Properties

Label 4598.2.a.cb.1.1
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 1.1
Root \(-1.93433\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.48823 q^{3} +1.00000 q^{4} +3.25915 q^{5} -2.48823 q^{6} +4.58972 q^{7} +1.00000 q^{8} +3.19127 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.48823 q^{3} +1.00000 q^{4} +3.25915 q^{5} -2.48823 q^{6} +4.58972 q^{7} +1.00000 q^{8} +3.19127 q^{9} +3.25915 q^{10} -2.48823 q^{12} -4.82326 q^{13} +4.58972 q^{14} -8.10950 q^{15} +1.00000 q^{16} +1.77745 q^{17} +3.19127 q^{18} -1.00000 q^{19} +3.25915 q^{20} -11.4203 q^{21} +6.51169 q^{23} -2.48823 q^{24} +5.62205 q^{25} -4.82326 q^{26} -0.475914 q^{27} +4.58972 q^{28} -0.0103559 q^{29} -8.10950 q^{30} -3.91892 q^{31} +1.00000 q^{32} +1.77745 q^{34} +14.9586 q^{35} +3.19127 q^{36} +0.688142 q^{37} -1.00000 q^{38} +12.0014 q^{39} +3.25915 q^{40} +3.27479 q^{41} -11.4203 q^{42} +1.38020 q^{43} +10.4008 q^{45} +6.51169 q^{46} -6.30125 q^{47} -2.48823 q^{48} +14.0655 q^{49} +5.62205 q^{50} -4.42270 q^{51} -4.82326 q^{52} +14.4636 q^{53} -0.475914 q^{54} +4.58972 q^{56} +2.48823 q^{57} -0.0103559 q^{58} -9.20149 q^{59} -8.10950 q^{60} +13.4651 q^{61} -3.91892 q^{62} +14.6470 q^{63} +1.00000 q^{64} -15.7197 q^{65} +5.08657 q^{67} +1.77745 q^{68} -16.2026 q^{69} +14.9586 q^{70} +9.43943 q^{71} +3.19127 q^{72} -15.2769 q^{73} +0.688142 q^{74} -13.9889 q^{75} -1.00000 q^{76} +12.0014 q^{78} +12.9416 q^{79} +3.25915 q^{80} -8.38962 q^{81} +3.27479 q^{82} +2.32216 q^{83} -11.4203 q^{84} +5.79298 q^{85} +1.38020 q^{86} +0.0257678 q^{87} -6.04905 q^{89} +10.4008 q^{90} -22.1374 q^{91} +6.51169 q^{92} +9.75117 q^{93} -6.30125 q^{94} -3.25915 q^{95} -2.48823 q^{96} -19.4094 q^{97} +14.0655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + 8 q^{12} - 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} - 4 q^{17} + 22 q^{18} - 8 q^{19} - 20 q^{21} + 14 q^{23} + 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} + 4 q^{28} - 2 q^{29} + 4 q^{30} + 8 q^{32} - 4 q^{34} + 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} + 16 q^{39} + 8 q^{41} - 20 q^{42} + 8 q^{43} + 16 q^{45} + 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} + 36 q^{50} + 18 q^{51} - 12 q^{52} + 36 q^{53} + 32 q^{54} + 4 q^{56} - 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} + 12 q^{61} + 24 q^{63} + 8 q^{64} + 16 q^{65} + 16 q^{67} - 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} + 22 q^{72} - 20 q^{73} + 24 q^{74} + 40 q^{75} - 8 q^{76} + 16 q^{78} - 12 q^{79} + 40 q^{81} + 8 q^{82} + 20 q^{83} - 20 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} - 16 q^{94} + 8 q^{96} + 4 q^{97} + 34 q^{98} + O(q^{100}) \)

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
\(3\) −2.48823 −1.43658 −0.718289 0.695745i \(-0.755074\pi\)
−0.718289 + 0.695745i \(0.755074\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.25915 1.45754 0.728768 0.684761i \(-0.240093\pi\)
0.728768 + 0.684761i \(0.240093\pi\)
\(6\) −2.48823 −1.01581
\(7\) 4.58972 1.73475 0.867376 0.497654i \(-0.165805\pi\)
0.867376 + 0.497654i \(0.165805\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.19127 1.06376
\(10\) 3.25915 1.03063
\(11\) 0 0
\(12\) −2.48823 −0.718289
\(13\) −4.82326 −1.33773 −0.668865 0.743384i \(-0.733220\pi\)
−0.668865 + 0.743384i \(0.733220\pi\)
\(14\) 4.58972 1.22665
\(15\) −8.10950 −2.09386
\(16\) 1.00000 0.250000
\(17\) 1.77745 0.431095 0.215548 0.976493i \(-0.430846\pi\)
0.215548 + 0.976493i \(0.430846\pi\)
\(18\) 3.19127 0.752189
\(19\) −1.00000 −0.229416
\(20\) 3.25915 0.728768
\(21\) −11.4203 −2.49210
\(22\) 0 0
\(23\) 6.51169 1.35778 0.678891 0.734239i \(-0.262461\pi\)
0.678891 + 0.734239i \(0.262461\pi\)
\(24\) −2.48823 −0.507907
\(25\) 5.62205 1.12441
\(26\) −4.82326 −0.945919
\(27\) −0.475914 −0.0915896
\(28\) 4.58972 0.867376
\(29\) −0.0103559 −0.00192304 −0.000961522 1.00000i \(-0.500306\pi\)
−0.000961522 1.00000i \(0.500306\pi\)
\(30\) −8.10950 −1.48059
\(31\) −3.91892 −0.703860 −0.351930 0.936026i \(-0.614474\pi\)
−0.351930 + 0.936026i \(0.614474\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.77745 0.304830
\(35\) 14.9586 2.52846
\(36\) 3.19127 0.531878
\(37\) 0.688142 0.113130 0.0565650 0.998399i \(-0.481985\pi\)
0.0565650 + 0.998399i \(0.481985\pi\)
\(38\) −1.00000 −0.162221
\(39\) 12.0014 1.92175
\(40\) 3.25915 0.515317
\(41\) 3.27479 0.511436 0.255718 0.966751i \(-0.417688\pi\)
0.255718 + 0.966751i \(0.417688\pi\)
\(42\) −11.4203 −1.76218
\(43\) 1.38020 0.210479 0.105240 0.994447i \(-0.466439\pi\)
0.105240 + 0.994447i \(0.466439\pi\)
\(44\) 0 0
\(45\) 10.4008 1.55046
\(46\) 6.51169 0.960096
\(47\) −6.30125 −0.919132 −0.459566 0.888144i \(-0.651995\pi\)
−0.459566 + 0.888144i \(0.651995\pi\)
\(48\) −2.48823 −0.359144
\(49\) 14.0655 2.00936
\(50\) 5.62205 0.795078
\(51\) −4.42270 −0.619302
\(52\) −4.82326 −0.668865
\(53\) 14.4636 1.98673 0.993365 0.115003i \(-0.0366877\pi\)
0.993365 + 0.115003i \(0.0366877\pi\)
\(54\) −0.475914 −0.0647636
\(55\) 0 0
\(56\) 4.58972 0.613327
