Properties

Label 230.4.a.h.1.1
Level $230$
Weight $4$
Character 230.1
Self dual yes
Analytic conductor $13.570$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 68 x^{2} - 111 x + 342\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.57209\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -7.57209 q^{3} +4.00000 q^{4} -5.00000 q^{5} +15.1442 q^{6} -35.4229 q^{7} -8.00000 q^{8} +30.3365 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -7.57209 q^{3} +4.00000 q^{4} -5.00000 q^{5} +15.1442 q^{6} -35.4229 q^{7} -8.00000 q^{8} +30.3365 q^{9} +10.0000 q^{10} -16.6298 q^{11} -30.2883 q^{12} -79.9132 q^{13} +70.8458 q^{14} +37.8604 q^{15} +16.0000 q^{16} -46.8219 q^{17} -60.6730 q^{18} -110.653 q^{19} -20.0000 q^{20} +268.225 q^{21} +33.2597 q^{22} -23.0000 q^{23} +60.5767 q^{24} +25.0000 q^{25} +159.826 q^{26} -25.2641 q^{27} -141.692 q^{28} -0.836422 q^{29} -75.7209 q^{30} -119.836 q^{31} -32.0000 q^{32} +125.923 q^{33} +93.6438 q^{34} +177.114 q^{35} +121.346 q^{36} +368.201 q^{37} +221.307 q^{38} +605.109 q^{39} +40.0000 q^{40} -95.7927 q^{41} -536.450 q^{42} +331.961 q^{43} -66.5193 q^{44} -151.682 q^{45} +46.0000 q^{46} -535.037 q^{47} -121.153 q^{48} +911.782 q^{49} -50.0000 q^{50} +354.539 q^{51} -319.653 q^{52} +409.345 q^{53} +50.5282 q^{54} +83.1492 q^{55} +283.383 q^{56} +837.876 q^{57} +1.67284 q^{58} -352.950 q^{59} +151.442 q^{60} -507.223 q^{61} +239.672 q^{62} -1074.61 q^{63} +64.0000 q^{64} +399.566 q^{65} -251.845 q^{66} -820.056 q^{67} -187.288 q^{68} +174.158 q^{69} -354.229 q^{70} -733.770 q^{71} -242.692 q^{72} -91.4599 q^{73} -736.402 q^{74} -189.302 q^{75} -442.613 q^{76} +589.077 q^{77} -1210.22 q^{78} +329.381 q^{79} -80.0000 q^{80} -627.783 q^{81} +191.585 q^{82} -753.834 q^{83} +1072.90 q^{84} +234.110 q^{85} -663.922 q^{86} +6.33346 q^{87} +133.039 q^{88} -1050.14 q^{89} +303.365 q^{90} +2830.76 q^{91} -92.0000 q^{92} +907.407 q^{93} +1070.07 q^{94} +553.267 q^{95} +242.307 q^{96} -271.928 q^{97} -1823.56 q^{98} -504.491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} - 32q^{8} + 32q^{9} + O(q^{10}) \) \( 4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} - 32q^{8} + 32q^{9} + 40q^{10} - 39q^{11} - 16q^{12} - 20q^{13} + 2q^{14} + 20q^{15} + 64q^{16} - 23q^{17} - 64q^{18} + 53q^{19} - 80q^{20} + 300q^{21} + 78q^{22} - 92q^{23} + 32q^{24} + 100q^{25} + 40q^{26} + 137q^{27} - 4q^{28} + 161q^{29} - 40q^{30} + 388q^{31} - 128q^{32} + 87q^{33} + 46q^{34} + 5q^{35} + 128q^{36} + 466q^{37} - 106q^{38} + 1047q^{39} + 160q^{40} + 484q^{41} - 600q^{42} + 894q^{43} - 156q^{44} - 160q^{45} + 184q^{46} - 265q^{47} - 64q^{48} + 1643q^{49} - 200q^{50} + 1825q^{51} - 80q^{52} + 576q^{53} - 274q^{54} + 195q^{55} + 8q^{56} + 178q^{57} - 322q^{58} - 94q^{59} + 80q^{60} + 1153q^{61} - 776q^{62} + 60q^{63} + 256q^{64} + 100q^{65} - 174q^{66} - 1472q^{67} - 92q^{68} + 92q^{69} - 10q^{70} + 200q^{71} - 256q^{72} + 1147q^{73} - 932q^{74} - 100q^{75} + 212q^{76} - 2176q^{77} - 2094q^{78} - 908q^{79} - 320q^{80} - 1056q^{81} - 968q^{82} - 1048q^{83} + 1200q^{84} + 115q^{85} - 1788q^{86} - 2167q^{87} + 312q^{88} - 1784q^{89} + 320q^{90} + 2329q^{91} - 368q^{92} + 1483q^{93} + 530q^{94} - 265q^{95} + 128q^{96} - 2047q^{97} - 3286q^{98} - 2665q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −7.57209 −1.45725 −0.728624 0.684914i \(-0.759840\pi\)
−0.728624 + 0.684914i \(0.759840\pi\)
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 15.1442 1.03043
\(7\) −35.4229 −1.91266 −0.956328 0.292294i \(-0.905581\pi\)
−0.956328 + 0.292294i \(0.905581\pi\)
\(8\) −8.00000 −0.353553
\(9\) 30.3365 1.12357
\(10\) 10.0000 0.316228
\(11\) −16.6298 −0.455826 −0.227913 0.973682i \(-0.573190\pi\)
−0.227913 + 0.973682i \(0.573190\pi\)
\(12\) −30.2883 −0.728624
\(13\) −79.9132 −1.70492 −0.852459 0.522795i \(-0.824889\pi\)
−0.852459 + 0.522795i \(0.824889\pi\)
\(14\) 70.8458 1.35245
\(15\) 37.8604 0.651701
\(16\) 16.0000 0.250000
\(17\) −46.8219 −0.667999 −0.333999 0.942573i \(-0.608398\pi\)
−0.333999 + 0.942573i \(0.608398\pi\)
\(18\) −60.6730 −0.794486
\(19\) −110.653 −1.33608 −0.668042 0.744123i \(-0.732868\pi\)
−0.668042 + 0.744123i \(0.732868\pi\)
\(20\) −20.0000 −0.223607
\(21\) 268.225 2.78722
\(22\) 33.2597 0.322318
\(23\) −23.0000 −0.208514
\(24\) 60.5767 0.515215
\(25\) 25.0000 0.200000
\(26\) 159.826 1.20556
\(27\) −25.2641 −0.180077
\(28\) −141.692 −0.956328
\(29\) −0.836422 −0.00535585 −0.00267793 0.999996i \(-0.500852\pi\)
−0.00267793 + 0.999996i \(0.500852\pi\)
\(30\) −75.7209 −0.460822
\(31\) −119.836 −0.694295 −0.347147 0.937811i \(-0.612850\pi\)
−0.347147 + 0.937811i \(0.612850\pi\)
\(32\) −32.0000 −0.176777
\(33\) 125.923 0.664252
\(34\) 93.6438 0.472346
\(35\) 177.114 0.855366
\(36\) 121.346 0.561787
\(37\) 368.201 1.63600 0.817998 0.575221i \(-0.195084\pi\)
0.817998 + 0.575221i \(0.195084\pi\)
\(38\) 221.307 0.944755
\(39\) 605.109 2.48449
\(40\) 40.0000 0.158114
\(41\) −95.7927 −0.364886 −0.182443 0.983216i \(-0.558400\pi\)
−0.182443 + 0.983216i \(0.558400\pi\)
\(42\) −536.450 −1.97086
\(43\) 331.961 1.17729 0.588647 0.808390i \(-0.299661\pi\)
0.588647 + 0.808390i \(0.299661\pi\)
\(44\) −66.5193 −0.227913
\(45\) −151.682 −0.502477
\(46\) 46.0000 0.147442
