Properties

Label 4598.2.a.cb.1.2
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.115899\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.11129 q^{3} +1.00000 q^{4} -2.75070 q^{5} -2.11129 q^{6} +3.66381 q^{7} +1.00000 q^{8} +1.45756 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.11129 q^{3} +1.00000 q^{4} -2.75070 q^{5} -2.11129 q^{6} +3.66381 q^{7} +1.00000 q^{8} +1.45756 q^{9} -2.75070 q^{10} -2.11129 q^{12} -3.81790 q^{13} +3.66381 q^{14} +5.80753 q^{15} +1.00000 q^{16} -4.31816 q^{17} +1.45756 q^{18} -1.00000 q^{19} -2.75070 q^{20} -7.73537 q^{21} -0.695478 q^{23} -2.11129 q^{24} +2.56634 q^{25} -3.81790 q^{26} +3.25654 q^{27} +3.66381 q^{28} +7.77089 q^{29} +5.80753 q^{30} +0.340036 q^{31} +1.00000 q^{32} -4.31816 q^{34} -10.0780 q^{35} +1.45756 q^{36} -0.692199 q^{37} -1.00000 q^{38} +8.06072 q^{39} -2.75070 q^{40} +9.21705 q^{41} -7.73537 q^{42} +2.01495 q^{43} -4.00932 q^{45} -0.695478 q^{46} -9.80308 q^{47} -2.11129 q^{48} +6.42347 q^{49} +2.56634 q^{50} +9.11690 q^{51} -3.81790 q^{52} -5.54956 q^{53} +3.25654 q^{54} +3.66381 q^{56} +2.11129 q^{57} +7.77089 q^{58} -8.22759 q^{59} +5.80753 q^{60} -0.344547 q^{61} +0.340036 q^{62} +5.34023 q^{63} +1.00000 q^{64} +10.5019 q^{65} -5.04040 q^{67} -4.31816 q^{68} +1.46836 q^{69} -10.0780 q^{70} -3.22075 q^{71} +1.45756 q^{72} +14.6020 q^{73} -0.692199 q^{74} -5.41830 q^{75} -1.00000 q^{76} +8.06072 q^{78} -4.44082 q^{79} -2.75070 q^{80} -11.2482 q^{81} +9.21705 q^{82} +12.0464 q^{83} -7.73537 q^{84} +11.8779 q^{85} +2.01495 q^{86} -16.4066 q^{87} -0.966434 q^{89} -4.00932 q^{90} -13.9881 q^{91} -0.695478 q^{92} -0.717915 q^{93} -9.80308 q^{94} +2.75070 q^{95} -2.11129 q^{96} +16.3170 q^{97} +6.42347 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + 8 q^{12} - 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} - 4 q^{17} + 22 q^{18} - 8 q^{19} - 20 q^{21} + 14 q^{23} + 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} + 4 q^{28} - 2 q^{29} + 4 q^{30} + 8 q^{32} - 4 q^{34} + 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} + 16 q^{39} + 8 q^{41} - 20 q^{42} + 8 q^{43} + 16 q^{45} + 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} + 36 q^{50} + 18 q^{51} - 12 q^{52} + 36 q^{53} + 32 q^{54} + 4 q^{56} - 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} + 12 q^{61} + 24 q^{63} + 8 q^{64} + 16 q^{65} + 16 q^{67} - 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} + 22 q^{72} - 20 q^{73} + 24 q^{74} + 40 q^{75} - 8 q^{76} + 16 q^{78} - 12 q^{79} + 40 q^{81} + 8 q^{82} + 20 q^{83} - 20 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} - 16 q^{94} + 8 q^{96} + 4 q^{97} + 34 q^{98} + O(q^{100}) \)

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
\(3\) −2.11129 −1.21896 −0.609478 0.792803i \(-0.708621\pi\)
−0.609478 + 0.792803i \(0.708621\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.75070 −1.23015 −0.615075 0.788469i \(-0.710874\pi\)
−0.615075 + 0.788469i \(0.710874\pi\)
\(6\) −2.11129 −0.861932
\(7\) 3.66381 1.38479 0.692394 0.721519i \(-0.256556\pi\)
0.692394 + 0.721519i \(0.256556\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.45756 0.485855
\(10\) −2.75070 −0.869847
\(11\) 0 0
\(12\) −2.11129 −0.609478
\(13\) −3.81790 −1.05890 −0.529448 0.848342i \(-0.677601\pi\)
−0.529448 + 0.848342i \(0.677601\pi\)
\(14\) 3.66381 0.979193
\(15\) 5.80753 1.49950
\(16\) 1.00000 0.250000
\(17\) −4.31816 −1.04731 −0.523653 0.851931i \(-0.675431\pi\)
−0.523653 + 0.851931i \(0.675431\pi\)
\(18\) 1.45756 0.343551
\(19\) −1.00000 −0.229416
\(20\) −2.75070 −0.615075
\(21\) −7.73537 −1.68800
\(22\) 0 0
\(23\) −0.695478 −0.145017 −0.0725086 0.997368i \(-0.523100\pi\)
−0.0725086 + 0.997368i \(0.523100\pi\)
\(24\) −2.11129 −0.430966
\(25\) 2.56634 0.513268
\(26\) −3.81790 −0.748753
\(27\) 3.25654 0.626721
\(28\) 3.66381 0.692394
\(29\) 7.77089 1.44302 0.721509 0.692405i \(-0.243449\pi\)
0.721509 + 0.692405i \(0.243449\pi\)
\(30\) 5.80753 1.06031
\(31\) 0.340036 0.0610722 0.0305361 0.999534i \(-0.490279\pi\)
0.0305361 + 0.999534i \(0.490279\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.31816 −0.740558
\(35\) −10.0780 −1.70350
\(36\) 1.45756 0.242927
\(37\) −0.692199 −0.113797 −0.0568984 0.998380i \(-0.518121\pi\)
−0.0568984 + 0.998380i \(0.518121\pi\)
\(38\) −1.00000 −0.162221
\(39\) 8.06072 1.29075
\(40\) −2.75070 −0.434924
\(41\) 9.21705 1.43946 0.719731 0.694253i \(-0.244265\pi\)
0.719731 + 0.694253i \(0.244265\pi\)
\(42\) −7.73537 −1.19359
\(43\) 2.01495 0.307277 0.153638 0.988127i \(-0.450901\pi\)
0.153638 + 0.988127i \(0.450901\pi\)
\(44\) 0 0
\(45\) −4.00932 −0.597674
\(46\) −0.695478 −0.102543
\(47\) −9.80308 −1.42993 −0.714963 0.699162i \(-0.753557\pi\)
−0.714963 + 0.699162i \(0.753557\pi\)
\(48\) −2.11129 −0.304739
\(49\) 6.42347 0.917639
\(50\) 2.56634 0.362935
\(51\) 9.11690 1.27662
\(52\) −3.81790 −0.529448
\(53\) −5.54956 −0.762290 −0.381145 0.924515i \(-0.624470\pi\)
−0.381145 + 0.924515i \(0.624470\pi\)
\(54\) 3.25654 0.443158
\(55\) 0 0
\(56\) 3.66381 0.489597