\(57\) 2.48823 0.329574
\(58\) −0.0103559 −0.00135980
\(59\) −9.20149 −1.19793 −0.598966 0.800775i \(-0.704421\pi\)
−0.598966 + 0.800775i \(0.704421\pi\)
\(60\) −8.10950 −1.04693
\(61\) 13.4651 1.72403 0.862017 0.506879i \(-0.169201\pi\)
0.862017 + 0.506879i \(0.169201\pi\)
\(62\) −3.91892 −0.497704
\(63\) 14.6470 1.84535
\(64\) 1.00000 0.125000
\(65\) −15.7197 −1.94979
\(66\) 0 0
\(67\) 5.08657 0.621424 0.310712 0.950504i \(-0.399433\pi\)
0.310712 + 0.950504i \(0.399433\pi\)
\(68\) 1.77745 0.215548
\(69\) −16.2026 −1.95056
\(70\) 14.9586 1.78789
\(71\) 9.43943 1.12025 0.560127 0.828406i \(-0.310752\pi\)
0.560127 + 0.828406i \(0.310752\pi\)
\(72\) 3.19127 0.376094
\(73\) −15.2769 −1.78803 −0.894014 0.448039i \(-0.852123\pi\)
−0.894014 + 0.448039i \(0.852123\pi\)
\(74\) 0.688142 0.0799949
\(75\) −13.9889 −1.61530
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 12.0014 1.35889
\(79\) 12.9416 1.45604 0.728019 0.685557i \(-0.240441\pi\)
0.728019 + 0.685557i \(0.240441\pi\)
\(80\) 3.25915 0.364384
\(81\) −8.38962 −0.932180
\(82\) 3.27479 0.361640
\(83\) 2.32216 0.254890 0.127445 0.991846i \(-0.459322\pi\)
0.127445 + 0.991846i \(0.459322\pi\)
\(84\) −11.4203 −1.24605
\(85\) 5.79298 0.628337
\(86\) 1.38020 0.148831
\(87\) 0.0257678 0.00276260
\(88\) 0 0
\(89\) −6.04905 −0.641198 −0.320599 0.947215i \(-0.603884\pi\)
−0.320599 + 0.947215i \(0.603884\pi\)
\(90\) 10.4008 1.09634
\(91\) −22.1374 −2.32063
\(92\) 6.51169 0.678891
\(93\) 9.75117 1.01115
\(94\) −6.30125 −0.649924
\(95\) −3.25915 −0.334382
\(96\) −2.48823 −0.253953
\(97\) −19.4094 −1.97073 −0.985364 0.170462i \(-0.945474\pi\)
−0.985364 + 0.170462i \(0.945474\pi\)
\(98\) 14.0655 1.42083
\(99\) 0 0
\(100\) 5.62205 0.562205
\(101\) −7.13850 −0.710308 −0.355154 0.934808i \(-0.615572\pi\)
−0.355154 + 0.934808i \(0.615572\pi\)
\(102\) −4.42270 −0.437912
\(103\) 8.65120 0.852428 0.426214 0.904622i \(-0.359847\pi\)
0.426214 + 0.904622i \(0.359847\pi\)
\(104\) −4.82326 −0.472959
\(105\) −37.2203 −3.63233
\(106\) 14.4636 1.40483
\(107\) −13.3149 −1.28720 −0.643599 0.765363i \(-0.722560\pi\)
−0.643599 + 0.765363i \(0.722560\pi\)
\(108\) −0.475914 −0.0457948
\(109\) −14.3467 −1.37416 −0.687080 0.726581i \(-0.741108\pi\)
−0.687080 + 0.726581i \(0.741108\pi\)
\(110\) 0 0
\(111\) −1.71225 −0.162520
\(112\) 4.58972 0.433688
\(113\) 17.6690 1.66216 0.831080 0.556153i \(-0.187723\pi\)
0.831080 + 0.556153i \(0.187723\pi\)
\(114\) 2.48823 0.233044
\(115\) 21.2226 1.97902
\(116\) −0.0103559 −0.000961522 0
\(117\) −15.3923 −1.42302
\(118\) −9.20149 −0.847065
\(119\) 8.15800 0.747843
\(120\) −8.10950 −0.740293
\(121\) 0 0
\(122\) 13.4651 1.21908
\(123\) −8.14842 −0.734718
\(124\) −3.91892 −0.351930
\(125\) 2.02737 0.181333
\(126\) 14.6470 1.30486
\(127\) −10.6201 −0.942381 −0.471191 0.882031i \(-0.656176\pi\)
−0.471191 + 0.882031i \(0.656176\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.43426 −0.302369
\(130\) −15.7197 −1.37871
\(131\) 16.9361 1.47971 0.739856 0.672765i \(-0.234894\pi\)
0.739856 + 0.672765i \(0.234894\pi\)
\(132\) 0 0
\(133\) −4.58972 −0.397979
\(134\) 5.08657 0.439413
\(135\) −1.55107 −0.133495
\(136\) 1.77745 0.152415
\(137\) −8.12858 −0.694472 −0.347236 0.937778i \(-0.612880\pi\)
−0.347236 + 0.937778i \(0.612880\pi\)
\(138\) −16.2026 −1.37925
\(139\) −0.909211 −0.0771183 −0.0385591 0.999256i \(-0.512277\pi\)
−0.0385591 + 0.999256i \(0.512277\pi\)
\(140\) 14.9586 1.26423
\(141\) 15.6789 1.32040
\(142\) 9.43943 0.792140
\(143\) 0 0
\(144\) 3.19127 0.265939
\(145\) −0.0337514 −0.00280290
\(146\) −15.2769 −1.26433
\(147\) −34.9982 −2.88660
\(148\) 0.688142 0.0565650
\(149\) 13.2547 1.08587 0.542933 0.839776i \(-0.317314\pi\)
0.542933 + 0.839776i \(0.317314\pi\)
\(150\) −13.9889 −1.14219
\(151\) 2.29362 0.186652 0.0933261 0.995636i \(-0.470250\pi\)
0.0933261 + 0.995636i \(0.470250\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.67232 0.458580
\(154\) 0 0
\(155\) −12.7724 −1.02590
\(156\) 12.0014 0.960877
\(157\) −11.2813 −0.900345 −0.450172 0.892942i \(-0.648637\pi\)
−0.450172 + 0.892942i \(0.648637\pi\)
\(158\) 12.9416 1.02957
\(159\) −35.9887 −2.85409
\(160\) 3.25915 0.257658
\(161\) 29.8868 2.35541
\(162\) −8.38962 −0.659151
\(163\) −6.52158 −0.510810 −0.255405 0.966834i \(-0.582209\pi\)
−0.255405 + 0.966834i \(0.582209\pi\)
\(164\) 3.27479 0.255718
\(165\) 0 0
\(166\) 2.32216 0.180235
\(167\) 20.7099 1.60258 0.801289 0.598278i \(-0.204148\pi\)
0.801289 + 0.598278i \(0.204148\pi\)
\(168\) −11.4203 −0.881092
\(169\) 10.2638 0.789524
\(170\) 5.79298 0.444301
\(171\) −3.19127 −0.244042
\(172\) 1.38020 0.105240
\(173\) −2.02529 −0.153980 −0.0769899 0.997032i \(-0.524531\pi\)
−0.0769899 + 0.997032i \(0.524531\pi\)
\(174\) 0.0257678 0.00195345
\(175\) 25.8037 1.95057
\(176\) 0 0
\(177\) 22.8954 1.72092
\(178\) −6.04905 −0.453395
\(179\) 20.7205 1.54873 0.774363 0.632742i \(-0.218071\pi\)
0.774363 + 0.632742i \(0.218071\pi\)
\(180\) 10.4008 0.775231
\(181\) −7.63695 −0.567650 −0.283825 0.958876i \(-0.591604\pi\)