\(47\) −535.037 −1.66049 −0.830246 0.557397i \(-0.811800\pi\)
−0.830246 + 0.557397i \(0.811800\pi\)
\(48\) −121.153 −0.364312
\(49\) 911.782 2.65826
\(50\) −50.0000 −0.141421
\(51\) 354.539 0.973440
\(52\) −319.653 −0.852459
\(53\) 409.345 1.06090 0.530452 0.847715i \(-0.322022\pi\)
0.530452 + 0.847715i \(0.322022\pi\)
\(54\) 50.5282 0.127334
\(55\) 83.1492 0.203852
\(56\) 283.383 0.676226
\(57\) 837.876 1.94701
\(58\) 1.67284 0.00378716
\(59\) −352.950 −0.778817 −0.389408 0.921065i \(-0.627321\pi\)
−0.389408 + 0.921065i \(0.627321\pi\)
\(60\) 151.442 0.325851
\(61\) −507.223 −1.06464 −0.532322 0.846542i \(-0.678680\pi\)
−0.532322 + 0.846542i \(0.678680\pi\)
\(62\) 239.672 0.490941
\(63\) −1074.61 −2.14901
\(64\) 64.0000 0.125000
\(65\) 399.566 0.762462
\(66\) −251.845 −0.469697
\(67\) −820.056 −1.49531 −0.747655 0.664087i \(-0.768820\pi\)
−0.747655 + 0.664087i \(0.768820\pi\)
\(68\) −187.288 −0.333999
\(69\) 174.158 0.303857
\(70\) −354.229 −0.604835
\(71\) −733.770 −1.22651 −0.613257 0.789884i \(-0.710141\pi\)
−0.613257 + 0.789884i \(0.710141\pi\)
\(72\) −242.692 −0.397243
\(73\) −91.4599 −0.146638 −0.0733190 0.997309i \(-0.523359\pi\)
−0.0733190 + 0.997309i \(0.523359\pi\)
\(74\) −736.402 −1.15682
\(75\) −189.302 −0.291450
\(76\) −442.613 −0.668042
\(77\) 589.077 0.871838
\(78\) −1210.22 −1.75680
\(79\) 329.381 0.469092 0.234546 0.972105i \(-0.424640\pi\)
0.234546 + 0.972105i \(0.424640\pi\)
\(80\) −80.0000 −0.111803
\(81\) −627.783 −0.861156
\(82\) 191.585 0.258013
\(83\) −753.834 −0.996916 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(84\) 1072.90 1.39361
\(85\) 234.110 0.298738
\(86\) −663.922 −0.832472
\(87\) 6.33346 0.00780481
\(88\) 133.039 0.161159
\(89\) −1050.14 −1.25073 −0.625365 0.780332i \(-0.715050\pi\)
−0.625365 + 0.780332i \(0.715050\pi\)
\(90\) 303.365 0.355305
\(91\) 2830.76 3.26092
\(92\) −92.0000 −0.104257
\(93\) 907.407 1.01176
\(94\) 1070.07 1.17415
\(95\) 553.267 0.597515
\(96\) 242.307 0.257608
\(97\) −271.928 −0.284641 −0.142320 0.989821i \(-0.545456\pi\)
−0.142320 + 0.989821i \(0.545456\pi\)
\(98\) −1823.56 −1.87967
\(99\) −504.491 −0.512154
\(100\) 100.000 0.100000
\(101\) 1658.68 1.63410 0.817052 0.576563i \(-0.195607\pi\)
0.817052 + 0.576563i \(0.195607\pi\)
\(102\) −709.079 −0.688326
\(103\) 1735.52 1.66025 0.830123 0.557580i \(-0.188270\pi\)
0.830123 + 0.557580i \(0.188270\pi\)
\(104\) 639.305 0.602779
\(105\) −1341.13 −1.24648
\(106\) −818.691 −0.750172
\(107\) −1629.88 −1.47258 −0.736292 0.676664i \(-0.763425\pi\)
−0.736292 + 0.676664i \(0.763425\pi\)
\(108\) −101.056 −0.0900385
\(109\) −432.151 −0.379748 −0.189874 0.981808i \(-0.560808\pi\)
−0.189874 + 0.981808i \(0.560808\pi\)
\(110\) −166.298 −0.144145
\(111\) −2788.05 −2.38405
\(112\) −566.766 −0.478164
\(113\) −240.115 −0.199895 −0.0999476 0.994993i \(-0.531868\pi\)
−0.0999476 + 0.994993i \(0.531868\pi\)
\(114\) −1675.75 −1.37674
\(115\) 115.000 0.0932505
\(116\) −3.34569 −0.00267793
\(117\) −2424.28 −1.91560
\(118\) 705.900 0.550707
\(119\) 1658.57 1.27765
\(120\) −302.883 −0.230411
\(121\) −1054.45 −0.792223
\(122\) 1014.45 0.752817
\(123\) 725.350 0.531729
\(124\) −479.343 −0.347147
\(125\) −125.000 −0.0894427
\(126\) 2149.21 1.51958
\(127\) −210.993 −0.147422 −0.0737110 0.997280i \(-0.523484\pi\)
−0.0737110 + 0.997280i \(0.523484\pi\)
\(128\) −128.000 −0.0883883
\(129\) −2513.64 −1.71561
\(130\) −799.132 −0.539142
\(131\) −746.178 −0.497663 −0.248832 0.968547i \(-0.580047\pi\)
−0.248832 + 0.968547i \(0.580047\pi\)
\(132\) 503.690 0.332126
\(133\) 3919.66 2.55547
\(134\) 1640.11 1.05734
\(135\) 126.320 0.0805329
\(136\) 374.575 0.236173
\(137\) −2420.07 −1.50920 −0.754601 0.656184i \(-0.772170\pi\)
−0.754601 + 0.656184i \(0.772170\pi\)
\(138\) −348.316 −0.214860
\(139\) 924.030 0.563850 0.281925 0.959436i \(-0.409027\pi\)
0.281925 + 0.959436i \(0.409027\pi\)
\(140\) 708.458 0.427683
\(141\) 4051.34 2.41975
\(142\) 1467.54 0.867276
\(143\) 1328.94 0.777145
\(144\) 485.384 0.280893
\(145\) 4.18211 0.00239521
\(146\) 182.920 0.103689
\(147\) −6904.09 −3.87374
\(148\) 1472.80 0.817998
\(149\) 430.614 0.236760 0.118380 0.992968i \(-0.462230\pi\)
0.118380 + 0.992968i \(0.462230\pi\)
\(150\) 378.604 0.206086
\(151\) −25.4118 −0.0136953 −0.00684763 0.999977i \(-0.502180\pi\)
−0.00684763 + 0.999977i \(0.502180\pi\)
\(152\) 885.226 0.472377
\(153\) −1420.41 −0.750546
\(154\) −1178.15 −0.616483
\(155\) 599.179 0.310498
\(156\) 2420.44 1.24224
\(157\) −1580.29 −0.803317 −0.401658 0.915790i \(-0.631566\pi\)
−0.401658 + 0.915790i \(0.631566\pi\)
\(158\) −658.762 −0.331698
\(159\) −3099.60 −1.54600
\(160\) 160.000 0.0790569
\(161\) 814.727 0.398817
\(162\) 1255.57 0.608930
\(163\) 458.739 0.220437 0.110218 0.993907i \(-0.464845\pi\)
0.110218 + 0.993907i \(0.464845\pi\)
\(164\) −383.171 −0.182443
\(165\) −629.613 −0.297062
\(166\) 1507.67 0.704926
\(167\) −561.275 −0.260076 −0.130038 0.991509i \(-0.541510\pi\)
−0.130038 + 0.991509i \(0.541510\pi\)
\(168\) −2145.80 −0.985430
\(169\) 4189.11 1.90674
\(170\) −468.219 −0.211240
\(171\) −3356.83 −1.50119
\(172\) 1327.84 0.588647
\(173\) 1573.95 0.691705 0.345852 0.938289i \(-0.387590\pi\)
0.345852 + 0.938289i \(0.387590\pi\)
\(174\) −12.6669 −0.00551883
\(175\) −885.572 −0.382531
\(176\) −266.077 −0.113956
