Properties

Label 2205.4.a.bz.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.241849\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +O(q^{10})\) \(q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +8.28031 q^{10} -69.5726 q^{11} -68.4326 q^{13} +5.70053 q^{16} +104.332 q^{17} -71.8929 q^{19} +26.2873 q^{20} +115.217 q^{22} +101.031 q^{23} +25.0000 q^{25} +113.329 q^{26} +114.661 q^{29} +73.6505 q^{31} -185.082 q^{32} -172.780 q^{34} -200.933 q^{37} +119.059 q^{38} -109.776 q^{40} +417.308 q^{41} +311.175 q^{43} +365.775 q^{44} -167.313 q^{46} +149.697 q^{47} -41.4016 q^{50} +359.781 q^{52} -271.474 q^{53} +347.863 q^{55} -189.885 q^{58} -518.028 q^{59} +219.926 q^{61} -121.970 q^{62} +260.903 q^{64} +342.163 q^{65} +80.6950 q^{67} -548.520 q^{68} +91.0463 q^{71} -882.282 q^{73} +332.758 q^{74} +377.974 q^{76} +599.877 q^{79} -28.5026 q^{80} -691.087 q^{82} +70.8820 q^{83} -521.659 q^{85} -515.325 q^{86} -1527.48 q^{88} +802.592 q^{89} -531.165 q^{92} -247.908 q^{94} +359.465 q^{95} -145.648 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.65606 −0.585506 −0.292753 0.956188i \(-0.594571\pi\)
−0.292753 + 0.956188i \(0.594571\pi\)
\(3\) 0 0
\(4\) −5.25746 −0.657182
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 21.9552 0.970291
\(9\) 0 0
\(10\) 8.28031 0.261846
\(11\) −69.5726 −1.90699 −0.953497 0.301402i \(-0.902545\pi\)
−0.953497 + 0.301402i \(0.902545\pi\)
\(12\) 0 0
\(13\) −68.4326 −1.45998 −0.729991 0.683456i \(-0.760476\pi\)
−0.729991 + 0.683456i \(0.760476\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.70053 0.0890707
\(17\) 104.332 1.48848 0.744240 0.667912i \(-0.232812\pi\)
0.744240 + 0.667912i \(0.232812\pi\)
\(18\) 0 0
\(19\) −71.8929 −0.868072 −0.434036 0.900896i \(-0.642911\pi\)
−0.434036 + 0.900896i \(0.642911\pi\)
\(20\) 26.2873 0.293901
\(21\) 0 0
\(22\) 115.217 1.11656
\(23\) 101.031 0.915929 0.457964 0.888970i \(-0.348579\pi\)
0.457964 + 0.888970i \(0.348579\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 113.329 0.854829
\(27\) 0 0
\(28\) 0 0
\(29\) 114.661 0.734205 0.367102 0.930181i \(-0.380350\pi\)
0.367102 + 0.930181i \(0.380350\pi\)
\(30\) 0 0
\(31\) 73.6505 0.426710 0.213355 0.976975i \(-0.431561\pi\)
0.213355 + 0.976975i \(0.431561\pi\)
\(32\) −185.082 −1.02244
\(33\) 0 0
\(34\) −172.780 −0.871514
\(35\) 0 0
\(36\) 0 0
\(37\) −200.933 −0.892790 −0.446395 0.894836i \(-0.647292\pi\)
−0.446395 + 0.894836i \(0.647292\pi\)
\(38\) 119.059 0.508262
\(39\) 0 0
\(40\) −109.776 −0.433927
\(41\) 417.308 1.58957 0.794786 0.606889i \(-0.207583\pi\)
0.794786 + 0.606889i \(0.207583\pi\)
\(42\) 0 0
\(43\) 311.175 1.10357 0.551787 0.833985i \(-0.313946\pi\)
0.551787 + 0.833985i \(0.313946\pi\)
\(44\) 365.775 1.25324
\(45\) 0 0
\(46\) −167.313 −0.536282
\(47\) 149.697 0.464586 0.232293 0.972646i \(-0.425377\pi\)
0.232293 + 0.972646i \(0.425377\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −41.4016 −0.117101
\(51\) 0 0
\(52\) 359.781 0.959475
\(53\) −271.474 −0.703582 −0.351791 0.936078i \(-0.614427\pi\)
−0.351791 + 0.936078i \(0.614427\pi\)
\(54\) 0 0
\(55\) 347.863 0.852834
\(56\) 0 0
\(57\) 0 0
\(58\) −189.885 −0.429882
\(59\) −518.028 −1.14308 −0.571538 0.820575i \(-0.693653\pi\)
−0.571538 + 0.820575i \(0.693653\pi\)
\(60\) 0 0
\(61\) 219.926 0.461616 0.230808 0.972999i \(-0.425863\pi\)
0.230808 + 0.972999i \(0.425863\pi\)
\(62\) −121.970 −0.249842
\(63\) 0 0
\(64\) 260.903 0.509576
\(65\) 342.163 0.652924
\(66\) 0 0
\(67\) 80.6950 0.147141 0.0735706 0.997290i \(-0.476561\pi\)
0.0735706 + 0.997290i \(0.476561\pi\)
\(68\) −548.520 −0.978202
\(69\) 0 0
\(70\) 0 0
\(71\) 91.0463 0.152186 0.0760930 0.997101i \(-0.475755\pi\)
0.0760930 + 0.997101i \(0.475755\pi\)
\(72\) 0 0
\(73\) −882.282 −1.41457 −0.707283 0.706931i \(-0.750079\pi\)
−0.707283 + 0.706931i \(0.750079\pi\)
\(74\) 332.758 0.522734
\(75\) 0 0
\(76\) 377.974 0.570481
\(77\) 0 0
\(78\) 0 0
\(79\) 599.877 0.854322 0.427161 0.904175i \(-0.359514\pi\)
0.427161 + 0.904175i \(0.359514\pi\)
\(80\) −28.5026 −0.0398336
\(81\) 0 0
\(82\) −691.087 −0.930705
\(83\) 70.8820 0.0937387 0.0468694 0.998901i \(-0.485076\pi\)
0.0468694 + 0.998901i \(0.485076\pi\)
\(84\) 0 0
\(85\) −521.659 −0.665668
\(86\) −515.325 −0.646150
\(87\) 0 0
\(88\) −1527.48 −1.85034
\(89\) 802.592 0.955894 0.477947 0.878389i \(-0.341381\pi\)
