Properties

Label 1143.3.b.a.890.11
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 890.11
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.74

$q$-expansion

\(f(q)\) \(=\) \(q-3.06240i q^{2} -5.37827 q^{4} -5.25232i q^{5} -2.64254 q^{7} +4.22081i q^{8} +O(q^{10})\) \(q-3.06240i q^{2} -5.37827 q^{4} -5.25232i q^{5} -2.64254 q^{7} +4.22081i q^{8} -16.0847 q^{10} -4.48093i q^{11} -6.53221 q^{13} +8.09251i q^{14} -8.58729 q^{16} -18.5009i q^{17} -4.12806 q^{19} +28.2484i q^{20} -13.7224 q^{22} -35.9663i q^{23} -2.58688 q^{25} +20.0042i q^{26} +14.2123 q^{28} +36.9550i q^{29} -31.4793 q^{31} +43.1809i q^{32} -56.6572 q^{34} +13.8795i q^{35} +60.1826 q^{37} +12.6418i q^{38} +22.1690 q^{40} -19.7137i q^{41} -19.2361 q^{43} +24.0996i q^{44} -110.143 q^{46} +61.4575i q^{47} -42.0170 q^{49} +7.92206i q^{50} +35.1320 q^{52} +46.6228i q^{53} -23.5353 q^{55} -11.1537i q^{56} +113.171 q^{58} -24.6150i q^{59} -43.2555 q^{61} +96.4020i q^{62} +97.8879 q^{64} +34.3093i q^{65} -6.75193 q^{67} +99.5030i q^{68} +42.5044 q^{70} -44.4482i q^{71} +8.96829 q^{73} -184.303i q^{74} +22.2018 q^{76} +11.8410i q^{77} -23.4298 q^{79} +45.1032i q^{80} -60.3710 q^{82} +60.5881i q^{83} -97.1728 q^{85} +58.9085i q^{86} +18.9131 q^{88} -43.3046i q^{89} +17.2616 q^{91} +193.436i q^{92} +188.207 q^{94} +21.6819i q^{95} +64.0922 q^{97} +128.673i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.06240i − 1.53120i −0.643318 0.765599i \(-0.722443\pi\)
0.643318 0.765599i \(-0.277557\pi\)
\(3\) 0 0
\(4\) −5.37827 −1.34457
\(5\) − 5.25232i − 1.05046i −0.850959 0.525232i \(-0.823978\pi\)
0.850959 0.525232i \(-0.176022\pi\)
\(6\) 0 0
\(7\) −2.64254 −0.377506 −0.188753 0.982025i \(-0.560445\pi\)
−0.188753 + 0.982025i \(0.560445\pi\)
\(8\) 4.22081i 0.527601i
\(9\) 0 0
\(10\) −16.0847 −1.60847
\(11\) − 4.48093i − 0.407357i −0.979038 0.203678i \(-0.934710\pi\)
0.979038 0.203678i \(-0.0652897\pi\)
\(12\) 0 0
\(13\) −6.53221 −0.502478 −0.251239 0.967925i \(-0.580838\pi\)
−0.251239 + 0.967925i \(0.580838\pi\)
\(14\) 8.09251i 0.578036i
\(15\) 0 0
\(16\) −8.58729 −0.536706
\(17\) − 18.5009i − 1.08829i −0.838991 0.544145i \(-0.816854\pi\)
0.838991 0.544145i \(-0.183146\pi\)
\(18\) 0 0
\(19\) −4.12806 −0.217266 −0.108633 0.994082i \(-0.534647\pi\)
−0.108633 + 0.994082i \(0.534647\pi\)
\(20\) 28.2484i 1.41242i
\(21\) 0 0
\(22\) −13.7224 −0.623744
\(23\) − 35.9663i − 1.56375i −0.623434 0.781876i \(-0.714263\pi\)
0.623434 0.781876i \(-0.285737\pi\)
\(24\) 0 0
\(25\) −2.58688 −0.103475
\(26\) 20.0042i 0.769393i
\(27\) 0 0
\(28\) 14.2123 0.507582
\(29\) 36.9550i 1.27431i 0.770735 + 0.637155i \(0.219889\pi\)
−0.770735 + 0.637155i \(0.780111\pi\)
\(30\) 0 0
\(31\) −31.4793 −1.01546 −0.507730 0.861516i \(-0.669515\pi\)
−0.507730 + 0.861516i \(0.669515\pi\)
\(32\) 43.1809i 1.34940i
\(33\) 0 0
\(34\) −56.6572 −1.66639
\(35\) 13.8795i 0.396556i
\(36\) 0 0
\(37\) 60.1826 1.62656 0.813278 0.581875i \(-0.197681\pi\)
0.813278 + 0.581875i \(0.197681\pi\)
\(38\) 12.6418i 0.332678i
\(39\) 0 0
\(40\) 22.1690 0.554226
\(41\) − 19.7137i − 0.480821i −0.970671 0.240410i \(-0.922718\pi\)
0.970671 0.240410i \(-0.0772820\pi\)
\(42\) 0 0
\(43\) −19.2361 −0.447351 −0.223676 0.974664i \(-0.571806\pi\)
−0.223676 + 0.974664i \(0.571806\pi\)
\(44\) 24.0996i 0.547719i
\(45\) 0 0
\(46\) −110.143 −2.39441
\(47\) 61.4575i 1.30761i 0.756664 + 0.653804i \(0.226828\pi\)
−0.756664 + 0.653804i \(0.773172\pi\)
\(48\) 0 0
\(49\) −42.0170 −0.857489
\(50\) 7.92206i 0.158441i
\(51\) 0 0
\(52\) 35.1320 0.675615
\(53\) 46.6228i 0.879676i 0.898077 + 0.439838i \(0.144964\pi\)
−0.898077 + 0.439838i \(0.855036\pi\)
\(54\) 0 0
\(55\) −23.5353 −0.427914
\(56\) − 11.1537i − 0.199172i
\(57\) 0 0
\(58\) 113.171 1.95122
\(59\) − 24.6150i − 0.417204i −0.978001 0.208602i \(-0.933109\pi\)
0.978001 0.208602i \(-0.0668913\pi\)
\(60\) 0 0
\(61\) −43.2555 −0.709107 −0.354553 0.935036i \(-0.615367\pi\)
−0.354553 + 0.935036i \(0.615367\pi\)
\(62\) 96.4020i 1.55487i
\(63\) 0 0
\(64\) 97.8879 1.52950
\(65\) 34.3093i 0.527835i
\(66\) 0 0
\(67\) −6.75193 −0.100775 −0.0503875 0.998730i \(-0.516046\pi\)
−0.0503875 + 0.998730i \(0.516046\pi\)
\(68\) 99.5030i 1.46328i
\(69\) 0 0
\(70\) 42.5044 0.607206
\(71\) − 44.4482i − 0.626030i −0.949748 0.313015i \(-0.898661\pi\)
0.949748 0.313015i \(-0.101339\pi\)
\(72\) 0 0
\(73\) 8.96829 0.122853 0.0614266 0.998112i \(-0.480435\pi\)
0.0614266 + 0.998112i \(0.480435\pi\)
\(74\) − 184.303i − 2.49058i
\(75\) 0 0
\(76\) 22.2018 0.292129
\(77\) 11.8410i 0.153780i
\(78\) 0 0
