Properties

Label 1150.4.a.p.1.1
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.8521965066\)
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: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.73081\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-expansion

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

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 \(-0.767025\pi\)
−0.743897 + 0.668294i \(0.767025\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −15.4616 −1.05203
\(7\) −23.5622 −1.27224 −0.636121 0.771589i \(-0.719462\pi\)
−0.636121 + 0.771589i \(0.719462\pi\)
\(8\) 8.00000 0.353553
\(9\) 32.7654 1.21353
\(10\) 0 0
\(11\) −32.1352 −0.880830 −0.440415 0.897794i \(-0.645169\pi\)
−0.440415 + 0.897794i \(0.645169\pi\)
\(12\) −30.9232 −0.743897
\(13\) −40.0811 −0.855115 −0.427557 0.903988i \(-0.640626\pi\)
−0.427557 + 0.903988i \(0.640626\pi\)
\(14\) −47.1245 −0.899611
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −126.165 −1.79997 −0.899987 0.435916i \(-0.856424\pi\)
−0.899987 + 0.435916i \(0.856424\pi\)
\(18\) 65.5308 0.858097
\(19\) 0.232742 0.00281024 0.00140512 0.999999i \(-0.499553\pi\)
0.00140512 + 0.999999i \(0.499553\pi\)
\(20\) 0 0
\(21\) 182.155 1.89283
\(22\) −64.2704 −0.622841
\(23\) 23.0000 0.208514
\(24\) −61.8465 −0.526015
\(25\) 0 0
\(26\) −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 \(0.644790\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) 131.062 0.606766
\(37\) −45.7057 −0.203080 −0.101540 0.994831i \(-0.532377\pi\)
−0.101540 + 0.994831i \(0.532377\pi\)
\(38\) 0.465483 0.00198714
\(39\) 309.859 1.27223
\(40\) 0 0
\(41\) −135.385 −0.515696 −0.257848 0.966186i \(-0.583013\pi\)
−0.257848 + 0.966186i \(0.583013\pi\)
\(42\) 364.310 1.33844
\(43\) −543.528 −1.92761 −0.963805 0.266608i \(-0.914097\pi\)
−0.963805 + 0.266608i \(0.914097\pi\)
\(44\) −128.541 −0.440415
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −26.4344 −0.0820394 −0.0410197 0.999158i \(-0.513061\pi\)
−0.0410197 + 0.999158i \(0.513061\pi\)
\(48\) −123.693 −0.371949
\(49\) 212.180 0.618599
\(50\) 0 0
\(51\) 975.359 2.67799
\(52\) −160.324 −0.427557
\(53\) −43.6958 −0.113247 −0.0566235 0.998396i \(-0.518033\pi\)
−0.0566235 + 0.998396i \(0.518033\pi\)
\(54\) −89.1422 −0.224643
\(55\) 0 0
\(56\) −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 \(0.428370\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) 496.862 0.926659
\(67\) 420.722 0.767155 0.383577 0.923509i \(-0.374692\pi\)
0.383577 + 0.923509i \(0.374692\pi\)
\(68\) −504.661 −0.899987
\(69\) −177.809 −0.310227
\(70\) 0 0
\(71\) 667.381 1.11554 0.557771 0.829995i \(-0.311657\pi\)
0.557771 + 0.829995i \(0.311657\pi\)
\(72\) 262.123 0.429049
\(73\) −602.960 −0.966728 −0.483364 0.875420i \(-0.660585\pi\)
−0.483364 + 0.875420i \(0.660585\pi\)
\(74\) −91.4115 −0.143600
\(75\) 0 0
\(76\) 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 \(0.939299\pi\)
\(80\) 0 0
\(81\) −540.095 −0.740871
\(82\) −270.769 −0.364652
\(83\) 485.178 0.641629 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(84\) 728.621 0.946417
\(85\) 0 0
\(86\) −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 \(0.734374\pi\)
\(90\) 0 0
\(91\) 944.400 1.08791
\(92\) 92.0000 0.104257
\(93\) −872.545 −0.972890
\(94\) −52.8688 −0.0580106
\(95\) 0 0
\(96\) −247.386 −0.263007
\(97\) 1486.24 1.55572 0.777858 0.628441i \(-0.216306\pi\)
0.777858 + 0.628441i \(0.216306\pi\)
\(98\) 424.359 0.437416
\(99\) −1052.92 −1.06892
\(100\) 0 0
\(101\) 1888.86 1.86087 0.930437 0.366451i \(-0.119427\pi\)
0.930437 + 0.366451i \(0.119427\pi\)
\(102\) 1950.72 1.89363
\(103\) 1497.17 1.43224 0.716118 0.697979i \(-0.245917\pi\)
0.716118 + 0.697979i \(0.245917\pi\)
\(104\) −320.649 −0.302329
\(105\) 0 0
\(106\) −87.3917 −0.0800777
\(107\) 585.150 0.528678 0.264339 0.964430i \(-0.414846\pi\)
0.264339 + 0.964430i \(0.414846\pi\)
\(108\) −178.284 −0.158846
