Properties

Label 1849.4.a.i.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

\(f(q)\) \(=\) \(q-1.26350 q^{2} -9.40007 q^{3} -6.40356 q^{4} -0.325119 q^{5} +11.8770 q^{6} -8.80478 q^{7} +18.1989 q^{8} +61.3613 q^{9} +O(q^{10})\) \(q-1.26350 q^{2} -9.40007 q^{3} -6.40356 q^{4} -0.325119 q^{5} +11.8770 q^{6} -8.80478 q^{7} +18.1989 q^{8} +61.3613 q^{9} +0.410789 q^{10} -23.6733 q^{11} +60.1939 q^{12} -23.2530 q^{13} +11.1249 q^{14} +3.05614 q^{15} +28.2341 q^{16} -125.680 q^{17} -77.5301 q^{18} -16.8288 q^{19} +2.08192 q^{20} +82.7656 q^{21} +29.9113 q^{22} +127.751 q^{23} -171.071 q^{24} -124.894 q^{25} +29.3802 q^{26} -322.998 q^{27} +56.3820 q^{28} +220.686 q^{29} -3.86145 q^{30} -303.073 q^{31} -181.265 q^{32} +222.531 q^{33} +158.797 q^{34} +2.86261 q^{35} -392.931 q^{36} -245.352 q^{37} +21.2632 q^{38} +218.580 q^{39} -5.91683 q^{40} +185.397 q^{41} -104.574 q^{42} +151.594 q^{44} -19.9497 q^{45} -161.414 q^{46} -1.86186 q^{47} -265.403 q^{48} -265.476 q^{49} +157.804 q^{50} +1181.40 q^{51} +148.902 q^{52} +261.722 q^{53} +408.109 q^{54} +7.69666 q^{55} -160.238 q^{56} +158.192 q^{57} -278.837 q^{58} -417.774 q^{59} -19.5702 q^{60} +699.788 q^{61} +382.934 q^{62} -540.273 q^{63} +3.15625 q^{64} +7.56001 q^{65} -281.168 q^{66} -279.849 q^{67} +804.800 q^{68} -1200.87 q^{69} -3.61691 q^{70} +675.761 q^{71} +1116.71 q^{72} +1029.66 q^{73} +310.002 q^{74} +1174.01 q^{75} +107.764 q^{76} +208.439 q^{77} -276.176 q^{78} -732.866 q^{79} -9.17946 q^{80} +1379.45 q^{81} -234.250 q^{82} +1094.07 q^{83} -529.994 q^{84} +40.8610 q^{85} -2074.46 q^{87} -430.829 q^{88} +200.368 q^{89} +25.2065 q^{90} +204.738 q^{91} -818.061 q^{92} +2848.91 q^{93} +2.35247 q^{94} +5.47136 q^{95} +1703.91 q^{96} +1442.40 q^{97} +335.429 q^{98} -1452.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26350 −0.446715 −0.223358 0.974737i \(-0.571702\pi\)
−0.223358 + 0.974737i \(0.571702\pi\)
\(3\) −9.40007 −1.80904 −0.904522 0.426427i \(-0.859772\pi\)
−0.904522 + 0.426427i \(0.859772\pi\)
\(4\) −6.40356 −0.800445
\(5\) −0.325119 −0.0290796 −0.0145398 0.999894i \(-0.504628\pi\)
−0.0145398 + 0.999894i \(0.504628\pi\)
\(6\) 11.8770 0.808128
\(7\) −8.80478 −0.475414 −0.237707 0.971337i \(-0.576396\pi\)
−0.237707 + 0.971337i \(0.576396\pi\)
\(8\) 18.1989 0.804287
\(9\) 61.3613 2.27264
\(10\) 0.410789 0.0129903
\(11\) −23.6733 −0.648889 −0.324444 0.945905i \(-0.605177\pi\)
−0.324444 + 0.945905i \(0.605177\pi\)
\(12\) 60.1939 1.44804
\(13\) −23.2530 −0.496095 −0.248047 0.968748i \(-0.579789\pi\)
−0.248047 + 0.968748i \(0.579789\pi\)
\(14\) 11.1249 0.212375
\(15\) 3.05614 0.0526062
\(16\) 28.2341 0.441158
\(17\) −125.680 −1.79305 −0.896526 0.442991i \(-0.853917\pi\)
−0.896526 + 0.442991i \(0.853917\pi\)
\(18\) −77.5301 −1.01522
\(19\) −16.8288 −0.203199 −0.101600 0.994825i \(-0.532396\pi\)
−0.101600 + 0.994825i \(0.532396\pi\)
\(20\) 2.08192 0.0232766
\(21\) 82.7656 0.860044
\(22\) 29.9113 0.289869
\(23\) 127.751 1.15817 0.579085 0.815267i \(-0.303410\pi\)
0.579085 + 0.815267i \(0.303410\pi\)
\(24\) −171.071 −1.45499
\(25\) −124.894 −0.999154
\(26\) 29.3802 0.221613
\(27\) −322.998 −2.30226
\(28\) 56.3820 0.380543
\(29\) 220.686 1.41311 0.706556 0.707657i \(-0.250248\pi\)
0.706556 + 0.707657i \(0.250248\pi\)
\(30\) −3.86145 −0.0235000
\(31\) −303.073 −1.75592 −0.877961 0.478733i \(-0.841096\pi\)
−0.877961 + 0.478733i \(0.841096\pi\)
\(32\) −181.265 −1.00136
\(33\) 222.531 1.17387
\(34\) 158.797 0.800984
\(35\) 2.86261 0.0138248
\(36\) −392.931 −1.81912
\(37\) −245.352 −1.09015 −0.545075 0.838387i \(-0.683499\pi\)
−0.545075 + 0.838387i \(0.683499\pi\)
\(38\) 21.2632 0.0907722
\(39\) 218.580 0.897457
\(40\) −5.91683 −0.0233883
\(41\) 185.397 0.706200 0.353100 0.935586i \(-0.385127\pi\)
0.353100 + 0.935586i \(0.385127\pi\)
\(42\) −104.574 −0.384195
\(43\) 0 0
\(44\) 151.594 0.519400
\(45\) −19.9497 −0.0660874
\(46\) −161.414 −0.517372
\(47\) −1.86186 −0.00577831 −0.00288915 0.999996i \(-0.500920\pi\)
−0.00288915 + 0.999996i \(0.500920\pi\)
\(48\) −265.403 −0.798074
\(49\) −265.476 −0.773982
\(50\) 157.804 0.446338
\(51\) 1181.40 3.24371
\(52\) 148.902 0.397097
\(53\) 261.722 0.678307 0.339153 0.940731i \(-0.389859\pi\)
0.339153 + 0.940731i \(0.389859\pi\)
\(54\) 408.109 1.02845
\(55\) 7.69666 0.0188694
\(56\) −160.238 −0.382369
\(57\) 158.192 0.367596
\(58\) −278.837 −0.631259
\(59\) −417.774 −0.921857 −0.460929 0.887437i \(-0.652484\pi\)
−0.460929 + 0.887437i \(0.652484\pi\)
\(60\) −19.5702 −0.0421084
\(61\) 699.788 1.46883 0.734416 0.678700i \(-0.237456\pi\)
0.734416 + 0.678700i \(0.237456\pi\)
\(62\) 382.934 0.784397
\(63\) −540.273 −1.08044
\(64\) 3.15625 0.00616454
\(65\) 7.56001 0.0144262
\(66\) −281.168 −0.524385
\(67\) −279.849 −0.510284 −0.255142 0.966904i \(-0.582122\pi\)
−0.255142 + 0.966904i \(0.582122\pi\)
\(68\) 804.800 1.43524
\(69\) −1200.87 −2.09518
\(70\) −3.61691 −0.00617576
\(71\) 675.761 1.12955 0.564776 0.825245i \(-0.308963\pi\)
0.564776 + 0.825245i \(0.308963\pi\)
\(72\) 1116.71 1.82785
