Properties

Label 285.2.a.g.1.2
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.27573645761\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 285.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -2.24264 q^{11} +3.82843 q^{12} -3.41421 q^{13} -3.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} +1.17157 q^{17} +2.41421 q^{18} -1.00000 q^{19} -3.82843 q^{20} -1.41421 q^{21} -5.41421 q^{22} +7.65685 q^{23} +4.41421 q^{24} +1.00000 q^{25} -8.24264 q^{26} +1.00000 q^{27} -5.41421 q^{28} +1.41421 q^{29} -2.41421 q^{30} -3.17157 q^{31} -1.58579 q^{32} -2.24264 q^{33} +2.82843 q^{34} +1.41421 q^{35} +3.82843 q^{36} -3.41421 q^{37} -2.41421 q^{38} -3.41421 q^{39} -4.41421 q^{40} -0.242641 q^{41} -3.41421 q^{42} +12.2426 q^{43} -8.58579 q^{44} -1.00000 q^{45} +18.4853 q^{46} -7.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} +2.41421 q^{50} +1.17157 q^{51} -13.0711 q^{52} +8.00000 q^{53} +2.41421 q^{54} +2.24264 q^{55} -6.24264 q^{56} -1.00000 q^{57} +3.41421 q^{58} +12.4853 q^{59} -3.82843 q^{60} +7.31371 q^{61} -7.65685 q^{62} -1.41421 q^{63} -9.82843 q^{64} +3.41421 q^{65} -5.41421 q^{66} -9.65685 q^{67} +4.48528 q^{68} +7.65685 q^{69} +3.41421 q^{70} -10.8284 q^{71} +4.41421 q^{72} -7.65685 q^{73} -8.24264 q^{74} +1.00000 q^{75} -3.82843 q^{76} +3.17157 q^{77} -8.24264 q^{78} -3.00000 q^{80} +1.00000 q^{81} -0.585786 q^{82} +12.8284 q^{83} -5.41421 q^{84} -1.17157 q^{85} +29.5563 q^{86} +1.41421 q^{87} -9.89949 q^{88} +5.89949 q^{89} -2.41421 q^{90} +4.82843 q^{91} +29.3137 q^{92} -3.17157 q^{93} -18.4853 q^{94} +1.00000 q^{95} -1.58579 q^{96} -9.75736 q^{97} -12.0711 q^{98} -2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 6q^{8} + 2q^{9} - 2q^{10} + 4q^{11} + 2q^{12} - 4q^{13} - 4q^{14} - 2q^{15} + 6q^{16} + 8q^{17} + 2q^{18} - 2q^{19} - 2q^{20} - 8q^{22} + 4q^{23} + 6q^{24} + 2q^{25} - 8q^{26} + 2q^{27} - 8q^{28} - 2q^{30} - 12q^{31} - 6q^{32} + 4q^{33} + 2q^{36} - 4q^{37} - 2q^{38} - 4q^{39} - 6q^{40} + 8q^{41} - 4q^{42} + 16q^{43} - 20q^{44} - 2q^{45} + 20q^{46} - 4q^{47} + 6q^{48} - 10q^{49} + 2q^{50} + 8q^{51} - 12q^{52} + 16q^{53} + 2q^{54} - 4q^{55} - 4q^{56} - 2q^{57} + 4q^{58} + 8q^{59} - 2q^{60} - 8q^{61} - 4q^{62} - 14q^{64} + 4q^{65} - 8q^{66} - 8q^{67} - 8q^{68} + 4q^{69} + 4q^{70} - 16q^{71} + 6q^{72} - 4q^{73} - 8q^{74} + 2q^{75} - 2q^{76} + 12q^{77} - 8q^{78} - 6q^{80} + 2q^{81} - 4q^{82} + 20q^{83} - 8q^{84} - 8q^{85} + 28q^{86} - 8q^{89} - 2q^{90} + 4q^{91} + 36q^{92} - 12q^{93} - 20q^{94} + 2q^{95} - 6q^{96} - 28q^{97} - 10q^{98} + 4q^{99} + O(q^{100}) \)

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 \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) 2.41421 0.985599
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) −2.41421 −0.763441
\(11\) −2.24264 −0.676182 −0.338091 0.941113i \(-0.609781\pi\)
−0.338091 + 0.941113i \(0.609781\pi\)
\(12\) 3.82843 1.10517
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) −3.41421 −0.912487
\(15\) −1.00000 −0.258199
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 2.41421 0.569036
\(19\) −1.00000 −0.229416
\(20\) −3.82843 −0.856062
\(21\) −1.41421 −0.308607
\(22\) −5.41421 −1.15431
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 4.41421 0.901048
\(25\) 1.00000 0.200000
\(26\) −8.24264 −1.61651
\(27\) 1.00000 0.192450
\(28\) −5.41421 −1.02319
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) −2.41421 −0.440773
\(31\) −3.17157 −0.569631 −0.284816 0.958582i \(-0.591932\pi\)
−0.284816 + 0.958582i \(0.591932\pi\)
\(32\) −1.58579 −0.280330
\(33\) −2.24264 −0.390394
\(34\) 2.82843 0.485071
\(35\) 1.41421 0.239046
\(36\) 3.82843 0.638071
\(37\) −3.41421 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) −2.41421 −0.391637
\(39\) −3.41421 −0.546712
\(40\) −4.41421 −0.697948
\(41\) −0.242641 −0.0378941 −0.0189471 0.999820i \(-0.506031\pi\)
−0.0189471 + 0.999820i \(0.506031\pi\)
\(42\) −3.41421 −0.526825
\(43\) 12.2426 1.86699 0.933493 0.358597i \(-0.116745\pi\)
0.933493 + 0.358597i \(0.116745\pi\)
\(44\) −8.58579 −1.29436
\(45\) −1.00000 −0.149071
\(46\) 18.4853 2.72551
\(47\) −7.65685 −1.11687 −0.558433 0.829549i \(-0.688597\pi\)
−0.558433 + 0.829549i \(0.688597\pi\)
\(48\) 3.00000 0.433013
\(49\) −5.00000 −0.714286
\(50\) 2.41421 0.341421
\(51\) 1.17157 0.164053
\(52\) −13.0711 −1.81263
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 2.41421 0.328533
\(55\) 2.24264 0.302398
\(56\) −6.24264 −0.834208
\(57\) −1.00000 −0.132453
\(58\) 3.41421 0.448308
\(59\) 12.4853 1.62545 0.812723 0.582651i \(-0.197984\pi\)
0.812723 + 0.582651i \(0.197984\pi\)
\(60\) −3.82843 −0.494248
\(61\) 7.31371 0.936424 0.468212 0.883616i \(-0.344898\pi\)
0.468212 + 0.883616i \(0.344898\pi\)
\(62\) −7.65685 −0.972421
\(63\) −1.41421 −0.178174
\(64\) −9.82843 −1.22855
\(65\) 3.41421 0.423481
\(66\) −5.41421 −0.666444
\(67\) −9.65685 −1.17977 −0.589886 0.807486i \(-0.700827\pi\)
−0.589886 + 0.807486i \(0.700827\pi\)
