Properties

Label 43.4.a.b.1.1
Level 43
Weight 4
Character 43.1
Self dual yes
Analytic conductor 2.537
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.31455\) of \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.31455 q^{2} -7.10409 q^{3} +2.98627 q^{4} +8.51910 q^{5} +23.5469 q^{6} +9.84502 q^{7} +16.6183 q^{8} +23.4681 q^{9} +O(q^{10})\) \(q-3.31455 q^{2} -7.10409 q^{3} +2.98627 q^{4} +8.51910 q^{5} +23.5469 q^{6} +9.84502 q^{7} +16.6183 q^{8} +23.4681 q^{9} -28.2370 q^{10} +0.597776 q^{11} -21.2147 q^{12} +34.9021 q^{13} -32.6318 q^{14} -60.5204 q^{15} -78.9723 q^{16} +91.3552 q^{17} -77.7863 q^{18} -24.0324 q^{19} +25.4403 q^{20} -69.9399 q^{21} -1.98136 q^{22} +127.719 q^{23} -118.058 q^{24} -52.4250 q^{25} -115.685 q^{26} +25.0908 q^{27} +29.3999 q^{28} +285.724 q^{29} +200.598 q^{30} -192.699 q^{31} +128.812 q^{32} -4.24666 q^{33} -302.802 q^{34} +83.8707 q^{35} +70.0821 q^{36} +363.399 q^{37} +79.6565 q^{38} -247.948 q^{39} +141.573 q^{40} +191.586 q^{41} +231.820 q^{42} -43.0000 q^{43} +1.78512 q^{44} +199.927 q^{45} -423.330 q^{46} -458.267 q^{47} +561.027 q^{48} -246.076 q^{49} +173.765 q^{50} -648.995 q^{51} +104.227 q^{52} -583.068 q^{53} -83.1648 q^{54} +5.09252 q^{55} +163.607 q^{56} +170.728 q^{57} -947.047 q^{58} +416.589 q^{59} -180.730 q^{60} -30.9376 q^{61} +638.710 q^{62} +231.044 q^{63} +204.825 q^{64} +297.334 q^{65} +14.0758 q^{66} +16.1466 q^{67} +272.811 q^{68} -907.324 q^{69} -277.994 q^{70} -219.461 q^{71} +390.000 q^{72} +1031.07 q^{73} -1204.50 q^{74} +372.432 q^{75} -71.7670 q^{76} +5.88512 q^{77} +821.836 q^{78} +445.438 q^{79} -672.773 q^{80} -811.887 q^{81} -635.022 q^{82} +298.511 q^{83} -208.859 q^{84} +778.264 q^{85} +142.526 q^{86} -2029.81 q^{87} +9.93402 q^{88} +846.185 q^{89} -662.669 q^{90} +343.612 q^{91} +381.402 q^{92} +1368.95 q^{93} +1518.95 q^{94} -204.734 q^{95} -915.091 q^{96} -259.068 q^{97} +815.631 q^{98} +14.0287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + 57q^{10} - 28q^{11} - 157q^{12} + 56q^{13} - 184q^{14} - 124q^{15} - 54q^{16} + 19q^{17} - 81q^{18} - 75q^{19} + 135q^{20} - 18q^{21} - 504q^{22} + 131q^{23} - 567q^{24} + 105q^{25} + 44q^{26} + 238q^{27} - 404q^{28} + 515q^{29} - 396q^{30} + 237q^{31} + 558q^{32} + 540q^{33} - 107q^{34} + 198q^{35} + 73q^{36} + 269q^{37} + 527q^{38} + 290q^{39} + 613q^{40} + 471q^{41} + 362q^{42} - 258q^{43} - 428q^{44} + 334q^{45} - 67q^{46} + 415q^{47} - 989q^{48} + 350q^{49} + 1335q^{50} - 1241q^{51} - 8q^{52} + 450q^{53} + 402q^{54} - 1732q^{55} - 780q^{56} - 1000q^{57} - 1055q^{58} + 356q^{59} - 2732q^{60} - 1328q^{61} + 1603q^{62} - 2290q^{63} + 466q^{64} - 62q^{65} + 156q^{66} - 632q^{67} + 571q^{68} - 1130q^{69} - 1902q^{70} - 144q^{71} + 567q^{72} + 864q^{73} + 1207q^{74} - 2494q^{75} + 1005q^{76} + 2660q^{77} + 2222q^{78} - 1613q^{79} + 2399q^{80} - 102q^{81} + 1673q^{82} - 682q^{83} + 3758q^{84} + 84q^{85} - 258q^{86} + 449q^{87} - 608q^{88} + 3378q^{89} + 930q^{90} - 3900q^{91} + 3491q^{92} + 1879q^{93} + 3197q^{94} - 79q^{95} - 591q^{96} - 55q^{97} + 2398q^{98} - 1612q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.31455 −1.17187 −0.585936 0.810357i \(-0.699273\pi\)
−0.585936 + 0.810357i \(0.699273\pi\)
\(3\) −7.10409 −1.36718 −0.683592 0.729865i \(-0.739583\pi\)
−0.683592 + 0.729865i \(0.739583\pi\)
\(4\) 2.98627 0.373283
\(5\) 8.51910 0.761971 0.380986 0.924581i \(-0.375585\pi\)
0.380986 + 0.924581i \(0.375585\pi\)
\(6\) 23.5469 1.60216
\(7\) 9.84502 0.531581 0.265791 0.964031i \(-0.414367\pi\)
0.265791 + 0.964031i \(0.414367\pi\)
\(8\) 16.6183 0.734432
\(9\) 23.4681 0.869190
\(10\) −28.2370 −0.892933
\(11\) 0.597776 0.0163851 0.00819256 0.999966i \(-0.497392\pi\)
0.00819256 + 0.999966i \(0.497392\pi\)
\(12\) −21.2147 −0.510347
\(13\) 34.9021 0.744623 0.372311 0.928108i \(-0.378565\pi\)
0.372311 + 0.928108i \(0.378565\pi\)
\(14\) −32.6318 −0.622945
\(15\) −60.5204 −1.04175
\(16\) −78.9723 −1.23394
\(17\) 91.3552 1.30335 0.651673 0.758500i \(-0.274067\pi\)
0.651673 + 0.758500i \(0.274067\pi\)
\(18\) −77.7863 −1.01858
\(19\) −24.0324 −0.290179 −0.145089 0.989419i \(-0.546347\pi\)
−0.145089 + 0.989419i \(0.546347\pi\)
\(20\) 25.4403 0.284431
\(21\) −69.9399 −0.726769
\(22\) −1.98136 −0.0192013
\(23\) 127.719 1.15788 0.578938 0.815371i \(-0.303467\pi\)
0.578938 + 0.815371i \(0.303467\pi\)
\(24\) −118.058 −1.00410
\(25\) −52.4250 −0.419400
\(26\) −115.685 −0.872602
\(27\) 25.0908 0.178842
\(28\) 29.3999 0.198430
\(29\) 285.724 1.82957 0.914786 0.403939i \(-0.132359\pi\)
0.914786 + 0.403939i \(0.132359\pi\)
\(30\) 200.598 1.22080
\(31\) −192.699 −1.11644 −0.558221 0.829692i \(-0.688516\pi\)
−0.558221 + 0.829692i \(0.688516\pi\)
\(32\) 128.812 0.711591
\(33\) −4.24666 −0.0224015
\(34\) −302.802 −1.52735
\(35\) 83.8707 0.405050
\(36\) 70.0821 0.324454
\(37\) 363.399 1.61466 0.807329 0.590101i \(-0.200912\pi\)
0.807329 + 0.590101i \(0.200912\pi\)
\(38\) 79.6565 0.340053
\(39\) −247.948 −1.01804
\(40\) 141.573 0.559616
\(41\) 191.586 0.729774 0.364887 0.931052i \(-0.381108\pi\)
0.364887 + 0.931052i \(0.381108\pi\)
\(42\) 231.820 0.851680
\(43\) −43.0000 −0.152499
\(44\) 1.78512 0.00611629
\(45\) 199.927 0.662297
\(46\) −423.330 −1.35688
\(47\) −458.267 −1.42223 −0.711117 0.703073i \(-0.751811\pi\)
