Properties

Label 1849.4.a.j.1.20
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.29424 q^{2} +3.36009 q^{3} -6.32495 q^{4} -1.78174 q^{5} -4.34876 q^{6} +27.5745 q^{7} +18.5399 q^{8} -15.7098 q^{9} +O(q^{10})\) \(q-1.29424 q^{2} +3.36009 q^{3} -6.32495 q^{4} -1.78174 q^{5} -4.34876 q^{6} +27.5745 q^{7} +18.5399 q^{8} -15.7098 q^{9} +2.30599 q^{10} +9.45755 q^{11} -21.2524 q^{12} +38.4667 q^{13} -35.6879 q^{14} -5.98680 q^{15} +26.6046 q^{16} +54.0990 q^{17} +20.3322 q^{18} -37.3489 q^{19} +11.2694 q^{20} +92.6528 q^{21} -12.2403 q^{22} -100.967 q^{23} +62.2958 q^{24} -121.825 q^{25} -49.7850 q^{26} -143.509 q^{27} -174.407 q^{28} -175.001 q^{29} +7.74835 q^{30} -247.586 q^{31} -182.752 q^{32} +31.7783 q^{33} -70.0170 q^{34} -49.1304 q^{35} +99.3634 q^{36} -234.362 q^{37} +48.3384 q^{38} +129.252 q^{39} -33.0332 q^{40} +325.321 q^{41} -119.915 q^{42} -59.8185 q^{44} +27.9907 q^{45} +130.675 q^{46} +148.643 q^{47} +89.3938 q^{48} +417.351 q^{49} +157.671 q^{50} +181.778 q^{51} -243.300 q^{52} +735.610 q^{53} +185.735 q^{54} -16.8509 q^{55} +511.228 q^{56} -125.496 q^{57} +226.493 q^{58} -275.860 q^{59} +37.8662 q^{60} -30.9134 q^{61} +320.436 q^{62} -433.188 q^{63} +23.6879 q^{64} -68.5375 q^{65} -41.1286 q^{66} -27.3051 q^{67} -342.173 q^{68} -339.259 q^{69} +63.5864 q^{70} +66.0822 q^{71} -291.257 q^{72} +511.898 q^{73} +303.320 q^{74} -409.345 q^{75} +236.230 q^{76} +260.787 q^{77} -167.282 q^{78} -1190.03 q^{79} -47.4023 q^{80} -58.0396 q^{81} -421.043 q^{82} -144.868 q^{83} -586.024 q^{84} -96.3902 q^{85} -588.021 q^{87} +175.342 q^{88} -769.313 q^{89} -36.2266 q^{90} +1060.70 q^{91} +638.611 q^{92} -831.914 q^{93} -192.379 q^{94} +66.5460 q^{95} -614.063 q^{96} +1881.67 q^{97} -540.151 q^{98} -148.576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29424 −0.457582 −0.228791 0.973476i \(-0.573477\pi\)
−0.228791 + 0.973476i \(0.573477\pi\)
\(3\) 3.36009 0.646650 0.323325 0.946288i \(-0.395199\pi\)
0.323325 + 0.946288i \(0.395199\pi\)
\(4\) −6.32495 −0.790618
\(5\) −1.78174 −0.159363 −0.0796817 0.996820i \(-0.525390\pi\)
−0.0796817 + 0.996820i \(0.525390\pi\)
\(6\) −4.34876 −0.295896
\(7\) 27.5745 1.48888 0.744440 0.667689i \(-0.232716\pi\)
0.744440 + 0.667689i \(0.232716\pi\)
\(8\) 18.5399 0.819355
\(9\) −15.7098 −0.581843
\(10\) 2.30599 0.0729218
\(11\) 9.45755 0.259233 0.129616 0.991564i \(-0.458625\pi\)
0.129616 + 0.991564i \(0.458625\pi\)
\(12\) −21.2524 −0.511254
\(13\) 38.4667 0.820672 0.410336 0.911934i \(-0.365411\pi\)
0.410336 + 0.911934i \(0.365411\pi\)
\(14\) −35.6879 −0.681285
\(15\) −5.98680 −0.103052
\(16\) 26.6046 0.415696
\(17\) 54.0990 0.771820 0.385910 0.922536i \(-0.373888\pi\)
0.385910 + 0.922536i \(0.373888\pi\)
\(18\) 20.3322 0.266241
\(19\) −37.3489 −0.450970 −0.225485 0.974247i \(-0.572397\pi\)
−0.225485 + 0.974247i \(0.572397\pi\)
\(20\) 11.2694 0.125996
\(21\) 92.6528 0.962786
\(22\) −12.2403 −0.118620
\(23\) −100.967 −0.915351 −0.457675 0.889119i \(-0.651318\pi\)
−0.457675 + 0.889119i \(0.651318\pi\)
\(24\) 62.2958 0.529836
\(25\) −121.825 −0.974603
\(26\) −49.7850 −0.375525
\(27\) −143.509 −1.02290
\(28\) −174.407 −1.17714
\(29\) −175.001 −1.12058 −0.560292 0.828295i \(-0.689311\pi\)
−0.560292 + 0.828295i \(0.689311\pi\)
\(30\) 7.74835 0.0471549
\(31\) −247.586 −1.43445 −0.717223 0.696844i \(-0.754587\pi\)
−0.717223 + 0.696844i \(0.754587\pi\)
\(32\) −182.752 −1.00957
\(33\) 31.7783 0.167633
\(34\) −70.0170 −0.353171
\(35\) −49.1304 −0.237273
\(36\) 99.3634 0.460016
\(37\) −234.362 −1.04132 −0.520660 0.853764i \(-0.674314\pi\)
−0.520660 + 0.853764i \(0.674314\pi\)
\(38\) 48.3384 0.206356
\(39\) 129.252 0.530688
\(40\) −33.0332 −0.130575
\(41\) 325.321 1.23919 0.619593 0.784923i \(-0.287298\pi\)
0.619593 + 0.784923i \(0.287298\pi\)
\(42\) −119.915 −0.440554
\(43\) 0 0
\(44\) −59.8185 −0.204954
\(45\) 27.9907 0.0927245
\(46\) 130.675 0.418848
\(47\) 148.643 0.461315 0.230657 0.973035i \(-0.425912\pi\)
0.230657 + 0.973035i \(0.425912\pi\)
\(48\) 89.3938 0.268810
\(49\) 417.351 1.21677
\(50\) 157.671 0.445961
\(51\) 181.778 0.499098
\(52\) −243.300 −0.648838
\(53\) 735.610 1.90649 0.953244 0.302202i \(-0.0977216\pi\)
0.953244 + 0.302202i \(0.0977216\pi\)
\(54\) 185.735 0.468061
\(55\) −16.8509 −0.0413122
\(56\) 511.228 1.21992
\(57\) −125.496 −0.291620
\(58\) 226.493 0.512760
\(59\) −275.860 −0.608710 −0.304355 0.952559i \(-0.598441\pi\)
−0.304355 + 0.952559i \(0.598441\pi\)
\(60\) 37.8662 0.0814751
\(61\) −30.9134 −0.0648861 −0.0324431 0.999474i \(-0.510329\pi\)
−0.0324431 + 0.999474i \(0.510329\pi\)
\(62\) 320.436 0.656377
\(63\) −433.188 −0.866295
\(64\) 23.6879 0.0462654
\(65\) −68.5375 −0.130785
\(66\) −41.1286 −0.0767058
\(67\) −27.3051 −0.0497888 −0.0248944 0.999690i \(-0.507925\pi\)
−0.0248944 + 0.999690i \(0.507925\pi\)
\(68\) −342.173 −0.610215
\(69\) −339.259 −0.591912
\(70\) 63.5864 0.108572
\(71\) 66.0822 0.110458 0.0552290 0.998474i \(-0.482411\pi\)
0.0552290 + 0.998474i \(0.482411\pi\)
\(72\) −291.257 −0.476736
\(73\) 511.898 0.820728 0.410364 0.911922i \(-0.365402\pi\)
