Properties

Label 230.4.a.h.1.4
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 1.4
Root \(8.73081\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +7.73081 q^{3} +4.00000 q^{4} -5.00000 q^{5} -15.4616 q^{6} +23.5622 q^{7} -8.00000 q^{8} +32.7654 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +7.73081 q^{3} +4.00000 q^{4} -5.00000 q^{5} -15.4616 q^{6} +23.5622 q^{7} -8.00000 q^{8} +32.7654 q^{9} +10.0000 q^{10} -32.1352 q^{11} +30.9232 q^{12} +40.0811 q^{13} -47.1245 q^{14} -38.6540 q^{15} +16.0000 q^{16} +126.165 q^{17} -65.5308 q^{18} +0.232742 q^{19} -20.0000 q^{20} +182.155 q^{21} +64.2704 q^{22} -23.0000 q^{23} -61.8465 q^{24} +25.0000 q^{25} -80.1621 q^{26} +44.5711 q^{27} +94.2490 q^{28} -137.226 q^{29} +77.3081 q^{30} +112.866 q^{31} -32.0000 q^{32} -248.431 q^{33} -252.331 q^{34} -117.811 q^{35} +131.062 q^{36} +45.7057 q^{37} -0.465483 q^{38} +309.859 q^{39} +40.0000 q^{40} -135.385 q^{41} -364.310 q^{42} +543.528 q^{43} -128.541 q^{44} -163.827 q^{45} +46.0000 q^{46} +26.4344 q^{47} +123.693 q^{48} +212.180 q^{49} -50.0000 q^{50} +975.359 q^{51} +160.324 q^{52} +43.6958 q^{53} -89.1422 q^{54} +160.676 q^{55} -188.498 q^{56} +1.79928 q^{57} +274.451 q^{58} +202.248 q^{59} -154.616 q^{60} +150.279 q^{61} -225.732 q^{62} +772.026 q^{63} +64.0000 q^{64} -200.405 q^{65} +496.862 q^{66} -420.722 q^{67} +504.661 q^{68} -177.809 q^{69} +235.622 q^{70} +667.381 q^{71} -262.123 q^{72} +602.960 q^{73} -91.4115 q^{74} +193.270 q^{75} +0.930966 q^{76} -757.178 q^{77} -619.718 q^{78} -1378.88 q^{79} -80.0000 q^{80} -540.095 q^{81} +270.769 q^{82} -485.178 q^{83} +728.621 q^{84} -630.826 q^{85} -1087.06 q^{86} -1060.86 q^{87} +257.082 q^{88} -1127.71 q^{89} +327.654 q^{90} +944.400 q^{91} -92.0000 q^{92} +872.545 q^{93} -52.8688 q^{94} -1.16371 q^{95} -247.386 q^{96} -1486.24 q^{97} -424.359 q^{98} -1052.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} - 32q^{8} + 32q^{9} + O(q^{10}) \) \( 4q - 8q^{2} - 4q^{3} + 16q^{4} - 20q^{5} + 8q^{6} - q^{7} - 32q^{8} + 32q^{9} + 40q^{10} - 39q^{11} - 16q^{12} - 20q^{13} + 2q^{14} + 20q^{15} + 64q^{16} - 23q^{17} - 64q^{18} + 53q^{19} - 80q^{20} + 300q^{21} + 78q^{22} - 92q^{23} + 32q^{24} + 100q^{25} + 40q^{26} + 137q^{27} - 4q^{28} + 161q^{29} - 40q^{30} + 388q^{31} - 128q^{32} + 87q^{33} + 46q^{34} + 5q^{35} + 128q^{36} + 466q^{37} - 106q^{38} + 1047q^{39} + 160q^{40} + 484q^{41} - 600q^{42} + 894q^{43} - 156q^{44} - 160q^{45} + 184q^{46} - 265q^{47} - 64q^{48} + 1643q^{49} - 200q^{50} + 1825q^{51} - 80q^{52} + 576q^{53} - 274q^{54} + 195q^{55} + 8q^{56} + 178q^{57} - 322q^{58} - 94q^{59} + 80q^{60} + 1153q^{61} - 776q^{62} + 60q^{63} + 256q^{64} + 100q^{65} - 174q^{66} - 1472q^{67} - 92q^{68} + 92q^{69} - 10q^{70} + 200q^{71} - 256q^{72} + 1147q^{73} - 932q^{74} - 100q^{75} + 212q^{76} - 2176q^{77} - 2094q^{78} - 908q^{79} - 320q^{80} - 1056q^{81} - 968q^{82} - 1048q^{83} + 1200q^{84} + 115q^{85} - 1788q^{86} - 2167q^{87} + 312q^{88} - 1784q^{89} + 320q^{90} + 2329q^{91} - 368q^{92} + 1483q^{93} + 530q^{94} - 265q^{95} + 128q^{96} - 2047q^{97} - 3286q^{98} - 2665q^{99} + O(q^{100}) \)

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
\(3\) 7.73081 1.48779 0.743897 0.668294i \(-0.232975\pi\)
0.743897 + 0.668294i \(0.232975\pi\)
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) −15.4616 −1.05203
\(7\) 23.5622 1.27224 0.636121 0.771589i \(-0.280538\pi\)
0.636121 + 0.771589i \(0.280538\pi\)
\(8\) −8.00000 −0.353553
\(9\) 32.7654 1.21353
\(10\) 10.0000 0.316228
\(11\) −32.1352 −0.880830 −0.440415 0.897794i \(-0.645169\pi\)
−0.440415 + 0.897794i \(0.645169\pi\)
\(12\) 30.9232 0.743897
\(13\) 40.0811 0.855115 0.427557 0.903988i \(-0.359374\pi\)
0.427557 + 0.903988i \(0.359374\pi\)
\(14\) −47.1245 −0.899611
\(15\) −38.6540 −0.665362
\(16\) 16.0000 0.250000
\(17\) 126.165 1.79997 0.899987 0.435916i \(-0.143576\pi\)
0.899987 + 0.435916i \(0.143576\pi\)
\(18\) −65.5308 −0.858097
\(19\) 0.232742 0.00281024 0.00140512 0.999999i \(-0.499553\pi\)
0.00140512 + 0.999999i \(0.499553\pi\)
\(20\) −20.0000 −0.223607
\(21\) 182.155 1.89283
\(22\) 64.2704 0.622841
\(23\) −23.0000 −0.208514
\(24\) −61.8465 −0.526015
\(25\) 25.0000 0.200000
\(26\) −80.1621 −0.604657
\(27\) 44.5711 0.317693
\(28\) 94.2490 0.636121
\(29\) −137.226 −0.878695 −0.439347 0.898317i \(-0.644790\pi\)
−0.439347 + 0.898317i \(0.644790\pi\)
\(30\) 77.3081 0.470482
\(31\) 112.866 0.653914 0.326957 0.945039i \(-0.393977\pi\)
0.326957 + 0.945039i \(0.393977\pi\)
\(32\) −32.0000 −0.176777
\(33\) −248.431 −1.31049
\(34\) −252.331 −1.27277
\(35\) −117.811 −0.568964
\(36\) 131.062 0.606766
\(37\) 45.7057 0.203080 0.101540 0.994831i \(-0.467623\pi\)
0.101540 + 0.994831i \(0.467623\pi\)
\(38\) −0.465483 −0.00198714
\(39\) 309.859 1.27223
\(40\) 40.0000 0.158114
\(41\) −135.385 −0.515696 −0.257848 0.966186i \(-0.583013\pi\)
−0.257848 + 0.966186i \(0.583013\pi\)
\(42\) −364.310 −1.33844
\(43\) 543.528 1.92761 0.963805 0.266608i \(-0.0859030\pi\)
0.963805 + 0.266608i \(0.0859030\pi\)
\(44\) −128.541 −0.440415
\(45\) −163.827 −0.542708
\(46\) 46.0000 0.147442
\(47\) 26.4344 0.0820394 0.0410197 0.999158i \(-0.486939\pi\)
0.0410197 + 0.999158i \(0.486939\pi\)
\(48\) 123.693 0.371949
