Properties

Label 1170.2.a.o.1.1
Level $1170$
Weight $2$
Character 1170.1
Self dual yes
Analytic conductor $9.342$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1170.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.65685 q^{11} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.00000 q^{20} +5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.82843 q^{28} +8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} +0.828427 q^{34} +2.82843 q^{35} -11.6569 q^{37} -2.82843 q^{38} +1.00000 q^{40} +7.65685 q^{41} +9.65685 q^{43} -5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -13.3137 q^{53} +5.65685 q^{55} +2.82843 q^{56} -8.82843 q^{58} +2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +5.65685 q^{67} -0.828427 q^{68} -2.82843 q^{70} +5.65685 q^{71} -14.4853 q^{73} +11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +2.34315 q^{79} -1.00000 q^{80} -7.65685 q^{82} -6.34315 q^{83} +0.828427 q^{85} -9.65685 q^{86} +5.65685 q^{88} +15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -8.00000 q^{94} -2.82843 q^{95} -3.17157 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + 2q^{10} - 2q^{13} + 2q^{16} + 4q^{17} - 2q^{20} + 2q^{25} + 2q^{26} + 12q^{29} + 8q^{31} - 2q^{32} - 4q^{34} - 12q^{37} + 2q^{40} + 4q^{41} + 8q^{43} + 16q^{47} + 2q^{49} - 2q^{50} - 2q^{52} - 4q^{53} - 12q^{58} + 16q^{59} + 12q^{61} - 8q^{62} + 2q^{64} + 2q^{65} + 4q^{68} - 12q^{73} + 12q^{74} + 32q^{77} + 16q^{79} - 2q^{80} - 4q^{82} - 24q^{83} - 4q^{85} - 8q^{86} + 20q^{89} - 16q^{94} - 12q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 2.82843 0.755929
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 5.65685 1.20605
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −2.82843 −0.534522
\(29\) 8.82843 1.63940 0.819699 0.572795i \(-0.194141\pi\)
0.819699 + 0.572795i \(0.194141\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.828427 0.142074
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −11.6569 −1.91638 −0.958188 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) −2.82843 −0.458831
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) −5.65685 −0.852803
\(45\) 0 0
\(46\) −8.48528 −1.25109
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 2.82843 0.377964
\(57\) 0 0
\(58\) −8.82843 −1.15923
\(59\) 2.34315 0.305052 0.152526 0.988299i \(-0.451259\pi\)
0.152526 + 0.988299i \(0.451259\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) −0.828427 −0.100462
\(69\) 0 0
\(70\) −2.82843 −0.338062
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) −14.4853 −1.69537 −0.847687 0.530497i \(-0.822005\pi\)
−0.847687 + 0.530497i \(0.822005\pi\)
\(74\) 11.6569 1.35508
\(75\) 0 0
\(76\) 2.82843 0.324443
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −7.65685 −0.845558
\(83\) −6.34315 −0.696251 −0.348125 0.937448i \(-0.613182\pi\)
−0.348125 + 0.937448i \(0.613182\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) −9.65685 −1.04133
\(87\) 0 0
\(88\) 5.65685 0.603023
\(89\) 15.6569 1.65962 0.829812 0.558044i \(-0.188448\pi\)
0.829812 + 0.558044i \(0.188448\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 8.48528 0.884652
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −3.17157 −0.322024 −0.161012 0.986952i \(-0.551476\pi\)
−0.161012 + 0.986952i \(0.551476\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −16.1421 −1.60620 −0.803101 0.595843i \(-0.796818\pi\)
−0.803101 + 0.595843i \(0.796818\pi\)
\(102\) 0 0
\(103\) −1.65685 −0.163255 −0.0816274 0.996663i \(-0.526012\pi\)
−0.0816274 + 0.996663i \(0.526012\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 13.3137 1.29314
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) −5.65685 −0.539360
\(111\) 0 0
\(112\) −2.82843 −0.267261
\(113\) −6.48528 −0.610084 −0.305042 0.952339i \(-0.598670\pi\)
−0.305042 + 0.952339i \(0.598670\pi\)