0.410364 + 0.911922i \(0.365402\pi\)
\(74\) 303.320 0.476490
\(75\) −409.345 −0.630228
\(76\) 236.230 0.356545
\(77\) 260.787 0.385967
\(78\) −167.282 −0.242833
\(79\) −1190.03 −1.69479 −0.847397 0.530959i \(-0.821832\pi\)
−0.847397 + 0.530959i \(0.821832\pi\)
\(80\) −47.4023 −0.0662467
\(81\) −58.0396 −0.0796154
\(82\) −421.043 −0.567030
\(83\) −144.868 −0.191582 −0.0957909 0.995401i \(-0.530538\pi\)
−0.0957909 + 0.995401i \(0.530538\pi\)
\(84\) −586.024 −0.761196
\(85\) −96.3902 −0.123000
\(86\) 0 0
\(87\) −588.021 −0.724626
\(88\) 175.342 0.212404
\(89\) −769.313 −0.916258 −0.458129 0.888886i \(-0.651480\pi\)
−0.458129 + 0.888886i \(0.651480\pi\)
\(90\) −36.2266 −0.0424291
\(91\) 1060.70 1.22188
\(92\) 638.611 0.723693
\(93\) −831.914 −0.927585
\(94\) −192.379 −0.211090
\(95\) 66.5460 0.0718681
\(96\) −614.063 −0.652839
\(97\) 1881.67 1.96964 0.984820 0.173577i \(-0.0555325\pi\)
0.984820 + 0.173577i \(0.0555325\pi\)
\(98\) −540.151 −0.556771
\(99\) −148.576 −0.150833
\(100\) 770.539 0.770539
\(101\) −664.679 −0.654832 −0.327416 0.944880i \(-0.606178\pi\)
−0.327416 + 0.944880i \(0.606178\pi\)
\(102\) −235.264 −0.228378
\(103\) −757.340 −0.724495 −0.362247 0.932082i \(-0.617990\pi\)
−0.362247 + 0.932082i \(0.617990\pi\)
\(104\) 713.168 0.672422
\(105\) −165.083 −0.153433
\(106\) −952.055 −0.872375
\(107\) −1312.68 −1.18599 −0.592996 0.805205i \(-0.702055\pi\)
−0.592996 + 0.805205i \(0.702055\pi\)
\(108\) 907.686 0.808723
\(109\) 314.072 0.275988 0.137994 0.990433i \(-0.455935\pi\)
0.137994 + 0.990433i \(0.455935\pi\)
\(110\) 21.8090 0.0189037
\(111\) −787.478 −0.673370
\(112\) 733.606 0.618922
\(113\) 2100.85 1.74895 0.874477 0.485068i \(-0.161205\pi\)
0.874477 + 0.485068i \(0.161205\pi\)
\(114\) 162.422 0.133440
\(115\) 179.896 0.145873
\(116\) 1106.87 0.885955
\(117\) −604.302 −0.477502
\(118\) 357.028 0.278535
\(119\) 1491.75 1.14915
\(120\) −110.995 −0.0844365
\(121\) −1241.55 −0.932798
\(122\) 40.0093 0.0296907
\(123\) 1093.11 0.801321
\(124\) 1565.97 1.13410
\(125\) 439.778 0.314679
\(126\) 560.649 0.396401
\(127\) −1771.00 −1.23741 −0.618703 0.785625i \(-0.712342\pi\)
−0.618703 + 0.785625i \(0.712342\pi\)
\(128\) 1431.36 0.988400
\(129\) 0 0
\(130\) 88.7038 0.0598449
\(131\) −2833.28 −1.88966 −0.944828 0.327567i \(-0.893771\pi\)
−0.944828 + 0.327567i \(0.893771\pi\)
\(132\) −200.996 −0.132534
\(133\) −1029.88 −0.671441
\(134\) 35.3393 0.0227825
\(135\) 255.695 0.163013
\(136\) 1002.99 0.632395
\(137\) −384.344 −0.239684 −0.119842 0.992793i \(-0.538239\pi\)
−0.119842 + 0.992793i \(0.538239\pi\)
\(138\) 439.081 0.270848
\(139\) 1099.42 0.670872 0.335436 0.942063i \(-0.391116\pi\)
0.335436 + 0.942063i \(0.391116\pi\)
\(140\) 310.747 0.187592
\(141\) 499.454 0.298310
\(142\) −85.5261 −0.0505436
\(143\) 363.800 0.212745
\(144\) −417.951 −0.241870
\(145\) 311.806 0.178580
\(146\) −662.518 −0.375550
\(147\) 1402.34 0.786823
\(148\) 1482.33 0.823287
\(149\) −2307.28 −1.26859 −0.634294 0.773092i \(-0.718709\pi\)
−0.634294 + 0.773092i \(0.718709\pi\)
\(150\) 529.790 0.288381
\(151\) −2158.29 −1.16317 −0.581586 0.813485i \(-0.697568\pi\)
−0.581586 + 0.813485i \(0.697568\pi\)
\(152\) −692.445 −0.369505
\(153\) −849.883 −0.449078
\(154\) −337.520 −0.176611
\(155\) 441.134 0.228598
\(156\) −817.510 −0.419572
\(157\) 2032.88 1.03338 0.516692 0.856171i \(-0.327163\pi\)
0.516692 + 0.856171i \(0.327163\pi\)
\(158\) 1540.18 0.775508
\(159\) 2471.72 1.23283
\(160\) 325.615 0.160888
\(161\) −2784.11 −1.36285
\(162\) 75.1171 0.0364306
\(163\) −41.5668 −0.0199740 −0.00998701 0.999950i \(-0.503179\pi\)
−0.00998701 + 0.999950i \(0.503179\pi\)
\(164\) −2057.64 −0.979724
\(165\) −56.6205 −0.0267145
\(166\) 187.493 0.0876644
\(167\) −3405.82 −1.57814 −0.789072 0.614301i \(-0.789438\pi\)
−0.789072 + 0.614301i \(0.789438\pi\)
\(168\) 1717.77 0.788863
\(169\) −717.315 −0.326498
\(170\) 124.752 0.0562825
\(171\) 586.743 0.262394
\(172\) 0 0
\(173\) 841.476 0.369805 0.184902 0.982757i \(-0.440803\pi\)
0.184902 + 0.982757i \(0.440803\pi\)
\(174\) 761.040 0.331576
\(175\) −3359.27 −1.45107
\(176\) 251.614 0.107762
\(177\) −926.915 −0.393622
\(178\) 995.674 0.419263
\(179\) −757.562 −0.316329 −0.158164 0.987413i \(-0.550558\pi\)
−0.158164 + 0.987413i \(0.550558\pi\)
\(180\) −177.039 −0.0733097
\(181\) −2451.87 −1.00688 −0.503442 0.864029i \(-0.667933\pi\)
−0.503442 + 0.864029i \(0.667933\pi\)
\(182\) −1372.80 −0.559112
\(183\) −103.872 −0.0419586
\(184\) −1871.92 −0.749997
\(185\) 417.571 0.165948
\(186\) 1076.69 0.424446
\(187\) 511.644 0.200081
\(188\) −940.159 −0.364724
\(189\) −3957.18 −1.52298
\(190\) −86.1263 −0.0328856
\(191\) −957.463 −0.362720 −0.181360 0.983417i \(-0.558050\pi\)
−0.181360 + 0.983417i \(0.558050\pi\)
\(192\) 79.5935 0.0299175
\(193\) 4925.30 1.83695 0.918474 0.395481i \(-0.129422\pi\)
0.918474 + 0.395481i \(0.129422\pi\)
\(194\) −2435.34 −0.901273
\(195\) −230.292 −0.0845722
\(196\) −2639.72 −0.961998
\(197\) −1146.48 −0.414636 −0.207318 0.978274i \(-0.566474\pi\)