\(73\) 1029.66 1.65086 0.825432 0.564502i \(-0.190932\pi\)
0.825432 + 0.564502i \(0.190932\pi\)
\(74\) 310.002 0.486987
\(75\) 1174.01 1.80751
\(76\) 107.764 0.162650
\(77\) 208.439 0.308491
\(78\) −276.176 −0.400908
\(79\) −732.866 −1.04372 −0.521860 0.853031i \(-0.674762\pi\)
−0.521860 + 0.853031i \(0.674762\pi\)
\(80\) −9.17946 −0.0128287
\(81\) 1379.45 1.89225
\(82\) −234.250 −0.315471
\(83\) 1094.07 1.44686 0.723431 0.690397i \(-0.242564\pi\)
0.723431 + 0.690397i \(0.242564\pi\)
\(84\) −529.994 −0.688418
\(85\) 40.8610 0.0521412
\(86\) 0 0
\(87\) −2074.46 −2.55638
\(88\) −430.829 −0.521893
\(89\) 200.368 0.238639 0.119320 0.992856i \(-0.461929\pi\)
0.119320 + 0.992856i \(0.461929\pi\)
\(90\) 25.2065 0.0295222
\(91\) 204.738 0.235850
\(92\) −818.061 −0.927052
\(93\) 2848.91 3.17654
\(94\) 2.35247 0.00258126
\(95\) 5.47136 0.00590895
\(96\) 1703.91 1.81150
\(97\) 1442.40 1.50983 0.754916 0.655822i \(-0.227678\pi\)
0.754916 + 0.655822i \(0.227678\pi\)
\(98\) 335.429 0.345750
\(99\) −1452.63 −1.47469
\(100\) 799.768 0.799768
\(101\) 909.111 0.895643 0.447822 0.894123i \(-0.352200\pi\)
0.447822 + 0.894123i \(0.352200\pi\)
\(102\) −1492.70 −1.44902
\(103\) 1263.28 1.20849 0.604245 0.796798i \(-0.293475\pi\)
0.604245 + 0.796798i \(0.293475\pi\)
\(104\) −423.180 −0.399002
\(105\) −26.9087 −0.0250097
\(106\) −330.686 −0.303010
\(107\) 333.716 0.301509 0.150755 0.988571i \(-0.451830\pi\)
0.150755 + 0.988571i \(0.451830\pi\)
\(108\) 2068.34 1.84283
\(109\) 115.351 0.101364 0.0506818 0.998715i \(-0.483861\pi\)
0.0506818 + 0.998715i \(0.483861\pi\)
\(110\) −9.72475 −0.00842926
\(111\) 2306.32 1.97213
\(112\) −248.595 −0.209733
\(113\) 719.984 0.599384 0.299692 0.954036i \(-0.403116\pi\)
0.299692 + 0.954036i \(0.403116\pi\)
\(114\) −199.875 −0.164211
\(115\) −41.5343 −0.0336791
\(116\) −1413.17 −1.13112
\(117\) −1426.83 −1.12744
\(118\) 527.859 0.411808
\(119\) 1106.59 0.852441
\(120\) 55.6186 0.0423105
\(121\) −770.573 −0.578943
\(122\) −884.184 −0.656150
\(123\) −1742.75 −1.27755
\(124\) 1940.75 1.40552
\(125\) 81.2455 0.0581345
\(126\) 682.636 0.482651
\(127\) 1823.69 1.27422 0.637111 0.770772i \(-0.280129\pi\)
0.637111 + 0.770772i \(0.280129\pi\)
\(128\) 1446.13 0.998605
\(129\) 0 0
\(130\) −9.55209 −0.00644441
\(131\) −827.828 −0.552120 −0.276060 0.961140i \(-0.589029\pi\)
−0.276060 + 0.961140i \(0.589029\pi\)
\(132\) −1424.99 −0.939617
\(133\) 148.174 0.0966037
\(134\) 353.590 0.227952
\(135\) 105.013 0.0669487
\(136\) −2287.24 −1.44213
\(137\) 1732.64 1.08051 0.540254 0.841502i \(-0.318328\pi\)
0.540254 + 0.841502i \(0.318328\pi\)
\(138\) 1517.30 0.935949
\(139\) −2409.59 −1.47035 −0.735177 0.677875i \(-0.762901\pi\)
−0.735177 + 0.677875i \(0.762901\pi\)
\(140\) −18.3309 −0.0110660
\(141\) 17.5016 0.0104532
\(142\) −853.826 −0.504588
\(143\) 550.477 0.321910
\(144\) 1732.48 1.00259
\(145\) −71.7492 −0.0410927
\(146\) −1300.98 −0.737466
\(147\) 2495.49 1.40017
\(148\) 1571.13 0.872606
\(149\) −1235.52 −0.679316 −0.339658 0.940549i \(-0.610311\pi\)
−0.339658 + 0.940549i \(0.610311\pi\)
\(150\) −1483.37 −0.807444
\(151\) −76.5960 −0.0412801 −0.0206400 0.999787i \(-0.506570\pi\)
−0.0206400 + 0.999787i \(0.506570\pi\)
\(152\) −306.266 −0.163430
\(153\) −7711.88 −4.07496
\(154\) −263.363 −0.137808
\(155\) 98.5350 0.0510614
\(156\) −1399.69 −0.718365
\(157\) −1307.88 −0.664840 −0.332420 0.943131i \(-0.607865\pi\)
−0.332420 + 0.943131i \(0.607865\pi\)
\(158\) 925.978 0.466246
\(159\) −2460.20 −1.22709
\(160\) 58.9329 0.0291191
\(161\) −1124.82 −0.550610
\(162\) −1742.94 −0.845297
\(163\) 2230.12 1.07164 0.535818 0.844334i \(-0.320003\pi\)
0.535818 + 0.844334i \(0.320003\pi\)
\(164\) −1187.20 −0.565275
\(165\) −72.3491 −0.0341356
\(166\) −1382.36 −0.646335
\(167\) −1109.52 −0.514115 −0.257058 0.966396i \(-0.582753\pi\)
−0.257058 + 0.966396i \(0.582753\pi\)
\(168\) 1506.24 0.691722
\(169\) −1656.30 −0.753890
\(170\) −51.6280 −0.0232923
\(171\) −1032.63 −0.461799
\(172\) 0 0
\(173\) 226.055 0.0993447 0.0496723 0.998766i \(-0.484182\pi\)
0.0496723 + 0.998766i \(0.484182\pi\)
\(174\) 2621.08 1.14198
\(175\) 1099.67 0.475012
\(176\) −668.395 −0.286262
\(177\) 3927.11 1.66768
\(178\) −253.165 −0.106604
\(179\) 1174.15 0.490280 0.245140 0.969488i \(-0.421166\pi\)
0.245140 + 0.969488i \(0.421166\pi\)
\(180\) 127.749 0.0528993
\(181\) −948.581 −0.389544 −0.194772 0.980849i \(-0.562397\pi\)
−0.194772 + 0.980849i \(0.562397\pi\)
\(182\) −258.687 −0.105358
\(183\) −6578.06 −2.65718
\(184\) 2324.93 0.931501
\(185\) 79.7686 0.0317011
\(186\) −3599.60 −1.41901
\(187\) 2975.26 1.16349
\(188\) 11.9225 0.00462522
\(189\) 2843.93 1.09453
\(190\) −6.91308 −0.00263962
\(191\) −241.015 −0.0913047 −0.0456524 0.998957i \(-0.514537\pi\)
−0.0456524 + 0.998957i \(0.514537\pi\)
\(192\) −29.6689 −0.0111519
\(193\) −754.139 −0.281265 −0.140632 0.990062i \(-0.544914\pi\)
−0.140632 + 0.990062i \(0.544914\pi\)
\(194\) −1822.48 −0.674465
\(195\) −71.0646 −0.0260977
\(196\) 1699.99 0.619530
\(197\) −4583.62 −1.65771 −0.828856 0.559462i \(-0.811008\pi\)
−0.828856 + 0.559462i \(0.811008\pi\)