0.477947 + 0.878389i \(0.341381\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −531.165 −0.601932
\(93\) 0 0
\(94\) −247.908 −0.272018
\(95\) 359.465 0.388214
\(96\) 0 0
\(97\) −145.648 −0.152457 −0.0762283 0.997090i \(-0.524288\pi\)
−0.0762283 + 0.997090i \(0.524288\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −131.436 −0.131436
\(101\) −619.435 −0.610259 −0.305129 0.952311i \(-0.598700\pi\)
−0.305129 + 0.952311i \(0.598700\pi\)
\(102\) 0 0
\(103\) 1822.08 1.74306 0.871528 0.490345i \(-0.163129\pi\)
0.871528 + 0.490345i \(0.163129\pi\)
\(104\) −1502.45 −1.41661
\(105\) 0 0
\(106\) 449.578 0.411952
\(107\) −1089.70 −0.984536 −0.492268 0.870444i \(-0.663832\pi\)
−0.492268 + 0.870444i \(0.663832\pi\)
\(108\) 0 0
\(109\) 589.667 0.518164 0.259082 0.965855i \(-0.416580\pi\)
0.259082 + 0.965855i \(0.416580\pi\)
\(110\) −576.083 −0.499340
\(111\) 0 0
\(112\) 0 0
\(113\) 900.358 0.749544 0.374772 0.927117i \(-0.377721\pi\)
0.374772 + 0.927117i \(0.377721\pi\)
\(114\) 0 0
\(115\) −505.154 −0.409616
\(116\) −602.823 −0.482506
\(117\) 0 0
\(118\) 857.887 0.669279
\(119\) 0 0
\(120\) 0 0
\(121\) 3509.35 2.63663
\(122\) −364.210 −0.270279
\(123\) 0 0
\(124\) −387.214 −0.280426
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1755.75 −1.22676 −0.613378 0.789790i \(-0.710190\pi\)
−0.613378 + 0.789790i \(0.710190\pi\)
\(128\) 1048.58 0.724082
\(129\) 0 0
\(130\) −566.643 −0.382291
\(131\) 1809.10 1.20658 0.603289 0.797523i \(-0.293857\pi\)
0.603289 + 0.797523i \(0.293857\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −133.636 −0.0861521
\(135\) 0 0
\(136\) 2290.62 1.44426
\(137\) 18.5134 0.0115453 0.00577265 0.999983i \(-0.498162\pi\)
0.00577265 + 0.999983i \(0.498162\pi\)
\(138\) 0 0
\(139\) 625.608 0.381751 0.190875 0.981614i \(-0.438867\pi\)
0.190875 + 0.981614i \(0.438867\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −150.778 −0.0891059
\(143\) 4761.03 2.78418
\(144\) 0 0
\(145\) −573.303 −0.328346
\(146\) 1461.11 0.828237
\(147\) 0 0
\(148\) 1056.40 0.586725
\(149\) 1028.63 0.565563 0.282782 0.959184i \(-0.408743\pi\)
0.282782 + 0.959184i \(0.408743\pi\)
\(150\) 0 0
\(151\) 71.0073 0.0382682 0.0191341 0.999817i \(-0.493909\pi\)
0.0191341 + 0.999817i \(0.493909\pi\)
\(152\) −1578.42 −0.842282
\(153\) 0 0
\(154\) 0 0
\(155\) −368.252 −0.190831
\(156\) 0 0
\(157\) 2061.32 1.04784 0.523922 0.851767i \(-0.324468\pi\)
0.523922 + 0.851767i \(0.324468\pi\)
\(158\) −993.434 −0.500211
\(159\) 0 0
\(160\) 925.409 0.457250
\(161\) 0 0
\(162\) 0 0
\(163\) −1963.80 −0.943660 −0.471830 0.881690i \(-0.656406\pi\)
−0.471830 + 0.881690i \(0.656406\pi\)
\(164\) −2193.98 −1.04464
\(165\) 0 0
\(166\) −117.385 −0.0548846
\(167\) 2855.04 1.32293 0.661467 0.749974i \(-0.269934\pi\)
0.661467 + 0.749974i \(0.269934\pi\)
\(168\) 0 0
\(169\) 2486.01 1.13155
\(170\) 863.899 0.389753
\(171\) 0 0
\(172\) −1635.99 −0.725249
\(173\) −1553.21 −0.682590 −0.341295 0.939956i \(-0.610866\pi\)
−0.341295 + 0.939956i \(0.610866\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −396.601 −0.169857
\(177\) 0 0
\(178\) −1329.14 −0.559682
\(179\) −269.841 −0.112675 −0.0563376 0.998412i \(-0.517942\pi\)
−0.0563376 + 0.998412i \(0.517942\pi\)
\(180\) 0 0
\(181\) −2229.61 −0.915613 −0.457806 0.889052i \(-0.651365\pi\)
−0.457806 + 0.889052i \(0.651365\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2218.15 0.888717
\(185\) 1004.67 0.399268
\(186\) 0 0
\(187\) −7258.63 −2.83852
\(188\) −787.026 −0.305318
\(189\) 0 0
\(190\) −595.296 −0.227302
\(191\) 465.920 0.176507 0.0882533 0.996098i \(-0.471871\pi\)
0.0882533 + 0.996098i \(0.471871\pi\)
\(192\) 0 0
\(193\) −4414.46 −1.64642 −0.823212 0.567734i \(-0.807820\pi\)
−0.823212 + 0.567734i \(0.807820\pi\)
\(194\) 241.202 0.0892643
\(195\) 0 0
\(196\) 0 0
\(197\) 289.812 0.104814 0.0524068 0.998626i \(-0.483311\pi\)
0.0524068 + 0.998626i \(0.483311\pi\)
\(198\) 0 0
\(199\) −4817.73 −1.71618 −0.858091 0.513498i \(-0.828349\pi\)
−0.858091 + 0.513498i \(0.828349\pi\)
\(200\) 548.879 0.194058
\(201\) 0 0
\(202\) 1025.82 0.357310
\(203\) 0 0
\(204\) 0 0
\(205\) −2086.54 −0.710879
\(206\) −3017.48 −1.02057
\(207\) 0 0
\(208\) −390.102 −0.130042
\(209\) 5001.78 1.65541
\(210\) 0 0
\(211\) 2022.01 0.659719 0.329859 0.944030i \(-0.392999\pi\)
0.329859 + 0.944030i \(0.392999\pi\)
