Properties

Label 1143.3.b.a.890.72
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.72
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.13

$q$-expansion

\(f(q)\) \(=\) \(q+2.99230i q^{2} -4.95386 q^{4} -7.40706i q^{5} -4.26180 q^{7} -2.85423i q^{8} +O(q^{10})\) \(q+2.99230i q^{2} -4.95386 q^{4} -7.40706i q^{5} -4.26180 q^{7} -2.85423i q^{8} +22.1641 q^{10} +11.1948i q^{11} -7.91934 q^{13} -12.7526i q^{14} -11.2747 q^{16} +14.3310i q^{17} +18.0911 q^{19} +36.6935i q^{20} -33.4981 q^{22} -33.5108i q^{23} -29.8645 q^{25} -23.6970i q^{26} +21.1123 q^{28} -23.5050i q^{29} +43.9625 q^{31} -45.1543i q^{32} -42.8826 q^{34} +31.5674i q^{35} +2.75938 q^{37} +54.1340i q^{38} -21.1415 q^{40} -16.3082i q^{41} +48.3645 q^{43} -55.4573i q^{44} +100.274 q^{46} -4.06398i q^{47} -30.8371 q^{49} -89.3635i q^{50} +39.2313 q^{52} -74.2719i q^{53} +82.9203 q^{55} +12.1642i q^{56} +70.3339 q^{58} +55.4624i q^{59} +45.3013 q^{61} +131.549i q^{62} +90.0162 q^{64} +58.6590i q^{65} +124.316 q^{67} -70.9936i q^{68} -94.4591 q^{70} +116.550i q^{71} +93.3854 q^{73} +8.25690i q^{74} -89.6207 q^{76} -47.7098i q^{77} -2.92284 q^{79} +83.5125i q^{80} +48.7991 q^{82} +72.3263i q^{83} +106.150 q^{85} +144.721i q^{86} +31.9525 q^{88} -19.0077i q^{89} +33.7506 q^{91} +166.008i q^{92} +12.1606 q^{94} -134.002i q^{95} +18.2337 q^{97} -92.2738i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.99230i 1.49615i 0.663614 + 0.748075i \(0.269022\pi\)
−0.663614 + 0.748075i \(0.730978\pi\)
\(3\) 0 0
\(4\) −4.95386 −1.23846
\(5\) − 7.40706i − 1.48141i −0.671830 0.740706i \(-0.734491\pi\)
0.671830 0.740706i \(-0.265509\pi\)
\(6\) 0 0
\(7\) −4.26180 −0.608828 −0.304414 0.952540i \(-0.598461\pi\)
−0.304414 + 0.952540i \(0.598461\pi\)
\(8\) − 2.85423i − 0.356779i
\(9\) 0 0
\(10\) 22.1641 2.21641
\(11\) 11.1948i 1.01771i 0.860853 + 0.508853i \(0.169930\pi\)
−0.860853 + 0.508853i \(0.830070\pi\)
\(12\) 0 0
\(13\) −7.91934 −0.609180 −0.304590 0.952484i \(-0.598519\pi\)
−0.304590 + 0.952484i \(0.598519\pi\)
\(14\) − 12.7526i − 0.910899i
\(15\) 0 0
\(16\) −11.2747 −0.704670
\(17\) 14.3310i 0.842999i 0.906829 + 0.421499i \(0.138496\pi\)
−0.906829 + 0.421499i \(0.861504\pi\)
\(18\) 0 0
\(19\) 18.0911 0.952163 0.476082 0.879401i \(-0.342057\pi\)
0.476082 + 0.879401i \(0.342057\pi\)
\(20\) 36.6935i 1.83468i
\(21\) 0 0
\(22\) −33.4981 −1.52264
\(23\) − 33.5108i − 1.45699i −0.685050 0.728496i \(-0.740220\pi\)
0.685050 0.728496i \(-0.259780\pi\)
\(24\) 0 0
\(25\) −29.8645 −1.19458
\(26\) − 23.6970i − 0.911425i
\(27\) 0 0
\(28\) 21.1123 0.754012
\(29\) − 23.5050i − 0.810516i −0.914202 0.405258i \(-0.867182\pi\)
0.914202 0.405258i \(-0.132818\pi\)
\(30\) 0 0
\(31\) 43.9625 1.41814 0.709072 0.705136i \(-0.249114\pi\)
0.709072 + 0.705136i \(0.249114\pi\)
\(32\) − 45.1543i − 1.41107i
\(33\) 0 0
\(34\) −42.8826 −1.26125
\(35\) 31.5674i 0.901925i
\(36\) 0 0
\(37\) 2.75938 0.0745779 0.0372890 0.999305i \(-0.488128\pi\)
0.0372890 + 0.999305i \(0.488128\pi\)
\(38\) 54.1340i 1.42458i
\(39\) 0 0
\(40\) −21.1415 −0.528536
\(41\) − 16.3082i − 0.397761i −0.980024 0.198881i \(-0.936269\pi\)
0.980024 0.198881i \(-0.0637306\pi\)
\(42\) 0 0
\(43\) 48.3645 1.12476 0.562378 0.826880i \(-0.309886\pi\)
0.562378 + 0.826880i \(0.309886\pi\)
\(44\) − 55.4573i − 1.26039i
\(45\) 0 0
\(46\) 100.274 2.17988
\(47\) − 4.06398i − 0.0864677i −0.999065 0.0432338i \(-0.986234\pi\)
0.999065 0.0432338i \(-0.0137660\pi\)
\(48\) 0 0
\(49\) −30.8371 −0.629328
\(50\) − 89.3635i − 1.78727i
\(51\) 0 0
\(52\) 39.2313 0.754448
\(53\) − 74.2719i − 1.40136i −0.713477 0.700678i \(-0.752881\pi\)
0.713477 0.700678i \(-0.247119\pi\)
\(54\) 0 0
\(55\) 82.9203 1.50764
\(56\) 12.1642i 0.217217i
\(57\) 0 0
\(58\) 70.3339 1.21265
\(59\) 55.4624i 0.940041i 0.882655 + 0.470021i \(0.155754\pi\)
−0.882655 + 0.470021i \(0.844246\pi\)
\(60\) 0 0
\(61\) 45.3013 0.742644 0.371322 0.928504i \(-0.378905\pi\)
0.371322 + 0.928504i \(0.378905\pi\)
\(62\) 131.549i 2.12176i
\(63\) 0 0
\(64\) 90.0162 1.40650
\(65\) 58.6590i 0.902446i
\(66\) 0 0
\(67\) 124.316 1.85546 0.927731 0.373250i \(-0.121757\pi\)
0.927731 + 0.373250i \(0.121757\pi\)
\(68\) − 70.9936i − 1.04402i
\(69\) 0 0
\(70\) −94.4591 −1.34942
\(71\) 116.550i 1.64155i 0.571253 + 0.820774i \(0.306458\pi\)
−0.571253 + 0.820774i \(0.693542\pi\)
\(72\) 0 0
\(73\) 93.3854 1.27925 0.639626 0.768686i \(-0.279089\pi\)
0.639626 + 0.768686i \(0.279089\pi\)
\(74\) 8.25690i 0.111580i
\(75\) 0 0
\(76\) −89.6207 −1.17922
\(77\) − 47.7098i − 0.619608i
\(78\) 0 0
