Properties

Label 4598.2.a.by.1.6
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 1.6
Root \(1.24609\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.79123 q^{3} +1.00000 q^{4} +4.02325 q^{5} -2.79123 q^{6} -0.274917 q^{7} -1.00000 q^{8} +4.79095 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.79123 q^{3} +1.00000 q^{4} +4.02325 q^{5} -2.79123 q^{6} -0.274917 q^{7} -1.00000 q^{8} +4.79095 q^{9} -4.02325 q^{10} +2.79123 q^{12} -0.597016 q^{13} +0.274917 q^{14} +11.2298 q^{15} +1.00000 q^{16} +2.62721 q^{17} -4.79095 q^{18} +1.00000 q^{19} +4.02325 q^{20} -0.767355 q^{21} +4.04740 q^{23} -2.79123 q^{24} +11.1866 q^{25} +0.597016 q^{26} +4.99894 q^{27} -0.274917 q^{28} +7.29532 q^{29} -11.2298 q^{30} -1.48056 q^{31} -1.00000 q^{32} -2.62721 q^{34} -1.10606 q^{35} +4.79095 q^{36} -8.31155 q^{37} -1.00000 q^{38} -1.66641 q^{39} -4.02325 q^{40} +12.3427 q^{41} +0.767355 q^{42} -5.58432 q^{43} +19.2752 q^{45} -4.04740 q^{46} -9.10733 q^{47} +2.79123 q^{48} -6.92442 q^{49} -11.1866 q^{50} +7.33314 q^{51} -0.597016 q^{52} +7.30127 q^{53} -4.99894 q^{54} +0.274917 q^{56} +2.79123 q^{57} -7.29532 q^{58} -7.53037 q^{59} +11.2298 q^{60} -4.83533 q^{61} +1.48056 q^{62} -1.31711 q^{63} +1.00000 q^{64} -2.40195 q^{65} -14.0217 q^{67} +2.62721 q^{68} +11.2972 q^{69} +1.10606 q^{70} -9.61823 q^{71} -4.79095 q^{72} +3.15625 q^{73} +8.31155 q^{74} +31.2242 q^{75} +1.00000 q^{76} +1.66641 q^{78} +2.42106 q^{79} +4.02325 q^{80} -0.419665 q^{81} -12.3427 q^{82} -15.7948 q^{83} -0.767355 q^{84} +10.5699 q^{85} +5.58432 q^{86} +20.3629 q^{87} +1.49871 q^{89} -19.2752 q^{90} +0.164130 q^{91} +4.04740 q^{92} -4.13259 q^{93} +9.10733 q^{94} +4.02325 q^{95} -2.79123 q^{96} +9.00939 q^{97} +6.92442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98} + O(q^{100}) \)

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
\(3\) 2.79123 1.61152 0.805758 0.592245i \(-0.201758\pi\)
0.805758 + 0.592245i \(0.201758\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.02325 1.79925 0.899627 0.436660i \(-0.143839\pi\)
0.899627 + 0.436660i \(0.143839\pi\)
\(6\) −2.79123 −1.13951
\(7\) −0.274917 −0.103909 −0.0519544 0.998649i \(-0.516545\pi\)
−0.0519544 + 0.998649i \(0.516545\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.79095 1.59698
\(10\) −4.02325 −1.27226
\(11\) 0 0
\(12\) 2.79123 0.805758
\(13\) −0.597016 −0.165583 −0.0827913 0.996567i \(-0.526383\pi\)
−0.0827913 + 0.996567i \(0.526383\pi\)
\(14\) 0.274917 0.0734746
\(15\) 11.2298 2.89952
\(16\) 1.00000 0.250000
\(17\) 2.62721 0.637192 0.318596 0.947891i \(-0.396789\pi\)
0.318596 + 0.947891i \(0.396789\pi\)
\(18\) −4.79095 −1.12924
\(19\) 1.00000 0.229416
\(20\) 4.02325 0.899627
\(21\) −0.767355 −0.167451
\(22\) 0 0
\(23\) 4.04740 0.843942 0.421971 0.906609i \(-0.361338\pi\)
0.421971 + 0.906609i \(0.361338\pi\)
\(24\) −2.79123 −0.569757
\(25\) 11.1866 2.23731
\(26\) 0.597016 0.117085
\(27\) 4.99894 0.962047
\(28\) −0.274917 −0.0519544
\(29\) 7.29532 1.35471 0.677354 0.735657i \(-0.263127\pi\)
0.677354 + 0.735657i \(0.263127\pi\)
\(30\) −11.2298 −2.05027
\(31\) −1.48056 −0.265917 −0.132959 0.991122i \(-0.542448\pi\)
−0.132959 + 0.991122i \(0.542448\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.62721 −0.450563
\(35\) −1.10606 −0.186958
\(36\) 4.79095 0.798491
\(37\) −8.31155 −1.36641 −0.683205 0.730227i \(-0.739414\pi\)
−0.683205 + 0.730227i \(0.739414\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.66641 −0.266839
\(40\) −4.02325 −0.636132
\(41\) 12.3427 1.92760 0.963800 0.266627i \(-0.0859093\pi\)
0.963800 + 0.266627i \(0.0859093\pi\)
\(42\) 0.767355 0.118405
\(43\) −5.58432 −0.851601 −0.425800 0.904817i \(-0.640007\pi\)
−0.425800 + 0.904817i \(0.640007\pi\)
\(44\) 0 0
\(45\) 19.2752 2.87338
\(46\) −4.04740 −0.596757
\(47\) −9.10733 −1.32844 −0.664221 0.747537i \(-0.731236\pi\)
−0.664221 + 0.747537i \(0.731236\pi\)
\(48\) 2.79123 0.402879
\(49\) −6.92442 −0.989203
\(50\) −11.1866 −1.58202
\(51\) 7.33314 1.02684
\(52\) −0.597016 −0.0827913
\(53\) 7.30127 1.00291 0.501454 0.865185i \(-0.332799\pi\)
0.501454 + 0.865185i \(0.332799\pi\)
\(54\) −4.99894 −0.680270
\(55\) 0 0
\(56\) 0.274917 0.0367373
\(57\) 2.79123 0.369707
\(58\) −7.29532 −0.957923
\(59\) −7.53037 −0.980370 −0.490185 0.871618i \(-0.663071\pi\)
−0.490185 + 0.871618i \(0.663071\pi\)
\(60\) 11.2298 1.44976
\(61\) −4.83533 −0.619100 −0.309550 0.950883i \(-0.600178\pi\)
−0.309550 + 0.950883i \(0.600178\pi\)
\(62\) 1.48056 0.188032
\(63\) −1.31711 −0.165940
\(64\) 1.00000 0.125000
\(65\) −2.40195 −0.297925
\(66\) 0 0
\(67\) −14.0217 −1.71303 −0.856514 0.516124i \(-0.827374\pi\)
−0.856514 + 0.516124i \(0.827374\pi\)
\(68\) 2.62721 0.318596
\(69\) 11.2972 1.36003
\(70\) 1.10606 0.132199
\(71\) −9.61823 −1.14147 −0.570737 0.821133i \(-0.693342\pi\)
−0.570737 + 0.821133i \(0.693342\pi\)
\(72\) −4.79095 −0.564619
\(73\) 3.15625 0.369411 0.184705 0.982794i \(-0.440867\pi\)