\(177\) 2672.57 1.13493
\(178\) 2100.29 0.884400
\(179\) 2215.59 0.925145 0.462572 0.886581i \(-0.346927\pi\)
0.462572 + 0.886581i \(0.346927\pi\)
\(180\) −606.730 −0.251239
\(181\) 873.497 0.358710 0.179355 0.983784i \(-0.442599\pi\)
0.179355 + 0.983784i \(0.442599\pi\)
\(182\) −5661.51 −2.30582
\(183\) 3840.74 1.55145
\(184\) 184.000 0.0737210
\(185\) −1841.00 −0.731640
\(186\) −1814.81 −0.715422
\(187\) 778.641 0.304491
\(188\) −2140.15 −0.830246
\(189\) 894.928 0.344425
\(190\) −1106.53 −0.422507
\(191\) −2497.74 −0.946230 −0.473115 0.881001i \(-0.656870\pi\)
−0.473115 + 0.881001i \(0.656870\pi\)
\(192\) −484.613 −0.182156
\(193\) 909.155 0.339080 0.169540 0.985523i \(-0.445772\pi\)
0.169540 + 0.985523i \(0.445772\pi\)
\(194\) 543.857 0.201271
\(195\) −3025.55 −1.11110
\(196\) 3647.13 1.32913
\(197\) 608.627 0.220116 0.110058 0.993925i \(-0.464896\pi\)
0.110058 + 0.993925i \(0.464896\pi\)
\(198\) 1008.98 0.362147
\(199\) −2304.98 −0.821083 −0.410542 0.911842i \(-0.634660\pi\)
−0.410542 + 0.911842i \(0.634660\pi\)
\(200\) −200.000 −0.0707107
\(201\) 6209.54 2.17904
\(202\) −3317.36 −1.15549
\(203\) 29.6285 0.0102439
\(204\) 1418.16 0.486720
\(205\) 478.963 0.163182
\(206\) −3471.03 −1.17397
\(207\) −697.739 −0.234281
\(208\) −1278.61 −0.426229
\(209\) 1840.15 0.609022
\(210\) 2682.25 0.881395
\(211\) −4373.37 −1.42690 −0.713448 0.700708i \(-0.752868\pi\)
−0.713448 + 0.700708i \(0.752868\pi\)
\(212\) 1637.38 0.530452
\(213\) 5556.17 1.78733
\(214\) 3259.76 1.04127
\(215\) −1659.81 −0.526502
\(216\) 202.113 0.0636668
\(217\) 4244.93 1.32795
\(218\) 864.303 0.268523
\(219\) 692.542 0.213688
\(220\) 332.597 0.101926
\(221\) 3741.69 1.13888
\(222\) 5576.10 1.68578
\(223\) 5439.50 1.63343 0.816717 0.577038i \(-0.195792\pi\)
0.816717 + 0.577038i \(0.195792\pi\)
\(224\) 1133.53 0.338113
\(225\) 758.412 0.224715
\(226\) 480.231 0.141347
\(227\) 216.498 0.0633017 0.0316508 0.999499i \(-0.489924\pi\)
0.0316508 + 0.999499i \(0.489924\pi\)
\(228\) 3351.51 0.973504
\(229\) −924.664 −0.266828 −0.133414 0.991060i \(-0.542594\pi\)
−0.133414 + 0.991060i \(0.542594\pi\)
\(230\) −230.000 −0.0659380
\(231\) −4460.54 −1.27049
\(232\) 6.69138 0.00189358
\(233\) −2369.69 −0.666280 −0.333140 0.942877i \(-0.608108\pi\)
−0.333140 + 0.942877i \(0.608108\pi\)
\(234\) 4848.57 1.35453
\(235\) 2675.18 0.742595
\(236\) −1411.80 −0.389408
\(237\) −2494.10 −0.683584
\(238\) −3317.14 −0.903437
\(239\) −2769.60 −0.749583 −0.374791 0.927109i \(-0.622286\pi\)
−0.374791 + 0.927109i \(0.622286\pi\)
\(240\) 605.767 0.162925
\(241\) −5334.10 −1.42572 −0.712862 0.701305i \(-0.752601\pi\)
−0.712862 + 0.701305i \(0.752601\pi\)
\(242\) 2108.90 0.560186
\(243\) 5435.76 1.43500
\(244\) −2028.89 −0.532322
\(245\) −4558.91 −1.18881
\(246\) −1450.70 −0.375989
\(247\) 8842.66 2.27791
\(248\) 958.686 0.245470
\(249\) 5708.10 1.45275
\(250\) 250.000 0.0632456
\(251\) 4677.13 1.17617 0.588084 0.808800i \(-0.299882\pi\)
0.588084 + 0.808800i \(0.299882\pi\)
\(252\) −4298.42 −1.07451
\(253\) 382.486 0.0950463
\(254\) 421.986 0.104243
\(255\) −1772.70 −0.435336
\(256\) 256.000 0.0625000
\(257\) −2602.99 −0.631789 −0.315895 0.948794i \(-0.602305\pi\)
−0.315895 + 0.948794i \(0.602305\pi\)
\(258\) 5027.28 1.21312
\(259\) −13042.7 −3.12910
\(260\) 1598.26 0.381231
\(261\) −25.3741 −0.00601769
\(262\) 1492.36 0.351901
\(263\) −3411.30 −0.799809 −0.399904 0.916557i \(-0.630957\pi\)
−0.399904 + 0.916557i \(0.630957\pi\)
\(264\) −1007.38 −0.234848
\(265\) −2046.73 −0.474451
\(266\) −7839.32 −1.80699
\(267\) 7951.77 1.82262
\(268\) −3280.22 −0.747655
\(269\) 4366.56 0.989718 0.494859 0.868973i \(-0.335220\pi\)
0.494859 + 0.868973i \(0.335220\pi\)
\(270\) −252.641 −0.0569453
\(271\) 5682.45 1.27374 0.636872 0.770970i \(-0.280228\pi\)
0.636872 + 0.770970i \(0.280228\pi\)
\(272\) −749.150 −0.167000
\(273\) −21434.7 −4.75197
\(274\) 4840.14 1.06717
\(275\) −415.746 −0.0911652
\(276\) 696.632 0.151929
\(277\) −1884.62 −0.408794 −0.204397 0.978888i \(-0.565523\pi\)
−0.204397 + 0.978888i \(0.565523\pi\)
\(278\) −1848.06 −0.398702
\(279\) −3635.39 −0.780091
\(280\) −1416.92 −0.302418
\(281\) −2706.42 −0.574561 −0.287281 0.957846i \(-0.592751\pi\)
−0.287281 + 0.957846i \(0.592751\pi\)
\(282\) −8102.69 −1.71102
\(283\) −5963.64 −1.25266 −0.626328 0.779560i \(-0.715443\pi\)
−0.626328 + 0.779560i \(0.715443\pi\)
\(284\) −2935.08 −0.613257
\(285\) −4189.38 −0.870728
\(286\) −2657.89 −0.549525
\(287\) 3393.26 0.697901
\(288\) −970.767 −0.198622
\(289\) −2720.71 −0.553778
\(290\) −8.36422 −0.00169367
\(291\) 2059.06 0.414792
\(292\) −365.840 −0.0733190
\(293\) −4735.85 −0.944272 −0.472136 0.881526i \(-0.656517\pi\)
−0.472136 + 0.881526i \(0.656517\pi\)
\(294\) 13808.2 2.73915
\(295\) 1764.75 0.348297
\(296\) −2945.61 −0.578412
\(297\) 420.138 0.0820837
\(298\) −861.227 −0.167415
\(299\) 1838.00 0.355500
\(300\) −757.209 −0.145725
\(301\) −11759.0 −2.25176
\(302\) 50.8236 0.00968401
\(303\) −12559.6 −2.38130
\(304\) −1770.45 −0.334021
\(305\) 2536.12 0.476123
\(306\) 2840.82 0.530716
\(307\) −769.241 −0.143006 −0.0715031 0.997440i \(-0.522780\pi\)
−0.0715031 + 0.997440i \(0.522780\pi\)
\(308\) 2356.31 0.435919
\(309\) −13141.5 −2.41939
\(310\) −1198.36 −0.219555
