Properties

Label 165.2.a.a.1.1
Level $165$
Weight $2$
Character 165.1
Self dual yes
Analytic conductor $1.318$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.31753163335\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -1.00000 q^{11} -3.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -6.82843 q^{19} -3.82843 q^{20} -0.828427 q^{21} +2.41421 q^{22} -4.00000 q^{23} +4.41421 q^{24} +1.00000 q^{25} +13.6569 q^{26} -1.00000 q^{27} +3.17157 q^{28} -4.82843 q^{29} -2.41421 q^{30} +1.58579 q^{32} +1.00000 q^{33} +2.82843 q^{34} -0.828427 q^{35} +3.82843 q^{36} +11.6569 q^{37} +16.4853 q^{38} +5.65685 q^{39} +4.41421 q^{40} +4.82843 q^{41} +2.00000 q^{42} -8.82843 q^{43} -3.82843 q^{44} -1.00000 q^{45} +9.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +1.17157 q^{51} -21.6569 q^{52} +9.31371 q^{53} +2.41421 q^{54} +1.00000 q^{55} -3.65685 q^{56} +6.82843 q^{57} +11.6569 q^{58} -4.00000 q^{59} +3.82843 q^{60} -11.6569 q^{61} +0.828427 q^{63} -9.82843 q^{64} +5.65685 q^{65} -2.41421 q^{66} -5.65685 q^{67} -4.48528 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.34315 q^{71} -4.41421 q^{72} +11.3137 q^{73} -28.1421 q^{74} -1.00000 q^{75} -26.1421 q^{76} -0.828427 q^{77} -13.6569 q^{78} +8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -10.0000 q^{83} -3.17157 q^{84} +1.17157 q^{85} +21.3137 q^{86} +4.82843 q^{87} +4.41421 q^{88} +3.65685 q^{89} +2.41421 q^{90} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} +6.82843 q^{95} -1.58579 q^{96} +11.6569 q^{97} +15.2426 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{10} - 2q^{11} - 2q^{12} - 4q^{14} + 2q^{15} + 6q^{16} - 8q^{17} - 2q^{18} - 8q^{19} - 2q^{20} + 4q^{21} + 2q^{22} - 8q^{23} + 6q^{24} + 2q^{25} + 16q^{26} - 2q^{27} + 12q^{28} - 4q^{29} - 2q^{30} + 6q^{32} + 2q^{33} + 4q^{35} + 2q^{36} + 12q^{37} + 16q^{38} + 6q^{40} + 4q^{41} + 4q^{42} - 12q^{43} - 2q^{44} - 2q^{45} + 8q^{46} - 8q^{47} - 6q^{48} + 10q^{49} - 2q^{50} + 8q^{51} - 32q^{52} - 4q^{53} + 2q^{54} + 2q^{55} + 4q^{56} + 8q^{57} + 12q^{58} - 8q^{59} + 2q^{60} - 12q^{61} - 4q^{63} - 14q^{64} - 2q^{66} + 8q^{68} + 8q^{69} + 4q^{70} + 16q^{71} - 6q^{72} - 28q^{74} - 2q^{75} - 24q^{76} + 4q^{77} - 16q^{78} - 6q^{80} + 2q^{81} - 12q^{82} - 20q^{83} - 12q^{84} + 8q^{85} + 20q^{86} + 4q^{87} + 6q^{88} - 4q^{89} + 2q^{90} - 32q^{91} - 8q^{92} + 8q^{94} + 8q^{95} - 6q^{96} + 12q^{97} + 22q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) 2.41421 0.985599
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 2.41421 0.763441
\(11\) −1.00000 −0.301511
\(12\) −3.82843 −1.10517
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) −1.17157 −0.284148 −0.142074 0.989856i \(-0.545377\pi\)
−0.142074 + 0.989856i \(0.545377\pi\)
\(18\) −2.41421 −0.569036
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) −3.82843 −0.856062
\(21\) −0.828427 −0.180778
\(22\) 2.41421 0.514712
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 4.41421 0.901048
\(25\) 1.00000 0.200000
\(26\) 13.6569 2.67833
\(27\) −1.00000 −0.192450
\(28\) 3.17157 0.599371
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) −2.41421 −0.440773
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.58579 0.280330
\(33\) 1.00000 0.174078
\(34\) 2.82843 0.485071
\(35\) −0.828427 −0.140030
\(36\) 3.82843 0.638071
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) 16.4853 2.67427
\(39\) 5.65685 0.905822
\(40\) 4.41421 0.697948
\(41\) 4.82843 0.754074 0.377037 0.926198i \(-0.376943\pi\)
0.377037 + 0.926198i \(0.376943\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.82843 −1.34632 −0.673161 0.739496i \(-0.735064\pi\)
−0.673161 + 0.739496i \(0.735064\pi\)
\(44\) −3.82843 −0.577157
\(45\) −1.00000 −0.149071
\(46\) 9.65685 1.42383
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −3.00000 −0.433013
\(49\) −6.31371 −0.901958
\(50\) −2.41421 −0.341421
\(51\) 1.17157 0.164053
\(52\) −21.6569 −3.00327
\(53\) 9.31371 1.27934 0.639668 0.768651i \(-0.279072\pi\)
0.639668 + 0.768651i \(0.279072\pi\)
\(54\) 2.41421 0.328533
\(55\) 1.00000 0.134840
\(56\) −3.65685 −0.488668
\(57\) 6.82843 0.904447
\(58\) 11.6569 1.53062
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 3.82843 0.494248
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) 0.828427 0.104372
\(64\) −9.82843 −1.22855
\(65\) 5.65685 0.701646
\(66\) −2.41421 −0.297169
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) −4.48528 −0.543920
\(69\) 4.00000 0.481543
\(70\) 2.00000 0.239046
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) −4.41421 −0.520220
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) −28.1421 −3.27146
\(75\) −1.00000 −0.115470
\(76\) −26.1421 −2.99871
\(77\) −0.828427 −0.0944080
\(78\) −13.6569 −1.54633