\(49\) 212.180 0.618599
\(50\) −50.0000 −0.141421
\(51\) 975.359 2.67799
\(52\) 160.324 0.427557
\(53\) 43.6958 0.113247 0.0566235 0.998396i \(-0.481967\pi\)
0.0566235 + 0.998396i \(0.481967\pi\)
\(54\) −89.1422 −0.224643
\(55\) 160.676 0.393919
\(56\) −188.498 −0.449805
\(57\) 1.79928 0.00418106
\(58\) 274.451 0.621331
\(59\) 202.248 0.446278 0.223139 0.974787i \(-0.428370\pi\)
0.223139 + 0.974787i \(0.428370\pi\)
\(60\) −154.616 −0.332681
\(61\) 150.279 0.315430 0.157715 0.987485i \(-0.449587\pi\)
0.157715 + 0.987485i \(0.449587\pi\)
\(62\) −225.732 −0.462387
\(63\) 772.026 1.54391
\(64\) 64.0000 0.125000
\(65\) −200.405 −0.382419
\(66\) 496.862 0.926659
\(67\) −420.722 −0.767155 −0.383577 0.923509i \(-0.625308\pi\)
−0.383577 + 0.923509i \(0.625308\pi\)
\(68\) 504.661 0.899987
\(69\) −177.809 −0.310227
\(70\) 235.622 0.402318
\(71\) 667.381 1.11554 0.557771 0.829995i \(-0.311657\pi\)
0.557771 + 0.829995i \(0.311657\pi\)
\(72\) −262.123 −0.429049
\(73\) 602.960 0.966728 0.483364 0.875420i \(-0.339415\pi\)
0.483364 + 0.875420i \(0.339415\pi\)
\(74\) −91.4115 −0.143600
\(75\) 193.270 0.297559
\(76\) 0.930966 0.00140512
\(77\) −757.178 −1.12063
\(78\) −619.718 −0.899606
\(79\) −1378.88 −1.96374 −0.981872 0.189545i \(-0.939299\pi\)
−0.981872 + 0.189545i \(0.939299\pi\)
\(80\) −80.0000 −0.111803
\(81\) −540.095 −0.740871
\(82\) 270.769 0.364652
\(83\) −485.178 −0.641629 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(84\) 728.621 0.946417
\(85\) −630.826 −0.804973
\(86\) −1087.06 −1.36303
\(87\) −1060.86 −1.30732
\(88\) 257.082 0.311420
\(89\) −1127.71 −1.34311 −0.671556 0.740954i \(-0.734374\pi\)
−0.671556 + 0.740954i \(0.734374\pi\)
\(90\) 327.654 0.383753
\(91\) 944.400 1.08791
\(92\) −92.0000 −0.104257
\(93\) 872.545 0.972890
\(94\) −52.8688 −0.0580106
\(95\) −1.16371 −0.00125678
\(96\) −247.386 −0.263007
\(97\) −1486.24 −1.55572 −0.777858 0.628441i \(-0.783694\pi\)
−0.777858 + 0.628441i \(0.783694\pi\)
\(98\) −424.359 −0.437416
\(99\) −1052.92 −1.06892
\(100\) 100.000 0.100000
\(101\) 1888.86 1.86087 0.930437 0.366451i \(-0.119427\pi\)
0.930437 + 0.366451i \(0.119427\pi\)
\(102\) −1950.72 −1.89363
\(103\) −1497.17 −1.43224 −0.716118 0.697979i \(-0.754083\pi\)
−0.716118 + 0.697979i \(0.754083\pi\)
\(104\) −320.649 −0.302329
\(105\) −910.776 −0.846501
\(106\) −87.3917 −0.0800777
\(107\) −585.150 −0.528678 −0.264339 0.964430i \(-0.585154\pi\)
−0.264339 + 0.964430i \(0.585154\pi\)
\(108\) 178.284 0.158846
\(109\) −139.166 −0.122290 −0.0611452 0.998129i \(-0.519475\pi\)
−0.0611452 + 0.998129i \(0.519475\pi\)
\(110\) −321.352 −0.278543
\(111\) 353.342 0.302142
\(112\) 376.996 0.318060
\(113\) −1293.18 −1.07656 −0.538282 0.842765i \(-0.680927\pi\)
−0.538282 + 0.842765i \(0.680927\pi\)
\(114\) −3.59856 −0.00295646
\(115\) 115.000 0.0932505
\(116\) −548.902 −0.439347
\(117\) 1313.27 1.03771
\(118\) −404.495 −0.315566
\(119\) 2972.74 2.29000
\(120\) 309.232 0.235241
\(121\) −298.328 −0.224138
\(122\) −300.557 −0.223043
\(123\) −1046.63 −0.767250
\(124\) 451.464 0.326957
\(125\) −125.000 −0.0894427
\(126\) −1544.05 −1.09171
\(127\) −1298.43 −0.907221 −0.453610 0.891200i \(-0.649864\pi\)
−0.453610 + 0.891200i \(0.649864\pi\)
\(128\) −128.000 −0.0883883
\(129\) 4201.91 2.86789
\(130\) 400.811 0.270411
\(131\) −525.247 −0.350313 −0.175157 0.984541i \(-0.556043\pi\)
−0.175157 + 0.984541i \(0.556043\pi\)
\(132\) −993.725 −0.655247
\(133\) 5.48392 0.00357531
\(134\) 841.444 0.542460
\(135\) −222.855 −0.142077
\(136\) −1009.32 −0.636387
\(137\) −2428.12 −1.51422 −0.757109 0.653288i \(-0.773389\pi\)
−0.757109 + 0.653288i \(0.773389\pi\)
\(138\) 355.617 0.219363
\(139\) 2456.46 1.49895 0.749476 0.662031i \(-0.230305\pi\)
0.749476 + 0.662031i \(0.230305\pi\)
\(140\) −471.245 −0.284482
\(141\) 204.359 0.122058
\(142\) −1334.76 −0.788808
\(143\) −1288.01 −0.753211
\(144\) 524.246 0.303383
\(145\) 686.128 0.392964
\(146\) −1205.92 −0.683580
\(147\) 1640.32 0.920349
\(148\) 182.823 0.101540
\(149\) −2405.39 −1.32253 −0.661266 0.750151i \(-0.729981\pi\)
−0.661266 + 0.750151i \(0.729981\pi\)
\(150\) −386.540 −0.210406
\(151\) −649.276 −0.349916 −0.174958 0.984576i \(-0.555979\pi\)
−0.174958 + 0.984576i \(0.555979\pi\)
\(152\) −1.86193 −0.000993570 0
\(153\) 4133.85 2.18433
\(154\) 1514.36 0.792404
\(155\) −564.330 −0.292439
\(156\) 1239.44 0.636117
\(157\) 3665.04 1.86307 0.931536 0.363650i \(-0.118470\pi\)
0.931536 + 0.363650i \(0.118470\pi\)
\(158\) 2757.75 1.38858
\(159\) 337.804 0.168488
\(160\) 160.000 0.0790569
\(161\) −541.932 −0.265281
\(162\) 1080.19 0.523875
\(163\) −1055.27 −0.507088 −0.253544 0.967324i \(-0.581596\pi\)
−0.253544 + 0.967324i \(0.581596\pi\)
\(164\) −541.539 −0.257848
\(165\) 1242.16 0.586071
\(166\) 970.356 0.453700
\(167\) −731.805 −0.339095 −0.169547 0.985522i \(-0.554231\pi\)
−0.169547 + 0.985522i \(0.554231\pi\)
\(168\) −1457.24 −0.669218
\(169\) −590.507 −0.268779
\(170\) 1261.65 0.569202
\(171\) 7.62587 0.00341032
\(172\) 2174.11 0.963805
\(173\) −521.773 −0.229304 −0.114652 0.993406i \(-0.536575\pi\)
−0.114652 + 0.993406i \(0.536575\pi\)
\(174\) 2121.73 0.924413
\(175\) 589.056 0.254448
\(176\) −514.163 −0.220208
\(177\) 1563.54 0.663970
\(178\) 2255.42 0.949723
