Properties

Label 1849.4.a.j.1.17
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 1.17
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.89933 q^{2} +6.15189 q^{3} -4.39254 q^{4} +1.87975 q^{5} -11.6845 q^{6} -2.77777 q^{7} +23.5375 q^{8} +10.8458 q^{9} +O(q^{10})\) \(q-1.89933 q^{2} +6.15189 q^{3} -4.39254 q^{4} +1.87975 q^{5} -11.6845 q^{6} -2.77777 q^{7} +23.5375 q^{8} +10.8458 q^{9} -3.57027 q^{10} +5.01316 q^{11} -27.0224 q^{12} -31.2928 q^{13} +5.27590 q^{14} +11.5640 q^{15} -9.56527 q^{16} +42.4153 q^{17} -20.5997 q^{18} -2.31817 q^{19} -8.25689 q^{20} -17.0885 q^{21} -9.52164 q^{22} -10.8509 q^{23} +144.800 q^{24} -121.467 q^{25} +59.4353 q^{26} -99.3790 q^{27} +12.2015 q^{28} +82.1722 q^{29} -21.9639 q^{30} +296.425 q^{31} -170.133 q^{32} +30.8404 q^{33} -80.5606 q^{34} -5.22151 q^{35} -47.6406 q^{36} -186.475 q^{37} +4.40297 q^{38} -192.510 q^{39} +44.2448 q^{40} +343.407 q^{41} +32.4568 q^{42} -22.0205 q^{44} +20.3874 q^{45} +20.6094 q^{46} -200.405 q^{47} -58.8445 q^{48} -335.284 q^{49} +230.705 q^{50} +260.934 q^{51} +137.455 q^{52} -500.508 q^{53} +188.754 q^{54} +9.42349 q^{55} -65.3818 q^{56} -14.2611 q^{57} -156.072 q^{58} +613.901 q^{59} -50.7955 q^{60} -661.557 q^{61} -563.010 q^{62} -30.1271 q^{63} +399.661 q^{64} -58.8227 q^{65} -58.5761 q^{66} +198.433 q^{67} -186.311 q^{68} -66.7534 q^{69} +9.91739 q^{70} +896.331 q^{71} +255.283 q^{72} -619.729 q^{73} +354.177 q^{74} -747.249 q^{75} +10.1826 q^{76} -13.9254 q^{77} +365.640 q^{78} +58.4019 q^{79} -17.9803 q^{80} -904.205 q^{81} -652.244 q^{82} +188.055 q^{83} +75.0620 q^{84} +79.7302 q^{85} +505.515 q^{87} +117.997 q^{88} -1323.82 q^{89} -38.7224 q^{90} +86.9240 q^{91} +47.6629 q^{92} +1823.58 q^{93} +380.636 q^{94} -4.35758 q^{95} -1046.64 q^{96} +1022.15 q^{97} +636.815 q^{98} +54.3716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89933 −0.671515 −0.335758 0.941948i \(-0.608992\pi\)
−0.335758 + 0.941948i \(0.608992\pi\)
\(3\) 6.15189 1.18393 0.591966 0.805963i \(-0.298352\pi\)
0.591966 + 0.805963i \(0.298352\pi\)
\(4\) −4.39254 −0.549068
\(5\) 1.87975 0.168130 0.0840651 0.996460i \(-0.473210\pi\)
0.0840651 + 0.996460i \(0.473210\pi\)
\(6\) −11.6845 −0.795028
\(7\) −2.77777 −0.149985 −0.0749927 0.997184i \(-0.523893\pi\)
−0.0749927 + 0.997184i \(0.523893\pi\)
\(8\) 23.5375 1.04022
\(9\) 10.8458 0.401696
\(10\) −3.57027 −0.112902
\(11\) 5.01316 0.137411 0.0687056 0.997637i \(-0.478113\pi\)
0.0687056 + 0.997637i \(0.478113\pi\)
\(12\) −27.0224 −0.650059
\(13\) −31.2928 −0.667620 −0.333810 0.942640i \(-0.608334\pi\)
−0.333810 + 0.942640i \(0.608334\pi\)
\(14\) 5.27590 0.100717
\(15\) 11.5640 0.199055
\(16\) −9.56527 −0.149457
\(17\) 42.4153 0.605130 0.302565 0.953129i \(-0.402157\pi\)
0.302565 + 0.953129i \(0.402157\pi\)
\(18\) −20.5997 −0.269745
\(19\) −2.31817 −0.0279908 −0.0139954 0.999902i \(-0.504455\pi\)
−0.0139954 + 0.999902i \(0.504455\pi\)
\(20\) −8.25689 −0.0923148
\(21\) −17.0885 −0.177572
\(22\) −9.52164 −0.0922737
\(23\) −10.8509 −0.0983723 −0.0491862 0.998790i \(-0.515663\pi\)
−0.0491862 + 0.998790i \(0.515663\pi\)
\(24\) 144.800 1.23155
\(25\) −121.467 −0.971732
\(26\) 59.4353 0.448317
\(27\) −99.3790 −0.708352
\(28\) 12.2015 0.0823521
\(29\) 82.1722 0.526173 0.263086 0.964772i \(-0.415260\pi\)
0.263086 + 0.964772i \(0.415260\pi\)
\(30\) −21.9639 −0.133668
\(31\) 296.425 1.71740 0.858702 0.512475i \(-0.171271\pi\)
0.858702 + 0.512475i \(0.171271\pi\)
\(32\) −170.133 −0.939859
\(33\) 30.8404 0.162686
\(34\) −80.5606 −0.406354
\(35\) −5.22151 −0.0252171
\(36\) −47.6406 −0.220558
\(37\) −186.475 −0.828547 −0.414273 0.910152i \(-0.635964\pi\)
−0.414273 + 0.910152i \(0.635964\pi\)
\(38\) 4.40297 0.0187962
\(39\) −192.510 −0.790416
\(40\) 44.2448 0.174893
\(41\) 343.407 1.30808 0.654039 0.756461i \(-0.273073\pi\)
0.654039 + 0.756461i \(0.273073\pi\)
\(42\) 32.4568 0.119243
\(43\) 0 0
\(44\) −22.0205 −0.0754480
\(45\) 20.3874 0.0675372
\(46\) 20.6094 0.0660585
\(47\) −200.405 −0.621960 −0.310980 0.950417i \(-0.600657\pi\)
−0.310980 + 0.950417i \(0.600657\pi\)
\(48\) −58.8445 −0.176947
\(49\) −335.284 −0.977504
\(50\) 230.705 0.652533
\(51\) 260.934 0.716433
\(52\) 137.455 0.366568
\(53\) −500.508 −1.29717 −0.648585 0.761142i \(-0.724639\pi\)
−0.648585 + 0.761142i \(0.724639\pi\)
\(54\) 188.754 0.475669
\(55\) 9.42349 0.0231030
\(56\) −65.3818 −0.156018
\(57\) −14.2611 −0.0331392
\(58\) −156.072 −0.353333
\(59\) 613.901 1.35463 0.677315 0.735693i \(-0.263144\pi\)
0.677315 + 0.735693i \(0.263144\pi\)
\(60\) −50.7955 −0.109295
\(61\) −661.557 −1.38858 −0.694292 0.719693i \(-0.744283\pi\)
−0.694292 + 0.719693i \(0.744283\pi\)
\(62\) −563.010 −1.15326
\(63\) −30.1271 −0.0602485
