Properties

Label 2151.2.a.j.1.20
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Root \(-2.66148\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.66148 q^{2} +5.08345 q^{4} -2.46126 q^{5} -5.08484 q^{7} +8.20653 q^{8} +O(q^{10})\) \(q+2.66148 q^{2} +5.08345 q^{4} -2.46126 q^{5} -5.08484 q^{7} +8.20653 q^{8} -6.55057 q^{10} -0.841814 q^{11} -4.31476 q^{13} -13.5332 q^{14} +11.6746 q^{16} -0.607296 q^{17} -7.12670 q^{19} -12.5117 q^{20} -2.24047 q^{22} +6.45037 q^{23} +1.05778 q^{25} -11.4836 q^{26} -25.8485 q^{28} -0.953320 q^{29} +1.79400 q^{31} +14.6586 q^{32} -1.61630 q^{34} +12.5151 q^{35} -4.38489 q^{37} -18.9675 q^{38} -20.1984 q^{40} -9.65928 q^{41} +5.60860 q^{43} -4.27932 q^{44} +17.1675 q^{46} -5.73361 q^{47} +18.8556 q^{49} +2.81525 q^{50} -21.9339 q^{52} +0.509397 q^{53} +2.07192 q^{55} -41.7289 q^{56} -2.53724 q^{58} -10.9292 q^{59} +10.9561 q^{61} +4.77468 q^{62} +15.6642 q^{64} +10.6197 q^{65} +2.55272 q^{67} -3.08716 q^{68} +33.3086 q^{70} +4.53464 q^{71} +8.87319 q^{73} -11.6703 q^{74} -36.2283 q^{76} +4.28048 q^{77} -1.69076 q^{79} -28.7341 q^{80} -25.7079 q^{82} -10.0976 q^{83} +1.49471 q^{85} +14.9271 q^{86} -6.90837 q^{88} -13.6428 q^{89} +21.9399 q^{91} +32.7901 q^{92} -15.2599 q^{94} +17.5406 q^{95} +16.1354 q^{97} +50.1836 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66148 1.88195 0.940974 0.338480i \(-0.109913\pi\)
0.940974 + 0.338480i \(0.109913\pi\)
\(3\) 0 0
\(4\) 5.08345 2.54173
\(5\) −2.46126 −1.10071 −0.550353 0.834932i \(-0.685507\pi\)
−0.550353 + 0.834932i \(0.685507\pi\)
\(6\) 0 0
\(7\) −5.08484 −1.92189 −0.960944 0.276744i \(-0.910744\pi\)
−0.960944 + 0.276744i \(0.910744\pi\)
\(8\) 8.20653 2.90145
\(9\) 0 0
\(10\) −6.55057 −2.07147
\(11\) −0.841814 −0.253816 −0.126908 0.991914i \(-0.540505\pi\)
−0.126908 + 0.991914i \(0.540505\pi\)
\(12\) 0 0
\(13\) −4.31476 −1.19670 −0.598350 0.801235i \(-0.704177\pi\)
−0.598350 + 0.801235i \(0.704177\pi\)
\(14\) −13.5332 −3.61689
\(15\) 0 0
\(16\) 11.6746 2.91865
\(17\) −0.607296 −0.147291 −0.0736454 0.997284i \(-0.523463\pi\)
−0.0736454 + 0.997284i \(0.523463\pi\)
\(18\) 0 0
\(19\) −7.12670 −1.63498 −0.817489 0.575944i \(-0.804635\pi\)
−0.817489 + 0.575944i \(0.804635\pi\)
\(20\) −12.5117 −2.79770
\(21\) 0 0
\(22\) −2.24047 −0.477669
\(23\) 6.45037 1.34499 0.672497 0.740100i \(-0.265222\pi\)
0.672497 + 0.740100i \(0.265222\pi\)
\(24\) 0 0
\(25\) 1.05778 0.211555
\(26\) −11.4836 −2.25213
\(27\) 0 0
\(28\) −25.8485 −4.88491
\(29\) −0.953320 −0.177027 −0.0885136 0.996075i \(-0.528212\pi\)
−0.0885136 + 0.996075i \(0.528212\pi\)
\(30\) 0 0
\(31\) 1.79400 0.322212 0.161106 0.986937i \(-0.448494\pi\)
0.161106 + 0.986937i \(0.448494\pi\)
\(32\) 14.6586 2.59129
\(33\) 0 0
\(34\) −1.61630 −0.277194
\(35\) 12.5151 2.11543
\(36\) 0 0
\(37\) −4.38489 −0.720872 −0.360436 0.932784i \(-0.617372\pi\)
−0.360436 + 0.932784i \(0.617372\pi\)
\(38\) −18.9675 −3.07694
\(39\) 0 0
\(40\) −20.1984 −3.19364
\(41\) −9.65928 −1.50853 −0.754263 0.656572i \(-0.772006\pi\)
−0.754263 + 0.656572i \(0.772006\pi\)
\(42\) 0 0
\(43\) 5.60860 0.855303 0.427652 0.903944i \(-0.359341\pi\)
0.427652 + 0.903944i \(0.359341\pi\)
\(44\) −4.27932 −0.645132
\(45\) 0 0
\(46\) 17.1675 2.53121
\(47\) −5.73361 −0.836332 −0.418166 0.908371i \(-0.637327\pi\)
−0.418166 + 0.908371i \(0.637327\pi\)
\(48\) 0 0
\(49\) 18.8556 2.69365
\(50\) 2.81525 0.398136
\(51\) 0 0
\(52\) −21.9339 −3.04168
\(53\) 0.509397 0.0699711 0.0349855 0.999388i \(-0.488861\pi\)
0.0349855 + 0.999388i \(0.488861\pi\)
\(54\) 0 0
\(55\) 2.07192 0.279377
\(56\) −41.7289 −5.57626
\(57\) 0 0
\(58\) −2.53724 −0.333156
\(59\) −10.9292 −1.42286 −0.711432 0.702755i \(-0.751953\pi\)
−0.711432 + 0.702755i \(0.751953\pi\)
\(60\) 0 0
\(61\) 10.9561 1.40278 0.701390 0.712778i \(-0.252563\pi\)
0.701390 + 0.712778i \(0.252563\pi\)
\(62\) 4.77468 0.606385
\(63\) 0 0
\(64\) 15.6642 1.95803
\(65\) 10.6197 1.31722
\(66\) 0 0
\(67\) 2.55272 0.311864 0.155932 0.987768i \(-0.450162\pi\)
0.155932 + 0.987768i \(0.450162\pi\)
\(68\) −3.08716 −0.374373
\(69\) 0 0
\(70\) 33.3086 3.98114
\(71\) 4.53464 0.538162 0.269081 0.963117i \(-0.413280\pi\)
0.269081 + 0.963117i \(0.413280\pi\)
\(72\) 0 0
\(73\) 8.87319 1.03853 0.519264 0.854614i \(-0.326206\pi\)
0.519264 + 0.854614i \(0.326206\pi\)
\(74\) −11.6703 −1.35664
\(75\) 0 0
\(76\) −36.2283 −4.15567
\(77\) 4.28048 0.487806
\(78\) 0 0
