Properties

Label 1150.4.b.n.599.6
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 599.6
Root \(-3.74869i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.n.599.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.74869i q^{3} -4.00000 q^{4} +9.49738 q^{6} -29.3684i q^{7} -8.00000i q^{8} +4.44993 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -4.74869i q^{3} -4.00000 q^{4} +9.49738 q^{6} -29.3684i q^{7} -8.00000i q^{8} +4.44993 q^{9} -38.1645 q^{11} +18.9948i q^{12} -22.5396i q^{13} +58.7368 q^{14} +16.0000 q^{16} +104.049i q^{17} +8.89986i q^{18} -142.000 q^{19} -139.461 q^{21} -76.3291i q^{22} -23.0000i q^{23} -37.9895 q^{24} +45.0792 q^{26} -149.346i q^{27} +117.474i q^{28} -241.429 q^{29} +99.2706 q^{31} +32.0000i q^{32} +181.232i q^{33} -208.097 q^{34} -17.7997 q^{36} -59.9452i q^{37} -284.000i q^{38} -107.034 q^{39} +249.248 q^{41} -278.923i q^{42} -163.863i q^{43} +152.658 q^{44} +46.0000 q^{46} +205.591i q^{47} -75.9791i q^{48} -519.502 q^{49} +494.095 q^{51} +90.1584i q^{52} +491.274i q^{53} +298.692 q^{54} -234.947 q^{56} +674.313i q^{57} -482.858i q^{58} -433.734 q^{59} +660.902 q^{61} +198.541i q^{62} -130.687i q^{63} -64.0000 q^{64} -362.463 q^{66} +323.564i q^{67} -416.195i q^{68} -109.220 q^{69} +893.243 q^{71} -35.5994i q^{72} +196.273i q^{73} +119.890 q^{74} +567.999 q^{76} +1120.83i q^{77} -214.067i q^{78} +500.211 q^{79} -589.050 q^{81} +498.496i q^{82} +800.944i q^{83} +557.846 q^{84} +327.726 q^{86} +1146.47i q^{87} +305.316i q^{88} +729.016 q^{89} -661.952 q^{91} +92.0000i q^{92} -471.405i q^{93} -411.181 q^{94} +151.958 q^{96} -1139.96i q^{97} -1039.00i q^{98} -169.829 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} + O(q^{10}) \) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} - 78 q^{11} - 4 q^{14} + 128 q^{16} - 106 q^{19} + 600 q^{21} - 64 q^{24} + 80 q^{26} - 322 q^{29} + 776 q^{31} - 92 q^{34} + 256 q^{36} - 2094 q^{39} + 968 q^{41} + 312 q^{44} + 368 q^{46} - 3286 q^{49} + 3650 q^{51} + 548 q^{54} + 16 q^{56} + 188 q^{59} + 2306 q^{61} - 512 q^{64} - 348 q^{66} - 184 q^{69} + 400 q^{71} + 1864 q^{74} + 424 q^{76} + 1816 q^{79} - 2112 q^{81} - 2400 q^{84} - 3576 q^{86} + 3568 q^{89} + 4658 q^{91} - 1060 q^{94} + 256 q^{96} + 5330 q^{99} + O(q^{100}) \)

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
\(3\) − 4.74869i − 0.913886i −0.889496 0.456943i \(-0.848944\pi\)
0.889496 0.456943i \(-0.151056\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 9.49738 0.646215
\(7\) − 29.3684i − 1.58574i −0.609388 0.792872i \(-0.708585\pi\)
0.609388 0.792872i \(-0.291415\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 4.44993 0.164812
\(10\) 0 0
\(11\) −38.1645 −1.04609 −0.523047 0.852304i \(-0.675205\pi\)
−0.523047 + 0.852304i \(0.675205\pi\)
\(12\) 18.9948i 0.456943i
\(13\) − 22.5396i − 0.480874i −0.970665 0.240437i \(-0.922709\pi\)
0.970665 0.240437i \(-0.0772907\pi\)
\(14\) 58.7368 1.12129
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 104.049i 1.48444i 0.670155 + 0.742221i \(0.266228\pi\)
−0.670155 + 0.742221i \(0.733772\pi\)
\(18\) 8.89986i 0.116540i
\(19\) −142.000 −1.71458 −0.857289 0.514835i \(-0.827853\pi\)
−0.857289 + 0.514835i \(0.827853\pi\)
\(20\) 0 0
\(21\) −139.461 −1.44919
\(22\) − 76.3291i − 0.739701i
\(23\) − 23.0000i − 0.208514i
\(24\) −37.9895 −0.323108
\(25\) 0 0
\(26\) 45.0792 0.340029
\(27\) − 149.346i − 1.06451i
\(28\) 117.474i 0.792872i
\(29\) −241.429 −1.54594 −0.772969 0.634444i \(-0.781229\pi\)
−0.772969 + 0.634444i \(0.781229\pi\)
\(30\) 0 0
\(31\) 99.2706 0.575146 0.287573 0.957759i \(-0.407152\pi\)
0.287573 + 0.957759i \(0.407152\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 181.232i 0.956011i
\(34\) −208.097 −1.04966
\(35\) 0 0
\(36\) −17.7997 −0.0824061
\(37\) − 59.9452i − 0.266349i −0.991093 0.133175i \(-0.957483\pi\)
0.991093 0.133175i \(-0.0425171\pi\)
\(38\) − 284.000i − 1.21239i
\(39\) −107.034 −0.439464
\(40\) 0 0
\(41\) 249.248 0.949414 0.474707 0.880144i \(-0.342554\pi\)
0.474707 + 0.880144i \(0.342554\pi\)
\(42\) − 278.923i − 1.02473i
\(43\) − 163.863i − 0.581136i −0.956854 0.290568i \(-0.906156\pi\)
0.956854 0.290568i \(-0.0938442\pi\)
\(44\) 152.658 0.523047
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 205.591i 0.638052i 0.947746 + 0.319026i \(0.103356\pi\)
−0.947746 + 0.319026i \(0.896644\pi\)
\(48\) − 75.9791i − 0.228472i
\(49\) −519.502 −1.51458
\(50\) 0 0
\(51\) 494.095 1.35661
\(52\) 90.1584i 0.240437i
\(53\) 491.274i 1.27324i 0.771178 + 0.636620i \(0.219668\pi\)
−0.771178 + 0.636620i \(0.780332\pi\)
\(54\) 298.692 0.752719
\(55\) 0 0
\(56\) −234.947 −0.560645
\(57\) 674.313i 1.56693i
\(58\) − 482.858i − 1.09314i
\(59\) −433.734 −0.957074 −0.478537 0.878067i \(-0.658833\pi\)
−0.478537 + 0.878067i \(0.658833\pi\)
\(60\) 0 0
\(61\) 660.902 1.38721 0.693605 0.720355i \(-0.256021\pi\)
0.693605 + 0.720355i \(0.256021\pi\)
\(62\) 198.541i 0.406690i
