Properties

Label 729.2.a.b.1.2
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 1.2
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 \(-0.770578\pi\)
−0.751311 + 0.659948i \(0.770578\pi\)
\(3\) 0 0
\(4\) 2.51575 1.25787
\(5\) −2.07094 −0.926153 −0.463076 0.886318i \(-0.653254\pi\)
−0.463076 + 0.886318i \(0.653254\pi\)
\(6\) 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 \(-0.284871\pi\)
0.625559 + 0.780177i \(0.284871\pi\)
\(12\) 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 \(-0.407464\pi\)
0.286633 + 0.958040i \(0.407464\pi\)
\(18\) 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 \(-0.648105\pi\)
−0.448678 + 0.893693i \(0.648105\pi\)
\(24\) 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 \(-0.410708\pi\)
0.276855 + 0.960912i \(0.410708\pi\)
\(30\) 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 \(-0.443517\pi\)
0.176517 + 0.984298i \(0.443517\pi\)
\(42\) 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 \(-0.324483\pi\)
0.523884 + 0.851790i \(0.324483\pi\)
\(48\) 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 \(-0.643114\pi\)
−0.434611 + 0.900618i \(0.643114\pi\)
\(54\) 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 \(-0.494566\pi\)
0.0170715 + 0.999854i \(0.494566\pi\)
\(60\) 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 \(-0.441390\pi\)
0.183091 + 0.983096i \(0.441390\pi\)
\(72\) 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 \(-0.653558\pi\)
−0.463922 + 0.885876i \(0.653558\pi\)
\(84\) 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 \(-0.144261\pi\)
0.899046 + 0.437854i \(0.144261\pi\)
\(90\) 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 \(-0.124580\pi\)
0.924384 + 0.381464i \(0.124580\pi\)
\(102\) 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 \(-0.383122\pi\)
0.358988 + 0.933342i \(0.383122\pi\)
\(108\) 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 \(-0.536050\pi\)
−0.113012 + 0.993594i \(0.536050\pi\)
\(114\) 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 \(-0.733743\pi\)
−0.670088 + 0.742282i \(0.733743\pi\)
\(132\) 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 \(-0.549751\pi\)
−0.155663 + 0.987810i \(0.549751\pi\)
\(138\) 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 \(-0.381066\pi\)
0.365008 + 0.931004i \(0.381066\pi\)
\(150\) 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 \(-0.876880\pi\)
−0.926124 + 0.377219i \(0.876880\pi\)
\(168\) 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 \(-0.612953\pi\)
−0.347451 + 0.937698i \(0.612953\pi\)
\(174\) 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 \(-0.370340\pi\)
0.396169 + 0.918178i \(0.370340\pi\)
\(180\) 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 \(-0.351630\pi\)
0.449422 + 0.893320i \(0.351630\pi\)
\(192\) 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 \(-0.668674\pi\)
−0.505450 + 0.862856i \(0.668674\pi\)
\(198\) 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 \(-0.771432\pi\)
−0.753079 + 0.657930i \(0.771432\pi\)
\(228\) 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 \(-0.781085\pi\)
−0.772682 + 0.634793i \(0.781085\pi\)
\(234\) 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 \(-0.395661\pi\)
0.321952 + 0.946756i \(0.395661\pi\)
\(240\) 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 \(-0.573892\pi\)
−0.230060 + 0.973176i \(0.573892\pi\)
\(252\) 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 \(-0.758129\pi\)
−0.724932 + 0.688821i \(0.758129\pi\)
\(258\) 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 \(-0.820260\pi\)
−0.844766 + 0.535136i \(0.820260\pi\)
\(264\) 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 \(-0.407287\pi\)
0.287166 + 0.957881i \(0.407287\pi\)
\(270\) 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 \(-0.363069\pi\)
0.417036 + 0.908890i \(0.363069\pi\)
