Properties

Label 585.2.a.m.1.2
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} -3.41421 q^{11} -1.00000 q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427 q^{17} +0.585786 q^{19} -3.82843 q^{20} -8.24264 q^{22} -1.41421 q^{23} +1.00000 q^{25} -2.41421 q^{26} +18.4853 q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579 q^{32} -2.00000 q^{34} -4.82843 q^{35} -8.48528 q^{37} +1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} -11.0711 q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843 q^{47} +16.3137 q^{49} +2.41421 q^{50} -3.82843 q^{52} -2.48528 q^{53} +3.41421 q^{55} +21.3137 q^{56} +13.6569 q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264 q^{62} -9.82843 q^{64} +1.00000 q^{65} -2.00000 q^{67} -3.17157 q^{68} -11.6569 q^{70} -11.8995 q^{71} +8.48528 q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} +7.65685 q^{82} +3.17157 q^{83} +0.828427 q^{85} -26.7279 q^{86} -15.0711 q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421 q^{92} +11.6569 q^{94} -0.585786 q^{95} -7.65685 q^{97} +39.3848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) −2.41421 −0.763441
\(11\) −3.41421 −1.02942 −0.514712 0.857363i \(-0.672101\pi\)
−0.514712 + 0.857363i \(0.672101\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 11.6569 3.11543
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) −3.82843 −0.856062
\(21\) 0 0
\(22\) −8.24264 −1.75734
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.41421 −0.473466
\(27\) 0 0
\(28\) 18.4853 3.49339
\(29\) 5.65685 1.05045 0.525226 0.850963i \(-0.323981\pi\)
0.525226 + 0.850963i \(0.323981\pi\)
\(30\) 0 0
\(31\) 1.75736 0.315631 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.82843 −0.816153
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 1.41421 0.229416
\(39\) 0 0
\(40\) −4.41421 −0.697948
\(41\) 3.17157 0.495316 0.247658 0.968847i \(-0.420339\pi\)
0.247658 + 0.968847i \(0.420339\pi\)
\(42\) 0 0
\(43\) −11.0711 −1.68832 −0.844161 0.536090i \(-0.819901\pi\)
−0.844161 + 0.536090i \(0.819901\pi\)
\(44\) −13.0711 −1.97054
\(45\) 0 0
\(46\) −3.41421 −0.503398
\(47\) 4.82843 0.704298 0.352149 0.935944i \(-0.385451\pi\)
0.352149 + 0.935944i \(0.385451\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 2.41421 0.341421
\(51\) 0 0
\(52\) −3.82843 −0.530907
\(53\) −2.48528 −0.341380 −0.170690 0.985325i \(-0.554600\pi\)
−0.170690 + 0.985325i \(0.554600\pi\)
\(54\) 0 0
\(55\) 3.41421 0.460372
\(56\) 21.3137 2.84816
\(57\) 0 0
\(58\) 13.6569 1.79323
\(59\) −1.75736 −0.228789 −0.114394 0.993435i \(-0.536493\pi\)
−0.114394 + 0.993435i \(0.536493\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.24264 0.538816
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.17157 −0.384610
\(69\) 0 0
\(70\) −11.6569 −1.39326
\(71\) −11.8995 −1.41221 −0.706105 0.708107i \(-0.749549\pi\)
−0.706105 + 0.708107i \(0.749549\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −20.4853 −2.38137
\(75\) 0 0
\(76\) 2.24264 0.257249
\(77\) −16.4853 −1.87867
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 7.65685 0.845558
\(83\) 3.17157 0.348125 0.174063 0.984735i \(-0.444310\pi\)
0.174063 + 0.984735i \(0.444310\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) −26.7279 −2.88215
\(87\) 0 0
\(88\) −15.0711 −1.60658
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −4.82843 −0.506157
\(92\) −5.41421 −0.564471
\(93\) 0 0
\(94\) 11.6569 1.20231
\(95\) −0.585786 −0.0601004
\(96\) 0 0
\(97\) −7.65685 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) 39.3848 3.97846
\(99\) 0 0
\(100\) 3.82843 0.382843
\(101\) 3.65685 0.363871 0.181935 0.983311i \(-0.441764\pi\)
0.181935 + 0.983311i \(0.441764\pi\)
\(102\) 0 0
\(103\) 14.5858 1.43718 0.718590 0.695434i \(-0.244788\pi\)
0.718590 + 0.695434i \(0.244788\pi\)
\(104\) −4.41421 −0.432849
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 9.41421 0.910106 0.455053 0.890464i \(-0.349620\pi\)
0.455053 + 0.890464i \(0.349620\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 8.24264 0.785905
\(111\) 0 0
\(112\) 14.4853 1.36873
\(113\) 8.82843 0.830509 0.415254 0.909705i \(-0.363693\pi\)
0.415254 + 0.909705i \(0.363693\pi\)
\(114\) 0 0
\(115\) 1.41421 0.131876
\(116\) 21.6569 2.01079
\(117\) 0 0
\(118\) −4.24264 −0.390567
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 0.656854 0.0597140