\(79\) −1.69076 −0.190226 −0.0951128 0.995466i \(-0.530321\pi\)
−0.0951128 + 0.995466i \(0.530321\pi\)
\(80\) −28.7341 −3.21257
\(81\) 0 0
\(82\) −25.7079 −2.83897
\(83\) −10.0976 −1.10836 −0.554179 0.832398i \(-0.686968\pi\)
−0.554179 + 0.832398i \(0.686968\pi\)
\(84\) 0 0
\(85\) 1.49471 0.162124
\(86\) 14.9271 1.60964
\(87\) 0 0
\(88\) −6.90837 −0.736435
\(89\) −13.6428 −1.44614 −0.723068 0.690777i \(-0.757269\pi\)
−0.723068 + 0.690777i \(0.757269\pi\)
\(90\) 0 0
\(91\) 21.9399 2.29992
\(92\) 32.7901 3.41861
\(93\) 0 0
\(94\) −15.2599 −1.57393
\(95\) 17.5406 1.79963
\(96\) 0 0
\(97\) 16.1354 1.63830 0.819149 0.573581i \(-0.194446\pi\)
0.819149 + 0.573581i \(0.194446\pi\)
\(98\) 50.1836 5.06931
\(99\) 0 0
\(100\) 5.37716 0.537716
\(101\) 12.4874 1.24254 0.621272 0.783595i \(-0.286616\pi\)
0.621272 + 0.783595i \(0.286616\pi\)
\(102\) 0 0
\(103\) −12.4503 −1.22677 −0.613384 0.789785i \(-0.710192\pi\)
−0.613384 + 0.789785i \(0.710192\pi\)
\(104\) −35.4093 −3.47216
\(105\) 0 0
\(106\) 1.35575 0.131682
\(107\) 3.48642 0.337045 0.168522 0.985698i \(-0.446100\pi\)
0.168522 + 0.985698i \(0.446100\pi\)
\(108\) 0 0
\(109\) −6.59512 −0.631698 −0.315849 0.948809i \(-0.602289\pi\)
−0.315849 + 0.948809i \(0.602289\pi\)
\(110\) 5.51436 0.525773
\(111\) 0 0
\(112\) −59.3633 −5.60931
\(113\) −14.2622 −1.34168 −0.670838 0.741604i \(-0.734066\pi\)
−0.670838 + 0.741604i \(0.734066\pi\)
\(114\) 0 0
\(115\) −15.8760 −1.48044
\(116\) −4.84616 −0.449955
\(117\) 0 0
\(118\) −29.0879 −2.67776
\(119\) 3.08800 0.283076
\(120\) 0 0
\(121\) −10.2913 −0.935577
\(122\) 29.1593 2.63996
\(123\) 0 0
\(124\) 9.11970 0.818974
\(125\) 9.70282 0.867846
\(126\) 0 0
\(127\) 16.2640 1.44320 0.721599 0.692311i \(-0.243407\pi\)
0.721599 + 0.692311i \(0.243407\pi\)
\(128\) 12.3728 1.09361
\(129\) 0 0
\(130\) 28.2642 2.47893
\(131\) 21.0513 1.83926 0.919631 0.392783i \(-0.128488\pi\)
0.919631 + 0.392783i \(0.128488\pi\)
\(132\) 0 0
\(133\) 36.2381 3.14224
\(134\) 6.79399 0.586912
\(135\) 0 0
\(136\) −4.98379 −0.427357
\(137\) −14.5332 −1.24166 −0.620830 0.783946i \(-0.713204\pi\)
−0.620830 + 0.783946i \(0.713204\pi\)
\(138\) 0 0
\(139\) −7.13395 −0.605094 −0.302547 0.953134i \(-0.597837\pi\)
−0.302547 + 0.953134i \(0.597837\pi\)
\(140\) 63.6198 5.37685
\(141\) 0 0
\(142\) 12.0688 1.01279
\(143\) 3.63223 0.303742
\(144\) 0 0
\(145\) 2.34636 0.194855
\(146\) 23.6158 1.95446
\(147\) 0 0
\(148\) −22.2904 −1.83226
\(149\) −5.29154 −0.433500 −0.216750 0.976227i \(-0.569546\pi\)
−0.216750 + 0.976227i \(0.569546\pi\)
\(150\) 0 0
\(151\) −0.990912 −0.0806393 −0.0403196 0.999187i \(-0.512838\pi\)
−0.0403196 + 0.999187i \(0.512838\pi\)
\(152\) −58.4855 −4.74380
\(153\) 0 0
\(154\) 11.3924 0.918026
\(155\) −4.41549 −0.354660
\(156\) 0 0
\(157\) −6.80762 −0.543307 −0.271654 0.962395i \(-0.587570\pi\)
−0.271654 + 0.962395i \(0.587570\pi\)
\(158\) −4.49992 −0.357995
\(159\) 0 0
\(160\) −36.0784 −2.85225
\(161\) −32.7990 −2.58493
\(162\) 0 0
\(163\) 7.21737 0.565308 0.282654 0.959222i \(-0.408785\pi\)
0.282654 + 0.959222i \(0.408785\pi\)
\(164\) −49.1025 −3.83426
\(165\) 0 0
\(166\) −26.8746 −2.08587
\(167\) −3.13902 −0.242904 −0.121452 0.992597i \(-0.538755\pi\)
−0.121452 + 0.992597i \(0.538755\pi\)
\(168\) 0 0
\(169\) 5.61719 0.432091
\(170\) 3.97813 0.305109
\(171\) 0 0
\(172\) 28.5110 2.17395
\(173\) 15.5000 1.17844 0.589222 0.807971i \(-0.299434\pi\)
0.589222 + 0.807971i \(0.299434\pi\)
\(174\) 0 0
\(175\) −5.37862 −0.406585
\(176\) −9.82782 −0.740800
\(177\) 0 0
\(178\) −36.3100 −2.72155
\(179\) −16.9058 −1.26360 −0.631799 0.775133i \(-0.717683\pi\)
−0.631799 + 0.775133i \(0.717683\pi\)
\(180\) 0 0
\(181\) −5.63874 −0.419125 −0.209562 0.977795i \(-0.567204\pi\)
−0.209562 + 0.977795i \(0.567204\pi\)
\(182\) 58.3924 4.32833
\(183\) 0 0
\(184\) 52.9351 3.90243
\(185\) 10.7923 0.793468
\(186\) 0 0
\(187\) 0.511230 0.0373848
\(188\) −29.1465 −2.12573
\(189\) 0 0
\(190\) 46.6840 3.38681
\(191\) −9.41280 −0.681087 −0.340543 0.940229i \(-0.610611\pi\)
−0.340543 + 0.940229i \(0.610611\pi\)
\(192\) 0 0
\(193\) −19.0149 −1.36873 −0.684363 0.729142i \(-0.739920\pi\)
−0.684363 + 0.729142i \(0.739920\pi\)
\(194\) 42.9439 3.08319
\(195\) 0 0
\(196\) 95.8513 6.84652
\(197\) −5.53908 −0.394643 −0.197321 0.980339i \(-0.563224\pi\)
−0.197321 + 0.980339i \(0.563224\pi\)
\(198\) 0 0
\(199\) −13.1741 −0.933884 −0.466942 0.884288i \(-0.654644\pi\)
