5579 1.09717
\(502\) 12.6048 0.562578
\(503\) −19.3795 −0.864089 −0.432044 0.901852i \(-0.642208\pi\)
−0.432044 + 0.901852i \(0.642208\pi\)
\(504\) −11.3879 −0.507258
\(505\) −0.909410 −0.0404682
\(506\) 0 0
\(507\) 13.1051 0.582019
\(508\) −11.3645 −0.504219
\(509\) −20.0485 −0.888633 −0.444317 0.895870i \(-0.646554\pi\)
−0.444317 + 0.895870i \(0.646554\pi\)
\(510\) 0.946913 0.0419300
\(511\) −3.10639 −0.137419
\(512\) −1.00000 −0.0441942
\(513\) −5.12326 −0.226197
\(514\) 2.53986 0.112029
\(515\) −0.225826 −0.00995110
\(516\) 7.74000 0.340735
\(517\) 0 0
\(518\) −24.8551 −1.09207
\(519\) −8.20819 −0.360299
\(520\) −0.334833 −0.0146834
\(521\) −37.7371 −1.65329 −0.826645 0.562723i \(-0.809754\pi\)
−0.826645 + 0.562723i \(0.809754\pi\)
\(522\) −24.8972 −1.08972
\(523\) −19.3222 −0.844900 −0.422450 0.906386i \(-0.638830\pi\)
−0.422450 + 0.906386i \(0.638830\pi\)
\(524\) −19.6481 −0.858333
\(525\) 32.8947 1.43564
\(526\) 31.0972 1.35590
\(527\) 24.2626 1.05690
\(528\) 0 0
\(529\) −20.9235 −0.909717
\(530\) −0.646035 −0.0280620
\(531\) −41.1159 −1.78428
\(532\) 2.35735 0.102204
\(533\) 28.8001 1.24747
\(534\) −25.9081 −1.12115
\(535\) −0.763189 −0.0329956
\(536\) −3.41044 −0.147309
\(537\) −20.3491 −0.878130
\(538\) 21.4294 0.923887
\(539\) 0 0
\(540\) −0.594833 −0.0255975
\(541\) 36.4375 1.56657 0.783284 0.621664i \(-0.213543\pi\)
0.783284 + 0.621664i \(0.213543\pi\)
\(542\) 18.0152 0.773819
\(543\) −44.0485 −1.89030
\(544\) −2.91446 −0.124957
\(545\) 0.351031 0.0150365
\(546\) 19.0242 0.814162
\(547\) 0.362012 0.0154785 0.00773926 0.999970i \(-0.497536\pi\)
0.00773926 + 0.999970i \(0.497536\pi\)
\(548\) −5.18101 −0.221322
\(549\) 13.1142 0.559701
\(550\) 0 0
\(551\) 5.15383 0.219560
\(552\) −4.03245 −0.171633
\(553\) −4.56171 −0.193984
\(554\) −27.1522 −1.15358
\(555\) −3.42564 −0.145410
\(556\) 21.5598 0.914340
\(557\) 16.2646 0.689153 0.344577 0.938758i \(-0.388022\pi\)
0.344577 + 0.938758i \(0.388022\pi\)
\(558\) −40.2160 −1.70248
\(559\) −7.97659 −0.337374
\(560\) 0.273699 0.0115659
\(561\) 0 0
\(562\) −16.1810 −0.682555
\(563\) −27.4256 −1.15585 −0.577926 0.816089i \(-0.696138\pi\)
−0.577926 + 0.816089i \(0.696138\pi\)
\(564\) −11.8432 −0.498690
\(565\) −0.0555824 −0.00233837
\(566\) 23.8947 1.00437
\(567\) −0.367065 −0.0154153
\(568\) 1.16919 0.0490582
\(569\) −29.5183 −1.23747 −0.618736 0.785599i \(-0.712355\pi\)
−0.618736 + 0.785599i \(0.712355\pi\)
\(570\) 0.324902 0.0136086
\(571\) 11.2449 0.470583 0.235291 0.971925i \(-0.424396\pi\)
0.235291 + 0.971925i \(0.424396\pi\)
\(572\) 0 0
\(573\) −67.9996 −2.84072
