Properties

Label 2205.4.a.x.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} +O(q^{10})\) \(q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} -21.5831 q^{10} -19.7335 q^{11} -71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} -136.332 q^{19} +53.1662 q^{20} +85.1821 q^{22} +100.865 q^{23} +25.0000 q^{25} +307.916 q^{26} +288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +135.251 q^{34} +309.931 q^{37} +588.496 q^{38} -56.8338 q^{40} -181.662 q^{41} -18.2005 q^{43} -209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -107.916 q^{50} -758.496 q^{52} +127.995 q^{53} -98.6675 q^{55} -1244.04 q^{58} +322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -356.662 q^{65} -84.3960 q^{67} -333.166 q^{68} +315.736 q^{71} +1093.32 q^{73} -1337.86 q^{74} -1449.66 q^{76} +1233.19 q^{79} -180.000 q^{80} +784.169 q^{82} -643.325 q^{83} -156.662 q^{85} +78.5647 q^{86} +224.306 q^{88} +1140.32 q^{89} +1072.53 q^{92} +637.414 q^{94} -681.662 q^{95} -1411.99 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.31662 −1.52616 −0.763079 0.646306i \(-0.776313\pi\)
−0.763079 + 0.646306i \(0.776313\pi\)
\(3\) 0 0
\(4\) 10.6332 1.32916
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −11.3668 −0.502344
\(9\) 0 0
\(10\) −21.5831 −0.682518
\(11\) −19.7335 −0.540898 −0.270449 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(12\) 0 0
\(13\) −71.3325 −1.52185 −0.760926 0.648839i \(-0.775255\pi\)
−0.760926 + 0.648839i \(0.775255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −36.0000 −0.562500
\(17\) −31.3325 −0.447014 −0.223507 0.974702i \(-0.571751\pi\)
−0.223507 + 0.974702i \(0.571751\pi\)
\(18\) 0 0
\(19\) −136.332 −1.64615 −0.823074 0.567934i \(-0.807743\pi\)
−0.823074 + 0.567934i \(0.807743\pi\)
\(20\) 53.1662 0.594417
\(21\) 0 0
\(22\) 85.1821 0.825495
\(23\) 100.865 0.914431 0.457215 0.889356i \(-0.348847\pi\)
0.457215 + 0.889356i \(0.348847\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 307.916 2.32259
\(27\) 0 0
\(28\) 0 0
\(29\) 288.198 1.84541 0.922707 0.385501i \(-0.125971\pi\)
0.922707 + 0.385501i \(0.125971\pi\)
\(30\) 0 0
\(31\) 208.997 1.21087 0.605436 0.795894i \(-0.292999\pi\)
0.605436 + 0.795894i \(0.292999\pi\)
\(32\) 246.332 1.36081
\(33\) 0 0
\(34\) 135.251 0.682214
\(35\) 0 0
\(36\) 0 0
\(37\) 309.931 1.37709 0.688546 0.725192i \(-0.258249\pi\)
0.688546 + 0.725192i \(0.258249\pi\)
\(38\) 588.496 2.51228
\(39\) 0 0
\(40\) −56.8338 −0.224655
\(41\) −181.662 −0.691973 −0.345987 0.938239i \(-0.612456\pi\)
−0.345987 + 0.938239i \(0.612456\pi\)
\(42\) 0 0
\(43\) −18.2005 −0.0645477 −0.0322738 0.999479i \(-0.510275\pi\)
−0.0322738 + 0.999479i \(0.510275\pi\)
\(44\) −209.831 −0.718937
\(45\) 0 0
\(46\) −435.398 −1.39557
\(47\) −147.665 −0.458280 −0.229140 0.973393i \(-0.573591\pi\)
−0.229140 + 0.973393i \(0.573591\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −107.916 −0.305231
\(51\) 0 0
\(52\) −758.496 −2.02278
\(53\) 127.995 0.331726 0.165863 0.986149i \(-0.446959\pi\)
0.165863 + 0.986149i \(0.446959\pi\)
\(54\) 0 0
\(55\) −98.6675 −0.241897
\(56\) 0 0
\(57\) 0 0
\(58\) −1244.04 −2.81639
\(59\) 322.665 0.711990 0.355995 0.934488i \(-0.384142\pi\)
0.355995 + 0.934488i \(0.384142\pi\)
\(60\) 0 0
\(61\) −341.003 −0.715752 −0.357876 0.933769i \(-0.616499\pi\)
−0.357876 + 0.933769i \(0.616499\pi\)
\(62\) −902.164 −1.84798
\(63\) 0 0
\(64\) −775.325 −1.51431
\(65\) −356.662 −0.680593
\(66\) 0 0
\(67\) −84.3960 −0.153890 −0.0769449 0.997035i \(-0.524517\pi\)
−0.0769449 + 0.997035i \(0.524517\pi\)
\(68\) −333.166 −0.594152
\(69\) 0 0
\(70\) 0 0
\(71\) 315.736 0.527760 0.263880 0.964555i \(-0.414998\pi\)
0.263880 + 0.964555i \(0.414998\pi\)
\(72\) 0 0
\(73\) 1093.32 1.75293 0.876466 0.481464i \(-0.159895\pi\)
0.876466 + 0.481464i \(0.159895\pi\)
\(74\) −1337.86 −2.10166
\(75\) 0 0
\(76\) −1449.66 −2.18799
\(77\) 0 0
\(78\) 0 0
\(79\) 1233.19 1.75626 0.878128 0.478426i \(-0.158793\pi\)
0.878128 + 0.478426i \(0.158793\pi\)
\(80\) −180.000 −0.251558
\(81\) 0 0
\(82\) 784.169 1.05606
\(83\) −643.325 −0.850772 −0.425386 0.905012i \(-0.639862\pi\)
−0.425386 + 0.905012i \(0.639862\pi\)
\(84\) 0 0
\(85\) −156.662 −0.199911
\(86\) 78.5647 0.0985099
\(87\) 0 0
\(88\) 224.306 0.271717
\(89\) 1140.32 1.35813 0.679064 0.734079i \(-0.262386\pi\)