\(282\) 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 \(-0.244471\pi\)
0.719281 + 0.694719i \(0.244471\pi\)
\(294\) 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 \(-0.283535\pi\)
0.628828 + 0.777544i \(0.283535\pi\)
\(312\) 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 \(-0.249528\pi\)
0.708154 + 0.706058i \(0.249528\pi\)
\(318\) 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 \(-0.469937\pi\)
0.0943040 + 0.995543i \(0.469937\pi\)
\(348\) 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 \(-0.732572\pi\)
−0.667352 + 0.744742i \(0.732572\pi\)
\(354\) 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 \(-0.464587\pi\)
0.111025 + 0.993818i \(0.464587\pi\)
\(360\) 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 \(-0.418632\pi\)
0.252851 + 0.967505i \(0.418632\pi\)
\(384\) 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 \(-0.679794\pi\)
−0.535281 + 0.844674i \(0.679794\pi\)
\(390\) 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 \(-0.658597\pi\)
−0.477886 + 0.878422i \(0.658597\pi\)
\(402\) 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 \(-0.454659\pi\)
0.141963 + 0.989872i \(0.454659\pi\)
\(420\) 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 \(-0.840501\pi\)
−0.877064 + 0.480374i \(0.840501\pi\)
\(432\) 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 \(-0.371597\pi\)
0.392537 + 0.919736i \(0.371597\pi\)
\(444\) 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 \(-0.535704\pi\)
−0.111933 + 0.993716i \(0.535704\pi\)
\(450\) 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 \(-0.612969\pi\)
−0.347499 + 0.937680i \(0.612969\pi\)
\(462\) 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 \(-0.615584\pi\)
−0.355190 + 0.934794i \(0.615584\pi\)
\(468\) 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 \(-0.423277\pi\)
0.238705 + 0.971092i \(0.423277\pi\)
\(480\) 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 \(-0.578193\pi\)
−0.243187 + 0.969979i \(0.578193\pi\)
\(492\) 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 \(-0.586391\pi\)
−0.268086 + 0.963395i \(0.586391\pi\)
\(504\) 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 \(-0.607518\pi\)
−0.331391 + 0.943494i \(0.607518\pi\)
\(510\) 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 \(-0.194155\pi\)
0.819673 + 0.572832i \(0.194155\pi\)
\(522\) 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 \(-0.609579\pi\)
−0.337494 + 0.941328i \(0.609579\pi\)
\(558\) 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 \(-0.675516\pi\)
−0.523880 + 0.851792i \(0.675516\pi\)
\(564\) 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 \(-0.368231\pi\)
0.402243 + 0.915533i \(0.368231\pi\)
\(570\) 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 \(-0.775637\pi\)
−0.761704 + 0.647925i \(0.775637\pi\)
\(588\) 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 \(-0.471412\pi\)
0.0896923 + 0.995970i \(0.471412\pi\)
\(594\) 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 \(-0.720037\pi\)
−0.637513 + 0.770439i \(0.720037\pi\)
\(600\) 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 \(-0.324575\pi\)
0.523636 + 0.851942i \(0.324575\pi\)
\(618\) 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 \(-0.355531\pi\)
0.438442 + 0.898760i \(0.355531\pi\)
\(642\) 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 \(-0.415131\pi\)
0.263476 + 0.964666i \(0.415131\pi\)
\(648\) 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 \(-0.620649\pi\)
−0.370020 + 0.929024i \(0.620649\pi\)
\(654\) 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 \(-0.500175\pi\)
−0.000548897 1.00000i \(0.500175\pi\)
\(660\) 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 \(-0.212361\pi\)
0.785588 + 0.618750i \(0.212361\pi\)
\(678\) 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 \(-0.818791\pi\)