\(114\) 0 0
\(115\) −8.48528 −0.791257
\(116\) 8.82843 0.819699
\(117\) 0 0
\(118\) −2.34315 −0.215704
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.65685 0.856907 0.428454 0.903564i \(-0.359059\pi\)
0.428454 + 0.903564i \(0.359059\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) 6.14214 0.536641 0.268320 0.963330i \(-0.413531\pi\)
0.268320 + 0.963330i \(0.413531\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −5.65685 −0.488678
\(135\) 0 0
\(136\) 0.828427 0.0710370
\(137\) 17.3137 1.47921 0.739605 0.673041i \(-0.235012\pi\)
0.739605 + 0.673041i \(0.235012\pi\)
\(138\) 0 0
\(139\) 6.34315 0.538019 0.269009 0.963138i \(-0.413304\pi\)
0.269009 + 0.963138i \(0.413304\pi\)
\(140\) 2.82843 0.239046
\(141\) 0 0
\(142\) −5.65685 −0.474713
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) −8.82843 −0.733161
\(146\) 14.4853 1.19881
\(147\) 0 0
\(148\) −11.6569 −0.958188
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −2.82843 −0.229416
\(153\) 0 0
\(154\) −16.0000 −1.28932
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) −2.34315 −0.186411
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) 7.65685 0.597900
\(165\) 0 0
\(166\) 6.34315 0.492324
\(167\) −8.97056 −0.694163 −0.347081 0.937835i \(-0.612827\pi\)
−0.347081 + 0.937835i \(0.612827\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −0.828427 −0.0635375
\(171\) 0 0
\(172\) 9.65685 0.736328
\(173\) 9.31371 0.708108 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(174\) 0 0
\(175\) −2.82843 −0.213809
\(176\) −5.65685 −0.426401
\(177\) 0 0
\(178\) −15.6569 −1.17353
\(179\) −7.51472 −0.561676 −0.280838 0.959755i \(-0.590612\pi\)
−0.280838 + 0.959755i \(0.590612\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) −2.82843 −0.209657
\(183\) 0 0
\(184\) −8.48528 −0.625543
\(185\) 11.6569 0.857029
\(186\) 0 0
\(187\) 4.68629 0.342696
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 2.82843 0.205196
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 0 0
\(193\) 2.48528 0.178894 0.0894472 0.995992i \(-0.471490\pi\)
0.0894472 + 0.995992i \(0.471490\pi\)
\(194\) 3.17157 0.227706
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 13.3137 0.948562 0.474281 0.880373i \(-0.342708\pi\)
0.474281 + 0.880373i \(0.342708\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 16.1421 1.13576
\(203\) −24.9706 −1.75259
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 1.65685 0.115439
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) −13.3137 −0.914389
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −9.65685 −0.658592
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) −8.82843 −0.597937
\(219\) 0 0
\(220\) 5.65685 0.381385
\(221\) 0.828427 0.0557260
\(222\) 0 0
\(223\) 10.8284 0.725125 0.362563 0.931959i \(-0.381902\pi\)
0.362563 + 0.931959i \(0.381902\pi\)
\(224\) 2.82843 0.188982
\(225\) 0 0
\(226\) 6.48528 0.431394
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 4.14214 0.273720 0.136860 0.990590i \(-0.456299\pi\)
0.136860 + 0.990590i \(0.456299\pi\)
\(230\) 8.48528 0.559503
\(231\) 0 0
\(232\) −8.82843 −0.579615
\(233\) −5.51472 −0.361281 −0.180641 0.983549i \(-0.557817\pi\)
−0.180641 + 0.983549i \(0.557817\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 2.34315 0.152526
\(237\) 0 0
\(238\) −2.34315 −0.151884
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) −21.0000 −1.34993
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −2.82843 −0.179969
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −10.8284 −0.683484 −0.341742 0.939794i \(-0.611017\pi\)
−0.341742 + 0.939794i \(0.611017\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) −9.65685 −0.605925
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.82843 0.301189 0.150595 0.988596i \(-0.451881\pi\)
