Properties

Label 4005.2.a.o.1.4
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.49803\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} +O(q^{10})\) \(q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} -0.755898 q^{10} -4.45116 q^{11} -2.43229 q^{13} +0.0376612 q^{14} +0.898186 q^{16} +2.48065 q^{17} -5.16842 q^{19} +1.42862 q^{20} -3.36463 q^{22} +4.99050 q^{23} +1.00000 q^{25} -1.83856 q^{26} -0.0711781 q^{28} -2.59152 q^{29} -7.31064 q^{31} +5.86231 q^{32} +1.87512 q^{34} -0.0498231 q^{35} +5.13825 q^{37} -3.90680 q^{38} +2.59169 q^{40} -9.11073 q^{41} -0.543007 q^{43} +6.35901 q^{44} +3.77231 q^{46} +9.63395 q^{47} -6.99752 q^{49} +0.755898 q^{50} +3.47482 q^{52} +10.7093 q^{53} +4.45116 q^{55} -0.129126 q^{56} -1.95893 q^{58} +12.6337 q^{59} -2.47399 q^{61} -5.52610 q^{62} +2.63494 q^{64} +2.43229 q^{65} -5.36128 q^{67} -3.54390 q^{68} -0.0376612 q^{70} -9.58146 q^{71} +2.86369 q^{73} +3.88400 q^{74} +7.38370 q^{76} -0.221771 q^{77} +6.62473 q^{79} -0.898186 q^{80} -6.88678 q^{82} +12.0061 q^{83} -2.48065 q^{85} -0.410458 q^{86} +11.5360 q^{88} -1.00000 q^{89} -0.121184 q^{91} -7.12952 q^{92} +7.28229 q^{94} +5.16842 q^{95} +4.82051 q^{97} -5.28941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.755898 0.534501 0.267250 0.963627i \(-0.413885\pi\)
0.267250 + 0.963627i \(0.413885\pi\)
\(3\) 0 0
\(4\) −1.42862 −0.714309
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.0498231 0.0188314 0.00941568 0.999956i \(-0.497003\pi\)
0.00941568 + 0.999956i \(0.497003\pi\)
\(8\) −2.59169 −0.916299
\(9\) 0 0
\(10\) −0.755898 −0.239036
\(11\) −4.45116 −1.34208 −0.671038 0.741423i \(-0.734151\pi\)
−0.671038 + 0.741423i \(0.734151\pi\)
\(12\) 0 0
\(13\) −2.43229 −0.674596 −0.337298 0.941398i \(-0.609513\pi\)
−0.337298 + 0.941398i \(0.609513\pi\)
\(14\) 0.0376612 0.0100654
\(15\) 0 0
\(16\) 0.898186 0.224546
\(17\) 2.48065 0.601646 0.300823 0.953680i \(-0.402739\pi\)
0.300823 + 0.953680i \(0.402739\pi\)
\(18\) 0 0
\(19\) −5.16842 −1.18572 −0.592858 0.805307i \(-0.702001\pi\)
−0.592858 + 0.805307i \(0.702001\pi\)
\(20\) 1.42862 0.319449
\(21\) 0 0
\(22\) −3.36463 −0.717341
\(23\) 4.99050 1.04059 0.520295 0.853986i \(-0.325822\pi\)
0.520295 + 0.853986i \(0.325822\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.83856 −0.360572
\(27\) 0 0
\(28\) −0.0711781 −0.0134514
\(29\) −2.59152 −0.481234 −0.240617 0.970620i \(-0.577350\pi\)
−0.240617 + 0.970620i \(0.577350\pi\)
\(30\) 0 0
\(31\) −7.31064 −1.31303 −0.656515 0.754313i \(-0.727970\pi\)
−0.656515 + 0.754313i \(0.727970\pi\)
\(32\) 5.86231 1.03632
\(33\) 0 0
\(34\) 1.87512 0.321580
\(35\) −0.0498231 −0.00842164
\(36\) 0 0
\(37\) 5.13825 0.844724 0.422362 0.906427i \(-0.361201\pi\)
0.422362 + 0.906427i \(0.361201\pi\)
\(38\) −3.90680 −0.633766
\(39\) 0 0
\(40\) 2.59169 0.409782
\(41\) −9.11073 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(42\) 0 0
\(43\) −0.543007 −0.0828078 −0.0414039 0.999142i \(-0.513183\pi\)
−0.0414039 + 0.999142i \(0.513183\pi\)
\(44\) 6.35901 0.958657
\(45\) 0 0
\(46\) 3.77231 0.556196
\(47\) 9.63395 1.40526 0.702628 0.711557i \(-0.252010\pi\)
0.702628 + 0.711557i \(0.252010\pi\)
\(48\) 0 0
\(49\) −6.99752 −0.999645
\(50\) 0.755898 0.106900
\(51\) 0 0
\(52\) 3.47482 0.481870
\(53\) 10.7093 1.47104 0.735519 0.677504i \(-0.236938\pi\)
0.735519 + 0.677504i \(0.236938\pi\)
\(54\) 0 0
\(55\) 4.45116 0.600195
\(56\) −0.129126 −0.0172552
\(57\) 0 0
\(58\) −1.95893 −0.257220
\(59\) 12.6337 1.64477 0.822383 0.568934i \(-0.192644\pi\)
0.822383 + 0.568934i \(0.192644\pi\)
\(60\) 0 0
\(61\) −2.47399 −0.316762 −0.158381 0.987378i \(-0.550627\pi\)
−0.158381 + 0.987378i \(0.550627\pi\)
\(62\) −5.52610 −0.701816
\(63\) 0 0
\(64\) 2.63494 0.329367
\(65\) 2.43229 0.301689
\(66\) 0 0
\(67\) −5.36128 −0.654984 −0.327492 0.944854i \(-0.606203\pi\)
−0.327492 + 0.944854i \(0.606203\pi\)
\(68\) −3.54390 −0.429761
\(69\) 0 0
\(70\) −0.0376612 −0.00450137
\(71\) −9.58146 −1.13711 −0.568555 0.822645i \(-0.692497\pi\)
−0.568555 + 0.822645i \(0.692497\pi\)
\(72\) 0 0
\(73\) 2.86369 0.335169 0.167585 0.985858i \(-0.446403\pi\)
0.167585 + 0.985858i \(0.446403\pi\)
\(74\) 3.88400 0.451505
\(75\) 0 0
\(76\) 7.38370 0.846968
\(77\) −0.221771 −0.0252731
\(78\) 0 0
\(79\) 6.62473 0.745340 0.372670 0.927964i \(-0.378442\pi\)
0.372670 + 0.927964i \(0.378442\pi\)
\(80\) −0.898186 −0.100420
\(81\) 0 0
\(82\) −6.88678 −0.760518
\(83\) 12.0061 1.31784 0.658919 0.752214i \(-0.271014\pi\)
