Properties

Label 2070.4.a.bj.1.3
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(122.133953712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 68 x^{2} - 111 x + 342\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(8.73081\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +10.0000 q^{10} +32.1352 q^{11} +40.0811 q^{13} +47.1245 q^{14} +16.0000 q^{16} -126.165 q^{17} +0.232742 q^{19} +20.0000 q^{20} +64.2704 q^{22} +23.0000 q^{23} +25.0000 q^{25} +80.1621 q^{26} +94.2490 q^{28} +137.226 q^{29} +112.866 q^{31} +32.0000 q^{32} -252.331 q^{34} +117.811 q^{35} +45.7057 q^{37} +0.465483 q^{38} +40.0000 q^{40} +135.385 q^{41} +543.528 q^{43} +128.541 q^{44} +46.0000 q^{46} -26.4344 q^{47} +212.180 q^{49} +50.0000 q^{50} +160.324 q^{52} -43.6958 q^{53} +160.676 q^{55} +188.498 q^{56} +274.451 q^{58} -202.248 q^{59} +150.279 q^{61} +225.732 q^{62} +64.0000 q^{64} +200.405 q^{65} -420.722 q^{67} -504.661 q^{68} +235.622 q^{70} -667.381 q^{71} +602.960 q^{73} +91.4115 q^{74} +0.930966 q^{76} +757.178 q^{77} -1378.88 q^{79} +80.0000 q^{80} +270.769 q^{82} +485.178 q^{83} -630.826 q^{85} +1087.06 q^{86} +257.082 q^{88} +1127.71 q^{89} +944.400 q^{91} +92.0000 q^{92} -52.8688 q^{94} +1.16371 q^{95} -1486.24 q^{97} +424.359 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + O(q^{10}) \) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 23.5622 1.27224 0.636121 0.771589i \(-0.280538\pi\)
0.636121 + 0.771589i \(0.280538\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 32.1352 0.880830 0.440415 0.897794i \(-0.354831\pi\)
0.440415 + 0.897794i \(0.354831\pi\)
\(12\) 0 0
\(13\) 40.0811 0.855115 0.427557 0.903988i \(-0.359374\pi\)
0.427557 + 0.903988i \(0.359374\pi\)
\(14\) 47.1245 0.899611
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −126.165 −1.79997 −0.899987 0.435916i \(-0.856424\pi\)
−0.899987 + 0.435916i \(0.856424\pi\)
\(18\) 0 0
\(19\) 0.232742 0.00281024 0.00140512 0.999999i \(-0.499553\pi\)
0.00140512 + 0.999999i \(0.499553\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 64.2704 0.622841
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 80.1621 0.604657
\(27\) 0 0
\(28\) 94.2490 0.636121
\(29\) 137.226 0.878695 0.439347 0.898317i \(-0.355210\pi\)
0.439347 + 0.898317i \(0.355210\pi\)
\(30\) 0 0
\(31\) 112.866 0.653914 0.326957 0.945039i \(-0.393977\pi\)
0.326957 + 0.945039i \(0.393977\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −252.331 −1.27277
\(35\) 117.811 0.568964
\(36\) 0 0
\(37\) 45.7057 0.203080 0.101540 0.994831i \(-0.467623\pi\)
0.101540 + 0.994831i \(0.467623\pi\)
\(38\) 0.465483 0.00198714
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) 135.385 0.515696 0.257848 0.966186i \(-0.416987\pi\)
0.257848 + 0.966186i \(0.416987\pi\)
\(42\) 0 0
\(43\) 543.528 1.92761 0.963805 0.266608i \(-0.0859030\pi\)
0.963805 + 0.266608i \(0.0859030\pi\)
\(44\) 128.541 0.440415
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −26.4344 −0.0820394 −0.0410197 0.999158i \(-0.513061\pi\)
−0.0410197 + 0.999158i \(0.513061\pi\)
\(48\) 0 0
\(49\) 212.180 0.618599
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) 160.324 0.427557
\(53\) −43.6958 −0.113247 −0.0566235 0.998396i \(-0.518033\pi\)
−0.0566235 + 0.998396i \(0.518033\pi\)
\(54\) 0 0
\(55\) 160.676 0.393919
\(56\) 188.498 0.449805
\(57\) 0 0
\(58\) 274.451 0.621331
\(59\) −202.248 −0.446278 −0.223139 0.974787i \(-0.571630\pi\)
−0.223139 + 0.974787i \(0.571630\pi\)
\(60\) 0 0
\(61\) 150.279 0.315430 0.157715 0.987485i \(-0.449587\pi\)
0.157715 + 0.987485i \(0.449587\pi\)
\(62\) 225.732 0.462387
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 200.405 0.382419
\(66\) 0 0
\(67\) −420.722 −0.767155 −0.383577 0.923509i \(-0.625308\pi\)
−0.383577 + 0.923509i \(0.625308\pi\)
\(68\) −504.661 −0.899987
\(69\) 0 0
\(70\) 235.622 0.402318
\(71\) −667.381 −1.11554 −0.557771 0.829995i \(-0.688343\pi\)
−0.557771 + 0.829995i \(0.688343\pi\)
\(72\) 0 0
\(73\) 602.960 0.966728 0.483364 0.875420i \(-0.339415\pi\)
0.483364 + 0.875420i \(0.339415\pi\)
\(74\) 91.4115 0.143600
\(75\) 0 0
\(76\) 0.930966 0.00140512
\(77\) 757.178 1.12063
\(78\) 0 0
\(79\) −1378.88 −1.96374 −0.981872 0.189545i \(-0.939299\pi\)
−0.981872 + 0.189545i \(0.939299\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 270.769 0.364652
\(83\) 485.178 0.641629 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(84\) 0 0
\(85\) −630.826 −0.804973