0.184705 + 0.982794i \(0.440867\pi\)
\(74\) 8.31155 0.966198
\(75\) 31.2242 3.60546
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 1.66641 0.188684
\(79\) 2.42106 0.272391 0.136195 0.990682i \(-0.456513\pi\)
0.136195 + 0.990682i \(0.456513\pi\)
\(80\) 4.02325 0.449813
\(81\) −0.419665 −0.0466294
\(82\) −12.3427 −1.36302
\(83\) −15.7948 −1.73370 −0.866852 0.498566i \(-0.833860\pi\)
−0.866852 + 0.498566i \(0.833860\pi\)
\(84\) −0.767355 −0.0837253
\(85\) 10.5699 1.14647
\(86\) 5.58432 0.602173
\(87\) 20.3629 2.18313
\(88\) 0 0
\(89\) 1.49871 0.158863 0.0794315 0.996840i \(-0.474689\pi\)
0.0794315 + 0.996840i \(0.474689\pi\)
\(90\) −19.2752 −2.03178
\(91\) 0.164130 0.0172055
\(92\) 4.04740 0.421971
\(93\) −4.13259 −0.428530
\(94\) 9.10733 0.939350
\(95\) 4.02325 0.412777
\(96\) −2.79123 −0.284878
\(97\) 9.00939 0.914765 0.457382 0.889270i \(-0.348787\pi\)
0.457382 + 0.889270i \(0.348787\pi\)
\(98\) 6.92442 0.699472
\(99\) 0 0
\(100\) 11.1866 1.11866
\(101\) −2.30943 −0.229797 −0.114898 0.993377i \(-0.536654\pi\)
−0.114898 + 0.993377i \(0.536654\pi\)
\(102\) −7.33314 −0.726089
\(103\) 10.3403 1.01886 0.509428 0.860513i \(-0.329857\pi\)
0.509428 + 0.860513i \(0.329857\pi\)
\(104\) 0.597016 0.0585423
\(105\) −3.08726 −0.301286
\(106\) −7.30127 −0.709162
\(107\) 9.00544 0.870589 0.435294 0.900288i \(-0.356644\pi\)
0.435294 + 0.900288i \(0.356644\pi\)
\(108\) 4.99894 0.481023
\(109\) −16.8995 −1.61867 −0.809337 0.587344i \(-0.800173\pi\)
−0.809337 + 0.587344i \(0.800173\pi\)
\(110\) 0 0
\(111\) −23.1994 −2.20199
\(112\) −0.274917 −0.0259772
\(113\) 2.84546 0.267678 0.133839 0.991003i \(-0.457269\pi\)
0.133839 + 0.991003i \(0.457269\pi\)
\(114\) −2.79123 −0.261422
\(115\) 16.2837 1.51847
\(116\) 7.29532 0.677354
\(117\) −2.86027 −0.264432
\(118\) 7.53037 0.693227
\(119\) −0.722264 −0.0662098
\(120\) −11.2298 −1.02514
\(121\) 0 0
\(122\) 4.83533 0.437770
\(123\) 34.4512 3.10636
\(124\) −1.48056 −0.132959
\(125\) 24.8901 2.22624
\(126\) 1.31711 0.117338
\(127\) −5.59695 −0.496649 −0.248324 0.968677i \(-0.579880\pi\)
−0.248324 + 0.968677i \(0.579880\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.5871 −1.37237
\(130\) 2.40195 0.210665
\(131\) 7.73690 0.675976 0.337988 0.941150i \(-0.390254\pi\)
0.337988 + 0.941150i \(0.390254\pi\)
\(132\) 0 0
\(133\) −0.274917 −0.0238383
\(134\) 14.0217 1.21129
\(135\) 20.1120 1.73097
\(136\) −2.62721 −0.225281
\(137\) −1.10706 −0.0945829 −0.0472914 0.998881i \(-0.515059\pi\)
−0.0472914 + 0.998881i \(0.515059\pi\)
\(138\) −11.2972 −0.961683
\(139\) −14.8792 −1.26204 −0.631020 0.775767i \(-0.717363\pi\)
−0.631020 + 0.775767i \(0.717363\pi\)
\(140\) −1.10606 −0.0934791
\(141\) −25.4206 −2.14080
\(142\) 9.61823 0.807144
\(143\) 0 0
\(144\) 4.79095 0.399246
\(145\) 29.3509 2.43746
\(146\) −3.15625 −0.261213
\(147\) −19.3276 −1.59412
\(148\) −8.31155 −0.683205
\(149\) −20.9576 −1.71691 −0.858457 0.512885i \(-0.828577\pi\)
−0.858457 + 0.512885i \(0.828577\pi\)
\(150\) −31.2242 −2.54945
\(151\) 4.07961 0.331994 0.165997 0.986126i \(-0.446916\pi\)
0.165997 + 0.986126i \(0.446916\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 12.5868 1.01758
\(154\) 0 0
\(155\) −5.95669 −0.478453
\(156\) −1.66641 −0.133419
\(157\) 19.7161 1.57352 0.786758 0.617262i \(-0.211758\pi\)
0.786758 + 0.617262i \(0.211758\pi\)
\(158\) −2.42106 −0.192609
\(159\) 20.3795 1.61620
\(160\) −4.02325 −0.318066
\(161\) −1.11270 −0.0876929
\(162\) 0.419665 0.0329720
\(163\) 16.9154 1.32492 0.662460 0.749098i \(-0.269513\pi\)
0.662460 + 0.749098i \(0.269513\pi\)
\(164\) 12.3427 0.963800
\(165\) 0 0
\(166\) 15.7948 1.22591
\(167\) 23.1451 1.79102 0.895512 0.445038i \(-0.146810\pi\)
0.895512 + 0.445038i \(0.146810\pi\)
\(168\) 0.767355 0.0592027
\(169\) −12.6436 −0.972582
\(170\) −10.5699 −0.810676
\(171\) 4.79095 0.366373
\(172\) −5.58432 −0.425800
\(173\) 23.5008 1.78673 0.893366 0.449330i \(-0.148337\pi\)
0.893366 + 0.449330i \(0.148337\pi\)
\(174\) −20.3629 −1.54371
\(175\) −3.07537 −0.232476
\(176\) 0 0
\(177\) −21.0190 −1.57988
\(178\) −1.49871 −0.112333
\(179\) 7.52415 0.562381 0.281191 0.959652i \(-0.409271\pi\)
0.281191 + 0.959652i \(0.409271\pi\)
\(180\) 19.2752 1.43669
\(181\) 19.6253 1.45874 0.729370 0.684120i \(-0.239813\pi\)
0.729370 + 0.684120i \(0.239813\pi\)
\(182\) −0.164130 −0.0121661
\(183\) −13.4965 −0.997690
\(184\) −4.04740 −0.298379
\(185\) −33.4395 −2.45852
\(186\) 4.13259 0.303016
\(187\) 0 0
\(188\) −9.10733 −0.664221
\(189\) −1.37429 −0.0999650
\(190\) −4.02325 −0.291877
\(191\) 20.9322 1.51460 0.757300 0.653067i \(-0.226518\pi\)
0.757300 + 0.653067i \(0.226518\pi\)
\(192\) 2.79123 0.201439
\(193\) −1.89921 −0.136708 −0.0683542 0.997661i \(-0.521775\pi\)
−0.0683542 + 0.997661i \(0.521775\pi\)
\(194\) −9.00939 −0.646836
\(195\) −6.70438 −0.480111
\(196\) −6.92442 −0.494601
\(197\) 1.40362 0.100004 0.0500020 0.998749i \(-0.484077\pi\)
