Properties

Label 1617.2.a.r.1.2
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
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 1.2
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.254102 q^{2} +1.00000 q^{3} -1.93543 q^{4} +3.68133 q^{5} -0.254102 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.254102 q^{2} +1.00000 q^{3} -1.93543 q^{4} +3.68133 q^{5} -0.254102 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.935432 q^{10} +1.00000 q^{11} -1.93543 q^{12} -2.18953 q^{13} +3.68133 q^{15} +3.61676 q^{16} -3.36266 q^{17} -0.254102 q^{18} +3.68133 q^{19} -7.12497 q^{20} -0.254102 q^{22} +1.00000 q^{24} +8.55220 q^{25} +0.556364 q^{26} +1.00000 q^{27} +10.0604 q^{29} -0.935432 q^{30} +8.37907 q^{31} -2.91903 q^{32} +1.00000 q^{33} +0.854458 q^{34} -1.93543 q^{36} +0.189534 q^{37} -0.935432 q^{38} -2.18953 q^{39} +3.68133 q^{40} -8.37907 q^{41} -8.37907 q^{43} -1.93543 q^{44} +3.68133 q^{45} +5.17313 q^{47} +3.61676 q^{48} -2.17313 q^{50} -3.36266 q^{51} +4.23769 q^{52} -1.36266 q^{53} -0.254102 q^{54} +3.68133 q^{55} +3.68133 q^{57} -2.55636 q^{58} -4.53579 q^{59} -7.12497 q^{60} -0.379068 q^{61} -2.12914 q^{62} -6.49180 q^{64} -8.06040 q^{65} -0.254102 q^{66} +14.4067 q^{67} +6.50820 q^{68} -14.0880 q^{71} +1.00000 q^{72} +13.9313 q^{73} -0.0481609 q^{74} +8.55220 q^{75} -7.12497 q^{76} +0.556364 q^{78} -2.98359 q^{79} +13.3145 q^{80} +1.00000 q^{81} +2.12914 q^{82} +4.37907 q^{83} -12.3791 q^{85} +2.12914 q^{86} +10.0604 q^{87} +1.00000 q^{88} +0.637339 q^{89} -0.935432 q^{90} +8.37907 q^{93} -1.31450 q^{94} +13.5522 q^{95} -2.91903 q^{96} +9.39547 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.254102 −0.179677 −0.0898385 0.995956i \(-0.528635\pi\)
−0.0898385 + 0.995956i \(0.528635\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.93543 −0.967716
\(5\) 3.68133 1.64634 0.823171 0.567794i \(-0.192203\pi\)
0.823171 + 0.567794i \(0.192203\pi\)
\(6\) −0.254102 −0.103737
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.935432 −0.295810
\(11\) 1.00000 0.301511
\(12\) −1.93543 −0.558711
\(13\) −2.18953 −0.607267 −0.303634 0.952789i \(-0.598200\pi\)
−0.303634 + 0.952789i \(0.598200\pi\)
\(14\) 0 0
\(15\) 3.68133 0.950515
\(16\) 3.61676 0.904191
\(17\) −3.36266 −0.815565 −0.407783 0.913079i \(-0.633698\pi\)
−0.407783 + 0.913079i \(0.633698\pi\)
\(18\) −0.254102 −0.0598923
\(19\) 3.68133 0.844555 0.422278 0.906467i \(-0.361231\pi\)
0.422278 + 0.906467i \(0.361231\pi\)
\(20\) −7.12497 −1.59319
\(21\) 0 0
\(22\) −0.254102 −0.0541747
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 8.55220 1.71044
\(26\) 0.556364 0.109112
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.0604 1.86817 0.934085 0.357052i \(-0.116218\pi\)
0.934085 + 0.357052i \(0.116218\pi\)
\(30\) −0.935432 −0.170786
\(31\) 8.37907 1.50493 0.752463 0.658635i \(-0.228866\pi\)
0.752463 + 0.658635i \(0.228866\pi\)
\(32\) −2.91903 −0.516016
\(33\) 1.00000 0.174078
\(34\) 0.854458 0.146538
\(35\) 0 0
\(36\) −1.93543 −0.322572
\(37\) 0.189534 0.0311592 0.0155796 0.999879i \(-0.495041\pi\)
0.0155796 + 0.999879i \(0.495041\pi\)
\(38\) −0.935432 −0.151747
\(39\) −2.18953 −0.350606
\(40\) 3.68133 0.582069
\(41\) −8.37907 −1.30859 −0.654295 0.756239i \(-0.727035\pi\)
−0.654295 + 0.756239i \(0.727035\pi\)
\(42\) 0 0
\(43\) −8.37907 −1.27780 −0.638898 0.769291i \(-0.720609\pi\)
−0.638898 + 0.769291i \(0.720609\pi\)
\(44\) −1.93543 −0.291777
\(45\) 3.68133 0.548780
\(46\) 0 0
\(47\) 5.17313 0.754578 0.377289 0.926096i \(-0.376856\pi\)
0.377289 + 0.926096i \(0.376856\pi\)
\(48\) 3.61676 0.522035
\(49\) 0 0
\(50\) −2.17313 −0.307327
\(51\) −3.36266 −0.470867
\(52\) 4.23769 0.587663
\(53\) −1.36266 −0.187176 −0.0935880 0.995611i \(-0.529834\pi\)
−0.0935880 + 0.995611i \(0.529834\pi\)
\(54\) −0.254102 −0.0345789
\(55\) 3.68133 0.496391
\(56\) 0 0
\(57\) 3.68133 0.487604
\(58\) −2.55636 −0.335667
\(59\) −4.53579 −0.590509 −0.295255 0.955419i \(-0.595405\pi\)
−0.295255 + 0.955419i \(0.595405\pi\)
\(60\) −7.12497 −0.919829
\(61\) −0.379068 −0.0485347 −0.0242673 0.999706i \(-0.507725\pi\)
−0.0242673 + 0.999706i \(0.507725\pi\)
\(62\) −2.12914 −0.270400
\(63\) 0 0
\(64\) −6.49180 −0.811475
\(65\) −8.06040 −0.999769
\(66\) −0.254102 −0.0312778
\(67\) 14.4067 1.76005 0.880026 0.474925i \(-0.157525\pi\)
0.880026 + 0.474925i \(0.157525\pi\)
\(68\) 6.50820 0.789236
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0880 −1.67194 −0.835968 0.548778i \(-0.815093\pi\)
−0.835968 + 0.548778i \(0.815093\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.9313 1.63053 0.815266 0.579087i \(-0.196591\pi\)
0.815266 + 0.579087i \(0.196591\pi\)
\(74\) −0.0481609 −0.00559859
\(75\) 8.55220 0.987522
\(76\) −7.12497 −0.817290
\(77\) 0 0
