Properties

Label 1045.4.a.c.1.4
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.91893\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +O(q^{10})\) \(q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +19.5947 q^{10} -11.0000 q^{11} -24.4335 q^{12} -57.3328 q^{13} +110.193 q^{14} +16.6033 q^{15} -68.7236 q^{16} +77.5195 q^{17} +62.5980 q^{18} +19.0000 q^{19} -36.7901 q^{20} +93.3709 q^{21} +43.1083 q^{22} +25.3511 q^{23} -8.35424 q^{24} +25.0000 q^{25} +224.683 q^{26} +142.699 q^{27} -206.895 q^{28} +66.9421 q^{29} -65.0671 q^{30} -288.879 q^{31} +249.197 q^{32} +36.5272 q^{33} -303.794 q^{34} +140.591 q^{35} -117.532 q^{36} +287.719 q^{37} -74.4597 q^{38} +190.383 q^{39} -12.5792 q^{40} -140.535 q^{41} -365.914 q^{42} -203.180 q^{43} -80.9383 q^{44} +79.8662 q^{45} -99.3493 q^{46} +190.852 q^{47} +228.208 q^{48} +447.633 q^{49} -97.9733 q^{50} -257.416 q^{51} -421.857 q^{52} +585.208 q^{53} -559.229 q^{54} +55.0000 q^{55} -70.7409 q^{56} -63.0925 q^{57} -262.342 q^{58} -369.162 q^{59} +122.167 q^{60} +11.4860 q^{61} +1132.10 q^{62} +449.139 q^{63} -426.795 q^{64} +286.664 q^{65} -143.148 q^{66} -291.136 q^{67} +570.391 q^{68} -84.1823 q^{69} -550.967 q^{70} +921.785 q^{71} -40.1861 q^{72} +1188.93 q^{73} -1127.55 q^{74} -83.0164 q^{75} +139.803 q^{76} +309.300 q^{77} -746.096 q^{78} -179.464 q^{79} +343.618 q^{80} -42.5781 q^{81} +550.746 q^{82} -184.309 q^{83} +687.026 q^{84} -387.597 q^{85} +796.248 q^{86} -222.292 q^{87} -27.6742 q^{88} -335.864 q^{89} -312.990 q^{90} +1612.10 q^{91} +186.534 q^{92} +959.266 q^{93} -747.938 q^{94} -95.0000 q^{95} -827.496 q^{96} -1155.10 q^{97} -1754.24 q^{98} +175.706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.91893 −1.38555 −0.692776 0.721153i \(-0.743612\pi\)
−0.692776 + 0.721153i \(0.743612\pi\)
\(3\) −3.32066 −0.639061 −0.319530 0.947576i \(-0.603525\pi\)
−0.319530 + 0.947576i \(0.603525\pi\)
\(4\) 7.35803 0.919754
\(5\) −5.00000 −0.447214
\(6\) 13.0134 0.885452
\(7\) −28.1182 −1.51824 −0.759120 0.650951i \(-0.774370\pi\)
−0.759120 + 0.650951i \(0.774370\pi\)
\(8\) 2.51584 0.111186
\(9\) −15.9732 −0.591601
\(10\) 19.5947 0.619638
\(11\) −11.0000 −0.301511
\(12\) −24.4335 −0.587778
\(13\) −57.3328 −1.22317 −0.611587 0.791177i \(-0.709469\pi\)
−0.611587 + 0.791177i \(0.709469\pi\)
\(14\) 110.193 2.10360
\(15\) 16.6033 0.285797
\(16\) −68.7236 −1.07381
\(17\) 77.5195 1.10596 0.552978 0.833196i \(-0.313492\pi\)
0.552978 + 0.833196i \(0.313492\pi\)
\(18\) 62.5980 0.819694
\(19\) 19.0000 0.229416
\(20\) −36.7901 −0.411326
\(21\) 93.3709 0.970247
\(22\) 43.1083 0.417760
\(23\) 25.3511 0.229829 0.114915 0.993375i \(-0.463341\pi\)
0.114915 + 0.993375i \(0.463341\pi\)
\(24\) −8.35424 −0.0710543
\(25\) 25.0000 0.200000
\(26\) 224.683 1.69477
\(27\) 142.699 1.01713
\(28\) −206.895 −1.39641
\(29\) 66.9421 0.428650 0.214325 0.976762i \(-0.431245\pi\)
0.214325 + 0.976762i \(0.431245\pi\)
\(30\) −65.0671 −0.395986
\(31\) −288.879 −1.67368 −0.836841 0.547447i \(-0.815600\pi\)
−0.836841 + 0.547447i \(0.815600\pi\)
\(32\) 249.197 1.37663
\(33\) 36.5272 0.192684
\(34\) −303.794 −1.53236
\(35\) 140.591 0.678978
\(36\) −117.532 −0.544128
\(37\) 287.719 1.27840 0.639199 0.769042i \(-0.279266\pi\)
0.639199 + 0.769042i \(0.279266\pi\)
\(38\) −74.4597 −0.317867
\(39\) 190.383 0.781682
\(40\) −12.5792 −0.0497237
\(41\) −140.535 −0.535313 −0.267656 0.963514i \(-0.586249\pi\)
−0.267656 + 0.963514i \(0.586249\pi\)
\(42\) −365.914 −1.34433
\(43\) −203.180 −0.720573 −0.360286 0.932842i \(-0.617321\pi\)
−0.360286 + 0.932842i \(0.617321\pi\)
\(44\) −80.9383 −0.277316
\(45\) 79.8662 0.264572
\(46\) −99.3493 −0.318440
\(47\) 190.852 0.592313 0.296156 0.955139i \(-0.404295\pi\)
0.296156 + 0.955139i \(0.404295\pi\)
\(48\) 228.208 0.686228
\(49\) 447.633 1.30505
\(50\) −97.9733 −0.277110
\(51\) −257.416 −0.706772
\(52\) −421.857 −1.12502
\(53\) 585.208 1.51669 0.758345 0.651854i \(-0.226008\pi\)
0.758345 + 0.651854i \(0.226008\pi\)
\(54\) −559.229 −1.40929
\(55\) 55.0000 0.134840
\(56\) −70.7409 −0.168806
\(57\) −63.0925 −0.146611
\(58\) −262.342 −0.593916
\(59\) −369.162 −0.814590 −0.407295 0.913297i \(-0.633528\pi\)
−0.407295 + 0.913297i \(0.633528\pi\)
\(60\) 122.167 0.262862
\(61\) 11.4860 0.0241088 0.0120544 0.999927i \(-0.496163\pi\)
0.0120544 + 0.999927i \(0.496163\pi\)
\(62\) 1132.10 2.31897
\(63\) 449.139 0.898193
\(64\) −426.795 −0.833585
\(65\) 286.664 0.547020
\(66\) −143.148 −0.266974
\(67\) −291.136 −0.530864 −0.265432 0.964130i \(-0.585515\pi\)
−0.265432 + 0.964130i \(0.585515\pi\)
\(68\) 570.391 1.01721
\(69\) −84.1823 −0.146875
\(70\) −550.967 −0.940759
\(71\) 921.785 1.54078 0.770392 0.637570i \(-0.220060\pi\)
0.770392 + 0.637570i \(0.220060\pi\)
\(72\) −40.1861 −0.0657775
\(73\) 1188.93 1.90622 0.953111 0.302622i \(-0.0978621\pi\)
0.953111 + 0.302622i \(0.0978621\pi\)
\(74\) −1127.55 −1.77129
\(75\) −83.0164 −0.127812
\(76\) 139.803 0.211006
\(77\) 309.300 0.457767
\(78\) −746.096 −1.08306
\(79\) −179.464 −0.255586 −0.127793 0.991801i \(-0.540789\pi\)
