Properties

Label 460.2.a.e.1.2
Level $460$
Weight $2$
Character 460.1
Self dual yes
Analytic conductor $3.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 460.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} +3.56155 q^{9} +2.00000 q^{11} +0.561553 q^{13} +2.56155 q^{15} -1.56155 q^{17} +6.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} -2.12311 q^{29} -9.24621 q^{31} +5.12311 q^{33} -1.56155 q^{35} -0.438447 q^{37} +1.43845 q^{39} -4.12311 q^{41} +3.56155 q^{45} -7.68466 q^{47} -4.56155 q^{49} -4.00000 q^{51} -0.438447 q^{53} +2.00000 q^{55} +15.3693 q^{57} +8.68466 q^{59} +1.12311 q^{61} -5.56155 q^{63} +0.561553 q^{65} -4.43845 q^{67} +2.56155 q^{69} +1.87689 q^{71} -8.56155 q^{73} +2.56155 q^{75} -3.12311 q^{77} +13.1231 q^{79} -7.00000 q^{81} +14.9309 q^{83} -1.56155 q^{85} -5.43845 q^{87} -2.24621 q^{89} -0.876894 q^{91} -23.6847 q^{93} +6.00000 q^{95} -4.87689 q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} + q^{17} + 12 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} + q^{35} - 5 q^{37} + 7 q^{39} + 3 q^{45} - 3 q^{47} - 5 q^{49} - 8 q^{51} - 5 q^{53} + 4 q^{55} + 6 q^{57} + 5 q^{59} - 6 q^{61} - 7 q^{63} - 3 q^{65} - 13 q^{67} + q^{69} + 12 q^{71} - 13 q^{73} + q^{75} + 2 q^{77} + 18 q^{79} - 14 q^{81} + q^{83} + q^{85} - 15 q^{87} + 12 q^{89} - 10 q^{91} - 35 q^{93} + 12 q^{95} - 18 q^{97} + 6 q^{99} + O(q^{100}) \)

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 0
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) −1.56155 −0.378732 −0.189366 0.981907i \(-0.560643\pi\)
−0.189366 + 0.981907i \(0.560643\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −2.12311 −0.394251 −0.197125 0.980378i \(-0.563161\pi\)
−0.197125 + 0.980378i \(0.563161\pi\)
\(30\) 0 0
\(31\) −9.24621 −1.66067 −0.830334 0.557266i \(-0.811851\pi\)
−0.830334 + 0.557266i \(0.811851\pi\)
\(32\) 0 0
\(33\) 5.12311 0.891818
\(34\) 0 0
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) −0.438447 −0.0720803 −0.0360401 0.999350i \(-0.511474\pi\)
−0.0360401 + 0.999350i \(0.511474\pi\)
\(38\) 0 0
\(39\) 1.43845 0.230336
\(40\) 0 0
\(41\) −4.12311 −0.643921 −0.321960 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) −7.68466 −1.12092 −0.560461 0.828181i \(-0.689376\pi\)
−0.560461 + 0.828181i \(0.689376\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −0.438447 −0.0602254 −0.0301127 0.999547i \(-0.509587\pi\)
−0.0301127 + 0.999547i \(0.509587\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 15.3693 2.03572
\(58\) 0 0
\(59\) 8.68466 1.13065 0.565323 0.824870i \(-0.308751\pi\)
0.565323 + 0.824870i \(0.308751\pi\)
\(60\) 0 0
\(61\) 1.12311 0.143799 0.0718995 0.997412i \(-0.477094\pi\)
0.0718995 + 0.997412i \(0.477094\pi\)
\(62\) 0 0
\(63\) −5.56155 −0.700690
\(64\) 0 0
\(65\) 0.561553 0.0696521
\(66\) 0 0
\(67\) −4.43845 −0.542243 −0.271121 0.962545i \(-0.587394\pi\)
−0.271121 + 0.962545i \(0.587394\pi\)
\(68\) 0 0
\(69\) 2.56155 0.308375
\(70\) 0 0
\(71\) 1.87689 0.222746 0.111373 0.993779i \(-0.464475\pi\)
0.111373 + 0.993779i \(0.464475\pi\)
\(72\) 0 0
\(73\) −8.56155 −1.00205 −0.501027 0.865432i \(-0.667044\pi\)
−0.501027 + 0.865432i \(0.667044\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) −3.12311 −0.355911
\(78\) 0 0
\(79\) 13.1231 1.47646 0.738232 0.674546i \(-0.235661\pi\)
0.738232 + 0.674546i \(0.235661\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 14.9309 1.63888 0.819438 0.573168i \(-0.194286\pi\)
0.819438 + 0.573168i \(0.194286\pi\)
\(84\) 0 0
\(85\) −1.56155 −0.169374
\(86\) 0 0
\(87\) −5.43845 −0.583063
\(88\) 0 0
\(89\) −2.24621 −0.238098 −0.119049 0.992888i \(-0.537985\pi\)
−0.119049 + 0.992888i \(0.537985\pi\)
\(90\) 0 0
\(91\) −0.876894 −0.0919235
\(92\) 0 0
\(93\) −23.6847 −2.45598
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −4.87689 −0.495174 −0.247587 0.968866i \(-0.579638\pi\)
−0.247587 + 0.968866i \(0.579638\pi\)
\(98\) 0 0
\(99\) 7.12311 0.715899