0.658919 + 0.752214i \(0.271014\pi\)
\(84\) 0 0
\(85\) −2.48065 −0.269064
\(86\) −0.410458 −0.0442608
\(87\) 0 0
\(88\) 11.5360 1.22974
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −0.121184 −0.0127036
\(92\) −7.12952 −0.743303
\(93\) 0 0
\(94\) 7.28229 0.751111
\(95\) 5.16842 0.530269
\(96\) 0 0
\(97\) 4.82051 0.489449 0.244724 0.969593i \(-0.421303\pi\)
0.244724 + 0.969593i \(0.421303\pi\)
\(98\) −5.28941 −0.534311
\(99\) 0 0
\(100\) −1.42862 −0.142862
\(101\) 11.5561 1.14988 0.574938 0.818197i \(-0.305026\pi\)
0.574938 + 0.818197i \(0.305026\pi\)
\(102\) 0 0
\(103\) −2.92606 −0.288314 −0.144157 0.989555i \(-0.546047\pi\)
−0.144157 + 0.989555i \(0.546047\pi\)
\(104\) 6.30374 0.618132
\(105\) 0 0
\(106\) 8.09516 0.786271
\(107\) 3.97982 0.384744 0.192372 0.981322i \(-0.438382\pi\)
0.192372 + 0.981322i \(0.438382\pi\)
\(108\) 0 0
\(109\) 5.30200 0.507840 0.253920 0.967225i \(-0.418280\pi\)
0.253920 + 0.967225i \(0.418280\pi\)
\(110\) 3.36463 0.320805
\(111\) 0 0
\(112\) 0.0447504 0.00422851
\(113\) 10.3888 0.977296 0.488648 0.872481i \(-0.337490\pi\)
0.488648 + 0.872481i \(0.337490\pi\)
\(114\) 0 0
\(115\) −4.99050 −0.465366
\(116\) 3.70230 0.343750
\(117\) 0 0
\(118\) 9.54978 0.879129
\(119\) 0.123594 0.0113298
\(120\) 0 0
\(121\) 8.81287 0.801170
\(122\) −1.87008 −0.169310
\(123\) 0 0
\(124\) 10.4441 0.937909
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.31244 0.471403 0.235701 0.971826i \(-0.424261\pi\)
0.235701 + 0.971826i \(0.424261\pi\)
\(128\) −9.73287 −0.860273
\(129\) 0 0
\(130\) 1.83856 0.161253
\(131\) 22.2268 1.94197 0.970984 0.239145i \(-0.0768671\pi\)
0.970984 + 0.239145i \(0.0768671\pi\)
\(132\) 0 0
\(133\) −0.257507 −0.0223287
\(134\) −4.05258 −0.350089
\(135\) 0 0
\(136\) −6.42906 −0.551288
\(137\) −8.08077 −0.690387 −0.345193 0.938532i \(-0.612187\pi\)
−0.345193 + 0.938532i \(0.612187\pi\)
\(138\) 0 0
\(139\) 0.763404 0.0647511 0.0323755 0.999476i \(-0.489693\pi\)
0.0323755 + 0.999476i \(0.489693\pi\)
\(140\) 0.0711781 0.00601565
\(141\) 0 0
\(142\) −7.24261 −0.607786
\(143\) 10.8265 0.905360
\(144\) 0 0
\(145\) 2.59152 0.215214
\(146\) 2.16466 0.179148
\(147\) 0 0
\(148\) −7.34060 −0.603394
\(149\) 0.645836 0.0529089 0.0264545 0.999650i \(-0.491578\pi\)
0.0264545 + 0.999650i \(0.491578\pi\)
\(150\) 0 0
\(151\) −16.4449 −1.33827 −0.669134 0.743142i \(-0.733335\pi\)
−0.669134 + 0.743142i \(0.733335\pi\)
\(152\) 13.3949 1.08647
\(153\) 0 0
\(154\) −0.167636 −0.0135085
\(155\) 7.31064 0.587205
\(156\) 0 0
\(157\) 8.23439 0.657176 0.328588 0.944473i \(-0.393427\pi\)
0.328588 + 0.944473i \(0.393427\pi\)
\(158\) 5.00762 0.398385
\(159\) 0 0
\(160\) −5.86231 −0.463456
\(161\) 0.248642 0.0195957
\(162\) 0 0
\(163\) 1.57747 0.123557 0.0617786 0.998090i \(-0.480323\pi\)
0.0617786 + 0.998090i \(0.480323\pi\)
\(164\) 13.0157 1.01636
\(165\) 0 0
\(166\) 9.07537 0.704385
\(167\) −1.78625 −0.138224 −0.0691121 0.997609i \(-0.522017\pi\)
−0.0691121 + 0.997609i \(0.522017\pi\)
\(168\) 0 0
\(169\) −7.08396 −0.544920
\(170\) −1.87512 −0.143815
\(171\) 0 0
\(172\) 0.775750 0.0591504
\(173\) −23.0808 −1.75480 −0.877399 0.479761i \(-0.840723\pi\)
−0.877399 + 0.479761i \(0.840723\pi\)
\(174\) 0 0
\(175\) 0.0498231 0.00376627
\(176\) −3.99797 −0.301359
\(177\) 0 0
\(178\) −0.755898 −0.0566570
\(179\) 9.63601 0.720229 0.360115 0.932908i \(-0.382738\pi\)
0.360115 + 0.932908i \(0.382738\pi\)
\(180\) 0 0
\(181\) −7.10071 −0.527792 −0.263896 0.964551i \(-0.585008\pi\)
−0.263896 + 0.964551i \(0.585008\pi\)
\(182\) −0.0916030 −0.00679006
\(183\) 0 0
\(184\) −12.9338 −0.953493
\(185\) −5.13825 −0.377772
\(186\) 0 0
\(187\) −11.0418 −0.807455
\(188\) −13.7632 −1.00379
\(189\) 0 0
\(190\) 3.90680 0.283429
\(191\) −9.47202 −0.685371 −0.342686 0.939450i \(-0.611337\pi\)
−0.342686 + 0.939450i \(0.611337\pi\)
\(192\) 0 0
\(193\) 11.7250 0.843985 0.421992 0.906599i \(-0.361331\pi\)
0.421992 + 0.906599i \(0.361331\pi\)
\(194\) 3.64382 0.261611
\(195\) 0 0
\(196\) 9.99678 0.714056
\(197\) 18.2857 1.30280 0.651402 0.758733i \(-0.274181\pi\)
0.651402 + 0.758733i \(0.274181\pi\)
\(198\) 0 0
\(199\) 25.0469 1.77553 0.887763 0.460302i \(-0.152259\pi\)
0.887763 + 0.460302i \(0.152259\pi\)
\(200\) −2.59169 −0.183260
\(201\) 0 0
\(202\) 8.73524 0.614609
\(203\) −0.129118 −0.00906229
\(204\) 0 0
\(205\) 9.11073 0.636321
\(206\) −2.21181 −0.154104
\(207\) 0 0
\(208\) −2.18465 −0.151478
\(209\) 23.0055 1.59132