\(212\) 1427.26 0.462382
\(213\) 0 0
\(214\) 1804.61 0.576452
\(215\) −1555.87 −0.493533
\(216\) 0 0
\(217\) 0 0
\(218\) −976.525 −0.303388
\(219\) 0 0
\(220\) −1828.88 −0.560467
\(221\) −7139.69 −2.17315
\(222\) 0 0
\(223\) −4343.86 −1.30442 −0.652211 0.758037i \(-0.726158\pi\)
−0.652211 + 0.758037i \(0.726158\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1491.05 −0.438863
\(227\) −2647.59 −0.774127 −0.387064 0.922053i \(-0.626511\pi\)
−0.387064 + 0.922053i \(0.626511\pi\)
\(228\) 0 0
\(229\) 1445.09 0.417006 0.208503 0.978022i \(-0.433141\pi\)
0.208503 + 0.978022i \(0.433141\pi\)
\(230\) 836.566 0.239833
\(231\) 0 0
\(232\) 2517.39 0.712392
\(233\) 6245.91 1.75615 0.878076 0.478522i \(-0.158827\pi\)
0.878076 + 0.478522i \(0.158827\pi\)
\(234\) 0 0
\(235\) −748.485 −0.207769
\(236\) 2723.51 0.751209
\(237\) 0 0
\(238\) 0 0
\(239\) −1340.24 −0.362731 −0.181366 0.983416i \(-0.558052\pi\)
−0.181366 + 0.983416i \(0.558052\pi\)
\(240\) 0 0
\(241\) −3369.92 −0.900729 −0.450364 0.892845i \(-0.648706\pi\)
−0.450364 + 0.892845i \(0.648706\pi\)
\(242\) −5811.70 −1.54376
\(243\) 0 0
\(244\) −1156.25 −0.303366
\(245\) 0 0
\(246\) 0 0
\(247\) 4919.82 1.26737
\(248\) 1617.01 0.414033
\(249\) 0 0
\(250\) 207.008 0.0523693
\(251\) −3592.64 −0.903449 −0.451724 0.892158i \(-0.649191\pi\)
−0.451724 + 0.892158i \(0.649191\pi\)
\(252\) 0 0
\(253\) −7028.97 −1.74667
\(254\) 2907.64 0.718273
\(255\) 0 0
\(256\) −3823.74 −0.933531
\(257\) −2.84763 −0.000691167 0 −0.000345584 1.00000i \(-0.500110\pi\)
−0.000345584 1.00000i \(0.500110\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1798.91 −0.429090
\(261\) 0 0
\(262\) −2995.98 −0.706459
\(263\) 2817.07 0.660488 0.330244 0.943896i \(-0.392869\pi\)
0.330244 + 0.943896i \(0.392869\pi\)
\(264\) 0 0
\(265\) 1357.37 0.314652
\(266\) 0 0
\(267\) 0 0
\(268\) −424.250 −0.0966986
\(269\) −1447.43 −0.328072 −0.164036 0.986454i \(-0.552451\pi\)
−0.164036 + 0.986454i \(0.552451\pi\)
\(270\) 0 0
\(271\) −8054.50 −1.80545 −0.902724 0.430221i \(-0.858436\pi\)
−0.902724 + 0.430221i \(0.858436\pi\)
\(272\) 594.746 0.132580
\(273\) 0 0
\(274\) −30.6593 −0.00675985
\(275\) −1739.32 −0.381399
\(276\) 0 0
\(277\) −571.300 −0.123921 −0.0619604 0.998079i \(-0.519735\pi\)
−0.0619604 + 0.998079i \(0.519735\pi\)
\(278\) −1036.05 −0.223518
\(279\) 0 0
\(280\) 0 0
\(281\) 1784.48 0.378837 0.189418 0.981896i \(-0.439340\pi\)
0.189418 + 0.981896i \(0.439340\pi\)
\(282\) 0 0
\(283\) −3321.05 −0.697582 −0.348791 0.937201i \(-0.613408\pi\)
−0.348791 + 0.937201i \(0.613408\pi\)
\(284\) −478.672 −0.100014
\(285\) 0 0
\(286\) −7884.57 −1.63015
\(287\) 0 0
\(288\) 0 0
\(289\) 5972.11 1.21557
\(290\) 949.425 0.192249
\(291\) 0 0
\(292\) 4638.56 0.929627
\(293\) −5049.54 −1.00682 −0.503408 0.864049i \(-0.667921\pi\)
−0.503408 + 0.864049i \(0.667921\pi\)
\(294\) 0 0
\(295\) 2590.14 0.511199
\(296\) −4411.52 −0.866266
\(297\) 0 0
\(298\) −1703.48 −0.331141
\(299\) −6913.79 −1.33724
\(300\) 0 0
\(301\) 0 0
\(302\) −117.593 −0.0224063
\(303\) 0 0
\(304\) −409.828 −0.0773198
\(305\) −1099.63 −0.206441
\(306\) 0 0
\(307\) 1535.73 0.285500 0.142750 0.989759i \(-0.454405\pi\)
0.142750 + 0.989759i \(0.454405\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 609.849 0.111733
\(311\) −9283.05 −1.69258 −0.846291 0.532720i \(-0.821170\pi\)
−0.846291 + 0.532720i \(0.821170\pi\)
\(312\) 0 0
\(313\) −6025.43 −1.08811 −0.544054 0.839050i \(-0.683111\pi\)
−0.544054 + 0.839050i \(0.683111\pi\)
\(314\) −3413.68 −0.613519
\(315\) 0 0
\(316\) −3153.83 −0.561445
\(317\) 6977.58 1.23628 0.618139 0.786069i \(-0.287887\pi\)
0.618139 + 0.786069i \(0.287887\pi\)
\(318\) 0 0
\(319\) −7977.24 −1.40012
\(320\) −1304.51 −0.227889
\(321\) 0 0
\(322\) 0 0
\(323\) −7500.71 −1.29211
\(324\) 0 0
\(325\) −1710.81 −0.291997
\(326\) 3252.17 0.552519
\(327\) 0 0
\(328\) 9162.06 1.54235
\(329\) 0 0
\(330\) 0 0
\(331\) 984.878 0.163546 0.0817731 0.996651i \(-0.473942\pi\)
0.0817731 + 0.996651i \(0.473942\pi\)
\(332\) −372.659 −0.0616034
\(333\) 0 0
\(334\) −4728.13 −0.774586
\(335\) −403.475 −0.0658035
\(336\) 0 0
\(337\) 51.9653 0.00839979 0.00419990 0.999991i \(-0.498663\pi\)
0.00419990 + 0.999991i \(0.498663\pi\)
\(338\) −4116.99 −0.662529
\(339\) 0 0
\(340\) 2742.60 0.437465
\(341\) −5124.06 −0.813734
\(342\) 0 0
\(343\) 0 0
\(344\) 6831.89 1.07079
\(345\) 0 0