\(63\) − 130.687i − 0.261350i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −362.463 −0.676002
\(67\) 323.564i 0.589995i 0.955498 + 0.294997i \(0.0953188\pi\)
−0.955498 + 0.294997i \(0.904681\pi\)
\(68\) − 416.195i − 0.742221i
\(69\) −109.220 −0.190558
\(70\) 0 0
\(71\) 893.243 1.49308 0.746538 0.665342i \(-0.231714\pi\)
0.746538 + 0.665342i \(0.231714\pi\)
\(72\) − 35.5994i − 0.0582699i
\(73\) 196.273i 0.314685i 0.987544 + 0.157343i \(0.0502927\pi\)
−0.987544 + 0.157343i \(0.949707\pi\)
\(74\) 119.890 0.188337
\(75\) 0 0
\(76\) 567.999 0.857289
\(77\) 1120.83i 1.65884i
\(78\) − 214.067i − 0.310748i
\(79\) 500.211 0.712381 0.356191 0.934413i \(-0.384075\pi\)
0.356191 + 0.934413i \(0.384075\pi\)
\(80\) 0 0
\(81\) −589.050 −0.808025
\(82\) 498.496i 0.671337i
\(83\) 800.944i 1.05922i 0.848242 + 0.529609i \(0.177661\pi\)
−0.848242 + 0.529609i \(0.822339\pi\)
\(84\) 557.846 0.724595
\(85\) 0 0
\(86\) 327.726 0.410925
\(87\) 1146.47i 1.41281i
\(88\) 305.316i 0.369850i
\(89\) 729.016 0.868264 0.434132 0.900849i \(-0.357055\pi\)
0.434132 + 0.900849i \(0.357055\pi\)
\(90\) 0 0
\(91\) −661.952 −0.762543
\(92\) 92.0000i 0.104257i
\(93\) − 471.405i − 0.525618i
\(94\) −411.181 −0.451171
\(95\) 0 0
\(96\) 151.958 0.161554
\(97\) − 1139.96i − 1.19325i −0.802520 0.596626i \(-0.796508\pi\)
0.802520 0.596626i \(-0.203492\pi\)
\(98\) − 1039.00i − 1.07097i
\(99\) −169.829 −0.172409
\(100\) 0 0
\(101\) −1669.38 −1.64465 −0.822326 0.569017i \(-0.807324\pi\)
−0.822326 + 0.569017i \(0.807324\pi\)
\(102\) 988.190i 0.959269i
\(103\) − 1110.63i − 1.06246i −0.847228 0.531229i \(-0.821730\pi\)
0.847228 0.531229i \(-0.178270\pi\)
\(104\) −180.317 −0.170015
\(105\) 0 0
\(106\) −982.549 −0.900317
\(107\) − 39.1892i − 0.0354071i −0.999843 0.0177036i \(-0.994364\pi\)
0.999843 0.0177036i \(-0.00563551\pi\)
\(108\) 597.384i 0.532253i
\(109\) −807.545 −0.709622 −0.354811 0.934938i \(-0.615455\pi\)
−0.354811 + 0.934938i \(0.615455\pi\)
\(110\) 0 0
\(111\) −284.661 −0.243413
\(112\) − 469.894i − 0.396436i
\(113\) 1066.58i 0.887923i 0.896046 + 0.443961i \(0.146427\pi\)
−0.896046 + 0.443961i \(0.853573\pi\)
\(114\) −1348.63 −1.10799
\(115\) 0 0
\(116\) 965.715 0.772969
\(117\) − 100.300i − 0.0792539i
\(118\) − 867.468i − 0.676753i
\(119\) 3055.74 2.35395
\(120\) 0 0
\(121\) 125.532 0.0943139
\(122\) 1321.80i 0.980906i
\(123\) − 1183.60i − 0.867657i
\(124\) −397.082 −0.287573
\(125\) 0 0
\(126\) 261.374 0.184802
\(127\) 641.707i 0.448364i 0.974547 + 0.224182i \(0.0719711\pi\)
−0.974547 + 0.224182i \(0.928029\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −778.134 −0.531092
\(130\) 0 0
\(131\) 2389.92 1.59396 0.796979 0.604007i \(-0.206430\pi\)
0.796979 + 0.604007i \(0.206430\pi\)
\(132\) − 724.926i − 0.478006i
\(133\) 4170.31i 2.71888i
\(134\) −647.128 −0.417189
\(135\) 0 0
\(136\) 832.390 0.524830
\(137\) − 899.291i − 0.560814i −0.959881 0.280407i \(-0.909530\pi\)
0.959881 0.280407i \(-0.0904695\pi\)
\(138\) − 218.440i − 0.134745i
\(139\) −309.341 −0.188762 −0.0943811 0.995536i \(-0.530087\pi\)
−0.0943811 + 0.995536i \(0.530087\pi\)
\(140\) 0 0
\(141\) 976.286 0.583107
\(142\) 1786.49i 1.05576i
\(143\) 860.214i 0.503040i
\(144\) 71.1989 0.0412030
\(145\) 0 0
\(146\) −392.546 −0.222516
\(147\) 2466.96i 1.38416i
\(148\) 239.781i 0.133175i
\(149\) −2560.87 −1.40802 −0.704010 0.710190i \(-0.748609\pi\)
−0.704010 + 0.710190i \(0.748609\pi\)
\(150\) 0 0
\(151\) −2463.15 −1.32747 −0.663737 0.747966i \(-0.731031\pi\)
−0.663737 + 0.747966i \(0.731031\pi\)
\(152\) 1136.00i 0.606195i
\(153\) 463.009i 0.244654i
\(154\) −2241.66 −1.17298
\(155\) 0 0
\(156\) 428.135 0.219732
\(157\) 566.720i 0.288084i 0.989572 + 0.144042i \(0.0460100\pi\)
−0.989572 + 0.144042i \(0.953990\pi\)
\(158\) 1000.42i 0.503730i
\(159\) 2332.91 1.16360
\(160\) 0 0
\(161\) −675.473 −0.330650
\(162\) − 1178.10i − 0.571360i
\(163\) 960.902i 0.461740i 0.972985 + 0.230870i \(0.0741573\pi\)
−0.972985 + 0.230870i \(0.925843\pi\)
\(164\) −996.992 −0.474707
\(165\) 0 0
\(166\) −1601.89 −0.748980
\(167\) 4224.70i 1.95759i 0.204847 + 0.978794i \(0.434330\pi\)
−0.204847 + 0.978794i \(0.565670\pi\)
\(168\) 1115.69i 0.512366i
\(169\) 1688.97 0.768760
\(170\) 0 0
\(171\) −631.889 −0.282583
\(172\) 655.451i 0.290568i
\(173\) 3448.98i 1.51573i 0.652412 + 0.757865i \(0.273757\pi\)
−0.652412 + 0.757865i \(0.726243\pi\)
\(174\) −2292.94 −0.999008
\(175\) 0 0
\(176\) −610.633 −0.261524
\(177\) 2059.67i 0.874657i
\(178\) 1458.03i 0.613955i
\(179\) −1457.82 −0.608728 −0.304364 0.952556i \(-0.598444\pi\)
−0.304364 + 0.952556i \(0.598444\pi\)
\(180\) 0 0
\(181\) −3634.97 −1.49274 −0.746368 0.665534i \(-0.768204\pi\)
−0.746368 + 0.665534i \(0.768204\pi\)
\(182\) − 1323.90i − 0.539199i
\(183\) − 3138.42i − 1.26775i
\(184\) −184.000 −0.0737210
\(185\) 0 0
\(186\) 942.811 0.371668
\(187\) − 3970.97i − 1.55287i
\(188\) − 822.362i − 0.319026i
\(189\) −4386.05 −1.68803
\(190\) 0 0
\(191\) 448.811 0.170025 0.0850127 0.996380i \(-0.472907\pi\)