−0.127793 + 0.991801i \(0.540789\pi\)
\(80\) 343.618 0.480221
\(81\) −42.5781 −0.0584062
\(82\) 550.746 0.741704
\(83\) −184.309 −0.243741 −0.121871 0.992546i \(-0.538889\pi\)
−0.121871 + 0.992546i \(0.538889\pi\)
\(84\) 687.026 0.892389
\(85\) −387.597 −0.494598
\(86\) 796.248 0.998391
\(87\) −222.292 −0.273933
\(88\) −27.6742 −0.0335237
\(89\) −335.864 −0.400017 −0.200008 0.979794i \(-0.564097\pi\)
−0.200008 + 0.979794i \(0.564097\pi\)
\(90\) −312.990 −0.366578
\(91\) 1612.10 1.85707
\(92\) 186.534 0.211386
\(93\) 959.266 1.06958
\(94\) −747.938 −0.820680
\(95\) −95.0000 −0.102598
\(96\) −827.496 −0.879750
\(97\) −1155.10 −1.20910 −0.604551 0.796566i \(-0.706648\pi\)
−0.604551 + 0.796566i \(0.706648\pi\)
\(98\) −1754.24 −1.80822
\(99\) 175.706 0.178375
\(100\) 183.951 0.183951
\(101\) 1827.61 1.80053 0.900265 0.435342i \(-0.143372\pi\)
0.900265 + 0.435342i \(0.143372\pi\)
\(102\) 1008.79 0.979270
\(103\) 902.201 0.863073 0.431537 0.902095i \(-0.357972\pi\)
0.431537 + 0.902095i \(0.357972\pi\)
\(104\) −144.240 −0.135999
\(105\) −466.854 −0.433908
\(106\) −2293.39 −2.10145
\(107\) 253.661 0.229181 0.114590 0.993413i \(-0.463444\pi\)
0.114590 + 0.993413i \(0.463444\pi\)
\(108\) 1049.99 0.935509
\(109\) −1165.42 −1.02410 −0.512050 0.858956i \(-0.671114\pi\)
−0.512050 + 0.858956i \(0.671114\pi\)
\(110\) −215.541 −0.186828
\(111\) −955.416 −0.816974
\(112\) 1932.38 1.63030
\(113\) 1107.31 0.921829 0.460914 0.887445i \(-0.347522\pi\)
0.460914 + 0.887445i \(0.347522\pi\)
\(114\) 247.255 0.203137
\(115\) −126.756 −0.102783
\(116\) 492.562 0.394252
\(117\) 915.791 0.723632
\(118\) 1446.72 1.12866
\(119\) −2179.71 −1.67911
\(120\) 41.7712 0.0317764
\(121\) 121.000 0.0909091
\(122\) −45.0129 −0.0334039
\(123\) 466.667 0.342097
\(124\) −2125.58 −1.53937
\(125\) −125.000 −0.0894427
\(126\) −1760.14 −1.24449
\(127\) 1360.23 0.950401 0.475200 0.879878i \(-0.342376\pi\)
0.475200 + 0.879878i \(0.342376\pi\)
\(128\) −320.991 −0.221655
\(129\) 674.691 0.460490
\(130\) −1123.42 −0.757925
\(131\) 1370.14 0.913815 0.456908 0.889514i \(-0.348957\pi\)
0.456908 + 0.889514i \(0.348957\pi\)
\(132\) 268.768 0.177222
\(133\) −534.246 −0.348308
\(134\) 1140.94 0.735540
\(135\) −713.497 −0.454874
\(136\) 195.027 0.122966
\(137\) −1593.95 −0.994014 −0.497007 0.867747i \(-0.665568\pi\)
−0.497007 + 0.867747i \(0.665568\pi\)
\(138\) 329.905 0.203503
\(139\) 2001.81 1.22152 0.610760 0.791816i \(-0.290864\pi\)
0.610760 + 0.791816i \(0.290864\pi\)
\(140\) 1034.47 0.624492
\(141\) −633.756 −0.378524
\(142\) −3612.41 −2.13484
\(143\) 630.661 0.368801
\(144\) 1097.74 0.635266
\(145\) −334.711 −0.191698
\(146\) −4659.35 −2.64117
\(147\) −1486.44 −0.834008
\(148\) 2117.04 1.17581
\(149\) −1191.38 −0.655047 −0.327523 0.944843i \(-0.606214\pi\)
−0.327523 + 0.944843i \(0.606214\pi\)
\(150\) 325.336 0.177090
\(151\) −1632.57 −0.879845 −0.439923 0.898036i \(-0.644994\pi\)
−0.439923 + 0.898036i \(0.644994\pi\)
\(152\) 47.8010 0.0255077
\(153\) −1238.24 −0.654285
\(154\) −1212.13 −0.634259
\(155\) 1444.39 0.748493
\(156\) 1400.84 0.718955
\(157\) 1812.99 0.921606 0.460803 0.887503i \(-0.347562\pi\)
0.460803 + 0.887503i \(0.347562\pi\)
\(158\) 703.306 0.354127
\(159\) −1943.28 −0.969256
\(160\) −1245.98 −0.615647
\(161\) −712.828 −0.348936
\(162\) 166.861 0.0809249
\(163\) 785.746 0.377573 0.188787 0.982018i \(-0.439545\pi\)
0.188787 + 0.982018i \(0.439545\pi\)
\(164\) −1034.06 −0.492356
\(165\) −182.636 −0.0861709
\(166\) 722.294 0.337716
\(167\) −2673.35 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(168\) 234.906 0.107877
\(169\) 1090.05 0.496155
\(170\) 1518.97 0.685291
\(171\) −303.492 −0.135723
\(172\) −1495.00 −0.662750
\(173\) −2380.16 −1.04601 −0.523007 0.852328i \(-0.675190\pi\)
−0.523007 + 0.852328i \(0.675190\pi\)
\(174\) 871.146 0.379548
\(175\) −702.955 −0.303648
\(176\) 755.960 0.323765
\(177\) 1225.86 0.520572
\(178\) 1316.23 0.554244
\(179\) −1090.90 −0.455520 −0.227760 0.973717i \(-0.573140\pi\)
−0.227760 + 0.973717i \(0.573140\pi\)
\(180\) 587.658 0.243341
\(181\) −3398.71 −1.39572 −0.697858 0.716237i \(-0.745863\pi\)
−0.697858 + 0.716237i \(0.745863\pi\)
\(182\) −6317.69 −2.57307
\(183\) −38.1411 −0.0154070
\(184\) 63.7794 0.0255537
\(185\) −1438.59 −0.571717
\(186\) −3759.30 −1.48196
\(187\) −852.714 −0.333458
\(188\) 1404.30 0.544782
\(189\) −4012.45 −1.54425
\(190\) 372.299 0.142155
\(191\) −1008.83 −0.382181 −0.191091 0.981572i \(-0.561202\pi\)
−0.191091 + 0.981572i \(0.561202\pi\)
\(192\) 1417.24 0.532711
\(193\) 809.763 0.302011 0.151005 0.988533i \(-0.451749\pi\)
0.151005 + 0.988533i \(0.451749\pi\)
\(194\) 4526.77 1.67527
\(195\) −951.913 −0.349579
\(196\) 3293.70 1.20033
\(197\) −2965.40 −1.07247 −0.536233 0.844070i \(-0.680153\pi\)
−0.536233 + 0.844070i \(0.680153\pi\)
\(198\) −688.578 −0.247147
\(199\) −1714.62 −0.610783 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(200\) 62.8960 0.0222371
\(201\) 966.762 0.339254
\(202\) −7162.26 −2.49473
