Properties

Label 1150.4.b.n.599.4
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 136 x^{6} + 5308 x^{4} + 58833 x^{2} + 116964\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.4
Root \(6.57209i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.n.599.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{2} +7.57209i q^{3} -4.00000 q^{4} +15.1442 q^{6} -35.4229i q^{7} +8.00000i q^{8} -30.3365 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +7.57209i q^{3} -4.00000 q^{4} +15.1442 q^{6} -35.4229i q^{7} +8.00000i q^{8} -30.3365 q^{9} -16.6298 q^{11} -30.2883i q^{12} +79.9132i q^{13} -70.8458 q^{14} +16.0000 q^{16} -46.8219i q^{17} +60.6730i q^{18} +110.653 q^{19} +268.225 q^{21} +33.2597i q^{22} +23.0000i q^{23} -60.5767 q^{24} +159.826 q^{26} -25.2641i q^{27} +141.692i q^{28} +0.836422 q^{29} -119.836 q^{31} -32.0000i q^{32} -125.923i q^{33} -93.6438 q^{34} +121.346 q^{36} +368.201i q^{37} -221.307i q^{38} -605.109 q^{39} -95.7927 q^{41} -536.450i q^{42} -331.961i q^{43} +66.5193 q^{44} +46.0000 q^{46} -535.037i q^{47} +121.153i q^{48} -911.782 q^{49} +354.539 q^{51} -319.653i q^{52} -409.345i q^{53} -50.5282 q^{54} +283.383 q^{56} +837.876i q^{57} -1.67284i q^{58} +352.950 q^{59} -507.223 q^{61} +239.672i q^{62} +1074.61i q^{63} -64.0000 q^{64} -251.845 q^{66} -820.056i q^{67} +187.288i q^{68} -174.158 q^{69} -733.770 q^{71} -242.692i q^{72} +91.4599i q^{73} +736.402 q^{74} -442.613 q^{76} +589.077i q^{77} +1210.22i q^{78} -329.381 q^{79} -627.783 q^{81} +191.585i q^{82} +753.834i q^{83} -1072.90 q^{84} -663.922 q^{86} +6.33346i q^{87} -133.039i q^{88} +1050.14 q^{89} +2830.76 q^{91} -92.0000i q^{92} -907.407i q^{93} -1070.07 q^{94} +242.307 q^{96} -271.928i q^{97} +1823.56i q^{98} +504.491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} + O(q^{10}) \) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} - 78 q^{11} - 4 q^{14} + 128 q^{16} - 106 q^{19} + 600 q^{21} - 64 q^{24} + 80 q^{26} - 322 q^{29} + 776 q^{31} - 92 q^{34} + 256 q^{36} - 2094 q^{39} + 968 q^{41} + 312 q^{44} + 368 q^{46} - 3286 q^{49} + 3650 q^{51} + 548 q^{54} + 16 q^{56} + 188 q^{59} + 2306 q^{61} - 512 q^{64} - 348 q^{66} - 184 q^{69} + 400 q^{71} + 1864 q^{74} + 424 q^{76} + 1816 q^{79} - 2112 q^{81} - 2400 q^{84} - 3576 q^{86} + 3568 q^{89} + 4658 q^{91} - 1060 q^{94} + 256 q^{96} + 5330 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 7.57209i 1.45725i 0.684914 + 0.728624i \(0.259840\pi\)
−0.684914 + 0.728624i \(0.740160\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 15.1442 1.03043
\(7\) − 35.4229i − 1.91266i −0.292294 0.956328i \(-0.594419\pi\)
0.292294 0.956328i \(-0.405581\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −30.3365 −1.12357
\(10\) 0 0
\(11\) −16.6298 −0.455826 −0.227913 0.973682i \(-0.573190\pi\)
−0.227913 + 0.973682i \(0.573190\pi\)
\(12\) − 30.2883i − 0.728624i
\(13\) 79.9132i 1.70492i 0.522795 + 0.852459i \(0.324889\pi\)
−0.522795 + 0.852459i \(0.675111\pi\)
\(14\) −70.8458 −1.35245
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 46.8219i − 0.667999i −0.942573 0.333999i \(-0.891602\pi\)
0.942573 0.333999i \(-0.108398\pi\)
\(18\) 60.6730i 0.794486i
\(19\) 110.653 1.33608 0.668042 0.744123i \(-0.267132\pi\)
0.668042 + 0.744123i \(0.267132\pi\)
\(20\) 0 0
\(21\) 268.225 2.78722
\(22\) 33.2597i 0.322318i
\(23\) 23.0000i 0.208514i
\(24\) −60.5767 −0.515215
\(25\) 0 0
\(26\) 159.826 1.20556
\(27\) − 25.2641i − 0.180077i
\(28\) 141.692i 0.956328i
\(29\) 0.836422 0.00535585 0.00267793 0.999996i \(-0.499148\pi\)
0.00267793 + 0.999996i \(0.499148\pi\)
\(30\) 0 0
\(31\) −119.836 −0.694295 −0.347147 0.937811i \(-0.612850\pi\)
−0.347147 + 0.937811i \(0.612850\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 125.923i − 0.664252i
\(34\) −93.6438 −0.472346
\(35\) 0 0
\(36\) 121.346 0.561787
\(37\) 368.201i 1.63600i 0.575221 + 0.817998i \(0.304916\pi\)
−0.575221 + 0.817998i \(0.695084\pi\)
\(38\) − 221.307i − 0.944755i
\(39\) −605.109 −2.48449
\(40\) 0 0
\(41\) −95.7927 −0.364886 −0.182443 0.983216i \(-0.558400\pi\)
−0.182443 + 0.983216i \(0.558400\pi\)
\(42\) − 536.450i − 1.97086i
\(43\) − 331.961i − 1.17729i −0.808390 0.588647i \(-0.799661\pi\)
0.808390 0.588647i \(-0.200339\pi\)
\(44\) 66.5193 0.227913
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) − 535.037i − 1.66049i −0.557397 0.830246i \(-0.688200\pi\)
0.557397 0.830246i \(-0.311800\pi\)
\(48\) 121.153i 0.364312i
\(49\) −911.782 −2.65826
\(50\) 0 0
\(51\) 354.539 0.973440
\(52\) − 319.653i − 0.852459i
\(53\) − 409.345i − 1.06090i −0.847715 0.530452i \(-0.822022\pi\)
0.847715 0.530452i \(-0.177978\pi\)
\(54\) −50.5282 −0.127334
\(55\) 0 0
\(56\) 283.383 0.676226
\(57\) 837.876i 1.94701i
\(58\) − 1.67284i − 0.00378716i
\(59\) 352.950 0.778817 0.389408 0.921065i \(-0.372679\pi\)
0.389408 + 0.921065i \(0.372679\pi\)
\(60\) 0 0
\(61\) −507.223 −1.06464 −0.532322 0.846542i \(-0.678680\pi\)
−0.532322 + 0.846542i \(0.678680\pi\)