\(57\) 2.11129 0.279648
\(58\) 7.77089 1.02037
\(59\) −8.22759 −1.07114 −0.535570 0.844491i \(-0.679903\pi\)
−0.535570 + 0.844491i \(0.679903\pi\)
\(60\) 5.80753 0.749749
\(61\) −0.344547 −0.0441148 −0.0220574 0.999757i \(-0.507022\pi\)
−0.0220574 + 0.999757i \(0.507022\pi\)
\(62\) 0.340036 0.0431846
\(63\) 5.34023 0.672806
\(64\) 1.00000 0.125000
\(65\) 10.5019 1.30260
\(66\) 0 0
\(67\) −5.04040 −0.615783 −0.307892 0.951421i \(-0.599623\pi\)
−0.307892 + 0.951421i \(0.599623\pi\)
\(68\) −4.31816 −0.523653
\(69\) 1.46836 0.176770
\(70\) −10.0780 −1.20455
\(71\) −3.22075 −0.382233 −0.191117 0.981567i \(-0.561211\pi\)
−0.191117 + 0.981567i \(0.561211\pi\)
\(72\) 1.45756 0.171776
\(73\) 14.6020 1.70903 0.854517 0.519423i \(-0.173853\pi\)
0.854517 + 0.519423i \(0.173853\pi\)
\(74\) −0.692199 −0.0804666
\(75\) −5.41830 −0.625651
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 8.06072 0.912697
\(79\) −4.44082 −0.499631 −0.249816 0.968293i \(-0.580370\pi\)
−0.249816 + 0.968293i \(0.580370\pi\)
\(80\) −2.75070 −0.307537
\(81\) −11.2482 −1.24980
\(82\) 9.21705 1.01785
\(83\) 12.0464 1.32227 0.661133 0.750269i \(-0.270076\pi\)
0.661133 + 0.750269i \(0.270076\pi\)
\(84\) −7.73537 −0.843998
\(85\) 11.8779 1.28834
\(86\) 2.01495 0.217277
\(87\) −16.4066 −1.75898
\(88\) 0 0
\(89\) −0.966434 −0.102442 −0.0512209 0.998687i \(-0.516311\pi\)
−0.0512209 + 0.998687i \(0.516311\pi\)
\(90\) −4.00932 −0.422619
\(91\) −13.9881 −1.46635
\(92\) −0.695478 −0.0725086
\(93\) −0.717915 −0.0744443
\(94\) −9.80308 −1.01111
\(95\) 2.75070 0.282216
\(96\) −2.11129 −0.215483
\(97\) 16.3170 1.65674 0.828369 0.560183i \(-0.189269\pi\)
0.828369 + 0.560183i \(0.189269\pi\)
\(98\) 6.42347 0.648869
\(99\) 0 0
\(100\) 2.56634 0.256634
\(101\) −14.1398 −1.40696 −0.703479 0.710716i \(-0.748371\pi\)
−0.703479 + 0.710716i \(0.748371\pi\)
\(102\) 9.11690 0.902708
\(103\) 0.880175 0.0867262 0.0433631 0.999059i \(-0.486193\pi\)
0.0433631 + 0.999059i \(0.486193\pi\)
\(104\) −3.81790 −0.374376
\(105\) 21.2777 2.07649
\(106\) −5.54956 −0.539020
\(107\) −1.76713 −0.170835 −0.0854174 0.996345i \(-0.527222\pi\)
−0.0854174 + 0.996345i \(0.527222\pi\)
\(108\) 3.25654 0.313360
\(109\) 11.6553 1.11638 0.558190 0.829713i \(-0.311496\pi\)
0.558190 + 0.829713i \(0.311496\pi\)
\(110\) 0 0
\(111\) 1.46144 0.138713
\(112\) 3.66381 0.346197
\(113\) 19.9467 1.87643 0.938215 0.346054i \(-0.112478\pi\)
0.938215 + 0.346054i \(0.112478\pi\)
\(114\) 2.11129 0.197741
\(115\) 1.91305 0.178393
\(116\) 7.77089 0.721509
\(117\) −5.56484 −0.514470
\(118\) −8.22759 −0.757411
\(119\) −15.8209 −1.45030
\(120\) 5.80753 0.530153
\(121\) 0 0
\(122\) −0.344547 −0.0311938
\(123\) −19.4599 −1.75464
\(124\) 0.340036 0.0305361
\(125\) 6.69426 0.598753
\(126\) 5.34023 0.475746
\(127\) −0.568592 −0.0504544 −0.0252272 0.999682i \(-0.508031\pi\)
−0.0252272 + 0.999682i \(0.508031\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.25415 −0.374557
\(130\) 10.5019 0.921078
\(131\) 11.8544 1.03572 0.517862 0.855464i \(-0.326728\pi\)
0.517862 + 0.855464i \(0.326728\pi\)
\(132\) 0 0
\(133\) −3.66381 −0.317692
\(134\) −5.04040 −0.435424
\(135\) −8.95775 −0.770960
\(136\) −4.31816 −0.370279
\(137\) 11.2562 0.961679 0.480840 0.876809i \(-0.340332\pi\)
0.480840 + 0.876809i \(0.340332\pi\)
\(138\) 1.46836 0.124995
\(139\) 1.69259 0.143564 0.0717820 0.997420i \(-0.477131\pi\)
0.0717820 + 0.997420i \(0.477131\pi\)
\(140\) −10.0780 −0.851749
\(141\) 20.6972 1.74302
\(142\) −3.22075 −0.270280
\(143\) 0 0
\(144\) 1.45756 0.121464
\(145\) −21.3754 −1.77513
\(146\) 14.6020 1.20847
\(147\) −13.5618 −1.11856
\(148\) −0.692199 −0.0568984
\(149\) −21.5429 −1.76487 −0.882433 0.470438i \(-0.844096\pi\)
−0.882433 + 0.470438i \(0.844096\pi\)
\(150\) −5.41830 −0.442402
\(151\) 19.9995 1.62754 0.813768 0.581190i \(-0.197413\pi\)
0.813768 + 0.581190i \(0.197413\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −6.29399 −0.508839
\(154\) 0 0
\(155\) −0.935335 −0.0751279
\(156\) 8.06072 0.645374
\(157\) 6.94839 0.554542 0.277271 0.960792i \(-0.410570\pi\)
0.277271 + 0.960792i \(0.410570\pi\)
\(158\) −4.44082 −0.353293
\(159\) 11.7167 0.929198
\(160\) −2.75070 −0.217462
\(161\) −2.54810 −0.200818
\(162\) −11.2482 −0.883742
\(163\) 16.0104 1.25403 0.627014 0.779008i \(-0.284277\pi\)
0.627014 + 0.779008i \(0.284277\pi\)
\(164\) 9.21705 0.719731
\(165\) 0 0
\(166\) 12.0464 0.934983
\(167\) 3.68048 0.284804 0.142402 0.989809i \(-0.454517\pi\)
0.142402 + 0.989809i \(0.454517\pi\)
\(168\) −7.73537 −0.596797
\(169\) 1.57639 0.121261
\(170\) 11.8779 0.910997
\(171\) −1.45756 −0.111463
\(172\) 2.01495 0.153638
\(173\) −12.9528 −0.984782 −0.492391 0.870374i \(-0.663877\pi\)
−0.492391 + 0.870374i \(0.663877\pi\)
\(174\) −16.4066 −1.24378
\(175\) 9.40257 0.710768
\(176\) 0 0
\(177\) 17.3709 1.30567
\(178\) −0.966434 −0.0724373
\(179\) −13.7864 −1.03044 −0.515222 0.857056i \(-0.672291\pi\)
−0.515222 + 0.857056i \(0.672291\pi\)
\(180\) −4.00932 −0.298837
\(181\) −15.2785 −1.13564 −0.567820 0.823153i \(-0.692213\pi\)