−0.207318 + 0.978274i \(0.566474\pi\)
\(198\) 192.293 0.0690184
\(199\) −3188.35 −1.13576 −0.567880 0.823111i \(-0.692236\pi\)
−0.567880 + 0.823111i \(0.692236\pi\)
\(200\) −2258.63 −0.798546
\(201\) −91.7478 −0.0321960
\(202\) 860.253 0.299640
\(203\) −4825.57 −1.66842
\(204\) −1149.74 −0.394596
\(205\) −579.637 −0.197481
\(206\) 980.179 0.331516
\(207\) 1586.17 0.532591
\(208\) 1023.39 0.341150
\(209\) −353.230 −0.116906
\(210\) 213.656 0.0702081
\(211\) 5892.40 1.92251 0.961256 0.275659i \(-0.0888960\pi\)
0.961256 + 0.275659i \(0.0888960\pi\)
\(212\) −4652.70 −1.50730
\(213\) 222.042 0.0714277
\(214\) 1698.92 0.542689
\(215\) 0 0
\(216\) −2660.64 −0.838118
\(217\) −6827.06 −2.13572
\(218\) −406.484 −0.126287
\(219\) 1720.03 0.530724
\(220\) 106.581 0.0326622
\(221\) 2081.01 0.633411
\(222\) 1019.18 0.308122
\(223\) 1185.81 0.356088 0.178044 0.984023i \(-0.443023\pi\)
0.178044 + 0.984023i \(0.443023\pi\)
\(224\) −5039.28 −1.50313
\(225\) 1913.85 0.567066
\(226\) −2719.01 −0.800290
\(227\) −4647.73 −1.35895 −0.679473 0.733700i \(-0.737792\pi\)
−0.679473 + 0.733700i \(0.737792\pi\)
\(228\) 793.756 0.230560
\(229\) −1956.95 −0.564711 −0.282355 0.959310i \(-0.591116\pi\)
−0.282355 + 0.959310i \(0.591116\pi\)
\(230\) −232.829 −0.0667490
\(231\) 876.269 0.249585
\(232\) −3244.51 −0.918157
\(233\) 2211.01 0.621665 0.310832 0.950465i \(-0.399392\pi\)
0.310832 + 0.950465i \(0.399392\pi\)
\(234\) 782.111 0.218497
\(235\) −264.842 −0.0735167
\(236\) 1744.80 0.481257
\(237\) −3998.61 −1.09594
\(238\) −1930.68 −0.525830
\(239\) 808.463 0.218808 0.109404 0.993997i \(-0.465106\pi\)
0.109404 + 0.993997i \(0.465106\pi\)
\(240\) −159.276 −0.0428385
\(241\) 3816.46 1.02008 0.510041 0.860150i \(-0.329630\pi\)
0.510041 + 0.860150i \(0.329630\pi\)
\(242\) 1606.87 0.426832
\(243\) 3679.72 0.971416
\(244\) 195.526 0.0513002
\(245\) −743.609 −0.193908
\(246\) −1414.74 −0.366670
\(247\) −1436.69 −0.370099
\(248\) −4590.22 −1.17532
\(249\) −486.769 −0.123886
\(250\) −569.177 −0.143992
\(251\) 4999.26 1.25717 0.628586 0.777740i \(-0.283634\pi\)
0.628586 + 0.777740i \(0.283634\pi\)
\(252\) 2739.89 0.684909
\(253\) −954.900 −0.237289
\(254\) 2292.09 0.566215
\(255\) −323.880 −0.0795379
\(256\) −2042.02 −0.498540
\(257\) 2878.94 0.698767 0.349383 0.936980i \(-0.386391\pi\)
0.349383 + 0.936980i \(0.386391\pi\)
\(258\) 0 0
\(259\) −6462.40 −1.55040
\(260\) 433.496 0.103401
\(261\) 2749.23 0.652004
\(262\) 3666.94 0.864673
\(263\) −7196.10 −1.68719 −0.843594 0.536982i \(-0.819564\pi\)
−0.843594 + 0.536982i \(0.819564\pi\)
\(264\) 589.166 0.137351
\(265\) −1310.66 −0.303824
\(266\) 1332.91 0.307239
\(267\) −2584.96 −0.592499
\(268\) 172.703 0.0393640
\(269\) 8400.53 1.90405 0.952025 0.306019i \(-0.0989972\pi\)
0.952025 + 0.306019i \(0.0989972\pi\)
\(270\) −330.930 −0.0745917
\(271\) −692.143 −0.155147 −0.0775733 0.996987i \(-0.524717\pi\)
−0.0775733 + 0.996987i \(0.524717\pi\)
\(272\) 1439.28 0.320843
\(273\) 3564.04 0.790131
\(274\) 497.432 0.109675
\(275\) −1152.17 −0.252649
\(276\) 2145.79 0.467977
\(277\) 2581.54 0.559963 0.279981 0.960005i \(-0.409672\pi\)
0.279981 + 0.960005i \(0.409672\pi\)
\(278\) −1422.91 −0.306979
\(279\) 3889.52 0.834623
\(280\) −910.873 −0.194411
\(281\) −5866.29 −1.24539 −0.622693 0.782466i \(-0.713961\pi\)
−0.622693 + 0.782466i \(0.713961\pi\)
\(282\) −646.413 −0.136501
\(283\) −7501.69 −1.57572 −0.787861 0.615854i \(-0.788811\pi\)
−0.787861 + 0.615854i \(0.788811\pi\)
\(284\) −417.966 −0.0873301
\(285\) 223.601 0.0464736
\(286\) −470.844 −0.0973483
\(287\) 8970.56 1.84500
\(288\) 2870.99 0.587412
\(289\) −1986.30 −0.404294
\(290\) −403.552 −0.0817151
\(291\) 6322.61 1.27367
\(292\) −3237.73 −0.648883
\(293\) −4809.72 −0.958999 −0.479500 0.877542i \(-0.659182\pi\)
−0.479500 + 0.877542i \(0.659182\pi\)
\(294\) −1814.96 −0.360036
\(295\) 491.509 0.0970060
\(296\) −4345.04 −0.853211
\(297\) −1357.24 −0.265169
\(298\) 2986.17 0.580483
\(299\) −3883.86 −0.751203
\(300\) 2589.09 0.498270
\(301\) 0 0
\(302\) 2793.34 0.532247
\(303\) −2233.39 −0.423448
\(304\) −993.652 −0.187467
\(305\) 55.0795 0.0103405
\(306\) 1099.95 0.205490
\(307\) −2532.95 −0.470890 −0.235445 0.971888i \(-0.575655\pi\)
−0.235445 + 0.971888i \(0.575655\pi\)
\(308\) −1649.46 −0.305152
\(309\) −2544.73 −0.468495
\(310\) −570.932 −0.104602
\(311\) −9641.95 −1.75802 −0.879011 0.476802i \(-0.841796\pi\)
−0.879011 + 0.476802i \(0.841796\pi\)
\(312\) 2396.31 0.434822
\(313\) −7799.67 −1.40851 −0.704254 0.709948i \(-0.748719\pi\)
−0.704254 + 0.709948i \(0.748719\pi\)
\(314\) −2631.03 −0.472858
\(315\) 771.827 0.138056
\(316\) 7526.87 1.33994
\(317\) −8558.52 −1.51639 −0.758193 0.652031i \(-0.773917\pi\)
−0.758193 + 0.652031i \(0.773917\pi\)
\(318\) −3198.99 −0.564122
\(319\) −1655.08 −0.290492
\(320\) −42.2055 −0.00737300
\(321\) −4410.72 −0.766923
\(322\) 3603.30 0.623615
\(323\) −2020.54 −0.348068
\(324\) 367.098 0.0629454
\(325\) −4686.22 −0.799830
\(326\) 53.7974 0.00913976
\(327\) 1055.31 0.178468
\(328\) 6031.42 1.01533
\(329\) 4098.75 0.686843
\(330\) 73.2804 0.0122241