\(574\) −23.5417 −0.982614
\(575\) 7.18560 0.299660
\(576\) 4.83081 0.201284
\(577\) 3.62933 0.151091 0.0755455 0.997142i \(-0.475930\pi\)
0.0755455 + 0.997142i \(0.475930\pi\)
\(578\) 8.50591 0.353799
\(579\) 30.5301 1.26879
\(580\) 0.598382 0.0248464
\(581\) 9.55011 0.396205
\(582\) 24.3763 1.01043
\(583\) 0 0
\(584\) 1.31775 0.0545287
\(585\) 1.61751 0.0668759
\(586\) −5.58052 −0.230529
\(587\) 3.55545 0.146749 0.0733745 0.997304i \(-0.476623\pi\)
0.0733745 + 0.997304i \(0.476623\pi\)
\(588\) 4.03772 0.166513
\(589\) 8.32490 0.343022
\(590\) 0.988185 0.0406829
\(591\) 9.89383 0.406978
\(592\) 10.5436 0.433341
\(593\) 40.0017 1.64267 0.821336 0.570445i \(-0.193229\pi\)
0.821336 + 0.570445i \(0.193229\pi\)
\(594\) 0 0
\(595\) 0.797685 0.0327019
\(596\) −14.4518 −0.591967
\(597\) −17.0443 −0.697575
\(598\) 4.15571 0.169940
\(599\) 5.82276 0.237912 0.118956 0.992900i \(-0.462045\pi\)
0.118956 + 0.992900i \(0.462045\pi\)
\(600\) −13.9541 −0.569672
\(601\) −38.8791 −1.58591 −0.792955 0.609280i \(-0.791459\pi\)
−0.792955 + 0.609280i \(0.791459\pi\)
\(602\) 6.52022 0.265744
\(603\) 16.4752 0.670921
\(604\) −23.3987 −0.952079
\(605\) 0 0
\(606\) −21.9187 −0.890386
\(607\) 5.26210 0.213582 0.106791 0.994281i \(-0.465942\pi\)
0.106791 + 0.994281i \(0.465942\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −33.9983 −1.37768
\(610\) −0.315189 −0.0127616
\(611\) 12.2052 0.493771
\(612\) 14.0792 0.569118
\(613\) −41.7855 −1.68770 −0.843851 0.536577i \(-0.819717\pi\)
−0.843851 + 0.536577i \(0.819717\pi\)
\(614\) 1.22416 0.0494031
\(615\) −3.24464 −0.130836
\(616\) 0 0
\(617\) 18.7661 0.755496 0.377748 0.925908i \(-0.376699\pi\)
0.377748 + 0.925908i \(0.376699\pi\)
\(618\) −5.44289 −0.218945
\(619\) −26.7901 −1.07679 −0.538394 0.842694i \(-0.680969\pi\)
−0.538394 + 0.842694i \(0.680969\pi\)
\(620\) 0.966557 0.0388179
\(621\) 7.38265 0.296255
\(622\) −28.1663 −1.12936
\(623\) −21.8252 −0.874406
\(624\) −8.07017 −0.323065
\(625\) 24.7980 0.991919
\(626\) −33.4796 −1.33811
\(627\) 0 0
\(628\) 19.0773 0.761268
\(629\) 30.7290 1.22525
\(630\) −1.32219 −0.0526772
\(631\) 10.0038 0.398244 0.199122 0.979975i \(-0.436191\pi\)
0.199122 + 0.979975i \(0.436191\pi\)
\(632\) 1.93510 0.0769741
\(633\) −65.9996 −2.62325
\(634\) −14.4644 −0.574453
\(635\) −1.31947 −0.0523615
\(636\) −15.5708 −0.617423
\(637\) −4.16114 −0.164871
\(638\) 0 0
\(639\) −5.64814 −0.223437
\(640\) −0.116104 −0.00458943
\(641\) −17.8470 −0.704914 −0.352457 0.935828i \(-0.614654\pi\)
−0.352457 + 0.935828i \(0.614654\pi\)
\(642\) −18.3945 −0.725972
\(643\) 37.1053 1.46329 0.731646 0.681685i \(-0.238752\pi\)
