Properties

Label 729.2.a.e.1.5
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7459857.1
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.77773\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.12503 q^{2} +2.51575 q^{4} +2.07094 q^{5} +4.84867 q^{7} +1.09598 q^{8} +O(q^{10})\) \(q+2.12503 q^{2} +2.51575 q^{4} +2.07094 q^{5} +4.84867 q^{7} +1.09598 q^{8} +4.40081 q^{10} -4.14949 q^{11} -1.21602 q^{13} +10.3036 q^{14} -2.70251 q^{16} -2.36364 q^{17} -1.83801 q^{19} +5.20996 q^{20} -8.81778 q^{22} +4.30357 q^{23} -0.711206 q^{25} -2.58407 q^{26} +12.1980 q^{28} -2.98182 q^{29} +1.47359 q^{31} -7.93487 q^{32} -5.02280 q^{34} +10.0413 q^{35} +8.97108 q^{37} -3.90582 q^{38} +2.26970 q^{40} -2.26052 q^{41} -5.48486 q^{43} -10.4391 q^{44} +9.14521 q^{46} -7.18313 q^{47} +16.5096 q^{49} -1.51133 q^{50} -3.05919 q^{52} +6.32803 q^{53} -8.59334 q^{55} +5.31404 q^{56} -6.33645 q^{58} -0.262257 q^{59} -4.45561 q^{61} +3.13141 q^{62} -11.4568 q^{64} -2.51830 q^{65} +4.13110 q^{67} -5.94632 q^{68} +21.3381 q^{70} -3.08551 q^{71} +12.7601 q^{73} +19.0638 q^{74} -4.62396 q^{76} -20.1195 q^{77} +4.55241 q^{79} -5.59674 q^{80} -4.80368 q^{82} +8.45306 q^{83} -4.89495 q^{85} -11.6555 q^{86} -4.54775 q^{88} -16.9632 q^{89} -5.89606 q^{91} +10.8267 q^{92} -15.2644 q^{94} -3.80640 q^{95} -5.10977 q^{97} +35.0834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28} + 21 q^{29} + 15 q^{31} + 30 q^{35} + 3 q^{37} + 15 q^{38} + 3 q^{40} - 12 q^{41} + 6 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 12 q^{49} - 24 q^{50} + 3 q^{52} - 9 q^{53} + 15 q^{55} + 12 q^{56} - 15 q^{58} + 6 q^{59} + 24 q^{61} - 30 q^{62} + 6 q^{64} - 15 q^{65} + 15 q^{67} + 36 q^{68} - 15 q^{70} + 12 q^{73} + 24 q^{74} + 9 q^{76} + 15 q^{77} + 24 q^{79} - 21 q^{80} - 21 q^{82} - 6 q^{83} - 18 q^{85} - 30 q^{86} - 21 q^{88} - 9 q^{89} + 18 q^{91} + 6 q^{92} - 6 q^{94} - 33 q^{95} - 21 q^{97} + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12503 1.50262 0.751311 0.659948i \(-0.229422\pi\)
0.751311 + 0.659948i \(0.229422\pi\)
\(3\) 0 0
\(4\) 2.51575 1.25787
\(5\) 2.07094 0.926153 0.463076 0.886318i \(-0.346746\pi\)
0.463076 + 0.886318i \(0.346746\pi\)
\(6\) 0 0
\(7\) 4.84867 1.83263 0.916313 0.400463i \(-0.131151\pi\)
0.916313 + 0.400463i \(0.131151\pi\)
\(8\) 1.09598 0.387487
\(9\) 0 0
\(10\) 4.40081 1.39166
\(11\) −4.14949 −1.25112 −0.625559 0.780177i \(-0.715129\pi\)
−0.625559 + 0.780177i \(0.715129\pi\)
\(12\) 0 0
\(13\) −1.21602 −0.337262 −0.168631 0.985679i \(-0.553935\pi\)
−0.168631 + 0.985679i \(0.553935\pi\)
\(14\) 10.3036 2.75374
\(15\) 0 0
\(16\) −2.70251 −0.675628
\(17\) −2.36364 −0.573266 −0.286633 0.958040i \(-0.592536\pi\)
−0.286633 + 0.958040i \(0.592536\pi\)
\(18\) 0 0
\(19\) −1.83801 −0.421668 −0.210834 0.977522i \(-0.567618\pi\)
−0.210834 + 0.977522i \(0.567618\pi\)
\(20\) 5.20996 1.16498
\(21\) 0 0
\(22\) −8.81778 −1.87996
\(23\) 4.30357 0.897356 0.448678 0.893693i \(-0.351895\pi\)
0.448678 + 0.893693i \(0.351895\pi\)
\(24\) 0 0
\(25\) −0.711206 −0.142241
\(26\) −2.58407 −0.506777
\(27\) 0 0
\(28\) 12.1980 2.30521
\(29\) −2.98182 −0.553710 −0.276855 0.960912i \(-0.589292\pi\)
−0.276855 + 0.960912i \(0.589292\pi\)
\(30\) 0 0
\(31\) 1.47359 0.264664 0.132332 0.991205i \(-0.457754\pi\)
0.132332 + 0.991205i \(0.457754\pi\)
\(32\) −7.93487 −1.40270
\(33\) 0 0
\(34\) −5.02280 −0.861403
\(35\) 10.0413 1.69729
\(36\) 0 0
\(37\) 8.97108 1.47484 0.737418 0.675436i \(-0.236045\pi\)
0.737418 + 0.675436i \(0.236045\pi\)
\(38\) −3.90582 −0.633607
\(39\) 0 0
\(40\) 2.26970 0.358872
\(41\) −2.26052 −0.353035 −0.176517 0.984298i \(-0.556483\pi\)
−0.176517 + 0.984298i \(0.556483\pi\)
\(42\) 0 0
\(43\) −5.48486 −0.836433 −0.418216 0.908347i \(-0.637345\pi\)
−0.418216 + 0.908347i \(0.637345\pi\)
\(44\) −10.4391 −1.57375
\(45\) 0 0
\(46\) 9.14521 1.34839
\(47\) −7.18313 −1.04777 −0.523884 0.851790i \(-0.675517\pi\)
−0.523884 + 0.851790i \(0.675517\pi\)
\(48\) 0 0
\(49\) 16.5096 2.35852
\(50\) −1.51133 −0.213735
\(51\) 0 0
\(52\) −3.05919 −0.424233
\(53\) 6.32803 0.869222 0.434611 0.900618i \(-0.356886\pi\)
0.434611 + 0.900618i \(0.356886\pi\)
\(54\) 0 0
\(55\) −8.59334 −1.15873
\(56\) 5.31404 0.710118
\(57\) 0 0
\(58\) −6.33645 −0.832017
\(59\) −0.262257 −0.0341429 −0.0170715 0.999854i \(-0.505434\pi\)
−0.0170715 + 0.999854i \(0.505434\pi\)
\(60\) 0 0
\(61\) −4.45561 −0.570482 −0.285241 0.958456i \(-0.592074\pi\)
−0.285241 + 0.958456i \(0.592074\pi\)
\(62\) 3.13141 0.397690
\(63\) 0 0
\(64\) −11.4568 −1.43210
\(65\) −2.51830 −0.312356
\(66\) 0 0
\(67\) 4.13110 0.504695 0.252347 0.967637i \(-0.418797\pi\)