\(78\) 0.556364 0.0629959
\(79\) −2.98359 −0.335680 −0.167840 0.985814i \(-0.553679\pi\)
−0.167840 + 0.985814i \(0.553679\pi\)
\(80\) 13.3145 1.48861
\(81\) 1.00000 0.111111
\(82\) 2.12914 0.235124
\(83\) 4.37907 0.480665 0.240333 0.970691i \(-0.422743\pi\)
0.240333 + 0.970691i \(0.422743\pi\)
\(84\) 0 0
\(85\) −12.3791 −1.34270
\(86\) 2.12914 0.229591
\(87\) 10.0604 1.07859
\(88\) 1.00000 0.106600
\(89\) 0.637339 0.0675578 0.0337789 0.999429i \(-0.489246\pi\)
0.0337789 + 0.999429i \(0.489246\pi\)
\(90\) −0.935432 −0.0986032
\(91\) 0 0
\(92\) 0 0
\(93\) 8.37907 0.868869
\(94\) −1.31450 −0.135580
\(95\) 13.5522 1.39043
\(96\) −2.91903 −0.297922
\(97\) 9.39547 0.953966 0.476983 0.878913i \(-0.341730\pi\)
0.476983 + 0.878913i \(0.341730\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −16.5522 −1.65522
\(101\) 7.10439 0.706913 0.353457 0.935451i \(-0.385006\pi\)
0.353457 + 0.935451i \(0.385006\pi\)
\(102\) 0.854458 0.0846039
\(103\) 0.258271 0.0254482 0.0127241 0.999919i \(-0.495950\pi\)
0.0127241 + 0.999919i \(0.495950\pi\)
\(104\) −2.18953 −0.214701
\(105\) 0 0
\(106\) 0.346255 0.0336312
\(107\) 12.5358 1.21188 0.605940 0.795510i \(-0.292797\pi\)
0.605940 + 0.795510i \(0.292797\pi\)
\(108\) −1.93543 −0.186237
\(109\) −17.7417 −1.69935 −0.849675 0.527307i \(-0.823202\pi\)
−0.849675 + 0.527307i \(0.823202\pi\)
\(110\) −0.935432 −0.0891900
\(111\) 0.189534 0.0179898
\(112\) 0 0
\(113\) 18.1208 1.70466 0.852331 0.523003i \(-0.175188\pi\)
0.852331 + 0.523003i \(0.175188\pi\)
\(114\) −0.935432 −0.0876113
\(115\) 0 0
\(116\) −19.4712 −1.80786
\(117\) −2.18953 −0.202422
\(118\) 1.15255 0.106101
\(119\) 0 0
\(120\) 3.68133 0.336058
\(121\) 1.00000 0.0909091
\(122\) 0.0963218 0.00872057
\(123\) −8.37907 −0.755515
\(124\) −16.2171 −1.45634
\(125\) 13.0768 1.16963
\(126\) 0 0
\(127\) −10.9836 −0.974636 −0.487318 0.873224i \(-0.662025\pi\)
−0.487318 + 0.873224i \(0.662025\pi\)
\(128\) 7.48763 0.661819
\(129\) −8.37907 −0.737736
\(130\) 2.04816 0.179636
\(131\) 16.1208 1.40848 0.704240 0.709962i \(-0.251288\pi\)
0.704240 + 0.709962i \(0.251288\pi\)
\(132\) −1.93543 −0.168458
\(133\) 0 0
\(134\) −3.66075 −0.316241
\(135\) 3.68133 0.316838
\(136\) −3.36266 −0.288346
\(137\) −4.98359 −0.425777 −0.212889 0.977076i \(-0.568287\pi\)
−0.212889 + 0.977076i \(0.568287\pi\)
\(138\) 0 0
\(139\) −21.4506 −1.81942 −0.909710 0.415244i \(-0.863696\pi\)
−0.909710 + 0.415244i \(0.863696\pi\)
\(140\) 0 0
\(141\) 5.17313 0.435656
\(142\) 3.57978 0.300409
\(143\) −2.18953 −0.183098
\(144\) 3.61676 0.301397
\(145\) 37.0357 3.07564
\(146\) −3.53996 −0.292969
\(147\) 0 0
\(148\) −0.366830 −0.0301533
\(149\) 5.30226 0.434378 0.217189 0.976130i \(-0.430311\pi\)
0.217189 + 0.976130i \(0.430311\pi\)
\(150\) −2.17313 −0.177435
\(151\) 0.637339 0.0518659 0.0259329 0.999664i \(-0.491744\pi\)
0.0259329 + 0.999664i \(0.491744\pi\)
\(152\) 3.68133 0.298595
\(153\) −3.36266 −0.271855
\(154\) 0 0
\(155\) 30.8461 2.47762
\(156\) 4.23769 0.339287
\(157\) −9.70892 −0.774856 −0.387428 0.921900i \(-0.626636\pi\)
−0.387428 + 0.921900i \(0.626636\pi\)
\(158\) 0.758136 0.0603141
\(159\) −1.36266 −0.108066
\(160\) −10.7459 −0.849538
\(161\) 0 0
\(162\) −0.254102 −0.0199641
\(163\) 7.68133 0.601648 0.300824 0.953680i \(-0.402738\pi\)
0.300824 + 0.953680i \(0.402738\pi\)
\(164\) 16.2171 1.26634
\(165\) 3.68133 0.286591
\(166\) −1.11273 −0.0863645
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −8.20594 −0.631226
\(170\) 3.14554 0.241252
\(171\) 3.68133 0.281518
\(172\) 16.2171 1.23654
\(173\) −13.7089 −1.04227 −0.521135 0.853474i \(-0.674491\pi\)
−0.521135 + 0.853474i \(0.674491\pi\)
\(174\) −2.55636 −0.193797
\(175\) 0 0
\(176\) 3.61676 0.272624
\(177\) −4.53579 −0.340931
\(178\) −0.161949 −0.0121386
\(179\) −7.74173 −0.578644 −0.289322 0.957232i \(-0.593430\pi\)
−0.289322 + 0.957232i \(0.593430\pi\)
\(180\) −7.12497 −0.531064
\(181\) 20.3791 1.51476 0.757382 0.652972i \(-0.226478\pi\)
0.757382 + 0.652972i \(0.226478\pi\)
\(182\) 0 0
\(183\) −0.379068 −0.0280215
\(184\) 0 0
\(185\) 0.697737 0.0512987
\(186\) −2.12914 −0.156116
\(187\) −3.36266 −0.245902
\(188\) −10.0122 −0.730217
\(189\) 0 0
\(190\) −3.44364 −0.249828
\(191\) 9.70892 0.702512 0.351256 0.936279i \(-0.385755\pi\)
0.351256 + 0.936279i \(0.385755\pi\)
\(192\) −6.49180 −0.468505
\(193\) −20.0880 −1.44596 −0.722982 0.690866i \(-0.757229\pi\)
−0.722982 + 0.690866i \(0.757229\pi\)
\(194\) −2.38741 −0.171406
\(195\) −8.06040 −0.577217
\(196\) 0 0
\(197\) 2.25827 0.160895 0.0804476 0.996759i \(-0.474365\pi\)
0.0804476 + 0.996759i \(0.474365\pi\)
\(198\) −0.254102 −0.0180582
\(199\) −17.1372 −1.21482 −0.607412 0.794387i \(-0.707792\pi\)
−0.607412 + 0.794387i \(0.707792\pi\)
\(200\) 8.55220 0.604732
\(201\) 14.4067 1.01617
