Properties

Label 5415.2.a.n.1.2
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(0\)
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: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +6.24264 q^{11} +1.82843 q^{12} +0.585786 q^{13} +0.585786 q^{14} +1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} +0.414214 q^{18} +1.82843 q^{20} -1.41421 q^{21} +2.58579 q^{22} -3.65685 q^{23} +1.58579 q^{24} +1.00000 q^{25} +0.242641 q^{26} -1.00000 q^{27} -2.58579 q^{28} +1.41421 q^{29} +0.414214 q^{30} +8.82843 q^{31} +4.41421 q^{32} -6.24264 q^{33} +2.82843 q^{34} -1.41421 q^{35} -1.82843 q^{36} +0.585786 q^{37} -0.585786 q^{39} +1.58579 q^{40} -8.24264 q^{41} -0.585786 q^{42} +3.75736 q^{43} -11.4142 q^{44} -1.00000 q^{45} -1.51472 q^{46} +3.65685 q^{47} -3.00000 q^{48} -5.00000 q^{49} +0.414214 q^{50} -6.82843 q^{51} -1.07107 q^{52} -8.00000 q^{53} -0.414214 q^{54} -6.24264 q^{55} -2.24264 q^{56} +0.585786 q^{58} +4.48528 q^{59} -1.82843 q^{60} -15.3137 q^{61} +3.65685 q^{62} +1.41421 q^{63} -4.17157 q^{64} -0.585786 q^{65} -2.58579 q^{66} -1.65685 q^{67} -12.4853 q^{68} +3.65685 q^{69} -0.585786 q^{70} +5.17157 q^{71} -1.58579 q^{72} +3.65685 q^{73} +0.242641 q^{74} -1.00000 q^{75} +8.82843 q^{77} -0.242641 q^{78} -3.00000 q^{80} +1.00000 q^{81} -3.41421 q^{82} +7.17157 q^{83} +2.58579 q^{84} -6.82843 q^{85} +1.55635 q^{86} -1.41421 q^{87} -9.89949 q^{88} +13.8995 q^{89} -0.414214 q^{90} +0.828427 q^{91} +6.68629 q^{92} -8.82843 q^{93} +1.51472 q^{94} -4.41421 q^{96} +18.2426 q^{97} -2.07107 q^{98} +6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{11} - 2 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 6 q^{16} + 8 q^{17} - 2 q^{18} - 2 q^{20} + 8 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 8 q^{26} - 2 q^{27} - 8 q^{28} - 2 q^{30} + 12 q^{31} + 6 q^{32} - 4 q^{33} + 2 q^{36} + 4 q^{37} - 4 q^{39} + 6 q^{40} - 8 q^{41} - 4 q^{42} + 16 q^{43} - 20 q^{44} - 2 q^{45} - 20 q^{46} - 4 q^{47} - 6 q^{48} - 10 q^{49} - 2 q^{50} - 8 q^{51} + 12 q^{52} - 16 q^{53} + 2 q^{54} - 4 q^{55} + 4 q^{56} + 4 q^{58} - 8 q^{59} + 2 q^{60} - 8 q^{61} - 4 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{66} + 8 q^{67} - 8 q^{68} - 4 q^{69} - 4 q^{70} + 16 q^{71} - 6 q^{72} - 4 q^{73} - 8 q^{74} - 2 q^{75} + 12 q^{77} + 8 q^{78} - 6 q^{80} + 2 q^{81} - 4 q^{82} + 20 q^{83} + 8 q^{84} - 8 q^{85} - 28 q^{86} + 8 q^{89} + 2 q^{90} - 4 q^{91} + 36 q^{92} - 12 q^{93} + 20 q^{94} - 6 q^{96} + 28 q^{97} + 10 q^{98} + 4 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) −0.414214 −0.169102
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −0.414214 −0.130986
\(11\) 6.24264 1.88223 0.941113 0.338091i \(-0.109781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) 1.82843 0.527821
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0.585786 0.156558
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0.414214 0.0976311
\(19\) 0 0
\(20\) 1.82843 0.408849
\(21\) −1.41421 −0.308607
\(22\) 2.58579 0.551292
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 1.58579 0.323697
\(25\) 1.00000 0.200000
\(26\) 0.242641 0.0475858
\(27\) −1.00000 −0.192450
\(28\) −2.58579 −0.488668
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0.414214 0.0756247
\(31\) 8.82843 1.58563 0.792816 0.609461i \(-0.208614\pi\)
0.792816 + 0.609461i \(0.208614\pi\)
\(32\) 4.41421 0.780330
\(33\) −6.24264 −1.08670
\(34\) 2.82843 0.485071
\(35\) −1.41421 −0.239046
\(36\) −1.82843 −0.304738
\(37\) 0.585786 0.0963027 0.0481513 0.998840i \(-0.484667\pi\)
0.0481513 + 0.998840i \(0.484667\pi\)
\(38\) 0 0
\(39\) −0.585786 −0.0938009
\(40\) 1.58579 0.250735
\(41\) −8.24264 −1.28728 −0.643642 0.765327i \(-0.722577\pi\)
−0.643642 + 0.765327i \(0.722577\pi\)
\(42\) −0.585786 −0.0903888
\(43\) 3.75736 0.572992 0.286496 0.958081i \(-0.407509\pi\)
0.286496 + 0.958081i \(0.407509\pi\)
\(44\) −11.4142 −1.72076
\(45\) −1.00000 −0.149071
\(46\) −1.51472 −0.223333
\(47\) 3.65685 0.533407 0.266704 0.963779i \(-0.414066\pi\)
0.266704 + 0.963779i \(0.414066\pi\)
\(48\) −3.00000 −0.433013
\(49\) −5.00000 −0.714286
\(50\) 0.414214 0.0585786
\(51\) −6.82843 −0.956171
\(52\) −1.07107 −0.148530
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) −0.414214 −0.0563673
\(55\) −6.24264 −0.841757
\(56\) −2.24264 −0.299685
\(57\) 0 0
\(58\) 0.585786 0.0769175
\(59\) 4.48528 0.583934 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(60\) −1.82843 −0.236049
\(61\) −15.3137 −1.96072 −0.980360 0.197218i \(-0.936809\pi\)
−0.980360 + 0.197218i \(0.936809\pi\)
\(62\) 3.65685 0.464421
\(63\) 1.41421 0.178174
\(64\) −4.17157 −0.521447
\(65\) −0.585786 −0.0726579
\(66\) −2.58579 −0.318288
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) −12.4853 −1.51406
\(69\) 3.65685 0.440234
\(70\) −0.585786 −0.0700149
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) −1.58579 −0.186887