0.679064 + 0.734079i \(0.262386\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1072.53 1.21542
\(93\) 0 0
\(94\) 637.414 0.699407
\(95\) −681.662 −0.736180
\(96\) 0 0
\(97\) −1411.99 −1.47800 −0.739001 0.673705i \(-0.764702\pi\)
−0.739001 + 0.673705i \(0.764702\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 265.831 0.265831
\(101\) −545.673 −0.537589 −0.268794 0.963198i \(-0.586625\pi\)
−0.268794 + 0.963198i \(0.586625\pi\)
\(102\) 0 0
\(103\) −780.990 −0.747119 −0.373559 0.927606i \(-0.621863\pi\)
−0.373559 + 0.927606i \(0.621863\pi\)
\(104\) 810.819 0.764493
\(105\) 0 0
\(106\) −552.506 −0.506266
\(107\) 620.660 0.560761 0.280381 0.959889i \(-0.409539\pi\)
0.280381 + 0.959889i \(0.409539\pi\)
\(108\) 0 0
\(109\) 4.18794 0.00368011 0.00184005 0.999998i \(-0.499414\pi\)
0.00184005 + 0.999998i \(0.499414\pi\)
\(110\) 425.911 0.369173
\(111\) 0 0
\(112\) 0 0
\(113\) −1413.53 −1.17676 −0.588379 0.808585i \(-0.700234\pi\)
−0.588379 + 0.808585i \(0.700234\pi\)
\(114\) 0 0
\(115\) 504.327 0.408946
\(116\) 3064.48 2.45284
\(117\) 0 0
\(118\) −1392.82 −1.08661
\(119\) 0 0
\(120\) 0 0
\(121\) −941.589 −0.707430
\(122\) 1471.98 1.09235
\(123\) 0 0
\(124\) 2222.32 1.60944
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2046.26 −1.42974 −0.714868 0.699259i \(-0.753513\pi\)
−0.714868 + 0.699259i \(0.753513\pi\)
\(128\) 1376.13 0.950262
\(129\) 0 0
\(130\) 1539.58 1.03869
\(131\) 1140.32 0.760534 0.380267 0.924877i \(-0.375832\pi\)
0.380267 + 0.924877i \(0.375832\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 364.306 0.234860
\(135\) 0 0
\(136\) 356.149 0.224555
\(137\) 490.343 0.305787 0.152893 0.988243i \(-0.451141\pi\)
0.152893 + 0.988243i \(0.451141\pi\)
\(138\) 0 0
\(139\) −2800.00 −1.70858 −0.854291 0.519795i \(-0.826008\pi\)
−0.854291 + 0.519795i \(0.826008\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1362.91 −0.805445
\(143\) 1407.64 0.823166
\(144\) 0 0
\(145\) 1440.99 0.825294
\(146\) −4719.47 −2.67525
\(147\) 0 0
\(148\) 3295.58 1.83037
\(149\) −1166.12 −0.641154 −0.320577 0.947222i \(-0.603877\pi\)
−0.320577 + 0.947222i \(0.603877\pi\)
\(150\) 0 0
\(151\) 959.581 0.517150 0.258575 0.965991i \(-0.416747\pi\)
0.258575 + 0.965991i \(0.416747\pi\)
\(152\) 1549.66 0.826933
\(153\) 0 0
\(154\) 0 0
\(155\) 1044.99 0.541519
\(156\) 0 0
\(157\) 1020.66 0.518838 0.259419 0.965765i \(-0.416469\pi\)
0.259419 + 0.965765i \(0.416469\pi\)
\(158\) −5323.20 −2.68032
\(159\) 0 0
\(160\) 1231.66 0.608572
\(161\) 0 0
\(162\) 0 0
\(163\) −1566.02 −0.752514 −0.376257 0.926515i \(-0.622789\pi\)
−0.376257 + 0.926515i \(0.622789\pi\)
\(164\) −1931.66 −0.919741
\(165\) 0 0
\(166\) 2776.99 1.29841
\(167\) −1130.30 −0.523746 −0.261873 0.965102i \(-0.584340\pi\)
−0.261873 + 0.965102i \(0.584340\pi\)
\(168\) 0 0
\(169\) 2891.32 1.31603
\(170\) 676.253 0.305096
\(171\) 0 0
\(172\) −193.530 −0.0857940
\(173\) 2543.34 1.11772 0.558862 0.829260i \(-0.311238\pi\)
0.558862 + 0.829260i \(0.311238\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 710.406 0.304255
\(177\) 0 0
\(178\) −4922.32 −2.07272
\(179\) 1210.65 0.505521 0.252760 0.967529i \(-0.418662\pi\)
0.252760 + 0.967529i \(0.418662\pi\)
\(180\) 0 0
\(181\) 3031.32 1.24484 0.622421 0.782683i \(-0.286149\pi\)
0.622421 + 0.782683i \(0.286149\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1146.51 −0.459359
\(185\) 1549.66 0.615854
\(186\) 0 0
\(187\) 618.300 0.241789
\(188\) −1570.16 −0.609125
\(189\) 0 0
\(190\) 2942.48 1.12353
\(191\) 2168.64 0.821555 0.410778 0.911736i \(-0.365257\pi\)
0.410778 + 0.911736i \(0.365257\pi\)
\(192\) 0 0
\(193\) 1490.48 0.555892 0.277946 0.960597i \(-0.410346\pi\)
0.277946 + 0.960597i \(0.410346\pi\)
\(194\) 6095.04 2.25566
\(195\) 0 0
\(196\) 0 0
\(197\) −3380.57 −1.22262 −0.611308 0.791393i \(-0.709356\pi\)
−0.611308 + 0.791393i \(0.709356\pi\)
\(198\) 0 0
\(199\) −4595.33 −1.63696 −0.818478 0.574538i \(-0.805182\pi\)
−0.818478 + 0.574538i \(0.805182\pi\)
\(200\) −284.169 −0.100469
\(201\) 0 0
\(202\) 2355.46 0.820445
\(203\) 0 0
\(204\) 0 0
\(205\) −908.312 −0.309460
\(206\) 3371.24 1.14022
\(207\) 0 0
\(208\) 2567.97 0.856042
\(209\) 2690.32 0.890398
\(210\) 0 0
\(211\) −4988.66 −1.62765 −0.813824 0.581112i \(-0.802618\pi\)
−0.813824 + 0.581112i \(0.802618\pi\)
\(212\) 1361.00 0.440915
\(213\) 0 0