\(210\) 0 0
\(211\) −17.6282 −1.21358 −0.606789 0.794863i \(-0.707543\pi\)
−0.606789 + 0.794863i \(0.707543\pi\)
\(212\) −15.2995 −1.05078
\(213\) 0 0
\(214\) 3.00834 0.205646
\(215\) 0.543007 0.0370328
\(216\) 0 0
\(217\) −0.364239 −0.0247261
\(218\) 4.00777 0.271441
\(219\) 0 0
\(220\) −6.35901 −0.428725
\(221\) −6.03366 −0.405868
\(222\) 0 0
\(223\) 27.8359 1.86403 0.932014 0.362421i \(-0.118050\pi\)
0.932014 + 0.362421i \(0.118050\pi\)
\(224\) 0.292078 0.0195153
\(225\) 0 0
\(226\) 7.85288 0.522366
\(227\) −10.7150 −0.711181 −0.355591 0.934642i \(-0.615720\pi\)
−0.355591 + 0.934642i \(0.615720\pi\)
\(228\) 0 0
\(229\) −20.0837 −1.32717 −0.663584 0.748102i \(-0.730966\pi\)
−0.663584 + 0.748102i \(0.730966\pi\)
\(230\) −3.77231 −0.248739
\(231\) 0 0
\(232\) 6.71642 0.440954
\(233\) −1.64318 −0.107648 −0.0538242 0.998550i \(-0.517141\pi\)
−0.0538242 + 0.998550i \(0.517141\pi\)
\(234\) 0 0
\(235\) −9.63395 −0.628450
\(236\) −18.0487 −1.17487
\(237\) 0 0
\(238\) 0.0934241 0.00605579
\(239\) 12.3117 0.796379 0.398189 0.917303i \(-0.369639\pi\)
0.398189 + 0.917303i \(0.369639\pi\)
\(240\) 0 0
\(241\) −10.7917 −0.695152 −0.347576 0.937652i \(-0.612995\pi\)
−0.347576 + 0.937652i \(0.612995\pi\)
\(242\) 6.66163 0.428226
\(243\) 0 0
\(244\) 3.53439 0.226266
\(245\) 6.99752 0.447055
\(246\) 0 0
\(247\) 12.5711 0.799880
\(248\) 18.9469 1.20313
\(249\) 0 0
\(250\) −0.755898 −0.0478072
\(251\) 31.0270 1.95840 0.979202 0.202888i \(-0.0650326\pi\)
0.979202 + 0.202888i \(0.0650326\pi\)
\(252\) 0 0
\(253\) −22.2135 −1.39655
\(254\) 4.01566 0.251965
\(255\) 0 0
\(256\) −12.6269 −0.789183
\(257\) −16.5586 −1.03290 −0.516449 0.856318i \(-0.672747\pi\)
−0.516449 + 0.856318i \(0.672747\pi\)
\(258\) 0 0
\(259\) 0.256004 0.0159073
\(260\) −3.47482 −0.215499
\(261\) 0 0
\(262\) 16.8012 1.03798
\(263\) 12.0270 0.741619 0.370810 0.928709i \(-0.379080\pi\)
0.370810 + 0.928709i \(0.379080\pi\)
\(264\) 0 0
\(265\) −10.7093 −0.657868
\(266\) −0.194649 −0.0119347
\(267\) 0 0
\(268\) 7.65922 0.467861
\(269\) 22.0871 1.34667 0.673336 0.739336i \(-0.264861\pi\)
0.673336 + 0.739336i \(0.264861\pi\)
\(270\) 0 0
\(271\) 13.8002 0.838302 0.419151 0.907917i \(-0.362328\pi\)
0.419151 + 0.907917i \(0.362328\pi\)
\(272\) 2.22808 0.135097
\(273\) 0 0
\(274\) −6.10824 −0.369012
\(275\) −4.45116 −0.268415
\(276\) 0 0
\(277\) 3.04290 0.182830 0.0914151 0.995813i \(-0.470861\pi\)
0.0914151 + 0.995813i \(0.470861\pi\)
\(278\) 0.577056 0.0346095
\(279\) 0 0
\(280\) 0.129126 0.00771674
\(281\) −21.1218 −1.26002 −0.630011 0.776586i \(-0.716950\pi\)
−0.630011 + 0.776586i \(0.716950\pi\)
\(282\) 0 0
\(283\) 32.4871 1.93116 0.965579 0.260111i \(-0.0837590\pi\)
0.965579 + 0.260111i \(0.0837590\pi\)
\(284\) 13.6882 0.812248
\(285\) 0 0
\(286\) 8.18376 0.483916
\(287\) −0.453924 −0.0267943
\(288\) 0 0
\(289\) −10.8464 −0.638022
\(290\) 1.95893 0.115032
\(291\) 0 0
\(292\) −4.09112 −0.239414
\(293\) −32.0101 −1.87005 −0.935026 0.354578i \(-0.884624\pi\)
−0.935026 + 0.354578i \(0.884624\pi\)
\(294\) 0 0
\(295\) −12.6337 −0.735562
\(296\) −13.3167 −0.774020
\(297\) 0 0
\(298\) 0.488186 0.0282798
\(299\) −12.1383 −0.701979
\(300\) 0 0
\(301\) −0.0270543 −0.00155938
\(302\) −12.4307 −0.715305
\(303\) 0 0
\(304\) −4.64220 −0.266248
\(305\) 2.47399 0.141660
\(306\) 0 0
\(307\) −19.6188 −1.11970 −0.559852 0.828593i \(-0.689142\pi\)
−0.559852 + 0.828593i \(0.689142\pi\)
\(308\) 0.316826 0.0180528
\(309\) 0 0
\(310\) 5.52610 0.313861
\(311\) −24.8863 −1.41117 −0.705586 0.708625i \(-0.749316\pi\)
−0.705586 + 0.708625i \(0.749316\pi\)
\(312\) 0 0
\(313\) −6.17328 −0.348934 −0.174467 0.984663i \(-0.555820\pi\)
−0.174467 + 0.984663i \(0.555820\pi\)
\(314\) 6.22436 0.351261
\(315\) 0 0
\(316\) −9.46420 −0.532403
\(317\) 32.9455 1.85041 0.925203 0.379473i \(-0.123895\pi\)
0.925203 + 0.379473i \(0.123895\pi\)
\(318\) 0 0
\(319\) 11.5353 0.645853
\(320\) −2.63494 −0.147297
\(321\) 0 0
\(322\) 0.187948 0.0104739
\(323\) −12.8210 −0.713381
\(324\) 0 0
\(325\) −2.43229 −0.134919
\(326\) 1.19241 0.0660414
\(327\) 0 0
\(328\) 23.6121 1.30376
\(329\) 0.479993 0.0264629
\(330\) 0 0
\(331\) 30.9785 1.70273 0.851367 0.524570i \(-0.175774\pi\)
0.851367 + 0.524570i \(0.175774\pi\)
\(332\) −17.1521 −0.941343
\(333\) 0 0
\(334\) −1.35022 −0.0738809
\(335\) 5.36128 0.292918
\(336\) 0 0
\(337\) −3.53709 −0.192678 −0.0963389 0.995349i \(-0.530713\pi\)
−0.0963389 + 0.995349i \(0.530713\pi\)
