Properties

Label 1840.2.a.m.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -3.56155 q^{13} +1.56155 q^{15} +2.56155 q^{17} -6.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} +6.12311 q^{29} -7.24621 q^{31} -3.12311 q^{33} -2.56155 q^{35} -4.56155 q^{37} -5.56155 q^{39} +4.12311 q^{41} -0.561553 q^{45} -4.68466 q^{47} -0.438447 q^{49} +4.00000 q^{51} -4.56155 q^{53} -2.00000 q^{55} -9.36932 q^{57} +3.68466 q^{59} -7.12311 q^{61} +1.43845 q^{63} -3.56155 q^{65} +8.56155 q^{67} -1.56155 q^{69} -10.1231 q^{71} -4.43845 q^{73} +1.56155 q^{75} +5.12311 q^{77} -4.87689 q^{79} -7.00000 q^{81} +13.9309 q^{83} +2.56155 q^{85} +9.56155 q^{87} +14.2462 q^{89} +9.12311 q^{91} -11.3153 q^{93} -6.00000 q^{95} -13.1231 q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{5} - q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{5} - q^{7} + 3q^{9} - 4q^{11} - 3q^{13} - q^{15} + q^{17} - 12q^{19} - 8q^{21} - 2q^{23} + 2q^{25} - 7q^{27} + 4q^{29} + 2q^{31} + 2q^{33} - q^{35} - 5q^{37} - 7q^{39} + 3q^{45} + 3q^{47} - 5q^{49} + 8q^{51} - 5q^{53} - 4q^{55} + 6q^{57} - 5q^{59} - 6q^{61} + 7q^{63} - 3q^{65} + 13q^{67} + q^{69} - 12q^{71} - 13q^{73} - q^{75} + 2q^{77} - 18q^{79} - 14q^{81} - q^{83} + q^{85} + 15q^{87} + 12q^{89} + 10q^{91} - 35q^{93} - 12q^{95} - 18q^{97} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.56155 −0.968176 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) 2.56155 0.621268 0.310634 0.950530i \(-0.399459\pi\)
0.310634 + 0.950530i \(0.399459\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 6.12311 1.13703 0.568516 0.822672i \(-0.307518\pi\)
0.568516 + 0.822672i \(0.307518\pi\)
\(30\) 0 0
\(31\) −7.24621 −1.30146 −0.650729 0.759310i \(-0.725537\pi\)
−0.650729 + 0.759310i \(0.725537\pi\)
\(32\) 0 0
\(33\) −3.12311 −0.543663
\(34\) 0 0
\(35\) −2.56155 −0.432981
\(36\) 0 0
\(37\) −4.56155 −0.749915 −0.374957 0.927042i \(-0.622343\pi\)
−0.374957 + 0.927042i \(0.622343\pi\)
\(38\) 0 0
\(39\) −5.56155 −0.890561
\(40\) 0 0
\(41\) 4.12311 0.643921 0.321960 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) −4.68466 −0.683328 −0.341664 0.939822i \(-0.610990\pi\)
−0.341664 + 0.939822i \(0.610990\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −4.56155 −0.626577 −0.313289 0.949658i \(-0.601431\pi\)
−0.313289 + 0.949658i \(0.601431\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −9.36932 −1.24100
\(58\) 0 0
\(59\) 3.68466 0.479702 0.239851 0.970810i \(-0.422901\pi\)
0.239851 + 0.970810i \(0.422901\pi\)
\(60\) 0 0
\(61\) −7.12311 −0.912020 −0.456010 0.889975i \(-0.650722\pi\)
−0.456010 + 0.889975i \(0.650722\pi\)
\(62\) 0 0
\(63\) 1.43845 0.181227
\(64\) 0 0
\(65\) −3.56155 −0.441756
\(66\) 0 0
\(67\) 8.56155 1.04596 0.522980 0.852345i \(-0.324820\pi\)
0.522980 + 0.852345i \(0.324820\pi\)
\(68\) 0 0
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) −10.1231 −1.20139 −0.600696 0.799478i \(-0.705110\pi\)
−0.600696 + 0.799478i \(0.705110\pi\)
\(72\) 0 0
\(73\) −4.43845 −0.519481 −0.259740 0.965678i \(-0.583637\pi\)
−0.259740 + 0.965678i \(0.583637\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) 5.12311 0.583832
\(78\) 0 0
\(79\) −4.87689 −0.548693 −0.274347 0.961631i \(-0.588462\pi\)
−0.274347 + 0.961631i \(0.588462\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 13.9309 1.52911 0.764556 0.644558i \(-0.222958\pi\)
0.764556 + 0.644558i \(0.222958\pi\)
\(84\) 0 0
\(85\) 2.56155 0.277839
\(86\) 0 0
\(87\) 9.56155 1.02511
\(88\) 0 0
\(89\) 14.2462 1.51010 0.755048 0.655670i \(-0.227614\pi\)
0.755048 + 0.655670i \(0.227614\pi\)
\(90\) 0 0
\(91\) 9.12311 0.956361
\(92\) 0 0
\(93\) −11.3153 −1.17335
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −13.1231 −1.33245 −0.666225 0.745751i \(-0.732091\pi\)
−0.666225 + 0.745751i \(0.732091\pi\)
\(98\) 0 0
\(99\) 1.12311 0.112876
\(100\) 0 0
\(101\) −17.6847 −1.75969 −0.879845 0.475261i \(-0.842353\pi\)