−0.711117 + 0.703073i \(0.751811\pi\)
\(48\) 561.027 1.68703
\(49\) −246.076 −0.717422
\(50\) 173.765 0.491483
\(51\) −648.995 −1.78191
\(52\) 104.227 0.277955
\(53\) −583.068 −1.51114 −0.755572 0.655066i \(-0.772641\pi\)
−0.755572 + 0.655066i \(0.772641\pi\)
\(54\) −83.1648 −0.209580
\(55\) 5.09252 0.0124850
\(56\) 163.607 0.390410
\(57\) 170.728 0.396728
\(58\) −947.047 −2.14402
\(59\) 416.589 0.919243 0.459621 0.888115i \(-0.347985\pi\)
0.459621 + 0.888115i \(0.347985\pi\)
\(60\) −180.730 −0.388869
\(61\) −30.9376 −0.0649369 −0.0324685 0.999473i \(-0.510337\pi\)
−0.0324685 + 0.999473i \(0.510337\pi\)
\(62\) 638.710 1.30833
\(63\) 231.044 0.462045
\(64\) 204.825 0.400049
\(65\) 297.334 0.567381
\(66\) 14.0758 0.0262516
\(67\) 16.1466 0.0294421 0.0147210 0.999892i \(-0.495314\pi\)
0.0147210 + 0.999892i \(0.495314\pi\)
\(68\) 272.811 0.486517
\(69\) −907.324 −1.58303
\(70\) −277.994 −0.474666
\(71\) −219.461 −0.366835 −0.183418 0.983035i \(-0.558716\pi\)
−0.183418 + 0.983035i \(0.558716\pi\)
\(72\) 390.000 0.638360
\(73\) 1031.07 1.65311 0.826557 0.562853i \(-0.190296\pi\)
0.826557 + 0.562853i \(0.190296\pi\)
\(74\) −1204.50 −1.89217
\(75\) 372.432 0.573396
\(76\) −71.7670 −0.108319
\(77\) 5.88512 0.00871002
\(78\) 821.836 1.19301
\(79\) 445.438 0.634375 0.317188 0.948363i \(-0.397262\pi\)
0.317188 + 0.948363i \(0.397262\pi\)
\(80\) −672.773 −0.940229
\(81\) −811.887 −1.11370
\(82\) −635.022 −0.855201
\(83\) 298.511 0.394769 0.197384 0.980326i \(-0.436755\pi\)
0.197384 + 0.980326i \(0.436755\pi\)
\(84\) −208.859 −0.271291
\(85\) 778.264 0.993112
\(86\) 142.526 0.178709
\(87\) −2029.81 −2.50136
\(88\) 9.93402 0.0120338
\(89\) 846.185 1.00781 0.503907 0.863758i \(-0.331895\pi\)
0.503907 + 0.863758i \(0.331895\pi\)
\(90\) −662.669 −0.776128
\(91\) 343.612 0.395827
\(92\) 381.402 0.432216
\(93\) 1368.95 1.52638
\(94\) 1518.95 1.66668
\(95\) −204.734 −0.221108
\(96\) −915.091 −0.972876
\(97\) −259.068 −0.271179 −0.135590 0.990765i \(-0.543293\pi\)
−0.135590 + 0.990765i \(0.543293\pi\)
\(98\) 815.631 0.840726
\(99\) 14.0287 0.0142418
\(100\) −156.555 −0.156555
\(101\) −1182.01 −1.16450 −0.582252 0.813009i \(-0.697828\pi\)
−0.582252 + 0.813009i \(0.697828\pi\)
\(102\) 2151.13 2.08817
\(103\) 319.565 0.305706 0.152853 0.988249i \(-0.451154\pi\)
0.152853 + 0.988249i \(0.451154\pi\)
\(104\) 580.013 0.546874
\(105\) −595.825 −0.553777
\(106\) 1932.61 1.77087
\(107\) 1647.14 1.48818 0.744089 0.668081i \(-0.232884\pi\)
0.744089 + 0.668081i \(0.232884\pi\)
\(108\) 74.9279 0.0667587
\(109\) −560.538 −0.492567 −0.246283 0.969198i \(-0.579209\pi\)
−0.246283 + 0.969198i \(0.579209\pi\)
\(110\) −16.8794 −0.0146308
\(111\) −2581.62 −2.20753
\(112\) −777.484 −0.655941
\(113\) −65.6852 −0.0546827 −0.0273414 0.999626i \(-0.508704\pi\)
−0.0273414 + 0.999626i \(0.508704\pi\)
\(114\) −565.887 −0.464914
\(115\) 1088.05 0.882269
\(116\) 853.248 0.682949
\(117\) 819.086 0.647218
\(118\) −1380.81 −1.07723
\(119\) 899.393 0.692834
\(120\) −1005.75 −0.765097
\(121\) −1330.64 −0.999732
\(122\) 102.544 0.0760977
\(123\) −1361.05 −0.997734
\(124\) −575.450 −0.416749
\(125\) −1511.50 −1.08154
\(126\) −765.808 −0.541457
\(127\) −1906.92 −1.33238 −0.666189 0.745783i \(-0.732076\pi\)
−0.666189 + 0.745783i \(0.732076\pi\)
\(128\) −1709.40 −1.18040
\(129\) 305.476 0.208493
\(130\) −985.530 −0.664898
\(131\) −1536.09 −1.02449 −0.512246 0.858839i \(-0.671186\pi\)
−0.512246 + 0.858839i \(0.671186\pi\)
\(132\) −12.6817 −0.00836209
\(133\) −236.599 −0.154254
\(134\) −53.5187 −0.0345023
\(135\) 213.751 0.136272
\(136\) 1518.17 0.957218
\(137\) −1695.38 −1.05727 −0.528634 0.848850i \(-0.677296\pi\)
−0.528634 + 0.848850i \(0.677296\pi\)
\(138\) 3007.38 1.85511
\(139\) 947.580 0.578221 0.289110 0.957296i \(-0.406641\pi\)
0.289110 + 0.957296i \(0.406641\pi\)
\(140\) 250.460 0.151198
\(141\) 3255.57 1.94446
\(142\) 727.417 0.429884
\(143\) 20.8636 0.0122007
\(144\) −1853.33 −1.07253
\(145\) 2434.11 1.39408
\(146\) −3417.53 −1.93724
\(147\) 1748.14 0.980847
\(148\) 1085.21 0.602725
\(149\) 3084.10 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(150\) −1234.45 −0.671947
\(151\) −2346.83 −1.26478 −0.632391 0.774649i \(-0.717926\pi\)
−0.632391 + 0.774649i \(0.717926\pi\)
\(152\) −399.377 −0.213117
\(153\) 2143.93 1.13285
\(154\) −19.5066 −0.0102070
\(155\) −1641.62 −0.850697
\(156\) −740.438 −0.380016
\(157\) 1582.73 0.804556 0.402278 0.915517i \(-0.368218\pi\)
0.402278 + 0.915517i \(0.368218\pi\)
\(158\) −1476.43 −0.743406
\(159\) 4142.17 2.06601
\(160\) 1097.36 0.542212
\(161\) 1257.39 0.615505
\(162\) 2691.04 1.30511
\(163\) −820.887 −0.394459 −0.197229 0.980357i \(-0.563194\pi\)
−0.197229 + 0.980357i \(0.563194\pi\)
\(164\) 572.127 0.272412
\(165\) −36.1777 −0.0170693
\(166\) −989.430 −0.462618
\(167\) 1690.58 0.783358 0.391679 0.920102i \(-0.371894\pi\)
0.391679 + 0.920102i \(0.371894\pi\)
\(168\) −1162.28 −0.533762
\(169\) −978.845 −0.445537
\(170\) −2579.60 −1.16380
\(171\) −563.994 −0.252221
\(172\) −128.409 −0.0569252
\(173\) 1390.24 0.610973 0.305486 0.952196i \(-0.401181\pi\)
0.305486 + 0.952196i \(0.401181\pi\)
\(174\) 6727.91 2.93127
\(175\) −516.125 −0.222945
\(176\) −47.2078 −0.0202183
\(177\) −2959.49 −1.25677