\(73\) 3.65685 0.428002 0.214001 0.976833i \(-0.431350\pi\)
0.214001 + 0.976833i \(0.431350\pi\)
\(74\) 0.242641 0.0282064
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.82843 1.00609
\(78\) −0.242641 −0.0274736
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −3.41421 −0.377037
\(83\) 7.17157 0.787182 0.393591 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(84\) 2.58579 0.282132
\(85\) −6.82843 −0.740647
\(86\) 1.55635 0.167825
\(87\) −1.41421 −0.151620
\(88\) −9.89949 −1.05529
\(89\) 13.8995 1.47334 0.736672 0.676250i \(-0.236396\pi\)
0.736672 + 0.676250i \(0.236396\pi\)
\(90\) −0.414214 −0.0436619
\(91\) 0.828427 0.0868428
\(92\) 6.68629 0.697094
\(93\) −8.82843 −0.915465
\(94\) 1.51472 0.156231
\(95\) 0 0
\(96\) −4.41421 −0.450524
\(97\) 18.2426 1.85226 0.926130 0.377205i \(-0.123115\pi\)
0.926130 + 0.377205i \(0.123115\pi\)
\(98\) −2.07107 −0.209209
\(99\) 6.24264 0.627409
\(100\) −1.82843 −0.182843
\(101\) −8.82843 −0.878461 −0.439231 0.898374i \(-0.644749\pi\)
−0.439231 + 0.898374i \(0.644749\pi\)
\(102\) −2.82843 −0.280056
\(103\) −15.3137 −1.50890 −0.754452 0.656355i \(-0.772097\pi\)
−0.754452 + 0.656355i \(0.772097\pi\)
\(104\) −0.928932 −0.0910893
\(105\) 1.41421 0.138013
\(106\) −3.31371 −0.321856
\(107\) 3.31371 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(108\) 1.82843 0.175940
\(109\) −10.4853 −1.00431 −0.502154 0.864778i \(-0.667459\pi\)
−0.502154 + 0.864778i \(0.667459\pi\)
\(110\) −2.58579 −0.246545
\(111\) −0.585786 −0.0556004
\(112\) 4.24264 0.400892
\(113\) 18.1421 1.70667 0.853334 0.521364i \(-0.174577\pi\)
0.853334 + 0.521364i \(0.174577\pi\)
\(114\) 0 0
\(115\) 3.65685 0.341003
\(116\) −2.58579 −0.240084
\(117\) 0.585786 0.0541560
\(118\) 1.85786 0.171030
\(119\) 9.65685 0.885242
\(120\) −1.58579 −0.144762
\(121\) 27.9706 2.54278
\(122\) −6.34315 −0.574281
\(123\) 8.24264 0.743214
\(124\) −16.1421 −1.44961
\(125\) −1.00000 −0.0894427
\(126\) 0.585786 0.0521860
\(127\) 3.31371 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(128\) −10.5563 −0.933058
\(129\) −3.75736 −0.330817
\(130\) −0.242641 −0.0212810
\(131\) −11.4142 −0.997264 −0.498632 0.866814i \(-0.666164\pi\)
−0.498632 + 0.866814i \(0.666164\pi\)
\(132\) 11.4142 0.993480
\(133\) 0 0
\(134\) −0.686292 −0.0592866
\(135\) 1.00000 0.0860663
\(136\) −10.8284 −0.928530
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 1.51472 0.128941
\(139\) 14.1421 1.19952 0.599760 0.800180i \(-0.295263\pi\)
0.599760 + 0.800180i \(0.295263\pi\)
\(140\) 2.58579 0.218539
\(141\) −3.65685 −0.307963
\(142\) 2.14214 0.179764
\(143\) 3.65685 0.305802
\(144\) 3.00000 0.250000
\(145\) −1.41421 −0.117444
\(146\) 1.51472 0.125359
\(147\) 5.00000 0.412393
\(148\) −1.07107 −0.0880412
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −0.414214 −0.0338204
\(151\) −10.4853 −0.853280 −0.426640 0.904422i \(-0.640303\pi\)
−0.426640 + 0.904422i \(0.640303\pi\)
\(152\) 0 0
\(153\) 6.82843 0.552046
\(154\) 3.65685 0.294678
\(155\) −8.82843 −0.709116
\(156\) 1.07107 0.0857541
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) −4.41421 −0.348974
\(161\) −5.17157 −0.407577
\(162\) 0.414214 0.0325437
\(163\) 2.10051 0.164524 0.0822621 0.996611i \(-0.473786\pi\)
0.0822621 + 0.996611i \(0.473786\pi\)
\(164\) 15.0711 1.17685
\(165\) 6.24264 0.485989
\(166\) 2.97056 0.230560
\(167\) −5.31371 −0.411187 −0.205594 0.978637i \(-0.565912\pi\)
−0.205594 + 0.978637i \(0.565912\pi\)
\(168\) 2.24264 0.173023
\(169\) −12.6569 −0.973604
\(170\) −2.82843 −0.216930
\(171\) 0 0
\(172\) −6.87006 −0.523837
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) −0.585786 −0.0444084
\(175\) 1.41421 0.106904
\(176\) 18.7279 1.41167
\(177\) −4.48528 −0.337134
\(178\) 5.75736 0.431532
\(179\) 0.485281 0.0362716 0.0181358 0.999836i \(-0.494227\pi\)
0.0181358 + 0.999836i \(0.494227\pi\)
\(180\) 1.82843 0.136283
\(181\) 15.1716 1.12769 0.563847 0.825879i \(-0.309321\pi\)
0.563847 + 0.825879i \(0.309321\pi\)
\(182\) 0.343146 0.0254357
\(183\) 15.3137 1.13202
\(184\) 5.79899 0.427507
\(185\) −0.585786 −0.0430679
\(186\) −3.65685 −0.268134
\(187\) 42.6274 3.11723
\(188\) −6.68629 −0.487648
\(189\) −1.41421 −0.102869
\(190\) 0 0
\(191\) 1.75736 0.127158 0.0635790 0.997977i \(-0.479749\pi\)
0.0635790 + 0.997977i \(0.479749\pi\)
\(192\) 4.17157 0.301057
\(193\) 9.07107 0.652950 0.326475 0.945206i \(-0.394139\pi\)
0.326475 + 0.945206i \(0.394139\pi\)
\(194\) 7.55635 0.542514
\(195\) 0.585786 0.0419490
\(196\) 9.14214 0.653010
\(197\) 1.17157 0.0834711 0.0417356 0.999129i \(-0.486711\pi\)
0.0417356 + 0.999129i \(0.486711\pi\)
\(198\) 2.58579 0.183764
\(199\) −10.1421 −0.718957 −0.359478 0.933153i \(-0.617045\pi\)
−0.359478 + 0.933153i \(0.617045\pi\)
\(200\) −1.58579 −0.112132
\(201\) 1.65685 0.116865
\(202\) −3.65685 −0.257295
\(203\) 2.00000 0.140372
\(204\) 12.4853 0.874145