\(64\) 399.661 0.780587
\(65\) −58.8227 −0.112247
\(66\) −58.5761 −0.109246
\(67\) 198.433 0.361828 0.180914 0.983499i \(-0.442094\pi\)
0.180914 + 0.983499i \(0.442094\pi\)
\(68\) −186.311 −0.332257
\(69\) −66.7534 −0.116466
\(70\) 9.91739 0.0169336
\(71\) 896.331 1.49824 0.749119 0.662435i \(-0.230477\pi\)
0.749119 + 0.662435i \(0.230477\pi\)
\(72\) 255.283 0.417853
\(73\) −619.729 −0.993614 −0.496807 0.867861i \(-0.665494\pi\)
−0.496807 + 0.867861i \(0.665494\pi\)
\(74\) 354.177 0.556382
\(75\) −747.249 −1.15047
\(76\) 10.1826 0.0153688
\(77\) −13.9254 −0.0206097
\(78\) 365.640 0.530777
\(79\) 58.4019 0.0831737 0.0415868 0.999135i \(-0.486759\pi\)
0.0415868 + 0.999135i \(0.486759\pi\)
\(80\) −17.9803 −0.0251283
\(81\) −904.205 −1.24034
\(82\) −652.244 −0.878394
\(83\) 188.055 0.248695 0.124348 0.992239i \(-0.460316\pi\)
0.124348 + 0.992239i \(0.460316\pi\)
\(84\) 75.0620 0.0974993
\(85\) 79.7302 0.101741
\(86\) 0 0
\(87\) 505.515 0.622953
\(88\) 117.997 0.142938
\(89\) −1323.82 −1.57669 −0.788343 0.615237i \(-0.789061\pi\)
−0.788343 + 0.615237i \(0.789061\pi\)
\(90\) −38.7224 −0.0453522
\(91\) 86.9240 0.100133
\(92\) 47.6629 0.0540130
\(93\) 1823.58 2.03329
\(94\) 380.636 0.417655
\(95\) −4.35758 −0.00470609
\(96\) −1046.64 −1.11273
\(97\) 1022.15 1.06993 0.534967 0.844873i \(-0.320324\pi\)
0.534967 + 0.844873i \(0.320324\pi\)
\(98\) 636.815 0.656409
\(99\) 54.3716 0.0551975
\(100\) 533.547 0.533547
\(101\) −532.767 −0.524874 −0.262437 0.964949i \(-0.584526\pi\)
−0.262437 + 0.964949i \(0.584526\pi\)
\(102\) −495.600 −0.481096
\(103\) −1188.97 −1.13740 −0.568701 0.822544i \(-0.692554\pi\)
−0.568701 + 0.822544i \(0.692554\pi\)
\(104\) −736.555 −0.694473
\(105\) −32.1222 −0.0298553
\(106\) 950.631 0.871070
\(107\) 1016.86 0.918729 0.459364 0.888248i \(-0.348077\pi\)
0.459364 + 0.888248i \(0.348077\pi\)
\(108\) 436.526 0.388933
\(109\) −1514.02 −1.33043 −0.665215 0.746652i \(-0.731660\pi\)
−0.665215 + 0.746652i \(0.731660\pi\)
\(110\) −17.8983 −0.0155140
\(111\) −1147.17 −0.980943
\(112\) 26.5701 0.0224164
\(113\) 1252.63 1.04281 0.521403 0.853311i \(-0.325409\pi\)
0.521403 + 0.853311i \(0.325409\pi\)
\(114\) 27.0866 0.0222534
\(115\) −20.3970 −0.0165394
\(116\) −360.945 −0.288904
\(117\) −339.395 −0.268180
\(118\) −1166.00 −0.909654
\(119\) −117.820 −0.0907606
\(120\) 272.189 0.207061
\(121\) −1305.87 −0.981118
\(122\) 1256.52 0.932456
\(123\) 2112.60 1.54868
\(124\) −1302.06 −0.942971
\(125\) −463.296 −0.331508
\(126\) 57.2213 0.0404578
\(127\) 1389.50 0.970855 0.485427 0.874277i \(-0.338664\pi\)
0.485427 + 0.874277i \(0.338664\pi\)
\(128\) 601.974 0.415683
\(129\) 0 0
\(130\) 111.724 0.0753755
\(131\) −801.787 −0.534752 −0.267376 0.963592i \(-0.586157\pi\)
−0.267376 + 0.963592i \(0.586157\pi\)
\(132\) −135.468 −0.0893254
\(133\) 6.43933 0.00419820
\(134\) −376.891 −0.242973
\(135\) −186.808 −0.119095
\(136\) 998.351 0.629470
\(137\) 161.623 0.100791 0.0503955 0.998729i \(-0.483952\pi\)
0.0503955 + 0.998729i \(0.483952\pi\)
\(138\) 126.787 0.0782088
\(139\) −2295.60 −1.40080 −0.700398 0.713752i \(-0.746994\pi\)
−0.700398 + 0.713752i \(0.746994\pi\)
\(140\) 22.9357 0.0138459
\(141\) −1232.87 −0.736358
\(142\) −1702.43 −1.00609
\(143\) −156.876 −0.0917384
\(144\) −103.743 −0.0600364
\(145\) 154.463 0.0884655
\(146\) 1177.07 0.667226
\(147\) −2062.63 −1.15730
\(148\) 819.097 0.454928
\(149\) 1133.28 0.623102 0.311551 0.950229i \(-0.399152\pi\)
0.311551 + 0.950229i \(0.399152\pi\)
\(150\) 1419.27 0.772555
\(151\) −1357.22 −0.731451 −0.365725 0.930723i \(-0.619179\pi\)
−0.365725 + 0.930723i \(0.619179\pi\)
\(152\) −54.5640 −0.0291166
\(153\) 460.027 0.243078
\(154\) 26.4489 0.0138397
\(155\) 557.206 0.288747
\(156\) 845.607 0.433992
\(157\) 263.487 0.133940 0.0669698 0.997755i \(-0.478667\pi\)
0.0669698 + 0.997755i \(0.478667\pi\)
\(158\) −110.924 −0.0558524
\(159\) −3079.07 −1.53576
\(160\) −319.807 −0.158019
\(161\) 30.1412 0.0147544
\(162\) 1717.39 0.832904
\(163\) −602.189 −0.289369 −0.144684 0.989478i \(-0.546217\pi\)
−0.144684 + 0.989478i \(0.546217\pi\)
\(164\) −1508.43 −0.718223
\(165\) 57.9723 0.0273524
\(166\) −357.179 −0.167003
\(167\) −3100.06 −1.43646 −0.718232 0.695804i \(-0.755048\pi\)
−0.718232 + 0.695804i \(0.755048\pi\)
\(168\) −402.222 −0.184715
\(169\) −1217.76 −0.554284
\(170\) −151.434 −0.0683204
\(171\) −25.1424 −0.0112438
\(172\) 0 0
\(173\) −793.385 −0.348670 −0.174335 0.984686i \(-0.555778\pi\)
−0.174335 + 0.984686i \(0.555778\pi\)
\(174\) −960.140 −0.418322
\(175\) 337.406 0.145746
\(176\) −47.9522 −0.0205371
\(177\) 3776.66 1.60379
\(178\) 2514.38 1.05877
\(179\) 1832.74 0.765283 0.382642 0.923897i \(-0.375014\pi\)
0.382642 + 0.923897i \(0.375014\pi\)