\(214\) −2679.16 −0.855810
\(215\) −91.0025 −0.0288666
\(216\) 0 0
\(217\) 0 0
\(218\) −18.0778 −0.00561643
\(219\) 0 0
\(220\) −1049.16 −0.321519
\(221\) 2235.03 0.680290
\(222\) 0 0
\(223\) −3792.97 −1.13900 −0.569498 0.821993i \(-0.692862\pi\)
−0.569498 + 0.821993i \(0.692862\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6101.68 1.79592
\(227\) 3910.96 1.14352 0.571762 0.820420i \(-0.306260\pi\)
0.571762 + 0.820420i \(0.306260\pi\)
\(228\) 0 0
\(229\) −354.327 −0.102247 −0.0511236 0.998692i \(-0.516280\pi\)
−0.0511236 + 0.998692i \(0.516280\pi\)
\(230\) −2176.99 −0.624116
\(231\) 0 0
\(232\) −3275.87 −0.927033
\(233\) −6492.48 −1.82548 −0.912739 0.408543i \(-0.866037\pi\)
−0.912739 + 0.408543i \(0.866037\pi\)
\(234\) 0 0
\(235\) −738.325 −0.204949
\(236\) 3430.98 0.946346
\(237\) 0 0
\(238\) 0 0
\(239\) 342.688 0.0927474 0.0463737 0.998924i \(-0.485234\pi\)
0.0463737 + 0.998924i \(0.485234\pi\)
\(240\) 0 0
\(241\) −2313.67 −0.618408 −0.309204 0.950996i \(-0.600063\pi\)
−0.309204 + 0.950996i \(0.600063\pi\)
\(242\) 4064.49 1.07965
\(243\) 0 0
\(244\) −3625.96 −0.951347
\(245\) 0 0
\(246\) 0 0
\(247\) 9724.94 2.50519
\(248\) −2375.62 −0.608275
\(249\) 0 0
\(250\) −539.578 −0.136504
\(251\) 3989.29 1.00319 0.501597 0.865101i \(-0.332746\pi\)
0.501597 + 0.865101i \(0.332746\pi\)
\(252\) 0 0
\(253\) −1990.43 −0.494614
\(254\) 8832.95 2.18200
\(255\) 0 0
\(256\) 262.376 0.0640566
\(257\) 2291.32 0.556142 0.278071 0.960560i \(-0.410305\pi\)
0.278071 + 0.960560i \(0.410305\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3792.48 −0.904614
\(261\) 0 0
\(262\) −4922.32 −1.16070
\(263\) −6360.47 −1.49127 −0.745634 0.666356i \(-0.767853\pi\)
−0.745634 + 0.666356i \(0.767853\pi\)
\(264\) 0 0
\(265\) 639.975 0.148352
\(266\) 0 0
\(267\) 0 0
\(268\) −897.404 −0.204543
\(269\) −991.345 −0.224697 −0.112348 0.993669i \(-0.535837\pi\)
−0.112348 + 0.993669i \(0.535837\pi\)
\(270\) 0 0
\(271\) −730.977 −0.163851 −0.0819257 0.996638i \(-0.526107\pi\)
−0.0819257 + 0.996638i \(0.526107\pi\)
\(272\) 1127.97 0.251446
\(273\) 0 0
\(274\) −2116.62 −0.466679
\(275\) −493.338 −0.108180
\(276\) 0 0
\(277\) −3538.63 −0.767566 −0.383783 0.923423i \(-0.625379\pi\)
−0.383783 + 0.923423i \(0.625379\pi\)
\(278\) 12086.5 2.60756
\(279\) 0 0
\(280\) 0 0
\(281\) 4663.20 0.989975 0.494988 0.868900i \(-0.335173\pi\)
0.494988 + 0.868900i \(0.335173\pi\)
\(282\) 0 0
\(283\) 2104.95 0.442142 0.221071 0.975258i \(-0.429045\pi\)
0.221071 + 0.975258i \(0.429045\pi\)
\(284\) 3357.30 0.701476
\(285\) 0 0
\(286\) −6076.25 −1.25628
\(287\) 0 0
\(288\) 0 0
\(289\) −3931.27 −0.800178
\(290\) −6220.21 −1.25953
\(291\) 0 0
\(292\) 11625.6 2.32992
\(293\) −6594.66 −1.31489 −0.657447 0.753501i \(-0.728364\pi\)
−0.657447 + 0.753501i \(0.728364\pi\)
\(294\) 0 0
\(295\) 1613.32 0.318412
\(296\) −3522.91 −0.691774
\(297\) 0 0
\(298\) 5033.69 0.978503
\(299\) −7194.99 −1.39163
\(300\) 0 0
\(301\) 0 0
\(302\) −4142.15 −0.789252
\(303\) 0 0
\(304\) 4907.97 0.925958
\(305\) −1705.01 −0.320094
\(306\) 0 0
\(307\) 2672.97 0.496920 0.248460 0.968642i \(-0.420076\pi\)
0.248460 + 0.968642i \(0.420076\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4510.82 −0.826443
\(311\) −855.698 −0.156020 −0.0780099 0.996953i \(-0.524857\pi\)
−0.0780099 + 0.996953i \(0.524857\pi\)
\(312\) 0 0
\(313\) 3349.99 0.604960 0.302480 0.953156i \(-0.402185\pi\)
0.302480 + 0.953156i \(0.402185\pi\)
\(314\) −4405.81 −0.791828
\(315\) 0 0
\(316\) 13112.8 2.33434
\(317\) −7633.05 −1.35241 −0.676206 0.736712i \(-0.736377\pi\)
−0.676206 + 0.736712i \(0.736377\pi\)
\(318\) 0 0
\(319\) −5687.16 −0.998180
\(320\) −3876.62 −0.677218
\(321\) 0 0
\(322\) 0 0
\(323\) 4271.64 0.735852
\(324\) 0 0
\(325\) −1783.31 −0.304370
\(326\) 6759.90 1.14845
\(327\) 0 0
\(328\) 2064.91 0.347609
\(329\) 0 0
\(330\) 0 0
\(331\) −3321.70 −0.551593 −0.275796 0.961216i \(-0.588942\pi\)
−0.275796 + 0.961216i \(0.588942\pi\)
\(332\) −6840.63 −1.13081
\(333\) 0 0
\(334\) 4879.10 0.799319
\(335\) −421.980 −0.0688216
\(336\) 0 0
\(337\) −2233.98 −0.361107 −0.180553 0.983565i \(-0.557789\pi\)
−0.180553 + 0.983565i \(0.557789\pi\)
\(338\) −12480.8 −2.00847
\(339\) 0 0
\(340\) −1665.83 −0.265713
\(341\) −4124.25 −0.654958
\(342\) 0 0
\(343\) 0 0
\(344\) 206.881 0.0324252
\(345\) 0 0
\(346\) −10978.6 −1.70582
