Properties

Label 729.2.a.d.1.6
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: 6.6.1397493.1
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.05432\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.40162 q^{2} +3.76778 q^{4} -0.0930834 q^{5} -0.579861 q^{7} +4.24555 q^{8} +O(q^{10})\) \(q+2.40162 q^{2} +3.76778 q^{4} -0.0930834 q^{5} -0.579861 q^{7} +4.24555 q^{8} -0.223551 q^{10} +3.09308 q^{11} +4.20173 q^{13} -1.39261 q^{14} +2.66063 q^{16} +1.99099 q^{17} -3.84542 q^{19} -0.350718 q^{20} +7.42841 q^{22} +4.45282 q^{23} -4.99134 q^{25} +10.0910 q^{26} -2.18479 q^{28} -6.39951 q^{29} +1.65750 q^{31} -2.10127 q^{32} +4.78159 q^{34} +0.0539755 q^{35} +4.03009 q^{37} -9.23525 q^{38} -0.395190 q^{40} +1.09616 q^{41} -6.90112 q^{43} +11.6541 q^{44} +10.6940 q^{46} -3.59319 q^{47} -6.66376 q^{49} -11.9873 q^{50} +15.8312 q^{52} -5.40034 q^{53} -0.287915 q^{55} -2.46183 q^{56} -15.3692 q^{58} +10.2847 q^{59} -13.1963 q^{61} +3.98069 q^{62} -10.3677 q^{64} -0.391112 q^{65} -8.83729 q^{67} +7.50161 q^{68} +0.129629 q^{70} -1.14495 q^{71} +0.195472 q^{73} +9.67876 q^{74} -14.4887 q^{76} -1.79356 q^{77} -7.20799 q^{79} -0.247661 q^{80} +2.63255 q^{82} +14.9004 q^{83} -0.185328 q^{85} -16.5739 q^{86} +13.1318 q^{88} +1.55313 q^{89} -2.43642 q^{91} +16.7772 q^{92} -8.62949 q^{94} +0.357945 q^{95} +5.29553 q^{97} -16.0038 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35} + 3 q^{37} - 3 q^{38} + 6 q^{40} + 15 q^{41} + 3 q^{44} + 3 q^{46} + 21 q^{47} - 12 q^{49} + 3 q^{50} + 12 q^{52} + 9 q^{53} - 6 q^{55} - 6 q^{56} - 12 q^{58} + 24 q^{59} - 9 q^{61} - 12 q^{62} - 12 q^{64} - 6 q^{65} - 9 q^{67} - 9 q^{68} + 15 q^{70} + 27 q^{71} - 6 q^{73} - 12 q^{74} + 6 q^{76} - 12 q^{77} - 21 q^{80} - 6 q^{82} + 12 q^{83} - 21 q^{86} + 12 q^{88} + 9 q^{89} - 6 q^{91} + 6 q^{92} + 6 q^{94} + 12 q^{95} - 45 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.40162 1.69820 0.849101 0.528230i \(-0.177144\pi\)
0.849101 + 0.528230i \(0.177144\pi\)
\(3\) 0 0
\(4\) 3.76778 1.88389
\(5\) −0.0930834 −0.0416282 −0.0208141 0.999783i \(-0.506626\pi\)
−0.0208141 + 0.999783i \(0.506626\pi\)
\(6\) 0 0
\(7\) −0.579861 −0.219167 −0.109583 0.993978i \(-0.534952\pi\)
−0.109583 + 0.993978i \(0.534952\pi\)
\(8\) 4.24555 1.50103
\(9\) 0 0
\(10\) −0.223551 −0.0706931
\(11\) 3.09308 0.932600 0.466300 0.884627i \(-0.345587\pi\)
0.466300 + 0.884627i \(0.345587\pi\)
\(12\) 0 0
\(13\) 4.20173 1.16535 0.582676 0.812705i \(-0.302006\pi\)
0.582676 + 0.812705i \(0.302006\pi\)
\(14\) −1.39261 −0.372190
\(15\) 0 0
\(16\) 2.66063 0.665158
\(17\) 1.99099 0.482885 0.241443 0.970415i \(-0.422379\pi\)
0.241443 + 0.970415i \(0.422379\pi\)
\(18\) 0 0
\(19\) −3.84542 −0.882201 −0.441100 0.897458i \(-0.645412\pi\)
−0.441100 + 0.897458i \(0.645412\pi\)
\(20\) −0.350718 −0.0784230
\(21\) 0 0
\(22\) 7.42841 1.58374
\(23\) 4.45282 0.928476 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(24\) 0 0
\(25\) −4.99134 −0.998267
\(26\) 10.0910 1.97900
\(27\) 0 0
\(28\) −2.18479 −0.412887
\(29\) −6.39951 −1.18836 −0.594179 0.804332i \(-0.702523\pi\)
−0.594179 + 0.804332i \(0.702523\pi\)
\(30\) 0 0
\(31\) 1.65750 0.297696 0.148848 0.988860i \(-0.452444\pi\)
0.148848 + 0.988860i \(0.452444\pi\)
\(32\) −2.10127 −0.371456
\(33\) 0 0
\(34\) 4.78159 0.820037
\(35\) 0.0539755 0.00912352
\(36\) 0 0
\(37\) 4.03009 0.662543 0.331272 0.943535i \(-0.392522\pi\)
0.331272 + 0.943535i \(0.392522\pi\)
\(38\) −9.23525 −1.49816
\(39\) 0 0
\(40\) −0.395190 −0.0624850
\(41\) 1.09616 0.171191 0.0855954 0.996330i \(-0.472721\pi\)
0.0855954 + 0.996330i \(0.472721\pi\)
\(42\) 0 0
\(43\) −6.90112 −1.05241 −0.526206 0.850357i \(-0.676386\pi\)
−0.526206 + 0.850357i \(0.676386\pi\)
\(44\) 11.6541 1.75692
\(45\) 0 0
\(46\) 10.6940 1.57674
\(47\) −3.59319 −0.524121 −0.262061 0.965051i \(-0.584402\pi\)
−0.262061 + 0.965051i \(0.584402\pi\)
\(48\) 0 0
\(49\) −6.66376 −0.951966
\(50\) −11.9873 −1.69526
\(51\) 0 0
\(52\) 15.8312 2.19540
\(53\) −5.40034 −0.741793 −0.370897 0.928674i \(-0.620950\pi\)
−0.370897 + 0.928674i \(0.620950\pi\)
\(54\) 0 0
\(55\) −0.287915 −0.0388224
\(56\) −2.46183 −0.328976
\(57\) 0 0
\(58\) −15.3692 −2.01807
\(59\) 10.2847 1.33895 0.669474 0.742835i \(-0.266520\pi\)
0.669474 + 0.742835i \(0.266520\pi\)
\(60\) 0 0
\(61\) −13.1963 −1.68962 −0.844808 0.535070i \(-0.820285\pi\)
−0.844808 + 0.535070i \(0.820285\pi\)
\(62\) 3.98069 0.505548
\(63\) 0 0
\(64\) −10.3677 −1.29596
\(65\) −0.391112 −0.0485114
\(66\) 0 0
\(67\) −8.83729 −1.07965 −0.539824 0.841778i \(-0.681509\pi\)
−0.539824 + 0.841778i \(0.681509\pi\)
\(68\) 7.50161 0.909703
\(69\) 0 0
\(70\) 0.129629 0.0154936
\(71\) −1.14495 −0.135880 −0.0679401 0.997689i \(-0.521643\pi\)