\(122\) −19.3137 −1.74858
\(123\) 0 0
\(124\) 6.72792 0.604185
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.58579 −0.584394 −0.292197 0.956358i \(-0.594386\pi\)
−0.292197 + 0.956358i \(0.594386\pi\)
\(128\) −20.5563 −1.81694
\(129\) 0 0
\(130\) 2.41421 0.211741
\(131\) 16.9706 1.48272 0.741362 0.671105i \(-0.234180\pi\)
0.741362 + 0.671105i \(0.234180\pi\)
\(132\) 0 0
\(133\) 2.82843 0.245256
\(134\) −4.82843 −0.417113
\(135\) 0 0
\(136\) −3.65685 −0.313573
\(137\) 17.3137 1.47921 0.739605 0.673041i \(-0.235012\pi\)
0.739605 + 0.673041i \(0.235012\pi\)
\(138\) 0 0
\(139\) 4.48528 0.380437 0.190218 0.981742i \(-0.439080\pi\)
0.190218 + 0.981742i \(0.439080\pi\)
\(140\) −18.4853 −1.56229
\(141\) 0 0
\(142\) −28.7279 −2.41079
\(143\) 3.41421 0.285511
\(144\) 0 0
\(145\) −5.65685 −0.469776
\(146\) 20.4853 1.69537
\(147\) 0 0
\(148\) −32.4853 −2.67027
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) 0 0
\(151\) 9.75736 0.794043 0.397021 0.917809i \(-0.370044\pi\)
0.397021 + 0.917809i \(0.370044\pi\)
\(152\) 2.58579 0.209735
\(153\) 0 0
\(154\) −39.7990 −3.20709
\(155\) −1.75736 −0.141154
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −20.4853 −1.62972
\(159\) 0 0
\(160\) 1.58579 0.125367
\(161\) −6.82843 −0.538155
\(162\) 0 0
\(163\) 18.9706 1.48589 0.742945 0.669353i \(-0.233429\pi\)
0.742945 + 0.669353i \(0.233429\pi\)
\(164\) 12.1421 0.948141
\(165\) 0 0
\(166\) 7.65685 0.594287
\(167\) 3.17157 0.245424 0.122712 0.992442i \(-0.460841\pi\)
0.122712 + 0.992442i \(0.460841\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −42.3848 −3.23181
\(173\) −16.8284 −1.27944 −0.639721 0.768607i \(-0.720950\pi\)
−0.639721 + 0.768607i \(0.720950\pi\)
\(174\) 0 0
\(175\) 4.82843 0.364995
\(176\) −10.2426 −0.772068
\(177\) 0 0
\(178\) −14.4853 −1.08572
\(179\) −5.65685 −0.422813 −0.211407 0.977398i \(-0.567804\pi\)
−0.211407 + 0.977398i \(0.567804\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) −11.6569 −0.864064
\(183\) 0 0
\(184\) −6.24264 −0.460214
\(185\) 8.48528 0.623850
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) 18.4853 1.34818
\(189\) 0 0
\(190\) −1.41421 −0.102598
\(191\) −2.34315 −0.169544 −0.0847720 0.996400i \(-0.527016\pi\)
−0.0847720 + 0.996400i \(0.527016\pi\)
\(192\) 0 0
\(193\) 4.34315 0.312626 0.156313 0.987708i \(-0.450039\pi\)
0.156313 + 0.987708i \(0.450039\pi\)
\(194\) −18.4853 −1.32717
\(195\) 0 0
\(196\) 62.4558 4.46113
\(197\) −10.9706 −0.781620 −0.390810 0.920471i \(-0.627805\pi\)
−0.390810 + 0.920471i \(0.627805\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 4.41421 0.312132
\(201\) 0 0
\(202\) 8.82843 0.621166
\(203\) 27.3137 1.91705
\(204\) 0 0
\(205\) −3.17157 −0.221512
\(206\) 35.2132 2.45342
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 3.31371 0.228125 0.114063 0.993474i \(-0.463614\pi\)
0.114063 + 0.993474i \(0.463614\pi\)
\(212\) −9.51472 −0.653474
\(213\) 0 0
\(214\) 22.7279 1.55365
\(215\) 11.0711 0.755041
\(216\) 0 0
\(217\) 8.48528 0.576018
\(218\) −4.82843 −0.327022
\(219\) 0 0
\(220\) 13.0711 0.881251
\(221\) 0.828427 0.0557260
\(222\) 0 0
\(223\) 9.51472 0.637153 0.318576 0.947897i \(-0.396795\pi\)
0.318576 + 0.947897i \(0.396795\pi\)
\(224\) −7.65685 −0.511595
\(225\) 0 0
\(226\) 21.3137 1.41777
\(227\) 16.3431 1.08473 0.542366 0.840142i \(-0.317528\pi\)
0.542366 + 0.840142i \(0.317528\pi\)
\(228\) 0 0
\(229\) −4.82843 −0.319071 −0.159536 0.987192i \(-0.551000\pi\)
−0.159536 + 0.987192i \(0.551000\pi\)
\(230\) 3.41421 0.225127
\(231\) 0 0
\(232\) 24.9706 1.63940
\(233\) 20.6274 1.35135 0.675674 0.737201i \(-0.263853\pi\)
0.675674 + 0.737201i \(0.263853\pi\)
\(234\) 0 0
\(235\) −4.82843 −0.314972
\(236\) −6.72792 −0.437950
\(237\) 0 0
\(238\) −9.65685 −0.625961
\(239\) 3.41421 0.220847 0.110424 0.993885i \(-0.464779\pi\)
0.110424 + 0.993885i \(0.464779\pi\)
\(240\) 0 0
\(241\) −14.4853 −0.933079 −0.466539 0.884500i \(-0.654499\pi\)
−0.466539 + 0.884500i \(0.654499\pi\)
\(242\) 1.58579 0.101938
\(243\) 0 0
\(244\) −30.6274 −1.96072
\(245\) −16.3137 −1.04224
\(246\) 0 0
\(247\) −0.585786 −0.0372727
\(248\) 7.75736 0.492593
\(249\) 0 0
\(250\) −2.41421 −0.152688
\(251\) −19.7990 −1.24970 −0.624851 0.780744i \(-0.714840\pi\)
−0.624851 + 0.780744i \(0.714840\pi\)
\(252\) 0 0