0.0500020 + 0.998749i \(0.484077\pi\)
\(198\) 0 0
\(199\) −3.60404 −0.255483 −0.127742 0.991807i \(-0.540773\pi\)
−0.127742 + 0.991807i \(0.540773\pi\)
\(200\) −11.1866 −0.791009
\(201\) −39.1379 −2.76057
\(202\) 2.30943 0.162491
\(203\) −2.00561 −0.140766
\(204\) 7.33314 0.513422
\(205\) 49.6576 3.46824
\(206\) −10.3403 −0.720441
\(207\) 19.3909 1.34776
\(208\) −0.597016 −0.0413956
\(209\) 0 0
\(210\) 3.08726 0.213041
\(211\) −18.7715 −1.29228 −0.646141 0.763218i \(-0.723618\pi\)
−0.646141 + 0.763218i \(0.723618\pi\)
\(212\) 7.30127 0.501454
\(213\) −26.8467 −1.83950
\(214\) −9.00544 −0.615599
\(215\) −22.4671 −1.53225
\(216\) −4.99894 −0.340135
\(217\) 0.407032 0.0276311
\(218\) 16.8995 1.14458
\(219\) 8.80980 0.595311
\(220\) 0 0
\(221\) −1.56849 −0.105508
\(222\) 23.1994 1.55704
\(223\) −14.6688 −0.982298 −0.491149 0.871075i \(-0.663423\pi\)
−0.491149 + 0.871075i \(0.663423\pi\)
\(224\) 0.274917 0.0183686
\(225\) 53.5942 3.57295
\(226\) −2.84546 −0.189277
\(227\) −3.54422 −0.235238 −0.117619 0.993059i \(-0.537526\pi\)
−0.117619 + 0.993059i \(0.537526\pi\)
\(228\) 2.79123 0.184854
\(229\) −16.7749 −1.10852 −0.554259 0.832344i \(-0.686998\pi\)
−0.554259 + 0.832344i \(0.686998\pi\)
\(230\) −16.2837 −1.07372
\(231\) 0 0
\(232\) −7.29532 −0.478961
\(233\) −25.7392 −1.68623 −0.843115 0.537734i \(-0.819281\pi\)
−0.843115 + 0.537734i \(0.819281\pi\)
\(234\) 2.86027 0.186982
\(235\) −36.6411 −2.39020
\(236\) −7.53037 −0.490185
\(237\) 6.75773 0.438962
\(238\) 0.722264 0.0468174
\(239\) −3.83627 −0.248148 −0.124074 0.992273i \(-0.539596\pi\)
−0.124074 + 0.992273i \(0.539596\pi\)
\(240\) 11.2298 0.724881
\(241\) 10.9064 0.702540 0.351270 0.936274i \(-0.385750\pi\)
0.351270 + 0.936274i \(0.385750\pi\)
\(242\) 0 0
\(243\) −16.1682 −1.03719
\(244\) −4.83533 −0.309550
\(245\) −27.8587 −1.77983
\(246\) −34.4512 −2.19653
\(247\) −0.597016 −0.0379872
\(248\) 1.48056 0.0940160
\(249\) −44.0868 −2.79389
\(250\) −24.8901 −1.57419
\(251\) 13.9463 0.880283 0.440141 0.897928i \(-0.354928\pi\)
0.440141 + 0.897928i \(0.354928\pi\)
\(252\) −1.31711 −0.0829702
\(253\) 0 0
\(254\) 5.59695 0.351184
\(255\) 29.5031 1.84755
\(256\) 1.00000 0.0625000
\(257\) 16.8311 1.04989 0.524947 0.851135i \(-0.324085\pi\)
0.524947 + 0.851135i \(0.324085\pi\)
\(258\) 15.5871 0.970410
\(259\) 2.28498 0.141982
\(260\) −2.40195 −0.148962
\(261\) 34.9515 2.16344
\(262\) −7.73690 −0.477987
\(263\) 1.71132 0.105525 0.0527623 0.998607i \(-0.483197\pi\)
0.0527623 + 0.998607i \(0.483197\pi\)
\(264\) 0 0
\(265\) 29.3749 1.80448
\(266\) 0.274917 0.0168562
\(267\) 4.18324 0.256010
\(268\) −14.0217 −0.856514
\(269\) 22.3017 1.35976 0.679880 0.733323i \(-0.262032\pi\)
0.679880 + 0.733323i \(0.262032\pi\)
\(270\) −20.1120 −1.22398
\(271\) 2.26292 0.137463 0.0687314 0.997635i \(-0.478105\pi\)
0.0687314 + 0.997635i \(0.478105\pi\)
\(272\) 2.62721 0.159298
\(273\) 0.458123 0.0277269
\(274\) 1.10706 0.0668802
\(275\) 0 0
\(276\) 11.2972 0.680013
\(277\) −9.83937 −0.591190 −0.295595 0.955313i \(-0.595518\pi\)
−0.295595 + 0.955313i \(0.595518\pi\)
\(278\) 14.8792 0.892397
\(279\) −7.09331 −0.424665
\(280\) 1.10606 0.0660997
\(281\) −17.9598 −1.07139 −0.535697 0.844410i \(-0.679951\pi\)
−0.535697 + 0.844410i \(0.679951\pi\)
\(282\) 25.4206 1.51378
\(283\) −26.5193 −1.57641 −0.788206 0.615412i \(-0.788990\pi\)
−0.788206 + 0.615412i \(0.788990\pi\)
\(284\) −9.61823 −0.570737
\(285\) 11.2298 0.665197
\(286\) 0 0
\(287\) −3.39320 −0.200294
\(288\) −4.79095 −0.282309
\(289\) −10.0978 −0.593986
\(290\) −29.3509 −1.72355
\(291\) 25.1472 1.47416
\(292\) 3.15625 0.184705
\(293\) 12.3885 0.723745 0.361872 0.932228i \(-0.382138\pi\)
0.361872 + 0.932228i \(0.382138\pi\)
\(294\) 19.3276 1.12721
\(295\) −30.2966 −1.76393
\(296\) 8.31155 0.483099
\(297\) 0 0
\(298\) 20.9576 1.21404
\(299\) −2.41637 −0.139742
\(300\) 31.2242 1.80273
\(301\) 1.53522 0.0884887
\(302\) −4.07961 −0.234755
\(303\) −6.44614 −0.370321
\(304\) 1.00000 0.0573539
\(305\) −19.4537 −1.11392
\(306\) −12.5868 −0.719541
\(307\) 29.2289 1.66818 0.834090 0.551628i \(-0.185993\pi\)
0.834090 + 0.551628i \(0.185993\pi\)
\(308\) 0 0
\(309\) 28.8620 1.64190
\(310\) 5.95669 0.338317
\(311\) 10.9986 0.623675 0.311837 0.950136i \(-0.399056\pi\)
0.311837 + 0.950136i \(0.399056\pi\)
\(312\) 1.66641 0.0943418
\(313\) 15.5213 0.877314 0.438657 0.898654i \(-0.355454\pi\)
0.438657 + 0.898654i \(0.355454\pi\)
\(314\) −19.7161 −1.11264
\(315\) −5.29907 −0.298569
\(316\) 2.42106 0.136195
\(317\) −23.9485 −1.34508 −0.672541 0.740060i \(-0.734797\pi\)
−0.672541 + 0.740060i \(0.734797\pi\)
\(318\) −20.3795 −1.14283
\(319\) 0 0
\(320\) 4.02325 0.224907
\(321\) 25.1362 1.40297
\(322\) 1.11270 0.0620083
\(323\) 2.62721 0.146182
\(324\) −0.419665 −0.0233147
\(325\) −6.67856 −0.370460
\(326\) −16.9154 −0.936859
\(327\) −47.1702 −2.60852
\(328\) −12.3427 −0.681509
\(329\) 2.50376 0.138037