\(178\) −2804.73 −1.18103
\(179\) 693.895 0.289744 0.144872 0.989450i \(-0.453723\pi\)
0.144872 + 0.989450i \(0.453723\pi\)
\(180\) 597.036 0.247225
\(181\) −2355.50 −0.967309 −0.483655 0.875259i \(-0.660691\pi\)
−0.483655 + 0.875259i \(0.660691\pi\)
\(182\) −1138.92 −0.463859
\(183\) 219.783 0.0887807
\(184\) 2122.46 0.850381
\(185\) 3095.83 1.23032
\(186\) −4537.46 −1.78872
\(187\) 54.6100 0.0213555
\(188\) −1368.51 −0.530897
\(189\) 247.020 0.0950689
\(190\) 678.602 0.259110
\(191\) −2168.09 −0.821349 −0.410674 0.911782i \(-0.634707\pi\)
−0.410674 + 0.911782i \(0.634707\pi\)
\(192\) −1455.10 −0.546940
\(193\) 1305.00 0.486717 0.243358 0.969936i \(-0.421751\pi\)
0.243358 + 0.969936i \(0.421751\pi\)
\(194\) 858.695 0.317787
\(195\) −2112.29 −0.775714
\(196\) −734.847 −0.267802
\(197\) −4800.27 −1.73607 −0.868033 0.496507i \(-0.834616\pi\)
−0.868033 + 0.496507i \(0.834616\pi\)
\(198\) −46.4988 −0.0166895
\(199\) −2113.34 −0.752816 −0.376408 0.926454i \(-0.622841\pi\)
−0.376408 + 0.926454i \(0.622841\pi\)
\(200\) −871.213 −0.308020
\(201\) −114.707 −0.0402527
\(202\) 3917.85 1.36465
\(203\) 2812.96 0.972566
\(204\) −1938.07 −0.665158
\(205\) 1632.14 0.556067
\(206\) −1059.22 −0.358248
\(207\) 2997.31 1.00641
\(208\) −2756.30 −0.918822
\(209\) −14.3660 −0.00475462
\(210\) 1974.89 0.648955
\(211\) 5564.31 1.81546 0.907732 0.419551i \(-0.137812\pi\)
0.907732 + 0.419551i \(0.137812\pi\)
\(212\) −1741.20 −0.564085
\(213\) 1559.07 0.501531
\(214\) −5459.53 −1.74395
\(215\) −366.321 −0.116200
\(216\) 416.966 0.131347
\(217\) −1897.12 −0.593480
\(218\) 1857.93 0.577225
\(219\) −7324.80 −2.26011
\(220\) 15.2076 0.00466044
\(221\) 3188.49 0.970501
\(222\) 8556.91 2.58695
\(223\) −1709.75 −0.513422 −0.256711 0.966488i \(-0.582639\pi\)
−0.256711 + 0.966488i \(0.582639\pi\)
\(224\) 1268.15 0.378268
\(225\) −1230.32 −0.364538
\(226\) 217.717 0.0640811
\(227\) 347.506 0.101607 0.0508035 0.998709i \(-0.483822\pi\)
0.0508035 + 0.998709i \(0.483822\pi\)
\(228\) 509.840 0.148092
\(229\) 713.508 0.205895 0.102947 0.994687i \(-0.467173\pi\)
0.102947 + 0.994687i \(0.467173\pi\)
\(230\) −3606.39 −1.03391
\(231\) −41.8084 −0.0119082
\(232\) 4748.24 1.34370
\(233\) −6539.93 −1.83882 −0.919410 0.393300i \(-0.871333\pi\)
−0.919410 + 0.393300i \(0.871333\pi\)
\(234\) −2714.91 −0.758457
\(235\) −3904.02 −1.08370
\(236\) 1244.05 0.343138
\(237\) −3164.43 −0.867307
\(238\) −2981.09 −0.811913
\(239\) −2561.41 −0.693238 −0.346619 0.938006i \(-0.612670\pi\)
−0.346619 + 0.938006i \(0.612670\pi\)
\(240\) 4779.44 1.28547
\(241\) −4656.80 −1.24469 −0.622346 0.782742i \(-0.713820\pi\)
−0.622346 + 0.782742i \(0.713820\pi\)
\(242\) 4410.49 1.17156
\(243\) 5090.27 1.34379
\(244\) −92.3879 −0.0242399
\(245\) −2096.34 −0.546655
\(246\) 4511.26 1.16922
\(247\) −838.779 −0.216074
\(248\) −3202.32 −0.819950
\(249\) −2120.65 −0.539721
\(250\) 5009.95 1.26743
\(251\) 6104.04 1.53499 0.767497 0.641053i \(-0.221502\pi\)
0.767497 + 0.641053i \(0.221502\pi\)
\(252\) 689.959 0.172474
\(253\) 76.3471 0.0189720
\(254\) 6320.60 1.56138
\(255\) −5528.86 −1.35777
\(256\) 4027.29 0.983225
\(257\) 682.439 0.165640 0.0828199 0.996565i \(-0.473607\pi\)
0.0828199 + 0.996565i \(0.473607\pi\)
\(258\) −1012.52 −0.244328
\(259\) 3577.67 0.858322
\(260\) 887.919 0.211794
\(261\) 6705.40 1.59024
\(262\) 5091.44 1.20057
\(263\) 1671.87 0.391984 0.195992 0.980605i \(-0.437207\pi\)
0.195992 + 0.980605i \(0.437207\pi\)
\(264\) −70.5722 −0.0164523
\(265\) −4967.22 −1.15145
\(266\) 784.220 0.180766
\(267\) −6011.38 −1.37787
\(268\) 48.2180 0.0109902
\(269\) −261.999 −0.0593843 −0.0296921 0.999559i \(-0.509453\pi\)
−0.0296921 + 0.999559i \(0.509453\pi\)
\(270\) −708.489 −0.159694
\(271\) 4620.04 1.03560 0.517800 0.855502i \(-0.326751\pi\)
0.517800 + 0.855502i \(0.326751\pi\)
\(272\) −7214.53 −1.60825
\(273\) −2441.05 −0.541168
\(274\) 5619.41 1.23898
\(275\) −31.3384 −0.00687192
\(276\) −2709.51 −0.590918
\(277\) 5163.93 1.12011 0.560055 0.828456i \(-0.310780\pi\)
0.560055 + 0.828456i \(0.310780\pi\)
\(278\) −3140.81 −0.677601
\(279\) −4522.28 −0.970400
\(280\) 1393.79 0.297481
\(281\) −5275.67 −1.12000 −0.560000 0.828493i \(-0.689199\pi\)
−0.560000 + 0.828493i \(0.689199\pi\)
\(282\) −10790.8 −2.27865
\(283\) −5028.34 −1.05620 −0.528098 0.849183i \(-0.677095\pi\)
−0.528098 + 0.849183i \(0.677095\pi\)
\(284\) −655.371 −0.136933
\(285\) 1454.45 0.302295
\(286\) −69.1537 −0.0142977
\(287\) 1886.17 0.387934
\(288\) 3022.97 0.618508
\(289\) 3432.77 0.698711
\(290\) −8067.99 −1.63368
\(291\) 1840.44 0.370751
\(292\) 3079.04 0.617080
\(293\) 8840.69 1.76273 0.881363 0.472439i \(-0.156626\pi\)
0.881363 + 0.472439i \(0.156626\pi\)
\(294\) −5794.32 −1.14943
\(295\) 3548.96 0.700436
\(296\) 6039.06 1.18586
\(297\) 14.9987 0.00293035
\(298\) −10222.4 −1.98714
\(299\) 4457.64 0.862181
\(300\) 1112.18 0.214039
\(301\) −423.336 −0.0810654
\(302\) 7778.68 1.48216
\(303\) 8397.14 1.59209
\(304\) 1897.89 0.358064
\(305\) −263.560 −0.0494801
\(306\) −7106.18 −1.32756
\(307\) 6565.15 1.22050 0.610249 0.792209i \(-0.291069\pi\)
0.610249 + 0.792209i \(0.291069\pi\)
\(308\) 17.5745 0.00325131
\(309\) −2270.22 −0.417956
\(310\) 5441.23 0.996908
\(311\) 1250.31 0.227969 0.113985 0.993483i \(-0.463639\pi\)