\(253\) 4.82843 0.303561
\(254\) −15.8995 −0.997623
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) −27.6569 −1.72519 −0.862594 0.505898i \(-0.831161\pi\)
−0.862594 + 0.505898i \(0.831161\pi\)
\(258\) 0 0
\(259\) −40.9706 −2.54579
\(260\) 3.82843 0.237429
\(261\) 0 0
\(262\) 40.9706 2.53117
\(263\) 10.5858 0.652748 0.326374 0.945241i \(-0.394173\pi\)
0.326374 + 0.945241i \(0.394173\pi\)
\(264\) 0 0
\(265\) 2.48528 0.152670
\(266\) 6.82843 0.418678
\(267\) 0 0
\(268\) −7.65685 −0.467717
\(269\) 25.3137 1.54340 0.771702 0.635984i \(-0.219406\pi\)
0.771702 + 0.635984i \(0.219406\pi\)
\(270\) 0 0
\(271\) 26.7279 1.62361 0.811803 0.583932i \(-0.198486\pi\)
0.811803 + 0.583932i \(0.198486\pi\)
\(272\) −2.48528 −0.150692
\(273\) 0 0
\(274\) 41.7990 2.52517
\(275\) −3.41421 −0.205885
\(276\) 0 0
\(277\) −12.8284 −0.770785 −0.385393 0.922753i \(-0.625934\pi\)
−0.385393 + 0.922753i \(0.625934\pi\)
\(278\) 10.8284 0.649446
\(279\) 0 0
\(280\) −21.3137 −1.27374
\(281\) −21.7990 −1.30042 −0.650209 0.759755i \(-0.725319\pi\)
−0.650209 + 0.759755i \(0.725319\pi\)
\(282\) 0 0
\(283\) −16.7279 −0.994372 −0.497186 0.867644i \(-0.665633\pi\)
−0.497186 + 0.867644i \(0.665633\pi\)
\(284\) −45.5563 −2.70327
\(285\) 0 0
\(286\) 8.24264 0.487398
\(287\) 15.3137 0.903940
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) −13.6569 −0.801958
\(291\) 0 0
\(292\) 32.4853 1.90106
\(293\) −26.1421 −1.52724 −0.763620 0.645666i \(-0.776580\pi\)
−0.763620 + 0.645666i \(0.776580\pi\)
\(294\) 0 0
\(295\) 1.75736 0.102317
\(296\) −37.4558 −2.17708
\(297\) 0 0
\(298\) 28.1421 1.63023
\(299\) 1.41421 0.0817861
\(300\) 0 0
\(301\) −53.4558 −3.08114
\(302\) 23.5563 1.35552
\(303\) 0 0
\(304\) 1.75736 0.100791
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 24.8284 1.41703 0.708517 0.705694i \(-0.249365\pi\)
0.708517 + 0.705694i \(0.249365\pi\)
\(308\) −63.1127 −3.59618
\(309\) 0 0
\(310\) −4.24264 −0.240966
\(311\) −8.48528 −0.481156 −0.240578 0.970630i \(-0.577337\pi\)
−0.240578 + 0.970630i \(0.577337\pi\)
\(312\) 0 0
\(313\) −4.82843 −0.272919 −0.136459 0.990646i \(-0.543572\pi\)
−0.136459 + 0.990646i \(0.543572\pi\)
\(314\) 43.4558 2.45236
\(315\) 0 0
\(316\) −32.4853 −1.82744
\(317\) 2.14214 0.120314 0.0601572 0.998189i \(-0.480840\pi\)
0.0601572 + 0.998189i \(0.480840\pi\)
\(318\) 0 0
\(319\) −19.3137 −1.08136
\(320\) 9.82843 0.549426
\(321\) 0 0
\(322\) −16.4853 −0.918689
\(323\) −0.485281 −0.0270018
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 45.7990 2.53657
\(327\) 0 0
\(328\) 14.0000 0.773021
\(329\) 23.3137 1.28533
\(330\) 0 0
\(331\) −26.0416 −1.43138 −0.715689 0.698419i \(-0.753887\pi\)
−0.715689 + 0.698419i \(0.753887\pi\)
\(332\) 12.1421 0.666386
\(333\) 0 0
\(334\) 7.65685 0.418964
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 12.8284 0.698809 0.349404 0.936972i \(-0.386384\pi\)
0.349404 + 0.936972i \(0.386384\pi\)
\(338\) 2.41421 0.131316
\(339\) 0 0
\(340\) 3.17157 0.172003
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) −48.8701 −2.63490
\(345\) 0 0
\(346\) −40.6274 −2.18414
\(347\) 4.24264 0.227757 0.113878 0.993495i \(-0.463673\pi\)
0.113878 + 0.993495i \(0.463673\pi\)
\(348\) 0 0
\(349\) 18.4853 0.989494 0.494747 0.869037i \(-0.335261\pi\)
0.494747 + 0.869037i \(0.335261\pi\)
\(350\) 11.6569 0.623085
\(351\) 0 0
\(352\) 5.41421 0.288579
\(353\) −14.8284 −0.789238 −0.394619 0.918845i \(-0.629123\pi\)
−0.394619 + 0.918845i \(0.629123\pi\)
\(354\) 0 0
\(355\) 11.8995 0.631560
\(356\) −22.9706 −1.21744
\(357\) 0 0
\(358\) −13.6569 −0.721787
\(359\) 8.10051 0.427528 0.213764 0.976885i \(-0.431428\pi\)
0.213764 + 0.976885i \(0.431428\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 0 0
\(363\) 0 0
\(364\) −18.4853 −0.968892
\(365\) −8.48528 −0.444140
\(366\) 0 0
\(367\) 35.5563 1.85603 0.928013 0.372547i \(-0.121516\pi\)
0.928013 + 0.372547i \(0.121516\pi\)
\(368\) −4.24264 −0.221163
\(369\) 0 0
\(370\) 20.4853 1.06498
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) 6.82843 0.353090
\(375\) 0 0
\(376\) 21.3137 1.09917
\(377\) −5.65685 −0.291343
\(378\) 0 0
\(379\) 29.0711 1.49328 0.746640 0.665228i \(-0.231666\pi\)
0.746640 + 0.665228i \(0.231666\pi\)
\(380\) −2.24264 −0.115045
\(381\) 0 0
\(382\) −5.65685 −0.289430
\(383\) 29.1127 1.48759 0.743795 0.668408i \(-0.233024\pi\)