\(179\) −1183.07 −0.494005 −0.247002 0.969015i \(-0.579446\pi\)
−0.247002 + 0.969015i \(0.579446\pi\)
\(180\) −655.308 −0.271354
\(181\) 1723.92 0.707946 0.353973 0.935256i \(-0.384830\pi\)
0.353973 + 0.935256i \(0.384830\pi\)
\(182\) −1888.80 −0.769270
\(183\) 1161.78 0.469295
\(184\) 184.000 0.0737210
\(185\) −228.529 −0.0908204
\(186\) −1745.09 −0.687937
\(187\) −4054.35 −1.58547
\(188\) 105.738 0.0410197
\(189\) 1050.20 0.404182
\(190\) 2.32742 0.000888676 0
\(191\) −2798.12 −1.06002 −0.530012 0.847990i \(-0.677813\pi\)
−0.530012 + 0.847990i \(0.677813\pi\)
\(192\) 494.772 0.185974
\(193\) −497.686 −0.185618 −0.0928089 0.995684i \(-0.529585\pi\)
−0.0928089 + 0.995684i \(0.529585\pi\)
\(194\) 2972.47 1.10006
\(195\) −1549.30 −0.568961
\(196\) 848.718 0.309300
\(197\) 296.489 0.107228 0.0536141 0.998562i \(-0.482926\pi\)
0.0536141 + 0.998562i \(0.482926\pi\)
\(198\) 2105.85 0.755838
\(199\) 2347.43 0.836207 0.418103 0.908399i \(-0.362695\pi\)
0.418103 + 0.908399i \(0.362695\pi\)
\(200\) −200.000 −0.0707107
\(201\) −3252.52 −1.14137
\(202\) −3777.71 −1.31584
\(203\) −3233.34 −1.11791
\(204\) 3901.44 1.33900
\(205\) 676.923 0.230626
\(206\) 2994.34 1.01274
\(207\) −753.604 −0.253039
\(208\) 641.297 0.213779
\(209\) −7.47920 −0.00247535
\(210\) 1821.55 0.598567
\(211\) 2838.61 0.926153 0.463076 0.886318i \(-0.346746\pi\)
0.463076 + 0.886318i \(0.346746\pi\)
\(212\) 174.783 0.0566235
\(213\) 5159.39 1.65970
\(214\) 1170.30 0.373832
\(215\) −2717.64 −0.862053
\(216\) −356.569 −0.112321
\(217\) 2659.38 0.831937
\(218\) 278.331 0.0864723
\(219\) 4661.37 1.43829
\(220\) 642.704 0.196960
\(221\) 5056.84 1.53918
\(222\) −706.685 −0.213647
\(223\) −2124.68 −0.638024 −0.319012 0.947751i \(-0.603351\pi\)
−0.319012 + 0.947751i \(0.603351\pi\)
\(224\) −753.992 −0.224903
\(225\) 819.135 0.242707
\(226\) 2586.35 0.761246
\(227\) 1551.97 0.453780 0.226890 0.973920i \(-0.427144\pi\)
0.226890 + 0.973920i \(0.427144\pi\)
\(228\) 7.19712 0.00209053
\(229\) −158.531 −0.0457469 −0.0228735 0.999738i \(-0.507281\pi\)
−0.0228735 + 0.999738i \(0.507281\pi\)
\(230\) −230.000 −0.0659380
\(231\) −5853.60 −1.66727
\(232\) 1097.80 0.310666
\(233\) −728.999 −0.204971 −0.102486 0.994734i \(-0.532680\pi\)
−0.102486 + 0.994734i \(0.532680\pi\)
\(234\) −2626.54 −0.733772
\(235\) −132.172 −0.0366891
\(236\) 808.991 0.223139
\(237\) −10659.8 −2.92165
\(238\) −5945.47 −1.61928
\(239\) −6201.60 −1.67844 −0.839222 0.543789i \(-0.816989\pi\)
−0.839222 + 0.543789i \(0.816989\pi\)
\(240\) −618.465 −0.166340
\(241\) −7223.04 −1.93061 −0.965305 0.261126i \(-0.915906\pi\)
−0.965305 + 0.261126i \(0.915906\pi\)
\(242\) 596.656 0.158490
\(243\) −5378.79 −1.41996
\(244\) 601.115 0.157715
\(245\) −1060.90 −0.276646
\(246\) 2093.27 0.542527
\(247\) 9.32853 0.00240308
\(248\) −902.928 −0.231193
\(249\) −3750.82 −0.954612
\(250\) 250.000 0.0632456
\(251\) 5042.78 1.26812 0.634059 0.773285i \(-0.281388\pi\)
0.634059 + 0.773285i \(0.281388\pi\)
\(252\) 3088.10 0.771954
\(253\) 739.110 0.183666
\(254\) 2596.86 0.641502
\(255\) −4876.80 −1.19763
\(256\) 256.000 0.0625000
\(257\) 2981.15 0.723577 0.361788 0.932260i \(-0.382166\pi\)
0.361788 + 0.932260i \(0.382166\pi\)
\(258\) −8403.82 −2.02790
\(259\) 1076.93 0.258367
\(260\) −801.621 −0.191209
\(261\) −4496.25 −1.06632
\(262\) 1050.49 0.247709
\(263\) 7242.86 1.69815 0.849076 0.528271i \(-0.177159\pi\)
0.849076 + 0.528271i \(0.177159\pi\)
\(264\) 1987.45 0.463330
\(265\) −218.479 −0.0506456
\(266\) −10.9678 −0.00252812
\(267\) −8718.10 −1.99827
\(268\) −1682.89 −0.383577
\(269\) −2567.58 −0.581962 −0.290981 0.956729i \(-0.593982\pi\)
−0.290981 + 0.956729i \(0.593982\pi\)
\(270\) 445.711 0.100463
\(271\) 8066.27 1.80808 0.904042 0.427443i \(-0.140586\pi\)
0.904042 + 0.427443i \(0.140586\pi\)
\(272\) 2018.64 0.449994
\(273\) 7300.98 1.61859
\(274\) 4856.23 1.07071
\(275\) −803.380 −0.176166
\(276\) −711.234 −0.155113
\(277\) 8991.54 1.95036 0.975179 0.221418i \(-0.0710684\pi\)
0.975179 + 0.221418i \(0.0710684\pi\)
\(278\) −4912.92 −1.05992
\(279\) 3698.10 0.793546
\(280\) 942.490 0.201159
\(281\) 968.130 0.205529 0.102765 0.994706i \(-0.467231\pi\)
0.102765 + 0.994706i \(0.467231\pi\)
\(282\) −408.718 −0.0863079
\(283\) −2252.65 −0.473167 −0.236583 0.971611i \(-0.576028\pi\)
−0.236583 + 0.971611i \(0.576028\pi\)
\(284\) 2669.52 0.557771
\(285\) −8.99640 −0.00186983
\(286\) 2576.03 0.532600
\(287\) −3189.97 −0.656090
\(288\) −1048.49 −0.214524
\(289\) 11004.7 2.23991
\(290\) −1372.26 −0.277868
\(291\) −11489.8 −2.31458
\(292\) 2411.84 0.483364
\(293\) 2734.35 0.545196 0.272598 0.962128i \(-0.412117\pi\)
0.272598 + 0.962128i \(0.412117\pi\)
\(294\) −3280.64 −0.650785
\(295\) −1011.24 −0.199582
\(296\) −365.646 −0.0717998
\(297\) −1432.30 −0.279834
\(298\) 4810.78 0.935172
\(299\) −921.865 −0.178304
\(300\) 773.081 0.148779
\(301\) 12806.7 2.45239
\(302\) 1298.55 0.247428
\(303\) 14602.4 2.76860
\(304\) 3.72387 0.000702560 0
\(305\) −751.394 −0.141065
\(306\) −8267.71 −1.54455
\(307\) −7977.09 −1.48299 −0.741493 0.670961i \(-0.765882\pi\)
−0.741493 + 0.670961i \(0.765882\pi\)
\(308\) −3028.71 −0.560314
\(309\) −11574.3 −2.13087
\(310\) 1128.66 0.206786
\(311\) 7729.43 1.40931 0.704655 0.709550i \(-0.251102\pi\)
0.704655 + 0.709550i \(0.251102\pi\)