−0.0679401 + 0.997689i \(0.521643\pi\)
\(72\) 0 0
\(73\) 0.195472 0.0228783 0.0114391 0.999935i \(-0.496359\pi\)
0.0114391 + 0.999935i \(0.496359\pi\)
\(74\) 9.67876 1.12513
\(75\) 0 0
\(76\) −14.4887 −1.66197
\(77\) −1.79356 −0.204395
\(78\) 0 0
\(79\) −7.20799 −0.810963 −0.405481 0.914103i \(-0.632896\pi\)
−0.405481 + 0.914103i \(0.632896\pi\)
\(80\) −0.247661 −0.0276893
\(81\) 0 0
\(82\) 2.63255 0.290717
\(83\) 14.9004 1.63554 0.817768 0.575548i \(-0.195211\pi\)
0.817768 + 0.575548i \(0.195211\pi\)
\(84\) 0 0
\(85\) −0.185328 −0.0201016
\(86\) −16.5739 −1.78721
\(87\) 0 0
\(88\) 13.1318 1.39986
\(89\) 1.55313 0.164631 0.0823155 0.996606i \(-0.473768\pi\)
0.0823155 + 0.996606i \(0.473768\pi\)
\(90\) 0 0
\(91\) −2.43642 −0.255406
\(92\) 16.7772 1.74915
\(93\) 0 0
\(94\) −8.62949 −0.890064
\(95\) 0.357945 0.0367244
\(96\) 0 0
\(97\) 5.29553 0.537680 0.268840 0.963185i \(-0.413360\pi\)
0.268840 + 0.963185i \(0.413360\pi\)
\(98\) −16.0038 −1.61663
\(99\) 0 0
\(100\) −18.8063 −1.88063
\(101\) 7.26898 0.723291 0.361645 0.932316i \(-0.382215\pi\)
0.361645 + 0.932316i \(0.382215\pi\)
\(102\) 0 0
\(103\) 6.40137 0.630746 0.315373 0.948968i \(-0.397870\pi\)
0.315373 + 0.948968i \(0.397870\pi\)
\(104\) 17.8387 1.74922
\(105\) 0 0
\(106\) −12.9696 −1.25972
\(107\) 5.54365 0.535925 0.267963 0.963429i \(-0.413650\pi\)
0.267963 + 0.963429i \(0.413650\pi\)
\(108\) 0 0
\(109\) −6.23137 −0.596857 −0.298428 0.954432i \(-0.596462\pi\)
−0.298428 + 0.954432i \(0.596462\pi\)
\(110\) −0.691462 −0.0659283
\(111\) 0 0
\(112\) −1.54280 −0.145781
\(113\) 11.8440 1.11419 0.557094 0.830449i \(-0.311916\pi\)
0.557094 + 0.830449i \(0.311916\pi\)
\(114\) 0 0
\(115\) −0.414483 −0.0386508
\(116\) −24.1120 −2.23874
\(117\) 0 0
\(118\) 24.6998 2.27381
\(119\) −1.15450 −0.105832
\(120\) 0 0
\(121\) −1.43283 −0.130258
\(122\) −31.6926 −2.86931
\(123\) 0 0
\(124\) 6.24510 0.560827
\(125\) 0.930028 0.0831842
\(126\) 0 0
\(127\) 11.5294 1.02307 0.511533 0.859263i \(-0.329078\pi\)
0.511533 + 0.859263i \(0.329078\pi\)
\(128\) −20.6968 −1.82935
\(129\) 0 0
\(130\) −0.939302 −0.0823822
\(131\) 9.01177 0.787362 0.393681 0.919247i \(-0.371201\pi\)
0.393681 + 0.919247i \(0.371201\pi\)
\(132\) 0 0
\(133\) 2.22981 0.193349
\(134\) −21.2238 −1.83346
\(135\) 0 0
\(136\) 8.45283 0.724824
\(137\) −11.5202 −0.984240 −0.492120 0.870527i \(-0.663778\pi\)
−0.492120 + 0.870527i \(0.663778\pi\)
\(138\) 0 0
\(139\) 1.70383 0.144517 0.0722587 0.997386i \(-0.476979\pi\)
0.0722587 + 0.997386i \(0.476979\pi\)
\(140\) 0.203368 0.0171877
\(141\) 0 0
\(142\) −2.74973 −0.230752
\(143\) 12.9963 1.08681
\(144\) 0 0
\(145\) 0.595688 0.0494692
\(146\) 0.469450 0.0388520
\(147\) 0 0
\(148\) 15.1845 1.24816
\(149\) 21.6528 1.77386 0.886932 0.461900i \(-0.152832\pi\)
0.886932 + 0.461900i \(0.152832\pi\)
\(150\) 0 0
\(151\) −4.74152 −0.385860 −0.192930 0.981213i \(-0.561799\pi\)
−0.192930 + 0.981213i \(0.561799\pi\)
\(152\) −16.3259 −1.32421
\(153\) 0 0
\(154\) −4.30745 −0.347104
\(155\) −0.154286 −0.0123925
\(156\) 0 0
\(157\) −0.209206 −0.0166964 −0.00834822 0.999965i \(-0.502657\pi\)
−0.00834822 + 0.999965i \(0.502657\pi\)
\(158\) −17.3109 −1.37718
\(159\) 0 0
\(160\) 0.195593 0.0154630
\(161\) −2.58202 −0.203491
\(162\) 0 0
\(163\) 5.62384 0.440493 0.220247 0.975444i \(-0.429314\pi\)
0.220247 + 0.975444i \(0.429314\pi\)
\(164\) 4.13008 0.322505
\(165\) 0 0
\(166\) 35.7852 2.77747
\(167\) −16.6805 −1.29078 −0.645388 0.763855i \(-0.723304\pi\)
−0.645388 + 0.763855i \(0.723304\pi\)
\(168\) 0 0
\(169\) 4.65456 0.358043
\(170\) −0.445087 −0.0341366
\(171\) 0 0
\(172\) −26.0019 −1.98263
\(173\) −19.0021 −1.44471 −0.722353 0.691524i \(-0.756939\pi\)
−0.722353 + 0.691524i \(0.756939\pi\)
\(174\) 0 0
\(175\) 2.89428 0.218787
\(176\) 8.22955 0.620326
\(177\) 0 0
\(178\) 3.73002 0.279577
\(179\) −16.2352 −1.21348 −0.606739 0.794901i \(-0.707523\pi\)
−0.606739 + 0.794901i \(0.707523\pi\)
\(180\) 0 0
\(181\) −2.99158 −0.222362 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(182\) −5.85136 −0.433732
\(183\) 0 0
\(184\) 18.9046 1.39367
\(185\) −0.375135 −0.0275805
\(186\) 0 0
\(187\) 6.15829 0.450338
\(188\) −13.5384 −0.987388
\(189\) 0 0
\(190\) 0.859649 0.0623655
\(191\) 2.25382 0.163080 0.0815402 0.996670i \(-0.474016\pi\)
0.0815402 + 0.996670i \(0.474016\pi\)
\(192\) 0 0
\(193\) −0.880273 −0.0633634 −0.0316817 0.999498i \(-0.510086\pi\)
−0.0316817 + 0.999498i \(0.510086\pi\)
\(194\) 12.7179 0.913090
\(195\) 0 0
\(196\) −25.1076 −1.79340
\(197\) −20.2766 −1.44464 −0.722322 0.691557i \(-0.756925\pi\)
−0.722322 + 0.691557i \(0.756925\pi\)
\(198\) 0 0
\(199\) −19.0094 −1.34754 −0.673772 0.738939i \(-0.735327\pi\)