\(79\) −2.92284 −0.0369980 −0.0184990 0.999829i \(-0.505889\pi\)
−0.0184990 + 0.999829i \(0.505889\pi\)
\(80\) 83.5125i 1.04391i
\(81\) 0 0
\(82\) 48.7991 0.595111
\(83\) 72.3263i 0.871401i 0.900092 + 0.435701i \(0.143499\pi\)
−0.900092 + 0.435701i \(0.856501\pi\)
\(84\) 0 0
\(85\) 106.150 1.24883
\(86\) 144.721i 1.68280i
\(87\) 0 0
\(88\) 31.9525 0.363096
\(89\) − 19.0077i − 0.213570i −0.994282 0.106785i \(-0.965944\pi\)
0.994282 0.106785i \(-0.0340556\pi\)
\(90\) 0 0
\(91\) 33.7506 0.370886
\(92\) 166.008i 1.80443i
\(93\) 0 0
\(94\) 12.1606 0.129369
\(95\) − 134.002i − 1.41055i
\(96\) 0 0
\(97\) 18.2337 0.187977 0.0939883 0.995573i \(-0.470038\pi\)
0.0939883 + 0.995573i \(0.470038\pi\)
\(98\) − 92.2738i − 0.941569i
\(99\) 0 0
\(100\) 147.944 1.47944
\(101\) − 136.191i − 1.34843i −0.738536 0.674214i \(-0.764482\pi\)
0.738536 0.674214i \(-0.235518\pi\)
\(102\) 0 0
\(103\) 64.9125 0.630219 0.315109 0.949055i \(-0.397959\pi\)
0.315109 + 0.949055i \(0.397959\pi\)
\(104\) 22.6036i 0.217343i
\(105\) 0 0
\(106\) 222.244 2.09664
\(107\) − 194.211i − 1.81505i −0.419995 0.907526i \(-0.637968\pi\)
0.419995 0.907526i \(-0.362032\pi\)
\(108\) 0 0
\(109\) 155.670 1.42816 0.714081 0.700063i \(-0.246845\pi\)
0.714081 + 0.700063i \(0.246845\pi\)
\(110\) 248.122i 2.25566i
\(111\) 0 0
\(112\) 48.0506 0.429023
\(113\) 12.0707i 0.106820i 0.998573 + 0.0534100i \(0.0170090\pi\)
−0.998573 + 0.0534100i \(0.982991\pi\)
\(114\) 0 0
\(115\) −248.217 −2.15840
\(116\) 116.440i 1.00380i
\(117\) 0 0
\(118\) −165.960 −1.40644
\(119\) − 61.0758i − 0.513242i
\(120\) 0 0
\(121\) −4.32277 −0.0357254
\(122\) 135.555i 1.11111i
\(123\) 0 0
\(124\) −217.784 −1.75632
\(125\) 36.0315i 0.288252i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) 88.7385i 0.693269i
\(129\) 0 0
\(130\) −175.525 −1.35019
\(131\) 13.1220i 0.100168i 0.998745 + 0.0500839i \(0.0159489\pi\)
−0.998745 + 0.0500839i \(0.984051\pi\)
\(132\) 0 0
\(133\) −77.1006 −0.579704
\(134\) 371.991i 2.77605i
\(135\) 0 0
\(136\) 40.9039 0.300764
\(137\) 57.7826i 0.421771i 0.977511 + 0.210885i \(0.0676347\pi\)
−0.977511 + 0.210885i \(0.932365\pi\)
\(138\) 0 0
\(139\) −91.4008 −0.657560 −0.328780 0.944407i \(-0.606637\pi\)
−0.328780 + 0.944407i \(0.606637\pi\)
\(140\) − 156.380i − 1.11700i
\(141\) 0 0
\(142\) −348.752 −2.45600
\(143\) − 88.6552i − 0.619966i
\(144\) 0 0
\(145\) −174.103 −1.20071
\(146\) 279.437i 1.91395i
\(147\) 0 0
\(148\) −13.6696 −0.0923621
\(149\) 6.68338i 0.0448549i 0.999748 + 0.0224274i \(0.00713948\pi\)
−0.999748 + 0.0224274i \(0.992861\pi\)
\(150\) 0 0
\(151\) 246.584 1.63301 0.816504 0.577339i \(-0.195909\pi\)
0.816504 + 0.577339i \(0.195909\pi\)
\(152\) − 51.6362i − 0.339712i
\(153\) 0 0
\(154\) 142.762 0.927027
\(155\) − 325.633i − 2.10086i
\(156\) 0 0
\(157\) 72.7382 0.463300 0.231650 0.972799i \(-0.425588\pi\)
0.231650 + 0.972799i \(0.425588\pi\)
\(158\) − 8.74602i − 0.0553545i
\(159\) 0 0
\(160\) −334.460 −2.09038
\(161\) 142.816i 0.887058i
\(162\) 0 0
\(163\) −165.119 −1.01300 −0.506499 0.862241i \(-0.669061\pi\)
−0.506499 + 0.862241i \(0.669061\pi\)
\(164\) 80.7886i 0.492614i
\(165\) 0 0
\(166\) −216.422 −1.30375
\(167\) − 126.920i − 0.760001i −0.924986 0.380001i \(-0.875924\pi\)
0.924986 0.380001i \(-0.124076\pi\)
\(168\) 0 0
\(169\) −106.284 −0.628900
\(170\) 317.634i 1.86843i
\(171\) 0 0
\(172\) −239.591 −1.39297
\(173\) − 216.681i − 1.25249i −0.779626 0.626245i \(-0.784591\pi\)
0.779626 0.626245i \(-0.215409\pi\)
\(174\) 0 0
\(175\) 127.276 0.727294
\(176\) − 126.218i − 0.717147i
\(177\) 0 0
\(178\) 56.8767 0.319532
\(179\) − 49.1906i − 0.274808i −0.990515 0.137404i \(-0.956124\pi\)
0.990515 0.137404i \(-0.0438758\pi\)
\(180\) 0 0
\(181\) −273.724 −1.51229 −0.756143 0.654407i \(-0.772918\pi\)
−0.756143 + 0.654407i \(0.772918\pi\)
\(182\) 100.992i 0.554901i
\(183\) 0 0
\(184\) −95.6476 −0.519824
\(185\) − 20.4389i − 0.110481i
\(186\) 0 0
\(187\) −160.432 −0.857925
\(188\) 20.1324i 0.107087i
\(189\) 0 0
\(190\) 400.974 2.11039
\(191\) − 28.6902i − 0.150211i −0.997176 0.0751054i \(-0.976071\pi\)
0.997176 0.0751054i \(-0.0239293\pi\)
\(192\) 0 0
\(193\) 198.332 1.02763 0.513814 0.857901i \(-0.328232\pi\)
0.513814 + 0.857901i \(0.328232\pi\)
\(194\) 54.5608i 0.281241i
\(195\) 0 0
\(196\) 152.762 0.779400
\(197\) 236.281i 1.19940i 0.800227 + 0.599698i \(0.204712\pi\)
−0.800227 + 0.599698i \(0.795288\pi\)
\(198\) 0 0
\(199\) 143.294 0.720070 0.360035 0.932939i \(-0.382765\pi\)
0.360035 + 0.932939i \(0.382765\pi\)
\(200\) 85.2401i 0.426201i
\(201\) 0 0
\(202\) 407.525 2.01745
\(203\) 100.173i 0.493465i