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −11.6569 −1.28728
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) −3.17157 −0.346047
\(85\) 1.17157 0.127075
\(86\) 21.3137 2.29832
\(87\) 4.82843 0.517662
\(88\) 4.41421 0.470557
\(89\) 3.65685 0.387626 0.193813 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) 2.41421 0.254480
\(91\) −4.68629 −0.491257
\(92\) −15.3137 −1.59656
\(93\) 0 0
\(94\) 9.65685 0.996028
\(95\) 6.82843 0.700582
\(96\) −1.58579 −0.161849
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) 15.2426 1.53974
\(99\) −1.00000 −0.100504
\(100\) 3.82843 0.382843
\(101\) −0.828427 −0.0824316 −0.0412158 0.999150i \(-0.513123\pi\)
−0.0412158 + 0.999150i \(0.513123\pi\)
\(102\) −2.82843 −0.280056
\(103\) −3.31371 −0.326509 −0.163255 0.986584i \(-0.552199\pi\)
−0.163255 + 0.986584i \(0.552199\pi\)
\(104\) 24.9706 2.44857
\(105\) 0.828427 0.0808462
\(106\) −22.4853 −2.18396
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) −3.82843 −0.368391
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) −2.41421 −0.230186
\(111\) −11.6569 −1.10642
\(112\) 2.48528 0.234837
\(113\) −18.9706 −1.78460 −0.892300 0.451442i \(-0.850910\pi\)
−0.892300 + 0.451442i \(0.850910\pi\)
\(114\) −16.4853 −1.54399
\(115\) 4.00000 0.373002
\(116\) −18.4853 −1.71632
\(117\) −5.65685 −0.522976
\(118\) 9.65685 0.888985
\(119\) −0.970563 −0.0889713
\(120\) −4.41421 −0.402961
\(121\) 1.00000 0.0909091
\(122\) 28.1421 2.54787
\(123\) −4.82843 −0.435365
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −14.4853 −1.28536 −0.642680 0.766134i \(-0.722178\pi\)
−0.642680 + 0.766134i \(0.722178\pi\)
\(128\) 20.5563 1.81694
\(129\) 8.82843 0.777300
\(130\) −13.6569 −1.19779
\(131\) 3.31371 0.289520 0.144760 0.989467i \(-0.453759\pi\)
0.144760 + 0.989467i \(0.453759\pi\)
\(132\) 3.82843 0.333222
\(133\) −5.65685 −0.490511
\(134\) 13.6569 1.17977
\(135\) 1.00000 0.0860663
\(136\) 5.17157 0.443459
\(137\) −13.3137 −1.13747 −0.568733 0.822522i \(-0.692566\pi\)
−0.568733 + 0.822522i \(0.692566\pi\)
\(138\) −9.65685 −0.822046
\(139\) 0.485281 0.0411610 0.0205805 0.999788i \(-0.493449\pi\)
0.0205805 + 0.999788i \(0.493449\pi\)
\(140\) −3.17157 −0.268047
\(141\) 4.00000 0.336861
\(142\) −5.65685 −0.474713
\(143\) 5.65685 0.473050
\(144\) 3.00000 0.250000
\(145\) 4.82843 0.400979
\(146\) −27.3137 −2.26050
\(147\) 6.31371 0.520746
\(148\) 44.6274 3.66835
\(149\) 1.51472 0.124091 0.0620453 0.998073i \(-0.480238\pi\)
0.0620453 + 0.998073i \(0.480238\pi\)
\(150\) 2.41421 0.197120
\(151\) 16.4853 1.34155 0.670777 0.741659i \(-0.265961\pi\)
0.670777 + 0.741659i \(0.265961\pi\)
\(152\) 30.1421 2.44485
\(153\) −1.17157 −0.0947161
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 21.6569 1.73394
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −20.4853 −1.62972
\(159\) −9.31371 −0.738625
\(160\) −1.58579 −0.125367
\(161\) −3.31371 −0.261157
\(162\) −2.41421 −0.189679
\(163\) −7.31371 −0.572854 −0.286427 0.958102i \(-0.592468\pi\)
−0.286427 + 0.958102i \(0.592468\pi\)
\(164\) 18.4853 1.44346
\(165\) −1.00000 −0.0778499
\(166\) 24.1421 1.87379
\(167\) −13.3137 −1.03025 −0.515123 0.857116i \(-0.672254\pi\)
−0.515123 + 0.857116i \(0.672254\pi\)
\(168\) 3.65685 0.282132
\(169\) 19.0000 1.46154
\(170\) −2.82843 −0.216930
\(171\) −6.82843 −0.522183
\(172\) −33.7990 −2.57715
\(173\) 2.82843 0.215041 0.107521 0.994203i \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) −11.6569 −0.883704
\(175\) 0.828427 0.0626232
\(176\) −3.00000 −0.226134
\(177\) 4.00000 0.300658
\(178\) −8.82843 −0.661719
\(179\) −17.6569 −1.31974 −0.659868 0.751382i \(-0.729388\pi\)
−0.659868 + 0.751382i \(0.729388\pi\)
\(180\) −3.82843 −0.285354
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 11.3137 0.838628
\(183\) 11.6569 0.861699
\(184\) 17.6569 1.30168
\(185\) −11.6569 −0.857029
\(186\) 0 0
\(187\) 1.17157 0.0856739
\(188\) −15.3137 −1.11687
\(189\) −0.828427 −0.0602592
\(190\) −16.4853 −1.19597
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 9.82843 0.709306
\(193\) 13.6569 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(194\) −28.1421 −2.02049
\(195\) −5.65685 −0.405096
\(196\) −24.1716 −1.72654
\(197\) −8.48528 −0.604551 −0.302276 0.953221i \(-0.597746\pi\)
−0.302276 + 0.953221i \(0.597746\pi\)
\(198\) 2.41421 0.171571
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) −4.41421 −0.312132
\(201\) 5.65685 0.399004
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) 4.48528 0.314033
\(205\) −4.82843 −0.337232
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) −16.9706 −1.17670
\(209\) 6.82843 0.472332
\(210\) −2.00000 −0.138013
\(211\) 1.17157 0.0806544 0.0403272 0.999187i \(-0.487160\pi\)
0.0403272 + 0.999187i \(0.487160\pi\)