0.150595 + 0.988596i \(0.451881\pi\)
\(258\) 0 0
\(259\) 32.9706 2.04869
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −6.14214 −0.379462
\(263\) −16.4853 −1.01653 −0.508263 0.861202i \(-0.669712\pi\)
−0.508263 + 0.861202i \(0.669712\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) 5.65685 0.345547
\(269\) 14.4853 0.883183 0.441592 0.897216i \(-0.354414\pi\)
0.441592 + 0.897216i \(0.354414\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) −0.828427 −0.0502308
\(273\) 0 0
\(274\) −17.3137 −1.04596
\(275\) −5.65685 −0.341121
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −6.34315 −0.380437
\(279\) 0 0
\(280\) −2.82843 −0.169031
\(281\) −8.34315 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(282\) 0 0
\(283\) −17.6569 −1.04959 −0.524796 0.851228i \(-0.675858\pi\)
−0.524796 + 0.851228i \(0.675858\pi\)
\(284\) 5.65685 0.335673
\(285\) 0 0
\(286\) −5.65685 −0.334497
\(287\) −21.6569 −1.27836
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 8.82843 0.518423
\(291\) 0 0
\(292\) −14.4853 −0.847687
\(293\) 16.6274 0.971384 0.485692 0.874130i \(-0.338568\pi\)
0.485692 + 0.874130i \(0.338568\pi\)
\(294\) 0 0
\(295\) −2.34315 −0.136423
\(296\) 11.6569 0.677541
\(297\) 0 0
\(298\) 3.65685 0.211836
\(299\) −8.48528 −0.490716
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 2.82843 0.162221
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −21.6569 −1.23602 −0.618011 0.786169i \(-0.712061\pi\)
−0.618011 + 0.786169i \(0.712061\pi\)
\(308\) 16.0000 0.911685
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −30.9706 −1.75056 −0.875280 0.483617i \(-0.839323\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(314\) −5.31371 −0.299870
\(315\) 0 0
\(316\) 2.34315 0.131812
\(317\) −25.3137 −1.42176 −0.710880 0.703314i \(-0.751703\pi\)
−0.710880 + 0.703314i \(0.751703\pi\)
\(318\) 0 0
\(319\) −49.9411 −2.79617
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 24.0000 1.33747
\(323\) −2.34315 −0.130376
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −11.3137 −0.626608
\(327\) 0 0
\(328\) −7.65685 −0.422779
\(329\) −22.6274 −1.24749
\(330\) 0 0
\(331\) −8.48528 −0.466393 −0.233197 0.972430i \(-0.574919\pi\)
−0.233197 + 0.972430i \(0.574919\pi\)
\(332\) −6.34315 −0.348125
\(333\) 0 0
\(334\) 8.97056 0.490847
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) 10.9706 0.597605 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 0.828427 0.0449278
\(341\) −22.6274 −1.22534
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) −9.65685 −0.520663
\(345\) 0 0
\(346\) −9.31371 −0.500708
\(347\) 9.65685 0.518407 0.259204 0.965823i \(-0.416540\pi\)
0.259204 + 0.965823i \(0.416540\pi\)
\(348\) 0 0
\(349\) 12.1421 0.649954 0.324977 0.945722i \(-0.394644\pi\)
0.324977 + 0.945722i \(0.394644\pi\)
\(350\) 2.82843 0.151186
\(351\) 0 0
\(352\) 5.65685 0.301511
\(353\) −5.31371 −0.282820 −0.141410 0.989951i \(-0.545164\pi\)
−0.141410 + 0.989951i \(0.545164\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) 15.6569 0.829812
\(357\) 0 0
\(358\) 7.51472 0.397165
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 7.65685 0.402435
\(363\) 0 0
\(364\) 2.82843 0.148250
\(365\) 14.4853 0.758194
\(366\) 0 0
\(367\) 25.6569 1.33928 0.669638 0.742687i \(-0.266449\pi\)
0.669638 + 0.742687i \(0.266449\pi\)
\(368\) 8.48528 0.442326
\(369\) 0 0
\(370\) −11.6569 −0.606011
\(371\) 37.6569 1.95505
\(372\) 0 0
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) −4.68629 −0.242322
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −8.82843 −0.454687
\(378\) 0 0
\(379\) −7.51472 −0.386005 −0.193003 0.981198i \(-0.561823\pi\)
−0.193003 + 0.981198i \(0.561823\pi\)
\(380\) −2.82843 −0.145095
\(381\) 0 0
\(382\) 11.3137 0.578860
\(383\) 29.6569 1.51539 0.757697 0.652606i \(-0.226324\pi\)
