Properties

Label 1150.4.b.n.599.8
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 599.8
Root \(8.73081i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.n.599.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} +7.73081i q^{3} -4.00000 q^{4} -15.4616 q^{6} -23.5622i q^{7} -8.00000i q^{8} -32.7654 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +7.73081i q^{3} -4.00000 q^{4} -15.4616 q^{6} -23.5622i q^{7} -8.00000i q^{8} -32.7654 q^{9} -32.1352 q^{11} -30.9232i q^{12} +40.0811i q^{13} +47.1245 q^{14} +16.0000 q^{16} -126.165i q^{17} -65.5308i q^{18} -0.232742 q^{19} +182.155 q^{21} -64.2704i q^{22} -23.0000i q^{23} +61.8465 q^{24} -80.1621 q^{26} -44.5711i q^{27} +94.2490i q^{28} +137.226 q^{29} +112.866 q^{31} +32.0000i q^{32} -248.431i q^{33} +252.331 q^{34} +131.062 q^{36} -45.7057i q^{37} -0.465483i q^{38} -309.859 q^{39} -135.385 q^{41} +364.310i q^{42} +543.528i q^{43} +128.541 q^{44} +46.0000 q^{46} -26.4344i q^{47} +123.693i q^{48} -212.180 q^{49} +975.359 q^{51} -160.324i q^{52} +43.6958i q^{53} +89.1422 q^{54} -188.498 q^{56} -1.79928i q^{57} +274.451i q^{58} -202.248 q^{59} +150.279 q^{61} +225.732i q^{62} +772.026i q^{63} -64.0000 q^{64} +496.862 q^{66} +420.722i q^{67} +504.661i q^{68} +177.809 q^{69} +667.381 q^{71} +262.123i q^{72} +602.960i q^{73} +91.4115 q^{74} +0.930966 q^{76} +757.178i q^{77} -619.718i q^{78} +1378.88 q^{79} -540.095 q^{81} -270.769i q^{82} -485.178i q^{83} -728.621 q^{84} -1087.06 q^{86} +1060.86i q^{87} +257.082i q^{88} +1127.71 q^{89} +944.400 q^{91} +92.0000i q^{92} +872.545i q^{93} +52.8688 q^{94} -247.386 q^{96} +1486.24i q^{97} -424.359i q^{98} +1052.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} + O(q^{10}) \) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} - 78 q^{11} - 4 q^{14} + 128 q^{16} - 106 q^{19} + 600 q^{21} - 64 q^{24} + 80 q^{26} - 322 q^{29} + 776 q^{31} - 92 q^{34} + 256 q^{36} - 2094 q^{39} + 968 q^{41} + 312 q^{44} + 368 q^{46} - 3286 q^{49} + 3650 q^{51} + 548 q^{54} + 16 q^{56} + 188 q^{59} + 2306 q^{61} - 512 q^{64} - 348 q^{66} - 184 q^{69} + 400 q^{71} + 1864 q^{74} + 424 q^{76} + 1816 q^{79} - 2112 q^{81} - 2400 q^{84} - 3576 q^{86} + 3568 q^{89} + 4658 q^{91} - 1060 q^{94} + 256 q^{96} + 5330 q^{99} + O(q^{100}) \)

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
\(3\) 7.73081i 1.48779i 0.668294 + 0.743897i \(0.267025\pi\)
−0.668294 + 0.743897i \(0.732975\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −15.4616 −1.05203
\(7\) − 23.5622i − 1.27224i −0.771589 0.636121i \(-0.780538\pi\)
0.771589 0.636121i \(-0.219462\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) −32.7654 −1.21353
\(10\) 0 0
\(11\) −32.1352 −0.880830 −0.440415 0.897794i \(-0.645169\pi\)
−0.440415 + 0.897794i \(0.645169\pi\)
\(12\) − 30.9232i − 0.743897i
\(13\) 40.0811i 0.855115i 0.903988 + 0.427557i \(0.140626\pi\)
−0.903988 + 0.427557i \(0.859374\pi\)
\(14\) 47.1245 0.899611
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 126.165i − 1.79997i −0.435916 0.899987i \(-0.643576\pi\)
0.435916 0.899987i \(-0.356424\pi\)
\(18\) − 65.5308i − 0.858097i
\(19\) −0.232742 −0.00281024 −0.00140512 0.999999i \(-0.500447\pi\)
−0.00140512 + 0.999999i \(0.500447\pi\)
\(20\) 0 0
\(21\) 182.155 1.89283
\(22\) − 64.2704i − 0.622841i
\(23\) − 23.0000i − 0.208514i
\(24\) 61.8465 0.526015
\(25\) 0 0
\(26\) −80.1621 −0.604657
\(27\) − 44.5711i − 0.317693i
\(28\) 94.2490i 0.636121i
\(29\) 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 \(0.393977\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 248.431i − 1.31049i
\(34\) 252.331 1.27277
\(35\) 0 0
\(36\) 131.062 0.606766
\(37\) − 45.7057i − 0.203080i −0.994831 0.101540i \(-0.967623\pi\)
0.994831 0.101540i \(-0.0323771\pi\)
\(38\) − 0.465483i − 0.00198714i
\(39\) −309.859 −1.27223
\(40\) 0 0
\(41\) −135.385 −0.515696 −0.257848 0.966186i \(-0.583013\pi\)
−0.257848 + 0.966186i \(0.583013\pi\)
\(42\) 364.310i 1.33844i
\(43\) 543.528i 1.92761i 0.266608 + 0.963805i \(0.414097\pi\)
−0.266608 + 0.963805i \(0.585903\pi\)
\(44\) 128.541 0.440415
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) − 26.4344i − 0.0820394i −0.999158 0.0410197i \(-0.986939\pi\)
0.999158 0.0410197i \(-0.0130606\pi\)
\(48\) 123.693i 0.371949i
\(49\) −212.180 −0.618599
\(50\) 0 0
\(51\) 975.359 2.67799
\(52\) − 160.324i − 0.427557i
\(53\) 43.6958i 0.113247i 0.998396 + 0.0566235i \(0.0180335\pi\)
−0.998396 + 0.0566235i \(0.981967\pi\)
\(54\) 89.1422 0.224643
\(55\) 0 0
\(56\) −188.498 −0.449805
\(57\) − 1.79928i − 0.00418106i
\(58\) 274.451i 0.621331i
\(59\) −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 \(0.449587\pi\)
\(62\) 225.732i 0.462387i
