Properties

Label 2205.4.a.ca.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
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: \(0\)
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 \(-0.239524\pi\)
0.729991 + 0.683456i \(0.239524\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.70053 0.0890707
\(17\) −104.332 −1.48848 −0.744240 0.667912i \(-0.767188\pi\)
−0.744240 + 0.667912i \(0.767188\pi\)
\(18\) 0 0
\(19\) 71.8929 0.868072 0.434036 0.900896i \(-0.357089\pi\)
0.434036 + 0.900896i \(0.357089\pi\)
\(20\) −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 \(-0.568439\pi\)
−0.213355 + 0.976975i \(0.568439\pi\)
\(32\) −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 \(-0.792417\pi\)
−0.794786 + 0.606889i \(0.792417\pi\)
\(42\) 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 \(-0.574623\pi\)
−0.232293 + 0.972646i \(0.574623\pi\)
\(48\) 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 \(-0.306347\pi\)
0.571538 + 0.820575i \(0.306347\pi\)
\(60\) 0 0
\(61\) −219.926 −0.461616 −0.230808 0.972999i \(-0.574137\pi\)
−0.230808 + 0.972999i \(0.574137\pi\)
\(62\) 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 \(-0.249921\pi\)
0.707283 + 0.706931i \(0.249921\pi\)
\(74\) 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 \(-0.514924\pi\)
−0.0468694 + 0.998901i \(0.514924\pi\)
\(84\) 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 \(-0.658619\pi\)
−0.477947 + 0.878389i \(0.658619\pi\)
\(90\) 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 \(-0.475712\pi\)
0.0762283 + 0.997090i \(0.475712\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −131.436 −0.131436
\(101\) 619.435 0.610259 0.305129 0.952311i \(-0.401300\pi\)
0.305129 + 0.952311i \(0.401300\pi\)
\(102\) 0 0
\(103\) −1822.08 −1.74306 −0.871528 0.490345i \(-0.836871\pi\)
−0.871528 + 0.490345i \(0.836871\pi\)
\(104\) 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 \(-0.706143\pi\)
−0.603289 + 0.797523i \(0.706143\pi\)
\(132\) 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 \(-0.561133\pi\)
−0.190875 + 0.981614i \(0.561133\pi\)
\(140\) 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 \(-0.675532\pi\)
−0.523922 + 0.851767i \(0.675532\pi\)
\(158\) −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 \(-0.730066\pi\)
−0.661467 + 0.749974i \(0.730066\pi\)
\(168\) 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 \(-0.389134\pi\)
0.341295 + 0.939956i \(0.389134\pi\)
\(174\) 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 \(-0.348635\pi\)
0.457806 + 0.889052i \(0.348635\pi\)
\(182\) 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 \(-0.171651\pi\)
0.858091 + 0.513498i \(0.171651\pi\)
\(200\) 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 \(-0.273842\pi\)
0.652211 + 0.758037i \(0.273842\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1491.05 −0.438863
\(227\) 2647.59 0.774127 0.387064 0.922053i \(-0.373489\pi\)
0.387064 + 0.922053i \(0.373489\pi\)
\(228\) 0 0
\(229\) −1445.09 −0.417006 −0.208503 0.978022i \(-0.566859\pi\)
−0.208503 + 0.978022i \(0.566859\pi\)
\(230\) −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 \(-0.351294\pi\)
0.450364 + 0.892845i \(0.351294\pi\)
\(242\) −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 \(-0.350809\pi\)
0.451724 + 0.892158i \(0.350809\pi\)
\(252\) 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 \(-0.499890\pi\)
0.000345584 1.00000i \(0.499890\pi\)
\(258\) 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 \(-0.447549\pi\)
0.164036 + 0.986454i \(0.447549\pi\)
\(270\) 0 0
\(271\) 8054.50 1.80545 0.902724 0.430221i \(-0.141564\pi\)
0.902724 + 0.430221i \(0.141564\pi\)
\(272\) −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 \(-0.386592\pi\)