−0.879845 + 0.475261i \(0.842353\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) −3.43845 −0.332407 −0.166204 0.986091i \(-0.553151\pi\)
−0.166204 + 0.986091i \(0.553151\pi\)
\(108\) 0 0
\(109\) 15.3693 1.47211 0.736057 0.676920i \(-0.236686\pi\)
0.736057 + 0.676920i \(0.236686\pi\)
\(110\) 0 0
\(111\) −7.12311 −0.676095
\(112\) 0 0
\(113\) 12.8078 1.20485 0.602427 0.798174i \(-0.294201\pi\)
0.602427 + 0.798174i \(0.294201\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −6.56155 −0.601497
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.43845 0.580535
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.80776 −0.692827 −0.346414 0.938082i \(-0.612601\pi\)
−0.346414 + 0.938082i \(0.612601\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.192236 −0.0167957 −0.00839787 0.999965i \(-0.502673\pi\)
−0.00839787 + 0.999965i \(0.502673\pi\)
\(132\) 0 0
\(133\) 15.3693 1.33269
\(134\) 0 0
\(135\) −5.56155 −0.478662
\(136\) 0 0
\(137\) 6.87689 0.587533 0.293766 0.955877i \(-0.405091\pi\)
0.293766 + 0.955877i \(0.405091\pi\)
\(138\) 0 0
\(139\) 11.2462 0.953891 0.476946 0.878933i \(-0.341744\pi\)
0.476946 + 0.878933i \(0.341744\pi\)
\(140\) 0 0
\(141\) −7.31534 −0.616063
\(142\) 0 0
\(143\) 7.12311 0.595664
\(144\) 0 0
\(145\) 6.12311 0.508496
\(146\) 0 0
\(147\) −0.684658 −0.0564697
\(148\) 0 0
\(149\) −9.36932 −0.767564 −0.383782 0.923424i \(-0.625379\pi\)
−0.383782 + 0.923424i \(0.625379\pi\)
\(150\) 0 0
\(151\) 14.0540 1.14370 0.571848 0.820359i \(-0.306227\pi\)
0.571848 + 0.820359i \(0.306227\pi\)
\(152\) 0 0
\(153\) −1.43845 −0.116292
\(154\) 0 0
\(155\) −7.24621 −0.582030
\(156\) 0 0
\(157\) −4.31534 −0.344402 −0.172201 0.985062i \(-0.555088\pi\)
−0.172201 + 0.985062i \(0.555088\pi\)
\(158\) 0 0
\(159\) −7.12311 −0.564899
\(160\) 0 0
\(161\) 2.56155 0.201879
\(162\) 0 0
\(163\) −16.9309 −1.32613 −0.663064 0.748563i \(-0.730744\pi\)
−0.663064 + 0.748563i \(0.730744\pi\)
\(164\) 0 0
\(165\) −3.12311 −0.243133
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) 3.36932 0.257658
\(172\) 0 0
\(173\) 5.36932 0.408222 0.204111 0.978948i \(-0.434570\pi\)
0.204111 + 0.978948i \(0.434570\pi\)
\(174\) 0 0
\(175\) −2.56155 −0.193635
\(176\) 0 0
\(177\) 5.75379 0.432481
\(178\) 0 0
\(179\) 6.93087 0.518038 0.259019 0.965872i \(-0.416601\pi\)
0.259019 + 0.965872i \(0.416601\pi\)
\(180\) 0 0
\(181\) −5.36932 −0.399098 −0.199549 0.979888i \(-0.563948\pi\)
−0.199549 + 0.979888i \(0.563948\pi\)
\(182\) 0 0
\(183\) −11.1231 −0.822244
\(184\) 0 0
\(185\) −4.56155 −0.335372
\(186\) 0 0
\(187\) −5.12311 −0.374639
\(188\) 0 0
\(189\) 14.2462 1.03626
\(190\) 0 0
\(191\) −17.3693 −1.25680 −0.628400 0.777891i \(-0.716290\pi\)
−0.628400 + 0.777891i \(0.716290\pi\)
\(192\) 0 0
\(193\) 2.43845 0.175523 0.0877616 0.996142i \(-0.472029\pi\)
0.0877616 + 0.996142i \(0.472029\pi\)
\(194\) 0 0
\(195\) −5.56155 −0.398271
\(196\) 0 0
\(197\) −18.6847 −1.33123 −0.665613 0.746297i \(-0.731830\pi\)
−0.665613 + 0.746297i \(0.731830\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 13.3693 0.942999
\(202\) 0 0
\(203\) −15.6847 −1.10085
\(204\) 0 0
\(205\) 4.12311 0.287970
\(206\) 0 0
\(207\) 0.561553 0.0390306
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −0.315342 −0.0217090 −0.0108545 0.999941i \(-0.503455\pi\)
−0.0108545 + 0.999941i \(0.503455\pi\)
\(212\) 0 0
\(213\) −15.8078 −1.08313
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.5616 1.26004
\(218\) 0 0
\(219\) −6.93087 −0.468345
\(220\) 0 0
\(221\) −9.12311 −0.613686
\(222\) 0 0
\(223\) 17.6155 1.17962 0.589812 0.807541i \(-0.299202\pi\)
0.589812 + 0.807541i \(0.299202\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −26.2462 −1.74202 −0.871011 0.491263i \(-0.836535\pi\)
−0.871011 + 0.491263i \(0.836535\pi\)
\(228\) 0 0
\(229\) −26.7386 −1.76694 −0.883469 0.468489i \(-0.844799\pi\)