\(311\) −5592.71 −1.01972 −0.509861 0.860257i \(-0.670303\pi\)
−0.509861 + 0.860257i \(0.670303\pi\)
\(312\) −4840.87 −0.878399
\(313\) 9777.19 1.76562 0.882811 0.469729i \(-0.155648\pi\)
0.882811 + 0.469729i \(0.155648\pi\)
\(314\) 3160.58 0.568031
\(315\) 5373.03 0.961066
\(316\) 1317.52 0.234546
\(317\) 1868.51 0.331060 0.165530 0.986205i \(-0.447066\pi\)
0.165530 + 0.986205i \(0.447066\pi\)
\(318\) 6199.20 1.09319
\(319\) 13.9096 0.00244134
\(320\) −320.000 −0.0559017
\(321\) 12341.6 2.14592
\(322\) −1629.45 −0.282006
\(323\) 5181.00 0.892503
\(324\) −2511.13 −0.430578
\(325\) −1997.83 −0.340983
\(326\) −917.477 −0.155872
\(327\) 3272.29 0.553388
\(328\) 766.342 0.129007
\(329\) 18952.6 3.17595
\(330\) 1259.23 0.210055
\(331\) 6966.91 1.15691 0.578454 0.815715i \(-0.303656\pi\)
0.578454 + 0.815715i \(0.303656\pi\)
\(332\) −3015.34 −0.498458
\(333\) 11169.9 1.83816
\(334\) 1122.55 0.183902
\(335\) 4100.28 0.668723
\(336\) 4291.60 0.696804
\(337\) −17.5487 −0.00283661 −0.00141830 0.999999i \(-0.500451\pi\)
−0.00141830 + 0.999999i \(0.500451\pi\)
\(338\) −8378.23 −1.34827
\(339\) 1818.17 0.291297
\(340\) 936.438 0.149369
\(341\) 1992.85 0.316477
\(342\) 6713.66 1.06150
\(343\) −20147.9 −3.17167
\(344\) −2655.69 −0.416236
\(345\) −870.790 −0.135889
\(346\) −3147.89 −0.489109
\(347\) −9859.30 −1.52529 −0.762644 0.646818i \(-0.776099\pi\)
−0.762644 + 0.646818i \(0.776099\pi\)
\(348\) 25.3338 0.00390240
\(349\) 7200.25 1.10436 0.552178 0.833726i \(-0.313797\pi\)
0.552178 + 0.833726i \(0.313797\pi\)
\(350\) 1771.14 0.270491
\(351\) 2018.93 0.307016
\(352\) 532.155 0.0805794
\(353\) −6054.58 −0.912897 −0.456449 0.889750i \(-0.650879\pi\)
−0.456449 + 0.889750i \(0.650879\pi\)
\(354\) −5345.14 −0.802516
\(355\) 3668.85 0.548513
\(356\) −4200.57 −0.625365
\(357\) −12558.8 −1.86186
\(358\) −4431.18 −0.654176
\(359\) 2734.26 0.401974 0.200987 0.979594i \(-0.435585\pi\)
0.200987 + 0.979594i \(0.435585\pi\)
\(360\) 1213.46 0.177653
\(361\) 5385.16 0.785123
\(362\) −1746.99 −0.253646
\(363\) 7984.37 1.15447
\(364\) 11323.0 1.63046
\(365\) 457.300 0.0655785
\(366\) −7681.47 −1.09704
\(367\) −2465.50 −0.350676 −0.175338 0.984508i \(-0.556102\pi\)
−0.175338 + 0.984508i \(0.556102\pi\)
\(368\) −368.000 −0.0521286
\(369\) −2906.01 −0.409976
\(370\) 3682.01 0.517347
\(371\) −14500.2 −2.02915
\(372\) 3629.63 0.505880
\(373\) 5704.45 0.791863 0.395932 0.918280i \(-0.370422\pi\)
0.395932 + 0.918280i \(0.370422\pi\)
\(374\) −1557.28 −0.215308
\(375\) 946.511 0.130340
\(376\) 4280.29 0.587073
\(377\) 66.8411 0.00913128
\(378\) −1789.86 −0.243546
\(379\) 10231.9 1.38675 0.693376 0.720576i \(-0.256122\pi\)
0.693376 + 0.720576i \(0.256122\pi\)
\(380\) 2213.07 0.298758
\(381\) 1597.66 0.214830
\(382\) 4995.47 0.669086
\(383\) 6321.34 0.843356 0.421678 0.906746i \(-0.361441\pi\)
0.421678 + 0.906746i \(0.361441\pi\)
\(384\) 969.227 0.128804
\(385\) −2945.38 −0.389898
\(386\) −1818.31 −0.239766
\(387\) 10070.5 1.32278
\(388\) −1087.71 −0.142320
\(389\) −925.594 −0.120641 −0.0603207 0.998179i \(-0.519212\pi\)
−0.0603207 + 0.998179i \(0.519212\pi\)
\(390\) 6051.09 0.785664
\(391\) 1076.90 0.139287
\(392\) −7294.25 −0.939835
\(393\) 5650.13 0.725219
\(394\) −1217.25 −0.155646
\(395\) −1646.91 −0.209784
\(396\) −2017.96 −0.256077
\(397\) −2706.75 −0.342187 −0.171093 0.985255i \(-0.554730\pi\)
−0.171093 + 0.985255i \(0.554730\pi\)
\(398\) 4609.96 0.580594
\(399\) −29680.0 −3.72396
\(400\) 400.000 0.0500000
\(401\) −14095.3 −1.75532 −0.877661 0.479281i \(-0.840897\pi\)
−0.877661 + 0.479281i \(0.840897\pi\)
\(402\) −12419.1 −1.54081
\(403\) 9576.45 1.18372
\(404\) 6634.71 0.817052
\(405\) 3138.92 0.385121
\(406\) −59.2570 −0.00724353
\(407\) −6123.12 −0.745729
\(408\) −2836.32 −0.344163
\(409\) −10846.0 −1.31125 −0.655625 0.755087i \(-0.727595\pi\)
−0.655625 + 0.755087i \(0.727595\pi\)
\(410\) −957.927 −0.115387
\(411\) 18325.0 2.19928
\(412\) 6942.06 0.830123
\(413\) 12502.5 1.48961
\(414\) 1395.48 0.165662
\(415\) 3769.17 0.445834
\(416\) 2557.22 0.301390
\(417\) −6996.84 −0.821670
\(418\) −3680.29 −0.430644
\(419\) −5626.55 −0.656026 −0.328013 0.944673i \(-0.606379\pi\)
−0.328013 + 0.944673i \(0.606379\pi\)
\(420\) −5364.50 −0.623240
\(421\) −7109.09 −0.822983 −0.411491 0.911414i \(-0.634992\pi\)
−0.411491 + 0.911414i \(0.634992\pi\)
\(422\) 8746.74 1.00897
\(423\) −16231.1 −1.86568
\(424\) −3274.76 −0.375086
\(425\) −1170.55 −0.133600
\(426\) −11112.3 −1.26384
\(427\) 17967.3 2.03630
\(428\) −6519.52 −0.736292
\(429\) −10062.9 −1.13249
\(430\) 3319.61 0.372293
\(431\) 4464.14 0.498909 0.249455 0.968387i \(-0.419749\pi\)
0.249455 + 0.968387i \(0.419749\pi\)
\(432\) −404.226 −0.0450192
\(433\) 11009.4 1.22189 0.610947 0.791672i \(-0.290789\pi\)
0.610947 + 0.791672i \(0.290789\pi\)
\(434\) −8489.86 −0.939001
\(435\) −31.6673 −0.00349042
\(436\) −1728.61 −0.189874
\(437\) 2545.03 0.278593
\(438\) −1385.08 −0.151100
\(439\) −3052.70 −0.331885 −0.165943 0.986135i \(-0.553067\pi\)
−0.165943 + 0.986135i \(0.553067\pi\)
\(440\) −665.193 −0.0720724
\(441\) 27660.2 2.98675
\(442\) −7483.37 −0.805312
\(443\) −4465.82 −0.478956 −0.239478 0.970902i \(-0.576976\pi\)
−0.239478 + 0.970902i \(0.576976\pi\)
\(444\) −11152.2 −1.19203
\(445\) 5250.72 0.559343
\(446\) −10879.0 −1.15501