\(346\) 2572.21 0.399661
\(347\) 11300.5 1.74825 0.874123 0.485704i \(-0.161437\pi\)
0.874123 + 0.485704i \(0.161437\pi\)
\(348\) 0 0
\(349\) 2016.91 0.309349 0.154674 0.987966i \(-0.450567\pi\)
0.154674 + 0.987966i \(0.450567\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12876.6 1.94979
\(353\) −7589.41 −1.14432 −0.572158 0.820143i \(-0.693894\pi\)
−0.572158 + 0.820143i \(0.693894\pi\)
\(354\) 0 0
\(355\) −455.232 −0.0680597
\(356\) −4219.59 −0.628196
\(357\) 0 0
\(358\) 446.873 0.0659720
\(359\) −8734.24 −1.28405 −0.642027 0.766682i \(-0.721906\pi\)
−0.642027 + 0.766682i \(0.721906\pi\)
\(360\) 0 0
\(361\) −1690.41 −0.246451
\(362\) 3692.38 0.536097
\(363\) 0 0
\(364\) 0 0
\(365\) 4411.41 0.632613
\(366\) 0 0
\(367\) 1890.54 0.268898 0.134449 0.990921i \(-0.457074\pi\)
0.134449 + 0.990921i \(0.457074\pi\)
\(368\) 575.928 0.0815825
\(369\) 0 0
\(370\) −1663.79 −0.233774
\(371\) 0 0
\(372\) 0 0
\(373\) 2713.88 0.376728 0.188364 0.982099i \(-0.439682\pi\)
0.188364 + 0.982099i \(0.439682\pi\)
\(374\) 12020.7 1.66197
\(375\) 0 0
\(376\) 3286.63 0.450784
\(377\) −7846.52 −1.07193
\(378\) 0 0
\(379\) 8941.19 1.21182 0.605908 0.795535i \(-0.292810\pi\)
0.605908 + 0.795535i \(0.292810\pi\)
\(380\) −1889.87 −0.255127
\(381\) 0 0
\(382\) −771.592 −0.103346
\(383\) −9293.88 −1.23994 −0.619968 0.784627i \(-0.712854\pi\)
−0.619968 + 0.784627i \(0.712854\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7310.62 0.963992
\(387\) 0 0
\(388\) 765.737 0.100192
\(389\) −10454.8 −1.36267 −0.681333 0.731974i \(-0.738599\pi\)
−0.681333 + 0.731974i \(0.738599\pi\)
\(390\) 0 0
\(391\) 10540.7 1.36334
\(392\) 0 0
\(393\) 0 0
\(394\) −479.947 −0.0613690
\(395\) −2999.39 −0.382065
\(396\) 0 0
\(397\) 3626.03 0.458401 0.229200 0.973379i \(-0.426389\pi\)
0.229200 + 0.973379i \(0.426389\pi\)
\(398\) 7978.47 1.00484
\(399\) 0 0
\(400\) 142.513 0.0178141
\(401\) 8422.52 1.04888 0.524440 0.851448i \(-0.324275\pi\)
0.524440 + 0.851448i \(0.324275\pi\)
\(402\) 0 0
\(403\) −5040.09 −0.622989
\(404\) 3256.65 0.401051
\(405\) 0 0
\(406\) 0 0
\(407\) 13979.5 1.70254
\(408\) 0 0
\(409\) 14580.7 1.76276 0.881379 0.472409i \(-0.156616\pi\)
0.881379 + 0.472409i \(0.156616\pi\)
\(410\) 3455.44 0.416224
\(411\) 0 0
\(412\) −9579.51 −1.14551
\(413\) 0 0
\(414\) 0 0
\(415\) −354.410 −0.0419212
\(416\) 12665.6 1.49275
\(417\) 0 0
\(418\) −8283.26 −0.969252
\(419\) −2537.53 −0.295863 −0.147931 0.988998i \(-0.547261\pi\)
−0.147931 + 0.988998i \(0.547261\pi\)
\(420\) 0 0
\(421\) 9649.52 1.11708 0.558538 0.829479i \(-0.311363\pi\)
0.558538 + 0.829479i \(0.311363\pi\)
\(422\) −3348.57 −0.386269
\(423\) 0 0
\(424\) −5960.27 −0.682680
\(425\) 2608.29 0.297696
\(426\) 0 0
\(427\) 0 0
\(428\) 5729.06 0.647020
\(429\) 0 0
\(430\) 2576.62 0.288967
\(431\) 7262.56 0.811660 0.405830 0.913949i \(-0.366983\pi\)
0.405830 + 0.913949i \(0.366983\pi\)
\(432\) 0 0
\(433\) −11345.0 −1.25914 −0.629570 0.776944i \(-0.716769\pi\)
−0.629570 + 0.776944i \(0.716769\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3100.15 −0.340528
\(437\) −7263.40 −0.795092
\(438\) 0 0
\(439\) −11705.9 −1.27265 −0.636323 0.771423i \(-0.719545\pi\)
−0.636323 + 0.771423i \(0.719545\pi\)
\(440\) 7637.40 0.827497
\(441\) 0 0
\(442\) 11823.8 1.27240
\(443\) −15078.0 −1.61710 −0.808551 0.588426i \(-0.799748\pi\)
−0.808551 + 0.588426i \(0.799748\pi\)
\(444\) 0 0
\(445\) −4012.96 −0.427489
\(446\) 7193.70 0.763747
\(447\) 0 0
\(448\) 0 0
\(449\) −1075.45 −0.113037 −0.0565185 0.998402i \(-0.518000\pi\)
−0.0565185 + 0.998402i \(0.518000\pi\)
\(450\) 0 0
\(451\) −29033.2 −3.03131
\(452\) −4733.59 −0.492587
\(453\) 0 0
\(454\) 4384.58 0.453257
\(455\) 0 0
\(456\) 0 0
\(457\) −10736.9 −1.09902 −0.549511 0.835487i \(-0.685186\pi\)
−0.549511 + 0.835487i \(0.685186\pi\)
\(458\) −2393.16 −0.244159
\(459\) 0 0
\(460\) 2655.82 0.269192
\(461\) 452.568 0.0457228 0.0228614 0.999739i \(-0.492722\pi\)
0.0228614 + 0.999739i \(0.492722\pi\)
\(462\) 0 0
\(463\) 7118.15 0.714489 0.357244 0.934011i \(-0.383716\pi\)
0.357244 + 0.934011i \(0.383716\pi\)
\(464\) 653.626 0.0653962
\(465\) 0 0
\(466\) −10343.6 −1.02824
\(467\) 973.800 0.0964927 0.0482463 0.998835i \(-0.484637\pi\)
0.0482463 + 0.998835i \(0.484637\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1239.54 0.121650
\(471\) 0 0
\(472\) −11373.4 −1.10912