0.113985 + 0.993483i \(0.463639\pi\)
\(312\) −4120.46 −0.747677
\(313\) −2968.85 −0.536132 −0.268066 0.963401i \(-0.586385\pi\)
−0.268066 + 0.963401i \(0.586385\pi\)
\(314\) −5246.03 −0.942837
\(315\) 1968.29 0.352065
\(316\) 1330.20 0.236802
\(317\) 1054.32 0.186803 0.0934013 0.995629i \(-0.470226\pi\)
0.0934013 + 0.995629i \(0.470226\pi\)
\(318\) −13729.4 −2.42110
\(319\) 170.799 0.0299778
\(320\) 1744.93 0.304826
\(321\) −11701.4 −2.03461
\(322\) −4167.69 −0.721293
\(323\) −2195.48 −0.378204
\(324\) −2424.51 −0.415725
\(325\) −1829.74 −0.312295
\(326\) 2720.87 0.462255
\(327\) 3982.11 0.673429
\(328\) 3183.83 0.535969
\(329\) −4511.64 −0.756033
\(330\) 119.913 0.0200030
\(331\) −2881.73 −0.478532 −0.239266 0.970954i \(-0.576907\pi\)
−0.239266 + 0.970954i \(0.576907\pi\)
\(332\) 891.432 0.147361
\(333\) 8528.28 1.40344
\(334\) −5603.51 −0.917995
\(335\) 137.554 0.0224340
\(336\) 5523.32 0.896791
\(337\) −6147.77 −0.993740 −0.496870 0.867825i \(-0.665517\pi\)
−0.496870 + 0.867825i \(0.665517\pi\)
\(338\) 3244.43 0.522112
\(339\) 466.634 0.0747613
\(340\) 2324.10 0.370712
\(341\) −115.191 −0.0182930
\(342\) 1869.39 0.295570
\(343\) −5799.46 −0.912949
\(344\) −714.586 −0.112000
\(345\) −7729.58 −1.20622
\(346\) −4608.04 −0.715982
\(347\) 1404.42 0.217271 0.108636 0.994082i \(-0.465352\pi\)
0.108636 + 0.994082i \(0.465352\pi\)
\(348\) −6061.55 −0.933716
\(349\) −4472.14 −0.685926 −0.342963 0.939349i \(-0.611430\pi\)
−0.342963 + 0.939349i \(0.611430\pi\)
\(350\) 1710.72 0.261263
\(351\) 875.721 0.133170
\(352\) 77.0007 0.0116595
\(353\) 2146.48 0.323642 0.161821 0.986820i \(-0.448263\pi\)
0.161821 + 0.986820i \(0.448263\pi\)
\(354\) 9809.38 1.47278
\(355\) −1869.61 −0.279518
\(356\) 2526.93 0.376200
\(357\) −6389.37 −0.947231
\(358\) −2299.95 −0.339543
\(359\) 1960.49 0.288219 0.144109 0.989562i \(-0.453968\pi\)
0.144109 + 0.989562i \(0.453968\pi\)
\(360\) 3322.45 0.486412
\(361\) −6281.45 −0.915796
\(362\) 7807.43 1.13356
\(363\) 9453.01 1.36682
\(364\) 1026.12 0.147756
\(365\) 8783.77 1.25963
\(366\) −728.484 −0.104040
\(367\) 13485.2 1.91804 0.959021 0.283337i \(-0.0914414\pi\)
0.959021 + 0.283337i \(0.0914414\pi\)
\(368\) −10086.2 −1.42875
\(369\) 4496.16 0.634312
\(370\) −10261.3 −1.44178
\(371\) −5740.32 −0.803295
\(372\) 4088.05 0.569773
\(373\) −12034.5 −1.67057 −0.835285 0.549817i \(-0.814697\pi\)
−0.835285 + 0.549817i \(0.814697\pi\)
\(374\) −181.008 −0.0250259
\(375\) 10737.8 1.47867
\(376\) −7615.60 −1.04453
\(377\) 9972.36 1.36234
\(378\) −818.760 −0.111409
\(379\) −10185.1 −1.38041 −0.690204 0.723615i \(-0.742479\pi\)
−0.690204 + 0.723615i \(0.742479\pi\)
\(380\) −611.390 −0.0825359
\(381\) 13546.9 1.82160
\(382\) 7186.26 0.962515
\(383\) 13508.3 1.80219 0.901097 0.433618i \(-0.142763\pi\)
0.901097 + 0.433618i \(0.142763\pi\)
\(384\) 12143.7 1.61382
\(385\) 50.1359 0.00663679
\(386\) −4325.51 −0.570369
\(387\) −1009.13 −0.132550
\(388\) −773.646 −0.101227
\(389\) −6651.52 −0.866954 −0.433477 0.901165i \(-0.642714\pi\)
−0.433477 + 0.901165i \(0.642714\pi\)
\(390\) 7001.30 0.909037
\(391\) 11667.7 1.50911
\(392\) −4089.35 −0.526897
\(393\) 10912.5 1.40067
\(394\) 15910.7 2.03445
\(395\) 3794.73 0.483376
\(396\) 41.8934 0.00531622
\(397\) −8308.29 −1.05033 −0.525165 0.851000i \(-0.675996\pi\)
−0.525165 + 0.851000i \(0.675996\pi\)
\(398\) 7004.77 0.882204
\(399\) 1680.82 0.210893
\(400\) 4140.12 0.517515
\(401\) 2963.57 0.369061 0.184530 0.982827i \(-0.440924\pi\)
0.184530 + 0.982827i \(0.440924\pi\)
\(402\) 380.202 0.0471710
\(403\) −6725.59 −0.831328
\(404\) −3529.81 −0.434690
\(405\) −6916.54 −0.848607
\(406\) −9323.70 −1.13972
\(407\) 217.231 0.0264564
\(408\) −10785.2 −1.30869
\(409\) −9951.28 −1.20308 −0.601540 0.798843i \(-0.705446\pi\)
−0.601540 + 0.798843i \(0.705446\pi\)
\(410\) −5409.82 −0.651639
\(411\) 12044.1 1.44548
\(412\) 954.308 0.114115
\(413\) 4101.33 0.488652
\(414\) −9934.76 −1.17939
\(415\) 2543.04 0.300802
\(416\) 4495.80 0.529867
\(417\) −6731.70 −0.790534
\(418\) 47.6168 0.00557180
\(419\) 5281.70 0.615818 0.307909 0.951416i \(-0.400371\pi\)
0.307909 + 0.951416i \(0.400371\pi\)
\(420\) −1779.29 −0.206716
\(421\) −1668.54 −0.193159 −0.0965795 0.995325i \(-0.530790\pi\)
−0.0965795 + 0.995325i \(0.530790\pi\)
\(422\) −18443.2 −2.12749
\(423\) −10754.7 −1.23619
\(424\) −9689.60 −1.10983
\(425\) −4789.29 −0.546623
\(426\) −5167.64 −0.587730
\(427\) −304.581 −0.0345192
\(428\) 4918.80 0.555512
\(429\) −148.217 −0.0166806
\(430\) 1214.19 0.136171
\(431\) −10348.7 −1.15657 −0.578284 0.815835i \(-0.696278\pi\)
−0.578284 + 0.815835i \(0.696278\pi\)
\(432\) −1981.48 −0.220681
\(433\) 2968.27 0.329436 0.164718 0.986341i \(-0.447329\pi\)
0.164718 + 0.986341i \(0.447329\pi\)
\(434\) 6288.12 0.695482
\(435\) −17292.1 −1.90596
\(436\) −1673.91 −0.183867
\(437\) −3069.38 −0.335991
\(438\) 24278.4 2.64856
\(439\) −9224.53 −1.00288 −0.501438 0.865194i \(-0.667196\pi\)
−0.501438 + 0.865194i \(0.667196\pi\)
\(440\) 84.6289 0.00916937
\(441\) −5774.93 −0.623575
\(442\) −10568.4 −1.13730
\(443\) −10115.5 −1.08488 −0.542438 0.840096i \(-0.682499\pi\)
−0.542438 + 0.840096i \(0.682499\pi\)
\(444\) −7709.40 −0.824036
\(445\) 7208.73 0.767925
\(446\) 5667.05 0.601665
\(447\) −21909.7 −2.31833