\(62\) 239.672i 0.490941i
\(63\) 1074.61i 2.14901i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −251.845 −0.469697
\(67\) − 820.056i − 1.49531i −0.664087 0.747655i \(-0.731180\pi\)
0.664087 0.747655i \(-0.268820\pi\)
\(68\) 187.288i 0.333999i
\(69\) −174.158 −0.303857
\(70\) 0 0
\(71\) −733.770 −1.22651 −0.613257 0.789884i \(-0.710141\pi\)
−0.613257 + 0.789884i \(0.710141\pi\)
\(72\) − 242.692i − 0.397243i
\(73\) 91.4599i 0.146638i 0.997309 + 0.0733190i \(0.0233591\pi\)
−0.997309 + 0.0733190i \(0.976641\pi\)
\(74\) 736.402 1.15682
\(75\) 0 0
\(76\) −442.613 −0.668042
\(77\) 589.077i 0.871838i
\(78\) 1210.22i 1.75680i
\(79\) −329.381 −0.469092 −0.234546 0.972105i \(-0.575360\pi\)
−0.234546 + 0.972105i \(0.575360\pi\)
\(80\) 0 0
\(81\) −627.783 −0.861156
\(82\) 191.585i 0.258013i
\(83\) 753.834i 0.996916i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(84\) −1072.90 −1.39361
\(85\) 0 0
\(86\) −663.922 −0.832472
\(87\) 6.33346i 0.00780481i
\(88\) − 133.039i − 0.161159i
\(89\) 1050.14 1.25073 0.625365 0.780332i \(-0.284950\pi\)
0.625365 + 0.780332i \(0.284950\pi\)
\(90\) 0 0
\(91\) 2830.76 3.26092
\(92\) − 92.0000i − 0.104257i
\(93\) − 907.407i − 1.01176i
\(94\) −1070.07 −1.17415
\(95\) 0 0
\(96\) 242.307 0.257608
\(97\) − 271.928i − 0.284641i −0.989821 0.142320i \(-0.954544\pi\)
0.989821 0.142320i \(-0.0454563\pi\)
\(98\) 1823.56i 1.87967i
\(99\) 504.491 0.512154
\(100\) 0 0
\(101\) 1658.68 1.63410 0.817052 0.576563i \(-0.195607\pi\)
0.817052 + 0.576563i \(0.195607\pi\)
\(102\) − 709.079i − 0.688326i
\(103\) − 1735.52i − 1.66025i −0.557580 0.830123i \(-0.688270\pi\)
0.557580 0.830123i \(-0.311730\pi\)
\(104\) −639.305 −0.602779
\(105\) 0 0
\(106\) −818.691 −0.750172
\(107\) − 1629.88i − 1.47258i −0.676664 0.736292i \(-0.736575\pi\)
0.676664 0.736292i \(-0.263425\pi\)
\(108\) 101.056i 0.0900385i
\(109\) 432.151 0.379748 0.189874 0.981808i \(-0.439192\pi\)
0.189874 + 0.981808i \(0.439192\pi\)
\(110\) 0 0
\(111\) −2788.05 −2.38405
\(112\) − 566.766i − 0.478164i
\(113\) 240.115i 0.199895i 0.994993 + 0.0999476i \(0.0318675\pi\)
−0.994993 + 0.0999476i \(0.968132\pi\)
\(114\) 1675.75 1.37674
\(115\) 0 0
\(116\) −3.34569 −0.00267793
\(117\) − 2424.28i − 1.91560i
\(118\) − 705.900i − 0.550707i
\(119\) −1658.57 −1.27765
\(120\) 0 0
\(121\) −1054.45 −0.792223
\(122\) 1014.45i 0.752817i
\(123\) − 725.350i − 0.531729i
\(124\) 479.343 0.347147
\(125\) 0 0
\(126\) 2149.21 1.51958
\(127\) − 210.993i − 0.147422i −0.997280 0.0737110i \(-0.976516\pi\)
0.997280 0.0737110i \(-0.0234842\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 2513.64 1.71561
\(130\) 0 0
\(131\) −746.178 −0.497663 −0.248832 0.968547i \(-0.580047\pi\)
−0.248832 + 0.968547i \(0.580047\pi\)
\(132\) 503.690i 0.332126i
\(133\) − 3919.66i − 2.55547i
\(134\) −1640.11 −1.05734
\(135\) 0 0
\(136\) 374.575 0.236173
\(137\) − 2420.07i − 1.50920i −0.656184 0.754601i \(-0.727830\pi\)
0.656184 0.754601i \(-0.272170\pi\)
\(138\) 348.316i 0.214860i
\(139\) −924.030 −0.563850 −0.281925 0.959436i \(-0.590973\pi\)
−0.281925 + 0.959436i \(0.590973\pi\)
\(140\) 0 0
\(141\) 4051.34 2.41975
\(142\) 1467.54i 0.867276i
\(143\) − 1328.94i − 0.777145i
\(144\) −485.384 −0.280893
\(145\) 0 0
\(146\) 182.920 0.103689
\(147\) − 6904.09i − 3.87374i
\(148\) − 1472.80i − 0.817998i
\(149\) −430.614 −0.236760 −0.118380 0.992968i \(-0.537770\pi\)
−0.118380 + 0.992968i \(0.537770\pi\)
\(150\) 0 0
\(151\) −25.4118 −0.0136953 −0.00684763 0.999977i \(-0.502180\pi\)
−0.00684763 + 0.999977i \(0.502180\pi\)
\(152\) 885.226i 0.472377i
\(153\) 1420.41i 0.750546i
\(154\) 1178.15 0.616483
\(155\) 0 0
\(156\) 2420.44 1.24224
\(157\) − 1580.29i − 0.803317i −0.915790 0.401658i \(-0.868434\pi\)
0.915790 0.401658i \(-0.131566\pi\)
\(158\) 658.762i 0.331698i
\(159\) 3099.60 1.54600
\(160\) 0 0
\(161\) 814.727 0.398817
\(162\) 1255.57i 0.608930i
\(163\) − 458.739i − 0.220437i −0.993907 0.110218i \(-0.964845\pi\)
0.993907 0.110218i \(-0.0351550\pi\)
\(164\) 383.171 0.182443
\(165\) 0 0
\(166\) 1507.67 0.704926
\(167\) − 561.275i − 0.260076i −0.991509 0.130038i \(-0.958490\pi\)
0.991509 0.130038i \(-0.0415100\pi\)
\(168\) 2145.80i 0.985430i
\(169\) −4189.11 −1.90674
\(170\) 0 0
\(171\) −3356.83 −1.50119
\(172\) 1327.84i 0.588647i
\(173\) − 1573.95i − 0.691705i −0.938289 0.345852i \(-0.887590\pi\)
0.938289 0.345852i \(-0.112410\pi\)
\(174\) 12.6669 0.00551883
\(175\) 0 0
\(176\) −266.077 −0.113956
\(177\) 2672.57i 1.13493i
\(178\) − 2100.29i − 0.884400i
\(179\) −2215.59 −0.925145 −0.462572 0.886581i \(-0.653073\pi\)
−0.462572 + 0.886581i \(0.653073\pi\)
\(180\) 0 0
\(181\) 873.497 0.358710 0.179355 0.983784i \(-0.442599\pi\)
0.179355 + 0.983784i \(0.442599\pi\)
\(182\) − 5661.51i − 2.30582i
\(183\) − 3840.74i − 1.55145i
\(184\) −184.000 −0.0737210
\(185\) 0 0
\(186\) −1814.81 −0.715422
\(187\) 778.641i 0.304491i
\(188\) 2140.15i 0.830246i
\(189\) −894.928 −0.344425
\(190\) 0 0