\(100\) 0 0
\(101\) −5.31534 −0.528896 −0.264448 0.964400i \(-0.585190\pi\)
−0.264448 + 0.964400i \(0.585190\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) 7.56155 0.731003 0.365501 0.930811i \(-0.380898\pi\)
0.365501 + 0.930811i \(0.380898\pi\)
\(108\) 0 0
\(109\) −9.36932 −0.897418 −0.448709 0.893678i \(-0.648116\pi\)
−0.448709 + 0.893678i \(0.648116\pi\)
\(110\) 0 0
\(111\) −1.12311 −0.106600
\(112\) 0 0
\(113\) −7.80776 −0.734493 −0.367246 0.930124i \(-0.619699\pi\)
−0.367246 + 0.930124i \(0.619699\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 2.43845 0.223532
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −10.5616 −0.952303
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.8078 −1.13651 −0.568253 0.822854i \(-0.692380\pi\)
−0.568253 + 0.822854i \(0.692380\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.8078 1.81798 0.908991 0.416815i \(-0.136854\pi\)
0.908991 + 0.416815i \(0.136854\pi\)
\(132\) 0 0
\(133\) −9.36932 −0.812423
\(134\) 0 0
\(135\) 1.43845 0.123802
\(136\) 0 0
\(137\) 15.1231 1.29205 0.646027 0.763315i \(-0.276429\pi\)
0.646027 + 0.763315i \(0.276429\pi\)
\(138\) 0 0
\(139\) 5.24621 0.444978 0.222489 0.974935i \(-0.428582\pi\)
0.222489 + 0.974935i \(0.428582\pi\)
\(140\) 0 0
\(141\) −19.6847 −1.65775
\(142\) 0 0
\(143\) 1.12311 0.0939188
\(144\) 0 0
\(145\) −2.12311 −0.176314
\(146\) 0 0
\(147\) −11.6847 −0.963734
\(148\) 0 0
\(149\) 15.3693 1.25910 0.629552 0.776959i \(-0.283239\pi\)
0.629552 + 0.776959i \(0.283239\pi\)
\(150\) 0 0
\(151\) 23.0540 1.87611 0.938053 0.346492i \(-0.112627\pi\)
0.938053 + 0.346492i \(0.112627\pi\)
\(152\) 0 0
\(153\) −5.56155 −0.449625
\(154\) 0 0
\(155\) −9.24621 −0.742674
\(156\) 0 0
\(157\) −16.6847 −1.33158 −0.665790 0.746139i \(-0.731905\pi\)
−0.665790 + 0.746139i \(0.731905\pi\)
\(158\) 0 0
\(159\) −1.12311 −0.0890681
\(160\) 0 0
\(161\) −1.56155 −0.123068
\(162\) 0 0
\(163\) −11.9309 −0.934498 −0.467249 0.884126i \(-0.654755\pi\)
−0.467249 + 0.884126i \(0.654755\pi\)
\(164\) 0 0
\(165\) 5.12311 0.398833
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) 0 0
\(171\) 21.3693 1.63415
\(172\) 0 0
\(173\) −19.3693 −1.47262 −0.736311 0.676643i \(-0.763434\pi\)
−0.736311 + 0.676643i \(0.763434\pi\)
\(174\) 0 0
\(175\) −1.56155 −0.118042
\(176\) 0 0
\(177\) 22.2462 1.67213
\(178\) 0 0
\(179\) 21.9309 1.63919 0.819595 0.572943i \(-0.194198\pi\)
0.819595 + 0.572943i \(0.194198\pi\)
\(180\) 0 0
\(181\) 19.3693 1.43971 0.719855 0.694124i \(-0.244208\pi\)
0.719855 + 0.694124i \(0.244208\pi\)
\(182\) 0 0
\(183\) 2.87689 0.212666
\(184\) 0 0
\(185\) −0.438447 −0.0322353
\(186\) 0 0
\(187\) −3.12311 −0.228384
\(188\) 0 0
\(189\) −2.24621 −0.163388
\(190\) 0 0
\(191\) −7.36932 −0.533225 −0.266613 0.963804i \(-0.585904\pi\)
−0.266613 + 0.963804i \(0.585904\pi\)
\(192\) 0 0
\(193\) 6.56155 0.472311 0.236155 0.971715i \(-0.424113\pi\)
0.236155 + 0.971715i \(0.424113\pi\)
\(194\) 0 0
\(195\) 1.43845 0.103009
\(196\) 0 0
\(197\) −6.31534 −0.449949 −0.224975 0.974365i \(-0.572230\pi\)
−0.224975 + 0.974365i \(0.572230\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −11.3693 −0.801930
\(202\) 0 0
\(203\) 3.31534 0.232691
\(204\) 0 0
\(205\) −4.12311 −0.287970
\(206\) 0 0
\(207\) 3.56155 0.247545
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 12.6847 0.873248 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(212\) 0 0
\(213\) 4.80776 0.329423
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 14.4384 0.980146
\(218\) 0 0
\(219\) −21.9309 −1.48195
\(220\) 0 0
\(221\) −0.876894 −0.0589863
\(222\) 0 0
\(223\) 23.6155 1.58141 0.790706 0.612196i \(-0.209713\pi\)
0.790706 + 0.612196i \(0.209713\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) 9.75379 0.647382 0.323691 0.946163i \(-0.395076\pi\)
0.323691 + 0.946163i \(0.395076\pi\)
\(228\) 0 0
\(229\) 22.7386 1.50261 0.751306 0.659954i \(-0.229424\pi\)