\(212\) 35.6569 2.44892
\(213\) −2.34315 −0.160550
\(214\) −41.7990 −2.85732
\(215\) 8.82843 0.602094
\(216\) 4.41421 0.300349
\(217\) 0 0
\(218\) −41.7990 −2.83098
\(219\) −11.3137 −0.764510
\(220\) 3.82843 0.258113
\(221\) 6.62742 0.445808
\(222\) 28.1421 1.88878
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) 1.31371 0.0877758
\(225\) 1.00000 0.0666667
\(226\) 45.7990 3.04650
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 26.1421 1.73131
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) −9.65685 −0.636754
\(231\) 0.828427 0.0545065
\(232\) 21.3137 1.39931
\(233\) −18.8284 −1.23349 −0.616746 0.787163i \(-0.711549\pi\)
−0.616746 + 0.787163i \(0.711549\pi\)
\(234\) 13.6569 0.892776
\(235\) 4.00000 0.260931
\(236\) −15.3137 −0.996838
\(237\) −8.48528 −0.551178
\(238\) 2.34315 0.151884
\(239\) 17.6569 1.14213 0.571063 0.820906i \(-0.306531\pi\)
0.571063 + 0.820906i \(0.306531\pi\)
\(240\) 3.00000 0.193649
\(241\) −12.3431 −0.795092 −0.397546 0.917582i \(-0.630138\pi\)
−0.397546 + 0.917582i \(0.630138\pi\)
\(242\) −2.41421 −0.155192
\(243\) −1.00000 −0.0641500
\(244\) −44.6274 −2.85698
\(245\) 6.31371 0.403368
\(246\) 11.6569 0.743214
\(247\) 38.6274 2.45780
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 2.41421 0.152688
\(251\) 20.9706 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(252\) 3.17157 0.199790
\(253\) 4.00000 0.251478
\(254\) 34.9706 2.19425
\(255\) −1.17157 −0.0733667
\(256\) −29.9706 −1.87316
\(257\) −16.3431 −1.01946 −0.509729 0.860335i \(-0.670254\pi\)
−0.509729 + 0.860335i \(0.670254\pi\)
\(258\) −21.3137 −1.32693
\(259\) 9.65685 0.600048
\(260\) 21.6569 1.34310
\(261\) −4.82843 −0.298872
\(262\) −8.00000 −0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) −4.41421 −0.271676
\(265\) −9.31371 −0.572137
\(266\) 13.6569 0.837355
\(267\) −3.65685 −0.223796
\(268\) −21.6569 −1.32290
\(269\) 20.6274 1.25768 0.628838 0.777536i \(-0.283531\pi\)
0.628838 + 0.777536i \(0.283531\pi\)
\(270\) −2.41421 −0.146924
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) −3.51472 −0.213111
\(273\) 4.68629 0.283627
\(274\) 32.1421 1.94178
\(275\) −1.00000 −0.0603023
\(276\) 15.3137 0.921777
\(277\) −2.34315 −0.140786 −0.0703930 0.997519i \(-0.522425\pi\)
−0.0703930 + 0.997519i \(0.522425\pi\)
\(278\) −1.17157 −0.0702663
\(279\) 0 0
\(280\) 3.65685 0.218539
\(281\) −11.1716 −0.666440 −0.333220 0.942849i \(-0.608135\pi\)
−0.333220 + 0.942849i \(0.608135\pi\)
\(282\) −9.65685 −0.575057
\(283\) −8.82843 −0.524796 −0.262398 0.964960i \(-0.584513\pi\)
−0.262398 + 0.964960i \(0.584513\pi\)
\(284\) 8.97056 0.532305
\(285\) −6.82843 −0.404481
\(286\) −13.6569 −0.807547
\(287\) 4.00000 0.236113
\(288\) 1.58579 0.0934434
\(289\) −15.6274 −0.919260
\(290\) −11.6569 −0.684514
\(291\) −11.6569 −0.683337
\(292\) 43.3137 2.53474
\(293\) −6.82843 −0.398921 −0.199460 0.979906i \(-0.563919\pi\)
−0.199460 + 0.979906i \(0.563919\pi\)
\(294\) −15.2426 −0.888969
\(295\) 4.00000 0.232889
\(296\) −51.4558 −2.99081
\(297\) 1.00000 0.0580259
\(298\) −3.65685 −0.211836
\(299\) 22.6274 1.30858
\(300\) −3.82843 −0.221034
\(301\) −7.31371 −0.421555
\(302\) −39.7990 −2.29017
\(303\) 0.828427 0.0475919
\(304\) −20.4853 −1.17491
\(305\) 11.6569 0.667470
\(306\) 2.82843 0.161690
\(307\) −3.17157 −0.181011 −0.0905056 0.995896i \(-0.528848\pi\)
−0.0905056 + 0.995896i \(0.528848\pi\)
\(308\) −3.17157 −0.180717
\(309\) 3.31371 0.188510
\(310\) 0 0
\(311\) −3.31371 −0.187903 −0.0939516 0.995577i \(-0.529950\pi\)
−0.0939516 + 0.995577i \(0.529950\pi\)
\(312\) −24.9706 −1.41368
\(313\) 15.6569 0.884978 0.442489 0.896774i \(-0.354096\pi\)
0.442489 + 0.896774i \(0.354096\pi\)
\(314\) −43.4558 −2.45236
\(315\) −0.828427 −0.0466766
\(316\) 32.4853 1.82744
\(317\) −26.2843 −1.47627 −0.738136 0.674652i \(-0.764294\pi\)
−0.738136 + 0.674652i \(0.764294\pi\)
\(318\) 22.4853 1.26091
\(319\) 4.82843 0.270340
\(320\) 9.82843 0.549426
\(321\) −17.3137 −0.966357
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) 3.82843 0.212690
\(325\) −5.65685 −0.313786
\(326\) 17.6569 0.977923
\(327\) −17.3137 −0.957450
\(328\) −21.3137 −1.17685
\(329\) −3.31371 −0.182691
\(330\) 2.41421 0.132898
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) −38.2843 −2.10112
\(333\) 11.6569 0.638792
\(334\) 32.1421 1.75874
\(335\) 5.65685 0.309067
\(336\) −2.48528 −0.135583
\(337\) 3.31371 0.180509 0.0902546 0.995919i \(-0.471232\pi\)
0.0902546 + 0.995919i \(0.471232\pi\)
\(338\) −45.8701 −2.49500
\(339\) 18.9706 1.03034
\(340\) 4.48528 0.243249
\(341\) 0 0
\(342\) 16.4853 0.891422
\(343\) −11.0294 −0.595534
\(344\) 38.9706 2.10115
\(345\) −4.00000 −0.215353
\(346\) −6.82843 −0.367099
\(347\) −29.3137 −1.57364 −0.786821 0.617181i \(-0.788275\pi\)