−0.283825 + 0.958876i \(0.591604\pi\)
\(182\) −22.1374 −1.64093
\(183\) −33.5043 −2.47671
\(184\) 6.51169 0.480048
\(185\) 2.24276 0.164891
\(186\) 9.75117 0.714990
\(187\) 0 0
\(188\) −6.30125 −0.459566
\(189\) −2.18431 −0.158885
\(190\) −3.25915 −0.236444
\(191\) 20.6340 1.49302 0.746512 0.665372i \(-0.231727\pi\)
0.746512 + 0.665372i \(0.231727\pi\)
\(192\) −2.48823 −0.179572
\(193\) 8.64747 0.622458 0.311229 0.950335i \(-0.399259\pi\)
0.311229 + 0.950335i \(0.399259\pi\)
\(194\) −19.4094 −1.39352
\(195\) 39.1142 2.80103
\(196\) 14.0655 1.00468
\(197\) −10.6030 −0.755430 −0.377715 0.925922i \(-0.623290\pi\)
−0.377715 + 0.925922i \(0.623290\pi\)
\(198\) 0 0
\(199\) 2.19833 0.155836 0.0779179 0.996960i \(-0.475173\pi\)
0.0779179 + 0.996960i \(0.475173\pi\)
\(200\) 5.62205 0.397539
\(201\) −12.6565 −0.892724
\(202\) −7.13850 −0.502263
\(203\) −0.0475307 −0.00333600
\(204\) −4.42270 −0.309651
\(205\) 10.6730 0.745437
\(206\) 8.65120 0.602758
\(207\) 20.7805 1.44435
\(208\) −4.82326 −0.334433
\(209\) 0 0
\(210\) −37.2203 −2.56845
\(211\) −13.9756 −0.962118 −0.481059 0.876688i \(-0.659748\pi\)
−0.481059 + 0.876688i \(0.659748\pi\)
\(212\) 14.4636 0.993365
\(213\) −23.4874 −1.60933
\(214\) −13.3149 −0.910187
\(215\) 4.49829 0.306781
\(216\) −0.475914 −0.0323818
\(217\) −17.9868 −1.22102
\(218\) −14.3467 −0.971678
\(219\) 38.0124 2.56864
\(220\) 0 0
\(221\) −8.57310 −0.576689
\(222\) −1.71225 −0.114919
\(223\) 8.82515 0.590976 0.295488 0.955346i \(-0.404518\pi\)
0.295488 + 0.955346i \(0.404518\pi\)
\(224\) 4.58972 0.306664
\(225\) 17.9415 1.19610
\(226\) 17.6690 1.17532
\(227\) −4.53586 −0.301055 −0.150528 0.988606i \(-0.548097\pi\)
−0.150528 + 0.988606i \(0.548097\pi\)
\(228\) 2.48823 0.164787
\(229\) −6.29059 −0.415694 −0.207847 0.978161i \(-0.566646\pi\)
−0.207847 + 0.978161i \(0.566646\pi\)
\(230\) 21.2226 1.39938
\(231\) 0 0
\(232\) −0.0103559 −0.000679898 0
\(233\) −16.6897 −1.09338 −0.546689 0.837336i \(-0.684112\pi\)
−0.546689 + 0.837336i \(0.684112\pi\)
\(234\) −15.3923 −1.00623
\(235\) −20.5367 −1.33967
\(236\) −9.20149 −0.598966
\(237\) −32.2015 −2.09171
\(238\) 8.15800 0.528805
\(239\) −7.23299 −0.467863 −0.233932 0.972253i \(-0.575159\pi\)
−0.233932 + 0.972253i \(0.575159\pi\)
\(240\) −8.10950 −0.523466
\(241\) 13.5156 0.870618 0.435309 0.900281i \(-0.356639\pi\)
0.435309 + 0.900281i \(0.356639\pi\)
\(242\) 0 0
\(243\) 22.3030 1.43074
\(244\) 13.4651 0.862017
\(245\) 45.8417 2.92872
\(246\) −8.14842 −0.519524
\(247\) 4.82326 0.306897
\(248\) −3.91892 −0.248852
\(249\) −5.77806 −0.366170
\(250\) 2.02737 0.128222
\(251\) 3.08810 0.194919 0.0974595 0.995239i \(-0.468928\pi\)
0.0974595 + 0.995239i \(0.468928\pi\)
\(252\) 14.6470 0.922676
\(253\) 0 0
\(254\) −10.6201 −0.666364
\(255\) −14.4142 −0.902654
\(256\) 1.00000 0.0625000
\(257\) 17.3396 1.08161 0.540806 0.841147i \(-0.318119\pi\)
0.540806 + 0.841147i \(0.318119\pi\)
\(258\) −3.43426 −0.213808
\(259\) 3.15838 0.196252
\(260\) −15.7197 −0.974895
\(261\) −0.0330484 −0.00204565
\(262\) 16.9361 1.04631
\(263\) −16.8383 −1.03830 −0.519148 0.854684i \(-0.673751\pi\)
−0.519148 + 0.854684i \(0.673751\pi\)
\(264\) 0 0
\(265\) 47.1391 2.89573
\(266\) −4.58972 −0.281414
\(267\) 15.0514 0.921130
\(268\) 5.08657 0.310712
\(269\) −4.84616 −0.295475 −0.147738 0.989027i \(-0.547199\pi\)
−0.147738 + 0.989027i \(0.547199\pi\)
\(270\) −1.55107 −0.0943953
\(271\) 5.89504 0.358098 0.179049 0.983840i \(-0.442698\pi\)
0.179049 + 0.983840i \(0.442698\pi\)
\(272\) 1.77745 0.107774
\(273\) 55.0828 3.33377
\(274\) −8.12858 −0.491066
\(275\) 0 0
\(276\) −16.2026 −0.975279
\(277\) 30.7017 1.84468 0.922342 0.386375i \(-0.126273\pi\)
0.922342 + 0.386375i \(0.126273\pi\)
\(278\) −0.909211 −0.0545309
\(279\) −12.5063 −0.748734
\(280\) 14.9586 0.893946
\(281\) −14.4108 −0.859678 −0.429839 0.902905i \(-0.641430\pi\)
−0.429839 + 0.902905i \(0.641430\pi\)
\(282\) 15.6789 0.933667
\(283\) 15.1624 0.901313 0.450657 0.892697i \(-0.351190\pi\)
0.450657 + 0.892697i \(0.351190\pi\)
\(284\) 9.43943 0.560127
\(285\) 8.10950 0.480365
\(286\) 0 0
\(287\) 15.0304 0.887215
\(288\) 3.19127 0.188047
\(289\) −13.8407 −0.814157
\(290\) −0.0337514 −0.00198195
\(291\) 48.2950 2.83110
\(292\) −15.2769 −0.894014
\(293\) −30.5088 −1.78234 −0.891172 0.453665i \(-0.850116\pi\)
−0.891172 + 0.453665i \(0.850116\pi\)
\(294\) −34.9982 −2.04114
\(295\) −29.9890 −1.74603
\(296\) 0.688142 0.0399975
\(297\) 0 0
\(298\) 13.2547 0.767823
\(299\) −31.4076 −1.81635
\(300\) −13.9889 −0.807652
\(301\) 6.33475 0.365129
\(302\) 2.29362 0.131983
\(303\) 17.7622 1.02041
\(304\) −1.00000 −0.0573539
\(305\) 43.8849 2.51284
\(306\) 5.67232 0.324265
\(307\) −10.4099 −0.594123 −0.297062 0.954858i \(-0.596007\pi\)
−0.297062 + 0.954858i \(0.596007\pi\)
\(308\) 0 0
\(309\) −21.5261 −1.22458
\(310\) −12.7724 −0.725421
\(311\) 28.6601 1.62516 0.812582 0.582847i \(-0.198061\pi\)
0.812582 + 0.582847i \(0.198061\pi\)
\(312\) 12.0014 0.679443
\(313\) 32.8105 1.85456 0.927279 0.374370i \(-0.122141\pi\)