0.751306 + 0.659954i \(0.229424\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 11.6847 0.765487 0.382744 0.923855i \(-0.374979\pi\)
0.382744 + 0.923855i \(0.374979\pi\)
\(234\) 0 0
\(235\) −7.68466 −0.501292
\(236\) 0 0
\(237\) 33.6155 2.18356
\(238\) 0 0
\(239\) 19.2462 1.24493 0.622467 0.782646i \(-0.286131\pi\)
0.622467 + 0.782646i \(0.286131\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) −4.56155 −0.291427
\(246\) 0 0
\(247\) 3.36932 0.214384
\(248\) 0 0
\(249\) 38.2462 2.42376
\(250\) 0 0
\(251\) 17.3693 1.09634 0.548171 0.836366i \(-0.315324\pi\)
0.548171 + 0.836366i \(0.315324\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 11.1922 0.698152 0.349076 0.937094i \(-0.386495\pi\)
0.349076 + 0.937094i \(0.386495\pi\)
\(258\) 0 0
\(259\) 0.684658 0.0425426
\(260\) 0 0
\(261\) −7.56155 −0.468048
\(262\) 0 0
\(263\) 28.9309 1.78395 0.891977 0.452081i \(-0.149318\pi\)
0.891977 + 0.452081i \(0.149318\pi\)
\(264\) 0 0
\(265\) −0.438447 −0.0269336
\(266\) 0 0
\(267\) −5.75379 −0.352126
\(268\) 0 0
\(269\) 6.75379 0.411786 0.205893 0.978575i \(-0.433990\pi\)
0.205893 + 0.978575i \(0.433990\pi\)
\(270\) 0 0
\(271\) −6.93087 −0.421020 −0.210510 0.977592i \(-0.567513\pi\)
−0.210510 + 0.977592i \(0.567513\pi\)
\(272\) 0 0
\(273\) −2.24621 −0.135947
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 5.68466 0.341558 0.170779 0.985309i \(-0.445372\pi\)
0.170779 + 0.985309i \(0.445372\pi\)
\(278\) 0 0
\(279\) −32.9309 −1.97152
\(280\) 0 0
\(281\) −15.1231 −0.902169 −0.451084 0.892481i \(-0.648963\pi\)
−0.451084 + 0.892481i \(0.648963\pi\)
\(282\) 0 0
\(283\) −5.80776 −0.345236 −0.172618 0.984989i \(-0.555223\pi\)
−0.172618 + 0.984989i \(0.555223\pi\)
\(284\) 0 0
\(285\) 15.3693 0.910400
\(286\) 0 0
\(287\) 6.43845 0.380050
\(288\) 0 0
\(289\) −14.5616 −0.856562
\(290\) 0 0
\(291\) −12.4924 −0.732319
\(292\) 0 0
\(293\) 3.80776 0.222452 0.111226 0.993795i \(-0.464522\pi\)
0.111226 + 0.993795i \(0.464522\pi\)
\(294\) 0 0
\(295\) 8.68466 0.505640
\(296\) 0 0
\(297\) 2.87689 0.166934
\(298\) 0 0
\(299\) 0.561553 0.0324754
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −13.6155 −0.782192
\(304\) 0 0
\(305\) 1.12311 0.0643088
\(306\) 0 0
\(307\) −12.8769 −0.734923 −0.367462 0.930039i \(-0.619773\pi\)
−0.367462 + 0.930039i \(0.619773\pi\)
\(308\) 0 0
\(309\) −40.9848 −2.33155
\(310\) 0 0
\(311\) −3.68466 −0.208938 −0.104469 0.994528i \(-0.533314\pi\)
−0.104469 + 0.994528i \(0.533314\pi\)
\(312\) 0 0
\(313\) 19.8078 1.11960 0.559801 0.828627i \(-0.310878\pi\)
0.559801 + 0.828627i \(0.310878\pi\)
\(314\) 0 0
\(315\) −5.56155 −0.313358
\(316\) 0 0
\(317\) 6.87689 0.386245 0.193122 0.981175i \(-0.438139\pi\)
0.193122 + 0.981175i \(0.438139\pi\)
\(318\) 0 0
\(319\) −4.24621 −0.237742
\(320\) 0 0
\(321\) 19.3693 1.08109
\(322\) 0 0
\(323\) −9.36932 −0.521323
\(324\) 0 0
\(325\) 0.561553 0.0311493
\(326\) 0 0
\(327\) −24.0000 −1.32720
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −12.6155 −0.693412 −0.346706 0.937974i \(-0.612700\pi\)
−0.346706 + 0.937974i \(0.612700\pi\)
\(332\) 0 0
\(333\) −1.56155 −0.0855726
\(334\) 0 0
\(335\) −4.43845 −0.242498
\(336\) 0 0
\(337\) −27.6155 −1.50431 −0.752157 0.658984i \(-0.770986\pi\)
−0.752157 + 0.658984i \(0.770986\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) −18.4924 −1.00142
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 0 0
\(345\) 2.56155 0.137909
\(346\) 0 0
\(347\) −16.4924 −0.885360 −0.442680 0.896680i \(-0.645972\pi\)
−0.442680 + 0.896680i \(0.645972\pi\)
\(348\) 0 0
\(349\) −1.63068 −0.0872885 −0.0436442 0.999047i \(-0.513897\pi\)
−0.0436442 + 0.999047i \(0.513897\pi\)
\(350\) 0 0
\(351\) 0.807764 0.0431153
\(352\) 0 0
\(353\) 35.0540 1.86573 0.932867 0.360220i \(-0.117298\pi\)
0.932867 + 0.360220i \(0.117298\pi\)
\(354\) 0 0
\(355\) 1.87689 0.0996152
\(356\) 0 0
\(357\) 6.24621 0.330585
\(358\) 0 0
\(359\) −25.6155 −1.35194 −0.675968 0.736931i \(-0.736274\pi\)