\(330\) 0 0
\(331\) 25.6584 1.41031 0.705156 0.709052i \(-0.250877\pi\)
0.705156 + 0.709052i \(0.250877\pi\)
\(332\) −15.7948 −0.866852
\(333\) −39.8202 −2.18213
\(334\) −23.1451 −1.26644
\(335\) −56.4130 −3.08217
\(336\) −0.767355 −0.0418626
\(337\) 2.94204 0.160263 0.0801314 0.996784i \(-0.474466\pi\)
0.0801314 + 0.996784i \(0.474466\pi\)
\(338\) 12.6436 0.687720
\(339\) 7.94233 0.431368
\(340\) 10.5699 0.573235
\(341\) 0 0
\(342\) −4.79095 −0.259065
\(343\) 3.82806 0.206696
\(344\) 5.58432 0.301086
\(345\) 45.4516 2.44703
\(346\) −23.5008 −1.26341
\(347\) 6.33884 0.340287 0.170144 0.985419i \(-0.445577\pi\)
0.170144 + 0.985419i \(0.445577\pi\)
\(348\) 20.3629 1.09157
\(349\) 10.5789 0.566276 0.283138 0.959079i \(-0.408625\pi\)
0.283138 + 0.959079i \(0.408625\pi\)
\(350\) 3.07537 0.164386
\(351\) −2.98445 −0.159298
\(352\) 0 0
\(353\) −35.1642 −1.87160 −0.935800 0.352532i \(-0.885321\pi\)
−0.935800 + 0.352532i \(0.885321\pi\)
\(354\) 21.0190 1.11715
\(355\) −38.6966 −2.05380
\(356\) 1.49871 0.0794315
\(357\) −2.01600 −0.106698
\(358\) −7.52415 −0.397664
\(359\) −24.1939 −1.27691 −0.638453 0.769661i \(-0.720425\pi\)
−0.638453 + 0.769661i \(0.720425\pi\)
\(360\) −19.2752 −1.01589
\(361\) 1.00000 0.0526316
\(362\) −19.6253 −1.03148
\(363\) 0 0
\(364\) 0.164130 0.00860273
\(365\) 12.6984 0.664664
\(366\) 13.4965 0.705473
\(367\) −23.2519 −1.21374 −0.606869 0.794802i \(-0.707575\pi\)
−0.606869 + 0.794802i \(0.707575\pi\)
\(368\) 4.04740 0.210986
\(369\) 59.1330 3.07834
\(370\) 33.4395 1.73843
\(371\) −2.00724 −0.104211
\(372\) −4.13259 −0.214265
\(373\) −8.15845 −0.422429 −0.211214 0.977440i \(-0.567742\pi\)
−0.211214 + 0.977440i \(0.567742\pi\)
\(374\) 0 0
\(375\) 69.4739 3.58762
\(376\) 9.10733 0.469675
\(377\) −4.35543 −0.224316
\(378\) 1.37429 0.0706860
\(379\) −18.9997 −0.975951 −0.487976 0.872857i \(-0.662265\pi\)
−0.487976 + 0.872857i \(0.662265\pi\)
\(380\) 4.02325 0.206388
\(381\) −15.6224 −0.800357
\(382\) −20.9322 −1.07098
\(383\) 12.1817 0.622456 0.311228 0.950335i \(-0.399260\pi\)
0.311228 + 0.950335i \(0.399260\pi\)
\(384\) −2.79123 −0.142439
\(385\) 0 0
\(386\) 1.89921 0.0966674
\(387\) −26.7542 −1.35999
\(388\) 9.00939 0.457382
\(389\) −24.1433 −1.22411 −0.612057 0.790814i \(-0.709658\pi\)
−0.612057 + 0.790814i \(0.709658\pi\)
\(390\) 6.70438 0.339489
\(391\) 10.6334 0.537753
\(392\) 6.92442 0.349736
\(393\) 21.5954 1.08935
\(394\) −1.40362 −0.0707135
\(395\) 9.74054 0.490100
\(396\) 0 0
\(397\) −10.1918 −0.511510 −0.255755 0.966742i \(-0.582324\pi\)
−0.255755 + 0.966742i \(0.582324\pi\)
\(398\) 3.60404 0.180654
\(399\) −0.767355 −0.0384158
\(400\) 11.1866 0.559328
\(401\) 12.2313 0.610800 0.305400 0.952224i \(-0.401210\pi\)
0.305400 + 0.952224i \(0.401210\pi\)
\(402\) 39.1379 1.95202
\(403\) 0.883921 0.0440313
\(404\) −2.30943 −0.114898
\(405\) −1.68842 −0.0838981
\(406\) 2.00561 0.0995365
\(407\) 0 0
\(408\) −7.33314 −0.363044
\(409\) 27.8419 1.37669 0.688347 0.725381i \(-0.258337\pi\)
0.688347 + 0.725381i \(0.258337\pi\)
\(410\) −49.6576 −2.45242
\(411\) −3.09007 −0.152422
\(412\) 10.3403 0.509428
\(413\) 2.07022 0.101869
\(414\) −19.3909 −0.953011
\(415\) −63.5464 −3.11937
\(416\) 0.597016 0.0292711
\(417\) −41.5313 −2.03380
\(418\) 0 0
\(419\) 2.74637 0.134169 0.0670846 0.997747i \(-0.478630\pi\)
0.0670846 + 0.997747i \(0.478630\pi\)
\(420\) −3.08726 −0.150643
\(421\) 3.45765 0.168516 0.0842579 0.996444i \(-0.473148\pi\)
0.0842579 + 0.996444i \(0.473148\pi\)
\(422\) 18.7715 0.913782
\(423\) −43.6328 −2.12150
\(424\) −7.30127 −0.354581
\(425\) 29.3894 1.42560
\(426\) 26.8467 1.30073
\(427\) 1.32931 0.0643299
\(428\) 9.00544 0.435294
\(429\) 0 0
\(430\) 22.4671 1.08346
\(431\) −33.1455 −1.59656 −0.798281 0.602285i \(-0.794257\pi\)
−0.798281 + 0.602285i \(0.794257\pi\)
\(432\) 4.99894 0.240512
\(433\) −17.0841 −0.821008 −0.410504 0.911859i \(-0.634647\pi\)
−0.410504 + 0.911859i \(0.634647\pi\)
\(434\) −0.407032 −0.0195382
\(435\) 81.9251 3.92801
\(436\) −16.8995 −0.809337
\(437\) 4.04740 0.193614
\(438\) −8.80980 −0.420949
\(439\) −36.1437 −1.72505 −0.862523 0.506018i \(-0.831117\pi\)
−0.862523 + 0.506018i \(0.831117\pi\)
\(440\) 0 0
\(441\) −33.1745 −1.57974
\(442\) 1.56849 0.0746053
\(443\) 19.4900 0.925996 0.462998 0.886359i \(-0.346774\pi\)
0.462998 + 0.886359i \(0.346774\pi\)
\(444\) −23.1994 −1.10100
\(445\) 6.02969 0.285835
\(446\) 14.6688 0.694590
\(447\) −58.4974 −2.76683
\(448\) −0.274917 −0.0129886
\(449\) −12.8197 −0.604998 −0.302499 0.953150i \(-0.597821\pi\)
−0.302499 + 0.953150i \(0.597821\pi\)
\(450\) −53.5942 −2.52646
\(451\) 0 0
\(452\) 2.84546 0.133839
\(453\) 11.3871 0.535014
\(454\) 3.54422 0.166338
\(455\) 0.660335 0.0309570
\(456\) −2.79123 −0.130711
\(457\) −2.20055 −0.102937 −0.0514687 0.998675i \(-0.516390\pi\)
−0.0514687 + 0.998675i \(0.516390\pi\)
\(458\) 16.7749 0.783841
\(459\) 13.1333 0.613008
\(460\) 16.2837 0.759233
