Properties

Label 729.2.a.a.1.1
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
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: \(1\)
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 1.1
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 \(-0.822856\pi\)
−0.849101 + 0.528230i \(0.822856\pi\)
\(3\) 0 0
\(4\) 3.76778 1.88389
\(5\) 0.0930834 0.0416282 0.0208141 0.999783i \(-0.493374\pi\)
0.0208141 + 0.999783i \(0.493374\pi\)
\(6\) 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 \(-0.654413\pi\)
−0.466300 + 0.884627i \(0.654413\pi\)
\(12\) 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 \(-0.577621\pi\)
−0.241443 + 0.970415i \(0.577621\pi\)
\(18\) 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 \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(24\) 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 \(-0.297477\pi\)
0.594179 + 0.804332i \(0.297477\pi\)
\(30\) 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 \(-0.527279\pi\)
−0.0855954 + 0.996330i \(0.527279\pi\)
\(42\) 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 \(-0.415598\pi\)
0.262061 + 0.965051i \(0.415598\pi\)
\(48\) 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 \(-0.379050\pi\)
0.370897 + 0.928674i \(0.379050\pi\)
\(54\) 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 \(-0.733480\pi\)
−0.669474 + 0.742835i \(0.733480\pi\)
\(60\) 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 \(-0.478357\pi\)
0.0679401 + 0.997689i \(0.478357\pi\)
\(72\) 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 \(-0.804789\pi\)
−0.817768 + 0.575548i \(0.804789\pi\)
\(84\) 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 \(-0.526232\pi\)
−0.0823155 + 0.996606i \(0.526232\pi\)
\(90\) 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 \(-0.617785\pi\)
−0.361645 + 0.932316i \(0.617785\pi\)
\(102\) 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 \(-0.586350\pi\)
−0.267963 + 0.963429i \(0.586350\pi\)
\(108\) 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 \(-0.688084\pi\)
−0.557094 + 0.830449i \(0.688084\pi\)
\(114\) 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 \(-0.628799\pi\)
−0.393681 + 0.919247i \(0.628799\pi\)
\(132\) 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 \(-0.336222\pi\)
0.492120 + 0.870527i \(0.336222\pi\)
\(138\) 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 \(-0.847168\pi\)
−0.886932 + 0.461900i \(0.847168\pi\)
\(150\) 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 \(-0.276696\pi\)
0.645388 + 0.763855i \(0.276696\pi\)
\(168\) 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 \(-0.243061\pi\)
0.722353 + 0.691524i \(0.243061\pi\)
\(174\) 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 \(-0.292477\pi\)
0.606739 + 0.794901i \(0.292477\pi\)
\(180\) 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 \(-0.525984\pi\)
−0.0815402 + 0.996670i \(0.525984\pi\)
\(192\) 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 \(-0.243075\pi\)
0.722322 + 0.691557i \(0.243075\pi\)
\(198\) 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 \(-0.718723\pi\)
−0.634329 + 0.773064i \(0.718723\pi\)
\(228\) 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 \(-0.696626\pi\)
−0.579176 + 0.815203i \(0.696626\pi\)
\(234\) 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 \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(240\) 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 \(-0.314756\pi\)
0.549663 + 0.835386i \(0.314756\pi\)
\(252\) 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 \(-0.386545\pi\)
0.348931 + 0.937148i \(0.386545\pi\)
\(258\) 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 \(-0.720463\pi\)
−0.638544 + 0.769585i \(0.720463\pi\)
\(264\) 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 \(-0.830199\pi\)
−0.861060 + 0.508504i \(0.830199\pi\)
\(270\) 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 \(-0.531348\pi\)
−0.0983246 + 0.995154i \(0.531348\pi\)
\(282\) 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 \(-0.526310\pi\)