\(68\) 4.48528 0.543920
\(69\) 7.65685 0.921777
\(70\) 3.41421 0.408077
\(71\) −10.8284 −1.28510 −0.642549 0.766245i \(-0.722123\pi\)
−0.642549 + 0.766245i \(0.722123\pi\)
\(72\) 4.41421 0.520220
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) −8.24264 −0.958188
\(75\) 1.00000 0.115470
\(76\) −3.82843 −0.439151
\(77\) 3.17157 0.361434
\(78\) −8.24264 −0.933295
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −0.585786 −0.0646893
\(83\) 12.8284 1.40810 0.704051 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(84\) −5.41421 −0.590739
\(85\) −1.17157 −0.127075
\(86\) 29.5563 3.18714
\(87\) 1.41421 0.151620
\(88\) −9.89949 −1.05529
\(89\) 5.89949 0.625345 0.312673 0.949861i \(-0.398776\pi\)
0.312673 + 0.949861i \(0.398776\pi\)
\(90\) −2.41421 −0.254480
\(91\) 4.82843 0.506157
\(92\) 29.3137 3.05617
\(93\) −3.17157 −0.328877
\(94\) −18.4853 −1.90661
\(95\) 1.00000 0.102598
\(96\) −1.58579 −0.161849
\(97\) −9.75736 −0.990710 −0.495355 0.868691i \(-0.664962\pi\)
−0.495355 + 0.868691i \(0.664962\pi\)
\(98\) −12.0711 −1.21936
\(99\) −2.24264 −0.225394
\(100\) 3.82843 0.382843
\(101\) −3.17157 −0.315583 −0.157792 0.987472i \(-0.550437\pi\)
−0.157792 + 0.987472i \(0.550437\pi\)
\(102\) 2.82843 0.280056
\(103\) −7.31371 −0.720641 −0.360321 0.932829i \(-0.617333\pi\)
−0.360321 + 0.932829i \(0.617333\pi\)
\(104\) −15.0711 −1.47784
\(105\) 1.41421 0.138013
\(106\) 19.3137 1.87591
\(107\) 19.3137 1.86713 0.933563 0.358412i \(-0.116682\pi\)
0.933563 + 0.358412i \(0.116682\pi\)
\(108\) 3.82843 0.368391
\(109\) −6.48528 −0.621177 −0.310589 0.950544i \(-0.600526\pi\)
−0.310589 + 0.950544i \(0.600526\pi\)
\(110\) 5.41421 0.516225
\(111\) −3.41421 −0.324063
\(112\) −4.24264 −0.400892
\(113\) 10.1421 0.954092 0.477046 0.878878i \(-0.341708\pi\)
0.477046 + 0.878878i \(0.341708\pi\)
\(114\) −2.41421 −0.226112
\(115\) −7.65685 −0.714005
\(116\) 5.41421 0.502697
\(117\) −3.41421 −0.315644
\(118\) 30.1421 2.77481
\(119\) −1.65685 −0.151884
\(120\) −4.41421 −0.402961
\(121\) −5.97056 −0.542778
\(122\) 17.6569 1.59858
\(123\) −0.242641 −0.0218782
\(124\) −12.1421 −1.09040
\(125\) −1.00000 −0.0894427
\(126\) −3.41421 −0.304162
\(127\) 19.3137 1.71381 0.856907 0.515471i \(-0.172383\pi\)
0.856907 + 0.515471i \(0.172383\pi\)
\(128\) −20.5563 −1.81694
\(129\) 12.2426 1.07790
\(130\) 8.24264 0.722927
\(131\) −8.58579 −0.750144 −0.375072 0.926996i \(-0.622382\pi\)
−0.375072 + 0.926996i \(0.622382\pi\)
\(132\) −8.58579 −0.747297
\(133\) 1.41421 0.122628
\(134\) −23.3137 −2.01400
\(135\) −1.00000 −0.0860663
\(136\) 5.17157 0.443459
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 18.4853 1.57357
\(139\) −14.1421 −1.19952 −0.599760 0.800180i \(-0.704737\pi\)
−0.599760 + 0.800180i \(0.704737\pi\)
\(140\) 5.41421 0.457585
\(141\) −7.65685 −0.644823
\(142\) −26.1421 −2.19380
\(143\) 7.65685 0.640298
\(144\) 3.00000 0.250000
\(145\) −1.41421 −0.117444
\(146\) −18.4853 −1.52985
\(147\) −5.00000 −0.412393
\(148\) −13.0711 −1.07444
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 2.41421 0.197120
\(151\) −6.48528 −0.527765 −0.263882 0.964555i \(-0.585003\pi\)
−0.263882 + 0.964555i \(0.585003\pi\)
\(152\) −4.41421 −0.358040
\(153\) 1.17157 0.0947161
\(154\) 7.65685 0.617007
\(155\) 3.17157 0.254747
\(156\) −13.0711 −1.04652
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 1.58579 0.125367
\(161\) −10.8284 −0.853400
\(162\) 2.41421 0.189679
\(163\) 21.8995 1.71530 0.857650 0.514233i \(-0.171923\pi\)
0.857650 + 0.514233i \(0.171923\pi\)
\(164\) −0.928932 −0.0725374
\(165\) 2.24264 0.174589
\(166\) 30.9706 2.40378
\(167\) −17.3137 −1.33977 −0.669887 0.742463i \(-0.733658\pi\)
−0.669887 + 0.742463i \(0.733658\pi\)
\(168\) −6.24264 −0.481630
\(169\) −1.34315 −0.103319
\(170\) −2.82843 −0.216930
\(171\) −1.00000 −0.0764719
\(172\) 46.8701 3.57381
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 3.41421 0.258831
\(175\) −1.41421 −0.106904
\(176\) −6.72792 −0.507136
\(177\) 12.4853 0.938451
\(178\) 14.2426 1.06753
\(179\) 16.4853 1.23217 0.616084 0.787681i \(-0.288718\pi\)
0.616084 + 0.787681i \(0.288718\pi\)
\(180\) −3.82843 −0.285354
\(181\) −20.8284 −1.54816 −0.774082 0.633085i \(-0.781788\pi\)
−0.774082 + 0.633085i \(0.781788\pi\)
\(182\) 11.6569 0.864064
\(183\) 7.31371 0.540645
\(184\) 33.7990 2.49169
\(185\) 3.41421 0.251018
\(186\) −7.65685 −0.561428
\(187\) −2.62742 −0.192136
\(188\) −29.3137 −2.13792
\(189\) −1.41421 −0.102869
\(190\) 2.41421 0.175145
\(191\) 10.2426 0.741131 0.370566 0.928806i \(-0.379164\pi\)
0.370566 + 0.928806i \(0.379164\pi\)
\(192\) −9.82843 −0.709306
\(193\) 5.07107 0.365023 0.182512 0.983204i \(-0.441577\pi\)
0.182512 + 0.983204i \(0.441577\pi\)
\(194\) −23.5563 −1.69125
\(195\) 3.41421 0.244497
\(196\) −19.1421 −1.36730
\(197\) 6.82843 0.486505 0.243253 0.969963i \(-0.421786\pi\)
0.243253 + 0.969963i \(0.421786\pi\)
\(198\) −5.41421 −0.384771
\(199\) 18.1421 1.28606 0.643031 0.765840i \(-0.277677\pi\)