\(203\) −1882.29 −0.650793
\(204\) −1894.07 −0.650057
\(205\) 702.673 0.239399
\(206\) −3535.67 −1.19583
\(207\) −404.939 −0.135967
\(208\) 3940.12 1.31345
\(209\) −209.000 −0.0691714
\(210\) 1829.57 0.601202
\(211\) 3132.39 1.02200 0.511002 0.859580i \(-0.329275\pi\)
0.511002 + 0.859580i \(0.329275\pi\)
\(212\) 4305.98 1.39498
\(213\) −3060.93 −0.984655
\(214\) −994.080 −0.317542
\(215\) 1015.90 0.322250
\(216\) 359.009 0.113090
\(217\) 8122.74 2.54105
\(218\) 4567.19 1.41894
\(219\) −3948.04 −1.21819
\(220\) 404.692 0.124020
\(221\) −4444.41 −1.35278
\(222\) 3744.21 1.13196
\(223\) −908.542 −0.272827 −0.136414 0.990652i \(-0.543558\pi\)
−0.136414 + 0.990652i \(0.543558\pi\)
\(224\) −7006.96 −2.09005
\(225\) −399.331 −0.118320
\(226\) −4339.46 −1.27724
\(227\) 774.695 0.226513 0.113256 0.993566i \(-0.463872\pi\)
0.113256 + 0.993566i \(0.463872\pi\)
\(228\) −464.236 −0.134846
\(229\) −5095.67 −1.47044 −0.735221 0.677827i \(-0.762922\pi\)
−0.735221 + 0.677827i \(0.762922\pi\)
\(230\) 496.747 0.142411
\(231\) −1027.08 −0.292541
\(232\) 168.416 0.0476596
\(233\) −1911.29 −0.537395 −0.268697 0.963225i \(-0.586593\pi\)
−0.268697 + 0.963225i \(0.586593\pi\)
\(234\) −3588.92 −1.00263
\(235\) −954.262 −0.264890
\(236\) −2716.31 −0.749222
\(237\) 595.938 0.163335
\(238\) 8542.13 2.32649
\(239\) −4410.63 −1.19372 −0.596861 0.802344i \(-0.703586\pi\)
−0.596861 + 0.802344i \(0.703586\pi\)
\(240\) −1141.04 −0.306890
\(241\) 250.078 0.0668421 0.0334211 0.999441i \(-0.489360\pi\)
0.0334211 + 0.999441i \(0.489360\pi\)
\(242\) −474.191 −0.125959
\(243\) −3711.50 −0.979805
\(244\) 84.5144 0.0221741
\(245\) −2238.17 −0.583637
\(246\) −1828.84 −0.473994
\(247\) −1089.32 −0.280615
\(248\) −726.772 −0.186089
\(249\) 612.026 0.155765
\(250\) 489.867 0.123928
\(251\) −3445.73 −0.866503 −0.433252 0.901273i \(-0.642634\pi\)
−0.433252 + 0.901273i \(0.642634\pi\)
\(252\) 3304.78 0.826116
\(253\) −278.862 −0.0692961
\(254\) −5330.65 −1.31683
\(255\) 1287.08 0.316078
\(256\) 4672.30 1.14070
\(257\) −3970.68 −0.963751 −0.481876 0.876240i \(-0.660044\pi\)
−0.481876 + 0.876240i \(0.660044\pi\)
\(258\) −2644.07 −0.638033
\(259\) −8090.14 −1.94091
\(260\) 2109.28 0.503124
\(261\) −1069.28 −0.253590
\(262\) −5369.49 −1.26614
\(263\) 4618.72 1.08290 0.541450 0.840733i \(-0.317876\pi\)
0.541450 + 0.840733i \(0.317876\pi\)
\(264\) 91.8967 0.0214237
\(265\) −2926.04 −0.678284
\(266\) 2093.67 0.482599
\(267\) 1115.29 0.255635
\(268\) −2142.18 −0.488264
\(269\) 4579.37 1.03795 0.518976 0.854789i \(-0.326313\pi\)
0.518976 + 0.854789i \(0.326313\pi\)
\(270\) 2796.15 0.630252
\(271\) 196.774 0.0441077 0.0220539 0.999757i \(-0.492979\pi\)
0.0220539 + 0.999757i \(0.492979\pi\)
\(272\) −5327.42 −1.18758
\(273\) −5353.22 −1.18678
\(274\) 6246.56 1.37726
\(275\) −275.000 −0.0603023
\(276\) −619.416 −0.135089
\(277\) 1689.16 0.366396 0.183198 0.983076i \(-0.441355\pi\)
0.183198 + 0.983076i \(0.441355\pi\)
\(278\) −7844.95 −1.69248
\(279\) 4614.33 0.990152
\(280\) 353.705 0.0754925
\(281\) 1164.31 0.247177 0.123588 0.992334i \(-0.460560\pi\)
0.123588 + 0.992334i \(0.460560\pi\)
\(282\) 2483.65 0.524464
\(283\) −6879.58 −1.44505 −0.722523 0.691346i \(-0.757018\pi\)
−0.722523 + 0.691346i \(0.757018\pi\)
\(284\) 6782.52 1.41714
\(285\) 315.462 0.0655662
\(286\) −2471.52 −0.510993
\(287\) 3951.58 0.812733
\(288\) −3980.48 −0.814416
\(289\) 1096.27 0.223137
\(290\) 1311.71 0.265607
\(291\) 3835.70 0.772690
\(292\) 8748.21 1.75325
\(293\) 7796.11 1.55445 0.777225 0.629223i \(-0.216627\pi\)
0.777225 + 0.629223i \(0.216627\pi\)
\(294\) 5825.24 1.15556
\(295\) 1845.81 0.364296
\(296\) 723.855 0.142139
\(297\) −1569.69 −0.306676
\(298\) 4668.95 0.907601
\(299\) −1453.45 −0.281121
\(300\) −610.837 −0.117556
\(301\) 5713.05 1.09400
\(302\) 6397.93 1.21907
\(303\) −6068.85 −1.15065
\(304\) −1305.75 −0.246348
\(305\) −57.4301 −0.0107818
\(306\) 4852.57 0.906545
\(307\) 1410.97 0.262307 0.131153 0.991362i \(-0.458132\pi\)
0.131153 + 0.991362i \(0.458132\pi\)
\(308\) 2275.84 0.421032
\(309\) −2995.90 −0.551556
\(310\) −5660.48 −1.03708
\(311\) −2507.83 −0.457254 −0.228627 0.973514i \(-0.573424\pi\)
−0.228627 + 0.973514i \(0.573424\pi\)
\(312\) 478.972 0.0869118
\(313\) 3904.38 0.705075 0.352537 0.935798i \(-0.385319\pi\)
0.352537 + 0.935798i \(0.385319\pi\)
\(314\) −7104.97 −1.27693
\(315\) −2245.69 −0.401684
\(316\) −1320.50 −0.235076
\(317\) 8171.74 1.44786 0.723928 0.689875i \(-0.242335\pi\)
0.723928 + 0.689875i \(0.242335\pi\)
\(318\) 7615.56 1.34295
\(319\) −736.363 −0.129243
\(320\) 2133.98 0.372790
\(321\) −842.321 −0.146460
\(322\) 2793.52 0.483469
\(323\) 1472.87 0.253724
\(324\) −313.291 −0.0537193
\(325\) −1433.32 −0.244635
\(326\) −3079.29 −0.523147
\(327\) 3869.95 0.654461
\(328\) −353.563 −0.0595190
\(329\) −5366.43 −0.899273
\(330\) 715.739 0.119394
\(331\) 7359.36 1.22208 0.611038 0.791602i \(-0.290752\pi\)
0.611038 + 0.791602i \(0.290752\pi\)
\(332\) −1356.15 −0.224182
\(333\) −4595.80 −0.756302
\(334\) 10476.7 1.71634
\(335\) 1455.68 0.237410