\(312\) −2478.87 −0.449803
\(313\) 5506.25 0.994350 0.497175 0.867650i \(-0.334371\pi\)
0.497175 + 0.867650i \(0.334371\pi\)
\(314\) −7330.08 −1.31739
\(315\) −3860.13 −0.690456
\(316\) −5515.51 −0.981872
\(317\) 5231.37 0.926887 0.463443 0.886126i \(-0.346614\pi\)
0.463443 + 0.886126i \(0.346614\pi\)
\(318\) −675.608 −0.119139
\(319\) 4409.77 0.773981
\(320\) −320.000 −0.0559017
\(321\) −4523.68 −0.786565
\(322\) 1083.86 0.187582
\(323\) 29.3639 0.00505836
\(324\) −2160.38 −0.370435
\(325\) 1002.03 0.171023
\(326\) 2110.55 0.358566
\(327\) −1075.86 −0.181943
\(328\) 1083.08 0.182326
\(329\) 622.854 0.104374
\(330\) −2484.31 −0.414415
\(331\) −3551.61 −0.589771 −0.294885 0.955533i \(-0.595281\pi\)
−0.294885 + 0.955533i \(0.595281\pi\)
\(332\) −1940.71 −0.320815
\(333\) 1497.57 0.246445
\(334\) 1463.61 0.239776
\(335\) 2103.61 0.343082
\(336\) 2914.48 0.473209
\(337\) −7002.99 −1.13198 −0.565990 0.824412i \(-0.691506\pi\)
−0.565990 + 0.824412i \(0.691506\pi\)
\(338\) 1181.01 0.190055
\(339\) −9997.29 −1.60171
\(340\) −2523.31 −0.402487
\(341\) −3626.97 −0.575987
\(342\) −15.2517 −0.00241146
\(343\) −3082.42 −0.485234
\(344\) −4348.22 −0.681513
\(345\) 889.043 0.138738
\(346\) 1043.55 0.162143
\(347\) −10268.2 −1.58854 −0.794272 0.607562i \(-0.792148\pi\)
−0.794272 + 0.607562i \(0.792148\pi\)
\(348\) −4243.46 −0.653659
\(349\) 4515.58 0.692588 0.346294 0.938126i \(-0.387440\pi\)
0.346294 + 0.938126i \(0.387440\pi\)
\(350\) −1178.11 −0.179922
\(351\) 1786.46 0.271664
\(352\) 1028.33 0.155710
\(353\) 7838.63 1.18189 0.590947 0.806711i \(-0.298754\pi\)
0.590947 + 0.806711i \(0.298754\pi\)
\(354\) −3127.08 −0.469498
\(355\) −3336.90 −0.498886
\(356\) −4510.84 −0.671556
\(357\) 22981.7 3.40705
\(358\) 2366.14 0.349314
\(359\) −12464.8 −1.83250 −0.916250 0.400606i \(-0.868799\pi\)
−0.916250 + 0.400606i \(0.868799\pi\)
\(360\) 1310.62 0.191876
\(361\) −6858.95 −0.999992
\(362\) −3447.85 −0.500594
\(363\) −2306.32 −0.333472
\(364\) 3777.60 0.543956
\(365\) −3014.80 −0.432334
\(366\) −2323.55 −0.331842
\(367\) 3936.14 0.559850 0.279925 0.960022i \(-0.409690\pi\)
0.279925 + 0.960022i \(0.409690\pi\)
\(368\) −368.000 −0.0521286
\(369\) −4435.93 −0.625814
\(370\) 457.057 0.0642197
\(371\) 1029.57 0.144077
\(372\) 3490.18 0.486445
\(373\) 1920.72 0.266625 0.133313 0.991074i \(-0.457439\pi\)
0.133313 + 0.991074i \(0.457439\pi\)
\(374\) 8108.69 1.12110
\(375\) −966.351 −0.133072
\(376\) −211.475 −0.0290053
\(377\) −5500.15 −0.751385
\(378\) −2100.39 −0.285800
\(379\) −13074.5 −1.77201 −0.886004 0.463678i \(-0.846530\pi\)
−0.886004 + 0.463678i \(0.846530\pi\)
\(380\) −4.65483 −0.000628389 0
\(381\) −10037.9 −1.34976
\(382\) 5596.23 0.749550
\(383\) 8838.22 1.17914 0.589571 0.807716i \(-0.299297\pi\)
0.589571 + 0.807716i \(0.299297\pi\)
\(384\) −989.543 −0.131504
\(385\) 3785.89 0.501160
\(386\) 995.372 0.131252
\(387\) 17808.9 2.33922
\(388\) −5944.94 −0.777858
\(389\) −12687.3 −1.65365 −0.826827 0.562456i \(-0.809856\pi\)
−0.826827 + 0.562456i \(0.809856\pi\)
\(390\) 3098.59 0.402316
\(391\) −2901.80 −0.375321
\(392\) −1697.44 −0.218708
\(393\) −4060.58 −0.521194
\(394\) −592.977 −0.0758218
\(395\) 6894.38 0.878213
\(396\) −4211.69 −0.534458
\(397\) 8060.49 1.01900 0.509501 0.860470i \(-0.329830\pi\)
0.509501 + 0.860470i \(0.329830\pi\)
\(398\) −4694.87 −0.591287
\(399\) 42.3951 0.00531932
\(400\) 400.000 0.0500000
\(401\) 12325.1 1.53488 0.767440 0.641121i \(-0.221530\pi\)
0.767440 + 0.641121i \(0.221530\pi\)
\(402\) 6505.04 0.807070
\(403\) 4523.79 0.559171
\(404\) 7555.43 0.930437
\(405\) 2700.47 0.331328
\(406\) 6466.69 0.790483
\(407\) −1468.76 −0.178879
\(408\) −7802.87 −0.946813
\(409\) −10806.7 −1.30650 −0.653250 0.757143i \(-0.726595\pi\)
−0.653250 + 0.757143i \(0.726595\pi\)
\(410\) −1353.85 −0.163077
\(411\) −18771.3 −2.25285
\(412\) −5988.67 −0.716118
\(413\) 4765.41 0.567774
\(414\) 1507.21 0.178926
\(415\) 2425.89 0.286945
\(416\) −1282.59 −0.151164
\(417\) 18990.4 2.23013
\(418\) 14.9584 0.00175033
\(419\) −1823.74 −0.212639 −0.106319 0.994332i \(-0.533907\pi\)
−0.106319 + 0.994332i \(0.533907\pi\)
\(420\) −3643.10 −0.423251
\(421\) 5995.43 0.694060 0.347030 0.937854i \(-0.387190\pi\)
0.347030 + 0.937854i \(0.387190\pi\)
\(422\) −5677.23 −0.654889
\(423\) 866.133 0.0995575
\(424\) −349.567 −0.0400388
\(425\) 3154.13 0.359995
\(426\) −10318.8 −1.17358
\(427\) 3540.91 0.401303
\(428\) −2340.60 −0.264339
\(429\) −9957.39 −1.12062
\(430\) 5435.28 0.609564
\(431\) 3887.80 0.434499 0.217249 0.976116i \(-0.430292\pi\)
0.217249 + 0.976116i \(0.430292\pi\)
\(432\) 713.137 0.0794232
\(433\) −4422.93 −0.490884 −0.245442 0.969411i \(-0.578933\pi\)
−0.245442 + 0.969411i \(0.578933\pi\)
\(434\) −5318.75 −0.588268
\(435\) 5304.32 0.584650
\(436\) −556.662 −0.0611452
\(437\) −5.35306 −0.000585976 0
\(438\) −9322.74 −1.01703
\(439\) −1748.08 −0.190048 −0.0950242 0.995475i \(-0.530293\pi\)
−0.0950242 + 0.995475i \(0.530293\pi\)
\(440\) −1285.41 −0.139271
\(441\) 6952.14 0.750690
\(442\) −10113.7 −1.08837
\(443\) 3371.31 0.361571 0.180785 0.983523i \(-0.442136\pi\)
0.180785 + 0.983523i \(0.442136\pi\)
\(444\) 1413.37 0.151071
\(445\) 5638.55 0.600658
\(446\) 4249.37 0.451151
\(447\) −18595.6 −1.96766