−0.673772 + 0.738939i \(0.735327\pi\)
\(200\) −21.1910 −1.49843
\(201\) 0 0
\(202\) 17.4573 1.22829
\(203\) 3.71083 0.260449
\(204\) 0 0
\(205\) −0.102034 −0.00712636
\(206\) 15.3737 1.07113
\(207\) 0 0
\(208\) 11.1793 0.775142
\(209\) −11.8942 −0.822740
\(210\) 0 0
\(211\) 16.1841 1.11416 0.557079 0.830460i \(-0.311922\pi\)
0.557079 + 0.830460i \(0.311922\pi\)
\(212\) −20.3473 −1.39746
\(213\) 0 0
\(214\) 13.3138 0.910110
\(215\) 0.642380 0.0438100
\(216\) 0 0
\(217\) −0.961120 −0.0652451
\(218\) −14.9654 −1.01358
\(219\) 0 0
\(220\) −1.08480 −0.0731373
\(221\) 8.36559 0.562731
\(222\) 0 0
\(223\) 21.4573 1.43689 0.718443 0.695585i \(-0.244855\pi\)
0.718443 + 0.695585i \(0.244855\pi\)
\(224\) 1.21845 0.0814108
\(225\) 0 0
\(226\) 28.4448 1.89212
\(227\) 19.1142 1.26866 0.634329 0.773064i \(-0.281277\pi\)
0.634329 + 0.773064i \(0.281277\pi\)
\(228\) 0 0
\(229\) 22.4702 1.48487 0.742435 0.669918i \(-0.233671\pi\)
0.742435 + 0.669918i \(0.233671\pi\)
\(230\) −0.995432 −0.0656368
\(231\) 0 0
\(232\) −27.1694 −1.78376
\(233\) 17.6815 1.15835 0.579176 0.815203i \(-0.303374\pi\)
0.579176 + 0.815203i \(0.303374\pi\)
\(234\) 0 0
\(235\) 0.334467 0.0218182
\(236\) 38.7504 2.52243
\(237\) 0 0
\(238\) −2.77266 −0.179725
\(239\) 15.4203 0.997457 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(240\) 0 0
\(241\) 13.1514 0.847159 0.423580 0.905859i \(-0.360773\pi\)
0.423580 + 0.905859i \(0.360773\pi\)
\(242\) −3.44113 −0.221204
\(243\) 0 0
\(244\) −49.7209 −3.18305
\(245\) 0.620286 0.0396286
\(246\) 0 0
\(247\) −16.1574 −1.02807
\(248\) 7.03700 0.446850
\(249\) 0 0
\(250\) 2.23357 0.141264
\(251\) −17.4166 −1.09933 −0.549663 0.835386i \(-0.685244\pi\)
−0.549663 + 0.835386i \(0.685244\pi\)
\(252\) 0 0
\(253\) 13.7729 0.865897
\(254\) 27.6892 1.73737
\(255\) 0 0
\(256\) −28.9704 −1.81065
\(257\) −11.1876 −0.697862 −0.348931 0.937148i \(-0.613455\pi\)
−0.348931 + 0.937148i \(0.613455\pi\)
\(258\) 0 0
\(259\) −2.33690 −0.145208
\(260\) −1.47362 −0.0913903
\(261\) 0 0
\(262\) 21.6429 1.33710
\(263\) 20.7109 1.27709 0.638544 0.769585i \(-0.279537\pi\)
0.638544 + 0.769585i \(0.279537\pi\)
\(264\) 0 0
\(265\) 0.502682 0.0308795
\(266\) 5.35516 0.328346
\(267\) 0 0
\(268\) −33.2970 −2.03394
\(269\) 28.2449 1.72212 0.861060 0.508504i \(-0.169801\pi\)
0.861060 + 0.508504i \(0.169801\pi\)
\(270\) 0 0
\(271\) 17.2626 1.04863 0.524316 0.851524i \(-0.324321\pi\)
0.524316 + 0.851524i \(0.324321\pi\)
\(272\) 5.29728 0.321195
\(273\) 0 0
\(274\) −27.6672 −1.67144
\(275\) −15.4386 −0.930984
\(276\) 0 0
\(277\) 5.16898 0.310574 0.155287 0.987869i \(-0.450370\pi\)
0.155287 + 0.987869i \(0.450370\pi\)
\(278\) 4.09197 0.245420
\(279\) 0 0
\(280\) 0.229155 0.0136947
\(281\) 3.29644 0.196649 0.0983246 0.995154i \(-0.468652\pi\)
0.0983246 + 0.995154i \(0.468652\pi\)
\(282\) 0 0
\(283\) −9.13072 −0.542765 −0.271382 0.962472i \(-0.587481\pi\)
−0.271382 + 0.962472i \(0.587481\pi\)
\(284\) −4.31391 −0.255984
\(285\) 0 0
\(286\) 31.2122 1.84562
\(287\) −0.635618 −0.0375194
\(288\) 0 0
\(289\) −13.0360 −0.766822
\(290\) 1.43062 0.0840087
\(291\) 0 0
\(292\) 0.736497 0.0431002
\(293\) 2.82643 0.165122 0.0825610 0.996586i \(-0.473690\pi\)
0.0825610 + 0.996586i \(0.473690\pi\)
\(294\) 0 0
\(295\) −0.957331 −0.0557380
\(296\) 17.1100 0.994496
\(297\) 0 0
\(298\) 52.0017 3.01238
\(299\) 18.7095 1.08200
\(300\) 0 0
\(301\) 4.00169 0.230654
\(302\) −11.3873 −0.655268
\(303\) 0 0
\(304\) −10.2312 −0.586802
\(305\) 1.22836 0.0703356
\(306\) 0 0
\(307\) 6.29446 0.359244 0.179622 0.983736i \(-0.442513\pi\)
0.179622 + 0.983736i \(0.442513\pi\)
\(308\) −6.75775 −0.385058
\(309\) 0 0
\(310\) −0.370536 −0.0210450
\(311\) 7.37289 0.418078 0.209039 0.977907i \(-0.432966\pi\)
0.209039 + 0.977907i \(0.432966\pi\)
\(312\) 0 0
\(313\) −4.27075 −0.241397 −0.120699 0.992689i \(-0.538513\pi\)
−0.120699 + 0.992689i \(0.538513\pi\)
\(314\) −0.502433 −0.0283539
\(315\) 0 0
\(316\) −27.1582 −1.52777
\(317\) −16.1476 −0.906938 −0.453469 0.891272i \(-0.649814\pi\)
−0.453469 + 0.891272i \(0.649814\pi\)
\(318\) 0 0
\(319\) −19.7942 −1.10826
\(320\) 0.965063 0.0539486
\(321\) 0 0
\(322\) −6.20102 −0.345570
\(323\) −7.65618 −0.426001
\(324\) 0 0
\(325\) −20.9723 −1.16333
\(326\) 13.5063 0.748047
\(327\) 0 0
\(328\) 4.65378 0.256962
\(329\) 2.08355 0.114870
\(330\) 0 0
\(331\) 19.2282 1.05688 0.528440 0.848971i \(-0.322777\pi\)
0.528440 + 0.848971i \(0.322777\pi\)
\(332\) 56.1417 3.08117
\(333\) 0 0
\(334\) −40.0602 −2.19200
\(335\) 0.822605 0.0449437
\(336\) 0 0
\(337\) −29.4628 −1.60494 −0.802469 0.596693i \(-0.796481\pi\)
−0.802469 + 0.596693i \(0.796481\pi\)
\(338\) 11.1785 0.608030
\(339\) 0 0