\(109\) −139.166 −0.122290 −0.0611452 0.998129i \(-0.519475\pi\)
−0.0611452 + 0.998129i \(0.519475\pi\)
\(110\) 0 0
\(111\) 353.342 0.302142
\(112\) −376.996 −0.318060
\(113\) 1293.18 1.07656 0.538282 0.842765i \(-0.319073\pi\)
0.538282 + 0.842765i \(0.319073\pi\)
\(114\) −3.59856 −0.00295646
\(115\) 0 0
\(116\) −548.902 −0.439347
\(117\) −1313.27 −1.03771
\(118\) 404.495 0.315566
\(119\) 2972.74 2.29000
\(120\) 0 0
\(121\) −298.328 −0.224138
\(122\) 300.557 0.223043
\(123\) 1046.63 0.767250
\(124\) 451.464 0.326957
\(125\) 0 0
\(126\) −1544.05 −1.09171
\(127\) 1298.43 0.907221 0.453610 0.891200i \(-0.350136\pi\)
0.453610 + 0.891200i \(0.350136\pi\)
\(128\) 128.000 0.0883883
\(129\) 4201.91 2.86789
\(130\) 0 0
\(131\) −525.247 −0.350313 −0.175157 0.984541i \(-0.556043\pi\)
−0.175157 + 0.984541i \(0.556043\pi\)
\(132\) 993.725 0.655247
\(133\) −5.48392 −0.00357531
\(134\) 841.444 0.542460
\(135\) 0 0
\(136\) −1009.32 −0.636387
\(137\) 2428.12 1.51422 0.757109 0.653288i \(-0.226611\pi\)
0.757109 + 0.653288i \(0.226611\pi\)
\(138\) −355.617 −0.219363
\(139\) 2456.46 1.49895 0.749476 0.662031i \(-0.230305\pi\)
0.749476 + 0.662031i \(0.230305\pi\)
\(140\) 0 0
\(141\) 204.359 0.122058
\(142\) 1334.76 0.788808
\(143\) 1288.01 0.753211
\(144\) 524.246 0.303383
\(145\) 0 0
\(146\) −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 \(0.729981\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) 1239.44 0.636117
\(157\) −3665.04 −1.86307 −0.931536 0.363650i \(-0.881530\pi\)
−0.931536 + 0.363650i \(0.881530\pi\)
\(158\) −2757.75 −1.38858
\(159\) 337.804 0.168488
\(160\) 0 0
\(161\) −541.932 −0.265281
\(162\) −1080.19 −0.523875
\(163\) 1055.27 0.507088 0.253544 0.967324i \(-0.418404\pi\)
0.253544 + 0.967324i \(0.418404\pi\)
\(164\) −541.539 −0.257848
\(165\) 0 0
\(166\) 970.356 0.453700
\(167\) 731.805 0.339095 0.169547 0.985522i \(-0.445769\pi\)
0.169547 + 0.985522i \(0.445769\pi\)
\(168\) 1457.24 0.669218
\(169\) −590.507 −0.268779
\(170\) 0 0
\(171\) 7.62587 0.00341032
\(172\) −2174.11 −0.963805
\(173\) 521.773 0.229304 0.114652 0.993406i \(-0.463425\pi\)
0.114652 + 0.993406i \(0.463425\pi\)
\(174\) 2121.73 0.924413
\(175\) 0 0
\(176\) −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 \(0.579446\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −1745.09 −0.687937
\(187\) 4054.35 1.58547
\(188\) −105.738 −0.0410197
\(189\) 1050.20 0.404182
\(190\) 0 0
\(191\) −2798.12 −1.06002 −0.530012 0.847990i \(-0.677813\pi\)
−0.530012 + 0.847990i \(0.677813\pi\)
\(192\) −494.772 −0.185974
\(193\) 497.686 0.185618 0.0928089 0.995684i \(-0.470415\pi\)
0.0928089 + 0.995684i \(0.470415\pi\)
\(194\) 2972.47 1.10006
\(195\) 0 0
\(196\) 848.718 0.309300
\(197\) −296.489 −0.107228 −0.0536141 0.998562i \(-0.517074\pi\)
−0.0536141 + 0.998562i \(0.517074\pi\)
\(198\) −2105.85 −0.755838
\(199\) 2347.43 0.836207 0.418103 0.908399i \(-0.362695\pi\)
0.418103 + 0.908399i \(0.362695\pi\)
\(200\) 0 0
\(201\) −3252.52 −1.14137
\(202\) 3777.71 1.31584
\(203\) 3233.34 1.11791
\(204\) 3901.44 1.33900
\(205\) 0 0
\(206\) 2994.34 1.01274
\(207\) 753.604 0.253039
\(208\) −641.297 −0.213779
\(209\) −7.47920 −0.00247535
\(210\) 0 0
\(211\) 2838.61 0.926153 0.463076 0.886318i \(-0.346746\pi\)
0.463076 + 0.886318i \(0.346746\pi\)
\(212\) −174.783 −0.0566235
\(213\) −5159.39 −1.65970
\(214\) 1170.30 0.373832
\(215\) 0 0
\(216\) −356.569 −0.112321
\(217\) −2659.38 −0.831937
\(218\) −278.331 −0.0864723
\(219\) 4661.37 1.43829
\(220\) 0 0
\(221\) 5056.84 1.53918
\(222\) 706.685 0.213647
\(223\) 2124.68 0.638024 0.319012 0.947751i \(-0.396649\pi\)
0.319012 + 0.947751i \(0.396649\pi\)
\(224\) −753.992 −0.224903
\(225\) 0 0
\(226\) 2586.35 0.761246
\(227\) −1551.97 −0.453780 −0.226890 0.973920i \(-0.572856\pi\)
−0.226890 + 0.973920i \(0.572856\pi\)
\(228\) −7.19712 −0.00209053
\(229\) −158.531 −0.0457469 −0.0228735 0.999738i \(-0.507281\pi\)
−0.0228735 + 0.999738i \(0.507281\pi\)
\(230\) 0 0
\(231\) −5853.60 −1.66727