\(63\) 772.026i 1.54391i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 496.862 0.926659
\(67\) 420.722i 0.767155i 0.923509 + 0.383577i \(0.125308\pi\)
−0.923509 + 0.383577i \(0.874692\pi\)
\(68\) 504.661i 0.899987i
\(69\) 177.809 0.310227
\(70\) 0 0
\(71\) 667.381 1.11554 0.557771 0.829995i \(-0.311657\pi\)
0.557771 + 0.829995i \(0.311657\pi\)
\(72\) 262.123i 0.429049i
\(73\) 602.960i 0.966728i 0.875420 + 0.483364i \(0.160585\pi\)
−0.875420 + 0.483364i \(0.839415\pi\)
\(74\) 91.4115 0.143600
\(75\) 0 0
\(76\) 0.930966 0.00140512
\(77\) 757.178i 1.12063i
\(78\) − 619.718i − 0.899606i
\(79\) 1378.88 1.96374 0.981872 0.189545i \(-0.0607013\pi\)
0.981872 + 0.189545i \(0.0607013\pi\)
\(80\) 0 0
\(81\) −540.095 −0.740871
\(82\) − 270.769i − 0.364652i
\(83\) − 485.178i − 0.641629i −0.947142 0.320815i \(-0.896043\pi\)
0.947142 0.320815i \(-0.103957\pi\)
\(84\) −728.621 −0.946417
\(85\) 0 0
\(86\) −1087.06 −1.36303
\(87\) 1060.86i 1.30732i
\(88\) 257.082i 0.311420i
\(89\) 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
\(92\) 92.0000i 0.104257i
\(93\) 872.545i 0.972890i
\(94\) 52.8688 0.0580106
\(95\) 0 0
\(96\) −247.386 −0.263007
\(97\) 1486.24i 1.55572i 0.628441 + 0.777858i \(0.283694\pi\)
−0.628441 + 0.777858i \(0.716306\pi\)
\(98\) − 424.359i − 0.437416i
\(99\) 1052.92 1.06892
\(100\) 0 0
\(101\) 1888.86 1.86087 0.930437 0.366451i \(-0.119427\pi\)
0.930437 + 0.366451i \(0.119427\pi\)
\(102\) 1950.72i 1.89363i
\(103\) − 1497.17i − 1.43224i −0.697979 0.716118i \(-0.745917\pi\)
0.697979 0.716118i \(-0.254083\pi\)
\(104\) 320.649 0.302329
\(105\) 0 0
\(106\) −87.3917 −0.0800777
\(107\) 585.150i 0.528678i 0.964430 + 0.264339i \(0.0851538\pi\)
−0.964430 + 0.264339i \(0.914846\pi\)
\(108\) 178.284i 0.158846i
\(109\) 139.166 0.122290 0.0611452 0.998129i \(-0.480525\pi\)
0.0611452 + 0.998129i \(0.480525\pi\)
\(110\) 0 0
\(111\) 353.342 0.302142
\(112\) − 376.996i − 0.318060i
\(113\) − 1293.18i − 1.07656i −0.842765 0.538282i \(-0.819073\pi\)
0.842765 0.538282i \(-0.180927\pi\)
\(114\) 3.59856 0.00295646
\(115\) 0 0
\(116\) −548.902 −0.439347
\(117\) − 1313.27i − 1.03771i
\(118\) − 404.495i − 0.315566i
\(119\) −2972.74 −2.29000
\(120\) 0 0
\(121\) −298.328 −0.224138
\(122\) 300.557i 0.223043i
\(123\) − 1046.63i − 0.767250i
\(124\) −451.464 −0.326957
\(125\) 0 0
\(126\) −1544.05 −1.09171
\(127\) 1298.43i 0.907221i 0.891200 + 0.453610i \(0.149864\pi\)
−0.891200 + 0.453610i \(0.850136\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −4201.91 −2.86789
\(130\) 0 0
\(131\) −525.247 −0.350313 −0.175157 0.984541i \(-0.556043\pi\)
−0.175157 + 0.984541i \(0.556043\pi\)
\(132\) 993.725i 0.655247i
\(133\) 5.48392i 0.00357531i
\(134\) −841.444 −0.542460
\(135\) 0 0
\(136\) −1009.32 −0.636387
\(137\) 2428.12i 1.51422i 0.653288 + 0.757109i \(0.273389\pi\)
−0.653288 + 0.757109i \(0.726611\pi\)
\(138\) 355.617i 0.219363i
\(139\) −2456.46 −1.49895 −0.749476 0.662031i \(-0.769695\pi\)
−0.749476 + 0.662031i \(0.769695\pi\)
\(140\) 0 0
\(141\) 204.359 0.122058
\(142\) 1334.76i 0.788808i
\(143\) − 1288.01i − 0.753211i
\(144\) −524.246 −0.303383
\(145\) 0 0
\(146\) −1205.92 −0.683580
\(147\) − 1640.32i − 0.920349i
\(148\) 182.823i 0.101540i
\(149\) 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 \(0.555979\pi\)
\(152\) 1.86193i 0 0.000993570i
\(153\) 4133.85i 2.18433i
\(154\) −1514.36 −0.792404
\(155\) 0 0
\(156\) 1239.44 0.636117
\(157\) − 3665.04i − 1.86307i −0.363650 0.931536i \(-0.618470\pi\)
0.363650 0.931536i \(-0.381530\pi\)
\(158\) 2757.75i 1.38858i
\(159\) −337.804 −0.168488
\(160\) 0 0
\(161\) −541.932 −0.265281
\(162\) − 1080.19i − 0.523875i
\(163\) − 1055.27i − 0.507088i −0.967324 0.253544i \(-0.918404\pi\)
0.967324 0.253544i \(-0.0815963\pi\)
\(164\) 541.539 0.257848
\(165\) 0 0
\(166\) 970.356 0.453700
\(167\) 731.805i 0.339095i 0.985522 + 0.169547i \(0.0542306\pi\)
−0.985522 + 0.169547i \(0.945769\pi\)
\(168\) − 1457.24i − 0.669218i
\(169\) 590.507 0.268779
\(170\) 0 0
\(171\) 7.62587 0.00341032
\(172\) − 2174.11i − 0.963805i
\(173\) − 521.773i − 0.229304i −0.993406 0.114652i \(-0.963425\pi\)
0.993406 0.114652i \(-0.0365754\pi\)
\(174\) −2121.73 −0.924413
\(175\) 0 0
\(176\) −514.163 −0.220208
\(177\) − 1563.54i − 0.663970i
\(178\) 2255.42i 0.949723i
\(179\) 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 \(0.384830\pi\)
\(182\) 1888.80i 0.769270i
\(183\) 1161.78i 0.469295i
\(184\) −184.000 −0.0737210
\(185\) 0 0
\(186\) −1745.09 −0.687937
\(187\) 4054.35i 1.58547i
\(188\) 105.738i 0.0410197i
\(189\) −1050.20 −0.404182
\(190\) 0 0
\(191\) −2798.12 −1.06002 −0.530012 0.847990i \(-0.677813\pi\)
−0.530012 + 0.847990i \(0.677813\pi\)
\(192\) − 494.772i − 0.185974i