−0.466942 + 0.884288i \(0.654644\pi\)
\(200\) 8.68068 0.613817
\(201\) 0 0
\(202\) 33.2350 2.33840
\(203\) 4.84748 0.340226
\(204\) 0 0
\(205\) 23.7740 1.66045
\(206\) −33.1363 −2.30871
\(207\) 0 0
\(208\) −50.3731 −3.49274
\(209\) 5.99936 0.414984
\(210\) 0 0
\(211\) 11.4645 0.789250 0.394625 0.918842i \(-0.370875\pi\)
0.394625 + 0.918842i \(0.370875\pi\)
\(212\) 2.58950 0.177847
\(213\) 0 0
\(214\) 9.27902 0.634301
\(215\) −13.8042 −0.941438
\(216\) 0 0
\(217\) −9.12219 −0.619254
\(218\) −17.5527 −1.18882
\(219\) 0 0
\(220\) 10.5325 0.710101
\(221\) 2.62034 0.176263
\(222\) 0 0
\(223\) 4.94910 0.331416 0.165708 0.986175i \(-0.447009\pi\)
0.165708 + 0.986175i \(0.447009\pi\)
\(224\) −74.5363 −4.98017
\(225\) 0 0
\(226\) −37.9586 −2.52497
\(227\) 27.3319 1.81408 0.907039 0.421046i \(-0.138337\pi\)
0.907039 + 0.421046i \(0.138337\pi\)
\(228\) 0 0
\(229\) 17.1182 1.13121 0.565603 0.824678i \(-0.308644\pi\)
0.565603 + 0.824678i \(0.308644\pi\)
\(230\) −42.2536 −2.78612
\(231\) 0 0
\(232\) −7.82346 −0.513635
\(233\) −5.09654 −0.333886 −0.166943 0.985967i \(-0.553390\pi\)
−0.166943 + 0.985967i \(0.553390\pi\)
\(234\) 0 0
\(235\) 14.1119 0.920557
\(236\) −55.5582 −3.61653
\(237\) 0 0
\(238\) 8.21863 0.532735
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −3.20240 −0.206285 −0.103142 0.994667i \(-0.532890\pi\)
−0.103142 + 0.994667i \(0.532890\pi\)
\(242\) −27.3902 −1.76071
\(243\) 0 0
\(244\) 55.6946 3.56548
\(245\) −46.4083 −2.96492
\(246\) 0 0
\(247\) 30.7500 1.95658
\(248\) 14.7225 0.934880
\(249\) 0 0
\(250\) 25.8238 1.63324
\(251\) 10.7171 0.676455 0.338228 0.941064i \(-0.390173\pi\)
0.338228 + 0.941064i \(0.390173\pi\)
\(252\) 0 0
\(253\) −5.43000 −0.341381
\(254\) 43.2863 2.71602
\(255\) 0 0
\(256\) 1.60154 0.100096
\(257\) −28.7630 −1.79418 −0.897092 0.441843i \(-0.854325\pi\)
−0.897092 + 0.441843i \(0.854325\pi\)
\(258\) 0 0
\(259\) 22.2964 1.38543
\(260\) 53.9849 3.34800
\(261\) 0 0
\(262\) 56.0276 3.46139
\(263\) −17.0337 −1.05034 −0.525170 0.850997i \(-0.675998\pi\)
−0.525170 + 0.850997i \(0.675998\pi\)
\(264\) 0 0
\(265\) −1.25376 −0.0770177
\(266\) 96.4469 5.91354
\(267\) 0 0
\(268\) 12.9766 0.792673
\(269\) 30.6001 1.86572 0.932861 0.360235i \(-0.117304\pi\)
0.932861 + 0.360235i \(0.117304\pi\)
\(270\) 0 0
\(271\) 30.8062 1.87134 0.935672 0.352871i \(-0.114795\pi\)
0.935672 + 0.352871i \(0.114795\pi\)
\(272\) −7.08992 −0.429890
\(273\) 0 0
\(274\) −38.6799 −2.33674
\(275\) −0.890451 −0.0536962
\(276\) 0 0
\(277\) 1.08078 0.0649377 0.0324688 0.999473i \(-0.489663\pi\)
0.0324688 + 0.999473i \(0.489663\pi\)
\(278\) −18.9868 −1.13875
\(279\) 0 0
\(280\) 102.705 6.13782
\(281\) −31.0662 −1.85325 −0.926627 0.375981i \(-0.877306\pi\)
−0.926627 + 0.375981i \(0.877306\pi\)
\(282\) 0 0
\(283\) −18.3082 −1.08831 −0.544154 0.838985i \(-0.683149\pi\)
−0.544154 + 0.838985i \(0.683149\pi\)
\(284\) 23.0516 1.36786
\(285\) 0 0
\(286\) 9.66708 0.571627
\(287\) 49.1159 2.89922
\(288\) 0 0
\(289\) −16.6312 −0.978305
\(290\) 6.24479 0.366707
\(291\) 0 0
\(292\) 45.1064 2.63965
\(293\) −13.8274 −0.807803 −0.403901 0.914802i \(-0.632346\pi\)
−0.403901 + 0.914802i \(0.632346\pi\)
\(294\) 0 0
\(295\) 26.8996 1.56616
\(296\) −35.9848 −2.09157
\(297\) 0 0
\(298\) −14.0833 −0.815824
\(299\) −27.8318 −1.60955
\(300\) 0 0
\(301\) −28.5188 −1.64380
\(302\) −2.63729 −0.151759
\(303\) 0 0
\(304\) −83.2013 −4.77192
\(305\) −26.9657 −1.54405
\(306\) 0 0
\(307\) 16.4835 0.940764 0.470382 0.882463i \(-0.344116\pi\)
0.470382 + 0.882463i \(0.344116\pi\)
\(308\) 21.7596 1.23987
\(309\) 0 0
\(310\) −11.7517 −0.667452
\(311\) 7.93968 0.450218 0.225109 0.974334i \(-0.427726\pi\)
0.225109 + 0.974334i \(0.427726\pi\)
\(312\) 0 0
\(313\) −24.0730 −1.36069 −0.680344 0.732893i \(-0.738170\pi\)
−0.680344 + 0.732893i \(0.738170\pi\)
\(314\) −18.1183 −1.02248
\(315\) 0 0
\(316\) −8.59491 −0.483502
\(317\) −2.33927 −0.131386 −0.0656932 0.997840i \(-0.520926\pi\)
−0.0656932 + 0.997840i \(0.520926\pi\)
\(318\) 0 0
\(319\) 0.802518 0.0449324
\(320\) −38.5536 −2.15521
\(321\) 0 0
\(322\) −87.2939 −4.86470
\(323\) 4.32802 0.240817
\(324\) 0 0
\(325\) −4.56406 −0.253168
\(326\) 19.2088 1.06388
\(327\) 0 0
\(328\) −79.2692 −4.37691
\(329\) 29.1544 1.60734
\(330\) 0 0
\(331\) −19.4068 −1.06669 −0.533347 0.845896i \(-0.679066\pi\)
−0.533347 + 0.845896i \(0.679066\pi\)
\(332\) −51.3308 −2.81714
\(333\) 0 0