0.743795 + 0.668408i \(0.233024\pi\)
\(384\) 0 0
\(385\) 16.4853 0.840168
\(386\) 10.4853 0.533687
\(387\) 0 0
\(388\) −29.3137 −1.48818
\(389\) −28.6274 −1.45147 −0.725734 0.687976i \(-0.758500\pi\)
−0.725734 + 0.687976i \(0.758500\pi\)
\(390\) 0 0
\(391\) 1.17157 0.0592490
\(392\) 72.0122 3.63717
\(393\) 0 0
\(394\) −26.4853 −1.33431
\(395\) 8.48528 0.426941
\(396\) 0 0
\(397\) 11.7990 0.592174 0.296087 0.955161i \(-0.404318\pi\)
0.296087 + 0.955161i \(0.404318\pi\)
\(398\) 9.65685 0.484054
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) 0 0
\(403\) −1.75736 −0.0875403
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 65.9411 3.27260
\(407\) 28.9706 1.43602
\(408\) 0 0
\(409\) 7.17157 0.354611 0.177306 0.984156i \(-0.443262\pi\)
0.177306 + 0.984156i \(0.443262\pi\)
\(410\) −7.65685 −0.378145
\(411\) 0 0
\(412\) 55.8406 2.75107
\(413\) −8.48528 −0.417533
\(414\) 0 0
\(415\) −3.17157 −0.155686
\(416\) 1.58579 0.0777496
\(417\) 0 0
\(418\) −4.82843 −0.236166
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 0 0
\(421\) −34.9706 −1.70436 −0.852180 0.523248i \(-0.824720\pi\)
−0.852180 + 0.523248i \(0.824720\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) −10.9706 −0.532778
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) −38.6274 −1.86931
\(428\) 36.0416 1.74214
\(429\) 0 0
\(430\) 26.7279 1.28893
\(431\) −40.3848 −1.94527 −0.972633 0.232346i \(-0.925360\pi\)
−0.972633 + 0.232346i \(0.925360\pi\)
\(432\) 0 0
\(433\) 7.65685 0.367965 0.183982 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(434\) 20.4853 0.983325
\(435\) 0 0
\(436\) −7.65685 −0.366697
\(437\) −0.828427 −0.0396290
\(438\) 0 0
\(439\) 0.970563 0.0463224 0.0231612 0.999732i \(-0.492627\pi\)
0.0231612 + 0.999732i \(0.492627\pi\)
\(440\) 15.0711 0.718485
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 9.41421 0.447283 0.223641 0.974671i \(-0.428206\pi\)
0.223641 + 0.974671i \(0.428206\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 22.9706 1.08769
\(447\) 0 0
\(448\) −47.4558 −2.24208
\(449\) 33.1127 1.56268 0.781342 0.624103i \(-0.214535\pi\)
0.781342 + 0.624103i \(0.214535\pi\)
\(450\) 0 0
\(451\) −10.8284 −0.509891
\(452\) 33.7990 1.58977
\(453\) 0 0
\(454\) 39.4558 1.85175
\(455\) 4.82843 0.226360
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −11.6569 −0.544689
\(459\) 0 0
\(460\) 5.41421 0.252439
\(461\) −9.51472 −0.443145 −0.221572 0.975144i \(-0.571119\pi\)
−0.221572 + 0.975144i \(0.571119\pi\)
\(462\) 0 0
\(463\) −4.34315 −0.201843 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(464\) 16.9706 0.787839
\(465\) 0 0
\(466\) 49.7990 2.30689
\(467\) 13.4142 0.620736 0.310368 0.950617i \(-0.399548\pi\)
0.310368 + 0.950617i \(0.399548\pi\)
\(468\) 0 0
\(469\) −9.65685 −0.445912
\(470\) −11.6569 −0.537691
\(471\) 0 0
\(472\) −7.75736 −0.357061
\(473\) 37.7990 1.73800
\(474\) 0 0
\(475\) 0.585786 0.0268777
\(476\) −15.3137 −0.701903
\(477\) 0 0
\(478\) 8.24264 0.377010
\(479\) 30.7279 1.40399 0.701997 0.712180i \(-0.252292\pi\)
0.701997 + 0.712180i \(0.252292\pi\)
\(480\) 0 0
\(481\) 8.48528 0.386896
\(482\) −34.9706 −1.59287
\(483\) 0 0
\(484\) 2.51472 0.114305
\(485\) 7.65685 0.347680
\(486\) 0 0
\(487\) −10.9706 −0.497124 −0.248562 0.968616i \(-0.579958\pi\)
−0.248562 + 0.968616i \(0.579958\pi\)
\(488\) −35.3137 −1.59858
\(489\) 0 0
\(490\) −39.3848 −1.77922
\(491\) −5.17157 −0.233390 −0.116695 0.993168i \(-0.537230\pi\)
−0.116695 + 0.993168i \(0.537230\pi\)
\(492\) 0 0
\(493\) −4.68629 −0.211060
\(494\) −1.41421 −0.0636285
\(495\) 0 0
\(496\) 5.27208 0.236723
\(497\) −57.4558 −2.57725
\(498\) 0 0
\(499\) 41.5563 1.86032 0.930159 0.367157i \(-0.119669\pi\)
0.930159 + 0.367157i \(0.119669\pi\)
\(500\) −3.82843 −0.171212
\(501\) 0 0
\(502\) −47.7990 −2.13337
\(503\) −37.8995 −1.68985 −0.844927 0.534881i \(-0.820356\pi\)
−0.844927 + 0.534881i \(0.820356\pi\)
\(504\) 0 0
\(505\) −3.65685 −0.162728
\(506\) 11.6569 0.518210
\(507\) 0 0
\(508\) −25.2132 −1.11866
\(509\) −41.1127 −1.82229 −0.911144 0.412088i \(-0.864800\pi\)
−0.911144 + 0.412088i \(0.864800\pi\)
\(510\) 0 0
\(511\) 40.9706 1.81243
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) −66.7696 −2.94508
\(515\) −14.5858 −0.642727
\(516\) 0 0
\(517\) −16.4853 −0.725022
\(518\) −98.9117 −4.34593