\(473\) −21649.2 −2.10451
\(474\) 0 0
\(475\) −1797.32 −0.173614
\(476\) 0 0
\(477\) 0 0
\(478\) 2219.52 0.212381
\(479\) 9714.00 0.926606 0.463303 0.886200i \(-0.346664\pi\)
0.463303 + 0.886200i \(0.346664\pi\)
\(480\) 0 0
\(481\) 13750.4 1.30346
\(482\) 5580.80 0.527383
\(483\) 0 0
\(484\) −18450.3 −1.73274
\(485\) 728.239 0.0681806
\(486\) 0 0
\(487\) 923.389 0.0859194 0.0429597 0.999077i \(-0.486321\pi\)
0.0429597 + 0.999077i \(0.486321\pi\)
\(488\) 4828.50 0.447902
\(489\) 0 0
\(490\) 0 0
\(491\) −1289.11 −0.118486 −0.0592430 0.998244i \(-0.518869\pi\)
−0.0592430 + 0.998244i \(0.518869\pi\)
\(492\) 0 0
\(493\) 11962.7 1.09285
\(494\) −8147.52 −0.742053
\(495\) 0 0
\(496\) 419.846 0.0380074
\(497\) 0 0
\(498\) 0 0
\(499\) −19338.3 −1.73487 −0.867436 0.497549i \(-0.834234\pi\)
−0.867436 + 0.497549i \(0.834234\pi\)
\(500\) 657.182 0.0587802
\(501\) 0 0
\(502\) 5949.64 0.528975
\(503\) −1772.84 −0.157151 −0.0785757 0.996908i \(-0.525037\pi\)
−0.0785757 + 0.996908i \(0.525037\pi\)
\(504\) 0 0
\(505\) 3097.18 0.272916
\(506\) 11640.4 1.02269
\(507\) 0 0
\(508\) 9230.80 0.806202
\(509\) 3151.75 0.274458 0.137229 0.990539i \(-0.456180\pi\)
0.137229 + 0.990539i \(0.456180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −2056.31 −0.177494
\(513\) 0 0
\(514\) 4.71585 0.000404683 0
\(515\) −9110.40 −0.779519
\(516\) 0 0
\(517\) −10414.8 −0.885964
\(518\) 0 0
\(519\) 0 0
\(520\) 7512.24 0.633526
\(521\) −11029.4 −0.927458 −0.463729 0.885977i \(-0.653489\pi\)
−0.463729 + 0.885977i \(0.653489\pi\)
\(522\) 0 0
\(523\) −14448.4 −1.20800 −0.604002 0.796983i \(-0.706428\pi\)
−0.604002 + 0.796983i \(0.706428\pi\)
\(524\) −9511.26 −0.792941
\(525\) 0 0
\(526\) −4665.25 −0.386720
\(527\) 7684.08 0.635149
\(528\) 0 0
\(529\) −1959.79 −0.161074
\(530\) −2247.89 −0.184231
\(531\) 0 0
\(532\) 0 0
\(533\) −28557.4 −2.32075
\(534\) 0 0
\(535\) 5448.51 0.440298
\(536\) 1771.67 0.142770
\(537\) 0 0
\(538\) 2397.04 0.192089
\(539\) 0 0
\(540\) 0 0
\(541\) −22274.8 −1.77018 −0.885091 0.465418i \(-0.845904\pi\)
−0.885091 + 0.465418i \(0.845904\pi\)
\(542\) 13338.8 1.05710
\(543\) 0 0
\(544\) −19309.9 −1.52188
\(545\) −2948.33 −0.231730
\(546\) 0 0
\(547\) 18642.6 1.45722 0.728609 0.684930i \(-0.240167\pi\)
0.728609 + 0.684930i \(0.240167\pi\)
\(548\) −97.3334 −0.00758737
\(549\) 0 0
\(550\) 2880.42 0.223311
\(551\) −8243.28 −0.637343
\(552\) 0 0
\(553\) 0 0
\(554\) 946.108 0.0725565
\(555\) 0 0
\(556\) −3289.11 −0.250880
\(557\) 21431.8 1.63033 0.815165 0.579229i \(-0.196646\pi\)
0.815165 + 0.579229i \(0.196646\pi\)
\(558\) 0 0
\(559\) −21294.5 −1.61120
\(560\) 0 0
\(561\) 0 0
\(562\) −2955.21 −0.221811
\(563\) −6154.85 −0.460739 −0.230370 0.973103i \(-0.573993\pi\)
−0.230370 + 0.973103i \(0.573993\pi\)
\(564\) 0 0
\(565\) −4501.79 −0.335206
\(566\) 5499.86 0.408439
\(567\) 0 0
\(568\) 1998.94 0.147665
\(569\) 8389.99 0.618149 0.309074 0.951038i \(-0.399981\pi\)
0.309074 + 0.951038i \(0.399981\pi\)
\(570\) 0 0
\(571\) −600.502 −0.0440109 −0.0220055 0.999758i \(-0.507005\pi\)
−0.0220055 + 0.999758i \(0.507005\pi\)
\(572\) −25030.9 −1.82971
\(573\) 0 0
\(574\) 0 0
\(575\) 2525.77 0.183186
\(576\) 0 0
\(577\) −4062.35 −0.293098 −0.146549 0.989203i \(-0.546817\pi\)
−0.146549 + 0.989203i \(0.546817\pi\)
\(578\) −9890.18 −0.711725
\(579\) 0 0
\(580\) 3014.12 0.215783
\(581\) 0 0
\(582\) 0 0
\(583\) 18887.2 1.34173
\(584\) −19370.6 −1.37254
\(585\) 0 0
\(586\) 8362.35 0.589498
\(587\) −8387.50 −0.589760 −0.294880 0.955534i \(-0.595280\pi\)
−0.294880 + 0.955534i \(0.595280\pi\)
\(588\) 0 0
\(589\) −5294.95 −0.370415
\(590\) −4289.43 −0.299310
\(591\) 0 0
\(592\) −1145.43 −0.0795214
\(593\) −15342.6 −1.06247 −0.531236 0.847224i \(-0.678272\pi\)
−0.531236 + 0.847224i \(0.678272\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5408.00 −0.371678
\(597\) 0 0
\(598\) 11449.7 0.782963
\(599\) 19708.5 1.34435 0.672175 0.740392i \(-0.265360\pi\)
0.672175 + 0.740392i \(0.265360\pi\)
\(600\) 0 0
\(601\) 19002.5 1.28973 0.644866 0.764295i \(-0.276913\pi\)
0.644866 + 0.764295i \(0.276913\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −373.318 −0.0251492
\(605\) −17546.8 −1.17914
\(606\) 0 0
\(607\) −29453.1 −1.96946 −0.984732 0.174076i \(-0.944306\pi\)
−0.984732 + 0.174076i \(0.944306\pi\)
\(608\) 13306.1 0.887553
\(609\) 0 0