−0.786821 + 0.617181i \(0.788275\pi\)
\(348\) 18.4853 0.990915
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) −2.00000 −0.106904
\(351\) 5.65685 0.301941
\(352\) −1.58579 −0.0845227
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −9.65685 −0.513256
\(355\) −2.34315 −0.124361
\(356\) 14.0000 0.741999
\(357\) 0.970563 0.0513676
\(358\) 42.6274 2.25293
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 4.41421 0.232649
\(361\) 27.6274 1.45407
\(362\) 33.7990 1.77644
\(363\) −1.00000 −0.0524864
\(364\) −17.9411 −0.940370
\(365\) −11.3137 −0.592187
\(366\) −28.1421 −1.47101
\(367\) 9.65685 0.504084 0.252042 0.967716i \(-0.418898\pi\)
0.252042 + 0.967716i \(0.418898\pi\)
\(368\) −12.0000 −0.625543
\(369\) 4.82843 0.251358
\(370\) 28.1421 1.46304
\(371\) 7.71573 0.400581
\(372\) 0 0
\(373\) 10.6274 0.550267 0.275133 0.961406i \(-0.411278\pi\)
0.275133 + 0.961406i \(0.411278\pi\)
\(374\) −2.82843 −0.146254
\(375\) 1.00000 0.0516398
\(376\) 17.6569 0.910583
\(377\) 27.3137 1.40673
\(378\) 2.00000 0.102869
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 26.1421 1.34106
\(381\) 14.4853 0.742103
\(382\) −13.6569 −0.698745
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0.828427 0.0422206
\(386\) −32.9706 −1.67816
\(387\) −8.82843 −0.448774
\(388\) 44.6274 2.26561
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 13.6569 0.691542
\(391\) 4.68629 0.236996
\(392\) 27.8701 1.40765
\(393\) −3.31371 −0.167154
\(394\) 20.4853 1.03203
\(395\) −8.48528 −0.426941
\(396\) −3.82843 −0.192386
\(397\) −14.9706 −0.751351 −0.375676 0.926751i \(-0.622589\pi\)
−0.375676 + 0.926751i \(0.622589\pi\)
\(398\) 52.2843 2.62077
\(399\) 5.65685 0.283197
\(400\) 3.00000 0.150000
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) −13.6569 −0.681142
\(403\) 0 0
\(404\) −3.17157 −0.157792
\(405\) −1.00000 −0.0496904
\(406\) 9.65685 0.479262
\(407\) −11.6569 −0.577809
\(408\) −5.17157 −0.256031
\(409\) −19.6569 −0.971969 −0.485984 0.873967i \(-0.661539\pi\)
−0.485984 + 0.873967i \(0.661539\pi\)
\(410\) 11.6569 0.575691
\(411\) 13.3137 0.656717
\(412\) −12.6863 −0.625009
\(413\) −3.31371 −0.163057
\(414\) 9.65685 0.474608
\(415\) 10.0000 0.490881
\(416\) −8.97056 −0.439818
\(417\) −0.485281 −0.0237643
\(418\) −16.4853 −0.806321
\(419\) −36.9706 −1.80613 −0.903065 0.429504i \(-0.858689\pi\)
−0.903065 + 0.429504i \(0.858689\pi\)
\(420\) 3.17157 0.154757
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −2.82843 −0.137686
\(423\) −4.00000 −0.194487
\(424\) −41.1127 −1.99661
\(425\) −1.17157 −0.0568296
\(426\) 5.65685 0.274075
\(427\) −9.65685 −0.467328
\(428\) 66.2843 3.20397
\(429\) −5.65685 −0.273115
\(430\) −21.3137 −1.02784
\(431\) 21.6569 1.04317 0.521587 0.853198i \(-0.325340\pi\)
0.521587 + 0.853198i \(0.325340\pi\)
\(432\) −3.00000 −0.144338
\(433\) −15.6569 −0.752420 −0.376210 0.926534i \(-0.622773\pi\)
−0.376210 + 0.926534i \(0.622773\pi\)
\(434\) 0 0
\(435\) −4.82843 −0.231505
\(436\) 66.2843 3.17444
\(437\) 27.3137 1.30659
\(438\) 27.3137 1.30510
\(439\) −20.4853 −0.977709 −0.488855 0.872365i \(-0.662585\pi\)
−0.488855 + 0.872365i \(0.662585\pi\)
\(440\) −4.41421 −0.210439
\(441\) −6.31371 −0.300653
\(442\) −16.0000 −0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −44.6274 −2.11792
\(445\) −3.65685 −0.173352
\(446\) 15.3137 0.725125
\(447\) −1.51472 −0.0716437
\(448\) −8.14214 −0.384680
\(449\) 30.9706 1.46159 0.730796 0.682596i \(-0.239149\pi\)
0.730796 + 0.682596i \(0.239149\pi\)
\(450\) −2.41421 −0.113807
\(451\) −4.82843 −0.227362
\(452\) −72.6274 −3.41611
\(453\) −16.4853 −0.774546
\(454\) 33.7990 1.58627
\(455\) 4.68629 0.219697
\(456\) −30.1421 −1.41153
\(457\) −23.3137 −1.09057 −0.545285 0.838251i \(-0.683578\pi\)
−0.545285 + 0.838251i \(0.683578\pi\)
\(458\) 4.82843 0.225618
\(459\) 1.17157 0.0546843
\(460\) 15.3137 0.714005
\(461\) 0.142136 0.00661992 0.00330996 0.999995i \(-0.498946\pi\)
0.00330996 + 0.999995i \(0.498946\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 4.97056 0.231002 0.115501 0.993307i \(-0.463153\pi\)
0.115501 + 0.993307i \(0.463153\pi\)
\(464\) −14.4853 −0.672462
\(465\) 0 0
\(466\) 45.4558 2.10570
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) −21.6569 −1.00109
\(469\) −4.68629 −0.216393
\(470\) −9.65685 −0.445437
\(471\) −18.0000 −0.829396
\(472\) 17.6569 0.812723
\(473\) 8.82843 0.405932
\(474\) 20.4853 0.940920
\(475\) −6.82843 −0.313310
\(476\) −3.71573 −0.170310
\(477\) 9.31371 0.426445
\(478\) −42.6274 −1.94973
\(479\) 36.9706 1.68923 0.844614 0.535376i \(-0.179830\pi\)
0.844614 + 0.535376i \(0.179830\pi\)
\(480\) 1.58579 0.0723809
\(481\) −65.9411 −3.00666
\(482\) 29.7990 1.35731
\(483\) 3.31371 0.150779
\(484\) 3.82843 0.174019
\(485\) −11.6569 −0.529310