\(331\) 3134.43 0.520495 0.260247 0.965542i \(-0.416196\pi\)
0.260247 + 0.965542i \(0.416196\pi\)
\(332\) 916.280 0.151468
\(333\) 3681.77 0.605885
\(334\) 4407.94 0.722131
\(335\) 48.6505 0.00793451
\(336\) 2464.99 0.400226
\(337\) −192.133 −0.0310569 −0.0155284 0.999879i \(-0.504943\pi\)
−0.0155284 + 0.999879i \(0.504943\pi\)
\(338\) 928.377 0.149400
\(339\) 7059.07 1.13096
\(340\) 609.663 0.0972459
\(341\) −2341.56 −0.371855
\(342\) −759.385 −0.120067
\(343\) 2050.19 0.322740
\(344\) 0 0
\(345\) 604.469 0.0943291
\(346\) −1089.07 −0.169216
\(347\) −4066.72 −0.629144 −0.314572 0.949234i \(-0.601861\pi\)
−0.314572 + 0.949234i \(0.601861\pi\)
\(348\) 3719.20 0.572903
\(349\) 9154.26 1.40406 0.702029 0.712148i \(-0.252277\pi\)
0.702029 + 0.712148i \(0.252277\pi\)
\(350\) 4347.70 0.663983
\(351\) −5520.31 −0.839465
\(352\) −1728.38 −0.261714
\(353\) 854.088 0.128778 0.0643888 0.997925i \(-0.479490\pi\)
0.0643888 + 0.997925i \(0.479490\pi\)
\(354\) 1199.65 0.180115
\(355\) −117.741 −0.0176029
\(356\) 4865.86 0.724411
\(357\) 5012.42 0.743097
\(358\) 980.465 0.144746
\(359\) 3079.71 0.452761 0.226380 0.974039i \(-0.427311\pi\)
0.226380 + 0.974039i \(0.427311\pi\)
\(360\) 518.944 0.0759743
\(361\) −5464.06 −0.796626
\(362\) 3173.30 0.460732
\(363\) −4171.74 −0.603195
\(364\) −6708.86 −0.966043
\(365\) −912.067 −0.130794
\(366\) 134.435 0.0191995
\(367\) 1288.57 0.183277 0.0916384 0.995792i \(-0.470790\pi\)
0.0916384 + 0.995792i \(0.470790\pi\)
\(368\) −2686.18 −0.380508
\(369\) −5110.72 −0.721012
\(370\) −540.436 −0.0759350
\(371\) 20284.1 2.83853
\(372\) 5261.81 0.733366
\(373\) 9257.47 1.28508 0.642538 0.766254i \(-0.277881\pi\)
0.642538 + 0.766254i \(0.277881\pi\)
\(374\) −662.189 −0.0915535
\(375\) 1477.69 0.203488
\(376\) 2755.82 0.377981
\(377\) −6731.72 −0.919632
\(378\) 5121.53 0.696887
\(379\) −7746.53 −1.04990 −0.524951 0.851133i \(-0.675916\pi\)
−0.524951 + 0.851133i \(0.675916\pi\)
\(380\) −420.900 −0.0568203
\(381\) −5950.72 −0.800169
\(382\) 1239.19 0.165974
\(383\) −247.436 −0.0330114 −0.0165057 0.999864i \(-0.505254\pi\)
−0.0165057 + 0.999864i \(0.505254\pi\)
\(384\) 4809.49 0.639149
\(385\) −464.653 −0.0615089
\(386\) −6374.51 −0.840555
\(387\) 0 0
\(388\) −11901.5 −1.55723
\(389\) −10078.8 −1.31366 −0.656831 0.754038i \(-0.728104\pi\)
−0.656831 + 0.754038i \(0.728104\pi\)
\(390\) 298.053 0.0386987
\(391\) −5462.21 −0.706486
\(392\) 7737.64 0.996964
\(393\) −9520.09 −1.22195
\(394\) 1483.82 0.189730
\(395\) 2120.32 0.270088
\(396\) 939.735 0.119251
\(397\) −58.9880 −0.00745723 −0.00372862 0.999993i \(-0.501187\pi\)
−0.00372862 + 0.999993i \(0.501187\pi\)
\(398\) 4126.49 0.519704
\(399\) −3460.48 −0.434188
\(400\) −3241.11 −0.405139
\(401\) 7494.39 0.933297 0.466648 0.884443i \(-0.345461\pi\)
0.466648 + 0.884443i \(0.345461\pi\)
\(402\) 118.743 0.0147323
\(403\) −9523.82 −1.17721
\(404\) 4204.06 0.517723
\(405\) 103.411 0.0126878
\(406\) 6245.44 0.763438
\(407\) −2216.49 −0.269944
\(408\) 3370.14 0.408938
\(409\) 9345.06 1.12979 0.564894 0.825163i \(-0.308917\pi\)
0.564894 + 0.825163i \(0.308917\pi\)
\(410\) 750.188 0.0903637
\(411\) −1291.43 −0.154992
\(412\) 4790.14 0.572799
\(413\) −7606.68 −0.906296
\(414\) −2052.88 −0.243704
\(415\) 258.116 0.0305311
\(416\) −7029.85 −0.828526
\(417\) 3694.14 0.433820
\(418\) 457.163 0.0534942
\(419\) −11194.4 −1.30521 −0.652607 0.757697i \(-0.726325\pi\)
−0.652607 + 0.757697i \(0.726325\pi\)
\(420\) 1044.14 0.121307
\(421\) −4066.28 −0.470732 −0.235366 0.971907i \(-0.575629\pi\)
−0.235366 + 0.971907i \(0.575629\pi\)
\(422\) −7626.17 −0.879707
\(423\) −2335.15 −0.268413
\(424\) 13638.1 1.56209
\(425\) −6590.63 −0.752218
\(426\) −287.376 −0.0326840
\(427\) −852.420 −0.0966077
\(428\) 8302.61 0.937668
\(429\) 1222.40 0.137572
\(430\) 0 0
\(431\) −8136.58 −0.909340 −0.454670 0.890660i \(-0.650243\pi\)
−0.454670 + 0.890660i \(0.650243\pi\)
\(432\) −3817.99 −0.425215
\(433\) 7039.42 0.781277 0.390638 0.920544i \(-0.372254\pi\)
0.390638 + 0.920544i \(0.372254\pi\)
\(434\) 8835.84 0.977267
\(435\) 1047.70 0.115479
\(436\) −1986.49 −0.218201
\(437\) 3771.01 0.412796
\(438\) −2226.12 −0.242850
\(439\) 12596.3 1.36945 0.684727 0.728800i \(-0.259922\pi\)
0.684727 + 0.728800i \(0.259922\pi\)
\(440\) −312.413 −0.0338493
\(441\) −6556.49 −0.707967
\(442\) −2693.32 −0.289838
\(443\) 17582.4 1.88570 0.942851 0.333216i \(-0.108134\pi\)
0.942851 + 0.333216i \(0.108134\pi\)
\(444\) 4980.76 0.532379
\(445\) 1370.71 0.146018
\(446\) −1534.72 −0.162939
\(447\) −7752.67 −0.820333
\(448\) 653.180 0.0688836
\(449\) −1356.00 −0.142524 −0.0712621 0.997458i \(-0.522703\pi\)
−0.0712621 + 0.997458i \(0.522703\pi\)
\(450\) −2476.98 −0.259479
\(451\) 3076.74 0.321238
\(452\) −13287.8 −1.38275
\(453\) −7252.05 −0.752166
\(454\) 6015.27 0.621830
\(455\) −1889.88 −0.194723
\(456\) −2326.68 −0.238940
\(457\) −9380.12 −0.960139 −0.480069 0.877230i \(-0.659389\pi\)
−0.480069 + 0.877230i \(0.659389\pi\)
\(458\) 2532.76 0.258402
\(459\) −7763.69 −0.789494
\(460\) −1137.84 −0.115330