0.927279 + 0.374370i \(0.122141\pi\)
\(314\) −11.2813 −0.636640
\(315\) 47.7368 2.68967
\(316\) 12.9416 0.728019
\(317\) −11.3871 −0.639564 −0.319782 0.947491i \(-0.603610\pi\)
−0.319782 + 0.947491i \(0.603610\pi\)
\(318\) −35.9887 −2.01815
\(319\) 0 0
\(320\) 3.25915 0.182192
\(321\) 33.1304 1.84916
\(322\) 29.8868 1.66553
\(323\) −1.77745 −0.0989000
\(324\) −8.38962 −0.466090
\(325\) −27.1166 −1.50416
\(326\) −6.52158 −0.361197
\(327\) 35.6977 1.97409
\(328\) 3.27479 0.180820
\(329\) −28.9210 −1.59446
\(330\) 0 0
\(331\) 13.1649 0.723609 0.361805 0.932254i \(-0.382161\pi\)
0.361805 + 0.932254i \(0.382161\pi\)
\(332\) 2.32216 0.127445
\(333\) 2.19605 0.120343
\(334\) 20.7099 1.13319
\(335\) 16.5779 0.905747
\(336\) −11.4203 −0.623026
\(337\) −17.3554 −0.945407 −0.472704 0.881221i \(-0.656722\pi\)
−0.472704 + 0.881221i \(0.656722\pi\)
\(338\) 10.2638 0.558278
\(339\) −43.9644 −2.38782
\(340\) 5.79298 0.314168
\(341\) 0 0
\(342\) −3.19127 −0.172564
\(343\) 32.4288 1.75099
\(344\) 1.38020 0.0744156
\(345\) −52.8065 −2.84301
\(346\) −2.02529 −0.108880
\(347\) 5.01412 0.269172 0.134586 0.990902i \(-0.457030\pi\)
0.134586 + 0.990902i \(0.457030\pi\)
\(348\) 0.0257678 0.00138130
\(349\) −36.1563 −1.93540 −0.967701 0.252100i \(-0.918879\pi\)
−0.967701 + 0.252100i \(0.918879\pi\)
\(350\) 25.8037 1.37926
\(351\) 2.29545 0.122522
\(352\) 0 0
\(353\) 16.1472 0.859429 0.429714 0.902965i \(-0.358614\pi\)
0.429714 + 0.902965i \(0.358614\pi\)
\(354\) 22.8954 1.21688
\(355\) 30.7645 1.63281
\(356\) −6.04905 −0.320599
\(357\) −20.2990 −1.07433
\(358\) 20.7205 1.09511
\(359\) 8.01403 0.422964 0.211482 0.977382i \(-0.432171\pi\)
0.211482 + 0.977382i \(0.432171\pi\)
\(360\) 10.4008 0.548171
\(361\) 1.00000 0.0526316
\(362\) −7.63695 −0.401389
\(363\) 0 0
\(364\) −22.1374 −1.16032
\(365\) −49.7898 −2.60612
\(366\) −33.5043 −1.75130
\(367\) 22.3199 1.16509 0.582544 0.812799i \(-0.302058\pi\)
0.582544 + 0.812799i \(0.302058\pi\)
\(368\) 6.51169 0.339445
\(369\) 10.4507 0.544043
\(370\) 2.24276 0.116595
\(371\) 66.3839 3.44648
\(372\) 9.75117 0.505574
\(373\) −11.0167 −0.570423 −0.285211 0.958465i \(-0.592064\pi\)
−0.285211 + 0.958465i \(0.592064\pi\)
\(374\) 0 0
\(375\) −5.04454 −0.260499
\(376\) −6.30125 −0.324962
\(377\) 0.0499492 0.00257251
\(378\) −2.18431 −0.112349
\(379\) 19.2812 0.990410 0.495205 0.868776i \(-0.335093\pi\)
0.495205 + 0.868776i \(0.335093\pi\)
\(380\) −3.25915 −0.167191
\(381\) 26.4252 1.35380
\(382\) 20.6340 1.05573
\(383\) 15.5005 0.792039 0.396019 0.918242i \(-0.370391\pi\)
0.396019 + 0.918242i \(0.370391\pi\)
\(384\) −2.48823 −0.126977
\(385\) 0 0
\(386\) 8.64747 0.440145
\(387\) 4.40460 0.223898
\(388\) −19.4094 −0.985364
\(389\) 2.57878 0.130749 0.0653746 0.997861i \(-0.479176\pi\)
0.0653746 + 0.997861i \(0.479176\pi\)
\(390\) 39.1142 1.98062
\(391\) 11.5742 0.585333
\(392\) 14.0655 0.710417
\(393\) −42.1408 −2.12572
\(394\) −10.6030 −0.534170
\(395\) 42.1784 2.12223
\(396\) 0 0
\(397\) 4.30951 0.216288 0.108144 0.994135i \(-0.465509\pi\)
0.108144 + 0.994135i \(0.465509\pi\)
\(398\) 2.19833 0.110193
\(399\) 11.4203 0.571728
\(400\) 5.62205 0.281103
\(401\) −12.0460 −0.601551 −0.300775 0.953695i \(-0.597245\pi\)
−0.300775 + 0.953695i \(0.597245\pi\)
\(402\) −12.6565 −0.631251
\(403\) 18.9020 0.941575
\(404\) −7.13850 −0.355154
\(405\) −27.3430 −1.35869
\(406\) −0.0475307 −0.00235891
\(407\) 0 0
\(408\) −4.42270 −0.218956
\(409\) −35.2394 −1.74248 −0.871239 0.490859i \(-0.836683\pi\)
−0.871239 + 0.490859i \(0.836683\pi\)
\(410\) 10.6730 0.527104
\(411\) 20.2257 0.997663
\(412\) 8.65120 0.426214
\(413\) −42.2322 −2.07811
\(414\) 20.7805 1.02131
\(415\) 7.56827 0.371512
\(416\) −4.82326 −0.236480
\(417\) 2.26232 0.110786
\(418\) 0 0
\(419\) −8.29185 −0.405083 −0.202542 0.979274i \(-0.564920\pi\)
−0.202542 + 0.979274i \(0.564920\pi\)
\(420\) −37.2203 −1.81617
\(421\) −25.2575 −1.23098 −0.615488 0.788146i \(-0.711041\pi\)
−0.615488 + 0.788146i \(0.711041\pi\)
\(422\) −13.9756 −0.680320
\(423\) −20.1090 −0.977731
\(424\) 14.4636 0.702415
\(425\) 9.99292 0.484728
\(426\) −23.4874 −1.13797
\(427\) 61.8012 2.99077
\(428\) −13.3149 −0.643599
\(429\) 0 0
\(430\) 4.49829 0.216927
\(431\) −5.62638 −0.271013 −0.135507 0.990776i \(-0.543266\pi\)
−0.135507 + 0.990776i \(0.543266\pi\)
\(432\) −0.475914 −0.0228974
\(433\) 24.2774 1.16670 0.583349 0.812222i \(-0.301742\pi\)
0.583349 + 0.812222i \(0.301742\pi\)
\(434\) −17.9868 −0.863392
\(435\) 0.0839812 0.00402659
\(436\) −14.3467 −0.687080
\(437\) −6.51169 −0.311496
\(438\) 38.0124 1.81630
\(439\) −23.6907 −1.13070 −0.565348 0.824853i \(-0.691258\pi\)
−0.565348 + 0.824853i \(0.691258\pi\)
\(440\) 0 0
\(441\) 44.8869 2.13747
\(442\) −8.57310 −0.407781
\(443\) −26.9144 −1.27874 −0.639371 0.768899i \(-0.720805\pi\)
−0.639371 + 0.768899i \(0.720805\pi\)
\(444\) −1.71225 −0.0812600
\(445\) −19.7147 −0.934569
\(446\) 8.82515 0.417883
\(447\) −32.9806 −1.55993
\(448\) 4.58972 0.216844
\(449\) 15.5953 0.735987 0.367993 0.929828i \(-0.380045\pi\)