−0.675968 + 0.736931i \(0.736274\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −17.9309 −0.941127
\(364\) 0 0
\(365\) −8.56155 −0.448132
\(366\) 0 0
\(367\) −18.4384 −0.962479 −0.481240 0.876589i \(-0.659813\pi\)
−0.481240 + 0.876589i \(0.659813\pi\)
\(368\) 0 0
\(369\) −14.6847 −0.764453
\(370\) 0 0
\(371\) 0.684658 0.0355457
\(372\) 0 0
\(373\) −3.75379 −0.194364 −0.0971819 0.995267i \(-0.530983\pi\)
−0.0971819 + 0.995267i \(0.530983\pi\)
\(374\) 0 0
\(375\) 2.56155 0.132278
\(376\) 0 0
\(377\) −1.19224 −0.0614033
\(378\) 0 0
\(379\) −3.50758 −0.180172 −0.0900861 0.995934i \(-0.528714\pi\)
−0.0900861 + 0.995934i \(0.528714\pi\)
\(380\) 0 0
\(381\) −32.8078 −1.68079
\(382\) 0 0
\(383\) 23.8078 1.21652 0.608260 0.793738i \(-0.291868\pi\)
0.608260 + 0.793738i \(0.291868\pi\)
\(384\) 0 0
\(385\) −3.12311 −0.159168
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.1231 −0.969580 −0.484790 0.874631i \(-0.661104\pi\)
−0.484790 + 0.874631i \(0.661104\pi\)
\(390\) 0 0
\(391\) −1.56155 −0.0789711
\(392\) 0 0
\(393\) 53.3002 2.68864
\(394\) 0 0
\(395\) 13.1231 0.660295
\(396\) 0 0
\(397\) −35.5464 −1.78402 −0.892011 0.452013i \(-0.850706\pi\)
−0.892011 + 0.452013i \(0.850706\pi\)
\(398\) 0 0
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −5.19224 −0.258644
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) −0.876894 −0.0434660
\(408\) 0 0
\(409\) 29.9848 1.48266 0.741328 0.671143i \(-0.234197\pi\)
0.741328 + 0.671143i \(0.234197\pi\)
\(410\) 0 0
\(411\) 38.7386 1.91084
\(412\) 0 0
\(413\) −13.5616 −0.667320
\(414\) 0 0
\(415\) 14.9309 0.732928
\(416\) 0 0
\(417\) 13.4384 0.658084
\(418\) 0 0
\(419\) −27.1231 −1.32505 −0.662525 0.749040i \(-0.730515\pi\)
−0.662525 + 0.749040i \(0.730515\pi\)
\(420\) 0 0
\(421\) 16.8769 0.822530 0.411265 0.911516i \(-0.365087\pi\)
0.411265 + 0.911516i \(0.365087\pi\)
\(422\) 0 0
\(423\) −27.3693 −1.33074
\(424\) 0 0
\(425\) −1.56155 −0.0757464
\(426\) 0 0
\(427\) −1.75379 −0.0848718
\(428\) 0 0
\(429\) 2.87689 0.138898
\(430\) 0 0
\(431\) 6.24621 0.300869 0.150435 0.988620i \(-0.451933\pi\)
0.150435 + 0.988620i \(0.451933\pi\)
\(432\) 0 0
\(433\) 11.0691 0.531948 0.265974 0.963980i \(-0.414306\pi\)
0.265974 + 0.963980i \(0.414306\pi\)
\(434\) 0 0
\(435\) −5.43845 −0.260754
\(436\) 0 0
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −0.807764 −0.0385525 −0.0192762 0.999814i \(-0.506136\pi\)
−0.0192762 + 0.999814i \(0.506136\pi\)
\(440\) 0 0
\(441\) −16.2462 −0.773629
\(442\) 0 0
\(443\) 8.80776 0.418469 0.209235 0.977865i \(-0.432903\pi\)
0.209235 + 0.977865i \(0.432903\pi\)
\(444\) 0 0
\(445\) −2.24621 −0.106481
\(446\) 0 0
\(447\) 39.3693 1.86210
\(448\) 0 0
\(449\) −9.31534 −0.439618 −0.219809 0.975543i \(-0.570543\pi\)
−0.219809 + 0.975543i \(0.570543\pi\)
\(450\) 0 0
\(451\) −8.24621 −0.388299
\(452\) 0 0
\(453\) 59.0540 2.77460
\(454\) 0 0
\(455\) −0.876894 −0.0411094
\(456\) 0 0
\(457\) 13.5616 0.634383 0.317191 0.948362i \(-0.397260\pi\)
0.317191 + 0.948362i \(0.397260\pi\)
\(458\) 0 0
\(459\) −2.24621 −0.104844
\(460\) 0 0
\(461\) −12.0691 −0.562115 −0.281058 0.959691i \(-0.590685\pi\)
−0.281058 + 0.959691i \(0.590685\pi\)
\(462\) 0 0
\(463\) −6.63068 −0.308154 −0.154077 0.988059i \(-0.549240\pi\)
−0.154077 + 0.988059i \(0.549240\pi\)
\(464\) 0 0
\(465\) −23.6847 −1.09835
\(466\) 0 0
\(467\) −34.6847 −1.60501 −0.802507 0.596642i \(-0.796501\pi\)
−0.802507 + 0.596642i \(0.796501\pi\)
\(468\) 0 0
\(469\) 6.93087 0.320038
\(470\) 0 0
\(471\) −42.7386 −1.96929
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −1.56155 −0.0714986
\(478\) 0 0
\(479\) 39.2311 1.79251 0.896256 0.443536i \(-0.146276\pi\)
0.896256 + 0.443536i \(0.146276\pi\)
\(480\) 0 0
\(481\) −0.246211 −0.0112263
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) −4.87689 −0.221448
\(486\) 0 0
\(487\) −12.8078 −0.580375 −0.290188 0.956970i \(-0.593718\pi\)
−0.290188 + 0.956970i \(0.593718\pi\)