\(519\) 0 0
\(520\) 4.41421 0.193576
\(521\) 17.6569 0.773561 0.386780 0.922172i \(-0.373587\pi\)
0.386780 + 0.922172i \(0.373587\pi\)
\(522\) 0 0
\(523\) −19.7574 −0.863929 −0.431965 0.901891i \(-0.642179\pi\)
−0.431965 + 0.901891i \(0.642179\pi\)
\(524\) 64.9706 2.83825
\(525\) 0 0
\(526\) 25.5563 1.11431
\(527\) −1.45584 −0.0634176
\(528\) 0 0
\(529\) −21.0000 −0.913043
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 10.8284 0.469472
\(533\) −3.17157 −0.137376
\(534\) 0 0
\(535\) −9.41421 −0.407012
\(536\) −8.82843 −0.381330
\(537\) 0 0
\(538\) 61.1127 2.63476
\(539\) −55.6985 −2.39910
\(540\) 0 0
\(541\) −7.17157 −0.308330 −0.154165 0.988045i \(-0.549269\pi\)
−0.154165 + 0.988045i \(0.549269\pi\)
\(542\) 64.5269 2.77167
\(543\) 0 0
\(544\) 1.31371 0.0563248
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 13.2132 0.564956 0.282478 0.959274i \(-0.408844\pi\)
0.282478 + 0.959274i \(0.408844\pi\)
\(548\) 66.2843 2.83152
\(549\) 0 0
\(550\) −8.24264 −0.351467
\(551\) 3.31371 0.141169
\(552\) 0 0
\(553\) −40.9706 −1.74225
\(554\) −30.9706 −1.31581
\(555\) 0 0
\(556\) 17.1716 0.728237
\(557\) 35.7990 1.51685 0.758426 0.651759i \(-0.225969\pi\)
0.758426 + 0.651759i \(0.225969\pi\)
\(558\) 0 0
\(559\) 11.0711 0.468256
\(560\) −14.4853 −0.612115
\(561\) 0 0
\(562\) −52.6274 −2.21995
\(563\) 7.75736 0.326934 0.163467 0.986549i \(-0.447732\pi\)
0.163467 + 0.986549i \(0.447732\pi\)
\(564\) 0 0
\(565\) −8.82843 −0.371415
\(566\) −40.3848 −1.69750
\(567\) 0 0
\(568\) −52.5269 −2.20398
\(569\) 10.3431 0.433607 0.216804 0.976215i \(-0.430437\pi\)
0.216804 + 0.976215i \(0.430437\pi\)
\(570\) 0 0
\(571\) −11.5147 −0.481876 −0.240938 0.970541i \(-0.577455\pi\)
−0.240938 + 0.970541i \(0.577455\pi\)
\(572\) 13.0711 0.546529
\(573\) 0 0
\(574\) 36.9706 1.54312
\(575\) −1.41421 −0.0589768
\(576\) 0 0
\(577\) −34.8284 −1.44993 −0.724963 0.688788i \(-0.758143\pi\)
−0.724963 + 0.688788i \(0.758143\pi\)
\(578\) −39.3848 −1.63819
\(579\) 0 0
\(580\) −21.6569 −0.899252
\(581\) 15.3137 0.635320
\(582\) 0 0
\(583\) 8.48528 0.351424
\(584\) 37.4558 1.54993
\(585\) 0 0
\(586\) −63.1127 −2.60716
\(587\) −20.3431 −0.839651 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(588\) 0 0
\(589\) 1.02944 0.0424172
\(590\) 4.24264 0.174667
\(591\) 0 0
\(592\) −25.4558 −1.04623
\(593\) 24.6274 1.01133 0.505663 0.862731i \(-0.331248\pi\)
0.505663 + 0.862731i \(0.331248\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 44.6274 1.82801
\(597\) 0 0
\(598\) 3.41421 0.139618
\(599\) −25.4558 −1.04010 −0.520049 0.854137i \(-0.674086\pi\)
−0.520049 + 0.854137i \(0.674086\pi\)
\(600\) 0 0
\(601\) −44.6274 −1.82039 −0.910195 0.414180i \(-0.864069\pi\)
−0.910195 + 0.414180i \(0.864069\pi\)
\(602\) −129.054 −5.25984
\(603\) 0 0
\(604\) 37.3553 1.51997
\(605\) −0.656854 −0.0267049
\(606\) 0 0
\(607\) 31.7574 1.28899 0.644496 0.764608i \(-0.277067\pi\)
0.644496 + 0.764608i \(0.277067\pi\)
\(608\) −0.928932 −0.0376732
\(609\) 0 0
\(610\) 19.3137 0.781989
\(611\) −4.82843 −0.195337
\(612\) 0 0
\(613\) −14.6863 −0.593174 −0.296587 0.955006i \(-0.595848\pi\)
−0.296587 + 0.955006i \(0.595848\pi\)
\(614\) 59.9411 2.41903
\(615\) 0 0
\(616\) −72.7696 −2.93197
\(617\) −10.9706 −0.441658 −0.220829 0.975313i \(-0.570876\pi\)
−0.220829 + 0.975313i \(0.570876\pi\)
\(618\) 0 0
\(619\) 1.75736 0.0706342 0.0353171 0.999376i \(-0.488756\pi\)
0.0353171 + 0.999376i \(0.488756\pi\)
\(620\) −6.72792 −0.270200
\(621\) 0 0
\(622\) −20.4853 −0.821385
\(623\) −28.9706 −1.16068
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.6569 −0.465902
\(627\) 0 0
\(628\) 68.9117 2.74988
\(629\) 7.02944 0.280282
\(630\) 0 0
\(631\) −9.75736 −0.388434 −0.194217 0.980959i \(-0.562217\pi\)
−0.194217 + 0.980959i \(0.562217\pi\)
\(632\) −37.4558 −1.48991
\(633\) 0 0
\(634\) 5.17157 0.205389
\(635\) 6.58579 0.261349
\(636\) 0 0
\(637\) −16.3137 −0.646373
\(638\) −46.6274 −1.84600
\(639\) 0 0
\(640\) 20.5563 0.812561
\(641\) −47.6569 −1.88233 −0.941166 0.337944i \(-0.890269\pi\)
−0.941166 + 0.337944i \(0.890269\pi\)
\(642\) 0 0
\(643\) 9.51472 0.375224 0.187612 0.982243i \(-0.439925\pi\)
0.187612 + 0.982243i \(0.439925\pi\)
\(644\) −26.1421 −1.03014
\(645\) 0 0
\(646\) −1.17157 −0.0460949
\(647\) −9.41421 −0.370111 −0.185055 0.982728i \(-0.559246\pi\)