−0.883469 + 0.468489i \(0.844799\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −0.684658 −0.0448535 −0.0224267 0.999748i \(-0.507139\pi\)
−0.0224267 + 0.999748i \(0.507139\pi\)
\(234\) 0 0
\(235\) −4.68466 −0.305593
\(236\) 0 0
\(237\) −7.61553 −0.494682
\(238\) 0 0
\(239\) −2.75379 −0.178128 −0.0890639 0.996026i \(-0.528388\pi\)
−0.0890639 + 0.996026i \(0.528388\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) −0.438447 −0.0280114
\(246\) 0 0
\(247\) 21.3693 1.35970
\(248\) 0 0
\(249\) 21.7538 1.37859
\(250\) 0 0
\(251\) 7.36932 0.465147 0.232574 0.972579i \(-0.425285\pi\)
0.232574 + 0.972579i \(0.425285\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 31.8078 1.98411 0.992057 0.125790i \(-0.0401465\pi\)
0.992057 + 0.125790i \(0.0401465\pi\)
\(258\) 0 0
\(259\) 11.6847 0.726049
\(260\) 0 0
\(261\) −3.43845 −0.212835
\(262\) 0 0
\(263\) −0.0691303 −0.00426276 −0.00213138 0.999998i \(-0.500678\pi\)
−0.00213138 + 0.999998i \(0.500678\pi\)
\(264\) 0 0
\(265\) −4.56155 −0.280214
\(266\) 0 0
\(267\) 22.2462 1.36145
\(268\) 0 0
\(269\) 23.2462 1.41735 0.708673 0.705537i \(-0.249294\pi\)
0.708673 + 0.705537i \(0.249294\pi\)
\(270\) 0 0
\(271\) −21.9309 −1.33221 −0.666103 0.745860i \(-0.732039\pi\)
−0.666103 + 0.745860i \(0.732039\pi\)
\(272\) 0 0
\(273\) 14.2462 0.862220
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −6.68466 −0.401642 −0.200821 0.979628i \(-0.564361\pi\)
−0.200821 + 0.979628i \(0.564361\pi\)
\(278\) 0 0
\(279\) 4.06913 0.243612
\(280\) 0 0
\(281\) −6.87689 −0.410241 −0.205121 0.978737i \(-0.565759\pi\)
−0.205121 + 0.978737i \(0.565759\pi\)
\(282\) 0 0
\(283\) −14.8078 −0.880230 −0.440115 0.897941i \(-0.645062\pi\)
−0.440115 + 0.897941i \(0.645062\pi\)
\(284\) 0 0
\(285\) −9.36932 −0.554990
\(286\) 0 0
\(287\) −10.5616 −0.623429
\(288\) 0 0
\(289\) −10.4384 −0.614026
\(290\) 0 0
\(291\) −20.4924 −1.20129
\(292\) 0 0
\(293\) −16.8078 −0.981920 −0.490960 0.871182i \(-0.663354\pi\)
−0.490960 + 0.871182i \(0.663354\pi\)
\(294\) 0 0
\(295\) 3.68466 0.214529
\(296\) 0 0
\(297\) 11.1231 0.645428
\(298\) 0 0
\(299\) 3.56155 0.205970
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −27.6155 −1.58647
\(304\) 0 0
\(305\) −7.12311 −0.407868
\(306\) 0 0
\(307\) 21.1231 1.20556 0.602780 0.797908i \(-0.294060\pi\)
0.602780 + 0.797908i \(0.294060\pi\)
\(308\) 0 0
\(309\) 24.9848 1.42134
\(310\) 0 0
\(311\) −8.68466 −0.492462 −0.246231 0.969211i \(-0.579192\pi\)
−0.246231 + 0.969211i \(0.579192\pi\)
\(312\) 0 0
\(313\) −0.807764 −0.0456575 −0.0228288 0.999739i \(-0.507267\pi\)
−0.0228288 + 0.999739i \(0.507267\pi\)
\(314\) 0 0
\(315\) 1.43845 0.0810473
\(316\) 0 0
\(317\) 15.1231 0.849398 0.424699 0.905335i \(-0.360380\pi\)
0.424699 + 0.905335i \(0.360380\pi\)
\(318\) 0 0
\(319\) −12.2462 −0.685656
\(320\) 0 0
\(321\) −5.36932 −0.299686
\(322\) 0 0
\(323\) −15.3693 −0.855172
\(324\) 0 0
\(325\) −3.56155 −0.197559
\(326\) 0 0
\(327\) 24.0000 1.32720
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.6155 −1.57285 −0.786426 0.617685i \(-0.788071\pi\)
−0.786426 + 0.617685i \(0.788071\pi\)
\(332\) 0 0
\(333\) 2.56155 0.140372
\(334\) 0 0
\(335\) 8.56155 0.467768
\(336\) 0 0
\(337\) 13.6155 0.741685 0.370843 0.928696i \(-0.379069\pi\)
0.370843 + 0.928696i \(0.379069\pi\)
\(338\) 0 0
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 14.4924 0.784809
\(342\) 0 0
\(343\) 19.0540 1.02882
\(344\) 0 0
\(345\) −1.56155 −0.0840712
\(346\) 0 0
\(347\) −16.4924 −0.885360 −0.442680 0.896680i \(-0.645972\pi\)
−0.442680 + 0.896680i \(0.645972\pi\)
\(348\) 0 0
\(349\) −26.3693 −1.41152 −0.705759 0.708452i \(-0.749394\pi\)
−0.705759 + 0.708452i \(0.749394\pi\)
\(350\) 0 0
\(351\) 19.8078 1.05726
\(352\) 0 0
\(353\) −2.05398 −0.109322 −0.0546610 0.998505i \(-0.517408\pi\)
−0.0546610 + 0.998505i \(0.517408\pi\)
\(354\) 0 0
\(355\) −10.1231 −0.537279
\(356\) 0 0
\(357\) −10.2462 −0.542287
\(358\) 0 0
\(359\) −15.6155 −0.824156 −0.412078 0.911149i \(-0.635197\pi\)