0.643031 + 0.765840i \(0.277677\pi\)
\(200\) 4.41421 0.312132
\(201\) −9.65685 −0.681142
\(202\) −7.65685 −0.538734
\(203\) −2.00000 −0.140372
\(204\) 4.48528 0.314033
\(205\) 0.242641 0.0169468
\(206\) −17.6569 −1.23021
\(207\) 7.65685 0.532188
\(208\) −10.2426 −0.710199
\(209\) 2.24264 0.155127
\(210\) 3.41421 0.235603
\(211\) 7.31371 0.503496 0.251748 0.967793i \(-0.418995\pi\)
0.251748 + 0.967793i \(0.418995\pi\)
\(212\) 30.6274 2.10350
\(213\) −10.8284 −0.741952
\(214\) 46.6274 3.18738
\(215\) −12.2426 −0.834941
\(216\) 4.41421 0.300349
\(217\) 4.48528 0.304481
\(218\) −15.6569 −1.06042
\(219\) −7.65685 −0.517402
\(220\) 8.58579 0.578854
\(221\) −4.00000 −0.269069
\(222\) −8.24264 −0.553210
\(223\) 18.6274 1.24738 0.623692 0.781670i \(-0.285632\pi\)
0.623692 + 0.781670i \(0.285632\pi\)
\(224\) 2.24264 0.149843
\(225\) 1.00000 0.0666667
\(226\) 24.4853 1.62874
\(227\) 14.9706 0.993631 0.496816 0.867856i \(-0.334503\pi\)
0.496816 + 0.867856i \(0.334503\pi\)
\(228\) −3.82843 −0.253544
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) −18.4853 −1.21888
\(231\) 3.17157 0.208674
\(232\) 6.24264 0.409849
\(233\) −0.343146 −0.0224802 −0.0112401 0.999937i \(-0.503578\pi\)
−0.0112401 + 0.999937i \(0.503578\pi\)
\(234\) −8.24264 −0.538838
\(235\) 7.65685 0.499478
\(236\) 47.7990 3.11145
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −26.7279 −1.72889 −0.864443 0.502731i \(-0.832329\pi\)
−0.864443 + 0.502731i \(0.832329\pi\)
\(240\) −3.00000 −0.193649
\(241\) −19.6569 −1.26621 −0.633105 0.774066i \(-0.718220\pi\)
−0.633105 + 0.774066i \(0.718220\pi\)
\(242\) −14.4142 −0.926581
\(243\) 1.00000 0.0641500
\(244\) 28.0000 1.79252
\(245\) 5.00000 0.319438
\(246\) −0.585786 −0.0373484
\(247\) 3.41421 0.217241
\(248\) −14.0000 −0.889001
\(249\) 12.8284 0.812969
\(250\) −2.41421 −0.152688
\(251\) −1.75736 −0.110924 −0.0554618 0.998461i \(-0.517663\pi\)
−0.0554618 + 0.998461i \(0.517663\pi\)
\(252\) −5.41421 −0.341063
\(253\) −17.1716 −1.07957
\(254\) 46.6274 2.92566
\(255\) −1.17157 −0.0733667
\(256\) −29.9706 −1.87316
\(257\) −4.48528 −0.279784 −0.139892 0.990167i \(-0.544676\pi\)
−0.139892 + 0.990167i \(0.544676\pi\)
\(258\) 29.5563 1.84010
\(259\) 4.82843 0.300024
\(260\) 13.0711 0.810633
\(261\) 1.41421 0.0875376
\(262\) −20.7279 −1.28058
\(263\) −23.4558 −1.44635 −0.723175 0.690665i \(-0.757318\pi\)
−0.723175 + 0.690665i \(0.757318\pi\)
\(264\) −9.89949 −0.609272
\(265\) −8.00000 −0.491436
\(266\) 3.41421 0.209339
\(267\) 5.89949 0.361043
\(268\) −36.9706 −2.25834
\(269\) −3.07107 −0.187246 −0.0936232 0.995608i \(-0.529845\pi\)
−0.0936232 + 0.995608i \(0.529845\pi\)
\(270\) −2.41421 −0.146924
\(271\) −10.8284 −0.657780 −0.328890 0.944368i \(-0.606675\pi\)
−0.328890 + 0.944368i \(0.606675\pi\)
\(272\) 3.51472 0.213111
\(273\) 4.82843 0.292230
\(274\) 24.1421 1.45848
\(275\) −2.24264 −0.135236
\(276\) 29.3137 1.76448
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −34.1421 −2.04771
\(279\) −3.17157 −0.189877
\(280\) 6.24264 0.373069
\(281\) −14.5858 −0.870115 −0.435058 0.900403i \(-0.643272\pi\)
−0.435058 + 0.900403i \(0.643272\pi\)
\(282\) −18.4853 −1.10078
\(283\) −10.3848 −0.617311 −0.308655 0.951174i \(-0.599879\pi\)
−0.308655 + 0.951174i \(0.599879\pi\)
\(284\) −41.4558 −2.45995
\(285\) 1.00000 0.0592349
\(286\) 18.4853 1.09306
\(287\) 0.343146 0.0202553
\(288\) −1.58579 −0.0934434
\(289\) −15.6274 −0.919260
\(290\) −3.41421 −0.200490
\(291\) −9.75736 −0.571987
\(292\) −29.3137 −1.71546
\(293\) −11.5147 −0.672697 −0.336349 0.941738i \(-0.609192\pi\)
−0.336349 + 0.941738i \(0.609192\pi\)
\(294\) −12.0711 −0.703999
\(295\) −12.4853 −0.726921
\(296\) −15.0711 −0.875988
\(297\) −2.24264 −0.130131
\(298\) −14.4853 −0.839110
\(299\) −26.1421 −1.51184
\(300\) 3.82843 0.221034
\(301\) −17.3137 −0.997946
\(302\) −15.6569 −0.900951
\(303\) −3.17157 −0.182202
\(304\) −3.00000 −0.172062
\(305\) −7.31371 −0.418782
\(306\) 2.82843 0.161690
\(307\) 21.1716 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(308\) 12.1421 0.691862
\(309\) −7.31371 −0.416062
\(310\) 7.65685 0.434880
\(311\) −6.24264 −0.353988 −0.176994 0.984212i \(-0.556637\pi\)
−0.176994 + 0.984212i \(0.556637\pi\)
\(312\) −15.0711 −0.853231
\(313\) 5.79899 0.327778 0.163889 0.986479i \(-0.447596\pi\)
0.163889 + 0.986479i \(0.447596\pi\)
\(314\) 25.3137 1.42854
\(315\) 1.41421 0.0796819
\(316\) 0 0
\(317\) −26.6274 −1.49554 −0.747772 0.663955i \(-0.768877\pi\)
−0.747772 + 0.663955i \(0.768877\pi\)
\(318\) 19.3137 1.08306
\(319\) −3.17157 −0.177574
\(320\) 9.82843 0.549426
\(321\) 19.3137 1.07799
\(322\) −26.1421 −1.45684
\(323\) −1.17157 −0.0651881
\(324\) 3.82843 0.212690
\(325\) −3.41421 −0.189386
\(326\) 52.8701 2.92820
\(327\) −6.48528 −0.358637
\(328\) −1.07107 −0.0591398
\(329\) 10.8284 0.596991
\(330\) 5.41421 0.298043
\(331\) 28.1421 1.54683 0.773416 0.633899i \(-0.218547\pi\)
0.773416 + 0.633899i \(0.218547\pi\)