\(204\) 0 0
\(205\) −120.796 −0.589248
\(206\) 194.238i 0.942901i
\(207\) 0 0
\(208\) 89.2883 0.429271
\(209\) 202.526i 0.969022i
\(210\) 0 0
\(211\) −24.6190 −0.116678 −0.0583388 0.998297i \(-0.518580\pi\)
−0.0583388 + 0.998297i \(0.518580\pi\)
\(212\) 367.933i 1.73553i
\(213\) 0 0
\(214\) 581.136 2.71559
\(215\) − 358.239i − 1.66623i
\(216\) 0 0
\(217\) −187.359 −0.863407
\(218\) 465.810i 2.13674i
\(219\) 0 0
\(220\) −410.775 −1.86716
\(221\) − 113.492i − 0.513538i
\(222\) 0 0
\(223\) 120.070 0.538431 0.269215 0.963080i \(-0.413236\pi\)
0.269215 + 0.963080i \(0.413236\pi\)
\(224\) 192.438i 0.859100i
\(225\) 0 0
\(226\) −36.1190 −0.159819
\(227\) 41.6972i 0.183688i 0.995773 + 0.0918442i \(0.0292762\pi\)
−0.995773 + 0.0918442i \(0.970724\pi\)
\(228\) 0 0
\(229\) −256.974 −1.12216 −0.561078 0.827763i \(-0.689613\pi\)
−0.561078 + 0.827763i \(0.689613\pi\)
\(230\) − 742.738i − 3.22930i
\(231\) 0 0
\(232\) −67.0886 −0.289175
\(233\) − 259.489i − 1.11369i −0.830617 0.556844i \(-0.812012\pi\)
0.830617 0.556844i \(-0.187988\pi\)
\(234\) 0 0
\(235\) −30.1021 −0.128094
\(236\) − 274.753i − 1.16421i
\(237\) 0 0
\(238\) 182.757 0.767886
\(239\) 180.742i 0.756242i 0.925756 + 0.378121i \(0.123430\pi\)
−0.925756 + 0.378121i \(0.876570\pi\)
\(240\) 0 0
\(241\) −293.575 −1.21816 −0.609078 0.793111i \(-0.708460\pi\)
−0.609078 + 0.793111i \(0.708460\pi\)
\(242\) − 12.9350i − 0.0534505i
\(243\) 0 0
\(244\) −224.416 −0.919739
\(245\) 228.412i 0.932294i
\(246\) 0 0
\(247\) −143.270 −0.580039
\(248\) − 125.479i − 0.505964i
\(249\) 0 0
\(250\) −107.817 −0.431268
\(251\) − 330.675i − 1.31743i −0.752393 0.658714i \(-0.771101\pi\)
0.752393 0.658714i \(-0.228899\pi\)
\(252\) 0 0
\(253\) 375.146 1.48279
\(254\) 33.7215i 0.132762i
\(255\) 0 0
\(256\) 94.5327 0.369268
\(257\) 281.301i 1.09456i 0.836950 + 0.547279i \(0.184336\pi\)
−0.836950 + 0.547279i \(0.815664\pi\)
\(258\) 0 0
\(259\) −11.7599 −0.0454052
\(260\) − 290.588i − 1.11765i
\(261\) 0 0
\(262\) −39.2649 −0.149866
\(263\) − 211.737i − 0.805085i −0.915401 0.402543i \(-0.868127\pi\)
0.915401 0.402543i \(-0.131873\pi\)
\(264\) 0 0
\(265\) −550.136 −2.07599
\(266\) − 230.708i − 0.867324i
\(267\) 0 0
\(268\) −615.844 −2.29792
\(269\) 72.9266i 0.271103i 0.990770 + 0.135551i \(0.0432806\pi\)
−0.990770 + 0.135551i \(0.956719\pi\)
\(270\) 0 0
\(271\) −435.095 −1.60552 −0.802759 0.596304i \(-0.796635\pi\)
−0.802759 + 0.596304i \(0.796635\pi\)
\(272\) − 161.578i − 0.594036i
\(273\) 0 0
\(274\) −172.903 −0.631032
\(275\) − 334.326i − 1.21573i
\(276\) 0 0
\(277\) 46.5294 0.167976 0.0839881 0.996467i \(-0.473234\pi\)
0.0839881 + 0.996467i \(0.473234\pi\)
\(278\) − 273.499i − 0.983808i
\(279\) 0 0
\(280\) 90.1006 0.321788
\(281\) 392.596i 1.39714i 0.715543 + 0.698569i \(0.246179\pi\)
−0.715543 + 0.698569i \(0.753821\pi\)
\(282\) 0 0
\(283\) 204.724 0.723405 0.361702 0.932294i \(-0.382196\pi\)
0.361702 + 0.932294i \(0.382196\pi\)
\(284\) − 577.372i − 2.03300i
\(285\) 0 0
\(286\) 265.283 0.927562
\(287\) 69.5024i 0.242168i
\(288\) 0 0
\(289\) 83.6230 0.289353
\(290\) − 520.967i − 1.79644i
\(291\) 0 0
\(292\) −462.618 −1.58431
\(293\) − 108.950i − 0.371843i −0.982565 0.185921i \(-0.940473\pi\)
0.982565 0.185921i \(-0.0595270\pi\)
\(294\) 0 0
\(295\) 410.813 1.39259
\(296\) − 7.87592i − 0.0266078i
\(297\) 0 0
\(298\) −19.9987 −0.0671096
\(299\) 265.384i 0.887570i
\(300\) 0 0
\(301\) −206.120 −0.684784
\(302\) 737.854i 2.44323i
\(303\) 0 0
\(304\) −203.972 −0.670961
\(305\) − 335.549i − 1.10016i
\(306\) 0 0
\(307\) −83.1949 −0.270993 −0.135497 0.990778i \(-0.543263\pi\)
−0.135497 + 0.990778i \(0.543263\pi\)
\(308\) 236.348i 0.767363i
\(309\) 0 0
\(310\) 974.390 3.14319
\(311\) − 157.785i − 0.507347i −0.967290 0.253673i \(-0.918361\pi\)
0.967290 0.253673i \(-0.0816389\pi\)
\(312\) 0 0
\(313\) 522.092 1.66802 0.834012 0.551746i \(-0.186038\pi\)
0.834012 + 0.551746i \(0.186038\pi\)
\(314\) 217.654i 0.693167i
\(315\) 0 0
\(316\) 14.4793 0.0458207
\(317\) 524.888i 1.65580i 0.560876 + 0.827900i \(0.310464\pi\)
−0.560876 + 0.827900i \(0.689536\pi\)
\(318\) 0 0
\(319\) 263.133 0.824867
\(320\) − 666.755i − 2.08361i
\(321\) 0 0
\(322\) −427.349 −1.32717
\(323\) 259.263i 0.802672i
\(324\) 0 0
\(325\) 236.507 0.727714
\(326\) − 494.084i − 1.51560i
\(327\) 0 0
\(328\) −46.5474 −0.141913
\(329\) 17.3199i 0.0526440i
\(330\) 0 0
\(331\) 65.0345 0.196479 0.0982394 0.995163i \(-0.468679\pi\)
0.0982394 + 0.995163i \(0.468679\pi\)
\(332\) − 358.294i − 1.07920i
\(333\) 0 0
\(334\) 379.783 1.13708