\(461\) 3.39457 0.158101 0.0790505 0.996871i \(-0.474811\pi\)
0.0790505 + 0.996871i \(0.474811\pi\)
\(462\) 0 0
\(463\) 18.5198 0.860690 0.430345 0.902665i \(-0.358392\pi\)
0.430345 + 0.902665i \(0.358392\pi\)
\(464\) 7.29532 0.338677
\(465\) −16.6265 −0.771034
\(466\) 25.7392 1.19234
\(467\) 29.7909 1.37856 0.689280 0.724495i \(-0.257927\pi\)
0.689280 + 0.724495i \(0.257927\pi\)
\(468\) −2.86027 −0.132216
\(469\) 3.85481 0.177999
\(470\) 36.6411 1.69013
\(471\) 55.0321 2.53574
\(472\) 7.53037 0.346613
\(473\) 0 0
\(474\) −6.75773 −0.310393
\(475\) 11.1866 0.513275
\(476\) −0.722264 −0.0331049
\(477\) 34.9800 1.60163
\(478\) 3.83627 0.175467
\(479\) −25.3380 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(480\) −11.2298 −0.512568
\(481\) 4.96213 0.226254
\(482\) −10.9064 −0.496771
\(483\) −3.10579 −0.141319
\(484\) 0 0
\(485\) 36.2470 1.64589
\(486\) 16.1682 0.733405
\(487\) −1.08718 −0.0492650 −0.0246325 0.999697i \(-0.507842\pi\)
−0.0246325 + 0.999697i \(0.507842\pi\)
\(488\) 4.83533 0.218885
\(489\) 47.2148 2.13513
\(490\) 27.8587 1.25853
\(491\) −17.1850 −0.775550 −0.387775 0.921754i \(-0.626756\pi\)
−0.387775 + 0.921754i \(0.626756\pi\)
\(492\) 34.4512 1.55318
\(493\) 19.1663 0.863209
\(494\) 0.597016 0.0268610
\(495\) 0 0
\(496\) −1.48056 −0.0664793
\(497\) 2.64421 0.118609
\(498\) 44.0868 1.97558
\(499\) −8.60896 −0.385390 −0.192695 0.981259i \(-0.561723\pi\)
−0.192695 + 0.981259i \(0.561723\pi\)
\(500\) 24.8901 1.11312
\(501\) 64.6033 2.88626
\(502\) −13.9463 −0.622454
\(503\) 7.29741 0.325376 0.162688 0.986678i \(-0.447984\pi\)
0.162688 + 0.986678i \(0.447984\pi\)
\(504\) 1.31711 0.0586688
\(505\) −9.29141 −0.413462
\(506\) 0 0
\(507\) −35.2911 −1.56733
\(508\) −5.59695 −0.248324
\(509\) 3.57411 0.158420 0.0792098 0.996858i \(-0.474760\pi\)
0.0792098 + 0.996858i \(0.474760\pi\)
\(510\) −29.5031 −1.30642
\(511\) −0.867705 −0.0383850
\(512\) −1.00000 −0.0441942
\(513\) 4.99894 0.220709
\(514\) −16.8311 −0.742388
\(515\) 41.6015 1.83318
\(516\) −15.5871 −0.686184
\(517\) 0 0
\(518\) −2.28498 −0.100396
\(519\) 65.5960 2.87935
\(520\) 2.40195 0.105332
\(521\) 20.7089 0.907275 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(522\) −34.9515 −1.52979
\(523\) −9.17755 −0.401306 −0.200653 0.979662i \(-0.564306\pi\)
−0.200653 + 0.979662i \(0.564306\pi\)
\(524\) 7.73690 0.337988
\(525\) −8.58406 −0.374639
\(526\) −1.71132 −0.0746171
\(527\) −3.88975 −0.169440
\(528\) 0 0
\(529\) −6.61852 −0.287762
\(530\) −29.3749 −1.27596
\(531\) −36.0776 −1.56563
\(532\) −0.274917 −0.0119191
\(533\) −7.36877 −0.319177
\(534\) −4.18324 −0.181027
\(535\) 36.2312 1.56641
\(536\) 14.0217 0.605647
\(537\) 21.0016 0.906286
\(538\) −22.3017 −0.961495
\(539\) 0 0
\(540\) 20.1120 0.865483
\(541\) 18.0117 0.774383 0.387191 0.921999i \(-0.373445\pi\)
0.387191 + 0.921999i \(0.373445\pi\)
\(542\) −2.26292 −0.0972008
\(543\) 54.7788 2.35078
\(544\) −2.62721 −0.112641
\(545\) −67.9908 −2.91240
\(546\) −0.458123 −0.0196059
\(547\) 18.2006 0.778200 0.389100 0.921196i \(-0.372786\pi\)
0.389100 + 0.921196i \(0.372786\pi\)
\(548\) −1.10706 −0.0472914
\(549\) −23.1658 −0.988692
\(550\) 0 0
\(551\) 7.29532 0.310791
\(552\) −11.2972 −0.480842
\(553\) −0.665590 −0.0283038
\(554\) 9.83937 0.418035
\(555\) −93.3371 −3.96194
\(556\) −14.8792 −0.631020
\(557\) 31.3931 1.33017 0.665083 0.746769i \(-0.268396\pi\)
0.665083 + 0.746769i \(0.268396\pi\)
\(558\) 7.09331 0.300284
\(559\) 3.33393 0.141010
\(560\) −1.10606 −0.0467395
\(561\) 0 0
\(562\) 17.9598 0.757590
\(563\) 11.9362 0.503051 0.251526 0.967851i \(-0.419068\pi\)
0.251526 + 0.967851i \(0.419068\pi\)
\(564\) −25.4206 −1.07040
\(565\) 11.4480 0.481621
\(566\) 26.5193 1.11469
\(567\) 0.115373 0.00484520
\(568\) 9.61823 0.403572
\(569\) 4.74045 0.198730 0.0993649 0.995051i \(-0.468319\pi\)
0.0993649 + 0.995051i \(0.468319\pi\)
\(570\) −11.2298 −0.470365
\(571\) −10.4663 −0.438001 −0.219001 0.975725i \(-0.570280\pi\)
−0.219001 + 0.975725i \(0.570280\pi\)
\(572\) 0 0
\(573\) 58.4265 2.44080
\(574\) 3.39320 0.141630
\(575\) 45.2765 1.88816
\(576\) 4.79095 0.199623
\(577\) −27.8884 −1.16101 −0.580504 0.814258i \(-0.697144\pi\)
−0.580504 + 0.814258i \(0.697144\pi\)
\(578\) 10.0978 0.420012
\(579\) −5.30113 −0.220308
\(580\) 29.3509 1.21873
\(581\) 4.34225 0.180147
\(582\) −25.1472 −1.04239
\(583\) 0 0
\(584\) −3.15625 −0.130606
\(585\) −11.5076 −0.475781
\(586\) −12.3885 −0.511765
\(587\) −0.355712 −0.0146818 −0.00734089 0.999973i \(-0.502337\pi\)
−0.00734089 + 0.999973i \(0.502337\pi\)
\(588\) −19.3276 −0.797058
\(589\) −1.48056 −0.0610056
\(590\) 30.2966 1.24729
\(591\) 3.91783 0.161158
\(592\) −8.31155 −0.341603
\(593\) 34.4326 1.41398 0.706990 0.707224i \(-0.250053\pi\)
0.706990 + 0.707224i \(0.250053\pi\)
\(594\) 0 0
\(595\) −2.90585 −0.119128
\(596\) −20.9576 −0.858457
\(597\) −10.0597 −0.411716
\(598\) 2.41637 0.0988126