\(180\) −89.5525 −0.0370825
\(181\) −3600.53 −1.47859 −0.739297 0.673380i \(-0.764842\pi\)
−0.739297 + 0.673380i \(0.764842\pi\)
\(182\) −165.098 −0.0672409
\(183\) −4069.83 −1.64399
\(184\) −255.403 −0.102329
\(185\) −350.526 −0.139304
\(186\) −3463.58 −1.36539
\(187\) 212.634 0.0831517
\(188\) 880.288 0.341498
\(189\) 276.052 0.106242
\(190\) 8.27649 0.00316021
\(191\) 1718.34 0.650967 0.325484 0.945548i \(-0.394473\pi\)
0.325484 + 0.945548i \(0.394473\pi\)
\(192\) 2458.67 0.924162
\(193\) −902.915 −0.336753 −0.168376 0.985723i \(-0.553852\pi\)
−0.168376 + 0.985723i \(0.553852\pi\)
\(194\) −1941.40 −0.718476
\(195\) −361.871 −0.132893
\(196\) 1472.75 0.536716
\(197\) −2564.17 −0.927360 −0.463680 0.886003i \(-0.653471\pi\)
−0.463680 + 0.886003i \(0.653471\pi\)
\(198\) −103.270 −0.0370660
\(199\) 4092.54 1.45785 0.728925 0.684593i \(-0.240020\pi\)
0.728925 + 0.684593i \(0.240020\pi\)
\(200\) −2859.02 −1.01082
\(201\) 1220.74 0.428380
\(202\) 1011.90 0.352461
\(203\) −228.255 −0.0789182
\(204\) −1146.16 −0.393370
\(205\) 645.520 0.219927
\(206\) 2258.24 0.763783
\(207\) −117.686 −0.0395157
\(208\) 299.324 0.0997806
\(209\) −11.6213 −0.00384624
\(210\) 61.0107 0.0200483
\(211\) −1558.96 −0.508640 −0.254320 0.967120i \(-0.581852\pi\)
−0.254320 + 0.967120i \(0.581852\pi\)
\(212\) 2198.50 0.712234
\(213\) 5514.13 1.77381
\(214\) −1931.36 −0.616940
\(215\) 0 0
\(216\) −2339.14 −0.736843
\(217\) −823.400 −0.257585
\(218\) 2875.63 0.893404
\(219\) −3812.51 −1.17637
\(220\) −41.3931 −0.0126851
\(221\) −1327.29 −0.403997
\(222\) 2178.86 0.658718
\(223\) −5191.82 −1.55906 −0.779529 0.626366i \(-0.784542\pi\)
−0.779529 + 0.626366i \(0.784542\pi\)
\(224\) 472.589 0.140965
\(225\) −1317.40 −0.390341
\(226\) −2379.15 −0.700260
\(227\) −6436.83 −1.88206 −0.941029 0.338325i \(-0.890140\pi\)
−0.941029 + 0.338325i \(0.890140\pi\)
\(228\) 62.6426 0.0181956
\(229\) −5961.17 −1.72020 −0.860099 0.510127i \(-0.829598\pi\)
−0.860099 + 0.510127i \(0.829598\pi\)
\(230\) 38.7406 0.0111064
\(231\) −85.6674 −0.0244005
\(232\) 1934.13 0.547336
\(233\) 1629.54 0.458174 0.229087 0.973406i \(-0.426426\pi\)
0.229087 + 0.973406i \(0.426426\pi\)
\(234\) 644.623 0.180087
\(235\) −376.712 −0.104570
\(236\) −2696.59 −0.743783
\(237\) 359.282 0.0984720
\(238\) 223.779 0.0609471
\(239\) 1788.54 0.484064 0.242032 0.970268i \(-0.422186\pi\)
0.242032 + 0.970268i \(0.422186\pi\)
\(240\) −110.613 −0.0297502
\(241\) −4933.89 −1.31875 −0.659377 0.751812i \(-0.729180\pi\)
−0.659377 + 0.751812i \(0.729180\pi\)
\(242\) 2480.28 0.658836
\(243\) −2879.34 −0.760123
\(244\) 2905.92 0.762427
\(245\) −630.251 −0.164348
\(246\) −4012.53 −1.03996
\(247\) 72.5419 0.0186872
\(248\) 6977.12 1.78648
\(249\) 1156.89 0.294439
\(250\) 879.953 0.222612
\(251\) 5862.79 1.47433 0.737163 0.675715i \(-0.236165\pi\)
0.737163 + 0.675715i \(0.236165\pi\)
\(252\) 132.334 0.0330805
\(253\) −54.3971 −0.0135175
\(254\) −2639.13 −0.651944
\(255\) 490.492 0.120454
\(256\) −4340.63 −1.05972
\(257\) −1027.25 −0.249332 −0.124666 0.992199i \(-0.539786\pi\)
−0.124666 + 0.992199i \(0.539786\pi\)
\(258\) 0 0
\(259\) 517.983 0.124270
\(260\) 258.381 0.0616312
\(261\) 891.223 0.211361
\(262\) 1522.86 0.359094
\(263\) 8377.92 1.96428 0.982138 0.188160i \(-0.0602524\pi\)
0.982138 + 0.188160i \(0.0602524\pi\)
\(264\) 725.907 0.169229
\(265\) −940.831 −0.218094
\(266\) −12.2304 −0.00281916
\(267\) −8144.02 −1.86669
\(268\) −871.626 −0.198668
\(269\) −2853.05 −0.646668 −0.323334 0.946285i \(-0.604804\pi\)
−0.323334 + 0.946285i \(0.604804\pi\)
\(270\) 354.810 0.0799743
\(271\) −4091.04 −0.917021 −0.458511 0.888689i \(-0.651617\pi\)
−0.458511 + 0.888689i \(0.651617\pi\)
\(272\) −405.713 −0.0904411
\(273\) 534.747 0.118551
\(274\) −306.975 −0.0676827
\(275\) −608.931 −0.133527
\(276\) 293.217 0.0639478
\(277\) 5795.51 1.25711 0.628553 0.777766i \(-0.283647\pi\)
0.628553 + 0.777766i \(0.283647\pi\)
\(278\) 4360.11 0.940656
\(279\) 3214.96 0.689874
\(280\) −122.902 −0.0262313
\(281\) 4930.69 1.04676 0.523382 0.852098i \(-0.324670\pi\)
0.523382 + 0.852098i \(0.324670\pi\)
\(282\) 2341.63 0.494476
\(283\) 3343.99 0.702402 0.351201 0.936300i \(-0.385773\pi\)
0.351201 + 0.936300i \(0.385773\pi\)
\(284\) −3937.17 −0.822634
\(285\) −26.8074 −0.00557169
\(286\) 297.959 0.0616037
\(287\) −953.905 −0.196192
\(288\) −1845.22 −0.377538
\(289\) −3113.95 −0.633818
\(290\) −293.377 −0.0594059
\(291\) 6288.15 1.26673
\(292\) 2722.18 0.545561
\(293\) 3791.10 0.755898 0.377949 0.925826i \(-0.376629\pi\)
0.377949 + 0.925826i \(0.376629\pi\)
\(294\) 3917.62 0.777144
\(295\) 1153.98 0.227754
\(296\) −4389.15 −0.861873
\(297\) −498.202 −0.0973355
\(298\) −2152.48 −0.418422
\(299\) 339.554 0.0656753