\(193\) − 497.686i − 0.185618i −0.995684 0.0928089i \(-0.970415\pi\)
0.995684 0.0928089i \(-0.0295846\pi\)
\(194\) −2972.47 −1.10006
\(195\) 0 0
\(196\) 848.718 0.309300
\(197\) − 296.489i − 0.107228i −0.998562 0.0536141i \(-0.982926\pi\)
0.998562 0.0536141i \(-0.0170741\pi\)
\(198\) 2105.85i 0.755838i
\(199\) −2347.43 −0.836207 −0.418103 0.908399i \(-0.637305\pi\)
−0.418103 + 0.908399i \(0.637305\pi\)
\(200\) 0 0
\(201\) −3252.52 −1.14137
\(202\) 3777.71i 1.31584i
\(203\) − 3233.34i − 1.11791i
\(204\) −3901.44 −1.33900
\(205\) 0 0
\(206\) 2994.34 1.01274
\(207\) 753.604i 0.253039i
\(208\) 641.297i 0.213779i
\(209\) 7.47920 0.00247535
\(210\) 0 0
\(211\) 2838.61 0.926153 0.463076 0.886318i \(-0.346746\pi\)
0.463076 + 0.886318i \(0.346746\pi\)
\(212\) − 174.783i − 0.0566235i
\(213\) 5159.39i 1.65970i
\(214\) −1170.30 −0.373832
\(215\) 0 0
\(216\) −356.569 −0.112321
\(217\) − 2659.38i − 0.831937i
\(218\) 278.331i 0.0864723i
\(219\) −4661.37 −1.43829
\(220\) 0 0
\(221\) 5056.84 1.53918
\(222\) 706.685i 0.213647i
\(223\) − 2124.68i − 0.638024i −0.947751 0.319012i \(-0.896649\pi\)
0.947751 0.319012i \(-0.103351\pi\)
\(224\) 753.992 0.224903
\(225\) 0 0
\(226\) 2586.35 0.761246
\(227\) − 1551.97i − 0.453780i −0.973920 0.226890i \(-0.927144\pi\)
0.973920 0.226890i \(-0.0728557\pi\)
\(228\) 7.19712i 0.00209053i
\(229\) 158.531 0.0457469 0.0228735 0.999738i \(-0.492719\pi\)
0.0228735 + 0.999738i \(0.492719\pi\)
\(230\) 0 0
\(231\) −5853.60 −1.66727
\(232\) − 1097.80i − 0.310666i
\(233\) − 728.999i − 0.204971i −0.994734 0.102486i \(-0.967320\pi\)
0.994734 0.102486i \(-0.0326796\pi\)
\(234\) 2626.54 0.733772
\(235\) 0 0
\(236\) 808.991 0.223139
\(237\) 10659.8i 2.92165i
\(238\) − 5945.47i − 1.61928i
\(239\) 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 \(0.915906\pi\)
\(242\) − 596.656i − 0.158490i
\(243\) − 5378.79i − 1.41996i
\(244\) −601.115 −0.157715
\(245\) 0 0
\(246\) 2093.27 0.542527
\(247\) − 9.32853i − 0.00240308i
\(248\) − 902.928i − 0.231193i
\(249\) 3750.82 0.954612
\(250\) 0 0
\(251\) 5042.78 1.26812 0.634059 0.773285i \(-0.281388\pi\)
0.634059 + 0.773285i \(0.281388\pi\)
\(252\) − 3088.10i − 0.771954i
\(253\) 739.110i 0.183666i
\(254\) −2596.86 −0.641502
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 2981.15i − 0.723577i −0.932260 0.361788i \(-0.882166\pi\)
0.932260 0.361788i \(-0.117834\pi\)
\(258\) − 8403.82i − 2.02790i
\(259\) −1076.93 −0.258367
\(260\) 0 0
\(261\) −4496.25 −1.06632
\(262\) − 1050.49i − 0.247709i
\(263\) 7242.86i 1.69815i 0.528271 + 0.849076i \(0.322841\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(264\) −1987.45 −0.463330
\(265\) 0 0
\(266\) −10.9678 −0.00252812
\(267\) 8718.10i 1.99827i
\(268\) − 1682.89i − 0.383577i
\(269\) 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 \(0.140586\pi\)
\(272\) − 2018.64i − 0.449994i
\(273\) 7300.98i 1.61859i
\(274\) −4856.23 −1.07071
\(275\) 0 0
\(276\) −711.234 −0.155113
\(277\) − 8991.54i − 1.95036i −0.221418 0.975179i \(-0.571068\pi\)
0.221418 0.975179i \(-0.428932\pi\)
\(278\) − 4912.92i − 1.05992i
\(279\) −3698.10 −0.793546
\(280\) 0 0
\(281\) 968.130 0.205529 0.102765 0.994706i \(-0.467231\pi\)
0.102765 + 0.994706i \(0.467231\pi\)
\(282\) 408.718i 0.0863079i
\(283\) − 2252.65i − 0.473167i −0.971611 0.236583i \(-0.923972\pi\)
0.971611 0.236583i \(-0.0760277\pi\)
\(284\) −2669.52 −0.557771
\(285\) 0 0
\(286\) 2576.03 0.532600
\(287\) 3189.97i 0.656090i
\(288\) − 1048.49i − 0.214524i
\(289\) −11004.7 −2.23991
\(290\) 0 0
\(291\) −11489.8 −2.31458
\(292\) − 2411.84i − 0.483364i
\(293\) 2734.35i 0.545196i 0.962128 + 0.272598i \(0.0878829\pi\)
−0.962128 + 0.272598i \(0.912117\pi\)
\(294\) 3280.64 0.650785
\(295\) 0 0
\(296\) −365.646 −0.0717998
\(297\) 1432.30i 0.279834i
\(298\) 4810.78i 0.935172i
\(299\) 921.865 0.178304
\(300\) 0 0
\(301\) 12806.7 2.45239
\(302\) − 1298.55i − 0.247428i
\(303\) 14602.4i 2.76860i
\(304\) −3.72387 −0.000702560 0
\(305\) 0 0
\(306\) −8267.71 −1.54455
\(307\) 7977.09i 1.48299i 0.670961 + 0.741493i \(0.265882\pi\)
−0.670961 + 0.741493i \(0.734118\pi\)
\(308\) − 3028.71i − 0.560314i
\(309\) 11574.3 2.13087
\(310\) 0 0
\(311\) 7729.43 1.40931 0.704655 0.709550i \(-0.251102\pi\)
0.704655 + 0.709550i \(0.251102\pi\)
\(312\) 2478.87i 0.449803i
\(313\) 5506.25i 0.994350i 0.867650 + 0.497175i \(0.165629\pi\)
−0.867650 + 0.497175i \(0.834371\pi\)
\(314\) 7330.08 1.31739
\(315\) 0 0
\(316\) −5515.51 −0.981872
\(317\) − 5231.37i − 0.926887i −0.886126 0.463443i \(-0.846614\pi\)
0.886126 0.463443i \(-0.153386\pi\)
\(318\) − 675.608i − 0.119139i
\(319\) −4409.77 −0.773981
\(320\) 0 0
\(321\) −4523.68 −0.786565
\(322\) − 1083.86i − 0.187582i