0.757697 + 0.652606i \(0.226324\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) −2.48528 −0.126497
\(387\) 0 0
\(388\) −3.17157 −0.161012
\(389\) 6.48528 0.328817 0.164408 0.986392i \(-0.447429\pi\)
0.164408 + 0.986392i \(0.447429\pi\)
\(390\) 0 0
\(391\) −7.02944 −0.355494
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −13.3137 −0.670735
\(395\) −2.34315 −0.117896
\(396\) 0 0
\(397\) 30.2843 1.51992 0.759962 0.649968i \(-0.225218\pi\)
0.759962 + 0.649968i \(0.225218\pi\)
\(398\) −10.3431 −0.518455
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 26.9706 1.34685 0.673423 0.739258i \(-0.264823\pi\)
0.673423 + 0.739258i \(0.264823\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −16.1421 −0.803101
\(405\) 0 0
\(406\) 24.9706 1.23927
\(407\) 65.9411 3.26858
\(408\) 0 0
\(409\) −3.65685 −0.180820 −0.0904099 0.995905i \(-0.528818\pi\)
−0.0904099 + 0.995905i \(0.528818\pi\)
\(410\) 7.65685 0.378145
\(411\) 0 0
\(412\) −1.65685 −0.0816274
\(413\) −6.62742 −0.326114
\(414\) 0 0
\(415\) 6.34315 0.311373
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) 10.8284 0.529003 0.264502 0.964385i \(-0.414793\pi\)
0.264502 + 0.964385i \(0.414793\pi\)
\(420\) 0 0
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) −0.686292 −0.0334081
\(423\) 0 0
\(424\) 13.3137 0.646571
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) −16.9706 −0.821263
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 9.65685 0.465695
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −22.9706 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(434\) 11.3137 0.543075
\(435\) 0 0
\(436\) 8.82843 0.422805
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 22.6274 1.07995 0.539974 0.841682i \(-0.318434\pi\)
0.539974 + 0.841682i \(0.318434\pi\)
\(440\) −5.65685 −0.269680
\(441\) 0 0
\(442\) −0.828427 −0.0394043
\(443\) −30.3431 −1.44165 −0.720823 0.693119i \(-0.756236\pi\)
−0.720823 + 0.693119i \(0.756236\pi\)
\(444\) 0 0
\(445\) −15.6569 −0.742206
\(446\) −10.8284 −0.512741
\(447\) 0 0
\(448\) −2.82843 −0.133631
\(449\) −26.2843 −1.24043 −0.620216 0.784431i \(-0.712955\pi\)
−0.620216 + 0.784431i \(0.712955\pi\)
\(450\) 0 0
\(451\) −43.3137 −2.03956
\(452\) −6.48528 −0.305042
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) −2.82843 −0.132599
\(456\) 0 0
\(457\) 20.8284 0.974313 0.487156 0.873315i \(-0.338034\pi\)
0.487156 + 0.873315i \(0.338034\pi\)
\(458\) −4.14214 −0.193549
\(459\) 0 0
\(460\) −8.48528 −0.395628
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −3.79899 −0.176554 −0.0882770 0.996096i \(-0.528136\pi\)
−0.0882770 + 0.996096i \(0.528136\pi\)
\(464\) 8.82843 0.409849
\(465\) 0 0
\(466\) 5.51472 0.255464
\(467\) −7.31371 −0.338438 −0.169219 0.985578i \(-0.554125\pi\)
−0.169219 + 0.985578i \(0.554125\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) −2.34315 −0.107852
\(473\) −54.6274 −2.51177
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 2.34315 0.107398
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) −5.31371 −0.242033
\(483\) 0 0
\(484\) 21.0000 0.954545
\(485\) 3.17157 0.144014
\(486\) 0 0
\(487\) 16.4853 0.747019 0.373510 0.927626i \(-0.378154\pi\)
0.373510 + 0.927626i \(0.378154\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 38.1421 1.72133 0.860665 0.509171i \(-0.170048\pi\)
0.860665 + 0.509171i \(0.170048\pi\)
\(492\) 0 0
\(493\) −7.31371 −0.329393
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 0.485281 0.0217242 0.0108621 0.999941i \(-0.496542\pi\)
0.0108621 + 0.999941i \(0.496542\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 10.8284 0.483296
\(503\) 23.5147 1.04847 0.524235 0.851574i \(-0.324351\pi\)
0.524235 + 0.851574i \(0.324351\pi\)
\(504\) 0 0
\(505\) 16.1421 0.718316
\(506\) 48.0000 2.13386
\(507\) 0 0
\(508\) 9.65685 0.428454
\(509\) 37.3137 1.65390 0.826951 0.562275i \(-0.190074\pi\)