0.252347 + 0.967637i \(0.418797\pi\)
\(68\) −5.94632 −0.721097
\(69\) 0 0
\(70\) 21.3381 2.55039
\(71\) −3.08551 −0.366183 −0.183091 0.983096i \(-0.558610\pi\)
−0.183091 + 0.983096i \(0.558610\pi\)
\(72\) 0 0
\(73\) 12.7601 1.49345 0.746726 0.665132i \(-0.231625\pi\)
0.746726 + 0.665132i \(0.231625\pi\)
\(74\) 19.0638 2.21612
\(75\) 0 0
\(76\) −4.62396 −0.530405
\(77\) −20.1195 −2.29283
\(78\) 0 0
\(79\) 4.55241 0.512186 0.256093 0.966652i \(-0.417565\pi\)
0.256093 + 0.966652i \(0.417565\pi\)
\(80\) −5.59674 −0.625734
\(81\) 0 0
\(82\) −4.80368 −0.530478
\(83\) 8.45306 0.927844 0.463922 0.885876i \(-0.346442\pi\)
0.463922 + 0.885876i \(0.346442\pi\)
\(84\) 0 0
\(85\) −4.89495 −0.530932
\(86\) −11.6555 −1.25684
\(87\) 0 0
\(88\) −4.54775 −0.484791
\(89\) −16.9632 −1.79809 −0.899046 0.437854i \(-0.855739\pi\)
−0.899046 + 0.437854i \(0.855739\pi\)
\(90\) 0 0
\(91\) −5.89606 −0.618075
\(92\) 10.8267 1.12876
\(93\) 0 0
\(94\) −15.2644 −1.57440
\(95\) −3.80640 −0.390529
\(96\) 0 0
\(97\) −5.10977 −0.518819 −0.259409 0.965767i \(-0.583528\pi\)
−0.259409 + 0.965767i \(0.583528\pi\)
\(98\) 35.0834 3.54396
\(99\) 0 0
\(100\) −1.78921 −0.178921
\(101\) −18.5799 −1.84877 −0.924384 0.381464i \(-0.875420\pi\)
−0.924384 + 0.381464i \(0.875420\pi\)
\(102\) 0 0
\(103\) 9.00577 0.887365 0.443682 0.896184i \(-0.353672\pi\)
0.443682 + 0.896184i \(0.353672\pi\)
\(104\) −1.33273 −0.130684
\(105\) 0 0
\(106\) 13.4472 1.30611
\(107\) −7.42680 −0.717976 −0.358988 0.933342i \(-0.616878\pi\)
−0.358988 + 0.933342i \(0.616878\pi\)
\(108\) 0 0
\(109\) −5.62396 −0.538678 −0.269339 0.963045i \(-0.586805\pi\)
−0.269339 + 0.963045i \(0.586805\pi\)
\(110\) −18.2611 −1.74113
\(111\) 0 0
\(112\) −13.1036 −1.23817
\(113\) 2.40267 0.226024 0.113012 0.993594i \(-0.463950\pi\)
0.113012 + 0.993594i \(0.463950\pi\)
\(114\) 0 0
\(115\) 8.91243 0.831089
\(116\) −7.50150 −0.696497
\(117\) 0 0
\(118\) −0.557303 −0.0513039
\(119\) −11.4605 −1.05058
\(120\) 0 0
\(121\) 6.21826 0.565296
\(122\) −9.46829 −0.857219
\(123\) 0 0
\(124\) 3.70717 0.332914
\(125\) −11.8276 −1.05789
\(126\) 0 0
\(127\) −9.23469 −0.819447 −0.409723 0.912210i \(-0.634375\pi\)
−0.409723 + 0.912210i \(0.634375\pi\)
\(128\) −8.47629 −0.749206
\(129\) 0 0
\(130\) −5.35145 −0.469353
\(131\) 15.3390 1.34018 0.670088 0.742282i \(-0.266257\pi\)
0.670088 + 0.742282i \(0.266257\pi\)
\(132\) 0 0
\(133\) −8.91189 −0.772759
\(134\) 8.77871 0.758365
\(135\) 0 0
\(136\) −2.59049 −0.222133
\(137\) 3.64397 0.311325 0.155663 0.987810i \(-0.450249\pi\)
0.155663 + 0.987810i \(0.450249\pi\)
\(138\) 0 0
\(139\) 13.2755 1.12602 0.563008 0.826451i \(-0.309644\pi\)
0.563008 + 0.826451i \(0.309644\pi\)
\(140\) 25.2614 2.13498
\(141\) 0 0
\(142\) −6.55680 −0.550234
\(143\) 5.04584 0.421954
\(144\) 0 0
\(145\) −6.17517 −0.512820
\(146\) 27.1155 2.24409
\(147\) 0 0
\(148\) 22.5690 1.85516
\(149\) −8.91098 −0.730016 −0.365008 0.931004i \(-0.618934\pi\)
−0.365008 + 0.931004i \(0.618934\pi\)
\(150\) 0 0
\(151\) −0.712404 −0.0579746 −0.0289873 0.999580i \(-0.509228\pi\)
−0.0289873 + 0.999580i \(0.509228\pi\)
\(152\) −2.01441 −0.163391
\(153\) 0 0
\(154\) −42.7545 −3.44526
\(155\) 3.05171 0.245119
\(156\) 0 0
\(157\) −13.8181 −1.10280 −0.551400 0.834241i \(-0.685906\pi\)
−0.551400 + 0.834241i \(0.685906\pi\)
\(158\) 9.67399 0.769622
\(159\) 0 0
\(160\) −16.4326 −1.29911
\(161\) 20.8666 1.64452
\(162\) 0 0
\(163\) −1.19321 −0.0934597 −0.0467298 0.998908i \(-0.514880\pi\)
−0.0467298 + 0.998908i \(0.514880\pi\)
\(164\) −5.68691 −0.444073
\(165\) 0 0
\(166\) 17.9630 1.39420
\(167\) 23.9363 1.85225 0.926124 0.377219i \(-0.123120\pi\)
0.926124 + 0.377219i \(0.123120\pi\)
\(168\) 0 0
\(169\) −11.5213 −0.886254
\(170\) −10.4019 −0.797791
\(171\) 0 0
\(172\) −13.7985 −1.05213
\(173\) 9.14000 0.694902 0.347451 0.937698i \(-0.387047\pi\)
0.347451 + 0.937698i \(0.387047\pi\)
\(174\) 0 0
\(175\) −3.44841 −0.260675
\(176\) 11.2140 0.845290
\(177\) 0 0
\(178\) −36.0472 −2.70185
\(179\) −10.6008 −0.792337 −0.396169 0.918178i \(-0.629660\pi\)
−0.396169 + 0.918178i \(0.629660\pi\)
\(180\) 0 0
\(181\) −1.46292 −0.108738 −0.0543690 0.998521i \(-0.517315\pi\)
−0.0543690 + 0.998521i \(0.517315\pi\)
\(182\) −12.5293 −0.928733
\(183\) 0 0
\(184\) 4.71661 0.347713
\(185\) 18.5786 1.36592
\(186\) 0 0
\(187\) 9.80789 0.717224
\(188\) −18.0709 −1.31796
\(189\) 0 0
\(190\) −8.08871 −0.586817
\(191\) −12.4223 −0.898844 −0.449422 0.893320i \(-0.648370\pi\)
−0.449422 + 0.893320i \(0.648370\pi\)
\(192\) 0 0
\(193\) 20.7733 1.49529 0.747647 0.664096i \(-0.231183\pi\)
0.747647 + 0.664096i \(0.231183\pi\)
\(194\) −10.8584 −0.779588