\(340\) −0.698275 −0.0378693
\(341\) 5.12679 0.277631
\(342\) 0 0
\(343\) 7.92309 0.427806
\(344\) −29.2990 −1.57970
\(345\) 0 0
\(346\) −45.6360 −2.45340
\(347\) −11.3970 −0.611825 −0.305912 0.952060i \(-0.598961\pi\)
−0.305912 + 0.952060i \(0.598961\pi\)
\(348\) 0 0
\(349\) 28.1616 1.50746 0.753728 0.657186i \(-0.228253\pi\)
0.753728 + 0.657186i \(0.228253\pi\)
\(350\) 6.95097 0.371545
\(351\) 0 0
\(352\) −6.49941 −0.346419
\(353\) −28.6541 −1.52510 −0.762552 0.646927i \(-0.776054\pi\)
−0.762552 + 0.646927i \(0.776054\pi\)
\(354\) 0 0
\(355\) 0.106576 0.00565644
\(356\) 5.85184 0.310147
\(357\) 0 0
\(358\) −38.9909 −2.06073
\(359\) 31.0322 1.63782 0.818909 0.573923i \(-0.194579\pi\)
0.818909 + 0.573923i \(0.194579\pi\)
\(360\) 0 0
\(361\) −4.21272 −0.221722
\(362\) −7.18464 −0.377616
\(363\) 0 0
\(364\) −9.17992 −0.481158
\(365\) −0.0181952 −0.000952381 0
\(366\) 0 0
\(367\) 24.1242 1.25927 0.629637 0.776889i \(-0.283204\pi\)
0.629637 + 0.776889i \(0.283204\pi\)
\(368\) 11.8473 0.617583
\(369\) 0 0
\(370\) −0.900932 −0.0468372
\(371\) 3.13145 0.162577
\(372\) 0 0
\(373\) −12.5856 −0.651658 −0.325829 0.945429i \(-0.605643\pi\)
−0.325829 + 0.945429i \(0.605643\pi\)
\(374\) 14.7899 0.764766
\(375\) 0 0
\(376\) −15.2551 −0.786721
\(377\) −26.8890 −1.38486
\(378\) 0 0
\(379\) −7.70522 −0.395790 −0.197895 0.980223i \(-0.563411\pi\)
−0.197895 + 0.980223i \(0.563411\pi\)
\(380\) 1.34866 0.0691848
\(381\) 0 0
\(382\) 5.41281 0.276944
\(383\) 17.8616 0.912687 0.456343 0.889804i \(-0.349159\pi\)
0.456343 + 0.889804i \(0.349159\pi\)
\(384\) 0 0
\(385\) 0.166951 0.00850859
\(386\) −2.11408 −0.107604
\(387\) 0 0
\(388\) 19.9524 1.01293
\(389\) −27.3885 −1.38865 −0.694325 0.719662i \(-0.744297\pi\)
−0.694325 + 0.719662i \(0.744297\pi\)
\(390\) 0 0
\(391\) 8.86549 0.448347
\(392\) −28.2913 −1.42893
\(393\) 0 0
\(394\) −48.6966 −2.45330
\(395\) 0.670945 0.0337589
\(396\) 0 0
\(397\) 4.21599 0.211594 0.105797 0.994388i \(-0.466261\pi\)
0.105797 + 0.994388i \(0.466261\pi\)
\(398\) −45.6535 −2.28840
\(399\) 0 0
\(400\) −13.2801 −0.664005
\(401\) −15.1725 −0.757678 −0.378839 0.925462i \(-0.623677\pi\)
−0.378839 + 0.925462i \(0.623677\pi\)
\(402\) 0 0
\(403\) 6.96437 0.346920
\(404\) 27.3880 1.36260
\(405\) 0 0
\(406\) 8.91200 0.442295
\(407\) 12.4654 0.617888
\(408\) 0 0
\(409\) 4.70961 0.232875 0.116438 0.993198i \(-0.462852\pi\)
0.116438 + 0.993198i \(0.462852\pi\)
\(410\) −0.245047 −0.0121020
\(411\) 0 0
\(412\) 24.1190 1.18826
\(413\) −5.96367 −0.293453
\(414\) 0 0
\(415\) −1.38698 −0.0680844
\(416\) −8.82898 −0.432876
\(417\) 0 0
\(418\) −28.5654 −1.39718
\(419\) −19.7911 −0.966860 −0.483430 0.875383i \(-0.660609\pi\)
−0.483430 + 0.875383i \(0.660609\pi\)
\(420\) 0 0
\(421\) 28.1857 1.37369 0.686844 0.726805i \(-0.258996\pi\)
0.686844 + 0.726805i \(0.258996\pi\)
\(422\) 38.8680 1.89206
\(423\) 0 0
\(424\) −22.9274 −1.11345
\(425\) −9.93768 −0.482048
\(426\) 0 0
\(427\) 7.65204 0.370308
\(428\) 20.8873 1.00963
\(429\) 0 0
\(430\) 1.54275 0.0743982
\(431\) 5.19681 0.250321 0.125161 0.992136i \(-0.460055\pi\)
0.125161 + 0.992136i \(0.460055\pi\)
\(432\) 0 0
\(433\) 25.3285 1.21721 0.608605 0.793473i \(-0.291730\pi\)
0.608605 + 0.793473i \(0.291730\pi\)
\(434\) −2.30825 −0.110799
\(435\) 0 0
\(436\) −23.4785 −1.12441
\(437\) −17.1230 −0.819102
\(438\) 0 0
\(439\) −15.6612 −0.747470 −0.373735 0.927536i \(-0.621923\pi\)
−0.373735 + 0.927536i \(0.621923\pi\)
\(440\) −1.22236 −0.0582735
\(441\) 0 0
\(442\) 20.0910 0.955631
\(443\) 18.2538 0.867266 0.433633 0.901089i \(-0.357231\pi\)
0.433633 + 0.901089i \(0.357231\pi\)
\(444\) 0 0
\(445\) −0.144570 −0.00685329
\(446\) 51.5323 2.44013
\(447\) 0 0
\(448\) 6.01184 0.284033
\(449\) −28.7216 −1.35546 −0.677729 0.735312i \(-0.737036\pi\)
−0.677729 + 0.735312i \(0.737036\pi\)
\(450\) 0 0
\(451\) 3.39050 0.159652
\(452\) 44.6256 2.09901
\(453\) 0 0
\(454\) 45.9052 2.15444
\(455\) 0.226791 0.0106321
\(456\) 0 0
\(457\) −35.3879 −1.65538 −0.827688 0.561188i \(-0.810344\pi\)
−0.827688 + 0.561188i \(0.810344\pi\)
\(458\) 53.9648 2.52161
\(459\) 0 0
\(460\) −1.56168 −0.0728139
\(461\) −2.27673 −0.106038 −0.0530189 0.998594i \(-0.516884\pi\)
−0.0530189 + 0.998594i \(0.516884\pi\)
\(462\) 0 0
\(463\) 18.3745 0.853937 0.426968 0.904267i \(-0.359582\pi\)
0.426968 + 0.904267i \(0.359582\pi\)
\(464\) −17.0267 −0.790446
\(465\) 0 0
\(466\) 42.4642 1.96712
\(467\) −4.65870 −0.215579 −0.107789 0.994174i \(-0.534377\pi\)
−0.107789 + 0.994174i \(0.534377\pi\)
\(468\) 0 0
\(469\) 5.12440 0.236623
\(470\) 0.803263 0.0370517
\(471\) 0 0
\(472\) 43.6640 2.00980
\(473\) −21.3457 −0.981478
\(474\) 0 0
\(475\) 19.1938 0.880672
\(476\) −4.34989 −0.199377
\(477\) 0 0