\(198\) 1835.40 0.658767
\(199\) −2564.26 −0.913445 −0.456723 0.889609i \(-0.650977\pi\)
−0.456723 + 0.889609i \(0.650977\pi\)
\(200\) −2272.94 −0.803607
\(201\) 2630.60 0.923125
\(202\) −1148.66 −0.400098
\(203\) −1943.09 −0.671813
\(204\) −7565.17 −2.59641
\(205\) −60.2763 −0.0205360
\(206\) −1596.15 −0.539851
\(207\) 7838.96 2.63210
\(208\) −656.528 −0.218856
\(209\) 398.393 0.131854
\(210\) 33.9992 0.0111722
\(211\) −1951.52 −0.636722 −0.318361 0.947970i \(-0.603132\pi\)
−0.318361 + 0.947970i \(0.603132\pi\)
\(212\) −1675.95 −0.542947
\(213\) −6352.20 −2.04341
\(214\) −421.650 −0.134689
\(215\) 0 0
\(216\) −5878.22 −1.85168
\(217\) 2668.49 0.834789
\(218\) −145.746 −0.0452807
\(219\) −9678.91 −2.98648
\(220\) −49.2860 −0.0151039
\(221\) 2922.44 0.889523
\(222\) −2914.04 −0.880981
\(223\) −2379.09 −0.714421 −0.357211 0.934024i \(-0.616272\pi\)
−0.357211 + 0.934024i \(0.616272\pi\)
\(224\) 1596.00 0.476060
\(225\) −7663.67 −2.27072
\(226\) −909.702 −0.267754
\(227\) −665.520 −0.194591 −0.0972955 0.995256i \(-0.531019\pi\)
−0.0972955 + 0.995256i \(0.531019\pi\)
\(228\) −1012.99 −0.294241
\(229\) 3250.55 0.938002 0.469001 0.883198i \(-0.344614\pi\)
0.469001 + 0.883198i \(0.344614\pi\)
\(230\) 52.4787 0.0150450
\(231\) −1959.34 −0.558073
\(232\) 4016.24 1.13655
\(233\) 5961.66 1.67623 0.838115 0.545494i \(-0.183658\pi\)
0.838115 + 0.545494i \(0.183658\pi\)
\(234\) 1802.81 0.503647
\(235\) 0.605327 0.000168031 0
\(236\) 2675.24 0.737896
\(237\) 6888.99 1.88814
\(238\) −1398.17 −0.380799
\(239\) 1651.94 0.447092 0.223546 0.974693i \(-0.428237\pi\)
0.223546 + 0.974693i \(0.428237\pi\)
\(240\) 86.2875 0.0232076
\(241\) 524.998 0.140324 0.0701621 0.997536i \(-0.477648\pi\)
0.0701621 + 0.997536i \(0.477648\pi\)
\(242\) 973.621 0.258623
\(243\) −4245.97 −1.12090
\(244\) −4481.14 −1.17572
\(245\) 86.3113 0.0225071
\(246\) 2201.97 0.570700
\(247\) 391.320 0.100806
\(248\) −5515.61 −1.41226
\(249\) −10284.3 −2.61744
\(250\) −102.654 −0.0259696
\(251\) 5790.68 1.45619 0.728096 0.685475i \(-0.240405\pi\)
0.728096 + 0.685475i \(0.240405\pi\)
\(252\) 3459.67 0.864836
\(253\) −3024.29 −0.751524
\(254\) −2304.24 −0.569215
\(255\) −384.096 −0.0943257
\(256\) −1852.44 −0.452257
\(257\) 4622.57 1.12198 0.560989 0.827823i \(-0.310421\pi\)
0.560989 + 0.827823i \(0.310421\pi\)
\(258\) 0 0
\(259\) 2160.27 0.518273
\(260\) −48.4110 −0.0115474
\(261\) 13541.5 3.21150
\(262\) 1045.96 0.246640
\(263\) 468.704 0.109892 0.0549458 0.998489i \(-0.482501\pi\)
0.0549458 + 0.998489i \(0.482501\pi\)
\(264\) 4049.82 0.944127
\(265\) −85.0908 −0.0197249
\(266\) −187.218 −0.0431544
\(267\) −1883.47 −0.431709
\(268\) 1792.03 0.408454
\(269\) −1162.74 −0.263546 −0.131773 0.991280i \(-0.542067\pi\)
−0.131773 + 0.991280i \(0.542067\pi\)
\(270\) −132.684 −0.0299070
\(271\) −5794.03 −1.29875 −0.649377 0.760467i \(-0.724970\pi\)
−0.649377 + 0.760467i \(0.724970\pi\)
\(272\) −3548.46 −0.791019
\(273\) −1924.55 −0.426663
\(274\) −2189.20 −0.482680
\(275\) 2956.66 0.648340
\(276\) 7689.83 1.67708
\(277\) −1078.12 −0.233855 −0.116928 0.993140i \(-0.537305\pi\)
−0.116928 + 0.993140i \(0.537305\pi\)
\(278\) 3044.53 0.656830
\(279\) −18597.0 −3.99058
\(280\) 52.0964 0.0111191
\(281\) 4395.46 0.933136 0.466568 0.884485i \(-0.345490\pi\)
0.466568 + 0.884485i \(0.345490\pi\)
\(282\) −22.1133 −0.00466961
\(283\) 1861.43 0.390990 0.195495 0.980705i \(-0.437369\pi\)
0.195495 + 0.980705i \(0.437369\pi\)
\(284\) −4327.28 −0.904144
\(285\) −51.4312 −0.0106895
\(286\) −695.528 −0.143802
\(287\) −1632.38 −0.335737
\(288\) −11122.7 −2.27573
\(289\) 10882.5 2.21504
\(290\) 90.6552 0.0183568
\(291\) −13558.7 −2.73135
\(292\) −6593.52 −1.32143
\(293\) −480.757 −0.0958572 −0.0479286 0.998851i \(-0.515262\pi\)
−0.0479286 + 0.998851i \(0.515262\pi\)
\(294\) −3153.06 −0.625476
\(295\) 135.827 0.0268072
\(296\) −4465.14 −0.876794
\(297\) 7646.44 1.49391
\(298\) 1561.09 0.303461
\(299\) −2970.60 −0.574562
\(300\) −7517.88 −1.44682
\(301\) 0 0
\(302\) 96.7792 0.0184404
\(303\) −8545.71 −1.62026
\(304\) −475.145 −0.0896430
\(305\) −227.515 −0.0427130
\(306\) 9743.98 1.82035
\(307\) 6933.15 1.28891 0.644456 0.764641i \(-0.277084\pi\)
0.644456 + 0.764641i \(0.277084\pi\)
\(308\) −1334.75 −0.246930
\(309\) −11874.9 −2.18621
\(310\) −124.499 −0.0228099
\(311\) −2059.92 −0.375586 −0.187793 0.982209i \(-0.560133\pi\)
−0.187793 + 0.982209i \(0.560133\pi\)
\(312\) 3977.92 0.721813
\(313\) −5359.79 −0.967901 −0.483950 0.875095i \(-0.660799\pi\)
−0.483950 + 0.875095i \(0.660799\pi\)
\(314\) 1652.50 0.296994
\(315\) 175.653 0.0314188
\(316\) 4692.96 0.835441
\(317\) −7019.47 −1.24370 −0.621849 0.783137i \(-0.713618\pi\)
−0.621849 + 0.783137i \(0.713618\pi\)
\(318\) 3108.47 0.548158
\(319\) −5224.36 −0.916953
\(320\) −1.02616 −0.000179262 0
\(321\) −3136.95 −0.545444
\(322\) 1421.21 0.245966
\(323\) 2115.04 0.364347
\(324\) −8833.39 −1.51464
\(325\) 2904.17 0.495675
\(326\) −2817.76 −0.478716
\(327\) −1084.31 −0.183371
\(328\) 3374.04 0.567988
\(329\) 16.3933 0.00274709