\(191\) −2497.74 −0.946230 −0.473115 0.881001i \(-0.656870\pi\)
−0.473115 + 0.881001i \(0.656870\pi\)
\(192\) − 484.613i − 0.182156i
\(193\) − 909.155i − 0.339080i −0.985523 0.169540i \(-0.945772\pi\)
0.985523 0.169540i \(-0.0542282\pi\)
\(194\) −543.857 −0.201271
\(195\) 0 0
\(196\) 3647.13 1.32913
\(197\) 608.627i 0.220116i 0.993925 + 0.110058i \(0.0351037\pi\)
−0.993925 + 0.110058i \(0.964896\pi\)
\(198\) − 1008.98i − 0.362147i
\(199\) 2304.98 0.821083 0.410542 0.911842i \(-0.365340\pi\)
0.410542 + 0.911842i \(0.365340\pi\)
\(200\) 0 0
\(201\) 6209.54 2.17904
\(202\) − 3317.36i − 1.15549i
\(203\) − 29.6285i − 0.0102439i
\(204\) −1418.16 −0.486720
\(205\) 0 0
\(206\) −3471.03 −1.17397
\(207\) − 697.739i − 0.234281i
\(208\) 1278.61i 0.426229i
\(209\) −1840.15 −0.609022
\(210\) 0 0
\(211\) −4373.37 −1.42690 −0.713448 0.700708i \(-0.752868\pi\)
−0.713448 + 0.700708i \(0.752868\pi\)
\(212\) 1637.38i 0.530452i
\(213\) − 5556.17i − 1.78733i
\(214\) −3259.76 −1.04127
\(215\) 0 0
\(216\) 202.113 0.0636668
\(217\) 4244.93i 1.32795i
\(218\) − 864.303i − 0.268523i
\(219\) −692.542 −0.213688
\(220\) 0 0
\(221\) 3741.69 1.13888
\(222\) 5576.10i 1.68578i
\(223\) − 5439.50i − 1.63343i −0.577038 0.816717i \(-0.695792\pi\)
0.577038 0.816717i \(-0.304208\pi\)
\(224\) −1133.53 −0.338113
\(225\) 0 0
\(226\) 480.231 0.141347
\(227\) 216.498i 0.0633017i 0.999499 + 0.0316508i \(0.0100765\pi\)
−0.999499 + 0.0316508i \(0.989924\pi\)
\(228\) − 3351.51i − 0.973504i
\(229\) 924.664 0.266828 0.133414 0.991060i \(-0.457406\pi\)
0.133414 + 0.991060i \(0.457406\pi\)
\(230\) 0 0
\(231\) −4460.54 −1.27049
\(232\) 6.69138i 0.00189358i
\(233\) 2369.69i 0.666280i 0.942877 + 0.333140i \(0.108108\pi\)
−0.942877 + 0.333140i \(0.891892\pi\)
\(234\) −4848.57 −1.35453
\(235\) 0 0
\(236\) −1411.80 −0.389408
\(237\) − 2494.10i − 0.683584i
\(238\) 3317.14i 0.903437i
\(239\) 2769.60 0.749583 0.374791 0.927109i \(-0.377714\pi\)
0.374791 + 0.927109i \(0.377714\pi\)
\(240\) 0 0
\(241\) −5334.10 −1.42572 −0.712862 0.701305i \(-0.752601\pi\)
−0.712862 + 0.701305i \(0.752601\pi\)
\(242\) 2108.90i 0.560186i
\(243\) − 5435.76i − 1.43500i
\(244\) 2028.89 0.532322
\(245\) 0 0
\(246\) −1450.70 −0.375989
\(247\) 8842.66i 2.27791i
\(248\) − 958.686i − 0.245470i
\(249\) −5708.10 −1.45275
\(250\) 0 0
\(251\) 4677.13 1.17617 0.588084 0.808800i \(-0.299882\pi\)
0.588084 + 0.808800i \(0.299882\pi\)
\(252\) − 4298.42i − 1.07451i
\(253\) − 382.486i − 0.0950463i
\(254\) −421.986 −0.104243
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 2602.99i − 0.631789i −0.948794 0.315895i \(-0.897695\pi\)
0.948794 0.315895i \(-0.102305\pi\)
\(258\) − 5027.28i − 1.21312i
\(259\) 13042.7 3.12910
\(260\) 0 0
\(261\) −25.3741 −0.00601769
\(262\) 1492.36i 0.351901i
\(263\) 3411.30i 0.799809i 0.916557 + 0.399904i \(0.130957\pi\)
−0.916557 + 0.399904i \(0.869043\pi\)
\(264\) 1007.38 0.234848
\(265\) 0 0
\(266\) −7839.32 −1.80699
\(267\) 7951.77i 1.82262i
\(268\) 3280.22i 0.747655i
\(269\) −4366.56 −0.989718 −0.494859 0.868973i \(-0.664780\pi\)
−0.494859 + 0.868973i \(0.664780\pi\)
\(270\) 0 0
\(271\) 5682.45 1.27374 0.636872 0.770970i \(-0.280228\pi\)
0.636872 + 0.770970i \(0.280228\pi\)
\(272\) − 749.150i − 0.167000i
\(273\) 21434.7i 4.75197i
\(274\) −4840.14 −1.06717
\(275\) 0 0
\(276\) 696.632 0.151929
\(277\) − 1884.62i − 0.408794i −0.978888 0.204397i \(-0.934477\pi\)
0.978888 0.204397i \(-0.0655234\pi\)
\(278\) 1848.06i 0.398702i
\(279\) 3635.39 0.780091
\(280\) 0 0
\(281\) −2706.42 −0.574561 −0.287281 0.957846i \(-0.592751\pi\)
−0.287281 + 0.957846i \(0.592751\pi\)
\(282\) − 8102.69i − 1.71102i
\(283\) 5963.64i 1.25266i 0.779560 + 0.626328i \(0.215443\pi\)
−0.779560 + 0.626328i \(0.784557\pi\)
\(284\) 2935.08 0.613257
\(285\) 0 0
\(286\) −2657.89 −0.549525
\(287\) 3393.26i 0.697901i
\(288\) 970.767i 0.198622i
\(289\) 2720.71 0.553778
\(290\) 0 0
\(291\) 2059.06 0.414792
\(292\) − 365.840i − 0.0733190i
\(293\) 4735.85i 0.944272i 0.881526 + 0.472136i \(0.156517\pi\)
−0.881526 + 0.472136i \(0.843483\pi\)
\(294\) −13808.2 −2.73915
\(295\) 0 0
\(296\) −2945.61 −0.578412
\(297\) 420.138i 0.0820837i
\(298\) 861.227i 0.167415i
\(299\) −1838.00 −0.355500
\(300\) 0 0
\(301\) −11759.0 −2.25176
\(302\) 50.8236i 0.00968401i
\(303\) 12559.6i 2.38130i
\(304\) 1770.45 0.334021
\(305\) 0 0
\(306\) 2840.82 0.530716
\(307\) − 769.241i − 0.143006i −0.997440 0.0715031i \(-0.977220\pi\)
0.997440 0.0715031i \(-0.0227796\pi\)
\(308\) − 2356.31i − 0.435919i
\(309\) 13141.5 2.41939
\(310\) 0 0
\(311\) −5592.71 −1.01972 −0.509861 0.860257i \(-0.670303\pi\)
−0.509861 + 0.860257i \(0.670303\pi\)
\(312\) − 4840.87i − 0.878399i
\(313\) − 9777.19i − 1.76562i −0.469729 0.882811i \(-0.655648\pi\)
0.469729 0.882811i \(-0.344352\pi\)
\(314\) −3160.58 −0.568031
\(315\) 0 0
\(316\) 1317.52 0.234546
\(317\) 1868.51i 0.331060i 0.986205 + 0.165530i \(0.0529335\pi\)
−0.986205 + 0.165530i \(0.947066\pi\)
\(318\) − 6199.20i − 1.09319i
\(319\) −13.9096 −0.00244134