0.348791 + 0.937201i \(0.386592\pi\)
\(284\) −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 \(-0.332079\pi\)
0.503408 + 0.864049i \(0.332079\pi\)
\(294\) 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 \(-0.545595\pi\)
−0.142750 + 0.989759i \(0.545595\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 609.849 0.111733
\(311\) 9283.05 1.69258 0.846291 0.532720i \(-0.178830\pi\)
0.846291 + 0.532720i \(0.178830\pi\)
\(312\) 0 0
\(313\) 6025.43 1.08811 0.544054 0.839050i \(-0.316889\pi\)
0.544054 + 0.839050i \(0.316889\pi\)
\(314\) 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 \(-0.549433\pi\)
−0.154674 + 0.987966i \(0.549433\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12876.6 1.94979
\(353\) 7589.41 1.14432 0.572158 0.820143i \(-0.306106\pi\)
0.572158 + 0.820143i \(0.306106\pi\)
\(354\) 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 \(-0.542926\pi\)
−0.134449 + 0.990921i \(0.542926\pi\)
\(368\) 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 \(-0.287146\pi\)
0.619968 + 0.784627i \(0.287146\pi\)
\(384\) 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 \(-0.573611\pi\)
−0.229200 + 0.973379i \(0.573611\pi\)
\(398\) −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 \(-0.843384\pi\)
−0.881379 + 0.472409i \(0.843384\pi\)
\(410\) 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 \(-0.452739\pi\)
0.147931 + 0.988998i \(0.452739\pi\)
\(420\) 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 \(-0.283231\pi\)
0.629570 + 0.776944i \(0.283231\pi\)
\(434\) 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 \(-0.280455\pi\)
0.636323 + 0.771423i \(0.280455\pi\)
\(440\) −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 \(-0.507278\pi\)
−0.0228614 + 0.999739i \(0.507278\pi\)
\(462\) 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 \(-0.515363\pi\)
−0.0482463 + 0.998835i \(0.515363\pi\)
\(468\) 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 \(-0.653336\pi\)
−0.463303 + 0.886200i \(0.653336\pi\)
\(480\) 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 \(-0.474963\pi\)
0.0785757 + 0.996908i \(0.474963\pi\)
\(504\) 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 \(-0.543820\pi\)
−0.137229 + 0.990539i \(0.543820\pi\)
\(510\) 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 \(-0.346511\pi\)
0.463729 + 0.885977i \(0.346511\pi\)
\(522\) 0 0
\(523\) 14448.4 1.20800 0.604002 0.796983i \(-0.293572\pi\)
0.604002 + 0.796983i \(0.293572\pi\)
\(524\) 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 \(-0.426007\pi\)
0.230370 + 0.973103i \(0.426007\pi\)
\(564\) 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 \(-0.453183\pi\)
0.146549 + 0.989203i \(0.453183\pi\)
\(578\) −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 \(-0.404720\pi\)
0.294880 + 0.955534i \(0.404720\pi\)
\(588\) 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 \(-0.321728\pi\)
0.531236 + 0.847224i \(0.321728\pi\)
\(594\) 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 \(-0.723087\pi\)
−0.644866 + 0.764295i \(0.723087\pi\)
\(602\) 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 \(-0.0556939\pi\)
0.984732 + 0.174076i \(0.0556939\pi\)
\(608\) −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 \(-0.503252\pi\)
−0.0102161 + 0.999948i \(0.503252\pi\)
\(620\) 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 \(-0.274391\pi\)
0.650902 + 0.759162i \(0.274391\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12421.6 0.756537
\(647\) 2740.35 0.166514 0.0832568 0.996528i \(-0.473468\pi\)
0.0832568 + 0.996528i \(0.473468\pi\)
\(648\) 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 \(-0.347823\pi\)