\(86\) 1087.06 1.36303
\(87\) 0 0
\(88\) 257.082 0.311420
\(89\) 1127.71 1.34311 0.671556 0.740954i \(-0.265626\pi\)
0.671556 + 0.740954i \(0.265626\pi\)
\(90\) 0 0
\(91\) 944.400 1.08791
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) −52.8688 −0.0580106
\(95\) 1.16371 0.00125678
\(96\) 0 0
\(97\) −1486.24 −1.55572 −0.777858 0.628441i \(-0.783694\pi\)
−0.777858 + 0.628441i \(0.783694\pi\)
\(98\) 424.359 0.437416
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −1888.86 −1.86087 −0.930437 0.366451i \(-0.880573\pi\)
−0.930437 + 0.366451i \(0.880573\pi\)
\(102\) 0 0
\(103\) −1497.17 −1.43224 −0.716118 0.697979i \(-0.754083\pi\)
−0.716118 + 0.697979i \(0.754083\pi\)
\(104\) 320.649 0.302329
\(105\) 0 0
\(106\) −87.3917 −0.0800777
\(107\) 585.150 0.528678 0.264339 0.964430i \(-0.414846\pi\)
0.264339 + 0.964430i \(0.414846\pi\)
\(108\) 0 0
\(109\) −139.166 −0.122290 −0.0611452 0.998129i \(-0.519475\pi\)
−0.0611452 + 0.998129i \(0.519475\pi\)
\(110\) 321.352 0.278543
\(111\) 0 0
\(112\) 376.996 0.318060
\(113\) 1293.18 1.07656 0.538282 0.842765i \(-0.319073\pi\)
0.538282 + 0.842765i \(0.319073\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 548.902 0.439347
\(117\) 0 0
\(118\) −404.495 −0.315566
\(119\) −2972.74 −2.29000
\(120\) 0 0
\(121\) −298.328 −0.224138
\(122\) 300.557 0.223043
\(123\) 0 0
\(124\) 451.464 0.326957
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1298.43 −0.907221 −0.453610 0.891200i \(-0.649864\pi\)
−0.453610 + 0.891200i \(0.649864\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 400.811 0.270411
\(131\) 525.247 0.350313 0.175157 0.984541i \(-0.443957\pi\)
0.175157 + 0.984541i \(0.443957\pi\)
\(132\) 0 0
\(133\) 5.48392 0.00357531
\(134\) −841.444 −0.542460
\(135\) 0 0
\(136\) −1009.32 −0.636387
\(137\) 2428.12 1.51422 0.757109 0.653288i \(-0.226611\pi\)
0.757109 + 0.653288i \(0.226611\pi\)
\(138\) 0 0
\(139\) 2456.46 1.49895 0.749476 0.662031i \(-0.230305\pi\)
0.749476 + 0.662031i \(0.230305\pi\)
\(140\) 471.245 0.284482
\(141\) 0 0
\(142\) −1334.76 −0.788808
\(143\) 1288.01 0.753211
\(144\) 0 0
\(145\) 686.128 0.392964
\(146\) 1205.92 0.683580
\(147\) 0 0
\(148\) 182.823 0.101540
\(149\) 2405.39 1.32253 0.661266 0.750151i \(-0.270019\pi\)
0.661266 + 0.750151i \(0.270019\pi\)
\(150\) 0 0
\(151\) −649.276 −0.349916 −0.174958 0.984576i \(-0.555979\pi\)
−0.174958 + 0.984576i \(0.555979\pi\)
\(152\) 1.86193 0.000993570 0
\(153\) 0 0
\(154\) 1514.36 0.792404
\(155\) 564.330 0.292439
\(156\) 0 0
\(157\) 3665.04 1.86307 0.931536 0.363650i \(-0.118470\pi\)
0.931536 + 0.363650i \(0.118470\pi\)
\(158\) −2757.75 −1.38858
\(159\) 0 0
\(160\) 160.000 0.0790569
\(161\) 541.932 0.265281
\(162\) 0 0
\(163\) −1055.27 −0.507088 −0.253544 0.967324i \(-0.581596\pi\)
−0.253544 + 0.967324i \(0.581596\pi\)
\(164\) 541.539 0.257848
\(165\) 0 0
\(166\) 970.356 0.453700
\(167\) 731.805 0.339095 0.169547 0.985522i \(-0.445769\pi\)
0.169547 + 0.985522i \(0.445769\pi\)
\(168\) 0 0
\(169\) −590.507 −0.268779
\(170\) −1261.65 −0.569202
\(171\) 0 0
\(172\) 2174.11 0.963805
\(173\) 521.773 0.229304 0.114652 0.993406i \(-0.463425\pi\)
0.114652 + 0.993406i \(0.463425\pi\)
\(174\) 0 0
\(175\) 589.056 0.254448
\(176\) 514.163 0.220208
\(177\) 0 0
\(178\) 2255.42 0.949723
\(179\) 1183.07 0.494005 0.247002 0.969015i \(-0.420554\pi\)
0.247002 + 0.969015i \(0.420554\pi\)
\(180\) 0 0
\(181\) 1723.92 0.707946 0.353973 0.935256i \(-0.384830\pi\)
0.353973 + 0.935256i \(0.384830\pi\)
\(182\) 1888.80 0.769270
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 228.529 0.0908204
\(186\) 0 0
\(187\) −4054.35 −1.58547
\(188\) −105.738 −0.0410197
\(189\) 0 0
\(190\) 2.32742 0.000888676 0
\(191\) 2798.12 1.06002 0.530012 0.847990i \(-0.322187\pi\)
0.530012 + 0.847990i \(0.322187\pi\)
\(192\) 0 0
\(193\) −497.686 −0.185618 −0.0928089 0.995684i \(-0.529585\pi\)
−0.0928089 + 0.995684i \(0.529585\pi\)
\(194\) −2972.47 −1.10006
\(195\) 0 0
\(196\) 848.718 0.309300
\(197\) −296.489 −0.107228 −0.0536141 0.998562i \(-0.517074\pi\)
−0.0536141 + 0.998562i \(0.517074\pi\)
\(198\) 0 0
\(199\) 2347.43 0.836207 0.418103 0.908399i \(-0.362695\pi\)
0.418103 + 0.908399i \(0.362695\pi\)
\(200\) 200.000 0.0707107
\(201\) 0 0
\(202\) −3777.71 −1.31584
\(203\) 3233.34 1.11791
\(204\) 0 0
\(205\) 676.923 0.230626
\(206\) −2994.34 −1.01274
\(207\) 0 0
\(208\) 641.297 0.213779
\(209\) 7.47920 0.00247535
\(210\) 0 0