0.731646 + 0.681685i \(0.238752\pi\)
\(644\) −3.39696 −0.133859
\(645\) 0.898648 0.0353842
\(646\) −2.91446 −0.114668
\(647\) 28.7099 1.12870 0.564351 0.825535i \(-0.309126\pi\)
0.564351 + 0.825535i \(0.309126\pi\)
\(648\) 0.155711 0.00611689
\(649\) 0 0
\(650\) 14.3806 0.564054
\(651\) −54.9170 −2.15237
\(652\) 14.2465 0.557937
\(653\) −23.2212 −0.908717 −0.454358 0.890819i \(-0.650131\pi\)
−0.454358 + 0.890819i \(0.650131\pi\)
\(654\) 8.46059 0.330835
\(655\) −2.28123 −0.0891352
\(656\) 9.98652 0.389908
\(657\) −6.36578 −0.248353
\(658\) −9.97681 −0.388936
\(659\) 9.79370 0.381508 0.190754 0.981638i \(-0.438907\pi\)
0.190754 + 0.981638i \(0.438907\pi\)
\(660\) 0 0
\(661\) 25.9368 1.00883 0.504413 0.863463i \(-0.331709\pi\)
0.504413 + 0.863463i \(0.331709\pi\)
\(662\) 5.36895 0.208670
\(663\) −23.5202 −0.913450
\(664\) −4.05120 −0.157217
\(665\) 0.273699 0.0106136
\(666\) −50.9343 −1.97366
\(667\) −7.42669 −0.287563
\(668\) −8.77584 −0.339547
\(669\) −5.11799 −0.197873
\(670\) −0.395967 −0.0152975
\(671\) 0 0
\(672\) 6.59672 0.254474
\(673\) −16.0695 −0.619434 −0.309717 0.950829i \(-0.600234\pi\)
−0.309717 + 0.950829i \(0.600234\pi\)
\(674\) −29.2373 −1.12618
\(675\) 25.5472 0.983314
\(676\) −4.68315 −0.180121
\(677\) 18.9254 0.727362 0.363681 0.931524i \(-0.381520\pi\)
0.363681 + 0.931524i \(0.381520\pi\)
\(678\) −1.33965 −0.0514491
\(679\) 20.5347 0.788051
\(680\) −0.338382 −0.0129763
\(681\) −24.9381 −0.955630
\(682\) 0 0
\(683\) 24.8175 0.949617 0.474809 0.880089i \(-0.342517\pi\)
0.474809 + 0.880089i \(0.342517\pi\)
\(684\) 4.83081 0.184711
\(685\) −0.601537 −0.0229836
\(686\) 19.9029 0.759895
\(687\) 26.4646 1.00969
\(688\) −2.76591 −0.105449
\(689\) 16.0468 0.611333
\(690\) −0.468185 −0.0178235
\(691\) −0.761528 −0.0289699 −0.0144849 0.999895i \(-0.504611\pi\)
−0.0144849 + 0.999895i \(0.504611\pi\)
\(692\) 2.93321 0.111504
\(693\) 0 0
\(694\) −26.6983 −1.01345
\(695\) 2.50319 0.0949514
\(696\) 14.4223 0.546674
\(697\) 29.1053 1.10244
\(698\) 9.89006 0.374344
\(699\) 48.7737 1.84479
\(700\) −11.7550 −0.444297
\(701\) −37.8593 −1.42993 −0.714964 0.699162i \(-0.753557\pi\)
−0.714964 + 0.699162i \(0.753557\pi\)
\(702\) 14.7749 0.557644
\(703\) 10.5436 0.397661
\(704\) 0 0
\(705\) −1.37505 −0.0517874
\(706\) −34.4402 −1.29617
\(707\) −18.4644 −0.694426
\(708\) 23.8173 0.895110
\(709\) 10.3898 0.390197 0.195099 0.980784i \(-0.437497\pi\)
0.195099 + 0.980784i \(0.437497\pi\)
\(710\) 0.135748 0.00509454
\(711\) −9.34809 −0.350581
\(712\) 9.25833 0.346971
\(713\) −11.9962 −0.449262
\(714\) 19.2259 0.719511
\(715\) 0 0