−0.842287 + 0.539029i \(0.818791\pi\)
\(684\) 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 \(-0.578036\pi\)
−0.242709 + 0.970099i \(0.578036\pi\)
\(702\) 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 \(-0.533523\pi\)
−0.105121 + 0.994459i \(0.533523\pi\)
\(720\) 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 \(-0.279667\pi\)
0.638229 + 0.769846i \(0.279667\pi\)
\(744\) 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 \(-0.572420\pi\)
−0.225555 + 0.974230i \(0.572420\pi\)
\(762\) 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 \(-0.727289\pi\)
−0.654900 + 0.755716i \(0.727289\pi\)
\(774\) 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 \(-0.480454\pi\)
0.0613670 + 0.998115i \(0.480454\pi\)
\(798\) 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 \(-0.643844\pi\)
−0.436674 + 0.899620i \(0.643844\pi\)
\(810\) 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 \(-0.252659\pi\)
0.701174 + 0.712990i \(0.252659\pi\)
\(822\) 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 \(-0.527709\pi\)
−0.0869419 + 0.996213i \(0.527709\pi\)
\(828\) 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 \(-0.647565\pi\)
−0.447162 + 0.894453i \(0.647565\pi\)
\(840\) 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 \(-0.580687\pi\)
−0.250781 + 0.968044i \(0.580687\pi\)
\(858\) 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 \(-0.465625\pi\)
0.107783 + 0.994174i \(0.465625\pi\)
\(864\) 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 \(-0.689954\pi\)
−0.561963 + 0.827163i \(0.689954\pi\)
\(882\) 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 \(-0.385040\pi\)
0.353357 + 0.935489i \(0.385040\pi\)
\(888\) 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 \(-0.403456\pi\)
0.298673 + 0.954356i \(0.403456\pi\)
\(912\) 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 \(-0.711715\pi\)
−0.617155 + 0.786842i \(0.711715\pi\)
\(930\) 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 \(-0.518327\pi\)
−0.0575427 + 0.998343i \(0.518327\pi\)
\(942\) 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 \(-0.574401\pi\)
−0.231614 + 0.972808i \(0.574401\pi\)
\(948\) 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 \(-0.439615\pi\)
0.188570 + 0.982060i \(0.439615\pi\)
\(954\) 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 \(-0.223507\pi\)
0.763444 + 0.645874i \(0.223507\pi\)
\(972\) 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 \(-0.468807\pi\)
0.0978389 + 0.995202i \(0.468807\pi\)
\(978\) 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 \(-0.445665\pi\)
0.169870 + 0.985466i \(0.445665\pi\)
\(984\) 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.b.1.2 6
3.2 odd 2 729.2.a.e.1.5 yes 6
9.2 odd 6 729.2.c.a.244.2 12
9.4 even 3 729.2.c.d.487.5 12
9.5 odd 6 729.2.c.a.487.2 12
9.7 even 3 729.2.c.d.244.5 12
27.2 odd 18 729.2.e.l.568.1 12
27.4 even 9 729.2.e.t.406.1 12
27.5 odd 18 729.2.e.u.649.2 12
27.7 even 9 729.2.e.t.325.1 12
27.11 odd 18 729.2.e.u.82.2 12
27.13 even 9 729.2.e.s.163.2 12
27.14 odd 18 729.2.e.l.163.1 12
27.16 even 9 729.2.e.j.82.1 12
27.20 odd 18 729.2.e.k.325.2 12
27.22 even 9 729.2.e.j.649.1 12
27.23 odd 18 729.2.e.k.406.2 12
27.25 even 9 729.2.e.s.568.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.2 6 1.1 even 1 trivial
729.2.a.e.1.5 yes 6 3.2 odd 2
729.2.c.a.244.2 12 9.2 odd 6
729.2.c.a.487.2 12 9.5 odd 6
729.2.c.d.244.5 12 9.7 even 3
729.2.c.d.487.5 12 9.4 even 3
729.2.e.j.82.1 12 27.16 even 9
729.2.e.j.649.1 12 27.22 even 9
729.2.e.k.325.2 12 27.20 odd 18
729.2.e.k.406.2 12 27.23 odd 18
729.2.e.l.163.1 12 27.14 odd 18
729.2.e.l.568.1 12 27.2 odd 18
729.2.e.s.163.2 12 27.13 even 9
729.2.e.s.568.2 12 27.25 even 9
729.2.e.t.325.1 12 27.7 even 9
729.2.e.t.406.1 12 27.4 even 9
729.2.e.u.82.2 12 27.11 odd 18
729.2.e.u.649.2 12 27.5 odd 18