\(486\) 2.41421 0.109511
\(487\) −12.9706 −0.587752 −0.293876 0.955844i \(-0.594945\pi\)
−0.293876 + 0.955844i \(0.594945\pi\)
\(488\) 51.4558 2.32930
\(489\) 7.31371 0.330737
\(490\) −15.2426 −0.688592
\(491\) 14.3431 0.647297 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(492\) −18.4853 −0.833381
\(493\) 5.65685 0.254772
\(494\) −93.2548 −4.19573
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 1.94113 0.0870714
\(498\) −24.1421 −1.08183
\(499\) −22.3431 −1.00022 −0.500108 0.865963i \(-0.666706\pi\)
−0.500108 + 0.865963i \(0.666706\pi\)
\(500\) −3.82843 −0.171212
\(501\) 13.3137 0.594813
\(502\) −50.6274 −2.25961
\(503\) 17.3137 0.771980 0.385990 0.922503i \(-0.373860\pi\)
0.385990 + 0.922503i \(0.373860\pi\)
\(504\) −3.65685 −0.162889
\(505\) 0.828427 0.0368645
\(506\) −9.65685 −0.429300
\(507\) −19.0000 −0.843820
\(508\) −55.4558 −2.46046
\(509\) 18.6863 0.828255 0.414128 0.910219i \(-0.364087\pi\)
0.414128 + 0.910219i \(0.364087\pi\)
\(510\) 2.82843 0.125245
\(511\) 9.37258 0.414619
\(512\) 31.2426 1.38074
\(513\) 6.82843 0.301482
\(514\) 39.4558 1.74032
\(515\) 3.31371 0.146019
\(516\) 33.7990 1.48792
\(517\) 4.00000 0.175920
\(518\) −23.3137 −1.02435
\(519\) −2.82843 −0.124154
\(520\) −24.9706 −1.09503
\(521\) −32.6274 −1.42943 −0.714717 0.699414i \(-0.753444\pi\)
−0.714717 + 0.699414i \(0.753444\pi\)
\(522\) 11.6569 0.510207
\(523\) −9.51472 −0.416050 −0.208025 0.978124i \(-0.566703\pi\)
−0.208025 + 0.978124i \(0.566703\pi\)
\(524\) 12.6863 0.554203
\(525\) −0.828427 −0.0361555
\(526\) −43.4558 −1.89476
\(527\) 0 0
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) 22.4853 0.976698
\(531\) −4.00000 −0.173585
\(532\) −21.6569 −0.938944
\(533\) −27.3137 −1.18309
\(534\) 8.82843 0.382043
\(535\) −17.3137 −0.748537
\(536\) 24.9706 1.07856
\(537\) 17.6569 0.761950
\(538\) −49.7990 −2.14699
\(539\) 6.31371 0.271951
\(540\) 3.82843 0.164749
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) 28.4853 1.22355
\(543\) 14.0000 0.600798
\(544\) −1.85786 −0.0796553
\(545\) −17.3137 −0.741638
\(546\) −11.3137 −0.484182
\(547\) −8.14214 −0.348133 −0.174066 0.984734i \(-0.555691\pi\)
−0.174066 + 0.984734i \(0.555691\pi\)
\(548\) −50.9706 −2.17735
\(549\) −11.6569 −0.497502
\(550\) 2.41421 0.102942
\(551\) 32.9706 1.40459
\(552\) −17.6569 −0.751526
\(553\) 7.02944 0.298922
\(554\) 5.65685 0.240337
\(555\) 11.6569 0.494806
\(556\) 1.85786 0.0787910
\(557\) 5.17157 0.219127 0.109563 0.993980i \(-0.465055\pi\)
0.109563 + 0.993980i \(0.465055\pi\)
\(558\) 0 0
\(559\) 49.9411 2.11228
\(560\) −2.48528 −0.105022
\(561\) −1.17157 −0.0494638
\(562\) 26.9706 1.13768
\(563\) 31.6569 1.33418 0.667089 0.744978i \(-0.267540\pi\)
0.667089 + 0.744978i \(0.267540\pi\)
\(564\) 15.3137 0.644823
\(565\) 18.9706 0.798098
\(566\) 21.3137 0.895882
\(567\) 0.828427 0.0347907
\(568\) −10.3431 −0.433989
\(569\) −35.4558 −1.48639 −0.743193 0.669077i \(-0.766690\pi\)
−0.743193 + 0.669077i \(0.766690\pi\)
\(570\) 16.4853 0.690492
\(571\) −16.4853 −0.689888 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(572\) 21.6569 0.905519
\(573\) −5.65685 −0.236318
\(574\) −9.65685 −0.403069
\(575\) −4.00000 −0.166812
\(576\) −9.82843 −0.409518
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 37.7279 1.56927
\(579\) −13.6569 −0.567559
\(580\) 18.4853 0.767560
\(581\) −8.28427 −0.343689
\(582\) 28.1421 1.16653
\(583\) −9.31371 −0.385734
\(584\) −49.9411 −2.06658
\(585\) 5.65685 0.233882
\(586\) 16.4853 0.681001
\(587\) 14.6274 0.603738 0.301869 0.953349i \(-0.402389\pi\)
0.301869 + 0.953349i \(0.402389\pi\)
\(588\) 24.1716 0.996819
\(589\) 0 0
\(590\) −9.65685 −0.397566
\(591\) 8.48528 0.349038
\(592\) 34.9706 1.43728
\(593\) −22.8284 −0.937451 −0.468726 0.883344i \(-0.655287\pi\)
−0.468726 + 0.883344i \(0.655287\pi\)
\(594\) −2.41421 −0.0990564
\(595\) 0.970563 0.0397892
\(596\) 5.79899 0.237536
\(597\) 21.6569 0.886356
\(598\) −54.6274 −2.23388
\(599\) 27.3137 1.11601 0.558004 0.829838i \(-0.311567\pi\)
0.558004 + 0.829838i \(0.311567\pi\)
\(600\) 4.41421 0.180210
\(601\) −5.31371 −0.216751 −0.108375 0.994110i \(-0.534565\pi\)
−0.108375 + 0.994110i \(0.534565\pi\)
\(602\) 17.6569 0.719640
\(603\) −5.65685 −0.230365
\(604\) 63.1127 2.56802
\(605\) −1.00000 −0.0406558
\(606\) −2.00000 −0.0812444
\(607\) 1.51472 0.0614805 0.0307403 0.999527i \(-0.490214\pi\)
0.0307403 + 0.999527i \(0.490214\pi\)
\(608\) −10.8284 −0.439151
\(609\) 4.00000 0.162088
\(610\) −28.1421 −1.13944
\(611\) 22.6274 0.915407
\(612\) −4.48528 −0.181307
\(613\) −45.9411 −1.85554 −0.927772 0.373147i \(-0.878279\pi\)
−0.927772 + 0.373147i \(0.878279\pi\)
\(614\) 7.65685 0.309005
\(615\) 4.82843 0.194701
\(616\) 3.65685 0.147339
\(617\) 0.343146 0.0138145 0.00690726 0.999976i \(-0.497801\pi\)