\(332\) 49.1127 2.69541
\(333\) −3.41421 −0.187098
\(334\) −41.7990 −2.28714
\(335\) 9.65685 0.527610
\(336\) −4.24264 −0.231455
\(337\) −24.5858 −1.33927 −0.669637 0.742689i \(-0.733550\pi\)
−0.669637 + 0.742689i \(0.733550\pi\)
\(338\) −3.24264 −0.176376
\(339\) 10.1421 0.550845
\(340\) −4.48528 −0.243249
\(341\) 7.11270 0.385174
\(342\) −2.41421 −0.130546
\(343\) 16.9706 0.916324
\(344\) 54.0416 2.91373
\(345\) −7.65685 −0.412231
\(346\) −47.7990 −2.56969
\(347\) −27.4558 −1.47391 −0.736953 0.675943i \(-0.763736\pi\)
−0.736953 + 0.675943i \(0.763736\pi\)
\(348\) 5.41421 0.290232
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) −3.41421 −0.182497
\(351\) −3.41421 −0.182237
\(352\) 3.55635 0.189554
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) 30.1421 1.60204
\(355\) 10.8284 0.574713
\(356\) 22.5858 1.19704
\(357\) −1.65685 −0.0876900
\(358\) 39.7990 2.10344
\(359\) 24.8701 1.31259 0.656296 0.754504i \(-0.272122\pi\)
0.656296 + 0.754504i \(0.272122\pi\)
\(360\) −4.41421 −0.232649
\(361\) 1.00000 0.0526316
\(362\) −50.2843 −2.64288
\(363\) −5.97056 −0.313373
\(364\) 18.4853 0.968892
\(365\) 7.65685 0.400778
\(366\) 17.6569 0.922939
\(367\) −14.3848 −0.750879 −0.375440 0.926847i \(-0.622508\pi\)
−0.375440 + 0.926847i \(0.622508\pi\)
\(368\) 22.9706 1.19742
\(369\) −0.242641 −0.0126314
\(370\) 8.24264 0.428514
\(371\) −11.3137 −0.587378
\(372\) −12.1421 −0.629540
\(373\) 0.585786 0.0303309 0.0151654 0.999885i \(-0.495173\pi\)
0.0151654 + 0.999885i \(0.495173\pi\)
\(374\) −6.34315 −0.327996
\(375\) −1.00000 −0.0516398
\(376\) −33.7990 −1.74305
\(377\) −4.82843 −0.248677
\(378\) −3.41421 −0.175608
\(379\) 3.17157 0.162913 0.0814564 0.996677i \(-0.474043\pi\)
0.0814564 + 0.996677i \(0.474043\pi\)
\(380\) 3.82843 0.196394
\(381\) 19.3137 0.989471
\(382\) 24.7279 1.26519
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) −20.5563 −1.04901
\(385\) −3.17157 −0.161638
\(386\) 12.2426 0.623134
\(387\) 12.2426 0.622328
\(388\) −37.3553 −1.89643
\(389\) 30.9706 1.57027 0.785135 0.619325i \(-0.212594\pi\)
0.785135 + 0.619325i \(0.212594\pi\)
\(390\) 8.24264 0.417382
\(391\) 8.97056 0.453661
\(392\) −22.0711 −1.11476
\(393\) −8.58579 −0.433096
\(394\) 16.4853 0.830516
\(395\) 0 0
\(396\) −8.58579 −0.431452
\(397\) 28.6274 1.43677 0.718384 0.695646i \(-0.244882\pi\)
0.718384 + 0.695646i \(0.244882\pi\)
\(398\) 43.7990 2.19544
\(399\) 1.41421 0.0707992
\(400\) 3.00000 0.150000
\(401\) −7.75736 −0.387384 −0.193692 0.981062i \(-0.562046\pi\)
−0.193692 + 0.981062i \(0.562046\pi\)
\(402\) −23.3137 −1.16278
\(403\) 10.8284 0.539402
\(404\) −12.1421 −0.604094
\(405\) −1.00000 −0.0496904
\(406\) −4.82843 −0.239631
\(407\) 7.65685 0.379536
\(408\) 5.17157 0.256031
\(409\) −12.8284 −0.634325 −0.317162 0.948371i \(-0.602730\pi\)
−0.317162 + 0.948371i \(0.602730\pi\)
\(410\) 0.585786 0.0289299
\(411\) 10.0000 0.493264
\(412\) −28.0000 −1.37946
\(413\) −17.6569 −0.868837
\(414\) 18.4853 0.908502
\(415\) −12.8284 −0.629723
\(416\) 5.41421 0.265454
\(417\) −14.1421 −0.692543
\(418\) 5.41421 0.264818
\(419\) 3.41421 0.166795 0.0833976 0.996516i \(-0.473423\pi\)
0.0833976 + 0.996516i \(0.473423\pi\)
\(420\) 5.41421 0.264187
\(421\) 9.31371 0.453922 0.226961 0.973904i \(-0.427121\pi\)
0.226961 + 0.973904i \(0.427121\pi\)
\(422\) 17.6569 0.859522
\(423\) −7.65685 −0.372289
\(424\) 35.3137 1.71499
\(425\) 1.17157 0.0568296
\(426\) −26.1421 −1.26659
\(427\) −10.3431 −0.500540
\(428\) 73.9411 3.57408
\(429\) 7.65685 0.369676
\(430\) −29.5563 −1.42533
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) 3.00000 0.144338
\(433\) −10.9289 −0.525211 −0.262605 0.964903i \(-0.584582\pi\)
−0.262605 + 0.964903i \(0.584582\pi\)
\(434\) 10.8284 0.519781
\(435\) −1.41421 −0.0678064
\(436\) −24.8284 −1.18907
\(437\) −7.65685 −0.366277
\(438\) −18.4853 −0.883261
\(439\) −21.6569 −1.03363 −0.516813 0.856099i \(-0.672882\pi\)
−0.516813 + 0.856099i \(0.672882\pi\)
\(440\) 9.89949 0.471940
\(441\) −5.00000 −0.238095
\(442\) −9.65685 −0.459330
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −13.0711 −0.620325
\(445\) −5.89949 −0.279663
\(446\) 44.9706 2.12942
\(447\) −6.00000 −0.283790
\(448\) 13.8995 0.656689
\(449\) −26.8701 −1.26808 −0.634038 0.773302i \(-0.718604\pi\)
−0.634038 + 0.773302i \(0.718604\pi\)
\(450\) 2.41421 0.113807
\(451\) 0.544156 0.0256233
\(452\) 38.8284 1.82634
\(453\) −6.48528 −0.304705
\(454\) 36.1421 1.69623
\(455\) −4.82843 −0.226360
\(456\) −4.41421 −0.206714
\(457\) 32.8284 1.53565 0.767825 0.640660i \(-0.221339\pi\)
0.767825 + 0.640660i \(0.221339\pi\)
\(458\) −54.6274 −2.55257
\(459\) 1.17157 0.0546843
\(460\) −29.3137 −1.36676
\(461\) 19.6569 0.915511 0.457755 0.889078i \(-0.348654\pi\)
0.457755 + 0.889078i \(0.348654\pi\)
\(462\) 7.65685 0.356229
\(463\) 24.2426 1.12665 0.563326 0.826235i \(-0.309522\pi\)
0.563326 + 0.826235i \(0.309522\pi\)
\(464\) 4.24264 0.196960
\(465\) 3.17157 0.147078