0.0850127 + 0.996380i \(0.472907\pi\)
\(192\) 303.916i 0.114236i
\(193\) 4259.35i 1.58857i 0.607544 + 0.794286i \(0.292155\pi\)
−0.607544 + 0.794286i \(0.707845\pi\)
\(194\) 2279.92 0.843756
\(195\) 0 0
\(196\) 2078.01 0.757292
\(197\) 1767.37i 0.639189i 0.947554 + 0.319594i \(0.103547\pi\)
−0.947554 + 0.319594i \(0.896453\pi\)
\(198\) − 339.659i − 0.121912i
\(199\) −1004.29 −0.357750 −0.178875 0.983872i \(-0.557246\pi\)
−0.178875 + 0.983872i \(0.557246\pi\)
\(200\) 0 0
\(201\) 1536.51 0.539188
\(202\) − 3338.76i − 1.16294i
\(203\) 7090.37i 2.45146i
\(204\) −1976.38 −0.678305
\(205\) 0 0
\(206\) 2221.25 0.751272
\(207\) − 102.348i − 0.0343657i
\(208\) − 360.634i − 0.120218i
\(209\) 5419.36 1.79361
\(210\) 0 0
\(211\) 5395.17 1.76028 0.880140 0.474714i \(-0.157449\pi\)
0.880140 + 0.474714i \(0.157449\pi\)
\(212\) − 1965.10i − 0.636620i
\(213\) − 4241.74i − 1.36450i
\(214\) 78.3783 0.0250366
\(215\) 0 0
\(216\) −1194.77 −0.376360
\(217\) − 2915.42i − 0.912034i
\(218\) − 1615.09i − 0.501779i
\(219\) 932.041 0.287587
\(220\) 0 0
\(221\) 2345.22 0.713830
\(222\) − 569.322i − 0.172119i
\(223\) − 1504.79i − 0.451876i −0.974142 0.225938i \(-0.927455\pi\)
0.974142 0.225938i \(-0.0725447\pi\)
\(224\) 939.788 0.280323
\(225\) 0 0
\(226\) −2133.16 −0.627856
\(227\) 1779.44i 0.520288i 0.965570 + 0.260144i \(0.0837700\pi\)
−0.965570 + 0.260144i \(0.916230\pi\)
\(228\) − 2697.25i − 0.783465i
\(229\) −3976.16 −1.14739 −0.573694 0.819070i \(-0.694490\pi\)
−0.573694 + 0.819070i \(0.694490\pi\)
\(230\) 0 0
\(231\) 5322.48 1.51599
\(232\) 1931.43i 0.546572i
\(233\) − 2311.12i − 0.649815i −0.945746 0.324907i \(-0.894667\pi\)
0.945746 0.324907i \(-0.105333\pi\)
\(234\) 200.599 0.0560410
\(235\) 0 0
\(236\) 1734.94 0.478537
\(237\) − 2375.35i − 0.651035i
\(238\) 6111.49i 1.66449i
\(239\) −824.267 −0.223085 −0.111543 0.993760i \(-0.535579\pi\)
−0.111543 + 0.993760i \(0.535579\pi\)
\(240\) 0 0
\(241\) 3641.83 0.973406 0.486703 0.873567i \(-0.338199\pi\)
0.486703 + 0.873567i \(0.338199\pi\)
\(242\) 251.064i 0.0666900i
\(243\) − 1235.13i − 0.326063i
\(244\) −2643.61 −0.693605
\(245\) 0 0
\(246\) 2367.20 0.613526
\(247\) 3200.62i 0.824496i
\(248\) − 794.165i − 0.203345i
\(249\) 3803.44 0.968004
\(250\) 0 0
\(251\) −7767.51 −1.95331 −0.976655 0.214813i \(-0.931086\pi\)
−0.976655 + 0.214813i \(0.931086\pi\)
\(252\) 522.749i 0.130675i
\(253\) 877.784i 0.218126i
\(254\) −1283.41 −0.317042
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1501.17i 0.364359i 0.983265 + 0.182179i \(0.0583151\pi\)
−0.983265 + 0.182179i \(0.941685\pi\)
\(258\) − 1556.27i − 0.375539i
\(259\) −1760.49 −0.422362
\(260\) 0 0
\(261\) −1074.34 −0.254789
\(262\) 4779.84i 1.12710i
\(263\) − 5430.98i − 1.27334i −0.771136 0.636671i \(-0.780311\pi\)
0.771136 0.636671i \(-0.219689\pi\)
\(264\) 1449.85 0.338001
\(265\) 0 0
\(266\) −8340.61 −1.92254
\(267\) − 3461.87i − 0.793494i
\(268\) − 1294.26i − 0.294997i
\(269\) 5014.20 1.13651 0.568255 0.822852i \(-0.307619\pi\)
0.568255 + 0.822852i \(0.307619\pi\)
\(270\) 0 0
\(271\) 7215.54 1.61739 0.808695 0.588228i \(-0.200174\pi\)
0.808695 + 0.588228i \(0.200174\pi\)
\(272\) 1664.78i 0.371111i
\(273\) 3143.41i 0.696877i
\(274\) 1798.58 0.396556
\(275\) 0 0
\(276\) 436.880 0.0952792
\(277\) 7439.60i 1.61373i 0.590738 + 0.806863i \(0.298837\pi\)
−0.590738 + 0.806863i \(0.701163\pi\)
\(278\) − 618.681i − 0.133475i
\(279\) 441.747 0.0947910
\(280\) 0 0
\(281\) −2605.59 −0.553155 −0.276578 0.960992i \(-0.589200\pi\)
−0.276578 + 0.960992i \(0.589200\pi\)
\(282\) 1952.57i 0.412319i
\(283\) − 7162.57i − 1.50449i −0.658884 0.752245i \(-0.728971\pi\)
0.658884 0.752245i \(-0.271029\pi\)
\(284\) −3572.97 −0.746538
\(285\) 0 0
\(286\) −1720.43 −0.355703
\(287\) − 7320.01i − 1.50553i
\(288\) 142.398i 0.0291349i
\(289\) −5913.13 −1.20357
\(290\) 0 0
\(291\) −5413.32 −1.09050
\(292\) − 785.093i − 0.157343i
\(293\) 9310.55i 1.85641i 0.372069 + 0.928205i \(0.378649\pi\)
−0.372069 + 0.928205i \(0.621351\pi\)
\(294\) −4933.91 −0.978747
\(295\) 0 0
\(296\) −479.562 −0.0941687
\(297\) 5699.72i 1.11357i
\(298\) − 5121.74i − 0.995620i
\(299\) −518.411 −0.100269
\(300\) 0 0
\(301\) −4812.39 −0.921533
\(302\) − 4926.31i − 0.938666i
\(303\) 7927.38i 1.50302i
\(304\) −2272.00 −0.428645
\(305\) 0 0
\(306\) −926.018 −0.172997
\(307\) − 3359.07i − 0.624470i −0.950005 0.312235i \(-0.898922\pi\)
0.950005 0.312235i \(-0.101078\pi\)
\(308\) − 4483.32i − 0.829419i
\(309\) −5274.02 −0.970966
\(310\) 0 0
\(311\) −5509.20 −1.00450 −0.502248 0.864724i \(-0.667493\pi\)
−0.502248 + 0.864724i \(0.667493\pi\)
\(312\) 856.269i 0.155374i
\(313\) 2401.42i 0.433661i 0.976209 + 0.216831i \(0.0695720\pi\)
−0.976209 + 0.216831i \(0.930428\pi\)
\(314\) −1133.44 −0.203706
\(315\) 0 0
\(316\) −2000.84 −0.356191
\(317\) 1057.36i 0.187341i 0.995603 + 0.0936704i \(0.0298600\pi\)
−0.995603 + 0.0936704i \(0.970140\pi\)
\(318\) 4665.82i 0.822787i
\(319\) 9214.02 1.61720
\(320\) 0 0
\(321\) −186.097 −0.0323581