0.00690726 + 0.999976i \(0.497801\pi\)
\(618\) −8.00000 −0.321807
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 3.02944 0.121372
\(624\) 16.9706 0.679366
\(625\) 1.00000 0.0400000
\(626\) −37.7990 −1.51075
\(627\) −6.82843 −0.272701
\(628\) 68.9117 2.74988
\(629\) −13.6569 −0.544534
\(630\) 2.00000 0.0796819
\(631\) 45.6569 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(632\) −37.4558 −1.48991
\(633\) −1.17157 −0.0465658
\(634\) 63.4558 2.52015
\(635\) 14.4853 0.574831
\(636\) −35.6569 −1.41389
\(637\) 35.7157 1.41511
\(638\) −11.6569 −0.461499
\(639\) 2.34315 0.0926934
\(640\) −20.5563 −0.812561
\(641\) 6.97056 0.275321 0.137660 0.990479i \(-0.456042\pi\)
0.137660 + 0.990479i \(0.456042\pi\)
\(642\) 41.7990 1.64967
\(643\) 37.9411 1.49625 0.748126 0.663557i \(-0.230954\pi\)
0.748126 + 0.663557i \(0.230954\pi\)
\(644\) −12.6863 −0.499910
\(645\) −8.82843 −0.347619
\(646\) −19.3137 −0.759888
\(647\) 4.68629 0.184237 0.0921186 0.995748i \(-0.470636\pi\)
0.0921186 + 0.995748i \(0.470636\pi\)
\(648\) −4.41421 −0.173407
\(649\) 4.00000 0.157014
\(650\) 13.6569 0.535666
\(651\) 0 0
\(652\) −28.0000 −1.09656
\(653\) 6.97056 0.272779 0.136390 0.990655i \(-0.456450\pi\)
0.136390 + 0.990655i \(0.456450\pi\)
\(654\) 41.7990 1.63447
\(655\) −3.31371 −0.129477
\(656\) 14.4853 0.565555
\(657\) 11.3137 0.441390
\(658\) 8.00000 0.311872
\(659\) −15.3137 −0.596537 −0.298269 0.954482i \(-0.596409\pi\)
−0.298269 + 0.954482i \(0.596409\pi\)
\(660\) −3.82843 −0.149021
\(661\) 9.31371 0.362261 0.181131 0.983459i \(-0.442024\pi\)
0.181131 + 0.983459i \(0.442024\pi\)
\(662\) −15.3137 −0.595184
\(663\) −6.62742 −0.257388
\(664\) 44.1421 1.71305
\(665\) 5.65685 0.219363
\(666\) −28.1421 −1.09049
\(667\) 19.3137 0.747830
\(668\) −50.9706 −1.97211
\(669\) 6.34315 0.245240
\(670\) −13.6569 −0.527610
\(671\) 11.6569 0.450008
\(672\) −1.31371 −0.0506774
\(673\) 18.3431 0.707076 0.353538 0.935420i \(-0.384978\pi\)
0.353538 + 0.935420i \(0.384978\pi\)
\(674\) −8.00000 −0.308148
\(675\) −1.00000 −0.0384900
\(676\) 72.7401 2.79770
\(677\) 29.4558 1.13208 0.566040 0.824378i \(-0.308475\pi\)
0.566040 + 0.824378i \(0.308475\pi\)
\(678\) −45.7990 −1.75890
\(679\) 9.65685 0.370596
\(680\) −5.17157 −0.198321
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −26.1421 −0.999570
\(685\) 13.3137 0.508691
\(686\) 26.6274 1.01664
\(687\) 2.00000 0.0763048
\(688\) −26.4853 −1.00974
\(689\) −52.6863 −2.00719
\(690\) 9.65685 0.367630
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 10.8284 0.411635
\(693\) −0.828427 −0.0314693
\(694\) 70.7696 2.68638
\(695\) −0.485281 −0.0184078
\(696\) −21.3137 −0.807894
\(697\) −5.65685 −0.214269
\(698\) −26.4853 −1.00248
\(699\) 18.8284 0.712157
\(700\) 3.17157 0.119874
\(701\) 36.1421 1.36507 0.682535 0.730853i \(-0.260878\pi\)
0.682535 + 0.730853i \(0.260878\pi\)
\(702\) −13.6569 −0.515445
\(703\) −79.5980 −3.00209
\(704\) 9.82843 0.370423
\(705\) −4.00000 −0.150649
\(706\) −62.7696 −2.36236
\(707\) −0.686292 −0.0258106
\(708\) 15.3137 0.575524
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) 5.65685 0.212298
\(711\) 8.48528 0.318223
\(712\) −16.1421 −0.604952
\(713\) 0 0
\(714\) −2.34315 −0.0876900
\(715\) −5.65685 −0.211554
\(716\) −67.5980 −2.52626
\(717\) −17.6569 −0.659407
\(718\) −28.9706 −1.08117
\(719\) −47.5980 −1.77511 −0.887553 0.460706i \(-0.847596\pi\)
−0.887553 + 0.460706i \(0.847596\pi\)
\(720\) −3.00000 −0.111803
\(721\) −2.74517 −0.102235
\(722\) −66.6985 −2.48226
\(723\) 12.3431 0.459047
\(724\) −53.5980 −1.99195
\(725\) −4.82843 −0.179323
\(726\) 2.41421 0.0895999
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 20.6863 0.766685
\(729\) 1.00000 0.0370370
\(730\) 27.3137 1.01093
\(731\) 10.3431 0.382555
\(732\) 44.6274 1.64948
\(733\) −6.34315 −0.234289 −0.117145 0.993115i \(-0.537374\pi\)
−0.117145 + 0.993115i \(0.537374\pi\)
\(734\) −23.3137 −0.860525
\(735\) −6.31371 −0.232885
\(736\) −6.34315 −0.233811
\(737\) 5.65685 0.208373
\(738\) −11.6569 −0.429095
\(739\) −15.1127 −0.555930 −0.277965 0.960591i \(-0.589660\pi\)
−0.277965 + 0.960591i \(0.589660\pi\)
\(740\) −44.6274 −1.64054
\(741\) −38.6274 −1.41901
\(742\) −18.6274 −0.683834
\(743\) 36.3431 1.33330 0.666650 0.745371i \(-0.267727\pi\)
0.666650 + 0.745371i \(0.267727\pi\)
\(744\) 0 0
\(745\) −1.51472 −0.0554950
\(746\) −25.6569 −0.939364
\(747\) −10.0000 −0.365881
\(748\) 4.48528 0.163998
\(749\) 14.3431 0.524087
\(750\) −2.41421 −0.0881546
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) −12.0000 −0.437595
\(753\) −20.9706 −0.764210
\(754\) −65.9411 −2.40143
\(755\) −16.4853 −0.599961
\(756\) −3.17157 −0.115349
\(757\) −36.6274 −1.33125 −0.665623 0.746288i \(-0.731834\pi\)