\(330\) 91.4133 0.0152489
\(331\) −5551.69 −0.921898 −0.460949 0.887427i \(-0.652491\pi\)
−0.460949 + 0.887427i \(0.652491\pi\)
\(332\) −7005.93 −1.15813
\(333\) −15055.1 −2.47752
\(334\) 1401.88 0.229663
\(335\) 90.9844 0.0148388
\(336\) 2336.81 0.379415
\(337\) 1460.34 0.236053 0.118026 0.993010i \(-0.462343\pi\)
0.118026 + 0.993010i \(0.462343\pi\)
\(338\) 2092.73 0.336774
\(339\) −6767.90 −1.08431
\(340\) −261.656 −0.0417362
\(341\) 7174.75 1.13940
\(342\) 1304.74 0.206293
\(343\) 5357.50 0.843375
\(344\) 0 0
\(345\) 390.425 0.0609269
\(346\) −285.621 −0.0443788
\(347\) 7150.17 1.10617 0.553085 0.833125i \(-0.313450\pi\)
0.553085 + 0.833125i \(0.313450\pi\)
\(348\) 13283.9 2.04624
\(349\) 5169.61 0.792902 0.396451 0.918056i \(-0.370242\pi\)
0.396451 + 0.918056i \(0.370242\pi\)
\(350\) −1389.43 −0.212195
\(351\) 7510.68 1.14214
\(352\) 4291.15 0.649771
\(353\) 1936.50 0.291981 0.145991 0.989286i \(-0.453363\pi\)
0.145991 + 0.989286i \(0.453363\pi\)
\(354\) −4961.91 −0.744978
\(355\) −219.703 −0.0328469
\(356\) −1283.07 −0.191018
\(357\) −10402.0 −1.54210
\(358\) −1483.54 −0.219016
\(359\) 1557.07 0.228911 0.114456 0.993428i \(-0.463488\pi\)
0.114456 + 0.993428i \(0.463488\pi\)
\(360\) −363.064 −0.0531532
\(361\) −6575.79 −0.958710
\(362\) 1198.53 0.174015
\(363\) 7243.44 1.04733
\(364\) −1311.05 −0.188785
\(365\) −334.764 −0.0480064
\(366\) 8311.39 1.18700
\(367\) 5890.68 0.837851 0.418925 0.908021i \(-0.362407\pi\)
0.418925 + 0.908021i \(0.362407\pi\)
\(368\) 3606.93 0.510936
\(369\) 11376.2 1.60494
\(370\) −100.788 −0.0141614
\(371\) −2304.40 −0.322476
\(372\) −18243.2 −2.54265
\(373\) 1361.91 0.189053 0.0945265 0.995522i \(-0.469866\pi\)
0.0945265 + 0.995522i \(0.469866\pi\)
\(374\) −3759.25 −0.519750
\(375\) −763.713 −0.105168
\(376\) −33.8839 −0.00464741
\(377\) −5131.61 −0.701038
\(378\) −3593.31 −0.488942
\(379\) −2673.20 −0.362304 −0.181152 0.983455i \(-0.557983\pi\)
−0.181152 + 0.983455i \(0.557983\pi\)
\(380\) −35.0362 −0.00472979
\(381\) −17142.8 −2.30512
\(382\) 304.522 0.0407872
\(383\) −3017.72 −0.402607 −0.201303 0.979529i \(-0.564518\pi\)
−0.201303 + 0.979529i \(0.564518\pi\)
\(384\) −13593.8 −1.80652
\(385\) −67.7674 −0.00897078
\(386\) 952.857 0.125645
\(387\) 0 0
\(388\) −9236.51 −1.20854
\(389\) 15220.5 1.98384 0.991918 0.126879i \(-0.0404959\pi\)
0.991918 + 0.126879i \(0.0404959\pi\)
\(390\) 89.7903 0.0116582
\(391\) −16055.7 −2.07666
\(392\) −4831.38 −0.622503
\(393\) 7781.64 0.998809
\(394\) 5791.41 0.740526
\(395\) 238.269 0.0303509
\(396\) 9301.98 1.18041
\(397\) −9599.48 −1.21356 −0.606781 0.794869i \(-0.707540\pi\)
−0.606781 + 0.794869i \(0.707540\pi\)
\(398\) 3239.95 0.408050
\(399\) −1392.84 −0.174760
\(400\) −3526.28 −0.440785
\(401\) 15516.8 1.93235 0.966177 0.257879i \(-0.0830236\pi\)
0.966177 + 0.257879i \(0.0830236\pi\)
\(402\) −3323.77 −0.412374
\(403\) 7047.37 0.871103
\(404\) −5821.55 −0.716913
\(405\) −448.486 −0.0550258
\(406\) 2455.10 0.300109
\(407\) 5808.29 0.707387
\(408\) 21500.2 2.60887
\(409\) −13082.4 −1.58162 −0.790812 0.612059i \(-0.790342\pi\)
−0.790812 + 0.612059i \(0.790342\pi\)
\(410\) 76.1593 0.00917375
\(411\) −16287.0 −1.95469
\(412\) −8089.48 −0.967330
\(413\) 3678.41 0.438263
\(414\) −9904.54 −1.17580
\(415\) −355.703 −0.0420741
\(416\) 4214.97 0.496769
\(417\) 22650.4 2.65993
\(418\) −503.371 −0.0589011
\(419\) 4000.38 0.466424 0.233212 0.972426i \(-0.425076\pi\)
0.233212 + 0.972426i \(0.425076\pi\)
\(420\) 172.312 0.0200189
\(421\) 14443.5 1.67204 0.836022 0.548695i \(-0.184875\pi\)
0.836022 + 0.548695i \(0.184875\pi\)
\(422\) 2465.75 0.284434
\(423\) −114.246 −0.0131320
\(424\) 4763.06 0.545553
\(425\) 15696.7 1.79154
\(426\) 8026.02 0.912822
\(427\) −6161.49 −0.698303
\(428\) −2136.97 −0.241342
\(429\) −5174.52 −0.582350
\(430\) 0 0
\(431\) 8704.07 0.972762 0.486381 0.873747i \(-0.338317\pi\)
0.486381 + 0.873747i \(0.338317\pi\)
\(432\) −9119.56 −1.01566
\(433\) −13149.7 −1.45943 −0.729717 0.683749i \(-0.760348\pi\)
−0.729717 + 0.683749i \(0.760348\pi\)
\(434\) −3371.65 −0.372913
\(435\) 674.447 0.0743385
\(436\) −738.658 −0.0811360
\(437\) −2149.89 −0.235339
\(438\) 12229.3 1.33411
\(439\) 11391.2 1.23844 0.619218 0.785219i \(-0.287450\pi\)
0.619218 + 0.785219i \(0.287450\pi\)
\(440\) 140.071 0.0151764
\(441\) −16289.9 −1.75898
\(442\) −3692.51 −0.397364
\(443\) −4305.83 −0.461797 −0.230898 0.972978i \(-0.574166\pi\)
−0.230898 + 0.972978i \(0.574166\pi\)
\(444\) −14768.7 −1.57858
\(445\) −65.1434 −0.00693953
\(446\) 3005.99 0.319143
\(447\) 11614.0 1.22891
\(448\) −27.7901 −0.00293071
\(449\) 3738.80 0.392973 0.196487 0.980506i \(-0.437047\pi\)
0.196487 + 0.980506i \(0.437047\pi\)
\(450\) 9683.06 1.01436
\(451\) −4388.98 −0.458246
\(452\) −4610.47 −0.479774
\(453\) 720.007 0.0746775
\(454\) 840.886 0.0869268
\(455\) −66.5643 −0.00685842
\(456\) 2878.92 0.295653
\(457\) −8908.27 −0.911841 −0.455920 0.890021i \(-0.650690\pi\)
−0.455920 + 0.890021i \(0.650690\pi\)
\(458\) −4107.08 −0.419020
\(459\) 40594.4 4.12807
\(460\) 265.968 0.0269583