\(205\) 8.24264 0.575691
\(206\) −6.34315 −0.441948
\(207\) −3.65685 −0.254169
\(208\) 1.75736 0.121851
\(209\) 0 0
\(210\) 0.585786 0.0404231
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) 14.6274 1.00462
\(213\) −5.17157 −0.354350
\(214\) 1.37258 0.0938278
\(215\) −3.75736 −0.256250
\(216\) 1.58579 0.107899
\(217\) 12.4853 0.847556
\(218\) −4.34315 −0.294155
\(219\) −3.65685 −0.247107
\(220\) 11.4142 0.769546
\(221\) 4.00000 0.269069
\(222\) −0.242641 −0.0162850
\(223\) 26.6274 1.78310 0.891552 0.452919i \(-0.149617\pi\)
0.891552 + 0.452919i \(0.149617\pi\)
\(224\) 6.24264 0.417104
\(225\) 1.00000 0.0666667
\(226\) 7.51472 0.499872
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) 0 0
\(229\) 22.6274 1.49526 0.747631 0.664114i \(-0.231191\pi\)
0.747631 + 0.664114i \(0.231191\pi\)
\(230\) 1.51472 0.0998776
\(231\) −8.82843 −0.580868
\(232\) −2.24264 −0.147237
\(233\) −11.6569 −0.763666 −0.381833 0.924231i \(-0.624707\pi\)
−0.381833 + 0.924231i \(0.624707\pi\)
\(234\) 0.242641 0.0158619
\(235\) −3.65685 −0.238547
\(236\) −8.20101 −0.533840
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) −1.27208 −0.0822839 −0.0411419 0.999153i \(-0.513100\pi\)
−0.0411419 + 0.999153i \(0.513100\pi\)
\(240\) 3.00000 0.193649
\(241\) 8.34315 0.537429 0.268715 0.963220i \(-0.413401\pi\)
0.268715 + 0.963220i \(0.413401\pi\)
\(242\) 11.5858 0.744763
\(243\) −1.00000 −0.0641500
\(244\) 28.0000 1.79252
\(245\) 5.00000 0.319438
\(246\) 3.41421 0.217682
\(247\) 0 0
\(248\) −14.0000 −0.889001
\(249\) −7.17157 −0.454480
\(250\) −0.414214 −0.0261972
\(251\) −10.2426 −0.646510 −0.323255 0.946312i \(-0.604777\pi\)
−0.323255 + 0.946312i \(0.604777\pi\)
\(252\) −2.58579 −0.162889
\(253\) −22.8284 −1.43521
\(254\) 1.37258 0.0861235
\(255\) 6.82843 0.427613
\(256\) 3.97056 0.248160
\(257\) −12.4853 −0.778810 −0.389405 0.921067i \(-0.627319\pi\)
−0.389405 + 0.921067i \(0.627319\pi\)
\(258\) −1.55635 −0.0968941
\(259\) 0.828427 0.0514760
\(260\) 1.07107 0.0664248
\(261\) 1.41421 0.0875376
\(262\) −4.72792 −0.292092
\(263\) 27.4558 1.69300 0.846500 0.532389i \(-0.178706\pi\)
0.846500 + 0.532389i \(0.178706\pi\)
\(264\) 9.89949 0.609272
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) −13.8995 −0.850635
\(268\) 3.02944 0.185052
\(269\) −11.0711 −0.675015 −0.337507 0.941323i \(-0.609584\pi\)
−0.337507 + 0.941323i \(0.609584\pi\)
\(270\) 0.414214 0.0252082
\(271\) −5.17157 −0.314151 −0.157075 0.987587i \(-0.550207\pi\)
−0.157075 + 0.987587i \(0.550207\pi\)
\(272\) 20.4853 1.24210
\(273\) −0.828427 −0.0501387
\(274\) 4.14214 0.250236
\(275\) 6.24264 0.376445
\(276\) −6.68629 −0.402467
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 5.85786 0.351331
\(279\) 8.82843 0.528544
\(280\) 2.24264 0.134023
\(281\) 17.4142 1.03884 0.519422 0.854518i \(-0.326147\pi\)
0.519422 + 0.854518i \(0.326147\pi\)
\(282\) −1.51472 −0.0902002
\(283\) 26.3848 1.56841 0.784206 0.620500i \(-0.213071\pi\)
0.784206 + 0.620500i \(0.213071\pi\)
\(284\) −9.45584 −0.561101
\(285\) 0 0
\(286\) 1.51472 0.0895672
\(287\) −11.6569 −0.688082
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) −0.585786 −0.0343986
\(291\) −18.2426 −1.06940
\(292\) −6.68629 −0.391286
\(293\) 28.4853 1.66413 0.832064 0.554680i \(-0.187159\pi\)
0.832064 + 0.554680i \(0.187159\pi\)
\(294\) 2.07107 0.120787
\(295\) −4.48528 −0.261143
\(296\) −0.928932 −0.0539931
\(297\) −6.24264 −0.362235
\(298\) −2.48528 −0.143968
\(299\) −2.14214 −0.123883
\(300\) 1.82843 0.105564
\(301\) 5.31371 0.306277
\(302\) −4.34315 −0.249920
\(303\) 8.82843 0.507180
\(304\) 0 0
\(305\) 15.3137 0.876860
\(306\) 2.82843 0.161690
\(307\) −26.8284 −1.53118 −0.765590 0.643329i \(-0.777553\pi\)
−0.765590 + 0.643329i \(0.777553\pi\)
\(308\) −16.1421 −0.919784
\(309\) 15.3137 0.871166
\(310\) −3.65685 −0.207695
\(311\) 2.24264 0.127168 0.0635842 0.997976i \(-0.479747\pi\)
0.0635842 + 0.997976i \(0.479747\pi\)
\(312\) 0.928932 0.0525904
\(313\) −33.7990 −1.91043 −0.955216 0.295910i \(-0.904377\pi\)
−0.955216 + 0.295910i \(0.904377\pi\)
\(314\) −2.68629 −0.151596
\(315\) −1.41421 −0.0796819
\(316\) 0 0
\(317\) −18.6274 −1.04622 −0.523110 0.852265i \(-0.675228\pi\)
−0.523110 + 0.852265i \(0.675228\pi\)
\(318\) 3.31371 0.185824
\(319\) 8.82843 0.494297
\(320\) 4.17157 0.233198
\(321\) −3.31371 −0.184953
\(322\) −2.14214 −0.119377
\(323\) 0 0
\(324\) −1.82843 −0.101579
\(325\) 0.585786 0.0324936
\(326\) 0.870058 0.0481880
\(327\) 10.4853 0.579837
\(328\) 13.0711 0.721729
\(329\) 5.17157 0.285118
\(330\) 2.58579 0.142343
\(331\) 0.142136 0.00781248 0.00390624 0.999992i \(-0.498757\pi\)
0.00390624 + 0.999992i \(0.498757\pi\)
\(332\) −13.1127 −0.719653
\(333\) 0.585786 0.0321009
\(334\) −2.20101 −0.120434
\(335\) 1.65685 0.0905236
\(336\) −4.24264 −0.231455
\(337\) 27.4142 1.49335 0.746674 0.665191i \(-0.231650\pi\)