\(488\) 0 0
\(489\) −30.5616 −1.38204
\(490\) 0 0
\(491\) −21.4924 −0.969939 −0.484970 0.874531i \(-0.661169\pi\)
−0.484970 + 0.874531i \(0.661169\pi\)
\(492\) 0 0
\(493\) 3.31534 0.149315
\(494\) 0 0
\(495\) 7.12311 0.320160
\(496\) 0 0
\(497\) −2.93087 −0.131467
\(498\) 0 0
\(499\) −18.6155 −0.833345 −0.416673 0.909057i \(-0.636804\pi\)
−0.416673 + 0.909057i \(0.636804\pi\)
\(500\) 0 0
\(501\) −20.4924 −0.915534
\(502\) 0 0
\(503\) 24.9309 1.11161 0.555806 0.831312i \(-0.312410\pi\)
0.555806 + 0.831312i \(0.312410\pi\)
\(504\) 0 0
\(505\) −5.31534 −0.236530
\(506\) 0 0
\(507\) −32.4924 −1.44304
\(508\) 0 0
\(509\) 18.1771 0.805685 0.402842 0.915269i \(-0.368022\pi\)
0.402842 + 0.915269i \(0.368022\pi\)
\(510\) 0 0
\(511\) 13.3693 0.591424
\(512\) 0 0
\(513\) 8.63068 0.381054
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −15.3693 −0.675942
\(518\) 0 0
\(519\) −49.6155 −2.17788
\(520\) 0 0
\(521\) 27.6155 1.20986 0.604929 0.796279i \(-0.293201\pi\)
0.604929 + 0.796279i \(0.293201\pi\)
\(522\) 0 0
\(523\) 34.7386 1.51901 0.759507 0.650499i \(-0.225440\pi\)
0.759507 + 0.650499i \(0.225440\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 14.4384 0.628949
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 30.9309 1.34229
\(532\) 0 0
\(533\) −2.31534 −0.100289
\(534\) 0 0
\(535\) 7.56155 0.326914
\(536\) 0 0
\(537\) 56.1771 2.42422
\(538\) 0 0
\(539\) −9.12311 −0.392960
\(540\) 0 0
\(541\) 9.68466 0.416376 0.208188 0.978089i \(-0.433243\pi\)
0.208188 + 0.978089i \(0.433243\pi\)
\(542\) 0 0
\(543\) 49.6155 2.12921
\(544\) 0 0
\(545\) −9.36932 −0.401337
\(546\) 0 0
\(547\) 24.1771 1.03374 0.516869 0.856065i \(-0.327098\pi\)
0.516869 + 0.856065i \(0.327098\pi\)
\(548\) 0 0
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −12.7386 −0.542684
\(552\) 0 0
\(553\) −20.4924 −0.871426
\(554\) 0 0
\(555\) −1.12311 −0.0476732
\(556\) 0 0
\(557\) −3.31534 −0.140476 −0.0702378 0.997530i \(-0.522376\pi\)
−0.0702378 + 0.997530i \(0.522376\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) −45.6695 −1.92474 −0.962370 0.271742i \(-0.912400\pi\)
−0.962370 + 0.271742i \(0.912400\pi\)
\(564\) 0 0
\(565\) −7.80776 −0.328475
\(566\) 0 0
\(567\) 10.9309 0.459053
\(568\) 0 0
\(569\) −32.7386 −1.37247 −0.686237 0.727378i \(-0.740739\pi\)
−0.686237 + 0.727378i \(0.740739\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) −18.8769 −0.788594
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −25.4384 −1.05902 −0.529508 0.848305i \(-0.677624\pi\)
−0.529508 + 0.848305i \(0.677624\pi\)
\(578\) 0 0
\(579\) 16.8078 0.698507
\(580\) 0 0
\(581\) −23.3153 −0.967283
\(582\) 0 0
\(583\) −0.876894 −0.0363173
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 35.9309 1.48303 0.741513 0.670939i \(-0.234109\pi\)
0.741513 + 0.670939i \(0.234109\pi\)
\(588\) 0 0
\(589\) −55.4773 −2.28590
\(590\) 0 0
\(591\) −16.1771 −0.665436
\(592\) 0 0
\(593\) −25.6155 −1.05190 −0.525952 0.850514i \(-0.676291\pi\)
−0.525952 + 0.850514i \(0.676291\pi\)
\(594\) 0 0
\(595\) 2.43845 0.0999666
\(596\) 0 0
\(597\) −25.6155 −1.04837
\(598\) 0 0
\(599\) 32.9848 1.34772 0.673862 0.738857i \(-0.264634\pi\)
0.673862 + 0.738857i \(0.264634\pi\)
\(600\) 0 0
\(601\) 11.6307 0.474425 0.237213 0.971458i \(-0.423766\pi\)
0.237213 + 0.971458i \(0.423766\pi\)
\(602\) 0 0
\(603\) −15.8078 −0.643742
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −24.4924 −0.994117 −0.497058 0.867717i \(-0.665587\pi\)
−0.497058 + 0.867717i \(0.665587\pi\)
\(608\) 0 0
\(609\) 8.49242 0.344130
\(610\) 0 0
\(611\) −4.31534 −0.174580
\(612\) 0 0
\(613\) 35.6155 1.43850 0.719249 0.694753i \(-0.244486\pi\)
0.719249 + 0.694753i \(0.244486\pi\)
\(614\) 0 0
\(615\) −10.5616 −0.425883
\(616\) 0 0
\(617\) −3.56155 −0.143383 −0.0716914 0.997427i \(-0.522840\pi\)
−0.0716914 + 0.997427i \(0.522840\pi\)
\(618\) 0 0
\(619\) −20.4924 −0.823660 −0.411830 0.911261i \(-0.635110\pi\)