\(336\) −6416.79 −1.04186
\(337\) −1322.59 −0.213787 −0.106894 0.994270i \(-0.534090\pi\)
−0.106894 + 0.994270i \(0.534090\pi\)
\(338\) −4271.84 −0.687448
\(339\) −3676.98 −0.589104
\(340\) −2851.95 −0.454908
\(341\) 3177.66 0.504634
\(342\) 1189.36 0.188051
\(343\) −2942.09 −0.463143
\(344\) −511.168 −0.0801173
\(345\) 420.912 0.0656844
\(346\) 9327.70 1.44931
\(347\) 1027.87 0.159017 0.0795083 0.996834i \(-0.474665\pi\)
0.0795083 + 0.996834i \(0.474665\pi\)
\(348\) −1635.63 −0.251951
\(349\) −511.102 −0.0783917 −0.0391958 0.999232i \(-0.512480\pi\)
−0.0391958 + 0.999232i \(0.512480\pi\)
\(350\) 2754.83 0.420720
\(351\) −8181.36 −1.24413
\(352\) −2741.16 −0.415069
\(353\) −5115.08 −0.771242 −0.385621 0.922657i \(-0.626013\pi\)
−0.385621 + 0.922657i \(0.626013\pi\)
\(354\) −4804.06 −0.721280
\(355\) −4608.92 −0.689060
\(356\) −2471.30 −0.367917
\(357\) 7238.06 1.07305
\(358\) 4275.18 0.631146
\(359\) 6241.86 0.917640 0.458820 0.888529i \(-0.348272\pi\)
0.458820 + 0.888529i \(0.348272\pi\)
\(360\) 200.931 0.0294166
\(361\) 361.000 0.0526316
\(362\) 13319.3 1.93384
\(363\) −401.799 −0.0580964
\(364\) 11861.8 1.70805
\(365\) −5944.67 −0.852488
\(366\) 149.472 0.0213471
\(367\) −2128.98 −0.302811 −0.151406 0.988472i \(-0.548380\pi\)
−0.151406 + 0.988472i \(0.548380\pi\)
\(368\) −1742.22 −0.246792
\(369\) 2244.79 0.316692
\(370\) 5637.76 0.792143
\(371\) −16455.0 −2.30270
\(372\) 7058.31 0.983754
\(373\) 11718.8 1.62674 0.813370 0.581746i \(-0.197630\pi\)
0.813370 + 0.581746i \(0.197630\pi\)
\(374\) 3341.73 0.462023
\(375\) 415.082 0.0571593
\(376\) 480.154 0.0658566
\(377\) −3837.98 −0.524313
\(378\) 15724.5 2.13963
\(379\) 551.770 0.0747824 0.0373912 0.999301i \(-0.488095\pi\)
0.0373912 + 0.999301i \(0.488095\pi\)
\(380\) −699.013 −0.0943647
\(381\) −4516.86 −0.607364
\(382\) 3953.55 0.529532
\(383\) 2797.83 0.373270 0.186635 0.982429i \(-0.440242\pi\)
0.186635 + 0.982429i \(0.440242\pi\)
\(384\) 1065.90 0.141651
\(385\) −1546.50 −0.204719
\(386\) −3173.41 −0.418451
\(387\) 3245.44 0.426292
\(388\) −8499.28 −1.11208
\(389\) −14952.0 −1.94883 −0.974415 0.224755i \(-0.927842\pi\)
−0.974415 + 0.224755i \(0.927842\pi\)
\(390\) 3730.48 0.484360
\(391\) 1965.21 0.254181
\(392\) 1126.17 0.145103
\(393\) −4549.77 −0.583983
\(394\) 11621.2 1.48596
\(395\) 897.319 0.114301
\(396\) 1292.85 0.164061
\(397\) 1991.30 0.251740 0.125870 0.992047i \(-0.459828\pi\)
0.125870 + 0.992047i \(0.459828\pi\)
\(398\) 6719.46 0.846272
\(399\) 1774.05 0.222590
\(400\) −1718.09 −0.214761
\(401\) 14488.0 1.80422 0.902112 0.431502i \(-0.142016\pi\)
0.902112 + 0.431502i \(0.142016\pi\)
\(402\) −3788.67 −0.470054
\(403\) 16562.2 2.04720
\(404\) 13447.6 1.65604
\(405\) 212.891 0.0261201
\(406\) 7376.57 0.901707
\(407\) −3164.91 −0.385451
\(408\) −647.617 −0.0785829
\(409\) −8769.62 −1.06022 −0.530110 0.847929i \(-0.677849\pi\)
−0.530110 + 0.847929i \(0.677849\pi\)
\(410\) −2753.73 −0.331700
\(411\) 5292.94 0.635235
\(412\) 6638.42 0.793815
\(413\) 10380.2 1.23674
\(414\) 1586.93 0.188390
\(415\) 921.544 0.109004
\(416\) −14287.1 −1.68386
\(417\) −6647.32 −0.780625
\(418\) 819.057 0.0958406
\(419\) −9576.93 −1.11662 −0.558310 0.829633i \(-0.688550\pi\)
−0.558310 + 0.829633i \(0.688550\pi\)
\(420\) −3435.13 −0.399088
\(421\) 15818.9 1.83127 0.915637 0.402005i \(-0.131687\pi\)
0.915637 + 0.402005i \(0.131687\pi\)
\(422\) −12275.6 −1.41604
\(423\) −3048.53 −0.350413
\(424\) 1472.29 0.168634
\(425\) 1937.99 0.221191
\(426\) 11995.6 1.36429
\(427\) −322.966 −0.0366029
\(428\) 1866.44 0.210790
\(429\) −2094.21 −0.235686
\(430\) −3981.24 −0.446494
\(431\) −12782.7 −1.42859 −0.714293 0.699846i \(-0.753252\pi\)
−0.714293 + 0.699846i \(0.753252\pi\)
\(432\) −9806.82 −1.09220
\(433\) −14205.7 −1.57663 −0.788315 0.615272i \(-0.789046\pi\)
−0.788315 + 0.615272i \(0.789046\pi\)
\(434\) −31832.5 −3.52076
\(435\) 1111.46 0.122507
\(436\) −8575.18 −0.941919
\(437\) 481.671 0.0527264
\(438\) 15472.1 1.68787
\(439\) 1601.33 0.174094 0.0870471 0.996204i \(-0.472257\pi\)
0.0870471 + 0.996204i \(0.472257\pi\)
\(440\) 138.371 0.0149923
\(441\) −7150.15 −0.772071
\(442\) 17417.3 1.87434
\(443\) −8342.54 −0.894732 −0.447366 0.894351i \(-0.647638\pi\)
−0.447366 + 0.894351i \(0.647638\pi\)
\(444\) −7029.98 −0.751414
\(445\) 1679.32 0.178893
\(446\) 3560.52 0.378016
\(447\) 3956.18 0.418615
\(448\) 12000.7 1.26558
\(449\) −752.975 −0.0791428 −0.0395714 0.999217i \(-0.512599\pi\)
−0.0395714 + 0.999217i \(0.512599\pi\)
\(450\) 1564.95 0.163939
\(451\) 1545.88 0.161403
\(452\) 8147.59 0.847855
\(453\) 5421.20 0.562274
\(454\) −3035.98 −0.313845
\(455\) −8060.48 −0.830508
\(456\) −158.731 −0.0163010
\(457\) 9358.93 0.957970 0.478985 0.877823i \(-0.341005\pi\)
0.478985 + 0.877823i \(0.341005\pi\)
\(458\) 19969.6 2.03737
\(459\) 11062.0 1.12490
\(460\) −932.671 −0.0945348
\(461\) 4841.74 0.489159 0.244579 0.969629i \(-0.421350\pi\)
0.244579 + 0.969629i \(0.421350\pi\)
\(462\) 4025.06 0.405330
\(463\) 17472.4 1.75381 0.876903 0.480668i \(-0.159606\pi\)