\(466\) −0.828427 −0.0383761
\(467\) 35.6569 1.65000 0.825001 0.565131i \(-0.191174\pi\)
0.825001 + 0.565131i \(0.191174\pi\)
\(468\) −13.0711 −0.604210
\(469\) 13.6569 0.630615
\(470\) 18.4853 0.852662
\(471\) 10.4853 0.483136
\(472\) 55.1127 2.53677
\(473\) −27.4558 −1.26242
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −6.34315 −0.290738
\(477\) 8.00000 0.366295
\(478\) −64.5269 −2.95139
\(479\) 17.0711 0.779997 0.389998 0.920815i \(-0.372476\pi\)
0.389998 + 0.920815i \(0.372476\pi\)
\(480\) 1.58579 0.0723809
\(481\) 11.6569 0.531507
\(482\) −47.4558 −2.16155
\(483\) −10.8284 −0.492710
\(484\) −22.8579 −1.03899
\(485\) 9.75736 0.443059
\(486\) 2.41421 0.109511
\(487\) 20.4853 0.928277 0.464138 0.885763i \(-0.346364\pi\)
0.464138 + 0.885763i \(0.346364\pi\)
\(488\) 32.2843 1.46144
\(489\) 21.8995 0.990329
\(490\) 12.0711 0.545315
\(491\) −1.75736 −0.0793085 −0.0396543 0.999213i \(-0.512626\pi\)
−0.0396543 + 0.999213i \(0.512626\pi\)
\(492\) −0.928932 −0.0418795
\(493\) 1.65685 0.0746210
\(494\) 8.24264 0.370854
\(495\) 2.24264 0.100799
\(496\) −9.51472 −0.427223
\(497\) 15.3137 0.686914
\(498\) 30.9706 1.38782
\(499\) −5.17157 −0.231511 −0.115756 0.993278i \(-0.536929\pi\)
−0.115756 + 0.993278i \(0.536929\pi\)
\(500\) −3.82843 −0.171212
\(501\) −17.3137 −0.773519
\(502\) −4.24264 −0.189358
\(503\) 4.82843 0.215289 0.107644 0.994189i \(-0.465669\pi\)
0.107644 + 0.994189i \(0.465669\pi\)
\(504\) −6.24264 −0.278069
\(505\) 3.17157 0.141133
\(506\) −41.4558 −1.84294
\(507\) −1.34315 −0.0596512
\(508\) 73.9411 3.28061
\(509\) −0.727922 −0.0322646 −0.0161323 0.999870i \(-0.505135\pi\)
−0.0161323 + 0.999870i \(0.505135\pi\)
\(510\) −2.82843 −0.125245
\(511\) 10.8284 0.479021
\(512\) −31.2426 −1.38074
\(513\) −1.00000 −0.0441511
\(514\) −10.8284 −0.477621
\(515\) 7.31371 0.322281
\(516\) 46.8701 2.06334
\(517\) 17.1716 0.755205
\(518\) 11.6569 0.512173
\(519\) −19.7990 −0.869079
\(520\) 15.0711 0.660910
\(521\) 7.75736 0.339856 0.169928 0.985456i \(-0.445646\pi\)
0.169928 + 0.985456i \(0.445646\pi\)
\(522\) 3.41421 0.149436
\(523\) −15.5147 −0.678411 −0.339206 0.940712i \(-0.610158\pi\)
−0.339206 + 0.940712i \(0.610158\pi\)
\(524\) −32.8701 −1.43594
\(525\) −1.41421 −0.0617213
\(526\) −56.6274 −2.46907
\(527\) −3.71573 −0.161860
\(528\) −6.72792 −0.292795
\(529\) 35.6274 1.54902
\(530\) −19.3137 −0.838934
\(531\) 12.4853 0.541815
\(532\) 5.41421 0.234736
\(533\) 0.828427 0.0358832
\(534\) 14.2426 0.616339
\(535\) −19.3137 −0.835004
\(536\) −42.6274 −1.84122
\(537\) 16.4853 0.711392
\(538\) −7.41421 −0.319649
\(539\) 11.2132 0.482987
\(540\) −3.82843 −0.164749
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −26.1421 −1.12290
\(543\) −20.8284 −0.893833
\(544\) −1.85786 −0.0796553
\(545\) 6.48528 0.277799
\(546\) 11.6569 0.498867
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) 38.2843 1.63542
\(549\) 7.31371 0.312141
\(550\) −5.41421 −0.230863
\(551\) −1.41421 −0.0602475
\(552\) 33.7990 1.43858
\(553\) 0 0
\(554\) −53.1127 −2.25654
\(555\) 3.41421 0.144925
\(556\) −54.1421 −2.29614
\(557\) −25.3137 −1.07258 −0.536288 0.844035i \(-0.680174\pi\)
−0.536288 + 0.844035i \(0.680174\pi\)
\(558\) −7.65685 −0.324140
\(559\) −41.7990 −1.76791
\(560\) 4.24264 0.179284
\(561\) −2.62742 −0.110930
\(562\) −35.2132 −1.48538
\(563\) −30.2843 −1.27633 −0.638165 0.769900i \(-0.720306\pi\)
−0.638165 + 0.769900i \(0.720306\pi\)
\(564\) −29.3137 −1.23433
\(565\) −10.1421 −0.426683
\(566\) −25.0711 −1.05382
\(567\) −1.41421 −0.0593914
\(568\) −47.7990 −2.00560
\(569\) 31.0711 1.30257 0.651283 0.758835i \(-0.274231\pi\)
0.651283 + 0.758835i \(0.274231\pi\)
\(570\) 2.41421 0.101120
\(571\) 9.17157 0.383818 0.191909 0.981413i \(-0.438532\pi\)
0.191909 + 0.981413i \(0.438532\pi\)
\(572\) 29.3137 1.22567
\(573\) 10.2426 0.427892
\(574\) 0.828427 0.0345779
\(575\) 7.65685 0.319313
\(576\) −9.82843 −0.409518
\(577\) 19.4558 0.809957 0.404979 0.914326i \(-0.367279\pi\)
0.404979 + 0.914326i \(0.367279\pi\)
\(578\) −37.7279 −1.56927
\(579\) 5.07107 0.210746
\(580\) −5.41421 −0.224813
\(581\) −18.1421 −0.752663
\(582\) −23.5563 −0.976442
\(583\) −17.9411 −0.743045
\(584\) −33.7990 −1.39861
\(585\) 3.41421 0.141160
\(586\) −27.7990 −1.14837
\(587\) −12.3431 −0.509456 −0.254728 0.967013i \(-0.581986\pi\)
−0.254728 + 0.967013i \(0.581986\pi\)
\(588\) −19.1421 −0.789408
\(589\) 3.17157 0.130682
\(590\) −30.1421 −1.24093
\(591\) 6.82843 0.280884
\(592\) −10.2426 −0.420970
\(593\) 36.6274 1.50411 0.752054 0.659102i \(-0.229063\pi\)
0.752054 + 0.659102i \(0.229063\pi\)
\(594\) −5.41421 −0.222148
\(595\) 1.65685 0.0679244
\(596\) −22.9706 −0.940911
\(597\) 18.1421 0.742508
\(598\) −63.1127 −2.58087
\(599\) −3.02944 −0.123779 −0.0618897 0.998083i \(-0.519713\pi\)
−0.0618897 + 0.998083i \(0.519713\pi\)
\(600\) 4.41421 0.180210
\(601\) 8.14214 0.332125 0.166062 0.986115i \(-0.446895\pi\)
0.166062 + 0.986115i \(0.446895\pi\)