\(478\) 37.0338 1.69388
\(479\) 14.1759 0.647713 0.323857 0.946106i \(-0.395020\pi\)
0.323857 + 0.946106i \(0.395020\pi\)
\(480\) 0 0
\(481\) 16.9334 0.772096
\(482\) 31.5848 1.43865
\(483\) 0 0
\(484\) −5.39861 −0.245392
\(485\) −0.492926 −0.0223826
\(486\) 0 0
\(487\) −21.4338 −0.971258 −0.485629 0.874165i \(-0.661409\pi\)
−0.485629 + 0.874165i \(0.661409\pi\)
\(488\) −56.0256 −2.53616
\(489\) 0 0
\(490\) 1.48969 0.0672974
\(491\) 14.0879 0.635776 0.317888 0.948128i \(-0.397026\pi\)
0.317888 + 0.948128i \(0.397026\pi\)
\(492\) 0 0
\(493\) −12.7413 −0.573841
\(494\) −38.8041 −1.74588
\(495\) 0 0
\(496\) 4.40999 0.198015
\(497\) 0.663910 0.0297804
\(498\) 0 0
\(499\) 14.9093 0.667433 0.333717 0.942673i \(-0.391697\pi\)
0.333717 + 0.942673i \(0.391697\pi\)
\(500\) 3.50414 0.156710
\(501\) 0 0
\(502\) −41.8281 −1.86688
\(503\) 15.8631 0.707299 0.353650 0.935378i \(-0.384941\pi\)
0.353650 + 0.935378i \(0.384941\pi\)
\(504\) 0 0
\(505\) −0.676622 −0.0301093
\(506\) 33.0774 1.47047
\(507\) 0 0
\(508\) 43.4402 1.92735
\(509\) −33.9267 −1.50377 −0.751887 0.659292i \(-0.770856\pi\)
−0.751887 + 0.659292i \(0.770856\pi\)
\(510\) 0 0
\(511\) −0.113347 −0.00501416
\(512\) −28.1824 −1.24550
\(513\) 0 0
\(514\) −26.8683 −1.18511
\(515\) −0.595862 −0.0262568
\(516\) 0 0
\(517\) −11.1140 −0.488795
\(518\) −5.61234 −0.246592
\(519\) 0 0
\(520\) −1.66048 −0.0728170
\(521\) 42.7798 1.87422 0.937108 0.349039i \(-0.113492\pi\)
0.937108 + 0.349039i \(0.113492\pi\)
\(522\) 0 0
\(523\) −2.77785 −0.121467 −0.0607335 0.998154i \(-0.519344\pi\)
−0.0607335 + 0.998154i \(0.519344\pi\)
\(524\) 33.9544 1.48331
\(525\) 0 0
\(526\) 49.7397 2.16875
\(527\) 3.30006 0.143753
\(528\) 0 0
\(529\) −3.17243 −0.137932
\(530\) 1.20725 0.0524396
\(531\) 0 0
\(532\) 8.40145 0.364249
\(533\) 4.60575 0.199497
\(534\) 0 0
\(535\) −0.516022 −0.0223096
\(536\) −37.5192 −1.62058
\(537\) 0 0
\(538\) 67.8335 2.92451
\(539\) −20.6116 −0.887803
\(540\) 0 0
\(541\) −3.59390 −0.154514 −0.0772570 0.997011i \(-0.524616\pi\)
−0.0772570 + 0.997011i \(0.524616\pi\)
\(542\) 41.4583 1.78079
\(543\) 0 0
\(544\) −4.18360 −0.179370
\(545\) 0.580037 0.0248461
\(546\) 0 0
\(547\) −39.5858 −1.69257 −0.846284 0.532732i \(-0.821165\pi\)
−0.846284 + 0.532732i \(0.821165\pi\)
\(548\) −43.4057 −1.85420
\(549\) 0 0
\(550\) −37.0777 −1.58100
\(551\) 24.6088 1.04837
\(552\) 0 0
\(553\) 4.17964 0.177736
\(554\) 12.4139 0.527417
\(555\) 0 0
\(556\) 6.41968 0.272255
\(557\) 11.4346 0.484501 0.242250 0.970214i \(-0.422114\pi\)
0.242250 + 0.970214i \(0.422114\pi\)
\(558\) 0 0
\(559\) −28.9967 −1.22643
\(560\) 0.143609 0.00606858
\(561\) 0 0
\(562\) 7.91680 0.333950
\(563\) −14.5108 −0.611557 −0.305778 0.952103i \(-0.598917\pi\)
−0.305778 + 0.952103i \(0.598917\pi\)
\(564\) 0 0
\(565\) −1.10248 −0.0463816
\(566\) −21.9285 −0.921725
\(567\) 0 0
\(568\) −4.86093 −0.203960
\(569\) 1.29918 0.0544646 0.0272323 0.999629i \(-0.491331\pi\)
0.0272323 + 0.999629i \(0.491331\pi\)
\(570\) 0 0
\(571\) 16.0573 0.671978 0.335989 0.941866i \(-0.390929\pi\)
0.335989 + 0.941866i \(0.390929\pi\)
\(572\) 48.9673 2.04743
\(573\) 0 0
\(574\) −1.52651 −0.0637155
\(575\) −22.2255 −0.926867
\(576\) 0 0
\(577\) −8.46033 −0.352208 −0.176104 0.984372i \(-0.556350\pi\)
−0.176104 + 0.984372i \(0.556350\pi\)
\(578\) −31.3075 −1.30222
\(579\) 0 0
\(580\) 2.24442 0.0931947
\(581\) −8.64019 −0.358456
\(582\) 0 0
\(583\) −16.7037 −0.691796
\(584\) 0.829886 0.0343409
\(585\) 0 0
\(586\) 6.78802 0.280411
\(587\) 18.3852 0.758838 0.379419 0.925225i \(-0.376124\pi\)
0.379419 + 0.925225i \(0.376124\pi\)
\(588\) 0 0
\(589\) −6.37379 −0.262627
\(590\) −2.29915 −0.0946544
\(591\) 0 0
\(592\) 10.7226 0.440696
\(593\) −13.5128 −0.554905 −0.277452 0.960739i \(-0.589490\pi\)
−0.277452 + 0.960739i \(0.589490\pi\)
\(594\) 0 0
\(595\) 0.107464 0.00440561
\(596\) 81.5830 3.34177
\(597\) 0 0
\(598\) 44.9332 1.83746
\(599\) −11.0277 −0.450580 −0.225290 0.974292i \(-0.572333\pi\)
−0.225290 + 0.974292i \(0.572333\pi\)
\(600\) 0 0
\(601\) 25.7979 1.05232 0.526160 0.850386i \(-0.323631\pi\)
0.526160 + 0.850386i \(0.323631\pi\)
\(602\) 9.61055 0.391697
\(603\) 0 0
\(604\) −17.8650 −0.726918
\(605\) 0.133373 0.00542239
\(606\) 0 0
\(607\) −14.7899 −0.600302 −0.300151 0.953892i \(-0.597037\pi\)
−0.300151 + 0.953892i \(0.597037\pi\)
\(608\) 8.08027 0.327698
\(609\) 0 0
\(610\) 2.95005 0.119444
\(611\) −15.0976 −0.610785
\(612\) 0 0
\(613\) 36.2739 1.46509 0.732545 0.680719i \(-0.238332\pi\)
0.732545 + 0.680719i \(0.238332\pi\)
\(614\) 15.1169 0.610069
\(615\) 0 0
\(616\) −7.61464 −0.306803
\(617\) 40.3735 1.62538 0.812689 0.582698i \(-0.198003\pi\)
0.812689 + 0.582698i \(0.198003\pi\)