\(599\) 9.20766 0.376215 0.188107 0.982148i \(-0.439765\pi\)
0.188107 + 0.982148i \(0.439765\pi\)
\(600\) −31.2242 −1.27472
\(601\) 40.2925 1.64357 0.821783 0.569801i \(-0.192980\pi\)
0.821783 + 0.569801i \(0.192980\pi\)
\(602\) −1.53522 −0.0625710
\(603\) −67.1774 −2.73568
\(604\) 4.07961 0.165997
\(605\) 0 0
\(606\) 6.44614 0.261856
\(607\) −21.8513 −0.886916 −0.443458 0.896295i \(-0.646249\pi\)
−0.443458 + 0.896295i \(0.646249\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −5.59810 −0.226847
\(610\) 19.4537 0.787659
\(611\) 5.43723 0.219967
\(612\) 12.5868 0.508792
\(613\) −18.0805 −0.730264 −0.365132 0.930956i \(-0.618976\pi\)
−0.365132 + 0.930956i \(0.618976\pi\)
\(614\) −29.2289 −1.17958
\(615\) 138.606 5.58912
\(616\) 0 0
\(617\) −45.2047 −1.81987 −0.909936 0.414749i \(-0.863870\pi\)
−0.909936 + 0.414749i \(0.863870\pi\)
\(618\) −28.8620 −1.16100
\(619\) 18.8902 0.759263 0.379631 0.925138i \(-0.376051\pi\)
0.379631 + 0.925138i \(0.376051\pi\)
\(620\) −5.95669 −0.239226
\(621\) 20.2327 0.811912
\(622\) −10.9986 −0.441004
\(623\) −0.412021 −0.0165073
\(624\) −1.66641 −0.0667097
\(625\) 44.2063 1.76825
\(626\) −15.5213 −0.620355
\(627\) 0 0
\(628\) 19.7161 0.786758
\(629\) −21.8362 −0.870666
\(630\) 5.29907 0.211120
\(631\) −6.89416 −0.274452 −0.137226 0.990540i \(-0.543819\pi\)
−0.137226 + 0.990540i \(0.543819\pi\)
\(632\) −2.42106 −0.0963046
\(633\) −52.3955 −2.08253
\(634\) 23.9485 0.951116
\(635\) −22.5179 −0.893597
\(636\) 20.3795 0.808100
\(637\) 4.13399 0.163795
\(638\) 0 0
\(639\) −46.0804 −1.82291
\(640\) −4.02325 −0.159033
\(641\) −15.7499 −0.622085 −0.311043 0.950396i \(-0.600678\pi\)
−0.311043 + 0.950396i \(0.600678\pi\)
\(642\) −25.1362 −0.992048
\(643\) 40.9900 1.61649 0.808245 0.588847i \(-0.200418\pi\)
0.808245 + 0.588847i \(0.200418\pi\)
\(644\) −1.11270 −0.0438465
\(645\) −62.7108 −2.46924
\(646\) −2.62721 −0.103366
\(647\) −5.82684 −0.229077 −0.114538 0.993419i \(-0.536539\pi\)
−0.114538 + 0.993419i \(0.536539\pi\)
\(648\) 0.419665 0.0164860
\(649\) 0 0
\(650\) 6.67856 0.261955
\(651\) 1.13612 0.0445280
\(652\) 16.9154 0.662460
\(653\) 18.8227 0.736591 0.368295 0.929709i \(-0.379942\pi\)
0.368295 + 0.929709i \(0.379942\pi\)
\(654\) 47.1702 1.84450
\(655\) 31.1275 1.21625
\(656\) 12.3427 0.481900
\(657\) 15.1214 0.589943
\(658\) −2.50376 −0.0976066
\(659\) −29.6469 −1.15488 −0.577440 0.816433i \(-0.695948\pi\)
−0.577440 + 0.816433i \(0.695948\pi\)
\(660\) 0 0
\(661\) 26.1657 1.01773 0.508864 0.860847i \(-0.330066\pi\)
0.508864 + 0.860847i \(0.330066\pi\)
\(662\) −25.6584 −0.997241
\(663\) −4.37800 −0.170028
\(664\) 15.7948 0.612957
\(665\) −1.10606 −0.0428911
\(666\) 39.8202 1.54300
\(667\) 29.5271 1.14329
\(668\) 23.1451 0.895512
\(669\) −40.9441 −1.58299
\(670\) 56.4130 2.17942
\(671\) 0 0
\(672\) 0.767355 0.0296014
\(673\) −30.9303 −1.19228 −0.596138 0.802882i \(-0.703299\pi\)
−0.596138 + 0.802882i \(0.703299\pi\)
\(674\) −2.94204 −0.113323
\(675\) 55.9210 2.15240
\(676\) −12.6436 −0.486291
\(677\) 21.2288 0.815888 0.407944 0.913007i \(-0.366246\pi\)
0.407944 + 0.913007i \(0.366246\pi\)
\(678\) −7.94233 −0.305023
\(679\) −2.47683 −0.0950520
\(680\) −10.5699 −0.405338
\(681\) −9.89271 −0.379090
\(682\) 0 0
\(683\) −8.46842 −0.324035 −0.162017 0.986788i \(-0.551800\pi\)
−0.162017 + 0.986788i \(0.551800\pi\)
\(684\) 4.79095 0.183186
\(685\) −4.45400 −0.170179
\(686\) −3.82806 −0.146156
\(687\) −46.8226 −1.78639
\(688\) −5.58432 −0.212900
\(689\) −4.35898 −0.166064
\(690\) −45.4516 −1.73031
\(691\) 0.920788 0.0350284 0.0175142 0.999847i \(-0.494425\pi\)
0.0175142 + 0.999847i \(0.494425\pi\)
\(692\) 23.5008 0.893366
\(693\) 0 0
\(694\) −6.33884 −0.240619
\(695\) −59.8629 −2.27073
\(696\) −20.3629 −0.771854
\(697\) 32.4268 1.22825
\(698\) −10.5789 −0.400418
\(699\) −71.8439 −2.71738
\(700\) −3.07537 −0.116238
\(701\) 10.4383 0.394249 0.197124 0.980379i \(-0.436840\pi\)
0.197124 + 0.980379i \(0.436840\pi\)
\(702\) 2.98445 0.112641
\(703\) −8.31155 −0.313476
\(704\) 0 0
\(705\) −102.274 −3.85185
\(706\) 35.1642 1.32342
\(707\) 0.634900 0.0238779
\(708\) −21.0190 −0.789941
\(709\) −13.2648 −0.498170 −0.249085 0.968482i \(-0.580130\pi\)
−0.249085 + 0.968482i \(0.580130\pi\)
\(710\) 38.6966 1.45226
\(711\) 11.5992 0.435003
\(712\) −1.49871 −0.0561666
\(713\) −5.99244 −0.224419
\(714\) 2.01600 0.0754470
\(715\) 0 0
\(716\) 7.52415 0.281191
\(717\) −10.7079 −0.399894
\(718\) 24.1939 0.902909
\(719\) −44.0301 −1.64205 −0.821023 0.570895i \(-0.806596\pi\)
−0.821023 + 0.570895i \(0.806596\pi\)
\(720\) 19.2752 0.718344
\(721\) −2.84271 −0.105868
\(722\) −1.00000 −0.0372161
\(723\) 30.4421 1.13215
\(724\) 19.6253 0.729370
\(725\) 81.6096 3.03090
\(726\) 0 0
\(727\) −14.8909 −0.552271 −0.276136 0.961119i \(-0.589054\pi\)
−0.276136 + 0.961119i \(0.589054\pi\)
\(728\) −0.164130 −0.00608305
\(729\) −43.8701 −1.62482
\(730\) −12.6984 −0.469988
\(731\) −14.6712 −0.542633