\(202\) −1.80524 −0.127016
\(203\) 0 0
\(204\) 6.50820 0.455665
\(205\) −30.8461 −2.15439
\(206\) −0.0656270 −0.00457245
\(207\) 0 0
\(208\) −7.91903 −0.549086
\(209\) 3.68133 0.254643
\(210\) 0 0
\(211\) −9.65375 −0.664591 −0.332296 0.943175i \(-0.607823\pi\)
−0.332296 + 0.943175i \(0.607823\pi\)
\(212\) 2.63734 0.181133
\(213\) −14.0880 −0.965293
\(214\) −3.18537 −0.217747
\(215\) −30.8461 −2.10369
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 4.50820 0.305334
\(219\) 13.9313 0.941388
\(220\) −7.12497 −0.480365
\(221\) 7.36266 0.495266
\(222\) −0.0481609 −0.00323235
\(223\) 1.65375 0.110743 0.0553715 0.998466i \(-0.482366\pi\)
0.0553715 + 0.998466i \(0.482366\pi\)
\(224\) 0 0
\(225\) 8.55220 0.570146
\(226\) −4.60453 −0.306288
\(227\) −4.69251 −0.311453 −0.155726 0.987800i \(-0.549772\pi\)
−0.155726 + 0.987800i \(0.549772\pi\)
\(228\) −7.12497 −0.471862
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.0604 0.660498
\(233\) −15.0164 −0.983758 −0.491879 0.870664i \(-0.663690\pi\)
−0.491879 + 0.870664i \(0.663690\pi\)
\(234\) 0.556364 0.0363707
\(235\) 19.0440 1.24229
\(236\) 8.77871 0.571445
\(237\) −2.98359 −0.193805
\(238\) 0 0
\(239\) −20.2775 −1.31164 −0.655822 0.754916i \(-0.727678\pi\)
−0.655822 + 0.754916i \(0.727678\pi\)
\(240\) 13.3145 0.859447
\(241\) −21.1731 −1.36388 −0.681941 0.731408i \(-0.738864\pi\)
−0.681941 + 0.731408i \(0.738864\pi\)
\(242\) −0.254102 −0.0163343
\(243\) 1.00000 0.0641500
\(244\) 0.733661 0.0469678
\(245\) 0 0
\(246\) 2.12914 0.135749
\(247\) −8.06040 −0.512871
\(248\) 8.37907 0.532071
\(249\) 4.37907 0.277512
\(250\) −3.32284 −0.210155
\(251\) 7.08514 0.447210 0.223605 0.974680i \(-0.428217\pi\)
0.223605 + 0.974680i \(0.428217\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.79095 0.175120
\(255\) −12.3791 −0.775207
\(256\) 11.0810 0.692561
\(257\) −23.5439 −1.46863 −0.734313 0.678811i \(-0.762495\pi\)
−0.734313 + 0.678811i \(0.762495\pi\)
\(258\) 2.12914 0.132554
\(259\) 0 0
\(260\) 15.6004 0.967493
\(261\) 10.0604 0.622723
\(262\) −4.09632 −0.253072
\(263\) −9.17313 −0.565639 −0.282820 0.959173i \(-0.591270\pi\)
−0.282820 + 0.959173i \(0.591270\pi\)
\(264\) 1.00000 0.0615457
\(265\) −5.01641 −0.308155
\(266\) 0 0
\(267\) 0.637339 0.0390045
\(268\) −27.8831 −1.70323
\(269\) −17.3955 −1.06062 −0.530310 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(270\) −0.935432 −0.0569286
\(271\) 22.1484 1.34542 0.672709 0.739907i \(-0.265131\pi\)
0.672709 + 0.739907i \(0.265131\pi\)
\(272\) −12.1619 −0.737426
\(273\) 0 0
\(274\) 1.26634 0.0765024
\(275\) 8.55220 0.515717
\(276\) 0 0
\(277\) −14.6925 −0.882787 −0.441394 0.897314i \(-0.645516\pi\)
−0.441394 + 0.897314i \(0.645516\pi\)
\(278\) 5.45065 0.326908
\(279\) 8.37907 0.501642
\(280\) 0 0
\(281\) −27.8901 −1.66378 −0.831892 0.554937i \(-0.812742\pi\)
−0.831892 + 0.554937i \(0.812742\pi\)
\(282\) −1.31450 −0.0782774
\(283\) 16.3738 0.973324 0.486662 0.873590i \(-0.338214\pi\)
0.486662 + 0.873590i \(0.338214\pi\)
\(284\) 27.2663 1.61796
\(285\) 13.5522 0.802763
\(286\) 0.556364 0.0328985
\(287\) 0 0
\(288\) −2.91903 −0.172005
\(289\) −5.69251 −0.334853
\(290\) −9.41082 −0.552623
\(291\) 9.39547 0.550772
\(292\) −26.9630 −1.57789
\(293\) 12.7581 0.745338 0.372669 0.927964i \(-0.378443\pi\)
0.372669 + 0.927964i \(0.378443\pi\)
\(294\) 0 0
\(295\) −16.6977 −0.972180
\(296\) 0.189534 0.0110164
\(297\) 1.00000 0.0580259
\(298\) −1.34731 −0.0780478
\(299\) 0 0
\(300\) −16.5522 −0.955641
\(301\) 0 0
\(302\) −0.161949 −0.00931911
\(303\) 7.10439 0.408137
\(304\) 13.3145 0.763639
\(305\) −1.39547 −0.0799047
\(306\) 0.854458 0.0488461
\(307\) 29.4506 1.68084 0.840419 0.541938i \(-0.182309\pi\)
0.840419 + 0.541938i \(0.182309\pi\)
\(308\) 0 0
\(309\) 0.258271 0.0146925
\(310\) −7.83805 −0.445171
\(311\) 12.7581 0.723448 0.361724 0.932285i \(-0.382188\pi\)
0.361724 + 0.932285i \(0.382188\pi\)
\(312\) −2.18953 −0.123958
\(313\) 18.3463 1.03699 0.518496 0.855080i \(-0.326492\pi\)
0.518496 + 0.855080i \(0.326492\pi\)
\(314\) 2.46705 0.139224
\(315\) 0 0
\(316\) 5.77454 0.324843
\(317\) 3.01641 0.169418 0.0847091 0.996406i \(-0.473004\pi\)
0.0847091 + 0.996406i \(0.473004\pi\)
\(318\) 0.346255 0.0194170
\(319\) 10.0604 0.563274
\(320\) −23.8984 −1.33596
\(321\) 12.5358 0.699679
\(322\) 0 0
\(323\) −12.3791 −0.688790
\(324\) −1.93543 −0.107524
\(325\) −18.7253 −1.03869
\(326\) −1.95184 −0.108102
\(327\) −17.7417 −0.981120
\(328\) −8.37907 −0.462657
\(329\) 0 0
\(330\) −0.935432 −0.0514939
\(331\) −14.7909 −0.812984 −0.406492 0.913654i \(-0.633248\pi\)
−0.406492 + 0.913654i \(0.633248\pi\)
\(332\) −8.47539 −0.465147
\(333\) 0.189534 0.0103864
\(334\) −4.06563 −0.222461
\(335\) 53.0357 2.89765
\(336\) 0 0