\(334\) −8.35441 −0.457133
\(335\) −6.28289 −0.343271
\(336\) 0 0
\(337\) 3.36874 0.183507 0.0917534 0.995782i \(-0.470753\pi\)
0.0917534 + 0.995782i \(0.470753\pi\)
\(338\) 14.9500 0.813173
\(339\) 0 0
\(340\) 7.59828 0.412075
\(341\) −1.51021 −0.0817826
\(342\) 0 0
\(343\) −60.2835 −3.25500
\(344\) 46.0271 2.48162
\(345\) 0 0
\(346\) 41.2529 2.21777
\(347\) −14.5068 −0.778767 −0.389383 0.921076i \(-0.627312\pi\)
−0.389383 + 0.921076i \(0.627312\pi\)
\(348\) 0 0
\(349\) 7.09523 0.379799 0.189900 0.981804i \(-0.439184\pi\)
0.189900 + 0.981804i \(0.439184\pi\)
\(350\) −14.3151 −0.765172
\(351\) 0 0
\(352\) −12.3398 −0.657712
\(353\) 9.42842 0.501824 0.250912 0.968010i \(-0.419270\pi\)
0.250912 + 0.968010i \(0.419270\pi\)
\(354\) 0 0
\(355\) −11.1609 −0.592359
\(356\) −69.3526 −3.67568
\(357\) 0 0
\(358\) −44.9943 −2.37802
\(359\) 20.1249 1.06215 0.531077 0.847324i \(-0.321788\pi\)
0.531077 + 0.847324i \(0.321788\pi\)
\(360\) 0 0
\(361\) 31.7899 1.67315
\(362\) −15.0074 −0.788770
\(363\) 0 0
\(364\) 111.530 5.84577
\(365\) −21.8392 −1.14311
\(366\) 0 0
\(367\) −20.3115 −1.06025 −0.530126 0.847919i \(-0.677855\pi\)
−0.530126 + 0.847919i \(0.677855\pi\)
\(368\) 75.3053 3.92556
\(369\) 0 0
\(370\) 28.7235 1.49327
\(371\) −2.59020 −0.134477
\(372\) 0 0
\(373\) 14.0769 0.728875 0.364438 0.931228i \(-0.381261\pi\)
0.364438 + 0.931228i \(0.381261\pi\)
\(374\) 1.36063 0.0703562
\(375\) 0 0
\(376\) −47.0530 −2.42657
\(377\) 4.11335 0.211848
\(378\) 0 0
\(379\) −32.4329 −1.66597 −0.832983 0.553299i \(-0.813369\pi\)
−0.832983 + 0.553299i \(0.813369\pi\)
\(380\) 89.1670 4.57417
\(381\) 0 0
\(382\) −25.0520 −1.28177
\(383\) −28.8970 −1.47657 −0.738283 0.674491i \(-0.764363\pi\)
−0.738283 + 0.674491i \(0.764363\pi\)
\(384\) 0 0
\(385\) −10.5354 −0.536932
\(386\) −50.6078 −2.57587
\(387\) 0 0
\(388\) 82.0233 4.16410
\(389\) −26.1348 −1.32509 −0.662543 0.749024i \(-0.730523\pi\)
−0.662543 + 0.749024i \(0.730523\pi\)
\(390\) 0 0
\(391\) −3.91728 −0.198105
\(392\) 154.739 7.81549
\(393\) 0 0
\(394\) −14.7421 −0.742697
\(395\) 4.16140 0.209383
\(396\) 0 0
\(397\) −18.8466 −0.945882 −0.472941 0.881094i \(-0.656808\pi\)
−0.472941 + 0.881094i \(0.656808\pi\)
\(398\) −35.0624 −1.75752
\(399\) 0 0
\(400\) 12.3491 0.617455
\(401\) −28.8908 −1.44274 −0.721370 0.692550i \(-0.756487\pi\)
−0.721370 + 0.692550i \(0.756487\pi\)
\(402\) 0 0
\(403\) −7.74068 −0.385591
\(404\) 63.4792 3.15821
\(405\) 0 0
\(406\) 12.9014 0.640288
\(407\) 3.69126 0.182969
\(408\) 0 0
\(409\) −29.0306 −1.43547 −0.717736 0.696316i \(-0.754821\pi\)
−0.717736 + 0.696316i \(0.754821\pi\)
\(410\) 63.2738 3.12487
\(411\) 0 0
\(412\) −63.2907 −3.11811
\(413\) 55.5733 2.73459
\(414\) 0 0
\(415\) 24.8528 1.21998
\(416\) −63.2482 −3.10100
\(417\) 0 0
\(418\) 15.9671 0.780978
\(419\) −6.92669 −0.338391 −0.169195 0.985583i \(-0.554117\pi\)
−0.169195 + 0.985583i \(0.554117\pi\)
\(420\) 0 0
\(421\) 31.8865 1.55405 0.777027 0.629467i \(-0.216727\pi\)
0.777027 + 0.629467i \(0.216727\pi\)
\(422\) 30.5125 1.48533
\(423\) 0 0
\(424\) 4.18039 0.203017
\(425\) −0.642383 −0.0311602
\(426\) 0 0
\(427\) −55.7098 −2.69599
\(428\) 17.7231 0.856676
\(429\) 0 0
\(430\) −36.7395 −1.77174
\(431\) −0.380253 −0.0183161 −0.00915807 0.999958i \(-0.502915\pi\)
−0.00915807 + 0.999958i \(0.502915\pi\)
\(432\) 0 0
\(433\) 14.4837 0.696042 0.348021 0.937487i \(-0.386854\pi\)
0.348021 + 0.937487i \(0.386854\pi\)
\(434\) −24.2785 −1.16540
\(435\) 0 0
\(436\) −33.5260 −1.60560
\(437\) −45.9698 −2.19904
\(438\) 0 0
\(439\) −4.62838 −0.220900 −0.110450 0.993882i \(-0.535229\pi\)
−0.110450 + 0.993882i \(0.535229\pi\)
\(440\) 17.0033 0.810599
\(441\) 0 0
\(442\) 6.97396 0.331718
\(443\) 25.3781 1.20575 0.602874 0.797837i \(-0.294022\pi\)
0.602874 + 0.797837i \(0.294022\pi\)
\(444\) 0 0
\(445\) 33.5785 1.59177
\(446\) 13.1719 0.623707
\(447\) 0 0
\(448\) −79.6500 −3.76311
\(449\) −7.89535 −0.372605 −0.186302 0.982492i \(-0.559650\pi\)
−0.186302 + 0.982492i \(0.559650\pi\)
\(450\) 0 0
\(451\) 8.13132 0.382889
\(452\) −72.5013 −3.41018
\(453\) 0 0
\(454\) 72.7431 3.41400
\(455\) −53.9996 −2.53154
\(456\) 0 0
\(457\) 24.6158 1.15148 0.575740 0.817633i \(-0.304714\pi\)
0.575740 + 0.817633i \(0.304714\pi\)
\(458\) 45.5598 2.12887
\(459\) 0 0
\(460\) −80.7049 −3.76288
\(461\) −26.5949 −1.23865 −0.619324 0.785136i \(-0.712593\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(462\) 0 0