\(338\) −5.35475 −0.291260
\(339\) 0 0
\(340\) 3.54390 0.192195
\(341\) 32.5409 1.76219
\(342\) 0 0
\(343\) −0.697399 −0.0376560
\(344\) 1.40730 0.0758768
\(345\) 0 0
\(346\) −17.4467 −0.937941
\(347\) 13.0198 0.698939 0.349470 0.936948i \(-0.386362\pi\)
0.349470 + 0.936948i \(0.386362\pi\)
\(348\) 0 0
\(349\) 27.6562 1.48040 0.740201 0.672385i \(-0.234730\pi\)
0.740201 + 0.672385i \(0.234730\pi\)
\(350\) 0.0376612 0.00201307
\(351\) 0 0
\(352\) −26.0941 −1.39082
\(353\) 26.3090 1.40029 0.700144 0.714002i \(-0.253119\pi\)
0.700144 + 0.714002i \(0.253119\pi\)
\(354\) 0 0
\(355\) 9.58146 0.508531
\(356\) 1.42862 0.0757166
\(357\) 0 0
\(358\) 7.28384 0.384963
\(359\) −14.0008 −0.738934 −0.369467 0.929244i \(-0.620460\pi\)
−0.369467 + 0.929244i \(0.620460\pi\)
\(360\) 0 0
\(361\) 7.71256 0.405924
\(362\) −5.36741 −0.282105
\(363\) 0 0
\(364\) 0.173126 0.00907427
\(365\) −2.86369 −0.149892
\(366\) 0 0
\(367\) 22.0304 1.14998 0.574989 0.818161i \(-0.305006\pi\)
0.574989 + 0.818161i \(0.305006\pi\)
\(368\) 4.48239 0.233661
\(369\) 0 0
\(370\) −3.88400 −0.201919
\(371\) 0.533571 0.0277016
\(372\) 0 0
\(373\) 13.6705 0.707830 0.353915 0.935278i \(-0.384850\pi\)
0.353915 + 0.935278i \(0.384850\pi\)
\(374\) −8.34646 −0.431585
\(375\) 0 0
\(376\) −24.9682 −1.28764
\(377\) 6.30334 0.324639
\(378\) 0 0
\(379\) −15.1907 −0.780292 −0.390146 0.920753i \(-0.627575\pi\)
−0.390146 + 0.920753i \(0.627575\pi\)
\(380\) −7.38370 −0.378776
\(381\) 0 0
\(382\) −7.15988 −0.366331
\(383\) 0.297784 0.0152160 0.00760802 0.999971i \(-0.497578\pi\)
0.00760802 + 0.999971i \(0.497578\pi\)
\(384\) 0 0
\(385\) 0.221771 0.0113025
\(386\) 8.86291 0.451110
\(387\) 0 0
\(388\) −6.88667 −0.349618
\(389\) 2.25933 0.114553 0.0572763 0.998358i \(-0.481758\pi\)
0.0572763 + 0.998358i \(0.481758\pi\)
\(390\) 0 0
\(391\) 12.3797 0.626067
\(392\) 18.1354 0.915974
\(393\) 0 0
\(394\) 13.8221 0.696349
\(395\) −6.62473 −0.333326
\(396\) 0 0
\(397\) −23.8373 −1.19636 −0.598180 0.801361i \(-0.704109\pi\)
−0.598180 + 0.801361i \(0.704109\pi\)
\(398\) 18.9329 0.949019
\(399\) 0 0
\(400\) 0.898186 0.0449093
\(401\) −26.1681 −1.30677 −0.653386 0.757025i \(-0.726652\pi\)
−0.653386 + 0.757025i \(0.726652\pi\)
\(402\) 0 0
\(403\) 17.7816 0.885766
\(404\) −16.5093 −0.821366
\(405\) 0 0
\(406\) −0.0975998 −0.00484380
\(407\) −22.8712 −1.13368
\(408\) 0 0
\(409\) −22.9348 −1.13406 −0.567028 0.823699i \(-0.691907\pi\)
−0.567028 + 0.823699i \(0.691907\pi\)
\(410\) 6.88678 0.340114
\(411\) 0 0
\(412\) 4.18023 0.205945
\(413\) 0.629449 0.0309732
\(414\) 0 0
\(415\) −12.0061 −0.589355
\(416\) −14.2588 −0.699097
\(417\) 0 0
\(418\) 17.3898 0.850563
\(419\) −5.25412 −0.256681 −0.128340 0.991730i \(-0.540965\pi\)
−0.128340 + 0.991730i \(0.540965\pi\)
\(420\) 0 0
\(421\) 18.0800 0.881166 0.440583 0.897712i \(-0.354772\pi\)
0.440583 + 0.897712i \(0.354772\pi\)
\(422\) −13.3251 −0.648658
\(423\) 0 0
\(424\) −27.7552 −1.34791
\(425\) 2.48065 0.120329
\(426\) 0 0
\(427\) −0.123262 −0.00596506
\(428\) −5.68564 −0.274826
\(429\) 0 0
\(430\) 0.410458 0.0197940
\(431\) −5.55910 −0.267773 −0.133886 0.990997i \(-0.542746\pi\)
−0.133886 + 0.990997i \(0.542746\pi\)
\(432\) 0 0
\(433\) 20.5795 0.988986 0.494493 0.869182i \(-0.335354\pi\)
0.494493 + 0.869182i \(0.335354\pi\)
\(434\) −0.275327 −0.0132161
\(435\) 0 0
\(436\) −7.57454 −0.362755
\(437\) −25.7930 −1.23385
\(438\) 0 0
\(439\) 26.6104 1.27004 0.635022 0.772494i \(-0.280991\pi\)
0.635022 + 0.772494i \(0.280991\pi\)
\(440\) −11.5360 −0.549958
\(441\) 0 0
\(442\) −4.56083 −0.216937
\(443\) −20.4411 −0.971188 −0.485594 0.874185i \(-0.661397\pi\)
−0.485594 + 0.874185i \(0.661397\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) 21.0411 0.996325
\(447\) 0 0
\(448\) 0.131281 0.00620243
\(449\) −5.61381 −0.264932 −0.132466 0.991188i \(-0.542290\pi\)
−0.132466 + 0.991188i \(0.542290\pi\)
\(450\) 0 0
\(451\) 40.5533 1.90958
\(452\) −14.8416 −0.698092
\(453\) 0 0
\(454\) −8.09947 −0.380127
\(455\) 0.121184 0.00568121
\(456\) 0 0
\(457\) −7.13528 −0.333774 −0.166887 0.985976i \(-0.553372\pi\)
−0.166887 + 0.985976i \(0.553372\pi\)
\(458\) −15.1812 −0.709372
\(459\) 0 0
\(460\) 7.12952 0.332415
\(461\) 24.0844 1.12172 0.560860 0.827911i \(-0.310471\pi\)
0.560860 + 0.827911i \(0.310471\pi\)
\(462\) 0 0
\(463\) −22.2301 −1.03312 −0.516561 0.856250i \(-0.672788\pi\)
−0.516561 + 0.856250i \(0.672788\pi\)
\(464\) −2.32767 −0.108059
\(465\) 0 0
\(466\) −1.24208 −0.0575382