\(447\) −3260.64 −0.345018
\(448\) −2267.07 −0.239082
\(449\) 9040.14 0.950179 0.475090 0.879937i \(-0.342416\pi\)
0.475090 + 0.879937i \(0.342416\pi\)
\(450\) −1516.82 −0.158897
\(451\) 1593.02 0.166324
\(452\) −960.462 −0.0999476
\(453\) 192.420 0.0199574
\(454\) −432.996 −0.0447610
\(455\) −14153.8 −1.45833
\(456\) −6703.01 −0.688371
\(457\) 7756.88 0.793986 0.396993 0.917822i \(-0.370054\pi\)
0.396993 + 0.917822i \(0.370054\pi\)
\(458\) 1849.33 0.188676
\(459\) 1182.91 0.120291
\(460\) 460.000 0.0466252
\(461\) 7064.19 0.713692 0.356846 0.934163i \(-0.383852\pi\)
0.356846 + 0.934163i \(0.383852\pi\)
\(462\) 8921.08 0.898369
\(463\) 12599.7 1.26470 0.632350 0.774682i \(-0.282090\pi\)
0.632350 + 0.774682i \(0.282090\pi\)
\(464\) −13.3828 −0.00133896
\(465\) −4537.03 −0.452473
\(466\) 4739.37 0.471131
\(467\) 14746.0 1.46117 0.730583 0.682824i \(-0.239249\pi\)
0.730583 + 0.682824i \(0.239249\pi\)
\(468\) −9697.14 −0.957800
\(469\) 29048.8 2.86002
\(470\) −5350.37 −0.525094
\(471\) 11966.1 1.17063
\(472\) 2823.60 0.275353
\(473\) −5520.46 −0.536641
\(474\) 4988.20 0.483367
\(475\) −2766.33 −0.267217
\(476\) 6634.27 0.638826
\(477\) 12418.1 1.19200
\(478\) 5539.19 0.530035
\(479\) −17411.2 −1.66084 −0.830418 0.557141i \(-0.811898\pi\)
−0.830418 + 0.557141i \(0.811898\pi\)
\(480\) −1211.53 −0.115206
\(481\) −29424.1 −2.78924
\(482\) 10668.2 1.00814
\(483\) −6169.18 −0.581175
\(484\) −4217.79 −0.396111
\(485\) 1359.64 0.127295
\(486\) −10871.5 −1.01470
\(487\) 1320.34 0.122855 0.0614276 0.998112i \(-0.480435\pi\)
0.0614276 + 0.998112i \(0.480435\pi\)
\(488\) 4057.78 0.376408
\(489\) −3473.61 −0.321231
\(490\) 9117.82 0.840614
\(491\) 17115.8 1.57317 0.786583 0.617485i \(-0.211848\pi\)
0.786583 + 0.617485i \(0.211848\pi\)
\(492\) 2901.40 0.265864
\(493\) 39.1629 0.00357770
\(494\) −17685.3 −1.61073
\(495\) 2522.45 0.229042
\(496\) −1917.37 −0.173574
\(497\) 25992.3 2.34590
\(498\) −11416.2 −1.02725
\(499\) 17540.6 1.57360 0.786798 0.617210i \(-0.211737\pi\)
0.786798 + 0.617210i \(0.211737\pi\)
\(500\) −500.000 −0.0447214
\(501\) 4250.02 0.378996
\(502\) −9354.27 −0.831676
\(503\) −4934.98 −0.437455 −0.218727 0.975786i \(-0.570191\pi\)
−0.218727 + 0.975786i \(0.570191\pi\)
\(504\) 8596.85 0.759790
\(505\) −8293.39 −0.730794
\(506\) −764.972 −0.0672079
\(507\) −31720.3 −2.77860
\(508\) −843.972 −0.0737110
\(509\) 11927.3 1.03865 0.519323 0.854578i \(-0.326184\pi\)
0.519323 + 0.854578i \(0.326184\pi\)
\(510\) 3545.39 0.307829
\(511\) 3239.78 0.280468
\(512\) −512.000 −0.0441942
\(513\) 2795.56 0.240598
\(514\) 5205.97 0.446743
\(515\) −8677.58 −0.742485
\(516\) −10054.6 −0.857804
\(517\) 8897.57 0.756895
\(518\) 26085.5 2.21261
\(519\) −11918.1 −1.00799
\(520\) −3196.53 −0.269571
\(521\) −9592.76 −0.806653 −0.403327 0.915056i \(-0.632146\pi\)
−0.403327 + 0.915056i \(0.632146\pi\)
\(522\) 50.7482 0.00425515
\(523\) 6525.20 0.545558 0.272779 0.962077i \(-0.412057\pi\)
0.272779 + 0.962077i \(0.412057\pi\)
\(524\) −2984.71 −0.248832
\(525\) 6705.63 0.557443
\(526\) 6822.60 0.565550
\(527\) 5610.94 0.463788
\(528\) 2014.76 0.166063
\(529\) 529.000 0.0434783
\(530\) 4093.45 0.335487
\(531\) −10707.3 −0.875058
\(532\) 15678.6 1.27774
\(533\) 7655.10 0.622100
\(534\) −15903.5 −1.28879
\(535\) 8149.40 0.658560
\(536\) 6560.45 0.528672
\(537\) −16776.6 −1.34817
\(538\) −8733.13 −0.699836
\(539\) −15162.8 −1.21170
\(540\) 505.282 0.0402664
\(541\) 5634.05 0.447739 0.223870 0.974619i \(-0.428131\pi\)
0.223870 + 0.974619i \(0.428131\pi\)
\(542\) −11364.9 −0.900673
\(543\) −6614.19 −0.522729
\(544\) 1498.30 0.118087
\(545\) 2160.76 0.169829
\(546\) 42869.5 3.36015
\(547\) 6093.08 0.476273 0.238136 0.971232i \(-0.423463\pi\)
0.238136 + 0.971232i \(0.423463\pi\)
\(548\) −9680.29 −0.754601
\(549\) −15387.4 −1.19620
\(550\) 831.492 0.0644635
\(551\) 92.5529 0.00715587
\(552\) −1393.26 −0.107430
\(553\) −11667.6 −0.897212
\(554\) 3769.24 0.289061
\(555\) 13940.2 1.06618
\(556\) 3696.12 0.281925
\(557\) 21461.9 1.63262 0.816311 0.577612i \(-0.196015\pi\)
0.816311 + 0.577612i \(0.196015\pi\)
\(558\) 7270.79 0.551608
\(559\) −26528.1 −2.00719
\(560\) 2833.83 0.213842
\(561\) −5895.93 −0.443719
\(562\) 5412.85 0.406276
\(563\) −11038.5 −0.826316 −0.413158 0.910659i \(-0.635574\pi\)
−0.413158 + 0.910659i \(0.635574\pi\)
\(564\) 16205.4 1.20988
\(565\) 1200.58 0.0893959
\(566\) 11927.3 0.885761
\(567\) 22237.9 1.64710
\(568\) 5870.16 0.433638
\(569\) −19210.3 −1.41535 −0.707677 0.706536i \(-0.750257\pi\)
−0.707677 + 0.706536i \(0.750257\pi\)
\(570\) 8378.76 0.615698
\(571\) −11967.0 −0.877066 −0.438533 0.898715i \(-0.644502\pi\)
−0.438533 + 0.898715i \(0.644502\pi\)
\(572\) 5315.77 0.388573
\(573\) 18913.1 1.37889
\(574\) −6786.51 −0.493490
\(575\) −575.000 −0.0417029
\(576\) 1941.53 0.140447
\(577\) −14982.3 −1.08097 −0.540487 0.841352i \(-0.681760\pi\)
−0.540487 + 0.841352i \(0.681760\pi\)
\(578\) 5441.42 0.391580
\(579\) −6884.20 −0.494124
\(580\) 16.7284 0.00119760
\(581\) 26703.0 1.90676
\(582\) −4118.13 −0.293302
\(583\) −6807.35 −0.483587
\(584\) 731.679 0.0518444
\(585\) 12121.4 0.856682
\(586\) 9471.71 0.667701
\(587\) 24002.9 1.68774 0.843871 0.536547i \(-0.180271\pi\)
0.843871 + 0.536547i \(0.180271\pi\)
\(588\) −27616.4 −1.93687