\(335\) − 920.815i − 2.74870i
\(336\) 0 0
\(337\) −290.810 −0.862937 −0.431469 0.902128i \(-0.642004\pi\)
−0.431469 + 0.902128i \(0.642004\pi\)
\(338\) − 318.034i − 0.940928i
\(339\) 0 0
\(340\) −525.854 −1.54663
\(341\) 492.150i 1.44325i
\(342\) 0 0
\(343\) 340.250 0.991981
\(344\) − 138.044i − 0.401289i
\(345\) 0 0
\(346\) 648.374 1.87391
\(347\) 80.9729i 0.233351i 0.993170 + 0.116676i \(0.0372238\pi\)
−0.993170 + 0.116676i \(0.962776\pi\)
\(348\) 0 0
\(349\) 388.559 1.11335 0.556675 0.830730i \(-0.312077\pi\)
0.556675 + 0.830730i \(0.312077\pi\)
\(350\) 380.849i 1.08814i
\(351\) 0 0
\(352\) 505.491 1.43606
\(353\) 529.417i 1.49977i 0.661570 + 0.749883i \(0.269890\pi\)
−0.661570 + 0.749883i \(0.730110\pi\)
\(354\) 0 0
\(355\) 863.292 2.43181
\(356\) 94.1614i 0.264498i
\(357\) 0 0
\(358\) 147.193 0.411154
\(359\) − 488.615i − 1.36105i −0.732727 0.680523i \(-0.761753\pi\)
0.732727 0.680523i \(-0.238247\pi\)
\(360\) 0 0
\(361\) −33.7122 −0.0933855
\(362\) − 819.063i − 2.26261i
\(363\) 0 0
\(364\) −167.196 −0.459329
\(365\) − 691.711i − 1.89510i
\(366\) 0 0
\(367\) −289.731 −0.789458 −0.394729 0.918798i \(-0.629162\pi\)
−0.394729 + 0.918798i \(0.629162\pi\)
\(368\) 377.825i 1.02670i
\(369\) 0 0
\(370\) 61.1593 0.165296
\(371\) 316.532i 0.853186i
\(372\) 0 0
\(373\) 523.958 1.40471 0.702356 0.711826i \(-0.252131\pi\)
0.702356 + 0.711826i \(0.252131\pi\)
\(374\) − 480.061i − 1.28358i
\(375\) 0 0
\(376\) −11.5995 −0.0308498
\(377\) 186.144i 0.493750i
\(378\) 0 0
\(379\) −716.027 −1.88925 −0.944626 0.328148i \(-0.893576\pi\)
−0.944626 + 0.328148i \(0.893576\pi\)
\(380\) 663.826i 1.74691i
\(381\) 0 0
\(382\) 85.8498 0.224738
\(383\) − 364.557i − 0.951847i −0.879487 0.475923i \(-0.842114\pi\)
0.879487 0.475923i \(-0.157886\pi\)
\(384\) 0 0
\(385\) −353.389 −0.917895
\(386\) 593.470i 1.53749i
\(387\) 0 0
\(388\) −90.3273 −0.232802
\(389\) − 13.8459i − 0.0355937i −0.999842 0.0177968i \(-0.994335\pi\)
0.999842 0.0177968i \(-0.00566521\pi\)
\(390\) 0 0
\(391\) 480.243 1.22824
\(392\) 88.0161i 0.224531i
\(393\) 0 0
\(394\) −707.023 −1.79447
\(395\) 21.6496i 0.0548092i
\(396\) 0 0
\(397\) −370.702 −0.933757 −0.466879 0.884321i \(-0.654622\pi\)
−0.466879 + 0.884321i \(0.654622\pi\)
\(398\) 428.779i 1.07733i
\(399\) 0 0
\(400\) 336.714 0.841784
\(401\) 95.5302i 0.238230i 0.992880 + 0.119115i \(0.0380057\pi\)
−0.992880 + 0.119115i \(0.961994\pi\)
\(402\) 0 0
\(403\) −348.154 −0.863905
\(404\) 674.672i 1.66998i
\(405\) 0 0
\(406\) −299.749 −0.738298
\(407\) 30.8906i 0.0758984i
\(408\) 0 0
\(409\) 182.733 0.446779 0.223390 0.974729i \(-0.428288\pi\)
0.223390 + 0.974729i \(0.428288\pi\)
\(410\) − 361.458i − 0.881604i
\(411\) 0 0
\(412\) −321.567 −0.780503
\(413\) − 236.370i − 0.572324i
\(414\) 0 0
\(415\) 535.725 1.29090
\(416\) 357.592i 0.859596i
\(417\) 0 0
\(418\) −606.017 −1.44980
\(419\) − 811.040i − 1.93566i −0.251611 0.967829i \(-0.580960\pi\)
0.251611 0.967829i \(-0.419040\pi\)
\(420\) 0 0
\(421\) −448.621 −1.06561 −0.532804 0.846239i \(-0.678862\pi\)
−0.532804 + 0.846239i \(0.678862\pi\)
\(422\) − 73.6673i − 0.174567i
\(423\) 0 0
\(424\) −211.989 −0.499975
\(425\) − 427.987i − 1.00703i
\(426\) 0 0
\(427\) −193.065 −0.452143
\(428\) 962.092i 2.24788i
\(429\) 0 0
\(430\) 1071.96 2.49293
\(431\) 205.939i 0.477816i 0.971042 + 0.238908i \(0.0767895\pi\)
−0.971042 + 0.238908i \(0.923211\pi\)
\(432\) 0 0
\(433\) 427.090 0.986350 0.493175 0.869930i \(-0.335836\pi\)
0.493175 + 0.869930i \(0.335836\pi\)
\(434\) − 560.635i − 1.29179i
\(435\) 0 0
\(436\) −771.165 −1.76873
\(437\) − 606.247i − 1.38729i
\(438\) 0 0
\(439\) −675.407 −1.53851 −0.769256 0.638941i \(-0.779373\pi\)
−0.769256 + 0.638941i \(0.779373\pi\)
\(440\) − 236.674i − 0.537895i
\(441\) 0 0
\(442\) 339.602 0.768330
\(443\) 85.6791i 0.193406i 0.995313 + 0.0967032i \(0.0308298\pi\)
−0.995313 + 0.0967032i \(0.969170\pi\)
\(444\) 0 0
\(445\) −140.791 −0.316384
\(446\) 359.285i 0.805573i
\(447\) 0 0
\(448\) −383.631 −0.856319
\(449\) − 416.580i − 0.927795i −0.885889 0.463897i \(-0.846451\pi\)
0.885889 0.463897i \(-0.153549\pi\)
\(450\) 0 0
\(451\) 182.567 0.404804
\(452\) − 59.7963i − 0.132293i
\(453\) 0 0
\(454\) −124.771 −0.274825
\(455\) − 249.993i − 0.549435i
\(456\) 0 0
\(457\) −185.518 −0.405947 −0.202974 0.979184i \(-0.565061\pi\)
−0.202974 + 0.979184i \(0.565061\pi\)
\(458\) − 768.942i − 1.67891i
\(459\) 0 0
\(460\) 1229.63 2.67311
\(461\) 334.030i 0.724578i 0.932066 + 0.362289i \(0.118005\pi\)
−0.932066 + 0.362289i \(0.881995\pi\)
\(462\) 0 0
\(463\) 490.710 1.05985 0.529924 0.848045i \(-0.322220\pi\)