\(322\) − 1350.95i − 0.233805i
\(323\) − 14774.9i − 2.54519i
\(324\) 2356.20 0.404012
\(325\) 0 0
\(326\) −1921.80 −0.326500
\(327\) 3834.78i 0.648514i
\(328\) − 1993.98i − 0.335669i
\(329\) 6037.86 1.01179
\(330\) 0 0
\(331\) −9680.21 −1.60747 −0.803735 0.594987i \(-0.797157\pi\)
−0.803735 + 0.594987i \(0.797157\pi\)
\(332\) − 3203.78i − 0.529609i
\(333\) − 266.752i − 0.0438976i
\(334\) −8449.40 −1.38422
\(335\) 0 0
\(336\) −2231.38 −0.362297
\(337\) − 467.694i − 0.0755991i −0.999285 0.0377996i \(-0.987965\pi\)
0.999285 0.0377996i \(-0.0120348\pi\)
\(338\) 3377.93i 0.543596i
\(339\) 5064.85 0.811460
\(340\) 0 0
\(341\) −3788.62 −0.601657
\(342\) − 1263.78i − 0.199817i
\(343\) 5183.59i 0.815998i
\(344\) −1310.90 −0.205463
\(345\) 0 0
\(346\) −6897.96 −1.07178
\(347\) − 3371.55i − 0.521597i −0.965393 0.260798i \(-0.916014\pi\)
0.965393 0.260798i \(-0.0839858\pi\)
\(348\) − 4585.88i − 0.706406i
\(349\) −10958.7 −1.68082 −0.840412 0.541947i \(-0.817687\pi\)
−0.840412 + 0.541947i \(0.817687\pi\)
\(350\) 0 0
\(351\) −3366.20 −0.511893
\(352\) − 1221.27i − 0.184925i
\(353\) − 1584.76i − 0.238947i −0.992837 0.119474i \(-0.961879\pi\)
0.992837 0.119474i \(-0.0381207\pi\)
\(354\) −4119.34 −0.618476
\(355\) 0 0
\(356\) −2916.06 −0.434132
\(357\) − 14510.8i − 2.15124i
\(358\) − 2915.63i − 0.430436i
\(359\) −5130.67 −0.754280 −0.377140 0.926156i \(-0.623092\pi\)
−0.377140 + 0.926156i \(0.623092\pi\)
\(360\) 0 0
\(361\) 13304.9 1.93978
\(362\) − 7269.94i − 1.05552i
\(363\) − 596.112i − 0.0861922i
\(364\) 2647.81 0.381271
\(365\) 0 0
\(366\) 6276.84 0.896437
\(367\) − 2811.27i − 0.399856i −0.979811 0.199928i \(-0.935929\pi\)
0.979811 0.199928i \(-0.0640708\pi\)
\(368\) − 368.000i − 0.0521286i
\(369\) 1109.14 0.156475
\(370\) 0 0
\(371\) 14427.9 2.01903
\(372\) 1885.62i 0.262809i
\(373\) − 1952.42i − 0.271026i −0.990776 0.135513i \(-0.956732\pi\)
0.990776 0.135513i \(-0.0432682\pi\)
\(374\) 7941.94 1.09804
\(375\) 0 0
\(376\) 1644.72 0.225586
\(377\) 5441.71i 0.743401i
\(378\) − 8772.10i − 1.19362i
\(379\) −9609.27 −1.30236 −0.651181 0.758923i \(-0.725726\pi\)
−0.651181 + 0.758923i \(0.725726\pi\)
\(380\) 0 0
\(381\) 3047.27 0.409754
\(382\) 897.623i 0.120226i
\(383\) 5027.44i 0.670732i 0.942088 + 0.335366i \(0.108860\pi\)
−0.942088 + 0.335366i \(0.891140\pi\)
\(384\) −607.833 −0.0807769
\(385\) 0 0
\(386\) −8518.70 −1.12329
\(387\) − 729.178i − 0.0957783i
\(388\) 4559.84i 0.596626i
\(389\) −5892.29 −0.767997 −0.383999 0.923334i \(-0.625453\pi\)
−0.383999 + 0.923334i \(0.625453\pi\)
\(390\) 0 0
\(391\) 2393.12 0.309528
\(392\) 4156.02i 0.535486i
\(393\) − 11349.0i − 1.45670i
\(394\) −3534.75 −0.451975
\(395\) 0 0
\(396\) 679.318 0.0862046
\(397\) 2454.41i 0.310285i 0.987892 + 0.155143i \(0.0495837\pi\)
−0.987892 + 0.155143i \(0.950416\pi\)
\(398\) − 2008.58i − 0.252967i
\(399\) 19803.5 2.48475
\(400\) 0 0
\(401\) 9458.39 1.17788 0.588939 0.808177i \(-0.299546\pi\)
0.588939 + 0.808177i \(0.299546\pi\)
\(402\) 3073.01i 0.381264i
\(403\) − 2237.52i − 0.276573i
\(404\) 6677.53 0.822326
\(405\) 0 0
\(406\) −14180.7 −1.73345
\(407\) 2287.78i 0.278627i
\(408\) − 3952.76i − 0.479634i
\(409\) 5857.25 0.708123 0.354062 0.935222i \(-0.384800\pi\)
0.354062 + 0.935222i \(0.384800\pi\)
\(410\) 0 0
\(411\) −4270.45 −0.512521
\(412\) 4442.51i 0.531229i
\(413\) 12738.1i 1.51767i
\(414\) 204.697 0.0243002
\(415\) 0 0
\(416\) 721.267 0.0850073
\(417\) 1468.96i 0.172507i
\(418\) 10838.7i 1.26827i
\(419\) 5252.61 0.612426 0.306213 0.951963i \(-0.400938\pi\)
0.306213 + 0.951963i \(0.400938\pi\)
\(420\) 0 0
\(421\) −203.517 −0.0235602 −0.0117801 0.999931i \(-0.503750\pi\)
−0.0117801 + 0.999931i \(0.503750\pi\)
\(422\) 10790.3i 1.24471i
\(423\) 914.863i 0.105159i
\(424\) 3930.20 0.450158
\(425\) 0 0
\(426\) 8483.47 0.964849
\(427\) − 19409.6i − 2.19976i
\(428\) 156.757i 0.0177036i
\(429\) 4084.89 0.459721
\(430\) 0 0
\(431\) 3107.72 0.347317 0.173659 0.984806i \(-0.444441\pi\)
0.173659 + 0.984806i \(0.444441\pi\)
\(432\) − 2389.54i − 0.266126i
\(433\) − 5713.46i − 0.634114i −0.948406 0.317057i \(-0.897305\pi\)
0.948406 0.317057i \(-0.102695\pi\)
\(434\) 5830.83 0.644905
\(435\) 0 0
\(436\) 3230.18 0.354811
\(437\) 3266.00i 0.357514i
\(438\) 1864.08i 0.203354i
\(439\) 4587.14 0.498706 0.249353 0.968413i \(-0.419782\pi\)
0.249353 + 0.968413i \(0.419782\pi\)
\(440\) 0 0
\(441\) −2311.75 −0.249622
\(442\) 4690.43i 0.504754i
\(443\) 5268.01i 0.564990i 0.959269 + 0.282495i \(0.0911621\pi\)
−0.959269 + 0.282495i \(0.908838\pi\)
\(444\) 1138.64 0.121707
\(445\) 0 0
\(446\) 3009.58 0.319525
\(447\) 12160.8i 1.28677i
\(448\) 1879.58i 0.198218i
\(449\) −16866.8 −1.77282 −0.886409 0.462902i \(-0.846808\pi\)
−0.886409 + 0.462902i \(0.846808\pi\)
\(450\) 0 0
\(451\) −9512.43 −0.993177
\(452\) − 4266.31i − 0.443961i
\(453\) 11696.8i 1.21316i
\(454\) −3558.87 −0.367899
\(455\) 0 0
\(456\) 5394.51 0.553993
\(457\) − 9124.25i − 0.933948i −0.884271 0.466974i \(-0.845344\pi\)