\(602\) −41.7990 −1.70360
\(603\) −9.65685 −0.393258
\(604\) −24.8284 −1.01025
\(605\) 5.97056 0.242738
\(606\) −7.65685 −0.311038
\(607\) 16.4853 0.669117 0.334558 0.942375i \(-0.391413\pi\)
0.334558 + 0.942375i \(0.391413\pi\)
\(608\) 1.58579 0.0643121
\(609\) −2.00000 −0.0810441
\(610\) −17.6569 −0.714905
\(611\) 26.1421 1.05760
\(612\) 4.48528 0.181307
\(613\) −10.4853 −0.423497 −0.211748 0.977324i \(-0.567916\pi\)
−0.211748 + 0.977324i \(0.567916\pi\)
\(614\) 51.1127 2.06274
\(615\) 0.242641 0.00978422
\(616\) 14.0000 0.564076
\(617\) 28.4853 1.14677 0.573387 0.819285i \(-0.305629\pi\)
0.573387 + 0.819285i \(0.305629\pi\)
\(618\) −17.6569 −0.710263
\(619\) −20.4853 −0.823373 −0.411686 0.911326i \(-0.635060\pi\)
−0.411686 + 0.911326i \(0.635060\pi\)
\(620\) 12.1421 0.487640
\(621\) 7.65685 0.307259
\(622\) −15.0711 −0.604295
\(623\) −8.34315 −0.334261
\(624\) −10.2426 −0.410034
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) 2.24264 0.0895624
\(628\) 40.1421 1.60185
\(629\) −4.00000 −0.159490
\(630\) 3.41421 0.136026
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 7.31371 0.290694
\(634\) −64.2843 −2.55305
\(635\) −19.3137 −0.766441
\(636\) 30.6274 1.21446
\(637\) 17.0711 0.676380
\(638\) −7.65685 −0.303138
\(639\) −10.8284 −0.428366
\(640\) 20.5563 0.812561
\(641\) −31.3553 −1.23846 −0.619231 0.785209i \(-0.712555\pi\)
−0.619231 + 0.785209i \(0.712555\pi\)
\(642\) 46.6274 1.84024
\(643\) 42.3848 1.67149 0.835746 0.549116i \(-0.185035\pi\)
0.835746 + 0.549116i \(0.185035\pi\)
\(644\) −41.4558 −1.63359
\(645\) −12.2426 −0.482054
\(646\) −2.82843 −0.111283
\(647\) −36.1421 −1.42089 −0.710447 0.703751i \(-0.751507\pi\)
−0.710447 + 0.703751i \(0.751507\pi\)
\(648\) 4.41421 0.173407
\(649\) −28.0000 −1.09910
\(650\) −8.24264 −0.323303
\(651\) 4.48528 0.175792
\(652\) 83.8406 3.28345
\(653\) 42.8284 1.67601 0.838003 0.545666i \(-0.183723\pi\)
0.838003 + 0.545666i \(0.183723\pi\)
\(654\) −15.6569 −0.612231
\(655\) 8.58579 0.335474
\(656\) −0.727922 −0.0284206
\(657\) −7.65685 −0.298722
\(658\) 26.1421 1.01913
\(659\) −18.6274 −0.725621 −0.362811 0.931863i \(-0.618183\pi\)
−0.362811 + 0.931863i \(0.618183\pi\)
\(660\) 8.58579 0.334201
\(661\) 7.45584 0.289999 0.144999 0.989432i \(-0.453682\pi\)
0.144999 + 0.989432i \(0.453682\pi\)
\(662\) 67.9411 2.64061
\(663\) −4.00000 −0.155347
\(664\) 56.6274 2.19757
\(665\) −1.41421 −0.0548408
\(666\) −8.24264 −0.319396
\(667\) 10.8284 0.419278
\(668\) −66.2843 −2.56462
\(669\) 18.6274 0.720178
\(670\) 23.3137 0.900687
\(671\) −16.4020 −0.633193
\(672\) 2.24264 0.0865117
\(673\) −44.1838 −1.70316 −0.851580 0.524225i \(-0.824355\pi\)
−0.851580 + 0.524225i \(0.824355\pi\)
\(674\) −59.3553 −2.28628
\(675\) 1.00000 0.0384900
\(676\) −5.14214 −0.197774
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) 24.4853 0.940352
\(679\) 13.7990 0.529557
\(680\) −5.17157 −0.198321
\(681\) 14.9706 0.573673
\(682\) 17.1716 0.657534
\(683\) 18.3431 0.701881 0.350940 0.936398i \(-0.385862\pi\)
0.350940 + 0.936398i \(0.385862\pi\)
\(684\) −3.82843 −0.146384
\(685\) −10.0000 −0.382080
\(686\) 40.9706 1.56426
\(687\) −22.6274 −0.863290
\(688\) 36.7279 1.40024
\(689\) −27.3137 −1.04057
\(690\) −18.4853 −0.703723
\(691\) 46.8284 1.78144 0.890719 0.454555i \(-0.150202\pi\)
0.890719 + 0.454555i \(0.150202\pi\)
\(692\) −75.7990 −2.88145
\(693\) 3.17157 0.120478
\(694\) −66.2843 −2.51612
\(695\) 14.1421 0.536442
\(696\) 6.24264 0.236627
\(697\) −0.284271 −0.0107675
\(698\) 43.4558 1.64483
\(699\) −0.343146 −0.0129790
\(700\) −5.41421 −0.204638
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) −8.24264 −0.311098
\(703\) 3.41421 0.128770
\(704\) 22.0416 0.830725
\(705\) 7.65685 0.288374
\(706\) 8.82843 0.332262
\(707\) 4.48528 0.168686
\(708\) 47.7990 1.79640
\(709\) 28.9706 1.08801 0.544006 0.839081i \(-0.316907\pi\)
0.544006 + 0.839081i \(0.316907\pi\)
\(710\) 26.1421 0.981097
\(711\) 0 0
\(712\) 26.0416 0.975951
\(713\) −24.2843 −0.909453
\(714\) −4.00000 −0.149696
\(715\) −7.65685 −0.286350
\(716\) 63.1127 2.35863
\(717\) −26.7279 −0.998173
\(718\) 60.0416 2.24073
\(719\) −1.07107 −0.0399441 −0.0199720 0.999801i \(-0.506358\pi\)
−0.0199720 + 0.999801i \(0.506358\pi\)
\(720\) −3.00000 −0.111803
\(721\) 10.3431 0.385199
\(722\) 2.41421 0.0898477
\(723\) −19.6569 −0.731046
\(724\) −79.7401 −2.96352
\(725\) 1.41421 0.0525226
\(726\) −14.4142 −0.534962
\(727\) 47.3553 1.75631 0.878156 0.478374i \(-0.158774\pi\)
0.878156 + 0.478374i \(0.158774\pi\)
\(728\) 21.3137 0.789939
\(729\) 1.00000 0.0370370
\(730\) 18.4853 0.684171
\(731\) 14.3431 0.530500
\(732\) 28.0000 1.03491
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −34.7279 −1.28183
\(735\) 5.00000 0.184428
\(736\) −12.1421 −0.447565
\(737\) 21.6569 0.797740
\(738\) −0.585786 −0.0215631
\(739\) 14.3431 0.527621 0.263811 0.964575i \(-0.415021\pi\)
0.263811 + 0.964575i \(0.415021\pi\)
\(740\) 13.0711 0.480502