\(195\) 0 0
\(196\) 41.5340 2.96672
\(197\) 14.1887 1.01090 0.505450 0.862856i \(-0.331326\pi\)
0.505450 + 0.862856i \(0.331326\pi\)
\(198\) 0 0
\(199\) 20.3286 1.44106 0.720529 0.693424i \(-0.243899\pi\)
0.720529 + 0.693424i \(0.243899\pi\)
\(200\) −0.779466 −0.0551165
\(201\) 0 0
\(202\) −39.4828 −2.77800
\(203\) −14.4579 −1.01474
\(204\) 0 0
\(205\) −4.68141 −0.326964
\(206\) 19.1375 1.33337
\(207\) 0 0
\(208\) 3.28629 0.227864
\(209\) 7.62679 0.527556
\(210\) 0 0
\(211\) −9.86332 −0.679019 −0.339509 0.940603i \(-0.610261\pi\)
−0.339509 + 0.940603i \(0.610261\pi\)
\(212\) 15.9197 1.09337
\(213\) 0 0
\(214\) −15.7822 −1.07885
\(215\) −11.3588 −0.774665
\(216\) 0 0
\(217\) 7.14494 0.485030
\(218\) −11.9511 −0.809429
\(219\) 0 0
\(220\) −21.6187 −1.45753
\(221\) 2.87422 0.193341
\(222\) 0 0
\(223\) 15.0714 1.00925 0.504627 0.863337i \(-0.331630\pi\)
0.504627 + 0.863337i \(0.331630\pi\)
\(224\) −38.4736 −2.57062
\(225\) 0 0
\(226\) 5.10574 0.339629
\(227\) 22.6926 1.50616 0.753079 0.657930i \(-0.228568\pi\)
0.753079 + 0.657930i \(0.228568\pi\)
\(228\) 0 0
\(229\) 8.71670 0.576016 0.288008 0.957628i \(-0.407007\pi\)
0.288008 + 0.957628i \(0.407007\pi\)
\(230\) 18.9392 1.24881
\(231\) 0 0
\(232\) −3.26801 −0.214555
\(233\) 23.5890 1.54536 0.772682 0.634793i \(-0.218915\pi\)
0.772682 + 0.634793i \(0.218915\pi\)
\(234\) 0 0
\(235\) −14.8758 −0.970393
\(236\) −0.659772 −0.0429475
\(237\) 0 0
\(238\) −24.3539 −1.57863
\(239\) −9.95452 −0.643904 −0.321952 0.946756i \(-0.604339\pi\)
−0.321952 + 0.946756i \(0.604339\pi\)
\(240\) 0 0
\(241\) 5.71345 0.368036 0.184018 0.982923i \(-0.441090\pi\)
0.184018 + 0.982923i \(0.441090\pi\)
\(242\) 13.2140 0.849427
\(243\) 0 0
\(244\) −11.2092 −0.717594
\(245\) 34.1905 2.18435
\(246\) 0 0
\(247\) 2.23504 0.142212
\(248\) 1.61502 0.102554
\(249\) 0 0
\(250\) −25.1339 −1.58961
\(251\) 7.28966 0.460120 0.230060 0.973176i \(-0.426108\pi\)
0.230060 + 0.973176i \(0.426108\pi\)
\(252\) 0 0
\(253\) −17.8576 −1.12270
\(254\) −19.6240 −1.23132
\(255\) 0 0
\(256\) 4.90123 0.306327
\(257\) 23.2431 1.44986 0.724932 0.688821i \(-0.241871\pi\)
0.724932 + 0.688821i \(0.241871\pi\)
\(258\) 0 0
\(259\) 43.4978 2.70282
\(260\) −6.33539 −0.392904
\(261\) 0 0
\(262\) 32.5958 2.01378
\(263\) 27.3996 1.68953 0.844766 0.535136i \(-0.179740\pi\)
0.844766 + 0.535136i \(0.179740\pi\)
\(264\) 0 0
\(265\) 13.1050 0.805032
\(266\) −18.9380 −1.16116
\(267\) 0 0
\(268\) 10.3928 0.634842
\(269\) −9.41973 −0.574331 −0.287166 0.957881i \(-0.592713\pi\)
−0.287166 + 0.957881i \(0.592713\pi\)
\(270\) 0 0
\(271\) 26.2797 1.59638 0.798189 0.602408i \(-0.205792\pi\)
0.798189 + 0.602408i \(0.205792\pi\)
\(272\) 6.38776 0.387315
\(273\) 0 0
\(274\) 7.74354 0.467804
\(275\) 2.95114 0.177961
\(276\) 0 0
\(277\) 0.374812 0.0225203 0.0112601 0.999937i \(-0.496416\pi\)
0.0112601 + 0.999937i \(0.496416\pi\)
\(278\) 28.2109 1.69198
\(279\) 0 0
\(280\) 11.0051 0.657678
\(281\) −13.9816 −0.834071 −0.417036 0.908890i \(-0.636931\pi\)
−0.417036 + 0.908890i \(0.636931\pi\)
\(282\) 0 0
\(283\) 15.5619 0.925058 0.462529 0.886604i \(-0.346942\pi\)
0.462529 + 0.886604i \(0.346942\pi\)
\(284\) −7.76236 −0.460612
\(285\) 0 0
\(286\) 10.7226 0.634038
\(287\) −10.9605 −0.646980
\(288\) 0 0
\(289\) −11.4132 −0.671366
\(290\) −13.1224 −0.770575
\(291\) 0 0
\(292\) 32.1011 1.87857
\(293\) −24.6242 −1.43856 −0.719281 0.694719i \(-0.755529\pi\)
−0.719281 + 0.694719i \(0.755529\pi\)
\(294\) 0 0
\(295\) −0.543118 −0.0316216
\(296\) 9.83210 0.571479
\(297\) 0 0
\(298\) −18.9361 −1.09694
\(299\) −5.23321 −0.302644
\(300\) 0 0
\(301\) −26.5943 −1.53287
\(302\) −1.51388 −0.0871139
\(303\) 0 0
\(304\) 4.96723 0.284890
\(305\) −9.22729 −0.528353
\(306\) 0 0
\(307\) −20.3912 −1.16379 −0.581893 0.813265i \(-0.697688\pi\)
−0.581893 + 0.813265i \(0.697688\pi\)
\(308\) −50.6156 −2.88409
\(309\) 0 0
\(310\) 6.48497 0.368322
\(311\) −22.1790 −1.25766 −0.628828 0.777544i \(-0.716465\pi\)
−0.628828 + 0.777544i \(0.716465\pi\)
\(312\) 0 0
\(313\) −11.1720 −0.631481 −0.315740 0.948846i \(-0.602253\pi\)
−0.315740 + 0.948846i \(0.602253\pi\)
\(314\) −29.3638 −1.65709
\(315\) 0 0
\(316\) 11.4527 0.644265
\(317\) −25.2167 −1.41631 −0.708154 0.706058i \(-0.750472\pi\)
−0.708154 + 0.706058i \(0.750472\pi\)
\(318\) 0 0
\(319\) 12.3730 0.692757
\(320\) −23.7264 −1.32634
\(321\) 0 0
\(322\) 44.3421 2.47109
\(323\) 4.34438 0.241728
\(324\) 0 0
\(325\) 0.864837 0.0479725
\(326\) −2.53561 −0.140435
\(327\) 0 0
\(328\) −2.47748 −0.136796
\(329\) −34.8287 −1.92017
\(330\) 0 0
\(331\) −26.1657 −1.43820 −0.719098 0.694908i \(-0.755445\pi\)