\(589\) 13260.2 0.927637
\(590\) −3529.50 −0.246283
\(591\) −4608.58 −0.320764
\(592\) 5891.21 0.408999
\(593\) 5124.67 0.354882 0.177441 0.984131i \(-0.443218\pi\)
0.177441 + 0.984131i \(0.443218\pi\)
\(594\) −840.276 −0.0580420
\(595\) −8292.84 −0.571384
\(596\) 1722.45 0.118380
\(597\) 17453.5 1.19652
\(598\) −3676.01 −0.251376
\(599\) −23776.8 −1.62186 −0.810928 0.585146i \(-0.801037\pi\)
−0.810928 + 0.585146i \(0.801037\pi\)
\(600\) 1514.42 0.103043
\(601\) −25435.3 −1.72633 −0.863166 0.504920i \(-0.831522\pi\)
−0.863166 + 0.504920i \(0.831522\pi\)
\(602\) 23518.1 1.59223
\(603\) −24877.6 −1.68009
\(604\) −101.647 −0.00684763
\(605\) 5272.24 0.354293
\(606\) 25119.3 1.68383
\(607\) −7445.94 −0.497894 −0.248947 0.968517i \(-0.580084\pi\)
−0.248947 + 0.968517i \(0.580084\pi\)
\(608\) 3540.91 0.236189
\(609\) −224.349 −0.0149279
\(610\) −5072.23 −0.336670
\(611\) 42756.5 2.83100
\(612\) −5681.65 −0.375273
\(613\) −12874.4 −0.848275 −0.424138 0.905598i \(-0.639423\pi\)
−0.424138 + 0.905598i \(0.639423\pi\)
\(614\) 1538.48 0.101121
\(615\) −3626.75 −0.237796
\(616\) −4712.62 −0.308241
\(617\) −19247.2 −1.25585 −0.627927 0.778272i \(-0.716096\pi\)
−0.627927 + 0.778272i \(0.716096\pi\)
\(618\) 26282.9 1.71077
\(619\) −14496.4 −0.941293 −0.470647 0.882322i \(-0.655979\pi\)
−0.470647 + 0.882322i \(0.655979\pi\)
\(620\) 2396.72 0.155249
\(621\) 581.074 0.0375486
\(622\) 11185.4 0.721052
\(623\) 37199.1 2.39222
\(624\) 9681.75 0.621122
\(625\) 625.000 0.0400000
\(626\) −19554.4 −1.24848
\(627\) −13933.7 −0.887496
\(628\) −6321.15 −0.401658
\(629\) −17239.9 −1.09284
\(630\) −10746.1 −0.679577
\(631\) −18030.9 −1.13756 −0.568779 0.822490i \(-0.692584\pi\)
−0.568779 + 0.822490i \(0.692584\pi\)
\(632\) −2635.05 −0.165849
\(633\) 33115.5 2.07934
\(634\) −3737.03 −0.234095
\(635\) 1054.96 0.0659291
\(636\) −12398.4 −0.773000
\(637\) −72863.4 −4.53211
\(638\) −27.8191 −0.00172628
\(639\) −22260.0 −1.37808
\(640\) 640.000 0.0395285
\(641\) 11776.4 0.725646 0.362823 0.931858i \(-0.381813\pi\)
0.362823 + 0.931858i \(0.381813\pi\)
\(642\) −24683.2 −1.51740
\(643\) 20207.4 1.23935 0.619676 0.784858i \(-0.287264\pi\)
0.619676 + 0.784858i \(0.287264\pi\)
\(644\) 3258.91 0.199408
\(645\) 12568.2 0.767244
\(646\) −10362.0 −0.631095
\(647\) 21452.8 1.30355 0.651776 0.758412i \(-0.274024\pi\)
0.651776 + 0.758412i \(0.274024\pi\)
\(648\) 5022.26 0.304465
\(649\) 5869.50 0.355005
\(650\) 3995.66 0.241112
\(651\) −32143.0 −1.93515
\(652\) 1834.95 0.110218
\(653\) −15095.2 −0.904624 −0.452312 0.891860i \(-0.649401\pi\)
−0.452312 + 0.891860i \(0.649401\pi\)
\(654\) −6544.57 −0.391304
\(655\) 3730.89 0.222562
\(656\) −1532.68 −0.0912214
\(657\) −2774.57 −0.164759
\(658\) −37905.1 −2.24574
\(659\) 6964.68 0.411693 0.205846 0.978584i \(-0.434005\pi\)
0.205846 + 0.978584i \(0.434005\pi\)
\(660\) −2518.45 −0.148531
\(661\) −17380.4 −1.02272 −0.511361 0.859366i \(-0.670858\pi\)
−0.511361 + 0.859366i \(0.670858\pi\)
\(662\) −13933.8 −0.818057
\(663\) −28332.4 −1.65964
\(664\) 6030.67 0.352463
\(665\) −19598.3 −1.14284
\(666\) −22339.8 −1.29978
\(667\) 19.2377 0.00111677
\(668\) −2245.10 −0.130038
\(669\) −41188.3 −2.38032
\(670\) −8200.56 −0.472859
\(671\) 8435.04 0.485292
\(672\) −8583.21 −0.492715
\(673\) 12383.2 0.709271 0.354635 0.935005i \(-0.384605\pi\)
0.354635 + 0.935005i \(0.384605\pi\)
\(674\) 35.0973 0.00200578
\(675\) −631.602 −0.0360154
\(676\) 16756.5 0.953371
\(677\) −8375.74 −0.475489 −0.237744 0.971328i \(-0.576408\pi\)
−0.237744 + 0.971328i \(0.576408\pi\)
\(678\) −3636.35 −0.205978
\(679\) 9632.49 0.544420
\(680\) −1872.88 −0.105620
\(681\) −1639.34 −0.0922463
\(682\) −3985.70 −0.223783
\(683\) −4434.66 −0.248444 −0.124222 0.992254i \(-0.539644\pi\)
−0.124222 + 0.992254i \(0.539644\pi\)
\(684\) −13427.3 −0.750595
\(685\) 12100.4 0.674936
\(686\) 40295.8 2.24271
\(687\) 7001.64 0.388834
\(688\) 5311.38 0.294323
\(689\) −32712.1 −1.80875
\(690\) 1741.58 0.0960881
\(691\) 11032.8 0.607389 0.303694 0.952770i \(-0.401780\pi\)
0.303694 + 0.952770i \(0.401780\pi\)
\(692\) 6295.79 0.345852
\(693\) 17870.5 0.979574
\(694\) 19718.6 1.07854
\(695\) −4620.15 −0.252162
\(696\) −50.6677 −0.00275942
\(697\) 4485.20 0.243743
\(698\) −14400.5 −0.780898
\(699\) 17943.5 0.970936
\(700\) −3542.29 −0.191266
\(701\) 3708.40 0.199806 0.0999032 0.994997i \(-0.468147\pi\)
0.0999032 + 0.994997i \(0.468147\pi\)
\(702\) −4037.87 −0.217093
\(703\) −40742.6 −2.18583
\(704\) −1064.31 −0.0569782
\(705\) −20256.7 −1.08215
\(706\) 12109.2 0.645516
\(707\) −58755.2 −3.12548
\(708\) 10690.3 0.567465
\(709\) −6315.04 −0.334508 −0.167254 0.985914i \(-0.553490\pi\)
−0.167254 + 0.985914i \(0.553490\pi\)
\(710\) −7337.70 −0.387858
\(711\) 9992.26 0.527059
\(712\) 8401.15 0.442200
\(713\) 2756.22 0.144770
\(714\) 25117.6 1.31653
\(715\) −6644.71 −0.347550
\(716\) 8862.36 0.462572
\(717\) 20971.6 1.09233
\(718\) −5468.52 −0.284239
\(719\) 24736.9 1.28307 0.641536 0.767093i \(-0.278297\pi\)
0.641536 + 0.767093i \(0.278297\pi\)
\(720\) −2426.92 −0.125619
\(721\) −61477.0 −3.17548
\(722\) −10770.3 −0.555165
\(723\) 40390.2 2.07763
\(724\) 3493.99 0.179355
\(725\) −20.9106 −0.00107117
\(726\) −15968.7 −0.816330
\(727\) 33283.6 1.69797 0.848983 0.528420i \(-0.177215\pi\)