\(323\) 29.3639i 0.00505836i
\(324\) 2160.38 0.370435
\(325\) 0 0
\(326\) 2110.55 0.358566
\(327\) 1075.86i 0.181943i
\(328\) 1083.08i 0.182326i
\(329\) −622.854 −0.104374
\(330\) 0 0
\(331\) −3551.61 −0.589771 −0.294885 0.955533i \(-0.595281\pi\)
−0.294885 + 0.955533i \(0.595281\pi\)
\(332\) 1940.71i 0.320815i
\(333\) 1497.57i 0.246445i
\(334\) −1463.61 −0.239776
\(335\) 0 0
\(336\) 2914.48 0.473209
\(337\) 7002.99i 1.13198i 0.824412 + 0.565990i \(0.191506\pi\)
−0.824412 + 0.565990i \(0.808494\pi\)
\(338\) 1181.01i 0.190055i
\(339\) 9997.29 1.60171
\(340\) 0 0
\(341\) −3626.97 −0.575987
\(342\) 15.2517i 0.00241146i
\(343\) − 3082.42i − 0.485234i
\(344\) 4348.22 0.681513
\(345\) 0 0
\(346\) 1043.55 0.162143
\(347\) 10268.2i 1.58854i 0.607562 + 0.794272i \(0.292148\pi\)
−0.607562 + 0.794272i \(0.707852\pi\)
\(348\) − 4243.46i − 0.653659i
\(349\) −4515.58 −0.692588 −0.346294 0.938126i \(-0.612560\pi\)
−0.346294 + 0.938126i \(0.612560\pi\)
\(350\) 0 0
\(351\) 1786.46 0.271664
\(352\) − 1028.33i − 0.155710i
\(353\) 7838.63i 1.18189i 0.806711 + 0.590947i \(0.201246\pi\)
−0.806711 + 0.590947i \(0.798754\pi\)
\(354\) 3127.08 0.469498
\(355\) 0 0
\(356\) −4510.84 −0.671556
\(357\) − 22981.7i − 3.40705i
\(358\) 2366.14i 0.349314i
\(359\) 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
\(362\) 3447.85i 0.500594i
\(363\) − 2306.32i − 0.333472i
\(364\) −3777.60 −0.543956
\(365\) 0 0
\(366\) −2323.55 −0.331842
\(367\) − 3936.14i − 0.559850i −0.960022 0.279925i \(-0.909690\pi\)
0.960022 0.279925i \(-0.0903096\pi\)
\(368\) − 368.000i − 0.0521286i
\(369\) 4435.93 0.625814
\(370\) 0 0
\(371\) 1029.57 0.144077
\(372\) − 3490.18i − 0.486445i
\(373\) 1920.72i 0.266625i 0.991074 + 0.133313i \(0.0425614\pi\)
−0.991074 + 0.133313i \(0.957439\pi\)
\(374\) −8108.69 −1.12110
\(375\) 0 0
\(376\) −211.475 −0.0290053
\(377\) 5500.15i 0.751385i
\(378\) − 2100.39i − 0.285800i
\(379\) 13074.5 1.77201 0.886004 0.463678i \(-0.153470\pi\)
0.886004 + 0.463678i \(0.153470\pi\)
\(380\) 0 0
\(381\) −10037.9 −1.34976
\(382\) − 5596.23i − 0.749550i
\(383\) 8838.22i 1.17914i 0.807716 + 0.589571i \(0.200703\pi\)
−0.807716 + 0.589571i \(0.799297\pi\)
\(384\) 989.543 0.131504
\(385\) 0 0
\(386\) 995.372 0.131252
\(387\) − 17808.9i − 2.33922i
\(388\) − 5944.94i − 0.777858i
\(389\) 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
\(392\) 1697.44i 0.218708i
\(393\) − 4060.58i − 0.521194i
\(394\) 592.977 0.0758218
\(395\) 0 0
\(396\) −4211.69 −0.534458
\(397\) − 8060.49i − 1.01900i −0.860470 0.509501i \(-0.829830\pi\)
0.860470 0.509501i \(-0.170170\pi\)
\(398\) − 4694.87i − 0.591287i
\(399\) −42.3951 −0.00531932
\(400\) 0 0
\(401\) 12325.1 1.53488 0.767440 0.641121i \(-0.221530\pi\)
0.767440 + 0.641121i \(0.221530\pi\)
\(402\) − 6505.04i − 0.807070i
\(403\) 4523.79i 0.559171i
\(404\) −7555.43 −0.930437
\(405\) 0 0
\(406\) 6466.69 0.790483
\(407\) 1468.76i 0.178879i
\(408\) − 7802.87i − 0.946813i
\(409\) 10806.7 1.30650 0.653250 0.757143i \(-0.273405\pi\)
0.653250 + 0.757143i \(0.273405\pi\)
\(410\) 0 0
\(411\) −18771.3 −2.25285
\(412\) 5988.67i 0.716118i
\(413\) 4765.41i 0.567774i
\(414\) −1507.21 −0.178926
\(415\) 0 0
\(416\) −1282.59 −0.151164
\(417\) − 18990.4i − 2.23013i
\(418\) 14.9584i 0.00175033i
\(419\) 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 \(0.387190\pi\)
\(422\) 5677.23i 0.654889i
\(423\) 866.133i 0.0995575i
\(424\) 349.567 0.0400388
\(425\) 0 0
\(426\) −10318.8 −1.17358
\(427\) − 3540.91i − 0.401303i
\(428\) − 2340.60i − 0.264339i
\(429\) 9957.39 1.12062
\(430\) 0 0
\(431\) 3887.80 0.434499 0.217249 0.976116i \(-0.430292\pi\)
0.217249 + 0.976116i \(0.430292\pi\)
\(432\) − 713.137i − 0.0794232i
\(433\) − 4422.93i − 0.490884i −0.969411 0.245442i \(-0.921067\pi\)
0.969411 0.245442i \(-0.0789330\pi\)
\(434\) 5318.75 0.588268
\(435\) 0 0
\(436\) −556.662 −0.0611452
\(437\) 5.35306i 0 0.000585976i
\(438\) − 9322.74i − 1.01703i
\(439\) 1748.08 0.190048 0.0950242 0.995475i \(-0.469707\pi\)
0.0950242 + 0.995475i \(0.469707\pi\)
\(440\) 0 0
\(441\) 6952.14 0.750690
\(442\) 10113.7i 1.08837i
\(443\) 3371.31i 0.361571i 0.983523 + 0.180785i \(0.0578639\pi\)
−0.983523 + 0.180785i \(0.942136\pi\)
\(444\) −1413.37 −0.151071
\(445\) 0 0
\(446\) 4249.37 0.451151
\(447\) 18595.6i 1.96766i
\(448\) 1507.98i 0.159030i
\(449\) −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
\(452\) 5172.70i 0.538282i
\(453\) − 5019.43i − 0.520603i
\(454\) 3103.94 0.320871
\(455\) 0 0
\(456\) −14.3942 −0.00147823
\(457\) − 3767.20i − 0.385607i −0.981237 0.192803i \(-0.938242\pi\)
0.981237 0.192803i \(-0.0617579\pi\)
\(458\) 317.063i 0.0323480i
\(459\) −5623.32 −0.571839
\(460\) 0 0