\(320\) 0 0
\(321\) 12341.6 2.14592
\(322\) − 1629.45i − 0.282006i
\(323\) − 5181.00i − 0.892503i
\(324\) 2511.13 0.430578
\(325\) 0 0
\(326\) −917.477 −0.155872
\(327\) 3272.29i 0.553388i
\(328\) − 766.342i − 0.129007i
\(329\) −18952.6 −3.17595
\(330\) 0 0
\(331\) 6966.91 1.15691 0.578454 0.815715i \(-0.303656\pi\)
0.578454 + 0.815715i \(0.303656\pi\)
\(332\) − 3015.34i − 0.498458i
\(333\) − 11169.9i − 1.83816i
\(334\) −1122.55 −0.183902
\(335\) 0 0
\(336\) 4291.60 0.696804
\(337\) − 17.5487i − 0.00283661i −0.999999 0.00141830i \(-0.999549\pi\)
0.999999 0.00141830i \(-0.000451460\pi\)
\(338\) 8378.23i 1.34827i
\(339\) −1818.17 −0.291297
\(340\) 0 0
\(341\) 1992.85 0.316477
\(342\) 6713.66i 1.06150i
\(343\) 20147.9i 3.17167i
\(344\) 2655.69 0.416236
\(345\) 0 0
\(346\) −3147.89 −0.489109
\(347\) − 9859.30i − 1.52529i −0.646818 0.762644i \(-0.723901\pi\)
0.646818 0.762644i \(-0.276099\pi\)
\(348\) − 25.3338i − 0.00390240i
\(349\) −7200.25 −1.10436 −0.552178 0.833726i \(-0.686203\pi\)
−0.552178 + 0.833726i \(0.686203\pi\)
\(350\) 0 0
\(351\) 2018.93 0.307016
\(352\) 532.155i 0.0805794i
\(353\) 6054.58i 0.912897i 0.889750 + 0.456449i \(0.150879\pi\)
−0.889750 + 0.456449i \(0.849121\pi\)
\(354\) 5345.14 0.802516
\(355\) 0 0
\(356\) −4200.57 −0.625365
\(357\) − 12558.8i − 1.86186i
\(358\) 4431.18i 0.654176i
\(359\) −2734.26 −0.401974 −0.200987 0.979594i \(-0.564415\pi\)
−0.200987 + 0.979594i \(0.564415\pi\)
\(360\) 0 0
\(361\) 5385.16 0.785123
\(362\) − 1746.99i − 0.253646i
\(363\) − 7984.37i − 1.15447i
\(364\) −11323.0 −1.63046
\(365\) 0 0
\(366\) −7681.47 −1.09704
\(367\) − 2465.50i − 0.350676i −0.984508 0.175338i \(-0.943898\pi\)
0.984508 0.175338i \(-0.0561018\pi\)
\(368\) 368.000i 0.0521286i
\(369\) 2906.01 0.409976
\(370\) 0 0
\(371\) −14500.2 −2.02915
\(372\) 3629.63i 0.505880i
\(373\) − 5704.45i − 0.791863i −0.918280 0.395932i \(-0.870422\pi\)
0.918280 0.395932i \(-0.129578\pi\)
\(374\) 1557.28 0.215308
\(375\) 0 0
\(376\) 4280.29 0.587073
\(377\) 66.8411i 0.00913128i
\(378\) 1789.86i 0.243546i
\(379\) −10231.9 −1.38675 −0.693376 0.720576i \(-0.743878\pi\)
−0.693376 + 0.720576i \(0.743878\pi\)
\(380\) 0 0
\(381\) 1597.66 0.214830
\(382\) 4995.47i 0.669086i
\(383\) − 6321.34i − 0.843356i −0.906746 0.421678i \(-0.861441\pi\)
0.906746 0.421678i \(-0.138559\pi\)
\(384\) −969.227 −0.128804
\(385\) 0 0
\(386\) −1818.31 −0.239766
\(387\) 10070.5i 1.32278i
\(388\) 1087.71i 0.142320i
\(389\) 925.594 0.120641 0.0603207 0.998179i \(-0.480788\pi\)
0.0603207 + 0.998179i \(0.480788\pi\)
\(390\) 0 0
\(391\) 1076.90 0.139287
\(392\) − 7294.25i − 0.939835i
\(393\) − 5650.13i − 0.725219i
\(394\) 1217.25 0.155646
\(395\) 0 0
\(396\) −2017.96 −0.256077
\(397\) − 2706.75i − 0.342187i −0.985255 0.171093i \(-0.945270\pi\)
0.985255 0.171093i \(-0.0547299\pi\)
\(398\) − 4609.96i − 0.580594i
\(399\) 29680.0 3.72396
\(400\) 0 0
\(401\) −14095.3 −1.75532 −0.877661 0.479281i \(-0.840897\pi\)
−0.877661 + 0.479281i \(0.840897\pi\)
\(402\) − 12419.1i − 1.54081i
\(403\) − 9576.45i − 1.18372i
\(404\) −6634.71 −0.817052
\(405\) 0 0
\(406\) −59.2570 −0.00724353
\(407\) − 6123.12i − 0.745729i
\(408\) 2836.32i 0.344163i
\(409\) 10846.0 1.31125 0.655625 0.755087i \(-0.272405\pi\)
0.655625 + 0.755087i \(0.272405\pi\)
\(410\) 0 0
\(411\) 18325.0 2.19928
\(412\) 6942.06i 0.830123i
\(413\) − 12502.5i − 1.48961i
\(414\) −1395.48 −0.165662
\(415\) 0 0
\(416\) 2557.22 0.301390
\(417\) − 6996.84i − 0.821670i
\(418\) 3680.29i 0.430644i
\(419\) 5626.55 0.656026 0.328013 0.944673i \(-0.393621\pi\)
0.328013 + 0.944673i \(0.393621\pi\)
\(420\) 0 0
\(421\) −7109.09 −0.822983 −0.411491 0.911414i \(-0.634992\pi\)
−0.411491 + 0.911414i \(0.634992\pi\)
\(422\) 8746.74i 1.00897i
\(423\) 16231.1i 1.86568i
\(424\) 3274.76 0.375086
\(425\) 0 0
\(426\) −11112.3 −1.26384
\(427\) 17967.3i 2.03630i
\(428\) 6519.52i 0.736292i
\(429\) 10062.9 1.13249
\(430\) 0 0
\(431\) 4464.14 0.498909 0.249455 0.968387i \(-0.419749\pi\)
0.249455 + 0.968387i \(0.419749\pi\)
\(432\) − 404.226i − 0.0450192i
\(433\) − 11009.4i − 1.22189i −0.791672 0.610947i \(-0.790789\pi\)
0.791672 0.610947i \(-0.209211\pi\)
\(434\) 8489.86 0.939001
\(435\) 0 0
\(436\) −1728.61 −0.189874
\(437\) 2545.03i 0.278593i
\(438\) 1385.08i 0.151100i
\(439\) 3052.70 0.331885 0.165943 0.986135i \(-0.446933\pi\)
0.165943 + 0.986135i \(0.446933\pi\)
\(440\) 0 0
\(441\) 27660.2 2.98675
\(442\) − 7483.37i − 0.805312i
\(443\) 4465.82i 0.478956i 0.970902 + 0.239478i \(0.0769764\pi\)
−0.970902 + 0.239478i \(0.923024\pi\)
\(444\) 11152.2 1.19203
\(445\) 0 0
\(446\) −10879.0 −1.15501
\(447\) − 3260.64i − 0.345018i
\(448\) 2267.07i 0.239082i
\(449\) −9040.14 −0.950179 −0.475090 0.879937i \(-0.657584\pi\)
−0.475090 + 0.879937i \(0.657584\pi\)
\(450\) 0 0
\(451\) 1593.02 0.166324
\(452\) − 960.462i − 0.0999476i
\(453\) − 192.420i − 0.0199574i
\(454\) 432.996 0.0447610
\(455\) 0 0
\(456\) −6703.01 −0.688371