0.367993 + 0.929828i \(0.380045\pi\)
\(450\) 17.9415 0.845769
\(451\) 0 0
\(452\) 17.6690 0.831080
\(453\) −5.70705 −0.268140
\(454\) −4.53586 −0.212878
\(455\) −72.1491 −3.38240
\(456\) 2.48823 0.116522
\(457\) −24.0214 −1.12367 −0.561837 0.827248i \(-0.689905\pi\)
−0.561837 + 0.827248i \(0.689905\pi\)
\(458\) −6.29059 −0.293940
\(459\) −0.845913 −0.0394838
\(460\) 21.2226 0.989508
\(461\) −19.8822 −0.926005 −0.463003 0.886357i \(-0.653228\pi\)
−0.463003 + 0.886357i \(0.653228\pi\)
\(462\) 0 0
\(463\) 37.7710 1.75537 0.877684 0.479240i \(-0.159087\pi\)
0.877684 + 0.479240i \(0.159087\pi\)
\(464\) −0.0103559 −0.000480761 0
\(465\) 31.7805 1.47379
\(466\) −16.6897 −0.773135
\(467\) −42.9098 −1.98563 −0.992815 0.119656i \(-0.961821\pi\)
−0.992815 + 0.119656i \(0.961821\pi\)
\(468\) −15.3923 −0.711509
\(469\) 23.3459 1.07802
\(470\) −20.5367 −0.947288
\(471\) 28.0704 1.29342
\(472\) −9.20149 −0.423533
\(473\) 0 0
\(474\) −32.2015 −1.47906
\(475\) −5.62205 −0.257958
\(476\) 8.15800 0.373921
\(477\) 46.1572 2.11340
\(478\) −7.23299 −0.330829
\(479\) 26.1271 1.19378 0.596888 0.802324i \(-0.296404\pi\)
0.596888 + 0.802324i \(0.296404\pi\)
\(480\) −8.10950 −0.370146
\(481\) −3.31909 −0.151337
\(482\) 13.5156 0.615620
\(483\) −74.3652 −3.38373
\(484\) 0 0
\(485\) −63.2582 −2.87241
\(486\) 22.3030 1.01168
\(487\) 13.0467 0.591204 0.295602 0.955311i \(-0.404480\pi\)
0.295602 + 0.955311i \(0.404480\pi\)
\(488\) 13.4651 0.609538
\(489\) 16.2272 0.733818
\(490\) 45.8417 2.07092
\(491\) −16.0017 −0.722145 −0.361072 0.932538i \(-0.617589\pi\)
−0.361072 + 0.932538i \(0.617589\pi\)
\(492\) −8.14842 −0.367359
\(493\) −0.0184071 −0.000829015 0
\(494\) 4.82326 0.217009
\(495\) 0 0
\(496\) −3.91892 −0.175965
\(497\) 43.3244 1.94336
\(498\) −5.77806 −0.258921
\(499\) −29.3850 −1.31545 −0.657727 0.753256i \(-0.728482\pi\)
−0.657727 + 0.753256i \(0.728482\pi\)
\(500\) 2.02737 0.0906665
\(501\) −51.5308 −2.30223
\(502\) 3.08810 0.137829
\(503\) −8.80839 −0.392747 −0.196373 0.980529i \(-0.562916\pi\)
−0.196373 + 0.980529i \(0.562916\pi\)
\(504\) 14.6470 0.652430
\(505\) −23.2655 −1.03530
\(506\) 0 0
\(507\) −25.5387 −1.13421
\(508\) −10.6201 −0.471191
\(509\) −15.7974 −0.700209 −0.350104 0.936711i \(-0.613854\pi\)
−0.350104 + 0.936711i \(0.613854\pi\)
\(510\) −14.4142 −0.638273
\(511\) −70.1168 −3.10178
\(512\) 1.00000 0.0441942
\(513\) 0.475914 0.0210121
\(514\) 17.3396 0.764815
\(515\) 28.1955 1.24244
\(516\) −3.43426 −0.151185
\(517\) 0 0
\(518\) 3.15838 0.138771
\(519\) 5.03938 0.221204
\(520\) −15.7197 −0.689355
\(521\) 27.2791 1.19512 0.597560 0.801824i \(-0.296137\pi\)
0.597560 + 0.801824i \(0.296137\pi\)
\(522\) −0.0330484 −0.00144649
\(523\) −24.3662 −1.06546 −0.532730 0.846285i \(-0.678834\pi\)
−0.532730 + 0.846285i \(0.678834\pi\)
\(524\) 16.9361 0.739856
\(525\) −64.2053 −2.80215
\(526\) −16.8383 −0.734186
\(527\) −6.96570 −0.303430
\(528\) 0 0
\(529\) 19.4021 0.843571
\(530\) 47.1391 2.04759
\(531\) −29.3644 −1.27431
\(532\) −4.58972 −0.198990
\(533\) −15.7952 −0.684164
\(534\) 15.0514 0.651337
\(535\) −43.3952 −1.87614
\(536\) 5.08657 0.219706
\(537\) −51.5574 −2.22486
\(538\) −4.84616 −0.208933
\(539\) 0 0
\(540\) −1.55107 −0.0667476
\(541\) 15.4651 0.664897 0.332449 0.943121i \(-0.392125\pi\)
0.332449 + 0.943121i \(0.392125\pi\)
\(542\) 5.89504 0.253214
\(543\) 19.0025 0.815474
\(544\) 1.77745 0.0762076
\(545\) −46.7579 −2.00289
\(546\) 55.0828 2.35733
\(547\) −1.33234 −0.0569667 −0.0284834 0.999594i \(-0.509068\pi\)
−0.0284834 + 0.999594i \(0.509068\pi\)
\(548\) −8.12858 −0.347236
\(549\) 42.9708 1.83395
\(550\) 0 0
\(551\) 0.0103559 0.000441176 0
\(552\) −16.2026 −0.689627
\(553\) 59.3981 2.52586
\(554\) 30.7017 1.30439
\(555\) −5.58049 −0.236879
\(556\) −0.909211 −0.0385591
\(557\) 26.6638 1.12978 0.564892 0.825165i \(-0.308918\pi\)
0.564892 + 0.825165i \(0.308918\pi\)
\(558\) −12.5063 −0.529435
\(559\) −6.65708 −0.281564
\(560\) 14.9586 0.632116
\(561\) 0 0
\(562\) −14.4108 −0.607884
\(563\) −25.1226 −1.05879 −0.529396 0.848375i \(-0.677582\pi\)
−0.529396 + 0.848375i \(0.677582\pi\)
\(564\) 15.6789 0.660202
\(565\) 57.5859 2.42266
\(566\) 15.1624 0.637325
\(567\) −38.5060 −1.61710
\(568\) 9.43943 0.396070
\(569\) −16.6061 −0.696162 −0.348081 0.937464i \(-0.613167\pi\)
−0.348081 + 0.937464i \(0.613167\pi\)
\(570\) 8.10950 0.339670
\(571\) −27.1795 −1.13743 −0.568713 0.822536i \(-0.692558\pi\)
−0.568713 + 0.822536i \(0.692558\pi\)
\(572\) 0 0
\(573\) −51.3421 −2.14485
\(574\) 15.0304 0.627356
\(575\) 36.6091 1.52670
\(576\) 3.19127 0.132969
\(577\) −20.5578 −0.855831 −0.427915 0.903819i \(-0.640752\pi\)
−0.427915 + 0.903819i \(0.640752\pi\)
\(578\) −13.8407 −0.575696
\(579\) −21.5168 −0.894210
\(580\) −0.0337514 −0.00140145
\(581\) 10.6581 0.442171
\(582\) 48.2950 2.00189
\(583\) 0 0
\(584\) −15.2769 −0.632163
\(585\) −50.1658 −2.07410
\(586\) −30.5088 −1.26031
\(587\) −13.6615 −0.563872 −0.281936 0.959433i \(-0.590977\pi\)
−0.281936 + 0.959433i \(0.590977\pi\)
\(588\) −34.9982 −1.44330