−0.567820 + 0.823153i \(0.692213\pi\)
\(182\) −13.9881 −1.03686
\(183\) 0.727441 0.0537740
\(184\) −0.695478 −0.0512713
\(185\) 1.90403 0.139987
\(186\) −0.717915 −0.0526401
\(187\) 0 0
\(188\) −9.80308 −0.714963
\(189\) 11.9313 0.867876
\(190\) 2.75070 0.199557
\(191\) 23.9456 1.73265 0.866323 0.499485i \(-0.166477\pi\)
0.866323 + 0.499485i \(0.166477\pi\)
\(192\) −2.11129 −0.152370
\(193\) 27.1307 1.95291 0.976456 0.215715i \(-0.0692082\pi\)
0.976456 + 0.215715i \(0.0692082\pi\)
\(194\) 16.3170 1.17149
\(195\) −22.1726 −1.58781
\(196\) 6.42347 0.458820
\(197\) 21.1960 1.51015 0.755076 0.655637i \(-0.227600\pi\)
0.755076 + 0.655637i \(0.227600\pi\)
\(198\) 0 0
\(199\) −3.03824 −0.215375 −0.107688 0.994185i \(-0.534345\pi\)
−0.107688 + 0.994185i \(0.534345\pi\)
\(200\) 2.56634 0.181468
\(201\) 10.6418 0.750613
\(202\) −14.1398 −0.994870
\(203\) 28.4710 1.99827
\(204\) 9.11690 0.638311
\(205\) −25.3533 −1.77075
\(206\) 0.880175 0.0613247
\(207\) −1.01370 −0.0704572
\(208\) −3.81790 −0.264724
\(209\) 0 0
\(210\) 21.2777 1.46830
\(211\) 27.4130 1.88719 0.943593 0.331108i \(-0.107422\pi\)
0.943593 + 0.331108i \(0.107422\pi\)
\(212\) −5.54956 −0.381145
\(213\) 6.79996 0.465925
\(214\) −1.76713 −0.120799
\(215\) −5.54251 −0.377996
\(216\) 3.25654 0.221579
\(217\) 1.24582 0.0845721
\(218\) 11.6553 0.789399
\(219\) −30.8291 −2.08324
\(220\) 0 0
\(221\) 16.4863 1.10899
\(222\) 1.46144 0.0980852
\(223\) 18.7060 1.25265 0.626323 0.779564i \(-0.284559\pi\)
0.626323 + 0.779564i \(0.284559\pi\)
\(224\) 3.66381 0.244798
\(225\) 3.74060 0.249374
\(226\) 19.9467 1.32684
\(227\) 12.6689 0.840862 0.420431 0.907324i \(-0.361879\pi\)
0.420431 + 0.907324i \(0.361879\pi\)
\(228\) 2.11129 0.139824
\(229\) 22.6558 1.49714 0.748568 0.663058i \(-0.230742\pi\)
0.748568 + 0.663058i \(0.230742\pi\)
\(230\) 1.91305 0.126143
\(231\) 0 0
\(232\) 7.77089 0.510184
\(233\) 20.0819 1.31561 0.657805 0.753189i \(-0.271485\pi\)
0.657805 + 0.753189i \(0.271485\pi\)
\(234\) −5.56484 −0.363785
\(235\) 26.9653 1.75902
\(236\) −8.22759 −0.535570
\(237\) 9.37587 0.609028
\(238\) −15.8209 −1.02552
\(239\) 19.7417 1.27699 0.638493 0.769627i \(-0.279558\pi\)
0.638493 + 0.769627i \(0.279558\pi\)
\(240\) 5.80753 0.374875
\(241\) −20.0092 −1.28890 −0.644452 0.764645i \(-0.722915\pi\)
−0.644452 + 0.764645i \(0.722915\pi\)
\(242\) 0 0
\(243\) 13.9787 0.896731
\(244\) −0.344547 −0.0220574
\(245\) −17.6690 −1.12883
\(246\) −19.4599 −1.24072
\(247\) 3.81790 0.242927
\(248\) 0.340036 0.0215923
\(249\) −25.4335 −1.61178
\(250\) 6.69426 0.423382
\(251\) −28.0383 −1.76976 −0.884880 0.465818i \(-0.845760\pi\)
−0.884880 + 0.465818i \(0.845760\pi\)
\(252\) 5.34023 0.336403
\(253\) 0 0
\(254\) −0.568592 −0.0356766
\(255\) −25.0778 −1.57044
\(256\) 1.00000 0.0625000
\(257\) −13.5564 −0.845628 −0.422814 0.906217i \(-0.638958\pi\)
−0.422814 + 0.906217i \(0.638958\pi\)
\(258\) −4.25415 −0.264852
\(259\) −2.53608 −0.157585
\(260\) 10.5019 0.651300
\(261\) 11.3266 0.701097
\(262\) 11.8544 0.732367
\(263\) −4.42685 −0.272971 −0.136486 0.990642i \(-0.543581\pi\)
−0.136486 + 0.990642i \(0.543581\pi\)
\(264\) 0 0
\(265\) 15.2652 0.937731
\(266\) −3.66381 −0.224642
\(267\) 2.04043 0.124872
\(268\) −5.04040 −0.307892
\(269\) −7.98617 −0.486926 −0.243463 0.969910i \(-0.578283\pi\)
−0.243463 + 0.969910i \(0.578283\pi\)
\(270\) −8.95775 −0.545151
\(271\) 17.9453 1.09010 0.545048 0.838405i \(-0.316511\pi\)
0.545048 + 0.838405i \(0.316511\pi\)
\(272\) −4.31816 −0.261827
\(273\) 29.5329 1.78741
\(274\) 11.2562 0.680010
\(275\) 0 0
\(276\) 1.46836 0.0883848
\(277\) −21.4997 −1.29179 −0.645896 0.763425i \(-0.723516\pi\)
−0.645896 + 0.763425i \(0.723516\pi\)
\(278\) 1.69259 0.101515
\(279\) 0.495624 0.0296722
\(280\) −10.0780 −0.602277
\(281\) 19.5998 1.16923 0.584614 0.811311i \(-0.301246\pi\)
0.584614 + 0.811311i \(0.301246\pi\)
\(282\) 20.6972 1.23250
\(283\) 13.0294 0.774518 0.387259 0.921971i \(-0.373422\pi\)
0.387259 + 0.921971i \(0.373422\pi\)
\(284\) −3.22075 −0.191117
\(285\) −5.80753 −0.344009
\(286\) 0 0
\(287\) 33.7695 1.99335
\(288\) 1.45756 0.0858878
\(289\) 1.64648 0.0968515
\(290\) −21.3754 −1.25520
\(291\) −34.4499 −2.01949
\(292\) 14.6020 0.854517
\(293\) −12.4424 −0.726895 −0.363448 0.931615i \(-0.618400\pi\)
−0.363448 + 0.931615i \(0.618400\pi\)
\(294\) −13.5618 −0.790943
\(295\) 22.6316 1.31766
\(296\) −0.692199 −0.0402333
\(297\) 0 0
\(298\) −21.5429 −1.24795
\(299\) 2.65527 0.153558
\(300\) −5.41830 −0.312826
\(301\) 7.38237 0.425513
\(302\) 19.9995 1.15084
\(303\) 29.8532 1.71502
\(304\) −1.00000 −0.0573539
\(305\) 0.947746 0.0542678
\(306\) −6.29399 −0.359803
\(307\) −18.8124 −1.07368 −0.536840 0.843684i \(-0.680382\pi\)
−0.536840 + 0.843684i \(0.680382\pi\)
\(308\) 0 0
\(309\) −1.85831 −0.105715
\(310\) −0.935335 −0.0531235
\(311\) −13.2168 −0.749455 −0.374727 0.927135i \(-0.622264\pi\)
−0.374727 + 0.927135i \(0.622264\pi\)
\(312\) 8.06072 0.456348
\(313\) −3.51233 −0.198529 −0.0992644 0.995061i \(-0.531649\pi\)