\(448\) 2016.51 0.212659
\(449\) 14386.0 1.51207 0.756034 0.654532i \(-0.227134\pi\)
0.756034 + 0.654532i \(0.227134\pi\)
\(450\) 4077.95 0.427192
\(451\) 114.526 0.0119574
\(452\) −196.154 −0.0204121
\(453\) 16672.1 1.72919
\(454\) −1151.83 −0.119070
\(455\) 2927.26 0.301609
\(456\) 2837.21 0.291369
\(457\) −10031.8 −1.02684 −0.513421 0.858137i \(-0.671622\pi\)
−0.513421 + 0.858137i \(0.671622\pi\)
\(458\) −2364.96 −0.241282
\(459\) 2292.18 0.233093
\(460\) 3249.20 0.329336
\(461\) −124.788 −0.0126072 −0.00630362 0.999980i \(-0.502007\pi\)
−0.00630362 + 0.999980i \(0.502007\pi\)
\(462\) 138.576 0.0139549
\(463\) 15438.9 1.54969 0.774847 0.632148i \(-0.217827\pi\)
0.774847 + 0.632148i \(0.217827\pi\)
\(464\) −22564.3 −2.25759
\(465\) 11662.2 1.16306
\(466\) 21677.0 2.15486
\(467\) −12014.7 −1.19052 −0.595259 0.803534i \(-0.702951\pi\)
−0.595259 + 0.803534i \(0.702951\pi\)
\(468\) 2446.01 0.241596
\(469\) 158.963 0.0156508
\(470\) 12940.1 1.26996
\(471\) −11243.8 −1.09998
\(472\) 6923.00 0.675121
\(473\) −25.7044 −0.00249871
\(474\) 10488.7 1.01637
\(475\) 1259.90 0.121701
\(476\) 2685.83 0.258623
\(477\) −13683.5 −1.31347
\(478\) 8489.93 0.812386
\(479\) 6241.39 0.595358 0.297679 0.954666i \(-0.403788\pi\)
0.297679 + 0.954666i \(0.403788\pi\)
\(480\) −7795.75 −0.741303
\(481\) 12683.4 1.20231
\(482\) 15435.2 1.45862
\(483\) −8932.63 −0.841508
\(484\) −3973.65 −0.373183
\(485\) −2207.03 −0.206631
\(486\) −16872.0 −1.57475
\(487\) 924.180 0.0859930 0.0429965 0.999075i \(-0.486310\pi\)
0.0429965 + 0.999075i \(0.486310\pi\)
\(488\) −514.130 −0.0476917
\(489\) 5831.65 0.539297
\(490\) 6948.44 0.640609
\(491\) −7044.15 −0.647451 −0.323725 0.946151i \(-0.604935\pi\)
−0.323725 + 0.946151i \(0.604935\pi\)
\(492\) −4064.44 −0.372438
\(493\) 26102.3 2.38457
\(494\) 2780.18 0.253211
\(495\) 119.512 0.0108518
\(496\) 15217.9 1.37763
\(497\) −2160.60 −0.195003
\(498\) 7029.00 0.632484
\(499\) −3981.76 −0.357210 −0.178605 0.983921i \(-0.557158\pi\)
−0.178605 + 0.983921i \(0.557158\pi\)
\(500\) −4513.74 −0.403722
\(501\) −12010.0 −1.07099
\(502\) −20232.2 −1.79882
\(503\) −11650.2 −1.03272 −0.516358 0.856373i \(-0.672712\pi\)
−0.516358 + 0.856373i \(0.672712\pi\)
\(504\) 3839.56 0.339340
\(505\) −10069.7 −0.887318
\(506\) −253.057 −0.0222327
\(507\) 6953.80 0.609131
\(508\) −5694.58 −0.497354
\(509\) 11408.7 0.993481 0.496740 0.867899i \(-0.334530\pi\)
0.496740 + 0.867899i \(0.334530\pi\)
\(510\) 18325.7 1.59113
\(511\) 10150.9 0.878764
\(512\) 326.513 0.0281836
\(513\) −602.991 −0.0518961
\(514\) −2261.98 −0.194108
\(515\) 2722.41 0.232939
\(516\) 912.233 0.0778271
\(517\) −273.941 −0.0233035
\(518\) −11858.4 −1.00584
\(519\) −9876.42 −0.835312
\(520\) 4941.19 0.416703
\(521\) 15272.4 1.28425 0.642127 0.766598i \(-0.278052\pi\)
0.642127 + 0.766598i \(0.278052\pi\)
\(522\) −22225.4 −1.86356
\(523\) −20831.4 −1.74167 −0.870835 0.491575i \(-0.836421\pi\)
−0.870835 + 0.491575i \(0.836421\pi\)
\(524\) −4587.16 −0.382426
\(525\) 3666.60 0.304807
\(526\) −5541.50 −0.459355
\(527\) −17604.0 −1.45511
\(528\) 335.369 0.0276421
\(529\) 4145.03 0.340678
\(530\) 16464.1 1.34935
\(531\) 9776.57 0.798996
\(532\) −706.548 −0.0575803
\(533\) 6686.75 0.543406
\(534\) 19925.0 1.61468
\(535\) 14032.1 1.13395
\(536\) 268.328 0.0216232
\(537\) −4929.50 −0.396133
\(538\) 868.410 0.0695907
\(539\) −147.098 −0.0117550
\(540\) 638.318 0.0508682
\(541\) −1411.76 −0.112192 −0.0560962 0.998425i \(-0.517865\pi\)
−0.0560962 + 0.998425i \(0.517865\pi\)
\(542\) −15313.4 −1.21359
\(543\) 16733.7 1.32249
\(544\) 11767.6 0.927450
\(545\) −4775.27 −0.375322
\(546\) 8090.99 0.634180
\(547\) −328.281 −0.0256605 −0.0128302 0.999918i \(-0.504084\pi\)
−0.0128302 + 0.999918i \(0.504084\pi\)
\(548\) −5062.84 −0.394661
\(549\) −726.047 −0.0564425
\(550\) 103.873 0.00805301
\(551\) −6866.62 −0.530903
\(552\) −15078.2 −1.16263
\(553\) 4385.34 0.337222
\(554\) −17116.1 −1.31262
\(555\) −21993.1 −1.68208
\(556\) 2829.73 0.215840
\(557\) −1908.49 −0.145180 −0.0725901 0.997362i \(-0.523126\pi\)
−0.0725901 + 0.997362i \(0.523126\pi\)
\(558\) 14989.3 1.13718
\(559\) −1500.79 −0.113554
\(560\) −6623.46 −0.499808
\(561\) −387.954 −0.0291969
\(562\) 17486.5 1.31250
\(563\) 3413.23 0.255507 0.127753 0.991806i \(-0.459223\pi\)
0.127753 + 0.991806i \(0.459223\pi\)
\(564\) 9721.99 0.725833
\(565\) −559.579 −0.0416667
\(566\) 16666.7 1.23773
\(567\) −7993.04 −0.592021
\(568\) −3647.07 −0.269415
\(569\) −15405.8 −1.13505 −0.567527 0.823355i \(-0.692100\pi\)
−0.567527 + 0.823355i \(0.692100\pi\)
\(570\) −4820.85 −0.354251
\(571\) 18819.2 1.37926 0.689632 0.724160i \(-0.257772\pi\)
0.689632 + 0.724160i \(0.257772\pi\)
\(572\) 62.3044 0.00455433
\(573\) 15402.3 1.12293
\(574\) −6251.81 −0.454609
\(575\) −6695.64 −0.485613
\(576\) 4806.86 0.347719
\(577\) 8098.53 0.584309 0.292154 0.956371i \(-0.405628\pi\)
0.292154 + 0.956371i \(0.405628\pi\)
\(578\) −11378.1 −0.818799
\(579\) −9270.87 −0.665431
\(580\) 7268.90 0.520387
\(581\) 2938.84 0.209852
\(582\) −6100.25 −0.434473
\(583\) −348.545 −0.0247603
\(584\) 17134.6 1.21410
\(585\) 6977.87 0.493162
\(586\) −29303.0 −2.06569
\(587\) −3631.44 −0.255342 −0.127671 0.991817i \(-0.540750\pi\)
−0.127671 + 0.991817i \(0.540750\pi\)