\(337\) −19.0716 −1.03890 −0.519448 0.854502i \(-0.673862\pi\)
−0.519448 + 0.854502i \(0.673862\pi\)
\(338\) 2.08514 0.113417
\(339\) 18.1208 0.984187
\(340\) 23.9588 1.29935
\(341\) 8.37907 0.453752
\(342\) −0.935432 −0.0505824
\(343\) 0 0
\(344\) −8.37907 −0.451769
\(345\) 0 0
\(346\) 3.48346 0.187272
\(347\) 9.45065 0.507337 0.253669 0.967291i \(-0.418363\pi\)
0.253669 + 0.967291i \(0.418363\pi\)
\(348\) −19.4712 −1.04377
\(349\) −28.9805 −1.55129 −0.775645 0.631170i \(-0.782575\pi\)
−0.775645 + 0.631170i \(0.782575\pi\)
\(350\) 0 0
\(351\) −2.18953 −0.116869
\(352\) −2.91903 −0.155585
\(353\) −18.7306 −0.996927 −0.498463 0.866911i \(-0.666102\pi\)
−0.498463 + 0.866911i \(0.666102\pi\)
\(354\) 1.15255 0.0612574
\(355\) −51.8625 −2.75258
\(356\) −1.23353 −0.0653767
\(357\) 0 0
\(358\) 1.96719 0.103969
\(359\) 7.48346 0.394962 0.197481 0.980307i \(-0.436724\pi\)
0.197481 + 0.980307i \(0.436724\pi\)
\(360\) 3.68133 0.194023
\(361\) −5.44780 −0.286727
\(362\) −5.17836 −0.272168
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 51.2856 2.68441
\(366\) 0.0963218 0.00503482
\(367\) 15.4283 0.805350 0.402675 0.915343i \(-0.368080\pi\)
0.402675 + 0.915343i \(0.368080\pi\)
\(368\) 0 0
\(369\) −8.37907 −0.436197
\(370\) −0.177296 −0.00921719
\(371\) 0 0
\(372\) −16.2171 −0.840818
\(373\) −26.1208 −1.35248 −0.676242 0.736680i \(-0.736393\pi\)
−0.676242 + 0.736680i \(0.736393\pi\)
\(374\) 0.854458 0.0441830
\(375\) 13.0768 0.675283
\(376\) 5.17313 0.266784
\(377\) −22.0276 −1.13448
\(378\) 0 0
\(379\) −13.6485 −0.701077 −0.350539 0.936548i \(-0.614001\pi\)
−0.350539 + 0.936548i \(0.614001\pi\)
\(380\) −26.2294 −1.34554
\(381\) −10.9836 −0.562707
\(382\) −2.46705 −0.126225
\(383\) 25.4506 1.30047 0.650234 0.759734i \(-0.274671\pi\)
0.650234 + 0.759734i \(0.274671\pi\)
\(384\) 7.48763 0.382101
\(385\) 0 0
\(386\) 5.10439 0.259807
\(387\) −8.37907 −0.425932
\(388\) −18.1843 −0.923168
\(389\) −24.7909 −1.25695 −0.628476 0.777829i \(-0.716321\pi\)
−0.628476 + 0.777829i \(0.716321\pi\)
\(390\) 2.04816 0.103713
\(391\) 0 0
\(392\) 0 0
\(393\) 16.1208 0.813187
\(394\) −0.573830 −0.0289092
\(395\) −10.9836 −0.552645
\(396\) −1.93543 −0.0972591
\(397\) −19.8625 −0.996872 −0.498436 0.866927i \(-0.666092\pi\)
−0.498436 + 0.866927i \(0.666092\pi\)
\(398\) 4.35459 0.218276
\(399\) 0 0
\(400\) 30.9313 1.54656
\(401\) 5.24186 0.261766 0.130883 0.991398i \(-0.458219\pi\)
0.130883 + 0.991398i \(0.458219\pi\)
\(402\) −3.66075 −0.182582
\(403\) −18.3463 −0.913892
\(404\) −13.7501 −0.684091
\(405\) 3.68133 0.182927
\(406\) 0 0
\(407\) 0.189534 0.00939485
\(408\) −3.36266 −0.166477
\(409\) 5.07158 0.250773 0.125387 0.992108i \(-0.459983\pi\)
0.125387 + 0.992108i \(0.459983\pi\)
\(410\) 7.83805 0.387094
\(411\) −4.98359 −0.245823
\(412\) −0.499865 −0.0246266
\(413\) 0 0
\(414\) 0 0
\(415\) 16.1208 0.791339
\(416\) 6.39131 0.313360
\(417\) −21.4506 −1.05044
\(418\) −0.935432 −0.0457535
\(419\) −6.25516 −0.305585 −0.152792 0.988258i \(-0.548827\pi\)
−0.152792 + 0.988258i \(0.548827\pi\)
\(420\) 0 0
\(421\) −10.2223 −0.498207 −0.249103 0.968477i \(-0.580136\pi\)
−0.249103 + 0.968477i \(0.580136\pi\)
\(422\) 2.45303 0.119412
\(423\) 5.17313 0.251526
\(424\) −1.36266 −0.0661767
\(425\) −28.7581 −1.39497
\(426\) 3.57978 0.173441
\(427\) 0 0
\(428\) −24.2622 −1.17276
\(429\) −2.18953 −0.105712
\(430\) 7.83805 0.377984
\(431\) 2.76125 0.133005 0.0665023 0.997786i \(-0.478816\pi\)
0.0665023 + 0.997786i \(0.478816\pi\)
\(432\) 3.61676 0.174012
\(433\) 27.7417 1.33318 0.666591 0.745423i \(-0.267753\pi\)
0.666591 + 0.745423i \(0.267753\pi\)
\(434\) 0 0
\(435\) 37.0357 1.77572
\(436\) 34.3379 1.64449
\(437\) 0 0
\(438\) −3.53996 −0.169146
\(439\) −35.4782 −1.69328 −0.846642 0.532163i \(-0.821379\pi\)
−0.846642 + 0.532163i \(0.821379\pi\)
\(440\) 3.68133 0.175501
\(441\) 0 0
\(442\) −1.87086 −0.0889880
\(443\) −3.68656 −0.175154 −0.0875769 0.996158i \(-0.527912\pi\)
−0.0875769 + 0.996158i \(0.527912\pi\)
\(444\) −0.366830 −0.0174090
\(445\) 2.34625 0.111223
\(446\) −0.420220 −0.0198980
\(447\) 5.30226 0.250788
\(448\) 0 0
\(449\) 4.22546 0.199412 0.0997058 0.995017i \(-0.468210\pi\)
0.0997058 + 0.995017i \(0.468210\pi\)
\(450\) −2.17313 −0.102442
\(451\) −8.37907 −0.394555
\(452\) −35.0716 −1.64963
\(453\) 0.637339 0.0299448
\(454\) 1.19237 0.0559609
\(455\) 0 0
\(456\) 3.68133 0.172394
\(457\) 28.5878 1.33728 0.668642 0.743585i \(-0.266876\pi\)
0.668642 + 0.743585i \(0.266876\pi\)
\(458\) 2.03281 0.0949871
\(459\) −3.36266 −0.156956
\(460\) 0 0
\(461\) −36.8789 −1.71762 −0.858812 0.512292i \(-0.828797\pi\)
−0.858812 + 0.512292i \(0.828797\pi\)
\(462\) 0 0
\(463\) −38.2692 −1.77852 −0.889260 0.457402i \(-0.848780\pi\)
−0.889260 + 0.457402i \(0.848780\pi\)