0.884271 0.466974i \(-0.154656\pi\)
\(458\) − 7952.31i − 0.811326i
\(459\) 15539.3 1.58020
\(460\) 0 0
\(461\) 6011.92 0.607382 0.303691 0.952771i \(-0.401781\pi\)
0.303691 + 0.952771i \(0.401781\pi\)
\(462\) 10645.0i 1.07197i
\(463\) − 8584.09i − 0.861634i −0.902439 0.430817i \(-0.858225\pi\)
0.902439 0.430817i \(-0.141775\pi\)
\(464\) −3862.86 −0.386484
\(465\) 0 0
\(466\) 4622.25 0.459488
\(467\) − 3954.09i − 0.391806i −0.980623 0.195903i \(-0.937236\pi\)
0.980623 0.195903i \(-0.0627639\pi\)
\(468\) 401.198i 0.0396269i
\(469\) 9502.56 0.935581
\(470\) 0 0
\(471\) 2691.18 0.263276
\(472\) 3469.87i 0.338377i
\(473\) 6253.75i 0.607923i
\(474\) 4750.69 0.460351
\(475\) 0 0
\(476\) −12223.0 −1.17697
\(477\) 2186.14i 0.209845i
\(478\) − 1648.53i − 0.157745i
\(479\) 95.3377 0.00909413 0.00454707 0.999990i \(-0.498553\pi\)
0.00454707 + 0.999990i \(0.498553\pi\)
\(480\) 0 0
\(481\) −1351.14 −0.128080
\(482\) 7283.66i 0.688302i
\(483\) 3207.61i 0.302177i
\(484\) −502.127 −0.0471570
\(485\) 0 0
\(486\) 2470.25 0.230561
\(487\) − 13915.2i − 1.29478i −0.762158 0.647391i \(-0.775860\pi\)
0.762158 0.647391i \(-0.224140\pi\)
\(488\) − 5287.22i − 0.490453i
\(489\) 4563.03 0.421978
\(490\) 0 0
\(491\) 6291.33 0.578256 0.289128 0.957290i \(-0.406635\pi\)
0.289128 + 0.957290i \(0.406635\pi\)
\(492\) 4734.41i 0.433828i
\(493\) − 25120.3i − 2.29486i
\(494\) −6401.24 −0.583007
\(495\) 0 0
\(496\) 1588.33 0.143786
\(497\) − 26233.1i − 2.36764i
\(498\) 7606.87i 0.684482i
\(499\) 638.332 0.0572658 0.0286329 0.999590i \(-0.490885\pi\)
0.0286329 + 0.999590i \(0.490885\pi\)
\(500\) 0 0
\(501\) 20061.8 1.78901
\(502\) − 15535.0i − 1.38120i
\(503\) 15063.4i 1.33527i 0.744487 + 0.667637i \(0.232694\pi\)
−0.744487 + 0.667637i \(0.767306\pi\)
\(504\) −1045.50 −0.0924011
\(505\) 0 0
\(506\) −1755.57 −0.154238
\(507\) − 8020.38i − 0.702559i
\(508\) − 2566.83i − 0.224182i
\(509\) 13623.1 1.18631 0.593157 0.805087i \(-0.297881\pi\)
0.593157 + 0.805087i \(0.297881\pi\)
\(510\) 0 0
\(511\) 5764.23 0.499010
\(512\) 512.000i 0.0441942i
\(513\) 21207.1i 1.82518i
\(514\) −3002.33 −0.257640
\(515\) 0 0
\(516\) 3112.54 0.265546
\(517\) − 7846.27i − 0.667463i
\(518\) − 3520.99i − 0.298655i
\(519\) 16378.2 1.38520
\(520\) 0 0
\(521\) −21659.4 −1.82133 −0.910666 0.413143i \(-0.864431\pi\)
−0.910666 + 0.413143i \(0.864431\pi\)
\(522\) − 2148.68i − 0.180163i
\(523\) − 3813.85i − 0.318868i −0.987209 0.159434i \(-0.949033\pi\)
0.987209 0.159434i \(-0.0509669\pi\)
\(524\) −9559.69 −0.796979
\(525\) 0 0
\(526\) 10862.0 0.900388
\(527\) 10329.0i 0.853771i
\(528\) 2899.71i 0.239003i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) −1930.09 −0.157737
\(532\) − 16681.2i − 1.35944i
\(533\) − 5617.95i − 0.456549i
\(534\) 6923.74 0.561085
\(535\) 0 0
\(536\) 2588.51 0.208595
\(537\) 6922.72i 0.556308i
\(538\) 10028.4i 0.803634i
\(539\) 19826.6 1.58440
\(540\) 0 0
\(541\) 9727.32 0.773031 0.386516 0.922283i \(-0.373679\pi\)
0.386516 + 0.922283i \(0.373679\pi\)
\(542\) 14431.1i 1.14367i
\(543\) 17261.3i 1.36419i
\(544\) −3329.56 −0.262415
\(545\) 0 0
\(546\) −6286.81 −0.492767
\(547\) − 782.647i − 0.0611766i −0.999532 0.0305883i \(-0.990262\pi\)
0.999532 0.0305883i \(-0.00973807\pi\)
\(548\) 3597.16i 0.280407i
\(549\) 2940.97 0.228629
\(550\) 0 0
\(551\) 34282.8 2.65063
\(552\) 873.759i 0.0673726i
\(553\) − 14690.4i − 1.12965i
\(554\) −14879.2 −1.14108
\(555\) 0 0
\(556\) 1237.36 0.0943811
\(557\) − 3472.87i − 0.264184i −0.991237 0.132092i \(-0.957831\pi\)
0.991237 0.132092i \(-0.0421694\pi\)
\(558\) 883.494i 0.0670274i
\(559\) −3693.40 −0.279453
\(560\) 0 0
\(561\) −18856.9 −1.41914
\(562\) − 5211.18i − 0.391140i
\(563\) 7544.75i 0.564784i 0.959299 + 0.282392i \(0.0911279\pi\)
−0.959299 + 0.282392i \(0.908872\pi\)
\(564\) −3905.14 −0.291554
\(565\) 0 0
\(566\) 14325.1 1.06383
\(567\) 17299.5i 1.28132i
\(568\) − 7145.94i − 0.527882i
\(569\) −13222.5 −0.974193 −0.487096 0.873348i \(-0.661944\pi\)
−0.487096 + 0.873348i \(0.661944\pi\)
\(570\) 0 0
\(571\) 4069.49 0.298254 0.149127 0.988818i \(-0.452354\pi\)
0.149127 + 0.988818i \(0.452354\pi\)
\(572\) − 3440.85i − 0.251520i
\(573\) − 2131.27i − 0.155384i
\(574\) 14640.0 1.06457
\(575\) 0 0
\(576\) −284.795 −0.0206015
\(577\) − 846.627i − 0.0610841i −0.999533 0.0305421i \(-0.990277\pi\)
0.999533 0.0305421i \(-0.00972336\pi\)
\(578\) − 11826.3i − 0.851051i
\(579\) 20226.3 1.45177
\(580\) 0 0
\(581\) 23522.4 1.67965
\(582\) − 10826.6i − 0.771097i
\(583\) − 18749.3i − 1.33193i
\(584\) 1570.19 0.111258
\(585\) 0 0
\(586\) −18621.1 −1.31268
\(587\) − 4702.77i − 0.330672i −0.986237 0.165336i \(-0.947129\pi\)
0.986237 0.165336i \(-0.0528708\pi\)
\(588\) − 9867.82i − 0.692079i
\(589\) −14096.4 −0.986133
\(590\) 0 0
\(591\) 8392.71 0.584146
\(592\) − 959.123i − 0.0665874i
\(593\) − 14016.5i − 0.970640i −0.874337 0.485320i \(-0.838703\pi\)
0.874337 0.485320i \(-0.161297\pi\)
\(594\) −11399.4 −0.787415
\(595\) 0 0
\(596\) 10243.5 0.704010