\(232\) −1097.80 −0.310666
\(233\) 728.999 0.204971 0.102486 0.994734i \(-0.467320\pi\)
0.102486 + 0.994734i \(0.467320\pi\)
\(234\) −2626.54 −0.733772
\(235\) 0 0
\(236\) 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 \(0.816989\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) 2093.27 0.542527
\(247\) −9.32853 −0.00240308
\(248\) 902.928 0.231193
\(249\) −3750.82 −0.954612
\(250\) 0 0
\(251\) 5042.78 1.26812 0.634059 0.773285i \(-0.281388\pi\)
0.634059 + 0.773285i \(0.281388\pi\)
\(252\) −3088.10 −0.771954
\(253\) −739.110 −0.183666
\(254\) 2596.86 0.641502
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2981.15 −0.723577 −0.361788 0.932260i \(-0.617834\pi\)
−0.361788 + 0.932260i \(0.617834\pi\)
\(258\) 8403.82 2.02790
\(259\) 1076.93 0.258367
\(260\) 0 0
\(261\) −4496.25 −1.06632
\(262\) −1050.49 −0.247709
\(263\) −7242.86 −1.69815 −0.849076 0.528271i \(-0.822841\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(264\) 1987.45 0.463330
\(265\) 0 0
\(266\) −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 \(0.593982\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) −711.234 −0.155113
\(277\) −8991.54 −1.95036 −0.975179 0.221418i \(-0.928932\pi\)
−0.975179 + 0.221418i \(0.928932\pi\)
\(278\) 4912.92 1.05992
\(279\) 3698.10 0.793546
\(280\) 0 0
\(281\) 968.130 0.205529 0.102765 0.994706i \(-0.467231\pi\)
0.102765 + 0.994706i \(0.467231\pi\)
\(282\) 408.718 0.0863079
\(283\) 2252.65 0.473167 0.236583 0.971611i \(-0.423972\pi\)
0.236583 + 0.971611i \(0.423972\pi\)
\(284\) 2669.52 0.557771
\(285\) 0 0
\(286\) 2576.03 0.532600
\(287\) 3189.97 0.656090
\(288\) 1048.49 0.214524
\(289\) 11004.7 2.23991
\(290\) 0 0
\(291\) −11489.8 −2.31458
\(292\) −2411.84 −0.483364
\(293\) −2734.35 −0.545196 −0.272598 0.962128i \(-0.587883\pi\)
−0.272598 + 0.962128i \(0.587883\pi\)
\(294\) −3280.64 −0.650785
\(295\) 0 0
\(296\) −365.646 −0.0717998
\(297\) 1432.30 0.279834
\(298\) −4810.78 −0.935172
\(299\) −921.865 −0.178304
\(300\) 0 0
\(301\) 12806.7 2.45239
\(302\) −1298.55 −0.247428
\(303\) −14602.4 −2.76860
\(304\) 3.72387 0.000702560 0
\(305\) 0 0
\(306\) −8267.71 −1.54455
\(307\) 7977.09 1.48299 0.741493 0.670961i \(-0.234118\pi\)
0.741493 + 0.670961i \(0.234118\pi\)
\(308\) 3028.71 0.560314
\(309\) −11574.3 −2.13087
\(310\) 0 0
\(311\) 7729.43 1.40931 0.704655 0.709550i \(-0.251102\pi\)
0.704655 + 0.709550i \(0.251102\pi\)
\(312\) 2478.87 0.449803
\(313\) −5506.25 −0.994350 −0.497175 0.867650i \(-0.665629\pi\)
−0.497175 + 0.867650i \(0.665629\pi\)
\(314\) −7330.08 −1.31739
\(315\) 0 0
\(316\) −5515.51 −0.981872
\(317\) −5231.37 −0.926887 −0.463443 0.886126i \(-0.653386\pi\)
−0.463443 + 0.886126i \(0.653386\pi\)
\(318\) 675.608 0.119139
\(319\) 4409.77 0.773981
\(320\) 0 0
\(321\) −4523.68 −0.786565
\(322\) −1083.86 −0.187582
\(323\) −29.3639 −0.00505836
\(324\) −2160.38 −0.370435
\(325\) 0 0
\(326\) 2110.55 0.358566
\(327\) 1075.86 0.181943
\(328\) −1083.08 −0.182326
\(329\) 622.854 0.104374
\(330\) 0 0
\(331\) −3551.61 −0.589771 −0.294885 0.955533i \(-0.595281\pi\)
−0.294885 + 0.955533i \(0.595281\pi\)
\(332\) 1940.71 0.320815
\(333\) −1497.57 −0.246445
\(334\) 1463.61 0.239776
\(335\) 0 0
\(336\) 2914.48 0.473209
\(337\) 7002.99 1.13198 0.565990 0.824412i \(-0.308494\pi\)
0.565990 + 0.824412i \(0.308494\pi\)
\(338\) −1181.01 −0.190055
\(339\) −9997.29 −1.60171
\(340\) 0 0
\(341\) −3626.97 −0.575987
\(342\) 15.2517 0.00241146
\(343\) 3082.42 0.485234
\(344\) −4348.22 −0.681513
\(345\) 0 0
\(346\) 1043.55 0.162143
\(347\) 10268.2 1.58854 0.794272 0.607562i \(-0.207852\pi\)
0.794272 + 0.607562i \(0.207852\pi\)
\(348\) 4243.46 0.653659
\(349\) 4515.58 0.692588 0.346294 0.938126i \(-0.387440\pi\)
0.346294 + 0.938126i \(0.387440\pi\)
\(350\) 0 0
\(351\) 1786.46 0.271664
\(352\) −1028.33 −0.155710
\(353\) −7838.63 −1.18189 −0.590947 0.806711i \(-0.701246\pi\)
−0.590947 + 0.806711i \(0.701246\pi\)
\(354\) −3127.08 −0.469498