\(467\) −10.4176 −0.482069 −0.241034 0.970517i \(-0.577487\pi\)
−0.241034 + 0.970517i \(0.577487\pi\)
\(468\) 0 0
\(469\) −0.267115 −0.0123342
\(470\) −7.28229 −0.335907
\(471\) 0 0
\(472\) −32.7426 −1.50710
\(473\) 2.41701 0.111134
\(474\) 0 0
\(475\) −5.16842 −0.237143
\(476\) −0.176568 −0.00809298
\(477\) 0 0
\(478\) 9.30640 0.425665
\(479\) 20.6366 0.942912 0.471456 0.881889i \(-0.343729\pi\)
0.471456 + 0.881889i \(0.343729\pi\)
\(480\) 0 0
\(481\) −12.4977 −0.569848
\(482\) −8.15740 −0.371559
\(483\) 0 0
\(484\) −12.5902 −0.572283
\(485\) −4.82051 −0.218888
\(486\) 0 0
\(487\) −29.5496 −1.33902 −0.669510 0.742803i \(-0.733496\pi\)
−0.669510 + 0.742803i \(0.733496\pi\)
\(488\) 6.41181 0.290249
\(489\) 0 0
\(490\) 5.28941 0.238951
\(491\) −37.8108 −1.70638 −0.853189 0.521602i \(-0.825335\pi\)
−0.853189 + 0.521602i \(0.825335\pi\)
\(492\) 0 0
\(493\) −6.42866 −0.289532
\(494\) 9.50247 0.427537
\(495\) 0 0
\(496\) −6.56632 −0.294836
\(497\) −0.477378 −0.0214133
\(498\) 0 0
\(499\) 7.09756 0.317730 0.158865 0.987300i \(-0.449217\pi\)
0.158865 + 0.987300i \(0.449217\pi\)
\(500\) 1.42862 0.0638897
\(501\) 0 0
\(502\) 23.4532 1.04677
\(503\) 25.0615 1.11744 0.558718 0.829358i \(-0.311294\pi\)
0.558718 + 0.829358i \(0.311294\pi\)
\(504\) 0 0
\(505\) −11.5561 −0.514240
\(506\) −16.7912 −0.746458
\(507\) 0 0
\(508\) −7.58945 −0.336727
\(509\) 1.59934 0.0708895 0.0354447 0.999372i \(-0.488715\pi\)
0.0354447 + 0.999372i \(0.488715\pi\)
\(510\) 0 0
\(511\) 0.142678 0.00631169
\(512\) 9.92107 0.438454
\(513\) 0 0
\(514\) −12.5166 −0.552085
\(515\) 2.92606 0.128938
\(516\) 0 0
\(517\) −42.8823 −1.88596
\(518\) 0.193513 0.00850246
\(519\) 0 0
\(520\) −6.30374 −0.276437
\(521\) −25.2660 −1.10693 −0.553463 0.832874i \(-0.686694\pi\)
−0.553463 + 0.832874i \(0.686694\pi\)
\(522\) 0 0
\(523\) −1.75283 −0.0766458 −0.0383229 0.999265i \(-0.512202\pi\)
−0.0383229 + 0.999265i \(0.512202\pi\)
\(524\) −31.7537 −1.38717
\(525\) 0 0
\(526\) 9.09122 0.396396
\(527\) −18.1351 −0.789979
\(528\) 0 0
\(529\) 1.90507 0.0828292
\(530\) −8.09516 −0.351631
\(531\) 0 0
\(532\) 0.367879 0.0159496
\(533\) 22.1599 0.959854
\(534\) 0 0
\(535\) −3.97982 −0.172063
\(536\) 13.8947 0.600161
\(537\) 0 0
\(538\) 16.6956 0.719797
\(539\) 31.1471 1.34160
\(540\) 0 0
\(541\) 5.15589 0.221669 0.110835 0.993839i \(-0.464648\pi\)
0.110835 + 0.993839i \(0.464648\pi\)
\(542\) 10.4315 0.448073
\(543\) 0 0
\(544\) 14.5423 0.623497
\(545\) −5.30200 −0.227113
\(546\) 0 0
\(547\) 13.3060 0.568925 0.284462 0.958687i \(-0.408185\pi\)
0.284462 + 0.958687i \(0.408185\pi\)
\(548\) 11.5443 0.493149
\(549\) 0 0
\(550\) −3.36463 −0.143468
\(551\) 13.3941 0.570607
\(552\) 0 0
\(553\) 0.330064 0.0140358
\(554\) 2.30012 0.0977229
\(555\) 0 0
\(556\) −1.09061 −0.0462523
\(557\) 32.2911 1.36822 0.684109 0.729380i \(-0.260191\pi\)
0.684109 + 0.729380i \(0.260191\pi\)
\(558\) 0 0
\(559\) 1.32075 0.0558619
\(560\) −0.0447504 −0.00189105
\(561\) 0 0
\(562\) −15.9659 −0.673482
\(563\) 1.17102 0.0493525 0.0246762 0.999695i \(-0.492145\pi\)
0.0246762 + 0.999695i \(0.492145\pi\)
\(564\) 0 0
\(565\) −10.3888 −0.437060
\(566\) 24.5569 1.03221
\(567\) 0 0
\(568\) 24.8321 1.04193
\(569\) −20.9355 −0.877662 −0.438831 0.898570i \(-0.644607\pi\)
−0.438831 + 0.898570i \(0.644607\pi\)
\(570\) 0 0
\(571\) −44.5105 −1.86271 −0.931354 0.364116i \(-0.881371\pi\)
−0.931354 + 0.364116i \(0.881371\pi\)
\(572\) −15.4670 −0.646707
\(573\) 0 0
\(574\) −0.343121 −0.0143216
\(575\) 4.99050 0.208118
\(576\) 0 0
\(577\) −44.7453 −1.86277 −0.931386 0.364034i \(-0.881399\pi\)
−0.931386 + 0.364034i \(0.881399\pi\)
\(578\) −8.19876 −0.341023
\(579\) 0 0
\(580\) −3.70230 −0.153730
\(581\) 0.598180 0.0248167
\(582\) 0 0
\(583\) −47.6690 −1.97425
\(584\) −7.42178 −0.307115
\(585\) 0 0
\(586\) −24.1964 −0.999544
\(587\) −14.8194 −0.611663 −0.305832 0.952086i \(-0.598934\pi\)
−0.305832 + 0.952086i \(0.598934\pi\)
\(588\) 0 0
\(589\) 37.7845 1.55688
\(590\) −9.54978 −0.393158
\(591\) 0 0
\(592\) 4.61511 0.189680
\(593\) 23.3625 0.959384 0.479692 0.877437i \(-0.340748\pi\)
0.479692 + 0.877437i \(0.340748\pi\)
\(594\) 0 0
\(595\) −0.123594 −0.00506684
\(596\) −0.922652 −0.0377933
\(597\) 0 0
\(598\) −9.17535 −0.375208
\(599\) 2.73683 0.111824 0.0559120 0.998436i \(-0.482193\pi\)
0.0559120 + 0.998436i \(0.482193\pi\)
\(600\) 0 0
\(601\) −14.7244 −0.600622 −0.300311 0.953841i \(-0.597090\pi\)
−0.300311 + 0.953841i \(0.597090\pi\)