\(732\) −13.4965 −0.498845
\(733\) −19.1078 −0.705763 −0.352882 0.935668i \(-0.614798\pi\)
−0.352882 + 0.935668i \(0.614798\pi\)
\(734\) 23.2519 0.858242
\(735\) −77.7599 −2.86822
\(736\) −4.04740 −0.149189
\(737\) 0 0
\(738\) −59.1330 −2.17672
\(739\) 18.8804 0.694527 0.347264 0.937768i \(-0.387111\pi\)
0.347264 + 0.937768i \(0.387111\pi\)
\(740\) −33.4395 −1.22926
\(741\) −1.66641 −0.0612170
\(742\) 2.00724 0.0736882
\(743\) −27.1802 −0.997147 −0.498573 0.866848i \(-0.666143\pi\)
−0.498573 + 0.866848i \(0.666143\pi\)
\(744\) 4.13259 0.151508
\(745\) −84.3178 −3.08916
\(746\) 8.15845 0.298702
\(747\) −75.6720 −2.76869
\(748\) 0 0
\(749\) −2.47575 −0.0904618
\(750\) −69.4739 −2.53683
\(751\) 12.7321 0.464601 0.232300 0.972644i \(-0.425375\pi\)
0.232300 + 0.972644i \(0.425375\pi\)
\(752\) −9.10733 −0.332110
\(753\) 38.9273 1.41859
\(754\) 4.35543 0.158615
\(755\) 16.4133 0.597341
\(756\) −1.37429 −0.0499825
\(757\) 36.4140 1.32349 0.661745 0.749729i \(-0.269816\pi\)
0.661745 + 0.749729i \(0.269816\pi\)
\(758\) 18.9997 0.690102
\(759\) 0 0
\(760\) −4.02325 −0.145939
\(761\) −19.2127 −0.696461 −0.348230 0.937409i \(-0.613217\pi\)
−0.348230 + 0.937409i \(0.613217\pi\)
\(762\) 15.6224 0.565938
\(763\) 4.64594 0.168194
\(764\) 20.9322 0.757300
\(765\) 50.6400 1.83089
\(766\) −12.1817 −0.440143
\(767\) 4.49575 0.162332
\(768\) 2.79123 0.100720
\(769\) 15.0316 0.542052 0.271026 0.962572i \(-0.412637\pi\)
0.271026 + 0.962572i \(0.412637\pi\)
\(770\) 0 0
\(771\) 46.9794 1.69192
\(772\) −1.89921 −0.0683542
\(773\) −25.2426 −0.907914 −0.453957 0.891023i \(-0.649988\pi\)
−0.453957 + 0.891023i \(0.649988\pi\)
\(774\) 26.7542 0.961659
\(775\) −16.5624 −0.594940
\(776\) −9.00939 −0.323418
\(777\) 6.37791 0.228806
\(778\) 24.1433 0.865579
\(779\) 12.3427 0.442222
\(780\) −6.70438 −0.240055
\(781\) 0 0
\(782\) −10.6334 −0.380249
\(783\) 36.4689 1.30329
\(784\) −6.92442 −0.247301
\(785\) 79.3228 2.83115
\(786\) −21.5954 −0.770284
\(787\) 32.2538 1.14972 0.574862 0.818250i \(-0.305056\pi\)
0.574862 + 0.818250i \(0.305056\pi\)
\(788\) 1.40362 0.0500020
\(789\) 4.77668 0.170054
\(790\) −9.74054 −0.346553
\(791\) −0.782265 −0.0278141
\(792\) 0 0
\(793\) 2.88677 0.102512
\(794\) 10.1918 0.361692
\(795\) 81.9919 2.90795
\(796\) −3.60404 −0.127742
\(797\) −17.7608 −0.629121 −0.314561 0.949237i \(-0.601857\pi\)
−0.314561 + 0.949237i \(0.601857\pi\)
\(798\) 0.767355 0.0271641
\(799\) −23.9269 −0.846472
\(800\) −11.1866 −0.395505
\(801\) 7.18025 0.253701
\(802\) −12.2313 −0.431901
\(803\) 0 0
\(804\) −39.1379 −1.38029
\(805\) −4.47667 −0.157782
\(806\) −0.883921 −0.0311348
\(807\) 62.2492 2.19127
\(808\) 2.30943 0.0812454
\(809\) −21.3809 −0.751714 −0.375857 0.926678i \(-0.622652\pi\)
−0.375857 + 0.926678i \(0.622652\pi\)
\(810\) 1.68842 0.0593249
\(811\) 9.48316 0.332999 0.166499 0.986042i \(-0.446754\pi\)
0.166499 + 0.986042i \(0.446754\pi\)
\(812\) −2.00561 −0.0703830
\(813\) 6.31633 0.221523
\(814\) 0 0
\(815\) 68.0551 2.38387
\(816\) 7.33314 0.256711
\(817\) −5.58432 −0.195371
\(818\) −27.8419 −0.973470
\(819\) 0.786337 0.0274768
\(820\) 49.6576 1.73412
\(821\) 3.84674 0.134252 0.0671261 0.997744i \(-0.478617\pi\)
0.0671261 + 0.997744i \(0.478617\pi\)
\(822\) 3.09007 0.107778
\(823\) −15.9227 −0.555031 −0.277516 0.960721i \(-0.589511\pi\)
−0.277516 + 0.960721i \(0.589511\pi\)
\(824\) −10.3403 −0.360220
\(825\) 0 0
\(826\) −2.07022 −0.0720323
\(827\) −21.3561 −0.742624 −0.371312 0.928508i \(-0.621092\pi\)
−0.371312 + 0.928508i \(0.621092\pi\)
\(828\) 19.3909 0.673880
\(829\) 22.3416 0.775957 0.387978 0.921668i \(-0.373174\pi\)
0.387978 + 0.921668i \(0.373174\pi\)
\(830\) 63.5464 2.20573
\(831\) −27.4639 −0.952712
\(832\) −0.597016 −0.0206978
\(833\) −18.1919 −0.630312
\(834\) 41.5313 1.43811
\(835\) 93.1187 3.22250
\(836\) 0 0
\(837\) −7.40126 −0.255825
\(838\) −2.74637 −0.0948719
\(839\) 0.807131 0.0278653 0.0139326 0.999903i \(-0.495565\pi\)
0.0139326 + 0.999903i \(0.495565\pi\)
\(840\) 3.08726 0.106521
\(841\) 24.2217 0.835233
\(842\) −3.45765 −0.119159
\(843\) −50.1300 −1.72657
\(844\) −18.7715 −0.646141
\(845\) −50.8683 −1.74992
\(846\) 43.6328 1.50013
\(847\) 0 0
\(848\) 7.30127 0.250727
\(849\) −74.0215 −2.54041
\(850\) −29.3894 −1.00805
\(851\) −33.6402 −1.15317
\(852\) −26.8467 −0.919752
\(853\) 35.6981 1.22228 0.611139 0.791523i \(-0.290711\pi\)
0.611139 + 0.791523i \(0.290711\pi\)
\(854\) −1.32931 −0.0454881
\(855\) 19.2752 0.659198
\(856\) −9.00544 −0.307800
\(857\) −53.2489 −1.81895 −0.909474 0.415760i \(-0.863516\pi\)
−0.909474 + 0.415760i \(0.863516\pi\)
\(858\) 0 0
\(859\) −57.5833 −1.96472 −0.982358 0.187009i \(-0.940121\pi\)
−0.982358 + 0.187009i \(0.940121\pi\)
\(860\) −22.4671 −0.766123
\(861\) −9.47120 −0.322778
\(862\) 33.1455 1.12894
\(863\) −41.2077 −1.40273 −0.701364 0.712804i \(-0.747425\pi\)
−0.701364 + 0.712804i \(0.747425\pi\)
\(864\) −4.99894 −0.170067
\(865\) 94.5496 3.21478