0.848983 + 0.528420i \(0.177215\pi\)
\(728\) −22646.0 −1.15291
\(729\) −24209.9 −1.22999
\(730\) −914.599 −0.0463710
\(731\) −15543.1 −0.786430
\(732\) 15362.9 0.775725
\(733\) −33636.5 −1.69494 −0.847471 0.530841i \(-0.821876\pi\)
−0.847471 + 0.530841i \(0.821876\pi\)
\(734\) 4931.00 0.247965
\(735\) 34520.4 1.73239
\(736\) 736.000 0.0368605
\(737\) 13637.4 0.681601
\(738\) 5812.03 0.289897
\(739\) 36575.5 1.82064 0.910319 0.413907i \(-0.135836\pi\)
0.910319 + 0.413907i \(0.135836\pi\)
\(740\) −7364.02 −0.365820
\(741\) −66957.3 −3.31949
\(742\) 29000.4 1.43482
\(743\) −17704.7 −0.874190 −0.437095 0.899415i \(-0.643993\pi\)
−0.437095 + 0.899415i \(0.643993\pi\)
\(744\) −7259.25 −0.357711
\(745\) −2153.07 −0.105882
\(746\) −11408.9 −0.559932
\(747\) −22868.7 −1.12011
\(748\) 3114.56 0.152246
\(749\) 57735.1 2.81655
\(750\) −1893.02 −0.0921645
\(751\) 8858.37 0.430421 0.215211 0.976568i \(-0.430956\pi\)
0.215211 + 0.976568i \(0.430956\pi\)
\(752\) −8560.59 −0.415123
\(753\) −35415.7 −1.71397
\(754\) −133.682 −0.00645679
\(755\) 127.059 0.00612470
\(756\) 3579.71 0.172213
\(757\) 7899.26 0.379265 0.189633 0.981855i \(-0.439270\pi\)
0.189633 + 0.981855i \(0.439270\pi\)
\(758\) −20463.9 −0.980582
\(759\) −2896.22 −0.138506
\(760\) −4426.13 −0.211254
\(761\) 23437.4 1.11643 0.558216 0.829696i \(-0.311486\pi\)
0.558216 + 0.829696i \(0.311486\pi\)
\(762\) −3195.31 −0.151908
\(763\) 15308.1 0.726329
\(764\) −9990.95 −0.473115
\(765\) 7102.06 0.335654
\(766\) −12642.7 −0.596343
\(767\) 28205.4 1.32782
\(768\) −1938.45 −0.0910780
\(769\) 31447.7 1.47468 0.737342 0.675519i \(-0.236081\pi\)
0.737342 + 0.675519i \(0.236081\pi\)
\(770\) 5890.77 0.275699
\(771\) 19710.0 0.920674
\(772\) 3636.62 0.169540
\(773\) 2397.42 0.111551 0.0557756 0.998443i \(-0.482237\pi\)
0.0557756 + 0.998443i \(0.482237\pi\)
\(774\) −20141.1 −0.935343
\(775\) −2995.89 −0.138859
\(776\) 2175.43 0.100636
\(777\) 98760.8 4.55987
\(778\) 1851.19 0.0853063
\(779\) 10599.8 0.487518
\(780\) −12102.2 −0.555548
\(781\) 12202.5 0.559077
\(782\) −2153.81 −0.0984911
\(783\) 21.1314 0.000964465 0
\(784\) 14588.5 0.664564
\(785\) 7901.44 0.359254
\(786\) −11300.3 −0.512807
\(787\) −19407.0 −0.879014 −0.439507 0.898239i \(-0.644847\pi\)
−0.439507 + 0.898239i \(0.644847\pi\)
\(788\) 2434.51 0.110058
\(789\) 25830.7 1.16552
\(790\) 3293.81 0.148340
\(791\) 8505.58 0.382331
\(792\) 4035.93 0.181074
\(793\) 40533.8 1.81513
\(794\) 5413.51 0.241962
\(795\) 15498.0 0.691393
\(796\) −9219.91 −0.410542
\(797\) −28631.1 −1.27248 −0.636240 0.771491i \(-0.719511\pi\)
−0.636240 + 0.771491i \(0.719511\pi\)
\(798\) 59360.0 2.63324
\(799\) 25051.4 1.10921
\(800\) −800.000 −0.0353553
\(801\) −31857.6 −1.40529
\(802\) 28190.6 1.24120
\(803\) 1520.96 0.0668414
\(804\) 24838.1 1.08952
\(805\) −4073.63 −0.178356
\(806\) −19152.9 −0.837013
\(807\) −33064.0 −1.44227
\(808\) −13269.4 −0.577743
\(809\) −9191.25 −0.399440 −0.199720 0.979853i \(-0.564003\pi\)
−0.199720 + 0.979853i \(0.564003\pi\)
\(810\) −6277.83 −0.272322
\(811\) −567.477 −0.0245706 −0.0122853 0.999925i \(-0.503911\pi\)
−0.0122853 + 0.999925i \(0.503911\pi\)
\(812\) 118.514 0.00512195
\(813\) −43028.0 −1.85616
\(814\) 12246.2 0.527310
\(815\) −2293.69 −0.0985823
\(816\) 5672.63 0.243360
\(817\) −36732.6 −1.57296
\(818\) 21692.0 0.927193
\(819\) 85875.2 3.66388
\(820\) 1915.85 0.0815909
\(821\) −6057.35 −0.257494 −0.128747 0.991677i \(-0.541096\pi\)
−0.128747 + 0.991677i \(0.541096\pi\)
\(822\) −36650.0 −1.55513
\(823\) −21327.0 −0.903294 −0.451647 0.892197i \(-0.649163\pi\)
−0.451647 + 0.892197i \(0.649163\pi\)
\(824\) −13884.1 −0.586986
\(825\) 3148.06 0.132850
\(826\) −25005.0 −1.05331
\(827\) −27553.8 −1.15857 −0.579286 0.815124i \(-0.696668\pi\)
−0.579286 + 0.815124i \(0.696668\pi\)
\(828\) −2790.96 −0.117141
\(829\) 11181.6 0.468461 0.234231 0.972181i \(-0.424743\pi\)
0.234231 + 0.972181i \(0.424743\pi\)
\(830\) −7538.34 −0.315253
\(831\) 14270.5 0.595715
\(832\) −5114.44 −0.213115
\(833\) −42691.4 −1.77571
\(834\) 13993.7 0.581009
\(835\) 2806.37 0.116310
\(836\) 7360.59 0.304511
\(837\) 3027.54 0.125026
\(838\) 11253.1 0.463881
\(839\) 9811.83 0.403745 0.201872 0.979412i \(-0.435297\pi\)
0.201872 + 0.979412i \(0.435297\pi\)
\(840\) 10729.0 0.440698
\(841\) −24388.3 −0.999971
\(842\) 14218.2 0.581937
\(843\) 20493.3 0.837279
\(844\) −17493.5 −0.713448
\(845\) −20945.6 −0.852721
\(846\) 32462.3 1.31924
\(847\) 37351.6 1.51525
\(848\) 6549.53 0.265226
\(849\) 45157.2 1.82543
\(850\) 2341.10 0.0944693
\(851\) −8468.62 −0.341129
\(852\) 22224.7 0.893667
\(853\) −24030.8 −0.964594 −0.482297 0.876008i \(-0.660197\pi\)
−0.482297 + 0.876008i \(0.660197\pi\)
\(854\) −35934.6 −1.43988
\(855\) 16784.2 0.671352
\(856\) 13039.0 0.520637
\(857\) −16934.8 −0.675008 −0.337504 0.941324i \(-0.609583\pi\)
−0.337504 + 0.941324i \(0.609583\pi\)
\(858\) 20125.7 0.800794
\(859\) −31057.0 −1.23359 −0.616793 0.787125i \(-0.711568\pi\)
−0.616793 + 0.787125i \(0.711568\pi\)
\(860\) −6639.22 −0.263251
\(861\) −25694.0 −1.01701
\(862\) −8928.27 −0.352782
\(863\) −16157.8 −0.637332 −0.318666 0.947867i \(-0.603235\pi\)
−0.318666 + 0.947867i \(0.603235\pi\)
\(864\) 808.451 0.0318334
\(865\) −7869.73 −0.309340
\(866\) −22018.9 −0.864009