0.746674 + 0.665191i \(0.231650\pi\)
\(338\) −5.24264 −0.285162
\(339\) −18.1421 −0.985346
\(340\) 12.4853 0.677109
\(341\) 55.1127 2.98452
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) −5.95837 −0.321254
\(345\) −3.65685 −0.196878
\(346\) −8.20101 −0.440889
\(347\) 23.4558 1.25918 0.629588 0.776929i \(-0.283224\pi\)
0.629588 + 0.776929i \(0.283224\pi\)
\(348\) 2.58579 0.138613
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0.585786 0.0313116
\(351\) −0.585786 −0.0312670
\(352\) 27.5563 1.46876
\(353\) −7.65685 −0.407533 −0.203767 0.979019i \(-0.565318\pi\)
−0.203767 + 0.979019i \(0.565318\pi\)
\(354\) −1.85786 −0.0987444
\(355\) −5.17157 −0.274479
\(356\) −25.4142 −1.34695
\(357\) −9.65685 −0.511095
\(358\) 0.201010 0.0106237
\(359\) −28.8701 −1.52370 −0.761852 0.647752i \(-0.775710\pi\)
−0.761852 + 0.647752i \(0.775710\pi\)
\(360\) 1.58579 0.0835783
\(361\) 0 0
\(362\) 6.28427 0.330294
\(363\) −27.9706 −1.46807
\(364\) −1.51472 −0.0793928
\(365\) −3.65685 −0.191408
\(366\) 6.34315 0.331562
\(367\) 22.3848 1.16848 0.584238 0.811582i \(-0.301394\pi\)
0.584238 + 0.811582i \(0.301394\pi\)
\(368\) −10.9706 −0.571880
\(369\) −8.24264 −0.429095
\(370\) −0.242641 −0.0126143
\(371\) −11.3137 −0.587378
\(372\) 16.1421 0.836931
\(373\) −3.41421 −0.176781 −0.0883906 0.996086i \(-0.528172\pi\)
−0.0883906 + 0.996086i \(0.528172\pi\)
\(374\) 17.6569 0.913014
\(375\) 1.00000 0.0516398
\(376\) −5.79899 −0.299060
\(377\) 0.828427 0.0426662
\(378\) −0.585786 −0.0301296
\(379\) −8.82843 −0.453486 −0.226743 0.973955i \(-0.572808\pi\)
−0.226743 + 0.973955i \(0.572808\pi\)
\(380\) 0 0
\(381\) −3.31371 −0.169766
\(382\) 0.727922 0.0372437
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 10.5563 0.538701
\(385\) −8.82843 −0.449938
\(386\) 3.75736 0.191245
\(387\) 3.75736 0.190997
\(388\) −33.3553 −1.69336
\(389\) −2.97056 −0.150614 −0.0753068 0.997160i \(-0.523994\pi\)
−0.0753068 + 0.997160i \(0.523994\pi\)
\(390\) 0.242641 0.0122866
\(391\) −24.9706 −1.26282
\(392\) 7.92893 0.400472
\(393\) 11.4142 0.575771
\(394\) 0.485281 0.0244481
\(395\) 0 0
\(396\) −11.4142 −0.573586
\(397\) −16.6274 −0.834506 −0.417253 0.908790i \(-0.637007\pi\)
−0.417253 + 0.908790i \(0.637007\pi\)
\(398\) −4.20101 −0.210578
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 16.2426 0.811119 0.405559 0.914069i \(-0.367077\pi\)
0.405559 + 0.914069i \(0.367077\pi\)
\(402\) 0.686292 0.0342291
\(403\) 5.17157 0.257614
\(404\) 16.1421 0.803101
\(405\) −1.00000 −0.0496904
\(406\) 0.828427 0.0411141
\(407\) 3.65685 0.181264
\(408\) 10.8284 0.536087
\(409\) 7.17157 0.354611 0.177306 0.984156i \(-0.443262\pi\)
0.177306 + 0.984156i \(0.443262\pi\)
\(410\) 3.41421 0.168616
\(411\) −10.0000 −0.493264
\(412\) 28.0000 1.37946
\(413\) 6.34315 0.312126
\(414\) −1.51472 −0.0744444
\(415\) −7.17157 −0.352039
\(416\) 2.58579 0.126779
\(417\) −14.1421 −0.692543
\(418\) 0 0
\(419\) 0.585786 0.0286175 0.0143088 0.999898i \(-0.495445\pi\)
0.0143088 + 0.999898i \(0.495445\pi\)
\(420\) −2.58579 −0.126173
\(421\) 13.3137 0.648870 0.324435 0.945908i \(-0.394826\pi\)
0.324435 + 0.945908i \(0.394826\pi\)
\(422\) 6.34315 0.308780
\(423\) 3.65685 0.177802
\(424\) 12.6863 0.616101
\(425\) 6.82843 0.331227
\(426\) −2.14214 −0.103787
\(427\) −21.6569 −1.04805
\(428\) −6.05887 −0.292867
\(429\) −3.65685 −0.176555
\(430\) −1.55635 −0.0750538
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) −3.00000 −0.144338
\(433\) 25.0711 1.20484 0.602419 0.798180i \(-0.294204\pi\)
0.602419 + 0.798180i \(0.294204\pi\)
\(434\) 5.17157 0.248243
\(435\) 1.41421 0.0678064
\(436\) 19.1716 0.918152
\(437\) 0 0
\(438\) −1.51472 −0.0723761
\(439\) 10.3431 0.493651 0.246826 0.969060i \(-0.420612\pi\)
0.246826 + 0.969060i \(0.420612\pi\)
\(440\) 9.89949 0.471940
\(441\) −5.00000 −0.238095
\(442\) 1.65685 0.0788085
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 1.07107 0.0508306
\(445\) −13.8995 −0.658899
\(446\) 11.0294 0.522259
\(447\) 6.00000 0.283790
\(448\) −5.89949 −0.278725
\(449\) −26.8701 −1.26808 −0.634038 0.773302i \(-0.718604\pi\)
−0.634038 + 0.773302i \(0.718604\pi\)
\(450\) 0.414214 0.0195262
\(451\) −51.4558 −2.42296
\(452\) −33.1716 −1.56026
\(453\) 10.4853 0.492641
\(454\) 7.85786 0.368788
\(455\) −0.828427 −0.0388373
\(456\) 0 0
\(457\) 27.1716 1.27103 0.635516 0.772087i \(-0.280787\pi\)
0.635516 + 0.772087i \(0.280787\pi\)
\(458\) 9.37258 0.437952
\(459\) −6.82843 −0.318724
\(460\) −6.68629 −0.311750
\(461\) 8.34315 0.388579 0.194290 0.980944i \(-0.437760\pi\)
0.194290 + 0.980944i \(0.437760\pi\)
\(462\) −3.65685 −0.170132
\(463\) 15.7574 0.732307 0.366153 0.930555i \(-0.380675\pi\)
0.366153 + 0.930555i \(0.380675\pi\)
\(464\) 4.24264 0.196960
\(465\) 8.82843 0.409409
\(466\) −4.82843 −0.223673
\(467\) 24.3431 1.12647 0.563233 0.826298i \(-0.309557\pi\)
0.563233 + 0.826298i \(0.309557\pi\)