\(866\) 17.0841 0.580540
\(867\) −28.1852 −0.957218
\(868\) 0.407032 0.0138156
\(869\) 0 0
\(870\) −81.9251 −2.77752
\(871\) 8.37121 0.283647
\(872\) 16.8995 0.572288
\(873\) 43.1635 1.46086
\(874\) −4.04740 −0.136905
\(875\) −6.84270 −0.231326
\(876\) 8.80980 0.297656
\(877\) 29.3504 0.991092 0.495546 0.868582i \(-0.334968\pi\)
0.495546 + 0.868582i \(0.334968\pi\)
\(878\) 36.1437 1.21979
\(879\) 34.5792 1.16633
\(880\) 0 0
\(881\) 1.36463 0.0459757 0.0229878 0.999736i \(-0.492682\pi\)
0.0229878 + 0.999736i \(0.492682\pi\)
\(882\) 33.1745 1.11704
\(883\) 11.8574 0.399034 0.199517 0.979894i \(-0.436063\pi\)
0.199517 + 0.979894i \(0.436063\pi\)
\(884\) −1.56849 −0.0527539
\(885\) −84.5646 −2.84261
\(886\) −19.4900 −0.654778
\(887\) 28.7149 0.964152 0.482076 0.876129i \(-0.339883\pi\)
0.482076 + 0.876129i \(0.339883\pi\)
\(888\) 23.1994 0.778521
\(889\) 1.53869 0.0516061
\(890\) −6.02969 −0.202116
\(891\) 0 0
\(892\) −14.6688 −0.491149
\(893\) −9.10733 −0.304765
\(894\) 58.4974 1.95645
\(895\) 30.2716 1.01187
\(896\) 0.274917 0.00918432
\(897\) −6.74463 −0.225196
\(898\) 12.8197 0.427798
\(899\) −10.8012 −0.360240
\(900\) 53.5942 1.78647
\(901\) 19.1820 0.639044
\(902\) 0 0
\(903\) 4.28515 0.142601
\(904\) −2.84546 −0.0946386
\(905\) 78.9577 2.62464
\(906\) −11.3871 −0.378312
\(907\) 20.8425 0.692063 0.346031 0.938223i \(-0.387529\pi\)
0.346031 + 0.938223i \(0.387529\pi\)
\(908\) −3.54422 −0.117619
\(909\) −11.0644 −0.366981
\(910\) −0.660335 −0.0218899
\(911\) −33.3467 −1.10483 −0.552413 0.833571i \(-0.686293\pi\)
−0.552413 + 0.833571i \(0.686293\pi\)
\(912\) 2.79123 0.0924268
\(913\) 0 0
\(914\) 2.20055 0.0727877
\(915\) −54.2998 −1.79510
\(916\) −16.7749 −0.554259
\(917\) −2.12700 −0.0702398
\(918\) −13.1333 −0.433462
\(919\) 15.9480 0.526076 0.263038 0.964786i \(-0.415276\pi\)
0.263038 + 0.964786i \(0.415276\pi\)
\(920\) −16.2837 −0.536859
\(921\) 81.5845 2.68830
\(922\) −3.39457 −0.111794
\(923\) 5.74224 0.189008
\(924\) 0 0
\(925\) −92.9776 −3.05709
\(926\) −18.5198 −0.608600
\(927\) 49.5397 1.62710
\(928\) −7.29532 −0.239481
\(929\) −34.2521 −1.12377 −0.561887 0.827214i \(-0.689924\pi\)
−0.561887 + 0.827214i \(0.689924\pi\)
\(930\) 16.6265 0.545203
\(931\) −6.92442 −0.226939
\(932\) −25.7392 −0.843115
\(933\) 30.6996 1.00506
\(934\) −29.7909 −0.974789
\(935\) 0 0
\(936\) 2.86027 0.0934910
\(937\) −32.5123 −1.06213 −0.531065 0.847331i \(-0.678208\pi\)
−0.531065 + 0.847331i \(0.678208\pi\)
\(938\) −3.85481 −0.125864
\(939\) 43.3234 1.41381
\(940\) −36.6411 −1.19510
\(941\) −12.8788 −0.419838 −0.209919 0.977719i \(-0.567320\pi\)
−0.209919 + 0.977719i \(0.567320\pi\)
\(942\) −55.0321 −1.79304
\(943\) 49.9557 1.62678
\(944\) −7.53037 −0.245093
\(945\) −5.52912 −0.179862
\(946\) 0 0
\(947\) −27.4064 −0.890589 −0.445295 0.895384i \(-0.646901\pi\)
−0.445295 + 0.895384i \(0.646901\pi\)
\(948\) 6.75773 0.219481
\(949\) −1.88433 −0.0611680
\(950\) −11.1866 −0.362940
\(951\) −66.8457 −2.16762
\(952\) 0.722264 0.0234087
\(953\) −13.5962 −0.440423 −0.220212 0.975452i \(-0.570675\pi\)
−0.220212 + 0.975452i \(0.570675\pi\)
\(954\) −34.9800 −1.13252
\(955\) 84.2155 2.72515
\(956\) −3.83627 −0.124074
\(957\) 0 0
\(958\) 25.3380 0.818632
\(959\) 0.304350 0.00982798
\(960\) 11.2298 0.362441
\(961\) −28.8079 −0.929288
\(962\) −4.96213 −0.159985
\(963\) 43.1446 1.39031
\(964\) 10.9064 0.351270
\(965\) −7.64101 −0.245973
\(966\) 3.10579 0.0999273
\(967\) 19.4593 0.625768 0.312884 0.949791i \(-0.398705\pi\)
0.312884 + 0.949791i \(0.398705\pi\)
\(968\) 0 0
\(969\) 7.33314 0.235574
\(970\) −36.2470 −1.16382
\(971\) −54.0585 −1.73482 −0.867410 0.497594i \(-0.834217\pi\)
−0.867410 + 0.497594i \(0.834217\pi\)
\(972\) −16.1682 −0.518595
\(973\) 4.09055 0.131137
\(974\) 1.08718 0.0348356
\(975\) −18.6414 −0.597002
\(976\) −4.83533 −0.154775
\(977\) −34.0503 −1.08937 −0.544684 0.838642i \(-0.683350\pi\)
−0.544684 + 0.838642i \(0.683350\pi\)
\(978\) −47.2148 −1.50976
\(979\) 0 0
\(980\) −27.8587 −0.889913
\(981\) −80.9644 −2.58499
\(982\) 17.1850 0.548397
\(983\) 51.6148 1.64626 0.823129 0.567855i \(-0.192226\pi\)
0.823129 + 0.567855i \(0.192226\pi\)
\(984\) −34.4512 −1.09826
\(985\) 5.64713 0.179932
\(986\) −19.1663 −0.610381
\(987\) 6.98856 0.222448
\(988\) −0.597016 −0.0189936
\(989\) −22.6020 −0.718702
\(990\) 0 0
\(991\) 7.22913 0.229641 0.114820 0.993386i \(-0.463371\pi\)
0.114820 + 0.993386i \(0.463371\pi\)
\(992\) 1.48056 0.0470080
\(993\) 71.6183 2.27274
\(994\) −2.64421 −0.0838693
\(995\) −14.5000 −0.459679
\(996\) −44.0868 −1.39695
\(997\) −40.4932 −1.28243 −0.641216 0.767360i \(-0.721570\pi\)
−0.641216 + 0.767360i \(0.721570\pi\)
\(998\) 8.60896 0.272512
\(999\) −41.5489 −1.31455
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.by.1.6 8
11.10 odd 2 4598.2.a.cb.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.6 8 1.1 even 1 trivial
4598.2.a.cb.1.6 yes 8 11.10 odd 2