\(211\) 2838.61 0.926153 0.463076 0.886318i \(-0.346746\pi\)
0.463076 + 0.886318i \(0.346746\pi\)
\(212\) −174.783 −0.0566235
\(213\) 0 0
\(214\) 1170.30 0.373832
\(215\) 2717.64 0.862053
\(216\) 0 0
\(217\) 2659.38 0.831937
\(218\) −278.331 −0.0864723
\(219\) 0 0
\(220\) 642.704 0.196960
\(221\) −5056.84 −1.53918
\(222\) 0 0
\(223\) −2124.68 −0.638024 −0.319012 0.947751i \(-0.603351\pi\)
−0.319012 + 0.947751i \(0.603351\pi\)
\(224\) 753.992 0.224903
\(225\) 0 0
\(226\) 2586.35 0.761246
\(227\) −1551.97 −0.453780 −0.226890 0.973920i \(-0.572856\pi\)
−0.226890 + 0.973920i \(0.572856\pi\)
\(228\) 0 0
\(229\) −158.531 −0.0457469 −0.0228735 0.999738i \(-0.507281\pi\)
−0.0228735 + 0.999738i \(0.507281\pi\)
\(230\) 230.000 0.0659380
\(231\) 0 0
\(232\) 1097.80 0.310666
\(233\) 728.999 0.204971 0.102486 0.994734i \(-0.467320\pi\)
0.102486 + 0.994734i \(0.467320\pi\)
\(234\) 0 0
\(235\) −132.172 −0.0366891
\(236\) −808.991 −0.223139
\(237\) 0 0
\(238\) −5945.47 −1.61928
\(239\) 6201.60 1.67844 0.839222 0.543789i \(-0.183011\pi\)
0.839222 + 0.543789i \(0.183011\pi\)
\(240\) 0 0
\(241\) −7223.04 −1.93061 −0.965305 0.261126i \(-0.915906\pi\)
−0.965305 + 0.261126i \(0.915906\pi\)
\(242\) −596.656 −0.158490
\(243\) 0 0
\(244\) 601.115 0.157715
\(245\) 1060.90 0.276646
\(246\) 0 0
\(247\) 9.32853 0.00240308
\(248\) 902.928 0.231193
\(249\) 0 0
\(250\) 250.000 0.0632456
\(251\) −5042.78 −1.26812 −0.634059 0.773285i \(-0.718612\pi\)
−0.634059 + 0.773285i \(0.718612\pi\)
\(252\) 0 0
\(253\) 739.110 0.183666
\(254\) −2596.86 −0.641502
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2981.15 −0.723577 −0.361788 0.932260i \(-0.617834\pi\)
−0.361788 + 0.932260i \(0.617834\pi\)
\(258\) 0 0
\(259\) 1076.93 0.258367
\(260\) 801.621 0.191209
\(261\) 0 0
\(262\) 1050.49 0.247709
\(263\) −7242.86 −1.69815 −0.849076 0.528271i \(-0.822841\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(264\) 0 0
\(265\) −218.479 −0.0506456
\(266\) 10.9678 0.00252812
\(267\) 0 0
\(268\) −1682.89 −0.383577
\(269\) 2567.58 0.581962 0.290981 0.956729i \(-0.406018\pi\)
0.290981 + 0.956729i \(0.406018\pi\)
\(270\) 0 0
\(271\) 8066.27 1.80808 0.904042 0.427443i \(-0.140586\pi\)
0.904042 + 0.427443i \(0.140586\pi\)
\(272\) −2018.64 −0.449994
\(273\) 0 0
\(274\) 4856.23 1.07071
\(275\) 803.380 0.176166
\(276\) 0 0
\(277\) 8991.54 1.95036 0.975179 0.221418i \(-0.0710684\pi\)
0.975179 + 0.221418i \(0.0710684\pi\)
\(278\) 4912.92 1.05992
\(279\) 0 0
\(280\) 942.490 0.201159
\(281\) −968.130 −0.205529 −0.102765 0.994706i \(-0.532769\pi\)
−0.102765 + 0.994706i \(0.532769\pi\)
\(282\) 0 0
\(283\) −2252.65 −0.473167 −0.236583 0.971611i \(-0.576028\pi\)
−0.236583 + 0.971611i \(0.576028\pi\)
\(284\) −2669.52 −0.557771
\(285\) 0 0
\(286\) 2576.03 0.532600
\(287\) 3189.97 0.656090
\(288\) 0 0
\(289\) 11004.7 2.23991
\(290\) 1372.26 0.277868
\(291\) 0 0
\(292\) 2411.84 0.483364
\(293\) −2734.35 −0.545196 −0.272598 0.962128i \(-0.587883\pi\)
−0.272598 + 0.962128i \(0.587883\pi\)
\(294\) 0 0
\(295\) −1011.24 −0.199582
\(296\) 365.646 0.0717998
\(297\) 0 0
\(298\) 4810.78 0.935172
\(299\) 921.865 0.178304
\(300\) 0 0
\(301\) 12806.7 2.45239
\(302\) −1298.55 −0.247428
\(303\) 0 0
\(304\) 3.72387 0.000702560 0
\(305\) 751.394 0.141065
\(306\) 0 0
\(307\) −7977.09 −1.48299 −0.741493 0.670961i \(-0.765882\pi\)
−0.741493 + 0.670961i \(0.765882\pi\)
\(308\) 3028.71 0.560314
\(309\) 0 0
\(310\) 1128.66 0.206786
\(311\) −7729.43 −1.40931 −0.704655 0.709550i \(-0.748898\pi\)
−0.704655 + 0.709550i \(0.748898\pi\)
\(312\) 0 0
\(313\) 5506.25 0.994350 0.497175 0.867650i \(-0.334371\pi\)
0.497175 + 0.867650i \(0.334371\pi\)
\(314\) 7330.08 1.31739
\(315\) 0 0
\(316\) −5515.51 −0.981872
\(317\) −5231.37 −0.926887 −0.463443 0.886126i \(-0.653386\pi\)
−0.463443 + 0.886126i \(0.653386\pi\)
\(318\) 0 0
\(319\) 4409.77 0.773981
\(320\) 320.000 0.0559017
\(321\) 0 0
\(322\) 1083.86 0.187582
\(323\) −29.3639 −0.00505836
\(324\) 0 0
\(325\) 1002.03 0.171023
\(326\) −2110.55 −0.358566
\(327\) 0 0
\(328\) 1083.08 0.182326
\(329\) −622.854 −0.104374
\(330\) 0 0
\(331\) −3551.61 −0.589771 −0.294885 0.955533i \(-0.595281\pi\)
−0.294885 + 0.955533i \(0.595281\pi\)
\(332\) 1940.71 0.320815
\(333\) 0 0
\(334\) 1463.61 0.239776
\(335\) −2103.61 −0.343082
\(336\) 0 0
\(337\) −7002.99 −1.13198 −0.565990 0.824412i \(-0.691506\pi\)
−0.565990 + 0.824412i \(0.691506\pi\)
\(338\) −1181.01 −0.190055
\(339\) 0 0