\(716\) 7.27182 0.271760
\(717\) 69.9690 2.61304
\(718\) 18.2006 0.679240
\(719\) −49.0175 −1.82805 −0.914023 0.405663i \(-0.867041\pi\)
−0.914023 + 0.405663i \(0.867041\pi\)
\(720\) 0.560878 0.0209027
\(721\) −4.58512 −0.170759
\(722\) −1.00000 −0.0372161
\(723\) −5.44289 −0.202423
\(724\) 15.7408 0.585003
\(725\) −25.6997 −0.954461
\(726\) 0 0
\(727\) −5.39418 −0.200059 −0.100030 0.994984i \(-0.531894\pi\)
−0.100030 + 0.994984i \(0.531894\pi\)
\(728\) −6.79836 −0.251964
\(729\) −43.7624 −1.62083
\(730\) 0.152996 0.00566264
\(731\) −8.06113 −0.298152
\(732\) −7.59672 −0.280783
\(733\) 19.9693 0.737582 0.368791 0.929512i \(-0.379772\pi\)
0.368791 + 0.929512i \(0.379772\pi\)
\(734\) 8.77961 0.324061
\(735\) 0.468797 0.0172918
\(736\) 1.44101 0.0531162
\(737\) 0 0
\(738\) −48.2430 −1.77585
\(739\) −48.7027 −1.79156 −0.895778 0.444501i \(-0.853381\pi\)
−0.895778 + 0.444501i \(0.853381\pi\)
\(740\) 1.22416 0.0450011
\(741\) −8.07017 −0.296465
\(742\) −13.1169 −0.481538
\(743\) 37.5929 1.37915 0.689576 0.724214i \(-0.257797\pi\)
0.689576 + 0.724214i \(0.257797\pi\)
\(744\) 23.2961 0.854075
\(745\) −1.67791 −0.0614740
\(746\) −36.2449 −1.32702
\(747\) 19.5706 0.716050
\(748\) 0 0
\(749\) −15.4956 −0.566197
\(750\) −3.24464 −0.118477
\(751\) −26.4442 −0.964964 −0.482482 0.875906i \(-0.660265\pi\)
−0.482482 + 0.875906i \(0.660265\pi\)
\(752\) 4.23221 0.154333
\(753\) 35.2726 1.28541
\(754\) −14.8631 −0.541282
\(755\) −2.71669 −0.0988704
\(756\) −12.0773 −0.439248
\(757\) 49.7039 1.80652 0.903260 0.429093i \(-0.141167\pi\)
0.903260 + 0.429093i \(0.141167\pi\)
\(758\) −21.2124 −0.770469
\(759\) 0 0
\(760\) −0.116104 −0.00421155
\(761\) 7.24019 0.262457 0.131228 0.991352i \(-0.458108\pi\)
0.131228 + 0.991352i \(0.458108\pi\)
\(762\) −31.8020 −1.15206
\(763\) 7.12725 0.258024
\(764\) 24.2998 0.879137
\(765\) 1.63466 0.0591012
\(766\) 13.4808 0.487082
\(767\) −24.5454 −0.886281
\(768\) −2.79836 −0.100977
\(769\) −31.5373 −1.13726 −0.568632 0.822592i \(-0.692527\pi\)
−0.568632 + 0.822592i \(0.692527\pi\)
\(770\) 0 0
\(771\) 7.10744 0.255968
\(772\) −10.9100 −0.392660
\(773\) −36.0430 −1.29638 −0.648188 0.761480i \(-0.724473\pi\)
−0.648188 + 0.761480i \(0.724473\pi\)
\(774\) 13.3616 0.480272
\(775\) −41.5123 −1.49117
\(776\) −8.71094 −0.312704
\(777\) −69.5533 −2.49521
\(778\) 5.06679 0.181653
\(779\) 9.98652 0.357804
\(780\) −0.936982 −0.0335493
\(781\) 0 0
\(782\) 4.19976 0.150183
\(783\) −26.4044 −0.943616
\(784\) −1.44289 −0.0515318
\(785\) 2.21496 0.0790553
\(786\) −54.9825 −1.96116
\(787\) 24.5894 0.876517 0.438259 0.898849i \(-0.355595\pi\)