\(347\) −2528.61 −0.391190 −0.195595 0.980685i \(-0.562664\pi\)
−0.195595 + 0.980685i \(0.562664\pi\)
\(348\) 0 0
\(349\) −1291.00 −0.198011 −0.0990054 0.995087i \(-0.531566\pi\)
−0.0990054 + 0.995087i \(0.531566\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4861.00 −0.736058
\(353\) −7768.64 −1.17134 −0.585670 0.810550i \(-0.699169\pi\)
−0.585670 + 0.810550i \(0.699169\pi\)
\(354\) 0 0
\(355\) 1578.68 0.236022
\(356\) 12125.3 1.80516
\(357\) 0 0
\(358\) −5225.92 −0.771504
\(359\) 2284.14 0.335800 0.167900 0.985804i \(-0.446301\pi\)
0.167900 + 0.985804i \(0.446301\pi\)
\(360\) 0 0
\(361\) 11727.5 1.70980
\(362\) −13085.1 −1.89982
\(363\) 0 0
\(364\) 0 0
\(365\) 5466.62 0.783935
\(366\) 0 0
\(367\) −10707.0 −1.52289 −0.761446 0.648229i \(-0.775510\pi\)
−0.761446 + 0.648229i \(0.775510\pi\)
\(368\) −3631.16 −0.514367
\(369\) 0 0
\(370\) −6689.29 −0.939891
\(371\) 0 0
\(372\) 0 0
\(373\) −830.429 −0.115276 −0.0576381 0.998338i \(-0.518357\pi\)
−0.0576381 + 0.998338i \(0.518357\pi\)
\(374\) −2668.97 −0.369008
\(375\) 0 0
\(376\) 1678.47 0.230214
\(377\) −20557.9 −2.80845
\(378\) 0 0
\(379\) 5253.17 0.711972 0.355986 0.934491i \(-0.384145\pi\)
0.355986 + 0.934491i \(0.384145\pi\)
\(380\) −7248.29 −0.978498
\(381\) 0 0
\(382\) −9361.19 −1.25382
\(383\) −11243.9 −1.50010 −0.750048 0.661383i \(-0.769970\pi\)
−0.750048 + 0.661383i \(0.769970\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6433.84 −0.848378
\(387\) 0 0
\(388\) −15014.1 −1.96449
\(389\) 8506.85 1.10878 0.554389 0.832258i \(-0.312952\pi\)
0.554389 + 0.832258i \(0.312952\pi\)
\(390\) 0 0
\(391\) −3160.37 −0.408764
\(392\) 0 0
\(393\) 0 0
\(394\) 14592.6 1.86590
\(395\) 6165.93 0.785421
\(396\) 0 0
\(397\) −3123.24 −0.394838 −0.197419 0.980319i \(-0.563256\pi\)
−0.197419 + 0.980319i \(0.563256\pi\)
\(398\) 19836.3 2.49825
\(399\) 0 0
\(400\) −900.000 −0.112500
\(401\) 11255.3 1.40166 0.700828 0.713330i \(-0.252814\pi\)
0.700828 + 0.713330i \(0.252814\pi\)
\(402\) 0 0
\(403\) −14908.3 −1.84277
\(404\) −5802.27 −0.714539
\(405\) 0 0
\(406\) 0 0
\(407\) −6116.03 −0.744866
\(408\) 0 0
\(409\) 7919.92 0.957494 0.478747 0.877953i \(-0.341091\pi\)
0.478747 + 0.877953i \(0.341091\pi\)
\(410\) 3920.84 0.472285
\(411\) 0 0
\(412\) −8304.46 −0.993037
\(413\) 0 0
\(414\) 0 0
\(415\) −3216.62 −0.380477
\(416\) −17571.5 −2.07095
\(417\) 0 0
\(418\) −11613.1 −1.35889
\(419\) −5257.28 −0.612972 −0.306486 0.951875i \(-0.599153\pi\)
−0.306486 + 0.951875i \(0.599153\pi\)
\(420\) 0 0
\(421\) 1457.36 0.168711 0.0843556 0.996436i \(-0.473117\pi\)
0.0843556 + 0.996436i \(0.473117\pi\)
\(422\) 21534.2 2.48405
\(423\) 0 0
\(424\) −1454.89 −0.166640
\(425\) −783.312 −0.0894029
\(426\) 0 0
\(427\) 0 0
\(428\) 6599.63 0.745339
\(429\) 0 0
\(430\) 392.824 0.0440550
\(431\) 15291.2 1.70893 0.854467 0.519506i \(-0.173884\pi\)
0.854467 + 0.519506i \(0.173884\pi\)
\(432\) 0 0
\(433\) −187.260 −0.0207832 −0.0103916 0.999946i \(-0.503308\pi\)
−0.0103916 + 0.999946i \(0.503308\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 44.5314 0.00489144
\(437\) −13751.2 −1.50529
\(438\) 0 0
\(439\) −3587.92 −0.390073 −0.195037 0.980796i \(-0.562483\pi\)
−0.195037 + 0.980796i \(0.562483\pi\)
\(440\) 1121.53 0.121515
\(441\) 0 0
\(442\) −9647.76 −1.03823
\(443\) −4915.65 −0.527200 −0.263600 0.964632i \(-0.584910\pi\)
−0.263600 + 0.964632i \(0.584910\pi\)
\(444\) 0 0
\(445\) 5701.59 0.607373
\(446\) 16372.8 1.73829
\(447\) 0 0
\(448\) 0 0
\(449\) −7091.12 −0.745324 −0.372662 0.927967i \(-0.621555\pi\)
−0.372662 + 0.927967i \(0.621555\pi\)
\(450\) 0 0
\(451\) 3584.84 0.374287
\(452\) −15030.4 −1.56410
\(453\) 0 0
\(454\) −16882.2 −1.74520
\(455\) 0 0
\(456\) 0 0
\(457\) 5051.81 0.517098 0.258549 0.965998i \(-0.416756\pi\)
0.258549 + 0.965998i \(0.416756\pi\)
\(458\) 1529.50 0.156045
\(459\) 0 0
\(460\) 5362.64 0.543553
\(461\) −16681.3 −1.68531 −0.842653 0.538456i \(-0.819008\pi\)
−0.842653 + 0.538456i \(0.819008\pi\)
\(462\) 0 0
\(463\) 15569.6 1.56280 0.781402 0.624027i \(-0.214505\pi\)
0.781402 + 0.624027i \(0.214505\pi\)
\(464\) −10375.1 −1.03805
\(465\) 0 0
\(466\) 28025.6 2.78597
\(467\) 3328.35 0.329802 0.164901 0.986310i \(-0.447269\pi\)
0.164901 + 0.986310i \(0.447269\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3187.07 0.312784