\(463\) 20.5795 0.956412 0.478206 0.878248i \(-0.341287\pi\)
0.478206 + 0.878248i \(0.341287\pi\)
\(464\) −11.1296 −0.516680
\(465\) 0 0
\(466\) −13.5643 −0.628355
\(467\) 4.02325 0.186174 0.0930869 0.995658i \(-0.470327\pi\)
0.0930869 + 0.995658i \(0.470327\pi\)
\(468\) 0 0
\(469\) −12.9801 −0.599367
\(470\) 37.5584 1.73244
\(471\) 0 0
\(472\) −89.6911 −4.12837
\(473\) −4.72139 −0.217090
\(474\) 0 0
\(475\) −7.53846 −0.345888
\(476\) 15.6977 0.719502
\(477\) 0 0
\(478\) −2.66148 −0.121733
\(479\) 11.1649 0.510138 0.255069 0.966923i \(-0.417902\pi\)
0.255069 + 0.966923i \(0.417902\pi\)
\(480\) 0 0
\(481\) 18.9198 0.862667
\(482\) −8.52310 −0.388217
\(483\) 0 0
\(484\) −52.3156 −2.37798
\(485\) −39.7132 −1.80329
\(486\) 0 0
\(487\) 20.4989 0.928894 0.464447 0.885601i \(-0.346253\pi\)
0.464447 + 0.885601i \(0.346253\pi\)
\(488\) 89.9113 4.07009
\(489\) 0 0
\(490\) −123.515 −5.57982
\(491\) −18.7828 −0.847657 −0.423828 0.905743i \(-0.639314\pi\)
−0.423828 + 0.905743i \(0.639314\pi\)
\(492\) 0 0
\(493\) 0.578947 0.0260745
\(494\) 81.8405 3.68218
\(495\) 0 0
\(496\) 20.9442 0.940422
\(497\) −23.0579 −1.03429
\(498\) 0 0
\(499\) 28.8795 1.29283 0.646413 0.762988i \(-0.276269\pi\)
0.646413 + 0.762988i \(0.276269\pi\)
\(500\) 49.3238 2.20583
\(501\) 0 0
\(502\) 28.5232 1.27305
\(503\) −4.25152 −0.189566 −0.0947828 0.995498i \(-0.530216\pi\)
−0.0947828 + 0.995498i \(0.530216\pi\)
\(504\) 0 0
\(505\) −30.7347 −1.36768
\(506\) −14.4518 −0.642462
\(507\) 0 0
\(508\) 82.6774 3.66822
\(509\) −32.0917 −1.42244 −0.711219 0.702970i \(-0.751857\pi\)
−0.711219 + 0.702970i \(0.751857\pi\)
\(510\) 0 0
\(511\) −45.1187 −1.99593
\(512\) −20.4832 −0.905238
\(513\) 0 0
\(514\) −76.5519 −3.37656
\(515\) 30.6434 1.35031
\(516\) 0 0
\(517\) 4.82663 0.212275
\(518\) 59.3415 2.60731
\(519\) 0 0
\(520\) 87.1512 3.82183
\(521\) 19.6524 0.860986 0.430493 0.902594i \(-0.358340\pi\)
0.430493 + 0.902594i \(0.358340\pi\)
\(522\) 0 0
\(523\) −4.99001 −0.218198 −0.109099 0.994031i \(-0.534797\pi\)
−0.109099 + 0.994031i \(0.534797\pi\)
\(524\) 107.013 4.67490
\(525\) 0 0
\(526\) −45.3347 −1.97668
\(527\) −1.08949 −0.0474588
\(528\) 0 0
\(529\) 18.6072 0.809009
\(530\) −3.33684 −0.144943
\(531\) 0 0
\(532\) 184.215 7.98672
\(533\) 41.6775 1.80525
\(534\) 0 0
\(535\) −8.58097 −0.370988
\(536\) 20.9490 0.904857
\(537\) 0 0
\(538\) 81.4415 3.51119
\(539\) −15.8729 −0.683692
\(540\) 0 0
\(541\) −1.40609 −0.0604526 −0.0302263 0.999543i \(-0.509623\pi\)
−0.0302263 + 0.999543i \(0.509623\pi\)
\(542\) 81.9900 3.52177
\(543\) 0 0
\(544\) −8.90208 −0.381673
\(545\) 16.2323 0.695314
\(546\) 0 0
\(547\) 3.50342 0.149795 0.0748977 0.997191i \(-0.476137\pi\)
0.0748977 + 0.997191i \(0.476137\pi\)
\(548\) −73.8791 −3.15596
\(549\) 0 0
\(550\) −2.36991 −0.101053
\(551\) 6.79403 0.289436
\(552\) 0 0
\(553\) 8.59725 0.365592
\(554\) 2.87647 0.122209
\(555\) 0 0
\(556\) −36.2651 −1.53798
\(557\) 29.7433 1.26026 0.630131 0.776489i \(-0.283001\pi\)
0.630131 + 0.776489i \(0.283001\pi\)
\(558\) 0 0
\(559\) −24.1998 −1.02354
\(560\) 146.108 6.17420
\(561\) 0 0
\(562\) −82.6819 −3.48773
\(563\) −17.7864 −0.749609 −0.374805 0.927104i \(-0.622290\pi\)
−0.374805 + 0.927104i \(0.622290\pi\)
\(564\) 0 0
\(565\) 35.1030 1.47679
\(566\) −48.7268 −2.04814
\(567\) 0 0
\(568\) 37.2137 1.56145
\(569\) −18.0597 −0.757101 −0.378551 0.925581i \(-0.623577\pi\)
−0.378551 + 0.925581i \(0.623577\pi\)
\(570\) 0 0
\(571\) −6.80579 −0.284813 −0.142407 0.989808i \(-0.545484\pi\)
−0.142407 + 0.989808i \(0.545484\pi\)
\(572\) 18.4643 0.772029
\(573\) 0 0
\(574\) 130.721 5.45618
\(575\) 6.82305 0.284541
\(576\) 0 0
\(577\) −0.745890 −0.0310518 −0.0155259 0.999879i \(-0.504942\pi\)
−0.0155259 + 0.999879i \(0.504942\pi\)
\(578\) −44.2635 −1.84112
\(579\) 0 0
\(580\) 11.9276 0.495268
\(581\) 51.3447 2.13014
\(582\) 0 0
\(583\) −0.428817 −0.0177598
\(584\) 72.8181 3.01324
\(585\) 0 0
\(586\) −36.8012 −1.52024
\(587\) −23.8937 −0.986199 −0.493099 0.869973i \(-0.664136\pi\)
−0.493099 + 0.869973i \(0.664136\pi\)
\(588\) 0 0
\(589\) −12.7853 −0.526809
\(590\) 71.5927 2.94742
\(591\) 0 0
\(592\) −51.1918 −2.10397
\(593\) 40.9714 1.68249 0.841247 0.540650i \(-0.181822\pi\)
0.841247 + 0.540650i \(0.181822\pi\)
\(594\) 0 0
\(595\) −7.60035 −0.311584
\(596\) −26.8993 −1.10184
\(597\) 0 0
\(598\) −74.0737 −3.02910
\(599\) 25.0423 1.02320 0.511601 0.859223i \(-0.329053\pi\)