\(461\) 8627.66 0.871649 0.435825 0.900032i \(-0.356457\pi\)
0.435825 + 0.900032i \(0.356457\pi\)
\(462\) −1134.10 −0.114206
\(463\) 2501.34 0.251073 0.125537 0.992089i \(-0.459935\pi\)
0.125537 + 0.992089i \(0.459935\pi\)
\(464\) −4655.83 −0.465823
\(465\) 1482.25 0.147823
\(466\) −2861.57 −0.284463
\(467\) −10147.1 −1.00546 −0.502731 0.864443i \(-0.667671\pi\)
−0.502731 + 0.864443i \(0.667671\pi\)
\(468\) 3822.18 0.377522
\(469\) −752.924 −0.0741296
\(470\) 342.769 0.0336399
\(471\) 6830.66 0.668238
\(472\) −5114.41 −0.498749
\(473\) 0 0
\(474\) 5175.15 0.501483
\(475\) 4550.05 0.439517
\(476\) −9435.25 −0.908538
\(477\) −11556.3 −1.10928
\(478\) −1046.34 −0.100123
\(479\) 20100.3 1.91735 0.958673 0.284511i \(-0.0918313\pi\)
0.958673 + 0.284511i \(0.0918313\pi\)
\(480\) 1094.10 0.104039
\(481\) −9015.12 −0.854582
\(482\) −4939.41 −0.466772
\(483\) −9354.87 −0.881287
\(484\) 7852.77 0.737488
\(485\) −3352.65 −0.313888
\(486\) −4762.43 −0.444503
\(487\) −8302.85 −0.772563 −0.386281 0.922381i \(-0.626241\pi\)
−0.386281 + 0.922381i \(0.626241\pi\)
\(488\) −573.131 −0.0531648
\(489\) −139.669 −0.0129162
\(490\) 962.407 0.0887288
\(491\) −988.707 −0.0908752 −0.0454376 0.998967i \(-0.514468\pi\)
−0.0454376 + 0.998967i \(0.514468\pi\)
\(492\) −6913.87 −0.633539
\(493\) −9467.40 −0.864889
\(494\) 1859.42 0.169351
\(495\) 264.723 0.0240372
\(496\) −6586.92 −0.596294
\(497\) 1822.18 0.164459
\(498\) 629.995 0.0566882
\(499\) 1586.04 0.142286 0.0711430 0.997466i \(-0.477335\pi\)
0.0711430 + 0.997466i \(0.477335\pi\)
\(500\) −2781.57 −0.248791
\(501\) −11443.9 −1.02051
\(502\) −6470.23 −0.575260
\(503\) −9056.14 −0.802770 −0.401385 0.915909i \(-0.631471\pi\)
−0.401385 + 0.915909i \(0.631471\pi\)
\(504\) −8031.26 −0.709803
\(505\) 1184.28 0.104356
\(506\) 1235.87 0.108579
\(507\) −2410.25 −0.211130
\(508\) 11201.5 0.978316
\(509\) 10268.3 0.894170 0.447085 0.894491i \(-0.352462\pi\)
0.447085 + 0.894491i \(0.352462\pi\)
\(510\) 419.178 0.0363951
\(511\) 14115.3 1.22197
\(512\) −8807.99 −0.760277
\(513\) 5359.90 0.461297
\(514\) −3726.03 −0.319743
\(515\) 1349.38 0.115458
\(516\) 0 0
\(517\) 1405.80 0.119588
\(518\) 8363.88 0.709436
\(519\) 2827.44 0.239134
\(520\) −1270.68 −0.107159
\(521\) −8690.28 −0.730764 −0.365382 0.930858i \(-0.619062\pi\)
−0.365382 + 0.930858i \(0.619062\pi\)
\(522\) −3558.16 −0.298346
\(523\) 11938.1 0.998124 0.499062 0.866566i \(-0.333678\pi\)
0.499062 + 0.866566i \(0.333678\pi\)
\(524\) 17920.4 1.49400
\(525\) −11287.5 −0.938334
\(526\) 9313.46 0.772027
\(527\) −13394.2 −1.10713
\(528\) 845.446 0.0696843
\(529\) −1972.67 −0.162133
\(530\) 1696.31 0.139025
\(531\) 4333.69 0.354174
\(532\) 6513.92 0.530854
\(533\) 12514.0 1.01697
\(534\) 3345.56 0.271117
\(535\) 2338.84 0.189004
\(536\) −506.234 −0.0407947
\(537\) −2545.48 −0.204554
\(538\) −10872.3 −0.871260
\(539\) 3947.12 0.315426
\(540\) −1617.26 −0.128881
\(541\) −11644.8 −0.925413 −0.462707 0.886511i \(-0.653122\pi\)
−0.462707 + 0.886511i \(0.653122\pi\)
\(542\) 895.798 0.0709923
\(543\) −8238.51 −0.651102
\(544\) −9886.69 −0.779206
\(545\) −559.594 −0.0439823
\(546\) −4612.72 −0.361550
\(547\) −16635.8 −1.30035 −0.650177 0.759783i \(-0.725305\pi\)
−0.650177 + 0.759783i \(0.725305\pi\)
\(548\) 2430.95 0.189498
\(549\) 485.642 0.0377536
\(550\) 1491.18 0.115608
\(551\) 6536.12 0.505350
\(552\) −6289.82 −0.484986
\(553\) −32814.4 −2.52335
\(554\) −3341.13 −0.256229
\(555\) 1403.08 0.107310
\(556\) −6953.75 −0.530404
\(557\) 310.333 0.0236073 0.0118036 0.999930i \(-0.496243\pi\)
0.0118036 + 0.999930i \(0.496243\pi\)
\(558\) −5033.97 −0.381908
\(559\) 0 0
\(560\) −1307.09 −0.0986335
\(561\) 1719.17 0.129382
\(562\) 7592.37 0.569866
\(563\) −5029.04 −0.376463 −0.188232 0.982125i \(-0.560276\pi\)
−0.188232 + 0.982125i \(0.560276\pi\)
\(564\) −3159.02 −0.235849
\(565\) −3743.17 −0.278719
\(566\) 9708.97 0.721022
\(567\) −1600.41 −0.118538
\(568\) 1225.16 0.0905043
\(569\) 11414.7 0.841001 0.420500 0.907292i \(-0.361855\pi\)
0.420500 + 0.907292i \(0.361855\pi\)
\(570\) −289.393 −0.0212655
\(571\) 5046.78 0.369879 0.184940 0.982750i \(-0.440791\pi\)
0.184940 + 0.982750i \(0.440791\pi\)
\(572\) −2301.02 −0.168200
\(573\) −3217.17 −0.234553
\(574\) −11610.0 −0.844240
\(575\) 12300.3 0.892104
\(576\) −372.131 −0.0269192
\(577\) 1338.61 0.0965807 0.0482904 0.998833i \(-0.484623\pi\)
0.0482904 + 0.998833i \(0.484623\pi\)
\(578\) 2570.74 0.184998
\(579\) 16549.5 1.18786
\(580\) −1972.16 −0.141189
\(581\) −3994.65 −0.285243
\(582\) −8182.96 −0.582808
\(583\) 6957.07 0.494224
\(584\) 9490.53 0.672468
\(585\) 1076.71 0.0760964
\(586\) 6224.92 0.438821
\(587\) 8253.72 0.580354 0.290177 0.956973i \(-0.406286\pi\)
0.290177 + 0.956973i \(0.406286\pi\)
\(588\) −8869.72 −0.622077
\(589\) 9247.09 0.646892
\(590\) −636.130 −0.0443882
\(591\) −3852.28 −0.268125
\(592\) −6235.09 −0.432873
\(593\) −17646.3 −1.22200 −0.610999 0.791631i \(-0.709232\pi\)
−0.610999 + 0.791631i \(0.709232\pi\)
\(594\) 1756.59 0.121337
\(595\) −2657.91 −0.183132
\(596\) 14593.4 1.00297