\(300\) 3282.32 0.631683
\(301\) 0 0
\(302\) 2577.81 0.491180
\(303\) −3277.52 −0.621415
\(304\) 22.1739 0.00418342
\(305\) −1243.56 −0.233463
\(306\) −873.743 −0.163231
\(307\) −4492.58 −0.835195 −0.417598 0.908632i \(-0.637128\pi\)
−0.417598 + 0.908632i \(0.637128\pi\)
\(308\) 61.1678 0.0113161
\(309\) −7314.40 −1.34661
\(310\) −1058.32 −0.193898
\(311\) 5187.14 0.945774 0.472887 0.881123i \(-0.343212\pi\)
0.472887 + 0.881123i \(0.343212\pi\)
\(312\) −4531.21 −0.822209
\(313\) −5096.23 −0.920307 −0.460153 0.887839i \(-0.652206\pi\)
−0.460153 + 0.887839i \(0.652206\pi\)
\(314\) −500.448 −0.0899425
\(315\) −56.6314 −0.0101296
\(316\) −256.533 −0.0456680
\(317\) −6988.00 −1.23812 −0.619062 0.785342i \(-0.712487\pi\)
−0.619062 + 0.785342i \(0.712487\pi\)
\(318\) 5848.18 1.03129
\(319\) 411.942 0.0723020
\(320\) 751.263 0.131240
\(321\) 6255.64 1.08771
\(322\) −57.2481 −0.00990780
\(323\) −98.3257 −0.0169380
\(324\) 3971.76 0.681028
\(325\) 3801.02 0.648747
\(326\) 1143.76 0.194315
\(327\) −9314.09 −1.57514
\(328\) 8082.96 1.36069
\(329\) 556.679 0.0932848
\(330\) −110.109 −0.0183675
\(331\) 2689.73 0.446650 0.223325 0.974744i \(-0.428309\pi\)
0.223325 + 0.974744i \(0.428309\pi\)
\(332\) −826.039 −0.136551
\(333\) −2022.46 −0.332824
\(334\) 5888.03 0.964607
\(335\) 373.006 0.0608342
\(336\) 163.456 0.0265395
\(337\) 9821.54 1.58758 0.793788 0.608194i \(-0.208106\pi\)
0.793788 + 0.608194i \(0.208106\pi\)
\(338\) 2312.93 0.372210
\(339\) 7706.02 1.23461
\(340\) −350.218 −0.0558625
\(341\) 1486.03 0.235991
\(342\) 47.7537 0.00755036
\(343\) 1884.12 0.296597
\(344\) 0 0
\(345\) −125.480 −0.0195815
\(346\) 1506.90 0.234137
\(347\) −1195.67 −0.184977 −0.0924885 0.995714i \(-0.529482\pi\)
−0.0924885 + 0.995714i \(0.529482\pi\)
\(348\) −2220.49 −0.342043
\(349\) −2146.24 −0.329186 −0.164593 0.986362i \(-0.552631\pi\)
−0.164593 + 0.986362i \(0.552631\pi\)
\(350\) −640.845 −0.0978703
\(351\) 3109.84 0.472909
\(352\) −852.902 −0.129147
\(353\) −11886.2 −1.79217 −0.896087 0.443879i \(-0.853602\pi\)
−0.896087 + 0.443879i \(0.853602\pi\)
\(354\) −7173.12 −1.07697
\(355\) 1684.88 0.251899
\(356\) 5814.95 0.865707
\(357\) −724.814 −0.107454
\(358\) −3480.99 −0.513899
\(359\) −6715.29 −0.987241 −0.493621 0.869677i \(-0.664327\pi\)
−0.493621 + 0.869677i \(0.664327\pi\)
\(360\) 479.869 0.0702537
\(361\) −6853.63 −0.999217
\(362\) 6838.60 0.992898
\(363\) −8033.56 −1.16158
\(364\) −381.817 −0.0549799
\(365\) −1164.94 −0.167056
\(366\) 7729.95 1.10396
\(367\) −7744.24 −1.10149 −0.550744 0.834674i \(-0.685656\pi\)
−0.550744 + 0.834674i \(0.685656\pi\)
\(368\) 103.791 0.0147025
\(369\) 3724.52 0.525449
\(370\) 665.765 0.0935445
\(371\) 1390.29 0.194557
\(372\) −8010.13 −1.11641
\(373\) 56.6698 0.00786662 0.00393331 0.999992i \(-0.498748\pi\)
0.00393331 + 0.999992i \(0.498748\pi\)
\(374\) −403.863 −0.0558376
\(375\) −2850.15 −0.392483
\(376\) −4717.05 −0.646976
\(377\) −2571.40 −0.351283
\(378\) −524.314 −0.0713433
\(379\) 2963.95 0.401709 0.200854 0.979621i \(-0.435628\pi\)
0.200854 + 0.979621i \(0.435628\pi\)
\(380\) 19.1409 0.00258396
\(381\) 8548.08 1.14943
\(382\) −3263.70 −0.437134
\(383\) −1125.17 −0.150114 −0.0750570 0.997179i \(-0.523914\pi\)
−0.0750570 + 0.997179i \(0.523914\pi\)
\(384\) 3703.28 0.492141
\(385\) −26.1763 −0.00346511
\(386\) 1714.93 0.226134
\(387\) 0 0
\(388\) −4489.83 −0.587466
\(389\) −11051.0 −1.44038 −0.720189 0.693777i \(-0.755945\pi\)
−0.720189 + 0.693777i \(0.755945\pi\)
\(390\) 687.313 0.0892395
\(391\) −460.242 −0.0595280
\(392\) −7891.76 −1.01682
\(393\) −4932.51 −0.633110
\(394\) 4870.22 0.622736
\(395\) 109.781 0.0139840
\(396\) −238.830 −0.0303072
\(397\) 6471.64 0.818142 0.409071 0.912503i \(-0.365853\pi\)
0.409071 + 0.912503i \(0.365853\pi\)
\(398\) −7773.08 −0.978968
\(399\) 39.6141 0.00497039
\(400\) 1161.86 0.145232
\(401\) −3098.52 −0.385867 −0.192934 0.981212i \(-0.561800\pi\)
−0.192934 + 0.981212i \(0.561800\pi\)
\(402\) −2318.59 −0.287664
\(403\) −9275.97 −1.14657
\(404\) 2340.20 0.288191
\(405\) −1699.68 −0.208538
\(406\) 433.533 0.0529947
\(407\) −934.826 −0.113852
\(408\) 6141.75 0.745249
\(409\) −12700.2 −1.53542 −0.767710 0.640798i \(-0.778604\pi\)
−0.767710 + 0.640798i \(0.778604\pi\)
\(410\) −1226.06 −0.147684
\(411\) 994.286 0.119330
\(412\) 5222.59 0.624511
\(413\) −1705.27 −0.203175
\(414\) 223.525 0.0265354
\(415\) 353.497 0.0418132
\(416\) 5323.92 0.627468
\(417\) −14122.3 −1.65845
\(418\) 22.0728 0.00258281
\(419\) 8277.25 0.965084 0.482542 0.875873i \(-0.339714\pi\)
0.482542 + 0.875873i \(0.339714\pi\)
\(420\) 141.098 0.0163926
\(421\) −6783.31 −0.785269 −0.392634 0.919695i \(-0.628436\pi\)
−0.392634 + 0.919695i \(0.628436\pi\)