−0.719098 + 0.694908i \(0.755445\pi\)
\(332\) 21.2658 1.16711
\(333\) 0 0
\(334\) 50.8654 2.78323
\(335\) 8.55527 0.467424
\(336\) 0 0
\(337\) 10.9460 0.596269 0.298135 0.954524i \(-0.403636\pi\)
0.298135 + 0.954524i \(0.403636\pi\)
\(338\) −24.4831 −1.33171
\(339\) 0 0
\(340\) −12.3145 −0.667846
\(341\) −6.11463 −0.331126
\(342\) 0 0
\(343\) 46.1091 2.48966
\(344\) −6.01128 −0.324106
\(345\) 0 0
\(346\) 19.4228 1.04417
\(347\) −3.51338 −0.188608 −0.0943040 0.995543i \(-0.530063\pi\)
−0.0943040 + 0.995543i \(0.530063\pi\)
\(348\) 0 0
\(349\) −29.1968 −1.56287 −0.781436 0.623986i \(-0.785512\pi\)
−0.781436 + 0.623986i \(0.785512\pi\)
\(350\) −7.32796 −0.391696
\(351\) 0 0
\(352\) 32.9256 1.75494
\(353\) 25.0768 1.33470 0.667352 0.744742i \(-0.267428\pi\)
0.667352 + 0.744742i \(0.267428\pi\)
\(354\) 0 0
\(355\) −6.38991 −0.339141
\(356\) −42.6750 −2.26177
\(357\) 0 0
\(358\) −22.5269 −1.19058
\(359\) −4.20724 −0.222050 −0.111025 0.993818i \(-0.535413\pi\)
−0.111025 + 0.993818i \(0.535413\pi\)
\(360\) 0 0
\(361\) −15.6217 −0.822196
\(362\) −3.10875 −0.163392
\(363\) 0 0
\(364\) −14.8330 −0.777460
\(365\) 26.4253 1.38316
\(366\) 0 0
\(367\) 17.4966 0.913317 0.456658 0.889642i \(-0.349046\pi\)
0.456658 + 0.889642i \(0.349046\pi\)
\(368\) −11.6304 −0.606279
\(369\) 0 0
\(370\) 39.4800 2.05247
\(371\) 30.6825 1.59296
\(372\) 0 0
\(373\) −29.6546 −1.53546 −0.767728 0.640776i \(-0.778613\pi\)
−0.767728 + 0.640776i \(0.778613\pi\)
\(374\) 20.8420 1.07772
\(375\) 0 0
\(376\) −7.87255 −0.405996
\(377\) 3.62594 0.186745
\(378\) 0 0
\(379\) 20.9523 1.07625 0.538124 0.842865i \(-0.319133\pi\)
0.538124 + 0.842865i \(0.319133\pi\)
\(380\) −9.57594 −0.491236
\(381\) 0 0
\(382\) −26.3977 −1.35062
\(383\) −9.89678 −0.505702 −0.252851 0.967505i \(-0.581368\pi\)
−0.252851 + 0.967505i \(0.581368\pi\)
\(384\) 0 0
\(385\) −41.6663 −2.12351
\(386\) 44.1439 2.24686
\(387\) 0 0
\(388\) −12.8549 −0.652608
\(389\) 21.1148 1.07056 0.535281 0.844674i \(-0.320206\pi\)
0.535281 + 0.844674i \(0.320206\pi\)
\(390\) 0 0
\(391\) −10.1721 −0.514424
\(392\) 18.0942 0.913894
\(393\) 0 0
\(394\) 30.1513 1.51900
\(395\) 9.42776 0.474362
\(396\) 0 0
\(397\) −9.77909 −0.490799 −0.245399 0.969422i \(-0.578919\pi\)
−0.245399 + 0.969422i \(0.578919\pi\)
\(398\) 43.1989 2.16537
\(399\) 0 0
\(400\) 1.92204 0.0961021
\(401\) 19.1393 0.955773 0.477886 0.878422i \(-0.341403\pi\)
0.477886 + 0.878422i \(0.341403\pi\)
\(402\) 0 0
\(403\) −1.79190 −0.0892611
\(404\) −46.7423 −2.32552
\(405\) 0 0
\(406\) −30.7234 −1.52478
\(407\) −37.2254 −1.84519
\(408\) 0 0
\(409\) 14.9956 0.741483 0.370741 0.928736i \(-0.379104\pi\)
0.370741 + 0.928736i \(0.379104\pi\)
\(410\) −9.94813 −0.491303
\(411\) 0 0
\(412\) 22.6562 1.11619
\(413\) −1.27160 −0.0625712
\(414\) 0 0
\(415\) 17.5058 0.859325
\(416\) 9.64892 0.473077
\(417\) 0 0
\(418\) 16.2071 0.792717
\(419\) −5.81180 −0.283925 −0.141963 0.989872i \(-0.545341\pi\)
−0.141963 + 0.989872i \(0.545341\pi\)
\(420\) 0 0
\(421\) −13.3135 −0.648861 −0.324431 0.945910i \(-0.605173\pi\)
−0.324431 + 0.945910i \(0.605173\pi\)
\(422\) −20.9598 −1.02031
\(423\) 0 0
\(424\) 6.93538 0.336812
\(425\) 1.68103 0.0815421
\(426\) 0 0
\(427\) −21.6038 −1.04548
\(428\) −18.6840 −0.903123
\(429\) 0 0
\(430\) −24.1378 −1.16403
\(431\) 36.4166 1.75413 0.877064 0.480374i \(-0.159499\pi\)
0.877064 + 0.480374i \(0.159499\pi\)
\(432\) 0 0
\(433\) −10.8761 −0.522674 −0.261337 0.965248i \(-0.584163\pi\)
−0.261337 + 0.965248i \(0.584163\pi\)
\(434\) 15.1832 0.728817
\(435\) 0 0
\(436\) −14.1485 −0.677589
\(437\) −7.90999 −0.378386
\(438\) 0 0
\(439\) 9.45326 0.451180 0.225590 0.974222i \(-0.427569\pi\)
0.225590 + 0.974222i \(0.427569\pi\)
\(440\) −9.41811 −0.448991
\(441\) 0 0
\(442\) 6.10780 0.290518
\(443\) −16.5239 −0.785075 −0.392537 0.919736i \(-0.628403\pi\)
−0.392537 + 0.919736i \(0.628403\pi\)
\(444\) 0 0
\(445\) −35.1297 −1.66531
\(446\) 32.0271 1.51653
\(447\) 0 0
\(448\) −55.5503 −2.62450
\(449\) 4.74362 0.223865 0.111933 0.993716i \(-0.464296\pi\)
0.111933 + 0.993716i \(0.464296\pi\)
\(450\) 0 0
\(451\) 9.38002 0.441688
\(452\) 6.04451 0.284310
\(453\) 0 0
\(454\) 48.2223 2.26319
\(455\) −12.2104 −0.572432
\(456\) 0 0
\(457\) −22.4085 −1.04823 −0.524114 0.851648i \(-0.675603\pi\)
−0.524114 + 0.851648i \(0.675603\pi\)
\(458\) 18.5232 0.865534
\(459\) 0 0
\(460\) 22.4214 1.04540
\(461\) 14.9223 0.694999 0.347499 0.937680i \(-0.387031\pi\)
0.347499 + 0.937680i \(0.387031\pi\)
\(462\) 0 0
\(463\) −14.8472 −0.690007 −0.345003 0.938601i \(-0.612122\pi\)
−0.345003 + 0.938601i \(0.612122\pi\)
\(464\) 8.05840 0.374102
\(465\) 0 0