0.438259 + 0.898849i \(0.355595\pi\)
\(788\) −3.53558 −0.125950
\(789\) 87.0211 3.09803
\(790\) 0.224673 0.00799352
\(791\) −1.12853 −0.0401260
\(792\) 0 0
\(793\) 7.82893 0.278013
\(794\) −5.76862 −0.204721
\(795\) −1.80784 −0.0641174
\(796\) 6.09081 0.215883
\(797\) −39.0060 −1.38166 −0.690832 0.723016i \(-0.742755\pi\)
−0.690832 + 0.723016i \(0.742755\pi\)
\(798\) 6.59672 0.233521
\(799\) 12.3346 0.436367
\(800\) 4.98652 0.176300
\(801\) −44.7252 −1.58029
\(802\) −18.9431 −0.668906
\(803\) 0 0
\(804\) −9.54363 −0.336578
\(805\) −0.394402 −0.0139008
\(806\) −24.0081 −0.845651
\(807\) 59.9672 2.11094
\(808\) 7.83269 0.275553
\(809\) −37.8563 −1.33096 −0.665478 0.746418i \(-0.731772\pi\)
−0.665478 + 0.746418i \(0.731772\pi\)
\(810\) 0.0180787 0.000635220 0
\(811\) 37.2754 1.30891 0.654457 0.756099i \(-0.272897\pi\)
0.654457 + 0.756099i \(0.272897\pi\)
\(812\) 12.1494 0.426360
\(813\) 50.4130 1.76806
\(814\) 0 0
\(815\) 1.65408 0.0579400
\(816\) −8.15571 −0.285507
\(817\) −2.76591 −0.0967669
\(818\) −28.7314 −1.00457
\(819\) 32.8416 1.14758
\(820\) 1.15948 0.0404907
\(821\) 1.94263 0.0677984 0.0338992 0.999425i \(-0.489207\pi\)
0.0338992 + 0.999425i \(0.489207\pi\)
\(822\) −14.4983 −0.505687
\(823\) 29.1781 1.01708 0.508542 0.861037i \(-0.330185\pi\)
0.508542 + 0.861037i \(0.330185\pi\)
\(824\) 1.94503 0.0677583
\(825\) 0 0
\(826\) 20.0638 0.698111
\(827\) −4.87930 −0.169670 −0.0848349 0.996395i \(-0.527036\pi\)
−0.0848349 + 0.996395i \(0.527036\pi\)
\(828\) −6.96122 −0.241919
\(829\) 15.4007 0.534889 0.267445 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(830\) −0.470362 −0.0163265
\(831\) −75.9815 −2.63577
\(832\) 2.88390 0.0999811
\(833\) −4.20525 −0.145703
\(834\) 60.3321 2.08913
\(835\) −1.01891 −0.0352609
\(836\) 0 0
\(837\) −42.6506 −1.47422
\(838\) 32.9954 1.13981
\(839\) −0.435061 −0.0150200 −0.00750999 0.999972i \(-0.502391\pi\)
−0.00750999 + 0.999972i \(0.502391\pi\)
\(840\) 0.765907 0.0264263
\(841\) −2.43807 −0.0840713
\(842\) −21.1197 −0.727833
\(843\) −45.2803 −1.55953
\(844\) 23.5851 0.811833
\(845\) −0.543734 −0.0187050
\(846\) −20.4450 −0.702913
\(847\) 0 0
\(848\) 5.56427 0.191078
\(849\) 66.8658 2.29483
\(850\) 14.5330 0.498479
\(851\) −15.1934 −0.520824
\(852\) 3.27182 0.112091
\(853\) 30.7594 1.05318 0.526591 0.850119i \(-0.323470\pi\)
0.526591 + 0.850119i \(0.323470\pi\)
\(854\) −6.39952 −0.218987
\(855\) 0.560878 0.0191816
\(856\) 6.57331 0.224671
\(857\) 19.0071 0.649270 0.324635 0.945839i \(-0.394759\pi\)
0.324635 + 0.945839i \(0.394759\pi\)
\(858\) 0 0
\(859\) −40.7564 −1.39059 −0.695295 0.718724i \(-0.744726\pi\)