\(588\) 5220.42 0.366134
\(589\) 4631.00 0.323968
\(590\) −11763.2 −0.820822
\(591\) 34101.5 2.37352
\(592\) −28698.4 −1.99240
\(593\) −14298.2 −0.990147 −0.495073 0.868851i \(-0.664859\pi\)
−0.495073 + 0.868851i \(0.664859\pi\)
\(594\) −49.7140 −0.00343399
\(595\) 7662.02 0.527920
\(596\) 9209.95 0.632977
\(597\) 15013.3 1.02924
\(598\) −14775.1 −1.01037
\(599\) −27823.6 −1.89790 −0.948949 0.315430i \(-0.897851\pi\)
−0.948949 + 0.315430i \(0.897851\pi\)
\(600\) 6189.18 0.421120
\(601\) 3557.15 0.241430 0.120715 0.992687i \(-0.461481\pi\)
0.120715 + 0.992687i \(0.461481\pi\)
\(602\) 1403.17 0.0949982
\(603\) 378.930 0.0255907
\(604\) −7008.25 −0.472122
\(605\) −11335.9 −0.761767
\(606\) −27832.8 −1.86572
\(607\) 19170.9 1.28192 0.640958 0.767576i \(-0.278537\pi\)
0.640958 + 0.767576i \(0.278537\pi\)
\(608\) −3095.65 −0.206489
\(609\) −19983.5 −1.32968
\(610\) 873.585 0.0579843
\(611\) −15994.5 −1.05903
\(612\) 6402.36 0.422876
\(613\) 4287.83 0.282518 0.141259 0.989973i \(-0.454885\pi\)
0.141259 + 0.989973i \(0.454885\pi\)
\(614\) −21760.6 −1.43027
\(615\) −11594.9 −0.760245
\(616\) 97.8006 0.00639692
\(617\) −10569.5 −0.689649 −0.344824 0.938667i \(-0.612062\pi\)
−0.344824 + 0.938667i \(0.612062\pi\)
\(618\) 7524.77 0.489791
\(619\) 19947.5 1.29525 0.647624 0.761960i \(-0.275763\pi\)
0.647624 + 0.761960i \(0.275763\pi\)
\(620\) −4902.31 −0.317551
\(621\) 3204.56 0.207077
\(622\) −4144.21 −0.267150
\(623\) 8330.71 0.535735
\(624\) 19581.0 1.25620
\(625\) −6323.50 −0.404704
\(626\) 9840.42 0.628278
\(627\) 102.057 0.00650043
\(628\) 4726.44 0.300328
\(629\) 33198.3 2.10446
\(630\) −6523.99 −0.412575
\(631\) 8854.29 0.558611 0.279306 0.960202i \(-0.409896\pi\)
0.279306 + 0.960202i \(0.409896\pi\)
\(632\) 7402.41 0.465905
\(633\) −39529.4 −2.48207
\(634\) −3494.60 −0.218909
\(635\) −16245.3 −1.01523
\(636\) 12369.6 0.771207
\(637\) −8588.55 −0.534208
\(638\) −566.122 −0.0351301
\(639\) −5150.35 −0.318849
\(640\) −14562.5 −0.899429
\(641\) 10022.5 0.617571 0.308786 0.951132i \(-0.400077\pi\)
0.308786 + 0.951132i \(0.400077\pi\)
\(642\) 38785.0 2.38430
\(643\) −5806.52 −0.356123 −0.178061 0.984019i \(-0.556983\pi\)
−0.178061 + 0.984019i \(0.556983\pi\)
\(644\) 3754.91 0.229758
\(645\) 2602.38 0.158866
\(646\) 7277.04 0.443206
\(647\) 8583.93 0.521591 0.260795 0.965394i \(-0.416015\pi\)
0.260795 + 0.965394i \(0.416015\pi\)
\(648\) −13492.2 −0.817936
\(649\) 249.027 0.0150619
\(650\) 6064.77 0.365969
\(651\) 13477.3 0.811395
\(652\) −2451.39 −0.147245
\(653\) 17048.8 1.02170 0.510849 0.859670i \(-0.329331\pi\)
0.510849 + 0.859670i \(0.329331\pi\)
\(654\) −13198.9 −0.789172
\(655\) −13086.1 −0.780633
\(656\) −15130.0 −0.900499
\(657\) 24197.2 1.43687
\(658\) 14954.1 0.885974
\(659\) −2694.69 −0.159287 −0.0796437 0.996823i \(-0.525378\pi\)
−0.0796437 + 0.996823i \(0.525378\pi\)
\(660\) −108.036 −0.00637168
\(661\) −29666.8 −1.74570 −0.872848 0.487992i \(-0.837730\pi\)
−0.872848 + 0.487992i \(0.837730\pi\)
\(662\) 9551.64 0.560778
\(663\) −22651.3 −1.32685
\(664\) 4960.74 0.289931
\(665\) −2015.61 −0.117537
\(666\) −28267.5 −1.64466
\(667\) 36492.2 2.11842
\(668\) 5048.52 0.292415
\(669\) 12146.2 0.701942
\(670\) −455.931 −0.0262898
\(671\) −18.4938 −0.00106400
\(672\) −9009.09 −0.517162
\(673\) 17932.8 1.02713 0.513564 0.858051i \(-0.328325\pi\)
0.513564 + 0.858051i \(0.328325\pi\)
\(674\) 20377.1 1.16454
\(675\) −1315.39 −0.0750062
\(676\) −2923.09 −0.166312
\(677\) 2440.74 0.138560 0.0692801 0.997597i \(-0.477930\pi\)
0.0692801 + 0.997597i \(0.477930\pi\)
\(678\) −1546.68 −0.0876106
\(679\) −2550.53 −0.144154
\(680\) 12933.4 0.729373
\(681\) −2468.72 −0.138915
\(682\) 381.806 0.0214371
\(683\) −12722.2 −0.712737 −0.356369 0.934345i \(-0.615985\pi\)
−0.356369 + 0.934345i \(0.615985\pi\)
\(684\) −1684.24 −0.0941497
\(685\) −14443.1 −0.805608
\(686\) 19222.6 1.06986
\(687\) −5068.83 −0.281496
\(688\) 3395.81 0.188175
\(689\) −20350.3 −1.12523
\(690\) 25620.1 1.41354
\(691\) −5584.80 −0.307461 −0.153731 0.988113i \(-0.549129\pi\)
−0.153731 + 0.988113i \(0.549129\pi\)
\(692\) 4151.64 0.228066
\(693\) 138.113 0.00757066
\(694\) −4655.02 −0.254614
\(695\) 8072.53 0.440588
\(696\) −33731.9 −1.83708
\(697\) 17502.4 0.951148
\(698\) 14823.1 0.803817
\(699\) 46460.3 2.51400
\(700\) −1541.29 −0.0832217
\(701\) 19891.2 1.07173 0.535864 0.844304i \(-0.319986\pi\)
0.535864 + 0.844304i \(0.319986\pi\)
\(702\) −2902.63 −0.156058
\(703\) −8733.33 −0.468540
\(704\) 122.440 0.00655486
\(705\) 27734.5 1.48162
\(706\) −7114.62 −0.379266
\(707\) −11637.0 −0.619028
\(708\) −8837.82 −0.469132
\(709\) −14613.5 −0.774078 −0.387039 0.922063i \(-0.626502\pi\)
−0.387039 + 0.922063i \(0.626502\pi\)
\(710\) 6196.94 0.327559
\(711\) 10453.6 0.551392
\(712\) 14062.1 0.740170
\(713\) −24611.2 −1.29270
\(714\) 21177.9 1.11003
\(715\) 177.739 0.00929661
\(716\) 2072.16 0.108157
\(717\) 18196.5 0.947783
\(718\) −6498.14 −0.337755
\(719\) −11027.7 −0.571995 −0.285998 0.958230i \(-0.592325\pi\)
−0.285998 + 0.958230i \(0.592325\pi\)
\(720\) −15788.7 −0.817237
\(721\) 3146.13 0.162508
\(722\) 20820.2 1.07320
\(723\) 33082.3 1.70172
\(724\) −7034.15 −0.361080
\(725\) −14979.1 −0.767322
\(726\) −31332.5 −1.60173
\(727\) 1650.84 0.0842178 0.0421089 0.999113i \(-0.486592\pi\)