\(355\) 0 0
\(356\) −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 \(0.868799\pi\)
\(360\) 0 0
\(361\) −6858.95 −0.999992
\(362\) 3447.85 0.500594
\(363\) 2306.32 0.333472
\(364\) 3777.60 0.543956
\(365\) 0 0
\(366\) −2323.55 −0.331842
\(367\) −3936.14 −0.559850 −0.279925 0.960022i \(-0.590310\pi\)
−0.279925 + 0.960022i \(0.590310\pi\)
\(368\) 368.000 0.0521286
\(369\) −4435.93 −0.625814
\(370\) 0 0
\(371\) 1029.57 0.144077
\(372\) −3490.18 −0.486445
\(373\) −1920.72 −0.266625 −0.133313 0.991074i \(-0.542561\pi\)
−0.133313 + 0.991074i \(0.542561\pi\)
\(374\) 8108.69 1.12110
\(375\) 0 0
\(376\) −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 \(0.846530\pi\)
\(380\) 0 0
\(381\) −10037.9 −1.34976
\(382\) −5596.23 −0.749550
\(383\) −8838.22 −1.17914 −0.589571 0.807716i \(-0.700703\pi\)
−0.589571 + 0.807716i \(0.700703\pi\)
\(384\) −989.543 −0.131504
\(385\) 0 0
\(386\) 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 \(0.809856\pi\)
\(390\) 0 0
\(391\) −2901.80 −0.375321
\(392\) 1697.44 0.218708
\(393\) 4060.58 0.521194
\(394\) −592.977 −0.0758218
\(395\) 0 0
\(396\) −4211.69 −0.534458
\(397\) −8060.49 −1.01900 −0.509501 0.860470i \(-0.670170\pi\)
−0.509501 + 0.860470i \(0.670170\pi\)
\(398\) 4694.87 0.591287
\(399\) 42.3951 0.00531932
\(400\) 0 0
\(401\) 12325.1 1.53488 0.767440 0.641121i \(-0.221530\pi\)
0.767440 + 0.641121i \(0.221530\pi\)
\(402\) −6505.04 −0.807070
\(403\) −4523.79 −0.559171
\(404\) 7555.43 0.930437
\(405\) 0 0
\(406\) 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 \(0.726595\pi\)
\(410\) 0 0
\(411\) −18771.3 −2.25285
\(412\) 5988.67 0.716118
\(413\) −4765.41 −0.567774
\(414\) 1507.21 0.178926
\(415\) 0 0
\(416\) −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 \(0.533907\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) −10318.8 −1.17358
\(427\) −3540.91 −0.401303
\(428\) 2340.60 0.264339
\(429\) −9957.39 −1.12062
\(430\) 0 0
\(431\) 3887.80 0.434499 0.217249 0.976116i \(-0.430292\pi\)
0.217249 + 0.976116i \(0.430292\pi\)
\(432\) −713.137 −0.0794232
\(433\) 4422.93 0.490884 0.245442 0.969411i \(-0.421067\pi\)
0.245442 + 0.969411i \(0.421067\pi\)
\(434\) −5318.75 −0.588268
\(435\) 0 0
\(436\) −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 \(0.530293\pi\)
\(440\) 0 0
\(441\) 6952.14 0.750690
\(442\) 10113.7 1.08837
\(443\) −3371.31 −0.361571 −0.180785 0.983523i \(-0.557864\pi\)
−0.180785 + 0.983523i \(0.557864\pi\)
\(444\) 1413.37 0.151071
\(445\) 0 0
\(446\) 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 \(0.253302\pi\)
\(450\) 0 0
\(451\) 4350.61 0.454240
\(452\) 5172.70 0.538282
\(453\) 5019.43 0.520603
\(454\) −3103.94 −0.320871
\(455\) 0 0
\(456\) −14.3942 −0.00147823
\(457\) −3767.20 −0.385607 −0.192803 0.981237i \(-0.561758\pi\)
−0.192803 + 0.981237i \(0.561758\pi\)
\(458\) −317.063 −0.0323480
\(459\) 5623.32 0.571839
\(460\) 0 0
\(461\) 9674.06 0.977366 0.488683 0.872461i \(-0.337477\pi\)
0.488683 + 0.872461i \(0.337477\pi\)
\(462\) −11707.2 −1.17893
\(463\) −2977.73 −0.298892 −0.149446 0.988770i \(-0.547749\pi\)
−0.149446 + 0.988770i \(0.547749\pi\)
\(464\) −2195.61 −0.219674
\(465\) 0 0
\(466\) 1458.00 0.144937
\(467\) 10701.0 1.06035 0.530176 0.847888i \(-0.322126\pi\)
0.530176 + 0.847888i \(0.322126\pi\)
\(468\) −5253.09 −0.518855
\(469\) −9913.16 −0.976006
\(470\) 0 0
\(471\) 28333.7 2.77187
\(472\) 1617.98 0.157783
\(473\) 17466.4 1.69790
\(474\) 21319.7 2.06592
\(475\) 0 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 \(0.464445\pi\)
\(480\) 0 0
\(481\) 1831.94 0.173657
\(482\) −14446.1 −1.36515
\(483\) 4189.57 0.394683
\(484\) −1193.31 −0.112069
\(485\) 0 0
\(486\) 10757.6 1.00406
\(487\) 7183.85 0.668442 0.334221 0.942495i \(-0.391527\pi\)
0.334221 + 0.942495i \(0.391527\pi\)
\(488\) 1202.23 0.111521
\(489\) −8158.12 −0.754443
\(490\) 0 0
\(491\) −12084.2 −1.11070 −0.555350 0.831617i \(-0.687416\pi\)
−0.555350 + 0.831617i \(0.687416\pi\)