−0.0825610 + 0.996586i \(0.526310\pi\)
\(294\) 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 \(-0.567034\pi\)
−0.209039 + 0.977907i \(0.567034\pi\)
\(312\) 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 \(-0.350186\pi\)
0.453469 + 0.891272i \(0.350186\pi\)
\(318\) 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 \(-0.401039\pi\)
0.305912 + 0.952060i \(0.401039\pi\)
\(348\) 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 \(-0.223946\pi\)
0.762552 + 0.646927i \(0.223946\pi\)
\(354\) 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 \(-0.805421\pi\)
−0.818909 + 0.573923i \(0.805421\pi\)
\(360\) 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 \(-0.650841\pi\)
−0.456343 + 0.889804i \(0.650841\pi\)
\(384\) 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 \(-0.255703\pi\)
0.694325 + 0.719662i \(0.255703\pi\)
\(390\) 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 \(-0.376323\pi\)
0.378839 + 0.925462i \(0.376323\pi\)
\(402\) 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 \(-0.339391\pi\)
0.483430 + 0.875383i \(0.339391\pi\)
\(420\) 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 \(-0.539945\pi\)
−0.125161 + 0.992136i \(0.539945\pi\)
\(432\) 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 \(-0.642769\pi\)
−0.433633 + 0.901089i \(0.642769\pi\)
\(444\) 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 \(-0.262964\pi\)
0.677729 + 0.735312i \(0.262964\pi\)
\(450\) 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 \(-0.483116\pi\)
0.0530189 + 0.998594i \(0.483116\pi\)
\(462\) 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 \(-0.465623\pi\)
0.107789 + 0.994174i \(0.465623\pi\)
\(468\) 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 \(-0.604980\pi\)
−0.323857 + 0.946106i \(0.604980\pi\)
\(480\) 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 \(-0.602974\pi\)
−0.317888 + 0.948128i \(0.602974\pi\)
\(492\) 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 \(-0.615059\pi\)
−0.353650 + 0.935378i \(0.615059\pi\)
\(504\) 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 \(-0.229144\pi\)
0.751887 + 0.659292i \(0.229144\pi\)
\(510\) 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 \(-0.886508\pi\)
−0.937108 + 0.349039i \(0.886508\pi\)
\(522\) 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 \(-0.577886\pi\)
−0.242250 + 0.970214i \(0.577886\pi\)
\(558\) 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 \(-0.401083\pi\)
0.305778 + 0.952103i \(0.401083\pi\)
\(564\) 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 \(-0.508669\pi\)
−0.0272323 + 0.999629i \(0.508669\pi\)
\(570\) 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 \(-0.623876\pi\)
−0.379419 + 0.925225i \(0.623876\pi\)
\(588\) 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 \(-0.410510\pi\)
0.277452 + 0.960739i \(0.410510\pi\)
\(594\) 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 \(-0.427667\pi\)
0.225290 + 0.974292i \(0.427667\pi\)
\(600\) 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 \(-0.801997\pi\)
−0.812689 + 0.582698i \(0.801997\pi\)
\(618\) 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 \(-0.178500\pi\)
0.846844 + 0.531842i \(0.178500\pi\)
\(642\) 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 \(-0.602912\pi\)
−0.317703 + 0.948190i \(0.602912\pi\)
\(648\) 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 \(-0.282666\pi\)
0.630947 + 0.775826i \(0.282666\pi\)
\(654\) 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 \(-0.317732\pi\)
0.541829 + 0.840489i \(0.317732\pi\)
\(660\) 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 \(-0.386305\pi\)
0.349636 + 0.936886i \(0.386305\pi\)
\(678\) 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 \(-0.351586\pi\)
0.449546 + 0.893257i \(0.351586\pi\)
\(684\) 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 \(-0.658264\pi\)
−0.476967 + 0.878921i \(0.658264\pi\)
\(702\) 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 \(-0.957934\pi\)
−0.991280 + 0.131771i \(0.957934\pi\)