\(422\) 2960.98 0.341560
\(423\) −2173.55 −0.249839
\(424\) −11780.7 −1.34935
\(425\) −5152.03 −0.588024
\(426\) −10473.2 −1.19114
\(427\) 1837.65 0.208267
\(428\) −4466.62 −0.504444
\(429\) −965.082 −0.108612
\(430\) 0 0
\(431\) −5654.52 −0.631946 −0.315973 0.948768i \(-0.602331\pi\)
−0.315973 + 0.948768i \(0.602331\pi\)
\(432\) 950.586 0.105868
\(433\) −2479.97 −0.275242 −0.137621 0.990485i \(-0.543946\pi\)
−0.137621 + 0.990485i \(0.543946\pi\)
\(434\) 1563.91 0.172973
\(435\) 950.243 0.104737
\(436\) 6650.40 0.730496
\(437\) 25.1541 0.00275351
\(438\) 7241.21 0.789951
\(439\) −6930.24 −0.753445 −0.376723 0.926326i \(-0.622949\pi\)
−0.376723 + 0.926326i \(0.622949\pi\)
\(440\) 221.806 0.0240322
\(441\) −3636.42 −0.392659
\(442\) 2520.97 0.271290
\(443\) −14683.8 −1.57482 −0.787412 0.616427i \(-0.788580\pi\)
−0.787412 + 0.616427i \(0.788580\pi\)
\(444\) 5039.00 0.538604
\(445\) −2488.46 −0.265088
\(446\) 9860.99 1.04693
\(447\) 6971.84 0.737711
\(448\) −1110.16 −0.117077
\(449\) −12567.9 −1.32097 −0.660487 0.750837i \(-0.729650\pi\)
−0.660487 + 0.750837i \(0.729650\pi\)
\(450\) 2502.18 0.262120
\(451\) 1721.55 0.179745
\(452\) −5502.21 −0.572571
\(453\) −8349.48 −0.865988
\(454\) 12225.7 1.26383
\(455\) 163.396 0.0168354
\(456\) −335.672 −0.0344721
\(457\) −3586.37 −0.367097 −0.183549 0.983011i \(-0.558758\pi\)
−0.183549 + 0.983011i \(0.558758\pi\)
\(458\) 11322.2 1.15514
\(459\) −4215.18 −0.428645
\(460\) 89.5944 0.00908122
\(461\) 11289.0 1.14053 0.570263 0.821462i \(-0.306841\pi\)
0.570263 + 0.821462i \(0.306841\pi\)
\(462\) 162.711 0.0163853
\(463\) −16287.2 −1.63484 −0.817419 0.576044i \(-0.804596\pi\)
−0.817419 + 0.576044i \(0.804596\pi\)
\(464\) −785.999 −0.0786403
\(465\) 3427.87 0.341858
\(466\) −3095.03 −0.307671
\(467\) 544.522 0.0539561 0.0269780 0.999636i \(-0.491412\pi\)
0.0269780 + 0.999636i \(0.491412\pi\)
\(468\) 1490.81 0.147249
\(469\) −551.202 −0.0542689
\(470\) 715.501 0.0702205
\(471\) 1620.94 0.158575
\(472\) 14449.7 1.40912
\(473\) 0 0
\(474\) −682.395 −0.0661254
\(475\) 281.580 0.0271995
\(476\) 517.528 0.0498337
\(477\) −5428.40 −0.521068
\(478\) −3397.04 −0.325056
\(479\) −3992.32 −0.380822 −0.190411 0.981704i \(-0.560982\pi\)
−0.190411 + 0.981704i \(0.560982\pi\)
\(480\) −1967.42 −0.187083
\(481\) 5835.31 0.553154
\(482\) 9371.09 0.885563
\(483\) 185.425 0.0174682
\(484\) 5736.08 0.538700
\(485\) 1921.39 0.179888
\(486\) 5468.82 0.510434
\(487\) −9965.60 −0.927278 −0.463639 0.886024i \(-0.653457\pi\)
−0.463639 + 0.886024i \(0.653457\pi\)
\(488\) −15571.4 −1.44444
\(489\) −3704.60 −0.342593
\(490\) 1197.06 0.110362
\(491\) −15698.8 −1.44293 −0.721463 0.692452i \(-0.756530\pi\)
−0.721463 + 0.692452i \(0.756530\pi\)
\(492\) −9279.70 −0.850327
\(493\) 3485.36 0.318403
\(494\) −137.781 −0.0125487
\(495\) 102.205 0.00928037
\(496\) −2835.39 −0.256679
\(497\) −2489.80 −0.224714
\(498\) −2197.33 −0.197720
\(499\) 2241.60 0.201098 0.100549 0.994932i \(-0.467940\pi\)
0.100549 + 0.994932i \(0.467940\pi\)
\(500\) 2035.05 0.182020
\(501\) −19071.2 −1.70068
\(502\) −11135.4 −0.990032
\(503\) 5388.26 0.477635 0.238818 0.971064i \(-0.423240\pi\)
0.238818 + 0.971064i \(0.423240\pi\)
\(504\) −709.117 −0.0626718
\(505\) −1001.47 −0.0882472
\(506\) 103.318 0.00907718
\(507\) −7491.54 −0.656235
\(508\) −6103.45 −0.533065
\(509\) −3055.68 −0.266092 −0.133046 0.991110i \(-0.542476\pi\)
−0.133046 + 0.991110i \(0.542476\pi\)
\(510\) −931.606 −0.0808867
\(511\) 1721.46 0.149027
\(512\) 3428.51 0.295938
\(513\) 230.377 0.0198273
\(514\) 1951.10 0.167430
\(515\) −2234.96 −0.191232
\(516\) 0 0
\(517\) −1004.66 −0.0854642
\(518\) −983.821 −0.0834491
\(519\) −4880.82 −0.412802
\(520\) −1384.54 −0.116762
\(521\) −13025.1 −1.09528 −0.547638 0.836716i \(-0.684473\pi\)
−0.547638 + 0.836716i \(0.684473\pi\)
\(522\) −1692.73 −0.141932
\(523\) −2116.73 −0.176976 −0.0884878 0.996077i \(-0.528203\pi\)
−0.0884878 + 0.996077i \(0.528203\pi\)
\(524\) 3521.88 0.293615
\(525\) 2075.68 0.172553
\(526\) −15912.4 −1.31904
\(527\) 12572.9 1.03925
\(528\) −294.997 −0.0243145
\(529\) −12049.3 −0.990323
\(530\) 1786.95 0.146453
\(531\) 6658.24 0.544149
\(532\) −28.2850 −0.00230510
\(533\) −10746.2 −0.873298
\(534\) 15468.2 1.25351
\(535\) 1911.45 0.154466
\(536\) 4670.63 0.376382
\(537\) 11274.8 0.906044
\(538\) 5418.89 0.434247
\(539\) −1680.83 −0.134320
\(540\) 820.561 0.0653914
\(541\) −346.282 −0.0275191 −0.0137596 0.999905i \(-0.504380\pi\)
−0.0137596 + 0.999905i \(0.504380\pi\)
\(542\) 7770.24 0.615794
\(543\) −22150.1 −1.75056
\(544\) −7216.22 −0.568737
\(545\) −2845.98 −0.223685
\(546\) −1015.66 −0.0796087
\(547\) 21019.5 1.64301 0.821507 0.570198i \(-0.193133\pi\)
0.821507 + 0.570198i \(0.193133\pi\)