0.826951 + 0.562275i \(0.190074\pi\)
\(510\) 0 0
\(511\) 40.9706 1.81243
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −4.82843 −0.212973
\(515\) 1.65685 0.0730097
\(516\) 0 0
\(517\) −45.2548 −1.99031
\(518\) −32.9706 −1.44864
\(519\) 0 0
\(520\) −1.00000 −0.0438529
\(521\) −26.9706 −1.18160 −0.590801 0.806817i \(-0.701188\pi\)
−0.590801 + 0.806817i \(0.701188\pi\)
\(522\) 0 0
\(523\) −10.6274 −0.464704 −0.232352 0.972632i \(-0.574642\pi\)
−0.232352 + 0.972632i \(0.574642\pi\)
\(524\) 6.14214 0.268320
\(525\) 0 0
\(526\) 16.4853 0.718792
\(527\) −3.31371 −0.144347
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) −13.3137 −0.578311
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −7.65685 −0.331655
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) −5.65685 −0.244339
\(537\) 0 0
\(538\) −14.4853 −0.624505
\(539\) −5.65685 −0.243658
\(540\) 0 0
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) −7.31371 −0.314151
\(543\) 0 0
\(544\) 0.828427 0.0355185
\(545\) −8.82843 −0.378168
\(546\) 0 0
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) 17.3137 0.739605
\(549\) 0 0
\(550\) 5.65685 0.241209
\(551\) 24.9706 1.06378
\(552\) 0 0
\(553\) −6.62742 −0.281826
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 6.34315 0.269009
\(557\) −10.6863 −0.452793 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(558\) 0 0
\(559\) −9.65685 −0.408441
\(560\) 2.82843 0.119523
\(561\) 0 0
\(562\) 8.34315 0.351934
\(563\) 30.3431 1.27881 0.639406 0.768870i \(-0.279181\pi\)
0.639406 + 0.768870i \(0.279181\pi\)
\(564\) 0 0
\(565\) 6.48528 0.272838
\(566\) 17.6569 0.742173
\(567\) 0 0
\(568\) −5.65685 −0.237356
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 5.65685 0.236525
\(573\) 0 0
\(574\) 21.6569 0.903940
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) −23.4558 −0.976480 −0.488240 0.872710i \(-0.662361\pi\)
−0.488240 + 0.872710i \(0.662361\pi\)
\(578\) 16.3137 0.678561
\(579\) 0 0
\(580\) −8.82843 −0.366580
\(581\) 17.9411 0.744323
\(582\) 0 0
\(583\) 75.3137 3.11918
\(584\) 14.4853 0.599405
\(585\) 0 0
\(586\) −16.6274 −0.686872
\(587\) −2.62742 −0.108445 −0.0542226 0.998529i \(-0.517268\pi\)
−0.0542226 + 0.998529i \(0.517268\pi\)
\(588\) 0 0
\(589\) 11.3137 0.466173
\(590\) 2.34315 0.0964658
\(591\) 0 0
\(592\) −11.6569 −0.479094
\(593\) 0.343146 0.0140913 0.00704565 0.999975i \(-0.497757\pi\)
0.00704565 + 0.999975i \(0.497757\pi\)
\(594\) 0 0
\(595\) −2.34315 −0.0960596
\(596\) −3.65685 −0.149791
\(597\) 0 0
\(598\) 8.48528 0.346989
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 27.3137 1.11322
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −21.0000 −0.853771
\(606\) 0 0
\(607\) 28.9706 1.17588 0.587939 0.808905i \(-0.299939\pi\)
0.587939 + 0.808905i \(0.299939\pi\)
\(608\) −2.82843 −0.114708
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 22.2843 0.900053 0.450027 0.893015i \(-0.351414\pi\)
0.450027 + 0.893015i \(0.351414\pi\)
\(614\) 21.6569 0.874000
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) −34.8284 −1.39987 −0.699936 0.714205i \(-0.746788\pi\)
−0.699936 + 0.714205i \(0.746788\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −44.2843 −1.77421
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.9706 1.23783
\(627\) 0 0
\(628\) 5.31371 0.212040
\(629\) 9.65685 0.385044
\(630\) 0 0
\(631\) −33.6569 −1.33986 −0.669929 0.742425i \(-0.733676\pi\)
−0.669929 + 0.742425i \(0.733676\pi\)
\(632\) −2.34315 −0.0932053
\(633\) 0 0
\(634\) 25.3137 1.00534
\(635\) −9.65685 −0.383221
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 49.9411 1.97719
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 4.62742 0.182772 0.0913860 0.995816i \(-0.470870\pi\)
0.0913860 + 0.995816i \(0.470870\pi\)
\(642\) 0 0
\(643\) 39.5980 1.56159 0.780796 0.624786i \(-0.214814\pi\)