\(464\) 36.3861 1.68918
\(465\) 30.8461 1.43045
\(466\) 3.81569 0.176759
\(467\) 22.8820 1.05885 0.529427 0.848355i \(-0.322407\pi\)
0.529427 + 0.848355i \(0.322407\pi\)
\(468\) 4.23769 0.195888
\(469\) 0 0
\(470\) −4.83911 −0.223212
\(471\) −9.70892 −0.447363
\(472\) −4.53579 −0.208777
\(473\) −8.37907 −0.385270
\(474\) 0.758136 0.0348223
\(475\) 31.4835 1.44456
\(476\) 0 0
\(477\) −1.36266 −0.0623920
\(478\) 5.15255 0.235672
\(479\) −5.77454 −0.263846 −0.131923 0.991260i \(-0.542115\pi\)
−0.131923 + 0.991260i \(0.542115\pi\)
\(480\) −10.7459 −0.490481
\(481\) −0.414991 −0.0189220
\(482\) 5.38013 0.245058
\(483\) 0 0
\(484\) −1.93543 −0.0879742
\(485\) 34.5878 1.57055
\(486\) −0.254102 −0.0115263
\(487\) −12.6925 −0.575152 −0.287576 0.957758i \(-0.592849\pi\)
−0.287576 + 0.957758i \(0.592849\pi\)
\(488\) −0.379068 −0.0171596
\(489\) 7.68133 0.347362
\(490\) 0 0
\(491\) 20.0192 0.903456 0.451728 0.892156i \(-0.350808\pi\)
0.451728 + 0.892156i \(0.350808\pi\)
\(492\) 16.2171 0.731124
\(493\) −33.8297 −1.52361
\(494\) 2.04816 0.0921511
\(495\) 3.68133 0.165464
\(496\) 30.3051 1.36074
\(497\) 0 0
\(498\) −1.11273 −0.0498626
\(499\) 17.0768 0.764463 0.382231 0.924067i \(-0.375156\pi\)
0.382231 + 0.924067i \(0.375156\pi\)
\(500\) −25.3093 −1.13187
\(501\) 16.0000 0.714827
\(502\) −1.80035 −0.0803534
\(503\) 0.313441 0.0139756 0.00698782 0.999976i \(-0.497776\pi\)
0.00698782 + 0.999976i \(0.497776\pi\)
\(504\) 0 0
\(505\) 26.1536 1.16382
\(506\) 0 0
\(507\) −8.20594 −0.364439
\(508\) 21.2580 0.943171
\(509\) −16.1208 −0.714542 −0.357271 0.934001i \(-0.616293\pi\)
−0.357271 + 0.934001i \(0.616293\pi\)
\(510\) 3.14554 0.139287
\(511\) 0 0
\(512\) −17.7909 −0.786256
\(513\) 3.68133 0.162535
\(514\) 5.98253 0.263878
\(515\) 0.950780 0.0418964
\(516\) 16.2171 0.713919
\(517\) 5.17313 0.227514
\(518\) 0 0
\(519\) −13.7089 −0.601755
\(520\) −8.06040 −0.353472
\(521\) −19.9948 −0.875987 −0.437993 0.898978i \(-0.644311\pi\)
−0.437993 + 0.898978i \(0.644311\pi\)
\(522\) −2.55636 −0.111889
\(523\) −6.98882 −0.305600 −0.152800 0.988257i \(-0.548829\pi\)
−0.152800 + 0.988257i \(0.548829\pi\)
\(524\) −31.2007 −1.36301
\(525\) 0 0
\(526\) 2.33091 0.101632
\(527\) −28.1760 −1.22736
\(528\) 3.61676 0.157399
\(529\) −23.0000 −1.00000
\(530\) 1.27468 0.0553684
\(531\) −4.53579 −0.196836
\(532\) 0 0
\(533\) 18.3463 0.794664
\(534\) −0.161949 −0.00700821
\(535\) 46.1484 1.99517
\(536\) 14.4067 0.622273
\(537\) −7.74173 −0.334080
\(538\) 4.42022 0.190569
\(539\) 0 0
\(540\) −7.12497 −0.306610
\(541\) 7.08203 0.304480 0.152240 0.988344i \(-0.451351\pi\)
0.152240 + 0.988344i \(0.451351\pi\)
\(542\) −5.62794 −0.241741
\(543\) 20.3791 0.874550
\(544\) 9.81569 0.420844
\(545\) −65.3132 −2.79771
\(546\) 0 0
\(547\) 31.5386 1.34849 0.674247 0.738506i \(-0.264468\pi\)
0.674247 + 0.738506i \(0.264468\pi\)
\(548\) 9.64541 0.412031
\(549\) −0.379068 −0.0161782
\(550\) −2.17313 −0.0926625
\(551\) 37.0357 1.57777
\(552\) 0 0
\(553\) 0 0
\(554\) 3.73339 0.158617
\(555\) 0.697737 0.0296173
\(556\) 41.5163 1.76068
\(557\) −14.8737 −0.630219 −0.315110 0.949055i \(-0.602041\pi\)
−0.315110 + 0.949055i \(0.602041\pi\)
\(558\) −2.12914 −0.0901335
\(559\) 18.3463 0.775964
\(560\) 0 0
\(561\) −3.36266 −0.141972
\(562\) 7.08692 0.298944
\(563\) −37.8953 −1.59710 −0.798549 0.601930i \(-0.794399\pi\)
−0.798549 + 0.601930i \(0.794399\pi\)
\(564\) −10.0122 −0.421591
\(565\) 66.7086 2.80645
\(566\) −4.16062 −0.174884
\(567\) 0 0
\(568\) −14.0880 −0.591119
\(569\) −41.9177 −1.75728 −0.878641 0.477484i \(-0.841549\pi\)
−0.878641 + 0.477484i \(0.841549\pi\)
\(570\) −3.44364 −0.144238
\(571\) −28.1208 −1.17682 −0.588409 0.808563i \(-0.700246\pi\)
−0.588409 + 0.808563i \(0.700246\pi\)
\(572\) 4.23769 0.177187
\(573\) 9.70892 0.405596
\(574\) 0 0
\(575\) 0 0
\(576\) −6.49180 −0.270492
\(577\) −47.6043 −1.98179 −0.990896 0.134633i \(-0.957015\pi\)
−0.990896 + 0.134633i \(0.957015\pi\)
\(578\) 1.44648 0.0601655
\(579\) −20.0880 −0.834828
\(580\) −71.6800 −2.97635
\(581\) 0 0
\(582\) −2.38741 −0.0989612
\(583\) −1.36266 −0.0564357
\(584\) 13.9313 0.576480
\(585\) −8.06040 −0.333256
\(586\) −3.24186 −0.133920
\(587\) 39.3267 1.62319 0.811594 0.584222i \(-0.198600\pi\)
0.811594 + 0.584222i \(0.198600\pi\)
\(588\) 0 0
\(589\) 30.8461 1.27099
\(590\) 4.24292 0.174678
\(591\) 2.25827 0.0928928
\(592\) 0.685500 0.0281739
\(593\) 29.3955 1.20713 0.603564 0.797315i \(-0.293747\pi\)
0.603564 + 0.797315i \(0.293747\pi\)
\(594\) −0.254102 −0.0104259
\(595\) 0 0
\(596\) −10.2622 −0.420355
\(597\) −17.1372 −0.701379
\(598\) 0 0
\(599\) 26.4671 1.08141 0.540707 0.841211i \(-0.318157\pi\)
0.540707 + 0.841211i \(0.318157\pi\)
\(600\) 8.55220 0.349142
\(601\) −33.5522 −1.36862 −0.684311 0.729190i \(-0.739897\pi\)