\(597\) 4769.07i 0.326943i
\(598\) − 1036.82i − 0.0709010i
\(599\) −12885.0 −0.878910 −0.439455 0.898265i \(-0.644828\pi\)
−0.439455 + 0.898265i \(0.644828\pi\)
\(600\) 0 0
\(601\) −4753.41 −0.322622 −0.161311 0.986904i \(-0.551572\pi\)
−0.161311 + 0.986904i \(0.551572\pi\)
\(602\) − 9624.77i − 0.651622i
\(603\) 1439.84i 0.0972383i
\(604\) 9852.61 0.663737
\(605\) 0 0
\(606\) −15854.8 −1.06280
\(607\) 6800.06i 0.454705i 0.973813 + 0.227352i \(0.0730070\pi\)
−0.973813 + 0.227352i \(0.926993\pi\)
\(608\) − 4543.99i − 0.303097i
\(609\) 33670.0 2.24036
\(610\) 0 0
\(611\) 4633.93 0.306823
\(612\) − 1852.04i − 0.122327i
\(613\) 20842.3i 1.37327i 0.727003 + 0.686634i \(0.240913\pi\)
−0.727003 + 0.686634i \(0.759087\pi\)
\(614\) 6718.14 0.441567
\(615\) 0 0
\(616\) 8966.65 0.586488
\(617\) − 18917.7i − 1.23436i −0.786823 0.617179i \(-0.788276\pi\)
0.786823 0.617179i \(-0.211724\pi\)
\(618\) − 10548.0i − 0.686577i
\(619\) 28044.8 1.82103 0.910513 0.413480i \(-0.135687\pi\)
0.910513 + 0.413480i \(0.135687\pi\)
\(620\) 0 0
\(621\) −3434.96 −0.221965
\(622\) − 11018.4i − 0.710285i
\(623\) − 21410.0i − 1.37684i
\(624\) −1712.54 −0.109866
\(625\) 0 0
\(626\) −4802.83 −0.306645
\(627\) − 25734.9i − 1.63916i
\(628\) − 2266.88i − 0.144042i
\(629\) 6237.22 0.395380
\(630\) 0 0
\(631\) −25796.2 −1.62746 −0.813732 0.581241i \(-0.802567\pi\)
−0.813732 + 0.581241i \(0.802567\pi\)
\(632\) − 4001.69i − 0.251865i
\(633\) − 25620.0i − 1.60870i
\(634\) −2114.71 −0.132470
\(635\) 0 0
\(636\) −9331.64 −0.581798
\(637\) 11709.4i 0.728324i
\(638\) 18428.0i 1.14353i
\(639\) 3974.87 0.246077
\(640\) 0 0
\(641\) −26482.3 −1.63181 −0.815903 0.578188i \(-0.803760\pi\)
−0.815903 + 0.578188i \(0.803760\pi\)
\(642\) − 372.195i − 0.0228806i
\(643\) 30458.1i 1.86804i 0.357219 + 0.934020i \(0.383725\pi\)
−0.357219 + 0.934020i \(0.616275\pi\)
\(644\) 2701.89 0.165325
\(645\) 0 0
\(646\) 29549.8 1.79972
\(647\) 6746.24i 0.409926i 0.978770 + 0.204963i \(0.0657074\pi\)
−0.978770 + 0.204963i \(0.934293\pi\)
\(648\) 4712.40i 0.285680i
\(649\) 16553.3 1.00119
\(650\) 0 0
\(651\) −13844.4 −0.833495
\(652\) − 3843.61i − 0.230870i
\(653\) 12976.2i 0.777640i 0.921314 + 0.388820i \(0.127117\pi\)
−0.921314 + 0.388820i \(0.872883\pi\)
\(654\) −7669.57 −0.458569
\(655\) 0 0
\(656\) 3987.97 0.237354
\(657\) 873.401i 0.0518640i
\(658\) 12075.7i 0.715442i
\(659\) 6538.56 0.386504 0.193252 0.981149i \(-0.438097\pi\)
0.193252 + 0.981149i \(0.438097\pi\)
\(660\) 0 0
\(661\) −25177.1 −1.48151 −0.740753 0.671777i \(-0.765531\pi\)
−0.740753 + 0.671777i \(0.765531\pi\)
\(662\) − 19360.4i − 1.13665i
\(663\) − 11136.7i − 0.652359i
\(664\) 6407.55 0.374490
\(665\) 0 0
\(666\) 533.504 0.0310403
\(667\) 5552.86i 0.322350i
\(668\) − 16898.8i − 0.978794i
\(669\) −7145.79 −0.412963
\(670\) 0 0
\(671\) −25223.0 −1.45115
\(672\) − 4462.77i − 0.256183i
\(673\) 18339.3i 1.05041i 0.850975 + 0.525207i \(0.176012\pi\)
−0.850975 + 0.525207i \(0.823988\pi\)
\(674\) 935.388 0.0534567
\(675\) 0 0
\(676\) −6755.86 −0.384380
\(677\) − 31876.9i − 1.80965i −0.425789 0.904823i \(-0.640003\pi\)
0.425789 0.904823i \(-0.359997\pi\)
\(678\) 10129.7i 0.573789i
\(679\) −33478.8 −1.89219
\(680\) 0 0
\(681\) 8450.00 0.475484
\(682\) − 7577.23i − 0.425436i
\(683\) − 28843.7i − 1.61592i −0.589239 0.807959i \(-0.700572\pi\)
0.589239 0.807959i \(-0.299428\pi\)
\(684\) 2527.56 0.141292
\(685\) 0 0
\(686\) −10367.2 −0.576998
\(687\) 18881.5i 1.04858i
\(688\) − 2621.80i − 0.145284i
\(689\) 11073.1 0.612268
\(690\) 0 0
\(691\) −18327.7 −1.00900 −0.504499 0.863413i \(-0.668323\pi\)
−0.504499 + 0.863413i \(0.668323\pi\)
\(692\) − 13795.9i − 0.757865i
\(693\) 4987.62i 0.273397i
\(694\) 6743.09 0.368825
\(695\) 0 0
\(696\) 9171.77 0.499504
\(697\) 25933.9i 1.40935i
\(698\) − 21917.5i − 1.18852i
\(699\) −10974.8 −0.593857
\(700\) 0 0
\(701\) −8329.48 −0.448788 −0.224394 0.974499i \(-0.572040\pi\)
−0.224394 + 0.974499i \(0.572040\pi\)
\(702\) − 6732.40i − 0.361963i
\(703\) 8512.20i 0.456677i
\(704\) 2442.53 0.130762
\(705\) 0 0
\(706\) 3169.52 0.168961
\(707\) 49027.1i 2.60800i
\(708\) − 8238.68i − 0.437328i
\(709\) 6169.85 0.326818 0.163409 0.986558i \(-0.447751\pi\)
0.163409 + 0.986558i \(0.447751\pi\)
\(710\) 0 0
\(711\) 2225.90 0.117409
\(712\) − 5832.12i − 0.306978i
\(713\) − 2283.22i − 0.119926i
\(714\) 29021.6 1.52115
\(715\) 0 0
\(716\) 5831.27 0.304364
\(717\) 3914.19i 0.203875i
\(718\) − 10261.3i − 0.533356i
\(719\) −28740.8 −1.49075 −0.745377 0.666644i \(-0.767730\pi\)
−0.745377 + 0.666644i \(0.767730\pi\)
\(720\) 0 0
\(721\) −32617.3 −1.68479
\(722\) 26609.9i 1.37163i
\(723\) − 17293.9i − 0.889583i
\(724\) 14539.9 0.746368
\(725\) 0 0
\(726\) 1192.22 0.0609471
\(727\) − 11174.0i − 0.570041i −0.958521 0.285021i \(-0.908000\pi\)
0.958521 0.285021i \(-0.0920004\pi\)
\(728\) 5295.62i 0.269600i
\(729\) −21769.6 −1.10601
\(730\) 0 0
\(731\) 17049.7 0.862663
\(732\) 12553.7i 0.633876i