\(340\) −2523.31 −0.402487
\(341\) 3626.97 0.575987
\(342\) 0 0
\(343\) −3082.42 −0.485234
\(344\) 4348.22 0.681513
\(345\) 0 0
\(346\) 1043.55 0.162143
\(347\) 10268.2 1.58854 0.794272 0.607562i \(-0.207852\pi\)
0.794272 + 0.607562i \(0.207852\pi\)
\(348\) 0 0
\(349\) 4515.58 0.692588 0.346294 0.938126i \(-0.387440\pi\)
0.346294 + 0.938126i \(0.387440\pi\)
\(350\) 1178.11 0.179922
\(351\) 0 0
\(352\) 1028.33 0.155710
\(353\) −7838.63 −1.18189 −0.590947 0.806711i \(-0.701246\pi\)
−0.590947 + 0.806711i \(0.701246\pi\)
\(354\) 0 0
\(355\) −3336.90 −0.498886
\(356\) 4510.84 0.671556
\(357\) 0 0
\(358\) 2366.14 0.349314
\(359\) 12464.8 1.83250 0.916250 0.400606i \(-0.131201\pi\)
0.916250 + 0.400606i \(0.131201\pi\)
\(360\) 0 0
\(361\) −6858.95 −0.999992
\(362\) 3447.85 0.500594
\(363\) 0 0
\(364\) 3777.60 0.543956
\(365\) 3014.80 0.432334
\(366\) 0 0
\(367\) 3936.14 0.559850 0.279925 0.960022i \(-0.409690\pi\)
0.279925 + 0.960022i \(0.409690\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) 457.057 0.0642197
\(371\) −1029.57 −0.144077
\(372\) 0 0
\(373\) 1920.72 0.266625 0.133313 0.991074i \(-0.457439\pi\)
0.133313 + 0.991074i \(0.457439\pi\)
\(374\) −8108.69 −1.12110
\(375\) 0 0
\(376\) −211.475 −0.0290053
\(377\) 5500.15 0.751385
\(378\) 0 0
\(379\) −13074.5 −1.77201 −0.886004 0.463678i \(-0.846530\pi\)
−0.886004 + 0.463678i \(0.846530\pi\)
\(380\) 4.65483 0.000628389 0
\(381\) 0 0
\(382\) 5596.23 0.749550
\(383\) −8838.22 −1.17914 −0.589571 0.807716i \(-0.700703\pi\)
−0.589571 + 0.807716i \(0.700703\pi\)
\(384\) 0 0
\(385\) 3785.89 0.501160
\(386\) −995.372 −0.131252
\(387\) 0 0
\(388\) −5944.94 −0.777858
\(389\) 12687.3 1.65365 0.826827 0.562456i \(-0.190144\pi\)
0.826827 + 0.562456i \(0.190144\pi\)
\(390\) 0 0
\(391\) −2901.80 −0.375321
\(392\) 1697.44 0.218708
\(393\) 0 0
\(394\) −592.977 −0.0758218
\(395\) −6894.38 −0.878213
\(396\) 0 0
\(397\) 8060.49 1.01900 0.509501 0.860470i \(-0.329830\pi\)
0.509501 + 0.860470i \(0.329830\pi\)
\(398\) 4694.87 0.591287
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) −12325.1 −1.53488 −0.767440 0.641121i \(-0.778470\pi\)
−0.767440 + 0.641121i \(0.778470\pi\)
\(402\) 0 0
\(403\) 4523.79 0.559171
\(404\) −7555.43 −0.930437
\(405\) 0 0
\(406\) 6466.69 0.790483
\(407\) 1468.76 0.178879
\(408\) 0 0
\(409\) −10806.7 −1.30650 −0.653250 0.757143i \(-0.726595\pi\)
−0.653250 + 0.757143i \(0.726595\pi\)
\(410\) 1353.85 0.163077
\(411\) 0 0
\(412\) −5988.67 −0.716118
\(413\) −4765.41 −0.567774
\(414\) 0 0
\(415\) 2425.89 0.286945
\(416\) 1282.59 0.151164
\(417\) 0 0
\(418\) 14.9584 0.00175033
\(419\) 1823.74 0.212639 0.106319 0.994332i \(-0.466093\pi\)
0.106319 + 0.994332i \(0.466093\pi\)
\(420\) 0 0
\(421\) 5995.43 0.694060 0.347030 0.937854i \(-0.387190\pi\)
0.347030 + 0.937854i \(0.387190\pi\)
\(422\) 5677.23 0.654889
\(423\) 0 0
\(424\) −349.567 −0.0400388
\(425\) −3154.13 −0.359995
\(426\) 0 0
\(427\) 3540.91 0.401303
\(428\) 2340.60 0.264339
\(429\) 0 0
\(430\) 5435.28 0.609564
\(431\) −3887.80 −0.434499 −0.217249 0.976116i \(-0.569708\pi\)
−0.217249 + 0.976116i \(0.569708\pi\)
\(432\) 0 0
\(433\) −4422.93 −0.490884 −0.245442 0.969411i \(-0.578933\pi\)
−0.245442 + 0.969411i \(0.578933\pi\)
\(434\) 5318.75 0.588268
\(435\) 0 0
\(436\) −556.662 −0.0611452
\(437\) 5.35306 0.000585976 0
\(438\) 0 0
\(439\) −1748.08 −0.190048 −0.0950242 0.995475i \(-0.530293\pi\)
−0.0950242 + 0.995475i \(0.530293\pi\)
\(440\) 1285.41 0.139271
\(441\) 0 0
\(442\) −10113.7 −1.08837
\(443\) −3371.31 −0.361571 −0.180785 0.983523i \(-0.557864\pi\)
−0.180785 + 0.983523i \(0.557864\pi\)
\(444\) 0 0
\(445\) 5638.55 0.600658
\(446\) −4249.37 −0.451151
\(447\) 0 0
\(448\) 1507.98 0.159030
\(449\) −13314.7 −1.39947 −0.699734 0.714404i \(-0.746698\pi\)
−0.699734 + 0.714404i \(0.746698\pi\)
\(450\) 0 0
\(451\) 4350.61 0.454240
\(452\) 5172.70 0.538282
\(453\) 0 0
\(454\) −3103.94 −0.320871
\(455\) 4722.00 0.486529
\(456\) 0 0
\(457\) 3767.20 0.385607 0.192803 0.981237i \(-0.438242\pi\)
0.192803 + 0.981237i \(0.438242\pi\)
\(458\) −317.063 −0.0323480
\(459\) 0 0
\(460\) 460.000 0.0466252
\(461\) −9674.06 −0.977366 −0.488683 0.872461i \(-0.662523\pi\)
−0.488683 + 0.872461i \(0.662523\pi\)
\(462\) 0 0
\(463\) 2977.73 0.298892 0.149446 0.988770i \(-0.452251\pi\)
0.149446 + 0.988770i \(0.452251\pi\)
\(464\) 2195.61 0.219674
\(465\) 0 0
\(466\) 1458.00 0.144937
\(467\) 10701.0 1.06035 0.530176 0.847888i \(-0.322126\pi\)