\(457\) 7756.88i 0.793986i 0.917822 + 0.396993i \(0.129946\pi\)
−0.917822 + 0.396993i \(0.870054\pi\)
\(458\) − 1849.33i − 0.188676i
\(459\) −1182.91 −0.120291
\(460\) 0 0
\(461\) 7064.19 0.713692 0.356846 0.934163i \(-0.383852\pi\)
0.356846 + 0.934163i \(0.383852\pi\)
\(462\) 8921.08i 0.898369i
\(463\) − 12599.7i − 1.26470i −0.774682 0.632350i \(-0.782090\pi\)
0.774682 0.632350i \(-0.217910\pi\)
\(464\) 13.3828 0.00133896
\(465\) 0 0
\(466\) 4739.37 0.471131
\(467\) 14746.0i 1.46117i 0.682824 + 0.730583i \(0.260751\pi\)
−0.682824 + 0.730583i \(0.739249\pi\)
\(468\) 9697.14i 0.957800i
\(469\) −29048.8 −2.86002
\(470\) 0 0
\(471\) 11966.1 1.17063
\(472\) 2823.60i 0.275353i
\(473\) 5520.46i 0.536641i
\(474\) −4988.20 −0.483367
\(475\) 0 0
\(476\) 6634.27 0.638826
\(477\) 12418.1i 1.19200i
\(478\) − 5539.19i − 0.530035i
\(479\) 17411.2 1.66084 0.830418 0.557141i \(-0.188102\pi\)
0.830418 + 0.557141i \(0.188102\pi\)
\(480\) 0 0
\(481\) −29424.1 −2.78924
\(482\) 10668.2i 1.00814i
\(483\) 6169.18i 0.581175i
\(484\) 4217.79 0.396111
\(485\) 0 0
\(486\) −10871.5 −1.01470
\(487\) 1320.34i 0.122855i 0.998112 + 0.0614276i \(0.0195653\pi\)
−0.998112 + 0.0614276i \(0.980435\pi\)
\(488\) − 4057.78i − 0.376408i
\(489\) 3473.61 0.321231
\(490\) 0 0
\(491\) 17115.8 1.57317 0.786583 0.617485i \(-0.211848\pi\)
0.786583 + 0.617485i \(0.211848\pi\)
\(492\) 2901.40i 0.265864i
\(493\) − 39.1629i − 0.00357770i
\(494\) 17685.3 1.61073
\(495\) 0 0
\(496\) −1917.37 −0.173574
\(497\) 25992.3i 2.34590i
\(498\) 11416.2i 1.02725i
\(499\) −17540.6 −1.57360 −0.786798 0.617210i \(-0.788263\pi\)
−0.786798 + 0.617210i \(0.788263\pi\)
\(500\) 0 0
\(501\) 4250.02 0.378996
\(502\) − 9354.27i − 0.831676i
\(503\) 4934.98i 0.437455i 0.975786 + 0.218727i \(0.0701905\pi\)
−0.975786 + 0.218727i \(0.929809\pi\)
\(504\) −8596.85 −0.759790
\(505\) 0 0
\(506\) −764.972 −0.0672079
\(507\) − 31720.3i − 2.77860i
\(508\) 843.972i 0.0737110i
\(509\) −11927.3 −1.03865 −0.519323 0.854578i \(-0.673816\pi\)
−0.519323 + 0.854578i \(0.673816\pi\)
\(510\) 0 0
\(511\) 3239.78 0.280468
\(512\) − 512.000i − 0.0441942i
\(513\) − 2795.56i − 0.240598i
\(514\) −5205.97 −0.446743
\(515\) 0 0
\(516\) −10054.6 −0.857804
\(517\) 8897.57i 0.756895i
\(518\) − 26085.5i − 2.21261i
\(519\) 11918.1 1.00799
\(520\) 0 0
\(521\) −9592.76 −0.806653 −0.403327 0.915056i \(-0.632146\pi\)
−0.403327 + 0.915056i \(0.632146\pi\)
\(522\) 50.7482i 0.00425515i
\(523\) − 6525.20i − 0.545558i −0.962077 0.272779i \(-0.912057\pi\)
0.962077 0.272779i \(-0.0879428\pi\)
\(524\) 2984.71 0.248832
\(525\) 0 0
\(526\) 6822.60 0.565550
\(527\) 5610.94i 0.463788i
\(528\) − 2014.76i − 0.166063i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) −10707.3 −0.875058
\(532\) 15678.6i 1.27774i
\(533\) − 7655.10i − 0.622100i
\(534\) 15903.5 1.28879
\(535\) 0 0
\(536\) 6560.45 0.528672
\(537\) − 16776.6i − 1.34817i
\(538\) 8733.13i 0.699836i
\(539\) 15162.8 1.21170
\(540\) 0 0
\(541\) 5634.05 0.447739 0.223870 0.974619i \(-0.428131\pi\)
0.223870 + 0.974619i \(0.428131\pi\)
\(542\) − 11364.9i − 0.900673i
\(543\) 6614.19i 0.522729i
\(544\) −1498.30 −0.118087
\(545\) 0 0
\(546\) 42869.5 3.36015
\(547\) 6093.08i 0.476273i 0.971232 + 0.238136i \(0.0765365\pi\)
−0.971232 + 0.238136i \(0.923463\pi\)
\(548\) 9680.29i 0.754601i
\(549\) 15387.4 1.19620
\(550\) 0 0
\(551\) 92.5529 0.00715587
\(552\) − 1393.26i − 0.107430i
\(553\) 11667.6i 0.897212i
\(554\) −3769.24 −0.289061
\(555\) 0 0
\(556\) 3696.12 0.281925
\(557\) 21461.9i 1.63262i 0.577612 + 0.816311i \(0.303985\pi\)
−0.577612 + 0.816311i \(0.696015\pi\)
\(558\) − 7270.79i − 0.551608i
\(559\) 26528.1 2.00719
\(560\) 0 0
\(561\) −5895.93 −0.443719
\(562\) 5412.85i 0.406276i
\(563\) 11038.5i 0.826316i 0.910659 + 0.413158i \(0.135574\pi\)
−0.910659 + 0.413158i \(0.864426\pi\)
\(564\) −16205.4 −1.20988
\(565\) 0 0
\(566\) 11927.3 0.885761
\(567\) 22237.9i 1.64710i
\(568\) − 5870.16i − 0.433638i
\(569\) 19210.3 1.41535 0.707677 0.706536i \(-0.249743\pi\)
0.707677 + 0.706536i \(0.249743\pi\)
\(570\) 0 0
\(571\) −11967.0 −0.877066 −0.438533 0.898715i \(-0.644502\pi\)
−0.438533 + 0.898715i \(0.644502\pi\)
\(572\) 5315.77i 0.388573i
\(573\) − 18913.1i − 1.37889i
\(574\) 6786.51 0.493490
\(575\) 0 0
\(576\) 1941.53 0.140447
\(577\) − 14982.3i − 1.08097i −0.841352 0.540487i \(-0.818240\pi\)
0.841352 0.540487i \(-0.181760\pi\)
\(578\) − 5441.42i − 0.391580i
\(579\) 6884.20 0.494124
\(580\) 0 0
\(581\) 26703.0 1.90676
\(582\) − 4118.13i − 0.293302i
\(583\) 6807.35i 0.483587i
\(584\) −731.679 −0.0518444
\(585\) 0 0
\(586\) 9471.71 0.667701
\(587\) 24002.9i 1.68774i 0.536547 + 0.843871i \(0.319729\pi\)
−0.536547 + 0.843871i \(0.680271\pi\)
\(588\) 27616.4i 1.93687i
\(589\) −13260.2 −0.927637
\(590\) 0 0
\(591\) −4608.58 −0.320764
\(592\) 5891.21i 0.408999i
\(593\) − 5124.67i − 0.354882i −0.984131 0.177441i \(-0.943218\pi\)
0.984131 0.177441i \(-0.0567819\pi\)
\(594\) 840.276 0.0580420
\(595\) 0 0
\(596\) 1722.45 0.118380