\(466\) 50.1272 2.32210
\(467\) 15.3514 0.710379 0.355190 0.934794i \(-0.384416\pi\)
0.355190 + 0.934794i \(0.384416\pi\)
\(468\) 0 0
\(469\) 20.0304 0.924917
\(470\) −31.6116 −1.45813
\(471\) 0 0
\(472\) −0.287427 −0.0132299
\(473\) 22.7594 1.04648
\(474\) 0 0
\(475\) 1.30720 0.0599785
\(476\) −28.8317 −1.32150
\(477\) 0 0
\(478\) −21.1536 −0.967545
\(479\) −10.4486 −0.477409 −0.238705 0.971092i \(-0.576723\pi\)
−0.238705 + 0.971092i \(0.576723\pi\)
\(480\) 0 0
\(481\) −10.9090 −0.497406
\(482\) 12.1412 0.553018
\(483\) 0 0
\(484\) 15.6436 0.711071
\(485\) −10.5820 −0.480505
\(486\) 0 0
\(487\) −16.1649 −0.732500 −0.366250 0.930516i \(-0.619359\pi\)
−0.366250 + 0.930516i \(0.619359\pi\)
\(488\) −4.88324 −0.221054
\(489\) 0 0
\(490\) 72.6557 3.28225
\(491\) 10.7773 0.486375 0.243187 0.969979i \(-0.421807\pi\)
0.243187 + 0.969979i \(0.421807\pi\)
\(492\) 0 0
\(493\) 7.04794 0.317423
\(494\) 4.74953 0.213692
\(495\) 0 0
\(496\) −3.98238 −0.178814
\(497\) −14.9606 −0.671076
\(498\) 0 0
\(499\) −19.1872 −0.858936 −0.429468 0.903082i \(-0.641299\pi\)
−0.429468 + 0.903082i \(0.641299\pi\)
\(500\) −29.7552 −1.33069
\(501\) 0 0
\(502\) 15.4907 0.691386
\(503\) 12.0251 0.536171 0.268086 0.963395i \(-0.413609\pi\)
0.268086 + 0.963395i \(0.413609\pi\)
\(504\) 0 0
\(505\) −38.4778 −1.71224
\(506\) −37.9479 −1.68699
\(507\) 0 0
\(508\) −23.2322 −1.03076
\(509\) 14.9530 0.662781 0.331391 0.943494i \(-0.392482\pi\)
0.331391 + 0.943494i \(0.392482\pi\)
\(510\) 0 0
\(511\) 61.8694 2.73694
\(512\) 27.3678 1.20950
\(513\) 0 0
\(514\) 49.3922 2.17860
\(515\) 18.6504 0.821835
\(516\) 0 0
\(517\) 29.8063 1.31088
\(518\) 92.4342 4.06132
\(519\) 0 0
\(520\) −2.75999 −0.121034
\(521\) −37.4188 −1.63935 −0.819673 0.572832i \(-0.805845\pi\)
−0.819673 + 0.572832i \(0.805845\pi\)
\(522\) 0 0
\(523\) −8.44979 −0.369483 −0.184742 0.982787i \(-0.559145\pi\)
−0.184742 + 0.982787i \(0.559145\pi\)
\(524\) 38.5891 1.68577
\(525\) 0 0
\(526\) 58.2250 2.53873
\(527\) −3.48303 −0.151723
\(528\) 0 0
\(529\) −4.47929 −0.194752
\(530\) 27.8484 1.20966
\(531\) 0 0
\(532\) −22.4201 −0.972033
\(533\) 2.74883 0.119065
\(534\) 0 0
\(535\) −15.3805 −0.664956
\(536\) 4.52760 0.195562
\(537\) 0 0
\(538\) −20.0172 −0.863003
\(539\) −68.5065 −2.95078
\(540\) 0 0
\(541\) 12.6259 0.542828 0.271414 0.962463i \(-0.412509\pi\)
0.271414 + 0.962463i \(0.412509\pi\)
\(542\) 55.8451 2.39875
\(543\) 0 0
\(544\) 18.7552 0.804121
\(545\) −11.6469 −0.498898
\(546\) 0 0
\(547\) 31.3951 1.34236 0.671178 0.741296i \(-0.265789\pi\)
0.671178 + 0.741296i \(0.265789\pi\)
\(548\) 9.16731 0.391608
\(549\) 0 0
\(550\) 6.27126 0.267407
\(551\) 5.48060 0.233482
\(552\) 0 0
\(553\) 22.0731 0.938645
\(554\) 0.796486 0.0338394
\(555\) 0 0
\(556\) 33.3979 1.41639
\(557\) 15.9303 0.674988 0.337494 0.941328i \(-0.390421\pi\)
0.337494 + 0.941328i \(0.390421\pi\)
\(558\) 0 0
\(559\) 6.66967 0.282097
\(560\) −27.1368 −1.14674
\(561\) 0 0
\(562\) −29.7113 −1.25329
\(563\) 24.8609 1.04776 0.523880 0.851792i \(-0.324484\pi\)
0.523880 + 0.851792i \(0.324484\pi\)
\(564\) 0 0
\(565\) 4.97578 0.209333
\(566\) 33.0695 1.39001
\(567\) 0 0
\(568\) −3.38165 −0.141891
\(569\) −19.1900 −0.804487 −0.402243 0.915533i \(-0.631769\pi\)
−0.402243 + 0.915533i \(0.631769\pi\)
\(570\) 0 0
\(571\) −20.3792 −0.852844 −0.426422 0.904524i \(-0.640226\pi\)
−0.426422 + 0.904524i \(0.640226\pi\)
\(572\) 12.6941 0.530765
\(573\) 0 0
\(574\) −23.2915 −0.972167
\(575\) −3.06072 −0.127641
\(576\) 0 0
\(577\) 23.2991 0.969953 0.484976 0.874527i \(-0.338828\pi\)
0.484976 + 0.874527i \(0.338828\pi\)
\(578\) −24.2534 −1.00881
\(579\) 0 0
\(580\) −15.5352 −0.645063
\(581\) 40.9861 1.70039
\(582\) 0 0
\(583\) −26.2581 −1.08750
\(584\) 13.9847 0.578693
\(585\) 0 0
\(586\) −52.3272 −2.16162
\(587\) 36.9093 1.52341 0.761704 0.647925i \(-0.224363\pi\)
0.761704 + 0.647925i \(0.224363\pi\)
\(588\) 0 0
\(589\) −2.70846 −0.111600
\(590\) −1.15414 −0.0475153
\(591\) 0 0
\(592\) −24.2444 −0.996440
\(593\) −4.36830 −0.179385 −0.0896923 0.995970i \(-0.528588\pi\)
−0.0896923 + 0.995970i \(0.528588\pi\)
\(594\) 0 0
\(595\) −23.7340 −0.973000
\(596\) −22.4178 −0.918268
\(597\) 0 0
\(598\) −11.1207 −0.454760
\(599\) 31.2056 1.27503 0.637513 0.770439i \(-0.279963\pi\)
0.637513 + 0.770439i \(0.279963\pi\)
\(600\) 0 0
\(601\) 43.9371 1.79223 0.896116 0.443820i \(-0.146377\pi\)
0.896116 + 0.443820i \(0.146377\pi\)
\(602\) −56.5136 −2.30332
\(603\) 0 0
\(604\) −1.79223 −0.0729247
\(605\) 12.8776 0.523551
\(606\) 0 0
\(607\) 17.3587 0.704566 0.352283 0.935894i \(-0.385405\pi\)
0.352283 + 0.935894i \(0.385405\pi\)
\(608\) 14.5843 0.591473