\(461\) 9674.06 0.977366 0.488683 0.872461i \(-0.337477\pi\)
0.488683 + 0.872461i \(0.337477\pi\)
\(462\) − 11707.2i − 1.17893i
\(463\) 2977.73i 0.298892i 0.988770 + 0.149446i \(0.0477490\pi\)
−0.988770 + 0.149446i \(0.952251\pi\)
\(464\) 2195.61 0.219674
\(465\) 0 0
\(466\) 1458.00 0.144937
\(467\) 10701.0i 1.06035i 0.847888 + 0.530176i \(0.177874\pi\)
−0.847888 + 0.530176i \(0.822126\pi\)
\(468\) 5253.09i 0.518855i
\(469\) 9913.16 0.976006
\(470\) 0 0
\(471\) 28333.7 2.77187
\(472\) 1617.98i 0.157783i
\(473\) − 17466.4i − 1.69790i
\(474\) −21319.7 −2.06592
\(475\) 0 0
\(476\) 11890.9 1.14500
\(477\) − 1431.71i − 0.137429i
\(478\) 12403.2i 1.18684i
\(479\) −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
\(482\) − 14446.1i − 1.36515i
\(483\) − 4189.57i − 0.394683i
\(484\) 1193.31 0.112069
\(485\) 0 0
\(486\) 10757.6 1.00406
\(487\) 7183.85i 0.668442i 0.942495 + 0.334221i \(0.108473\pi\)
−0.942495 + 0.334221i \(0.891527\pi\)
\(488\) − 1202.23i − 0.111521i
\(489\) 8158.12 0.754443
\(490\) 0 0
\(491\) −12084.2 −1.11070 −0.555350 0.831617i \(-0.687416\pi\)
−0.555350 + 0.831617i \(0.687416\pi\)
\(492\) 4186.53i 0.383625i
\(493\) − 17313.1i − 1.58163i
\(494\) 18.6571 0.00169923
\(495\) 0 0
\(496\) 1805.86 0.163478
\(497\) − 15725.0i − 1.41924i
\(498\) 7501.64i 0.675013i
\(499\) −7145.78 −0.641060 −0.320530 0.947238i \(-0.603861\pi\)
−0.320530 + 0.947238i \(0.603861\pi\)
\(500\) 0 0
\(501\) −5657.45 −0.504503
\(502\) 10085.6i 0.896695i
\(503\) 20436.2i 1.81154i 0.423771 + 0.905770i \(0.360706\pi\)
−0.423771 + 0.905770i \(0.639294\pi\)
\(504\) 6176.21 0.545854
\(505\) 0 0
\(506\) −1478.22 −0.129871
\(507\) 4565.10i 0.399888i
\(508\) − 5193.72i − 0.453610i
\(509\) −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
\(512\) 512.000i 0.0441942i
\(513\) 10.3735i 0 0.000892794i
\(514\) 5962.30 0.511646
\(515\) 0 0
\(516\) 16807.6 1.43394
\(517\) 849.475i 0.0722628i
\(518\) − 2153.86i − 0.182693i
\(519\) 4033.73 0.341158
\(520\) 0 0
\(521\) 3483.23 0.292904 0.146452 0.989218i \(-0.453215\pi\)
0.146452 + 0.989218i \(0.453215\pi\)
\(522\) − 8992.50i − 0.754006i
\(523\) 15689.0i 1.31172i 0.754881 + 0.655862i \(0.227695\pi\)
−0.754881 + 0.655862i \(0.772305\pi\)
\(524\) 2100.99 0.175157
\(525\) 0 0
\(526\) −14485.7 −1.20077
\(527\) − 14239.8i − 1.17703i
\(528\) − 3974.90i − 0.327624i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 6626.72 0.541573
\(532\) − 21.9357i − 0.00178765i
\(533\) − 5426.36i − 0.440979i
\(534\) −17436.2 −1.41299
\(535\) 0 0
\(536\) 3365.78 0.271230
\(537\) 9146.09i 0.734977i
\(538\) 5135.15i 0.411510i
\(539\) 6818.43 0.544881
\(540\) 0 0
\(541\) 7620.73 0.605621 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(542\) 16132.5i 1.27851i
\(543\) 13327.3i 1.05328i
\(544\) 4037.29 0.318194
\(545\) 0 0
\(546\) −14602.0 −1.14452
\(547\) − 5925.62i − 0.463183i −0.972813 0.231592i \(-0.925607\pi\)
0.972813 0.231592i \(-0.0743933\pi\)
\(548\) − 9712.47i − 0.757109i
\(549\) −4923.94 −0.382784
\(550\) 0 0
\(551\) −31.9381 −0.00246934
\(552\) − 1422.47i − 0.109682i
\(553\) − 32489.4i − 2.49836i
\(554\) 17983.1 1.37911
\(555\) 0 0
\(556\) 9825.85 0.749476
\(557\) 5456.16i 0.415054i 0.978229 + 0.207527i \(0.0665414\pi\)
−0.978229 + 0.207527i \(0.933459\pi\)
\(558\) − 7396.20i − 0.561122i
\(559\) −21785.2 −1.64833
\(560\) 0 0
\(561\) −31343.4 −2.35886
\(562\) 1936.26i 0.145331i
\(563\) − 9194.29i − 0.688265i −0.938921 0.344132i \(-0.888173\pi\)
0.938921 0.344132i \(-0.111827\pi\)
\(564\) −817.437 −0.0610289
\(565\) 0 0
\(566\) 4505.30 0.334580
\(567\) 12725.8i 0.942567i
\(568\) − 5339.05i − 0.394404i
\(569\) −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 \(0.520139\pi\)
\(572\) 5152.06i 0.376605i
\(573\) − 21631.7i − 1.57710i
\(574\) −6379.93 −0.463926
\(575\) 0 0
\(576\) 2096.98 0.151692
\(577\) − 23300.0i − 1.68109i −0.541738 0.840547i \(-0.682234\pi\)
0.541738 0.840547i \(-0.317766\pi\)
\(578\) − 22009.3i − 1.58385i
\(579\) 3847.52 0.276161
\(580\) 0 0
\(581\) −11431.9 −0.816307
\(582\) − 22979.6i − 1.63666i
\(583\) − 1404.18i − 0.0997513i
\(584\) 4823.68 0.341790
\(585\) 0 0
\(586\) −5468.70 −0.385512
\(587\) − 15653.3i − 1.10065i −0.834950 0.550325i \(-0.814504\pi\)
0.834950 0.550325i \(-0.185496\pi\)
\(588\) 6561.28i 0.460174i
\(589\) −26.2686 −0.00183766
\(590\) 0 0
\(591\) 2292.10 0.159533
\(592\) − 731.292i − 0.0507701i
\(593\) − 2658.08i − 0.184072i −0.995756 0.0920358i \(-0.970663\pi\)
0.995756 0.0920358i \(-0.0293374\pi\)
\(594\) −2864.60 −0.197872
\(595\) 0 0
\(596\) −9621.57 −0.661266
\(597\) − 18147.6i − 1.24410i
\(598\) 1843.73i 0.126080i
\(599\) −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 \(0.715619\pi\)