\(867\) 20601.4 0.806992
\(868\) 16979.7 0.663974
\(869\) −5477.55 −0.213824
\(870\) 63.3346 0.00246810
\(871\) 65533.3 2.54938
\(872\) 3457.21 0.134261
\(873\) −8249.35 −0.319815
\(874\) −5090.05 −0.196995
\(875\) 4427.86 0.171073
\(876\) 2770.17 0.106844
\(877\) −32473.9 −1.25036 −0.625180 0.780480i \(-0.714975\pi\)
−0.625180 + 0.780480i \(0.714975\pi\)
\(878\) 6105.41 0.234678
\(879\) 35860.3 1.37604
\(880\) 1330.39 0.0509629
\(881\) −15703.2 −0.600517 −0.300258 0.953858i \(-0.597073\pi\)
−0.300258 + 0.953858i \(0.597073\pi\)
\(882\) −55320.5 −2.11195
\(883\) −14330.4 −0.546155 −0.273078 0.961992i \(-0.588042\pi\)
−0.273078 + 0.961992i \(0.588042\pi\)
\(884\) 14966.7 0.569441
\(885\) −13362.8 −0.507556
\(886\) 8931.65 0.338673
\(887\) 23238.2 0.879664 0.439832 0.898080i \(-0.355038\pi\)
0.439832 + 0.898080i \(0.355038\pi\)
\(888\) 22304.4 0.842890
\(889\) 7473.98 0.281968
\(890\) −10501.4 −0.395515
\(891\) 10439.9 0.392537
\(892\) 21758.0 0.816717
\(893\) 59203.6 2.21856
\(894\) 6521.29 0.243965
\(895\) −11077.9 −0.413737
\(896\) 4534.13 0.169057
\(897\) −13917.5 −0.518052
\(898\) −18080.3 −0.671878
\(899\) 100.233 0.00371854
\(900\) 3033.65 0.112357
\(901\) −19166.3 −0.708683
\(902\) −3186.03 −0.117609
\(903\) 89040.4 3.28137
\(904\) 1920.92 0.0706736
\(905\) −4367.48 −0.160420
\(906\) −384.841 −0.0141120
\(907\) −3667.46 −0.134263 −0.0671313 0.997744i \(-0.521385\pi\)
−0.0671313 + 0.997744i \(0.521385\pi\)
\(908\) 865.992 0.0316508
\(909\) 50318.4 1.83604
\(910\) 28307.6 1.03119
\(911\) 21457.9 0.780385 0.390192 0.920733i \(-0.372409\pi\)
0.390192 + 0.920733i \(0.372409\pi\)
\(912\) 13406.0 0.486752
\(913\) 12536.1 0.454420
\(914\) −15513.8 −0.561433
\(915\) −19203.7 −0.693830
\(916\) −3698.66 −0.133414
\(917\) 26431.8 0.951859
\(918\) −2365.83 −0.0850587
\(919\) −11109.3 −0.398762 −0.199381 0.979922i \(-0.563893\pi\)
−0.199381 + 0.979922i \(0.563893\pi\)
\(920\) −920.000 −0.0329690
\(921\) 5824.76 0.208396
\(922\) −14128.4 −0.504657
\(923\) 58637.9 2.09110
\(924\) −17842.2 −0.635243
\(925\) 9205.02 0.327199
\(926\) −25199.3 −0.894279
\(927\) 52649.4 1.86541
\(928\) 26.7655 0.000946790 0
\(929\) −20804.3 −0.734732 −0.367366 0.930076i \(-0.619740\pi\)
−0.367366 + 0.930076i \(0.619740\pi\)
\(930\) 9074.07 0.319947
\(931\) −100892. −3.55166
\(932\) −9478.75 −0.333140
\(933\) 42348.5 1.48599
\(934\) −29492.0 −1.03320
\(935\) −3893.20 −0.136173
\(936\) 19394.3 0.677267
\(937\) −35550.5 −1.23947 −0.619735 0.784811i \(-0.712760\pi\)
−0.619735 + 0.784811i \(0.712760\pi\)
\(938\) −58097.5 −2.02234
\(939\) −74033.7 −2.57295
\(940\) 10700.7 0.371297
\(941\) 49674.4 1.72087 0.860436 0.509558i \(-0.170191\pi\)
0.860436 + 0.509558i \(0.170191\pi\)
\(942\) −23932.2 −0.827762
\(943\) 2203.23 0.0760839
\(944\) −5647.20 −0.194704
\(945\) −4474.64 −0.154032
\(946\) 11040.9 0.379462
\(947\) 33466.2 1.14837 0.574184 0.818726i \(-0.305319\pi\)
0.574184 + 0.818726i \(0.305319\pi\)
\(948\) −9976.41 −0.341792
\(949\) 7308.85 0.250006
\(950\) 5532.67 0.188951
\(951\) −14148.5 −0.482437
\(952\) −13268.5 −0.451718
\(953\) −17298.4 −0.587984 −0.293992 0.955808i \(-0.594984\pi\)
−0.293992 + 0.955808i \(0.594984\pi\)
\(954\) −24836.2 −0.842874
\(955\) 12488.7 0.423167
\(956\) −11078.4 −0.374791
\(957\) −105.324 −0.00355763
\(958\) 34822.5 1.17439
\(959\) 85726.0 2.88659
\(960\) 2423.07 0.0814627
\(961\) −15430.4 −0.517955
\(962\) 58848.2 1.97229
\(963\) −49444.8 −1.65456
\(964\) −21336.4 −0.712862
\(965\) −4545.78 −0.151641
\(966\) 12338.4 0.410953
\(967\) −14869.7 −0.494495 −0.247248 0.968952i \(-0.579526\pi\)
−0.247248 + 0.968952i \(0.579526\pi\)
\(968\) 8435.59 0.280093
\(969\) −39231.0 −1.30060
\(970\) −2719.28 −0.0900113
\(971\) 26634.9 0.880283 0.440142 0.897928i \(-0.354928\pi\)
0.440142 + 0.897928i \(0.354928\pi\)
\(972\) 21743.0 0.717498
\(973\) −32731.8 −1.07845
\(974\) −2640.69 −0.0868717
\(975\) 15127.7 0.496898
\(976\) −8115.57 −0.266161
\(977\) 40803.6 1.33615 0.668077 0.744092i \(-0.267118\pi\)
0.668077 + 0.744092i \(0.267118\pi\)
\(978\) 6947.22 0.227145
\(979\) 17463.7 0.570115
\(980\) −18235.6 −0.594404
\(981\) −13109.9 −0.426675
\(982\) −34231.6 −1.11240
\(983\) −18615.8 −0.604021 −0.302011 0.953305i \(-0.597658\pi\)
−0.302011 + 0.953305i \(0.597658\pi\)
\(984\) −5802.80 −0.187995
\(985\) −3043.14 −0.0984389
\(986\) −78.3257 −0.00252982
\(987\) −143510. −4.62815
\(988\) 35370.6 1.13896
\(989\) −7635.11 −0.245483
\(990\) −5044.91 −0.161957
\(991\) 46185.1 1.48044 0.740220 0.672364i \(-0.234721\pi\)
0.740220 + 0.672364i \(0.234721\pi\)
\(992\) 3834.74 0.122735
\(993\) −52754.1 −1.68590
\(994\) −51984.5 −1.65880
\(995\) 11524.9 0.367200
\(996\) 22832.4 0.726377
\(997\) 16544.3 0.525541 0.262771 0.964858i \(-0.415364\pi\)
0.262771 + 0.964858i \(0.415364\pi\)
\(998\) −35081.2 −1.11270
\(999\) −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 230.4.a.h.1.1 4
3.2 odd 2 2070.4.a.bj.1.1 4
4.3 odd 2 1840.4.a.m.1.4 4
5.2 odd 4 1150.4.b.n.599.4 8
5.3 odd 4 1150.4.b.n.599.5 8
5.4 even 2 1150.4.a.p.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.1 4 1.1 even 1 trivial
1150.4.a.p.1.4 4 5.4 even 2
1150.4.b.n.599.4 8 5.2 odd 4
1150.4.b.n.599.5 8 5.3 odd 4
1840.4.a.m.1.4 4 4.3 odd 2
2070.4.a.bj.1.1 4 3.2 odd 2