\(609\) 0 0
\(610\) −19.6083 −0.793915
\(611\) 8.73480 0.353372
\(612\) 0 0
\(613\) −1.19805 −0.0483887 −0.0241944 0.999707i \(-0.507702\pi\)
−0.0241944 + 0.999707i \(0.507702\pi\)
\(614\) −43.3318 −1.74873
\(615\) 0 0
\(616\) −22.0505 −0.888441
\(617\) −26.0137 −1.04727 −0.523636 0.851942i \(-0.675425\pi\)
−0.523636 + 0.851942i \(0.675425\pi\)
\(618\) 0 0
\(619\) 9.63135 0.387117 0.193558 0.981089i \(-0.437997\pi\)
0.193558 + 0.981089i \(0.437997\pi\)
\(620\) 7.67733 0.308329
\(621\) 0 0
\(622\) −47.1311 −1.88978
\(623\) −82.2488 −3.29523
\(624\) 0 0
\(625\) −20.9382 −0.837526
\(626\) −23.7409 −0.948877
\(627\) 0 0
\(628\) −34.7627 −1.38718
\(629\) −21.2044 −0.845474
\(630\) 0 0
\(631\) −14.1673 −0.563992 −0.281996 0.959416i \(-0.590997\pi\)
−0.281996 + 0.959416i \(0.590997\pi\)
\(632\) 4.98933 0.198465
\(633\) 0 0
\(634\) −53.5861 −2.12818
\(635\) −19.1245 −0.758933
\(636\) 0 0
\(637\) −20.0760 −0.795438
\(638\) 26.2930 1.04095
\(639\) 0 0
\(640\) −17.5539 −0.693879
\(641\) −22.2009 −0.876883 −0.438442 0.898760i \(-0.644469\pi\)
−0.438442 + 0.898760i \(0.644469\pi\)
\(642\) 0 0
\(643\) 21.5368 0.849328 0.424664 0.905351i \(-0.360392\pi\)
0.424664 + 0.905351i \(0.360392\pi\)
\(644\) 52.4951 2.06860
\(645\) 0 0
\(646\) 9.23194 0.363226
\(647\) −13.4037 −0.526952 −0.263476 0.964666i \(-0.584869\pi\)
−0.263476 + 0.964666i \(0.584869\pi\)
\(648\) 0 0
\(649\) 1.08823 0.0427168
\(650\) 1.83780 0.0720846
\(651\) 0 0
\(652\) −3.00182 −0.117560
\(653\) 18.9109 0.740039 0.370020 0.929024i \(-0.379351\pi\)
0.370020 + 0.929024i \(0.379351\pi\)
\(654\) 0 0
\(655\) 31.7662 1.24121
\(656\) 6.10909 0.238520
\(657\) 0 0
\(658\) −74.0119 −2.88528
\(659\) 0.0281814 0.00109779 0.000548897 1.00000i \(-0.499825\pi\)
0.000548897 1.00000i \(0.499825\pi\)
\(660\) 0 0
\(661\) −45.0417 −1.75192 −0.875960 0.482383i \(-0.839771\pi\)
−0.875960 + 0.482383i \(0.839771\pi\)
\(662\) −55.6028 −2.16107
\(663\) 0 0
\(664\) 9.26436 0.359527
\(665\) −18.4560 −0.715693
\(666\) 0 0
\(667\) −12.8325 −0.496875
\(668\) 60.2177 2.32989
\(669\) 0 0
\(670\) 18.1802 0.702362
\(671\) 18.4885 0.713740
\(672\) 0 0
\(673\) −9.75969 −0.376208 −0.188104 0.982149i \(-0.560234\pi\)
−0.188104 + 0.982149i \(0.560234\pi\)
\(674\) 23.2607 0.895968
\(675\) 0 0
\(676\) −28.9847 −1.11480
\(677\) −40.8808 −1.57118 −0.785588 0.618750i \(-0.787639\pi\)
−0.785588 + 0.618750i \(0.787639\pi\)
\(678\) 0 0
\(679\) −24.7756 −0.950801
\(680\) −5.36476 −0.205729
\(681\) 0 0
\(682\) −12.9938 −0.497557
\(683\) 44.0251 1.68457 0.842287 0.539029i \(-0.181209\pi\)
0.842287 + 0.539029i \(0.181209\pi\)
\(684\) 0 0
\(685\) 7.54644 0.288335
\(686\) 97.9831 3.74101
\(687\) 0 0
\(688\) 14.8229 0.565117
\(689\) −7.69498 −0.293155
\(690\) 0 0
\(691\) 21.5516 0.819862 0.409931 0.912117i \(-0.365553\pi\)
0.409931 + 0.912117i \(0.365553\pi\)
\(692\) 22.9939 0.874098
\(693\) 0 0
\(694\) −7.46603 −0.283407
\(695\) 27.4928 1.04286
\(696\) 0 0
\(697\) 5.34306 0.202383
\(698\) −62.0441 −2.34841
\(699\) 0 0
\(700\) −8.67532 −0.327896
\(701\) 12.8521 0.485419 0.242709 0.970099i \(-0.421964\pi\)
0.242709 + 0.970099i \(0.421964\pi\)
\(702\) 0 0
\(703\) −16.4889 −0.621891
\(704\) 47.5399 1.79173
\(705\) 0 0
\(706\) 53.2889 2.00556
\(707\) −90.0878 −3.38810
\(708\) 0 0
\(709\) −49.7117 −1.86696 −0.933481 0.358627i \(-0.883245\pi\)
−0.933481 + 0.358627i \(0.883245\pi\)
\(710\) −13.5787 −0.509601
\(711\) 0 0
\(712\) −18.5912 −0.696737
\(713\) 6.34168 0.237498
\(714\) 0 0
\(715\) 10.4496 0.390794
\(716\) −26.6688 −0.996660
\(717\) 0 0
\(718\) −8.94051 −0.333657
\(719\) 5.63745 0.210242 0.105121 0.994459i \(-0.466477\pi\)
0.105121 + 0.994459i \(0.466477\pi\)
\(720\) 0 0
\(721\) 43.6660 1.62621
\(722\) −33.1966 −1.23545
\(723\) 0 0
\(724\) −3.68034 −0.136779
\(725\) 2.12069 0.0787604
\(726\) 0 0
\(727\) −45.4143 −1.68432 −0.842161 0.539226i \(-0.818717\pi\)
−0.842161 + 0.539226i \(0.818717\pi\)
\(728\) −6.46195 −0.239496
\(729\) 0 0
\(730\) 56.1546 2.07837
\(731\) 12.9642 0.479499
\(732\) 0 0
\(733\) −25.6669 −0.948027 −0.474013 0.880518i \(-0.657195\pi\)
−0.474013 + 0.880518i \(0.657195\pi\)
\(734\) 37.1808 1.37237
\(735\) 0 0
\(736\) −34.1483 −1.25872
\(737\) −17.1420 −0.631433
\(738\) 0 0
\(739\) −14.4553 −0.531745 −0.265873 0.964008i \(-0.585660\pi\)
−0.265873 + 0.964008i \(0.585660\pi\)
\(740\) 46.7390 1.71816
\(741\) 0 0
\(742\) 65.2013 2.39361
\(743\) −34.7937 −1.27646 −0.638229 0.769846i \(-0.720333\pi\)
−0.638229 + 0.769846i \(0.720333\pi\)
\(744\) 0 0
\(745\) −18.4541 −0.676107
\(746\) −63.0168 −2.30721
\(747\) 0 0
\(748\) 24.6742 0.902177
\(749\) −36.0101 −1.31578