−0.0992644 + 0.995061i \(0.531649\pi\)
\(314\) 6.94839 0.392120
\(315\) −14.6894 −0.827652
\(316\) −4.44082 −0.249816
\(317\) 8.53754 0.479516 0.239758 0.970833i \(-0.422932\pi\)
0.239758 + 0.970833i \(0.422932\pi\)
\(318\) 11.7167 0.657042
\(319\) 0 0
\(320\) −2.75070 −0.153769
\(321\) 3.73093 0.208240
\(322\) −2.54810 −0.142000
\(323\) 4.31816 0.240269
\(324\) −11.2482 −0.624900
\(325\) −9.79804 −0.543497
\(326\) 16.0104 0.886732
\(327\) −24.6079 −1.36082
\(328\) 9.21705 0.508927
\(329\) −35.9166 −1.98015
\(330\) 0 0
\(331\) −3.43770 −0.188953 −0.0944766 0.995527i \(-0.530118\pi\)
−0.0944766 + 0.995527i \(0.530118\pi\)
\(332\) 12.0464 0.661133
\(333\) −1.00893 −0.0552888
\(334\) 3.68048 0.201387
\(335\) 13.8646 0.757505
\(336\) −7.73537 −0.421999
\(337\) 23.8752 1.30057 0.650283 0.759692i \(-0.274650\pi\)
0.650283 + 0.759692i \(0.274650\pi\)
\(338\) 1.57639 0.0857444
\(339\) −42.1134 −2.28729
\(340\) 11.8779 0.644172
\(341\) 0 0
\(342\) −1.45756 −0.0788160
\(343\) −2.11228 −0.114052
\(344\) 2.01495 0.108639
\(345\) −4.03901 −0.217453
\(346\) −12.9528 −0.696346
\(347\) 7.32500 0.393227 0.196613 0.980481i \(-0.437006\pi\)
0.196613 + 0.980481i \(0.437006\pi\)
\(348\) −16.4066 −0.879488
\(349\) 8.11645 0.434464 0.217232 0.976120i \(-0.430297\pi\)
0.217232 + 0.976120i \(0.430297\pi\)
\(350\) 9.40257 0.502589
\(351\) −12.4331 −0.663632
\(352\) 0 0
\(353\) −5.08395 −0.270591 −0.135296 0.990805i \(-0.543198\pi\)
−0.135296 + 0.990805i \(0.543198\pi\)
\(354\) 17.3709 0.923251
\(355\) 8.85932 0.470204
\(356\) −0.966434 −0.0512209
\(357\) 33.4026 1.76785
\(358\) −13.7864 −0.728635
\(359\) 15.2156 0.803050 0.401525 0.915848i \(-0.368480\pi\)
0.401525 + 0.915848i \(0.368480\pi\)
\(360\) −4.00932 −0.211310
\(361\) 1.00000 0.0526316
\(362\) −15.2785 −0.803019
\(363\) 0 0
\(364\) −13.9881 −0.733174
\(365\) −40.1657 −2.10237
\(366\) 0.727441 0.0380239
\(367\) 29.9084 1.56120 0.780602 0.625028i \(-0.214913\pi\)
0.780602 + 0.625028i \(0.214913\pi\)
\(368\) −0.695478 −0.0362543
\(369\) 13.4344 0.699369
\(370\) 1.90403 0.0989859
\(371\) −20.3325 −1.05561
\(372\) −0.717915 −0.0372222
\(373\) −32.7204 −1.69420 −0.847100 0.531434i \(-0.821653\pi\)
−0.847100 + 0.531434i \(0.821653\pi\)
\(374\) 0 0
\(375\) −14.1336 −0.729854
\(376\) −9.80308 −0.505555
\(377\) −29.6685 −1.52801
\(378\) 11.9313 0.613681
\(379\) 21.4522 1.10193 0.550963 0.834530i \(-0.314261\pi\)
0.550963 + 0.834530i \(0.314261\pi\)
\(380\) 2.75070 0.141108
\(381\) 1.20046 0.0615017
\(382\) 23.9456 1.22517
\(383\) −17.1641 −0.877044 −0.438522 0.898720i \(-0.644498\pi\)
−0.438522 + 0.898720i \(0.644498\pi\)
\(384\) −2.11129 −0.107742
\(385\) 0 0
\(386\) 27.1307 1.38092
\(387\) 2.93691 0.149292
\(388\) 16.3170 0.828369
\(389\) 28.8568 1.46310 0.731550 0.681788i \(-0.238797\pi\)
0.731550 + 0.681788i \(0.238797\pi\)
\(390\) −22.1726 −1.12275
\(391\) 3.00318 0.151877
\(392\) 6.42347 0.324434
\(393\) −25.0281 −1.26250
\(394\) 21.1960 1.06784
\(395\) 12.2153 0.614621
\(396\) 0 0
\(397\) −5.83003 −0.292601 −0.146300 0.989240i \(-0.546737\pi\)
−0.146300 + 0.989240i \(0.546737\pi\)
\(398\) −3.03824 −0.152293
\(399\) 7.73537 0.387253
\(400\) 2.56634 0.128317
\(401\) 13.2388 0.661113 0.330556 0.943786i \(-0.392764\pi\)
0.330556 + 0.943786i \(0.392764\pi\)
\(402\) 10.6418 0.530763
\(403\) −1.29822 −0.0646691
\(404\) −14.1398 −0.703479
\(405\) 30.9404 1.53744
\(406\) 28.4710 1.41299
\(407\) 0 0
\(408\) 9.11690 0.451354
\(409\) −17.6372 −0.872102 −0.436051 0.899922i \(-0.643623\pi\)
−0.436051 + 0.899922i \(0.643623\pi\)
\(410\) −25.3533 −1.25211
\(411\) −23.7651 −1.17224
\(412\) 0.880175 0.0433631
\(413\) −30.1443 −1.48330
\(414\) −1.01370 −0.0498208
\(415\) −33.1360 −1.62658
\(416\) −3.81790 −0.187188
\(417\) −3.57357 −0.174998
\(418\) 0 0
\(419\) −28.5638 −1.39543 −0.697715 0.716375i \(-0.745800\pi\)
−0.697715 + 0.716375i \(0.745800\pi\)
\(420\) 21.2777 1.03824
\(421\) 10.1449 0.494432 0.247216 0.968960i \(-0.420484\pi\)
0.247216 + 0.968960i \(0.420484\pi\)
\(422\) 27.4130 1.33444
\(423\) −14.2886 −0.694737
\(424\) −5.54956 −0.269510
\(425\) −11.0819 −0.537549
\(426\) 6.79996 0.329459
\(427\) −1.26235 −0.0610896
\(428\) −1.76713 −0.0854174
\(429\) 0 0
\(430\) −5.54251 −0.267284
\(431\) 22.6879 1.09284 0.546418 0.837512i \(-0.315991\pi\)
0.546418 + 0.837512i \(0.315991\pi\)
\(432\) 3.25654 0.156680
\(433\) 31.0778 1.49350 0.746751 0.665104i \(-0.231613\pi\)
0.746751 + 0.665104i \(0.231613\pi\)
\(434\) 1.24582 0.0598015
\(435\) 45.1297 2.16380
\(436\) 11.6553 0.558190
\(437\) 0.695478 0.0332692
\(438\) −30.8291 −1.47307
\(439\) −30.3380 −1.44795 −0.723977 0.689824i \(-0.757688\pi\)
−0.723977 + 0.689824i \(0.757688\pi\)
\(440\) 0 0
\(441\) 9.36263 0.445839
\(442\) 16.4863 0.784174
\(443\) 5.08267 0.241485 0.120742 0.992684i \(-0.461472\pi\)
0.120742 + 0.992684i \(0.461472\pi\)
\(444\) 1.46144 0.0693567
\(445\) 2.65837 0.126019
\(446\) 18.7060 0.885754
\(447\) 45.4835 2.15129
\(448\) 3.66381 0.173099
\(449\) −34.4255 −1.62464 −0.812321 0.583211i \(-0.801796\pi\)