−0.185055 + 0.982728i \(0.559246\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) −2.41421 −0.0946932
\(651\) 0 0
\(652\) 72.6274 2.84431
\(653\) −46.9706 −1.83810 −0.919050 0.394141i \(-0.871042\pi\)
−0.919050 + 0.394141i \(0.871042\pi\)
\(654\) 0 0
\(655\) −16.9706 −0.663095
\(656\) 9.51472 0.371487
\(657\) 0 0
\(658\) 56.2843 2.19419
\(659\) −17.8579 −0.695644 −0.347822 0.937561i \(-0.613079\pi\)
−0.347822 + 0.937561i \(0.613079\pi\)
\(660\) 0 0
\(661\) 29.5980 1.15123 0.575614 0.817722i \(-0.304763\pi\)
0.575614 + 0.817722i \(0.304763\pi\)
\(662\) −62.8701 −2.44351
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) −2.82843 −0.109682
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 12.1421 0.469793
\(669\) 0 0
\(670\) 4.82843 0.186538
\(671\) 27.3137 1.05443
\(672\) 0 0
\(673\) 6.48528 0.249989 0.124995 0.992157i \(-0.460109\pi\)
0.124995 + 0.992157i \(0.460109\pi\)
\(674\) 30.9706 1.19294
\(675\) 0 0
\(676\) 3.82843 0.147247
\(677\) 20.1421 0.774125 0.387063 0.922053i \(-0.373490\pi\)
0.387063 + 0.922053i \(0.373490\pi\)
\(678\) 0 0
\(679\) −36.9706 −1.41880
\(680\) 3.65685 0.140234
\(681\) 0 0
\(682\) −14.4853 −0.554670
\(683\) −10.6863 −0.408900 −0.204450 0.978877i \(-0.565541\pi\)
−0.204450 + 0.978877i \(0.565541\pi\)
\(684\) 0 0
\(685\) −17.3137 −0.661523
\(686\) 108.569 4.14517
\(687\) 0 0
\(688\) −33.2132 −1.26624
\(689\) 2.48528 0.0946817
\(690\) 0 0
\(691\) 6.92893 0.263589 0.131795 0.991277i \(-0.457926\pi\)
0.131795 + 0.991277i \(0.457926\pi\)
\(692\) −64.4264 −2.44912
\(693\) 0 0
\(694\) 10.2426 0.388805
\(695\) −4.48528 −0.170136
\(696\) 0 0
\(697\) −2.62742 −0.0995205
\(698\) 44.6274 1.68917
\(699\) 0 0
\(700\) 18.4853 0.698678
\(701\) −14.6863 −0.554694 −0.277347 0.960770i \(-0.589455\pi\)
−0.277347 + 0.960770i \(0.589455\pi\)
\(702\) 0 0
\(703\) −4.97056 −0.187468
\(704\) 33.5563 1.26470
\(705\) 0 0
\(706\) −35.7990 −1.34731
\(707\) 17.6569 0.664054
\(708\) 0 0
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) 28.7279 1.07814
\(711\) 0 0
\(712\) −26.4853 −0.992578
\(713\) −2.48528 −0.0930745
\(714\) 0 0
\(715\) −3.41421 −0.127684
\(716\) −21.6569 −0.809355
\(717\) 0 0
\(718\) 19.5563 0.729836
\(719\) −28.9706 −1.08042 −0.540210 0.841530i \(-0.681655\pi\)
−0.540210 + 0.841530i \(0.681655\pi\)
\(720\) 0 0
\(721\) 70.4264 2.62282
\(722\) −45.0416 −1.67628
\(723\) 0 0
\(724\) 0 0
\(725\) 5.65685 0.210090
\(726\) 0 0
\(727\) −51.3553 −1.90466 −0.952332 0.305063i \(-0.901322\pi\)
−0.952332 + 0.305063i \(0.901322\pi\)
\(728\) −21.3137 −0.789939
\(729\) 0 0
\(730\) −20.4853 −0.758194
\(731\) 9.17157 0.339223
\(732\) 0 0
\(733\) 21.3137 0.787240 0.393620 0.919273i \(-0.371223\pi\)
0.393620 + 0.919273i \(0.371223\pi\)
\(734\) 85.8406 3.16844
\(735\) 0 0
\(736\) 2.24264 0.0826648
\(737\) 6.82843 0.251528
\(738\) 0 0
\(739\) 5.27208 0.193937 0.0969683 0.995287i \(-0.469085\pi\)
0.0969683 + 0.995287i \(0.469085\pi\)
\(740\) 32.4853 1.19418
\(741\) 0 0
\(742\) −28.9706 −1.06354
\(743\) 21.5147 0.789298 0.394649 0.918832i \(-0.370866\pi\)
0.394649 + 0.918832i \(0.370866\pi\)
\(744\) 0 0
\(745\) −11.6569 −0.427074
\(746\) −6.48528 −0.237443
\(747\) 0 0
\(748\) 10.8284 0.395927
\(749\) 45.4558 1.66092
\(750\) 0 0
\(751\) −27.5147 −1.00403 −0.502013 0.864860i \(-0.667407\pi\)
−0.502013 + 0.864860i \(0.667407\pi\)
\(752\) 14.4853 0.528224
\(753\) 0 0
\(754\) −13.6569 −0.497353
\(755\) −9.75736 −0.355107
\(756\) 0 0
\(757\) −24.1421 −0.877461 −0.438730 0.898619i \(-0.644572\pi\)
−0.438730 + 0.898619i \(0.644572\pi\)
\(758\) 70.1838 2.54919
\(759\) 0 0
\(760\) −2.58579 −0.0937963
\(761\) −8.62742 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(762\) 0 0
\(763\) −9.65685 −0.349602
\(764\) −8.97056 −0.324544
\(765\) 0 0
\(766\) 70.2843 2.53947
\(767\) 1.75736 0.0634546
\(768\) 0 0
\(769\) −22.9706 −0.828340 −0.414170 0.910200i \(-0.635928\pi\)
−0.414170 + 0.910200i \(0.635928\pi\)
\(770\) 39.7990 1.43426
\(771\) 0 0
\(772\) 16.6274 0.598434
\(773\) 22.1421 0.796397 0.398199 0.917299i \(-0.369635\pi\)
0.398199 + 0.917299i \(0.369635\pi\)
\(774\) 0 0
\(775\) 1.75736 0.0631262
\(776\) −33.7990 −1.21331
\(777\) 0 0
\(778\) −69.1127 −2.47781
\(779\) 1.85786 0.0665649
\(780\) 0 0
\(781\) 40.6274 1.45376
\(782\) 2.82843 0.101144