\(79\) −23.4298 −0.296579 −0.148290 0.988944i \(-0.547377\pi\)
−0.148290 + 0.988944i \(0.547377\pi\)
\(80\) 45.1032i 0.563790i
\(81\) 0 0
\(82\) −60.3710 −0.736232
\(83\) 60.5881i 0.729977i 0.931012 + 0.364989i \(0.118927\pi\)
−0.931012 + 0.364989i \(0.881073\pi\)
\(84\) 0 0
\(85\) −97.1728 −1.14321
\(86\) 58.9085i 0.684983i
\(87\) 0 0
\(88\) 18.9131 0.214922
\(89\) − 43.3046i − 0.486569i −0.969955 0.243284i \(-0.921775\pi\)
0.969955 0.243284i \(-0.0782249\pi\)
\(90\) 0 0
\(91\) 17.2616 0.189688
\(92\) 193.436i 2.10257i
\(93\) 0 0
\(94\) 188.207 2.00221
\(95\) 21.6819i 0.228231i
\(96\) 0 0
\(97\) 64.0922 0.660744 0.330372 0.943851i \(-0.392826\pi\)
0.330372 + 0.943851i \(0.392826\pi\)
\(98\) 128.673i 1.31299i
\(99\) 0 0
\(100\) 13.9130 0.139130
\(101\) 84.6780i 0.838396i 0.907895 + 0.419198i \(0.137689\pi\)
−0.907895 + 0.419198i \(0.862311\pi\)
\(102\) 0 0
\(103\) 81.0163 0.786566 0.393283 0.919417i \(-0.371339\pi\)
0.393283 + 0.919417i \(0.371339\pi\)
\(104\) − 27.5712i − 0.265108i
\(105\) 0 0
\(106\) 142.778 1.34696
\(107\) − 166.803i − 1.55891i −0.626460 0.779454i \(-0.715497\pi\)
0.626460 0.779454i \(-0.284503\pi\)
\(108\) 0 0
\(109\) −6.11967 −0.0561437 −0.0280719 0.999606i \(-0.508937\pi\)
−0.0280719 + 0.999606i \(0.508937\pi\)
\(110\) 72.0743i 0.655221i
\(111\) 0 0
\(112\) 22.6923 0.202610
\(113\) 225.544i 1.99597i 0.0634585 + 0.997984i \(0.479787\pi\)
−0.0634585 + 0.997984i \(0.520213\pi\)
\(114\) 0 0
\(115\) −188.907 −1.64267
\(116\) − 198.754i − 1.71340i
\(117\) 0 0
\(118\) −75.3810 −0.638822
\(119\) 48.8895i 0.410836i
\(120\) 0 0
\(121\) 100.921 0.834060
\(122\) 132.465i 1.08578i
\(123\) 0 0
\(124\) 169.304 1.36536
\(125\) − 117.721i − 0.941767i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 127.048i − 0.992562i
\(129\) 0 0
\(130\) 105.069 0.808220
\(131\) 232.707i 1.77639i 0.459469 + 0.888194i \(0.348040\pi\)
−0.459469 + 0.888194i \(0.651960\pi\)
\(132\) 0 0
\(133\) 10.9086 0.0820193
\(134\) 20.6771i 0.154307i
\(135\) 0 0
\(136\) 78.0889 0.574183
\(137\) − 131.074i − 0.956747i −0.878157 0.478373i \(-0.841227\pi\)
0.878157 0.478373i \(-0.158773\pi\)
\(138\) 0 0
\(139\) 212.317 1.52746 0.763732 0.645534i \(-0.223365\pi\)
0.763732 + 0.645534i \(0.223365\pi\)
\(140\) − 74.6476i − 0.533197i
\(141\) 0 0
\(142\) −136.118 −0.958577
\(143\) 29.2704i 0.204688i
\(144\) 0 0
\(145\) 194.100 1.33862
\(146\) − 27.4645i − 0.188113i
\(147\) 0 0
\(148\) −323.678 −2.18702
\(149\) 6.04907i 0.0405978i 0.999794 + 0.0202989i \(0.00646178\pi\)
−0.999794 + 0.0202989i \(0.993538\pi\)
\(150\) 0 0
\(151\) 114.743 0.759889 0.379945 0.925009i \(-0.375943\pi\)
0.379945 + 0.925009i \(0.375943\pi\)
\(152\) − 17.4238i − 0.114630i
\(153\) 0 0
\(154\) 36.2619 0.235467
\(155\) 165.339i 1.06671i
\(156\) 0 0
\(157\) −239.194 −1.52353 −0.761765 0.647854i \(-0.775667\pi\)
−0.761765 + 0.647854i \(0.775667\pi\)
\(158\) 71.7512i 0.454122i
\(159\) 0 0
\(160\) 226.800 1.41750
\(161\) 95.0424i 0.590325i
\(162\) 0 0
\(163\) −125.077 −0.767346 −0.383673 0.923469i \(-0.625341\pi\)
−0.383673 + 0.923469i \(0.625341\pi\)
\(164\) 106.025i 0.646496i
\(165\) 0 0
\(166\) 185.545 1.11774
\(167\) − 14.8310i − 0.0888085i −0.999014 0.0444043i \(-0.985861\pi\)
0.999014 0.0444043i \(-0.0141390\pi\)
\(168\) 0 0
\(169\) −126.330 −0.747516
\(170\) 297.582i 1.75048i
\(171\) 0 0
\(172\) 103.457 0.601494
\(173\) 22.0241i 0.127307i 0.997972 + 0.0636535i \(0.0202752\pi\)
−0.997972 + 0.0636535i \(0.979725\pi\)
\(174\) 0 0
\(175\) 6.83595 0.0390625
\(176\) 38.4790i 0.218631i
\(177\) 0 0
\(178\) −132.616 −0.745033
\(179\) − 335.906i − 1.87657i −0.345862 0.938285i \(-0.612414\pi\)
0.345862 0.938285i \(-0.387586\pi\)
\(180\) 0 0
\(181\) −271.379 −1.49933 −0.749665 0.661818i \(-0.769785\pi\)
−0.749665 + 0.661818i \(0.769785\pi\)
\(182\) − 52.8620i − 0.290450i
\(183\) 0 0
\(184\) 151.807 0.825037
\(185\) − 316.098i − 1.70864i
\(186\) 0 0
\(187\) −82.9013 −0.443322
\(188\) − 330.535i − 1.75817i
\(189\) 0 0
\(190\) 66.3986 0.349466
\(191\) − 114.291i − 0.598382i −0.954193 0.299191i \(-0.903283\pi\)
0.954193 0.299191i \(-0.0967168\pi\)
\(192\) 0 0
\(193\) −119.740 −0.620413 −0.310206 0.950669i \(-0.600398\pi\)
−0.310206 + 0.950669i \(0.600398\pi\)
\(194\) − 196.276i − 1.01173i
\(195\) 0 0
\(196\) 225.979 1.15295
\(197\) − 272.581i − 1.38366i −0.722060 0.691830i \(-0.756805\pi\)
0.722060 0.691830i \(-0.243195\pi\)
\(198\) 0 0
\(199\) 86.4883 0.434615 0.217307 0.976103i \(-0.430273\pi\)
0.217307 + 0.976103i \(0.430273\pi\)
\(200\) − 10.9187i − 0.0545937i
\(201\) 0 0
\(202\) 259.318 1.28375