0.0421089 + 0.999113i \(0.486592\pi\)
\(728\) 5710.24 0.290708
\(729\) −14240.8 −0.723506
\(730\) −29114.3 −1.47612
\(731\) −3928.27 −0.198758
\(732\) 656.332 0.0331403
\(733\) −28314.8 −1.42678 −0.713391 0.700767i \(-0.752841\pi\)
−0.713391 + 0.700767i \(0.752841\pi\)
\(734\) −44697.4 −2.24770
\(735\) 14892.6 0.747377
\(736\) 16451.7 0.823935
\(737\) 9.65204 0.000482412 0
\(738\) −14902.8 −0.743332
\(739\) −28963.4 −1.44172 −0.720862 0.693078i \(-0.756254\pi\)
−0.720862 + 0.693078i \(0.756254\pi\)
\(740\) 9244.97 0.459259
\(741\) 5958.76 0.295413
\(742\) 19026.6 0.941359
\(743\) −15510.4 −0.765844 −0.382922 0.923781i \(-0.625082\pi\)
−0.382922 + 0.923781i \(0.625082\pi\)
\(744\) 22749.6 1.12102
\(745\) 26273.8 1.29208
\(746\) 39889.0 1.95769
\(747\) 7005.48 0.343129
\(748\) 163.080 0.00797165
\(749\) 16216.1 0.791087
\(750\) −35591.1 −1.73281
\(751\) −2901.63 −0.140988 −0.0704941 0.997512i \(-0.522458\pi\)
−0.0704941 + 0.997512i \(0.522458\pi\)
\(752\) 36190.4 1.75496
\(753\) −43363.6 −2.09862
\(754\) −33053.9 −1.59649
\(755\) −19992.8 −0.963727
\(756\) 737.666 0.0354876
\(757\) 9260.00 0.444598 0.222299 0.974979i \(-0.428644\pi\)
0.222299 + 0.974979i \(0.428644\pi\)
\(758\) 33759.2 1.61766
\(759\) −542.377 −0.0259381
\(760\) −3402.33 −0.162389
\(761\) −6418.29 −0.305733 −0.152866 0.988247i \(-0.548850\pi\)
−0.152866 + 0.988247i \(0.548850\pi\)
\(762\) −44902.1 −2.13469
\(763\) −5518.50 −0.261839
\(764\) −6474.50 −0.306596
\(765\) 18264.4 0.863203
\(766\) −44773.9 −2.11194
\(767\) 14539.8 0.684489
\(768\) −28610.2 −1.34425
\(769\) 5335.88 0.250217 0.125108 0.992143i \(-0.460072\pi\)
0.125108 + 0.992143i \(0.460072\pi\)
\(770\) −166.178 −0.00777746
\(771\) −4848.11 −0.226460
\(772\) 3897.09 0.181683
\(773\) 20062.9 0.933520 0.466760 0.884384i \(-0.345421\pi\)
0.466760 + 0.884384i \(0.345421\pi\)
\(774\) 3344.81 0.155332
\(775\) 10102.2 0.468236
\(776\) −4305.27 −0.199162
\(777\) −25416.1 −1.17348
\(778\) 22046.8 1.01596
\(779\) −4604.26 −0.211765
\(780\) −6307.86 −0.289561
\(781\) −131.189 −0.00601064
\(782\) −38673.4 −1.76849
\(783\) 7169.04 0.327204
\(784\) 19433.2 0.885257
\(785\) 13483.4 0.613049
\(786\) −36170.1 −1.64140
\(787\) 17182.2 0.778248 0.389124 0.921185i \(-0.372778\pi\)
0.389124 + 0.921185i \(0.372778\pi\)
\(788\) −14334.9 −0.648045
\(789\) −11877.1 −0.535914
\(790\) −12577.8 −0.566454
\(791\) −646.672 −0.0290683
\(792\) 233.133 0.0104596
\(793\) −1079.79 −0.0483535
\(794\) 27538.3 1.23085
\(795\) 35287.6 1.57424
\(796\) −6310.99 −0.281014
\(797\) −3342.90 −0.148572 −0.0742859 0.997237i \(-0.523668\pi\)
−0.0742859 + 0.997237i \(0.523668\pi\)
\(798\) −5571.17 −0.247140
\(799\) −41865.0 −1.85366
\(800\) −6752.95 −0.298441
\(801\) 19858.4 0.875981
\(802\) −9822.90 −0.432492
\(803\) 616.348 0.0270865
\(804\) −342.545 −0.0150257
\(805\) 10711.8 0.468997
\(806\) 22292.3 0.974210
\(807\) 1861.27 0.0811892
\(808\) −19643.1 −0.855248
\(809\) −10782.5 −0.468596 −0.234298 0.972165i \(-0.575279\pi\)
−0.234298 + 0.972165i \(0.575279\pi\)
\(810\) 22925.2 0.994458
\(811\) −22440.1 −0.971612 −0.485806 0.874067i \(-0.661474\pi\)
−0.485806 + 0.874067i \(0.661474\pi\)
\(812\) 8400.24 0.363043
\(813\) −32821.2 −1.41586
\(814\) −720.024 −0.0310035
\(815\) −6993.21 −0.300566
\(816\) 51252.7 2.19878
\(817\) 1033.39 0.0442519
\(818\) 32984.1 1.40985
\(819\) 8063.92 0.344049
\(820\) 4874.01 0.207570
\(821\) −10938.7 −0.464996 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(822\) −39920.8 −1.69392
\(823\) −37928.5 −1.60645 −0.803223 0.595678i \(-0.796883\pi\)
−0.803223 + 0.595678i \(0.796883\pi\)
\(824\) 5310.63 0.224520
\(825\) 222.631 0.00939517
\(826\) −13594.1 −0.572637
\(827\) 36038.1 1.51532 0.757660 0.652650i \(-0.226343\pi\)
0.757660 + 0.652650i \(0.226343\pi\)
\(828\) 8950.78 0.375678
\(829\) 29221.5 1.22425 0.612127 0.790760i \(-0.290314\pi\)
0.612127 + 0.790760i \(0.290314\pi\)
\(830\) −8429.05 −0.352502
\(831\) −36685.0 −1.53139
\(832\) 7148.82 0.297886
\(833\) −22480.3 −0.935048
\(834\) 22312.6 0.926404
\(835\) 14402.2 0.596896
\(836\) −42.9006 −0.00177482
\(837\) −4834.97 −0.199667
\(838\) −17506.5 −0.721660
\(839\) −6285.44 −0.258638 −0.129319 0.991603i \(-0.541279\pi\)
−0.129319 + 0.991603i \(0.541279\pi\)
\(840\) −9901.59 −0.406711
\(841\) 57249.1 2.34733
\(842\) 5530.48 0.226357
\(843\) 37478.8 1.53125
\(844\) 16616.5 0.677682
\(845\) −8338.87 −0.339486
\(846\) 35646.9 1.44866
\(847\) −13100.2 −0.531438
\(848\) 46046.3 1.86466
\(849\) 35721.8 1.44401
\(850\) 15874.4 0.640572
\(851\) 46412.8 1.86958
\(852\) 4655.81 0.187213
\(853\) −35129.3 −1.41009 −0.705044 0.709164i \(-0.749073\pi\)
−0.705044 + 0.709164i \(0.749073\pi\)
\(854\) 1009.55 0.0404521
\(855\) −4804.72 −0.192185
\(856\) 27372.6 1.09296
\(857\) 32983.7 1.31470 0.657352 0.753584i \(-0.271676\pi\)
0.657352 + 0.753584i \(0.271676\pi\)
\(858\) 491.274 0.0195476
\(859\) −35152.9 −1.39628 −0.698139 0.715962i \(-0.745988\pi\)
−0.698139 + 0.715962i \(0.745988\pi\)
\(860\) −1093.93 −0.0433753
\(861\) −13399.5 −0.530377
\(862\) 34301.4 1.35535
\(863\) 24925.6 0.983173 0.491587 0.870829i \(-0.336417\pi\)
0.491587 + 0.870829i \(0.336417\pi\)
\(864\) 3231.99 0.127262
\(865\) 11843.6 0.465544
\(866\) −9838.48 −0.386057