\(468\) −1.07107 −0.0495101
\(469\) −2.34315 −0.108196
\(470\) −1.51472 −0.0698688
\(471\) 6.48528 0.298826
\(472\) −7.11270 −0.327388
\(473\) 23.4558 1.07850
\(474\) 0 0
\(475\) 0 0
\(476\) −17.6569 −0.809301
\(477\) −8.00000 −0.366295
\(478\) −0.526912 −0.0241004
\(479\) 2.92893 0.133826 0.0669132 0.997759i \(-0.478685\pi\)
0.0669132 + 0.997759i \(0.478685\pi\)
\(480\) 4.41421 0.201480
\(481\) 0.343146 0.0156461
\(482\) 3.45584 0.157409
\(483\) 5.17157 0.235315
\(484\) −51.1421 −2.32464
\(485\) −18.2426 −0.828356
\(486\) −0.414214 −0.0187891
\(487\) −3.51472 −0.159267 −0.0796336 0.996824i \(-0.525375\pi\)
−0.0796336 + 0.996824i \(0.525375\pi\)
\(488\) 24.2843 1.09930
\(489\) −2.10051 −0.0949881
\(490\) 2.07107 0.0935613
\(491\) −10.2426 −0.462244 −0.231122 0.972925i \(-0.574240\pi\)
−0.231122 + 0.972925i \(0.574240\pi\)
\(492\) −15.0711 −0.679456
\(493\) 9.65685 0.434923
\(494\) 0 0
\(495\) −6.24264 −0.280586
\(496\) 26.4853 1.18922
\(497\) 7.31371 0.328065
\(498\) −2.97056 −0.133114
\(499\) −10.8284 −0.484747 −0.242373 0.970183i \(-0.577926\pi\)
−0.242373 + 0.970183i \(0.577926\pi\)
\(500\) 1.82843 0.0817697
\(501\) 5.31371 0.237399
\(502\) −4.24264 −0.189358
\(503\) −0.828427 −0.0369377 −0.0184689 0.999829i \(-0.505879\pi\)
−0.0184689 + 0.999829i \(0.505879\pi\)
\(504\) −2.24264 −0.0998952
\(505\) 8.82843 0.392860
\(506\) −9.45584 −0.420364
\(507\) 12.6569 0.562111
\(508\) −6.05887 −0.268819
\(509\) −24.7279 −1.09605 −0.548023 0.836463i \(-0.684619\pi\)
−0.548023 + 0.836463i \(0.684619\pi\)
\(510\) 2.82843 0.125245
\(511\) 5.17157 0.228777
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) −5.17157 −0.228108
\(515\) 15.3137 0.674803
\(516\) 6.87006 0.302437
\(517\) 22.8284 1.00399
\(518\) 0.343146 0.0150770
\(519\) 19.7990 0.869079
\(520\) 0.928932 0.0407364
\(521\) −16.2426 −0.711603 −0.355802 0.934562i \(-0.615792\pi\)
−0.355802 + 0.934562i \(0.615792\pi\)
\(522\) 0.585786 0.0256392
\(523\) 32.4853 1.42048 0.710241 0.703959i \(-0.248586\pi\)
0.710241 + 0.703959i \(0.248586\pi\)
\(524\) 20.8701 0.911713
\(525\) −1.41421 −0.0617213
\(526\) 11.3726 0.495868
\(527\) 60.2843 2.62602
\(528\) −18.7279 −0.815028
\(529\) −9.62742 −0.418583
\(530\) 3.31371 0.143938
\(531\) 4.48528 0.194645
\(532\) 0 0
\(533\) −4.82843 −0.209142
\(534\) −5.75736 −0.249145
\(535\) −3.31371 −0.143264
\(536\) 2.62742 0.113487
\(537\) −0.485281 −0.0209414
\(538\) −4.58579 −0.197707
\(539\) −31.2132 −1.34445
\(540\) −1.82843 −0.0786830
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −2.14214 −0.0920126
\(543\) −15.1716 −0.651075
\(544\) 30.1421 1.29233
\(545\) 10.4853 0.449140
\(546\) −0.343146 −0.0146853
\(547\) −7.51472 −0.321306 −0.160653 0.987011i \(-0.551360\pi\)
−0.160653 + 0.987011i \(0.551360\pi\)
\(548\) −18.2843 −0.781065
\(549\) −15.3137 −0.653573
\(550\) 2.58579 0.110258
\(551\) 0 0
\(552\) −5.79899 −0.246821
\(553\) 0 0
\(554\) −9.11270 −0.387161
\(555\) 0.585786 0.0248652
\(556\) −25.8579 −1.09662
\(557\) −2.68629 −0.113822 −0.0569109 0.998379i \(-0.518125\pi\)
−0.0569109 + 0.998379i \(0.518125\pi\)
\(558\) 3.65685 0.154807
\(559\) 2.20101 0.0930928
\(560\) −4.24264 −0.179284
\(561\) −42.6274 −1.79973
\(562\) 7.21320 0.304271
\(563\) −26.2843 −1.10775 −0.553875 0.832600i \(-0.686851\pi\)
−0.553875 + 0.832600i \(0.686851\pi\)
\(564\) 6.68629 0.281544
\(565\) −18.1421 −0.763245
\(566\) 10.9289 0.459377
\(567\) 1.41421 0.0593914
\(568\) −8.20101 −0.344107
\(569\) −16.9289 −0.709698 −0.354849 0.934924i \(-0.615468\pi\)
−0.354849 + 0.934924i \(0.615468\pi\)
\(570\) 0 0
\(571\) 14.8284 0.620550 0.310275 0.950647i \(-0.399579\pi\)
0.310275 + 0.950647i \(0.399579\pi\)
\(572\) −6.68629 −0.279568
\(573\) −1.75736 −0.0734147
\(574\) −4.82843 −0.201535
\(575\) −3.65685 −0.152501
\(576\) −4.17157 −0.173816
\(577\) −31.4558 −1.30952 −0.654762 0.755835i \(-0.727231\pi\)
−0.654762 + 0.755835i \(0.727231\pi\)
\(578\) 12.2721 0.510451
\(579\) −9.07107 −0.376981
\(580\) 2.58579 0.107369
\(581\) 10.1421 0.420767
\(582\) −7.55635 −0.313221
\(583\) −49.9411 −2.06835
\(584\) −5.79899 −0.239964
\(585\) −0.585786 −0.0242193
\(586\) 11.7990 0.487412
\(587\) −23.6569 −0.976423 −0.488211 0.872725i \(-0.662351\pi\)
−0.488211 + 0.872725i \(0.662351\pi\)
\(588\) −9.14214 −0.377015
\(589\) 0 0
\(590\) −1.85786 −0.0764871
\(591\) −1.17157 −0.0481921
\(592\) 1.75736 0.0722270
\(593\) −8.62742 −0.354286 −0.177143 0.984185i \(-0.556685\pi\)
−0.177143 + 0.984185i \(0.556685\pi\)
\(594\) −2.58579 −0.106096
\(595\) −9.65685 −0.395892
\(596\) 10.9706 0.449372
\(597\) 10.1421 0.415090
\(598\) −0.887302 −0.0362845
\(599\) 36.9706 1.51058 0.755288 0.655393i \(-0.227497\pi\)
0.755288 + 0.655393i \(0.227497\pi\)
\(600\) 1.58579 0.0647395
\(601\) 20.1421 0.821615 0.410807 0.911722i \(-0.365247\pi\)
0.410807 + 0.911722i \(0.365247\pi\)
\(602\) 2.20101 0.0897065
\(603\) −1.65685 −0.0674723