\(203\) − 97.6551i − 0.481060i
\(204\) 0 0
\(205\) −103.542 −0.505085
\(206\) − 248.104i − 1.20439i
\(207\) 0 0
\(208\) 56.0940 0.269683
\(209\) 18.4975i 0.0885050i
\(210\) 0 0
\(211\) −38.3542 −0.181774 −0.0908868 0.995861i \(-0.528970\pi\)
−0.0908868 + 0.995861i \(0.528970\pi\)
\(212\) − 250.750i − 1.18278i
\(213\) 0 0
\(214\) −510.817 −2.38700
\(215\) 101.034i 0.469926i
\(216\) 0 0
\(217\) 83.1853 0.383342
\(218\) 18.7408i 0.0859672i
\(219\) 0 0
\(220\) 126.579 0.575359
\(221\) 120.852i 0.546842i
\(222\) 0 0
\(223\) −79.5135 −0.356563 −0.178281 0.983980i \(-0.557054\pi\)
−0.178281 + 0.983980i \(0.557054\pi\)
\(224\) − 114.107i − 0.509408i
\(225\) 0 0
\(226\) 690.707 3.05622
\(227\) − 241.685i − 1.06469i −0.846528 0.532345i \(-0.821311\pi\)
0.846528 0.532345i \(-0.178689\pi\)
\(228\) 0 0
\(229\) −341.127 −1.48964 −0.744820 0.667266i \(-0.767464\pi\)
−0.744820 + 0.667266i \(0.767464\pi\)
\(230\) 578.507i 2.51525i
\(231\) 0 0
\(232\) −155.980 −0.672328
\(233\) 237.837i 1.02076i 0.859950 + 0.510379i \(0.170495\pi\)
−0.859950 + 0.510379i \(0.829505\pi\)
\(234\) 0 0
\(235\) 322.795 1.37359
\(236\) 132.386i 0.560959i
\(237\) 0 0
\(238\) 149.719 0.629071
\(239\) 65.3419i 0.273397i 0.990613 + 0.136698i \(0.0436491\pi\)
−0.990613 + 0.136698i \(0.956351\pi\)
\(240\) 0 0
\(241\) −139.424 −0.578521 −0.289261 0.957250i \(-0.593409\pi\)
−0.289261 + 0.957250i \(0.593409\pi\)
\(242\) − 309.061i − 1.27711i
\(243\) 0 0
\(244\) 232.640 0.953441
\(245\) 220.687i 0.900762i
\(246\) 0 0
\(247\) 26.9654 0.109172
\(248\) − 132.868i − 0.535758i
\(249\) 0 0
\(250\) −360.508 −1.44203
\(251\) − 250.839i − 0.999357i −0.866211 0.499679i \(-0.833451\pi\)
0.866211 0.499679i \(-0.166549\pi\)
\(252\) 0 0
\(253\) −161.162 −0.637005
\(254\) − 34.5115i − 0.135872i
\(255\) 0 0
\(256\) 2.48077 0.00969049
\(257\) 91.9826i 0.357909i 0.983857 + 0.178954i \(0.0572715\pi\)
−0.983857 + 0.178954i \(0.942729\pi\)
\(258\) 0 0
\(259\) −159.035 −0.614035
\(260\) − 184.525i − 0.709710i
\(261\) 0 0
\(262\) 712.640 2.72000
\(263\) − 358.057i − 1.36143i −0.732547 0.680716i \(-0.761669\pi\)
0.732547 0.680716i \(-0.238331\pi\)
\(264\) 0 0
\(265\) 244.878 0.924069
\(266\) − 33.4064i − 0.125588i
\(267\) 0 0
\(268\) 36.3137 0.135499
\(269\) − 116.606i − 0.433479i −0.976229 0.216739i \(-0.930458\pi\)
0.976229 0.216739i \(-0.0695422\pi\)
\(270\) 0 0
\(271\) −48.0667 −0.177368 −0.0886839 0.996060i \(-0.528266\pi\)
−0.0886839 + 0.996060i \(0.528266\pi\)
\(272\) 158.873i 0.584092i
\(273\) 0 0
\(274\) −401.401 −1.46497
\(275\) 11.5916i 0.0421514i
\(276\) 0 0
\(277\) 81.2173 0.293203 0.146602 0.989196i \(-0.453166\pi\)
0.146602 + 0.989196i \(0.453166\pi\)
\(278\) − 650.200i − 2.33885i
\(279\) 0 0
\(280\) −58.5826 −0.209224
\(281\) − 115.455i − 0.410870i −0.978671 0.205435i \(-0.934139\pi\)
0.978671 0.205435i \(-0.0658610\pi\)
\(282\) 0 0
\(283\) −174.042 −0.614991 −0.307495 0.951550i \(-0.599491\pi\)
−0.307495 + 0.951550i \(0.599491\pi\)
\(284\) 239.054i 0.841740i
\(285\) 0 0
\(286\) 89.6374 0.313418
\(287\) 52.0941i 0.181513i
\(288\) 0 0
\(289\) −53.2844 −0.184375
\(290\) − 594.410i − 2.04969i
\(291\) 0 0
\(292\) −48.2339 −0.165185
\(293\) 330.591i 1.12830i 0.825673 + 0.564149i \(0.190795\pi\)
−0.825673 + 0.564149i \(0.809205\pi\)
\(294\) 0 0
\(295\) −129.286 −0.438258
\(296\) 254.019i 0.858173i
\(297\) 0 0
\(298\) 18.5246 0.0621632
\(299\) 234.939i 0.785751i
\(300\) 0 0
\(301\) 50.8322 0.168878
\(302\) − 351.389i − 1.16354i
\(303\) 0 0
\(304\) 35.4489 0.116608
\(305\) 227.192i 0.744891i
\(306\) 0 0
\(307\) −240.533 −0.783494 −0.391747 0.920073i \(-0.628129\pi\)
−0.391747 + 0.920073i \(0.628129\pi\)
\(308\) − 63.6843i − 0.206767i
\(309\) 0 0
\(310\) 506.334 1.63334
\(311\) − 103.308i − 0.332179i −0.986111 0.166089i \(-0.946886\pi\)
0.986111 0.166089i \(-0.0531140\pi\)
\(312\) 0 0
\(313\) −173.122 −0.553104 −0.276552 0.960999i \(-0.589192\pi\)
−0.276552 + 0.960999i \(0.589192\pi\)
\(314\) 732.507i 2.33283i
\(315\) 0 0
\(316\) 126.012 0.398771
\(317\) 301.974i 0.952600i 0.879283 + 0.476300i \(0.158022\pi\)
−0.879283 + 0.476300i \(0.841978\pi\)
\(318\) 0 0
\(319\) 165.593 0.519099
\(320\) − 514.139i − 1.60668i
\(321\) 0 0
\(322\) 291.057 0.903905
\(323\) 76.3730i 0.236449i
\(324\) 0 0
\(325\) 16.8981 0.0519941
\(326\) 383.036i 1.17496i
\(327\) 0 0
\(328\) 83.2075 0.253682
\(329\) − 162.404i − 0.493629i
\(330\) 0 0
\(331\) −437.919 −1.32302 −0.661509 0.749938i \(-0.730083\pi\)
−0.661509 + 0.749938i \(0.730083\pi\)
\(332\) − 325.859i − 0.981503i
\(333\) 0 0
\(334\) −45.4185 −0.135983