−0.665623 + 0.746288i \(0.731834\pi\)
\(758\) 56.2843 2.04434
\(759\) −4.00000 −0.145191
\(760\) −30.1421 −1.09337
\(761\) −28.8284 −1.04503 −0.522515 0.852630i \(-0.675006\pi\)
−0.522515 + 0.852630i \(0.675006\pi\)
\(762\) −34.9706 −1.26685
\(763\) 14.3431 0.519257
\(764\) 21.6569 0.783517
\(765\) 1.17157 0.0423583
\(766\) 19.3137 0.697833
\(767\) 22.6274 0.817029
\(768\) 29.9706 1.08147
\(769\) 10.6863 0.385358 0.192679 0.981262i \(-0.438282\pi\)
0.192679 + 0.981262i \(0.438282\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 16.3431 0.588584
\(772\) 52.2843 1.88175
\(773\) 3.65685 0.131528 0.0657640 0.997835i \(-0.479052\pi\)
0.0657640 + 0.997835i \(0.479052\pi\)
\(774\) 21.3137 0.766105
\(775\) 0 0
\(776\) −51.4558 −1.84716
\(777\) −9.65685 −0.346438
\(778\) 57.1127 2.04759
\(779\) −32.9706 −1.18129
\(780\) −21.6569 −0.775440
\(781\) −2.34315 −0.0838443
\(782\) −11.3137 −0.404577
\(783\) 4.82843 0.172554
\(784\) −18.9411 −0.676469
\(785\) −18.0000 −0.642448
\(786\) 8.00000 0.285351
\(787\) −20.1421 −0.717990 −0.358995 0.933340i \(-0.616880\pi\)
−0.358995 + 0.933340i \(0.616880\pi\)
\(788\) −32.4853 −1.15724
\(789\) −18.0000 −0.640817
\(790\) 20.4853 0.728834
\(791\) −15.7157 −0.558787
\(792\) 4.41421 0.156852
\(793\) 65.9411 2.34164
\(794\) 36.1421 1.28264
\(795\) 9.31371 0.330323
\(796\) −82.9117 −2.93873
\(797\) 34.9706 1.23872 0.619360 0.785107i \(-0.287392\pi\)
0.619360 + 0.785107i \(0.287392\pi\)
\(798\) −13.6569 −0.483447
\(799\) 4.68629 0.165789
\(800\) 1.58579 0.0560660
\(801\) 3.65685 0.129209
\(802\) 16.1421 0.569999
\(803\) −11.3137 −0.399252
\(804\) 21.6569 0.763778
\(805\) 3.31371 0.116793
\(806\) 0 0
\(807\) −20.6274 −0.726119
\(808\) 3.65685 0.128648
\(809\) 28.4264 0.999419 0.499710 0.866193i \(-0.333440\pi\)
0.499710 + 0.866193i \(0.333440\pi\)
\(810\) 2.41421 0.0848268
\(811\) 0.485281 0.0170405 0.00852027 0.999964i \(-0.497288\pi\)
0.00852027 + 0.999964i \(0.497288\pi\)
\(812\) −15.3137 −0.537406
\(813\) 11.7990 0.413809
\(814\) 28.1421 0.986381
\(815\) 7.31371 0.256188
\(816\) 3.51472 0.123040
\(817\) 60.2843 2.10908
\(818\) 47.4558 1.65925
\(819\) −4.68629 −0.163752
\(820\) −18.4853 −0.645534
\(821\) −12.8284 −0.447715 −0.223858 0.974622i \(-0.571865\pi\)
−0.223858 + 0.974622i \(0.571865\pi\)
\(822\) −32.1421 −1.12109
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 14.6274 0.509570
\(825\) 1.00000 0.0348155
\(826\) 8.00000 0.278356
\(827\) 41.3137 1.43662 0.718309 0.695724i \(-0.244916\pi\)
0.718309 + 0.695724i \(0.244916\pi\)
\(828\) −15.3137 −0.532188
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −24.1421 −0.837986
\(831\) 2.34315 0.0812828
\(832\) 55.5980 1.92751
\(833\) 7.39697 0.256290
\(834\) 1.17157 0.0405683
\(835\) 13.3137 0.460740
\(836\) 26.1421 0.904145
\(837\) 0 0
\(838\) 89.2548 3.08326
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) −3.65685 −0.126173
\(841\) −5.68629 −0.196079
\(842\) 14.4853 0.499196
\(843\) 11.1716 0.384769
\(844\) 4.48528 0.154390
\(845\) −19.0000 −0.653620
\(846\) 9.65685 0.332009
\(847\) 0.828427 0.0284651
\(848\) 27.9411 0.959502
\(849\) 8.82843 0.302991
\(850\) 2.82843 0.0970143
\(851\) −46.6274 −1.59837
\(852\) −8.97056 −0.307326
\(853\) −8.68629 −0.297413 −0.148706 0.988881i \(-0.547511\pi\)
−0.148706 + 0.988881i \(0.547511\pi\)
\(854\) 23.3137 0.797779
\(855\) 6.82843 0.233527
\(856\) −76.4264 −2.61220
\(857\) −28.4853 −0.973039 −0.486519 0.873670i \(-0.661734\pi\)
−0.486519 + 0.873670i \(0.661734\pi\)
\(858\) 13.6569 0.466237
\(859\) −52.9706 −1.80733 −0.903666 0.428238i \(-0.859135\pi\)
−0.903666 + 0.428238i \(0.859135\pi\)
\(860\) 33.7990 1.15254
\(861\) −4.00000 −0.136320
\(862\) −52.2843 −1.78081
\(863\) −20.6863 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(864\) −1.58579 −0.0539496
\(865\) −2.82843 −0.0961694
\(866\) 37.7990 1.28446
\(867\) 15.6274 0.530735
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 11.6569 0.395204
\(871\) 32.0000 1.08428
\(872\) −76.4264 −2.58812
\(873\) 11.6569 0.394525
\(874\) −65.9411 −2.23049
\(875\) −0.828427 −0.0280059
\(876\) −43.3137 −1.46343
\(877\) 2.62742 0.0887216 0.0443608 0.999016i \(-0.485875\pi\)
0.0443608 + 0.999016i \(0.485875\pi\)
\(878\) 49.4558 1.66905
\(879\) 6.82843 0.230317
\(880\) 3.00000 0.101130
\(881\) −46.9706 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(882\) 15.2426 0.513246
\(883\) −5.37258 −0.180802 −0.0904009 0.995905i \(-0.528815\pi\)
−0.0904009 + 0.995905i \(0.528815\pi\)
\(884\) 25.3726 0.853372
\(885\) −4.00000 −0.134459
\(886\) −28.9706 −0.973285
\(887\) −15.6569 −0.525706 −0.262853 0.964836i \(-0.584663\pi\)
−0.262853 + 0.964836i \(0.584663\pi\)
\(888\) 51.4558 1.72675
\(889\) −12.0000 −0.402467
\(890\) 8.82843 0.295930