\(448\) 1507.98 0.159030
\(449\) 13314.7 1.39947 0.699734 0.714404i \(-0.253302\pi\)
0.699734 + 0.714404i \(0.253302\pi\)
\(450\) −1638.27 −0.171619
\(451\) 4350.61 0.454240
\(452\) −5172.70 −0.538282
\(453\) −5019.43 −0.520603
\(454\) −3103.94 −0.320871
\(455\) −4722.00 −0.486529
\(456\) −14.3942 −0.00147823
\(457\) 3767.20 0.385607 0.192803 0.981237i \(-0.438242\pi\)
0.192803 + 0.981237i \(0.438242\pi\)
\(458\) 317.063 0.0323480
\(459\) 5623.32 0.571839
\(460\) 460.000 0.0466252
\(461\) 9674.06 0.977366 0.488683 0.872461i \(-0.337477\pi\)
0.488683 + 0.872461i \(0.337477\pi\)
\(462\) 11707.2 1.17893
\(463\) 2977.73 0.298892 0.149446 0.988770i \(-0.452251\pi\)
0.149446 + 0.988770i \(0.452251\pi\)
\(464\) −2195.61 −0.219674
\(465\) −4362.73 −0.435089
\(466\) 1458.00 0.144937
\(467\) −10701.0 −1.06035 −0.530176 0.847888i \(-0.677874\pi\)
−0.530176 + 0.847888i \(0.677874\pi\)
\(468\) 5253.09 0.518855
\(469\) −9913.16 −0.976006
\(470\) 264.344 0.0259431
\(471\) 28333.7 2.77187
\(472\) −1617.98 −0.157783
\(473\) −17466.4 −1.69790
\(474\) 21319.7 2.06592
\(475\) 5.81854 0.000562048 0
\(476\) 11890.9 1.14500
\(477\) 1431.71 0.137429
\(478\) 12403.2 1.18684
\(479\) 2337.15 0.222937 0.111469 0.993768i \(-0.464445\pi\)
0.111469 + 0.993768i \(0.464445\pi\)
\(480\) 1236.93 0.117620
\(481\) 1831.94 0.173657
\(482\) 14446.1 1.36515
\(483\) −4189.57 −0.394683
\(484\) −1193.31 −0.112069
\(485\) 7431.18 0.695737
\(486\) 10757.6 1.00406
\(487\) −7183.85 −0.668442 −0.334221 0.942495i \(-0.608473\pi\)
−0.334221 + 0.942495i \(0.608473\pi\)
\(488\) −1202.23 −0.111521
\(489\) −8158.12 −0.754443
\(490\) 2121.80 0.195618
\(491\) −12084.2 −1.11070 −0.555350 0.831617i \(-0.687416\pi\)
−0.555350 + 0.831617i \(0.687416\pi\)
\(492\) −4186.53 −0.383625
\(493\) −17313.1 −1.58163
\(494\) −18.6571 −0.00169923
\(495\) 5264.61 0.478034
\(496\) 1805.86 0.163478
\(497\) 15725.0 1.41924
\(498\) 7501.64 0.675013
\(499\) 7145.78 0.641060 0.320530 0.947238i \(-0.396139\pi\)
0.320530 + 0.947238i \(0.396139\pi\)
\(500\) −500.000 −0.0447214
\(501\) −5657.45 −0.504503
\(502\) −10085.6 −0.896695
\(503\) 20436.2 1.81154 0.905770 0.423771i \(-0.139294\pi\)
0.905770 + 0.423771i \(0.139294\pi\)
\(504\) −6176.21 −0.545854
\(505\) −9444.29 −0.832208
\(506\) −1478.22 −0.129871
\(507\) −4565.10 −0.399888
\(508\) −5193.72 −0.453610
\(509\) 19721.7 1.71738 0.858690 0.512495i \(-0.171279\pi\)
0.858690 + 0.512495i \(0.171279\pi\)
\(510\) 9753.59 0.846856
\(511\) 14207.1 1.22991
\(512\) −512.000 −0.0441942
\(513\) 10.3735 0.000892794 0
\(514\) −5962.30 −0.511646
\(515\) 7485.84 0.640516
\(516\) 16807.6 1.43394
\(517\) −849.475 −0.0722628
\(518\) −2153.86 −0.182693
\(519\) −4033.73 −0.341158
\(520\) 1603.24 0.135205
\(521\) 3483.23 0.292904 0.146452 0.989218i \(-0.453215\pi\)
0.146452 + 0.989218i \(0.453215\pi\)
\(522\) 8992.50 0.754006
\(523\) 15689.0 1.31172 0.655862 0.754881i \(-0.272305\pi\)
0.655862 + 0.754881i \(0.272305\pi\)
\(524\) −2100.99 −0.175157
\(525\) 4553.88 0.378567
\(526\) −14485.7 −1.20077
\(527\) 14239.8 1.17703
\(528\) −3974.90 −0.327624
\(529\) 529.000 0.0434783
\(530\) 436.958 0.0358118
\(531\) 6626.72 0.541573
\(532\) 21.9357 0.00178765
\(533\) −5426.36 −0.440979
\(534\) 17436.2 1.41299
\(535\) 2925.75 0.236432
\(536\) 3365.78 0.271230
\(537\) −9146.09 −0.734977
\(538\) 5135.15 0.411510
\(539\) −6818.43 −0.544881
\(540\) −891.422 −0.0710383
\(541\) 7620.73 0.605621 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(542\) −16132.5 −1.27851
\(543\) 13327.3 1.05328
\(544\) −4037.29 −0.318194
\(545\) 695.828 0.0546899
\(546\) −14602.0 −1.14452
\(547\) 5925.62 0.463183 0.231592 0.972813i \(-0.425607\pi\)
0.231592 + 0.972813i \(0.425607\pi\)
\(548\) −9712.47 −0.757109
\(549\) 4923.94 0.382784
\(550\) 1606.76 0.124568
\(551\) −31.9381 −0.00246934
\(552\) 1422.47 0.109682
\(553\) −32489.4 −2.49836
\(554\) −17983.1 −1.37911
\(555\) −1766.71 −0.135122
\(556\) 9825.85 0.749476
\(557\) −5456.16 −0.415054 −0.207527 0.978229i \(-0.566541\pi\)
−0.207527 + 0.978229i \(0.566541\pi\)
\(558\) −7396.20 −0.561122
\(559\) 21785.2 1.64833
\(560\) −1884.98 −0.142241
\(561\) −31343.4 −2.35886
\(562\) −1936.26 −0.145331
\(563\) −9194.29 −0.688265 −0.344132 0.938921i \(-0.611827\pi\)
−0.344132 + 0.938921i \(0.611827\pi\)
\(564\) 817.437 0.0610289
\(565\) 6465.88 0.481454
\(566\) 4505.30 0.334580
\(567\) −12725.8 −0.942567
\(568\) −5339.05 −0.394404
\(569\) 338.831 0.0249640 0.0124820 0.999922i \(-0.496027\pi\)
0.0124820 + 0.999922i \(0.496027\pi\)
\(570\) 17.9928 0.00132217
\(571\) −1725.34 −0.126451 −0.0632254 0.997999i \(-0.520139\pi\)
−0.0632254 + 0.997999i \(0.520139\pi\)
\(572\) −5152.06 −0.376605
\(573\) −21631.7 −1.57710
\(574\) 6379.93 0.463926
\(575\) −575.000 −0.0417029
\(576\) 2096.98 0.151692
\(577\) 23300.0 1.68109 0.840547 0.541738i \(-0.182234\pi\)
0.840547 + 0.541738i \(0.182234\pi\)
\(578\) −22009.3 −1.58385
\(579\) −3847.52 −0.276161
\(580\) 2744.51 0.196482
\(581\) −11431.9 −0.816307
\(582\) 22979.6 1.63666
\(583\) −1404.18 −0.0997513
\(584\) −4823.68 −0.341790
\(585\) −6566.36 −0.464078
\(586\) −5468.70 −0.385512
\(587\) 15653.3 1.10065 0.550325 0.834950i \(-0.314504\pi\)
0.550325 + 0.834950i \(0.314504\pi\)
\(588\) 6561.28 0.460174