\(618\) 0 0
\(619\) −6.89330 −0.277065 −0.138533 0.990358i \(-0.544239\pi\)
−0.138533 + 0.990358i \(0.544239\pi\)
\(620\) −0.581315 −0.0233462
\(621\) 0 0
\(622\) 17.7069 0.709981
\(623\) −0.900598 −0.0360817
\(624\) 0 0
\(625\) 24.8701 0.994804
\(626\) −10.2567 −0.409941
\(627\) 0 0
\(628\) −0.788242 −0.0314543
\(629\) 8.02386 0.319932
\(630\) 0 0
\(631\) 29.8191 1.18708 0.593539 0.804805i \(-0.297730\pi\)
0.593539 + 0.804805i \(0.297730\pi\)
\(632\) −30.6019 −1.21728
\(633\) 0 0
\(634\) −38.7804 −1.54016
\(635\) −1.07319 −0.0425884
\(636\) 0 0
\(637\) −27.9993 −1.10937
\(638\) −47.5382 −1.88206
\(639\) 0 0
\(640\) 1.92653 0.0761527
\(641\) −42.8807 −1.69369 −0.846844 0.531842i \(-0.821500\pi\)
−0.846844 + 0.531842i \(0.821500\pi\)
\(642\) 0 0
\(643\) −27.4133 −1.08107 −0.540537 0.841320i \(-0.681779\pi\)
−0.540537 + 0.841320i \(0.681779\pi\)
\(644\) −9.72848 −0.383356
\(645\) 0 0
\(646\) −18.3873 −0.723437
\(647\) 16.1623 0.635407 0.317703 0.948190i \(-0.397088\pi\)
0.317703 + 0.948190i \(0.397088\pi\)
\(648\) 0 0
\(649\) 31.8113 1.24870
\(650\) −50.3674 −1.97557
\(651\) 0 0
\(652\) 21.1894 0.829842
\(653\) −32.2463 −1.26189 −0.630947 0.775826i \(-0.717334\pi\)
−0.630947 + 0.775826i \(0.717334\pi\)
\(654\) 0 0
\(655\) −0.838847 −0.0327765
\(656\) 2.91647 0.113869
\(657\) 0 0
\(658\) 5.00391 0.195073
\(659\) −27.8186 −1.08366 −0.541829 0.840489i \(-0.682268\pi\)
−0.541829 + 0.840489i \(0.682268\pi\)
\(660\) 0 0
\(661\) 30.7886 1.19754 0.598769 0.800922i \(-0.295657\pi\)
0.598769 + 0.800922i \(0.295657\pi\)
\(662\) 46.1789 1.79480
\(663\) 0 0
\(664\) 63.2606 2.45499
\(665\) −0.207559 −0.00804877
\(666\) 0 0
\(667\) −28.4958 −1.10336
\(668\) −62.8485 −2.43168
\(669\) 0 0
\(670\) 1.97559 0.0763236
\(671\) −40.8173 −1.57574
\(672\) 0 0
\(673\) −26.1306 −1.00726 −0.503631 0.863919i \(-0.668003\pi\)
−0.503631 + 0.863919i \(0.668003\pi\)
\(674\) −70.7584 −2.72551
\(675\) 0 0
\(676\) 17.5374 0.674515
\(677\) −18.1945 −0.699271 −0.349636 0.936886i \(-0.613695\pi\)
−0.349636 + 0.936886i \(0.613695\pi\)
\(678\) 0 0
\(679\) −3.07068 −0.117842
\(680\) −0.786818 −0.0301731
\(681\) 0 0
\(682\) 12.3126 0.471474
\(683\) −23.4971 −0.899092 −0.449546 0.893257i \(-0.648414\pi\)
−0.449546 + 0.893257i \(0.648414\pi\)
\(684\) 0 0
\(685\) 1.07234 0.0409721
\(686\) 19.0283 0.726502
\(687\) 0 0
\(688\) −18.3613 −0.700019
\(689\) −22.6908 −0.864449
\(690\) 0 0
\(691\) −44.5379 −1.69430 −0.847151 0.531352i \(-0.821684\pi\)
−0.847151 + 0.531352i \(0.821684\pi\)
\(692\) −71.5960 −2.72167
\(693\) 0 0
\(694\) −27.3714 −1.03900
\(695\) −0.158599 −0.00601599
\(696\) 0 0
\(697\) 2.18243 0.0826655
\(698\) 67.6335 2.55997
\(699\) 0 0
\(700\) 10.9050 0.412171
\(701\) 25.2567 0.953934 0.476967 0.878921i \(-0.341736\pi\)
0.476967 + 0.878921i \(0.341736\pi\)
\(702\) 0 0
\(703\) −15.4974 −0.584496
\(704\) −32.0682 −1.20862
\(705\) 0 0
\(706\) −68.8163 −2.58994
\(707\) −4.21500 −0.158521
\(708\) 0 0
\(709\) −15.6840 −0.589026 −0.294513 0.955648i \(-0.595157\pi\)
−0.294513 + 0.955648i \(0.595157\pi\)
\(710\) 0.255954 0.00960579
\(711\) 0 0
\(712\) 6.59387 0.247116
\(713\) 7.38054 0.276403
\(714\) 0 0
\(715\) −1.20974 −0.0452417
\(716\) −61.1708 −2.28606
\(717\) 0 0
\(718\) 74.5277 2.78135
\(719\) 53.1607 1.98256 0.991280 0.131771i \(-0.0420664\pi\)
0.991280 + 0.131771i \(0.0420664\pi\)
\(720\) 0 0
\(721\) −3.71191 −0.138239
\(722\) −10.1174 −0.376529
\(723\) 0 0
\(724\) −11.2716 −0.418907
\(725\) 31.9421 1.18630
\(726\) 0 0
\(727\) 0.469172 0.0174006 0.00870032 0.999962i \(-0.497231\pi\)
0.00870032 + 0.999962i \(0.497231\pi\)
\(728\) −10.3440 −0.383372
\(729\) 0 0
\(730\) −0.0436980 −0.00161734
\(731\) −13.7400 −0.508194
\(732\) 0 0
\(733\) −46.4063 −1.71406 −0.857028 0.515271i \(-0.827691\pi\)
−0.857028 + 0.515271i \(0.827691\pi\)
\(734\) 57.9372 2.13850
\(735\) 0 0
\(736\) −9.35657 −0.344888
\(737\) −27.3345 −1.00688
\(738\) 0 0
\(739\) 25.8093 0.949411 0.474706 0.880145i \(-0.342555\pi\)
0.474706 + 0.880145i \(0.342555\pi\)
\(740\) −1.41343 −0.0519586
\(741\) 0 0
\(742\) 7.52055 0.276088
\(743\) −34.7096 −1.27337 −0.636687 0.771123i \(-0.719695\pi\)
−0.636687 + 0.771123i \(0.719695\pi\)
\(744\) 0 0
\(745\) −2.01551 −0.0738427
\(746\) −30.2259 −1.10665
\(747\) 0 0
\(748\) 23.2031 0.848389
\(749\) −3.21455 −0.117457
\(750\) 0 0
\(751\) 23.9858 0.875256 0.437628 0.899156i \(-0.355819\pi\)
0.437628 + 0.899156i \(0.355819\pi\)
\(752\) −9.56016 −0.348623
\(753\) 0 0
\(754\) −64.5773 −2.35177
\(755\) 0.441357 0.0160626
\(756\) 0 0
\(757\) −8.78780 −0.319398 −0.159699 0.987166i \(-0.551052\pi\)
−0.159699 + 0.987166i \(0.551052\pi\)
\(758\) −18.5050 −0.672132
\(759\) 0 0
\(760\) 1.51967 0.0551243