\(471\) 0 0
\(472\) −3667.65 −0.357664
\(473\) 359.160 0.0349137
\(474\) 0 0
\(475\) −3408.31 −0.329230
\(476\) 0 0
\(477\) 0 0
\(478\) −1479.25 −0.141547
\(479\) −14607.5 −1.39339 −0.696695 0.717367i \(-0.745347\pi\)
−0.696695 + 0.717367i \(0.745347\pi\)
\(480\) 0 0
\(481\) −22108.2 −2.09573
\(482\) 9987.23 0.943789
\(483\) 0 0
\(484\) −10012.2 −0.940285
\(485\) −7059.96 −0.660982
\(486\) 0 0
\(487\) −1879.49 −0.174882 −0.0874412 0.996170i \(-0.527869\pi\)
−0.0874412 + 0.996170i \(0.527869\pi\)
\(488\) 3876.09 0.359554
\(489\) 0 0
\(490\) 0 0
\(491\) −3221.13 −0.296064 −0.148032 0.988983i \(-0.547294\pi\)
−0.148032 + 0.988983i \(0.547294\pi\)
\(492\) 0 0
\(493\) −9029.96 −0.824927
\(494\) −41978.9 −3.82332
\(495\) 0 0
\(496\) −7523.91 −0.681116
\(497\) 0 0
\(498\) 0 0
\(499\) 9713.81 0.871443 0.435721 0.900082i \(-0.356493\pi\)
0.435721 + 0.900082i \(0.356493\pi\)
\(500\) 1329.16 0.118883
\(501\) 0 0
\(502\) −17220.3 −1.53103
\(503\) 2078.32 0.184230 0.0921152 0.995748i \(-0.470637\pi\)
0.0921152 + 0.995748i \(0.470637\pi\)
\(504\) 0 0
\(505\) −2728.36 −0.240417
\(506\) 8591.94 0.754858
\(507\) 0 0
\(508\) −21758.4 −1.90034
\(509\) −18974.5 −1.65232 −0.826158 0.563439i \(-0.809478\pi\)
−0.826158 + 0.563439i \(0.809478\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −12141.6 −1.04802
\(513\) 0 0
\(514\) −9890.77 −0.848761
\(515\) −3904.95 −0.334122
\(516\) 0 0
\(517\) 2913.95 0.247883
\(518\) 0 0
\(519\) 0 0
\(520\) 4054.09 0.341892
\(521\) 17523.6 1.47355 0.736777 0.676136i \(-0.236347\pi\)
0.736777 + 0.676136i \(0.236347\pi\)
\(522\) 0 0
\(523\) −15218.6 −1.27239 −0.636197 0.771527i \(-0.719493\pi\)
−0.636197 + 0.771527i \(0.719493\pi\)
\(524\) 12125.3 1.01087
\(525\) 0 0
\(526\) 27455.8 2.27591
\(527\) −6548.41 −0.541278
\(528\) 0 0
\(529\) −1993.15 −0.163816
\(530\) −2762.53 −0.226409
\(531\) 0 0
\(532\) 0 0
\(533\) 12958.4 1.05308
\(534\) 0 0
\(535\) 3103.30 0.250780
\(536\) 959.308 0.0773056
\(537\) 0 0
\(538\) 4279.26 0.342922
\(539\) 0 0
\(540\) 0 0
\(541\) −15559.3 −1.23650 −0.618249 0.785983i \(-0.712158\pi\)
−0.618249 + 0.785983i \(0.712158\pi\)
\(542\) 3155.36 0.250063
\(543\) 0 0
\(544\) −7718.21 −0.608301
\(545\) 20.9397 0.00164579
\(546\) 0 0
\(547\) −8690.70 −0.679319 −0.339660 0.940548i \(-0.610312\pi\)
−0.339660 + 0.940548i \(0.610312\pi\)
\(548\) 5213.93 0.406438
\(549\) 0 0
\(550\) 2129.55 0.165099
\(551\) −39290.8 −3.03783
\(552\) 0 0
\(553\) 0 0
\(554\) 15274.9 1.17143
\(555\) 0 0
\(556\) −29773.1 −2.27097
\(557\) 7376.26 0.561117 0.280559 0.959837i \(-0.409480\pi\)
0.280559 + 0.959837i \(0.409480\pi\)
\(558\) 0 0
\(559\) 1298.29 0.0982320
\(560\) 0 0
\(561\) 0 0
\(562\) −20129.3 −1.51086
\(563\) 12875.4 0.963825 0.481913 0.876219i \(-0.339942\pi\)
0.481913 + 0.876219i \(0.339942\pi\)
\(564\) 0 0
\(565\) −7067.65 −0.526263
\(566\) −9086.28 −0.674779
\(567\) 0 0
\(568\) −3588.89 −0.265117
\(569\) 12064.6 0.888882 0.444441 0.895808i \(-0.353402\pi\)
0.444441 + 0.895808i \(0.353402\pi\)
\(570\) 0 0
\(571\) 23745.6 1.74032 0.870158 0.492772i \(-0.164016\pi\)
0.870158 + 0.492772i \(0.164016\pi\)
\(572\) 14967.8 1.09412
\(573\) 0 0
\(574\) 0 0
\(575\) 2521.64 0.182886
\(576\) 0 0
\(577\) 9846.08 0.710394 0.355197 0.934791i \(-0.384414\pi\)
0.355197 + 0.934791i \(0.384414\pi\)
\(578\) 16969.8 1.22120
\(579\) 0 0
\(580\) 15322.4 1.09695
\(581\) 0 0
\(582\) 0 0
\(583\) −2525.79 −0.179430
\(584\) −12427.6 −0.880575
\(585\) 0 0
\(586\) 28466.7 2.00674
\(587\) 10074.7 0.708392 0.354196 0.935171i \(-0.384755\pi\)
0.354196 + 0.935171i \(0.384755\pi\)
\(588\) 0 0
\(589\) −28493.1 −1.99328
\(590\) −6964.12 −0.485946
\(591\) 0 0
\(592\) −11157.5 −0.774615
\(593\) −7387.25 −0.511565 −0.255782 0.966734i \(-0.582333\pi\)
−0.255782 + 0.966734i \(0.582333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12399.6 −0.852194
\(597\) 0 0
\(598\) 31058.1 2.12384
\(599\) −1252.73 −0.0854510 −0.0427255 0.999087i \(-0.513604\pi\)
−0.0427255 + 0.999087i \(0.513604\pi\)
\(600\) 0 0
\(601\) 1800.81 0.122224 0.0611120 0.998131i \(-0.480535\pi\)
0.0611120 + 0.998131i \(0.480535\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 10203.5 0.687373
\(605\) −4707.94 −0.316372
\(606\) 0 0
\(607\) −2497.06 −0.166973 −0.0834863 0.996509i \(-0.526605\pi\)
−0.0834863 + 0.996509i \(0.526605\pi\)