\(602\) 25613.5i 1.73410i
\(603\) − 13785.1i − 0.930968i
\(604\) 2597.10 0.174958
\(605\) 0 0
\(606\) −29204.8 −1.95770
\(607\) − 3968.56i − 0.265369i −0.991158 0.132684i \(-0.957640\pi\)
0.991158 0.132684i \(-0.0423596\pi\)
\(608\) − 7.44773i 0 0.000496785i
\(609\) 24996.4 1.66322
\(610\) 0 0
\(611\) 1059.52 0.0701531
\(612\) − 16535.4i − 1.09216i
\(613\) − 11478.7i − 0.756311i −0.925742 0.378156i \(-0.876558\pi\)
0.925742 0.378156i \(-0.123442\pi\)
\(614\) −15954.2 −1.04863
\(615\) 0 0
\(616\) 6057.42 0.396202
\(617\) 15691.1i 1.02382i 0.859039 + 0.511911i \(0.171062\pi\)
−0.859039 + 0.511911i \(0.828938\pi\)
\(618\) 23148.6i 1.50676i
\(619\) 18249.4 1.18499 0.592494 0.805575i \(-0.298144\pi\)
0.592494 + 0.805575i \(0.298144\pi\)
\(620\) 0 0
\(621\) −1025.14 −0.0662436
\(622\) 15458.9i 0.996533i
\(623\) − 26571.4i − 1.70876i
\(624\) −4957.75 −0.318059
\(625\) 0 0
\(626\) −11012.5 −0.703111
\(627\) 57.8203i 0.00368281i
\(628\) 14660.2i 0.931536i
\(629\) −5766.48 −0.365540
\(630\) 0 0
\(631\) −23528.8 −1.48442 −0.742208 0.670169i \(-0.766222\pi\)
−0.742208 + 0.670169i \(0.766222\pi\)
\(632\) − 11031.0i − 0.694288i
\(633\) 21944.8i 1.37793i
\(634\) 10462.7 0.655408
\(635\) 0 0
\(636\) 1351.22 0.0842441
\(637\) − 8504.38i − 0.528973i
\(638\) − 8819.55i − 0.547287i
\(639\) −21867.0 −1.35375
\(640\) 0 0
\(641\) 3748.79 0.230996 0.115498 0.993308i \(-0.463154\pi\)
0.115498 + 0.993308i \(0.463154\pi\)
\(642\) − 9047.36i − 0.556185i
\(643\) 15924.3i 0.976660i 0.872659 + 0.488330i \(0.162394\pi\)
−0.872659 + 0.488330i \(0.837606\pi\)
\(644\) 2167.73 0.132640
\(645\) 0 0
\(646\) −58.7278 −0.00357680
\(647\) 2854.11i 0.173426i 0.996233 + 0.0867129i \(0.0276363\pi\)
−0.996233 + 0.0867129i \(0.972364\pi\)
\(648\) 4320.76i 0.261937i
\(649\) 6499.27 0.393095
\(650\) 0 0
\(651\) 20559.1 1.23775
\(652\) 4221.09i 0.253544i
\(653\) 7925.97i 0.474988i 0.971389 + 0.237494i \(0.0763260\pi\)
−0.971389 + 0.237494i \(0.923674\pi\)
\(654\) −2151.72 −0.128653
\(655\) 0 0
\(656\) −2166.15 −0.128924
\(657\) − 19756.2i − 1.17316i
\(658\) − 1245.71i − 0.0738035i
\(659\) 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 \(0.271896\pi\)
\(662\) − 7103.22i − 0.417031i
\(663\) 39093.4i 2.28999i
\(664\) −3881.42 −0.226850
\(665\) 0 0
\(666\) −2995.13 −0.174263
\(667\) − 3156.19i − 0.183221i
\(668\) − 2927.22i − 0.169547i
\(669\) 16425.5 0.949249
\(670\) 0 0
\(671\) −4829.24 −0.277840
\(672\) 5828.97i 0.334609i
\(673\) − 23253.5i − 1.33188i −0.746005 0.665940i \(-0.768031\pi\)
0.746005 0.665940i \(-0.231969\pi\)
\(674\) −14006.0 −0.800430
\(675\) 0 0
\(676\) −2362.03 −0.134389
\(677\) 19003.7i 1.07884i 0.842038 + 0.539419i \(0.181356\pi\)
−0.842038 + 0.539419i \(0.818644\pi\)
\(678\) 19994.6i 1.13258i
\(679\) 35019.1 1.97925
\(680\) 0 0
\(681\) 11998.0 0.675131
\(682\) − 7253.94i − 0.407284i
\(683\) − 11222.5i − 0.628722i −0.949304 0.314361i \(-0.898210\pi\)
0.949304 0.314361i \(-0.101790\pi\)
\(684\) −30.5035 −0.00170516
\(685\) 0 0
\(686\) 6164.85 0.343112
\(687\) 1225.58i 0.0680620i
\(688\) 8696.45i 0.481902i
\(689\) −1751.38 −0.0968391
\(690\) 0 0
\(691\) 4297.68 0.236601 0.118301 0.992978i \(-0.462255\pi\)
0.118301 + 0.992978i \(0.462255\pi\)
\(692\) 2087.09i 0.114652i
\(693\) − 24809.2i − 1.35992i
\(694\) −20536.4 −1.12327
\(695\) 0 0
\(696\) 8486.92 0.462206
\(697\) 17080.8i 0.928240i
\(698\) − 9031.15i − 0.489734i
\(699\) 5635.75 0.304955
\(700\) 0 0
\(701\) 25769.6 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(702\) 3572.91i 0.192095i
\(703\) 10.6376i 0 0.000570705i
\(704\) 2056.65 0.110104
\(705\) 0 0
\(706\) −15677.3 −0.835725
\(707\) − 44505.7i − 2.36748i
\(708\) 6254.15i 0.331985i
\(709\) 4900.01 0.259554 0.129777 0.991543i \(-0.458574\pi\)
0.129777 + 0.991543i \(0.458574\pi\)
\(710\) 0 0
\(711\) −45179.4 −2.38307
\(712\) − 9021.67i − 0.474862i
\(713\) − 2595.92i − 0.136350i
\(714\) 45963.3 2.40915
\(715\) 0 0
\(716\) −4732.28 −0.247002
\(717\) 47943.4i 2.49718i
\(718\) 24929.6i 1.29577i
\(719\) −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
\(722\) − 13717.9i − 0.707101i
\(723\) − 55839.9i − 2.87235i
\(724\) −6895.70 −0.353973
\(725\) 0 0
\(726\) 4612.64 0.235800
\(727\) 36985.6i 1.88682i 0.331628 + 0.943410i \(0.392402\pi\)
−0.331628 + 0.943410i \(0.607598\pi\)
\(728\) − 7555.20i − 0.384635i
\(729\) 26999.8 1.37173
\(730\) 0 0
\(731\) 68574.3 3.46965
\(732\) − 4647.10i − 0.234647i
\(733\) − 22227.3i − 1.12003i −0.828482 0.560015i \(-0.810795\pi\)
0.828482 0.560015i \(-0.189205\pi\)
\(734\) 7872.29 0.395874
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) − 13520.0i − 0.675733i
\(738\) 8871.86i 0.442517i
\(739\) 17899.1 0.890974 0.445487 0.895288i \(-0.353030\pi\)