−0.695295 + 0.718724i \(0.744726\pi\)
\(860\) −0.321134 −0.0109506
\(861\) −65.8782 −2.24512
\(862\) −38.9916 −1.32806
\(863\) −40.9208 −1.39296 −0.696480 0.717576i \(-0.745251\pi\)
−0.696480 + 0.717576i \(0.745251\pi\)
\(864\) 5.12326 0.174297
\(865\) 0.340559 0.0115794
\(866\) −15.8146 −0.537402
\(867\) 23.8026 0.808378
\(868\) 19.6247 0.666107
\(869\) 0 0
\(870\) 1.67449 0.0567704
\(871\) 9.83535 0.333258
\(872\) −3.02341 −0.102386
\(873\) 42.0809 1.42422
\(874\) 1.44101 0.0487428
\(875\) −2.73330 −0.0924024
\(876\) 3.68753 0.124590
\(877\) −41.0999 −1.38785 −0.693923 0.720050i \(-0.744119\pi\)
−0.693923 + 0.720050i \(0.744119\pi\)
\(878\) 14.6697 0.495077
\(879\) −15.6163 −0.526725
\(880\) 0 0
\(881\) −28.0388 −0.944650 −0.472325 0.881424i \(-0.656585\pi\)
−0.472325 + 0.881424i \(0.656585\pi\)
\(882\) 6.97032 0.234703
\(883\) −9.46358 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(884\) 8.40501 0.282691
\(885\) 2.76530 0.0929544
\(886\) −18.4375 −0.619418
\(887\) 11.8774 0.398804 0.199402 0.979918i \(-0.436100\pi\)
0.199402 + 0.979918i \(0.436100\pi\)
\(888\) 29.5049 0.990118
\(889\) −26.7901 −0.898513
\(890\) 1.07493 0.0360318
\(891\) 0 0
\(892\) 1.82893 0.0612369
\(893\) 4.23221 0.141625
\(894\) −40.4412 −1.35256
\(895\) 0.844289 0.0282215
\(896\) −2.35735 −0.0787536
\(897\) 11.6292 0.388287
\(898\) −26.6266 −0.888541
\(899\) 42.9051 1.43097
\(900\) −24.0889 −0.802964
\(901\) 16.2168 0.540261
\(902\) 0 0
\(903\) 18.2459 0.607186
\(904\) 0.478728 0.0159223
\(905\) 1.82758 0.0607508
\(906\) −65.4779 −2.17536
\(907\) 36.1192 1.19932 0.599659 0.800256i \(-0.295303\pi\)
0.599659 + 0.800256i \(0.295303\pi\)
\(908\) 8.91169 0.295745
\(909\) −37.8382 −1.25502
\(910\) −0.789319 −0.0261657
\(911\) 27.7855 0.920576 0.460288 0.887770i \(-0.347746\pi\)
0.460288 + 0.887770i \(0.347746\pi\)
\(912\) −2.79836 −0.0926629
\(913\) 0 0
\(914\) −5.75520 −0.190365
\(915\) −0.882012 −0.0291584
\(916\) −9.45720 −0.312475
\(917\) −46.3176 −1.52954
\(918\) 14.9315 0.492815
\(919\) 46.5464 1.53542 0.767712 0.640795i \(-0.221395\pi\)
0.767712 + 0.640795i \(0.221395\pi\)
\(920\) 0.167307 0.00551595
\(921\) 3.42564 0.112879
\(922\) −18.3114 −0.603055
\(923\) −3.37182 −0.110985
\(924\) 0 0
\(925\) −52.5760 −1.72869
\(926\) −1.12176 −0.0368632
\(927\) −9.39607 −0.308607
\(928\) −5.15383 −0.169183
\(929\) 28.6551 0.940144 0.470072 0.882628i \(-0.344228\pi\)
0.470072 + 0.882628i \(0.344228\pi\)
\(930\) 2.70477 0.0886930
\(931\) −1.44289 −0.0472888
\(932\) −17.4294 −0.570919
\(933\) −78.8193 −2.58043
\(934\) 19.1053 0.625146