−0.412078 + 0.911149i \(0.635197\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −10.9309 −0.573722
\(364\) 0 0
\(365\) −4.43845 −0.232319
\(366\) 0 0
\(367\) 22.5616 1.17770 0.588852 0.808241i \(-0.299580\pi\)
0.588852 + 0.808241i \(0.299580\pi\)
\(368\) 0 0
\(369\) −2.31534 −0.120532
\(370\) 0 0
\(371\) 11.6847 0.606637
\(372\) 0 0
\(373\) −20.2462 −1.04831 −0.524155 0.851623i \(-0.675619\pi\)
−0.524155 + 0.851623i \(0.675619\pi\)
\(374\) 0 0
\(375\) 1.56155 0.0806382
\(376\) 0 0
\(377\) −21.8078 −1.12316
\(378\) 0 0
\(379\) 36.4924 1.87449 0.937245 0.348672i \(-0.113367\pi\)
0.937245 + 0.348672i \(0.113367\pi\)
\(380\) 0 0
\(381\) −12.1922 −0.624627
\(382\) 0 0
\(383\) −3.19224 −0.163116 −0.0815578 0.996669i \(-0.525990\pi\)
−0.0815578 + 0.996669i \(0.525990\pi\)
\(384\) 0 0
\(385\) 5.12311 0.261098
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.8769 −0.551480 −0.275740 0.961232i \(-0.588923\pi\)
−0.275740 + 0.961232i \(0.588923\pi\)
\(390\) 0 0
\(391\) −2.56155 −0.129543
\(392\) 0 0
\(393\) −0.300187 −0.0151424
\(394\) 0 0
\(395\) −4.87689 −0.245383
\(396\) 0 0
\(397\) 34.5464 1.73383 0.866917 0.498453i \(-0.166098\pi\)
0.866917 + 0.498453i \(0.166098\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 25.8078 1.28558
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) 9.12311 0.452216
\(408\) 0 0
\(409\) −35.9848 −1.77934 −0.889668 0.456608i \(-0.849064\pi\)
−0.889668 + 0.456608i \(0.849064\pi\)
\(410\) 0 0
\(411\) 10.7386 0.529698
\(412\) 0 0
\(413\) −9.43845 −0.464436
\(414\) 0 0
\(415\) 13.9309 0.683839
\(416\) 0 0
\(417\) 17.5616 0.859993
\(418\) 0 0
\(419\) 18.8769 0.922197 0.461098 0.887349i \(-0.347456\pi\)
0.461098 + 0.887349i \(0.347456\pi\)
\(420\) 0 0
\(421\) 25.1231 1.22443 0.612213 0.790693i \(-0.290280\pi\)
0.612213 + 0.790693i \(0.290280\pi\)
\(422\) 0 0
\(423\) 2.63068 0.127908
\(424\) 0 0
\(425\) 2.56155 0.124254
\(426\) 0 0
\(427\) 18.2462 0.882996
\(428\) 0 0
\(429\) 11.1231 0.537029
\(430\) 0 0
\(431\) 10.2462 0.493543 0.246771 0.969074i \(-0.420630\pi\)
0.246771 + 0.969074i \(0.420630\pi\)
\(432\) 0 0
\(433\) 39.9309 1.91896 0.959478 0.281785i \(-0.0909265\pi\)
0.959478 + 0.281785i \(0.0909265\pi\)
\(434\) 0 0
\(435\) 9.56155 0.458441
\(436\) 0 0
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −19.8078 −0.945373 −0.472686 0.881231i \(-0.656716\pi\)
−0.472686 + 0.881231i \(0.656716\pi\)
\(440\) 0 0
\(441\) 0.246211 0.0117243
\(442\) 0 0
\(443\) 11.8078 0.561004 0.280502 0.959853i \(-0.409499\pi\)
0.280502 + 0.959853i \(0.409499\pi\)
\(444\) 0 0
\(445\) 14.2462 0.675335
\(446\) 0 0
\(447\) −14.6307 −0.692008
\(448\) 0 0
\(449\) −21.6847 −1.02336 −0.511681 0.859175i \(-0.670977\pi\)
−0.511681 + 0.859175i \(0.670977\pi\)
\(450\) 0 0
\(451\) −8.24621 −0.388299
\(452\) 0 0
\(453\) 21.9460 1.03111
\(454\) 0 0
\(455\) 9.12311 0.427698
\(456\) 0 0
\(457\) 9.43845 0.441512 0.220756 0.975329i \(-0.429148\pi\)
0.220756 + 0.975329i \(0.429148\pi\)
\(458\) 0 0
\(459\) −14.2462 −0.664956
\(460\) 0 0
\(461\) −40.9309 −1.90634 −0.953170 0.302434i \(-0.902201\pi\)
−0.953170 + 0.302434i \(0.902201\pi\)
\(462\) 0 0
\(463\) 31.3693 1.45786 0.728928 0.684590i \(-0.240019\pi\)
0.728928 + 0.684590i \(0.240019\pi\)
\(464\) 0 0
\(465\) −11.3153 −0.524736
\(466\) 0 0
\(467\) 22.3153 1.03263 0.516315 0.856398i \(-0.327303\pi\)
0.516315 + 0.856398i \(0.327303\pi\)
\(468\) 0 0
\(469\) −21.9309 −1.01267
\(470\) 0 0
\(471\) −6.73863 −0.310500
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 2.56155 0.117285
\(478\) 0 0
\(479\) 43.2311 1.97528 0.987639 0.156748i \(-0.0501009\pi\)
0.987639 + 0.156748i \(0.0501009\pi\)
\(480\) 0 0
\(481\) 16.2462 0.740763
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −13.1231 −0.595890
\(486\) 0 0
\(487\) −7.80776 −0.353804 −0.176902 0.984229i \(-0.556607\pi\)
−0.176902 + 0.984229i \(0.556607\pi\)
\(488\) 0 0
\(489\) −26.4384 −1.19559
\(490\) 0 0