\(720\) 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 \(-0.280305\pi\)
0.636687 + 0.771123i \(0.280305\pi\)
\(744\) 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 \(-0.419803\pi\)
0.249289 + 0.968429i \(0.419803\pi\)
\(762\) 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 \(-0.331224\pi\)
0.505729 + 0.862693i \(0.331224\pi\)
\(774\) 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 \(-0.675652\pi\)
−0.524245 + 0.851568i \(0.675652\pi\)
\(798\) 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 \(-0.532283\pi\)
−0.101245 + 0.994861i \(0.532283\pi\)
\(810\) 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 \(-0.229321\pi\)
0.751520 + 0.659710i \(0.229321\pi\)
\(822\) 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 \(-0.466207\pi\)
0.105965 + 0.994370i \(0.466207\pi\)
\(828\) 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 \(-0.759971\pi\)
−0.728905 + 0.684615i \(0.759971\pi\)
\(840\) 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 \(-0.456872\pi\)
0.135077 + 0.990835i \(0.456872\pi\)
\(858\) 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 \(-0.627543\pi\)
−0.390052 + 0.920793i \(0.627543\pi\)
\(864\) 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 \(-0.446862\pi\)
0.166164 + 0.986098i \(0.446862\pi\)
\(882\) 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 \(-0.428008\pi\)
0.224247 + 0.974532i \(0.428008\pi\)
\(888\) 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 \(-0.796212\pi\)
−0.801966 + 0.597370i \(0.796212\pi\)
\(912\) 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 \(-0.532662\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(930\) 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 \(-0.522602\pi\)
−0.0709459 + 0.997480i \(0.522602\pi\)
\(942\) 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 \(-0.489568\pi\)
0.0327675 + 0.999463i \(0.489568\pi\)
\(948\) 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 \(-0.696433\pi\)
−0.578684 + 0.815552i \(0.696433\pi\)
\(954\) 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 \(-0.224178\pi\)
0.762081 + 0.647482i \(0.224178\pi\)
\(972\) 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 \(-0.564617\pi\)
−0.201608 + 0.979466i \(0.564617\pi\)
\(978\) 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 \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(984\) 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.a.1.1 6
3.2 odd 2 729.2.a.d.1.6 6
9.2 odd 6 729.2.c.b.244.1 12
9.4 even 3 729.2.c.e.487.6 12
9.5 odd 6 729.2.c.b.487.1 12
9.7 even 3 729.2.c.e.244.6 12
27.2 odd 18 243.2.e.a.190.1 12
27.4 even 9 243.2.e.c.136.1 12
27.5 odd 18 81.2.e.a.73.2 12
27.7 even 9 243.2.e.c.109.1 12
27.11 odd 18 81.2.e.a.10.2 12
27.13 even 9 243.2.e.d.55.2 12
27.14 odd 18 243.2.e.a.55.1 12
27.16 even 9 27.2.e.a.13.1 12
27.20 odd 18 243.2.e.b.109.2 12
27.22 even 9 27.2.e.a.25.1 yes 12
27.23 odd 18 243.2.e.b.136.2 12
27.25 even 9 243.2.e.d.190.2 12
108.43 odd 18 432.2.u.c.337.1 12
108.103 odd 18 432.2.u.c.241.1 12
135.22 odd 36 675.2.u.b.349.1 24
135.43 odd 36 675.2.u.b.499.1 24
135.49 even 18 675.2.l.c.376.2 12
135.97 odd 36 675.2.u.b.499.4 24
135.103 odd 36 675.2.u.b.349.4 24
135.124 even 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 27.16 even 9
27.2.e.a.25.1 yes 12 27.22 even 9
81.2.e.a.10.2 12 27.11 odd 18
81.2.e.a.73.2 12 27.5 odd 18
243.2.e.a.55.1 12 27.14 odd 18
243.2.e.a.190.1 12 27.2 odd 18
243.2.e.b.109.2 12 27.20 odd 18
243.2.e.b.136.2 12 27.23 odd 18
243.2.e.c.109.1 12 27.7 even 9
243.2.e.c.136.1 12 27.4 even 9
243.2.e.d.55.2 12 27.13 even 9
243.2.e.d.190.2 12 27.25 even 9
432.2.u.c.241.1 12 108.103 odd 18
432.2.u.c.337.1 12 108.43 odd 18
675.2.l.c.376.2 12 135.49 even 18
675.2.l.c.526.2 12 135.124 even 18
675.2.u.b.349.1 24 135.22 odd 36
675.2.u.b.349.4 24 135.103 odd 36
675.2.u.b.499.1 24 135.43 odd 36
675.2.u.b.499.4 24 135.97 odd 36
729.2.a.a.1.1 6 1.1 even 1 trivial
729.2.a.d.1.6 6 3.2 odd 2
729.2.c.b.244.1 12 9.2 odd 6
729.2.c.b.487.1 12 9.5 odd 6
729.2.c.e.244.6 12 9.7 even 3
729.2.c.e.487.6 12 9.4 even 3