−0.812321 + 0.583211i \(0.801796\pi\)
\(450\) 3.74060 0.176334
\(451\) 0 0
\(452\) 19.9467 0.938215
\(453\) −42.2248 −1.98390
\(454\) 12.6689 0.594579
\(455\) 38.4769 1.80383
\(456\) 2.11129 0.0988704
\(457\) −5.43151 −0.254075 −0.127038 0.991898i \(-0.540547\pi\)
−0.127038 + 0.991898i \(0.540547\pi\)
\(458\) 22.6558 1.05864
\(459\) −14.0622 −0.656369
\(460\) 1.91305 0.0891964
\(461\) 0.796700 0.0371060 0.0185530 0.999828i \(-0.494094\pi\)
0.0185530 + 0.999828i \(0.494094\pi\)
\(462\) 0 0
\(463\) 24.1337 1.12159 0.560795 0.827955i \(-0.310496\pi\)
0.560795 + 0.827955i \(0.310496\pi\)
\(464\) 7.77089 0.360754
\(465\) 1.97477 0.0915777
\(466\) 20.0819 0.930276
\(467\) 8.96049 0.414642 0.207321 0.978273i \(-0.433526\pi\)
0.207321 + 0.978273i \(0.433526\pi\)
\(468\) −5.56484 −0.257235
\(469\) −18.4671 −0.852729
\(470\) 26.9653 1.24382
\(471\) −14.6701 −0.675962
\(472\) −8.22759 −0.378705
\(473\) 0 0
\(474\) 9.37587 0.430648
\(475\) −2.56634 −0.117752
\(476\) −15.8209 −0.725149
\(477\) −8.08883 −0.370362
\(478\) 19.7417 0.902966
\(479\) −13.5637 −0.619743 −0.309871 0.950778i \(-0.600286\pi\)
−0.309871 + 0.950778i \(0.600286\pi\)
\(480\) 5.80753 0.265076
\(481\) 2.64275 0.120499
\(482\) −20.0092 −0.911393
\(483\) 5.37978 0.244788
\(484\) 0 0
\(485\) −44.8831 −2.03804
\(486\) 13.9787 0.634084
\(487\) 1.14086 0.0516975 0.0258488 0.999666i \(-0.491771\pi\)
0.0258488 + 0.999666i \(0.491771\pi\)
\(488\) −0.344547 −0.0155969
\(489\) −33.8026 −1.52861
\(490\) −17.6690 −0.798206
\(491\) −16.4537 −0.742545 −0.371272 0.928524i \(-0.621078\pi\)
−0.371272 + 0.928524i \(0.621078\pi\)
\(492\) −19.4599 −0.877320
\(493\) −33.5559 −1.51128
\(494\) 3.81790 0.171776
\(495\) 0 0
\(496\) 0.340036 0.0152680
\(497\) −11.8002 −0.529312
\(498\) −25.4335 −1.13970
\(499\) 27.3412 1.22396 0.611980 0.790873i \(-0.290373\pi\)
0.611980 + 0.790873i \(0.290373\pi\)
\(500\) 6.69426 0.299377
\(501\) −7.77057 −0.347163
\(502\) −28.0383 −1.25141
\(503\) 20.2121 0.901213 0.450607 0.892723i \(-0.351208\pi\)
0.450607 + 0.892723i \(0.351208\pi\)
\(504\) 5.34023 0.237873
\(505\) 38.8942 1.73077
\(506\) 0 0
\(507\) −3.32823 −0.147812
\(508\) −0.568592 −0.0252272
\(509\) −0.629459 −0.0279003 −0.0139502 0.999903i \(-0.504441\pi\)
−0.0139502 + 0.999903i \(0.504441\pi\)
\(510\) −25.0778 −1.11047
\(511\) 53.4989 2.36665
\(512\) 1.00000 0.0441942
\(513\) −3.25654 −0.143780
\(514\) −13.5564 −0.597949
\(515\) −2.42109 −0.106686
\(516\) −4.25415 −0.187278
\(517\) 0 0
\(518\) −2.53608 −0.111429
\(519\) 27.3471 1.20041
\(520\) 10.5019 0.460539
\(521\) −25.0746 −1.09854 −0.549270 0.835645i \(-0.685094\pi\)
−0.549270 + 0.835645i \(0.685094\pi\)
\(522\) 11.3266 0.495750
\(523\) −13.1638 −0.575611 −0.287806 0.957689i \(-0.592926\pi\)
−0.287806 + 0.957689i \(0.592926\pi\)
\(524\) 11.8544 0.517862
\(525\) −19.8516 −0.866395
\(526\) −4.42685 −0.193020
\(527\) −1.46833 −0.0639613
\(528\) 0 0
\(529\) −22.5163 −0.978970
\(530\) 15.2652 0.663076
\(531\) −11.9922 −0.520419
\(532\) −3.66381 −0.158846
\(533\) −35.1898 −1.52424
\(534\) 2.04043 0.0882979
\(535\) 4.86084 0.210152
\(536\) −5.04040 −0.217712
\(537\) 29.1072 1.25607
\(538\) −7.98617 −0.344308
\(539\) 0 0
\(540\) −8.95775 −0.385480
\(541\) 29.6788 1.27599 0.637995 0.770041i \(-0.279764\pi\)
0.637995 + 0.770041i \(0.279764\pi\)
\(542\) 17.9453 0.770815
\(543\) 32.2574 1.38430
\(544\) −4.31816 −0.185139
\(545\) −32.0603 −1.37331
\(546\) 29.5329 1.26389
\(547\) −38.2994 −1.63756 −0.818781 0.574106i \(-0.805350\pi\)
−0.818781 + 0.574106i \(0.805350\pi\)
\(548\) 11.2562 0.480840
\(549\) −0.502200 −0.0214334
\(550\) 0 0
\(551\) −7.77089 −0.331051
\(552\) 1.46836 0.0624975
\(553\) −16.2703 −0.691883
\(554\) −21.4997 −0.913435
\(555\) −4.01997 −0.170638
\(556\) 1.69259 0.0717820
\(557\) −7.05263 −0.298829 −0.149415 0.988775i \(-0.547739\pi\)
−0.149415 + 0.988775i \(0.547739\pi\)
\(558\) 0.495624 0.0209814
\(559\) −7.69287 −0.325374
\(560\) −10.0780 −0.425874
\(561\) 0 0
\(562\) 19.5998 0.826769
\(563\) −17.3112 −0.729581 −0.364790 0.931090i \(-0.618859\pi\)
−0.364790 + 0.931090i \(0.618859\pi\)
\(564\) 20.6972 0.871509
\(565\) −54.8674 −2.30829
\(566\) 13.0294 0.547667
\(567\) −41.2112 −1.73071
\(568\) −3.22075 −0.135140
\(569\) −15.7749 −0.661318 −0.330659 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(570\) −5.80753 −0.243251
\(571\) 24.8790 1.04116 0.520578 0.853814i \(-0.325717\pi\)
0.520578 + 0.853814i \(0.325717\pi\)
\(572\) 0 0
\(573\) −50.5563 −2.11202
\(574\) 33.7695 1.40951
\(575\) −1.78483 −0.0744326
\(576\) 1.45756 0.0607318
\(577\) 12.6925 0.528395 0.264197 0.964469i \(-0.414893\pi\)
0.264197 + 0.964469i \(0.414893\pi\)
\(578\) 1.64648 0.0684844
\(579\) −57.2810 −2.38052
\(580\) −21.3754 −0.887564
\(581\) 44.1357 1.83106
\(582\) −34.4499 −1.42800
\(583\) 0 0
\(584\) 14.6020 0.604235
\(585\) 15.3072 0.632875
\(586\) −12.4424 −0.513993
\(587\) −37.8091 −1.56055 −0.780275 0.625437i \(-0.784921\pi\)
−0.780275 + 0.625437i \(0.784921\pi\)
\(588\) −13.5618 −0.559281