−0.684311 + 0.729190i \(0.739897\pi\)
\(602\) 0 0
\(603\) 14.4067 0.586684
\(604\) −1.23353 −0.0501915
\(605\) 3.68133 0.149667
\(606\) −1.80524 −0.0733328
\(607\) 17.9724 0.729478 0.364739 0.931110i \(-0.381158\pi\)
0.364739 + 0.931110i \(0.381158\pi\)
\(608\) −10.7459 −0.435804
\(609\) 0 0
\(610\) 0.354593 0.0143570
\(611\) −11.3267 −0.458231
\(612\) 6.50820 0.263079
\(613\) 28.5878 1.15465 0.577326 0.816514i \(-0.304096\pi\)
0.577326 + 0.816514i \(0.304096\pi\)
\(614\) −7.48346 −0.302008
\(615\) −30.8461 −1.24384
\(616\) 0 0
\(617\) −31.0164 −1.24867 −0.624337 0.781155i \(-0.714631\pi\)
−0.624337 + 0.781155i \(0.714631\pi\)
\(618\) −0.0656270 −0.00263991
\(619\) −11.4835 −0.461559 −0.230780 0.973006i \(-0.574128\pi\)
−0.230780 + 0.973006i \(0.574128\pi\)
\(620\) −59.7006 −2.39763
\(621\) 0 0
\(622\) −3.24186 −0.129987
\(623\) 0 0
\(624\) −7.91903 −0.317015
\(625\) 5.37907 0.215163
\(626\) −4.66181 −0.186324
\(627\) 3.68133 0.147018
\(628\) 18.7909 0.749841
\(629\) −0.637339 −0.0254124
\(630\) 0 0
\(631\) −10.6597 −0.424356 −0.212178 0.977231i \(-0.568056\pi\)
−0.212178 + 0.977231i \(0.568056\pi\)
\(632\) −2.98359 −0.118681
\(633\) −9.65375 −0.383702
\(634\) −0.766474 −0.0304406
\(635\) −40.4342 −1.60458
\(636\) 2.63734 0.104577
\(637\) 0 0
\(638\) −2.55636 −0.101207
\(639\) −14.0880 −0.557312
\(640\) 27.5644 1.08958
\(641\) 2.69251 0.106348 0.0531739 0.998585i \(-0.483066\pi\)
0.0531739 + 0.998585i \(0.483066\pi\)
\(642\) −3.18537 −0.125716
\(643\) 40.3791 1.59239 0.796197 0.605037i \(-0.206842\pi\)
0.796197 + 0.605037i \(0.206842\pi\)
\(644\) 0 0
\(645\) −30.8461 −1.21456
\(646\) 3.14554 0.123760
\(647\) 18.0687 0.710355 0.355178 0.934799i \(-0.384420\pi\)
0.355178 + 0.934799i \(0.384420\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.53579 −0.178045
\(650\) 4.75814 0.186629
\(651\) 0 0
\(652\) −14.8667 −0.582225
\(653\) 29.1044 1.13894 0.569471 0.822011i \(-0.307148\pi\)
0.569471 + 0.822011i \(0.307148\pi\)
\(654\) 4.50820 0.176285
\(655\) 59.3460 2.31884
\(656\) −30.3051 −1.18322
\(657\) 13.9313 0.543510
\(658\) 0 0
\(659\) 38.5686 1.50242 0.751210 0.660064i \(-0.229471\pi\)
0.751210 + 0.660064i \(0.229471\pi\)
\(660\) −7.12497 −0.277339
\(661\) 23.1596 0.900803 0.450401 0.892826i \(-0.351281\pi\)
0.450401 + 0.892826i \(0.351281\pi\)
\(662\) 3.75841 0.146075
\(663\) 7.36266 0.285942
\(664\) 4.37907 0.169941
\(665\) 0 0
\(666\) −0.0481609 −0.00186620
\(667\) 0 0
\(668\) −30.9669 −1.19815
\(669\) 1.65375 0.0639375
\(670\) −13.4764 −0.520641
\(671\) −0.379068 −0.0146338
\(672\) 0 0
\(673\) −29.5491 −1.13903 −0.569517 0.821980i \(-0.692870\pi\)
−0.569517 + 0.821980i \(0.692870\pi\)
\(674\) 4.84612 0.186666
\(675\) 8.55220 0.329174
\(676\) 15.8820 0.610848
\(677\) 37.9505 1.45856 0.729278 0.684217i \(-0.239856\pi\)
0.729278 + 0.684217i \(0.239856\pi\)
\(678\) −4.60453 −0.176836
\(679\) 0 0
\(680\) −12.3791 −0.474716
\(681\) −4.69251 −0.179817
\(682\) −2.12914 −0.0815288
\(683\) −42.9669 −1.64408 −0.822042 0.569427i \(-0.807165\pi\)
−0.822042 + 0.569427i \(0.807165\pi\)
\(684\) −7.12497 −0.272430
\(685\) −18.3463 −0.700974
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) −30.3051 −1.15537
\(689\) 2.98359 0.113666
\(690\) 0 0
\(691\) −18.0880 −0.688099 −0.344050 0.938951i \(-0.611799\pi\)
−0.344050 + 0.938951i \(0.611799\pi\)
\(692\) 26.5327 1.00862
\(693\) 0 0
\(694\) −2.40142 −0.0911568
\(695\) −78.9669 −2.99539
\(696\) 10.0604 0.381338
\(697\) 28.1760 1.06724
\(698\) 7.36399 0.278731
\(699\) −15.0164 −0.567973
\(700\) 0 0
\(701\) −16.4671 −0.621952 −0.310976 0.950418i \(-0.600656\pi\)
−0.310976 + 0.950418i \(0.600656\pi\)
\(702\) 0.556364 0.0209986
\(703\) 0.697737 0.0263157
\(704\) −6.49180 −0.244669
\(705\) 19.0440 0.717238
\(706\) 4.75946 0.179125
\(707\) 0 0
\(708\) 8.77871 0.329924
\(709\) 13.1955 0.495567 0.247783 0.968815i \(-0.420298\pi\)
0.247783 + 0.968815i \(0.420298\pi\)
\(710\) 13.1784 0.494575
\(711\) −2.98359 −0.111893
\(712\) 0.637339 0.0238853
\(713\) 0 0
\(714\) 0 0
\(715\) −8.06040 −0.301442
\(716\) 14.9836 0.559963
\(717\) −20.2775 −0.757278
\(718\) −1.90156 −0.0709656
\(719\) 41.7938 1.55865 0.779323 0.626623i \(-0.215563\pi\)
0.779323 + 0.626623i \(0.215563\pi\)
\(720\) 13.3145 0.496202
\(721\) 0 0
\(722\) 1.38430 0.0515182
\(723\) −21.1731 −0.787437
\(724\) −39.4423 −1.46586
\(725\) 86.0385 3.19539
\(726\) −0.254102 −0.00943060
\(727\) −46.9013 −1.73947 −0.869736 0.493517i \(-0.835711\pi\)
−0.869736 + 0.493517i \(0.835711\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −13.0318 −0.482327
\(731\) 28.1760 1.04213
\(732\) 0.733661 0.0271169
\(733\) −5.07158 −0.187323 −0.0936615 0.995604i \(-0.529857\pi\)
−0.0936615 + 0.995604i \(0.529857\pi\)