\(491\) −11.4924 −0.518646 −0.259323 0.965791i \(-0.583499\pi\)
−0.259323 + 0.965791i \(0.583499\pi\)
\(492\) 0 0
\(493\) 15.6847 0.706401
\(494\) 0 0
\(495\) 1.12311 0.0504798
\(496\) 0 0
\(497\) 25.9309 1.16316
\(498\) 0 0
\(499\) −22.6155 −1.01241 −0.506205 0.862413i \(-0.668952\pi\)
−0.506205 + 0.862413i \(0.668952\pi\)
\(500\) 0 0
\(501\) 12.4924 0.558120
\(502\) 0 0
\(503\) 3.93087 0.175269 0.0876344 0.996153i \(-0.472069\pi\)
0.0876344 + 0.996153i \(0.472069\pi\)
\(504\) 0 0
\(505\) −17.6847 −0.786957
\(506\) 0 0
\(507\) −0.492423 −0.0218693
\(508\) 0 0
\(509\) −27.1771 −1.20460 −0.602301 0.798269i \(-0.705749\pi\)
−0.602301 + 0.798269i \(0.705749\pi\)
\(510\) 0 0
\(511\) 11.3693 0.502949
\(512\) 0 0
\(513\) 33.3693 1.47329
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 9.36932 0.412062
\(518\) 0 0
\(519\) 8.38447 0.368037
\(520\) 0 0
\(521\) −13.6155 −0.596507 −0.298254 0.954487i \(-0.596404\pi\)
−0.298254 + 0.954487i \(0.596404\pi\)
\(522\) 0 0
\(523\) 14.7386 0.644475 0.322238 0.946659i \(-0.395565\pi\)
0.322238 + 0.946659i \(0.395565\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −18.5616 −0.808554
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.06913 −0.0897926
\(532\) 0 0
\(533\) −14.6847 −0.636063
\(534\) 0 0
\(535\) −3.43845 −0.148657
\(536\) 0 0
\(537\) 10.8229 0.467043
\(538\) 0 0
\(539\) 0.876894 0.0377705
\(540\) 0 0
\(541\) −2.68466 −0.115422 −0.0577112 0.998333i \(-0.518380\pi\)
−0.0577112 + 0.998333i \(0.518380\pi\)
\(542\) 0 0
\(543\) −8.38447 −0.359812
\(544\) 0 0
\(545\) 15.3693 0.658349
\(546\) 0 0
\(547\) 21.1771 0.905467 0.452733 0.891646i \(-0.350449\pi\)
0.452733 + 0.891646i \(0.350449\pi\)
\(548\) 0 0
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −36.7386 −1.56512
\(552\) 0 0
\(553\) 12.4924 0.531232
\(554\) 0 0
\(555\) −7.12311 −0.302359
\(556\) 0 0
\(557\) −15.6847 −0.664580 −0.332290 0.943177i \(-0.607821\pi\)
−0.332290 + 0.943177i \(0.607821\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) −32.6695 −1.37686 −0.688428 0.725305i \(-0.741699\pi\)
−0.688428 + 0.725305i \(0.741699\pi\)
\(564\) 0 0
\(565\) 12.8078 0.538827
\(566\) 0 0
\(567\) 17.9309 0.753026
\(568\) 0 0
\(569\) 16.7386 0.701720 0.350860 0.936428i \(-0.385889\pi\)
0.350860 + 0.936428i \(0.385889\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) −27.1231 −1.13308
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −29.5616 −1.23066 −0.615332 0.788268i \(-0.710978\pi\)
−0.615332 + 0.788268i \(0.710978\pi\)
\(578\) 0 0
\(579\) 3.80776 0.158245
\(580\) 0 0
\(581\) −35.6847 −1.48045
\(582\) 0 0
\(583\) 9.12311 0.377840
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) −7.06913 −0.291774 −0.145887 0.989301i \(-0.546604\pi\)
−0.145887 + 0.989301i \(0.546604\pi\)
\(588\) 0 0
\(589\) 43.4773 1.79145
\(590\) 0 0
\(591\) −29.1771 −1.20018
\(592\) 0 0
\(593\) 15.6155 0.641253 0.320626 0.947206i \(-0.396107\pi\)
0.320626 + 0.947206i \(0.396107\pi\)
\(594\) 0 0
\(595\) −6.56155 −0.268997
\(596\) 0 0
\(597\) 15.6155 0.639101
\(598\) 0 0
\(599\) 32.9848 1.34772 0.673862 0.738857i \(-0.264634\pi\)
0.673862 + 0.738857i \(0.264634\pi\)
\(600\) 0 0
\(601\) 36.3693 1.48354 0.741768 0.670657i \(-0.233988\pi\)
0.741768 + 0.670657i \(0.233988\pi\)
\(602\) 0 0
\(603\) −4.80776 −0.195787
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −8.49242 −0.344697 −0.172348 0.985036i \(-0.555135\pi\)
−0.172348 + 0.985036i \(0.555135\pi\)
\(608\) 0 0
\(609\) −24.4924 −0.992483
\(610\) 0 0
\(611\) 16.6847 0.674989
\(612\) 0 0
\(613\) −5.61553 −0.226809 −0.113405 0.993549i \(-0.536176\pi\)
−0.113405 + 0.993549i \(0.536176\pi\)
\(614\) 0 0
\(615\) 6.43845 0.259623
\(616\) 0 0
\(617\) 0.561553 0.0226073 0.0113036 0.999936i \(-0.496402\pi\)
0.0113036 + 0.999936i \(0.496402\pi\)
\(618\) 0 0
\(619\) −12.4924 −0.502113 −0.251056 0.967972i \(-0.580778\pi\)