0.780796 + 0.624786i \(0.214814\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 2.34315 0.0921898
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) 0 0
\(649\) −13.2548 −0.520298
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 11.3137 0.443079
\(653\) 42.2843 1.65471 0.827356 0.561678i \(-0.189844\pi\)
0.827356 + 0.561678i \(0.189844\pi\)
\(654\) 0 0
\(655\) −6.14214 −0.239993
\(656\) 7.65685 0.298950
\(657\) 0 0
\(658\) 22.6274 0.882109
\(659\) 7.51472 0.292732 0.146366 0.989231i \(-0.453242\pi\)
0.146366 + 0.989231i \(0.453242\pi\)
\(660\) 0 0
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) 8.48528 0.329790
\(663\) 0 0
\(664\) 6.34315 0.246162
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 74.9117 2.90059
\(668\) −8.97056 −0.347081
\(669\) 0 0
\(670\) 5.65685 0.218543
\(671\) −33.9411 −1.31028
\(672\) 0 0
\(673\) 32.6274 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(674\) −10.9706 −0.422570
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −12.3431 −0.474386 −0.237193 0.971463i \(-0.576227\pi\)
−0.237193 + 0.971463i \(0.576227\pi\)
\(678\) 0 0
\(679\) 8.97056 0.344259
\(680\) −0.828427 −0.0317687
\(681\) 0 0
\(682\) 22.6274 0.866449
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) 0 0
\(685\) −17.3137 −0.661523
\(686\) −16.9706 −0.647939
\(687\) 0 0
\(688\) 9.65685 0.368164
\(689\) 13.3137 0.507212
\(690\) 0 0
\(691\) −27.7990 −1.05752 −0.528762 0.848770i \(-0.677343\pi\)
−0.528762 + 0.848770i \(0.677343\pi\)
\(692\) 9.31371 0.354054
\(693\) 0 0
\(694\) −9.65685 −0.366569
\(695\) −6.34315 −0.240609
\(696\) 0 0
\(697\) −6.34315 −0.240264
\(698\) −12.1421 −0.459587
\(699\) 0 0
\(700\) −2.82843 −0.106904
\(701\) −0.142136 −0.00536839 −0.00268419 0.999996i \(-0.500854\pi\)
−0.00268419 + 0.999996i \(0.500854\pi\)
\(702\) 0 0
\(703\) −32.9706 −1.24351
\(704\) −5.65685 −0.213201
\(705\) 0 0
\(706\) 5.31371 0.199984
\(707\) 45.6569 1.71710
\(708\) 0 0
\(709\) −7.17157 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(710\) 5.65685 0.212298
\(711\) 0 0
\(712\) −15.6569 −0.586765
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) −7.51472 −0.280838
\(717\) 0 0
\(718\) −28.2843 −1.05556
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 11.0000 0.409378
\(723\) 0 0
\(724\) −7.65685 −0.284565
\(725\) 8.82843 0.327880
\(726\) 0 0
\(727\) −45.9411 −1.70386 −0.851931 0.523654i \(-0.824568\pi\)
−0.851931 + 0.523654i \(0.824568\pi\)
\(728\) −2.82843 −0.104828
\(729\) 0 0
\(730\) −14.4853 −0.536124
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −0.343146 −0.0126744 −0.00633719 0.999980i \(-0.502017\pi\)
−0.00633719 + 0.999980i \(0.502017\pi\)
\(734\) −25.6569 −0.947012
\(735\) 0 0
\(736\) −8.48528 −0.312772
\(737\) −32.0000 −1.17874
\(738\) 0 0
\(739\) −14.1421 −0.520227 −0.260113 0.965578i \(-0.583760\pi\)
−0.260113 + 0.965578i \(0.583760\pi\)
\(740\) 11.6569 0.428514
\(741\) 0 0
\(742\) −37.6569 −1.38243
\(743\) 36.2843 1.33114 0.665570 0.746335i \(-0.268188\pi\)
0.665570 + 0.746335i \(0.268188\pi\)
\(744\) 0 0
\(745\) 3.65685 0.133977
\(746\) 2.68629 0.0983521
\(747\) 0 0
\(748\) 4.68629 0.171348
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 11.3137 0.412843 0.206422 0.978463i \(-0.433818\pi\)
0.206422 + 0.978463i \(0.433818\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 8.82843 0.321512
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 19.9411 0.724773 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(758\) 7.51472 0.272947
\(759\) 0 0
\(760\) 2.82843 0.102598
\(761\) −27.6569 −1.00256 −0.501280 0.865285i \(-0.667137\pi\)
−0.501280 + 0.865285i \(0.667137\pi\)
\(762\) 0 0
\(763\) −24.9706 −0.903995
\(764\) −11.3137 −0.409316
\(765\) 0 0
\(766\) −29.6569 −1.07155
\(767\) −2.34315 −0.0846061