\(761\) −13.7539 −0.498578 −0.249289 0.968429i \(-0.580197\pi\)
−0.249289 + 0.968429i \(0.580197\pi\)
\(762\) 0 0
\(763\) 3.61333 0.130811
\(764\) 8.49190 0.307226
\(765\) 0 0
\(766\) 42.8969 1.54993
\(767\) 43.2134 1.56034
\(768\) 0 0
\(769\) −31.3579 −1.13080 −0.565398 0.824818i \(-0.691277\pi\)
−0.565398 + 0.824818i \(0.691277\pi\)
\(770\) 0.400952 0.0144493
\(771\) 0 0
\(772\) −3.31668 −0.119370
\(773\) −28.1214 −1.01146 −0.505729 0.862693i \(-0.668776\pi\)
−0.505729 + 0.862693i \(0.668776\pi\)
\(774\) 0 0
\(775\) −8.27314 −0.297180
\(776\) 22.4824 0.807073
\(777\) 0 0
\(778\) −65.7767 −2.35821
\(779\) −4.21518 −0.151025
\(780\) 0 0
\(781\) −3.54142 −0.126722
\(782\) 21.2916 0.761385
\(783\) 0 0
\(784\) −17.7298 −0.633207
\(785\) 0.0194736 0.000695042 0
\(786\) 0 0
\(787\) −36.2002 −1.29040 −0.645198 0.764015i \(-0.723225\pi\)
−0.645198 + 0.764015i \(0.723225\pi\)
\(788\) −76.3977 −2.72155
\(789\) 0 0
\(790\) 1.61136 0.0573294
\(791\) −6.86787 −0.244193
\(792\) 0 0
\(793\) −55.4474 −1.96900
\(794\) 10.1252 0.359330
\(795\) 0 0
\(796\) −71.6235 −2.53863
\(797\) 29.6001 1.04849 0.524245 0.851568i \(-0.324348\pi\)
0.524245 + 0.851568i \(0.324348\pi\)
\(798\) 0 0
\(799\) −7.15400 −0.253090
\(800\) 10.4881 0.370812
\(801\) 0 0
\(802\) −36.4386 −1.28669
\(803\) 0.604612 0.0213363
\(804\) 0 0
\(805\) 0.240343 0.00847097
\(806\) 16.7258 0.589141
\(807\) 0 0
\(808\) 30.8608 1.08568
\(809\) 5.75943 0.202491 0.101245 0.994861i \(-0.467717\pi\)
0.101245 + 0.994861i \(0.467717\pi\)
\(810\) 0 0
\(811\) 12.4896 0.438569 0.219284 0.975661i \(-0.429628\pi\)
0.219284 + 0.975661i \(0.429628\pi\)
\(812\) 13.9816 0.490658
\(813\) 0 0
\(814\) 29.9372 1.04930
\(815\) −0.523487 −0.0183369
\(816\) 0 0
\(817\) 26.5377 0.928438
\(818\) 11.3107 0.395470
\(819\) 0 0
\(820\) −0.384442 −0.0134253
\(821\) −43.0668 −1.50304 −0.751520 0.659710i \(-0.770679\pi\)
−0.751520 + 0.659710i \(0.770679\pi\)
\(822\) 0 0
\(823\) 10.2841 0.358481 0.179240 0.983805i \(-0.442636\pi\)
0.179240 + 0.983805i \(0.442636\pi\)
\(824\) 27.1773 0.946767
\(825\) 0 0
\(826\) −14.3225 −0.498343
\(827\) −6.09463 −0.211931 −0.105965 0.994370i \(-0.533793\pi\)
−0.105965 + 0.994370i \(0.533793\pi\)
\(828\) 0 0
\(829\) −33.6979 −1.17038 −0.585188 0.810898i \(-0.698979\pi\)
−0.585188 + 0.810898i \(0.698979\pi\)
\(830\) −3.33101 −0.115621
\(831\) 0 0
\(832\) −43.5624 −1.51025
\(833\) −13.2675 −0.459690
\(834\) 0 0
\(835\) 1.55268 0.0537326
\(836\) −44.8148 −1.54995
\(837\) 0 0
\(838\) −47.5308 −1.64192
\(839\) 42.2262 1.45781 0.728905 0.684615i \(-0.240029\pi\)
0.728905 + 0.684615i \(0.240029\pi\)
\(840\) 0 0
\(841\) 11.9537 0.412197
\(842\) 67.6914 2.33280
\(843\) 0 0
\(844\) 60.9781 2.09895
\(845\) −0.433262 −0.0149047
\(846\) 0 0
\(847\) 0.830846 0.0285482
\(848\) −14.3683 −0.493409
\(849\) 0 0
\(850\) −23.8665 −0.818616
\(851\) 17.9453 0.615156
\(852\) 0 0
\(853\) −35.7296 −1.22336 −0.611679 0.791106i \(-0.709506\pi\)
−0.611679 + 0.791106i \(0.709506\pi\)
\(854\) 18.3773 0.628858
\(855\) 0 0
\(856\) 23.5358 0.804439
\(857\) −7.90865 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(858\) 0 0
\(859\) −44.7191 −1.52580 −0.762899 0.646518i \(-0.776225\pi\)
−0.762899 + 0.646518i \(0.776225\pi\)
\(860\) 2.42035 0.0825332
\(861\) 0 0
\(862\) 12.4808 0.425096
\(863\) 22.9170 0.780103 0.390052 0.920793i \(-0.372457\pi\)
0.390052 + 0.920793i \(0.372457\pi\)
\(864\) 0 0
\(865\) 1.76878 0.0601405
\(866\) 60.8294 2.06707
\(867\) 0 0
\(868\) −3.62129 −0.122915
\(869\) −22.2949 −0.756304
\(870\) 0 0
\(871\) −37.1319 −1.25817
\(872\) −26.4556 −0.895899
\(873\) 0 0
\(874\) −41.1229 −1.39100
\(875\) −0.539287 −0.0182312
\(876\) 0 0
\(877\) −22.1376 −0.747535 −0.373767 0.927522i \(-0.621934\pi\)
−0.373767 + 0.927522i \(0.621934\pi\)
\(878\) −37.6124 −1.26936
\(879\) 0 0
\(880\) −0.766035 −0.0258230
\(881\) −9.86404 −0.332328 −0.166164 0.986098i \(-0.553138\pi\)
−0.166164 + 0.986098i \(0.553138\pi\)
\(882\) 0 0
\(883\) 47.5731 1.60096 0.800481 0.599358i \(-0.204578\pi\)
0.800481 + 0.599358i \(0.204578\pi\)
\(884\) 31.5197 1.06012
\(885\) 0 0
\(886\) 43.8388 1.47279
\(887\) −13.3573 −0.448494 −0.224247 0.974532i \(-0.571992\pi\)
−0.224247 + 0.974532i \(0.571992\pi\)
\(888\) 0 0
\(889\) −6.68544 −0.224222
\(890\) −0.347203 −0.0116383
\(891\) 0 0
\(892\) 80.8465 2.70694
\(893\) 13.8174 0.462380
\(894\) 0 0
\(895\) 1.51123 0.0505149
\(896\) 12.0013 0.400934
\(897\) 0 0
\(898\) −68.9785 −2.30184
\(899\) −10.6072 −0.353769
\(900\) 0 0
\(901\) −10.7520 −0.358201
\(902\) 8.14270 0.271122
\(903\) 0 0
\(904\) 50.2842 1.67243
\(905\) 0.278467 0.00925654
\(906\) 0 0
\(907\) 36.9606 1.22726 0.613629 0.789595i \(-0.289709\pi\)