\(548\) −709.935 −0.0553411
\(549\) −7175.10 −0.557789
\(550\) 1156.56 0.0896653
\(551\) −190.489 −0.0147280
\(552\) −1571.21 −0.121151
\(553\) −162.227 −0.0124748
\(554\) −11007.6 −0.844166
\(555\) −2156.40 −0.164926
\(556\) 10083.5 0.769132
\(557\) 16058.5 1.22158 0.610790 0.791792i \(-0.290852\pi\)
0.610790 + 0.791792i \(0.290852\pi\)
\(558\) −6106.28 −0.463261
\(559\) 0 0
\(560\) 49.9452 0.00376887
\(561\) 1308.10 0.0984459
\(562\) −9365.02 −0.702917
\(563\) −9252.27 −0.692605 −0.346302 0.938123i \(-0.612563\pi\)
−0.346302 + 0.938123i \(0.612563\pi\)
\(564\) 5415.44 0.404310
\(565\) 2354.63 0.175327
\(566\) −6351.35 −0.471673
\(567\) 2511.67 0.186032
\(568\) 21097.4 1.55850
\(569\) 11700.9 0.862090 0.431045 0.902330i \(-0.358145\pi\)
0.431045 + 0.902330i \(0.358145\pi\)
\(570\) 50.9161 0.00374148
\(571\) 21685.1 1.58931 0.794653 0.607064i \(-0.207653\pi\)
0.794653 + 0.607064i \(0.207653\pi\)
\(572\) 689.082 0.0503706
\(573\) 10571.0 0.770701
\(574\) 1811.78 0.131746
\(575\) 1318.02 0.0955915
\(576\) 4334.63 0.313559
\(577\) −6485.51 −0.467929 −0.233965 0.972245i \(-0.575170\pi\)
−0.233965 + 0.972245i \(0.575170\pi\)
\(578\) 5914.42 0.425618
\(579\) −5554.64 −0.398692
\(580\) −678.487 −0.0485735
\(581\) −522.373 −0.0373007
\(582\) −11943.3 −0.850628
\(583\) −2509.12 −0.178246
\(584\) −14586.9 −1.03358
\(585\) −637.978 −0.0450892
\(586\) −7200.55 −0.507597
\(587\) 22400.2 1.57505 0.787524 0.616284i \(-0.211363\pi\)
0.787524 + 0.616284i \(0.211363\pi\)
\(588\) 9060.19 0.635435
\(589\) −687.163 −0.0480714
\(590\) −2191.80 −0.152940
\(591\) −15774.5 −1.09793
\(592\) 1783.68 0.123832
\(593\) 1760.43 0.121909 0.0609546 0.998141i \(-0.480586\pi\)
0.0609546 + 0.998141i \(0.480586\pi\)
\(594\) 946.251 0.0653622
\(595\) −221.472 −0.0152596
\(596\) −4977.99 −0.342125
\(597\) 25176.8 1.72600
\(598\) −644.925 −0.0441019
\(599\) −8806.31 −0.600695 −0.300347 0.953830i \(-0.597103\pi\)
−0.300347 + 0.953830i \(0.597103\pi\)
\(600\) −17588.4 −1.19674
\(601\) 3892.88 0.264216 0.132108 0.991235i \(-0.457825\pi\)
0.132108 + 0.991235i \(0.457825\pi\)
\(602\) 0 0
\(603\) 2152.17 0.145345
\(604\) 5961.65 0.401616
\(605\) −2454.71 −0.164956
\(606\) 6225.10 0.417290
\(607\) 14131.4 0.944938 0.472469 0.881347i \(-0.343363\pi\)
0.472469 + 0.881347i \(0.343363\pi\)
\(608\) 394.396 0.0263074
\(609\) −1404.20 −0.0934338
\(610\) 2361.94 0.156774
\(611\) 6271.24 0.415232
\(612\) −2020.69 −0.133466
\(613\) 21940.2 1.44561 0.722804 0.691053i \(-0.242853\pi\)
0.722804 + 0.691053i \(0.242853\pi\)
\(614\) 8532.89 0.560846
\(615\) 3971.17 0.260379
\(616\) −327.769 −0.0214386
\(617\) 14064.3 0.917678 0.458839 0.888519i \(-0.348265\pi\)
0.458839 + 0.888519i \(0.348265\pi\)
\(618\) 13892.5 0.904267
\(619\) −12577.8 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(620\) −2447.55 −0.158542
\(621\) 1078.35 0.0696822
\(622\) −9852.10 −0.635102
\(623\) 3677.27 0.236480
\(624\) 1841.41 0.118133
\(625\) 14312.4 0.915996
\(626\) 9679.43 0.618000
\(627\) −71.4932 −0.00455369
\(628\) −1157.38 −0.0735419
\(629\) −7909.37 −0.501378
\(630\) 107.562 0.00680217
\(631\) 4073.63 0.257003 0.128501 0.991709i \(-0.458983\pi\)
0.128501 + 0.991709i \(0.458983\pi\)
\(632\) 1374.64 0.0865191
\(633\) −9590.55 −0.602196
\(634\) 13272.5 0.831419
\(635\) 2611.92 0.163230
\(636\) 13524.9 0.843237
\(637\) 10492.0 0.652601
\(638\) −782.415 −0.0485519
\(639\) 9721.42 0.601836
\(640\) 1131.56 0.0698889
\(641\) 25739.7 1.58605 0.793023 0.609192i \(-0.208506\pi\)
0.793023 + 0.609192i \(0.208506\pi\)
\(642\) −11881.5 −0.730415
\(643\) −6453.80 −0.395821 −0.197911 0.980220i \(-0.563416\pi\)
−0.197911 + 0.980220i \(0.563416\pi\)
\(644\) −132.396 −0.00810116
\(645\) 0 0
\(646\) 186.753 0.0113742
\(647\) 3686.03 0.223977 0.111988 0.993710i \(-0.464278\pi\)
0.111988 + 0.993710i \(0.464278\pi\)
\(648\) −21282.8 −1.29023
\(649\) 3077.58 0.186141
\(650\) −7219.41 −0.435644
\(651\) −5065.47 −0.304964
\(652\) 2645.14 0.158883
\(653\) −4652.32 −0.278804 −0.139402 0.990236i \(-0.544518\pi\)
−0.139402 + 0.990236i \(0.544518\pi\)
\(654\) 17690.5 1.05773
\(655\) −1507.16 −0.0899079
\(656\) −3284.78 −0.195502
\(657\) −6721.45 −0.399130
\(658\) −1057.32 −0.0626422
\(659\) 19533.7 1.15467 0.577334 0.816508i \(-0.304093\pi\)
0.577334 + 0.816508i \(0.304093\pi\)
\(660\) −254.646 −0.0150183
\(661\) 10301.4 0.606169 0.303085 0.952964i \(-0.401984\pi\)
0.303085 + 0.952964i \(0.401984\pi\)
\(662\) −5108.69 −0.299932
\(663\) −8165.35 −0.478305
\(664\) 4426.35 0.258698
\(665\) 12.1043 0.000705844 0
\(666\) 3841.33 0.223496
\(667\) −891.641 −0.0517608
\(668\) 13617.1 0.788716
\(669\) −31939.5 −1.84582
\(670\) −708.461 −0.0408511
\(671\) −3316.49 −0.190807