\(597\) −10713.2 −0.734440
\(598\) 5026.64 0.343737
\(599\) −19407.0 −1.32379 −0.661894 0.749598i \(-0.730247\pi\)
−0.661894 + 0.749598i \(0.730247\pi\)
\(600\) −7589.21 −0.516380
\(601\) −2556.29 −0.173500 −0.0867499 0.996230i \(-0.527648\pi\)
−0.0867499 + 0.996230i \(0.527648\pi\)
\(602\) 0 0
\(603\) 428.957 0.0289693
\(604\) 13651.1 0.919625
\(605\) 2212.12 0.148654
\(606\) 2890.53 0.193762
\(607\) −14431.3 −0.964988 −0.482494 0.875899i \(-0.660269\pi\)
−0.482494 + 0.875899i \(0.660269\pi\)
\(608\) 6825.58 0.455286
\(609\) −16214.4 −1.07888
\(610\) −71.2860 −0.00473162
\(611\) 5717.80 0.378588
\(612\) 5375.46 0.355049
\(613\) −10358.6 −0.682512 −0.341256 0.939970i \(-0.610852\pi\)
−0.341256 + 0.939970i \(0.610852\pi\)
\(614\) 3278.25 0.215471
\(615\) −1947.63 −0.127701
\(616\) 4834.96 0.316244
\(617\) −25620.7 −1.67172 −0.835860 0.548942i \(-0.815031\pi\)
−0.835860 + 0.548942i \(0.815031\pi\)
\(618\) 3293.49 0.214375
\(619\) 7346.23 0.477011 0.238505 0.971141i \(-0.423343\pi\)
0.238505 + 0.971141i \(0.423343\pi\)
\(620\) −2790.15 −0.180734
\(621\) 14489.7 0.936312
\(622\) 12479.0 0.804440
\(623\) −21213.4 −1.36420
\(624\) 3438.68 0.220605
\(625\) 14444.6 0.924455
\(626\) 10094.6 0.644509
\(627\) −1186.88 −0.0755975
\(628\) −12857.8 −0.817012
\(629\) −12678.7 −0.803711
\(630\) −998.928 −0.0631718
\(631\) 10677.3 0.673626 0.336813 0.941572i \(-0.390651\pi\)
0.336813 + 0.941572i \(0.390651\pi\)
\(632\) −22063.0 −1.38864
\(633\) 19799.0 1.24319
\(634\) 11076.8 0.693871
\(635\) 3155.45 0.197197
\(636\) −15633.5 −0.974699
\(637\) 16054.1 0.998566
\(638\) 2142.07 0.132924
\(639\) −1038.14 −0.0642692
\(640\) −2550.30 −0.157515
\(641\) −1114.95 −0.0687018 −0.0343509 0.999410i \(-0.510936\pi\)
−0.0343509 + 0.999410i \(0.510936\pi\)
\(642\) 5708.52 0.350930
\(643\) 9253.91 0.567556 0.283778 0.958890i \(-0.408412\pi\)
0.283778 + 0.958890i \(0.408412\pi\)
\(644\) 17609.3 1.07749
\(645\) 0 0
\(646\) 2615.06 0.159270
\(647\) 18799.8 1.14234 0.571172 0.820831i \(-0.306489\pi\)
0.571172 + 0.820831i \(0.306489\pi\)
\(648\) −1076.05 −0.0652333
\(649\) −2608.96 −0.157797
\(650\) 6065.08 0.365988
\(651\) −22939.6 −1.38106
\(652\) 262.908 0.0157918
\(653\) −24082.8 −1.44324 −0.721618 0.692291i \(-0.756601\pi\)
−0.721618 + 0.692291i \(0.756601\pi\)
\(654\) −1365.82 −0.0816636
\(655\) 5048.16 0.301142
\(656\) 8655.03 0.515125
\(657\) −8041.80 −0.477535
\(658\) −5304.76 −0.314287
\(659\) −23306.6 −1.37769 −0.688843 0.724911i \(-0.741881\pi\)
−0.688843 + 0.724911i \(0.741881\pi\)
\(660\) 358.122 0.0211210
\(661\) 8490.11 0.499587 0.249793 0.968299i \(-0.419637\pi\)
0.249793 + 0.968299i \(0.419637\pi\)
\(662\) −4056.69 −0.238169
\(663\) 6992.39 0.409595
\(664\) −2685.83 −0.156974
\(665\) 1834.97 0.107003
\(666\) −4765.09 −0.277242
\(667\) 17669.4 1.02573
\(668\) 21541.6 1.24771
\(669\) 3984.43 0.230264
\(670\) −62.9654 −0.00363069
\(671\) −292.365 −0.0168206
\(672\) −16932.5 −0.972000
\(673\) −6648.43 −0.380800 −0.190400 0.981707i \(-0.560978\pi\)
−0.190400 + 0.981707i \(0.560978\pi\)
\(674\) 248.666 0.0142111
\(675\) 17483.0 0.996921
\(676\) 4536.98 0.258135
\(677\) 18154.8 1.03064 0.515320 0.856998i \(-0.327673\pi\)
0.515320 + 0.856998i \(0.327673\pi\)
\(678\) −9136.11 −0.517508
\(679\) 51886.2 2.93256
\(680\) −1787.06 −0.100781
\(681\) −15616.8 −0.878763
\(682\) 3030.54 0.170154
\(683\) 1949.71 0.109229 0.0546147 0.998508i \(-0.482607\pi\)
0.0546147 + 0.998508i \(0.482607\pi\)
\(684\) −3711.12 −0.207454
\(685\) 684.799 0.0381968
\(686\) −2653.43 −0.147680
\(687\) −6575.53 −0.365170
\(688\) 0 0
\(689\) 28296.5 1.56460
\(690\) −782.327 −0.0431633
\(691\) −6988.90 −0.384762 −0.192381 0.981320i \(-0.561621\pi\)
−0.192381 + 0.981320i \(0.561621\pi\)
\(692\) −5322.29 −0.292374
\(693\) −4096.90 −0.224572
\(694\) 5263.31 0.287885
\(695\) −1958.87 −0.106912
\(696\) −10901.9 −0.593726
\(697\) 17599.6 0.956429
\(698\) −11847.8 −0.642472
\(699\) 7429.19 0.402000
\(700\) 21247.2 1.14724
\(701\) −6698.64 −0.360919 −0.180459 0.983582i \(-0.557758\pi\)
−0.180459 + 0.983582i \(0.557758\pi\)
\(702\) 7144.59 0.384124
\(703\) 8753.17 0.469604
\(704\) 224.029 0.0119935
\(705\) −889.896 −0.0475396
\(706\) −1105.39 −0.0589263
\(707\) −18328.2 −0.974968
\(708\) 5862.69 0.311205
\(709\) −33993.1 −1.80062 −0.900309 0.435251i \(-0.856659\pi\)
−0.900309 + 0.435251i \(0.856659\pi\)
\(710\) 152.385 0.00805479
\(711\) 18695.1 0.986105
\(712\) −14263.0 −0.750741
\(713\) 24998.0 1.31302
\(714\) −6487.27 −0.340028
\(715\) −648.196 −0.0339037
\(716\) 4791.54 0.250095
\(717\) 2716.51 0.141492
\(718\) −3985.88 −0.207175
\(719\) 6671.11 0.346023 0.173011 0.984920i \(-0.444650\pi\)
0.173011 + 0.984920i \(0.444650\pi\)
\(720\) 744.679 0.0385452
\(721\) −20883.3 −1.07869
\(722\) 7071.79 0.364522
\(723\) 12823.7 0.659637
\(724\) 15507.9 0.796061
\(725\) 21319.6 1.09213
\(726\) 5399.23 0.276011
\(727\) −20828.5 −1.06257 −0.531285 0.847193i \(-0.678290\pi\)
−0.531285 + 0.847193i \(0.678290\pi\)
\(728\) 19665.2 1.00116
\(729\) 13931.3 0.707782
\(730\) 1180.43 0.0598490