\(608\) −33583.1 −2.24009
\(609\) 0 0
\(610\) 7359.90 0.488514
\(611\) 10533.3 0.697434
\(612\) 0 0
\(613\) −19750.8 −1.30135 −0.650674 0.759357i \(-0.725513\pi\)
−0.650674 + 0.759357i \(0.725513\pi\)
\(614\) −11538.2 −0.758378
\(615\) 0 0
\(616\) 0 0
\(617\) −16797.4 −1.09601 −0.548004 0.836476i \(-0.684612\pi\)
−0.548004 + 0.836476i \(0.684612\pi\)
\(618\) 0 0
\(619\) −26547.4 −1.72379 −0.861897 0.507084i \(-0.830723\pi\)
−0.861897 + 0.507084i \(0.830723\pi\)
\(620\) 11111.6 0.719763
\(621\) 0 0
\(622\) 3693.73 0.238111
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −14460.6 −0.923264
\(627\) 0 0
\(628\) 10852.9 0.689616
\(629\) −9710.93 −0.615580
\(630\) 0 0
\(631\) 5394.86 0.340358 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(632\) −14017.3 −0.882245
\(633\) 0 0
\(634\) 32949.0 2.06399
\(635\) −10231.3 −0.639398
\(636\) 0 0
\(637\) 0 0
\(638\) 24549.3 1.52338
\(639\) 0 0
\(640\) 6880.63 0.424970
\(641\) −2452.41 −0.151114 −0.0755572 0.997141i \(-0.524074\pi\)
−0.0755572 + 0.997141i \(0.524074\pi\)
\(642\) 0 0
\(643\) −7074.97 −0.433919 −0.216959 0.976181i \(-0.569614\pi\)
−0.216959 + 0.976181i \(0.569614\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −18439.1 −1.12303
\(647\) 3341.37 0.203034 0.101517 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(648\) 0 0
\(649\) −6367.31 −0.385114
\(650\) 7697.89 0.464517
\(651\) 0 0
\(652\) −16651.8 −1.00021
\(653\) 23061.6 1.38204 0.691019 0.722837i \(-0.257162\pi\)
0.691019 + 0.722837i \(0.257162\pi\)
\(654\) 0 0
\(655\) 5701.59 0.340121
\(656\) 6539.85 0.389235
\(657\) 0 0
\(658\) 0 0
\(659\) −1742.64 −0.103010 −0.0515049 0.998673i \(-0.516402\pi\)
−0.0515049 + 0.998673i \(0.516402\pi\)
\(660\) 0 0
\(661\) 12576.5 0.740046 0.370023 0.929023i \(-0.379350\pi\)
0.370023 + 0.929023i \(0.379350\pi\)
\(662\) 14338.5 0.841817
\(663\) 0 0
\(664\) 7312.51 0.427380
\(665\) 0 0
\(666\) 0 0
\(667\) 29069.2 1.68750
\(668\) −12018.8 −0.696141
\(669\) 0 0
\(670\) 1821.53 0.105033
\(671\) 6729.17 0.387149
\(672\) 0 0
\(673\) −10680.8 −0.611760 −0.305880 0.952070i \(-0.598951\pi\)
−0.305880 + 0.952070i \(0.598951\pi\)
\(674\) 9643.27 0.551105
\(675\) 0 0
\(676\) 30744.2 1.74921
\(677\) −29559.1 −1.67806 −0.839032 0.544082i \(-0.816878\pi\)
−0.839032 + 0.544082i \(0.816878\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1780.74 0.100424
\(681\) 0 0
\(682\) 17802.8 0.999569
\(683\) −10250.4 −0.574263 −0.287132 0.957891i \(-0.592702\pi\)
−0.287132 + 0.957891i \(0.592702\pi\)
\(684\) 0 0
\(685\) 2451.71 0.136752
\(686\) 0 0
\(687\) 0 0
\(688\) 655.218 0.0363081
\(689\) −9130.20 −0.504837
\(690\) 0 0
\(691\) 8874.04 0.488544 0.244272 0.969707i \(-0.421451\pi\)
0.244272 + 0.969707i \(0.421451\pi\)
\(692\) 27043.9 1.48563
\(693\) 0 0
\(694\) 10915.1 0.597018
\(695\) −14000.0 −0.764101
\(696\) 0 0
\(697\) 5691.94 0.309322
\(698\) 5572.77 0.302196
\(699\) 0 0
\(700\) 0 0
\(701\) 22086.2 1.18999 0.594996 0.803729i \(-0.297154\pi\)
0.594996 + 0.803729i \(0.297154\pi\)
\(702\) 0 0
\(703\) −42253.7 −2.26690
\(704\) 15299.9 0.819085
\(705\) 0 0
\(706\) 33534.3 1.78765
\(707\) 0 0
\(708\) 0 0
\(709\) −27878.9 −1.47675 −0.738373 0.674392i \(-0.764406\pi\)
−0.738373 + 0.674392i \(0.764406\pi\)
\(710\) −6814.57 −0.360206
\(711\) 0 0
\(712\) −12961.7 −0.682248
\(713\) 21080.6 1.10726
\(714\) 0 0
\(715\) 7038.20 0.368131
\(716\) 12873.1 0.671916
\(717\) 0 0
\(718\) −9859.76 −0.512483
\(719\) 25863.3 1.34150 0.670750 0.741684i \(-0.265973\pi\)
0.670750 + 0.741684i \(0.265973\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −50623.4 −2.60943
\(723\) 0 0
\(724\) 32232.8 1.65459
\(725\) 7204.95 0.369083
\(726\) 0 0
\(727\) −29157.0 −1.48744 −0.743722 0.668489i \(-0.766941\pi\)
−0.743722 + 0.668489i \(0.766941\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −23597.4 −1.19641
\(731\) 570.267 0.0288538
\(732\) 0 0
\(733\) −11006.1 −0.554595 −0.277297 0.960784i \(-0.589439\pi\)
−0.277297 + 0.960784i \(0.589439\pi\)
\(734\) 46218.1 2.32417
\(735\) 0 0
\(736\) 24846.4 1.24436
\(737\) 1665.43 0.0832386
\(738\) 0 0
\(739\) −37214.4 −1.85244 −0.926221 0.376982i \(-0.876962\pi\)
−0.926221 + 0.376982i \(0.876962\pi\)
\(740\) 16477.9 0.818567
\(741\) 0 0
\(742\) 0 0
\(743\) −11214.5 −0.553730 −0.276865 0.960909i \(-0.589296\pi\)
−0.276865 + 0.960909i \(0.589296\pi\)