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) 22.4853 0.801514 0.400757 0.916184i \(-0.368747\pi\)
0.400757 + 0.916184i \(0.368747\pi\)
\(788\) −42.0000 −1.49619
\(789\) 0 0
\(790\) 20.4853 0.728834
\(791\) 42.6274 1.51566
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 28.4853 1.01090
\(795\) 0 0
\(796\) 15.3137 0.542780
\(797\) 22.9706 0.813659 0.406830 0.913504i \(-0.366634\pi\)
0.406830 + 0.913504i \(0.366634\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) −1.58579 −0.0560660
\(801\) 0 0
\(802\) 12.8284 0.452988
\(803\) −28.9706 −1.02235
\(804\) 0 0
\(805\) 6.82843 0.240670
\(806\) −4.24264 −0.149441
\(807\) 0 0
\(808\) 16.1421 0.567878
\(809\) −45.2548 −1.59108 −0.795538 0.605904i \(-0.792811\pi\)
−0.795538 + 0.605904i \(0.792811\pi\)
\(810\) 0 0
\(811\) −28.3848 −0.996724 −0.498362 0.866969i \(-0.666065\pi\)
−0.498362 + 0.866969i \(0.666065\pi\)
\(812\) 104.569 3.66964
\(813\) 0 0
\(814\) 69.9411 2.45144
\(815\) −18.9706 −0.664510
\(816\) 0 0
\(817\) −6.48528 −0.226891
\(818\) 17.3137 0.605360
\(819\) 0 0
\(820\) −12.1421 −0.424022
\(821\) 51.2548 1.78881 0.894403 0.447262i \(-0.147601\pi\)
0.894403 + 0.447262i \(0.147601\pi\)
\(822\) 0 0
\(823\) −2.38478 −0.0831281 −0.0415640 0.999136i \(-0.513234\pi\)
−0.0415640 + 0.999136i \(0.513234\pi\)
\(824\) 64.3848 2.24295
\(825\) 0 0
\(826\) −20.4853 −0.712774
\(827\) −56.1421 −1.95225 −0.976127 0.217202i \(-0.930307\pi\)
−0.976127 + 0.217202i \(0.930307\pi\)
\(828\) 0 0
\(829\) 40.9706 1.42297 0.711483 0.702703i \(-0.248024\pi\)
0.711483 + 0.702703i \(0.248024\pi\)
\(830\) −7.65685 −0.265773
\(831\) 0 0
\(832\) 9.82843 0.340739
\(833\) −13.5147 −0.468257
\(834\) 0 0
\(835\) −3.17157 −0.109757
\(836\) −7.65685 −0.264818
\(837\) 0 0
\(838\) −26.1421 −0.903065
\(839\) −6.72792 −0.232274 −0.116137 0.993233i \(-0.537051\pi\)
−0.116137 + 0.993233i \(0.537051\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) −84.4264 −2.90953
\(843\) 0 0
\(844\) 12.6863 0.436680
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 3.17157 0.108977
\(848\) −7.45584 −0.256035
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −13.4558 −0.460719 −0.230360 0.973106i \(-0.573990\pi\)
−0.230360 + 0.973106i \(0.573990\pi\)
\(854\) −93.2548 −3.19111
\(855\) 0 0
\(856\) 41.5563 1.42037
\(857\) −11.6569 −0.398191 −0.199095 0.979980i \(-0.563800\pi\)
−0.199095 + 0.979980i \(0.563800\pi\)
\(858\) 0 0
\(859\) −27.7990 −0.948489 −0.474245 0.880393i \(-0.657279\pi\)
−0.474245 + 0.880393i \(0.657279\pi\)
\(860\) 42.3848 1.44531
\(861\) 0 0
\(862\) −97.4975 −3.32078
\(863\) 31.4558 1.07077 0.535385 0.844608i \(-0.320167\pi\)
0.535385 + 0.844608i \(0.320167\pi\)
\(864\) 0 0
\(865\) 16.8284 0.572184
\(866\) 18.4853 0.628155
\(867\) 0 0
\(868\) 32.4853 1.10262
\(869\) 28.9706 0.982759
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) −8.82843 −0.298968
\(873\) 0 0
\(874\) −2.00000 −0.0676510
\(875\) −4.82843 −0.163231
\(876\) 0 0
\(877\) 25.3137 0.854783 0.427392 0.904067i \(-0.359433\pi\)
0.427392 + 0.904067i \(0.359433\pi\)
\(878\) 2.34315 0.0790773
\(879\) 0 0
\(880\) 10.2426 0.345279
\(881\) −19.0294 −0.641118 −0.320559 0.947229i \(-0.603871\pi\)
−0.320559 + 0.947229i \(0.603871\pi\)
\(882\) 0 0
\(883\) −23.7574 −0.799499 −0.399749 0.916624i \(-0.630903\pi\)
−0.399749 + 0.916624i \(0.630903\pi\)
\(884\) 3.17157 0.106672
\(885\) 0 0
\(886\) 22.7279 0.763559
\(887\) 22.3848 0.751607 0.375804 0.926699i \(-0.377367\pi\)
0.375804 + 0.926699i \(0.377367\pi\)
\(888\) 0 0
\(889\) −31.7990 −1.06650
\(890\) 14.4853 0.485548
\(891\) 0 0
\(892\) 36.4264 1.21965
\(893\) 2.82843 0.0946497
\(894\) 0 0
\(895\) 5.65685 0.189088
\(896\) −99.2548 −3.31587
\(897\) 0 0
\(898\) 79.9411 2.66767
\(899\) 9.94113 0.331555
\(900\) 0 0
\(901\) 2.05887 0.0685911
\(902\) −26.1421 −0.870438
\(903\) 0 0
\(904\) 38.9706 1.29614
\(905\) 0 0
\(906\) 0 0
\(907\) 9.21320 0.305919 0.152960 0.988232i \(-0.451120\pi\)
0.152960 + 0.988232i \(0.451120\pi\)
\(908\) 62.5685 2.07641
\(909\) 0 0
\(910\) 11.6569 0.386421
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −10.8284 −0.358369
\(914\) −43.4558 −1.43739
\(915\) 0 0
\(916\) −18.4853 −0.610771
\(917\) 81.9411 2.70593
\(918\) 0 0
\(919\) 0.485281 0.0160080 0.00800398 0.999968i \(-0.497452\pi\)
0.00800398 + 0.999968i \(0.497452\pi\)
\(920\) 6.24264 0.205814