\(672\) 2907.32 0.166893
\(673\) 32709.1 1.87347 0.936734 0.350041i \(-0.113832\pi\)
0.936734 + 0.350041i \(0.113832\pi\)
\(674\) −18654.4 −1.06608
\(675\) 12071.2 0.688328
\(676\) 5349.07 0.304339
\(677\) −6223.46 −0.353304 −0.176652 0.984273i \(-0.556527\pi\)
−0.176652 + 0.984273i \(0.556527\pi\)
\(678\) −14636.3 −0.829060
\(679\) −2839.29 −0.160474
\(680\) 1876.65 0.105833
\(681\) −39598.7 −2.22823
\(682\) −2822.46 −0.158471
\(683\) −14250.9 −0.798380 −0.399190 0.916868i \(-0.630709\pi\)
−0.399190 + 0.916868i \(0.630709\pi\)
\(684\) 110.439 0.00617359
\(685\) 303.811 0.0169460
\(686\) −3578.56 −0.199169
\(687\) −36672.5 −2.03660
\(688\) 0 0
\(689\) 15662.3 0.866017
\(690\) 238.328 0.0131493
\(691\) 18569.4 1.02230 0.511152 0.859490i \(-0.329219\pi\)
0.511152 + 0.859490i \(0.329219\pi\)
\(692\) 3484.98 0.191444
\(693\) −151.032 −0.00827882
\(694\) 2270.98 0.124215
\(695\) −4315.17 −0.235516
\(696\) 11898.6 0.648009
\(697\) 14565.7 0.791557
\(698\) 4076.43 0.221053
\(699\) 10024.7 0.542447
\(700\) −1482.07 −0.0800242
\(701\) −3684.53 −0.198520 −0.0992602 0.995062i \(-0.531648\pi\)
−0.0992602 + 0.995062i \(0.531648\pi\)
\(702\) −5906.62 −0.317566
\(703\) 432.279 0.0231916
\(704\) 2003.56 0.107261
\(705\) −2317.49 −0.123804
\(706\) 22575.8 1.20347
\(707\) 1479.90 0.0787234
\(708\) −16589.1 −0.880589
\(709\) −28799.5 −1.52551 −0.762757 0.646685i \(-0.776155\pi\)
−0.762757 + 0.646685i \(0.776155\pi\)
\(710\) −3200.15 −0.169154
\(711\) 633.414 0.0334105
\(712\) −31159.5 −1.64010
\(713\) −3216.47 −0.168945
\(714\) 1376.66 0.0721573
\(715\) −294.887 −0.0154240
\(716\) −8050.40 −0.420192
\(717\) 11002.9 0.573099
\(718\) 12754.6 0.662947
\(719\) −3244.36 −0.168281 −0.0841406 0.996454i \(-0.526814\pi\)
−0.0841406 + 0.996454i \(0.526814\pi\)
\(720\) −195.011 −0.0100939
\(721\) 3302.67 0.170594
\(722\) 13017.3 0.670989
\(723\) −30352.8 −1.56132
\(724\) 15815.5 0.811848
\(725\) −9981.18 −0.511299
\(726\) 15258.4 0.780017
\(727\) 26113.0 1.33216 0.666079 0.745881i \(-0.267971\pi\)
0.666079 + 0.745881i \(0.267971\pi\)
\(728\) 2045.98 0.104161
\(729\) 6700.14 0.340403
\(730\) 2212.60 0.112181
\(731\) 0 0
\(732\) 17876.9 0.902662
\(733\) 8054.45 0.405863 0.202932 0.979193i \(-0.434953\pi\)
0.202932 + 0.979193i \(0.434953\pi\)
\(734\) 14708.9 0.739666
\(735\) −3877.24 −0.194577
\(736\) 1846.09 0.0924561
\(737\) 994.777 0.0497193
\(738\) −7074.10 −0.352847
\(739\) −19543.4 −0.972822 −0.486411 0.873730i \(-0.661694\pi\)
−0.486411 + 0.873730i \(0.661694\pi\)
\(740\) 1539.70 0.0764871
\(741\) 446.270 0.0221243
\(742\) −2640.63 −0.130648
\(743\) −6207.63 −0.306508 −0.153254 0.988187i \(-0.548975\pi\)
−0.153254 + 0.988187i \(0.548975\pi\)
\(744\) 42922.5 2.11507
\(745\) 2130.29 0.104762
\(746\) −107.635 −0.00528256
\(747\) 2039.60 0.0998999
\(748\) −934.005 −0.0456559
\(749\) −2824.61 −0.137796
\(750\) 5413.38 0.263558
\(751\) −11820.4 −0.574346 −0.287173 0.957879i \(-0.592715\pi\)
−0.287173 + 0.957879i \(0.592715\pi\)
\(752\) 1916.93 0.0929564
\(753\) 36067.2 1.74550
\(754\) 4883.94 0.235892
\(755\) −2551.24 −0.122979
\(756\) −1212.57 −0.0583342
\(757\) 23218.6 1.11479 0.557393 0.830249i \(-0.311802\pi\)
0.557393 + 0.830249i \(0.311802\pi\)
\(758\) −5629.52 −0.269754
\(759\) −334.645 −0.0160038
\(760\) −102.567 −0.00489538
\(761\) 14404.3 0.686144 0.343072 0.939309i \(-0.388532\pi\)
0.343072 + 0.939309i \(0.388532\pi\)
\(762\) −16235.6 −0.771857
\(763\) 4205.60 0.199545
\(764\) −7547.88 −0.357425
\(765\) 864.737 0.0408688
\(766\) 2137.08 0.100804
\(767\) −19210.7 −0.904377
\(768\) −26703.1 −1.25464
\(769\) 2924.07 0.137119 0.0685597 0.997647i \(-0.478160\pi\)
0.0685597 + 0.997647i \(0.478160\pi\)
\(770\) 49.7174 0.00232687
\(771\) −6319.56 −0.295192
\(772\) 3966.09 0.184900
\(773\) 1040.49 0.0484137 0.0242068 0.999707i \(-0.492294\pi\)
0.0242068 + 0.999707i \(0.492294\pi\)
\(774\) 0 0
\(775\) −36005.7 −1.66886
\(776\) 24058.9 1.11297
\(777\) 3186.58 0.147127
\(778\) 20989.5 0.967236
\(779\) −796.075 −0.0366141
\(780\) 1589.53 0.0729672
\(781\) 4493.45 0.205875
\(782\) 874.153 0.0399740
\(783\) −8166.19 −0.372715
\(784\) 3207.08 0.146095
\(785\) 495.290 0.0225193
\(786\) 9368.47 0.425143
\(787\) 35796.2 1.62134 0.810671 0.585501i \(-0.199102\pi\)
0.810671 + 0.585501i \(0.199102\pi\)
\(788\) 11263.2 0.509183
\(789\) 51540.1 2.32557
\(790\) −208.511 −0.00939047
\(791\) −3479.50 −0.156406
\(792\) 1279.77 0.0574177
\(793\) 20701.9 0.927046
\(794\) −12291.8 −0.549395
\(795\) −5787.89 −0.258208
\(796\) −17976.6 −0.800458
\(797\) 15694.6 0.697531 0.348766 0.937210i \(-0.386601\pi\)
0.348766 + 0.937210i \(0.386601\pi\)
\(798\) −75.2403 −0.00333769
\(799\) −8500.24 −0.376366
\(800\) 20665.4 0.913292
\(801\) −14357.9 −0.633348