0.511601 + 0.859223i \(0.329053\pi\)
\(600\) 0 0
\(601\) 36.5737 1.49187 0.745937 0.666017i \(-0.232002\pi\)
0.745937 + 0.666017i \(0.232002\pi\)
\(602\) −75.9021 −3.09354
\(603\) 0 0
\(604\) −5.03725 −0.204963
\(605\) 25.3296 1.02980
\(606\) 0 0
\(607\) −3.60649 −0.146383 −0.0731915 0.997318i \(-0.523318\pi\)
−0.0731915 + 0.997318i \(0.523318\pi\)
\(608\) −104.467 −4.23670
\(609\) 0 0
\(610\) −71.7685 −2.90582
\(611\) 24.7392 1.00084
\(612\) 0 0
\(613\) 30.2711 1.22264 0.611319 0.791384i \(-0.290639\pi\)
0.611319 + 0.791384i \(0.290639\pi\)
\(614\) 43.8705 1.77047
\(615\) 0 0
\(616\) 35.1279 1.41534
\(617\) 44.0450 1.77319 0.886593 0.462551i \(-0.153066\pi\)
0.886593 + 0.462551i \(0.153066\pi\)
\(618\) 0 0
\(619\) 46.1839 1.85629 0.928144 0.372221i \(-0.121404\pi\)
0.928144 + 0.372221i \(0.121404\pi\)
\(620\) −22.4459 −0.901450
\(621\) 0 0
\(622\) 21.1313 0.847286
\(623\) 69.3715 2.77931
\(624\) 0 0
\(625\) −29.1700 −1.16680
\(626\) −64.0698 −2.56074
\(627\) 0 0
\(628\) −34.6062 −1.38094
\(629\) 2.66292 0.106178
\(630\) 0 0
\(631\) −12.9537 −0.515678 −0.257839 0.966188i \(-0.583010\pi\)
−0.257839 + 0.966188i \(0.583010\pi\)
\(632\) −13.8753 −0.551930
\(633\) 0 0
\(634\) −6.22590 −0.247262
\(635\) −40.0299 −1.58854
\(636\) 0 0
\(637\) −81.3573 −3.22349
\(638\) 2.13588 0.0845604
\(639\) 0 0
\(640\) −30.4527 −1.20375
\(641\) −33.2322 −1.31259 −0.656297 0.754502i \(-0.727878\pi\)
−0.656297 + 0.754502i \(0.727878\pi\)
\(642\) 0 0
\(643\) 28.4401 1.12157 0.560784 0.827962i \(-0.310500\pi\)
0.560784 + 0.827962i \(0.310500\pi\)
\(644\) −166.732 −6.57018
\(645\) 0 0
\(646\) 11.5189 0.453205
\(647\) −37.1337 −1.45988 −0.729938 0.683513i \(-0.760451\pi\)
−0.729938 + 0.683513i \(0.760451\pi\)
\(648\) 0 0
\(649\) 9.20037 0.361146
\(650\) −12.1471 −0.476449
\(651\) 0 0
\(652\) 36.6891 1.43686
\(653\) −5.54976 −0.217179 −0.108590 0.994087i \(-0.534633\pi\)
−0.108590 + 0.994087i \(0.534633\pi\)
\(654\) 0 0
\(655\) −51.8127 −2.02449
\(656\) −112.768 −4.40286
\(657\) 0 0
\(658\) 77.5938 3.02492
\(659\) 26.9825 1.05109 0.525545 0.850766i \(-0.323862\pi\)
0.525545 + 0.850766i \(0.323862\pi\)
\(660\) 0 0
\(661\) 19.9018 0.774089 0.387044 0.922061i \(-0.373496\pi\)
0.387044 + 0.922061i \(0.373496\pi\)
\(662\) −51.6508 −2.00746
\(663\) 0 0
\(664\) −82.8665 −3.21584
\(665\) −89.1913 −3.45869
\(666\) 0 0
\(667\) −6.14927 −0.238101
\(668\) −15.9570 −0.617396
\(669\) 0 0
\(670\) −16.7218 −0.646018
\(671\) −9.22296 −0.356048
\(672\) 0 0
\(673\) −5.01717 −0.193398 −0.0966988 0.995314i \(-0.530828\pi\)
−0.0966988 + 0.995314i \(0.530828\pi\)
\(674\) 8.96581 0.345350
\(675\) 0 0
\(676\) 28.5547 1.09826
\(677\) −23.7408 −0.912431 −0.456216 0.889869i \(-0.650795\pi\)
−0.456216 + 0.889869i \(0.650795\pi\)
\(678\) 0 0
\(679\) −82.0457 −3.14862
\(680\) 12.2664 0.470394
\(681\) 0 0
\(682\) −4.01939 −0.153910
\(683\) 34.8003 1.33160 0.665799 0.746132i \(-0.268091\pi\)
0.665799 + 0.746132i \(0.268091\pi\)
\(684\) 0 0
\(685\) 35.7700 1.36670
\(686\) −160.443 −6.12575
\(687\) 0 0
\(688\) 65.4780 2.49633
\(689\) −2.19793 −0.0837344
\(690\) 0 0
\(691\) 10.4812 0.398724 0.199362 0.979926i \(-0.436113\pi\)
0.199362 + 0.979926i \(0.436113\pi\)
\(692\) 78.7936 2.99528
\(693\) 0 0
\(694\) −38.6095 −1.46560
\(695\) 17.5585 0.666031
\(696\) 0 0
\(697\) 5.86604 0.222192
\(698\) 18.8838 0.714762
\(699\) 0 0
\(700\) −27.3420 −1.03343
\(701\) 3.42304 0.129286 0.0646432 0.997908i \(-0.479409\pi\)
0.0646432 + 0.997908i \(0.479409\pi\)
\(702\) 0 0
\(703\) 31.2498 1.17861
\(704\) −13.1864 −0.496979
\(705\) 0 0
\(706\) 25.0935 0.944406
\(707\) −63.4965 −2.38803
\(708\) 0 0
\(709\) −50.4436 −1.89445 −0.947225 0.320571i \(-0.896125\pi\)
−0.947225 + 0.320571i \(0.896125\pi\)
\(710\) −29.7045 −1.11479
\(711\) 0 0
\(712\) −111.960 −4.19589
\(713\) 11.5719 0.433373
\(714\) 0 0
\(715\) −8.93984 −0.334331
\(716\) −85.9397 −3.21172
\(717\) 0 0
\(718\) 53.5620 1.99892
\(719\) 9.37934 0.349791 0.174895 0.984587i \(-0.444041\pi\)
0.174895 + 0.984587i \(0.444041\pi\)
\(720\) 0 0
\(721\) 63.3079 2.35771
\(722\) 84.6081 3.14879
\(723\) 0 0
\(724\) −28.6643 −1.06530
\(725\) −1.00840 −0.0374510
\(726\) 0 0
\(727\) −4.55094 −0.168785 −0.0843926 0.996433i \(-0.526895\pi\)
−0.0843926 + 0.996433i \(0.526895\pi\)
\(728\) 180.050 6.67311
\(729\) 0 0
\(730\) −58.1244 −2.15128
\(731\) −3.40608 −0.125978
\(732\) 0 0
\(733\) −44.9913 −1.66179 −0.830896 0.556427i \(-0.812172\pi\)