\(734\) −3.92035 −0.144703
\(735\) 0 0
\(736\) 0 0
\(737\) 14.4067 0.530676
\(738\) 2.12914 0.0783746
\(739\) −23.4178 −0.861439 −0.430719 0.902486i \(-0.641740\pi\)
−0.430719 + 0.902486i \(0.641740\pi\)
\(740\) −1.35042 −0.0496426
\(741\) −8.06040 −0.296106
\(742\) 0 0
\(743\) −11.2059 −0.411106 −0.205553 0.978646i \(-0.565899\pi\)
−0.205553 + 0.978646i \(0.565899\pi\)
\(744\) 8.37907 0.307192
\(745\) 19.5194 0.715135
\(746\) 6.63734 0.243010
\(747\) 4.37907 0.160222
\(748\) 6.50820 0.237963
\(749\) 0 0
\(750\) −3.32284 −0.121333
\(751\) 44.2468 1.61459 0.807295 0.590148i \(-0.200931\pi\)
0.807295 + 0.590148i \(0.200931\pi\)
\(752\) 18.7100 0.682283
\(753\) 7.08514 0.258197
\(754\) 5.59725 0.203840
\(755\) 2.34625 0.0853889
\(756\) 0 0
\(757\) 20.2552 0.736186 0.368093 0.929789i \(-0.380011\pi\)
0.368093 + 0.929789i \(0.380011\pi\)
\(758\) 3.46811 0.125967
\(759\) 0 0
\(760\) 13.5522 0.491590
\(761\) 41.7745 1.51433 0.757163 0.653226i \(-0.226585\pi\)
0.757163 + 0.653226i \(0.226585\pi\)
\(762\) 2.79095 0.101105
\(763\) 0 0
\(764\) −18.7909 −0.679833
\(765\) −12.3791 −0.447566
\(766\) −6.46705 −0.233664
\(767\) 9.93126 0.358597
\(768\) 11.0810 0.399850
\(769\) −0.997160 −0.0359585 −0.0179792 0.999838i \(-0.505723\pi\)
−0.0179792 + 0.999838i \(0.505723\pi\)
\(770\) 0 0
\(771\) −23.5439 −0.847911
\(772\) 38.8789 1.39928
\(773\) 45.8245 1.64819 0.824096 0.566450i \(-0.191684\pi\)
0.824096 + 0.566450i \(0.191684\pi\)
\(774\) 2.12914 0.0765302
\(775\) 71.6594 2.57408
\(776\) 9.39547 0.337278
\(777\) 0 0
\(778\) 6.29942 0.225845
\(779\) −30.8461 −1.10518
\(780\) 15.6004 0.558582
\(781\) −14.0880 −0.504108
\(782\) 0 0
\(783\) 10.0604 0.359529
\(784\) 0 0
\(785\) −35.7417 −1.27568
\(786\) −4.09632 −0.146111
\(787\) 19.8021 0.705870 0.352935 0.935648i \(-0.385184\pi\)
0.352935 + 0.935648i \(0.385184\pi\)
\(788\) −4.37073 −0.155701
\(789\) −9.17313 −0.326572
\(790\) 2.79095 0.0992975
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 0.829982 0.0294735
\(794\) 5.04710 0.179115
\(795\) −5.01641 −0.177914
\(796\) 33.1679 1.17561
\(797\) 29.8349 1.05681 0.528404 0.848993i \(-0.322791\pi\)
0.528404 + 0.848993i \(0.322791\pi\)
\(798\) 0 0
\(799\) −17.3955 −0.615408
\(800\) −24.9641 −0.882613
\(801\) 0.637339 0.0225193
\(802\) −1.33197 −0.0470334
\(803\) 13.9313 0.491624
\(804\) −27.8831 −0.983361
\(805\) 0 0
\(806\) 4.66181 0.164205
\(807\) −17.3955 −0.612350
\(808\) 7.10439 0.249932
\(809\) 19.0112 0.668397 0.334199 0.942503i \(-0.391534\pi\)
0.334199 + 0.942503i \(0.391534\pi\)
\(810\) −0.935432 −0.0328677
\(811\) −20.4395 −0.717727 −0.358863 0.933390i \(-0.616836\pi\)
−0.358863 + 0.933390i \(0.616836\pi\)
\(812\) 0 0
\(813\) 22.1484 0.776778
\(814\) −0.0481609 −0.00168804
\(815\) 28.2775 0.990518
\(816\) −12.1619 −0.425753
\(817\) −30.8461 −1.07917
\(818\) −1.28870 −0.0450582
\(819\) 0 0
\(820\) 59.7006 2.08483
\(821\) −28.0932 −0.980460 −0.490230 0.871593i \(-0.663087\pi\)
−0.490230 + 0.871593i \(0.663087\pi\)
\(822\) 1.26634 0.0441687
\(823\) −28.1260 −0.980412 −0.490206 0.871607i \(-0.663078\pi\)
−0.490206 + 0.871607i \(0.663078\pi\)
\(824\) 0.258271 0.00899729
\(825\) 8.55220 0.297749
\(826\) 0 0
\(827\) −0.914857 −0.0318127 −0.0159063 0.999873i \(-0.505063\pi\)
−0.0159063 + 0.999873i \(0.505063\pi\)
\(828\) 0 0
\(829\) 29.6922 1.03125 0.515627 0.856813i \(-0.327559\pi\)
0.515627 + 0.856813i \(0.327559\pi\)
\(830\) −4.09632 −0.142185
\(831\) −14.6925 −0.509677
\(832\) 14.2140 0.492782
\(833\) 0 0
\(834\) 5.45065 0.188740
\(835\) 58.9013 2.03836
\(836\) −7.12497 −0.246422
\(837\) 8.37907 0.289623
\(838\) 1.58945 0.0549065
\(839\) −32.2775 −1.11434 −0.557172 0.830397i \(-0.688114\pi\)
−0.557172 + 0.830397i \(0.688114\pi\)
\(840\) 0 0
\(841\) 72.2116 2.49006
\(842\) 2.59752 0.0895163
\(843\) −27.8901 −0.960586
\(844\) 18.6842 0.643136
\(845\) −30.2088 −1.03921
\(846\) −1.31450 −0.0451935
\(847\) 0 0
\(848\) −4.92842 −0.169243
\(849\) 16.3738 0.561949
\(850\) 7.30749 0.250645
\(851\) 0 0
\(852\) 27.2663 0.934129
\(853\) −9.65375 −0.330538 −0.165269 0.986249i \(-0.552849\pi\)
−0.165269 + 0.986249i \(0.552849\pi\)
\(854\) 0 0
\(855\) 13.5522 0.463475
\(856\) 12.5358 0.428464
\(857\) 17.1372 0.585396 0.292698 0.956205i \(-0.405447\pi\)
0.292698 + 0.956205i \(0.405447\pi\)
\(858\) 0.556364 0.0189940
\(859\) −29.7089 −1.01366 −0.506828 0.862047i \(-0.669182\pi\)
−0.506828 + 0.862047i \(0.669182\pi\)
\(860\) 59.7006 2.03577
\(861\) 0 0
\(862\) −0.701637 −0.0238979
\(863\) −49.0716 −1.67042 −0.835208 0.549934i \(-0.814653\pi\)
−0.835208 + 0.549934i \(0.814653\pi\)
\(864\) −2.91903 −0.0993073
\(865\) −50.4671 −1.71593
\(866\) −7.04922 −0.239542
\(867\) −5.69251 −0.193328
\(868\) 0 0
\(869\) −2.98359 −0.101211
\(870\) −9.41082 −0.319057