0.876903 + 0.480668i \(0.159606\pi\)
\(464\) −4600.50 −0.460287
\(465\) −4796.33 −0.478332
\(466\) 7490.23 0.744588
\(467\) −5971.48 −0.591707 −0.295854 0.955233i \(-0.595604\pi\)
−0.295854 + 0.955233i \(0.595604\pi\)
\(468\) 6738.42 0.665563
\(469\) 8186.21 0.805979
\(470\) 3739.69 0.367019
\(471\) −6020.31 −0.588962
\(472\) −928.753 −0.0905706
\(473\) 2234.98 0.217261
\(474\) −2335.44 −0.226309
\(475\) 475.000 0.0458831
\(476\) −16038.4 −1.54436
\(477\) −9347.67 −0.897275
\(478\) 17284.9 1.65396
\(479\) −5237.71 −0.499618 −0.249809 0.968295i \(-0.580368\pi\)
−0.249809 + 0.968295i \(0.580368\pi\)
\(480\) 4137.48 0.393436
\(481\) −16495.7 −1.56370
\(482\) −980.040 −0.0926133
\(483\) 2367.06 0.222991
\(484\) 890.322 0.0836140
\(485\) 5775.52 0.540727
\(486\) 14545.1 1.35757
\(487\) −14711.5 −1.36888 −0.684438 0.729071i \(-0.739952\pi\)
−0.684438 + 0.729071i \(0.739952\pi\)
\(488\) 28.8970 0.00268054
\(489\) −2609.19 −0.241292
\(490\) 8771.22 0.808660
\(491\) −6658.64 −0.612016 −0.306008 0.952029i \(-0.598994\pi\)
−0.306008 + 0.952029i \(0.598994\pi\)
\(492\) 3433.75 0.314645
\(493\) 5189.32 0.474067
\(494\) 4268.98 0.388807
\(495\) −878.528 −0.0797715
\(496\) 19852.8 1.79721
\(497\) −25918.9 −2.33928
\(498\) −2398.49 −0.215821
\(499\) 10280.8 0.922306 0.461153 0.887321i \(-0.347436\pi\)
0.461153 + 0.887321i \(0.347436\pi\)
\(500\) −919.754 −0.0822653
\(501\) 8877.28 0.791632
\(502\) 13503.6 1.20058
\(503\) 5869.41 0.520286 0.260143 0.965570i \(-0.416230\pi\)
0.260143 + 0.965570i \(0.416230\pi\)
\(504\) 1129.96 0.0998660
\(505\) −9138.03 −0.805222
\(506\) 1092.84 0.0960134
\(507\) −3619.69 −0.317073
\(508\) 10008.6 0.874134
\(509\) 4264.71 0.371375 0.185688 0.982609i \(-0.440549\pi\)
0.185688 + 0.982609i \(0.440549\pi\)
\(510\) −5043.97 −0.437943
\(511\) −33430.7 −2.89410
\(512\) −15742.5 −1.35884
\(513\) 2711.29 0.233346
\(514\) 15560.8 1.33533
\(515\) −4511.01 −0.385978
\(516\) 4964.39 0.423537
\(517\) −2099.38 −0.178589
\(518\) 31704.7 2.68924
\(519\) 7903.70 0.668467
\(520\) 721.201 0.0608207
\(521\) 21955.6 1.84624 0.923122 0.384508i \(-0.125629\pi\)
0.923122 + 0.384508i \(0.125629\pi\)
\(522\) 4190.44 0.351362
\(523\) 22503.6 1.88148 0.940742 0.339124i \(-0.110131\pi\)
0.940742 + 0.339124i \(0.110131\pi\)
\(524\) 10081.5 0.840485
\(525\) 2334.27 0.194049
\(526\) −18100.5 −1.50041
\(527\) −22393.7 −1.85102
\(528\) −2510.28 −0.206905
\(529\) −11524.3 −0.947179
\(530\) 11467.0 0.939798
\(531\) 5896.71 0.481913
\(532\) −3931.00 −0.320358
\(533\) 8057.25 0.654781
\(534\) −4370.74 −0.354196
\(535\) −1268.30 −0.102493
\(536\) −732.451 −0.0590244
\(537\) 3622.52 0.291105
\(538\) −17946.2 −1.43814
\(539\) −4923.96 −0.393488
\(540\) −5249.93 −0.418372
\(541\) 1278.17 0.101577 0.0507883 0.998709i \(-0.483827\pi\)
0.0507883 + 0.998709i \(0.483827\pi\)
\(542\) −771.145 −0.0611135
\(543\) 11286.0 0.891947
\(544\) 19317.6 1.52249
\(545\) 5827.09 0.457991
\(546\) 20978.9 1.64435
\(547\) −20428.7 −1.59684 −0.798418 0.602103i \(-0.794330\pi\)
−0.798418 + 0.602103i \(0.794330\pi\)
\(548\) −11728.3 −0.914248
\(549\) −183.469 −0.0142628
\(550\) 1077.71 0.0835519
\(551\) 1271.90 0.0983389
\(552\) −211.789 −0.0163304
\(553\) 5046.20 0.388040
\(554\) −6619.69 −0.507660
\(555\) 4777.08 0.365362
\(556\) 14729.4 1.12350
\(557\) 11799.0 0.897561 0.448781 0.893642i \(-0.351858\pi\)
0.448781 + 0.893642i \(0.351858\pi\)
\(558\) −18083.2 −1.37191
\(559\) 11648.9 0.881386
\(560\) −9661.92 −0.729091
\(561\) 2831.57 0.213100
\(562\) −4562.84 −0.342476
\(563\) 402.180 0.0301064 0.0150532 0.999887i \(-0.495208\pi\)
0.0150532 + 0.999887i \(0.495208\pi\)
\(564\) −4663.19 −0.348149
\(565\) −5536.53 −0.412254
\(566\) 26960.6 2.00219
\(567\) 1197.22 0.0886747
\(568\) 2319.06 0.171313
\(569\) 25457.3 1.87561 0.937807 0.347157i \(-0.112853\pi\)
0.937807 + 0.347157i \(0.112853\pi\)
\(570\) −1236.28 −0.0908454
\(571\) 9738.70 0.713751 0.356876 0.934152i \(-0.383842\pi\)
0.356876 + 0.934152i \(0.383842\pi\)
\(572\) 4640.42 0.339206
\(573\) 3349.99 0.244237
\(574\) −15486.0 −1.12608
\(575\) 633.778 0.0459659
\(576\) 6817.30 0.493150
\(577\) 2334.56 0.168438 0.0842192 0.996447i \(-0.473160\pi\)
0.0842192 + 0.996447i \(0.473160\pi\)
\(578\) −4296.21 −0.309168
\(579\) −2688.95 −0.193003
\(580\) −2462.81 −0.176315
\(581\) 5182.43 0.370058
\(582\) −15031.9 −1.07060
\(583\) −6437.29 −0.457299
\(584\) 2991.17 0.211944
\(585\) −4578.95 −0.323618
\(586\) −30552.4 −2.15377
\(587\) −2351.89 −0.165371 −0.0826856 0.996576i \(-0.526350\pi\)
−0.0826856 + 0.996576i \(0.526350\pi\)
\(588\) −10937.2 −0.767082
\(589\) −5488.69 −0.383969
\(590\) −7233.61 −0.504751
\(591\) 9847.07 0.685371
\(592\) −19773.1 −1.37275
\(593\) 2161.67 0.149695 0.0748476 0.997195i \(-0.476153\pi\)
0.0748476 + 0.997195i \(0.476153\pi\)
\(594\) 6151.52 0.424916
\(595\) 10898.5 0.750919
\(596\) −8766.24 −0.602482
\(597\) 5693.65 0.390328
\(598\) 5695.98 0.389508
\(599\) 596.820 0.0407102 0.0203551 0.999793i \(-0.493520\pi\)
0.0203551 + 0.999793i \(0.493520\pi\)