\(741\) 3.41421 0.125424
\(742\) −27.3137 −1.00272
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) −14.0000 −0.513265
\(745\) 6.00000 0.219823
\(746\) 1.41421 0.0517780
\(747\) 12.8284 0.469368
\(748\) −10.0589 −0.367789
\(749\) −27.3137 −0.998021
\(750\) −2.41421 −0.0881546
\(751\) −24.1421 −0.880959 −0.440480 0.897763i \(-0.645192\pi\)
−0.440480 + 0.897763i \(0.645192\pi\)
\(752\) −22.9706 −0.837650
\(753\) −1.75736 −0.0640417
\(754\) −11.6569 −0.424518
\(755\) 6.48528 0.236024
\(756\) −5.41421 −0.196913
\(757\) −32.4264 −1.17856 −0.589279 0.807930i \(-0.700588\pi\)
−0.589279 + 0.807930i \(0.700588\pi\)
\(758\) 7.65685 0.278109
\(759\) −17.1716 −0.623289
\(760\) 4.41421 0.160120
\(761\) 43.9411 1.59286 0.796432 0.604728i \(-0.206718\pi\)
0.796432 + 0.604728i \(0.206718\pi\)
\(762\) 46.6274 1.68913
\(763\) 9.17157 0.332033
\(764\) 39.2132 1.41868
\(765\) −1.17157 −0.0423583
\(766\) 67.5980 2.44241
\(767\) −42.6274 −1.53919
\(768\) −29.9706 −1.08147
\(769\) 42.9706 1.54956 0.774779 0.632232i \(-0.217861\pi\)
0.774779 + 0.632232i \(0.217861\pi\)
\(770\) −7.65685 −0.275934
\(771\) −4.48528 −0.161533
\(772\) 19.4142 0.698733
\(773\) −25.6569 −0.922813 −0.461406 0.887189i \(-0.652655\pi\)
−0.461406 + 0.887189i \(0.652655\pi\)
\(774\) 29.5563 1.06238
\(775\) −3.17157 −0.113926
\(776\) −43.0711 −1.54616
\(777\) 4.82843 0.173219
\(778\) 74.7696 2.68062
\(779\) 0.242641 0.00869350
\(780\) 13.0711 0.468019
\(781\) 24.2843 0.868960
\(782\) 21.6569 0.774448
\(783\) 1.41421 0.0505399
\(784\) −15.0000 −0.535714
\(785\) −10.4853 −0.374236
\(786\) −20.7279 −0.739340
\(787\) −42.8284 −1.52667 −0.763334 0.646004i \(-0.776439\pi\)
−0.763334 + 0.646004i \(0.776439\pi\)
\(788\) 26.1421 0.931275
\(789\) −23.4558 −0.835050
\(790\) 0 0
\(791\) −14.3431 −0.509984
\(792\) −9.89949 −0.351763
\(793\) −24.9706 −0.886731
\(794\) 69.1127 2.45272
\(795\) −8.00000 −0.283731
\(796\) 69.4558 2.46180
\(797\) −14.1421 −0.500940 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(798\) 3.41421 0.120862
\(799\) −8.97056 −0.317356
\(800\) −1.58579 −0.0560660
\(801\) 5.89949 0.208448
\(802\) −18.7279 −0.661306
\(803\) 17.1716 0.605972
\(804\) −36.9706 −1.30385
\(805\) 10.8284 0.381652
\(806\) 26.1421 0.920817
\(807\) −3.07107 −0.108107
\(808\) −14.0000 −0.492518
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −2.41421 −0.0848268
\(811\) 8.68629 0.305017 0.152508 0.988302i \(-0.451265\pi\)
0.152508 + 0.988302i \(0.451265\pi\)
\(812\) −7.65685 −0.268703
\(813\) −10.8284 −0.379770
\(814\) 18.4853 0.647909
\(815\) −21.8995 −0.767106
\(816\) 3.51472 0.123040
\(817\) −12.2426 −0.428316
\(818\) −30.9706 −1.08286
\(819\) 4.82843 0.168719
\(820\) 0.928932 0.0324397
\(821\) 14.4853 0.505540 0.252770 0.967526i \(-0.418658\pi\)
0.252770 + 0.967526i \(0.418658\pi\)
\(822\) 24.1421 0.842054
\(823\) −25.0122 −0.871870 −0.435935 0.899978i \(-0.643582\pi\)
−0.435935 + 0.899978i \(0.643582\pi\)
\(824\) −32.2843 −1.12468
\(825\) −2.24264 −0.0780787
\(826\) −42.6274 −1.48320
\(827\) −2.68629 −0.0934115 −0.0467058 0.998909i \(-0.514872\pi\)
−0.0467058 + 0.998909i \(0.514872\pi\)
\(828\) 29.3137 1.01872
\(829\) 46.4853 1.61450 0.807250 0.590209i \(-0.200955\pi\)
0.807250 + 0.590209i \(0.200955\pi\)
\(830\) −30.9706 −1.07500
\(831\) −22.0000 −0.763172
\(832\) 33.5563 1.16336
\(833\) −5.85786 −0.202963
\(834\) −34.1421 −1.18225
\(835\) 17.3137 0.599166
\(836\) 8.58579 0.296946
\(837\) −3.17157 −0.109626
\(838\) 8.24264 0.284737
\(839\) −9.85786 −0.340331 −0.170166 0.985415i \(-0.554430\pi\)
−0.170166 + 0.985415i \(0.554430\pi\)
\(840\) 6.24264 0.215392
\(841\) −27.0000 −0.931034
\(842\) 22.4853 0.774894
\(843\) −14.5858 −0.502361
\(844\) 28.0000 0.963800
\(845\) 1.34315 0.0462056
\(846\) −18.4853 −0.635537
\(847\) 8.44365 0.290127
\(848\) 24.0000 0.824163
\(849\) −10.3848 −0.356405
\(850\) 2.82843 0.0970143
\(851\) −26.1421 −0.896141
\(852\) −41.4558 −1.42025
\(853\) −16.8284 −0.576194 −0.288097 0.957601i \(-0.593023\pi\)
−0.288097 + 0.957601i \(0.593023\pi\)
\(854\) −24.9706 −0.854475
\(855\) 1.00000 0.0341993
\(856\) 85.2548 2.91395
\(857\) 12.6863 0.433355 0.216678 0.976243i \(-0.430478\pi\)
0.216678 + 0.976243i \(0.430478\pi\)
\(858\) 18.4853 0.631077
\(859\) −49.9411 −1.70397 −0.851985 0.523567i \(-0.824601\pi\)
−0.851985 + 0.523567i \(0.824601\pi\)
\(860\) −46.8701 −1.59826
\(861\) 0.343146 0.0116944
\(862\) −75.1127 −2.55835
\(863\) 8.68629 0.295685 0.147842 0.989011i \(-0.452767\pi\)
0.147842 + 0.989011i \(0.452767\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 19.7990 0.673186
\(866\) −26.3848 −0.896591
\(867\) −15.6274 −0.530735
\(868\) 17.1716 0.582841
\(869\) 0 0
\(870\) −3.41421 −0.115753
\(871\) 32.9706 1.11716
\(872\) −28.6274 −0.969447
\(873\) −9.75736 −0.330237
\(874\) −18.4853 −0.625274
\(875\) 1.41421 0.0478091
\(876\) −29.3137 −0.990418
\(877\) −34.9289 −1.17947 −0.589733 0.807598i \(-0.700767\pi\)
−0.589733 + 0.807598i \(0.700767\pi\)