\(744\) 0 0
\(745\) −5830.58 −0.286733
\(746\) 3584.65 0.175930
\(747\) 0 0
\(748\) 6574.54 0.321375
\(749\) 0 0
\(750\) 0 0
\(751\) 6965.26 0.338437 0.169218 0.985579i \(-0.445876\pi\)
0.169218 + 0.985579i \(0.445876\pi\)
\(752\) 5315.94 0.257782
\(753\) 0 0
\(754\) 88740.7 4.28613
\(755\) 4797.91 0.231276
\(756\) 0 0
\(757\) −19352.8 −0.929180 −0.464590 0.885526i \(-0.653798\pi\)
−0.464590 + 0.885526i \(0.653798\pi\)
\(758\) −22676.0 −1.08658
\(759\) 0 0
\(760\) 7748.29 0.369816
\(761\) 32383.6 1.54258 0.771291 0.636483i \(-0.219612\pi\)
0.771291 + 0.636483i \(0.219612\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 23059.7 1.09198
\(765\) 0 0
\(766\) 48535.7 2.28938
\(767\) −23016.5 −1.08354
\(768\) 0 0
\(769\) 25353.9 1.18893 0.594463 0.804123i \(-0.297365\pi\)
0.594463 + 0.804123i \(0.297365\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15848.6 0.738867
\(773\) 26117.0 1.21522 0.607610 0.794236i \(-0.292128\pi\)
0.607610 + 0.794236i \(0.292128\pi\)
\(774\) 0 0
\(775\) 5224.94 0.242175
\(776\) 16049.8 0.742465
\(777\) 0 0
\(778\) −36720.9 −1.69217
\(779\) 24766.5 1.13909
\(780\) 0 0
\(781\) −6230.58 −0.285464
\(782\) 13642.1 0.623838
\(783\) 0 0
\(784\) 0 0
\(785\) 5103.30 0.232031
\(786\) 0 0
\(787\) 2273.38 0.102970 0.0514849 0.998674i \(-0.483605\pi\)
0.0514849 + 0.998674i \(0.483605\pi\)
\(788\) −35946.4 −1.62505
\(789\) 0 0
\(790\) −26616.0 −1.19868
\(791\) 0 0
\(792\) 0 0
\(793\) 24324.6 1.08927
\(794\) 13481.8 0.602585
\(795\) 0 0
\(796\) −48863.3 −2.17577
\(797\) 2937.42 0.130551 0.0652753 0.997867i \(-0.479207\pi\)
0.0652753 + 0.997867i \(0.479207\pi\)
\(798\) 0 0
\(799\) 4626.71 0.204858
\(800\) 6158.31 0.272162
\(801\) 0 0
\(802\) −48585.0 −2.13915
\(803\) −21575.1 −0.948157
\(804\) 0 0
\(805\) 0 0
\(806\) 64353.6 2.81236
\(807\) 0 0
\(808\) 6202.52 0.270054
\(809\) 4317.51 0.187634 0.0938169 0.995589i \(-0.470093\pi\)
0.0938169 + 0.995589i \(0.470093\pi\)
\(810\) 0 0
\(811\) −1286.12 −0.0556863 −0.0278432 0.999612i \(-0.508864\pi\)
−0.0278432 + 0.999612i \(0.508864\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 26400.6 1.13678
\(815\) −7830.08 −0.336534
\(816\) 0 0
\(817\) 2481.32 0.106255
\(818\) −34187.3 −1.46129
\(819\) 0 0
\(820\) −9658.31 −0.411321
\(821\) −26350.2 −1.12013 −0.560066 0.828448i \(-0.689224\pi\)
−0.560066 + 0.828448i \(0.689224\pi\)
\(822\) 0 0
\(823\) 9820.05 0.415924 0.207962 0.978137i \(-0.433317\pi\)
0.207962 + 0.978137i \(0.433317\pi\)
\(824\) 8877.32 0.375311
\(825\) 0 0
\(826\) 0 0
\(827\) 30370.7 1.27702 0.638509 0.769615i \(-0.279552\pi\)
0.638509 + 0.769615i \(0.279552\pi\)
\(828\) 0 0
\(829\) 30817.7 1.29112 0.645562 0.763708i \(-0.276623\pi\)
0.645562 + 0.763708i \(0.276623\pi\)
\(830\) 13885.0 0.580668
\(831\) 0 0
\(832\) 55305.9 2.30455
\(833\) 0 0
\(834\) 0 0
\(835\) −5651.52 −0.234226
\(836\) 28606.8 1.18348
\(837\) 0 0
\(838\) 22693.7 0.935491
\(839\) −24746.0 −1.01827 −0.509134 0.860688i \(-0.670034\pi\)
−0.509134 + 0.860688i \(0.670034\pi\)
\(840\) 0 0
\(841\) 58669.1 2.40556
\(842\) −6290.88 −0.257480
\(843\) 0 0
\(844\) −53045.7 −2.16340
\(845\) 14456.6 0.588548
\(846\) 0 0
\(847\) 0 0
\(848\) −4607.82 −0.186596
\(849\) 0 0
\(850\) 3381.27 0.136443
\(851\) 31261.4 1.25926
\(852\) 0 0
\(853\) 11812.1 0.474135 0.237067 0.971493i \(-0.423814\pi\)
0.237067 + 0.971493i \(0.423814\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7054.89 −0.281695
\(857\) −23440.6 −0.934322 −0.467161 0.884172i \(-0.654723\pi\)
−0.467161 + 0.884172i \(0.654723\pi\)
\(858\) 0 0
\(859\) −8945.58 −0.355319 −0.177660 0.984092i \(-0.556853\pi\)
−0.177660 + 0.984092i \(0.556853\pi\)
\(860\) −967.652 −0.0383682
\(861\) 0 0
\(862\) −66006.3 −2.60810
\(863\) −19313.5 −0.761806 −0.380903 0.924615i \(-0.624387\pi\)
−0.380903 + 0.924615i \(0.624387\pi\)
\(864\) 0 0
\(865\) 12716.7 0.499862
\(866\) 808.330 0.0317184
\(867\) 0 0
\(868\) 0 0
\(869\) −24335.1 −0.949955
\(870\) 0 0
\(871\) 6020.18 0.234197
\(872\) −47.6033 −0.00184868
\(873\) 0 0
\(874\) 59359.0 2.29731
\(875\) 0 0
\(876\) 0 0
\(877\) −12154.8 −0.468001 −0.234001 0.972236i \(-0.575182\pi\)
−0.234001 + 0.972236i \(0.575182\pi\)
\(878\) 15487.7 0.595313
\(879\) 0 0
\(880\) 3552.03 0.136067
\(881\) −29390.4 −1.12394 −0.561968 0.827159i \(-0.689956\pi\)