\(335\) 35.4633i 0.105861i
\(336\) 0 0
\(337\) −359.907 −1.06797 −0.533986 0.845493i \(-0.679306\pi\)
−0.533986 + 0.845493i \(0.679306\pi\)
\(338\) 386.873i 1.14459i
\(339\) 0 0
\(340\) 522.622 1.53712
\(341\) 141.056i 0.413655i
\(342\) 0 0
\(343\) 240.516 0.701213
\(344\) − 81.1918i − 0.236023i
\(345\) 0 0
\(346\) 67.4465 0.194932
\(347\) − 220.886i − 0.636560i −0.947997 0.318280i \(-0.896895\pi\)
0.947997 0.318280i \(-0.103105\pi\)
\(348\) 0 0
\(349\) −354.027 −1.01440 −0.507201 0.861828i \(-0.669320\pi\)
−0.507201 + 0.861828i \(0.669320\pi\)
\(350\) − 20.9344i − 0.0598125i
\(351\) 0 0
\(352\) 193.491 0.549689
\(353\) 620.179i 1.75688i 0.477851 + 0.878441i \(0.341416\pi\)
−0.477851 + 0.878441i \(0.658584\pi\)
\(354\) 0 0
\(355\) −233.456 −0.657623
\(356\) 232.904i 0.654225i
\(357\) 0 0
\(358\) −1028.68 −2.87340
\(359\) − 589.579i − 1.64228i −0.570726 0.821141i \(-0.693338\pi\)
0.570726 0.821141i \(-0.306662\pi\)
\(360\) 0 0
\(361\) −343.959 −0.952795
\(362\) 831.069i 2.29577i
\(363\) 0 0
\(364\) −92.8377 −0.255049
\(365\) − 47.1043i − 0.129053i
\(366\) 0 0
\(367\) −371.572 −1.01246 −0.506228 0.862399i \(-0.668961\pi\)
−0.506228 + 0.862399i \(0.668961\pi\)
\(368\) 308.853i 0.839275i
\(369\) 0 0
\(370\) −968.019 −2.61627
\(371\) − 123.203i − 0.332083i
\(372\) 0 0
\(373\) −57.7415 −0.154803 −0.0774015 0.997000i \(-0.524662\pi\)
−0.0774015 + 0.997000i \(0.524662\pi\)
\(374\) 253.877i 0.678814i
\(375\) 0 0
\(376\) −259.400 −0.689895
\(377\) − 241.398i − 0.640313i
\(378\) 0 0
\(379\) −368.555 −0.972441 −0.486220 0.873836i \(-0.661625\pi\)
−0.486220 + 0.873836i \(0.661625\pi\)
\(380\) − 116.611i − 0.306871i
\(381\) 0 0
\(382\) −350.004 −0.916242
\(383\) − 165.786i − 0.432861i −0.976298 0.216431i \(-0.930558\pi\)
0.976298 0.216431i \(-0.0694415\pi\)
\(384\) 0 0
\(385\) 62.1929 0.161540
\(386\) 366.690i 0.949975i
\(387\) 0 0
\(388\) −344.705 −0.888415
\(389\) − 24.7882i − 0.0637228i −0.999492 0.0318614i \(-0.989856\pi\)
0.999492 0.0318614i \(-0.0101435\pi\)
\(390\) 0 0
\(391\) −665.410 −1.70182
\(392\) − 177.346i − 0.452412i
\(393\) 0 0
\(394\) −834.751 −2.11866
\(395\) 123.061i 0.311546i
\(396\) 0 0
\(397\) 397.699 1.00176 0.500881 0.865516i \(-0.333010\pi\)
0.500881 + 0.865516i \(0.333010\pi\)
\(398\) − 264.861i − 0.665481i
\(399\) 0 0
\(400\) 22.2143 0.0555358
\(401\) − 336.736i − 0.839742i −0.907584 0.419871i \(-0.862075\pi\)
0.907584 0.419871i \(-0.137925\pi\)
\(402\) 0 0
\(403\) 205.629 0.510247
\(404\) − 455.421i − 1.12728i
\(405\) 0 0
\(406\) −299.059 −0.736598
\(407\) − 269.674i − 0.662589i
\(408\) 0 0
\(409\) 334.028 0.816695 0.408347 0.912827i \(-0.366105\pi\)
0.408347 + 0.912827i \(0.366105\pi\)
\(410\) 317.088i 0.773385i
\(411\) 0 0
\(412\) −435.728 −1.05759
\(413\) 65.0462i 0.157497i
\(414\) 0 0
\(415\) 318.228 0.766815
\(416\) − 282.067i − 0.678046i
\(417\) 0 0
\(418\) 56.6468 0.135519
\(419\) 293.225i 0.699821i 0.936783 + 0.349910i \(0.113788\pi\)
−0.936783 + 0.349910i \(0.886212\pi\)
\(420\) 0 0
\(421\) −154.980 −0.368123 −0.184061 0.982915i \(-0.558924\pi\)
−0.184061 + 0.982915i \(0.558924\pi\)
\(422\) 117.456i 0.278331i
\(423\) 0 0
\(424\) −196.786 −0.464118
\(425\) 47.8597i 0.112611i
\(426\) 0 0
\(427\) 114.304 0.267692
\(428\) 897.112i 2.09606i
\(429\) 0 0
\(430\) 309.407 0.719550
\(431\) 273.927i 0.635562i 0.948164 + 0.317781i \(0.102938\pi\)
−0.948164 + 0.317781i \(0.897062\pi\)
\(432\) 0 0
\(433\) −505.428 −1.16727 −0.583635 0.812016i \(-0.698370\pi\)
−0.583635 + 0.812016i \(0.698370\pi\)
\(434\) − 254.746i − 0.586973i
\(435\) 0 0
\(436\) 32.9132 0.0754890
\(437\) 148.471i 0.339751i
\(438\) 0 0
\(439\) −60.9146 −0.138758 −0.0693789 0.997590i \(-0.522102\pi\)
−0.0693789 + 0.997590i \(0.522102\pi\)
\(440\) − 99.3378i − 0.225768i
\(441\) 0 0
\(442\) 370.097 0.837323
\(443\) − 24.2180i − 0.0546682i −0.999626 0.0273341i \(-0.991298\pi\)
0.999626 0.0273341i \(-0.00870180\pi\)
\(444\) 0 0
\(445\) −227.450 −0.511123
\(446\) 243.502i 0.545968i
\(447\) 0 0
\(448\) −258.673 −0.577395
\(449\) − 29.2631i − 0.0651740i −0.999469 0.0325870i \(-0.989625\pi\)
0.999469 0.0325870i \(-0.0103746\pi\)
\(450\) 0 0
\(451\) −88.3354 −0.195866
\(452\) − 1213.04i − 2.68371i
\(453\) 0 0
\(454\) −740.134 −1.63025
\(455\) − 90.6637i − 0.199261i
\(456\) 0 0
\(457\) 284.619 0.622799 0.311399 0.950279i \(-0.399202\pi\)
0.311399 + 0.950279i \(0.399202\pi\)
\(458\) 1044.67i 2.28093i
\(459\) 0 0
\(460\) 1015.99 2.20867
\(461\) − 246.275i − 0.534218i −0.963666 0.267109i \(-0.913932\pi\)
0.963666 0.267109i \(-0.0860684\pi\)
\(462\) 0 0