\(600\) −208.856 −0.0142109
\(601\) 22746.4 1.54384 0.771918 0.635722i \(-0.219298\pi\)
0.771918 + 0.635722i \(0.219298\pi\)
\(602\) −22389.1 −1.51580
\(603\) 4650.38 0.314060
\(604\) −12012.5 −0.809241
\(605\) −605.000 −0.0406558
\(606\) 23783.4 1.59428
\(607\) −669.883 −0.0447936 −0.0223968 0.999749i \(-0.507130\pi\)
−0.0223968 + 0.999749i \(0.507130\pi\)
\(608\) 4734.73 0.315820
\(609\) 6250.44 0.415896
\(610\) 225.065 0.0149387
\(611\) −10942.1 −0.724502
\(612\) −9110.99 −0.601781
\(613\) 11588.7 0.763562 0.381781 0.924253i \(-0.375311\pi\)
0.381781 + 0.924253i \(0.375311\pi\)
\(614\) −5529.49 −0.363440
\(615\) −2333.34 −0.152991
\(616\) 778.150 0.0508970
\(617\) −3803.57 −0.248178 −0.124089 0.992271i \(-0.539601\pi\)
−0.124089 + 0.992271i \(0.539601\pi\)
\(618\) 11740.7 0.764210
\(619\) −16334.9 −1.06067 −0.530337 0.847787i \(-0.677934\pi\)
−0.530337 + 0.847787i \(0.677934\pi\)
\(620\) 10627.9 0.688429
\(621\) 3617.59 0.233766
\(622\) 9828.02 0.633550
\(623\) 9443.89 0.607321
\(624\) −13083.8 −0.839376
\(625\) 625.000 0.0400000
\(626\) −15301.0 −0.976917
\(627\) 694.017 0.0442048
\(628\) 13340.0 0.847650
\(629\) 22303.8 1.41385
\(630\) 8800.72 0.556554
\(631\) −10215.1 −0.644466 −0.322233 0.946660i \(-0.604433\pi\)
−0.322233 + 0.946660i \(0.604433\pi\)
\(632\) −451.502 −0.0284174
\(633\) −10401.6 −0.653122
\(634\) −32024.5 −2.00608
\(635\) −6801.15 −0.425032
\(636\) −14298.7 −0.891477
\(637\) −25664.1 −1.59631
\(638\) 2885.76 0.179072
\(639\) −14723.9 −0.911531
\(640\) 1604.95 0.0991271
\(641\) 5783.20 0.356353 0.178177 0.983999i \(-0.442980\pi\)
0.178177 + 0.983999i \(0.442980\pi\)
\(642\) 3301.00 0.202928
\(643\) −22633.8 −1.38816 −0.694081 0.719897i \(-0.744189\pi\)
−0.694081 + 0.719897i \(0.744189\pi\)
\(644\) −5245.01 −0.320935
\(645\) −3373.45 −0.205937
\(646\) −5772.08 −0.351547
\(647\) −17532.2 −1.06532 −0.532660 0.846330i \(-0.678807\pi\)
−0.532660 + 0.846330i \(0.678807\pi\)
\(648\) −107.120 −0.00649393
\(649\) 4060.78 0.245608
\(650\) 5617.09 0.338954
\(651\) −26972.8 −1.62388
\(652\) 5781.55 0.347274
\(653\) −13881.6 −0.831898 −0.415949 0.909388i \(-0.636551\pi\)
−0.415949 + 0.909388i \(0.636551\pi\)
\(654\) −15166.1 −0.906790
\(655\) −6850.70 −0.408670
\(656\) 9658.05 0.574823
\(657\) −18991.1 −1.12772
\(658\) 21030.7 1.24599
\(659\) −10563.2 −0.624407 −0.312203 0.950015i \(-0.601067\pi\)
−0.312203 + 0.950015i \(0.601067\pi\)
\(660\) −1343.84 −0.0792560
\(661\) −28208.8 −1.65990 −0.829952 0.557835i \(-0.811632\pi\)
−0.829952 + 0.557835i \(0.811632\pi\)
\(662\) −28840.8 −1.69325
\(663\) 14758.4 0.864506
\(664\) −463.692 −0.0271005
\(665\) 2671.23 0.155768
\(666\) 18010.6 1.04790
\(667\) 1697.06 0.0985162
\(668\) −19670.6 −1.13934
\(669\) 3016.96 0.174353
\(670\) −5704.71 −0.328943
\(671\) −126.346 −0.00726906
\(672\) 23267.7 1.33567
\(673\) −869.218 −0.0497859 −0.0248929 0.999690i \(-0.507924\pi\)
−0.0248929 + 0.999690i \(0.507924\pi\)
\(674\) 5183.15 0.296213
\(675\) 3567.48 0.203426
\(676\) 8020.63 0.456340
\(677\) 2529.80 0.143616 0.0718082 0.997418i \(-0.477123\pi\)
0.0718082 + 0.997418i \(0.477123\pi\)
\(678\) 14409.9 0.816235
\(679\) 32479.4 1.83571
\(680\) −975.133 −0.0549921
\(681\) −2572.50 −0.144755
\(682\) −12453.0 −0.699196
\(683\) −26204.9 −1.46808 −0.734042 0.679104i \(-0.762368\pi\)
−0.734042 + 0.679104i \(0.762368\pi\)
\(684\) −2233.10 −0.124831
\(685\) 7969.73 0.444537
\(686\) 11529.9 0.641708
\(687\) 16921.0 0.939702
\(688\) 13963.3 0.773756
\(689\) −33551.6 −1.85517
\(690\) −1649.52 −0.0910092
\(691\) −24451.0 −1.34610 −0.673052 0.739595i \(-0.735017\pi\)
−0.673052 + 0.739595i \(0.735017\pi\)
\(692\) −17513.3 −0.962075
\(693\) −4940.53 −0.270815
\(694\) −4028.14 −0.220326
\(695\) −10009.0 −0.546280
\(696\) −559.251 −0.0304574
\(697\) −10894.2 −0.592032
\(698\) 2002.98 0.108616
\(699\) 6346.75 0.343428
\(700\) −5172.36 −0.279281
\(701\) 27478.4 1.48052 0.740260 0.672320i \(-0.234702\pi\)
0.740260 + 0.672320i \(0.234702\pi\)
\(702\) 32062.2 1.72380
\(703\) 5466.66 0.293285
\(704\) 4694.75 0.251335
\(705\) 3168.78 0.169281
\(706\) 20045.7 1.06860
\(707\) −51389.0 −2.73364
\(708\) 9019.92 0.478798
\(709\) −20299.2 −1.07525 −0.537625 0.843184i \(-0.680678\pi\)
−0.537625 + 0.843184i \(0.680678\pi\)
\(710\) 18062.1 0.954728
\(711\) 2866.62 0.151205
\(712\) −844.980 −0.0444761
\(713\) −7323.39 −0.384661
\(714\) −28365.5 −1.48677
\(715\) −3153.30 −0.164933
\(716\) −8026.90 −0.418966
\(717\) 14646.2 0.762861
\(718\) −24461.4 −1.27144
\(719\) 36000.9 1.86732 0.933662 0.358155i \(-0.116594\pi\)
0.933662 + 0.358155i \(0.116594\pi\)
\(720\) −5488.70 −0.284099
\(721\) −25368.3 −1.31035
\(722\) −1414.73 −0.0729238
\(723\) −830.424 −0.0427162
\(724\) −25007.8 −1.28371
\(725\) 1673.55 0.0857299
\(726\) 1574.62 0.0804956
\(727\) 14499.3 0.739683 0.369842 0.929095i \(-0.379412\pi\)
0.369842 + 0.929095i \(0.379412\pi\)
\(728\) 4055.78 0.206479
\(729\) 13474.2 0.684561
\(730\) 23296.7 1.18117
\(731\) −15750.4 −0.796921
\(732\) −280.643 −0.0141706