\(589\) 26.2686 0.00183766
\(590\) 2022.48 0.141126
\(591\) 2292.10 0.159533
\(592\) 731.292 0.0507701
\(593\) −2658.08 −0.184072 −0.0920358 0.995756i \(-0.529337\pi\)
−0.0920358 + 0.995756i \(0.529337\pi\)
\(594\) 2864.60 0.197872
\(595\) −14863.7 −1.02412
\(596\) −9621.57 −0.661266
\(597\) 18147.6 1.24410
\(598\) 1843.73 0.126080
\(599\) 19417.6 1.32451 0.662256 0.749278i \(-0.269599\pi\)
0.662256 + 0.749278i \(0.269599\pi\)
\(600\) −1546.16 −0.105203
\(601\) −18469.0 −1.25352 −0.626760 0.779213i \(-0.715619\pi\)
−0.626760 + 0.779213i \(0.715619\pi\)
\(602\) −25613.5 −1.73410
\(603\) −13785.1 −0.930968
\(604\) −2597.10 −0.174958
\(605\) 1491.64 0.100238
\(606\) −29204.8 −1.95770
\(607\) 3968.56 0.265369 0.132684 0.991158i \(-0.457640\pi\)
0.132684 + 0.991158i \(0.457640\pi\)
\(608\) −7.44773 −0.000496785 0
\(609\) −24996.4 −1.66322
\(610\) 1502.79 0.0997477
\(611\) 1059.52 0.0701531
\(612\) 16535.4 1.09216
\(613\) −11478.7 −0.756311 −0.378156 0.925742i \(-0.623442\pi\)
−0.378156 + 0.925742i \(0.623442\pi\)
\(614\) 15954.2 1.04863
\(615\) 5233.16 0.343124
\(616\) 6057.42 0.396202
\(617\) −15691.1 −1.02382 −0.511911 0.859039i \(-0.671062\pi\)
−0.511911 + 0.859039i \(0.671062\pi\)
\(618\) 23148.6 1.50676
\(619\) −18249.4 −1.18499 −0.592494 0.805575i \(-0.701856\pi\)
−0.592494 + 0.805575i \(0.701856\pi\)
\(620\) −2257.32 −0.146220
\(621\) −1025.14 −0.0662436
\(622\) −15458.9 −0.996533
\(623\) −26571.4 −1.70876
\(624\) 4957.75 0.318059
\(625\) 625.000 0.0400000
\(626\) −11012.5 −0.703111
\(627\) −57.8203 −0.00368281
\(628\) 14660.2 0.931536
\(629\) 5766.48 0.365540
\(630\) 7720.26 0.488226
\(631\) −23528.8 −1.48442 −0.742208 0.670169i \(-0.766222\pi\)
−0.742208 + 0.670169i \(0.766222\pi\)
\(632\) 11031.0 0.694288
\(633\) 21944.8 1.37793
\(634\) −10462.7 −0.655408
\(635\) 6492.15 0.405722
\(636\) 1351.22 0.0842441
\(637\) 8504.38 0.528973
\(638\) −8819.55 −0.547287
\(639\) 21867.0 1.35375
\(640\) 640.000 0.0395285
\(641\) 3748.79 0.230996 0.115498 0.993308i \(-0.463154\pi\)
0.115498 + 0.993308i \(0.463154\pi\)
\(642\) 9047.36 0.556185
\(643\) 15924.3 0.976660 0.488330 0.872659i \(-0.337606\pi\)
0.488330 + 0.872659i \(0.337606\pi\)
\(644\) −2167.73 −0.132640
\(645\) −21009.5 −1.28256
\(646\) −58.7278 −0.00357680
\(647\) −2854.11 −0.173426 −0.0867129 0.996233i \(-0.527636\pi\)
−0.0867129 + 0.996233i \(0.527636\pi\)
\(648\) 4320.76 0.261937
\(649\) −6499.27 −0.393095
\(650\) −2004.05 −0.120931
\(651\) 20559.1 1.23775
\(652\) −4221.09 −0.253544
\(653\) 7925.97 0.474988 0.237494 0.971389i \(-0.423674\pi\)
0.237494 + 0.971389i \(0.423674\pi\)
\(654\) 2151.72 0.128653
\(655\) 2626.24 0.156665
\(656\) −2166.15 −0.128924
\(657\) 19756.2 1.17316
\(658\) −1245.71 −0.0738035
\(659\) −5799.43 −0.342813 −0.171406 0.985200i \(-0.554831\pi\)
−0.171406 + 0.985200i \(0.554831\pi\)
\(660\) 4968.62 0.293035
\(661\) 22324.8 1.31367 0.656833 0.754036i \(-0.271896\pi\)
0.656833 + 0.754036i \(0.271896\pi\)
\(662\) 7103.22 0.417031
\(663\) 39093.4 2.28999
\(664\) 3881.42 0.226850
\(665\) −27.4196 −0.00159893
\(666\) −2995.13 −0.174263
\(667\) 3156.19 0.183221
\(668\) −2927.22 −0.169547
\(669\) −16425.5 −0.949249
\(670\) −4207.22 −0.242596
\(671\) −4829.24 −0.277840
\(672\) −5828.97 −0.334609
\(673\) −23253.5 −1.33188 −0.665940 0.746005i \(-0.731969\pi\)
−0.665940 + 0.746005i \(0.731969\pi\)
\(674\) 14006.0 0.800430
\(675\) 1114.28 0.0635386
\(676\) −2362.03 −0.134389
\(677\) −19003.7 −1.07884 −0.539419 0.842038i \(-0.681356\pi\)
−0.539419 + 0.842038i \(0.681356\pi\)
\(678\) 19994.6 1.13258
\(679\) −35019.1 −1.97925
\(680\) 5046.61 0.284601
\(681\) 11998.0 0.675131
\(682\) 7253.94 0.407284
\(683\) −11222.5 −0.628722 −0.314361 0.949304i \(-0.601790\pi\)
−0.314361 + 0.949304i \(0.601790\pi\)
\(684\) 30.5035 0.00170516
\(685\) 12140.6 0.677179
\(686\) 6164.85 0.343112
\(687\) −1225.58 −0.0680620
\(688\) 8696.45 0.481902
\(689\) 1751.38 0.0968391
\(690\) −1778.09 −0.0981023
\(691\) 4297.68 0.236601 0.118301 0.992978i \(-0.462255\pi\)
0.118301 + 0.992978i \(0.462255\pi\)
\(692\) −2087.09 −0.114652
\(693\) −24809.2 −1.35992
\(694\) 20536.4 1.12327
\(695\) −12282.3 −0.670352
\(696\) 8486.92 0.462206
\(697\) −17080.8 −0.928240
\(698\) −9031.15 −0.489734
\(699\) −5635.75 −0.304955
\(700\) 2356.22 0.127224
\(701\) 25769.6 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(702\) −3572.91 −0.192095
\(703\) 10.6376 0.000570705 0
\(704\) −2056.65 −0.110104
\(705\) −1021.80 −0.0545859
\(706\) −15677.3 −0.835725
\(707\) 44505.7 2.36748
\(708\) 6254.15 0.331985
\(709\) −4900.01 −0.259554 −0.129777 0.991543i \(-0.541426\pi\)
−0.129777 + 0.991543i \(0.541426\pi\)
\(710\) 6673.81 0.352766
\(711\) −45179.4 −2.38307
\(712\) 9021.67 0.474862
\(713\) −2595.92 −0.136350
\(714\) −45963.3 −2.40915
\(715\) 6440.07 0.336846
\(716\) −4732.28 −0.247002
\(717\) −47943.4 −2.49718
\(718\) 24929.6 1.29577
\(719\) 7631.85 0.395855 0.197928 0.980217i \(-0.436579\pi\)
0.197928 + 0.980217i \(0.436579\pi\)
\(720\) −2621.23 −0.135677
\(721\) −35276.6 −1.82215
\(722\) 13717.9 0.707101
\(723\) −55839.9 −2.87235
\(724\) 6895.70 0.353973
\(725\) −3430.64 −0.175739
\(726\) 4612.64 0.235800
\(727\) −36985.6 −1.88682 −0.943410 0.331628i \(-0.892402\pi\)
−0.943410 + 0.331628i \(0.892402\pi\)