−0.561968 + 0.827159i \(0.689956\pi\)
\(882\) 0 0
\(883\) 4180.02 0.159308 0.0796540 0.996823i \(-0.474618\pi\)
0.0796540 + 0.996823i \(0.474618\pi\)
\(884\) 23765.6 0.904211
\(885\) 0 0
\(886\) 21219.0 0.804590
\(887\) 21825.8 0.826198 0.413099 0.910686i \(-0.364446\pi\)
0.413099 + 0.910686i \(0.364446\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −24611.6 −0.926947
\(891\) 0 0
\(892\) −40331.6 −1.51390
\(893\) 20131.5 0.754397
\(894\) 0 0
\(895\) 6053.25 0.226076
\(896\) 0 0
\(897\) 0 0
\(898\) 30609.7 1.13748
\(899\) 60232.7 2.23456
\(900\) 0 0
\(901\) −4010.40 −0.148286
\(902\) −15474.4 −0.571221
\(903\) 0 0
\(904\) 16067.2 0.591138
\(905\) 15156.6 0.556710
\(906\) 0 0
\(907\) −8356.11 −0.305910 −0.152955 0.988233i \(-0.548879\pi\)
−0.152955 + 0.988233i \(0.548879\pi\)
\(908\) 41586.3 1.51992
\(909\) 0 0
\(910\) 0 0
\(911\) 4419.80 0.160740 0.0803701 0.996765i \(-0.474390\pi\)
0.0803701 + 0.996765i \(0.474390\pi\)
\(912\) 0 0
\(913\) 12695.1 0.460181
\(914\) −21806.8 −0.789173
\(915\) 0 0
\(916\) −3767.65 −0.135903
\(917\) 0 0
\(918\) 0 0
\(919\) −39257.6 −1.40913 −0.704563 0.709641i \(-0.748857\pi\)
−0.704563 + 0.709641i \(0.748857\pi\)
\(920\) −5732.56 −0.205432
\(921\) 0 0
\(922\) 72007.0 2.57204
\(923\) −22522.2 −0.803173
\(924\) 0 0
\(925\) 7748.29 0.275419
\(926\) −67207.9 −2.38509
\(927\) 0 0
\(928\) 70992.5 2.51125
\(929\) −13399.9 −0.473235 −0.236618 0.971603i \(-0.576039\pi\)
−0.236618 + 0.971603i \(0.576039\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −69036.1 −2.42635
\(933\) 0 0
\(934\) −14367.2 −0.503330
\(935\) 3091.50 0.108131
\(936\) 0 0
\(937\) −27539.8 −0.960176 −0.480088 0.877220i \(-0.659395\pi\)
−0.480088 + 0.877220i \(0.659395\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −7850.79 −0.272409
\(941\) 14363.8 0.497605 0.248802 0.968554i \(-0.419963\pi\)
0.248802 + 0.968554i \(0.419963\pi\)
\(942\) 0 0
\(943\) −18323.5 −0.632762
\(944\) −11615.9 −0.400494
\(945\) 0 0
\(946\) −1550.36 −0.0532838
\(947\) 6372.12 0.218655 0.109327 0.994006i \(-0.465130\pi\)
0.109327 + 0.994006i \(0.465130\pi\)
\(948\) 0 0
\(949\) −77989.6 −2.66770
\(950\) 14712.4 0.502456
\(951\) 0 0
\(952\) 0 0
\(953\) −958.776 −0.0325895 −0.0162948 0.999867i \(-0.505187\pi\)
−0.0162948 + 0.999867i \(0.505187\pi\)
\(954\) 0 0
\(955\) 10843.2 0.367411
\(956\) 3643.88 0.123276
\(957\) 0 0
\(958\) 63055.1 2.12653
\(959\) 0 0
\(960\) 0 0
\(961\) 13888.9 0.466213
\(962\) 95432.7 3.19842
\(963\) 0 0
\(964\) −24601.8 −0.821961
\(965\) 7452.40 0.248602
\(966\) 0 0
\(967\) −40104.5 −1.33369 −0.666843 0.745198i \(-0.732355\pi\)
−0.666843 + 0.745198i \(0.732355\pi\)
\(968\) 10702.8 0.355373
\(969\) 0 0
\(970\) 30475.2 1.00876
\(971\) 12397.4 0.409732 0.204866 0.978790i \(-0.434324\pi\)
0.204866 + 0.978790i \(0.434324\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8113.05 0.266898
\(975\) 0 0
\(976\) 12276.1 0.402611
\(977\) −44982.9 −1.47301 −0.736506 0.676432i \(-0.763525\pi\)
−0.736506 + 0.676432i \(0.763525\pi\)
\(978\) 0 0
\(979\) −22502.5 −0.734608
\(980\) 0 0
\(981\) 0 0
\(982\) 13904.4 0.451841
\(983\) −7895.76 −0.256191 −0.128095 0.991762i \(-0.540886\pi\)
−0.128095 + 0.991762i \(0.540886\pi\)
\(984\) 0 0
\(985\) −16902.8 −0.546771
\(986\) 38979.0 1.25897
\(987\) 0 0
\(988\) 103408. 3.32979
\(989\) −1835.80 −0.0590244
\(990\) 0 0
\(991\) 54534.9 1.74809 0.874046 0.485844i \(-0.161488\pi\)
0.874046 + 0.485844i \(0.161488\pi\)
\(992\) 51482.9 1.64776
\(993\) 0 0
\(994\) 0 0
\(995\) −22976.6 −0.732069
\(996\) 0 0
\(997\) 6028.06 0.191485 0.0957425 0.995406i \(-0.469477\pi\)
0.0957425 + 0.995406i \(0.469477\pi\)
\(998\) −41930.9 −1.32996
\(999\) 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.x.1.1 2
3.2 odd 2 245.4.a.i.1.2 2
7.6 odd 2 2205.4.a.w.1.1 2
15.14 odd 2 1225.4.a.q.1.1 2
21.2 odd 6 245.4.e.k.116.1 4
21.5 even 6 245.4.e.j.116.1 4
21.11 odd 6 245.4.e.k.226.1 4
21.17 even 6 245.4.e.j.226.1 4
21.20 even 2 245.4.a.j.1.2 yes 2
105.104 even 2 1225.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.2 2 3.2 odd 2
245.4.a.j.1.2 yes 2 21.20 even 2
245.4.e.j.116.1 4 21.5 even 6
245.4.e.j.226.1 4 21.17 even 6
245.4.e.k.116.1 4 21.2 odd 6
245.4.e.k.226.1 4 21.11 odd 6
1225.4.a.p.1.1 2 105.104 even 2
1225.4.a.q.1.1 2 15.14 odd 2
2205.4.a.w.1.1 2 7.6 odd 2
2205.4.a.x.1.1 2 1.1 even 1 trivial