\(463\) 47.9854 0.103640 0.0518201 0.998656i \(-0.483498\pi\)
0.0518201 + 0.998656i \(0.483498\pi\)
\(464\) − 317.344i − 0.683930i
\(465\) 0 0
\(466\) 728.350 1.56298
\(467\) − 376.578i − 0.806377i −0.915117 0.403189i \(-0.867902\pi\)
0.915117 0.403189i \(-0.132098\pi\)
\(468\) 0 0
\(469\) 17.8422 0.0380432
\(470\) − 988.526i − 2.10325i
\(471\) 0 0
\(472\) 103.895 0.220117
\(473\) 86.1955i 0.182232i
\(474\) 0 0
\(475\) 10.6788 0.0224817
\(476\) − 262.941i − 0.552396i
\(477\) 0 0
\(478\) 200.103 0.418625
\(479\) − 930.115i − 1.94179i −0.239514 0.970893i \(-0.576988\pi\)
0.239514 0.970893i \(-0.423012\pi\)
\(480\) 0 0
\(481\) −393.126 −0.817309
\(482\) 426.970i 0.885830i
\(483\) 0 0
\(484\) −542.782 −1.12145
\(485\) − 336.633i − 0.694088i
\(486\) 0 0
\(487\) 109.016 0.223853 0.111926 0.993717i \(-0.464298\pi\)
0.111926 + 0.993717i \(0.464298\pi\)
\(488\) − 182.573i − 0.374125i
\(489\) 0 0
\(490\) 675.830 1.37925
\(491\) − 16.1679i − 0.0329286i −0.999864 0.0164643i \(-0.994759\pi\)
0.999864 0.0164643i \(-0.00524098\pi\)
\(492\) 0 0
\(493\) 683.702 1.38682
\(494\) − 82.5787i − 0.167163i
\(495\) 0 0
\(496\) 270.322 0.545004
\(497\) 117.456i 0.236330i
\(498\) 0 0
\(499\) −496.964 −0.995919 −0.497960 0.867200i \(-0.665917\pi\)
−0.497960 + 0.867200i \(0.665917\pi\)
\(500\) 633.135i 1.26627i
\(501\) 0 0
\(502\) −768.167 −1.53021
\(503\) 72.7868i 0.144705i 0.997379 + 0.0723527i \(0.0230507\pi\)
−0.997379 + 0.0723527i \(0.976949\pi\)
\(504\) 0 0
\(505\) 444.756 0.880706
\(506\) 493.543i 0.975381i
\(507\) 0 0
\(508\) −60.6100 −0.119311
\(509\) 804.529i 1.58061i 0.612716 + 0.790303i \(0.290077\pi\)
−0.612716 + 0.790303i \(0.709923\pi\)
\(510\) 0 0
\(511\) −23.6991 −0.0463778
\(512\) − 515.789i − 1.00740i
\(513\) 0 0
\(514\) 281.687 0.548029
\(515\) − 425.524i − 0.826260i
\(516\) 0 0
\(517\) 275.387 0.532663
\(518\) 487.028i 0.940209i
\(519\) 0 0
\(520\) −144.813 −0.278486
\(521\) − 367.880i − 0.706103i −0.935604 0.353052i \(-0.885144\pi\)
0.935604 0.353052i \(-0.114856\pi\)
\(522\) 0 0
\(523\) −252.753 −0.483276 −0.241638 0.970366i \(-0.577685\pi\)
−0.241638 + 0.970366i \(0.577685\pi\)
\(524\) − 1251.56i − 2.38847i
\(525\) 0 0
\(526\) −1096.51 −2.08462
\(527\) 582.396i 1.10512i
\(528\) 0 0
\(529\) −764.574 −1.44532
\(530\) − 749.914i − 1.41493i
\(531\) 0 0
\(532\) −58.6692 −0.110281
\(533\) 128.774i 0.241602i
\(534\) 0 0
\(535\) −876.104 −1.63758
\(536\) − 28.4986i − 0.0531690i
\(537\) 0 0
\(538\) −357.093 −0.663742
\(539\) 188.275i 0.349304i
\(540\) 0 0
\(541\) 709.058 1.31064 0.655321 0.755350i \(-0.272533\pi\)
0.655321 + 0.755350i \(0.272533\pi\)
\(542\) 147.199i 0.271585i
\(543\) 0 0
\(544\) 798.887 1.46854
\(545\) 32.1425i 0.0589770i
\(546\) 0 0
\(547\) −84.1095 −0.153765 −0.0768826 0.997040i \(-0.524497\pi\)
−0.0768826 + 0.997040i \(0.524497\pi\)
\(548\) 704.953i 1.28641i
\(549\) 0 0
\(550\) 35.4982 0.0645421
\(551\) − 152.553i − 0.276865i
\(552\) 0 0
\(553\) 61.9141 0.111960
\(554\) − 248.720i − 0.448952i
\(555\) 0 0
\(556\) −1141.90 −2.05378
\(557\) 17.5000i 0.0314183i 0.999877 + 0.0157091i \(0.00500058\pi\)
−0.999877 + 0.0157091i \(0.994999\pi\)
\(558\) 0 0
\(559\) 125.654 0.224784
\(560\) − 119.187i − 0.212834i
\(561\) 0 0
\(562\) −353.568 −0.629124
\(563\) 123.655i 0.219635i 0.993952 + 0.109818i \(0.0350266\pi\)
−0.993952 + 0.109818i \(0.964973\pi\)
\(564\) 0 0
\(565\) 1184.63 2.09669
\(566\) 532.987i 0.941673i
\(567\) 0 0
\(568\) 187.607 0.330294
\(569\) − 587.125i − 1.03185i −0.856632 0.515927i \(-0.827447\pi\)
0.856632 0.515927i \(-0.172553\pi\)
\(570\) 0 0
\(571\) 531.233 0.930356 0.465178 0.885217i \(-0.345990\pi\)
0.465178 + 0.885217i \(0.345990\pi\)
\(572\) − 157.424i − 0.275217i
\(573\) 0 0
\(574\) 159.533 0.277932
\(575\) 93.0406i 0.161810i
\(576\) 0 0
\(577\) 89.0333 0.154304 0.0771519 0.997019i \(-0.475417\pi\)
0.0771519 + 0.997019i \(0.475417\pi\)
\(578\) 163.178i 0.282315i
\(579\) 0 0
\(580\) −1043.92 −1.79986
\(581\) − 160.107i − 0.275571i
\(582\) 0 0
\(583\) 208.914 0.358342
\(584\) 37.8534i 0.0648175i
\(585\) 0 0
\(586\) 1012.40 1.72765
\(587\) − 165.262i − 0.281536i −0.990043 0.140768i \(-0.955043\pi\)
0.990043 0.140768i \(-0.0449572\pi\)
\(588\) 0 0
\(589\) 129.948 0.220625
\(590\) 395.925i 0.671060i
\(591\) 0 0
\(592\) −516.806 −0.872983
\(593\) − 1069.09i − 1.80285i −0.432935 0.901425i \(-0.642522\pi\)
0.432935 0.901425i \(-0.357478\pi\)
\(594\) 0 0
\(595\) 256.783 0.431568
\(596\) − 32.5335i − 0.0545865i
\(597\) 0 0
\(598\) 719.478 1.20314
\(599\) 541.081i 0.903308i 0.892193 + 0.451654i \(0.149166\pi\)