0.529924 + 0.848045i \(0.322220\pi\)
\(464\) 265.012i 0.571146i
\(465\) 0 0
\(466\) 776.470 1.66624
\(467\) − 656.431i − 1.40563i −0.711371 0.702817i \(-0.751925\pi\)
0.711371 0.702817i \(-0.248075\pi\)
\(468\) 0 0
\(469\) −529.810 −1.12966
\(470\) − 90.0746i − 0.191648i
\(471\) 0 0
\(472\) 158.303 0.335387
\(473\) 541.430i 1.14467i
\(474\) 0 0
\(475\) −540.281 −1.13743
\(476\) 302.561i 0.635632i
\(477\) 0 0
\(478\) −540.834 −1.13145
\(479\) 376.859i 0.786762i 0.919375 + 0.393381i \(0.128695\pi\)
−0.919375 + 0.393381i \(0.871305\pi\)
\(480\) 0 0
\(481\) −21.8525 −0.0454314
\(482\) − 878.466i − 1.82254i
\(483\) 0 0
\(484\) 21.4144 0.0442446
\(485\) − 135.058i − 0.278471i
\(486\) 0 0
\(487\) 138.419 0.284228 0.142114 0.989850i \(-0.454610\pi\)
0.142114 + 0.989850i \(0.454610\pi\)
\(488\) − 129.300i − 0.264960i
\(489\) 0 0
\(490\) −683.477 −1.39485
\(491\) − 833.517i − 1.69759i −0.528722 0.848795i \(-0.677329\pi\)
0.528722 0.848795i \(-0.322671\pi\)
\(492\) 0 0
\(493\) 336.849 0.683264
\(494\) − 428.705i − 0.867825i
\(495\) 0 0
\(496\) −495.665 −0.999324
\(497\) − 496.712i − 0.999422i
\(498\) 0 0
\(499\) −557.277 −1.11679 −0.558394 0.829576i \(-0.688582\pi\)
−0.558394 + 0.829576i \(0.688582\pi\)
\(500\) − 178.495i − 0.356990i
\(501\) 0 0
\(502\) 989.478 1.97107
\(503\) − 32.4506i − 0.0645140i −0.999480 0.0322570i \(-0.989730\pi\)
0.999480 0.0322570i \(-0.0102695\pi\)
\(504\) 0 0
\(505\) −1008.78 −1.99758
\(506\) 1122.55i 2.21848i
\(507\) 0 0
\(508\) −55.8272 −0.109896
\(509\) − 411.171i − 0.807802i −0.914803 0.403901i \(-0.867654\pi\)
0.914803 0.403901i \(-0.132346\pi\)
\(510\) 0 0
\(511\) −397.990 −0.778845
\(512\) 637.824i 1.24575i
\(513\) 0 0
\(514\) −841.738 −1.63762
\(515\) − 480.811i − 0.933613i
\(516\) 0 0
\(517\) 45.4953 0.0879987
\(518\) − 35.1893i − 0.0679329i
\(519\) 0 0
\(520\) 167.426 0.321974
\(521\) 740.959i 1.42219i 0.703098 + 0.711093i \(0.251800\pi\)
−0.703098 + 0.711093i \(0.748200\pi\)
\(522\) 0 0
\(523\) 398.655 0.762248 0.381124 0.924524i \(-0.375537\pi\)
0.381124 + 0.924524i \(0.375537\pi\)
\(524\) − 65.0044i − 0.124054i
\(525\) 0 0
\(526\) 633.582 1.20453
\(527\) 630.025i 1.19549i
\(528\) 0 0
\(529\) −593.975 −1.12283
\(530\) − 1646.17i − 3.10599i
\(531\) 0 0
\(532\) 381.946 0.717943
\(533\) 129.150i 0.242308i
\(534\) 0 0
\(535\) −1438.53 −2.68884
\(536\) − 354.826i − 0.661990i
\(537\) 0 0
\(538\) −218.218 −0.405610
\(539\) − 345.214i − 0.640471i
\(540\) 0 0
\(541\) 164.327 0.303747 0.151873 0.988400i \(-0.451469\pi\)
0.151873 + 0.988400i \(0.451469\pi\)
\(542\) − 1301.94i − 2.40209i
\(543\) 0 0
\(544\) 647.105 1.18953
\(545\) − 1153.05i − 2.11569i
\(546\) 0 0
\(547\) −503.882 −0.921174 −0.460587 0.887614i \(-0.652361\pi\)
−0.460587 + 0.887614i \(0.652361\pi\)
\(548\) − 286.247i − 0.522348i
\(549\) 0 0
\(550\) 1000.40 1.81891
\(551\) − 425.231i − 0.771743i
\(552\) 0 0
\(553\) 12.4566 0.0225254
\(554\) 139.230i 0.251318i
\(555\) 0 0
\(556\) 452.787 0.814365
\(557\) 217.504i 0.390492i 0.980754 + 0.195246i \(0.0625505\pi\)
−0.980754 + 0.195246i \(0.937450\pi\)
\(558\) 0 0
\(559\) −383.015 −0.685179
\(560\) − 355.913i − 0.635560i
\(561\) 0 0
\(562\) −1174.76 −2.09033
\(563\) 522.367i 0.927827i 0.885880 + 0.463914i \(0.153555\pi\)
−0.885880 + 0.463914i \(0.846445\pi\)
\(564\) 0 0
\(565\) 89.4080 0.158244
\(566\) 612.594i 1.08232i
\(567\) 0 0
\(568\) 332.661 0.585670
\(569\) 28.4538i 0.0500066i 0.999687 + 0.0250033i \(0.00795963\pi\)
−0.999687 + 0.0250033i \(0.992040\pi\)
\(570\) 0 0
\(571\) −302.891 −0.530457 −0.265228 0.964186i \(-0.585447\pi\)
−0.265228 + 0.964186i \(0.585447\pi\)
\(572\) 439.185i 0.767806i
\(573\) 0 0
\(574\) −207.972 −0.362320
\(575\) 1000.78i 1.74049i
\(576\) 0 0
\(577\) 1061.82 1.84024 0.920121 0.391634i \(-0.128090\pi\)
0.920121 + 0.391634i \(0.128090\pi\)
\(578\) 250.225i 0.432915i
\(579\) 0 0
\(580\) 862.480 1.48703
\(581\) − 308.240i − 0.530534i
\(582\) 0 0
\(583\) 831.457 1.42617
\(584\) − 266.544i − 0.456410i
\(585\) 0 0
\(586\) 326.011 0.556333
\(587\) 718.632i 1.22425i 0.790763 + 0.612123i \(0.209684\pi\)
−0.790763 + 0.612123i \(0.790316\pi\)
\(588\) 0 0
\(589\) 795.330 1.35030
\(590\) 1229.28i 2.08352i
\(591\) 0 0
\(592\) −31.1113 −0.0525528
\(593\) 88.2527i 0.148824i 0.997228 + 0.0744121i \(0.0237080\pi\)
−0.997228 + 0.0744121i \(0.976292\pi\)
\(594\) 0 0
\(595\) −452.392 −0.760322
\(596\) − 33.1085i − 0.0555512i
\(597\) 0 0
\(598\) −794.107 −1.32794
\(599\) − 719.971i − 1.20196i −0.799266 0.600978i \(-0.794778\pi\)
0.799266 0.600978i \(-0.205222\pi\)