\(589\) 3.91892 0.161476
\(590\) −29.9890 −1.23463
\(591\) 26.3826 1.08523
\(592\) 0.688142 0.0282825
\(593\) 8.26443 0.339380 0.169690 0.985498i \(-0.445723\pi\)
0.169690 + 0.985498i \(0.445723\pi\)
\(594\) 0 0
\(595\) 26.5881 1.09001
\(596\) 13.2547 0.542933
\(597\) −5.46995 −0.223870
\(598\) −31.4076 −1.28435
\(599\) −11.3663 −0.464413 −0.232207 0.972666i \(-0.574595\pi\)
−0.232207 + 0.972666i \(0.574595\pi\)
\(600\) −13.9889 −0.571096
\(601\) −17.2129 −0.702130 −0.351065 0.936351i \(-0.614180\pi\)
−0.351065 + 0.936351i \(0.614180\pi\)
\(602\) 6.33475 0.258185
\(603\) 16.2326 0.661043
\(604\) 2.29362 0.0933261
\(605\) 0 0
\(606\) 17.7622 0.721540
\(607\) −2.71425 −0.110168 −0.0550841 0.998482i \(-0.517543\pi\)
−0.0550841 + 0.998482i \(0.517543\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0.118267 0.00479242
\(610\) 43.8849 1.77685
\(611\) 30.3925 1.22955
\(612\) 5.67232 0.229290
\(613\) −0.970155 −0.0391842 −0.0195921 0.999808i \(-0.506237\pi\)
−0.0195921 + 0.999808i \(0.506237\pi\)
\(614\) −10.4099 −0.420109
\(615\) −26.5569 −1.07088
\(616\) 0 0
\(617\) 0.377035 0.0151789 0.00758944 0.999971i \(-0.497584\pi\)
0.00758944 + 0.999971i \(0.497584\pi\)
\(618\) −21.5261 −0.865908
\(619\) −11.6033 −0.466377 −0.233188 0.972432i \(-0.574916\pi\)
−0.233188 + 0.972432i \(0.574916\pi\)
\(620\) −12.7724 −0.512950
\(621\) −3.09900 −0.124359
\(622\) 28.6601 1.14916
\(623\) −27.7634 −1.11232
\(624\) 12.0014 0.480439
\(625\) −21.5028 −0.860111
\(626\) 32.8105 1.31137
\(627\) 0 0
\(628\) −11.2813 −0.450172
\(629\) 1.22314 0.0487698
\(630\) 47.7368 1.90188
\(631\) 13.9531 0.555464 0.277732 0.960659i \(-0.410417\pi\)
0.277732 + 0.960659i \(0.410417\pi\)
\(632\) 12.9416 0.514787
\(633\) 34.7744 1.38216
\(634\) −11.3871 −0.452240
\(635\) −34.6125 −1.37355
\(636\) −35.9887 −1.42705
\(637\) −67.8417 −2.68799
\(638\) 0 0
\(639\) 30.1238 1.19168
\(640\) 3.25915 0.128829
\(641\) 3.80584 0.150322 0.0751608 0.997171i \(-0.476053\pi\)
0.0751608 + 0.997171i \(0.476053\pi\)
\(642\) 33.1304 1.30755
\(643\) 8.46396 0.333786 0.166893 0.985975i \(-0.446627\pi\)
0.166893 + 0.985975i \(0.446627\pi\)
\(644\) 29.8868 1.17771
\(645\) −11.1928 −0.440714
\(646\) −1.77745 −0.0699329
\(647\) −10.1179 −0.397775 −0.198888 0.980022i \(-0.563733\pi\)
−0.198888 + 0.980022i \(0.563733\pi\)
\(648\) −8.38962 −0.329575
\(649\) 0 0
\(650\) −27.1166 −1.06360
\(651\) 44.7551 1.75409
\(652\) −6.52158 −0.255405
\(653\) 7.24197 0.283400 0.141700 0.989910i \(-0.454743\pi\)
0.141700 + 0.989910i \(0.454743\pi\)
\(654\) 35.6977 1.39589
\(655\) 55.1972 2.15673
\(656\) 3.27479 0.127859
\(657\) −48.7527 −1.90202
\(658\) −28.9210 −1.12746
\(659\) −11.1870 −0.435783 −0.217891 0.975973i \(-0.569918\pi\)
−0.217891 + 0.975973i \(0.569918\pi\)
\(660\) 0 0
\(661\) 6.73868 0.262104 0.131052 0.991375i \(-0.458164\pi\)
0.131052 + 0.991375i \(0.458164\pi\)
\(662\) 13.1649 0.511669
\(663\) 21.3318 0.828459
\(664\) 2.32216 0.0901173
\(665\) −14.9586 −0.580069
\(666\) 2.19605 0.0850950
\(667\) −0.0674344 −0.00261107
\(668\) 20.7099 0.801289
\(669\) −21.9590 −0.848983
\(670\) 16.5779 0.640460
\(671\) 0 0
\(672\) −11.4203 −0.440546
\(673\) 3.80683 0.146743 0.0733713 0.997305i \(-0.476624\pi\)
0.0733713 + 0.997305i \(0.476624\pi\)
\(674\) −17.3554 −0.668504
\(675\) −2.67561 −0.102984
\(676\) 10.2638 0.394762
\(677\) 14.9987 0.576446 0.288223 0.957563i \(-0.406936\pi\)
0.288223 + 0.957563i \(0.406936\pi\)
\(678\) −43.9644 −1.68844
\(679\) −89.0838 −3.41872
\(680\) 5.79298 0.222151
\(681\) 11.2862 0.432489
\(682\) 0 0
\(683\) −12.7603 −0.488260 −0.244130 0.969742i \(-0.578502\pi\)
−0.244130 + 0.969742i \(0.578502\pi\)
\(684\) −3.19127 −0.122021
\(685\) −26.4923 −1.01222
\(686\) 32.4288 1.23814
\(687\) 15.6524 0.597177
\(688\) 1.38020 0.0526198
\(689\) −69.7617 −2.65771
\(690\) −52.8065 −2.01031
\(691\) 29.5740 1.12505 0.562525 0.826780i \(-0.309830\pi\)
0.562525 + 0.826780i \(0.309830\pi\)
\(692\) −2.02529 −0.0769899
\(693\) 0 0
\(694\) 5.01412 0.190333
\(695\) −2.96326 −0.112403
\(696\) 0.0257678 0.000976727 0
\(697\) 5.82078 0.220478
\(698\) −36.1563 −1.36854
\(699\) 41.5277 1.57072
\(700\) 25.8037 0.975286
\(701\) −35.4860 −1.34029 −0.670143 0.742232i \(-0.733767\pi\)
−0.670143 + 0.742232i \(0.733767\pi\)
\(702\) 2.29545 0.0866363
\(703\) −0.688142 −0.0259538
\(704\) 0 0
\(705\) 51.1000 1.92454
\(706\) 16.1472 0.607708
\(707\) −32.7637 −1.23221
\(708\) 22.8954 0.860461
\(709\) 7.58291 0.284782 0.142391 0.989810i \(-0.454521\pi\)
0.142391 + 0.989810i \(0.454521\pi\)
\(710\) 30.7645 1.15457
\(711\) 41.2999 1.54887
\(712\) −6.04905 −0.226698
\(713\) −25.5188 −0.955687
\(714\) −20.2990 −0.759669
\(715\) 0 0
\(716\) 20.7205 0.774363
\(717\) 17.9973 0.672122
\(718\) 8.01403 0.299081
\(719\) 10.5644 0.393985 0.196993 0.980405i \(-0.436883\pi\)
0.196993 + 0.980405i \(0.436883\pi\)
\(720\) 10.4008 0.387615
\(721\) 39.7066 1.47875
\(722\) 1.00000 0.0372161
\(723\) −33.6299 −1.25071
\(724\) −7.63695 −0.283825
\(725\) −0.0582214 −0.00216229
\(726\) 0 0