\(728\) −7555.20 −0.384635
\(729\) −26999.8 −1.37173
\(730\) 6029.60 0.305706
\(731\) 68574.3 3.46965
\(732\) 4647.10 0.234647
\(733\) −22227.3 −1.12003 −0.560015 0.828482i \(-0.689205\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(734\) −7872.29 −0.395874
\(735\) −8201.60 −0.411592
\(736\) 736.000 0.0368605
\(737\) 13520.0 0.675733
\(738\) 8871.86 0.442517
\(739\) −17899.1 −0.890974 −0.445487 0.895288i \(-0.646970\pi\)
−0.445487 + 0.895288i \(0.646970\pi\)
\(740\) −914.115 −0.0454102
\(741\) 72.1171 0.00357529
\(742\) −2059.14 −0.101878
\(743\) −29843.4 −1.47355 −0.736776 0.676137i \(-0.763653\pi\)
−0.736776 + 0.676137i \(0.763653\pi\)
\(744\) −6980.36 −0.343968
\(745\) 12027.0 0.591455
\(746\) −3841.44 −0.188532
\(747\) −15897.0 −0.778638
\(748\) −16217.4 −0.792736
\(749\) −13787.4 −0.672606
\(750\) 1932.70 0.0940964
\(751\) −9399.65 −0.456722 −0.228361 0.973577i \(-0.573337\pi\)
−0.228361 + 0.973577i \(0.573337\pi\)
\(752\) 422.950 0.0205098
\(753\) 38984.8 1.88670
\(754\) 11000.3 0.531309
\(755\) 3246.38 0.156487
\(756\) 4200.78 0.202091
\(757\) 29871.6 1.43422 0.717109 0.696961i \(-0.245465\pi\)
0.717109 + 0.696961i \(0.245465\pi\)
\(758\) 26149.0 1.25300
\(759\) 5713.92 0.273257
\(760\) 9.30966 0.000444338 0
\(761\) −273.955 −0.0130497 −0.00652487 0.999979i \(-0.502077\pi\)
−0.00652487 + 0.999979i \(0.502077\pi\)
\(762\) 20075.8 0.954423
\(763\) −3279.05 −0.155583
\(764\) −11192.5 −0.530012
\(765\) −20669.3 −0.976861
\(766\) −17676.4 −0.833780
\(767\) 8106.31 0.381619
\(768\) 1979.09 0.0929872
\(769\) −13273.8 −0.622450 −0.311225 0.950336i \(-0.600739\pi\)
−0.311225 + 0.950336i \(0.600739\pi\)
\(770\) −7571.78 −0.354374
\(771\) 23046.7 1.07653
\(772\) −1990.74 −0.0928089
\(773\) 34161.1 1.58951 0.794753 0.606933i \(-0.207600\pi\)
0.794753 + 0.606933i \(0.207600\pi\)
\(774\) −35617.8 −1.65408
\(775\) 2821.65 0.130783
\(776\) 11889.9 0.550028
\(777\) 8325.54 0.384398
\(778\) 25374.6 1.16931
\(779\) −31.5096 −0.00144923
\(780\) −6197.18 −0.284480
\(781\) −21446.4 −0.982603
\(782\) 5803.60 0.265392
\(783\) −6116.29 −0.279155
\(784\) 3394.87 0.154650
\(785\) −18325.2 −0.833191
\(786\) 8121.17 0.368540
\(787\) −38489.6 −1.74334 −0.871669 0.490095i \(-0.836962\pi\)
−0.871669 + 0.490095i \(0.836962\pi\)
\(788\) 1185.95 0.0536141
\(789\) 55993.2 2.52650
\(790\) −13788.8 −0.620990
\(791\) −30470.1 −1.36965
\(792\) 8423.38 0.377919
\(793\) 6023.33 0.269729
\(794\) −16121.0 −0.720544
\(795\) −1689.02 −0.0753502
\(796\) 9389.73 0.418103
\(797\) 17176.8 0.763405 0.381702 0.924285i \(-0.375338\pi\)
0.381702 + 0.924285i \(0.375338\pi\)
\(798\) −84.7902 −0.00376133
\(799\) 3335.10 0.147669
\(800\) −800.000 −0.0353553
\(801\) −36949.8 −1.62991
\(802\) −24650.2 −1.08532
\(803\) −19376.2 −0.851523
\(804\) −13010.1 −0.570684
\(805\) 2709.66 0.118637
\(806\) −9047.58 −0.395394
\(807\) −19849.4 −0.865841
\(808\) −15110.9 −0.657918
\(809\) 8226.61 0.357518 0.178759 0.983893i \(-0.442792\pi\)
0.178759 + 0.983893i \(0.442792\pi\)
\(810\) −5400.95 −0.234284
\(811\) 27867.9 1.20662 0.603312 0.797505i \(-0.293847\pi\)
0.603312 + 0.797505i \(0.293847\pi\)
\(812\) −12933.4 −0.558956
\(813\) 62358.8 2.69006
\(814\) 2937.53 0.126487
\(815\) 5276.37 0.226777
\(816\) 15605.7 0.669498
\(817\) 126.502 0.00541705
\(818\) 21613.5 0.923835
\(819\) 30943.6 1.32022
\(820\) 2707.69 0.115313
\(821\) −17637.5 −0.749761 −0.374880 0.927073i \(-0.622316\pi\)
−0.374880 + 0.927073i \(0.622316\pi\)
\(822\) 37542.6 1.59300
\(823\) 28048.6 1.18799 0.593993 0.804470i \(-0.297551\pi\)
0.593993 + 0.804470i \(0.297551\pi\)
\(824\) 11977.3 0.506372
\(825\) −6210.78 −0.262099
\(826\) −9530.82 −0.401477
\(827\) −32592.9 −1.37046 −0.685228 0.728329i \(-0.740297\pi\)
−0.685228 + 0.728329i \(0.740297\pi\)
\(828\) −3014.42 −0.126520
\(829\) 18815.9 0.788303 0.394152 0.919045i \(-0.371038\pi\)
0.394152 + 0.919045i \(0.371038\pi\)
\(830\) −4851.78 −0.202901
\(831\) 69511.9 2.90173
\(832\) 2565.19 0.106889
\(833\) 26769.7 1.11346
\(834\) −37980.9 −1.57694
\(835\) 3659.03 0.151648
\(836\) −29.9168 −0.00123767
\(837\) 5030.56 0.207744
\(838\) 3647.48 0.150358
\(839\) 7612.72 0.313254 0.156627 0.987658i \(-0.449938\pi\)
0.156627 + 0.987658i \(0.449938\pi\)
\(840\) 7286.21 0.299283
\(841\) −5558.14 −0.227896
\(842\) −11990.9 −0.490775
\(843\) 7484.43 0.305786
\(844\) 11354.5 0.463076
\(845\) 2952.54 0.120202
\(846\) −1732.27 −0.0703978
\(847\) −7029.28 −0.285158
\(848\) 699.134 0.0283117
\(849\) −17414.8 −0.703975
\(850\) −6308.26 −0.254555
\(851\) −1051.23 −0.0423452
\(852\) 20637.6 0.829849
\(853\) −31421.4 −1.26125 −0.630627 0.776086i \(-0.717202\pi\)
−0.630627 + 0.776086i \(0.717202\pi\)
\(854\) −7081.81 −0.283764
\(855\) −38.1293 −0.00152514
\(856\) 4681.20 0.186916
\(857\) 20909.5 0.833435 0.416718 0.909036i \(-0.363180\pi\)
0.416718 + 0.909036i \(0.363180\pi\)
\(858\) 19914.8 0.792400
\(859\) 23304.5 0.925655 0.462828 0.886448i \(-0.346835\pi\)
0.462828 + 0.886448i \(0.346835\pi\)
\(860\) −10870.6 −0.431027
\(861\) −24661.0 −0.976127
\(862\) −7775.61 −0.307237
\(863\) −19146.3 −0.755211 −0.377606 0.925966i \(-0.623253\pi\)
−0.377606 + 0.925966i \(0.623253\pi\)
\(864\) −1426.27 −0.0561607
\(865\) 2608.86 0.102548
\(866\) 8845.87 0.347107