0.530176 + 0.847888i \(0.322126\pi\)
\(468\) 0 0
\(469\) −9913.16 −0.976006
\(470\) −264.344 −0.0259431
\(471\) 0 0
\(472\) −1617.98 −0.157783
\(473\) 17466.4 1.69790
\(474\) 0 0
\(475\) 5.81854 0.000562048 0
\(476\) −11890.9 −1.14500
\(477\) 0 0
\(478\) 12403.2 1.18684
\(479\) −2337.15 −0.222937 −0.111469 0.993768i \(-0.535555\pi\)
−0.111469 + 0.993768i \(0.535555\pi\)
\(480\) 0 0
\(481\) 1831.94 0.173657
\(482\) −14446.1 −1.36515
\(483\) 0 0
\(484\) −1193.31 −0.112069
\(485\) −7431.18 −0.695737
\(486\) 0 0
\(487\) −7183.85 −0.668442 −0.334221 0.942495i \(-0.608473\pi\)
−0.334221 + 0.942495i \(0.608473\pi\)
\(488\) 1202.23 0.111521
\(489\) 0 0
\(490\) 2121.80 0.195618
\(491\) 12084.2 1.11070 0.555350 0.831617i \(-0.312584\pi\)
0.555350 + 0.831617i \(0.312584\pi\)
\(492\) 0 0
\(493\) −17313.1 −1.58163
\(494\) 18.6571 0.00169923
\(495\) 0 0
\(496\) 1805.86 0.163478
\(497\) −15725.0 −1.41924
\(498\) 0 0
\(499\) 7145.78 0.641060 0.320530 0.947238i \(-0.396139\pi\)
0.320530 + 0.947238i \(0.396139\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) −10085.6 −0.896695
\(503\) −20436.2 −1.81154 −0.905770 0.423771i \(-0.860706\pi\)
−0.905770 + 0.423771i \(0.860706\pi\)
\(504\) 0 0
\(505\) −9444.29 −0.832208
\(506\) 1478.22 0.129871
\(507\) 0 0
\(508\) −5193.72 −0.453610
\(509\) −19721.7 −1.71738 −0.858690 0.512495i \(-0.828721\pi\)
−0.858690 + 0.512495i \(0.828721\pi\)
\(510\) 0 0
\(511\) 14207.1 1.22991
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −5962.30 −0.511646
\(515\) −7485.84 −0.640516
\(516\) 0 0
\(517\) −849.475 −0.0722628
\(518\) 2153.86 0.182693
\(519\) 0 0
\(520\) 1603.24 0.135205
\(521\) −3483.23 −0.292904 −0.146452 0.989218i \(-0.546785\pi\)
−0.146452 + 0.989218i \(0.546785\pi\)
\(522\) 0 0
\(523\) 15689.0 1.31172 0.655862 0.754881i \(-0.272305\pi\)
0.655862 + 0.754881i \(0.272305\pi\)
\(524\) 2100.99 0.175157
\(525\) 0 0
\(526\) −14485.7 −1.20077
\(527\) −14239.8 −1.17703
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −436.958 −0.0358118
\(531\) 0 0
\(532\) 21.9357 0.00178765
\(533\) 5426.36 0.440979
\(534\) 0 0
\(535\) 2925.75 0.236432
\(536\) −3365.78 −0.271230
\(537\) 0 0
\(538\) 5135.15 0.411510
\(539\) 6818.43 0.544881
\(540\) 0 0
\(541\) 7620.73 0.605621 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(542\) 16132.5 1.27851
\(543\) 0 0
\(544\) −4037.29 −0.318194
\(545\) −695.828 −0.0546899
\(546\) 0 0
\(547\) 5925.62 0.463183 0.231592 0.972813i \(-0.425607\pi\)
0.231592 + 0.972813i \(0.425607\pi\)
\(548\) 9712.47 0.757109
\(549\) 0 0
\(550\) 1606.76 0.124568
\(551\) 31.9381 0.00246934
\(552\) 0 0
\(553\) −32489.4 −2.49836
\(554\) 17983.1 1.37911
\(555\) 0 0
\(556\) 9825.85 0.749476
\(557\) 5456.16 0.415054 0.207527 0.978229i \(-0.433459\pi\)
0.207527 + 0.978229i \(0.433459\pi\)
\(558\) 0 0
\(559\) 21785.2 1.64833
\(560\) 1884.98 0.142241
\(561\) 0 0
\(562\) −1936.26 −0.145331
\(563\) 9194.29 0.688265 0.344132 0.938921i \(-0.388173\pi\)
0.344132 + 0.938921i \(0.388173\pi\)
\(564\) 0 0
\(565\) 6465.88 0.481454
\(566\) −4505.30 −0.334580
\(567\) 0 0
\(568\) −5339.05 −0.394404
\(569\) −338.831 −0.0249640 −0.0124820 0.999922i \(-0.503973\pi\)
−0.0124820 + 0.999922i \(0.503973\pi\)
\(570\) 0 0
\(571\) −1725.34 −0.126451 −0.0632254 0.997999i \(-0.520139\pi\)
−0.0632254 + 0.997999i \(0.520139\pi\)
\(572\) 5152.06 0.376605
\(573\) 0 0
\(574\) 6379.93 0.463926
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 23300.0 1.68109 0.840547 0.541738i \(-0.182234\pi\)
0.840547 + 0.541738i \(0.182234\pi\)
\(578\) 22009.3 1.58385
\(579\) 0 0
\(580\) 2744.51 0.196482
\(581\) 11431.9 0.816307
\(582\) 0 0
\(583\) −1404.18 −0.0997513
\(584\) 4823.68 0.341790
\(585\) 0 0
\(586\) −5468.70 −0.385512
\(587\) −15653.3 −1.10065 −0.550325 0.834950i \(-0.685496\pi\)
−0.550325 + 0.834950i \(0.685496\pi\)
\(588\) 0 0
\(589\) 26.2686 0.00183766
\(590\) −2022.48 −0.141126
\(591\) 0 0
\(592\) 731.292 0.0507701
\(593\) 2658.08 0.184072 0.0920358 0.995756i \(-0.470663\pi\)
0.0920358 + 0.995756i \(0.470663\pi\)
\(594\) 0 0
\(595\) −14863.7 −1.02412
\(596\) 9621.57 0.661266
\(597\) 0 0
\(598\) 1843.73 0.126080
\(599\) −19417.6 −1.32451 −0.662256 0.749278i \(-0.730401\pi\)
−0.662256 + 0.749278i \(0.730401\pi\)
\(600\) 0 0
\(601\) −18469.0 −1.25352 −0.626760 0.779213i \(-0.715619\pi\)
−0.626760 + 0.779213i \(0.715619\pi\)
\(602\) 25613.5 1.73410
\(603\) 0 0
\(604\) −2597.10 −0.174958