\(610\) 1821.05 0.120873
\(611\) −10244.2 −0.678288
\(612\) 0 0
\(613\) 9987.27 0.658045 0.329023 0.944322i \(-0.393281\pi\)
0.329023 + 0.944322i \(0.393281\pi\)
\(614\) −2543.26 −0.167162
\(615\) 0 0
\(616\) 0 0
\(617\) 21076.1 1.37519 0.687593 0.726096i \(-0.258667\pi\)
0.687593 + 0.726096i \(0.258667\pi\)
\(618\) 0 0
\(619\) 314.668 0.0204323 0.0102161 0.999948i \(-0.496748\pi\)
0.0102161 + 0.999948i \(0.496748\pi\)
\(620\) 1936.07 0.125410
\(621\) 0 0
\(622\) 15373.3 0.991018
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 9978.49 0.637094
\(627\) 0 0
\(628\) −10837.3 −0.688624
\(629\) −20963.7 −1.32890
\(630\) 0 0
\(631\) 3314.96 0.209138 0.104569 0.994518i \(-0.466654\pi\)
0.104569 + 0.994518i \(0.466654\pi\)
\(632\) 13170.4 0.828941
\(633\) 0 0
\(634\) −11555.3 −0.723849
\(635\) 8778.77 0.548622
\(636\) 0 0
\(637\) 0 0
\(638\) 13210.8 0.819782
\(639\) 0 0
\(640\) −5242.92 −0.323820
\(641\) −3005.12 −0.185172 −0.0925858 0.995705i \(-0.529513\pi\)
−0.0925858 + 0.995705i \(0.529513\pi\)
\(642\) 0 0
\(643\) −21225.7 −1.30180 −0.650902 0.759162i \(-0.725609\pi\)
−0.650902 + 0.759162i \(0.725609\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12421.6 0.756537
\(647\) −2740.35 −0.166514 −0.0832568 0.996528i \(-0.526532\pi\)
−0.0832568 + 0.996528i \(0.526532\pi\)
\(648\) 0 0
\(649\) 36040.6 2.17984
\(650\) 2833.21 0.170966
\(651\) 0 0
\(652\) 10324.6 0.620157
\(653\) −22790.7 −1.36580 −0.682900 0.730511i \(-0.739282\pi\)
−0.682900 + 0.730511i \(0.739282\pi\)
\(654\) 0 0
\(655\) −9045.49 −0.539598
\(656\) 2378.87 0.141584
\(657\) 0 0
\(658\) 0 0
\(659\) −19405.1 −1.14706 −0.573532 0.819183i \(-0.694427\pi\)
−0.573532 + 0.819183i \(0.694427\pi\)
\(660\) 0 0
\(661\) −15637.3 −0.920150 −0.460075 0.887880i \(-0.652177\pi\)
−0.460075 + 0.887880i \(0.652177\pi\)
\(662\) −1631.02 −0.0957574
\(663\) 0 0
\(664\) 1556.23 0.0909538
\(665\) 0 0
\(666\) 0 0
\(667\) 11584.2 0.672479
\(668\) −15010.3 −0.869409
\(669\) 0 0
\(670\) 668.179 0.0385284
\(671\) −15300.8 −0.880299
\(672\) 0 0
\(673\) −2579.54 −0.147747 −0.0738735 0.997268i \(-0.523536\pi\)
−0.0738735 + 0.997268i \(0.523536\pi\)
\(674\) −86.0578 −0.00491813
\(675\) 0 0
\(676\) −13070.1 −0.743634
\(677\) 8159.56 0.463216 0.231608 0.972809i \(-0.425601\pi\)
0.231608 + 0.972809i \(0.425601\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −11453.1 −0.645892
\(681\) 0 0
\(682\) 8485.76 0.476446
\(683\) 20529.3 1.15012 0.575059 0.818112i \(-0.304979\pi\)
0.575059 + 0.818112i \(0.304979\pi\)
\(684\) 0 0
\(685\) −92.5670 −0.00516321
\(686\) 0 0
\(687\) 0 0
\(688\) 1773.86 0.0982961
\(689\) 18577.7 1.02722
\(690\) 0 0
\(691\) 917.295 0.0505001 0.0252500 0.999681i \(-0.491962\pi\)
0.0252500 + 0.999681i \(0.491962\pi\)
\(692\) 8165.92 0.448586
\(693\) 0 0
\(694\) −18714.3 −1.02361
\(695\) −3128.04 −0.170724
\(696\) 0 0
\(697\) 43538.4 2.36605
\(698\) −3340.13 −0.181126
\(699\) 0 0
\(700\) 0 0
\(701\) 10491.3 0.565266 0.282633 0.959228i \(-0.408792\pi\)
0.282633 + 0.959228i \(0.408792\pi\)
\(702\) 0 0
\(703\) 14445.7 0.775006
\(704\) −18151.7 −0.971758
\(705\) 0 0
\(706\) 12568.5 0.670005
\(707\) 0 0
\(708\) 0 0
\(709\) 23837.6 1.26268 0.631339 0.775507i \(-0.282506\pi\)
0.631339 + 0.775507i \(0.282506\pi\)
\(710\) 753.892 0.0398494
\(711\) 0 0
\(712\) 17621.0 0.927495
\(713\) 7440.96 0.390836
\(714\) 0 0
\(715\) −23805.2 −1.24512
\(716\) 1418.68 0.0740481
\(717\) 0 0
\(718\) 14464.4 0.751822
\(719\) −3926.19 −0.203647 −0.101824 0.994802i \(-0.532468\pi\)
−0.101824 + 0.994802i \(0.532468\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2799.42 0.144299
\(723\) 0 0
\(724\) 11722.1 0.601724
\(725\) 2866.51 0.146841
\(726\) 0 0
\(727\) 21071.2 1.07495 0.537474 0.843281i \(-0.319379\pi\)
0.537474 + 0.843281i \(0.319379\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7305.57 −0.370399
\(731\) 32465.4 1.64265
\(732\) 0 0
\(733\) 22840.7 1.15094 0.575470 0.817823i \(-0.304819\pi\)
0.575470 + 0.817823i \(0.304819\pi\)
\(734\) −3130.86 −0.157441
\(735\) 0 0
\(736\) −18699.0 −0.936485
\(737\) −5614.16 −0.280597
\(738\) 0 0
\(739\) 10837.3 0.539456 0.269728 0.962937i \(-0.413066\pi\)
0.269728 + 0.962937i \(0.413066\pi\)
\(740\) −5281.99 −0.262392
\(741\) 0 0
\(742\) 0 0
\(743\) 21631.9 1.06810 0.534050 0.845453i \(-0.320669\pi\)
0.534050 + 0.845453i \(0.320669\pi\)
\(744\) 0 0