\(461\) 2694.69 0.272244 0.136122 0.990692i \(-0.456536\pi\)
0.136122 + 0.990692i \(0.456536\pi\)
\(462\) 2475.63 0.249300
\(463\) 6709.80 0.673500 0.336750 0.941594i \(-0.390672\pi\)
0.336750 + 0.941594i \(0.390672\pi\)
\(464\) 6230.86 0.623406
\(465\) −926.236 −0.0923724
\(466\) −7532.57 −0.748798
\(467\) −16767.9 −1.66151 −0.830754 0.556639i \(-0.812091\pi\)
−0.830754 + 0.556639i \(0.812091\pi\)
\(468\) 9136.83 0.902457
\(469\) 2464.01 0.242596
\(470\) −0.764832 −7.50619e−5 0
\(471\) 12294.1 1.20272
\(472\) −7603.04 −0.741437
\(473\) 0 0
\(474\) −8704.26 −0.843460
\(475\) 2101.82 0.203027
\(476\) −7086.09 −0.682333
\(477\) 16059.6 1.54155
\(478\) −2087.23 −0.199723
\(479\) −13204.9 −1.25959 −0.629797 0.776760i \(-0.716862\pi\)
−0.629797 + 0.776760i \(0.716862\pi\)
\(480\) −553.973 −0.0526777
\(481\) 5705.17 0.540818
\(482\) −663.336 −0.0626849
\(483\) 10573.4 0.996077
\(484\) 4934.41 0.463412
\(485\) −468.953 −0.0439052
\(486\) 5364.79 0.500724
\(487\) −2291.19 −0.213191 −0.106595 0.994302i \(-0.533995\pi\)
−0.106595 + 0.994302i \(0.533995\pi\)
\(488\) 12735.4 1.18136
\(489\) −20963.3 −1.93864
\(490\) −109.055 −0.0100543
\(491\) −8817.80 −0.810472 −0.405236 0.914212i \(-0.632811\pi\)
−0.405236 + 0.914212i \(0.632811\pi\)
\(492\) 11159.8 1.02261
\(493\) −27735.8 −2.53378
\(494\) −494.434 −0.0450316
\(495\) 472.277 0.0428834
\(496\) −8557.00 −0.774639
\(497\) −5949.93 −0.537004
\(498\) 12994.2 1.16925
\(499\) −16011.1 −1.43638 −0.718192 0.695845i \(-0.755030\pi\)
−0.718192 + 0.695845i \(0.755030\pi\)
\(500\) −520.261 −0.0465335
\(501\) 10429.6 0.930057
\(502\) −7316.53 −0.650504
\(503\) 55.8134 0.00494751 0.00247375 0.999997i \(-0.499213\pi\)
0.00247375 + 0.999997i \(0.499213\pi\)
\(504\) −9832.38 −0.868987
\(505\) −295.570 −0.0260449
\(506\) 3821.20 0.335717
\(507\) 15569.3 1.36382
\(508\) −11678.1 −1.01995
\(509\) 9212.25 0.802212 0.401106 0.916032i \(-0.368626\pi\)
0.401106 + 0.916032i \(0.368626\pi\)
\(510\) 485.307 0.0421367
\(511\) −9065.97 −0.784843
\(512\) −9228.51 −0.796575
\(513\) 5435.66 0.467817
\(514\) −5840.63 −0.501205
\(515\) −410.716 −0.0351424
\(516\) 0 0
\(517\) 44.0765 0.00374948
\(518\) −2729.50 −0.231520
\(519\) −2124.93 −0.179719
\(520\) 137.584 0.0116028
\(521\) −14038.9 −1.18053 −0.590266 0.807209i \(-0.700977\pi\)
−0.590266 + 0.807209i \(0.700977\pi\)
\(522\) −17109.8 −1.43462
\(523\) 2476.00 0.207013 0.103507 0.994629i \(-0.466994\pi\)
0.103507 + 0.994629i \(0.466994\pi\)
\(524\) 5301.05 0.441942
\(525\) −10336.9 −0.859317
\(526\) −592.208 −0.0490903
\(527\) 38090.3 3.14846
\(528\) 6282.96 0.517861
\(529\) 4153.30 0.341358
\(530\) 107.512 0.00881140
\(531\) −25635.1 −2.09505
\(532\) −948.840 −0.0773260
\(533\) −4311.05 −0.350342
\(534\) 2379.77 0.192851
\(535\) −108.497 −0.00876776
\(536\) −5092.95 −0.410414
\(537\) −11037.1 −0.886938
\(538\) 1469.13 0.117730
\(539\) 6284.70 0.502228
\(540\) −672.457 −0.0535888
\(541\) 5185.20 0.412069 0.206034 0.978545i \(-0.433944\pi\)
0.206034 + 0.978545i \(0.433944\pi\)
\(542\) 7320.77 0.580173
\(543\) 8916.73 0.704702
\(544\) 22781.4 1.79549
\(545\) −37.5029 −0.00294761
\(546\) 2431.67 0.190597
\(547\) −7098.82 −0.554888 −0.277444 0.960742i \(-0.589487\pi\)
−0.277444 + 0.960742i \(0.589487\pi\)
\(548\) −11095.1 −0.864888
\(549\) 42939.9 3.33812
\(550\) −3735.75 −0.289624
\(551\) −3713.87 −0.287143
\(552\) −21854.5 −1.68513
\(553\) 6452.73 0.496199
\(554\) 1362.21 0.104467
\(555\) −749.830 −0.0573487
\(556\) 15430.0 1.17694
\(557\) −1358.66 −0.103354 −0.0516772 0.998664i \(-0.516457\pi\)
−0.0516772 + 0.998664i \(0.516457\pi\)
\(558\) 23497.3 1.78265
\(559\) 0 0
\(560\) 80.8232 0.00609893
\(561\) −27967.7 −2.10481
\(562\) −5553.67 −0.416846
\(563\) −13603.7 −1.01834 −0.509172 0.860665i \(-0.670048\pi\)
−0.509172 + 0.860665i \(0.670048\pi\)
\(564\) −112.073 −0.00836722
\(565\) −234.081 −0.0174298
\(566\) −2351.91 −0.174661
\(567\) −12145.8 −0.899601
\(568\) 12298.1 0.908483
\(569\) 3316.81 0.244373 0.122186 0.992507i \(-0.461009\pi\)
0.122186 + 0.992507i \(0.461009\pi\)
\(570\) 64.9834 0.00477518
\(571\) −7745.82 −0.567693 −0.283846 0.958870i \(-0.591611\pi\)
−0.283846 + 0.958870i \(0.591611\pi\)
\(572\) −3525.01 −0.257672
\(573\) 2265.55 0.165174
\(574\) 2062.52 0.149979
\(575\) −15955.4 −1.15719
\(576\) 193.671 0.0140098
\(577\) 24031.9 1.73390 0.866952 0.498391i \(-0.166076\pi\)
0.866952 + 0.498391i \(0.166076\pi\)
\(578\) −13750.0 −0.989491
\(579\) 7088.96 0.508821
\(580\) 459.450 0.0328925
\(581\) −9633.03 −0.687858
\(582\) 17131.4 1.22014
\(583\) −6195.83 −0.440146
\(584\) 18738.8 1.32777
\(585\) 463.892 0.0327856
\(586\) 607.438 0.0428209
\(587\) 2119.09 0.149002 0.0745011 0.997221i \(-0.476264\pi\)
0.0745011 + 0.997221i \(0.476264\pi\)
\(588\) −15980.0 −1.12076
\(589\) 5100.35 0.356802
\(590\) −171.617 −0.0119752
\(591\) 43086.3 2.99887
\(592\) −6927.29 −0.480929
\(593\) −6253.12 −0.433027 −0.216513 0.976280i \(-0.569469\pi\)
−0.216513 + 0.976280i \(0.569469\pi\)
\(594\) −9661.29 −0.667353
\(595\) −359.773 −0.0247886
\(596\) 7911.76 0.543756