\(750\) 0 0
\(751\) 18.1184 0.661152 0.330576 0.943779i \(-0.392757\pi\)
0.330576 + 0.943779i \(0.392757\pi\)
\(752\) 19.4125 0.707901
\(753\) 0 0
\(754\) 7.70522 0.280608
\(755\) −1.47535 −0.0536933
\(756\) 0 0
\(757\) −37.0045 −1.34495 −0.672475 0.740120i \(-0.734769\pi\)
−0.672475 + 0.740120i \(0.734769\pi\)
\(758\) 44.5243 1.61720
\(759\) 0 0
\(760\) −4.17173 −0.151325
\(761\) 12.4444 0.451110 0.225555 0.974230i \(-0.427580\pi\)
0.225555 + 0.974230i \(0.427580\pi\)
\(762\) 0 0
\(763\) −27.2688 −0.987195
\(764\) −31.2513 −1.13063
\(765\) 0 0
\(766\) −21.0309 −0.759878
\(767\) 0.318908 0.0115151
\(768\) 0 0
\(769\) −40.8856 −1.47437 −0.737186 0.675690i \(-0.763846\pi\)
−0.737186 + 0.675690i \(0.763846\pi\)
\(770\) −88.5421 −3.19084
\(771\) 0 0
\(772\) 52.2604 1.88089
\(773\) 36.4162 1.30980 0.654900 0.755716i \(-0.272711\pi\)
0.654900 + 0.755716i \(0.272711\pi\)
\(774\) 0 0
\(775\) −1.04802 −0.0376461
\(776\) −5.60019 −0.201035
\(777\) 0 0
\(778\) 44.8695 1.60865
\(779\) 4.15486 0.148863
\(780\) 0 0
\(781\) 12.8033 0.458138
\(782\) −21.6160 −0.772985
\(783\) 0 0
\(784\) −44.6174 −1.59348
\(785\) −28.6164 −1.02136
\(786\) 0 0
\(787\) 18.3472 0.654005 0.327003 0.945023i \(-0.393961\pi\)
0.327003 + 0.945023i \(0.393961\pi\)
\(788\) 35.6951 1.27158
\(789\) 0 0
\(790\) 20.0343 0.712787
\(791\) 11.6498 0.414218
\(792\) 0 0
\(793\) 5.41808 0.192402
\(794\) −20.7808 −0.737485
\(795\) 0 0
\(796\) 51.1417 1.81267
\(797\) −3.46492 −0.122734 −0.0613670 0.998115i \(-0.519546\pi\)
−0.0613670 + 0.998115i \(0.519546\pi\)
\(798\) 0 0
\(799\) 16.9783 0.600650
\(800\) 5.64333 0.199522
\(801\) 0 0
\(802\) 40.6716 1.43617
\(803\) −52.9477 −1.86849
\(804\) 0 0
\(805\) 43.2135 1.52307
\(806\) −3.80785 −0.134126
\(807\) 0 0
\(808\) −20.3631 −0.716372
\(809\) 24.8406 0.873348 0.436674 0.899620i \(-0.356156\pi\)
0.436674 + 0.899620i \(0.356156\pi\)
\(810\) 0 0
\(811\) −40.3286 −1.41613 −0.708063 0.706149i \(-0.750431\pi\)
−0.708063 + 0.706149i \(0.750431\pi\)
\(812\) −36.3723 −1.27642
\(813\) 0 0
\(814\) −79.1050 −2.77263
\(815\) −2.47107 −0.0865579
\(816\) 0 0
\(817\) 10.0812 0.352697
\(818\) 31.8660 1.11417
\(819\) 0 0
\(820\) −11.7772 −0.411279
\(821\) −40.1816 −1.40235 −0.701174 0.712990i \(-0.747341\pi\)
−0.701174 + 0.712990i \(0.747341\pi\)
\(822\) 0 0
\(823\) −47.9554 −1.67162 −0.835810 0.549019i \(-0.815001\pi\)
−0.835810 + 0.549019i \(0.815001\pi\)
\(824\) 9.87012 0.343842
\(825\) 0 0
\(826\) −2.70218 −0.0940209
\(827\) 5.00048 0.173884 0.0869419 0.996213i \(-0.472291\pi\)
0.0869419 + 0.996213i \(0.472291\pi\)
\(828\) 0 0
\(829\) 29.7037 1.03165 0.515826 0.856693i \(-0.327485\pi\)
0.515826 + 0.856693i \(0.327485\pi\)
\(830\) 37.2003 1.29124
\(831\) 0 0
\(832\) 13.9316 0.482993
\(833\) −39.0228 −1.35206
\(834\) 0 0
\(835\) 49.5707 1.71546
\(836\) 19.1871 0.663599
\(837\) 0 0
\(838\) −12.3502 −0.426632
\(839\) 25.9045 0.894323 0.447162 0.894453i \(-0.352435\pi\)
0.447162 + 0.894453i \(0.352435\pi\)
\(840\) 0 0
\(841\) −20.1088 −0.693405
\(842\) −28.2916 −0.974993
\(843\) 0 0
\(844\) −24.8136 −0.854120
\(845\) −23.8599 −0.820807
\(846\) 0 0
\(847\) 30.1503 1.03598
\(848\) −17.1016 −0.587270
\(849\) 0 0
\(850\) 3.57224 0.122527
\(851\) 38.6077 1.32345
\(852\) 0 0
\(853\) −4.99166 −0.170911 −0.0854556 0.996342i \(-0.527235\pi\)
−0.0854556 + 0.996342i \(0.527235\pi\)
\(854\) −45.9086 −1.57096
\(855\) 0 0
\(856\) −8.13961 −0.278206
\(857\) 14.6830 0.501562 0.250781 0.968044i \(-0.419313\pi\)
0.250781 + 0.968044i \(0.419313\pi\)
\(858\) 0 0
\(859\) −17.4849 −0.596576 −0.298288 0.954476i \(-0.596416\pi\)
−0.298288 + 0.954476i \(0.596416\pi\)
\(860\) −28.5759 −0.974430
\(861\) 0 0
\(862\) 77.3864 2.63579
\(863\) −6.33263 −0.215565 −0.107783 0.994174i \(-0.534375\pi\)
−0.107783 + 0.994174i \(0.534375\pi\)
\(864\) 0 0
\(865\) 18.9284 0.643585
\(866\) −23.1121 −0.785381
\(867\) 0 0
\(868\) 17.9749 0.610106
\(869\) −18.8902 −0.640805
\(870\) 0 0
\(871\) −5.02349 −0.170214
\(872\) −6.16374 −0.208730
\(873\) 0 0
\(874\) −16.8090 −0.568571
\(875\) −57.3480 −1.93872
\(876\) 0 0
\(877\) 32.5310 1.09849 0.549247 0.835660i \(-0.314915\pi\)
0.549247 + 0.835660i \(0.314915\pi\)
\(878\) 20.0885 0.677953
\(879\) 0 0
\(880\) 23.2236 0.782868
\(881\) 33.3599 1.12393 0.561963 0.827163i \(-0.310046\pi\)
0.561963 + 0.827163i \(0.310046\pi\)
\(882\) 0 0
\(883\) −54.8511 −1.84589 −0.922944 0.384934i \(-0.874224\pi\)
−0.922944 + 0.384934i \(0.874224\pi\)
\(884\) 7.23081 0.243198
\(885\) 0 0
\(886\) −35.1138 −1.17967
\(887\) −21.0477 −0.706714 −0.353357 0.935489i \(-0.614960\pi\)
−0.353357 + 0.935489i \(0.614960\pi\)
\(888\) 0 0