\(733\) 12240.5 0.616801 0.308400 0.951257i \(-0.400206\pi\)
0.308400 + 0.951257i \(0.400206\pi\)
\(734\) 8343.32 0.419561
\(735\) 7432.18 0.372980
\(736\) 6317.41 0.316390
\(737\) 3202.49 0.160062
\(738\) −8797.19 −0.438793
\(739\) −27133.2 −1.35062 −0.675311 0.737533i \(-0.735991\pi\)
−0.675311 + 0.737533i \(0.735991\pi\)
\(740\) −10585.2 −0.525839
\(741\) 3617.27 0.179330
\(742\) 64486.0 3.19051
\(743\) 34391.1 1.69810 0.849049 0.528314i \(-0.177176\pi\)
0.849049 + 0.528314i \(0.177176\pi\)
\(744\) 2413.36 0.118922
\(745\) 5956.92 0.292946
\(746\) −45925.0 −2.25393
\(747\) 2944.01 0.144198
\(748\) −6274.30 −0.306699
\(749\) −7132.49 −0.347951
\(750\) −1626.68 −0.0791972
\(751\) −14379.3 −0.698680 −0.349340 0.936996i \(-0.613594\pi\)
−0.349340 + 0.936996i \(0.613594\pi\)
\(752\) −13116.1 −0.636029
\(753\) 11442.1 0.553748
\(754\) 15040.8 0.726463
\(755\) 8162.85 0.393479
\(756\) −29523.7 −1.42033
\(757\) 30444.4 1.46172 0.730859 0.682528i \(-0.239120\pi\)
0.730859 + 0.682528i \(0.239120\pi\)
\(758\) −2162.35 −0.103615
\(759\) 926.006 0.0442844
\(760\) −239.005 −0.0114074
\(761\) −5965.31 −0.284156 −0.142078 0.989855i \(-0.545378\pi\)
−0.142078 + 0.989855i \(0.545378\pi\)
\(762\) 17701.3 0.841534
\(763\) 32769.5 1.55483
\(764\) −7423.03 −0.351513
\(765\) 6191.19 0.292605
\(766\) −10964.5 −0.517185
\(767\) 21165.1 0.996385
\(768\) −15515.1 −0.728976
\(769\) 14686.6 0.688701 0.344351 0.938841i \(-0.388099\pi\)
0.344351 + 0.938841i \(0.388099\pi\)
\(770\) 6060.63 0.283649
\(771\) 13185.3 0.615895
\(772\) 5958.26 0.277775
\(773\) −15133.4 −0.704155 −0.352077 0.935971i \(-0.614525\pi\)
−0.352077 + 0.935971i \(0.614525\pi\)
\(774\) −12718.7 −0.590650
\(775\) −7221.96 −0.334736
\(776\) −2906.06 −0.134435
\(777\) 26864.6 1.24036
\(778\) 58595.8 2.70021
\(779\) −2670.16 −0.122809
\(780\) −7004.20 −0.321527
\(781\) −10139.6 −0.464564
\(782\) −7701.51 −0.352181
\(783\) 9552.60 0.435992
\(784\) −30763.0 −1.40137
\(785\) −9064.93 −0.412155
\(786\) 17830.2 0.809139
\(787\) 32972.8 1.49346 0.746730 0.665127i \(-0.231623\pi\)
0.746730 + 0.665127i \(0.231623\pi\)
\(788\) −21819.5 −0.986404
\(789\) −15337.2 −0.692039
\(790\) −3516.53 −0.158370
\(791\) −31135.5 −1.39956
\(792\) 442.047 0.0198327
\(793\) −658.526 −0.0294892
\(794\) −7803.79 −0.348798
\(795\) 9716.38 0.433465
\(796\) −12616.2 −0.561770
\(797\) 18614.0 0.827279 0.413640 0.910441i \(-0.364257\pi\)
0.413640 + 0.910441i \(0.364257\pi\)
\(798\) −6952.37 −0.308410
\(799\) 14794.8 0.655071
\(800\) 6229.91 0.275326
\(801\) 5364.83 0.236651
\(802\) −56777.3 −2.49985
\(803\) −13078.3 −0.574747
\(804\) 7113.46 0.312030
\(805\) 3564.14 0.156049
\(806\) −64906.2 −2.83651
\(807\) −15206.5 −0.663314
\(808\) 4597.97 0.200193
\(809\) −14371.3 −0.624560 −0.312280 0.949990i \(-0.601093\pi\)
−0.312280 + 0.949990i \(0.601093\pi\)
\(810\) −834.304 −0.0361907
\(811\) −17584.7 −0.761385 −0.380693 0.924702i \(-0.624314\pi\)
−0.380693 + 0.924702i \(0.624314\pi\)
\(812\) −13850.0 −0.598569
\(813\) −653.420 −0.0281875
\(814\) 12403.1 0.534063
\(815\) −3928.73 −0.168856
\(816\) 17690.5 0.758937
\(817\) −3860.42 −0.165311
\(818\) 34367.6 1.46899
\(819\) −25750.4 −1.09865
\(820\) 5170.29 0.220188
\(821\) 7026.99 0.298713 0.149357 0.988783i \(-0.452280\pi\)
0.149357 + 0.988783i \(0.452280\pi\)
\(822\) −20742.7 −0.880151
\(823\) −37636.6 −1.59408 −0.797040 0.603926i \(-0.793602\pi\)
−0.797040 + 0.603926i \(0.793602\pi\)
\(824\) 2269.79 0.0959612
\(825\) 913.181 0.0385368
\(826\) −40679.2 −1.71357
\(827\) −4436.77 −0.186556 −0.0932778 0.995640i \(-0.529734\pi\)
−0.0932778 + 0.995640i \(0.529734\pi\)
\(828\) −2979.56 −0.125056
\(829\) −8504.57 −0.356304 −0.178152 0.984003i \(-0.557012\pi\)
−0.178152 + 0.984003i \(0.557012\pi\)
\(830\) −3611.47 −0.151031
\(831\) −5609.11 −0.234149
\(832\) 24469.4 1.01962
\(833\) 34700.3 1.44333
\(834\) 26050.4 1.08160
\(835\) 13366.8 0.553983
\(836\) −1537.83 −0.0636207
\(837\) −41222.8 −1.70235
\(838\) 37531.3 1.54713
\(839\) −18663.1 −0.767965 −0.383983 0.923340i \(-0.625448\pi\)
−0.383983 + 0.923340i \(0.625448\pi\)
\(840\) −1174.53 −0.0482443
\(841\) −19907.8 −0.816260
\(842\) −61993.3 −2.53733
\(843\) −3866.26 −0.157961
\(844\) 23048.2 0.939991
\(845\) −5450.26 −0.221887
\(846\) 11947.0 0.485515
\(847\) −3402.30 −0.138022
\(848\) −40217.6 −1.62863
\(849\) 22844.7 0.923473
\(850\) −7594.84 −0.306472
\(851\) 7294.00 0.293813
\(852\) −22522.4 −0.905640
\(853\) 24903.2 0.999611 0.499806 0.866138i \(-0.333405\pi\)
0.499806 + 0.866138i \(0.333405\pi\)
\(854\) 1265.68 0.0507152
\(855\) 1517.46 0.0606970
\(856\) 638.170 0.0254816
\(857\) 31032.0 1.23691 0.618456 0.785820i \(-0.287759\pi\)
0.618456 + 0.785820i \(0.287759\pi\)
\(858\) 8207.06 0.326555
\(859\) 35115.5 1.39479 0.697396 0.716686i \(-0.254342\pi\)
0.697396 + 0.716686i \(0.254342\pi\)
\(860\) 7475.02 0.296391
\(861\) −13121.8 −0.519386
\(862\) 50094.5 1.97938
\(863\) 25464.6 1.00443 0.502217 0.864742i \(-0.332518\pi\)
0.502217 + 0.864742i \(0.332518\pi\)
\(864\) 35560.2 1.40021