\(597\) 24104.2 1.65246
\(598\) 3753.35 0.256666
\(599\) 10339.6 0.705284 0.352642 0.935758i \(-0.385283\pi\)
0.352642 + 0.935758i \(0.385283\pi\)
\(600\) 21365.8 1.45376
\(601\) −5782.91 −0.392495 −0.196248 0.980554i \(-0.562876\pi\)
−0.196248 + 0.980554i \(0.562876\pi\)
\(602\) 0 0
\(603\) −17171.9 −1.15969
\(604\) 490.487 0.0330424
\(605\) 250.528 0.0168354
\(606\) 10797.5 0.723794
\(607\) −20822.9 −1.39238 −0.696191 0.717856i \(-0.745123\pi\)
−0.696191 + 0.717856i \(0.745123\pi\)
\(608\) 3050.47 0.203475
\(609\) 18265.2 1.21534
\(610\) 287.465 0.0190806
\(611\) 43.2939 0.00286659
\(612\) 49383.5 3.26178
\(613\) 11051.2 0.728148 0.364074 0.931370i \(-0.381386\pi\)
0.364074 + 0.931370i \(0.381386\pi\)
\(614\) −8760.05 −0.575777
\(615\) 566.601 0.0371505
\(616\) 3793.36 0.248115
\(617\) 1118.45 0.0729775 0.0364888 0.999334i \(-0.488383\pi\)
0.0364888 + 0.999334i \(0.488383\pi\)
\(618\) 15004.0 0.976615
\(619\) −15760.9 −1.02340 −0.511700 0.859164i \(-0.670984\pi\)
−0.511700 + 0.859164i \(0.670984\pi\)
\(620\) −630.975 −0.0408719
\(621\) −41263.3 −2.66641
\(622\) 2602.71 0.167780
\(623\) −1764.19 −0.113452
\(624\) 6171.41 0.395920
\(625\) 15585.4 0.997464
\(626\) 6772.10 0.432376
\(627\) −3744.92 −0.238529
\(628\) 8375.06 0.532168
\(629\) 30835.8 1.95470
\(630\) −221.938 −0.0140353
\(631\) −17527.6 −1.10581 −0.552904 0.833245i \(-0.686480\pi\)
−0.552904 + 0.833245i \(0.686480\pi\)
\(632\) −13337.4 −0.839451
\(633\) 18344.4 1.15186
\(634\) 8869.11 0.555579
\(635\) −592.917 −0.0370539
\(636\) 15754.1 0.982215
\(637\) 6173.11 0.383968
\(638\) 6600.99 0.409617
\(639\) 41465.6 2.56706
\(640\) −470.167 −0.0290390
\(641\) −9258.01 −0.570467 −0.285233 0.958458i \(-0.592071\pi\)
−0.285233 + 0.958458i \(0.592071\pi\)
\(642\) 3963.54 0.243658
\(643\) 3332.29 0.204374 0.102187 0.994765i \(-0.467416\pi\)
0.102187 + 0.994765i \(0.467416\pi\)
\(644\) 7202.85 0.440733
\(645\) 0 0
\(646\) −2672.36 −0.162759
\(647\) 4208.18 0.255704 0.127852 0.991793i \(-0.459192\pi\)
0.127852 + 0.991793i \(0.459192\pi\)
\(648\) 25104.5 1.52191
\(649\) 9890.11 0.598183
\(650\) −3669.43 −0.221426
\(651\) −25084.0 −1.51017
\(652\) −14280.7 −0.857786
\(653\) 25071.4 1.50248 0.751241 0.660028i \(-0.229456\pi\)
0.751241 + 0.660028i \(0.229456\pi\)
\(654\) 1370.03 0.0819147
\(655\) 269.143 0.0160554
\(656\) 5234.53 0.311546
\(657\) 63181.5 3.75182
\(658\) −20.7130 −0.00122717
\(659\) −16578.9 −0.980001 −0.490000 0.871722i \(-0.663003\pi\)
−0.490000 + 0.871722i \(0.663003\pi\)
\(660\) 463.292 0.0273237
\(661\) −13221.0 −0.777966 −0.388983 0.921245i \(-0.627174\pi\)
−0.388983 + 0.921245i \(0.627174\pi\)
\(662\) 7014.57 0.411826
\(663\) −27471.1 −1.60919
\(664\) 19910.9 1.16369
\(665\) −48.1742 −0.00280919
\(666\) 19022.1 1.10675
\(667\) 28192.8 1.63662
\(668\) 7104.88 0.411521
\(669\) 22363.6 1.29242
\(670\) −114.959 −0.00662873
\(671\) −16566.3 −0.953108
\(672\) −15002.5 −0.861213
\(673\) 9297.40 0.532524 0.266262 0.963901i \(-0.414211\pi\)
0.266262 + 0.963901i \(0.414211\pi\)
\(674\) −1845.14 −0.105449
\(675\) 40340.6 2.30031
\(676\) 10606.2 0.603448
\(677\) 5546.69 0.314884 0.157442 0.987528i \(-0.449675\pi\)
0.157442 + 0.987528i \(0.449675\pi\)
\(678\) 8551.26 0.484379
\(679\) −12700.0 −0.717795
\(680\) 743.627 0.0419365
\(681\) 6255.94 0.352023
\(682\) −9065.32 −0.508987
\(683\) −13290.9 −0.744603 −0.372301 0.928112i \(-0.621431\pi\)
−0.372301 + 0.928112i \(0.621431\pi\)
\(684\) 6612.54 0.369644
\(685\) −563.316 −0.0314207
\(686\) −6769.21 −0.376749
\(687\) −30555.4 −1.69689
\(688\) 0 0
\(689\) −6085.82 −0.336504
\(690\) −493.303 −0.0272170
\(691\) −16442.2 −0.905199 −0.452599 0.891714i \(-0.649503\pi\)
−0.452599 + 0.891714i \(0.649503\pi\)
\(692\) −1447.56 −0.0795200
\(693\) 12790.1 0.701088
\(694\) −9034.25 −0.494144
\(695\) 783.406 0.0427573
\(696\) −37752.9 −2.05607
\(697\) −23300.8 −1.26625
\(698\) −6531.81 −0.354201
\(699\) −56040.0 −3.03237
\(700\) −7041.79 −0.380221
\(701\) 25096.3 1.35217 0.676087 0.736822i \(-0.263675\pi\)
0.676087 + 0.736822i \(0.263675\pi\)
\(702\) −9489.76 −0.510211
\(703\) 4128.97 0.221518
\(704\) −74.7189 −0.00400010
\(705\) −5.69012 −0.000303975 0
\(706\) −2446.77 −0.130433
\(707\) −8004.53 −0.425801
\(708\) −25147.5 −1.33489
\(709\) 15705.5 0.831920 0.415960 0.909383i \(-0.363446\pi\)
0.415960 + 0.909383i \(0.363446\pi\)
\(710\) 277.595 0.0146732
\(711\) −44969.6 −2.37200
\(712\) 3646.47 0.191935
\(713\) −38717.9 −2.03366
\(714\) 13142.9 0.688882
\(715\) −178.971 −0.00936101
\(716\) −7518.74 −0.392442
\(717\) −15528.3 −0.808809
\(718\) −1967.36 −0.102258
\(719\) −37095.9 −1.92412 −0.962061 0.272836i \(-0.912038\pi\)
−0.962061 + 0.272836i \(0.912038\pi\)
\(720\) −563.263 −0.0291550
\(721\) −11122.9 −0.574533
\(722\) 8308.53 0.428271
\(723\) −4935.02 −0.253852
\(724\) 6074.30 0.311809
\(725\) −27562.4 −1.41192
\(726\) −9152.10 −0.467860
\(727\) −29835.3 −1.52205 −0.761024 0.648724i \(-0.775303\pi\)
−0.761024 + 0.648724i \(0.775303\pi\)
\(728\) 3726.01 0.189691
\(729\) 2667.26 0.135511
\(730\) 422.975 0.0214452