\(602\) −0.0204503 −0.000833491 0
\(603\) 0 0
\(604\) 23.4935 0.955937
\(605\) −8.81287 −0.358294
\(606\) 0 0
\(607\) −7.93966 −0.322261 −0.161131 0.986933i \(-0.551514\pi\)
−0.161131 + 0.986933i \(0.551514\pi\)
\(608\) −30.2989 −1.22878
\(609\) 0 0
\(610\) 1.87008 0.0757175
\(611\) −23.4326 −0.947981
\(612\) 0 0
\(613\) −5.63827 −0.227728 −0.113864 0.993496i \(-0.536323\pi\)
−0.113864 + 0.993496i \(0.536323\pi\)
\(614\) −14.8298 −0.598483
\(615\) 0 0
\(616\) 0.574760 0.0231577
\(617\) −24.0408 −0.967848 −0.483924 0.875110i \(-0.660789\pi\)
−0.483924 + 0.875110i \(0.660789\pi\)
\(618\) 0 0
\(619\) 18.0743 0.726469 0.363234 0.931698i \(-0.381672\pi\)
0.363234 + 0.931698i \(0.381672\pi\)
\(620\) −10.4441 −0.419446
\(621\) 0 0
\(622\) −18.8115 −0.754272
\(623\) −0.0498231 −0.00199612
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.66637 −0.186506
\(627\) 0 0
\(628\) −11.7638 −0.469427
\(629\) 12.7462 0.508224
\(630\) 0 0
\(631\) −3.58541 −0.142733 −0.0713665 0.997450i \(-0.522736\pi\)
−0.0713665 + 0.997450i \(0.522736\pi\)
\(632\) −17.1692 −0.682955
\(633\) 0 0
\(634\) 24.9035 0.989043
\(635\) −5.31244 −0.210818
\(636\) 0 0
\(637\) 17.0200 0.674357
\(638\) 8.71951 0.345209
\(639\) 0 0
\(640\) 9.73287 0.384726
\(641\) 19.3761 0.765311 0.382656 0.923891i \(-0.375010\pi\)
0.382656 + 0.923891i \(0.375010\pi\)
\(642\) 0 0
\(643\) 26.7275 1.05403 0.527015 0.849856i \(-0.323311\pi\)
0.527015 + 0.849856i \(0.323311\pi\)
\(644\) −0.355214 −0.0139974
\(645\) 0 0
\(646\) −9.69139 −0.381303
\(647\) −20.6868 −0.813281 −0.406641 0.913588i \(-0.633300\pi\)
−0.406641 + 0.913588i \(0.633300\pi\)
\(648\) 0 0
\(649\) −56.2346 −2.20740
\(650\) −1.83856 −0.0721144
\(651\) 0 0
\(652\) −2.25361 −0.0882580
\(653\) 33.7364 1.32021 0.660104 0.751174i \(-0.270512\pi\)
0.660104 + 0.751174i \(0.270512\pi\)
\(654\) 0 0
\(655\) −22.2268 −0.868474
\(656\) −8.18313 −0.319497
\(657\) 0 0
\(658\) 0.362826 0.0141444
\(659\) 9.44930 0.368092 0.184046 0.982918i \(-0.441080\pi\)
0.184046 + 0.982918i \(0.441080\pi\)
\(660\) 0 0
\(661\) −36.6486 −1.42547 −0.712733 0.701435i \(-0.752543\pi\)
−0.712733 + 0.701435i \(0.752543\pi\)
\(662\) 23.4166 0.910113
\(663\) 0 0
\(664\) −31.1160 −1.20753
\(665\) 0.257507 0.00998568
\(666\) 0 0
\(667\) −12.9330 −0.500768
\(668\) 2.55187 0.0987348
\(669\) 0 0
\(670\) 4.05258 0.156565
\(671\) 11.0121 0.425119
\(672\) 0 0
\(673\) 42.5711 1.64099 0.820497 0.571651i \(-0.193697\pi\)
0.820497 + 0.571651i \(0.193697\pi\)
\(674\) −2.67368 −0.102986
\(675\) 0 0
\(676\) 10.1203 0.389241
\(677\) 29.6301 1.13878 0.569389 0.822068i \(-0.307180\pi\)
0.569389 + 0.822068i \(0.307180\pi\)
\(678\) 0 0
\(679\) 0.240173 0.00921698
\(680\) 6.42906 0.246543
\(681\) 0 0
\(682\) 24.5976 0.941890
\(683\) −18.4389 −0.705545 −0.352772 0.935709i \(-0.614761\pi\)
−0.352772 + 0.935709i \(0.614761\pi\)
\(684\) 0 0
\(685\) 8.08077 0.308750
\(686\) −0.527163 −0.0201272
\(687\) 0 0
\(688\) −0.487721 −0.0185942
\(689\) −26.0482 −0.992357
\(690\) 0 0
\(691\) −39.7294 −1.51138 −0.755689 0.654930i \(-0.772698\pi\)
−0.755689 + 0.654930i \(0.772698\pi\)
\(692\) 32.9736 1.25347
\(693\) 0 0
\(694\) 9.84164 0.373584
\(695\) −0.763404 −0.0289576
\(696\) 0 0
\(697\) −22.6005 −0.856056
\(698\) 20.9053 0.791276
\(699\) 0 0
\(700\) −0.0711781 −0.00269028
\(701\) 8.13101 0.307104 0.153552 0.988141i \(-0.450929\pi\)
0.153552 + 0.988141i \(0.450929\pi\)
\(702\) 0 0
\(703\) −26.5567 −1.00160
\(704\) −11.7285 −0.442036
\(705\) 0 0
\(706\) 19.8869 0.748455
\(707\) 0.575761 0.0216537
\(708\) 0 0
\(709\) 24.2930 0.912343 0.456171 0.889892i \(-0.349220\pi\)
0.456171 + 0.889892i \(0.349220\pi\)
\(710\) 7.24261 0.271810
\(711\) 0 0
\(712\) 2.59169 0.0971275
\(713\) −36.4838 −1.36633
\(714\) 0 0
\(715\) −10.8265 −0.404889
\(716\) −13.7662 −0.514466
\(717\) 0 0
\(718\) −10.5832 −0.394961
\(719\) −35.3355 −1.31779 −0.658896 0.752234i \(-0.728976\pi\)
−0.658896 + 0.752234i \(0.728976\pi\)
\(720\) 0 0
\(721\) −0.145785 −0.00542933
\(722\) 5.82991 0.216967
\(723\) 0 0
\(724\) 10.1442 0.377006
\(725\) −2.59152 −0.0962468
\(726\) 0 0
\(727\) −27.6135 −1.02413 −0.512065 0.858947i \(-0.671119\pi\)
−0.512065 + 0.858947i \(0.671119\pi\)
\(728\) 0.314072 0.0116403
\(729\) 0 0
\(730\) −2.16466 −0.0801175
\(731\) −1.34701 −0.0498210
\(732\) 0 0
\(733\) −19.3580 −0.715005 −0.357503 0.933912i \(-0.616372\pi\)
−0.357503 + 0.933912i \(0.616372\pi\)
\(734\) 16.6527 0.614664
\(735\) 0 0