\(731\) 0 0
\(732\) 656.984 0.0331733
\(733\) −19082.3 −0.961556 −0.480778 0.876842i \(-0.659646\pi\)
−0.480778 + 0.876842i \(0.659646\pi\)
\(734\) −1667.71 −0.0838642
\(735\) −2498.60 −0.125391
\(736\) 18451.9 0.924111
\(737\) −258.240 −0.0129069
\(738\) 6614.49 0.329922
\(739\) 21641.2 1.07725 0.538623 0.842547i \(-0.318945\pi\)
0.538623 + 0.842547i \(0.318945\pi\)
\(740\) −2641.11 −0.131202
\(741\) −4827.41 −0.239324
\(742\) −26252.4 −1.29886
\(743\) 22127.2 1.09255 0.546277 0.837605i \(-0.316045\pi\)
0.546277 + 0.837605i \(0.316045\pi\)
\(744\) −15423.6 −0.760022
\(745\) 4110.96 0.202166
\(746\) −11981.4 −0.588028
\(747\) 2275.84 0.111471
\(748\) −3236.12 −0.158188
\(749\) −36196.3 −1.76580
\(750\) −1912.49 −0.0931123
\(751\) 18774.4 0.912234 0.456117 0.889920i \(-0.349240\pi\)
0.456117 + 0.889920i \(0.349240\pi\)
\(752\) 3954.58 0.191767
\(753\) 16798.0 0.812952
\(754\) 8712.45 0.420807
\(755\) 3845.50 0.185367
\(756\) 25029.0 1.20409
\(757\) 39416.8 1.89251 0.946253 0.323429i \(-0.104836\pi\)
0.946253 + 0.323429i \(0.104836\pi\)
\(758\) 10025.9 0.480416
\(759\) −3208.55 −0.153443
\(760\) 1233.75 0.0588855
\(761\) 562.546 0.0267967 0.0133983 0.999910i \(-0.495735\pi\)
0.0133983 + 0.999910i \(0.495735\pi\)
\(762\) 7701.64 0.366143
\(763\) 8660.37 0.410913
\(764\) 6055.91 0.286774
\(765\) 1514.27 0.0715666
\(766\) 320.241 0.0151054
\(767\) −10611.4 −0.499551
\(768\) −6861.38 −0.322381
\(769\) −25134.3 −1.17863 −0.589315 0.807903i \(-0.700602\pi\)
−0.589315 + 0.807903i \(0.700602\pi\)
\(770\) 601.372 0.0281454
\(771\) 9673.49 0.451858
\(772\) −31152.3 −1.45233
\(773\) −22440.0 −1.04413 −0.522063 0.852907i \(-0.674837\pi\)
−0.522063 + 0.852907i \(0.674837\pi\)
\(774\) 0 0
\(775\) 30162.3 1.39802
\(776\) 34886.1 1.61384
\(777\) −21714.3 −1.00257
\(778\) 13044.3 0.601108
\(779\) −12150.4 −0.558836
\(780\) 1456.59 0.0668643
\(781\) 624.976 0.0286343
\(782\) 7069.40 0.323275
\(783\) 25114.3 1.14625
\(784\) 11103.4 0.505805
\(785\) −3622.05 −0.164683
\(786\) 12321.3 0.559141
\(787\) 12520.6 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(788\) 7251.43 0.327819
\(789\) −24179.6 −1.09102
\(790\) −2744.20 −0.123588
\(791\) 57929.9 2.60398
\(792\) −2754.58 −0.123586
\(793\) −1189.14 −0.0532502
\(794\) 76.3445 0.00341230
\(795\) −4403.95 −0.196468
\(796\) 20166.2 0.897954
\(797\) −682.158 −0.0303178 −0.0151589 0.999885i \(-0.504825\pi\)
−0.0151589 + 0.999885i \(0.504825\pi\)
\(798\) 4478.69 0.198677
\(799\) 8041.44 0.356052
\(800\) 22263.8 0.983931
\(801\) 12085.7 0.533118
\(802\) −9699.52 −0.427060
\(803\) 4841.30 0.212759
\(804\) 580.300 0.0254547
\(805\) 4960.55 0.217188
\(806\) 12326.1 0.538670
\(807\) 28226.6 1.23126
\(808\) −12323.1 −0.536540
\(809\) −20230.9 −0.879211 −0.439605 0.898191i \(-0.644882\pi\)
−0.439605 + 0.898191i \(0.644882\pi\)
\(810\) −133.839 −0.00580570
\(811\) 30575.2 1.32385 0.661924 0.749571i \(-0.269740\pi\)
0.661924 + 0.749571i \(0.269740\pi\)
\(812\) 30521.5 1.31908
\(813\) −2325.67 −0.100326
\(814\) 2868.66 0.123522
\(815\) 74.0611 0.00318313
\(816\) 4836.12 0.207473
\(817\) 0 0
\(818\) −12094.7 −0.516971
\(819\) −16663.3 −0.710944
\(820\) 3666.17 0.156132
\(821\) −24080.7 −1.02365 −0.511827 0.859088i \(-0.671031\pi\)
−0.511827 + 0.859088i \(0.671031\pi\)
\(822\) 1671.42 0.0709214
\(823\) −37732.6 −1.59815 −0.799073 0.601234i \(-0.794676\pi\)
−0.799073 + 0.601234i \(0.794676\pi\)
\(824\) −14041.0 −0.593619
\(825\) −3871.40 −0.163376
\(826\) 9844.86 0.414705
\(827\) −4683.21 −0.196918 −0.0984590 0.995141i \(-0.531391\pi\)
−0.0984590 + 0.995141i \(0.531391\pi\)
\(828\) −10032.4 −0.421076
\(829\) 32363.7 1.35590 0.677948 0.735110i \(-0.262870\pi\)
0.677948 + 0.735110i \(0.262870\pi\)
\(830\) −334.063 −0.0139705
\(831\) 8674.22 0.362100
\(832\) 911.193 0.0379687
\(833\) 22578.3 0.939125
\(834\) −4781.10 −0.198508
\(835\) 6068.27 0.251498
\(836\) 2234.16 0.0924282
\(837\) 35530.8 1.46729
\(838\) 14488.3 0.597243
\(839\) 41429.4 1.70477 0.852385 0.522915i \(-0.175155\pi\)
0.852385 + 0.522915i \(0.175155\pi\)
\(840\) −3060.62 −0.125716
\(841\) 6236.50 0.255710
\(842\) 5262.73 0.215399
\(843\) −19711.3 −0.805329
\(844\) −37269.2 −1.51997
\(845\) 1278.07 0.0520317
\(846\) 3022.23 0.122821
\(847\) −34235.2 −1.38883
\(848\) 19570.6 0.792520
\(849\) −25206.4 −1.01894
\(850\) 8529.85 0.344202
\(851\) 23662.8 0.953173
\(852\) −1404.41 −0.0564720
\(853\) 3138.27 0.125970 0.0629849 0.998014i \(-0.479938\pi\)
0.0629849 + 0.998014i \(0.479938\pi\)
\(854\) 1103.23 0.0442060
\(855\) −1045.42 −0.0418160
\(856\) −24336.9 −0.971749
\(857\) 28519.2 1.13675 0.568376 0.822769i \(-0.307572\pi\)
0.568376 + 0.822769i \(0.307572\pi\)
\(858\) −1582.08 −0.0629503
\(859\) −11047.9 −0.438823 −0.219411 0.975632i \(-0.570414\pi\)
−0.219411 + 0.975632i \(0.570414\pi\)
\(860\) 0 0
\(861\) 30141.9 1.19307
\(862\) 10530.7 0.416098
\(863\) 38205.6 1.50699 0.753496 0.657452i \(-0.228366\pi\)
0.753496 + 0.657452i \(0.228366\pi\)
\(864\) 26226.5 1.03269
\(865\) −1499.29 −0.0589333
\(866\) −9110.68 −0.357498