0.460075 + 0.887880i \(0.347823\pi\)
\(662\) −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 \(-0.574399\pi\)
−0.231608 + 0.972809i \(0.574399\pi\)
\(678\) 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 \(-0.508038\pi\)
−0.0252500 + 0.999681i \(0.508038\pi\)
\(692\) −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 \(-0.467532\pi\)
0.101824 + 0.994802i \(0.467532\pi\)
\(720\) 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 \(-0.680621\pi\)
−0.537474 + 0.843281i \(0.680621\pi\)
\(728\) 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 \(-0.695181\pi\)
−0.575470 + 0.817823i \(0.695181\pi\)
\(734\) 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 \(-0.524162\pi\)
−0.0758337 + 0.997120i \(0.524162\pi\)
\(762\) 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 \(-0.211267\pi\)
0.787708 + 0.616048i \(0.211267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23208.8 1.08200
\(773\) −34386.0 −1.59997 −0.799986 0.600019i \(-0.795160\pi\)
−0.799986 + 0.600019i \(0.795160\pi\)
\(774\) 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 \(-0.559548\pi\)
−0.185986 + 0.982552i \(0.559548\pi\)
\(788\) −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 \(-0.804769\pi\)
−0.817732 + 0.575600i \(0.804769\pi\)
\(798\) 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 \(-0.348246\pi\)
0.458893 + 0.888492i \(0.348246\pi\)
\(812\) 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 \(-0.437912\pi\)
0.193821 + 0.981037i \(0.437912\pi\)
\(830\) 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 \(-0.248614\pi\)
0.710178 + 0.704022i \(0.248614\pi\)
\(840\) 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 \(-0.509575\pi\)
−0.0300763 + 0.999548i \(0.509575\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −23924.6 −0.955287
\(857\) 9357.02 0.372963 0.186482 0.982458i \(-0.440291\pi\)
0.186482 + 0.982458i \(0.440291\pi\)
\(858\) 0 0
\(859\) −29960.9 −1.19005 −0.595025 0.803707i \(-0.702858\pi\)
−0.595025 + 0.803707i \(0.702858\pi\)
\(860\) −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 \(-0.712759\pi\)
−0.619732 + 0.784813i \(0.712759\pi\)
\(882\) 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 \(-0.495856\pi\)
0.0130180 + 0.999915i \(0.495856\pi\)
\(888\) 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 \(-0.457052\pi\)
0.134515 + 0.990912i \(0.457052\pi\)
\(930\) 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 \(-0.862454\pi\)
−0.908082 + 0.418792i \(0.862454\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3935.13 0.136542
\(941\) 56765.1 1.96651 0.983257 0.182225i \(-0.0583299\pi\)
0.983257 + 0.182225i \(0.0583299\pi\)
\(942\) 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 \(-0.660725\pi\)
−0.483749 + 0.875207i \(0.660725\pi\)
\(972\) 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 \(-0.555039\pi\)
−0.172049 + 0.985088i \(0.555039\pi\)
\(984\) 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 \(-0.789405\pi\)
−0.789008 + 0.614383i \(0.789405\pi\)
\(998\) 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.ca.1.2 6
3.2 odd 2 245.4.a.p.1.5 yes 6
7.6 odd 2 2205.4.a.bz.1.2 6
15.14 odd 2 1225.4.a.bi.1.2 6
21.2 odd 6 245.4.e.p.116.2 12
21.5 even 6 245.4.e.q.116.2 12
21.11 odd 6 245.4.e.p.226.2 12
21.17 even 6 245.4.e.q.226.2 12
21.20 even 2 245.4.a.o.1.5 6
105.104 even 2 1225.4.a.bj.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.5 6 21.20 even 2
245.4.a.p.1.5 yes 6 3.2 odd 2
245.4.e.p.116.2 12 21.2 odd 6
245.4.e.p.226.2 12 21.11 odd 6
245.4.e.q.116.2 12 21.5 even 6
245.4.e.q.226.2 12 21.17 even 6
1225.4.a.bi.1.2 6 15.14 odd 2
1225.4.a.bj.1.2 6 105.104 even 2
2205.4.a.bz.1.2 6 7.6 odd 2
2205.4.a.ca.1.2 6 1.1 even 1 trivial