\(492\) 4186.53 0.383625
\(493\) 17313.1 1.58163
\(494\) −18.6571 −0.00169923
\(495\) 0 0
\(496\) 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 \(0.396139\pi\)
\(500\) 0 0
\(501\) −5657.45 −0.504503
\(502\) 10085.6 0.896695
\(503\) −20436.2 −1.81154 −0.905770 0.423771i \(-0.860706\pi\)
−0.905770 + 0.423771i \(0.860706\pi\)
\(504\) −6176.21 −0.545854
\(505\) 0 0
\(506\) −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 \(0.171279\pi\)
\(510\) 0 0
\(511\) 14207.1 1.22991
\(512\) 512.000 0.0441942
\(513\) −10.3735 −0.000892794 0
\(514\) −5962.30 −0.511646
\(515\) 0 0
\(516\) 16807.6 1.43394
\(517\) 849.475 0.0722628
\(518\) 2153.86 0.182693
\(519\) −4033.73 −0.341158
\(520\) 0 0
\(521\) 3483.23 0.292904 0.146452 0.989218i \(-0.453215\pi\)
0.146452 + 0.989218i \(0.453215\pi\)
\(522\) −8992.50 −0.754006
\(523\) −15689.0 −1.31172 −0.655862 0.754881i \(-0.727695\pi\)
−0.655862 + 0.754881i \(0.727695\pi\)
\(524\) −2100.99 −0.175157
\(525\) 0 0
\(526\) −14485.7 −1.20077
\(527\) −14239.8 −1.17703
\(528\) 3974.90 0.327624
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) 6626.72 0.541573
\(532\) −21.9357 −0.00178765
\(533\) 5426.36 0.440979
\(534\) 17436.2 1.41299
\(535\) 0 0
\(536\) 3365.78 0.271230
\(537\) 9146.09 0.734977
\(538\) −5135.15 −0.411510
\(539\) −6818.43 −0.544881
\(540\) 0 0
\(541\) 7620.73 0.605621 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(542\) 16132.5 1.27851
\(543\) −13327.3 −1.05328
\(544\) −4037.29 −0.318194
\(545\) 0 0
\(546\) −14602.0 −1.14452
\(547\) −5925.62 −0.463183 −0.231592 0.972813i \(-0.574393\pi\)
−0.231592 + 0.972813i \(0.574393\pi\)
\(548\) 9712.47 0.757109
\(549\) 4923.94 0.382784
\(550\) 0 0
\(551\) −31.9381 −0.00246934
\(552\) −1422.47 −0.109682
\(553\) 32489.4 2.49836
\(554\) −17983.1 −1.37911
\(555\) 0 0
\(556\) 9825.85 0.749476
\(557\) 5456.16 0.415054 0.207527 0.978229i \(-0.433459\pi\)
0.207527 + 0.978229i \(0.433459\pi\)
\(558\) 7396.20 0.561122
\(559\) 21785.2 1.64833
\(560\) 0 0
\(561\) −31343.4 −2.35886
\(562\) 1936.26 0.145331
\(563\) 9194.29 0.688265 0.344132 0.938921i \(-0.388173\pi\)
0.344132 + 0.938921i \(0.388173\pi\)
\(564\) 817.437 0.0610289
\(565\) 0 0
\(566\) 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 \(0.496027\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 2096.98 0.151692
\(577\) −23300.0 −1.68109 −0.840547 0.541738i \(-0.817766\pi\)
−0.840547 + 0.541738i \(0.817766\pi\)
\(578\) 22009.3 1.58385
\(579\) −3847.52 −0.276161
\(580\) 0 0
\(581\) −11431.9 −0.816307
\(582\) −22979.6 −1.63666
\(583\) 1404.18 0.0997513
\(584\) −4823.68 −0.341790
\(585\) 0 0
\(586\) −5468.70 −0.385512
\(587\) −15653.3 −1.10065 −0.550325 0.834950i \(-0.685496\pi\)
−0.550325 + 0.834950i \(0.685496\pi\)
\(588\) −6561.28 −0.460174
\(589\) 26.2686 0.00183766
\(590\) 0 0
\(591\) 2292.10 0.159533
\(592\) −731.292 −0.0507701
\(593\) 2658.08 0.184072 0.0920358 0.995756i \(-0.470663\pi\)
0.0920358 + 0.995756i \(0.470663\pi\)
\(594\) 2864.60 0.197872
\(595\) 0 0
\(596\) −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 \(0.269599\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) −29204.8 −1.95770
\(607\) −3968.56 −0.265369 −0.132684 0.991158i \(-0.542360\pi\)
−0.132684 + 0.991158i \(0.542360\pi\)
\(608\) 7.44773 0.000496785 0
\(609\) −24996.4 −1.66322
\(610\) 0 0
\(611\) 1059.52 0.0701531
\(612\) −16535.4 −1.09216
\(613\) 11478.7 0.756311 0.378156 0.925742i \(-0.376558\pi\)
0.378156 + 0.925742i \(0.376558\pi\)
\(614\) 15954.2 1.04863
\(615\) 0 0
\(616\) 6057.42 0.396202
\(617\) 15691.1 1.02382 0.511911 0.859039i \(-0.328938\pi\)
0.511911 + 0.859039i \(0.328938\pi\)
\(618\) −23148.6 −1.50676
\(619\) −18249.4 −1.18499 −0.592494 0.805575i \(-0.701856\pi\)
−0.592494 + 0.805575i \(0.701856\pi\)
\(620\) 0 0
\(621\) −1025.14 −0.0662436
\(622\) 15458.9 0.996533
\(623\) 26571.4 1.70876
\(624\) 4957.75 0.318059
\(625\) 0 0
\(626\) −11012.5 −0.703111
\(627\) 57.8203 0.00368281
\(628\) −14660.2 −0.931536