\(731\) 0 0
\(732\) 42123.0 2.12693
\(733\) 33242.9 1.67511 0.837553 0.546356i \(-0.183985\pi\)
0.837553 + 0.546356i \(0.183985\pi\)
\(734\) −7442.89 −0.374281
\(735\) −811.332 −0.0407163
\(736\) −23156.8 −1.15974
\(737\) 6624.96 0.331117
\(738\) −14373.9 −0.716951
\(739\) 19988.7 0.994990 0.497495 0.867467i \(-0.334253\pi\)
0.497495 + 0.867467i \(0.334253\pi\)
\(740\) −510.803 −0.0253750
\(741\) −3678.43 −0.182363
\(742\) 2911.62 0.144055
\(743\) −19510.4 −0.963346 −0.481673 0.876351i \(-0.659971\pi\)
−0.481673 + 0.876351i \(0.659971\pi\)
\(744\) 51847.1 2.55485
\(745\) 401.693 0.0197542
\(746\) −1720.77 −0.0844529
\(747\) 67133.3 3.28819
\(748\) −19052.3 −0.931311
\(749\) −2938.29 −0.143342
\(750\) 964.953 0.0469801
\(751\) 3423.45 0.166343 0.0831714 0.996535i \(-0.473495\pi\)
0.0831714 + 0.996535i \(0.473495\pi\)
\(752\) −52.5680 −0.00254915
\(753\) −54432.7 −2.63432
\(754\) 6483.80 0.313164
\(755\) 24.9028 0.00120041
\(756\) −18211.3 −0.876108
\(757\) 27084.2 1.30039 0.650193 0.759769i \(-0.274688\pi\)
0.650193 + 0.759769i \(0.274688\pi\)
\(758\) 3377.60 0.161847
\(759\) 28428.5 1.35954
\(760\) 99.5730 0.00475249
\(761\) −14059.4 −0.669714 −0.334857 0.942269i \(-0.608688\pi\)
−0.334857 + 0.942269i \(0.608688\pi\)
\(762\) 21660.0 1.02973
\(763\) −1015.64 −0.0481896
\(764\) 1543.35 0.0730844
\(765\) 2507.28 0.118498
\(766\) 3812.90 0.179851
\(767\) 9714.51 0.457328
\(768\) 17413.1 0.818152
\(769\) −37181.7 −1.74357 −0.871786 0.489886i \(-0.837038\pi\)
−0.871786 + 0.489886i \(0.837038\pi\)
\(770\) 85.6243 0.00400738
\(771\) −43452.5 −2.02971
\(772\) 4829.18 0.225137
\(773\) 4717.31 0.219495 0.109748 0.993959i \(-0.464996\pi\)
0.109748 + 0.993959i \(0.464996\pi\)
\(774\) 0 0
\(775\) 37852.1 1.75444
\(776\) 26250.2 1.21434
\(777\) −20306.7 −0.937578
\(778\) −19231.2 −0.886210
\(779\) −3120.01 −0.143499
\(780\) 455.067 0.0208897
\(781\) −15997.5 −0.732953
\(782\) 20286.5 0.927676
\(783\) −71281.0 −3.25335
\(784\) −7495.47 −0.341448
\(785\) 425.216 0.0193333
\(786\) −9832.12 −0.446183
\(787\) −28567.6 −1.29393 −0.646966 0.762519i \(-0.723963\pi\)
−0.646966 + 0.762519i \(0.723963\pi\)
\(788\) 29351.5 1.32691
\(789\) −4405.85 −0.198799
\(790\) −301.054 −0.0135582
\(791\) −6339.31 −0.284956
\(792\) −26436.2 −1.18607
\(793\) −16272.2 −0.728679
\(794\) 12129.0 0.542117
\(795\) 799.860 0.0356831
\(796\) 16420.4 0.731163
\(797\) −25388.9 −1.12838 −0.564191 0.825644i \(-0.690812\pi\)
−0.564191 + 0.825644i \(0.690812\pi\)
\(798\) 1759.86 0.0780681
\(799\) 233.999 0.0103608
\(800\) 22639.0 1.00051
\(801\) 12294.8 0.542341
\(802\) −19605.6 −0.863213
\(803\) −24375.6 −1.07123
\(804\) −16845.2 −0.738911
\(805\) 365.701 0.0160115
\(806\) −8904.37 −0.389135
\(807\) 10929.9 0.476765
\(808\) 16544.9 0.720354
\(809\) −6922.63 −0.300849 −0.150424 0.988622i \(-0.548064\pi\)
−0.150424 + 0.988622i \(0.548064\pi\)
\(810\) 566.663 0.0245809
\(811\) −5155.99 −0.223245 −0.111622 0.993751i \(-0.535605\pi\)
−0.111622 + 0.993751i \(0.535605\pi\)
\(812\) 12442.7 0.537750
\(813\) 54464.3 2.34950
\(814\) −7338.79 −0.316001
\(815\) −725.056 −0.0311627
\(816\) 33355.8 1.43099
\(817\) 0 0
\(818\) 16529.7 0.706536
\(819\) 12563.0 0.536002
\(820\) 385.983 0.0164379
\(821\) −28808.7 −1.22464 −0.612320 0.790610i \(-0.709764\pi\)
−0.612320 + 0.790610i \(0.709764\pi\)
\(822\) 20578.6 0.873189
\(823\) 14065.1 0.595720 0.297860 0.954610i \(-0.403727\pi\)
0.297860 + 0.954610i \(0.403727\pi\)
\(824\) 22990.3 0.971973
\(825\) −27792.8 −1.17288
\(826\) −4647.68 −0.195779
\(827\) −20288.1 −0.853068 −0.426534 0.904471i \(-0.640266\pi\)
−0.426534 + 0.904471i \(0.640266\pi\)
\(828\) −50197.2 −2.10685
\(829\) −2531.77 −0.106070 −0.0530349 0.998593i \(-0.516889\pi\)
−0.0530349 + 0.998593i \(0.516889\pi\)
\(830\) 449.431 0.0187952
\(831\) 10134.4 0.423055
\(832\) −73.3923 −0.00305820
\(833\) 33365.0 1.38779
\(834\) −28618.8 −1.18823
\(835\) 360.726 0.0149502
\(836\) −2551.14 −0.105542
\(837\) 97892.1 4.04259
\(838\) −5054.49 −0.208359
\(839\) −46374.3 −1.90824 −0.954122 0.299417i \(-0.903208\pi\)
−0.954122 + 0.299417i \(0.903208\pi\)
\(840\) −489.710 −0.0201150
\(841\) 24313.1 0.996888
\(842\) −18249.3 −0.746928
\(843\) −41317.6 −1.68808
\(844\) 12496.7 0.509661
\(845\) 538.494 0.0219228
\(846\) 144.350 0.00586627
\(847\) 6784.73 0.275238
\(848\) 7389.48 0.299240
\(849\) −17497.5 −0.707318
\(850\) −19832.8 −0.800307
\(851\) −31343.9 −1.26258
\(852\) 40676.7 1.63564
\(853\) 3436.56 0.137943 0.0689716 0.997619i \(-0.478028\pi\)
0.0689716 + 0.997619i \(0.478028\pi\)
\(854\) 7785.05 0.311943
\(855\) 335.730 0.0134289
\(856\) 6073.27 0.242500
\(857\) −29311.5 −1.16833 −0.584167 0.811633i \(-0.698579\pi\)
−0.584167 + 0.811633i \(0.698579\pi\)
\(858\) 6538.01 0.260145
\(859\) −31911.1 −1.26751 −0.633756 0.773533i \(-0.718488\pi\)
−0.633756 + 0.773533i \(0.718488\pi\)
\(860\) 0 0
\(861\) 15344.5 0.607364
\(862\) −10997.6 −0.434548
\(863\) −16307.5 −0.643239 −0.321619 0.946869i \(-0.604227\pi\)
−0.321619 + 0.946869i \(0.604227\pi\)
\(864\) 58548.3 2.30539