\(600\) 0 0
\(601\) 251.466 0.418413 0.209206 0.977872i \(-0.432912\pi\)
0.209206 + 0.977872i \(0.432912\pi\)
\(602\) − 616.773i − 1.02454i
\(603\) 0 0
\(604\) −1221.54 −2.02242
\(605\) 32.0190i 0.0529239i
\(606\) 0 0
\(607\) −124.195 −0.204605 −0.102302 0.994753i \(-0.532621\pi\)
−0.102302 + 0.994753i \(0.532621\pi\)
\(608\) − 816.890i − 1.34357i
\(609\) 0 0
\(610\) 1004.06 1.64601
\(611\) 32.1840i 0.0526744i
\(612\) 0 0
\(613\) 1001.06 1.63305 0.816525 0.577310i \(-0.195898\pi\)
0.816525 + 0.577310i \(0.195898\pi\)
\(614\) − 248.944i − 0.405447i
\(615\) 0 0
\(616\) −136.175 −0.221063
\(617\) 222.659i 0.360873i 0.983587 + 0.180437i \(0.0577511\pi\)
−0.983587 + 0.180437i \(0.942249\pi\)
\(618\) 0 0
\(619\) 1015.62 1.64074 0.820372 0.571830i \(-0.193766\pi\)
0.820372 + 0.571830i \(0.193766\pi\)
\(620\) 1613.14i 2.60183i
\(621\) 0 0
\(622\) 472.140 0.759067
\(623\) 81.0070i 0.130027i
\(624\) 0 0
\(625\) −479.725 −0.767560
\(626\) 1562.25i 2.49561i
\(627\) 0 0
\(628\) −360.335 −0.573781
\(629\) 39.5447i 0.0628691i
\(630\) 0 0
\(631\) 1155.66 1.83148 0.915740 0.401772i \(-0.131605\pi\)
0.915740 + 0.401772i \(0.131605\pi\)
\(632\) 8.34247i 0.0132001i
\(633\) 0 0
\(634\) −1570.62 −2.47732
\(635\) − 83.4733i − 0.131454i
\(636\) 0 0
\(637\) 244.209 0.383374
\(638\) 787.372i 1.23412i
\(639\) 0 0
\(640\) 657.291 1.02702
\(641\) − 805.647i − 1.25686i −0.777867 0.628430i \(-0.783698\pi\)
0.777867 0.628430i \(-0.216302\pi\)
\(642\) 0 0
\(643\) −949.145 −1.47612 −0.738060 0.674736i \(-0.764258\pi\)
−0.738060 + 0.674736i \(0.764258\pi\)
\(644\) − 707.492i − 1.09859i
\(645\) 0 0
\(646\) −775.793 −1.20092
\(647\) 816.742i 1.26235i 0.775639 + 0.631176i \(0.217428\pi\)
−0.775639 + 0.631176i \(0.782572\pi\)
\(648\) 0 0
\(649\) −620.889 −0.956686
\(650\) 707.700i 1.08877i
\(651\) 0 0
\(652\) 817.974 1.25456
\(653\) 1086.25i 1.66347i 0.555174 + 0.831735i \(0.312652\pi\)
−0.555174 + 0.831735i \(0.687348\pi\)
\(654\) 0 0
\(655\) 97.1952 0.148390
\(656\) 183.871i 0.280291i
\(657\) 0 0
\(658\) −51.8262 −0.0787633
\(659\) 277.890i 0.421685i 0.977520 + 0.210842i \(0.0676207\pi\)
−0.977520 + 0.210842i \(0.932379\pi\)
\(660\) 0 0
\(661\) 161.745 0.244697 0.122349 0.992487i \(-0.460957\pi\)
0.122349 + 0.992487i \(0.460957\pi\)
\(662\) 194.603i 0.293962i
\(663\) 0 0
\(664\) 206.436 0.310898
\(665\) 571.089i 0.858780i
\(666\) 0 0
\(667\) −787.671 −1.18092
\(668\) 628.745i 0.941235i
\(669\) 0 0
\(670\) 2755.36 4.11247
\(671\) 507.138i 0.755794i
\(672\) 0 0
\(673\) −1234.51 −1.83434 −0.917168 0.398501i \(-0.869531\pi\)
−0.917168 + 0.398501i \(0.869531\pi\)
\(674\) − 870.190i − 1.29108i
\(675\) 0 0
\(676\) 526.516 0.778870
\(677\) − 1105.68i − 1.63321i −0.577196 0.816606i \(-0.695853\pi\)
0.577196 0.816606i \(-0.304147\pi\)
\(678\) 0 0
\(679\) −77.7085 −0.114445
\(680\) − 302.978i − 0.445555i
\(681\) 0 0
\(682\) −1472.66 −2.15932
\(683\) 276.581i 0.404950i 0.979287 + 0.202475i \(0.0648985\pi\)
−0.979287 + 0.202475i \(0.935101\pi\)
\(684\) 0 0
\(685\) 427.999 0.624816
\(686\) 1018.13i 1.48415i
\(687\) 0 0
\(688\) −545.297 −0.792582
\(689\) 588.185i 0.853679i
\(690\) 0 0
\(691\) −540.511 −0.782215 −0.391108 0.920345i \(-0.627908\pi\)
−0.391108 + 0.920345i \(0.627908\pi\)
\(692\) 1073.41i 1.55117i
\(693\) 0 0
\(694\) −242.295 −0.349128
\(695\) 677.011i 0.974117i
\(696\) 0 0
\(697\) 233.713 0.335312
\(698\) 1162.69i 1.66574i
\(699\) 0 0
\(700\) −630.509 −0.900728
\(701\) − 133.789i − 0.190855i −0.995436 0.0954276i \(-0.969578\pi\)
0.995436 0.0954276i \(-0.0304218\pi\)
\(702\) 0 0
\(703\) 49.9203 0.0710103
\(704\) 1007.71i 1.43141i
\(705\) 0 0
\(706\) −1584.18 −2.24388
\(707\) 580.420i 0.820962i
\(708\) 0 0
\(709\) 376.567 0.531124 0.265562 0.964094i \(-0.414443\pi\)
0.265562 + 0.964094i \(0.414443\pi\)
\(710\) 2583.23i 3.63835i
\(711\) 0 0
\(712\) −54.2523 −0.0761971
\(713\) − 1473.22i − 2.06623i
\(714\) 0 0
\(715\) −656.674 −0.918425
\(716\) 243.683i 0.340340i
\(717\) 0 0
\(718\) 1462.08 2.03633
\(719\) − 57.6348i − 0.0801596i −0.999196 0.0400798i \(-0.987239\pi\)
0.999196 0.0400798i \(-0.0127612\pi\)
\(720\) 0 0
\(721\) −276.644 −0.383695
\(722\) − 100.877i − 0.139719i
\(723\) 0 0
\(724\) 1355.99 1.87291
\(725\) 701.964i 0.968226i
\(726\) 0 0
\(727\) 341.495 0.469732 0.234866 0.972028i \(-0.424535\pi\)
0.234866 + 0.972028i \(0.424535\pi\)
\(728\) − 96.3321i − 0.132324i
\(729\) 0 0
\(730\) 2069.81 2.83535
\(731\) 693.111i 0.948169i
\(732\) 0 0
\(733\) −442.745 −0.604018 −0.302009 0.953305i \(-0.597657\pi\)
−0.302009 + 0.953305i \(0.597657\pi\)