\(629\) 5766.48 0.365540
\(630\) 0 0
\(631\) −23528.8 −1.48442 −0.742208 0.670169i \(-0.766222\pi\)
−0.742208 + 0.670169i \(0.766222\pi\)
\(632\) −11031.0 −0.694288
\(633\) −21944.8 −1.37793
\(634\) −10462.7 −0.655408
\(635\) 0 0
\(636\) 1351.22 0.0842441
\(637\) −8504.38 −0.528973
\(638\) 8819.55 0.547287
\(639\) 21867.0 1.35375
\(640\) 0 0
\(641\) 3748.79 0.230996 0.115498 0.993308i \(-0.463154\pi\)
0.115498 + 0.993308i \(0.463154\pi\)
\(642\) −9047.36 −0.556185
\(643\) −15924.3 −0.976660 −0.488330 0.872659i \(-0.662394\pi\)
−0.488330 + 0.872659i \(0.662394\pi\)
\(644\) −2167.73 −0.132640
\(645\) 0 0
\(646\) −58.7278 −0.00357680
\(647\) 2854.11 0.173426 0.0867129 0.996233i \(-0.472364\pi\)
0.0867129 + 0.996233i \(0.472364\pi\)
\(648\) −4320.76 −0.261937
\(649\) −6499.27 −0.393095
\(650\) 0 0
\(651\) 20559.1 1.23775
\(652\) 4221.09 0.253544
\(653\) −7925.97 −0.474988 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(654\) 2151.72 0.128653
\(655\) 0 0
\(656\) −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 \(0.554831\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −2995.13 −0.174263
\(667\) −3156.19 −0.183221
\(668\) 2927.22 0.169547
\(669\) −16425.5 −0.949249
\(670\) 0 0
\(671\) −4829.24 −0.277840
\(672\) 5828.97 0.334609
\(673\) 23253.5 1.33188 0.665940 0.746005i \(-0.268031\pi\)
0.665940 + 0.746005i \(0.268031\pi\)
\(674\) 14006.0 0.800430
\(675\) 0 0
\(676\) −2362.03 −0.134389
\(677\) 19003.7 1.07884 0.539419 0.842038i \(-0.318644\pi\)
0.539419 + 0.842038i \(0.318644\pi\)
\(678\) −19994.6 −1.13258
\(679\) −35019.1 −1.97925
\(680\) 0 0
\(681\) 11998.0 0.675131
\(682\) −7253.94 −0.407284
\(683\) 11222.5 0.628722 0.314361 0.949304i \(-0.398210\pi\)
0.314361 + 0.949304i \(0.398210\pi\)
\(684\) 30.5035 0.00170516
\(685\) 0 0
\(686\) 6164.85 0.343112
\(687\) 1225.58 0.0680620
\(688\) −8696.45 −0.481902
\(689\) 1751.38 0.0968391
\(690\) 0 0
\(691\) 4297.68 0.236601 0.118301 0.992978i \(-0.462255\pi\)
0.118301 + 0.992978i \(0.462255\pi\)
\(692\) 2087.09 0.114652
\(693\) 24809.2 1.35992
\(694\) 20536.4 1.12327
\(695\) 0 0
\(696\) 8486.92 0.462206
\(697\) 17080.8 0.928240
\(698\) 9031.15 0.489734
\(699\) −5635.75 −0.304955
\(700\) 0 0
\(701\) 25769.6 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(702\) 3572.91 0.192095
\(703\) −10.6376 −0.000570705 0
\(704\) −2056.65 −0.110104
\(705\) 0 0
\(706\) −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 \(0.541426\pi\)
\(710\) 0 0
\(711\) −45179.4 −2.38307
\(712\) −9021.67 −0.474862
\(713\) 2595.92 0.136350
\(714\) −45963.3 −2.40915
\(715\) 0 0
\(716\) −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 \(0.436579\pi\)
\(720\) 0 0
\(721\) −35276.6 −1.82215
\(722\) −13717.9 −0.707101
\(723\) 55839.9 2.87235
\(724\) 6895.70 0.353973
\(725\) 0 0
\(726\) 4612.64 0.235800
\(727\) 36985.6 1.88682 0.943410 0.331628i \(-0.107598\pi\)
0.943410 + 0.331628i \(0.107598\pi\)
\(728\) 7555.20 0.384635
\(729\) −26999.8 −1.37173
\(730\) 0 0
\(731\) 68574.3 3.46965
\(732\) −4647.10 −0.234647
\(733\) 22227.3 1.12003 0.560015 0.828482i \(-0.310795\pi\)
0.560015 + 0.828482i \(0.310795\pi\)
\(734\) −7872.29 −0.395874
\(735\) 0 0
\(736\) 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 \(0.646970\pi\)
\(740\) 0 0
\(741\) 72.1171 0.00357529
\(742\) 2059.14 0.101878
\(743\) 29843.4 1.47355 0.736776 0.676137i \(-0.236347\pi\)
0.736776 + 0.676137i \(0.236347\pi\)
\(744\) −6980.36 −0.343968
\(745\) 0 0
\(746\) −3841.44 −0.188532
\(747\) 15897.0 0.778638
\(748\) 16217.4 0.792736
\(749\) −13787.4 −0.672606
\(750\) 0 0
\(751\) −9399.65 −0.456722 −0.228361 0.973577i \(-0.573337\pi\)
−0.228361 + 0.973577i \(0.573337\pi\)
\(752\) −422.950 −0.0205098
\(753\) −38984.8 −1.88670
\(754\) 11000.3 0.531309
\(755\) 0 0
\(756\) 4200.78 0.202091
\(757\) −29871.6 −1.43422 −0.717109 0.696961i \(-0.754535\pi\)
−0.717109 + 0.696961i \(0.754535\pi\)
\(758\) −26149.0 −1.25300
\(759\) 5713.92 0.273257