\(867\) 85075.0 3.33252
\(868\) 10637.5 0.415968
\(869\) 44310.5 1.72972
\(870\) −10608.6 −0.413410
\(871\) −16863.0 −0.656005
\(872\) 1113.32 0.0432362
\(873\) −48697.1 −1.88791
\(874\) 10.7061 0.000414347 0
\(875\) −2945.28 −0.113793
\(876\) 18645.5 0.719146
\(877\) 31389.0 1.20859 0.604293 0.796762i \(-0.293456\pi\)
0.604293 + 0.796762i \(0.293456\pi\)
\(878\) 3496.16 0.134384
\(879\) 21138.7 0.811139
\(880\) 2570.82 0.0984798
\(881\) 44496.3 1.70161 0.850806 0.525480i \(-0.176114\pi\)
0.850806 + 0.525480i \(0.176114\pi\)
\(882\) −13904.3 −0.530818
\(883\) 8906.65 0.339448 0.169724 0.985492i \(-0.445712\pi\)
0.169724 + 0.985492i \(0.445712\pi\)
\(884\) 20227.4 0.769592
\(885\) −7817.69 −0.296936
\(886\) −6742.62 −0.255669
\(887\) −36584.7 −1.38489 −0.692444 0.721472i \(-0.743466\pi\)
−0.692444 + 0.721472i \(0.743466\pi\)
\(888\) −2826.74 −0.106823
\(889\) −30593.9 −1.15420
\(890\) −11277.1 −0.424729
\(891\) 17356.1 0.652581
\(892\) −8498.74 −0.319012
\(893\) 6.15238 0.000230551 0
\(894\) 37191.2 1.39134
\(895\) 5915.35 0.220926
\(896\) −3015.97 −0.112451
\(897\) −7126.76 −0.265279
\(898\) −26629.5 −0.989573
\(899\) −15488.1 −0.574591
\(900\) 3276.54 0.121353
\(901\) 5512.90 0.203842
\(902\) −8701.23 −0.321197
\(903\) 99006.4 3.64865
\(904\) 10345.4 0.380623
\(905\) −8619.62 −0.316603
\(906\) 10038.9 0.368122
\(907\) −21346.1 −0.781463 −0.390731 0.920505i \(-0.627778\pi\)
−0.390731 + 0.920505i \(0.627778\pi\)
\(908\) 6207.89 0.226890
\(909\) 61889.1 2.25823
\(910\) 9444.00 0.344028
\(911\) −10208.0 −0.371246 −0.185623 0.982621i \(-0.559430\pi\)
−0.185623 + 0.982621i \(0.559430\pi\)
\(912\) 28.7885 0.00104527
\(913\) 15591.3 0.565166
\(914\) −7534.40 −0.272665
\(915\) −5808.88 −0.209875
\(916\) −634.125 −0.0228735
\(917\) −12376.0 −0.445683
\(918\) −11246.6 −0.404351
\(919\) 1758.00 0.0631025 0.0315513 0.999502i \(-0.489955\pi\)
0.0315513 + 0.999502i \(0.489955\pi\)
\(920\) −920.000 −0.0329690
\(921\) −61669.3 −2.20638
\(922\) −19348.1 −0.691102
\(923\) 26749.3 0.953917
\(924\) −23414.4 −0.833633
\(925\) 1142.64 0.0406161
\(926\) −5955.46 −0.211348
\(927\) −49055.3 −1.73807
\(928\) 4391.22 0.155333
\(929\) −845.008 −0.0298426 −0.0149213 0.999889i \(-0.504750\pi\)
−0.0149213 + 0.999889i \(0.504750\pi\)
\(930\) 8725.45 0.307655
\(931\) 49.3830 0.00173841
\(932\) −2915.99 −0.102486
\(933\) 59754.7 2.09676
\(934\) 21402.0 0.749782
\(935\) 20271.7 0.709045
\(936\) −10506.2 −0.366886
\(937\) 3851.61 0.134287 0.0671433 0.997743i \(-0.478612\pi\)
0.0671433 + 0.997743i \(0.478612\pi\)
\(938\) 19826.3 0.690141
\(939\) 42567.7 1.47939
\(940\) −528.688 −0.0183446
\(941\) −4379.36 −0.151714 −0.0758571 0.997119i \(-0.524169\pi\)
−0.0758571 + 0.997119i \(0.524169\pi\)
\(942\) −56667.5 −1.96001
\(943\) 3113.85 0.107530
\(944\) 3235.96 0.111570
\(945\) −5250.98 −0.180756
\(946\) 34932.8 1.20059
\(947\) 29746.9 1.02074 0.510371 0.859954i \(-0.329508\pi\)
0.510371 + 0.859954i \(0.329508\pi\)
\(948\) −42639.3 −1.46082
\(949\) 24167.3 0.826663
\(950\) −11.6371 −0.000397428 0
\(951\) 40442.7 1.37902
\(952\) −23781.9 −0.809638
\(953\) 4306.61 0.146385 0.0731924 0.997318i \(-0.476681\pi\)
0.0731924 + 0.997318i \(0.476681\pi\)
\(954\) −2863.42 −0.0971769
\(955\) 13990.6 0.474057
\(956\) −24806.4 −0.839222
\(957\) 34091.1 1.15152
\(958\) −4674.29 −0.157640
\(959\) −57211.9 −1.92645
\(960\) −2473.86 −0.0831702
\(961\) −17052.3 −0.572397
\(962\) −3663.87 −0.122794
\(963\) −19172.7 −0.641568
\(964\) −28892.2 −0.965305
\(965\) 2488.43 0.0830108
\(966\) 8379.14 0.279083
\(967\) −12233.4 −0.406824 −0.203412 0.979093i \(-0.565203\pi\)
−0.203412 + 0.979093i \(0.565203\pi\)
\(968\) 2386.63 0.0792449
\(969\) 227.007 0.00752581
\(970\) −14862.4 −0.491960
\(971\) 48207.7 1.59326 0.796631 0.604465i \(-0.206613\pi\)
0.796631 + 0.604465i \(0.206613\pi\)
\(972\) −21515.2 −0.709978
\(973\) 57879.8 1.90703
\(974\) 14367.7 0.472660
\(975\) 7746.48 0.254447
\(976\) 2404.46 0.0788575
\(977\) 28662.9 0.938595 0.469298 0.883040i \(-0.344507\pi\)
0.469298 + 0.883040i \(0.344507\pi\)
\(978\) 16316.2 0.533472
\(979\) 36239.2 1.18305
\(980\) −4243.59 −0.138323
\(981\) −4559.81 −0.148403
\(982\) 24168.4 0.785383
\(983\) −20944.1 −0.679567 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(984\) 8373.06 0.271264
\(985\) −1482.44 −0.0479539
\(986\) 34626.2 1.11838
\(987\) 4815.16 0.155287
\(988\) 37.3141 0.00120154
\(989\) −12501.1 −0.401934
\(990\) −10529.2 −0.338021
\(991\) −36440.2 −1.16808 −0.584038 0.811727i \(-0.698528\pi\)
−0.584038 + 0.811727i \(0.698528\pi\)
\(992\) −3611.71 −0.115597
\(993\) −27456.8 −0.877458
\(994\) −31450.0 −1.00355
\(995\) −11737.2 −0.373963
\(996\) −15003.3 −0.477306
\(997\) −11945.0 −0.379439 −0.189720 0.981838i \(-0.560758\pi\)
−0.189720 + 0.981838i \(0.560758\pi\)
\(998\) −14291.6 −0.453298
\(999\) 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 230.4.a.h.1.4 4
3.2 odd 2 2070.4.a.bj.1.3 4
4.3 odd 2 1840.4.a.m.1.1 4
5.2 odd 4 1150.4.b.n.599.1 8
5.3 odd 4 1150.4.b.n.599.8 8
5.4 even 2 1150.4.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 1.1 even 1 trivial
1150.4.a.p.1.1 4 5.4 even 2
1150.4.b.n.599.1 8 5.2 odd 4
1150.4.b.n.599.8 8 5.3 odd 4
1840.4.a.m.1.1 4 4.3 odd 2
2070.4.a.bj.1.3 4 3.2 odd 2