\(597\) 17453.5i 1.19652i
\(598\) 3676.01i 0.251376i
\(599\) 23776.8 1.62186 0.810928 0.585146i \(-0.198963\pi\)
0.810928 + 0.585146i \(0.198963\pi\)
\(600\) 0 0
\(601\) −25435.3 −1.72633 −0.863166 0.504920i \(-0.831522\pi\)
−0.863166 + 0.504920i \(0.831522\pi\)
\(602\) 23518.1i 1.59223i
\(603\) 24877.6i 1.68009i
\(604\) 101.647 0.00684763
\(605\) 0 0
\(606\) 25119.3 1.68383
\(607\) − 7445.94i − 0.497894i −0.968517 0.248947i \(-0.919916\pi\)
0.968517 0.248947i \(-0.0800844\pi\)
\(608\) − 3540.91i − 0.236189i
\(609\) 224.349 0.0149279
\(610\) 0 0
\(611\) 42756.5 2.83100
\(612\) − 5681.65i − 0.375273i
\(613\) 12874.4i 0.848275i 0.905598 + 0.424138i \(0.139423\pi\)
−0.905598 + 0.424138i \(0.860577\pi\)
\(614\) −1538.48 −0.101121
\(615\) 0 0
\(616\) −4712.62 −0.308241
\(617\) − 19247.2i − 1.25585i −0.778272 0.627927i \(-0.783904\pi\)
0.778272 0.627927i \(-0.216096\pi\)
\(618\) − 26282.9i − 1.71077i
\(619\) 14496.4 0.941293 0.470647 0.882322i \(-0.344021\pi\)
0.470647 + 0.882322i \(0.344021\pi\)
\(620\) 0 0
\(621\) 581.074 0.0375486
\(622\) 11185.4i 0.721052i
\(623\) − 37199.1i − 2.39222i
\(624\) −9681.75 −0.621122
\(625\) 0 0
\(626\) −19554.4 −1.24848
\(627\) − 13933.7i − 0.887496i
\(628\) 6321.15i 0.401658i
\(629\) 17239.9 1.09284
\(630\) 0 0
\(631\) −18030.9 −1.13756 −0.568779 0.822490i \(-0.692584\pi\)
−0.568779 + 0.822490i \(0.692584\pi\)
\(632\) − 2635.05i − 0.165849i
\(633\) − 33115.5i − 2.07934i
\(634\) 3737.03 0.234095
\(635\) 0 0
\(636\) −12398.4 −0.773000
\(637\) − 72863.4i − 4.53211i
\(638\) 27.8191i 0.00172628i
\(639\) 22260.0 1.37808
\(640\) 0 0
\(641\) 11776.4 0.725646 0.362823 0.931858i \(-0.381813\pi\)
0.362823 + 0.931858i \(0.381813\pi\)
\(642\) − 24683.2i − 1.51740i
\(643\) − 20207.4i − 1.23935i −0.784858 0.619676i \(-0.787264\pi\)
0.784858 0.619676i \(-0.212736\pi\)
\(644\) −3258.91 −0.199408
\(645\) 0 0
\(646\) −10362.0 −0.631095
\(647\) 21452.8i 1.30355i 0.758412 + 0.651776i \(0.225976\pi\)
−0.758412 + 0.651776i \(0.774024\pi\)
\(648\) − 5022.26i − 0.304465i
\(649\) −5869.50 −0.355005
\(650\) 0 0
\(651\) −32143.0 −1.93515
\(652\) 1834.95i 0.110218i
\(653\) 15095.2i 0.904624i 0.891860 + 0.452312i \(0.149401\pi\)
−0.891860 + 0.452312i \(0.850599\pi\)
\(654\) 6544.57 0.391304
\(655\) 0 0
\(656\) −1532.68 −0.0912214
\(657\) − 2774.57i − 0.164759i
\(658\) 37905.1i 2.24574i
\(659\) −6964.68 −0.411693 −0.205846 0.978584i \(-0.565995\pi\)
−0.205846 + 0.978584i \(0.565995\pi\)
\(660\) 0 0
\(661\) −17380.4 −1.02272 −0.511361 0.859366i \(-0.670858\pi\)
−0.511361 + 0.859366i \(0.670858\pi\)
\(662\) − 13933.8i − 0.818057i
\(663\) 28332.4i 1.65964i
\(664\) −6030.67 −0.352463
\(665\) 0 0
\(666\) −22339.8 −1.29978
\(667\) 19.2377i 0.00111677i
\(668\) 2245.10i 0.130038i
\(669\) 41188.3 2.38032
\(670\) 0 0
\(671\) 8435.04 0.485292
\(672\) − 8583.21i − 0.492715i
\(673\) − 12383.2i − 0.709271i −0.935005 0.354635i \(-0.884605\pi\)
0.935005 0.354635i \(-0.115395\pi\)
\(674\) −35.0973 −0.00200578
\(675\) 0 0
\(676\) 16756.5 0.953371
\(677\) − 8375.74i − 0.475489i −0.971328 0.237744i \(-0.923592\pi\)
0.971328 0.237744i \(-0.0764080\pi\)
\(678\) 3636.35i 0.205978i
\(679\) −9632.49 −0.544420
\(680\) 0 0
\(681\) −1639.34 −0.0922463
\(682\) − 3985.70i − 0.223783i
\(683\) 4434.66i 0.248444i 0.992254 + 0.124222i \(0.0396436\pi\)
−0.992254 + 0.124222i \(0.960356\pi\)
\(684\) 13427.3 0.750595
\(685\) 0 0
\(686\) 40295.8 2.24271
\(687\) 7001.64i 0.388834i
\(688\) − 5311.38i − 0.294323i
\(689\) 32712.1 1.80875
\(690\) 0 0
\(691\) 11032.8 0.607389 0.303694 0.952770i \(-0.401780\pi\)
0.303694 + 0.952770i \(0.401780\pi\)
\(692\) 6295.79i 0.345852i
\(693\) − 17870.5i − 0.979574i
\(694\) −19718.6 −1.07854
\(695\) 0 0
\(696\) −50.6677 −0.00275942
\(697\) 4485.20i 0.243743i
\(698\) 14400.5i 0.780898i
\(699\) −17943.5 −0.970936
\(700\) 0 0
\(701\) 3708.40 0.199806 0.0999032 0.994997i \(-0.468147\pi\)
0.0999032 + 0.994997i \(0.468147\pi\)
\(702\) − 4037.87i − 0.217093i
\(703\) 40742.6i 2.18583i
\(704\) 1064.31 0.0569782
\(705\) 0 0
\(706\) 12109.2 0.645516
\(707\) − 58755.2i − 3.12548i
\(708\) − 10690.3i − 0.567465i
\(709\) 6315.04 0.334508 0.167254 0.985914i \(-0.446510\pi\)
0.167254 + 0.985914i \(0.446510\pi\)
\(710\) 0 0
\(711\) 9992.26 0.527059
\(712\) 8401.15i 0.442200i
\(713\) − 2756.22i − 0.144770i
\(714\) −25117.6 −1.31653
\(715\) 0 0
\(716\) 8862.36 0.462572
\(717\) 20971.6i 1.09233i
\(718\) 5468.52i 0.284239i
\(719\) −24736.9 −1.28307 −0.641536 0.767093i \(-0.721703\pi\)
−0.641536 + 0.767093i \(0.721703\pi\)
\(720\) 0 0
\(721\) −61477.0 −3.17548
\(722\) − 10770.3i − 0.555165i
\(723\) − 40390.2i − 2.07763i
\(724\) −3493.99 −0.179355
\(725\) 0 0
\(726\) −15968.7 −0.816330
\(727\) 33283.6i 1.69797i 0.528420 + 0.848983i \(0.322785\pi\)
−0.528420 + 0.848983i \(0.677215\pi\)
\(728\) 22646.0i 1.15291i
\(729\) 24209.9 1.22999
\(730\) 0 0
\(731\) −15543.1 −0.786430
\(732\) 15362.9i 0.775725i
\(733\) 33636.5i 1.69494i 0.530841 + 0.847471i \(0.321876\pi\)