−0.892193 + 0.451654i \(0.850834\pi\)
\(600\) 0 0
\(601\) 964.958 1.60559 0.802793 0.596257i \(-0.203346\pi\)
0.802793 + 0.596257i \(0.203346\pi\)
\(602\) − 155.668i − 0.258585i
\(603\) 0 0
\(604\) −617.120 −1.02172
\(605\) − 530.071i − 0.876151i
\(606\) 0 0
\(607\) −270.756 −0.446057 −0.223028 0.974812i \(-0.571594\pi\)
−0.223028 + 0.974812i \(0.571594\pi\)
\(608\) − 178.254i − 0.293180i
\(609\) 0 0
\(610\) 695.751 1.14058
\(611\) − 401.454i − 0.657044i
\(612\) 0 0
\(613\) 374.707 0.611267 0.305634 0.952149i \(-0.401132\pi\)
0.305634 + 0.952149i \(0.401132\pi\)
\(614\) 736.606i 1.19968i
\(615\) 0 0
\(616\) −49.9787 −0.0811343
\(617\) − 458.820i − 0.743630i −0.928307 0.371815i \(-0.878736\pi\)
0.928307 0.371815i \(-0.121264\pi\)
\(618\) 0 0
\(619\) −470.296 −0.759767 −0.379884 0.925034i \(-0.624036\pi\)
−0.379884 + 0.925034i \(0.624036\pi\)
\(620\) − 889.239i − 1.43426i
\(621\) 0 0
\(622\) −316.369 −0.508631
\(623\) 114.434i 0.183683i
\(624\) 0 0
\(625\) −682.980 −1.09277
\(626\) 530.167i 0.846912i
\(627\) 0 0
\(628\) 1286.45 2.04849
\(629\) − 1113.43i − 1.77017i
\(630\) 0 0
\(631\) 949.552 1.50484 0.752418 0.658685i \(-0.228887\pi\)
0.752418 + 0.658685i \(0.228887\pi\)
\(632\) − 98.8925i − 0.156476i
\(633\) 0 0
\(634\) 924.765 1.45862
\(635\) − 59.1907i − 0.0932136i
\(636\) 0 0
\(637\) 274.464 0.430869
\(638\) − 507.110i − 0.794844i
\(639\) 0 0
\(640\) −667.296 −1.04265
\(641\) 163.550i 0.255149i 0.991829 + 0.127574i \(0.0407192\pi\)
−0.991829 + 0.127574i \(0.959281\pi\)
\(642\) 0 0
\(643\) 547.664 0.851732 0.425866 0.904786i \(-0.359970\pi\)
0.425866 + 0.904786i \(0.359970\pi\)
\(644\) − 511.164i − 0.793732i
\(645\) 0 0
\(646\) 233.884 0.362050
\(647\) 151.339i 0.233909i 0.993137 + 0.116954i \(0.0373131\pi\)
−0.993137 + 0.116954i \(0.962687\pi\)
\(648\) 0 0
\(649\) −110.298 −0.169951
\(650\) − 51.7486i − 0.0796132i
\(651\) 0 0
\(652\) 672.700 1.03175
\(653\) 935.358i 1.43240i 0.697894 + 0.716201i \(0.254121\pi\)
−0.697894 + 0.716201i \(0.745879\pi\)
\(654\) 0 0
\(655\) 1222.25 1.86603
\(656\) 169.287i 0.258059i
\(657\) 0 0
\(658\) −497.346 −0.755844
\(659\) − 577.471i − 0.876283i −0.898906 0.438142i \(-0.855637\pi\)
0.898906 0.438142i \(-0.144363\pi\)
\(660\) 0 0
\(661\) 518.169 0.783916 0.391958 0.919983i \(-0.371798\pi\)
0.391958 + 0.919983i \(0.371798\pi\)
\(662\) 1341.08i 2.02580i
\(663\) 0 0
\(664\) −255.731 −0.385137
\(665\) − 57.2953i − 0.0861584i
\(666\) 0 0
\(667\) 1329.14 1.99271
\(668\) 79.7652i 0.119409i
\(669\) 0 0
\(670\) 108.603 0.162094
\(671\) 193.825i 0.288859i
\(672\) 0 0
\(673\) −212.932 −0.316393 −0.158196 0.987408i \(-0.550568\pi\)
−0.158196 + 0.987408i \(0.550568\pi\)
\(674\) 1102.18i 1.63528i
\(675\) 0 0
\(676\) 679.438 1.00509
\(677\) − 293.836i − 0.434026i −0.976169 0.217013i \(-0.930369\pi\)
0.976169 0.217013i \(-0.0696315\pi\)
\(678\) 0 0
\(679\) −169.366 −0.249435
\(680\) − 410.148i − 0.603159i
\(681\) 0 0
\(682\) 431.970 0.633388
\(683\) − 800.471i − 1.17199i −0.810314 0.585996i \(-0.800703\pi\)
0.810314 0.585996i \(-0.199297\pi\)
\(684\) 0 0
\(685\) −688.444 −1.00503
\(686\) − 736.555i − 1.07370i
\(687\) 0 0
\(688\) 165.186 0.240096
\(689\) − 304.550i − 0.442018i
\(690\) 0 0
\(691\) 1089.35 1.57649 0.788244 0.615363i \(-0.210991\pi\)
0.788244 + 0.615363i \(0.210991\pi\)
\(692\) − 118.452i − 0.171173i
\(693\) 0 0
\(694\) −676.441 −0.974699
\(695\) − 1115.16i − 1.60455i
\(696\) 0 0
\(697\) −364.721 −0.523273
\(698\) 1084.17i 1.55325i
\(699\) 0 0
\(700\) −36.7656 −0.0525222
\(701\) 1214.68i 1.73278i 0.499370 + 0.866389i \(0.333565\pi\)
−0.499370 + 0.866389i \(0.666435\pi\)
\(702\) 0 0
\(703\) −248.437 −0.353396
\(704\) − 438.629i − 0.623052i
\(705\) 0 0
\(706\) 1899.23 2.69013
\(707\) − 223.765i − 0.316500i
\(708\) 0 0
\(709\) 571.377 0.805891 0.402945 0.915224i \(-0.367986\pi\)
0.402945 + 0.915224i \(0.367986\pi\)
\(710\) 714.935i 1.00695i
\(711\) 0 0
\(712\) 182.781 0.256714
\(713\) 1132.19i 1.58793i
\(714\) 0 0
\(715\) 153.737 0.215017
\(716\) 1806.59i 2.52318i
\(717\) 0 0
\(718\) −1805.52 −2.51466
\(719\) − 618.040i − 0.859583i −0.902928 0.429791i \(-0.858587\pi\)
0.902928 0.429791i \(-0.141413\pi\)
\(720\) 0 0
\(721\) −214.089 −0.296933
\(722\) 1053.34i 1.45892i
\(723\) 0 0
\(724\) 1459.55 2.01595
\(725\) − 95.5983i − 0.131860i
\(726\) 0 0
\(727\) −533.575 −0.733940 −0.366970 0.930233i \(-0.619605\pi\)
−0.366970 + 0.930233i \(0.619605\pi\)
\(728\) 72.8580i 0.100080i
\(729\) 0 0
\(730\) −144.252 −0.197606
\(731\) 355.886i 0.486848i
\(732\) 0 0
\(733\) −569.444 −0.776867 −0.388434 0.921477i \(-0.626984\pi\)