0.445487 + 0.895288i \(0.353030\pi\)
\(740\) 0 0
\(741\) 72.1171 0.00357529
\(742\) 2059.14i 0.101878i
\(743\) − 29843.4i − 1.47355i −0.676137 0.736776i \(-0.736347\pi\)
0.676137 0.736776i \(-0.263653\pi\)
\(744\) 6980.36 0.343968
\(745\) 0 0
\(746\) −3841.44 −0.188532
\(747\) 15897.0i 0.778638i
\(748\) − 16217.4i − 0.792736i
\(749\) 13787.4 0.672606
\(750\) 0 0
\(751\) −9399.65 −0.456722 −0.228361 0.973577i \(-0.573337\pi\)
−0.228361 + 0.973577i \(0.573337\pi\)
\(752\) − 422.950i − 0.0205098i
\(753\) 38984.8i 1.88670i
\(754\) −11000.3 −0.531309
\(755\) 0 0
\(756\) 4200.78 0.202091
\(757\) − 29871.6i − 1.43422i −0.696961 0.717109i \(-0.745465\pi\)
0.696961 0.717109i \(-0.254535\pi\)
\(758\) 26149.0i 1.25300i
\(759\) −5713.92 −0.273257
\(760\) 0 0
\(761\) −273.955 −0.0130497 −0.00652487 0.999979i \(-0.502077\pi\)
−0.00652487 + 0.999979i \(0.502077\pi\)
\(762\) − 20075.8i − 0.954423i
\(763\) − 3279.05i − 0.155583i
\(764\) 11192.5 0.530012
\(765\) 0 0
\(766\) −17676.4 −0.833780
\(767\) − 8106.31i − 0.381619i
\(768\) 1979.09i 0.0929872i
\(769\) 13273.8 0.622450 0.311225 0.950336i \(-0.399261\pi\)
0.311225 + 0.950336i \(0.399261\pi\)
\(770\) 0 0
\(771\) 23046.7 1.07653
\(772\) 1990.74i 0.0928089i
\(773\) 34161.1i 1.58951i 0.606933 + 0.794753i \(0.292400\pi\)
−0.606933 + 0.794753i \(0.707600\pi\)
\(774\) 35617.8 1.65408
\(775\) 0 0
\(776\) 11889.9 0.550028
\(777\) − 8325.54i − 0.384398i
\(778\) 25374.6i 1.16931i
\(779\) 31.5096 0.00144923
\(780\) 0 0
\(781\) −21446.4 −0.982603
\(782\) − 5803.60i − 0.265392i
\(783\) − 6116.29i − 0.279155i
\(784\) −3394.87 −0.154650
\(785\) 0 0
\(786\) 8121.17 0.368540
\(787\) 38489.6i 1.74334i 0.490095 + 0.871669i \(0.336962\pi\)
−0.490095 + 0.871669i \(0.663038\pi\)
\(788\) 1185.95i 0.0536141i
\(789\) −55993.2 −2.52650
\(790\) 0 0
\(791\) −30470.1 −1.36965
\(792\) − 8423.38i − 0.377919i
\(793\) 6023.33i 0.269729i
\(794\) 16121.0 0.720544
\(795\) 0 0
\(796\) 9389.73 0.418103
\(797\) − 17176.8i − 0.763405i −0.924285 0.381702i \(-0.875338\pi\)
0.924285 0.381702i \(-0.124662\pi\)
\(798\) − 84.7902i − 0.00376133i
\(799\) −3335.10 −0.147669
\(800\) 0 0
\(801\) −36949.8 −1.62991
\(802\) 24650.2i 1.08532i
\(803\) − 19376.2i − 0.851523i
\(804\) 13010.1 0.570684
\(805\) 0 0
\(806\) −9047.58 −0.395394
\(807\) 19849.4i 0.865841i
\(808\) − 15110.9i − 0.657918i
\(809\) −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 \(0.293847\pi\)
\(812\) 12933.4i 0.558956i
\(813\) 62358.8i 2.69006i
\(814\) −2937.53 −0.126487
\(815\) 0 0
\(816\) 15605.7 0.669498
\(817\) − 126.502i − 0.00541705i
\(818\) 21613.5i 0.923835i
\(819\) −30943.6 −1.32022
\(820\) 0 0
\(821\) −17637.5 −0.749761 −0.374880 0.927073i \(-0.622316\pi\)
−0.374880 + 0.927073i \(0.622316\pi\)
\(822\) − 37542.6i − 1.59300i
\(823\) 28048.6i 1.18799i 0.804470 + 0.593993i \(0.202449\pi\)
−0.804470 + 0.593993i \(0.797551\pi\)
\(824\) −11977.3 −0.506372
\(825\) 0 0
\(826\) −9530.82 −0.401477
\(827\) 32592.9i 1.37046i 0.728329 + 0.685228i \(0.240297\pi\)
−0.728329 + 0.685228i \(0.759703\pi\)
\(828\) − 3014.42i − 0.126520i
\(829\) −18815.9 −0.788303 −0.394152 0.919045i \(-0.628962\pi\)
−0.394152 + 0.919045i \(0.628962\pi\)
\(830\) 0 0
\(831\) 69511.9 2.90173
\(832\) − 2565.19i − 0.106889i
\(833\) 26769.7i 1.11346i
\(834\) 37980.9 1.57694
\(835\) 0 0
\(836\) −29.9168 −0.00123767
\(837\) − 5030.56i − 0.207744i
\(838\) 3647.48i 0.150358i
\(839\) −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
\(842\) 11990.9i 0.490775i
\(843\) 7484.43i 0.305786i
\(844\) −11354.5 −0.463076
\(845\) 0 0
\(846\) −1732.27 −0.0703978
\(847\) 7029.28i 0.285158i
\(848\) 699.134i 0.0283117i
\(849\) 17414.8 0.703975
\(850\) 0 0
\(851\) −1051.23 −0.0423452
\(852\) − 20637.6i − 0.829849i
\(853\) − 31421.4i − 1.26125i −0.776086 0.630627i \(-0.782798\pi\)
0.776086 0.630627i \(-0.217202\pi\)
\(854\) 7081.81 0.283764
\(855\) 0 0
\(856\) 4681.20 0.186916
\(857\) − 20909.5i − 0.833435i −0.909036 0.416718i \(-0.863180\pi\)
0.909036 0.416718i \(-0.136820\pi\)
\(858\) 19914.8i 0.792400i
\(859\) −23304.5 −0.925655 −0.462828 0.886448i \(-0.653165\pi\)
−0.462828 + 0.886448i \(0.653165\pi\)
\(860\) 0 0
\(861\) −24661.0 −0.976127
\(862\) 7775.61i 0.307237i
\(863\) − 19146.3i − 0.755211i −0.925966 0.377606i \(-0.876747\pi\)
0.925966 0.377606i \(-0.123253\pi\)
\(864\) 1426.27 0.0561607
\(865\) 0 0
\(866\) 8845.87 0.347107
\(867\) − 85075.0i − 3.33252i
\(868\) 10637.5i 0.415968i
\(869\) −44310.5 −1.72972
\(870\) 0 0
\(871\) −16863.0 −0.656005
\(872\) − 1113.32i − 0.0432362i
\(873\) − 48697.1i − 1.88791i
\(874\) −10.7061 −0.000414347 0
\(875\) 0 0
\(876\) 18645.5 0.719146
\(877\) − 31389.0i − 1.20859i −0.796762 0.604293i \(-0.793456\pi\)