\(605\) −1491.64 −0.100238
\(606\) 0 0
\(607\) 3968.56 0.265369 0.132684 0.991158i \(-0.457640\pi\)
0.132684 + 0.991158i \(0.457640\pi\)
\(608\) 7.44773 0.000496785 0
\(609\) 0 0
\(610\) 1502.79 0.0997477
\(611\) −1059.52 −0.0701531
\(612\) 0 0
\(613\) −11478.7 −0.756311 −0.378156 0.925742i \(-0.623442\pi\)
−0.378156 + 0.925742i \(0.623442\pi\)
\(614\) −15954.2 −1.04863
\(615\) 0 0
\(616\) 6057.42 0.396202
\(617\) 15691.1 1.02382 0.511911 0.859039i \(-0.328938\pi\)
0.511911 + 0.859039i \(0.328938\pi\)
\(618\) 0 0
\(619\) −18249.4 −1.18499 −0.592494 0.805575i \(-0.701856\pi\)
−0.592494 + 0.805575i \(0.701856\pi\)
\(620\) 2257.32 0.146220
\(621\) 0 0
\(622\) −15458.9 −0.996533
\(623\) 26571.4 1.70876
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 11012.5 0.703111
\(627\) 0 0
\(628\) 14660.2 0.931536
\(629\) −5766.48 −0.365540
\(630\) 0 0
\(631\) −23528.8 −1.48442 −0.742208 0.670169i \(-0.766222\pi\)
−0.742208 + 0.670169i \(0.766222\pi\)
\(632\) −11031.0 −0.694288
\(633\) 0 0
\(634\) −10462.7 −0.655408
\(635\) −6492.15 −0.405722
\(636\) 0 0
\(637\) 8504.38 0.528973
\(638\) 8819.55 0.547287
\(639\) 0 0
\(640\) 640.000 0.0395285
\(641\) −3748.79 −0.230996 −0.115498 0.993308i \(-0.536846\pi\)
−0.115498 + 0.993308i \(0.536846\pi\)
\(642\) 0 0
\(643\) 15924.3 0.976660 0.488330 0.872659i \(-0.337606\pi\)
0.488330 + 0.872659i \(0.337606\pi\)
\(644\) 2167.73 0.132640
\(645\) 0 0
\(646\) −58.7278 −0.00357680
\(647\) 2854.11 0.173426 0.0867129 0.996233i \(-0.472364\pi\)
0.0867129 + 0.996233i \(0.472364\pi\)
\(648\) 0 0
\(649\) −6499.27 −0.393095
\(650\) 2004.05 0.120931
\(651\) 0 0
\(652\) −4221.09 −0.253544
\(653\) −7925.97 −0.474988 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(654\) 0 0
\(655\) 2626.24 0.156665
\(656\) 2166.15 0.128924
\(657\) 0 0
\(658\) −1245.71 −0.0738035
\(659\) 5799.43 0.342813 0.171406 0.985200i \(-0.445169\pi\)
0.171406 + 0.985200i \(0.445169\pi\)
\(660\) 0 0
\(661\) 22324.8 1.31367 0.656833 0.754036i \(-0.271896\pi\)
0.656833 + 0.754036i \(0.271896\pi\)
\(662\) −7103.22 −0.417031
\(663\) 0 0
\(664\) 3881.42 0.226850
\(665\) 27.4196 0.00159893
\(666\) 0 0
\(667\) 3156.19 0.183221
\(668\) 2927.22 0.169547
\(669\) 0 0
\(670\) −4207.22 −0.242596
\(671\) 4829.24 0.277840
\(672\) 0 0
\(673\) −23253.5 −1.33188 −0.665940 0.746005i \(-0.731969\pi\)
−0.665940 + 0.746005i \(0.731969\pi\)
\(674\) −14006.0 −0.800430
\(675\) 0 0
\(676\) −2362.03 −0.134389
\(677\) 19003.7 1.07884 0.539419 0.842038i \(-0.318644\pi\)
0.539419 + 0.842038i \(0.318644\pi\)
\(678\) 0 0
\(679\) −35019.1 −1.97925
\(680\) −5046.61 −0.284601
\(681\) 0 0
\(682\) 7253.94 0.407284
\(683\) 11222.5 0.628722 0.314361 0.949304i \(-0.398210\pi\)
0.314361 + 0.949304i \(0.398210\pi\)
\(684\) 0 0
\(685\) 12140.6 0.677179
\(686\) −6164.85 −0.343112
\(687\) 0 0
\(688\) 8696.45 0.481902
\(689\) −1751.38 −0.0968391
\(690\) 0 0
\(691\) 4297.68 0.236601 0.118301 0.992978i \(-0.462255\pi\)
0.118301 + 0.992978i \(0.462255\pi\)
\(692\) 2087.09 0.114652
\(693\) 0 0
\(694\) 20536.4 1.12327
\(695\) 12282.3 0.670352
\(696\) 0 0
\(697\) −17080.8 −0.928240
\(698\) 9031.15 0.489734
\(699\) 0 0
\(700\) 2356.22 0.127224
\(701\) −25769.6 −1.38845 −0.694225 0.719758i \(-0.744253\pi\)
−0.694225 + 0.719758i \(0.744253\pi\)
\(702\) 0 0
\(703\) 10.6376 0.000570705 0
\(704\) 2056.65 0.110104
\(705\) 0 0
\(706\) −15677.3 −0.835725
\(707\) −44505.7 −2.36748
\(708\) 0 0
\(709\) −4900.01 −0.259554 −0.129777 0.991543i \(-0.541426\pi\)
−0.129777 + 0.991543i \(0.541426\pi\)
\(710\) −6673.81 −0.352766
\(711\) 0 0
\(712\) 9021.67 0.474862
\(713\) 2595.92 0.136350
\(714\) 0 0
\(715\) 6440.07 0.336846
\(716\) 4732.28 0.247002
\(717\) 0 0
\(718\) 24929.6 1.29577
\(719\) −7631.85 −0.395855 −0.197928 0.980217i \(-0.563421\pi\)
−0.197928 + 0.980217i \(0.563421\pi\)
\(720\) 0 0
\(721\) −35276.6 −1.82215
\(722\) −13717.9 −0.707101
\(723\) 0 0
\(724\) 6895.70 0.353973
\(725\) 3430.64 0.175739
\(726\) 0 0
\(727\) −36985.6 −1.88682 −0.943410 0.331628i \(-0.892402\pi\)
−0.943410 + 0.331628i \(0.892402\pi\)
\(728\) 7555.20 0.384635
\(729\) 0 0
\(730\) 6029.60 0.305706
\(731\) −68574.3 −3.46965
\(732\) 0 0
\(733\) −22227.3 −1.12003 −0.560015 0.828482i \(-0.689205\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(734\) 7872.29 0.395874
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) −13520.0 −0.675733
\(738\) 0 0
\(739\) −17899.1 −0.890974 −0.445487 0.895288i \(-0.646970\pi\)