−0.388434 + 0.921477i \(0.626984\pi\)
\(734\) 1137.90i 1.55027i
\(735\) 0 0
\(736\) 1553.06 2.11013
\(737\) 30.2549i 0.0410514i
\(738\) 0 0
\(739\) 52.8399 0.0715019 0.0357510 0.999361i \(-0.488618\pi\)
0.0357510 + 0.999361i \(0.488618\pi\)
\(740\) 1700.06i 2.29738i
\(741\) 0 0
\(742\) −377.296 −0.508485
\(743\) − 980.221i − 1.31927i −0.751584 0.659637i \(-0.770710\pi\)
0.751584 0.659637i \(-0.229290\pi\)
\(744\) 0 0
\(745\) 31.7717 0.0426465
\(746\) 176.827i 0.237034i
\(747\) 0 0
\(748\) 445.866 0.596077
\(749\) 440.784i 0.588497i
\(750\) 0 0
\(751\) 932.837 1.24213 0.621063 0.783760i \(-0.286701\pi\)
0.621063 + 0.783760i \(0.286701\pi\)
\(752\) − 527.754i − 0.701801i
\(753\) 0 0
\(754\) −739.256 −0.980446
\(755\) − 602.669i − 0.798236i
\(756\) 0 0
\(757\) −701.478 −0.926655 −0.463328 0.886187i \(-0.653345\pi\)
−0.463328 + 0.886187i \(0.653345\pi\)
\(758\) 1128.66i 1.48900i
\(759\) 0 0
\(760\) −91.5151 −0.120415
\(761\) 248.730i 0.326846i 0.986556 + 0.163423i \(0.0522535\pi\)
−0.986556 + 0.163423i \(0.947747\pi\)
\(762\) 0 0
\(763\) 16.1715 0.0211946
\(764\) 614.688i 0.804565i
\(765\) 0 0
\(766\) −507.702 −0.662796
\(767\) 160.791i 0.209636i
\(768\) 0 0
\(769\) −645.918 −0.839945 −0.419972 0.907537i \(-0.637960\pi\)
−0.419972 + 0.907537i \(0.637960\pi\)
\(770\) − 190.459i − 0.247350i
\(771\) 0 0
\(772\) 643.992 0.834187
\(773\) − 233.335i − 0.301857i −0.988545 0.150928i \(-0.951774\pi\)
0.988545 0.150928i \(-0.0482263\pi\)
\(774\) 0 0
\(775\) 81.4332 0.105075
\(776\) 270.521i 0.348609i
\(777\) 0 0
\(778\) −75.9112 −0.0975722
\(779\) 81.3792i 0.104466i
\(780\) 0 0
\(781\) −199.169 −0.255018
\(782\) 2037.75i 2.60582i
\(783\) 0 0
\(784\) 360.812 0.460220
\(785\) 1256.32i 1.60041i
\(786\) 0 0
\(787\) 1346.42 1.71082 0.855411 0.517950i \(-0.173305\pi\)
0.855411 + 0.517950i \(0.173305\pi\)
\(788\) 1466.01i 1.86043i
\(789\) 0 0
\(790\) 376.861 0.477039
\(791\) − 596.010i − 0.753490i
\(792\) 0 0
\(793\) 282.554 0.356310
\(794\) − 1217.91i − 1.53389i
\(795\) 0 0
\(796\) −465.157 −0.584369
\(797\) − 605.301i − 0.759474i −0.925095 0.379737i \(-0.876015\pi\)
0.925095 0.379737i \(-0.123985\pi\)
\(798\) 0 0
\(799\) 1137.02 1.42306
\(800\) − 111.704i − 0.139630i
\(801\) 0 0
\(802\) −1031.22 −1.28581
\(803\) − 40.1862i − 0.0500451i
\(804\) 0 0
\(805\) 499.193 0.620116
\(806\) − 629.719i − 0.781289i
\(807\) 0 0
\(808\) −357.410 −0.442339
\(809\) − 1066.82i − 1.31869i −0.751841 0.659344i \(-0.770834\pi\)
0.751841 0.659344i \(-0.229166\pi\)
\(810\) 0 0
\(811\) −528.604 −0.651793 −0.325897 0.945405i \(-0.605666\pi\)
−0.325897 + 0.945405i \(0.605666\pi\)
\(812\) 525.216i 0.646817i
\(813\) 0 0
\(814\) −825.848 −1.01456
\(815\) 656.946i 0.806069i
\(816\) 0 0
\(817\) 79.4078 0.0971943
\(818\) − 1022.93i − 1.25052i
\(819\) 0 0
\(820\) 556.879 0.679121
\(821\) 881.520i 1.07371i 0.843673 + 0.536857i \(0.180389\pi\)
−0.843673 + 0.536857i \(0.819611\pi\)
\(822\) 0 0
\(823\) −141.552 −0.171995 −0.0859977 0.996295i \(-0.527408\pi\)
−0.0859977 + 0.996295i \(0.527408\pi\)
\(824\) 341.954i 0.414993i
\(825\) 0 0
\(826\) 199.197 0.241159
\(827\) 1063.38i 1.28583i 0.765938 + 0.642914i \(0.222275\pi\)
−0.765938 + 0.642914i \(0.777725\pi\)
\(828\) 0 0
\(829\) 1374.71 1.65828 0.829139 0.559043i \(-0.188831\pi\)
0.829139 + 0.559043i \(0.188831\pi\)
\(830\) − 974.541i − 1.17415i
\(831\) 0 0
\(832\) −639.425 −0.768539
\(833\) 777.353i 0.933197i
\(834\) 0 0
\(835\) −77.8973 −0.0932902
\(836\) − 99.4847i − 0.119001i
\(837\) 0 0
\(838\) 897.971 1.07156
\(839\) − 65.4450i − 0.0780036i −0.999239 0.0390018i \(-0.987582\pi\)
0.999239 0.0390018i \(-0.0124178\pi\)
\(840\) 0 0
\(841\) −524.673 −0.623868
\(842\) 474.609i 0.563669i
\(843\) 0 0
\(844\) 206.279 0.244407
\(845\) 663.527i 0.785239i
\(846\) 0 0
\(847\) −266.689 −0.314863
\(848\) − 400.364i − 0.472127i
\(849\) 0 0
\(850\) 146.566 0.172430
\(851\) − 2164.55i − 2.54353i
\(852\) 0 0
\(853\) 250.955 0.294203 0.147101 0.989121i \(-0.453006\pi\)
0.147101 + 0.989121i \(0.453006\pi\)
\(854\) − 350.045i − 0.409889i
\(855\) 0 0
\(856\) 704.044 0.822481
\(857\) − 506.000i − 0.590432i −0.955431 0.295216i \(-0.904608\pi\)
0.955431 0.295216i \(-0.0953916\pi\)
\(858\) 0 0
\(859\) −1358.56 −1.58156 −0.790780 0.612101i \(-0.790325\pi\)
−0.790780 + 0.612101i \(0.790325\pi\)
\(860\) − 543.389i − 0.631848i
\(861\) 0 0
\(862\) 838.874 0.973172
\(863\) − 932.140i − 1.08012i −0.841628 0.540058i \(-0.818402\pi\)
0.841628 0.540058i \(-0.181598\pi\)
\(864\) 0 0
\(865\) 115.678 0.133731
\(866\) 1547.82i 1.78732i
\(867\) 0 0