\(604\) 19.1716 0.780080
\(605\) −27.9706 −1.13717
\(606\) 3.65685 0.148550
\(607\) 0.485281 0.0196970 0.00984848 0.999952i \(-0.496865\pi\)
0.00984848 + 0.999952i \(0.496865\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 6.34315 0.256826
\(611\) 2.14214 0.0866615
\(612\) −12.4853 −0.504688
\(613\) 6.48528 0.261938 0.130969 0.991386i \(-0.458191\pi\)
0.130969 + 0.991386i \(0.458191\pi\)
\(614\) −11.1127 −0.448472
\(615\) −8.24264 −0.332375
\(616\) −14.0000 −0.564076
\(617\) 11.5147 0.463565 0.231783 0.972768i \(-0.425544\pi\)
0.231783 + 0.972768i \(0.425544\pi\)
\(618\) 6.34315 0.255159
\(619\) −3.51472 −0.141268 −0.0706342 0.997502i \(-0.522502\pi\)
−0.0706342 + 0.997502i \(0.522502\pi\)
\(620\) 16.1421 0.648284
\(621\) 3.65685 0.146745
\(622\) 0.928932 0.0372468
\(623\) 19.6569 0.787535
\(624\) −1.75736 −0.0703507
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 11.8579 0.473180
\(629\) 4.00000 0.159490
\(630\) −0.585786 −0.0233383
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −15.3137 −0.608665
\(634\) −7.71573 −0.306431
\(635\) −3.31371 −0.131501
\(636\) −14.6274 −0.580015
\(637\) −2.92893 −0.116049
\(638\) 3.65685 0.144776
\(639\) 5.17157 0.204584
\(640\) 10.5563 0.417276
\(641\) −39.3553 −1.55444 −0.777221 0.629227i \(-0.783371\pi\)
−0.777221 + 0.629227i \(0.783371\pi\)
\(642\) −1.37258 −0.0541715
\(643\) 5.61522 0.221443 0.110721 0.993851i \(-0.464684\pi\)
0.110721 + 0.993851i \(0.464684\pi\)
\(644\) 9.45584 0.372612
\(645\) 3.75736 0.147946
\(646\) 0 0
\(647\) −7.85786 −0.308925 −0.154462 0.987999i \(-0.549365\pi\)
−0.154462 + 0.987999i \(0.549365\pi\)
\(648\) −1.58579 −0.0622956
\(649\) 28.0000 1.09910
\(650\) 0.242641 0.00951715
\(651\) −12.4853 −0.489337
\(652\) −3.84062 −0.150410
\(653\) 37.1716 1.45464 0.727318 0.686301i \(-0.240767\pi\)
0.727318 + 0.686301i \(0.240767\pi\)
\(654\) 4.34315 0.169830
\(655\) 11.4142 0.445990
\(656\) −24.7279 −0.965463
\(657\) 3.65685 0.142667
\(658\) 2.14214 0.0835091
\(659\) −26.6274 −1.03726 −0.518628 0.855000i \(-0.673557\pi\)
−0.518628 + 0.855000i \(0.673557\pi\)
\(660\) −11.4142 −0.444298
\(661\) 43.4558 1.69024 0.845118 0.534579i \(-0.179530\pi\)
0.845118 + 0.534579i \(0.179530\pi\)
\(662\) 0.0588745 0.00228822
\(663\) −4.00000 −0.155347
\(664\) −11.3726 −0.441342
\(665\) 0 0
\(666\) 0.242641 0.00940214
\(667\) −5.17157 −0.200244
\(668\) 9.71573 0.375913
\(669\) −26.6274 −1.02948
\(670\) 0.686292 0.0265138
\(671\) −95.5980 −3.69052
\(672\) −6.24264 −0.240815
\(673\) −32.1838 −1.24059 −0.620297 0.784367i \(-0.712988\pi\)
−0.620297 + 0.784367i \(0.712988\pi\)
\(674\) 11.3553 0.437391
\(675\) −1.00000 −0.0384900
\(676\) 23.1421 0.890082
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) −7.51472 −0.288601
\(679\) 25.7990 0.990074
\(680\) 10.8284 0.415251
\(681\) −18.9706 −0.726954
\(682\) 22.8284 0.874146
\(683\) −29.6569 −1.13479 −0.567394 0.823446i \(-0.692048\pi\)
−0.567394 + 0.823446i \(0.692048\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −7.02944 −0.268385
\(687\) −22.6274 −0.863290
\(688\) 11.2721 0.429744
\(689\) −4.68629 −0.178533
\(690\) −1.51472 −0.0576644
\(691\) 41.1716 1.56624 0.783120 0.621870i \(-0.213627\pi\)
0.783120 + 0.621870i \(0.213627\pi\)
\(692\) 36.2010 1.37616
\(693\) 8.82843 0.335364
\(694\) 9.71573 0.368804
\(695\) −14.1421 −0.536442
\(696\) 2.24264 0.0850071
\(697\) −56.2843 −2.13192
\(698\) 7.45584 0.282208
\(699\) 11.6569 0.440903
\(700\) −2.58579 −0.0977335
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) −0.242641 −0.00915788
\(703\) 0 0
\(704\) −26.0416 −0.981481
\(705\) 3.65685 0.137725
\(706\) −3.17157 −0.119364
\(707\) −12.4853 −0.469557
\(708\) 8.20101 0.308213
\(709\) −4.97056 −0.186673 −0.0933367 0.995635i \(-0.529753\pi\)
−0.0933367 + 0.995635i \(0.529753\pi\)
\(710\) −2.14214 −0.0803929
\(711\) 0 0
\(712\) −22.0416 −0.826045
\(713\) −32.2843 −1.20906
\(714\) −4.00000 −0.149696
\(715\) −3.65685 −0.136759
\(716\) −0.887302 −0.0331600
\(717\) 1.27208 0.0475066
\(718\) −11.9584 −0.446282
\(719\) 13.0711 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(720\) −3.00000 −0.111803
\(721\) −21.6569 −0.806543
\(722\) 0 0
\(723\) −8.34315 −0.310285
\(724\) −27.7401 −1.03095
\(725\) 1.41421 0.0525226
\(726\) −11.5858 −0.429989
\(727\) −23.3553 −0.866202 −0.433101 0.901345i \(-0.642581\pi\)
−0.433101 + 0.901345i \(0.642581\pi\)
\(728\) −1.31371 −0.0486893
\(729\) 1.00000 0.0370370
\(730\) −1.51472 −0.0560623
\(731\) 25.6569 0.948953
\(732\) −28.0000 −1.03491
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 9.27208 0.342239
\(735\) −5.00000 −0.184428
\(736\) −16.1421 −0.595007
\(737\) −10.3431 −0.380995
\(738\) −3.41421 −0.125679
\(739\) 25.6569 0.943803 0.471901 0.881651i \(-0.343568\pi\)
0.471901 + 0.881651i \(0.343568\pi\)
\(740\) 1.07107 0.0393732
\(741\) 0 0
\(742\) −4.68629 −0.172039