\(867\) −6674.15 −0.261437
\(868\) 43180.8 1.68854
\(869\) −11254.8 −0.439346
\(870\) −1355.97 −0.0528411
\(871\) −1050.34 −0.0408603
\(872\) 5822.86 0.226132
\(873\) −29560.7 −1.14602
\(874\) −4880.58 −0.188888
\(875\) 12126.6 0.468520
\(876\) −10879.1 −0.419600
\(877\) 17849.2 0.687258 0.343629 0.939106i \(-0.388344\pi\)
0.343629 + 0.939106i \(0.388344\pi\)
\(878\) −16302.6 −0.626637
\(879\) −16161.1 −0.620137
\(880\) −448.310 −0.0171733
\(881\) −3455.04 −0.132126 −0.0660632 0.997815i \(-0.521044\pi\)
−0.0660632 + 0.997815i \(0.521044\pi\)
\(882\) 8485.65 0.323953
\(883\) −2949.64 −0.112416 −0.0562081 0.998419i \(-0.517901\pi\)
−0.0562081 + 0.998419i \(0.517901\pi\)
\(884\) −13162.3 −0.500786
\(885\) 1651.52 0.0627290
\(886\) −22755.8 −0.862863
\(887\) −16846.9 −0.637728 −0.318864 0.947800i \(-0.603301\pi\)
−0.318864 + 0.947800i \(0.603301\pi\)
\(888\) −14599.8 −0.551729
\(889\) −48834.3 −1.84235
\(890\) −1774.03 −0.0668152
\(891\) −548.913 −0.0206389
\(892\) −7500.17 −0.281530
\(893\) −5551.66 −0.208039
\(894\) 10033.8 0.375370
\(895\) 1349.78 0.0504112
\(896\) 39468.9 1.47161
\(897\) −13050.1 −0.485766
\(898\) 1754.98 0.0652165
\(899\) 43328.0 1.60742
\(900\) −12105.0 −0.448333
\(901\) 39795.8 1.47147
\(902\) −3982.04 −0.146993
\(903\) 0 0
\(904\) 38949.6 1.43301
\(905\) 4368.58 0.160460
\(906\) 9385.88 0.344178
\(907\) −12483.3 −0.457004 −0.228502 0.973543i \(-0.573383\pi\)
−0.228502 + 0.973543i \(0.573383\pi\)
\(908\) 29396.7 1.07441
\(909\) 10442.0 0.381010
\(910\) 2445.96 0.0891019
\(911\) −30326.7 −1.10293 −0.551465 0.834198i \(-0.685931\pi\)
−0.551465 + 0.834198i \(0.685931\pi\)
\(912\) −3338.76 −0.121225
\(913\) −1370.09 −0.0496643
\(914\) 12140.1 0.439343
\(915\) 185.072 0.00668667
\(916\) 12377.6 0.446471
\(917\) −78126.2 −2.81347
\(918\) 10048.1 0.361259
\(919\) 23434.7 0.841175 0.420587 0.907252i \(-0.361824\pi\)
0.420587 + 0.907252i \(0.361824\pi\)
\(920\) 3335.26 0.119522
\(921\) −8510.96 −0.304501
\(922\) −11166.2 −0.398851
\(923\) 2541.96 0.0906497
\(924\) −5542.35 −0.197327
\(925\) 28551.2 1.01487
\(926\) −3237.32 −0.114887
\(927\) 11897.6 0.421542
\(928\) 31981.8 1.13131
\(929\) 8192.13 0.289317 0.144658 0.989482i \(-0.453792\pi\)
0.144658 + 0.989482i \(0.453792\pi\)
\(930\) −1918.38 −0.0676412
\(931\) −15587.6 −0.548726
\(932\) −13984.5 −0.491500
\(933\) −32397.9 −1.13683
\(934\) 13132.7 0.460082
\(935\) −911.615 −0.0318856
\(936\) −11203.7 −0.391244
\(937\) −12436.4 −0.433596 −0.216798 0.976216i \(-0.569561\pi\)
−0.216798 + 0.976216i \(0.569561\pi\)
\(938\) 974.463 0.0339204
\(939\) −26207.6 −0.910813
\(940\) 1675.11 0.0581236
\(941\) −15625.9 −0.541327 −0.270664 0.962674i \(-0.587243\pi\)
−0.270664 + 0.962674i \(0.587243\pi\)
\(942\) −8840.50 −0.305774
\(943\) −32846.7 −1.13429
\(944\) −7339.12 −0.253038
\(945\) 7050.65 0.242706
\(946\) 0 0
\(947\) −9888.65 −0.339322 −0.169661 0.985502i \(-0.554267\pi\)
−0.169661 + 0.985502i \(0.554267\pi\)
\(948\) 25291.0 0.866470
\(949\) 19691.0 0.673548
\(950\) −5888.85 −0.201115
\(951\) −28757.4 −0.980571
\(952\) 27656.9 0.941560
\(953\) −7170.55 −0.243732 −0.121866 0.992547i \(-0.538888\pi\)
−0.121866 + 0.992547i \(0.538888\pi\)
\(954\) 14956.6 0.507585
\(955\) 1705.95 0.0578043
\(956\) −5113.48 −0.172994
\(957\) −5561.24 −0.187847
\(958\) −26014.6 −0.877343
\(959\) −10598.1 −0.356861
\(960\) −141.815 −0.00476775
\(961\) 31508.0 1.05764
\(962\) 11667.7 0.391042
\(963\) 20621.8 0.690062
\(964\) −24138.9 −0.806496
\(965\) −8775.59 −0.292742
\(966\) 12107.4 0.403261
\(967\) 314.417 0.0104560 0.00522801 0.999986i \(-0.498336\pi\)
0.00522801 + 0.999986i \(0.498336\pi\)
\(968\) −23018.3 −0.764293
\(969\) −6789.21 −0.225078
\(970\) 4339.12 0.143630
\(971\) 4278.97 0.141420 0.0707100 0.997497i \(-0.477474\pi\)
0.0707100 + 0.997497i \(0.477474\pi\)
\(972\) −23274.0 −0.768020
\(973\) 30315.8 0.998849
\(974\) 10745.9 0.353511
\(975\) −15746.1 −0.517210
\(976\) −822.437 −0.0269729
\(977\) −23808.5 −0.779634 −0.389817 0.920892i \(-0.627462\pi\)
−0.389817 + 0.920892i \(0.627462\pi\)
\(978\) 180.764 0.00591023
\(979\) −7275.81 −0.237524
\(980\) 4703.29 0.153307
\(981\) −4934.00 −0.160582
\(982\) 1279.62 0.0415829
\(983\) −26887.8 −0.872418 −0.436209 0.899845i \(-0.643679\pi\)
−0.436209 + 0.899845i \(0.643679\pi\)
\(984\) 20266.1 0.656566
\(985\) 2042.73 0.0660778
\(986\) 12253.1 0.395758
\(987\) 13772.2 0.444147
\(988\) 9086.99 0.292607
\(989\) 0 0
\(990\) −342.615 −0.0109990
\(991\) 17203.2 0.551441 0.275720 0.961238i \(-0.411084\pi\)
0.275720 + 0.961238i \(0.411084\pi\)
\(992\) 45246.8 1.44817
\(993\) 10532.0 0.336578
\(994\) −2358.34 −0.0752534
\(995\) 5680.81 0.180999
\(996\) 3078.79 0.0979469
\(997\) 41515.7 1.31877 0.659385 0.751805i \(-0.270817\pi\)
0.659385 + 0.751805i \(0.270817\pi\)
\(998\) −2052.71 −0.0651076
\(999\) 33633.0 1.06517
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 1849.4.a.j.1.20 yes 50
43.42 odd 2 1849.4.a.i.1.31 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.31 50 43.42 odd 2
1849.4.a.j.1.20 yes 50 1.1 even 1 trivial