−0.251056 + 0.967972i \(0.580778\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) −36.4924 −1.46204
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 18.7386 0.748349
\(628\) 0 0
\(629\) −11.6847 −0.465898
\(630\) 0 0
\(631\) −30.2462 −1.20408 −0.602041 0.798465i \(-0.705646\pi\)
−0.602041 + 0.798465i \(0.705646\pi\)
\(632\) 0 0
\(633\) −0.492423 −0.0195720
\(634\) 0 0
\(635\) −7.80776 −0.309842
\(636\) 0 0
\(637\) 1.56155 0.0618710
\(638\) 0 0
\(639\) 5.68466 0.224882
\(640\) 0 0
\(641\) 6.87689 0.271621 0.135810 0.990735i \(-0.456636\pi\)
0.135810 + 0.990735i \(0.456636\pi\)
\(642\) 0 0
\(643\) −20.4233 −0.805416 −0.402708 0.915328i \(-0.631931\pi\)
−0.402708 + 0.915328i \(0.631931\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.31534 −0.208968 −0.104484 0.994527i \(-0.533319\pi\)
−0.104484 + 0.994527i \(0.533319\pi\)
\(648\) 0 0
\(649\) −7.36932 −0.289271
\(650\) 0 0
\(651\) 28.9848 1.13601
\(652\) 0 0
\(653\) 39.6695 1.55239 0.776194 0.630494i \(-0.217148\pi\)
0.776194 + 0.630494i \(0.217148\pi\)
\(654\) 0 0
\(655\) −0.192236 −0.00751128
\(656\) 0 0
\(657\) 2.49242 0.0972387
\(658\) 0 0
\(659\) −15.3693 −0.598704 −0.299352 0.954143i \(-0.596770\pi\)
−0.299352 + 0.954143i \(0.596770\pi\)
\(660\) 0 0
\(661\) −6.49242 −0.252526 −0.126263 0.991997i \(-0.540298\pi\)
−0.126263 + 0.991997i \(0.540298\pi\)
\(662\) 0 0
\(663\) −14.2462 −0.553277
\(664\) 0 0
\(665\) 15.3693 0.595997
\(666\) 0 0
\(667\) −6.12311 −0.237088
\(668\) 0 0
\(669\) 27.5076 1.06350
\(670\) 0 0
\(671\) 14.2462 0.549969
\(672\) 0 0
\(673\) 34.5464 1.33167 0.665833 0.746101i \(-0.268076\pi\)
0.665833 + 0.746101i \(0.268076\pi\)
\(674\) 0 0
\(675\) −5.56155 −0.214064
\(676\) 0 0
\(677\) 26.8078 1.03031 0.515153 0.857098i \(-0.327735\pi\)
0.515153 + 0.857098i \(0.327735\pi\)
\(678\) 0 0
\(679\) 33.6155 1.29005
\(680\) 0 0
\(681\) −40.9848 −1.57054
\(682\) 0 0
\(683\) −42.0540 −1.60915 −0.804575 0.593851i \(-0.797607\pi\)
−0.804575 + 0.593851i \(0.797607\pi\)
\(684\) 0 0
\(685\) 6.87689 0.262753
\(686\) 0 0
\(687\) −41.7538 −1.59301
\(688\) 0 0
\(689\) 16.2462 0.618931
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) 0 0
\(693\) −2.87689 −0.109284
\(694\) 0 0
\(695\) 11.2462 0.426593
\(696\) 0 0
\(697\) 10.5616 0.400047
\(698\) 0 0
\(699\) −1.06913 −0.0404382
\(700\) 0 0
\(701\) −1.75379 −0.0662397 −0.0331198 0.999451i \(-0.510544\pi\)
−0.0331198 + 0.999451i \(0.510544\pi\)
\(702\) 0 0
\(703\) 27.3693 1.03225
\(704\) 0 0
\(705\) −7.31534 −0.275512
\(706\) 0 0
\(707\) 45.3002 1.70369
\(708\) 0 0
\(709\) 29.7538 1.11743 0.558713 0.829361i \(-0.311295\pi\)
0.558713 + 0.829361i \(0.311295\pi\)
\(710\) 0 0
\(711\) 2.73863 0.102707
\(712\) 0 0
\(713\) 7.24621 0.271373
\(714\) 0 0
\(715\) 7.12311 0.266389
\(716\) 0 0
\(717\) −4.30019 −0.160593
\(718\) 0 0
\(719\) −19.0540 −0.710593 −0.355297 0.934754i \(-0.615620\pi\)
−0.355297 + 0.934754i \(0.615620\pi\)
\(720\) 0 0
\(721\) −40.9848 −1.52636
\(722\) 0 0
\(723\) 9.36932 0.348449
\(724\) 0 0
\(725\) 6.12311 0.227406
\(726\) 0 0
\(727\) 21.1922 0.785977 0.392988 0.919543i \(-0.371441\pi\)
0.392988 + 0.919543i \(0.371441\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.315342 0.0116474 0.00582370 0.999983i \(-0.498146\pi\)
0.00582370 + 0.999983i \(0.498146\pi\)
\(734\) 0 0
\(735\) −0.684658 −0.0252540
\(736\) 0 0
\(737\) −17.1231 −0.630738
\(738\) 0 0
\(739\) −0.615528 −0.0226426 −0.0113213 0.999936i \(-0.503604\pi\)
−0.0113213 + 0.999936i \(0.503604\pi\)
\(740\) 0 0
\(741\) 33.3693 1.22585
\(742\) 0 0
\(743\) −3.50758 −0.128681 −0.0643403 0.997928i \(-0.520494\pi\)
−0.0643403 + 0.997928i \(0.520494\pi\)
\(744\) 0 0
\(745\) −9.36932 −0.343265
\(746\) 0 0
\(747\) −7.82292 −0.286226
\(748\) 0 0
\(749\) 8.80776 0.321829
\(750\) 0 0
\(751\) 13.1231 0.478869 0.239434 0.970913i \(-0.423038\pi\)
0.239434 + 0.970913i \(0.423038\pi\)
\(752\) 0 0
\(753\) 11.5076 0.419359
\(754\) 0 0
\(755\) 14.0540 0.511477
\(756\) 0 0