\(760\) 0 0
\(761\) −273.955 −0.0130497 −0.00652487 0.999979i \(-0.502077\pi\)
−0.00652487 + 0.999979i \(0.502077\pi\)
\(762\) −20075.8 −0.954423
\(763\) 3279.05 0.155583
\(764\) −11192.5 −0.530012
\(765\) 0 0
\(766\) −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 \(0.600739\pi\)
\(770\) 0 0
\(771\) 23046.7 1.07653
\(772\) 1990.74 0.0928089
\(773\) −34161.1 −1.58951 −0.794753 0.606933i \(-0.792400\pi\)
−0.794753 + 0.606933i \(0.792400\pi\)
\(774\) −35617.8 −1.65408
\(775\) 0 0
\(776\) 11889.9 0.550028
\(777\) −8325.54 −0.384398
\(778\) −25374.6 −1.16931
\(779\) −31.5096 −0.00144923
\(780\) 0 0
\(781\) −21446.4 −0.982603
\(782\) −5803.60 −0.265392
\(783\) 6116.29 0.279155
\(784\) 3394.87 0.154650
\(785\) 0 0
\(786\) 8121.17 0.368540
\(787\) 38489.6 1.74334 0.871669 0.490095i \(-0.163038\pi\)
0.871669 + 0.490095i \(0.163038\pi\)
\(788\) −1185.95 −0.0536141
\(789\) 55993.2 2.52650
\(790\) 0 0
\(791\) −30470.1 −1.36965
\(792\) −8423.38 −0.377919
\(793\) −6023.33 −0.269729
\(794\) −16121.0 −0.720544
\(795\) 0 0
\(796\) 9389.73 0.418103
\(797\) −17176.8 −0.763405 −0.381702 0.924285i \(-0.624662\pi\)
−0.381702 + 0.924285i \(0.624662\pi\)
\(798\) 84.7902 0.00376133
\(799\) 3335.10 0.147669
\(800\) 0 0
\(801\) −36949.8 −1.62991
\(802\) 24650.2 1.08532
\(803\) 19376.2 0.851523
\(804\) −13010.1 −0.570684
\(805\) 0 0
\(806\) −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 \(0.442792\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 15605.7 0.669498
\(817\) −126.502 −0.00541705
\(818\) −21613.5 −0.923835
\(819\) 30943.6 1.32022
\(820\) 0 0
\(821\) −17637.5 −0.749761 −0.374880 0.927073i \(-0.622316\pi\)
−0.374880 + 0.927073i \(0.622316\pi\)
\(822\) −37542.6 −1.59300
\(823\) −28048.6 −1.18799 −0.593993 0.804470i \(-0.702449\pi\)
−0.593993 + 0.804470i \(0.702449\pi\)
\(824\) 11977.3 0.506372
\(825\) 0 0
\(826\) −9530.82 −0.401477
\(827\) 32592.9 1.37046 0.685228 0.728329i \(-0.259703\pi\)
0.685228 + 0.728329i \(0.259703\pi\)
\(828\) 3014.42 0.126520
\(829\) 18815.9 0.788303 0.394152 0.919045i \(-0.371038\pi\)
0.394152 + 0.919045i \(0.371038\pi\)
\(830\) 0 0
\(831\) 69511.9 2.90173
\(832\) −2565.19 −0.106889
\(833\) −26769.7 −1.11346
\(834\) −37980.9 −1.57694
\(835\) 0 0
\(836\) −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 \(0.449938\pi\)
\(840\) 0 0
\(841\) −5558.14 −0.227896
\(842\) 11990.9 0.490775
\(843\) −7484.43 −0.305786
\(844\) 11354.5 0.463076
\(845\) 0 0
\(846\) −1732.27 −0.0703978
\(847\) 7029.28 0.285158
\(848\) −699.134 −0.0283117
\(849\) −17414.8 −0.703975
\(850\) 0 0
\(851\) −1051.23 −0.0423452
\(852\) −20637.6 −0.829849
\(853\) 31421.4 1.26125 0.630627 0.776086i \(-0.282798\pi\)
0.630627 + 0.776086i \(0.282798\pi\)
\(854\) −7081.81 −0.283764
\(855\) 0 0
\(856\) 4681.20 0.186916
\(857\) −20909.5 −0.833435 −0.416718 0.909036i \(-0.636820\pi\)
−0.416718 + 0.909036i \(0.636820\pi\)
\(858\) −19914.8 −0.792400
\(859\) 23304.5 0.925655 0.462828 0.886448i \(-0.346835\pi\)
0.462828 + 0.886448i \(0.346835\pi\)
\(860\) 0 0
\(861\) −24661.0 −0.976127
\(862\) 7775.61 0.307237
\(863\) 19146.3 0.755211 0.377606 0.925966i \(-0.376747\pi\)
0.377606 + 0.925966i \(0.376747\pi\)
\(864\) −1426.27 −0.0561607
\(865\) 0 0
\(866\) 8845.87 0.347107
\(867\) −85075.0 −3.33252
\(868\) −10637.5 −0.415968
\(869\) 44310.5 1.72972
\(870\) 0 0
\(871\) −16863.0 −0.656005
\(872\) −1113.32 −0.0432362
\(873\) 48697.1 1.88791
\(874\) 10.7061 0.000414347 0
\(875\) 0 0
\(876\) 18645.5 0.719146
\(877\) −31389.0 −1.20859 −0.604293 0.796762i \(-0.706544\pi\)
−0.604293 + 0.796762i \(0.706544\pi\)
\(878\) −3496.16 −0.134384
\(879\) 21138.7 0.811139
\(880\) 0 0
\(881\) 44496.3 1.70161 0.850806 0.525480i \(-0.176114\pi\)
0.850806 + 0.525480i \(0.176114\pi\)
\(882\) 13904.3 0.530818
\(883\) −8906.65 −0.339448 −0.169724 0.985492i \(-0.554288\pi\)
−0.169724 + 0.985492i \(0.554288\pi\)
\(884\) 20227.4 0.769592
\(885\) 0 0
\(886\) −6742.62 −0.255669