\(865\) −73.4948 −0.00288890
\(866\) 16614.7 0.651952
\(867\) −102296. −4.00710
\(868\) −17087.9 −0.668203
\(869\) 17349.4 0.677259
\(870\) −852.165 −0.0332082
\(871\) 6507.34 0.253149
\(872\) 2099.27 0.0815254
\(873\) 88507.5 3.43130
\(874\) 2716.39 0.105130
\(875\) −715.349 −0.0276380
\(876\) 61979.5 2.39052
\(877\) 34825.8 1.34092 0.670458 0.741948i \(-0.266098\pi\)
0.670458 + 0.741948i \(0.266098\pi\)
\(878\) −14392.8 −0.553229
\(879\) 4519.15 0.173410
\(880\) 217.308 0.00832439
\(881\) 3505.74 0.134065 0.0670326 0.997751i \(-0.478647\pi\)
0.0670326 + 0.997751i \(0.478647\pi\)
\(882\) 20582.4 0.785764
\(883\) 23731.6 0.904451 0.452226 0.891904i \(-0.350630\pi\)
0.452226 + 0.891904i \(0.350630\pi\)
\(884\) −18714.0 −0.712015
\(885\) −1276.78 −0.0484954
\(886\) 5440.42 0.206292
\(887\) −15367.8 −0.581736 −0.290868 0.956763i \(-0.593944\pi\)
−0.290868 + 0.956763i \(0.593944\pi\)
\(888\) 41972.6 1.58616
\(889\) −16057.2 −0.605783
\(890\) 82.3088 0.00310000
\(891\) −32656.2 −1.22786
\(892\) 15234.7 0.571855
\(893\) 31.3328 0.00117415
\(894\) −14674.3 −0.548974
\(895\) −381.739 −0.0142571
\(896\) −12732.9 −0.474751
\(897\) 27923.8 1.03941
\(898\) −4723.99 −0.175547
\(899\) −66883.9 −2.48132
\(900\) 49074.8 1.81758
\(901\) −32893.2 −1.21624
\(902\) 5545.48 0.204705
\(903\) 0 0
\(904\) 13102.9 0.482077
\(905\) 308.402 0.0113278
\(906\) −909.731 −0.0333596
\(907\) 36989.5 1.35415 0.677076 0.735913i \(-0.263247\pi\)
0.677076 + 0.735913i \(0.263247\pi\)
\(908\) 4261.70 0.155759
\(909\) 55784.2 2.03547
\(910\) 84.1041 0.00306376
\(911\) −52269.2 −1.90094 −0.950469 0.310819i \(-0.899397\pi\)
−0.950469 + 0.310819i \(0.899397\pi\)
\(912\) 4466.40 0.162168
\(913\) −25900.2 −0.938852
\(914\) 11255.6 0.407333
\(915\) 2138.65 0.0772697
\(916\) −20815.1 −0.750819
\(917\) 7288.85 0.262485
\(918\) −51291.1 −1.84407
\(919\) 17187.9 0.616948 0.308474 0.951233i \(-0.400182\pi\)
0.308474 + 0.951233i \(0.400182\pi\)
\(920\) −755.880 −0.0270876
\(921\) −65172.1 −2.33170
\(922\) −3404.75 −0.121615
\(923\) −15713.5 −0.560364
\(924\) 12546.7 0.446707
\(925\) 30643.0 1.08923
\(926\) −8477.84 −0.300863
\(927\) 77516.3 2.74646
\(928\) −40002.6 −1.41503
\(929\) −18064.1 −0.637960 −0.318980 0.947761i \(-0.603340\pi\)
−0.318980 + 0.947761i \(0.603340\pi\)
\(930\) 1170.30 0.0412642
\(931\) 4467.63 0.157273
\(932\) −38175.9 −1.34173
\(933\) 19363.4 0.679451
\(934\) 21186.2 0.742222
\(935\) −967.317 −0.0338338
\(936\) −25966.9 −0.906788
\(937\) −12776.9 −0.445467 −0.222734 0.974879i \(-0.571498\pi\)
−0.222734 + 0.974879i \(0.571498\pi\)
\(938\) −3113.28 −0.108371
\(939\) 50382.3 1.75098
\(940\) −3.87625 −0.000134499 0
\(941\) 12576.8 0.435698 0.217849 0.975982i \(-0.430096\pi\)
0.217849 + 0.975982i \(0.430096\pi\)
\(942\) −15533.6 −0.537276
\(943\) 23684.7 0.817900
\(944\) −11795.5 −0.406685
\(945\) −924.617 −0.0318283
\(946\) 0 0
\(947\) 9064.67 0.311048 0.155524 0.987832i \(-0.450293\pi\)
0.155524 + 0.987832i \(0.450293\pi\)
\(948\) −44114.1 −1.51135
\(949\) −23942.8 −0.818984
\(950\) −2655.65 −0.0906955
\(951\) 65983.4 2.24990
\(952\) 20138.7 0.685607
\(953\) 24942.4 0.847811 0.423905 0.905707i \(-0.360659\pi\)
0.423905 + 0.905707i \(0.360659\pi\)
\(954\) −20291.3 −0.688632
\(955\) 78.3585 0.00265510
\(956\) −10578.3 −0.357873
\(957\) 49109.3 1.65881
\(958\) 16684.4 0.562680
\(959\) −15255.5 −0.513689
\(960\) 9.64595 0.000324293 0
\(961\) 62062.4 2.08326
\(962\) −7208.49 −0.241592
\(963\) 20477.2 0.685222
\(964\) −3361.86 −0.112322
\(965\) 245.185 0.00817906
\(966\) −13359.5 −0.444963
\(967\) −8748.78 −0.290943 −0.145471 0.989362i \(-0.546470\pi\)
−0.145471 + 0.989362i \(0.546470\pi\)
\(968\) −14023.6 −0.465636
\(969\) −19881.5 −0.659119
\(970\) 592.523 0.0196132
\(971\) 14600.8 0.482557 0.241279 0.970456i \(-0.422433\pi\)
0.241279 + 0.970456i \(0.422433\pi\)
\(972\) 27189.3 0.897220
\(973\) 21216.0 0.699026
\(974\) 2894.93 0.0952356
\(975\) −27299.4 −0.896698
\(976\) 19757.9 0.647987
\(977\) 22071.1 0.722741 0.361371 0.932422i \(-0.382309\pi\)
0.361371 + 0.932422i \(0.382309\pi\)
\(978\) 26487.2 0.866019
\(979\) −4743.37 −0.154851
\(980\) −552.700 −0.0180157
\(981\) 7078.09 0.230363
\(982\) 11141.3 0.362050
\(983\) 36744.9 1.19225 0.596125 0.802892i \(-0.296706\pi\)
0.596125 + 0.802892i \(0.296706\pi\)
\(984\) −31716.2 −1.02751
\(985\) 1490.22 0.0482056
\(986\) 35044.2 1.13188
\(987\) −154.098 −0.00496960
\(988\) −2505.84 −0.0806897
\(989\) 0 0
\(990\) −596.723 −0.0191567
\(991\) −3840.07 −0.123092 −0.0615458 0.998104i \(-0.519603\pi\)
−0.0615458 + 0.998104i \(0.519603\pi\)
\(992\) 54936.7 1.75831
\(993\) 52186.2 1.66775
\(994\) 7517.75 0.239888
\(995\) 833.691 0.0265626
\(996\) 65856.2 2.09511
\(997\) 11059.7 0.351318 0.175659 0.984451i \(-0.443794\pi\)
0.175659 + 0.984451i \(0.443794\pi\)
\(998\) 20230.1 0.641655
\(999\) 79248.1 2.50981
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.i.1.20 50
43.42 odd 2 1849.4.a.j.1.31 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.20 50 1.1 even 1 trivial
1849.4.a.j.1.31 yes 50 43.42 odd 2