−0.411830 + 0.911261i \(0.635110\pi\)
\(620\) 0 0
\(621\) 1.43845 0.0577229
\(622\) 0 0
\(623\) 3.50758 0.140528
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 30.7386 1.22758
\(628\) 0 0
\(629\) 0.684658 0.0272991
\(630\) 0 0
\(631\) 13.7538 0.547530 0.273765 0.961797i \(-0.411731\pi\)
0.273765 + 0.961797i \(0.411731\pi\)
\(632\) 0 0
\(633\) 32.4924 1.29146
\(634\) 0 0
\(635\) −12.8078 −0.508261
\(636\) 0 0
\(637\) −2.56155 −0.101492
\(638\) 0 0
\(639\) 6.68466 0.264441
\(640\) 0 0
\(641\) 15.1231 0.597327 0.298663 0.954359i \(-0.403459\pi\)
0.298663 + 0.954359i \(0.403459\pi\)
\(642\) 0 0
\(643\) −41.4233 −1.63358 −0.816788 0.576939i \(-0.804247\pi\)
−0.816788 + 0.576939i \(0.804247\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.6847 0.695256 0.347628 0.937633i \(-0.386987\pi\)
0.347628 + 0.937633i \(0.386987\pi\)
\(648\) 0 0
\(649\) 17.3693 0.681805
\(650\) 0 0
\(651\) 36.9848 1.44955
\(652\) 0 0
\(653\) −38.6695 −1.51325 −0.756627 0.653846i \(-0.773154\pi\)
−0.756627 + 0.653846i \(0.773154\pi\)
\(654\) 0 0
\(655\) 20.8078 0.813027
\(656\) 0 0
\(657\) −30.4924 −1.18962
\(658\) 0 0
\(659\) −9.36932 −0.364977 −0.182488 0.983208i \(-0.558415\pi\)
−0.182488 + 0.983208i \(0.558415\pi\)
\(660\) 0 0
\(661\) 26.4924 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(662\) 0 0
\(663\) −2.24621 −0.0872356
\(664\) 0 0
\(665\) −9.36932 −0.363327
\(666\) 0 0
\(667\) −2.12311 −0.0822070
\(668\) 0 0
\(669\) 60.4924 2.33877
\(670\) 0 0
\(671\) 2.24621 0.0867140
\(672\) 0 0
\(673\) −35.5464 −1.37021 −0.685106 0.728443i \(-0.740244\pi\)
−0.685106 + 0.728443i \(0.740244\pi\)
\(674\) 0 0
\(675\) 1.43845 0.0553659
\(676\) 0 0
\(677\) 6.19224 0.237987 0.118993 0.992895i \(-0.462033\pi\)
0.118993 + 0.992895i \(0.462033\pi\)
\(678\) 0 0
\(679\) 7.61553 0.292257
\(680\) 0 0
\(681\) 24.9848 0.957421
\(682\) 0 0
\(683\) 4.94602 0.189254 0.0946272 0.995513i \(-0.469834\pi\)
0.0946272 + 0.995513i \(0.469834\pi\)
\(684\) 0 0
\(685\) 15.1231 0.577824
\(686\) 0 0
\(687\) 58.2462 2.22223
\(688\) 0 0
\(689\) −0.246211 −0.00937990
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) 0 0
\(693\) −11.1231 −0.422532
\(694\) 0 0
\(695\) 5.24621 0.199000
\(696\) 0 0
\(697\) 6.43845 0.243874
\(698\) 0 0
\(699\) 29.9309 1.13209
\(700\) 0 0
\(701\) −18.2462 −0.689150 −0.344575 0.938759i \(-0.611977\pi\)
−0.344575 + 0.938759i \(0.611977\pi\)
\(702\) 0 0
\(703\) −2.63068 −0.0992181
\(704\) 0 0
\(705\) −19.6847 −0.741367
\(706\) 0 0
\(707\) 8.30019 0.312161
\(708\) 0 0
\(709\) 46.2462 1.73681 0.868406 0.495853i \(-0.165145\pi\)
0.868406 + 0.495853i \(0.165145\pi\)
\(710\) 0 0
\(711\) 46.7386 1.75284
\(712\) 0 0
\(713\) −9.24621 −0.346273
\(714\) 0 0
\(715\) 1.12311 0.0420018
\(716\) 0 0
\(717\) 49.3002 1.84115
\(718\) 0 0
\(719\) −18.0540 −0.673300 −0.336650 0.941630i \(-0.609294\pi\)
−0.336650 + 0.941630i \(0.609294\pi\)
\(720\) 0 0
\(721\) 24.9848 0.930484
\(722\) 0 0
\(723\) 15.3693 0.571591
\(724\) 0 0
\(725\) −2.12311 −0.0788502
\(726\) 0 0
\(727\) −41.8078 −1.55056 −0.775282 0.631615i \(-0.782392\pi\)
−0.775282 + 0.631615i \(0.782392\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 12.6847 0.468519 0.234259 0.972174i \(-0.424734\pi\)
0.234259 + 0.972174i \(0.424734\pi\)
\(734\) 0 0
\(735\) −11.6847 −0.430995
\(736\) 0 0
\(737\) −8.87689 −0.326985
\(738\) 0 0
\(739\) −40.6155 −1.49407 −0.747033 0.664787i \(-0.768522\pi\)
−0.747033 + 0.664787i \(0.768522\pi\)
\(740\) 0 0
\(741\) 8.63068 0.317056
\(742\) 0 0
\(743\) 36.4924 1.33878 0.669389 0.742912i \(-0.266556\pi\)
0.669389 + 0.742912i \(0.266556\pi\)
\(744\) 0 0
\(745\) 15.3693 0.563088
\(746\) 0 0
\(747\) 53.1771 1.94565
\(748\) 0 0
\(749\) −11.8078 −0.431446
\(750\) 0 0
\(751\) −4.87689 −0.177960 −0.0889802 0.996033i \(-0.528361\pi\)
−0.0889802 + 0.996033i \(0.528361\pi\)
\(752\) 0 0
\(753\) 44.4924 1.62139
\(754\) 0 0
\(755\) 23.0540 0.839020
\(756\) 0 0