\(745\) −5143.17 −0.252928
\(746\) −4494.36 −0.220577
\(747\) 0 0
\(748\) 38161.9 1.86543
\(749\) 0 0
\(750\) 0 0
\(751\) −16179.2 −0.786133 −0.393067 0.919510i \(-0.628586\pi\)
−0.393067 + 0.919510i \(0.628586\pi\)
\(752\) 853.352 0.0413811
\(753\) 0 0
\(754\) 12994.3 0.627620
\(755\) −355.037 −0.0171140
\(756\) 0 0
\(757\) −40930.9 −1.96520 −0.982601 0.185727i \(-0.940536\pi\)
−0.982601 + 0.185727i \(0.940536\pi\)
\(758\) −14807.2 −0.709526
\(759\) 0 0
\(760\) 7892.11 0.376680
\(761\) 3183.97 0.151667 0.0758337 0.997120i \(-0.475838\pi\)
0.0758337 + 0.997120i \(0.475838\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2449.55 −0.115997
\(765\) 0 0
\(766\) 15391.2 0.725990
\(767\) 35450.0 1.66887
\(768\) 0 0
\(769\) −33595.8 −1.57542 −0.787708 0.616048i \(-0.788733\pi\)
−0.787708 + 0.616048i \(0.788733\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23208.8 1.08200
\(773\) 34386.0 1.59997 0.799986 0.600019i \(-0.204840\pi\)
0.799986 + 0.600019i \(0.204840\pi\)
\(774\) 0 0
\(775\) 1841.26 0.0853420
\(776\) −3197.72 −0.147927
\(777\) 0 0
\(778\) 17313.7 0.797849
\(779\) −30001.5 −1.37986
\(780\) 0 0
\(781\) −6334.33 −0.290218
\(782\) −17456.1 −0.798245
\(783\) 0 0
\(784\) 0 0
\(785\) −10306.6 −0.468610
\(786\) 0 0
\(787\) 8212.43 0.371972 0.185986 0.982552i \(-0.440452\pi\)
0.185986 + 0.982552i \(0.440452\pi\)
\(788\) −1523.68 −0.0688816
\(789\) 0 0
\(790\) 4967.17 0.223701
\(791\) 0 0
\(792\) 0 0
\(793\) −15050.1 −0.673951
\(794\) −6004.92 −0.268396
\(795\) 0 0
\(796\) 25329.0 1.12784
\(797\) 36798.3 1.63546 0.817732 0.575600i \(-0.195231\pi\)
0.817732 + 0.575600i \(0.195231\pi\)
\(798\) 0 0
\(799\) 15618.2 0.691528
\(800\) −4627.05 −0.204488
\(801\) 0 0
\(802\) −13948.2 −0.614125
\(803\) 61382.7 2.69757
\(804\) 0 0
\(805\) 0 0
\(806\) 8346.70 0.364764
\(807\) 0 0
\(808\) −13599.8 −0.592128
\(809\) −10186.2 −0.442678 −0.221339 0.975197i \(-0.571043\pi\)
−0.221339 + 0.975197i \(0.571043\pi\)
\(810\) 0 0
\(811\) −21196.9 −0.917786 −0.458893 0.888492i \(-0.651754\pi\)
−0.458893 + 0.888492i \(0.651754\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −23150.8 −0.996851
\(815\) 9818.99 0.422018
\(816\) 0 0
\(817\) −22371.3 −0.957982
\(818\) −24146.5 −1.03211
\(819\) 0 0
\(820\) 10969.9 0.467177
\(821\) −7563.94 −0.321539 −0.160769 0.986992i \(-0.551398\pi\)
−0.160769 + 0.986992i \(0.551398\pi\)
\(822\) 0 0
\(823\) 8399.80 0.355770 0.177885 0.984051i \(-0.443075\pi\)
0.177885 + 0.984051i \(0.443075\pi\)
\(824\) 40004.1 1.69127
\(825\) 0 0
\(826\) 0 0
\(827\) −5479.54 −0.230402 −0.115201 0.993342i \(-0.536751\pi\)
−0.115201 + 0.993342i \(0.536751\pi\)
\(828\) 0 0
\(829\) −9252.56 −0.387641 −0.193821 0.981037i \(-0.562088\pi\)
−0.193821 + 0.981037i \(0.562088\pi\)
\(830\) 586.925 0.0245452
\(831\) 0 0
\(832\) −17854.2 −0.743972
\(833\) 0 0
\(834\) 0 0
\(835\) −14275.2 −0.591634
\(836\) −26296.6 −1.08790
\(837\) 0 0
\(838\) 4202.31 0.173230
\(839\) −34517.6 −1.42036 −0.710178 0.704022i \(-0.751386\pi\)
−0.710178 + 0.704022i \(0.751386\pi\)
\(840\) 0 0
\(841\) −11241.9 −0.460943
\(842\) −15980.2 −0.654055
\(843\) 0 0
\(844\) −10630.6 −0.433555
\(845\) −12430.1 −0.506044
\(846\) 0 0
\(847\) 0 0
\(848\) −1547.55 −0.0626686
\(849\) 0 0
\(850\) −4319.50 −0.174303
\(851\) −20300.4 −0.817732
\(852\) 0 0
\(853\) 1498.57 0.0601525 0.0300763 0.999548i \(-0.490425\pi\)
0.0300763 + 0.999548i \(0.490425\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −23924.6 −0.955287
\(857\) −9357.02 −0.372963 −0.186482 0.982458i \(-0.559709\pi\)
−0.186482 + 0.982458i \(0.559709\pi\)
\(858\) 0 0
\(859\) 29960.9 1.19005 0.595025 0.803707i \(-0.297142\pi\)
0.595025 + 0.803707i \(0.297142\pi\)
\(860\) 8179.94 0.324341
\(861\) 0 0
\(862\) −12027.3 −0.475232
\(863\) 33941.0 1.33878 0.669389 0.742912i \(-0.266556\pi\)
0.669389 + 0.742912i \(0.266556\pi\)
\(864\) 0 0
\(865\) 7766.03 0.305264
\(866\) 18788.1 0.737235
\(867\) 0 0
\(868\) 0 0
\(869\) −41735.0 −1.62919
\(870\) 0 0
\(871\) −5522.16 −0.214824
\(872\) 12946.2 0.502769
\(873\) 0 0
\(874\) 12028.6 0.465532
\(875\) 0 0
\(876\) 0 0
\(877\) −39436.6 −1.51845 −0.759224 0.650830i \(-0.774421\pi\)
−0.759224 + 0.650830i \(0.774421\pi\)
\(878\) 19385.7 0.745142
\(879\) 0 0
\(880\) 1983.00 0.0759625
\(881\) 32411.4 1.23946 0.619732 0.784813i \(-0.287241\pi\)
0.619732 + 0.784813i \(0.287241\pi\)