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 14.0000 0.513265
\(745\) 6.00000 0.219823
\(746\) −1.41421 −0.0517780
\(747\) 7.17157 0.262394
\(748\) −77.9411 −2.84981
\(749\) 4.68629 0.171233
\(750\) 0.414214 0.0151249
\(751\) −4.14214 −0.151149 −0.0755743 0.997140i \(-0.524079\pi\)
−0.0755743 + 0.997140i \(0.524079\pi\)
\(752\) 10.9706 0.400055
\(753\) 10.2426 0.373263
\(754\) 0.343146 0.0124966
\(755\) 10.4853 0.381598
\(756\) 2.58579 0.0940441
\(757\) 52.4264 1.90547 0.952735 0.303802i \(-0.0982562\pi\)
0.952735 + 0.303802i \(0.0982562\pi\)
\(758\) −3.65685 −0.132823
\(759\) 22.8284 0.828619
\(760\) 0 0
\(761\) −23.9411 −0.867865 −0.433933 0.900945i \(-0.642874\pi\)
−0.433933 + 0.900945i \(0.642874\pi\)
\(762\) −1.37258 −0.0497234
\(763\) −14.8284 −0.536825
\(764\) −3.21320 −0.116250
\(765\) −6.82843 −0.246882
\(766\) −11.5980 −0.419052
\(767\) 2.62742 0.0948705
\(768\) −3.97056 −0.143275
\(769\) 9.02944 0.325610 0.162805 0.986658i \(-0.447946\pi\)
0.162805 + 0.986658i \(0.447946\pi\)
\(770\) −3.65685 −0.131784
\(771\) 12.4853 0.449646
\(772\) −16.5858 −0.596936
\(773\) 14.3431 0.515887 0.257944 0.966160i \(-0.416955\pi\)
0.257944 + 0.966160i \(0.416955\pi\)
\(774\) 1.55635 0.0559418
\(775\) 8.82843 0.317126
\(776\) −28.9289 −1.03849
\(777\) −0.828427 −0.0297197
\(778\) −1.23045 −0.0441137
\(779\) 0 0
\(780\) −1.07107 −0.0383504
\(781\) 32.2843 1.15522
\(782\) −10.3431 −0.369870
\(783\) −1.41421 −0.0505399
\(784\) −15.0000 −0.535714
\(785\) 6.48528 0.231470
\(786\) 4.72792 0.168639
\(787\) 37.1716 1.32502 0.662512 0.749052i \(-0.269490\pi\)
0.662512 + 0.749052i \(0.269490\pi\)
\(788\) −2.14214 −0.0763104
\(789\) −27.4558 −0.977454
\(790\) 0 0
\(791\) 25.6569 0.912253
\(792\) −9.89949 −0.351763
\(793\) −8.97056 −0.318554
\(794\) −6.88730 −0.244421
\(795\) −8.00000 −0.283731
\(796\) 18.5442 0.657280
\(797\) −14.1421 −0.500940 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(798\) 0 0
\(799\) 24.9706 0.883395
\(800\) 4.41421 0.156066
\(801\) 13.8995 0.491115
\(802\) 6.72792 0.237571
\(803\) 22.8284 0.805598
\(804\) −3.02944 −0.106840
\(805\) 5.17157 0.182274
\(806\) 2.14214 0.0754535
\(807\) 11.0711 0.389720
\(808\) 14.0000 0.492518
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −0.414214 −0.0145540
\(811\) −31.3137 −1.09957 −0.549787 0.835305i \(-0.685291\pi\)
−0.549787 + 0.835305i \(0.685291\pi\)
\(812\) −3.65685 −0.128330
\(813\) 5.17157 0.181375
\(814\) 1.51472 0.0530909
\(815\) −2.10051 −0.0735775
\(816\) −20.4853 −0.717128
\(817\) 0 0
\(818\) 2.97056 0.103863
\(819\) 0.828427 0.0289476
\(820\) −15.0711 −0.526305
\(821\) −2.48528 −0.0867369 −0.0433685 0.999059i \(-0.513809\pi\)
−0.0433685 + 0.999059i \(0.513809\pi\)
\(822\) −4.14214 −0.144474
\(823\) 57.0122 1.98732 0.993660 0.112426i \(-0.0358622\pi\)
0.993660 + 0.112426i \(0.0358622\pi\)
\(824\) 24.2843 0.845983
\(825\) −6.24264 −0.217341
\(826\) 2.62742 0.0914195
\(827\) 25.3137 0.880244 0.440122 0.897938i \(-0.354935\pi\)
0.440122 + 0.897938i \(0.354935\pi\)
\(828\) 6.68629 0.232365
\(829\) −29.5147 −1.02509 −0.512544 0.858661i \(-0.671297\pi\)
−0.512544 + 0.858661i \(0.671297\pi\)
\(830\) −2.97056 −0.103110
\(831\) 22.0000 0.763172
\(832\) −2.44365 −0.0847183
\(833\) −34.1421 −1.18295
\(834\) −5.85786 −0.202841
\(835\) 5.31371 0.183888
\(836\) 0 0
\(837\) −8.82843 −0.305155
\(838\) 0.242641 0.00838188
\(839\) 38.1421 1.31681 0.658406 0.752663i \(-0.271231\pi\)
0.658406 + 0.752663i \(0.271231\pi\)
\(840\) −2.24264 −0.0773785
\(841\) −27.0000 −0.931034
\(842\) 5.51472 0.190050
\(843\) −17.4142 −0.599777
\(844\) −28.0000 −0.963800
\(845\) 12.6569 0.435409
\(846\) 1.51472 0.0520771
\(847\) 39.5563 1.35917
\(848\) −24.0000 −0.824163
\(849\) −26.3848 −0.905523
\(850\) 2.82843 0.0970143
\(851\) −2.14214 −0.0734315
\(852\) 9.45584 0.323952
\(853\) −11.1716 −0.382507 −0.191254 0.981541i \(-0.561255\pi\)
−0.191254 + 0.981541i \(0.561255\pi\)
\(854\) −8.97056 −0.306966
\(855\) 0 0
\(856\) −5.25483 −0.179607
\(857\) −35.3137 −1.20629 −0.603147 0.797630i \(-0.706087\pi\)
−0.603147 + 0.797630i \(0.706087\pi\)
\(858\) −1.51472 −0.0517116
\(859\) 17.9411 0.612143 0.306072 0.952008i \(-0.400985\pi\)
0.306072 + 0.952008i \(0.400985\pi\)
\(860\) 6.87006 0.234267
\(861\) 11.6569 0.397265
\(862\) −12.8873 −0.438943
\(863\) −31.3137 −1.06593 −0.532966 0.846137i \(-0.678922\pi\)
−0.532966 + 0.846137i \(0.678922\pi\)
\(864\) −4.41421 −0.150175
\(865\) 19.7990 0.673186
\(866\) 10.3848 0.352889
\(867\) −29.6274 −1.00620
\(868\) −22.8284 −0.774847
\(869\) 0 0
\(870\) 0.585786 0.0198600
\(871\) −0.970563 −0.0328863
\(872\) 16.6274 0.563075
\(873\) 18.2426 0.617420
\(874\) 0 0
\(875\) −1.41421 −0.0478091
\(876\) 6.68629 0.225909
\(877\) 49.0711 1.65701 0.828506 0.559980i \(-0.189191\pi\)
0.828506 + 0.559980i \(0.189191\pi\)
\(878\) 4.28427 0.144587
\(879\) −28.4853 −0.960785