0.796762 0.604293i \(-0.206544\pi\)
\(878\) 3496.16i 0.134384i
\(879\) −21138.7 −0.811139
\(880\) 0 0
\(881\) 44496.3 1.70161 0.850806 0.525480i \(-0.176114\pi\)
0.850806 + 0.525480i \(0.176114\pi\)
\(882\) 13904.3i 0.530818i
\(883\) 8906.65i 0.339448i 0.985492 + 0.169724i \(0.0542876\pi\)
−0.985492 + 0.169724i \(0.945712\pi\)
\(884\) −20227.4 −0.769592
\(885\) 0 0
\(886\) −6742.62 −0.255669
\(887\) 36584.7i 1.38489i 0.721472 + 0.692444i \(0.243466\pi\)
−0.721472 + 0.692444i \(0.756534\pi\)
\(888\) − 2826.74i − 0.106823i
\(889\) 30593.9 1.15420
\(890\) 0 0
\(891\) 17356.1 0.652581
\(892\) 8498.74i 0.319012i
\(893\) 6.15238i 0 0.000230551i
\(894\) −37191.2 −1.39134
\(895\) 0 0
\(896\) −3015.97 −0.112451
\(897\) 7126.76i 0.265279i
\(898\) − 26629.5i − 0.989573i
\(899\) 15488.1 0.574591
\(900\) 0 0
\(901\) 5512.90 0.203842
\(902\) 8701.23i 0.321197i
\(903\) 99006.4i 3.64865i
\(904\) −10345.4 −0.380623
\(905\) 0 0
\(906\) 10038.9 0.368122
\(907\) 21346.1i 0.781463i 0.920505 + 0.390731i \(0.127778\pi\)
−0.920505 + 0.390731i \(0.872222\pi\)
\(908\) 6207.89i 0.226890i
\(909\) −61889.1 −2.25823
\(910\) 0 0
\(911\) −10208.0 −0.371246 −0.185623 0.982621i \(-0.559430\pi\)
−0.185623 + 0.982621i \(0.559430\pi\)
\(912\) − 28.7885i − 0.00104527i
\(913\) 15591.3i 0.565166i
\(914\) 7534.40 0.272665
\(915\) 0 0
\(916\) −634.125 −0.0228735
\(917\) 12376.0i 0.445683i
\(918\) − 11246.6i − 0.404351i
\(919\) −1758.00 −0.0631025 −0.0315513 0.999502i \(-0.510045\pi\)
−0.0315513 + 0.999502i \(0.510045\pi\)
\(920\) 0 0
\(921\) −61669.3 −2.20638
\(922\) 19348.1i 0.691102i
\(923\) 26749.3i 0.953917i
\(924\) 23414.4 0.833633
\(925\) 0 0
\(926\) −5955.46 −0.211348
\(927\) 49055.3i 1.73807i
\(928\) 4391.22i 0.155333i
\(929\) 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
\(932\) 2915.99i 0.102486i
\(933\) 59754.7i 2.09676i
\(934\) −21402.0 −0.749782
\(935\) 0 0
\(936\) −10506.2 −0.366886
\(937\) − 3851.61i − 0.134287i −0.997743 0.0671433i \(-0.978612\pi\)
0.997743 0.0671433i \(-0.0213885\pi\)
\(938\) 19826.3i 0.690141i
\(939\) −42567.7 −1.47939
\(940\) 0 0
\(941\) −4379.36 −0.151714 −0.0758571 0.997119i \(-0.524169\pi\)
−0.0758571 + 0.997119i \(0.524169\pi\)
\(942\) 56667.5i 1.96001i
\(943\) 3113.85i 0.107530i
\(944\) −3235.96 −0.111570
\(945\) 0 0
\(946\) 34932.8 1.20059
\(947\) − 29746.9i − 1.02074i −0.859954 0.510371i \(-0.829508\pi\)
0.859954 0.510371i \(-0.170492\pi\)
\(948\) − 42639.3i − 1.46082i
\(949\) −24167.3 −0.826663
\(950\) 0 0
\(951\) 40442.7 1.37902
\(952\) 23781.9i 0.809638i
\(953\) 4306.61i 0.146385i 0.997318 + 0.0731924i \(0.0233187\pi\)
−0.997318 + 0.0731924i \(0.976681\pi\)
\(954\) 2863.42 0.0971769
\(955\) 0 0
\(956\) −24806.4 −0.839222
\(957\) − 34091.1i − 1.15152i
\(958\) − 4674.29i − 0.157640i
\(959\) 57211.9 1.92645
\(960\) 0 0
\(961\) −17052.3 −0.572397
\(962\) 3663.87i 0.122794i
\(963\) − 19172.7i − 0.641568i
\(964\) 28892.2 0.965305
\(965\) 0 0
\(966\) 8379.14 0.279083
\(967\) 12233.4i 0.406824i 0.979093 + 0.203412i \(0.0652031\pi\)
−0.979093 + 0.203412i \(0.934797\pi\)
\(968\) 2386.63i 0.0792449i
\(969\) −227.007 −0.00752581
\(970\) 0 0
\(971\) 48207.7 1.59326 0.796631 0.604465i \(-0.206613\pi\)
0.796631 + 0.604465i \(0.206613\pi\)
\(972\) 21515.2i 0.709978i
\(973\) 57879.8i 1.90703i
\(974\) −14367.7 −0.472660
\(975\) 0 0
\(976\) 2404.46 0.0788575
\(977\) − 28662.9i − 0.938595i −0.883040 0.469298i \(-0.844507\pi\)
0.883040 0.469298i \(-0.155493\pi\)
\(978\) 16316.2i 0.533472i
\(979\) −36239.2 −1.18305
\(980\) 0 0
\(981\) −4559.81 −0.148403
\(982\) − 24168.4i − 0.785383i
\(983\) − 20944.1i − 0.679567i −0.940504 0.339783i \(-0.889646\pi\)
0.940504 0.339783i \(-0.110354\pi\)
\(984\) −8373.06 −0.271264
\(985\) 0 0
\(986\) 34626.2 1.11838
\(987\) − 4815.16i − 0.155287i
\(988\) 37.3141i 0.00120154i
\(989\) 12501.1 0.401934
\(990\) 0 0
\(991\) −36440.2 −1.16808 −0.584038 0.811727i \(-0.698528\pi\)
−0.584038 + 0.811727i \(0.698528\pi\)
\(992\) 3611.71i 0.115597i
\(993\) − 27456.8i − 0.877458i
\(994\) 31450.0 1.00355
\(995\) 0 0
\(996\) −15003.3 −0.477306
\(997\) 11945.0i 0.379439i 0.981838 + 0.189720i \(0.0607579\pi\)
−0.981838 + 0.189720i \(0.939242\pi\)
\(998\) − 14291.6i − 0.453298i
\(999\) −2037.15 −0.0645172
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.599.8 8
5.2 odd 4 230.4.a.h.1.4 4
5.3 odd 4 1150.4.a.p.1.1 4
5.4 even 2 inner 1150.4.b.n.599.1 8
15.2 even 4 2070.4.a.bj.1.3 4
20.7 even 4 1840.4.a.m.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 5.2 odd 4
1150.4.a.p.1.1 4 5.3 odd 4
1150.4.b.n.599.1 8 5.4 even 2 inner
1150.4.b.n.599.8 8 1.1 even 1 trivial
1840.4.a.m.1.1 4 20.7 even 4
2070.4.a.bj.1.3 4 15.2 even 4