\(589\) −0.340036 −0.0140109
\(590\) 22.6316 0.931729
\(591\) −44.7510 −1.84081
\(592\) −0.692199 −0.0284492
\(593\) −11.7942 −0.484331 −0.242166 0.970235i \(-0.577858\pi\)
−0.242166 + 0.970235i \(0.577858\pi\)
\(594\) 0 0
\(595\) 43.5185 1.78408
\(596\) −21.5429 −0.882433
\(597\) 6.41462 0.262533
\(598\) 2.65527 0.108582
\(599\) −2.24280 −0.0916382 −0.0458191 0.998950i \(-0.514590\pi\)
−0.0458191 + 0.998950i \(0.514590\pi\)
\(600\) −5.41830 −0.221201
\(601\) −34.5443 −1.40909 −0.704545 0.709659i \(-0.748849\pi\)
−0.704545 + 0.709659i \(0.748849\pi\)
\(602\) 7.38237 0.300883
\(603\) −7.34671 −0.299181
\(604\) 19.9995 0.813768
\(605\) 0 0
\(606\) 29.8532 1.21270
\(607\) −0.623652 −0.0253133 −0.0126566 0.999920i \(-0.504029\pi\)
−0.0126566 + 0.999920i \(0.504029\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −60.1107 −2.43581
\(610\) 0.947746 0.0383731
\(611\) 37.4272 1.51414
\(612\) −6.29399 −0.254419
\(613\) −19.0977 −0.771347 −0.385673 0.922635i \(-0.626031\pi\)
−0.385673 + 0.922635i \(0.626031\pi\)
\(614\) −18.8124 −0.759206
\(615\) 53.5283 2.15847
\(616\) 0 0
\(617\) −12.8189 −0.516070 −0.258035 0.966136i \(-0.583075\pi\)
−0.258035 + 0.966136i \(0.583075\pi\)
\(618\) −1.85831 −0.0747521
\(619\) 18.3684 0.738288 0.369144 0.929372i \(-0.379651\pi\)
0.369144 + 0.929372i \(0.379651\pi\)
\(620\) −0.935335 −0.0375640
\(621\) −2.26485 −0.0908852
\(622\) −13.2168 −0.529944
\(623\) −3.54083 −0.141860
\(624\) 8.06072 0.322687
\(625\) −31.2456 −1.24982
\(626\) −3.51233 −0.140381
\(627\) 0 0
\(628\) 6.94839 0.277271
\(629\) 2.98903 0.119180
\(630\) −14.6894 −0.585238
\(631\) 5.04337 0.200774 0.100387 0.994948i \(-0.467992\pi\)
0.100387 + 0.994948i \(0.467992\pi\)
\(632\) −4.44082 −0.176646
\(633\) −57.8768 −2.30040
\(634\) 8.53754 0.339069
\(635\) 1.56402 0.0620664
\(636\) 11.7167 0.464599
\(637\) −24.5242 −0.971685
\(638\) 0 0
\(639\) −4.69445 −0.185710
\(640\) −2.75070 −0.108731
\(641\) 46.3321 1.83001 0.915003 0.403446i \(-0.132188\pi\)
0.915003 + 0.403446i \(0.132188\pi\)
\(642\) 3.73093 0.147248
\(643\) −25.3374 −0.999210 −0.499605 0.866253i \(-0.666522\pi\)
−0.499605 + 0.866253i \(0.666522\pi\)
\(644\) −2.54810 −0.100409
\(645\) 11.7019 0.460761
\(646\) 4.31816 0.169896
\(647\) −21.2531 −0.835545 −0.417773 0.908552i \(-0.637189\pi\)
−0.417773 + 0.908552i \(0.637189\pi\)
\(648\) −11.2482 −0.441871
\(649\) 0 0
\(650\) −9.79804 −0.384311
\(651\) −2.63030 −0.103090
\(652\) 16.0104 0.627014
\(653\) 6.92969 0.271180 0.135590 0.990765i \(-0.456707\pi\)
0.135590 + 0.990765i \(0.456707\pi\)
\(654\) −24.6079 −0.962243
\(655\) −32.6079 −1.27409
\(656\) 9.21705 0.359865
\(657\) 21.2834 0.830343
\(658\) −35.9166 −1.40017
\(659\) −34.0324 −1.32571 −0.662857 0.748746i \(-0.730656\pi\)
−0.662857 + 0.748746i \(0.730656\pi\)
\(660\) 0 0
\(661\) 12.8692 0.500552 0.250276 0.968175i \(-0.419479\pi\)
0.250276 + 0.968175i \(0.419479\pi\)
\(662\) −3.43770 −0.133610
\(663\) −34.8074 −1.35181
\(664\) 12.0464 0.467491
\(665\) 10.0780 0.390809
\(666\) −1.00893 −0.0390951
\(667\) −5.40448 −0.209262
\(668\) 3.68048 0.142402
\(669\) −39.4938 −1.52692
\(670\) 13.8646 0.535637
\(671\) 0 0
\(672\) −7.73537 −0.298398
\(673\) −32.0632 −1.23595 −0.617973 0.786199i \(-0.712046\pi\)
−0.617973 + 0.786199i \(0.712046\pi\)
\(674\) 23.8752 0.919638
\(675\) 8.35738 0.321676
\(676\) 1.57639 0.0606304
\(677\) −34.6314 −1.33099 −0.665496 0.746402i \(-0.731780\pi\)
−0.665496 + 0.746402i \(0.731780\pi\)
\(678\) −42.1134 −1.61736
\(679\) 59.7823 2.29423
\(680\) 11.8779 0.455498
\(681\) −26.7477 −1.02497
\(682\) 0 0
\(683\) −11.1244 −0.425663 −0.212831 0.977089i \(-0.568269\pi\)
−0.212831 + 0.977089i \(0.568269\pi\)
\(684\) −1.45756 −0.0557314
\(685\) −30.9623 −1.18301
\(686\) −2.11228 −0.0806471
\(687\) −47.8330 −1.82494
\(688\) 2.01495 0.0768191
\(689\) 21.1877 0.807186
\(690\) −4.03901 −0.153762
\(691\) 30.1726 1.14782 0.573910 0.818919i \(-0.305426\pi\)
0.573910 + 0.818919i \(0.305426\pi\)
\(692\) −12.9528 −0.492391
\(693\) 0 0
\(694\) 7.32500 0.278053
\(695\) −4.65582 −0.176605
\(696\) −16.4066 −0.621892
\(697\) −39.8007 −1.50756
\(698\) 8.11645 0.307212
\(699\) −42.3988 −1.60367
\(700\) 9.40257 0.355384
\(701\) −20.1483 −0.760991 −0.380495 0.924783i \(-0.624247\pi\)
−0.380495 + 0.924783i \(0.624247\pi\)
\(702\) −12.4331 −0.469259
\(703\) 0.692199 0.0261068
\(704\) 0 0
\(705\) −56.9317 −2.14417
\(706\) −5.08395 −0.191337
\(707\) −51.8053 −1.94834
\(708\) 17.3709 0.652837
\(709\) −1.20513 −0.0452597 −0.0226298 0.999744i \(-0.507204\pi\)
−0.0226298 + 0.999744i \(0.507204\pi\)
\(710\) 8.85932 0.332484
\(711\) −6.47278 −0.242748
\(712\) −0.966434 −0.0362186
\(713\) −0.236487 −0.00885651
\(714\) 33.4026 1.25006
\(715\) 0 0
\(716\) −13.7864 −0.515222
\(717\) −41.6806 −1.55659
\(718\) 15.2156 0.567842
\(719\) −49.8350 −1.85853 −0.929267 0.369409i \(-0.879560\pi\)
−0.929267 + 0.369409i \(0.879560\pi\)
\(720\) −4.00932 −0.149418
\(721\) 3.22479 0.120097
\(722\) 1.00000 0.0372161
\(723\) 42.2453 1.57112
\(724\) −15.2785 −0.567820