\(935\) 0 0
\(936\) −13.9315 −0.455367
\(937\) 37.8593 1.23681 0.618404 0.785860i \(-0.287779\pi\)
0.618404 + 0.785860i \(0.287779\pi\)
\(938\) −8.03961 −0.262502
\(939\) −93.6878 −3.05739
\(940\) 0.491378 0.0160270
\(941\) 4.46036 0.145403 0.0727017 0.997354i \(-0.476838\pi\)
0.0727017 + 0.997354i \(0.476838\pi\)
\(942\) 53.3852 1.73938
\(943\) −14.3906 −0.468624
\(944\) −8.51118 −0.277015
\(945\) −1.40223 −0.0456146
\(946\) 0 0
\(947\) 29.2314 0.949892 0.474946 0.880015i \(-0.342468\pi\)
0.474946 + 0.880015i \(0.342468\pi\)
\(948\) 5.41510 0.175874
\(949\) −3.80024 −0.123361
\(950\) 4.98652 0.161784
\(951\) −40.4765 −1.31254
\(952\) −6.87042 −0.222671
\(953\) 5.00428 0.162105 0.0810523 0.996710i \(-0.474172\pi\)
0.0810523 + 0.996710i \(0.474172\pi\)
\(954\) −26.8799 −0.870269
\(955\) 2.82132 0.0912956
\(956\) −25.0036 −0.808674
\(957\) 0 0
\(958\) −16.7568 −0.541388
\(959\) −12.2135 −0.394393
\(960\) −0.324902 −0.0104862
\(961\) 38.3040 1.23561
\(962\) −30.4067 −0.980352
\(963\) −31.7544 −1.02327
\(964\) 1.94503 0.0626452
\(965\) −1.26670 −0.0407765
\(966\) −9.50591 −0.305848
\(967\) −33.5154 −1.07778 −0.538891 0.842375i \(-0.681157\pi\)
−0.538891 + 0.842375i \(0.681157\pi\)
\(968\) 0 0
\(969\) −8.15571 −0.261999
\(970\) −1.01138 −0.0324734
\(971\) 55.5902 1.78398 0.891988 0.452060i \(-0.149311\pi\)
0.891988 + 0.452060i \(0.149311\pi\)
\(972\) 15.8055 0.506962
\(973\) 50.8241 1.62935
\(974\) 1.61042 0.0516010
\(975\) 40.2421 1.28878
\(976\) 2.71470 0.0868956
\(977\) 55.0510 1.76124 0.880618 0.473826i \(-0.157127\pi\)
0.880618 + 0.473826i \(0.157127\pi\)
\(978\) 39.8669 1.27480
\(979\) 0 0
\(980\) −0.167526 −0.00535141
\(981\) 14.6055 0.466318
\(982\) 0.200650 0.00640299
\(983\) −42.1648 −1.34485 −0.672424 0.740166i \(-0.734747\pi\)
−0.672424 + 0.740166i \(0.734747\pi\)
\(984\) 27.9459 0.890881
\(985\) −0.410497 −0.0130795
\(986\) −15.0206 −0.478354
\(987\) −27.9187 −0.888661
\(988\) 2.88390 0.0917489
\(989\) 3.98569 0.126738
\(990\) 0 0
\(991\) −50.2222 −1.59536 −0.797680 0.603081i \(-0.793940\pi\)
−0.797680 + 0.603081i \(0.793940\pi\)
\(992\) −8.32490 −0.264316
\(993\) 15.0242 0.476780
\(994\) 2.75619 0.0874212
\(995\) 0.707169 0.0224188
\(996\) −11.3367 −0.359218
\(997\) −41.3157 −1.30848 −0.654241 0.756286i \(-0.727012\pi\)
−0.654241 + 0.756286i \(0.727012\pi\)
\(998\) 22.9954 0.727907
\(999\) −54.0178 −1.70905
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 4598.2.a.br.1.1 4
11.10 odd 2 4598.2.a.bu.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.br.1.1 4 1.1 even 1 trivial
4598.2.a.bu.1.1 yes 4 11.10 odd 2