0.613629 + 0.789595i \(0.289709\pi\)
\(908\) 72.0184 2.39001
\(909\) 0 0
\(910\) 0.544665 0.0180555
\(911\) 48.4111 1.60393 0.801966 0.597370i \(-0.203788\pi\)
0.801966 + 0.597370i \(0.203788\pi\)
\(912\) 0 0
\(913\) 46.0883 1.52530
\(914\) −84.9884 −2.81117
\(915\) 0 0
\(916\) 84.6627 2.79733
\(917\) −5.22558 −0.172564
\(918\) 0 0
\(919\) 8.93459 0.294725 0.147363 0.989083i \(-0.452922\pi\)
0.147363 + 0.989083i \(0.452922\pi\)
\(920\) −1.75971 −0.0580159
\(921\) 0 0
\(922\) −5.46784 −0.180074
\(923\) −4.81076 −0.158348
\(924\) 0 0
\(925\) −20.1156 −0.661395
\(926\) 44.1287 1.45016
\(927\) 0 0
\(928\) 13.4471 0.441423
\(929\) 6.24415 0.204864 0.102432 0.994740i \(-0.467338\pi\)
0.102432 + 0.994740i \(0.467338\pi\)
\(930\) 0 0
\(931\) 25.6250 0.839825
\(932\) 66.6200 2.18221
\(933\) 0 0
\(934\) −11.1884 −0.366097
\(935\) −0.573234 −0.0187468
\(936\) 0 0
\(937\) 45.8424 1.49760 0.748802 0.662794i \(-0.230629\pi\)
0.748802 + 0.662794i \(0.230629\pi\)
\(938\) 12.3069 0.401834
\(939\) 0 0
\(940\) 1.26020 0.0411032
\(941\) 4.35263 0.141892 0.0709459 0.997480i \(-0.477398\pi\)
0.0709459 + 0.997480i \(0.477398\pi\)
\(942\) 0 0
\(943\) 4.88098 0.158947
\(944\) 27.3637 0.890612
\(945\) 0 0
\(946\) −51.2644 −1.66675
\(947\) −2.01674 −0.0655351 −0.0327675 0.999463i \(-0.510432\pi\)
−0.0327675 + 0.999463i \(0.510432\pi\)
\(948\) 0 0
\(949\) 0.821322 0.0266612
\(950\) 46.0962 1.49556
\(951\) 0 0
\(952\) −4.90147 −0.158857
\(953\) 35.7287 1.15737 0.578684 0.815552i \(-0.303567\pi\)
0.578684 + 0.815552i \(0.303567\pi\)
\(954\) 0 0
\(955\) −0.209793 −0.00678874
\(956\) 58.1004 1.87910
\(957\) 0 0
\(958\) 34.0451 1.09995
\(959\) 6.68014 0.215713
\(960\) 0 0
\(961\) −28.2527 −0.911377
\(962\) 40.6676 1.31117
\(963\) 0 0
\(964\) 49.5518 1.59596
\(965\) 0.0819388 0.00263770
\(966\) 0 0
\(967\) −0.693317 −0.0222956 −0.0111478 0.999938i \(-0.503549\pi\)
−0.0111478 + 0.999938i \(0.503549\pi\)
\(968\) −6.08317 −0.195520
\(969\) 0 0
\(970\) −1.18382 −0.0380103
\(971\) −47.4942 −1.52416 −0.762081 0.647482i \(-0.775822\pi\)
−0.762081 + 0.647482i \(0.775822\pi\)
\(972\) 0 0
\(973\) −0.987988 −0.0316734
\(974\) −51.4759 −1.64939
\(975\) 0 0
\(976\) −35.1105 −1.12386
\(977\) 12.6033 0.403216 0.201608 0.979466i \(-0.435383\pi\)
0.201608 + 0.979466i \(0.435383\pi\)
\(978\) 0 0
\(979\) 4.80395 0.153535
\(980\) 2.33710 0.0746560
\(981\) 0 0
\(982\) 33.8337 1.07968
\(983\) −11.5450 −0.368229 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(984\) 0 0
\(985\) 1.88741 0.0601379
\(986\) −30.5999 −0.974498
\(987\) 0 0
\(988\) −60.8778 −1.93678
\(989\) −30.7294 −0.977139
\(990\) 0 0
\(991\) 18.6935 0.593819 0.296910 0.954906i \(-0.404044\pi\)
0.296910 + 0.954906i \(0.404044\pi\)
\(992\) −3.48286 −0.110581
\(993\) 0 0
\(994\) 1.59446 0.0505732
\(995\) 1.76946 0.0560958
\(996\) 0 0
\(997\) 3.35880 0.106374 0.0531872 0.998585i \(-0.483062\pi\)
0.0531872 + 0.998585i \(0.483062\pi\)
\(998\) 35.8066 1.13344
\(999\) 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.d.1.6 6
3.2 odd 2 729.2.a.a.1.1 6
9.2 odd 6 729.2.c.e.244.6 12
9.4 even 3 729.2.c.b.487.1 12
9.5 odd 6 729.2.c.e.487.6 12
9.7 even 3 729.2.c.b.244.1 12
27.2 odd 18 243.2.e.d.190.2 12
27.4 even 9 243.2.e.b.136.2 12
27.5 odd 18 27.2.e.a.25.1 yes 12
27.7 even 9 243.2.e.b.109.2 12
27.11 odd 18 27.2.e.a.13.1 12
27.13 even 9 243.2.e.a.55.1 12
27.14 odd 18 243.2.e.d.55.2 12
27.16 even 9 81.2.e.a.10.2 12
27.20 odd 18 243.2.e.c.109.1 12
27.22 even 9 81.2.e.a.73.2 12
27.23 odd 18 243.2.e.c.136.1 12
27.25 even 9 243.2.e.a.190.1 12
108.11 even 18 432.2.u.c.337.1 12
108.59 even 18 432.2.u.c.241.1 12
135.32 even 36 675.2.u.b.349.1 24
135.38 even 36 675.2.u.b.499.1 24
135.59 odd 18 675.2.l.c.376.2 12
135.92 even 36 675.2.u.b.499.4 24
135.113 even 36 675.2.u.b.349.4 24
135.119 odd 18 675.2.l.c.526.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.13.1 12 27.11 odd 18
27.2.e.a.25.1 yes 12 27.5 odd 18
81.2.e.a.10.2 12 27.16 even 9
81.2.e.a.73.2 12 27.22 even 9
243.2.e.a.55.1 12 27.13 even 9
243.2.e.a.190.1 12 27.25 even 9
243.2.e.b.109.2 12 27.7 even 9
243.2.e.b.136.2 12 27.4 even 9
243.2.e.c.109.1 12 27.20 odd 18
243.2.e.c.136.1 12 27.23 odd 18
243.2.e.d.55.2 12 27.14 odd 18
243.2.e.d.190.2 12 27.2 odd 18
432.2.u.c.241.1 12 108.59 even 18
432.2.u.c.337.1 12 108.11 even 18
675.2.l.c.376.2 12 135.59 odd 18
675.2.l.c.526.2 12 135.119 odd 18
675.2.u.b.349.1 24 135.32 even 36
675.2.u.b.349.4 24 135.113 even 36
675.2.u.b.499.1 24 135.38 even 36
675.2.u.b.499.4 24 135.92 even 36
729.2.a.a.1.1 6 3.2 odd 2
729.2.a.d.1.6 6 1.1 even 1 trivial
729.2.c.b.244.1 12 9.7 even 3
729.2.c.b.487.1 12 9.4 even 3
729.2.c.e.244.6 12 9.2 odd 6
729.2.c.e.487.6 12 9.5 odd 6