\(889\) −44.7760 −1.50174
\(890\) −74.6516 −2.50233
\(891\) 0 0
\(892\) 37.9158 1.26951
\(893\) 13.2026 0.441810
\(894\) 0 0
\(895\) −21.9535 −0.733825
\(896\) −41.0988 −1.37301
\(897\) 0 0
\(898\) 10.0803 0.336385
\(899\) −4.39397 −0.146547
\(900\) 0 0
\(901\) −14.9572 −0.498296
\(902\) 19.9328 0.663690
\(903\) 0 0
\(904\) 2.63327 0.0875813
\(905\) −3.02962 −0.100708
\(906\) 0 0
\(907\) 14.8380 0.492689 0.246344 0.969182i \(-0.420771\pi\)
0.246344 + 0.969182i \(0.420771\pi\)
\(908\) 57.0887 1.89456
\(909\) 0 0
\(910\) −25.9474 −0.860149
\(911\) −18.0295 −0.597345 −0.298673 0.954356i \(-0.596544\pi\)
−0.298673 + 0.954356i \(0.596544\pi\)
\(912\) 0 0
\(913\) −35.0759 −1.16084
\(914\) −47.6188 −1.57509
\(915\) 0 0
\(916\) 21.9290 0.724555
\(917\) 74.3738 2.45604
\(918\) 0 0
\(919\) 13.4881 0.444932 0.222466 0.974940i \(-0.428589\pi\)
0.222466 + 0.974940i \(0.428589\pi\)
\(920\) 9.76783 0.322036
\(921\) 0 0
\(922\) 31.7102 1.04432
\(923\) 3.75203 0.123499
\(924\) 0 0
\(925\) −6.38029 −0.209783
\(926\) −31.5507 −1.03682
\(927\) 0 0
\(928\) 23.6603 0.776689
\(929\) 37.6211 1.23431 0.617155 0.786842i \(-0.288285\pi\)
0.617155 + 0.786842i \(0.288285\pi\)
\(930\) 0 0
\(931\) −30.3448 −0.994511
\(932\) 59.3439 1.94387
\(933\) 0 0
\(934\) 32.6222 1.06743
\(935\) 20.3116 0.664259
\(936\) 0 0
\(937\) 4.14458 0.135398 0.0676988 0.997706i \(-0.478434\pi\)
0.0676988 + 0.997706i \(0.478434\pi\)
\(938\) 42.5651 1.38980
\(939\) 0 0
\(940\) −37.4238 −1.22063
\(941\) 3.53033 0.115085 0.0575427 0.998343i \(-0.481673\pi\)
0.0575427 + 0.998343i \(0.481673\pi\)
\(942\) 0 0
\(943\) −9.72832 −0.316798
\(944\) 0.708752 0.0230679
\(945\) 0 0
\(946\) 48.3643 1.57246
\(947\) 14.2551 0.463228 0.231614 0.972808i \(-0.425599\pi\)
0.231614 + 0.972808i \(0.425599\pi\)
\(948\) 0 0
\(949\) −15.5164 −0.503685
\(950\) 2.77784 0.0901250
\(951\) 0 0
\(952\) −12.5605 −0.407087
\(953\) −11.6426 −0.377141 −0.188570 0.982060i \(-0.560385\pi\)
−0.188570 + 0.982060i \(0.560385\pi\)
\(954\) 0 0
\(955\) −25.7258 −0.832467
\(956\) −25.0430 −0.809950
\(957\) 0 0
\(958\) −22.2036 −0.717365
\(959\) 17.6684 0.570543
\(960\) 0 0
\(961\) −28.8285 −0.929953
\(962\) −23.1819 −0.747414
\(963\) 0 0
\(964\) 14.3736 0.462942
\(965\) 43.0203 1.38487
\(966\) 0 0
\(967\) 29.0681 0.934768 0.467384 0.884054i \(-0.345197\pi\)
0.467384 + 0.884054i \(0.345197\pi\)
\(968\) 6.81507 0.219045
\(969\) 0 0
\(970\) −22.4871 −0.722018
\(971\) −47.5792 −1.52689 −0.763444 0.645874i \(-0.776493\pi\)
−0.763444 + 0.645874i \(0.776493\pi\)
\(972\) 0 0
\(973\) 64.3687 2.06357
\(974\) −34.3508 −1.10067
\(975\) 0 0
\(976\) 12.0413 0.385433
\(977\) −6.11630 −0.195678 −0.0978389 0.995202i \(-0.531193\pi\)
−0.0978389 + 0.995202i \(0.531193\pi\)
\(978\) 0 0
\(979\) 70.3885 2.24963
\(980\) 86.0145 2.74763
\(981\) 0 0
\(982\) 22.9022 0.730837
\(983\) −10.6518 −0.339740 −0.169870 0.985466i \(-0.554335\pi\)
−0.169870 + 0.985466i \(0.554335\pi\)
\(984\) 0 0
\(985\) 29.3839 0.936248
\(986\) 14.9771 0.476967
\(987\) 0 0
\(988\) 5.62281 0.178885
\(989\) −23.6045 −0.750578
\(990\) 0 0
\(991\) 23.9856 0.761928 0.380964 0.924590i \(-0.375592\pi\)
0.380964 + 0.924590i \(0.375592\pi\)
\(992\) −11.6927 −0.371244
\(993\) 0 0
\(994\) −31.7918 −1.00837
\(995\) 42.0994 1.33464
\(996\) 0 0
\(997\) 4.31418 0.136631 0.0683157 0.997664i \(-0.478237\pi\)
0.0683157 + 0.997664i \(0.478237\pi\)
\(998\) −40.7733 −1.29066
\(999\) 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 729.2.a.e.1.5 yes 6
3.2 odd 2 729.2.a.b.1.2 6
9.2 odd 6 729.2.c.d.244.5 12
9.4 even 3 729.2.c.a.487.2 12
9.5 odd 6 729.2.c.d.487.5 12
9.7 even 3 729.2.c.a.244.2 12
27.2 odd 18 729.2.e.s.568.2 12
27.4 even 9 729.2.e.k.406.2 12
27.5 odd 18 729.2.e.j.649.1 12
27.7 even 9 729.2.e.k.325.2 12
27.11 odd 18 729.2.e.j.82.1 12
27.13 even 9 729.2.e.l.163.1 12
27.14 odd 18 729.2.e.s.163.2 12
27.16 even 9 729.2.e.u.82.2 12
27.20 odd 18 729.2.e.t.325.1 12
27.22 even 9 729.2.e.u.649.2 12
27.23 odd 18 729.2.e.t.406.1 12
27.25 even 9 729.2.e.l.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.2 6 3.2 odd 2
729.2.a.e.1.5 yes 6 1.1 even 1 trivial
729.2.c.a.244.2 12 9.7 even 3
729.2.c.a.487.2 12 9.4 even 3
729.2.c.d.244.5 12 9.2 odd 6
729.2.c.d.487.5 12 9.5 odd 6
729.2.e.j.82.1 12 27.11 odd 18
729.2.e.j.649.1 12 27.5 odd 18
729.2.e.k.325.2 12 27.7 even 9
729.2.e.k.406.2 12 27.4 even 9
729.2.e.l.163.1 12 27.13 even 9
729.2.e.l.568.1 12 27.25 even 9
729.2.e.s.163.2 12 27.14 odd 18
729.2.e.s.568.2 12 27.2 odd 18
729.2.e.t.325.1 12 27.20 odd 18
729.2.e.t.406.1 12 27.23 odd 18
729.2.e.u.82.2 12 27.16 even 9
729.2.e.u.649.2 12 27.22 even 9