\(757\) 8.43845 0.306701 0.153350 0.988172i \(-0.450994\pi\)
0.153350 + 0.988172i \(0.450994\pi\)
\(758\) 0 0
\(759\) 5.12311 0.185957
\(760\) 0 0
\(761\) 21.9848 0.796950 0.398475 0.917179i \(-0.369540\pi\)
0.398475 + 0.917179i \(0.369540\pi\)
\(762\) 0 0
\(763\) 14.6307 0.529666
\(764\) 0 0
\(765\) −5.56155 −0.201078
\(766\) 0 0
\(767\) 4.87689 0.176094
\(768\) 0 0
\(769\) −35.3693 −1.27545 −0.637725 0.770264i \(-0.720124\pi\)
−0.637725 + 0.770264i \(0.720124\pi\)
\(770\) 0 0
\(771\) 28.6695 1.03251
\(772\) 0 0
\(773\) −32.8769 −1.18250 −0.591250 0.806488i \(-0.701365\pi\)
−0.591250 + 0.806488i \(0.701365\pi\)
\(774\) 0 0
\(775\) −9.24621 −0.332134
\(776\) 0 0
\(777\) 1.75379 0.0629168
\(778\) 0 0
\(779\) −24.7386 −0.886354
\(780\) 0 0
\(781\) 3.75379 0.134321
\(782\) 0 0
\(783\) −3.05398 −0.109140
\(784\) 0 0
\(785\) −16.6847 −0.595501
\(786\) 0 0
\(787\) −27.3153 −0.973687 −0.486843 0.873489i \(-0.661852\pi\)
−0.486843 + 0.873489i \(0.661852\pi\)
\(788\) 0 0
\(789\) 74.1080 2.63831
\(790\) 0 0
\(791\) 12.1922 0.433506
\(792\) 0 0
\(793\) 0.630683 0.0223962
\(794\) 0 0
\(795\) −1.12311 −0.0398325
\(796\) 0 0
\(797\) 9.42329 0.333790 0.166895 0.985975i \(-0.446626\pi\)
0.166895 + 0.985975i \(0.446626\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) −17.1231 −0.604261
\(804\) 0 0
\(805\) −1.56155 −0.0550375
\(806\) 0 0
\(807\) 17.3002 0.608995
\(808\) 0 0
\(809\) −8.82292 −0.310197 −0.155099 0.987899i \(-0.549570\pi\)
−0.155099 + 0.987899i \(0.549570\pi\)
\(810\) 0 0
\(811\) 30.7538 1.07991 0.539956 0.841693i \(-0.318441\pi\)
0.539956 + 0.841693i \(0.318441\pi\)
\(812\) 0 0
\(813\) −17.7538 −0.622653
\(814\) 0 0
\(815\) −11.9309 −0.417920
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −3.12311 −0.109130
\(820\) 0 0
\(821\) −1.50758 −0.0526148 −0.0263074 0.999654i \(-0.508375\pi\)
−0.0263074 + 0.999654i \(0.508375\pi\)
\(822\) 0 0
\(823\) −0.946025 −0.0329763 −0.0164882 0.999864i \(-0.505249\pi\)
−0.0164882 + 0.999864i \(0.505249\pi\)
\(824\) 0 0
\(825\) 5.12311 0.178364
\(826\) 0 0
\(827\) 7.31534 0.254379 0.127190 0.991878i \(-0.459404\pi\)
0.127190 + 0.991878i \(0.459404\pi\)
\(828\) 0 0
\(829\) 40.5464 1.40823 0.704117 0.710084i \(-0.251343\pi\)
0.704117 + 0.710084i \(0.251343\pi\)
\(830\) 0 0
\(831\) 14.5616 0.505135
\(832\) 0 0
\(833\) 7.12311 0.246801
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) −13.3002 −0.459722
\(838\) 0 0
\(839\) −51.1231 −1.76497 −0.882483 0.470345i \(-0.844130\pi\)
−0.882483 + 0.470345i \(0.844130\pi\)
\(840\) 0 0
\(841\) −24.4924 −0.844566
\(842\) 0 0
\(843\) −38.7386 −1.33423
\(844\) 0 0
\(845\) −12.6847 −0.436366
\(846\) 0 0
\(847\) 10.9309 0.375589
\(848\) 0 0
\(849\) −14.8769 −0.510574
\(850\) 0 0
\(851\) −0.438447 −0.0150298
\(852\) 0 0
\(853\) −3.75379 −0.128527 −0.0642636 0.997933i \(-0.520470\pi\)
−0.0642636 + 0.997933i \(0.520470\pi\)
\(854\) 0 0
\(855\) 21.3693 0.730815
\(856\) 0 0
\(857\) −45.6847 −1.56056 −0.780279 0.625431i \(-0.784923\pi\)
−0.780279 + 0.625431i \(0.784923\pi\)
\(858\) 0 0
\(859\) −47.4924 −1.62042 −0.810210 0.586139i \(-0.800647\pi\)
−0.810210 + 0.586139i \(0.800647\pi\)
\(860\) 0 0
\(861\) 16.4924 0.562060
\(862\) 0 0
\(863\) −3.43845 −0.117046 −0.0585231 0.998286i \(-0.518639\pi\)
−0.0585231 + 0.998286i \(0.518639\pi\)
\(864\) 0 0
\(865\) −19.3693 −0.658577
\(866\) 0 0
\(867\) −37.3002 −1.26678
\(868\) 0 0
\(869\) 26.2462 0.890342
\(870\) 0 0
\(871\) −2.49242 −0.0844525
\(872\) 0 0
\(873\) −17.3693 −0.587862
\(874\) 0 0
\(875\) −1.56155 −0.0527901
\(876\) 0 0
\(877\) 46.9848 1.58657 0.793283 0.608853i \(-0.208370\pi\)
0.793283 + 0.608853i \(0.208370\pi\)
\(878\) 0 0
\(879\) 9.75379 0.328987
\(880\) 0 0
\(881\) −57.4773 −1.93646 −0.968229 0.250065i \(-0.919548\pi\)
−0.968229 + 0.250065i \(0.919548\pi\)
\(882\) 0 0
\(883\) −38.7386 −1.30366 −0.651829 0.758366i \(-0.725998\pi\)
−0.651829 + 0.758366i \(0.725998\pi\)
\(884\) 0 0
\(885\) 22.2462 0.747798