−0.830896 + 0.556427i \(0.812172\pi\)
\(734\) −54.0585 −1.99534
\(735\) 0 0
\(736\) 94.5530 3.48527
\(737\) −2.14891 −0.0791562
\(738\) 0 0
\(739\) −28.7428 −1.05732 −0.528660 0.848834i \(-0.677305\pi\)
−0.528660 + 0.848834i \(0.677305\pi\)
\(740\) 54.8623 2.01678
\(741\) 0 0
\(742\) −6.89376 −0.253078
\(743\) −17.5762 −0.644808 −0.322404 0.946602i \(-0.604491\pi\)
−0.322404 + 0.946602i \(0.604491\pi\)
\(744\) 0 0
\(745\) 13.0238 0.477156
\(746\) 37.4654 1.37171
\(747\) 0 0
\(748\) 2.59881 0.0950220
\(749\) −17.7279 −0.647762
\(750\) 0 0
\(751\) −11.6670 −0.425734 −0.212867 0.977081i \(-0.568280\pi\)
−0.212867 + 0.977081i \(0.568280\pi\)
\(752\) −66.9375 −2.44096
\(753\) 0 0
\(754\) 10.9476 0.398688
\(755\) 2.43889 0.0887602
\(756\) 0 0
\(757\) 40.2970 1.46462 0.732309 0.680972i \(-0.238443\pi\)
0.732309 + 0.680972i \(0.238443\pi\)
\(758\) −86.3194 −3.13526
\(759\) 0 0
\(760\) 143.948 5.22154
\(761\) −1.07339 −0.0389104 −0.0194552 0.999811i \(-0.506193\pi\)
−0.0194552 + 0.999811i \(0.506193\pi\)
\(762\) 0 0
\(763\) 33.5351 1.21405
\(764\) −47.8495 −1.73114
\(765\) 0 0
\(766\) −76.9086 −2.77882
\(767\) 47.1570 1.70274
\(768\) 0 0
\(769\) −53.2765 −1.92120 −0.960600 0.277935i \(-0.910350\pi\)
−0.960600 + 0.277935i \(0.910350\pi\)
\(770\) −28.0396 −1.01048
\(771\) 0 0
\(772\) −96.6616 −3.47893
\(773\) 6.87421 0.247248 0.123624 0.992329i \(-0.460548\pi\)
0.123624 + 0.992329i \(0.460548\pi\)
\(774\) 0 0
\(775\) 1.89765 0.0681656
\(776\) 132.415 4.75344
\(777\) 0 0
\(778\) −69.5571 −2.49374
\(779\) 68.8389 2.46641
\(780\) 0 0
\(781\) −3.81732 −0.136594
\(782\) −10.4257 −0.372824
\(783\) 0 0
\(784\) 220.131 7.86181
\(785\) 16.7553 0.598022
\(786\) 0 0
\(787\) 30.4891 1.08682 0.543410 0.839467i \(-0.317133\pi\)
0.543410 + 0.839467i \(0.317133\pi\)
\(788\) −28.1576 −1.00307
\(789\) 0 0
\(790\) 11.0755 0.394047
\(791\) 72.5210 2.57855
\(792\) 0 0
\(793\) −47.2728 −1.67871
\(794\) −50.1597 −1.78010
\(795\) 0 0
\(796\) −66.9697 −2.37368
\(797\) −16.2226 −0.574633 −0.287316 0.957836i \(-0.592763\pi\)
−0.287316 + 0.957836i \(0.592763\pi\)
\(798\) 0 0
\(799\) 3.48199 0.123184
\(800\) 15.5055 0.548201
\(801\) 0 0
\(802\) −76.8922 −2.71516
\(803\) −7.46957 −0.263595
\(804\) 0 0
\(805\) 80.7268 2.84525
\(806\) −20.6016 −0.725661
\(807\) 0 0
\(808\) 102.478 3.60518
\(809\) −29.3350 −1.03136 −0.515681 0.856781i \(-0.672461\pi\)
−0.515681 + 0.856781i \(0.672461\pi\)
\(810\) 0 0
\(811\) 10.7325 0.376867 0.188434 0.982086i \(-0.439659\pi\)
0.188434 + 0.982086i \(0.439659\pi\)
\(812\) 24.6419 0.864762
\(813\) 0 0
\(814\) 9.82420 0.344338
\(815\) −17.7638 −0.622238
\(816\) 0 0
\(817\) −39.9708 −1.39840
\(818\) −77.2643 −2.70148
\(819\) 0 0
\(820\) 120.854 4.22040
\(821\) −2.60579 −0.0909428 −0.0454714 0.998966i \(-0.514479\pi\)
−0.0454714 + 0.998966i \(0.514479\pi\)
\(822\) 0 0
\(823\) −18.0974 −0.630837 −0.315418 0.948953i \(-0.602145\pi\)
−0.315418 + 0.948953i \(0.602145\pi\)
\(824\) −102.174 −3.55940
\(825\) 0 0
\(826\) 147.907 5.14635
\(827\) 33.0633 1.14972 0.574862 0.818250i \(-0.305056\pi\)
0.574862 + 0.818250i \(0.305056\pi\)
\(828\) 0 0
\(829\) −33.0572 −1.14813 −0.574063 0.818811i \(-0.694633\pi\)
−0.574063 + 0.818811i \(0.694633\pi\)
\(830\) 66.1452 2.29593
\(831\) 0 0
\(832\) −67.5874 −2.34317
\(833\) −11.4509 −0.396750
\(834\) 0 0
\(835\) 7.72592 0.267366
\(836\) 30.4974 1.05478
\(837\) 0 0
\(838\) −18.4352 −0.636834
\(839\) −13.8010 −0.476462 −0.238231 0.971209i \(-0.576568\pi\)
−0.238231 + 0.971209i \(0.576568\pi\)
\(840\) 0 0
\(841\) −28.0912 −0.968661
\(842\) 84.8652 2.92465
\(843\) 0 0
\(844\) 58.2793 2.00606
\(845\) −13.8253 −0.475606
\(846\) 0 0
\(847\) 52.3298 1.79807
\(848\) 5.94700 0.204221
\(849\) 0 0
\(850\) −1.70969 −0.0586418
\(851\) −28.2841 −0.969568
\(852\) 0 0
\(853\) −19.4019 −0.664310 −0.332155 0.943225i \(-0.607776\pi\)
−0.332155 + 0.943225i \(0.607776\pi\)
\(854\) −148.270 −5.07370
\(855\) 0 0
\(856\) 28.6114 0.977918
\(857\) 7.72283 0.263807 0.131903 0.991263i \(-0.457891\pi\)
0.131903 + 0.991263i \(0.457891\pi\)
\(858\) 0 0
\(859\) −26.9600 −0.919865 −0.459932 0.887954i \(-0.652126\pi\)
−0.459932 + 0.887954i \(0.652126\pi\)
\(860\) −70.1729 −2.39288
\(861\) 0 0
\(862\) −1.01203 −0.0344700
\(863\) 37.1997 1.26629 0.633146 0.774032i \(-0.281763\pi\)
0.633146 + 0.774032i \(0.281763\pi\)
\(864\) 0 0
\(865\) −38.1495 −1.29712