\(891\) −1.00000 −0.0335013
\(892\) −24.2843 −0.813098
\(893\) 27.3137 0.914018
\(894\) 3.65685 0.122304
\(895\) 17.6569 0.590204
\(896\) 17.0294 0.568914
\(897\) −22.6274 −0.755507
\(898\) −74.7696 −2.49509
\(899\) 0 0
\(900\) 3.82843 0.127614
\(901\) −10.9117 −0.363521
\(902\) 11.6569 0.388131
\(903\) 7.31371 0.243385
\(904\) 83.7401 2.78515
\(905\) 14.0000 0.465376
\(906\) 39.7990 1.32223
\(907\) 40.9706 1.36041 0.680203 0.733024i \(-0.261892\pi\)
0.680203 + 0.733024i \(0.261892\pi\)
\(908\) −53.5980 −1.77871
\(909\) −0.828427 −0.0274772
\(910\) −11.3137 −0.375046
\(911\) −48.9706 −1.62247 −0.811234 0.584722i \(-0.801203\pi\)
−0.811234 + 0.584722i \(0.801203\pi\)
\(912\) 20.4853 0.678335
\(913\) 10.0000 0.330952
\(914\) 56.2843 1.86172
\(915\) −11.6569 −0.385364
\(916\) −7.65685 −0.252990
\(917\) 2.74517 0.0906534
\(918\) −2.82843 −0.0933520
\(919\) 11.5147 0.379836 0.189918 0.981800i \(-0.439178\pi\)
0.189918 + 0.981800i \(0.439178\pi\)
\(920\) −17.6569 −0.582129
\(921\) 3.17157 0.104507
\(922\) −0.343146 −0.0113009
\(923\) −13.2548 −0.436288
\(924\) 3.17157 0.104337
\(925\) 11.6569 0.383275
\(926\) −12.0000 −0.394344
\(927\) −3.31371 −0.108836
\(928\) −7.65685 −0.251349
\(929\) −45.5980 −1.49602 −0.748011 0.663687i \(-0.768991\pi\)
−0.748011 + 0.663687i \(0.768991\pi\)
\(930\) 0 0
\(931\) 43.1127 1.41296
\(932\) −72.0833 −2.36117
\(933\) 3.31371 0.108486
\(934\) 54.6274 1.78746
\(935\) −1.17157 −0.0383145
\(936\) 24.9706 0.816188
\(937\) 11.0294 0.360316 0.180158 0.983638i \(-0.442339\pi\)
0.180158 + 0.983638i \(0.442339\pi\)
\(938\) 11.3137 0.369406
\(939\) −15.6569 −0.510942
\(940\) 15.3137 0.499478
\(941\) −34.7696 −1.13346 −0.566728 0.823905i \(-0.691791\pi\)
−0.566728 + 0.823905i \(0.691791\pi\)
\(942\) 43.4558 1.41587
\(943\) −19.3137 −0.628941
\(944\) −12.0000 −0.390567
\(945\) 0.828427 0.0269487
\(946\) −21.3137 −0.692968
\(947\) 6.62742 0.215362 0.107681 0.994185i \(-0.465657\pi\)
0.107681 + 0.994185i \(0.465657\pi\)
\(948\) −32.4853 −1.05507
\(949\) −64.0000 −2.07753
\(950\) 16.4853 0.534853
\(951\) 26.2843 0.852326
\(952\) 4.28427 0.138854
\(953\) −11.7990 −0.382207 −0.191103 0.981570i \(-0.561207\pi\)
−0.191103 + 0.981570i \(0.561207\pi\)
\(954\) −22.4853 −0.727988
\(955\) −5.65685 −0.183052
\(956\) 67.5980 2.18627
\(957\) −4.82843 −0.156081
\(958\) −89.2548 −2.88369
\(959\) −11.0294 −0.356159
\(960\) −9.82843 −0.317211
\(961\) −31.0000 −1.00000
\(962\) 159.196 5.13268
\(963\) 17.3137 0.557926
\(964\) −47.2548 −1.52198
\(965\) −13.6569 −0.439630
\(966\) −8.00000 −0.257396
\(967\) 11.4558 0.368395 0.184198 0.982889i \(-0.441031\pi\)
0.184198 + 0.982889i \(0.441031\pi\)
\(968\) −4.41421 −0.141878
\(969\) −8.00000 −0.256997
\(970\) 28.1421 0.903590
\(971\) −34.6274 −1.11125 −0.555623 0.831434i \(-0.687520\pi\)
−0.555623 + 0.831434i \(0.687520\pi\)
\(972\) −3.82843 −0.122797
\(973\) 0.402020 0.0128882
\(974\) 31.3137 1.00336
\(975\) 5.65685 0.181164
\(976\) −34.9706 −1.11938
\(977\) 2.68629 0.0859421 0.0429710 0.999076i \(-0.486318\pi\)
0.0429710 + 0.999076i \(0.486318\pi\)
\(978\) −17.6569 −0.564604
\(979\) −3.65685 −0.116874
\(980\) 24.1716 0.772133
\(981\) 17.3137 0.552784
\(982\) −34.6274 −1.10501
\(983\) −30.6274 −0.976863 −0.488431 0.872602i \(-0.662431\pi\)
−0.488431 + 0.872602i \(0.662431\pi\)
\(984\) 21.3137 0.679456
\(985\) 8.48528 0.270364
\(986\) −13.6569 −0.434923
\(987\) 3.31371 0.105477
\(988\) 147.882 4.70476
\(989\) 35.3137 1.12291
\(990\) −2.41421 −0.0767287
\(991\) −30.6274 −0.972912 −0.486456 0.873705i \(-0.661711\pi\)
−0.486456 + 0.873705i \(0.661711\pi\)
\(992\) 0 0
\(993\) −6.34315 −0.201294
\(994\) −4.68629 −0.148640
\(995\) 21.6569 0.686568
\(996\) 38.2843 1.21308
\(997\) −39.3137 −1.24508 −0.622539 0.782589i \(-0.713899\pi\)
−0.622539 + 0.782589i \(0.713899\pi\)
\(998\) 53.9411 1.70748
\(999\) −11.6569 −0.368807
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 165.2.a.a.1.1 2
3.2 odd 2 495.2.a.d.1.2 2
4.3 odd 2 2640.2.a.bb.1.1 2
5.2 odd 4 825.2.c.e.199.1 4
5.3 odd 4 825.2.c.e.199.4 4
5.4 even 2 825.2.a.g.1.2 2
7.6 odd 2 8085.2.a.ba.1.1 2
11.10 odd 2 1815.2.a.k.1.2 2
12.11 even 2 7920.2.a.cg.1.1 2
15.2 even 4 2475.2.c.m.199.4 4
15.8 even 4 2475.2.c.m.199.1 4
15.14 odd 2 2475.2.a.m.1.1 2
33.32 even 2 5445.2.a.m.1.1 2
55.54 odd 2 9075.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 1.1 even 1 trivial
495.2.a.d.1.2 2 3.2 odd 2
825.2.a.g.1.2 2 5.4 even 2
825.2.c.e.199.1 4 5.2 odd 4
825.2.c.e.199.4 4 5.3 odd 4
1815.2.a.k.1.2 2 11.10 odd 2
2475.2.a.m.1.1 2 15.14 odd 2
2475.2.c.m.199.1 4 15.8 even 4
2475.2.c.m.199.4 4 15.2 even 4
2640.2.a.bb.1.1 2 4.3 odd 2
5445.2.a.m.1.1 2 33.32 even 2
7920.2.a.cg.1.1 2 12.11 even 2
8085.2.a.ba.1.1 2 7.6 odd 2
9075.2.a.v.1.1 2 55.54 odd 2