\(868\) −447.393 −0.515430
\(869\) 104.987i 0.120814i
\(870\) 0 0
\(871\) 44.1050 0.0506372
\(872\) − 25.8299i − 0.0296215i
\(873\) 0 0
\(874\) 454.677 0.520226
\(875\) 311.082i 0.355523i
\(876\) 0 0
\(877\) 714.424 0.814623 0.407311 0.913289i \(-0.366466\pi\)
0.407311 + 0.913289i \(0.366466\pi\)
\(878\) 186.545i 0.212466i
\(879\) 0 0
\(880\) 202.104 0.229664
\(881\) − 1279.83i − 1.45270i −0.687325 0.726350i \(-0.741215\pi\)
0.687325 0.726350i \(-0.258785\pi\)
\(882\) 0 0
\(883\) −13.5483 −0.0153435 −0.00767175 0.999971i \(-0.502442\pi\)
−0.00767175 + 0.999971i \(0.502442\pi\)
\(884\) − 649.975i − 0.735265i
\(885\) 0 0
\(886\) −74.1652 −0.0837079
\(887\) − 294.505i − 0.332024i −0.986124 0.166012i \(-0.946911\pi\)
0.986124 0.166012i \(-0.0530890\pi\)
\(888\) 0 0
\(889\) −29.7799 −0.0334982
\(890\) 696.542i 0.782631i
\(891\) 0 0
\(892\) 427.645 0.479422
\(893\) − 253.701i − 0.284099i
\(894\) 0 0
\(895\) −1764.29 −1.97127
\(896\) 335.729i 0.374698i
\(897\) 0 0
\(898\) −89.6153 −0.0997943
\(899\) − 1163.32i − 1.29401i
\(900\) 0 0
\(901\) 862.566 0.957343
\(902\) 270.518i 0.299909i
\(903\) 0 0
\(904\) −951.980 −1.05308
\(905\) 1425.37i 1.57499i
\(906\) 0 0
\(907\) −1050.45 −1.15816 −0.579078 0.815272i \(-0.696587\pi\)
−0.579078 + 0.815272i \(0.696587\pi\)
\(908\) 1299.85i 1.43155i
\(909\) 0 0
\(910\) −277.648 −0.305108
\(911\) 1613.61i 1.77125i 0.464404 + 0.885623i \(0.346269\pi\)
−0.464404 + 0.885623i \(0.653731\pi\)
\(912\) 0 0
\(913\) 271.491 0.297361
\(914\) − 871.616i − 0.953628i
\(915\) 0 0
\(916\) 1834.67 2.00292
\(917\) − 614.937i − 0.670597i
\(918\) 0 0
\(919\) −442.772 −0.481797 −0.240899 0.970550i \(-0.577442\pi\)
−0.240899 + 0.970550i \(0.577442\pi\)
\(920\) − 797.338i − 0.866672i
\(921\) 0 0
\(922\) −754.191 −0.817994
\(923\) 290.345i 0.314566i
\(924\) 0 0
\(925\) −155.685 −0.168309
\(926\) − 146.950i − 0.158694i
\(927\) 0 0
\(928\) −1595.75 −1.71956
\(929\) 1413.29i 1.52131i 0.649159 + 0.760653i \(0.275121\pi\)
−0.649159 + 0.760653i \(0.724879\pi\)
\(930\) 0 0
\(931\) 173.449 0.186304
\(932\) − 1279.15i − 1.37248i
\(933\) 0 0
\(934\) −1153.23 −1.23472
\(935\) 435.424i 0.465694i
\(936\) 0 0
\(937\) 310.345 0.331212 0.165606 0.986192i \(-0.447042\pi\)
0.165606 + 0.986192i \(0.447042\pi\)
\(938\) − 54.6400i − 0.0582516i
\(939\) 0 0
\(940\) −1736.08 −1.84689
\(941\) − 1123.68i − 1.19413i −0.802191 0.597067i \(-0.796333\pi\)
0.802191 0.597067i \(-0.203667\pi\)
\(942\) 0 0
\(943\) −709.027 −0.751885
\(944\) 211.377i 0.223916i
\(945\) 0 0
\(946\) 263.965 0.279033
\(947\) 817.493i 0.863245i 0.902054 + 0.431622i \(0.142059\pi\)
−0.902054 + 0.431622i \(0.857941\pi\)
\(948\) 0 0
\(949\) −58.5828 −0.0617311
\(950\) − 32.7028i − 0.0344240i
\(951\) 0 0
\(952\) −206.353 −0.216757
\(953\) 637.458i 0.668896i 0.942414 + 0.334448i \(0.108550\pi\)
−0.942414 + 0.334448i \(0.891450\pi\)
\(954\) 0 0
\(955\) −600.293 −0.628579
\(956\) − 351.426i − 0.367601i
\(957\) 0 0
\(958\) −2848.38 −2.97326
\(959\) 346.369i 0.361177i
\(960\) 0 0
\(961\) 29.9452 0.0311604
\(962\) 1203.91i 1.25146i
\(963\) 0 0
\(964\) 749.858 0.777861
\(965\) 628.911i 0.651722i
\(966\) 0 0
\(967\) −796.666 −0.823853 −0.411927 0.911217i \(-0.635144\pi\)
−0.411927 + 0.911217i \(0.635144\pi\)
\(968\) 425.969i 0.440051i
\(969\) 0 0
\(970\) −1030.90 −1.06279
\(971\) 459.354i 0.473073i 0.971623 + 0.236536i \(0.0760122\pi\)
−0.971623 + 0.236536i \(0.923988\pi\)
\(972\) 0 0
\(973\) −561.057 −0.576626
\(974\) − 333.851i − 0.342763i
\(975\) 0 0
\(976\) 371.448 0.380582
\(977\) − 1545.88i − 1.58227i −0.611639 0.791137i \(-0.709489\pi\)
0.611639 0.791137i \(-0.290511\pi\)
\(978\) 0 0
\(979\) −194.045 −0.198207
\(980\) − 1186.91i − 1.21114i
\(981\) 0 0
\(982\) −49.5126 −0.0504201
\(983\) − 842.380i − 0.856948i −0.903554 0.428474i \(-0.859051\pi\)
0.903554 0.428474i \(-0.140949\pi\)
\(984\) 0 0
\(985\) −1431.68 −1.45349
\(986\) − 2093.77i − 2.12350i
\(987\) 0 0
\(988\) −145.027 −0.146789
\(989\) 691.851i 0.699546i
\(990\) 0 0
\(991\) −965.398 −0.974166 −0.487083 0.873356i \(-0.661939\pi\)
−0.487083 + 0.873356i \(0.661939\pi\)
\(992\) − 1359.30i − 1.37027i
\(993\) 0 0
\(994\) 359.697 0.361868
\(995\) − 454.264i − 0.456547i
\(996\) 0 0
\(997\) 1834.22 1.83974 0.919871 0.392221i \(-0.128293\pi\)
0.919871 + 0.392221i \(0.128293\pi\)
\(998\) 1521.90i 1.52495i
\(999\) 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.890.11 84
3.2 odd 2 inner 1143.3.b.a.890.74 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.11 84 1.1 even 1 trivial
1143.3.b.a.890.74 yes 84 3.2 odd 2 inner