\(866\) 38.5480 1.30991
\(867\) 0 0
\(868\) −46.3722 −1.57397
\(869\) 1.42331 0.0482824
\(870\) 0 0
\(871\) −11.0144 −0.373208
\(872\) −54.1231 −1.83284
\(873\) 0 0
\(874\) −122.348 −4.13847
\(875\) −49.3372 −1.66790
\(876\) 0 0
\(877\) −3.19507 −0.107890 −0.0539449 0.998544i \(-0.517180\pi\)
−0.0539449 + 0.998544i \(0.517180\pi\)
\(878\) −12.3183 −0.415723
\(879\) 0 0
\(880\) 24.1888 0.815404
\(881\) −3.64187 −0.122698 −0.0613489 0.998116i \(-0.519540\pi\)
−0.0613489 + 0.998116i \(0.519540\pi\)
\(882\) 0 0
\(883\) 50.0188 1.68327 0.841633 0.540050i \(-0.181595\pi\)
0.841633 + 0.540050i \(0.181595\pi\)
\(884\) 13.3204 0.448012
\(885\) 0 0
\(886\) 67.5431 2.26915
\(887\) 11.3702 0.381775 0.190888 0.981612i \(-0.438863\pi\)
0.190888 + 0.981612i \(0.438863\pi\)
\(888\) 0 0
\(889\) −82.6999 −2.77366
\(890\) 89.3682 2.99563
\(891\) 0 0
\(892\) 25.1585 0.842369
\(893\) 40.8617 1.36739
\(894\) 0 0
\(895\) 41.6094 1.39085
\(896\) −62.9138 −2.10180
\(897\) 0 0
\(898\) −21.0133 −0.701222
\(899\) −1.71026 −0.0570402
\(900\) 0 0
\(901\) −0.309355 −0.0103061
\(902\) 21.6413 0.720576
\(903\) 0 0
\(904\) −117.043 −3.89281
\(905\) 13.8784 0.461333
\(906\) 0 0
\(907\) 24.2253 0.804389 0.402195 0.915554i \(-0.368248\pi\)
0.402195 + 0.915554i \(0.368248\pi\)
\(908\) 138.940 4.61089
\(909\) 0 0
\(910\) −143.719 −4.76423
\(911\) 17.0859 0.566081 0.283040 0.959108i \(-0.408657\pi\)
0.283040 + 0.959108i \(0.408657\pi\)
\(912\) 0 0
\(913\) 8.50031 0.281319
\(914\) 65.5144 2.16702
\(915\) 0 0
\(916\) 87.0198 2.87521
\(917\) −107.042 −3.53485
\(918\) 0 0
\(919\) 10.2142 0.336935 0.168467 0.985707i \(-0.446118\pi\)
0.168467 + 0.985707i \(0.446118\pi\)
\(920\) −130.287 −4.29543
\(921\) 0 0
\(922\) −70.7817 −2.33107
\(923\) −19.5659 −0.644019
\(924\) 0 0
\(925\) −4.63824 −0.152504
\(926\) 54.7719 1.79992
\(927\) 0 0
\(928\) −13.9743 −0.458729
\(929\) −4.19595 −0.137665 −0.0688324 0.997628i \(-0.521927\pi\)
−0.0688324 + 0.997628i \(0.521927\pi\)
\(930\) 0 0
\(931\) −134.378 −4.40406
\(932\) −25.9080 −0.848646
\(933\) 0 0
\(934\) 10.7078 0.350369
\(935\) −1.25827 −0.0411497
\(936\) 0 0
\(937\) 20.6095 0.673284 0.336642 0.941633i \(-0.390709\pi\)
0.336642 + 0.941633i \(0.390709\pi\)
\(938\) −34.5463 −1.12798
\(939\) 0 0
\(940\) 71.7370 2.33980
\(941\) 16.4231 0.535379 0.267689 0.963505i \(-0.413740\pi\)
0.267689 + 0.963505i \(0.413740\pi\)
\(942\) 0 0
\(943\) −62.3059 −2.02896
\(944\) −127.594 −4.15284
\(945\) 0 0
\(946\) −12.5659 −0.408552
\(947\) −46.2414 −1.50264 −0.751322 0.659936i \(-0.770583\pi\)
−0.751322 + 0.659936i \(0.770583\pi\)
\(948\) 0 0
\(949\) −38.2857 −1.24281
\(950\) −20.0634 −0.650944
\(951\) 0 0
\(952\) 25.3418 0.821331
\(953\) 17.1233 0.554678 0.277339 0.960772i \(-0.410547\pi\)
0.277339 + 0.960772i \(0.410547\pi\)
\(954\) 0 0
\(955\) 23.1673 0.749677
\(956\) −5.08345 −0.164411
\(957\) 0 0
\(958\) 29.7151 0.960053
\(959\) 73.8992 2.38633
\(960\) 0 0
\(961\) −27.7816 −0.896180
\(962\) 50.3545 1.62349
\(963\) 0 0
\(964\) −16.2792 −0.524319
\(965\) 46.8006 1.50657
\(966\) 0 0
\(967\) 9.08972 0.292306 0.146153 0.989262i \(-0.453311\pi\)
0.146153 + 0.989262i \(0.453311\pi\)
\(968\) −84.4563 −2.71453
\(969\) 0 0
\(970\) −105.696 −3.39369
\(971\) 17.5375 0.562806 0.281403 0.959590i \(-0.409200\pi\)
0.281403 + 0.959590i \(0.409200\pi\)
\(972\) 0 0
\(973\) 36.2750 1.16292
\(974\) 54.5573 1.74813
\(975\) 0 0
\(976\) 127.907 4.09422
\(977\) −43.9565 −1.40629 −0.703147 0.711044i \(-0.748223\pi\)
−0.703147 + 0.711044i \(0.748223\pi\)
\(978\) 0 0
\(979\) 11.4847 0.367053
\(980\) −235.914 −7.53601
\(981\) 0 0
\(982\) −49.9900 −1.59525
\(983\) −0.659283 −0.0210279 −0.0105139 0.999945i \(-0.503347\pi\)
−0.0105139 + 0.999945i \(0.503347\pi\)
\(984\) 0 0
\(985\) 13.6331 0.434386
\(986\) 1.54085 0.0490708
\(987\) 0 0
\(988\) 156.316 4.97309
\(989\) 36.1775 1.15038
\(990\) 0 0
\(991\) 25.6899 0.816067 0.408033 0.912967i \(-0.366215\pi\)
0.408033 + 0.912967i \(0.366215\pi\)
\(992\) 26.2974 0.834944
\(993\) 0 0
\(994\) −61.3680 −1.94647
\(995\) 32.4247 1.02793
\(996\) 0 0
\(997\) 18.2490 0.577953 0.288977 0.957336i \(-0.406685\pi\)
0.288977 + 0.957336i \(0.406685\pi\)
\(998\) 76.8621 2.43303
\(999\) 0 0
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 2151.2.a.j.1.20 20
3.2 odd 2 2151.2.a.k.1.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.20 20 1.1 even 1 trivial
2151.2.a.k.1.1 yes 20 3.2 odd 2