\(867\) −24386.7 −0.955266
\(868\) −5665.31 −0.221536
\(869\) 266.272 0.0103943
\(870\) 57315.7 2.23355
\(871\) 563.549 0.0219232
\(872\) −9315.17 −0.361756
\(873\) −6079.84 −0.235706
\(874\) 10173.6 0.393739
\(875\) −14880.8 −0.574927
\(876\) −21873.8 −0.843661
\(877\) 2057.63 0.0792261 0.0396131 0.999215i \(-0.487387\pi\)
0.0396131 + 0.999215i \(0.487387\pi\)
\(878\) 30575.2 1.17524
\(879\) −62805.1 −2.40997
\(880\) −402.168 −0.0154058
\(881\) 17912.5 0.685004 0.342502 0.939517i \(-0.388726\pi\)
0.342502 + 0.939517i \(0.388726\pi\)
\(882\) 19141.3 0.730750
\(883\) −27215.2 −1.03722 −0.518610 0.855011i \(-0.673550\pi\)
−0.518610 + 0.855011i \(0.673550\pi\)
\(884\) 9521.67 0.362272
\(885\) −25212.2 −0.957625
\(886\) 33528.2 1.27133
\(887\) 25313.0 0.958206 0.479103 0.877759i \(-0.340962\pi\)
0.479103 + 0.877759i \(0.340962\pi\)
\(888\) −42902.1 −1.62128
\(889\) −18773.7 −0.708267
\(890\) −23893.7 −0.899910
\(891\) −485.327 −0.0182481
\(892\) −5105.76 −0.191652
\(893\) 11013.2 0.412703
\(894\) 72621.0 2.71679
\(895\) 5911.36 0.220777
\(896\) −16829.1 −0.627477
\(897\) −31667.5 −1.17876
\(898\) −47683.3 −1.77195
\(899\) −55058.6 −2.04261
\(900\) −3674.05 −0.136076
\(901\) −53266.3 −1.96954
\(902\) −379.601 −0.0140126
\(903\) 3007.42 0.110831
\(904\) −1091.58 −0.0401607
\(905\) −20066.7 −0.737062
\(906\) −55260.5 −2.02639
\(907\) 45998.9 1.68398 0.841989 0.539495i \(-0.181385\pi\)
0.841989 + 0.539495i \(0.181385\pi\)
\(908\) 1037.75 0.0379282
\(909\) −27739.7 −1.01217
\(910\) −9702.57 −0.353447
\(911\) −46592.7 −1.69449 −0.847247 0.531199i \(-0.821742\pi\)
−0.847247 + 0.531199i \(0.821742\pi\)
\(912\) −13482.8 −0.489539
\(913\) 178.443 0.00646833
\(914\) 33250.9 1.20333
\(915\) 1872.36 0.0676483
\(916\) 2130.72 0.0768571
\(917\) −15122.8 −0.544601
\(918\) −7597.54 −0.273155
\(919\) 33168.1 1.19055 0.595274 0.803523i \(-0.297043\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(920\) 18081.5 0.647966
\(921\) −46639.5 −1.66865
\(922\) 413.615 0.0147741
\(923\) −7659.66 −0.273154
\(924\) −124.851 −0.00444513
\(925\) −19051.2 −0.677188
\(926\) −51173.2 −1.81604
\(927\) 7499.60 0.265716
\(928\) 36804.6 1.30191
\(929\) 19909.9 0.703148 0.351574 0.936160i \(-0.385647\pi\)
0.351574 + 0.936160i \(0.385647\pi\)
\(930\) −38655.0 −1.36296
\(931\) 5913.78 0.208181
\(932\) −19530.0 −0.686401
\(933\) −8882.29 −0.311675
\(934\) 39823.2 1.39514
\(935\) 465.228 0.0162723
\(936\) 13611.8 0.475338
\(937\) 26262.2 0.915634 0.457817 0.889046i \(-0.348631\pi\)
0.457817 + 0.889046i \(0.348631\pi\)
\(938\) −526.893 −0.0183408
\(939\) 21091.0 0.732991
\(940\) −11658.4 −0.404528
\(941\) 42227.8 1.46290 0.731449 0.681896i \(-0.238845\pi\)
0.731449 + 0.681896i \(0.238845\pi\)
\(942\) 37268.3 1.28903
\(943\) 24469.1 0.844988
\(944\) −32899.0 −1.13429
\(945\) 2104.38 0.0724398
\(946\) 85.1986 0.00292817
\(947\) −35227.0 −1.20879 −0.604394 0.796685i \(-0.706585\pi\)
−0.604394 + 0.796685i \(0.706585\pi\)
\(948\) −9449.83 −0.323751
\(949\) 35986.4 1.23095
\(950\) −4175.99 −0.142618
\(951\) −7489.98 −0.255393
\(952\) 14946.4 0.508839
\(953\) 49209.9 1.67268 0.836340 0.548211i \(-0.184691\pi\)
0.836340 + 0.548211i \(0.184691\pi\)
\(954\) 45354.8 1.53922
\(955\) −18470.2 −0.625844
\(956\) −7649.06 −0.258774
\(957\) −1213.37 −0.0409851
\(958\) −20687.4 −0.697683
\(959\) −16691.0 −0.562024
\(960\) −12396.1 −0.416753
\(961\) 7341.79 0.246443
\(962\) −42039.7 −1.40895
\(963\) 38655.3 1.29351
\(964\) −13906.4 −0.464623
\(965\) 11117.5 0.370864
\(966\) 29607.7 0.986140
\(967\) 13452.5 0.447366 0.223683 0.974662i \(-0.428192\pi\)
0.223683 + 0.974662i \(0.428192\pi\)
\(968\) −22113.0 −0.734234
\(969\) 15596.9 0.517074
\(970\) 7315.30 0.242145
\(971\) 37322.8 1.23352 0.616758 0.787152i \(-0.288446\pi\)
0.616758 + 0.787152i \(0.288446\pi\)
\(972\) 15200.9 0.501614
\(973\) 9328.95 0.307371
\(974\) −3063.24 −0.100773
\(975\) 12998.6 0.426964
\(976\) 2443.21 0.0801285
\(977\) −45153.4 −1.47859 −0.739297 0.673380i \(-0.764842\pi\)
−0.739297 + 0.673380i \(0.764842\pi\)
\(978\) −19329.3 −0.631987
\(979\) 505.829 0.0165132
\(980\) −6260.24 −0.204057
\(981\) −13154.8 −0.428134
\(982\) 23348.2 0.758729
\(983\) −15802.7 −0.512745 −0.256372 0.966578i \(-0.582527\pi\)
−0.256372 + 0.966578i \(0.582527\pi\)
\(984\) −22618.2 −0.732767
\(985\) −40894.0 −1.32283
\(986\) −86517.6 −2.79440
\(987\) 32051.1 1.03364
\(988\) −2504.82 −0.0806568
\(989\) −5491.90 −0.176575
\(990\) −396.128 −0.0127169
\(991\) 27570.0 0.883744 0.441872 0.897078i \(-0.354315\pi\)
0.441872 + 0.897078i \(0.354315\pi\)
\(992\) −24821.9 −0.794451
\(993\) 20472.1 0.654241
\(994\) 7161.43 0.228518
\(995\) −18003.7 −0.573624
\(996\) −6332.82 −0.201469
\(997\) −14969.8 −0.475526 −0.237763 0.971323i \(-0.576414\pi\)
−0.237763 + 0.971323i \(0.576414\pi\)
\(998\) 13197.7 0.418605
\(999\) 9117.97 0.288768
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 43.4.a.b.1.1 6
3.2 odd 2 387.4.a.h.1.6 6
4.3 odd 2 688.4.a.i.1.6 6
5.4 even 2 1075.4.a.b.1.6 6
7.6 odd 2 2107.4.a.c.1.1 6
43.42 odd 2 1849.4.a.c.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.1 6 1.1 even 1 trivial
387.4.a.h.1.6 6 3.2 odd 2
688.4.a.i.1.6 6 4.3 odd 2
1075.4.a.b.1.6 6 5.4 even 2
1849.4.a.c.1.6 6 43.42 odd 2
2107.4.a.c.1.1 6 7.6 odd 2