\(736\) 29.2558 1.07838
\(737\) 23.8639 0.879039
\(738\) 0 0
\(739\) 28.8545 1.06143 0.530715 0.847550i \(-0.321923\pi\)
0.530715 + 0.847550i \(0.321923\pi\)
\(740\) 7.34060 0.269846
\(741\) 0 0
\(742\) 0.403326 0.0148065
\(743\) −25.5760 −0.938292 −0.469146 0.883121i \(-0.655438\pi\)
−0.469146 + 0.883121i \(0.655438\pi\)
\(744\) 0 0
\(745\) −0.645836 −0.0236616
\(746\) 10.3335 0.378336
\(747\) 0 0
\(748\) 15.7745 0.576772
\(749\) 0.198287 0.00724525
\(750\) 0 0
\(751\) −26.6217 −0.971440 −0.485720 0.874114i \(-0.661443\pi\)
−0.485720 + 0.874114i \(0.661443\pi\)
\(752\) 8.65308 0.315545
\(753\) 0 0
\(754\) 4.76468 0.173520
\(755\) 16.4449 0.598492
\(756\) 0 0
\(757\) 16.8066 0.610848 0.305424 0.952216i \(-0.401202\pi\)
0.305424 + 0.952216i \(0.401202\pi\)
\(758\) −11.4826 −0.417067
\(759\) 0 0
\(760\) −13.3949 −0.485885
\(761\) 23.6421 0.857026 0.428513 0.903536i \(-0.359038\pi\)
0.428513 + 0.903536i \(0.359038\pi\)
\(762\) 0 0
\(763\) 0.264162 0.00956331
\(764\) 13.5319 0.489567
\(765\) 0 0
\(766\) 0.225094 0.00813299
\(767\) −30.7288 −1.10955
\(768\) 0 0
\(769\) 20.1859 0.727921 0.363960 0.931414i \(-0.381424\pi\)
0.363960 + 0.931414i \(0.381424\pi\)
\(770\) 0.167636 0.00604118
\(771\) 0 0
\(772\) −16.7506 −0.602866
\(773\) 14.8504 0.534132 0.267066 0.963678i \(-0.413946\pi\)
0.267066 + 0.963678i \(0.413946\pi\)
\(774\) 0 0
\(775\) −7.31064 −0.262606
\(776\) −12.4933 −0.448482
\(777\) 0 0
\(778\) 1.70782 0.0612285
\(779\) 47.0881 1.68710
\(780\) 0 0
\(781\) 42.6487 1.52609
\(782\) 9.35777 0.334633
\(783\) 0 0
\(784\) −6.28507 −0.224467
\(785\) −8.23439 −0.293898
\(786\) 0 0
\(787\) 22.9126 0.816746 0.408373 0.912815i \(-0.366096\pi\)
0.408373 + 0.912815i \(0.366096\pi\)
\(788\) −26.1233 −0.930604
\(789\) 0 0
\(790\) −5.00762 −0.178163
\(791\) 0.517602 0.0184038
\(792\) 0 0
\(793\) 6.01747 0.213687
\(794\) −18.0186 −0.639456
\(795\) 0 0
\(796\) −35.7824 −1.26827
\(797\) −32.7951 −1.16166 −0.580831 0.814024i \(-0.697272\pi\)
−0.580831 + 0.814024i \(0.697272\pi\)
\(798\) 0 0
\(799\) 23.8985 0.845467
\(800\) 5.86231 0.207264
\(801\) 0 0
\(802\) −19.7804 −0.698470
\(803\) −12.7467 −0.449823
\(804\) 0 0
\(805\) −0.248642 −0.00876348
\(806\) 13.4411 0.473442
\(807\) 0 0
\(808\) −29.9498 −1.05363
\(809\) 25.8540 0.908978 0.454489 0.890752i \(-0.349822\pi\)
0.454489 + 0.890752i \(0.349822\pi\)
\(810\) 0 0
\(811\) −5.12583 −0.179992 −0.0899962 0.995942i \(-0.528685\pi\)
−0.0899962 + 0.995942i \(0.528685\pi\)
\(812\) 0.184460 0.00647327
\(813\) 0 0
\(814\) −17.2883 −0.605955
\(815\) −1.57747 −0.0552565
\(816\) 0 0
\(817\) 2.80649 0.0981866
\(818\) −17.3364 −0.606153
\(819\) 0 0
\(820\) −13.0157 −0.454530
\(821\) 15.9798 0.557701 0.278850 0.960335i \(-0.410047\pi\)
0.278850 + 0.960335i \(0.410047\pi\)
\(822\) 0 0
\(823\) 37.5936 1.31043 0.655216 0.755442i \(-0.272578\pi\)
0.655216 + 0.755442i \(0.272578\pi\)
\(824\) 7.58344 0.264182
\(825\) 0 0
\(826\) 0.475799 0.0165552
\(827\) 53.0600 1.84508 0.922538 0.385906i \(-0.126111\pi\)
0.922538 + 0.385906i \(0.126111\pi\)
\(828\) 0 0
\(829\) 7.43536 0.258241 0.129120 0.991629i \(-0.458785\pi\)
0.129120 + 0.991629i \(0.458785\pi\)
\(830\) −9.07537 −0.315011
\(831\) 0 0
\(832\) −6.40893 −0.222190
\(833\) −17.3584 −0.601432
\(834\) 0 0
\(835\) 1.78625 0.0618157
\(836\) −32.8661 −1.13670
\(837\) 0 0
\(838\) −3.97158 −0.137196
\(839\) 38.6435 1.33412 0.667061 0.745003i \(-0.267552\pi\)
0.667061 + 0.745003i \(0.267552\pi\)
\(840\) 0 0
\(841\) −22.2840 −0.768414
\(842\) 13.6667 0.470984
\(843\) 0 0
\(844\) 25.1840 0.866869
\(845\) 7.08396 0.243695
\(846\) 0 0
\(847\) 0.439084 0.0150871
\(848\) 9.61896 0.330316
\(849\) 0 0
\(850\) 1.87512 0.0643160
\(851\) 25.6424 0.879012
\(852\) 0 0
\(853\) 34.8883 1.19455 0.597277 0.802035i \(-0.296249\pi\)
0.597277 + 0.802035i \(0.296249\pi\)
\(854\) −0.0931734 −0.00318833
\(855\) 0 0
\(856\) −10.3144 −0.352540
\(857\) 5.99133 0.204660 0.102330 0.994751i \(-0.467370\pi\)
0.102330 + 0.994751i \(0.467370\pi\)
\(858\) 0 0
\(859\) 36.8422 1.25704 0.628519 0.777794i \(-0.283661\pi\)
0.628519 + 0.777794i \(0.283661\pi\)
\(860\) −0.775750 −0.0264529
\(861\) 0 0
\(862\) −4.20212 −0.143125
\(863\) −51.2726 −1.74534 −0.872669 0.488312i \(-0.837613\pi\)
−0.872669 + 0.488312i \(0.837613\pi\)
\(864\) 0 0
\(865\) 23.0808 0.784770
\(866\) 15.5560 0.528614
\(867\) 0 0
\(868\) 0.520358 0.0176621
\(869\) −29.4877 −1.00030
\(870\) 0 0
\(871\) 13.0402 0.441850
\(872\) −13.7411 −0.465333