\(865\) 11900.8 0.467792
\(866\) 55671.1 2.18450
\(867\) −3640.34 −0.142598
\(868\) 59767.4 2.33714
\(869\) 1974.10 0.0770619
\(870\) −4355.73 −0.169739
\(871\) 16691.6 0.649339
\(872\) −2932.01 −0.113865
\(873\) 18450.7 0.715307
\(874\) −1887.64 −0.0730552
\(875\) 3514.77 0.135796
\(876\) −29049.8 −1.12044
\(877\) −22982.2 −0.884894 −0.442447 0.896795i \(-0.645890\pi\)
−0.442447 + 0.896795i \(0.645890\pi\)
\(878\) −6275.50 −0.241216
\(879\) −25888.2 −0.993387
\(880\) −3779.80 −0.144792
\(881\) −9389.36 −0.359064 −0.179532 0.983752i \(-0.557458\pi\)
−0.179532 + 0.983752i \(0.557458\pi\)
\(882\) 28020.9 1.06974
\(883\) −27191.0 −1.03630 −0.518148 0.855291i \(-0.673378\pi\)
−0.518148 + 0.855291i \(0.673378\pi\)
\(884\) −32702.1 −1.24422
\(885\) −6129.30 −0.232807
\(886\) 32693.9 1.23970
\(887\) −36800.3 −1.39305 −0.696523 0.717534i \(-0.745271\pi\)
−0.696523 + 0.717534i \(0.745271\pi\)
\(888\) −2403.67 −0.0908356
\(889\) −38247.2 −1.44294
\(890\) −6581.14 −0.247865
\(891\) 468.360 0.0176101
\(892\) −6685.08 −0.250934
\(893\) 3626.20 0.135886
\(894\) −15504.0 −0.580012
\(895\) 5454.52 0.203715
\(896\) 9025.68 0.336526
\(897\) 4826.41 0.179653
\(898\) 2950.86 0.109656
\(899\) −19338.1 −0.717423
\(900\) −2938.29 −0.108826
\(901\) 45365.0 1.67739
\(902\) −6058.20 −0.223632
\(903\) −18971.1 −0.699134
\(904\) 2785.81 0.102494
\(905\) 16993.6 0.624183
\(906\) −21245.3 −0.779060
\(907\) −2340.27 −0.0856753 −0.0428376 0.999082i \(-0.513640\pi\)
−0.0428376 + 0.999082i \(0.513640\pi\)
\(908\) 5700.23 0.208336
\(909\) −29192.8 −1.06520
\(910\) 31588.5 1.15071
\(911\) 14580.2 0.530256 0.265128 0.964213i \(-0.414586\pi\)
0.265128 + 0.964213i \(0.414586\pi\)
\(912\) 4335.94 0.157431
\(913\) 2027.40 0.0734908
\(914\) −36677.0 −1.32732
\(915\) 190.706 0.00689020
\(916\) −37494.1 −1.35244
\(917\) −38525.9 −1.38739
\(918\) −43351.2 −1.55861
\(919\) −13094.4 −0.470016 −0.235008 0.971993i \(-0.575512\pi\)
−0.235008 + 0.971993i \(0.575512\pi\)
\(920\) −318.897 −0.0114280
\(921\) −4685.34 −0.167630
\(922\) −18974.4 −0.677755
\(923\) −52848.5 −1.88465
\(924\) −7557.28 −0.269065
\(925\) 7192.97 0.255680
\(926\) −68473.2 −2.42999
\(927\) −14411.1 −0.510595
\(928\) 16681.7 0.590092
\(929\) −38888.5 −1.37340 −0.686701 0.726940i \(-0.740942\pi\)
−0.686701 + 0.726940i \(0.740942\pi\)
\(930\) 18796.5 0.662754
\(931\) 8505.03 0.299400
\(932\) −14063.4 −0.494271
\(933\) 8327.65 0.292213
\(934\) 23401.8 0.819841
\(935\) 4263.57 0.149127
\(936\) 2303.98 0.0804573
\(937\) 10205.2 0.355806 0.177903 0.984048i \(-0.443069\pi\)
0.177903 + 0.984048i \(0.443069\pi\)
\(938\) −32081.2 −1.11673
\(939\) −12965.1 −0.450585
\(940\) −7021.49 −0.243634
\(941\) −11322.2 −0.392234 −0.196117 0.980580i \(-0.562833\pi\)
−0.196117 + 0.980580i \(0.562833\pi\)
\(942\) 23593.2 0.816037
\(943\) −3562.71 −0.123031
\(944\) 25370.2 0.874712
\(945\) 20062.2 0.690608
\(946\) −8758.73 −0.301026
\(947\) −52171.9 −1.79024 −0.895122 0.445822i \(-0.852911\pi\)
−0.895122 + 0.445822i \(0.852911\pi\)
\(948\) 4384.93 0.150228
\(949\) −68164.9 −2.33164
\(950\) −1861.49 −0.0635735
\(951\) −27135.5 −0.925268
\(952\) −5483.80 −0.186692
\(953\) 47269.8 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(954\) 36632.9 1.24322
\(955\) 5044.17 0.170917
\(956\) −32453.5 −1.09793
\(957\) 2445.21 0.0825939
\(958\) 20526.2 0.692247
\(959\) 44818.9 1.50915
\(960\) −7086.20 −0.238236
\(961\) 53659.8 1.80121
\(962\) 64645.7 2.16659
\(963\) −4051.79 −0.135584
\(964\) 1840.08 0.0614783
\(965\) −4048.82 −0.135063
\(966\) −9276.33 −0.308966
\(967\) −9102.45 −0.302704 −0.151352 0.988480i \(-0.548363\pi\)
−0.151352 + 0.988480i \(0.548363\pi\)
\(968\) 304.417 0.0101078
\(969\) −4890.90 −0.162145
\(970\) −22633.9 −0.749205
\(971\) −35620.1 −1.17724 −0.588622 0.808409i \(-0.700329\pi\)
−0.588622 + 0.808409i \(0.700329\pi\)
\(972\) −27309.3 −0.901179
\(973\) −56287.2 −1.85456
\(974\) 57653.5 1.89665
\(975\) 4759.56 0.156336
\(976\) −789.361 −0.0258881
\(977\) −49668.6 −1.62645 −0.813224 0.581950i \(-0.802290\pi\)
−0.813224 + 0.581950i \(0.802290\pi\)
\(978\) 10225.3 0.334323
\(979\) 3694.50 0.120610
\(980\) −16468.5 −0.536802
\(981\) 18615.5 0.605859
\(982\) 26094.7 0.847980
\(983\) −17171.1 −0.557145 −0.278573 0.960415i \(-0.589861\pi\)
−0.278573 + 0.960415i \(0.589861\pi\)
\(984\) 1174.06 0.0380363
\(985\) 14827.0 0.479621
\(986\) −20336.6 −0.656845
\(987\) 17820.1 0.574690
\(988\) −8015.27 −0.258097
\(989\) −5150.84 −0.165609
\(990\) 3442.89 0.110528
\(991\) 43853.7 1.40571 0.702855 0.711333i \(-0.251908\pi\)
0.702855 + 0.711333i \(0.251908\pi\)
\(992\) −71987.5 −2.30404
\(993\) −24437.9 −0.780980
\(994\) 101575. 3.24120
\(995\) 8573.08 0.273151
\(996\) 4503.31 0.143266
\(997\) 37818.2 1.20132 0.600659 0.799505i \(-0.294905\pi\)
0.600659 + 0.799505i \(0.294905\pi\)
\(998\) −40289.6 −1.27790
\(999\) 41057.3 1.30030
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 1045.4.a.c.1.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.4 20 1.1 even 1 trivial