\(880\) −18.7279 −0.631318
\(881\) 33.7990 1.13872 0.569358 0.822089i \(-0.307192\pi\)
0.569358 + 0.822089i \(0.307192\pi\)
\(882\) −2.07107 −0.0697365
\(883\) 36.0416 1.21290 0.606449 0.795123i \(-0.292594\pi\)
0.606449 + 0.795123i \(0.292594\pi\)
\(884\) −7.31371 −0.245987
\(885\) 4.48528 0.150771
\(886\) 7.45584 0.250484
\(887\) 41.9411 1.40825 0.704123 0.710078i \(-0.251341\pi\)
0.704123 + 0.710078i \(0.251341\pi\)
\(888\) 0.928932 0.0311729
\(889\) 4.68629 0.157173
\(890\) −5.75736 −0.192987
\(891\) 6.24264 0.209136
\(892\) −48.6863 −1.63014
\(893\) 0 0
\(894\) 2.48528 0.0831202
\(895\) −0.485281 −0.0162212
\(896\) −14.9289 −0.498741
\(897\) 2.14214 0.0715238
\(898\) −11.1299 −0.371411
\(899\) 12.4853 0.416407
\(900\) −1.82843 −0.0609476
\(901\) −54.6274 −1.81990
\(902\) −21.3137 −0.709669
\(903\) −5.31371 −0.176829
\(904\) −28.7696 −0.956861
\(905\) −15.1716 −0.504320
\(906\) 4.34315 0.144291
\(907\) −10.1421 −0.336764 −0.168382 0.985722i \(-0.553854\pi\)
−0.168382 + 0.985722i \(0.553854\pi\)
\(908\) −34.6863 −1.15111
\(909\) −8.82843 −0.292820
\(910\) −0.343146 −0.0113752
\(911\) −31.3137 −1.03747 −0.518735 0.854935i \(-0.673597\pi\)
−0.518735 + 0.854935i \(0.673597\pi\)
\(912\) 0 0
\(913\) 44.7696 1.48166
\(914\) 11.2548 0.372277
\(915\) −15.3137 −0.506256
\(916\) −41.3726 −1.36699
\(917\) −16.1421 −0.533060
\(918\) −2.82843 −0.0933520
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) −5.79899 −0.191187
\(921\) 26.8284 0.884027
\(922\) 3.45584 0.113812
\(923\) 3.02944 0.0997151
\(924\) 16.1421 0.531037
\(925\) 0.585786 0.0192605
\(926\) 6.52691 0.214488
\(927\) −15.3137 −0.502968
\(928\) 6.24264 0.204925
\(929\) −29.1127 −0.955157 −0.477578 0.878589i \(-0.658485\pi\)
−0.477578 + 0.878589i \(0.658485\pi\)
\(930\) 3.65685 0.119913
\(931\) 0 0
\(932\) 21.3137 0.698154
\(933\) −2.24264 −0.0734208
\(934\) 10.0833 0.329934
\(935\) −42.6274 −1.39407
\(936\) −0.928932 −0.0303631
\(937\) 9.79899 0.320119 0.160060 0.987107i \(-0.448831\pi\)
0.160060 + 0.987107i \(0.448831\pi\)
\(938\) −0.970563 −0.0316900
\(939\) 33.7990 1.10299
\(940\) 6.68629 0.218083
\(941\) 30.8701 1.00634 0.503168 0.864189i \(-0.332168\pi\)
0.503168 + 0.864189i \(0.332168\pi\)
\(942\) 2.68629 0.0875241
\(943\) 30.1421 0.981563
\(944\) 13.4558 0.437950
\(945\) 1.41421 0.0460044
\(946\) 9.71573 0.315886
\(947\) −52.8284 −1.71669 −0.858347 0.513070i \(-0.828508\pi\)
−0.858347 + 0.513070i \(0.828508\pi\)
\(948\) 0 0
\(949\) 2.14214 0.0695367
\(950\) 0 0
\(951\) 18.6274 0.604035
\(952\) −15.3137 −0.496320
\(953\) 2.54416 0.0824133 0.0412066 0.999151i \(-0.486880\pi\)
0.0412066 + 0.999151i \(0.486880\pi\)
\(954\) −3.31371 −0.107285
\(955\) −1.75736 −0.0568668
\(956\) 2.32590 0.0752250
\(957\) −8.82843 −0.285383
\(958\) 1.21320 0.0391968
\(959\) 14.1421 0.456673
\(960\) −4.17157 −0.134637
\(961\) 46.9411 1.51423
\(962\) 0.142136 0.00458264
\(963\) 3.31371 0.106783
\(964\) −15.2548 −0.491325
\(965\) −9.07107 −0.292008
\(966\) 2.14214 0.0689221
\(967\) −40.0416 −1.28765 −0.643826 0.765172i \(-0.722654\pi\)
−0.643826 + 0.765172i \(0.722654\pi\)
\(968\) −44.3553 −1.42563
\(969\) 0 0
\(970\) −7.55635 −0.242620
\(971\) −33.6569 −1.08010 −0.540050 0.841633i \(-0.681595\pi\)
−0.540050 + 0.841633i \(0.681595\pi\)
\(972\) 1.82843 0.0586468
\(973\) 20.0000 0.641171
\(974\) −1.45584 −0.0466483
\(975\) −0.585786 −0.0187602
\(976\) −45.9411 −1.47054
\(977\) 24.2843 0.776923 0.388461 0.921465i \(-0.373007\pi\)
0.388461 + 0.921465i \(0.373007\pi\)
\(978\) −0.870058 −0.0278214
\(979\) 86.7696 2.77317
\(980\) −9.14214 −0.292035
\(981\) −10.4853 −0.334769
\(982\) −4.24264 −0.135388
\(983\) −20.6274 −0.657912 −0.328956 0.944345i \(-0.606697\pi\)
−0.328956 + 0.944345i \(0.606697\pi\)
\(984\) −13.0711 −0.416690
\(985\) −1.17157 −0.0373294
\(986\) 4.00000 0.127386
\(987\) −5.17157 −0.164613
\(988\) 0 0
\(989\) −13.7401 −0.436910
\(990\) −2.58579 −0.0821817
\(991\) −13.6569 −0.433824 −0.216912 0.976191i \(-0.569599\pi\)
−0.216912 + 0.976191i \(0.569599\pi\)
\(992\) 38.9706 1.23732
\(993\) −0.142136 −0.00451054
\(994\) 3.02944 0.0960879
\(995\) 10.1421 0.321527
\(996\) 13.1127 0.415492
\(997\) 58.0833 1.83952 0.919758 0.392487i \(-0.128385\pi\)
0.919758 + 0.392487i \(0.128385\pi\)
\(998\) −4.48528 −0.141979
\(999\) −0.585786 −0.0185335
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 5415.2.a.n.1.2 2
19.18 odd 2 285.2.a.g.1.1 2
57.56 even 2 855.2.a.d.1.2 2
76.75 even 2 4560.2.a.bf.1.1 2
95.18 even 4 1425.2.c.l.799.3 4
95.37 even 4 1425.2.c.l.799.2 4
95.94 odd 2 1425.2.a.k.1.2 2
285.284 even 2 4275.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.1 2 19.18 odd 2
855.2.a.d.1.2 2 57.56 even 2
1425.2.a.k.1.2 2 95.94 odd 2
1425.2.c.l.799.2 4 95.37 even 4
1425.2.c.l.799.3 4 95.18 even 4
4275.2.a.y.1.1 2 285.284 even 2
4560.2.a.bf.1.1 2 76.75 even 2
5415.2.a.n.1.2 2 1.1 even 1 trivial