\(921\) 0 0
\(922\) −22.9706 −0.756495
\(923\) 11.8995 0.391677
\(924\) 0 0
\(925\) −8.48528 −0.278994
\(926\) −10.4853 −0.344568
\(927\) 0 0
\(928\) −8.97056 −0.294473
\(929\) −16.8284 −0.552123 −0.276061 0.961140i \(-0.589029\pi\)
−0.276061 + 0.961140i \(0.589029\pi\)
\(930\) 0 0
\(931\) 9.55635 0.313197
\(932\) 78.9706 2.58677
\(933\) 0 0
\(934\) 32.3848 1.05966
\(935\) −2.82843 −0.0924995
\(936\) 0 0
\(937\) −22.9706 −0.750416 −0.375208 0.926941i \(-0.622429\pi\)
−0.375208 + 0.926941i \(0.622429\pi\)
\(938\) −23.3137 −0.761220
\(939\) 0 0
\(940\) −18.4853 −0.602923
\(941\) −18.7696 −0.611870 −0.305935 0.952052i \(-0.598969\pi\)
−0.305935 + 0.952052i \(0.598969\pi\)
\(942\) 0 0
\(943\) −4.48528 −0.146061
\(944\) −5.27208 −0.171592
\(945\) 0 0
\(946\) 91.2548 2.96695
\(947\) −17.1127 −0.556088 −0.278044 0.960568i \(-0.589686\pi\)
−0.278044 + 0.960568i \(0.589686\pi\)
\(948\) 0 0
\(949\) −8.48528 −0.275444
\(950\) 1.41421 0.0458831
\(951\) 0 0
\(952\) −17.6569 −0.572262
\(953\) −35.2548 −1.14202 −0.571008 0.820944i \(-0.693447\pi\)
−0.571008 + 0.820944i \(0.693447\pi\)
\(954\) 0 0
\(955\) 2.34315 0.0758224
\(956\) 13.0711 0.422749
\(957\) 0 0
\(958\) 74.1838 2.39677
\(959\) 83.5980 2.69952
\(960\) 0 0
\(961\) −27.9117 −0.900377
\(962\) 20.4853 0.660472
\(963\) 0 0
\(964\) −55.4558 −1.78611
\(965\) −4.34315 −0.139811
\(966\) 0 0
\(967\) 47.9411 1.54168 0.770841 0.637027i \(-0.219836\pi\)
0.770841 + 0.637027i \(0.219836\pi\)
\(968\) 2.89949 0.0931933
\(969\) 0 0
\(970\) 18.4853 0.593527
\(971\) −44.2843 −1.42115 −0.710575 0.703622i \(-0.751565\pi\)
−0.710575 + 0.703622i \(0.751565\pi\)
\(972\) 0 0
\(973\) 21.6569 0.694287
\(974\) −26.4853 −0.848643
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) 39.5147 1.26419 0.632094 0.774892i \(-0.282196\pi\)
0.632094 + 0.774892i \(0.282196\pi\)
\(978\) 0 0
\(979\) 20.4853 0.654712
\(980\) −62.4558 −1.99508
\(981\) 0 0
\(982\) −12.4853 −0.398421
\(983\) 1.02944 0.0328339 0.0164170 0.999865i \(-0.494774\pi\)
0.0164170 + 0.999865i \(0.494774\pi\)
\(984\) 0 0
\(985\) 10.9706 0.349551
\(986\) −11.3137 −0.360302
\(987\) 0 0
\(988\) −2.24264 −0.0713479
\(989\) 15.6569 0.497859
\(990\) 0 0
\(991\) 48.9706 1.55560 0.777801 0.628511i \(-0.216335\pi\)
0.777801 + 0.628511i \(0.216335\pi\)
\(992\) −2.78680 −0.0884809
\(993\) 0 0
\(994\) −138.711 −4.39964
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 28.8284 0.913005 0.456503 0.889722i \(-0.349102\pi\)
0.456503 + 0.889722i \(0.349102\pi\)
\(998\) 100.326 3.17576
\(999\) 0 0
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 585.2.a.m.1.2 2
3.2 odd 2 65.2.a.b.1.1 2
4.3 odd 2 9360.2.a.cd.1.1 2
5.2 odd 4 2925.2.c.r.2224.4 4
5.3 odd 4 2925.2.c.r.2224.1 4
5.4 even 2 2925.2.a.u.1.1 2
12.11 even 2 1040.2.a.j.1.2 2
13.12 even 2 7605.2.a.x.1.1 2
15.2 even 4 325.2.b.f.274.1 4
15.8 even 4 325.2.b.f.274.4 4
15.14 odd 2 325.2.a.i.1.2 2
21.20 even 2 3185.2.a.j.1.1 2
24.5 odd 2 4160.2.a.bf.1.2 2
24.11 even 2 4160.2.a.z.1.1 2
33.32 even 2 7865.2.a.j.1.2 2
39.2 even 12 845.2.m.f.316.1 8
39.5 even 4 845.2.c.b.506.4 4
39.8 even 4 845.2.c.b.506.1 4
39.11 even 12 845.2.m.f.316.4 8
39.17 odd 6 845.2.e.c.146.1 4
39.20 even 12 845.2.m.f.361.1 8
39.23 odd 6 845.2.e.c.191.1 4
39.29 odd 6 845.2.e.h.191.2 4
39.32 even 12 845.2.m.f.361.4 8
39.35 odd 6 845.2.e.h.146.2 4
39.38 odd 2 845.2.a.g.1.2 2
60.59 even 2 5200.2.a.bu.1.1 2
195.194 odd 2 4225.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.b.1.1 2 3.2 odd 2
325.2.a.i.1.2 2 15.14 odd 2
325.2.b.f.274.1 4 15.2 even 4
325.2.b.f.274.4 4 15.8 even 4
585.2.a.m.1.2 2 1.1 even 1 trivial
845.2.a.g.1.2 2 39.38 odd 2
845.2.c.b.506.1 4 39.8 even 4
845.2.c.b.506.4 4 39.5 even 4
845.2.e.c.146.1 4 39.17 odd 6
845.2.e.c.191.1 4 39.23 odd 6
845.2.e.h.146.2 4 39.35 odd 6
845.2.e.h.191.2 4 39.29 odd 6
845.2.m.f.316.1 8 39.2 even 12
845.2.m.f.316.4 8 39.11 even 12
845.2.m.f.361.1 8 39.20 even 12
845.2.m.f.361.4 8 39.32 even 12
1040.2.a.j.1.2 2 12.11 even 2
2925.2.a.u.1.1 2 5.4 even 2
2925.2.c.r.2224.1 4 5.3 odd 4
2925.2.c.r.2224.4 4 5.2 odd 4
3185.2.a.j.1.1 2 21.20 even 2
4160.2.a.z.1.1 2 24.11 even 2
4160.2.a.bf.1.2 2 24.5 odd 2
4225.2.a.r.1.1 2 195.194 odd 2
5200.2.a.bu.1.1 2 60.59 even 2
7605.2.a.x.1.1 2 13.12 even 2
7865.2.a.j.1.2 2 33.32 even 2
9360.2.a.cd.1.1 2 4.3 odd 2