\(878\) −52.2843 −1.76451
\(879\) −11.5147 −0.388382
\(880\) 6.72792 0.226798
\(881\) −5.79899 −0.195373 −0.0976865 0.995217i \(-0.531144\pi\)
−0.0976865 + 0.995217i \(0.531144\pi\)
\(882\) −12.0711 −0.406454
\(883\) −12.0416 −0.405233 −0.202617 0.979258i \(-0.564945\pi\)
−0.202617 + 0.979258i \(0.564945\pi\)
\(884\) −15.3137 −0.515056
\(885\) −12.4853 −0.419688
\(886\) 43.4558 1.45993
\(887\) 25.9411 0.871018 0.435509 0.900184i \(-0.356568\pi\)
0.435509 + 0.900184i \(0.356568\pi\)
\(888\) −15.0711 −0.505752
\(889\) −27.3137 −0.916072
\(890\) −14.2426 −0.477414
\(891\) −2.24264 −0.0751313
\(892\) 71.3137 2.38776
\(893\) 7.65685 0.256227
\(894\) −14.4853 −0.484460
\(895\) −16.4853 −0.551042
\(896\) 29.0711 0.971196
\(897\) −26.1421 −0.872861
\(898\) −64.8701 −2.16474
\(899\) −4.48528 −0.149593
\(900\) 3.82843 0.127614
\(901\) 9.37258 0.312246
\(902\) 1.31371 0.0437417
\(903\) −17.3137 −0.576164
\(904\) 44.7696 1.48901
\(905\) 20.8284 0.692360
\(906\) −15.6569 −0.520164
\(907\) −18.1421 −0.602400 −0.301200 0.953561i \(-0.597387\pi\)
−0.301200 + 0.953561i \(0.597387\pi\)
\(908\) 57.3137 1.90202
\(909\) −3.17157 −0.105194
\(910\) −11.6569 −0.386421
\(911\) 8.68629 0.287790 0.143895 0.989593i \(-0.454037\pi\)
0.143895 + 0.989593i \(0.454037\pi\)
\(912\) −3.00000 −0.0993399
\(913\) −28.7696 −0.952133
\(914\) 79.2548 2.62152
\(915\) −7.31371 −0.241784
\(916\) −86.6274 −2.86225
\(917\) 12.1421 0.400969
\(918\) 2.82843 0.0933520
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) −33.7990 −1.11432
\(921\) 21.1716 0.697627
\(922\) 47.4558 1.56287
\(923\) 36.9706 1.21690
\(924\) 12.1421 0.399447
\(925\) −3.41421 −0.112259
\(926\) 58.5269 1.92331
\(927\) −7.31371 −0.240214
\(928\) −2.24264 −0.0736183
\(929\) 33.1127 1.08639 0.543196 0.839606i \(-0.317214\pi\)
0.543196 + 0.839606i \(0.317214\pi\)
\(930\) 7.65685 0.251078
\(931\) 5.00000 0.163868
\(932\) −1.31371 −0.0430320
\(933\) −6.24264 −0.204375
\(934\) 86.0833 2.81673
\(935\) 2.62742 0.0859257
\(936\) −15.0711 −0.492613
\(937\) −29.7990 −0.973491 −0.486745 0.873544i \(-0.661816\pi\)
−0.486745 + 0.873544i \(0.661816\pi\)
\(938\) 32.9706 1.07653
\(939\) 5.79899 0.189243
\(940\) 29.3137 0.956108
\(941\) 22.8701 0.745543 0.372771 0.927923i \(-0.378408\pi\)
0.372771 + 0.927923i \(0.378408\pi\)
\(942\) 25.3137 0.824765
\(943\) −1.85786 −0.0605004
\(944\) 37.4558 1.21908
\(945\) 1.41421 0.0460044
\(946\) −66.2843 −2.15509
\(947\) −47.1716 −1.53287 −0.766435 0.642322i \(-0.777971\pi\)
−0.766435 + 0.642322i \(0.777971\pi\)
\(948\) 0 0
\(949\) 26.1421 0.848610
\(950\) −2.41421 −0.0783274
\(951\) −26.6274 −0.863453
\(952\) −7.31371 −0.237039
\(953\) −53.4558 −1.73160 −0.865802 0.500386i \(-0.833191\pi\)
−0.865802 + 0.500386i \(0.833191\pi\)
\(954\) 19.3137 0.625304
\(955\) −10.2426 −0.331444
\(956\) −102.326 −3.30946
\(957\) −3.17157 −0.102522
\(958\) 41.2132 1.33154
\(959\) −14.1421 −0.456673
\(960\) 9.82843 0.317211
\(961\) −20.9411 −0.675520
\(962\) 28.1421 0.907339
\(963\) 19.3137 0.622376
\(964\) −75.2548 −2.42379
\(965\) −5.07107 −0.163243
\(966\) −26.1421 −0.841109
\(967\) 8.04163 0.258601 0.129301 0.991605i \(-0.458727\pi\)
0.129301 + 0.991605i \(0.458727\pi\)
\(968\) −26.3553 −0.847093
\(969\) −1.17157 −0.0376363
\(970\) 23.5563 0.756349
\(971\) 22.3431 0.717026 0.358513 0.933525i \(-0.383284\pi\)
0.358513 + 0.933525i \(0.383284\pi\)
\(972\) 3.82843 0.122797
\(973\) 20.0000 0.641171
\(974\) 49.4558 1.58467
\(975\) −3.41421 −0.109342
\(976\) 21.9411 0.702318
\(977\) 32.2843 1.03287 0.516433 0.856328i \(-0.327260\pi\)
0.516433 + 0.856328i \(0.327260\pi\)
\(978\) 52.8701 1.69060
\(979\) −13.2304 −0.422847
\(980\) 19.1421 0.611473
\(981\) −6.48528 −0.207059
\(982\) −4.24264 −0.135388
\(983\) −24.6274 −0.785493 −0.392746 0.919647i \(-0.628475\pi\)
−0.392746 + 0.919647i \(0.628475\pi\)
\(984\) −1.07107 −0.0341444
\(985\) −6.82843 −0.217572
\(986\) 4.00000 0.127386
\(987\) 10.8284 0.344673
\(988\) 13.0711 0.415846
\(989\) 93.7401 2.98076
\(990\) 5.41421 0.172075
\(991\) 2.34315 0.0744325 0.0372162 0.999307i \(-0.488151\pi\)
0.0372162 + 0.999307i \(0.488151\pi\)
\(992\) 5.02944 0.159685
\(993\) 28.1421 0.893064
\(994\) 36.9706 1.17264
\(995\) −18.1421 −0.575144
\(996\) 49.1127 1.55620
\(997\) −38.0833 −1.20611 −0.603054 0.797700i \(-0.706050\pi\)
−0.603054 + 0.797700i \(0.706050\pi\)
\(998\) −12.4853 −0.395215
\(999\) −3.41421 −0.108021
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 285.2.a.g.1.2 2
3.2 odd 2 855.2.a.d.1.1 2
4.3 odd 2 4560.2.a.bf.1.2 2
5.2 odd 4 1425.2.c.l.799.4 4
5.3 odd 4 1425.2.c.l.799.1 4
5.4 even 2 1425.2.a.k.1.1 2
15.14 odd 2 4275.2.a.y.1.2 2
19.18 odd 2 5415.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.2 2 1.1 even 1 trivial
855.2.a.d.1.1 2 3.2 odd 2
1425.2.a.k.1.1 2 5.4 even 2
1425.2.c.l.799.1 4 5.3 odd 4
1425.2.c.l.799.4 4 5.2 odd 4
4275.2.a.y.1.2 2 15.14 odd 2
4560.2.a.bf.1.2 2 4.3 odd 2
5415.2.a.n.1.1 2 19.18 odd 2