\(727\) −13.9031 −0.515638 −0.257819 0.966193i \(-0.583004\pi\)
−0.257819 + 0.966193i \(0.583004\pi\)
\(728\) −22.1374 −0.820467
\(729\) −30.3260 −1.12319
\(730\) −49.7898 −1.84280
\(731\) 2.45324 0.0907365
\(732\) −33.5043 −1.23835
\(733\) 24.9256 0.920649 0.460325 0.887751i \(-0.347733\pi\)
0.460325 + 0.887751i \(0.347733\pi\)
\(734\) 22.3199 0.823841
\(735\) −114.064 −4.20733
\(736\) 6.51169 0.240024
\(737\) 0 0
\(738\) 10.4507 0.384697
\(739\) −9.43289 −0.346995 −0.173497 0.984834i \(-0.555507\pi\)
−0.173497 + 0.984834i \(0.555507\pi\)
\(740\) 2.24276 0.0824455
\(741\) −12.0014 −0.440881
\(742\) 66.3839 2.43703
\(743\) 50.6746 1.85907 0.929535 0.368735i \(-0.120209\pi\)
0.929535 + 0.368735i \(0.120209\pi\)
\(744\) 9.75117 0.357495
\(745\) 43.1990 1.58269
\(746\) −11.0167 −0.403350
\(747\) 7.41063 0.271141
\(748\) 0 0
\(749\) −61.1116 −2.23297
\(750\) −5.04454 −0.184201
\(751\) 53.0962 1.93751 0.968754 0.248025i \(-0.0797815\pi\)
0.968754 + 0.248025i \(0.0797815\pi\)
\(752\) −6.30125 −0.229783
\(753\) −7.68388 −0.280016
\(754\) 0.0499492 0.00181904
\(755\) 7.47525 0.272052
\(756\) −2.18431 −0.0794426
\(757\) −30.2934 −1.10103 −0.550517 0.834824i \(-0.685569\pi\)
−0.550517 + 0.834824i \(0.685569\pi\)
\(758\) 19.2812 0.700326
\(759\) 0 0
\(760\) −3.25915 −0.118222
\(761\) −4.83983 −0.175444 −0.0877219 0.996145i \(-0.527959\pi\)
−0.0877219 + 0.996145i \(0.527959\pi\)
\(762\) 26.4252 0.957284
\(763\) −65.8471 −2.38383
\(764\) 20.6340 0.746512
\(765\) 18.4869 0.668397
\(766\) 15.5005 0.560056
\(767\) 44.3811 1.60251
\(768\) −2.48823 −0.0897861
\(769\) −30.9675 −1.11672 −0.558358 0.829600i \(-0.688568\pi\)
−0.558358 + 0.829600i \(0.688568\pi\)
\(770\) 0 0
\(771\) −43.1447 −1.55382
\(772\) 8.64747 0.311229
\(773\) −30.9296 −1.11246 −0.556229 0.831029i \(-0.687752\pi\)
−0.556229 + 0.831029i \(0.687752\pi\)
\(774\) 4.40460 0.158320
\(775\) −22.0324 −0.791427
\(776\) −19.4094 −0.696758
\(777\) −7.85877 −0.281932
\(778\) 2.57878 0.0924536
\(779\) −3.27479 −0.117332
\(780\) 39.1142 1.40051
\(781\) 0 0
\(782\) 11.5742 0.413893
\(783\) 0.00492852 0.000176131 0
\(784\) 14.0655 0.502340
\(785\) −36.7674 −1.31228
\(786\) −42.1408 −1.50311
\(787\) −7.58212 −0.270274 −0.135137 0.990827i \(-0.543147\pi\)
−0.135137 + 0.990827i \(0.543147\pi\)
\(788\) −10.6030 −0.377715
\(789\) 41.8975 1.49159
\(790\) 42.1784 1.50064
\(791\) 81.0957 2.88343
\(792\) 0 0
\(793\) −64.9458 −2.30629
\(794\) 4.30951 0.152939
\(795\) −117.293 −4.15994
\(796\) 2.19833 0.0779179
\(797\) −1.66912 −0.0591232 −0.0295616 0.999563i \(-0.509411\pi\)
−0.0295616 + 0.999563i \(0.509411\pi\)
\(798\) 11.4203 0.404273
\(799\) −11.2002 −0.396233
\(800\) 5.62205 0.198770
\(801\) −19.3041 −0.682078
\(802\) −12.0460 −0.425361
\(803\) 0 0
\(804\) −12.6565 −0.446362
\(805\) 97.4057 3.43310
\(806\) 18.9020 0.665794
\(807\) 12.0583 0.424473
\(808\) −7.13850 −0.251132
\(809\) 27.4251 0.964215 0.482107 0.876112i \(-0.339871\pi\)
0.482107 + 0.876112i \(0.339871\pi\)
\(810\) −27.3430 −0.960736
\(811\) −52.1412 −1.83093 −0.915463 0.402401i \(-0.868176\pi\)
−0.915463 + 0.402401i \(0.868176\pi\)
\(812\) −0.0475307 −0.00166800
\(813\) −14.6682 −0.514436
\(814\) 0 0
\(815\) −21.2548 −0.744523
\(816\) −4.42270 −0.154825
\(817\) −1.38020 −0.0482872
\(818\) −35.2394 −1.23212
\(819\) −70.6463 −2.46858
\(820\) 10.6730 0.372718
\(821\) −19.2406 −0.671502 −0.335751 0.941951i \(-0.608990\pi\)
−0.335751 + 0.941951i \(0.608990\pi\)
\(822\) 20.2257 0.705454
\(823\) 23.7426 0.827614 0.413807 0.910365i \(-0.364199\pi\)
0.413807 + 0.910365i \(0.364199\pi\)
\(824\) 8.65120 0.301379
\(825\) 0 0
\(826\) −42.2322 −1.46945
\(827\) −19.7786 −0.687769 −0.343885 0.939012i \(-0.611743\pi\)
−0.343885 + 0.939012i \(0.611743\pi\)
\(828\) 20.7805 0.722174
\(829\) 9.35784 0.325011 0.162506 0.986708i \(-0.448042\pi\)
0.162506 + 0.986708i \(0.448042\pi\)
\(830\) 7.56827 0.262698
\(831\) −76.3926 −2.65003
\(832\) −4.82326 −0.167216
\(833\) 25.0008 0.866226
\(834\) 2.26232 0.0783378
\(835\) 67.4965 2.33581
\(836\) 0 0
\(837\) 1.86507 0.0644662
\(838\) −8.29185 −0.286437
\(839\) −7.65538 −0.264293 −0.132147 0.991230i \(-0.542187\pi\)
−0.132147 + 0.991230i \(0.542187\pi\)
\(840\) −37.2203 −1.28422
\(841\) −28.9999 −0.999996
\(842\) −25.2575 −0.870431
\(843\) 35.8574 1.23499
\(844\) −13.9756 −0.481059
\(845\) 33.4513 1.15076
\(846\) −20.1090 −0.691360
\(847\) 0 0
\(848\) 14.4636 0.496683
\(849\) −37.7276 −1.29481
\(850\) 9.99292 0.342754
\(851\) 4.48097 0.153606
\(852\) −23.4874 −0.804667
\(853\) 27.9084 0.955565 0.477783 0.878478i \(-0.341441\pi\)
0.477783 + 0.878478i \(0.341441\pi\)
\(854\) 61.8012 2.11479
\(855\) −10.4008 −0.355700
\(856\) −13.3149 −0.455093
\(857\) 38.8157 1.32592 0.662960 0.748655i \(-0.269300\pi\)
0.662960 + 0.748655i \(0.269300\pi\)
\(858\) 0 0
\(859\) −12.5352 −0.427697 −0.213848 0.976867i \(-0.568600\pi\)
−0.213848 + 0.976867i \(0.568600\pi\)
\(860\) 4.49829 0.153390
\(861\) −37.3990 −1.27455
\(862\) −5.62638 −0.191635
\(863\) −12.4527 −0.423896 −0.211948 0.977281i \(-0.567981\pi\)