\(873\) 0 0
\(874\) −19.4969 −0.659491
\(875\) −0.0498231 −0.00168433
\(876\) 0 0
\(877\) −5.47506 −0.184880 −0.0924399 0.995718i \(-0.529467\pi\)
−0.0924399 + 0.995718i \(0.529467\pi\)
\(878\) 20.1147 0.678840
\(879\) 0 0
\(880\) 3.99797 0.134772
\(881\) 16.6965 0.562521 0.281260 0.959632i \(-0.409248\pi\)
0.281260 + 0.959632i \(0.409248\pi\)
\(882\) 0 0
\(883\) −20.7620 −0.698696 −0.349348 0.936993i \(-0.613597\pi\)
−0.349348 + 0.936993i \(0.613597\pi\)
\(884\) 8.61980 0.289915
\(885\) 0 0
\(886\) −15.4514 −0.519100
\(887\) 30.8479 1.03577 0.517886 0.855450i \(-0.326719\pi\)
0.517886 + 0.855450i \(0.326719\pi\)
\(888\) 0 0
\(889\) 0.264682 0.00887715
\(890\) 0.755898 0.0253378
\(891\) 0 0
\(892\) −39.7669 −1.33149
\(893\) −49.7923 −1.66624
\(894\) 0 0
\(895\) −9.63601 −0.322096
\(896\) −0.484922 −0.0162001
\(897\) 0 0
\(898\) −4.24347 −0.141606
\(899\) 18.9457 0.631875
\(900\) 0 0
\(901\) 26.5661 0.885044
\(902\) 30.6542 1.02067
\(903\) 0 0
\(904\) −26.9245 −0.895496
\(905\) 7.10071 0.236036
\(906\) 0 0
\(907\) 30.8544 1.02450 0.512251 0.858836i \(-0.328812\pi\)
0.512251 + 0.858836i \(0.328812\pi\)
\(908\) 15.3077 0.508003
\(909\) 0 0
\(910\) 0.0916030 0.00303661
\(911\) −34.6247 −1.14717 −0.573584 0.819147i \(-0.694447\pi\)
−0.573584 + 0.819147i \(0.694447\pi\)
\(912\) 0 0
\(913\) −53.4410 −1.76864
\(914\) −5.39354 −0.178403
\(915\) 0 0
\(916\) 28.6919 0.948008
\(917\) 1.10741 0.0365699
\(918\) 0 0
\(919\) −3.03636 −0.100160 −0.0500802 0.998745i \(-0.515948\pi\)
−0.0500802 + 0.998745i \(0.515948\pi\)
\(920\) 12.9338 0.426415
\(921\) 0 0
\(922\) 18.2053 0.599560
\(923\) 23.3049 0.767090
\(924\) 0 0
\(925\) 5.13825 0.168945
\(926\) −16.8037 −0.552204
\(927\) 0 0
\(928\) −15.1923 −0.498712
\(929\) 17.2687 0.566568 0.283284 0.959036i \(-0.408576\pi\)
0.283284 + 0.959036i \(0.408576\pi\)
\(930\) 0 0
\(931\) 36.1661 1.18530
\(932\) 2.34748 0.0768943
\(933\) 0 0
\(934\) −7.87464 −0.257666
\(935\) 11.0418 0.361105
\(936\) 0 0
\(937\) 41.8282 1.36647 0.683234 0.730199i \(-0.260573\pi\)
0.683234 + 0.730199i \(0.260573\pi\)
\(938\) −0.201912 −0.00659266
\(939\) 0 0
\(940\) 13.7632 0.448907
\(941\) −55.0411 −1.79429 −0.897145 0.441736i \(-0.854363\pi\)
−0.897145 + 0.441736i \(0.854363\pi\)
\(942\) 0 0
\(943\) −45.4671 −1.48061
\(944\) 11.3474 0.369326
\(945\) 0 0
\(946\) 1.82702 0.0594014
\(947\) −5.20266 −0.169064 −0.0845318 0.996421i \(-0.526939\pi\)
−0.0845318 + 0.996421i \(0.526939\pi\)
\(948\) 0 0
\(949\) −6.96532 −0.226104
\(950\) −3.90680 −0.126753
\(951\) 0 0
\(952\) −0.320316 −0.0103815
\(953\) −23.9232 −0.774950 −0.387475 0.921880i \(-0.626653\pi\)
−0.387475 + 0.921880i \(0.626653\pi\)
\(954\) 0 0
\(955\) 9.47202 0.306507
\(956\) −17.5887 −0.568860
\(957\) 0 0
\(958\) 15.5992 0.503987
\(959\) −0.402609 −0.0130009
\(960\) 0 0
\(961\) 22.4455 0.724049
\(962\) −9.44701 −0.304584
\(963\) 0 0
\(964\) 15.4172 0.496553
\(965\) −11.7250 −0.377441
\(966\) 0 0
\(967\) 47.9852 1.54310 0.771550 0.636168i \(-0.219482\pi\)
0.771550 + 0.636168i \(0.219482\pi\)
\(968\) −22.8402 −0.734111
\(969\) 0 0
\(970\) −3.64382 −0.116996
\(971\) −4.06546 −0.130467 −0.0652334 0.997870i \(-0.520779\pi\)
−0.0652334 + 0.997870i \(0.520779\pi\)
\(972\) 0 0
\(973\) 0.0380351 0.00121935
\(974\) −22.3365 −0.715707
\(975\) 0 0
\(976\) −2.22210 −0.0711278
\(977\) −4.40304 −0.140866 −0.0704328 0.997517i \(-0.522438\pi\)
−0.0704328 + 0.997517i \(0.522438\pi\)
\(978\) 0 0
\(979\) 4.45116 0.142260
\(980\) −9.99678 −0.319335
\(981\) 0 0
\(982\) −28.5811 −0.912060
\(983\) 8.02508 0.255960 0.127980 0.991777i \(-0.459151\pi\)
0.127980 + 0.991777i \(0.459151\pi\)
\(984\) 0 0
\(985\) −18.2857 −0.582631
\(986\) −4.85941 −0.154755
\(987\) 0 0
\(988\) −17.9593 −0.571362
\(989\) −2.70988 −0.0861691
\(990\) 0 0
\(991\) 28.4040 0.902282 0.451141 0.892453i \(-0.351017\pi\)
0.451141 + 0.892453i \(0.351017\pi\)
\(992\) −42.8572 −1.36072
\(993\) 0 0
\(994\) −0.360849 −0.0114454
\(995\) −25.0469 −0.794039
\(996\) 0 0
\(997\) −6.03403 −0.191100 −0.0955499 0.995425i \(-0.530461\pi\)
−0.0955499 + 0.995425i \(0.530461\pi\)
\(998\) 5.36503 0.169827
\(999\) 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 4005.2.a.o.1.4 7
3.2 odd 2 445.2.a.f.1.4 7
12.11 even 2 7120.2.a.bj.1.6 7
15.14 odd 2 2225.2.a.k.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.4 7 3.2 odd 2
2225.2.a.k.1.4 7 15.14 odd 2
4005.2.a.o.1.4 7 1.1 even 1 trivial
7120.2.a.bj.1.6 7 12.11 even 2