−0.530841 + 0.847471i \(0.678124\pi\)
\(734\) −4931.00 −0.247965
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) 13637.4i 0.681601i
\(738\) − 5812.03i − 0.289897i
\(739\) −36575.5 −1.82064 −0.910319 0.413907i \(-0.864164\pi\)
−0.910319 + 0.413907i \(0.864164\pi\)
\(740\) 0 0
\(741\) −66957.3 −3.31949
\(742\) 29000.4i 1.43482i
\(743\) 17704.7i 0.874190i 0.899415 + 0.437095i \(0.143993\pi\)
−0.899415 + 0.437095i \(0.856007\pi\)
\(744\) 7259.25 0.357711
\(745\) 0 0
\(746\) −11408.9 −0.559932
\(747\) − 22868.7i − 1.12011i
\(748\) − 3114.56i − 0.152246i
\(749\) −57735.1 −2.81655
\(750\) 0 0
\(751\) 8858.37 0.430421 0.215211 0.976568i \(-0.430956\pi\)
0.215211 + 0.976568i \(0.430956\pi\)
\(752\) − 8560.59i − 0.415123i
\(753\) 35415.7i 1.71397i
\(754\) 133.682 0.00645679
\(755\) 0 0
\(756\) 3579.71 0.172213
\(757\) 7899.26i 0.379265i 0.981855 + 0.189633i \(0.0607297\pi\)
−0.981855 + 0.189633i \(0.939270\pi\)
\(758\) 20463.9i 0.980582i
\(759\) 2896.22 0.138506
\(760\) 0 0
\(761\) 23437.4 1.11643 0.558216 0.829696i \(-0.311486\pi\)
0.558216 + 0.829696i \(0.311486\pi\)
\(762\) − 3195.31i − 0.151908i
\(763\) − 15308.1i − 0.726329i
\(764\) 9990.95 0.473115
\(765\) 0 0
\(766\) −12642.7 −0.596343
\(767\) 28205.4i 1.32782i
\(768\) 1938.45i 0.0910780i
\(769\) −31447.7 −1.47468 −0.737342 0.675519i \(-0.763919\pi\)
−0.737342 + 0.675519i \(0.763919\pi\)
\(770\) 0 0
\(771\) 19710.0 0.920674
\(772\) 3636.62i 0.169540i
\(773\) − 2397.42i − 0.111551i −0.998443 0.0557756i \(-0.982237\pi\)
0.998443 0.0557756i \(-0.0177632\pi\)
\(774\) 20141.1 0.935343
\(775\) 0 0
\(776\) 2175.43 0.100636
\(777\) 98760.8i 4.55987i
\(778\) − 1851.19i − 0.0853063i
\(779\) −10599.8 −0.487518
\(780\) 0 0
\(781\) 12202.5 0.559077
\(782\) − 2153.81i − 0.0984911i
\(783\) − 21.1314i 0 0.000964465i
\(784\) −14588.5 −0.664564
\(785\) 0 0
\(786\) −11300.3 −0.512807
\(787\) − 19407.0i − 0.879014i −0.898239 0.439507i \(-0.855153\pi\)
0.898239 0.439507i \(-0.144847\pi\)
\(788\) − 2434.51i − 0.110058i
\(789\) −25830.7 −1.16552
\(790\) 0 0
\(791\) 8505.58 0.382331
\(792\) 4035.93i 0.181074i
\(793\) − 40533.8i − 1.81513i
\(794\) −5413.51 −0.241962
\(795\) 0 0
\(796\) −9219.91 −0.410542
\(797\) − 28631.1i − 1.27248i −0.771491 0.636240i \(-0.780489\pi\)
0.771491 0.636240i \(-0.219511\pi\)
\(798\) − 59360.0i − 2.63324i
\(799\) −25051.4 −1.10921
\(800\) 0 0
\(801\) −31857.6 −1.40529
\(802\) 28190.6i 1.24120i
\(803\) − 1520.96i − 0.0668414i
\(804\) −24838.1 −1.08952
\(805\) 0 0
\(806\) −19152.9 −0.837013
\(807\) − 33064.0i − 1.44227i
\(808\) 13269.4i 0.577743i
\(809\) 9191.25 0.399440 0.199720 0.979853i \(-0.435997\pi\)
0.199720 + 0.979853i \(0.435997\pi\)
\(810\) 0 0
\(811\) −567.477 −0.0245706 −0.0122853 0.999925i \(-0.503911\pi\)
−0.0122853 + 0.999925i \(0.503911\pi\)
\(812\) 118.514i 0.00512195i
\(813\) 43028.0i 1.85616i
\(814\) −12246.2 −0.527310
\(815\) 0 0
\(816\) 5672.63 0.243360
\(817\) − 36732.6i − 1.57296i
\(818\) − 21692.0i − 0.927193i
\(819\) −85875.2 −3.66388
\(820\) 0 0
\(821\) −6057.35 −0.257494 −0.128747 0.991677i \(-0.541096\pi\)
−0.128747 + 0.991677i \(0.541096\pi\)
\(822\) − 36650.0i − 1.55513i
\(823\) 21327.0i 0.903294i 0.892197 + 0.451647i \(0.149163\pi\)
−0.892197 + 0.451647i \(0.850837\pi\)
\(824\) 13884.1 0.586986
\(825\) 0 0
\(826\) −25005.0 −1.05331
\(827\) − 27553.8i − 1.15857i −0.815124 0.579286i \(-0.803332\pi\)
0.815124 0.579286i \(-0.196668\pi\)
\(828\) 2790.96i 0.117141i
\(829\) −11181.6 −0.468461 −0.234231 0.972181i \(-0.575257\pi\)
−0.234231 + 0.972181i \(0.575257\pi\)
\(830\) 0 0
\(831\) 14270.5 0.595715
\(832\) − 5114.44i − 0.213115i
\(833\) 42691.4i 1.77571i
\(834\) −13993.7 −0.581009
\(835\) 0 0
\(836\) 7360.59 0.304511
\(837\) 3027.54i 0.125026i
\(838\) − 11253.1i − 0.463881i
\(839\) −9811.83 −0.403745 −0.201872 0.979412i \(-0.564703\pi\)
−0.201872 + 0.979412i \(0.564703\pi\)
\(840\) 0 0
\(841\) −24388.3 −0.999971
\(842\) 14218.2i 0.581937i
\(843\) − 20493.3i − 0.837279i
\(844\) 17493.5 0.713448
\(845\) 0 0
\(846\) 32462.3 1.31924
\(847\) 37351.6i 1.51525i
\(848\) − 6549.53i − 0.265226i
\(849\) −45157.2 −1.82543
\(850\) 0 0
\(851\) −8468.62 −0.341129
\(852\) 22224.7i 0.893667i
\(853\) 24030.8i 0.964594i 0.876008 + 0.482297i \(0.160197\pi\)
−0.876008 + 0.482297i \(0.839803\pi\)
\(854\) 35934.6 1.43988
\(855\) 0 0
\(856\) 13039.0 0.520637
\(857\) − 16934.8i − 0.675008i −0.941324 0.337504i \(-0.890417\pi\)
0.941324 0.337504i \(-0.109583\pi\)
\(858\) − 20125.7i − 0.800794i
\(859\) 31057.0 1.23359 0.616793 0.787125i \(-0.288432\pi\)
0.616793 + 0.787125i \(0.288432\pi\)
\(860\) 0 0
\(861\) −25694.0 −1.01701
\(862\) − 8928.27i − 0.352782i
\(863\) 16157.8i 0.637332i 0.947867 + 0.318666i \(0.103235\pi\)
−0.947867 + 0.318666i \(0.896765\pi\)
\(864\) −808.451 −0.0318334
\(865\) 0 0
\(866\) −22018.9 −0.864009
\(867\) 20601.4i 0.806992i
\(868\) − 16979.7i − 0.663974i
\(869\) 5477.55 0.213824
\(870\) 0 0
\(871\) 65533.3 2.54938