\(886\) 0 0
\(887\) −43.7926 −1.47041 −0.735206 0.677844i \(-0.762915\pi\)
−0.735206 + 0.677844i \(0.762915\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) −14.0000 −0.469018
\(892\) 0 0
\(893\) −46.1080 −1.54294
\(894\) 0 0
\(895\) 21.9309 0.733068
\(896\) 0 0
\(897\) 1.43845 0.0480284
\(898\) 0 0
\(899\) 19.6307 0.654720
\(900\) 0 0
\(901\) 0.684658 0.0228093
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.3693 0.643858
\(906\) 0 0
\(907\) 9.31534 0.309311 0.154655 0.987968i \(-0.450573\pi\)
0.154655 + 0.987968i \(0.450573\pi\)
\(908\) 0 0
\(909\) −18.9309 −0.627897
\(910\) 0 0
\(911\) 5.12311 0.169736 0.0848680 0.996392i \(-0.472953\pi\)
0.0848680 + 0.996392i \(0.472953\pi\)
\(912\) 0 0
\(913\) 29.8617 0.988279
\(914\) 0 0
\(915\) 2.87689 0.0951072
\(916\) 0 0
\(917\) −32.4924 −1.07299
\(918\) 0 0
\(919\) −2.73863 −0.0903392 −0.0451696 0.998979i \(-0.514383\pi\)
−0.0451696 + 0.998979i \(0.514383\pi\)
\(920\) 0 0
\(921\) −32.9848 −1.08689
\(922\) 0 0
\(923\) 1.05398 0.0346920
\(924\) 0 0
\(925\) −0.438447 −0.0144161
\(926\) 0 0
\(927\) −56.9848 −1.87163
\(928\) 0 0
\(929\) 58.7235 1.92665 0.963327 0.268329i \(-0.0864713\pi\)
0.963327 + 0.268329i \(0.0864713\pi\)
\(930\) 0 0
\(931\) −27.3693 −0.896993
\(932\) 0 0
\(933\) −9.43845 −0.309001
\(934\) 0 0
\(935\) −3.12311 −0.102136
\(936\) 0 0
\(937\) 0.246211 0.00804337 0.00402169 0.999992i \(-0.498720\pi\)
0.00402169 + 0.999992i \(0.498720\pi\)
\(938\) 0 0
\(939\) 50.7386 1.65579
\(940\) 0 0
\(941\) −54.2462 −1.76838 −0.884188 0.467131i \(-0.845288\pi\)
−0.884188 + 0.467131i \(0.845288\pi\)
\(942\) 0 0
\(943\) −4.12311 −0.134267
\(944\) 0 0
\(945\) −2.24621 −0.0730693
\(946\) 0 0
\(947\) 44.3153 1.44006 0.720028 0.693945i \(-0.244129\pi\)
0.720028 + 0.693945i \(0.244129\pi\)
\(948\) 0 0
\(949\) −4.80776 −0.156067
\(950\) 0 0
\(951\) 17.6155 0.571223
\(952\) 0 0
\(953\) −5.50758 −0.178408 −0.0892040 0.996013i \(-0.528432\pi\)
−0.0892040 + 0.996013i \(0.528432\pi\)
\(954\) 0 0
\(955\) −7.36932 −0.238465
\(956\) 0 0
\(957\) −10.8769 −0.351600
\(958\) 0 0
\(959\) −23.6155 −0.762585
\(960\) 0 0
\(961\) 54.4924 1.75782
\(962\) 0 0
\(963\) 26.9309 0.867835
\(964\) 0 0
\(965\) 6.56155 0.211224
\(966\) 0 0
\(967\) −44.3153 −1.42509 −0.712543 0.701629i \(-0.752457\pi\)
−0.712543 + 0.701629i \(0.752457\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −47.3693 −1.52015 −0.760077 0.649833i \(-0.774839\pi\)
−0.760077 + 0.649833i \(0.774839\pi\)
\(972\) 0 0
\(973\) −8.19224 −0.262631
\(974\) 0 0
\(975\) 1.43845 0.0460672
\(976\) 0 0
\(977\) −26.7926 −0.857172 −0.428586 0.903501i \(-0.640988\pi\)
−0.428586 + 0.903501i \(0.640988\pi\)
\(978\) 0 0
\(979\) −4.49242 −0.143578
\(980\) 0 0
\(981\) −33.3693 −1.06540
\(982\) 0 0
\(983\) 0.0539753 0.00172155 0.000860773 1.00000i \(-0.499726\pi\)
0.000860773 1.00000i \(0.499726\pi\)
\(984\) 0 0
\(985\) −6.31534 −0.201224
\(986\) 0 0
\(987\) 30.7386 0.978421
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.300187 −0.00953574 −0.00476787 0.999989i \(-0.501518\pi\)
−0.00476787 + 0.999989i \(0.501518\pi\)
\(992\) 0 0
\(993\) −32.3153 −1.02550
\(994\) 0 0
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) 14.3845 0.455561 0.227780 0.973713i \(-0.426853\pi\)
0.227780 + 0.973713i \(0.426853\pi\)
\(998\) 0 0
\(999\) −0.630683 −0.0199539
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 460.2.a.e.1.2 2
3.2 odd 2 4140.2.a.m.1.1 2
4.3 odd 2 1840.2.a.m.1.1 2
5.2 odd 4 2300.2.c.h.1749.1 4
5.3 odd 4 2300.2.c.h.1749.4 4
5.4 even 2 2300.2.a.i.1.1 2
8.3 odd 2 7360.2.a.bo.1.2 2
8.5 even 2 7360.2.a.bi.1.1 2
20.19 odd 2 9200.2.a.bv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.2 2 1.1 even 1 trivial
1840.2.a.m.1.1 2 4.3 odd 2
2300.2.a.i.1.1 2 5.4 even 2
2300.2.c.h.1749.1 4 5.2 odd 4
2300.2.c.h.1749.4 4 5.3 odd 4
4140.2.a.m.1.1 2 3.2 odd 2
7360.2.a.bi.1.1 2 8.5 even 2
7360.2.a.bo.1.2 2 8.3 odd 2
9200.2.a.bv.1.2 2 20.19 odd 2