\(871\) −31.5439 −1.06882
\(872\) −17.7417 −0.600811
\(873\) 9.39547 0.317989
\(874\) 0 0
\(875\) 0 0
\(876\) −26.9630 −0.910996
\(877\) −26.3791 −0.890758 −0.445379 0.895342i \(-0.646931\pi\)
−0.445379 + 0.895342i \(0.646931\pi\)
\(878\) 9.01508 0.304244
\(879\) 12.7581 0.430321
\(880\) 13.3145 0.448832
\(881\) 16.9393 0.570701 0.285350 0.958423i \(-0.407890\pi\)
0.285350 + 0.958423i \(0.407890\pi\)
\(882\) 0 0
\(883\) 33.0768 1.11312 0.556562 0.830806i \(-0.312120\pi\)
0.556562 + 0.830806i \(0.312120\pi\)
\(884\) −14.2499 −0.479277
\(885\) −16.6977 −0.561288
\(886\) 0.936761 0.0314711
\(887\) 0.516541 0.0173438 0.00867188 0.999962i \(-0.497240\pi\)
0.00867188 + 0.999962i \(0.497240\pi\)
\(888\) 0.189534 0.00636035
\(889\) 0 0
\(890\) −0.596187 −0.0199842
\(891\) 1.00000 0.0335013
\(892\) −3.20071 −0.107168
\(893\) 19.0440 0.637283
\(894\) −1.34731 −0.0450609
\(895\) −28.4999 −0.952645
\(896\) 0 0
\(897\) 0 0
\(898\) −1.07370 −0.0358297
\(899\) 84.2968 2.81145
\(900\) −16.5522 −0.551740
\(901\) 4.58217 0.152654
\(902\) 2.12914 0.0708925
\(903\) 0 0
\(904\) 18.1208 0.602689
\(905\) 75.0221 2.49382
\(906\) −0.161949 −0.00538039
\(907\) −2.72532 −0.0904929 −0.0452464 0.998976i \(-0.514407\pi\)
−0.0452464 + 0.998976i \(0.514407\pi\)
\(908\) 9.08203 0.301398
\(909\) 7.10439 0.235638
\(910\) 0 0
\(911\) 42.4671 1.40700 0.703498 0.710697i \(-0.251620\pi\)
0.703498 + 0.710697i \(0.251620\pi\)
\(912\) 13.3145 0.440887
\(913\) 4.37907 0.144926
\(914\) −7.26422 −0.240279
\(915\) −1.39547 −0.0461330
\(916\) 15.4835 0.511588
\(917\) 0 0
\(918\) 0.854458 0.0282013
\(919\) 7.56576 0.249571 0.124786 0.992184i \(-0.460176\pi\)
0.124786 + 0.992184i \(0.460176\pi\)
\(920\) 0 0
\(921\) 29.4506 0.970432
\(922\) 9.37100 0.308617
\(923\) 30.8461 1.01531
\(924\) 0 0
\(925\) 1.62093 0.0532959
\(926\) 9.72426 0.319559
\(927\) 0.258271 0.00848272
\(928\) −29.3666 −0.964005
\(929\) 9.33508 0.306274 0.153137 0.988205i \(-0.451062\pi\)
0.153137 + 0.988205i \(0.451062\pi\)
\(930\) −7.83805 −0.257020
\(931\) 0 0
\(932\) 29.0632 0.951998
\(933\) 12.7581 0.417683
\(934\) −5.81437 −0.190252
\(935\) −12.3791 −0.404839
\(936\) −2.18953 −0.0715672
\(937\) 0.379068 0.0123836 0.00619181 0.999981i \(-0.498029\pi\)
0.00619181 + 0.999981i \(0.498029\pi\)
\(938\) 0 0
\(939\) 18.3463 0.598707
\(940\) −36.8584 −1.20219
\(941\) 50.6430 1.65092 0.825458 0.564464i \(-0.190917\pi\)
0.825458 + 0.564464i \(0.190917\pi\)
\(942\) 2.46705 0.0803809
\(943\) 0 0
\(944\) −16.4049 −0.533933
\(945\) 0 0
\(946\) 2.12914 0.0692242
\(947\) 42.6535 1.38605 0.693026 0.720913i \(-0.256277\pi\)
0.693026 + 0.720913i \(0.256277\pi\)
\(948\) 5.77454 0.187548
\(949\) −30.5030 −0.990168
\(950\) −8.00000 −0.259554
\(951\) 3.01641 0.0978137
\(952\) 0 0
\(953\) 14.5603 0.471653 0.235827 0.971795i \(-0.424220\pi\)
0.235827 + 0.971795i \(0.424220\pi\)
\(954\) 0.346255 0.0112104
\(955\) 35.7417 1.15658
\(956\) 39.2458 1.26930
\(957\) 10.0604 0.325207
\(958\) 1.46732 0.0474070
\(959\) 0 0
\(960\) −23.8984 −0.771319
\(961\) 39.2088 1.26480
\(962\) 0.105450 0.00339984
\(963\) 12.5358 0.403960
\(964\) 40.9792 1.31985
\(965\) −73.9505 −2.38055
\(966\) 0 0
\(967\) 10.4671 0.336598 0.168299 0.985736i \(-0.446173\pi\)
0.168299 + 0.985736i \(0.446173\pi\)
\(968\) 1.00000 0.0321412
\(969\) −12.3791 −0.397673
\(970\) −8.78883 −0.282192
\(971\) −5.49702 −0.176408 −0.0882040 0.996102i \(-0.528113\pi\)
−0.0882040 + 0.996102i \(0.528113\pi\)
\(972\) −1.93543 −0.0620790
\(973\) 0 0
\(974\) 3.22519 0.103342
\(975\) −18.7253 −0.599690
\(976\) −1.37100 −0.0438846
\(977\) 39.2028 1.25421 0.627105 0.778935i \(-0.284240\pi\)
0.627105 + 0.778935i \(0.284240\pi\)
\(978\) −1.95184 −0.0624129
\(979\) 0.637339 0.0203694
\(980\) 0 0
\(981\) −17.7417 −0.566450
\(982\) −5.08692 −0.162330
\(983\) 18.7253 0.597245 0.298623 0.954371i \(-0.403473\pi\)
0.298623 + 0.954371i \(0.403473\pi\)
\(984\) −8.37907 −0.267115
\(985\) 8.31344 0.264888
\(986\) 8.59619 0.273758
\(987\) 0 0
\(988\) 15.6004 0.496313
\(989\) 0 0
\(990\) −0.935432 −0.0297300
\(991\) −18.8513 −0.598833 −0.299416 0.954123i \(-0.596792\pi\)
−0.299416 + 0.954123i \(0.596792\pi\)
\(992\) −24.4587 −0.776565
\(993\) −14.7909 −0.469377
\(994\) 0 0
\(995\) −63.0877 −2.00002
\(996\) −8.47539 −0.268553
\(997\) −13.0716 −0.413981 −0.206991 0.978343i \(-0.566367\pi\)
−0.206991 + 0.978343i \(0.566367\pi\)
\(998\) −4.33925 −0.137356
\(999\) 0.189534 0.00599659
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 1617.2.a.r.1.2 yes 3
3.2 odd 2 4851.2.a.bl.1.2 3
7.6 odd 2 1617.2.a.q.1.2 3
21.20 even 2 4851.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.2 3 7.6 odd 2
1617.2.a.r.1.2 yes 3 1.1 even 1 trivial
4851.2.a.bl.1.2 3 3.2 odd 2
4851.2.a.bm.1.2 3 21.20 even 2