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) 2.48528 0.0894472
\(773\) 53.3137 1.91756 0.958780 0.284148i \(-0.0917107\pi\)
0.958780 + 0.284148i \(0.0917107\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 3.17157 0.113853
\(777\) 0 0
\(778\) −6.48528 −0.232509
\(779\) 21.6569 0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 7.02944 0.251372
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −5.31371 −0.189654
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) 13.3137 0.474281
\(789\) 0 0
\(790\) 2.34315 0.0833654
\(791\) 18.3431 0.652207
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) −30.2843 −1.07475
\(795\) 0 0
\(796\) 10.3431 0.366603
\(797\) −16.6274 −0.588973 −0.294487 0.955656i \(-0.595149\pi\)
−0.294487 + 0.955656i \(0.595149\pi\)
\(798\) 0 0
\(799\) −6.62742 −0.234461
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −26.9706 −0.952364
\(803\) 81.9411 2.89164
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 16.1421 0.567878
\(809\) −13.3137 −0.468085 −0.234043 0.972226i \(-0.575196\pi\)
−0.234043 + 0.972226i \(0.575196\pi\)
\(810\) 0 0
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) −24.9706 −0.876295
\(813\) 0 0
\(814\) −65.9411 −2.31124
\(815\) −11.3137 −0.396302
\(816\) 0 0
\(817\) 27.3137 0.955586
\(818\) 3.65685 0.127859
\(819\) 0 0
\(820\) −7.65685 −0.267389
\(821\) −34.2843 −1.19653 −0.598265 0.801299i \(-0.704143\pi\)
−0.598265 + 0.801299i \(0.704143\pi\)
\(822\) 0 0
\(823\) −52.9706 −1.84644 −0.923219 0.384275i \(-0.874452\pi\)
−0.923219 + 0.384275i \(0.874452\pi\)
\(824\) 1.65685 0.0577193
\(825\) 0 0
\(826\) 6.62742 0.230597
\(827\) −9.65685 −0.335802 −0.167901 0.985804i \(-0.553699\pi\)
−0.167901 + 0.985804i \(0.553699\pi\)
\(828\) 0 0
\(829\) −53.3137 −1.85166 −0.925831 0.377938i \(-0.876633\pi\)
−0.925831 + 0.377938i \(0.876633\pi\)
\(830\) −6.34315 −0.220174
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −0.828427 −0.0287033
\(834\) 0 0
\(835\) 8.97056 0.310439
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) −10.8284 −0.374062
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) 24.1421 0.831993
\(843\) 0 0
\(844\) 0.686292 0.0236231
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −59.3970 −2.04090
\(848\) −13.3137 −0.457195
\(849\) 0 0
\(850\) 0.828427 0.0284148
\(851\) −98.9117 −3.39065
\(852\) 0 0
\(853\) 38.2843 1.31083 0.655414 0.755270i \(-0.272494\pi\)
0.655414 + 0.755270i \(0.272494\pi\)
\(854\) 16.9706 0.580721
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 20.8284 0.711486 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(858\) 0 0
\(859\) 37.9411 1.29453 0.647267 0.762263i \(-0.275912\pi\)
0.647267 + 0.762263i \(0.275912\pi\)
\(860\) −9.65685 −0.329296
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 28.2843 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(864\) 0 0
\(865\) −9.31371 −0.316676
\(866\) 22.9706 0.780571
\(867\) 0 0
\(868\) −11.3137 −0.384012
\(869\) −13.2548 −0.449639
\(870\) 0 0
\(871\) −5.65685 −0.191675
\(872\) −8.82843 −0.298968
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) −51.2548 −1.73075 −0.865376 0.501122i \(-0.832921\pi\)
−0.865376 + 0.501122i \(0.832921\pi\)
\(878\) −22.6274 −0.763638
\(879\) 0 0
\(880\) 5.65685 0.190693
\(881\) 10.2843 0.346486 0.173243 0.984879i \(-0.444575\pi\)
0.173243 + 0.984879i \(0.444575\pi\)
\(882\) 0 0
\(883\) 31.3137 1.05379 0.526895 0.849930i \(-0.323356\pi\)
0.526895 + 0.849930i \(0.323356\pi\)
\(884\) 0.828427 0.0278630
\(885\) 0 0
\(886\) 30.3431 1.01940
\(887\) 40.4853 1.35936 0.679681 0.733508i \(-0.262118\pi\)
0.679681 + 0.733508i \(0.262118\pi\)
\(888\) 0 0
\(889\) −27.3137 −0.916072
\(890\) 15.6569 0.524819
\(891\) 0 0
\(892\) 10.8284 0.362563
\(893\) 22.6274 0.757198
\(894\) 0 0
\(895\) 7.51472 0.251189
\(896\) 2.82843 0.0944911
\(897\) 0 0