\(757\) 12.5616 0.456557 0.228279 0.973596i \(-0.426690\pi\)
0.228279 + 0.973596i \(0.426690\pi\)
\(758\) 0 0
\(759\) 3.12311 0.113362
\(760\) 0 0
\(761\) −43.9848 −1.59445 −0.797225 0.603683i \(-0.793699\pi\)
−0.797225 + 0.603683i \(0.793699\pi\)
\(762\) 0 0
\(763\) −39.3693 −1.42526
\(764\) 0 0
\(765\) −1.43845 −0.0520072
\(766\) 0 0
\(767\) −13.1231 −0.473848
\(768\) 0 0
\(769\) −10.6307 −0.383352 −0.191676 0.981458i \(-0.561392\pi\)
−0.191676 + 0.981458i \(0.561392\pi\)
\(770\) 0 0
\(771\) 49.6695 1.78880
\(772\) 0 0
\(773\) −41.1231 −1.47910 −0.739548 0.673104i \(-0.764961\pi\)
−0.739548 + 0.673104i \(0.764961\pi\)
\(774\) 0 0
\(775\) −7.24621 −0.260292
\(776\) 0 0
\(777\) 18.2462 0.654579
\(778\) 0 0
\(779\) −24.7386 −0.886354
\(780\) 0 0
\(781\) 20.2462 0.724466
\(782\) 0 0
\(783\) −34.0540 −1.21699
\(784\) 0 0
\(785\) −4.31534 −0.154021
\(786\) 0 0
\(787\) 39.6847 1.41461 0.707303 0.706911i \(-0.249912\pi\)
0.707303 + 0.706911i \(0.249912\pi\)
\(788\) 0 0
\(789\) −0.107951 −0.00384314
\(790\) 0 0
\(791\) −32.8078 −1.16651
\(792\) 0 0
\(793\) 25.3693 0.900891
\(794\) 0 0
\(795\) −7.12311 −0.252631
\(796\) 0 0
\(797\) −52.4233 −1.85693 −0.928464 0.371422i \(-0.878870\pi\)
−0.928464 + 0.371422i \(0.878870\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) 8.87689 0.313259
\(804\) 0 0
\(805\) 2.56155 0.0902829
\(806\) 0 0
\(807\) 36.3002 1.27783
\(808\) 0 0
\(809\) −54.1771 −1.90476 −0.952382 0.304906i \(-0.901375\pi\)
−0.952382 + 0.304906i \(0.901375\pi\)
\(810\) 0 0
\(811\) −47.2462 −1.65904 −0.829519 0.558478i \(-0.811386\pi\)
−0.829519 + 0.558478i \(0.811386\pi\)
\(812\) 0 0
\(813\) −34.2462 −1.20107
\(814\) 0 0
\(815\) −16.9309 −0.593062
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −5.12311 −0.179016
\(820\) 0 0
\(821\) −34.4924 −1.20379 −0.601897 0.798574i \(-0.705588\pi\)
−0.601897 + 0.798574i \(0.705588\pi\)
\(822\) 0 0
\(823\) 38.0540 1.32648 0.663239 0.748408i \(-0.269181\pi\)
0.663239 + 0.748408i \(0.269181\pi\)
\(824\) 0 0
\(825\) −3.12311 −0.108733
\(826\) 0 0
\(827\) −19.6847 −0.684503 −0.342251 0.939608i \(-0.611189\pi\)
−0.342251 + 0.939608i \(0.611189\pi\)
\(828\) 0 0
\(829\) −29.5464 −1.02619 −0.513094 0.858332i \(-0.671501\pi\)
−0.513094 + 0.858332i \(0.671501\pi\)
\(830\) 0 0
\(831\) −10.4384 −0.362106
\(832\) 0 0
\(833\) −1.12311 −0.0389133
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 40.3002 1.39298
\(838\) 0 0
\(839\) 42.8769 1.48027 0.740137 0.672456i \(-0.234760\pi\)
0.740137 + 0.672456i \(0.234760\pi\)
\(840\) 0 0
\(841\) 8.49242 0.292842
\(842\) 0 0
\(843\) −10.7386 −0.369858
\(844\) 0 0
\(845\) −0.315342 −0.0108481
\(846\) 0 0
\(847\) 17.9309 0.616112
\(848\) 0 0
\(849\) −23.1231 −0.793583
\(850\) 0 0
\(851\) 4.56155 0.156368
\(852\) 0 0
\(853\) −20.2462 −0.693217 −0.346609 0.938010i \(-0.612667\pi\)
−0.346609 + 0.938010i \(0.612667\pi\)
\(854\) 0 0
\(855\) 3.36932 0.115228
\(856\) 0 0
\(857\) −33.3153 −1.13803 −0.569015 0.822327i \(-0.692675\pi\)
−0.569015 + 0.822327i \(0.692675\pi\)
\(858\) 0 0
\(859\) 14.5076 0.494992 0.247496 0.968889i \(-0.420392\pi\)
0.247496 + 0.968889i \(0.420392\pi\)
\(860\) 0 0
\(861\) −16.4924 −0.562060
\(862\) 0 0
\(863\) 7.56155 0.257398 0.128699 0.991684i \(-0.458920\pi\)
0.128699 + 0.991684i \(0.458920\pi\)
\(864\) 0 0
\(865\) 5.36932 0.182562
\(866\) 0 0
\(867\) −16.3002 −0.553583
\(868\) 0 0
\(869\) 9.75379 0.330875
\(870\) 0 0
\(871\) −30.4924 −1.03320
\(872\) 0 0
\(873\) 7.36932 0.249414
\(874\) 0 0
\(875\) −2.56155 −0.0865963
\(876\) 0 0
\(877\) −18.9848 −0.641073 −0.320536 0.947236i \(-0.603863\pi\)
−0.320536 + 0.947236i \(0.603863\pi\)
\(878\) 0 0
\(879\) −26.2462 −0.885263
\(880\) 0 0
\(881\) 41.4773 1.39740 0.698702 0.715413i \(-0.253761\pi\)
0.698702 + 0.715413i \(0.253761\pi\)
\(882\) 0 0
\(883\) −10.7386 −0.361384 −0.180692 0.983540i \(-0.557834\pi\)
−0.180692 + 0.983540i \(0.557834\pi\)
\(884\) 0 0
\(885\) 5.75379 0.193411