\(734\) − 866.962i − 1.18115i
\(735\) 0 0
\(736\) −1513.16 −2.05592
\(737\) 1391.69i 1.88831i
\(738\) 0 0
\(739\) −152.397 −0.206220 −0.103110 0.994670i \(-0.532879\pi\)
−0.103110 + 0.994670i \(0.532879\pi\)
\(740\) 101.251i 0.136826i
\(741\) 0 0
\(742\) −947.159 −1.27649
\(743\) 143.763i 0.193490i 0.995309 + 0.0967449i \(0.0308431\pi\)
−0.995309 + 0.0967449i \(0.969157\pi\)
\(744\) 0 0
\(745\) 49.5042 0.0664485
\(746\) 1567.84i 2.10166i
\(747\) 0 0
\(748\) 794.757 1.06251
\(749\) 827.687i 1.10506i
\(750\) 0 0
\(751\) −721.144 −0.960244 −0.480122 0.877202i \(-0.659408\pi\)
−0.480122 + 0.877202i \(0.659408\pi\)
\(752\) 45.8202i 0.0609312i
\(753\) 0 0
\(754\) −556.998 −0.738724
\(755\) − 1826.46i − 2.41916i
\(756\) 0 0
\(757\) 483.778 0.639073 0.319536 0.947574i \(-0.396473\pi\)
0.319536 + 0.947574i \(0.396473\pi\)
\(758\) − 2142.57i − 2.82661i
\(759\) 0 0
\(760\) −382.472 −0.503253
\(761\) 418.522i 0.549963i 0.961450 + 0.274981i \(0.0886717\pi\)
−0.961450 + 0.274981i \(0.911328\pi\)
\(762\) 0 0
\(763\) −663.433 −0.869505
\(764\) 142.127i 0.186031i
\(765\) 0 0
\(766\) 1090.86 1.42411
\(767\) − 439.226i − 0.572654i
\(768\) 0 0
\(769\) −617.073 −0.802436 −0.401218 0.915983i \(-0.631413\pi\)
−0.401218 + 0.915983i \(0.631413\pi\)
\(770\) − 1057.45i − 1.37331i
\(771\) 0 0
\(772\) −982.510 −1.27268
\(773\) − 82.0130i − 0.106097i −0.998592 0.0530485i \(-0.983106\pi\)
0.998592 0.0530485i \(-0.0168938\pi\)
\(774\) 0 0
\(775\) −1312.92 −1.69409
\(776\) − 52.0433i − 0.0670661i
\(777\) 0 0
\(778\) 41.4312 0.0532535
\(779\) − 295.034i − 0.378734i
\(780\) 0 0
\(781\) −1304.75 −1.67061
\(782\) 1437.03i 1.83764i
\(783\) 0 0
\(784\) 347.679 0.443468
\(785\) − 538.776i − 0.686339i
\(786\) 0 0
\(787\) −384.669 −0.488779 −0.244389 0.969677i \(-0.578588\pi\)
−0.244389 + 0.969677i \(0.578588\pi\)
\(788\) − 1170.50i − 1.48541i
\(789\) 0 0
\(790\) −64.7822 −0.0820028
\(791\) − 51.4427i − 0.0650350i
\(792\) 0 0
\(793\) −358.756 −0.452404
\(794\) − 1109.25i − 1.39704i
\(795\) 0 0
\(796\) −709.858 −0.891782
\(797\) 4.35189i 0.00546033i 0.999996 + 0.00273017i \(0.000869040\pi\)
−0.999996 + 0.00273017i \(0.999131\pi\)
\(798\) 0 0
\(799\) 58.2408 0.0728921
\(800\) 1348.51i 1.68564i
\(801\) 0 0
\(802\) −285.855 −0.356428
\(803\) 1045.43i 1.30190i
\(804\) 0 0
\(805\) 1057.85 1.31410
\(806\) − 1041.78i − 1.29253i
\(807\) 0 0
\(808\) −388.721 −0.481091
\(809\) 147.197i 0.181949i 0.995853 + 0.0909745i \(0.0289982\pi\)
−0.995853 + 0.0909745i \(0.971002\pi\)
\(810\) 0 0
\(811\) 428.815 0.528749 0.264375 0.964420i \(-0.414835\pi\)
0.264375 + 0.964420i \(0.414835\pi\)
\(812\) − 496.245i − 0.611139i
\(813\) 0 0
\(814\) −92.4341 −0.113555
\(815\) 1223.04i 1.50067i
\(816\) 0 0
\(817\) 874.968 1.07095
\(818\) 546.791i 0.668449i
\(819\) 0 0
\(820\) 598.406 0.729763
\(821\) 1041.06i 1.26803i 0.773319 + 0.634017i \(0.218595\pi\)
−0.773319 + 0.634017i \(0.781405\pi\)
\(822\) 0 0
\(823\) 688.322 0.836357 0.418178 0.908365i \(-0.362669\pi\)
0.418178 + 0.908365i \(0.362669\pi\)
\(824\) − 185.275i − 0.224849i
\(825\) 0 0
\(826\) 707.289 0.856282
\(827\) − 1196.75i − 1.44710i −0.690274 0.723548i \(-0.742510\pi\)
0.690274 0.723548i \(-0.257490\pi\)
\(828\) 0 0
\(829\) 534.150 0.644330 0.322165 0.946683i \(-0.395589\pi\)
0.322165 + 0.946683i \(0.395589\pi\)
\(830\) 1603.05i 1.93139i
\(831\) 0 0
\(832\) −712.869 −0.856814
\(833\) − 441.925i − 0.530523i
\(834\) 0 0
\(835\) −940.105 −1.12587
\(836\) − 1003.28i − 1.20010i
\(837\) 0 0
\(838\) 2426.88 2.89603
\(839\) − 1019.14i − 1.21471i −0.794430 0.607356i \(-0.792230\pi\)
0.794430 0.607356i \(-0.207770\pi\)
\(840\) 0 0
\(841\) 288.517 0.343064
\(842\) − 1342.41i − 1.59431i
\(843\) 0 0
\(844\) 121.959 0.144501
\(845\) 787.252i 0.931659i
\(846\) 0 0
\(847\) 18.4228 0.0217506
\(848\) 837.395i 0.987494i
\(849\) 0 0
\(850\) 1280.67 1.50667
\(851\) − 92.4692i − 0.108659i
\(852\) 0 0
\(853\) 1551.01 1.81830 0.909152 0.416465i \(-0.136731\pi\)
0.909152 + 0.416465i \(0.136731\pi\)
\(854\) − 577.709i − 0.676474i
\(855\) 0 0
\(856\) −554.322 −0.647573
\(857\) − 59.4182i − 0.0693328i −0.999399 0.0346664i \(-0.988963\pi\)
0.999399 0.0346664i \(-0.0110369\pi\)
\(858\) 0 0
\(859\) 1306.85 1.52136 0.760681 0.649126i \(-0.224865\pi\)
0.760681 + 0.649126i \(0.224865\pi\)
\(860\) 1774.66i 2.06356i
\(861\) 0 0
\(862\) −616.231 −0.714885
\(863\) 183.653i 0.212807i 0.994323 + 0.106404i \(0.0339335\pi\)
−0.994323 + 0.106404i \(0.966066\pi\)
\(864\) 0 0
\(865\) −1604.97 −1.85545
\(866\) 1277.98i 1.47573i
\(867\) 0 0
\(868\) 928.151 1.06930