\(898\) 26.2843 0.877117
\(899\) 35.3137 1.17778
\(900\) 0 0
\(901\) 11.0294 0.367444
\(902\) 43.3137 1.44219
\(903\) 0 0
\(904\) 6.48528 0.215697
\(905\) 7.65685 0.254522
\(906\) 0 0
\(907\) 8.28427 0.275075 0.137537 0.990497i \(-0.456081\pi\)
0.137537 + 0.990497i \(0.456081\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) 2.82843 0.0937614
\(911\) −24.9706 −0.827312 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(912\) 0 0
\(913\) 35.8823 1.18753
\(914\) −20.8284 −0.688943
\(915\) 0 0
\(916\) 4.14214 0.136860
\(917\) −17.3726 −0.573693
\(918\) 0 0
\(919\) 41.9411 1.38351 0.691755 0.722132i \(-0.256838\pi\)
0.691755 + 0.722132i \(0.256838\pi\)
\(920\) 8.48528 0.279751
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) −5.65685 −0.186198
\(924\) 0 0
\(925\) −11.6569 −0.383275
\(926\) 3.79899 0.124843
\(927\) 0 0
\(928\) −8.82843 −0.289807
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) −5.51472 −0.180641
\(933\) 0 0
\(934\) 7.31371 0.239312
\(935\) −4.68629 −0.153258
\(936\) 0 0
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −54.9706 −1.79199 −0.895995 0.444065i \(-0.853536\pi\)
−0.895995 + 0.444065i \(0.853536\pi\)
\(942\) 0 0
\(943\) 64.9706 2.11573
\(944\) 2.34315 0.0762629
\(945\) 0 0
\(946\) 54.6274 1.77609
\(947\) −30.3431 −0.986020 −0.493010 0.870024i \(-0.664103\pi\)
−0.493010 + 0.870024i \(0.664103\pi\)
\(948\) 0 0
\(949\) 14.4853 0.470212
\(950\) −2.82843 −0.0917663
\(951\) 0 0
\(952\) −2.34315 −0.0759418
\(953\) 27.8579 0.902405 0.451202 0.892422i \(-0.350995\pi\)
0.451202 + 0.892422i \(0.350995\pi\)
\(954\) 0 0
\(955\) 11.3137 0.366103
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −11.3137 −0.365529
\(959\) −48.9706 −1.58134
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −11.6569 −0.375832
\(963\) 0 0
\(964\) 5.31371 0.171143
\(965\) −2.48528 −0.0800040
\(966\) 0 0
\(967\) −7.51472 −0.241657 −0.120829 0.992673i \(-0.538555\pi\)
−0.120829 + 0.992673i \(0.538555\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) −3.17157 −0.101833
\(971\) −15.5147 −0.497891 −0.248946 0.968517i \(-0.580084\pi\)
−0.248946 + 0.968517i \(0.580084\pi\)
\(972\) 0 0
\(973\) −17.9411 −0.575166
\(974\) −16.4853 −0.528222
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 8.34315 0.266921 0.133460 0.991054i \(-0.457391\pi\)
0.133460 + 0.991054i \(0.457391\pi\)
\(978\) 0 0
\(979\) −88.5685 −2.83066
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −38.1421 −1.21716
\(983\) 2.34315 0.0747347 0.0373674 0.999302i \(-0.488103\pi\)
0.0373674 + 0.999302i \(0.488103\pi\)
\(984\) 0 0
\(985\) −13.3137 −0.424210
\(986\) 7.31371 0.232916
\(987\) 0 0
\(988\) −2.82843 −0.0899843
\(989\) 81.9411 2.60558
\(990\) 0 0
\(991\) −42.9117 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −10.3431 −0.327900
\(996\) 0 0
\(997\) 61.3137 1.94182 0.970912 0.239435i \(-0.0769623\pi\)
0.970912 + 0.239435i \(0.0769623\pi\)
\(998\) −0.485281 −0.0153613
\(999\) 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 1170.2.a.o.1.1 2
3.2 odd 2 390.2.a.h.1.1 2
4.3 odd 2 9360.2.a.ch.1.2 2
5.2 odd 4 5850.2.e.bk.5149.1 4
5.3 odd 4 5850.2.e.bk.5149.4 4
5.4 even 2 5850.2.a.cl.1.2 2
12.11 even 2 3120.2.a.bc.1.2 2
15.2 even 4 1950.2.e.o.1249.3 4
15.8 even 4 1950.2.e.o.1249.2 4
15.14 odd 2 1950.2.a.bd.1.2 2
39.5 even 4 5070.2.b.q.1351.1 4
39.8 even 4 5070.2.b.q.1351.4 4
39.38 odd 2 5070.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 3.2 odd 2
1170.2.a.o.1.1 2 1.1 even 1 trivial
1950.2.a.bd.1.2 2 15.14 odd 2
1950.2.e.o.1249.2 4 15.8 even 4
1950.2.e.o.1249.3 4 15.2 even 4
3120.2.a.bc.1.2 2 12.11 even 2
5070.2.a.bc.1.2 2 39.38 odd 2
5070.2.b.q.1351.1 4 39.5 even 4
5070.2.b.q.1351.4 4 39.8 even 4
5850.2.a.cl.1.2 2 5.4 even 2
5850.2.e.bk.5149.1 4 5.2 odd 4
5850.2.e.bk.5149.4 4 5.3 odd 4
9360.2.a.ch.1.2 2 4.3 odd 2