\(733\) − 23560.7i − 1.18722i −0.804751 0.593612i \(-0.797701\pi\)
0.804751 0.593612i \(-0.202299\pi\)
\(734\) 5622.54 0.282741
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) − 12348.7i − 0.617191i
\(738\) 2218.27i 0.110645i
\(739\) −22713.6 −1.13063 −0.565313 0.824877i \(-0.691245\pi\)
−0.565313 + 0.824877i \(0.691245\pi\)
\(740\) 0 0
\(741\) 15198.8 0.753495
\(742\) 28855.9i 1.42767i
\(743\) − 15882.7i − 0.784224i −0.919917 0.392112i \(-0.871745\pi\)
0.919917 0.392112i \(-0.128255\pi\)
\(744\) −3771.24 −0.185834
\(745\) 0 0
\(746\) 3904.84 0.191644
\(747\) 3564.14i 0.174572i
\(748\) 15883.9i 0.776433i
\(749\) −1150.92 −0.0561466
\(750\) 0 0
\(751\) −30250.6 −1.46985 −0.734927 0.678146i \(-0.762784\pi\)
−0.734927 + 0.678146i \(0.762784\pi\)
\(752\) 3289.45i 0.159513i
\(753\) 36885.5i 1.78510i
\(754\) −10883.4 −0.525664
\(755\) 0 0
\(756\) 17544.2 0.844017
\(757\) 9611.05i 0.461453i 0.973019 + 0.230726i \(0.0741102\pi\)
−0.973019 + 0.230726i \(0.925890\pi\)
\(758\) − 19218.5i − 0.920908i
\(759\) 4168.33 0.199342
\(760\) 0 0
\(761\) 5892.64 0.280694 0.140347 0.990102i \(-0.455178\pi\)
0.140347 + 0.990102i \(0.455178\pi\)
\(762\) 6094.54i 0.289740i
\(763\) 23716.3i 1.12528i
\(764\) −1795.25 −0.0850127
\(765\) 0 0
\(766\) −10054.9 −0.474279
\(767\) 9776.19i 0.460232i
\(768\) − 1215.67i − 0.0571179i
\(769\) −1683.72 −0.0789553 −0.0394777 0.999220i \(-0.512569\pi\)
−0.0394777 + 0.999220i \(0.512569\pi\)
\(770\) 0 0
\(771\) 7128.57 0.332982
\(772\) − 17037.4i − 0.794286i
\(773\) 19880.8i 0.925048i 0.886607 + 0.462524i \(0.153056\pi\)
−0.886607 + 0.462524i \(0.846944\pi\)
\(774\) 1458.36 0.0677255
\(775\) 0 0
\(776\) −9119.68 −0.421878
\(777\) 8360.04i 0.385991i
\(778\) − 11784.6i − 0.543056i
\(779\) −35393.2 −1.62785
\(780\) 0 0
\(781\) −34090.2 −1.56190
\(782\) 4786.24i 0.218869i
\(783\) 36056.4i 1.64566i
\(784\) −8312.04 −0.378646
\(785\) 0 0
\(786\) 22698.0 1.03004
\(787\) 7622.64i 0.345258i 0.984987 + 0.172629i \(0.0552261\pi\)
−0.984987 + 0.172629i \(0.944774\pi\)
\(788\) − 7069.50i − 0.319594i
\(789\) −25790.1 −1.16369
\(790\) 0 0
\(791\) 31323.7 1.40802
\(792\) 1358.64i 0.0609558i
\(793\) − 14896.5i − 0.667074i
\(794\) −4908.82 −0.219405
\(795\) 0 0
\(796\) 4017.16 0.178875
\(797\) 22380.9i 0.994693i 0.867552 + 0.497347i \(0.165692\pi\)
−0.867552 + 0.497347i \(0.834308\pi\)
\(798\) 39607.0i 1.75698i
\(799\) −21391.4 −0.947152
\(800\) 0 0
\(801\) 3244.07 0.143100
\(802\) 18916.8i 0.832886i
\(803\) − 7490.67i − 0.329191i
\(804\) −6146.03 −0.269594
\(805\) 0 0
\(806\) 4475.04 0.195566
\(807\) − 23810.9i − 1.03864i
\(808\) 13355.1i 0.581472i
\(809\) −8117.18 −0.352762 −0.176381 0.984322i \(-0.556439\pi\)
−0.176381 + 0.984322i \(0.556439\pi\)
\(810\) 0 0
\(811\) 7221.05 0.312658 0.156329 0.987705i \(-0.450034\pi\)
0.156329 + 0.987705i \(0.450034\pi\)
\(812\) − 28361.5i − 1.22573i
\(813\) − 34264.4i − 1.47811i
\(814\) −4575.56 −0.197019
\(815\) 0 0
\(816\) 7905.52 0.339153
\(817\) 23268.5i 0.996403i
\(818\) 11714.5i 0.500719i
\(819\) −2945.64 −0.125676
\(820\) 0 0
\(821\) 34803.3 1.47947 0.739735 0.672899i \(-0.234951\pi\)
0.739735 + 0.672899i \(0.234951\pi\)
\(822\) − 8540.91i − 0.362407i
\(823\) 490.883i 0.0207911i 0.999946 + 0.0103956i \(0.00330907\pi\)
−0.999946 + 0.0103956i \(0.996691\pi\)
\(824\) −8885.01 −0.375636
\(825\) 0 0
\(826\) −25476.1 −1.07316
\(827\) − 30952.0i − 1.30146i −0.759310 0.650729i \(-0.774463\pi\)
0.759310 0.650729i \(-0.225537\pi\)
\(828\) 409.393i 0.0171829i
\(829\) −8587.63 −0.359784 −0.179892 0.983686i \(-0.557575\pi\)
−0.179892 + 0.983686i \(0.557575\pi\)
\(830\) 0 0
\(831\) 35328.4 1.47476
\(832\) 1442.53i 0.0601092i
\(833\) − 54053.5i − 2.24831i
\(834\) −2937.93 −0.121981
\(835\) 0 0
\(836\) −21677.4 −0.896806
\(837\) − 14825.7i − 0.612246i
\(838\) 10505.2i 0.433051i
\(839\) −3077.94 −0.126653 −0.0633267 0.997993i \(-0.520171\pi\)
−0.0633267 + 0.997993i \(0.520171\pi\)
\(840\) 0 0
\(841\) 33898.9 1.38992
\(842\) − 407.035i − 0.0166596i
\(843\) 12373.2i 0.505521i
\(844\) −21580.7 −0.880140
\(845\) 0 0
\(846\) −1829.73 −0.0743585
\(847\) − 3686.67i − 0.149558i
\(848\) 7860.39i 0.318310i
\(849\) −34012.8 −1.37493
\(850\) 0 0
\(851\) −1378.74 −0.0555377
\(852\) 16966.9i 0.682251i
\(853\) − 9193.14i − 0.369012i −0.982831 0.184506i \(-0.940932\pi\)
0.982831 0.184506i \(-0.0590685\pi\)
\(854\) 38819.3 1.55547
\(855\) 0 0
\(856\) −313.513 −0.0125183
\(857\) − 21445.8i − 0.854813i −0.904059 0.427407i \(-0.859427\pi\)
0.904059 0.427407i \(-0.140573\pi\)
\(858\) 8169.78i 0.325072i
\(859\) 40276.7 1.59979 0.799897 0.600137i \(-0.204887\pi\)
0.799897 + 0.600137i \(0.204887\pi\)
\(860\) 0 0
\(861\) −34760.5 −1.37588
\(862\) 6215.44i 0.245590i
\(863\) 43713.4i 1.72424i 0.506703 + 0.862120i \(0.330864\pi\)
−0.506703 + 0.862120i \(0.669136\pi\)
\(864\) 4779.07 0.188180
\(865\) 0 0
\(866\) 11426.9 0.448386
\(867\) 28079.6i 1.09992i
\(868\) 11661.7i 0.456017i
\(869\) −19090.3 −0.745218
\(870\) 0 0
\(871\) 7293.01 0.283713