\(882\) 0 0
\(883\) −17121.4 −0.652526 −0.326263 0.945279i \(-0.605790\pi\)
−0.326263 + 0.945279i \(0.605790\pi\)
\(884\) 37536.6 1.42816
\(885\) 0 0
\(886\) 24970.1 0.946824
\(887\) −687.797 −0.0260360 −0.0130180 0.999915i \(-0.504144\pi\)
−0.0130180 + 0.999915i \(0.504144\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6645.71 0.250297
\(891\) 0 0
\(892\) 22837.6 0.857243
\(893\) −10762.2 −0.403294
\(894\) 0 0
\(895\) 1349.20 0.0503898
\(896\) 0 0
\(897\) 0 0
\(898\) 1781.01 0.0661839
\(899\) 8444.81 0.313293
\(900\) 0 0
\(901\) −28323.4 −1.04727
\(902\) 48080.8 1.77485
\(903\) 0 0
\(904\) 19767.5 0.727276
\(905\) 11148.1 0.409474
\(906\) 0 0
\(907\) 19104.2 0.699388 0.349694 0.936864i \(-0.386286\pi\)
0.349694 + 0.936864i \(0.386286\pi\)
\(908\) 13919.6 0.508743
\(909\) 0 0
\(910\) 0 0
\(911\) −23135.3 −0.841390 −0.420695 0.907202i \(-0.638214\pi\)
−0.420695 + 0.907202i \(0.638214\pi\)
\(912\) 0 0
\(913\) −4931.45 −0.178759
\(914\) 17781.0 0.643484
\(915\) 0 0
\(916\) −7597.50 −0.274049
\(917\) 0 0
\(918\) 0 0
\(919\) 47373.9 1.70046 0.850230 0.526412i \(-0.176463\pi\)
0.850230 + 0.526412i \(0.176463\pi\)
\(920\) −11090.7 −0.397447
\(921\) 0 0
\(922\) −749.481 −0.0267710
\(923\) −6230.53 −0.222189
\(924\) 0 0
\(925\) −5023.33 −0.178558
\(926\) −11788.1 −0.418338
\(927\) 0 0
\(928\) −21221.6 −0.750682
\(929\) −7617.72 −0.269031 −0.134515 0.990912i \(-0.542948\pi\)
−0.134515 + 0.990912i \(0.542948\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −32837.6 −1.15411
\(933\) 0 0
\(934\) −1612.67 −0.0564971
\(935\) 36293.2 1.26943
\(936\) 0 0
\(937\) 52091.2 1.81616 0.908082 0.418792i \(-0.137546\pi\)
0.908082 + 0.418792i \(0.137546\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3935.13 0.136542
\(941\) −56765.1 −1.96651 −0.983257 0.182225i \(-0.941670\pi\)
−0.983257 + 0.182225i \(0.941670\pi\)
\(942\) 0 0
\(943\) 42160.9 1.45594
\(944\) −2953.03 −0.101815
\(945\) 0 0
\(946\) 35852.5 1.23220
\(947\) 47629.2 1.63436 0.817181 0.576381i \(-0.195535\pi\)
0.817181 + 0.576381i \(0.195535\pi\)
\(948\) 0 0
\(949\) 60376.8 2.06524
\(950\) 2976.48 0.101652
\(951\) 0 0
\(952\) 0 0
\(953\) −12893.5 −0.438259 −0.219129 0.975696i \(-0.570322\pi\)
−0.219129 + 0.975696i \(0.570322\pi\)
\(954\) 0 0
\(955\) −2329.60 −0.0789362
\(956\) 7046.24 0.238380
\(957\) 0 0
\(958\) −16087.0 −0.542534
\(959\) 0 0
\(960\) 0 0
\(961\) −24366.6 −0.817918
\(962\) −22771.5 −0.763183
\(963\) 0 0
\(964\) 17717.2 0.591943
\(965\) 22072.3 0.736303
\(966\) 0 0
\(967\) −28420.0 −0.945114 −0.472557 0.881300i \(-0.656669\pi\)
−0.472557 + 0.881300i \(0.656669\pi\)
\(968\) 77048.4 2.55830
\(969\) 0 0
\(970\) −1206.01 −0.0399202
\(971\) 29273.8 0.967497 0.483749 0.875207i \(-0.339275\pi\)
0.483749 + 0.875207i \(0.339275\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1529.19 −0.0503063
\(975\) 0 0
\(976\) 1253.69 0.0411165
\(977\) −12055.0 −0.394753 −0.197377 0.980328i \(-0.563242\pi\)
−0.197377 + 0.980328i \(0.563242\pi\)
\(978\) 0 0
\(979\) −55838.4 −1.82288
\(980\) 0 0
\(981\) 0 0
\(982\) 2134.84 0.0693743
\(983\) 10605.1 0.344099 0.172049 0.985088i \(-0.444961\pi\)
0.172049 + 0.985088i \(0.444961\pi\)
\(984\) 0 0
\(985\) −1449.06 −0.0468740
\(986\) −19811.0 −0.639870
\(987\) 0 0
\(988\) −25865.7 −0.832893
\(989\) 31438.2 1.01080
\(990\) 0 0
\(991\) 45229.1 1.44980 0.724898 0.688856i \(-0.241887\pi\)
0.724898 + 0.688856i \(0.241887\pi\)
\(992\) −13631.4 −0.436287
\(993\) 0 0
\(994\) 0 0
\(995\) 24088.7 0.767500
\(996\) 0 0
\(997\) 49676.8 1.57802 0.789008 0.614383i \(-0.210595\pi\)
0.789008 + 0.614383i \(0.210595\pi\)
\(998\) 32025.4 1.01578
\(999\) 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 2205.4.a.bz.1.2 6
3.2 odd 2 245.4.a.o.1.5 6
7.6 odd 2 2205.4.a.ca.1.2 6
15.14 odd 2 1225.4.a.bj.1.2 6
21.2 odd 6 245.4.e.q.116.2 12
21.5 even 6 245.4.e.p.116.2 12
21.11 odd 6 245.4.e.q.226.2 12
21.17 even 6 245.4.e.p.226.2 12
21.20 even 2 245.4.a.p.1.5 yes 6
105.104 even 2 1225.4.a.bi.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.5 6 3.2 odd 2
245.4.a.p.1.5 yes 6 21.20 even 2
245.4.e.p.116.2 12 21.5 even 6
245.4.e.p.226.2 12 21.17 even 6
245.4.e.q.116.2 12 21.2 odd 6
245.4.e.q.226.2 12 21.11 odd 6
1225.4.a.bi.1.2 6 105.104 even 2
1225.4.a.bj.1.2 6 15.14 odd 2
2205.4.a.bz.1.2 6 1.1 even 1 trivial
2205.4.a.ca.1.2 6 7.6 odd 2