\(725\) 19.9427 0.740655
\(726\) 0 0
\(727\) 6.51901 0.241777 0.120888 0.992666i \(-0.461426\pi\)
0.120888 + 0.992666i \(0.461426\pi\)
\(728\) −13.9881 −0.518432
\(729\) 4.23155 0.156724
\(730\) −40.1657 −1.48660
\(731\) −8.70086 −0.321813
\(732\) 0.727441 0.0268870
\(733\) −1.19268 −0.0440526 −0.0220263 0.999757i \(-0.507012\pi\)
−0.0220263 + 0.999757i \(0.507012\pi\)
\(734\) 29.9084 1.10394
\(735\) 37.3045 1.37600
\(736\) −0.695478 −0.0256356
\(737\) 0 0
\(738\) 13.4344 0.494529
\(739\) 21.4841 0.790305 0.395152 0.918616i \(-0.370692\pi\)
0.395152 + 0.918616i \(0.370692\pi\)
\(740\) 1.90403 0.0699936
\(741\) −8.06072 −0.296118
\(742\) −20.3325 −0.746429
\(743\) −33.2777 −1.22084 −0.610421 0.792077i \(-0.709000\pi\)
−0.610421 + 0.792077i \(0.709000\pi\)
\(744\) −0.717915 −0.0263201
\(745\) 59.2581 2.17105
\(746\) −32.7204 −1.19798
\(747\) 17.5584 0.642429
\(748\) 0 0
\(749\) −6.47442 −0.236570
\(750\) −14.1336 −0.516085
\(751\) 23.0796 0.842185 0.421092 0.907018i \(-0.361647\pi\)
0.421092 + 0.907018i \(0.361647\pi\)
\(752\) −9.80308 −0.357482
\(753\) 59.1971 2.15726
\(754\) −29.6685 −1.08046
\(755\) −55.0126 −2.00211
\(756\) 11.9313 0.433938
\(757\) 25.7018 0.934149 0.467074 0.884218i \(-0.345308\pi\)
0.467074 + 0.884218i \(0.345308\pi\)
\(758\) 21.4522 0.779179
\(759\) 0 0
\(760\) 2.75070 0.0997783
\(761\) 45.5538 1.65132 0.825661 0.564166i \(-0.190802\pi\)
0.825661 + 0.564166i \(0.190802\pi\)
\(762\) 1.20046 0.0434882
\(763\) 42.7029 1.54595
\(764\) 23.9456 0.866323
\(765\) 17.3129 0.625948
\(766\) −17.1641 −0.620164
\(767\) 31.4121 1.13423
\(768\) −2.11129 −0.0761848
\(769\) 50.7820 1.83125 0.915623 0.402038i \(-0.131698\pi\)
0.915623 + 0.402038i \(0.131698\pi\)
\(770\) 0 0
\(771\) 28.6216 1.03078
\(772\) 27.1307 0.976456
\(773\) 33.7448 1.21372 0.606858 0.794810i \(-0.292430\pi\)
0.606858 + 0.794810i \(0.292430\pi\)
\(774\) 2.93691 0.105565
\(775\) 0.872647 0.0313464
\(776\) 16.3170 0.585745
\(777\) 5.35442 0.192089
\(778\) 28.8568 1.03457
\(779\) −9.21705 −0.330235
\(780\) −22.1726 −0.793907
\(781\) 0 0
\(782\) 3.00318 0.107394
\(783\) 25.3062 0.904369
\(784\) 6.42347 0.229410
\(785\) −19.1129 −0.682169
\(786\) −25.0281 −0.892723
\(787\) 21.9550 0.782611 0.391306 0.920261i \(-0.372024\pi\)
0.391306 + 0.920261i \(0.372024\pi\)
\(788\) 21.1960 0.755076
\(789\) 9.34639 0.332740
\(790\) 12.2153 0.434603
\(791\) 73.0809 2.59846
\(792\) 0 0
\(793\) 1.31545 0.0467129
\(794\) −5.83003 −0.206900
\(795\) −32.2292 −1.14305
\(796\) −3.03824 −0.107688
\(797\) 32.3449 1.14571 0.572857 0.819655i \(-0.305835\pi\)
0.572857 + 0.819655i \(0.305835\pi\)
\(798\) 7.73537 0.273829
\(799\) 42.3313 1.49757
\(800\) 2.56634 0.0907338
\(801\) −1.40864 −0.0497718
\(802\) 13.2388 0.467477
\(803\) 0 0
\(804\) 10.6418 0.375306
\(805\) 7.00904 0.247036
\(806\) −1.29822 −0.0457280
\(807\) 16.8612 0.593541
\(808\) −14.1398 −0.497435
\(809\) −9.49040 −0.333665 −0.166832 0.985985i \(-0.553354\pi\)
−0.166832 + 0.985985i \(0.553354\pi\)
\(810\) 30.9404 1.08713
\(811\) 35.3671 1.24191 0.620953 0.783847i \(-0.286746\pi\)
0.620953 + 0.783847i \(0.286746\pi\)
\(812\) 28.4710 0.999137
\(813\) −37.8877 −1.32878
\(814\) 0 0
\(815\) −44.0397 −1.54264
\(816\) 9.11690 0.319155
\(817\) −2.01495 −0.0704941
\(818\) −17.6372 −0.616669
\(819\) −20.3885 −0.712432
\(820\) −25.3533 −0.885377
\(821\) −14.6841 −0.512478 −0.256239 0.966613i \(-0.582483\pi\)
−0.256239 + 0.966613i \(0.582483\pi\)
\(822\) −23.7651 −0.828902
\(823\) 28.4915 0.993152 0.496576 0.867993i \(-0.334590\pi\)
0.496576 + 0.867993i \(0.334590\pi\)
\(824\) 0.880175 0.0306623
\(825\) 0 0
\(826\) −30.1443 −1.04885
\(827\) −20.6508 −0.718098 −0.359049 0.933319i \(-0.616899\pi\)
−0.359049 + 0.933319i \(0.616899\pi\)
\(828\) −1.01370 −0.0352286
\(829\) 10.7947 0.374916 0.187458 0.982273i \(-0.439975\pi\)
0.187458 + 0.982273i \(0.439975\pi\)
\(830\) −33.1360 −1.15017
\(831\) 45.3922 1.57464
\(832\) −3.81790 −0.132362
\(833\) −27.7376 −0.961050
\(834\) −3.57357 −0.123742
\(835\) −10.1239 −0.350351
\(836\) 0 0
\(837\) 1.10734 0.0382752
\(838\) −28.5638 −0.986719
\(839\) −26.2123 −0.904950 −0.452475 0.891777i \(-0.649459\pi\)
−0.452475 + 0.891777i \(0.649459\pi\)
\(840\) 21.2777 0.734150
\(841\) 31.3867 1.08230
\(842\) 10.1449 0.349616
\(843\) −41.3810 −1.42524
\(844\) 27.4130 0.943593
\(845\) −4.33618 −0.149169
\(846\) −14.2886 −0.491253
\(847\) 0 0
\(848\) −5.54956 −0.190573
\(849\) −27.5089 −0.944104
\(850\) −11.0819 −0.380105
\(851\) 0.481409 0.0165025
\(852\) 6.79996 0.232963
\(853\) −7.01542 −0.240203 −0.120102 0.992762i \(-0.538322\pi\)
−0.120102 + 0.992762i \(0.538322\pi\)
\(854\) −1.26235 −0.0431969
\(855\) 4.00932 0.137116
\(856\) −1.76713 −0.0603993
\(857\) 22.6511 0.773746 0.386873 0.922133i \(-0.373555\pi\)
0.386873 + 0.922133i \(0.373555\pi\)
\(858\) 0 0
\(859\) 36.4532 1.24377 0.621883 0.783110i \(-0.286368\pi\)
0.621883 + 0.783110i \(0.286368\pi\)
\(860\) −5.54251 −0.188998
\(861\) −71.2973 −2.42981
\(862\) 22.6879 0.772752