\(872\) 6460.36i 0.250889i
\(873\) − 5072.74i − 0.196662i
\(874\) −6531.99 −0.252801
\(875\) 0 0
\(876\) −3728.16 −0.143793
\(877\) − 24740.5i − 0.952596i −0.879284 0.476298i \(-0.841978\pi\)
0.879284 0.476298i \(-0.158022\pi\)
\(878\) 9174.27i 0.352639i
\(879\) 44212.9 1.69655
\(880\) 0 0
\(881\) −44027.2 −1.68367 −0.841835 0.539735i \(-0.818524\pi\)
−0.841835 + 0.539735i \(0.818524\pi\)
\(882\) − 4623.50i − 0.176509i
\(883\) 30245.0i 1.15269i 0.817207 + 0.576344i \(0.195521\pi\)
−0.817207 + 0.576344i \(0.804479\pi\)
\(884\) −9380.87 −0.356915
\(885\) 0 0
\(886\) −10536.0 −0.399509
\(887\) 7317.82i 0.277010i 0.990362 + 0.138505i \(0.0442298\pi\)
−0.990362 + 0.138505i \(0.955770\pi\)
\(888\) 2277.29i 0.0860595i
\(889\) 18845.9 0.710991
\(890\) 0 0
\(891\) 22480.8 0.845270
\(892\) 6019.17i 0.225938i
\(893\) − 29193.8i − 1.09399i
\(894\) −24321.6 −0.909883
\(895\) 0 0
\(896\) −3759.15 −0.140161
\(897\) 2461.77i 0.0916346i
\(898\) − 33733.7i − 1.25357i
\(899\) −23966.8 −0.889140
\(900\) 0 0
\(901\) −51116.5 −1.89005
\(902\) − 19024.9i − 0.702282i
\(903\) 22852.5i 0.842176i
\(904\) 8532.63 0.313928
\(905\) 0 0
\(906\) −23393.5 −0.857834
\(907\) 35436.0i 1.29728i 0.761096 + 0.648640i \(0.224662\pi\)
−0.761096 + 0.648640i \(0.775338\pi\)
\(908\) − 7117.75i − 0.260144i
\(909\) −7428.63 −0.271058
\(910\) 0 0
\(911\) −11359.1 −0.413112 −0.206556 0.978435i \(-0.566225\pi\)
−0.206556 + 0.978435i \(0.566225\pi\)
\(912\) 10789.0i 0.391732i
\(913\) − 30567.7i − 1.10804i
\(914\) 18248.5 0.660401
\(915\) 0 0
\(916\) 15904.6 0.573694
\(917\) − 70188.2i − 2.52761i
\(918\) 31078.5i 1.11737i
\(919\) −40517.5 −1.45435 −0.727175 0.686452i \(-0.759167\pi\)
−0.727175 + 0.686452i \(0.759167\pi\)
\(920\) 0 0
\(921\) −15951.2 −0.570695
\(922\) 12023.8i 0.429484i
\(923\) − 20133.3i − 0.717982i
\(924\) −21289.9 −0.757995
\(925\) 0 0
\(926\) 17168.2 0.609267
\(927\) − 4942.21i − 0.175106i
\(928\) − 7725.72i − 0.273286i
\(929\) −16242.2 −0.573614 −0.286807 0.957988i \(-0.592594\pi\)
−0.286807 + 0.957988i \(0.592594\pi\)
\(930\) 0 0
\(931\) 73769.2 2.59687
\(932\) 9244.50i 0.324907i
\(933\) 26161.5i 0.917994i
\(934\) 7908.19 0.277049
\(935\) 0 0
\(936\) −802.397 −0.0280205
\(937\) 47445.0i 1.65417i 0.562074 + 0.827087i \(0.310004\pi\)
−0.562074 + 0.827087i \(0.689996\pi\)
\(938\) 19005.1i 0.661556i
\(939\) 11403.6 0.396317
\(940\) 0 0
\(941\) −48063.5 −1.66506 −0.832532 0.553976i \(-0.813110\pi\)
−0.832532 + 0.553976i \(0.813110\pi\)
\(942\) 5382.36i 0.186164i
\(943\) − 5732.70i − 0.197967i
\(944\) −6939.74 −0.239268
\(945\) 0 0
\(946\) −12507.5 −0.429867
\(947\) − 13745.4i − 0.471663i −0.971794 0.235831i \(-0.924219\pi\)
0.971794 0.235831i \(-0.0757813\pi\)
\(948\) 9501.39i 0.325518i
\(949\) 4423.92 0.151324
\(950\) 0 0
\(951\) 5021.06 0.171208
\(952\) − 24445.9i − 0.832245i
\(953\) − 13999.1i − 0.475839i −0.971285 0.237920i \(-0.923535\pi\)
0.971285 0.237920i \(-0.0764655\pi\)
\(954\) −4372.27 −0.148383
\(955\) 0 0
\(956\) 3297.07 0.111543
\(957\) − 43754.5i − 1.47793i
\(958\) 190.675i 0.00643052i
\(959\) −26410.7 −0.889308
\(960\) 0 0
\(961\) −19936.4 −0.669207
\(962\) − 2702.28i − 0.0905666i
\(963\) − 174.389i − 0.00583552i
\(964\) −14567.3 −0.486703
\(965\) 0 0
\(966\) −6415.23 −0.213671
\(967\) 800.187i 0.0266104i 0.999911 + 0.0133052i \(0.00423530\pi\)
−0.999911 + 0.0133052i \(0.995765\pi\)
\(968\) − 1004.25i − 0.0333450i
\(969\) −70161.4 −2.32602
\(970\) 0 0
\(971\) −11363.5 −0.375563 −0.187782 0.982211i \(-0.560130\pi\)
−0.187782 + 0.982211i \(0.560130\pi\)
\(972\) 4940.50i 0.163032i
\(973\) 9084.84i 0.299328i
\(974\) 27830.4 0.915549
\(975\) 0 0
\(976\) 10574.4 0.346803
\(977\) 17444.0i 0.571220i 0.958346 + 0.285610i \(0.0921962\pi\)
−0.958346 + 0.285610i \(0.907804\pi\)
\(978\) 9126.06i 0.298384i
\(979\) −27822.5 −0.908286
\(980\) 0 0
\(981\) −3593.52 −0.116954
\(982\) 12582.7i 0.408889i
\(983\) − 54032.7i − 1.75318i −0.481237 0.876590i \(-0.659812\pi\)
0.481237 0.876590i \(-0.340188\pi\)
\(984\) −9468.81 −0.306763
\(985\) 0 0
\(986\) 50240.7 1.62271
\(987\) − 28671.9i − 0.924659i
\(988\) − 12802.5i − 0.412248i
\(989\) −3768.84 −0.121175
\(990\) 0 0
\(991\) 28869.3 0.925392 0.462696 0.886517i \(-0.346882\pi\)
0.462696 + 0.886517i \(0.346882\pi\)
\(992\) 3176.66i 0.101672i
\(993\) 45968.3i 1.46904i
\(994\) 52466.2 1.67417
\(995\) 0 0
\(996\) −15213.7 −0.484002
\(997\) − 3358.78i − 0.106694i −0.998576 0.0533469i \(-0.983011\pi\)
0.998576 0.0533469i \(-0.0169889\pi\)
\(998\) 1276.66i 0.0404931i
\(999\) −8952.57 −0.283530
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.599.6 8
5.2 odd 4 230.4.a.h.1.2 4
5.3 odd 4 1150.4.a.p.1.3 4
5.4 even 2 inner 1150.4.b.n.599.3 8
15.2 even 4 2070.4.a.bj.1.4 4
20.7 even 4 1840.4.a.m.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.2 4 5.2 odd 4
1150.4.a.p.1.3 4 5.3 odd 4
1150.4.b.n.599.3 8 5.4 even 2 inner
1150.4.b.n.599.6 8 1.1 even 1 trivial
1840.4.a.m.1.3 4 20.7 even 4
2070.4.a.bj.1.4 4 15.2 even 4