\(886\) 0 0
\(887\) −42.7926 −1.43684 −0.718418 0.695612i \(-0.755133\pi\)
−0.718418 + 0.695612i \(0.755133\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 14.0000 0.469018
\(892\) 0 0
\(893\) 28.1080 0.940597
\(894\) 0 0
\(895\) 6.93087 0.231673
\(896\) 0 0
\(897\) 5.56155 0.185695
\(898\) 0 0
\(899\) −44.3693 −1.47980
\(900\) 0 0
\(901\) −11.6847 −0.389272
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.36932 −0.178482
\(906\) 0 0
\(907\) −21.6847 −0.720027 −0.360014 0.932947i \(-0.617228\pi\)
−0.360014 + 0.932947i \(0.617228\pi\)
\(908\) 0 0
\(909\) 9.93087 0.329386
\(910\) 0 0
\(911\) 3.12311 0.103473 0.0517366 0.998661i \(-0.483524\pi\)
0.0517366 + 0.998661i \(0.483524\pi\)
\(912\) 0 0
\(913\) −27.8617 −0.922089
\(914\) 0 0
\(915\) −11.1231 −0.367719
\(916\) 0 0
\(917\) 0.492423 0.0162612
\(918\) 0 0
\(919\) −46.7386 −1.54177 −0.770883 0.636977i \(-0.780185\pi\)
−0.770883 + 0.636977i \(0.780185\pi\)
\(920\) 0 0
\(921\) 32.9848 1.08689
\(922\) 0 0
\(923\) 36.0540 1.18673
\(924\) 0 0
\(925\) −4.56155 −0.149983
\(926\) 0 0
\(927\) −8.98485 −0.295101
\(928\) 0 0
\(929\) −56.7235 −1.86104 −0.930518 0.366245i \(-0.880643\pi\)
−0.930518 + 0.366245i \(0.880643\pi\)
\(930\) 0 0
\(931\) 2.63068 0.0862172
\(932\) 0 0
\(933\) −13.5616 −0.443985
\(934\) 0 0
\(935\) −5.12311 −0.167543
\(936\) 0 0
\(937\) −16.2462 −0.530741 −0.265370 0.964147i \(-0.585494\pi\)
−0.265370 + 0.964147i \(0.585494\pi\)
\(938\) 0 0
\(939\) −1.26137 −0.0411631
\(940\) 0 0
\(941\) −37.7538 −1.23074 −0.615369 0.788239i \(-0.710993\pi\)
−0.615369 + 0.788239i \(0.710993\pi\)
\(942\) 0 0
\(943\) −4.12311 −0.134267
\(944\) 0 0
\(945\) 14.2462 0.463429
\(946\) 0 0
\(947\) −56.6847 −1.84200 −0.921002 0.389558i \(-0.872628\pi\)
−0.921002 + 0.389558i \(0.872628\pi\)
\(948\) 0 0
\(949\) 15.8078 0.513142
\(950\) 0 0
\(951\) 23.6155 0.765786
\(952\) 0 0
\(953\) −38.4924 −1.24689 −0.623446 0.781866i \(-0.714268\pi\)
−0.623446 + 0.781866i \(0.714268\pi\)
\(954\) 0 0
\(955\) −17.3693 −0.562058
\(956\) 0 0
\(957\) −19.1231 −0.618162
\(958\) 0 0
\(959\) −17.6155 −0.568835
\(960\) 0 0
\(961\) 21.5076 0.693793
\(962\) 0 0
\(963\) 1.93087 0.0622214
\(964\) 0 0
\(965\) 2.43845 0.0784964
\(966\) 0 0
\(967\) 56.6847 1.82286 0.911428 0.411460i \(-0.134981\pi\)
0.911428 + 0.411460i \(0.134981\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 22.6307 0.726253 0.363127 0.931740i \(-0.381709\pi\)
0.363127 + 0.931740i \(0.381709\pi\)
\(972\) 0 0
\(973\) −28.8078 −0.923535
\(974\) 0 0
\(975\) −5.56155 −0.178112
\(976\) 0 0
\(977\) 59.7926 1.91294 0.956468 0.291839i \(-0.0942671\pi\)
0.956468 + 0.291839i \(0.0942671\pi\)
\(978\) 0 0
\(979\) −28.4924 −0.910622
\(980\) 0 0
\(981\) −8.63068 −0.275557
\(982\) 0 0
\(983\) 37.0540 1.18184 0.590919 0.806731i \(-0.298765\pi\)
0.590919 + 0.806731i \(0.298765\pi\)
\(984\) 0 0
\(985\) −18.6847 −0.595343
\(986\) 0 0
\(987\) 18.7386 0.596457
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −53.3002 −1.69314 −0.846568 0.532280i \(-0.821335\pi\)
−0.846568 + 0.532280i \(0.821335\pi\)
\(992\) 0 0
\(993\) −44.6847 −1.41802
\(994\) 0 0
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 55.6155 1.76136 0.880681 0.473710i \(-0.157086\pi\)
0.880681 + 0.473710i \(0.157086\pi\)
\(998\) 0 0
\(999\) 25.3693 0.802650
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 1840.2.a.m.1.2 2
4.3 odd 2 460.2.a.e.1.1 2
5.4 even 2 9200.2.a.bv.1.1 2
8.3 odd 2 7360.2.a.bi.1.2 2
8.5 even 2 7360.2.a.bo.1.1 2
12.11 even 2 4140.2.a.m.1.2 2
20.3 even 4 2300.2.c.h.1749.2 4
20.7 even 4 2300.2.c.h.1749.3 4
20.19 odd 2 2300.2.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.1 2 4.3 odd 2
1840.2.a.m.1.2 2 1.1 even 1 trivial
2300.2.a.i.1.2 2 20.19 odd 2
2300.2.c.h.1749.2 4 20.3 even 4
2300.2.c.h.1749.3 4 20.7 even 4
4140.2.a.m.1.2 2 12.11 even 2
7360.2.a.bi.1.2 2 8.3 odd 2
7360.2.a.bo.1.1 2 8.5 even 2
9200.2.a.bv.1.1 2 5.4 even 2