\(863\) 31.8436 1.08397 0.541984 0.840389i \(-0.317673\pi\)
0.541984 + 0.840389i \(0.317673\pi\)
\(864\) 3.25654 0.110790
\(865\) 35.6292 1.21143
\(866\) 31.0778 1.05607
\(867\) −3.47620 −0.118058
\(868\) 1.24582 0.0422860
\(869\) 0 0
\(870\) 45.1297 1.53004
\(871\) 19.2438 0.652050
\(872\) 11.6553 0.394700
\(873\) 23.7830 0.804934
\(874\) 0.695478 0.0235249
\(875\) 24.5265 0.829147
\(876\) −30.8291 −1.04162
\(877\) −32.0478 −1.08218 −0.541089 0.840966i \(-0.681988\pi\)
−0.541089 + 0.840966i \(0.681988\pi\)
\(878\) −30.3380 −1.02386
\(879\) 26.2697 0.886054
\(880\) 0 0
\(881\) −34.2368 −1.15347 −0.576733 0.816932i \(-0.695673\pi\)
−0.576733 + 0.816932i \(0.695673\pi\)
\(882\) 9.36263 0.315256
\(883\) 38.2142 1.28601 0.643005 0.765862i \(-0.277687\pi\)
0.643005 + 0.765862i \(0.277687\pi\)
\(884\) 16.4863 0.554495
\(885\) −47.7820 −1.60617
\(886\) 5.08267 0.170756
\(887\) 38.7150 1.29992 0.649961 0.759968i \(-0.274785\pi\)
0.649961 + 0.759968i \(0.274785\pi\)
\(888\) 1.46144 0.0490426
\(889\) −2.08321 −0.0698686
\(890\) 2.65837 0.0891087
\(891\) 0 0
\(892\) 18.7060 0.626323
\(893\) 9.80308 0.328048
\(894\) 45.4835 1.52120
\(895\) 37.9223 1.26760
\(896\) 3.66381 0.122399
\(897\) −5.60605 −0.187181
\(898\) −34.4255 −1.14880
\(899\) 2.64238 0.0881282
\(900\) 3.74060 0.124687
\(901\) 23.9638 0.798352
\(902\) 0 0
\(903\) −15.5864 −0.518682
\(904\) 19.9467 0.663418
\(905\) 42.0265 1.39701
\(906\) −42.2248 −1.40283
\(907\) −30.2793 −1.00541 −0.502704 0.864458i \(-0.667662\pi\)
−0.502704 + 0.864458i \(0.667662\pi\)
\(908\) 12.6689 0.420431
\(909\) −20.6096 −0.683577
\(910\) 38.4769 1.27550
\(911\) −38.8426 −1.28691 −0.643456 0.765483i \(-0.722500\pi\)
−0.643456 + 0.765483i \(0.722500\pi\)
\(912\) 2.11129 0.0699119
\(913\) 0 0
\(914\) −5.43151 −0.179658
\(915\) −2.00097 −0.0661500
\(916\) 22.6558 0.748568
\(917\) 43.4322 1.43426
\(918\) −14.0622 −0.464123
\(919\) −40.6344 −1.34041 −0.670203 0.742178i \(-0.733793\pi\)
−0.670203 + 0.742178i \(0.733793\pi\)
\(920\) 1.91305 0.0630714
\(921\) 39.7185 1.30877
\(922\) 0.796700 0.0262379
\(923\) 12.2965 0.404745
\(924\) 0 0
\(925\) −1.77642 −0.0584083
\(926\) 24.1337 0.793083
\(927\) 1.28291 0.0421363
\(928\) 7.77089 0.255092
\(929\) −24.9807 −0.819591 −0.409796 0.912177i \(-0.634400\pi\)
−0.409796 + 0.912177i \(0.634400\pi\)
\(930\) 1.97477 0.0647552
\(931\) −6.42347 −0.210521
\(932\) 20.0819 0.657805
\(933\) 27.9045 0.913552
\(934\) 8.96049 0.293196
\(935\) 0 0
\(936\) −5.56484 −0.181892
\(937\) 9.69911 0.316856 0.158428 0.987371i \(-0.449357\pi\)
0.158428 + 0.987371i \(0.449357\pi\)
\(938\) −18.4671 −0.602971
\(939\) 7.41557 0.241998
\(940\) 26.9653 0.879512
\(941\) −20.4974 −0.668195 −0.334098 0.942539i \(-0.608431\pi\)
−0.334098 + 0.942539i \(0.608431\pi\)
\(942\) −14.6701 −0.477978
\(943\) −6.41025 −0.208747
\(944\) −8.22759 −0.267785
\(945\) −32.8195 −1.06762
\(946\) 0 0
\(947\) 52.0636 1.69184 0.845920 0.533309i \(-0.179052\pi\)
0.845920 + 0.533309i \(0.179052\pi\)
\(948\) 9.37587 0.304514
\(949\) −55.7490 −1.80969
\(950\) −2.56634 −0.0832631
\(951\) −18.0253 −0.584509
\(952\) −15.8209 −0.512758
\(953\) −2.51516 −0.0814739 −0.0407370 0.999170i \(-0.512971\pi\)
−0.0407370 + 0.999170i \(0.512971\pi\)
\(954\) −8.08883 −0.261886
\(955\) −65.8672 −2.13141
\(956\) 19.7417 0.638493
\(957\) 0 0
\(958\) −13.5637 −0.438224
\(959\) 41.2404 1.33172
\(960\) 5.80753 0.187437
\(961\) −30.8844 −0.996270
\(962\) 2.64275 0.0852057
\(963\) −2.57570 −0.0830009
\(964\) −20.0092 −0.644452
\(965\) −74.6284 −2.40237
\(966\) 5.37978 0.173092
\(967\) 16.5427 0.531979 0.265989 0.963976i \(-0.414301\pi\)
0.265989 + 0.963976i \(0.414301\pi\)
\(968\) 0 0
\(969\) −9.11690 −0.292877
\(970\) −44.8831 −1.44111
\(971\) 40.7237 1.30688 0.653442 0.756976i \(-0.273324\pi\)
0.653442 + 0.756976i \(0.273324\pi\)
\(972\) 13.9787 0.448365
\(973\) 6.20134 0.198806
\(974\) 1.14086 0.0365557
\(975\) 20.6865 0.662500
\(976\) −0.344547 −0.0110287
\(977\) 0.602998 0.0192916 0.00964580 0.999953i \(-0.496930\pi\)
0.00964580 + 0.999953i \(0.496930\pi\)
\(978\) −33.8026 −1.08089
\(979\) 0 0
\(980\) −17.6690 −0.564417
\(981\) 16.9884 0.542398
\(982\) −16.4537 −0.525058
\(983\) 23.4122 0.746734 0.373367 0.927684i \(-0.378203\pi\)
0.373367 + 0.927684i \(0.378203\pi\)
\(984\) −19.4599 −0.620359
\(985\) −58.3038 −1.85771
\(986\) −33.5559 −1.06864
\(987\) 75.8305 2.41371
\(988\) 3.81790 0.121464
\(989\) −1.40135 −0.0445603
\(990\) 0 0
\(991\) −34.2320 −1.08742 −0.543708 0.839275i \(-0.682980\pi\)
−0.543708 + 0.839275i \(0.682980\pi\)
\(992\) 0.340036 0.0107961
\(993\) 7.25800 0.230326
\(994\) −11.8002 −0.374280
\(995\) 8.35728 0.264944
\(996\) −25.4335 −0.805892
\(997\) 21.6379 0.685280 0.342640 0.939467i \(-0.388679\pi\)
0.342640 + 0.939467i \(0.388679\pi\)
\(998\) 27.3412 0.865471
\(999\) −2.25417 −0.0713189
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.cb.1.2 yes 8
11.10 odd 2 4598.2.a.by.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.2 8 11.10 odd 2
4598.2.a.cb.1.2 yes 8 1.1 even 1 trivial