−0.211948 + 0.977281i \(0.567981\pi\)
\(864\) −0.475914 −0.0161909
\(865\) −6.60072 −0.224431
\(866\) 24.2774 0.824980
\(867\) 34.4387 1.16960
\(868\) −17.9868 −0.610511
\(869\) 0 0
\(870\) 0.0839812 0.00284723
\(871\) −24.5338 −0.831298
\(872\) −14.3467 −0.485839
\(873\) −61.9406 −2.09637
\(874\) −6.51169 −0.220261
\(875\) 9.30504 0.314568
\(876\) 38.0124 1.28432
\(877\) −11.4582 −0.386917 −0.193458 0.981108i \(-0.561970\pi\)
−0.193458 + 0.981108i \(0.561970\pi\)
\(878\) −23.6907 −0.799523
\(879\) 75.9128 2.56048
\(880\) 0 0
\(881\) 26.3649 0.888257 0.444129 0.895963i \(-0.353513\pi\)
0.444129 + 0.895963i \(0.353513\pi\)
\(882\) 44.8869 1.51142
\(883\) 52.5648 1.76895 0.884474 0.466590i \(-0.154517\pi\)
0.884474 + 0.466590i \(0.154517\pi\)
\(884\) −8.57310 −0.288345
\(885\) 74.6194 2.50830
\(886\) −26.9144 −0.904207
\(887\) 29.5635 0.992644 0.496322 0.868138i \(-0.334683\pi\)
0.496322 + 0.868138i \(0.334683\pi\)
\(888\) −1.71225 −0.0574595
\(889\) −48.7433 −1.63480
\(890\) −19.7147 −0.660840
\(891\) 0 0
\(892\) 8.82515 0.295488
\(893\) 6.30125 0.210863
\(894\) −32.9806 −1.10304
\(895\) 67.5313 2.25732
\(896\) 4.58972 0.153332
\(897\) 78.1491 2.60932
\(898\) 15.5953 0.520421
\(899\) 0.0405840 0.00135355
\(900\) 17.9415 0.598049
\(901\) 25.7084 0.856470
\(902\) 0 0
\(903\) −15.7623 −0.524536
\(904\) 17.6690 0.587662
\(905\) −24.8900 −0.827371
\(906\) −5.70705 −0.189604
\(907\) 49.2004 1.63367 0.816836 0.576870i \(-0.195726\pi\)
0.816836 + 0.576870i \(0.195726\pi\)
\(908\) −4.53586 −0.150528
\(909\) −22.7809 −0.755594
\(910\) −72.1491 −2.39172
\(911\) −11.8309 −0.391976 −0.195988 0.980606i \(-0.562791\pi\)
−0.195988 + 0.980606i \(0.562791\pi\)
\(912\) 2.48823 0.0823934
\(913\) 0 0
\(914\) −24.0214 −0.794557
\(915\) −109.196 −3.60989
\(916\) −6.29059 −0.207847
\(917\) 77.7319 2.56693
\(918\) −0.845913 −0.0279193
\(919\) 8.32413 0.274588 0.137294 0.990530i \(-0.456160\pi\)
0.137294 + 0.990530i \(0.456160\pi\)
\(920\) 21.2226 0.699688
\(921\) 25.9021 0.853504
\(922\) −19.8822 −0.654784
\(923\) −45.5288 −1.49860
\(924\) 0 0
\(925\) 3.86877 0.127204
\(926\) 37.7710 1.24123
\(927\) 27.6083 0.906775
\(928\) −0.0103559 −0.000339949 0
\(929\) −56.0858 −1.84011 −0.920057 0.391784i \(-0.871858\pi\)
−0.920057 + 0.391784i \(0.871858\pi\)
\(930\) 31.7805 1.04212
\(931\) −14.0655 −0.460979
\(932\) −16.6897 −0.546689
\(933\) −71.3127 −2.33467
\(934\) −42.9098 −1.40405
\(935\) 0 0
\(936\) −15.3923 −0.503113
\(937\) 25.2464 0.824764 0.412382 0.911011i \(-0.364697\pi\)
0.412382 + 0.911011i \(0.364697\pi\)
\(938\) 23.3459 0.762272
\(939\) −81.6399 −2.66422
\(940\) −20.5367 −0.669834
\(941\) −2.72172 −0.0887257 −0.0443628 0.999015i \(-0.514126\pi\)
−0.0443628 + 0.999015i \(0.514126\pi\)
\(942\) 28.0704 0.914583
\(943\) 21.3244 0.694419
\(944\) −9.20149 −0.299483
\(945\) −7.11899 −0.231581
\(946\) 0 0
\(947\) −28.7838 −0.935347 −0.467674 0.883901i \(-0.654908\pi\)
−0.467674 + 0.883901i \(0.654908\pi\)
\(948\) −32.2015 −1.04586
\(949\) 73.6845 2.39190
\(950\) −5.62205 −0.182404
\(951\) 28.3337 0.918784
\(952\) 8.15800 0.264402
\(953\) −19.3594 −0.627113 −0.313556 0.949570i \(-0.601520\pi\)
−0.313556 + 0.949570i \(0.601520\pi\)
\(954\) 46.1572 1.49440
\(955\) 67.2493 2.17614
\(956\) −7.23299 −0.233932
\(957\) 0 0
\(958\) 26.1271 0.844127
\(959\) −37.3079 −1.20474
\(960\) −8.10950 −0.261733
\(961\) −15.6420 −0.504582
\(962\) −3.31909 −0.107012
\(963\) −42.4913 −1.36926
\(964\) 13.5156 0.435309
\(965\) 28.1834 0.907255
\(966\) −74.3652 −2.39266
\(967\) −16.2018 −0.521016 −0.260508 0.965472i \(-0.583890\pi\)
−0.260508 + 0.965472i \(0.583890\pi\)
\(968\) 0 0
\(969\) 4.42270 0.142078
\(970\) −63.2582 −2.03110
\(971\) −38.0152 −1.21996 −0.609982 0.792415i \(-0.708824\pi\)
−0.609982 + 0.792415i \(0.708824\pi\)
\(972\) 22.3030 0.715369
\(973\) −4.17303 −0.133781
\(974\) 13.0467 0.418044
\(975\) 67.4722 2.16084
\(976\) 13.4651 0.431009
\(977\) 11.0980 0.355055 0.177528 0.984116i \(-0.443190\pi\)
0.177528 + 0.984116i \(0.443190\pi\)
\(978\) 16.2272 0.518888
\(979\) 0 0
\(980\) 45.8417 1.46436
\(981\) −45.7840 −1.46177
\(982\) −16.0017 −0.510633
\(983\) −29.1596 −0.930045 −0.465023 0.885299i \(-0.653954\pi\)
−0.465023 + 0.885299i \(0.653954\pi\)
\(984\) −8.14842 −0.259762
\(985\) −34.5566 −1.10107
\(986\) −0.0184071 −0.000586202 0
\(987\) 71.9619 2.29057
\(988\) 4.82326 0.153448
\(989\) 8.98746 0.285785
\(990\) 0 0
\(991\) 7.72841 0.245501 0.122751 0.992438i \(-0.460828\pi\)
0.122751 + 0.992438i \(0.460828\pi\)
\(992\) −3.91892 −0.124426
\(993\) −32.7573 −1.03952
\(994\) 43.3244 1.37417
\(995\) 7.16470 0.227136
\(996\) −5.77806 −0.183085
\(997\) −43.8825 −1.38977 −0.694886 0.719120i \(-0.744545\pi\)
−0.694886 + 0.719120i \(0.744545\pi\)
\(998\) −29.3850 −0.930166
\(999\) −0.327496 −0.0103615
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.1.1 yes 8
11.10 odd 2 4598.2.a.by.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.1 8 11.10 odd 2
4598.2.a.cb.1.1 yes 8 1.1 even 1 trivial