−0.445487 + 0.895288i \(0.646970\pi\)
\(740\) 914.115 0.0454102
\(741\) 0 0
\(742\) −2059.14 −0.101878
\(743\) 29843.4 1.47355 0.736776 0.676137i \(-0.236347\pi\)
0.736776 + 0.676137i \(0.236347\pi\)
\(744\) 0 0
\(745\) 12027.0 0.591455
\(746\) 3841.44 0.188532
\(747\) 0 0
\(748\) −16217.4 −0.792736
\(749\) 13787.4 0.672606
\(750\) 0 0
\(751\) −9399.65 −0.456722 −0.228361 0.973577i \(-0.573337\pi\)
−0.228361 + 0.973577i \(0.573337\pi\)
\(752\) −422.950 −0.0205098
\(753\) 0 0
\(754\) 11000.3 0.531309
\(755\) −3246.38 −0.156487
\(756\) 0 0
\(757\) 29871.6 1.43422 0.717109 0.696961i \(-0.245465\pi\)
0.717109 + 0.696961i \(0.245465\pi\)
\(758\) −26149.0 −1.25300
\(759\) 0 0
\(760\) 9.30966 0.000444338 0
\(761\) 273.955 0.0130497 0.00652487 0.999979i \(-0.497923\pi\)
0.00652487 + 0.999979i \(0.497923\pi\)
\(762\) 0 0
\(763\) −3279.05 −0.155583
\(764\) 11192.5 0.530012
\(765\) 0 0
\(766\) −17676.4 −0.833780
\(767\) −8106.31 −0.381619
\(768\) 0 0
\(769\) −13273.8 −0.622450 −0.311225 0.950336i \(-0.600739\pi\)
−0.311225 + 0.950336i \(0.600739\pi\)
\(770\) 7571.78 0.354374
\(771\) 0 0
\(772\) −1990.74 −0.0928089
\(773\) −34161.1 −1.58951 −0.794753 0.606933i \(-0.792400\pi\)
−0.794753 + 0.606933i \(0.792400\pi\)
\(774\) 0 0
\(775\) 2821.65 0.130783
\(776\) −11889.9 −0.550028
\(777\) 0 0
\(778\) 25374.6 1.16931
\(779\) 31.5096 0.00144923
\(780\) 0 0
\(781\) −21446.4 −0.982603
\(782\) −5803.60 −0.265392
\(783\) 0 0
\(784\) 3394.87 0.154650
\(785\) 18325.2 0.833191
\(786\) 0 0
\(787\) −38489.6 −1.74334 −0.871669 0.490095i \(-0.836962\pi\)
−0.871669 + 0.490095i \(0.836962\pi\)
\(788\) −1185.95 −0.0536141
\(789\) 0 0
\(790\) −13788.8 −0.620990
\(791\) 30470.1 1.36965
\(792\) 0 0
\(793\) 6023.33 0.269729
\(794\) 16121.0 0.720544
\(795\) 0 0
\(796\) 9389.73 0.418103
\(797\) −17176.8 −0.763405 −0.381702 0.924285i \(-0.624662\pi\)
−0.381702 + 0.924285i \(0.624662\pi\)
\(798\) 0 0
\(799\) 3335.10 0.147669
\(800\) 800.000 0.0353553
\(801\) 0 0
\(802\) −24650.2 −1.08532
\(803\) 19376.2 0.851523
\(804\) 0 0
\(805\) 2709.66 0.118637
\(806\) 9047.58 0.395394
\(807\) 0 0
\(808\) −15110.9 −0.657918
\(809\) −8226.61 −0.357518 −0.178759 0.983893i \(-0.557208\pi\)
−0.178759 + 0.983893i \(0.557208\pi\)
\(810\) 0 0
\(811\) 27867.9 1.20662 0.603312 0.797505i \(-0.293847\pi\)
0.603312 + 0.797505i \(0.293847\pi\)
\(812\) 12933.4 0.558956
\(813\) 0 0
\(814\) 2937.53 0.126487
\(815\) −5276.37 −0.226777
\(816\) 0 0
\(817\) 126.502 0.00541705
\(818\) −21613.5 −0.923835
\(819\) 0 0
\(820\) 2707.69 0.115313
\(821\) 17637.5 0.749761 0.374880 0.927073i \(-0.377684\pi\)
0.374880 + 0.927073i \(0.377684\pi\)
\(822\) 0 0
\(823\) 28048.6 1.18799 0.593993 0.804470i \(-0.297551\pi\)
0.593993 + 0.804470i \(0.297551\pi\)
\(824\) −11977.3 −0.506372
\(825\) 0 0
\(826\) −9530.82 −0.401477
\(827\) 32592.9 1.37046 0.685228 0.728329i \(-0.259703\pi\)
0.685228 + 0.728329i \(0.259703\pi\)
\(828\) 0 0
\(829\) 18815.9 0.788303 0.394152 0.919045i \(-0.371038\pi\)
0.394152 + 0.919045i \(0.371038\pi\)
\(830\) 4851.78 0.202901
\(831\) 0 0
\(832\) 2565.19 0.106889
\(833\) −26769.7 −1.11346
\(834\) 0 0
\(835\) 3659.03 0.151648
\(836\) 29.9168 0.00123767
\(837\) 0 0
\(838\) 3647.48 0.150358
\(839\) −7612.72 −0.313254 −0.156627 0.987658i \(-0.550062\pi\)
−0.156627 + 0.987658i \(0.550062\pi\)
\(840\) 0 0
\(841\) −5558.14 −0.227896
\(842\) 11990.9 0.490775
\(843\) 0 0
\(844\) 11354.5 0.463076
\(845\) −2952.54 −0.120202
\(846\) 0 0
\(847\) −7029.28 −0.285158
\(848\) −699.134 −0.0283117
\(849\) 0 0
\(850\) −6308.26 −0.254555
\(851\) 1051.23 0.0423452
\(852\) 0 0
\(853\) −31421.4 −1.26125 −0.630627 0.776086i \(-0.717202\pi\)
−0.630627 + 0.776086i \(0.717202\pi\)
\(854\) 7081.81 0.283764
\(855\) 0 0
\(856\) 4681.20 0.186916
\(857\) −20909.5 −0.833435 −0.416718 0.909036i \(-0.636820\pi\)
−0.416718 + 0.909036i \(0.636820\pi\)
\(858\) 0 0
\(859\) 23304.5 0.925655 0.462828 0.886448i \(-0.346835\pi\)
0.462828 + 0.886448i \(0.346835\pi\)
\(860\) 10870.6 0.431027
\(861\) 0 0
\(862\) −7775.61 −0.307237
\(863\) 19146.3 0.755211 0.377606 0.925966i \(-0.376747\pi\)
0.377606 + 0.925966i \(0.376747\pi\)
\(864\) 0 0
\(865\) 2608.86 0.102548
\(866\) −8845.87 −0.347107
\(867\) 0 0
\(868\) 10637.5 0.415968
\(869\) −44310.5 −1.72972
\(870\) 0 0
\(871\) −16863.0 −0.656005
\(872\) −1113.32 −0.0432362
\(873\) 0 0
\(874\) 10.7061 0.000414347 0
\(875\) 2945.28 0.113793
\(876\) 0 0