\(869\) − 32.7205i − 0.0376531i
\(870\) 0 0
\(871\) −984.500 −1.13031
\(872\) − 444.317i − 0.509538i
\(873\) 0 0
\(874\) 1814.07 2.07560
\(875\) − 153.559i − 0.175496i
\(876\) 0 0
\(877\) −313.408 −0.357364 −0.178682 0.983907i \(-0.557183\pi\)
−0.178682 + 0.983907i \(0.557183\pi\)
\(878\) − 2021.02i − 2.30184i
\(879\) 0 0
\(880\) −934.903 −1.06239
\(881\) 1370.61i 1.55575i 0.628421 + 0.777874i \(0.283702\pi\)
−0.628421 + 0.777874i \(0.716298\pi\)
\(882\) 0 0
\(883\) 1403.11 1.58903 0.794514 0.607246i \(-0.207726\pi\)
0.794514 + 0.607246i \(0.207726\pi\)
\(884\) 562.223i 0.635999i
\(885\) 0 0
\(886\) −256.377 −0.289365
\(887\) 1565.83i 1.76531i 0.470018 + 0.882657i \(0.344248\pi\)
−0.470018 + 0.882657i \(0.655752\pi\)
\(888\) 0 0
\(889\) −48.0280 −0.0540248
\(890\) − 421.289i − 0.473358i
\(891\) 0 0
\(892\) −594.810 −0.666827
\(893\) − 73.5219i − 0.0823313i
\(894\) 0 0
\(895\) −364.357 −0.407103
\(896\) − 378.186i − 0.422082i
\(897\) 0 0
\(898\) 1246.53 1.38812
\(899\) − 1033.34i − 1.14943i
\(900\) 0 0
\(901\) 1064.39 1.18134
\(902\) 546.294i 0.605648i
\(903\) 0 0
\(904\) 34.4524 0.0381111
\(905\) 2027.49i 2.24032i
\(906\) 0 0
\(907\) 1160.24 1.27921 0.639606 0.768703i \(-0.279098\pi\)
0.639606 + 0.768703i \(0.279098\pi\)
\(908\) − 206.562i − 0.227491i
\(909\) 0 0
\(910\) 748.054 0.822037
\(911\) − 1150.44i − 1.26283i −0.775445 0.631415i \(-0.782475\pi\)
0.775445 0.631415i \(-0.217525\pi\)
\(912\) 0 0
\(913\) −809.676 −0.886830
\(914\) − 555.125i − 0.607358i
\(915\) 0 0
\(916\) 1273.01 1.38975
\(917\) − 55.9232i − 0.0609850i
\(918\) 0 0
\(919\) −950.545 −1.03433 −0.517163 0.855887i \(-0.673012\pi\)
−0.517163 + 0.855887i \(0.673012\pi\)
\(920\) 708.467i 0.770073i
\(921\) 0 0
\(922\) −999.519 −1.08408
\(923\) − 922.999i − 0.999999i
\(924\) 0 0
\(925\) −82.4076 −0.0890892
\(926\) 1468.35i 1.58569i
\(927\) 0 0
\(928\) −1061.35 −1.14370
\(929\) − 40.1479i − 0.0432163i −0.999767 0.0216082i \(-0.993121\pi\)
0.999767 0.0216082i \(-0.00687862\pi\)
\(930\) 0 0
\(931\) −557.876 −0.599223
\(932\) 1285.47i 1.37926i
\(933\) 0 0
\(934\) 1964.24 2.10304
\(935\) 1188.33i 1.27094i
\(936\) 0 0
\(937\) −392.262 −0.418637 −0.209318 0.977848i \(-0.567124\pi\)
−0.209318 + 0.977848i \(0.567124\pi\)
\(938\) − 1585.35i − 1.69014i
\(939\) 0 0
\(940\) 149.122 0.158640
\(941\) 409.879i 0.435578i 0.975996 + 0.217789i \(0.0698845\pi\)
−0.975996 + 0.217789i \(0.930116\pi\)
\(942\) 0 0
\(943\) −546.502 −0.579535
\(944\) − 625.323i − 0.662419i
\(945\) 0 0
\(946\) −1620.12 −1.71260
\(947\) 54.8522i 0.0579221i 0.999581 + 0.0289610i \(0.00921988\pi\)
−0.999581 + 0.0289610i \(0.990780\pi\)
\(948\) 0 0
\(949\) −739.551 −0.779295
\(950\) − 1616.68i − 1.70177i
\(951\) 0 0
\(952\) −174.324 −0.183114
\(953\) − 1108.85i − 1.16354i −0.813354 0.581770i \(-0.802360\pi\)
0.813354 0.581770i \(-0.197640\pi\)
\(954\) 0 0
\(955\) −212.510 −0.222524
\(956\) − 895.370i − 0.936579i
\(957\) 0 0
\(958\) −1127.68 −1.17711
\(959\) − 246.258i − 0.256786i
\(960\) 0 0
\(961\) 971.700 1.01113
\(962\) − 65.3892i − 0.0679722i
\(963\) 0 0
\(964\) 1454.33 1.50864
\(965\) − 1469.06i − 1.52234i
\(966\) 0 0
\(967\) 1039.03 1.07449 0.537243 0.843427i \(-0.319466\pi\)
0.537243 + 0.843427i \(0.319466\pi\)
\(968\) 12.3382i 0.0127461i
\(969\) 0 0
\(970\) 404.135 0.416634
\(971\) 1405.38i 1.44735i 0.690140 + 0.723676i \(0.257549\pi\)
−0.690140 + 0.723676i \(0.742451\pi\)
\(972\) 0 0
\(973\) 389.532 0.400341
\(974\) 414.191i 0.425247i
\(975\) 0 0
\(976\) −510.760 −0.523319
\(977\) − 385.735i − 0.394816i −0.980321 0.197408i \(-0.936748\pi\)
0.980321 0.197408i \(-0.0632524\pi\)
\(978\) 0 0
\(979\) 212.787 0.217351
\(980\) − 1131.52i − 1.15461i
\(981\) 0 0
\(982\) 2494.13 2.53985
\(983\) 504.637i 0.513364i 0.966496 + 0.256682i \(0.0826293\pi\)
−0.966496 + 0.256682i \(0.917371\pi\)
\(984\) 0 0
\(985\) 1750.15 1.77680
\(986\) 1007.95i 1.02227i
\(987\) 0 0
\(988\) 709.737 0.718357
\(989\) − 1620.74i − 1.63876i
\(990\) 0 0
\(991\) −1595.30 −1.60978 −0.804892 0.593421i \(-0.797777\pi\)
−0.804892 + 0.593421i \(0.797777\pi\)
\(992\) − 1985.09i − 2.00110i
\(993\) 0 0
\(994\) 1486.31 1.49528
\(995\) − 1061.39i − 1.06672i
\(996\) 0 0
\(997\) 461.034 0.462421 0.231210 0.972904i \(-0.425731\pi\)
0.231210 + 0.972904i \(0.425731\pi\)
\(998\) − 1667.54i − 1.67088i
\(999\) 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 1143.3.b.a.890.72 yes 84
3.2 odd 2 inner 1143.3.b.a.890.13 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.13 84 3.2 odd 2 inner
1143.3.b.a.890.72 yes 84 1.1 even 1 trivial