\(872\) 3457.21i 0.134261i
\(873\) 8249.35i 0.319815i
\(874\) 5090.05 0.196995
\(875\) 0 0
\(876\) 2770.17 0.106844
\(877\) − 32473.9i − 1.25036i −0.780480 0.625180i \(-0.785025\pi\)
0.780480 0.625180i \(-0.214975\pi\)
\(878\) − 6105.41i − 0.234678i
\(879\) −35860.3 −1.37604
\(880\) 0 0
\(881\) −15703.2 −0.600517 −0.300258 0.953858i \(-0.597073\pi\)
−0.300258 + 0.953858i \(0.597073\pi\)
\(882\) − 55320.5i − 2.11195i
\(883\) 14330.4i 0.546155i 0.961992 + 0.273078i \(0.0880416\pi\)
−0.961992 + 0.273078i \(0.911958\pi\)
\(884\) −14966.7 −0.569441
\(885\) 0 0
\(886\) 8931.65 0.338673
\(887\) 23238.2i 0.879664i 0.898080 + 0.439832i \(0.144962\pi\)
−0.898080 + 0.439832i \(0.855038\pi\)
\(888\) − 22304.4i − 0.842890i
\(889\) −7473.98 −0.281968
\(890\) 0 0
\(891\) 10439.9 0.392537
\(892\) 21758.0i 0.816717i
\(893\) − 59203.6i − 2.21856i
\(894\) −6521.29 −0.243965
\(895\) 0 0
\(896\) 4534.13 0.169057
\(897\) − 13917.5i − 0.518052i
\(898\) 18080.3i 0.671878i
\(899\) −100.233 −0.00371854
\(900\) 0 0
\(901\) −19166.3 −0.708683
\(902\) − 3186.03i − 0.117609i
\(903\) − 89040.4i − 3.28137i
\(904\) −1920.92 −0.0706736
\(905\) 0 0
\(906\) −384.841 −0.0141120
\(907\) − 3667.46i − 0.134263i −0.997744 0.0671313i \(-0.978615\pi\)
0.997744 0.0671313i \(-0.0213846\pi\)
\(908\) − 865.992i − 0.0316508i
\(909\) −50318.4 −1.83604
\(910\) 0 0
\(911\) 21457.9 0.780385 0.390192 0.920733i \(-0.372409\pi\)
0.390192 + 0.920733i \(0.372409\pi\)
\(912\) 13406.0i 0.486752i
\(913\) − 12536.1i − 0.454420i
\(914\) 15513.8 0.561433
\(915\) 0 0
\(916\) −3698.66 −0.133414
\(917\) 26431.8i 0.951859i
\(918\) 2365.83i 0.0850587i
\(919\) 11109.3 0.398762 0.199381 0.979922i \(-0.436107\pi\)
0.199381 + 0.979922i \(0.436107\pi\)
\(920\) 0 0
\(921\) 5824.76 0.208396
\(922\) − 14128.4i − 0.504657i
\(923\) − 58637.9i − 2.09110i
\(924\) 17842.2 0.635243
\(925\) 0 0
\(926\) −25199.3 −0.894279
\(927\) 52649.4i 1.86541i
\(928\) − 26.7655i 0 0.000946790i
\(929\) 20804.3 0.734732 0.367366 0.930076i \(-0.380260\pi\)
0.367366 + 0.930076i \(0.380260\pi\)
\(930\) 0 0
\(931\) −100892. −3.55166
\(932\) − 9478.75i − 0.333140i
\(933\) − 42348.5i − 1.48599i
\(934\) 29492.0 1.03320
\(935\) 0 0
\(936\) 19394.3 0.677267
\(937\) − 35550.5i − 1.23947i −0.784811 0.619735i \(-0.787240\pi\)
0.784811 0.619735i \(-0.212760\pi\)
\(938\) 58097.5i 2.02234i
\(939\) 74033.7 2.57295
\(940\) 0 0
\(941\) 49674.4 1.72087 0.860436 0.509558i \(-0.170191\pi\)
0.860436 + 0.509558i \(0.170191\pi\)
\(942\) − 23932.2i − 0.827762i
\(943\) − 2203.23i − 0.0760839i
\(944\) 5647.20 0.194704
\(945\) 0 0
\(946\) 11040.9 0.379462
\(947\) 33466.2i 1.14837i 0.818726 + 0.574184i \(0.194681\pi\)
−0.818726 + 0.574184i \(0.805319\pi\)
\(948\) 9976.41i 0.341792i
\(949\) −7308.85 −0.250006
\(950\) 0 0
\(951\) −14148.5 −0.482437
\(952\) − 13268.5i − 0.451718i
\(953\) 17298.4i 0.587984i 0.955808 + 0.293992i \(0.0949839\pi\)
−0.955808 + 0.293992i \(0.905016\pi\)
\(954\) 24836.2 0.842874
\(955\) 0 0
\(956\) −11078.4 −0.374791
\(957\) − 105.324i − 0.00355763i
\(958\) − 34822.5i − 1.17439i
\(959\) −85726.0 −2.88659
\(960\) 0 0
\(961\) −15430.4 −0.517955
\(962\) 58848.2i 1.97229i
\(963\) 49444.8i 1.65456i
\(964\) 21336.4 0.712862
\(965\) 0 0
\(966\) 12338.4 0.410953
\(967\) − 14869.7i − 0.494495i −0.968952 0.247248i \(-0.920474\pi\)
0.968952 0.247248i \(-0.0795261\pi\)
\(968\) − 8435.59i − 0.280093i
\(969\) 39231.0 1.30060
\(970\) 0 0
\(971\) 26634.9 0.880283 0.440142 0.897928i \(-0.354928\pi\)
0.440142 + 0.897928i \(0.354928\pi\)
\(972\) 21743.0i 0.717498i
\(973\) 32731.8i 1.07845i
\(974\) 2640.69 0.0868717
\(975\) 0 0
\(976\) −8115.57 −0.266161
\(977\) 40803.6i 1.33615i 0.744092 + 0.668077i \(0.232882\pi\)
−0.744092 + 0.668077i \(0.767118\pi\)
\(978\) − 6947.22i − 0.227145i
\(979\) −17463.7 −0.570115
\(980\) 0 0
\(981\) −13109.9 −0.426675
\(982\) − 34231.6i − 1.11240i
\(983\) 18615.8i 0.604021i 0.953305 + 0.302011i \(0.0976578\pi\)
−0.953305 + 0.302011i \(0.902342\pi\)
\(984\) 5802.80 0.187995
\(985\) 0 0
\(986\) −78.3257 −0.00252982
\(987\) − 143510.i − 4.62815i
\(988\) − 35370.6i − 1.13896i
\(989\) 7635.11 0.245483
\(990\) 0 0
\(991\) 46185.1 1.48044 0.740220 0.672364i \(-0.234721\pi\)
0.740220 + 0.672364i \(0.234721\pi\)
\(992\) 3834.74i 0.122735i
\(993\) 52754.1i 1.68590i
\(994\) 51984.5 1.65880
\(995\) 0 0
\(996\) 22832.4 0.726377
\(997\) 16544.3i 0.525541i 0.964858 + 0.262771i \(0.0846363\pi\)
−0.964858 + 0.262771i \(0.915364\pi\)
\(998\) 35081.2i 1.11270i
\(999\) 9302.26 0.294605
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 1150.4.b.n.599.4 8
5.2 odd 4 1150.4.a.p.1.4 4
5.3 odd 4 230.4.a.h.1.1 4
5.4 even 2 inner 1150.4.b.n.599.5 8
15.8 even 4 2070.4.a.bj.1.1 4
20.3 even 4 1840.4.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.1 4 5.3 odd 4
1150.4.a.p.1.4 4 5.2 odd 4
1150.4.b.n.599.4 8 1.1 even 1 trivial
1150.4.b.n.599.5 8 5.4 even 2 inner
1840.4.a.m.1.4 4 20.3 even 4
2070.4.a.bj.1.1 4 15.8 even 4