\(877\) 31389.0 1.20859 0.604293 0.796762i \(-0.293456\pi\)
0.604293 + 0.796762i \(0.293456\pi\)
\(878\) −3496.16 −0.134384
\(879\) 0 0
\(880\) 2570.82 0.0984798
\(881\) −44496.3 −1.70161 −0.850806 0.525480i \(-0.823886\pi\)
−0.850806 + 0.525480i \(0.823886\pi\)
\(882\) 0 0
\(883\) 8906.65 0.339448 0.169724 0.985492i \(-0.445712\pi\)
0.169724 + 0.985492i \(0.445712\pi\)
\(884\) −20227.4 −0.769592
\(885\) 0 0
\(886\) −6742.62 −0.255669
\(887\) 36584.7 1.38489 0.692444 0.721472i \(-0.256534\pi\)
0.692444 + 0.721472i \(0.256534\pi\)
\(888\) 0 0
\(889\) −30593.9 −1.15420
\(890\) 11277.1 0.424729
\(891\) 0 0
\(892\) −8498.74 −0.319012
\(893\) −6.15238 −0.000230551 0
\(894\) 0 0
\(895\) 5915.35 0.220926
\(896\) 3015.97 0.112451
\(897\) 0 0
\(898\) −26629.5 −0.989573
\(899\) 15488.1 0.574591
\(900\) 0 0
\(901\) 5512.90 0.203842
\(902\) 8701.23 0.321197
\(903\) 0 0
\(904\) 10345.4 0.380623
\(905\) 8619.62 0.316603
\(906\) 0 0
\(907\) −21346.1 −0.781463 −0.390731 0.920505i \(-0.627778\pi\)
−0.390731 + 0.920505i \(0.627778\pi\)
\(908\) −6207.89 −0.226890
\(909\) 0 0
\(910\) 9444.00 0.344028
\(911\) 10208.0 0.371246 0.185623 0.982621i \(-0.440570\pi\)
0.185623 + 0.982621i \(0.440570\pi\)
\(912\) 0 0
\(913\) 15591.3 0.565166
\(914\) 7534.40 0.272665
\(915\) 0 0
\(916\) −634.125 −0.0228735
\(917\) 12376.0 0.445683
\(918\) 0 0
\(919\) 1758.00 0.0631025 0.0315513 0.999502i \(-0.489955\pi\)
0.0315513 + 0.999502i \(0.489955\pi\)
\(920\) 920.000 0.0329690
\(921\) 0 0
\(922\) −19348.1 −0.691102
\(923\) −26749.3 −0.953917
\(924\) 0 0
\(925\) 1142.64 0.0406161
\(926\) 5955.46 0.211348
\(927\) 0 0
\(928\) 4391.22 0.155333
\(929\) 845.008 0.0298426 0.0149213 0.999889i \(-0.495250\pi\)
0.0149213 + 0.999889i \(0.495250\pi\)
\(930\) 0 0
\(931\) 49.3830 0.00173841
\(932\) 2915.99 0.102486
\(933\) 0 0
\(934\) 21402.0 0.749782
\(935\) −20271.7 −0.709045
\(936\) 0 0
\(937\) 3851.61 0.134287 0.0671433 0.997743i \(-0.478612\pi\)
0.0671433 + 0.997743i \(0.478612\pi\)
\(938\) −19826.3 −0.690141
\(939\) 0 0
\(940\) −528.688 −0.0183446
\(941\) 4379.36 0.151714 0.0758571 0.997119i \(-0.475831\pi\)
0.0758571 + 0.997119i \(0.475831\pi\)
\(942\) 0 0
\(943\) 3113.85 0.107530
\(944\) −3235.96 −0.111570
\(945\) 0 0
\(946\) 34932.8 1.20059
\(947\) −29746.9 −1.02074 −0.510371 0.859954i \(-0.670492\pi\)
−0.510371 + 0.859954i \(0.670492\pi\)
\(948\) 0 0
\(949\) 24167.3 0.826663
\(950\) 11.6371 0.000397428 0
\(951\) 0 0
\(952\) −23781.9 −0.809638
\(953\) −4306.61 −0.146385 −0.0731924 0.997318i \(-0.523319\pi\)
−0.0731924 + 0.997318i \(0.523319\pi\)
\(954\) 0 0
\(955\) 13990.6 0.474057
\(956\) 24806.4 0.839222
\(957\) 0 0
\(958\) −4674.29 −0.157640
\(959\) 57211.9 1.92645
\(960\) 0 0
\(961\) −17052.3 −0.572397
\(962\) 3663.87 0.122794
\(963\) 0 0
\(964\) −28892.2 −0.965305
\(965\) −2488.43 −0.0830108
\(966\) 0 0
\(967\) −12233.4 −0.406824 −0.203412 0.979093i \(-0.565203\pi\)
−0.203412 + 0.979093i \(0.565203\pi\)
\(968\) −2386.63 −0.0792449
\(969\) 0 0
\(970\) −14862.4 −0.491960
\(971\) −48207.7 −1.59326 −0.796631 0.604465i \(-0.793387\pi\)
−0.796631 + 0.604465i \(0.793387\pi\)
\(972\) 0 0
\(973\) 57879.8 1.90703
\(974\) −14367.7 −0.472660
\(975\) 0 0
\(976\) 2404.46 0.0788575
\(977\) −28662.9 −0.938595 −0.469298 0.883040i \(-0.655493\pi\)
−0.469298 + 0.883040i \(0.655493\pi\)
\(978\) 0 0
\(979\) 36239.2 1.18305
\(980\) 4243.59 0.138323
\(981\) 0 0
\(982\) 24168.4 0.785383
\(983\) 20944.1 0.679567 0.339783 0.940504i \(-0.389646\pi\)
0.339783 + 0.940504i \(0.389646\pi\)
\(984\) 0 0
\(985\) −1482.44 −0.0479539
\(986\) −34626.2 −1.11838
\(987\) 0 0
\(988\) 37.3141 0.00120154
\(989\) 12501.1 0.401934
\(990\) 0 0
\(991\) −36440.2 −1.16808 −0.584038 0.811727i \(-0.698528\pi\)
−0.584038 + 0.811727i \(0.698528\pi\)
\(992\) 3611.71 0.115597
\(993\) 0 0
\(994\) −31450.0 −1.00355
\(995\) 11737.2 0.373963
\(996\) 0 0
\(997\) −11945.0 −0.379439 −0.189720 0.981838i \(-0.560758\pi\)
−0.189720 + 0.981838i \(0.560758\pi\)
\(998\) 14291.6 0.453298
\(999\) 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 2070.4.a.bj.1.3 4
3.2 odd 2 230.4.a.h.1.4 4
12.11 even 2 1840.4.a.m.1.1 4
15.2 even 4 1150.4.b.n.599.1 8
15.8 even 4 1150.4.b.n.599.8 8
15.14 odd 2 1150.4.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 3.2 odd 2
1150.4.a.p.1.1 4 15.14 odd 2
1150.4.b.n.599.1 8 15.2 even 4
1150.4.b.n.599.8 8 15.8 even 4
1840.4.a.m.1.1 4 12.11 even 2
2070.4.a.bj.1.3 4 1.1 even 1 trivial