\(887\) 36584.7 1.38489 0.692444 0.721472i \(-0.256534\pi\)
0.692444 + 0.721472i \(0.256534\pi\)
\(888\) 2826.74 0.106823
\(889\) −30593.9 −1.15420
\(890\) 0 0
\(891\) 17356.1 0.652581
\(892\) 8498.74 0.319012
\(893\) −6.15238 −0.000230551 0
\(894\) 37191.2 1.39134
\(895\) 0 0
\(896\) −3015.97 −0.112451
\(897\) 7126.76 0.265279
\(898\) 26629.5 0.989573
\(899\) −15488.1 −0.574591
\(900\) 0 0
\(901\) 5512.90 0.203842
\(902\) 8701.23 0.321197
\(903\) −99006.4 −3.64865
\(904\) 10345.4 0.380623
\(905\) 0 0
\(906\) 10038.9 0.368122
\(907\) 21346.1 0.781463 0.390731 0.920505i \(-0.372222\pi\)
0.390731 + 0.920505i \(0.372222\pi\)
\(908\) −6207.89 −0.226890
\(909\) 61889.1 2.25823
\(910\) 0 0
\(911\) −10208.0 −0.371246 −0.185623 0.982621i \(-0.559430\pi\)
−0.185623 + 0.982621i \(0.559430\pi\)
\(912\) −28.7885 −0.00104527
\(913\) −15591.3 −0.565166
\(914\) −7534.40 −0.272665
\(915\) 0 0
\(916\) −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 \(0.489955\pi\)
\(920\) 0 0
\(921\) −61669.3 −2.20638
\(922\) 19348.1 0.691102
\(923\) −26749.3 −0.953917
\(924\) −23414.4 −0.833633
\(925\) 0 0
\(926\) −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 \(0.504750\pi\)
\(930\) 0 0
\(931\) 49.3830 0.00173841
\(932\) 2915.99 0.102486
\(933\) −59754.7 −2.09676
\(934\) 21402.0 0.749782
\(935\) 0 0
\(936\) −10506.2 −0.366886
\(937\) −3851.61 −0.134287 −0.0671433 0.997743i \(-0.521388\pi\)
−0.0671433 + 0.997743i \(0.521388\pi\)
\(938\) −19826.3 −0.690141
\(939\) 42567.7 1.47939
\(940\) 0 0
\(941\) −4379.36 −0.151714 −0.0758571 0.997119i \(-0.524169\pi\)
−0.0758571 + 0.997119i \(0.524169\pi\)
\(942\) 56667.5 1.96001
\(943\) −3113.85 −0.107530
\(944\) 3235.96 0.111570
\(945\) 0 0
\(946\) 34932.8 1.20059
\(947\) −29746.9 −1.02074 −0.510371 0.859954i \(-0.670492\pi\)
−0.510371 + 0.859954i \(0.670492\pi\)
\(948\) 42639.3 1.46082
\(949\) 24167.3 0.826663
\(950\) 0 0
\(951\) 40442.7 1.37902
\(952\) 23781.9 0.809638
\(953\) −4306.61 −0.146385 −0.0731924 0.997318i \(-0.523319\pi\)
−0.0731924 + 0.997318i \(0.523319\pi\)
\(954\) −2863.42 −0.0971769
\(955\) 0 0
\(956\) −24806.4 −0.839222
\(957\) −34091.1 −1.15152
\(958\) 4674.29 0.157640
\(959\) −57211.9 −1.92645
\(960\) 0 0
\(961\) −17052.3 −0.572397
\(962\) 3663.87 0.122794
\(963\) 19172.7 0.641568
\(964\) −28892.2 −0.965305
\(965\) 0 0
\(966\) 8379.14 0.279083
\(967\) 12233.4 0.406824 0.203412 0.979093i \(-0.434797\pi\)
0.203412 + 0.979093i \(0.434797\pi\)
\(968\) −2386.63 −0.0792449
\(969\) 227.007 0.00752581
\(970\) 0 0
\(971\) 48207.7 1.59326 0.796631 0.604465i \(-0.206613\pi\)
0.796631 + 0.604465i \(0.206613\pi\)
\(972\) 21515.2 0.709978
\(973\) −57879.8 −1.90703
\(974\) 14367.7 0.472660
\(975\) 0 0
\(976\) 2404.46 0.0788575
\(977\) −28662.9 −0.938595 −0.469298 0.883040i \(-0.655493\pi\)
−0.469298 + 0.883040i \(0.655493\pi\)
\(978\) −16316.2 −0.533472
\(979\) 36239.2 1.18305
\(980\) 0 0
\(981\) −4559.81 −0.148403
\(982\) −24168.4 −0.785383
\(983\) 20944.1 0.679567 0.339783 0.940504i \(-0.389646\pi\)
0.339783 + 0.940504i \(0.389646\pi\)
\(984\) 8373.06 0.271264
\(985\) 0 0
\(986\) 34626.2 1.11838
\(987\) −4815.16 −0.155287
\(988\) −37.3141 −0.00120154
\(989\) −12501.1 −0.401934
\(990\) 0 0
\(991\) −36440.2 −1.16808 −0.584038 0.811727i \(-0.698528\pi\)
−0.584038 + 0.811727i \(0.698528\pi\)
\(992\) 3611.71 0.115597
\(993\) 27456.8 0.877458
\(994\) −31450.0 −1.00355
\(995\) 0 0
\(996\) −15003.3 −0.477306
\(997\) 11945.0 0.379439 0.189720 0.981838i \(-0.439242\pi\)
0.189720 + 0.981838i \(0.439242\pi\)
\(998\) 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 1150.4.a.p.1.1 4
5.2 odd 4 1150.4.b.n.599.8 8
5.3 odd 4 1150.4.b.n.599.1 8
5.4 even 2 230.4.a.h.1.4 4
15.14 odd 2 2070.4.a.bj.1.3 4
20.19 odd 2 1840.4.a.m.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 5.4 even 2
1150.4.a.p.1.1 4 1.1 even 1 trivial
1150.4.b.n.599.1 8 5.3 odd 4
1150.4.b.n.599.8 8 5.2 odd 4
1840.4.a.m.1.1 4 20.19 odd 2
2070.4.a.bj.1.3 4 15.14 odd 2