\(802\) 5885.12 0.259116
\(803\) −3106.80 −0.136534
\(804\) −5362.15 −0.235210
\(805\) 56.6580 0.00248066
\(806\) 17618.1 0.769941
\(807\) −17551.7 −0.765611
\(808\) −12540.0 −0.545986
\(809\) −8749.05 −0.380223 −0.190111 0.981763i \(-0.560885\pi\)
−0.190111 + 0.981763i \(0.560885\pi\)
\(810\) 3228.26 0.140036
\(811\) 24508.1 1.06115 0.530577 0.847637i \(-0.321975\pi\)
0.530577 + 0.847637i \(0.321975\pi\)
\(812\) 1002.62 0.0433314
\(813\) −25167.6 −1.08569
\(814\) 1775.54 0.0764531
\(815\) −1131.97 −0.0486516
\(816\) −2495.90 −0.107076
\(817\) 0 0
\(818\) 24122.0 1.03106
\(819\) 942.760 0.0402231
\(820\) −2835.47 −0.120755
\(821\) 43792.7 1.86160 0.930801 0.365527i \(-0.119111\pi\)
0.930801 + 0.365527i \(0.119111\pi\)
\(822\) −1888.48 −0.0801317
\(823\) 39701.8 1.68155 0.840776 0.541384i \(-0.182099\pi\)
0.840776 + 0.541384i \(0.182099\pi\)
\(824\) −27985.4 −1.18315
\(825\) −3746.08 −0.158087
\(826\) 3238.88 0.136435
\(827\) 40351.2 1.69668 0.848338 0.529456i \(-0.177604\pi\)
0.848338 + 0.529456i \(0.177604\pi\)
\(828\) 516.942 0.0216968
\(829\) −22880.5 −0.958593 −0.479297 0.877653i \(-0.659108\pi\)
−0.479297 + 0.877653i \(0.659108\pi\)
\(830\) −671.408 −0.0280782
\(831\) 35653.4 1.48833
\(832\) −12506.5 −0.521135
\(833\) −14221.2 −0.591517
\(834\) 26823.0 1.11367
\(835\) −5827.34 −0.241513
\(836\) 51.0472 0.00211185
\(837\) −29458.4 −1.21653
\(838\) −15721.2 −0.648069
\(839\) 9266.09 0.381288 0.190644 0.981659i \(-0.438942\pi\)
0.190644 + 0.981659i \(0.438942\pi\)
\(840\) −756.078 −0.0310561
\(841\) −17636.7 −0.723142
\(842\) 12883.7 0.527320
\(843\) 30333.1 1.23930
\(844\) 6847.79 0.279278
\(845\) −2289.09 −0.0931919
\(846\) 4128.30 0.167770
\(847\) 3627.40 0.147153
\(848\) 4787.49 0.193872
\(849\) 20571.9 0.831596
\(850\) 9785.42 0.394867
\(851\) 2023.41 0.0815061
\(852\) −24221.1 −0.973944
\(853\) −21969.4 −0.881852 −0.440926 0.897544i \(-0.645350\pi\)
−0.440926 + 0.897544i \(0.645350\pi\)
\(854\) −3490.31 −0.139855
\(855\) −47.2614 −0.00189042
\(856\) 23934.5 0.955682
\(857\) −29898.7 −1.19174 −0.595869 0.803082i \(-0.703192\pi\)
−0.595869 + 0.803082i \(0.703192\pi\)
\(858\) 1833.01 0.0729346
\(859\) 7922.96 0.314701 0.157350 0.987543i \(-0.449705\pi\)
0.157350 + 0.987543i \(0.449705\pi\)
\(860\) 0 0
\(861\) −5868.32 −0.232279
\(862\) 10739.8 0.424361
\(863\) 48962.3 1.93128 0.965642 0.259877i \(-0.0836821\pi\)
0.965642 + 0.259877i \(0.0836821\pi\)
\(864\) 16907.6 0.665751
\(865\) −1491.37 −0.0586220
\(866\) 4710.28 0.184829
\(867\) −19156.7 −0.750397
\(868\) 3616.82 0.141432
\(869\) 292.778 0.0114290
\(870\) −1804.83 −0.0703326
\(871\) −6209.53 −0.241564
\(872\) −35636.3 −1.38394
\(873\) 11086.0 0.429788
\(874\) −47.7761 −0.00184903
\(875\) 1286.93 0.0497213
\(876\) 16746.6 0.645907
\(877\) −2093.85 −0.0806206 −0.0403103 0.999187i \(-0.512835\pi\)
−0.0403103 + 0.999187i \(0.512835\pi\)
\(878\) 13162.8 0.505950
\(879\) 23322.4 0.894933
\(880\) −90.1382 −0.00345291
\(881\) −48729.2 −1.86348 −0.931742 0.363121i \(-0.881711\pi\)
−0.931742 + 0.363121i \(0.881711\pi\)
\(882\) 6906.76 0.263677
\(883\) −12474.4 −0.475422 −0.237711 0.971336i \(-0.576397\pi\)
−0.237711 + 0.971336i \(0.576397\pi\)
\(884\) 5830.18 0.221821
\(885\) 7099.18 0.269645
\(886\) 27889.4 1.05752
\(887\) 24736.2 0.936371 0.468186 0.883630i \(-0.344908\pi\)
0.468186 + 0.883630i \(0.344908\pi\)
\(888\) −27001.6 −1.02040
\(889\) −3859.72 −0.145614
\(890\) 4726.41 0.178011
\(891\) −4532.92 −0.170436
\(892\) 22805.3 0.856029
\(893\) 464.573 0.0174091
\(894\) −13241.8 −0.495384
\(895\) 3445.10 0.128667
\(896\) −1672.14 −0.0623464
\(897\) 2088.90 0.0777551
\(898\) 23870.7 0.887054
\(899\) 24357.9 0.903651
\(900\) 5786.73 0.214323
\(901\) −21229.2 −0.784957
\(902\) −3269.80 −0.120701
\(903\) 0 0
\(904\) 29483.7 1.08475
\(905\) −6768.11 −0.248596
\(906\) 15858.4 0.581524
\(907\) −21133.9 −0.773693 −0.386846 0.922144i \(-0.626436\pi\)
−0.386846 + 0.922144i \(0.626436\pi\)
\(908\) 28274.0 1.03338
\(909\) −5778.28 −0.210840
\(910\) −310.343 −0.0113052
\(911\) 43910.9 1.59696 0.798480 0.602021i \(-0.205638\pi\)
0.798480 + 0.602021i \(0.205638\pi\)
\(912\) 136.411 0.00495289
\(913\) 942.749 0.0341735
\(914\) 6811.71 0.246511
\(915\) −7650.27 −0.276404
\(916\) 26184.7 0.944505
\(917\) 2227.18 0.0802049
\(918\) 8006.03 0.287841
\(919\) −24074.8 −0.864151 −0.432075 0.901837i \(-0.642219\pi\)
−0.432075 + 0.901837i \(0.642219\pi\)
\(920\) −480.094 −0.0172046
\(921\) −27637.8 −0.988814
\(922\) −21441.6 −0.765880
\(923\) −28048.7 −1.00025
\(924\) 376.298 0.0133975
\(925\) 22650.4 0.805126
\(926\) 30934.8 1.09782
\(927\) −12895.3 −0.456890
\(928\) −13980.2 −0.494528
\(929\) 4596.72 0.162340 0.0811699 0.996700i \(-0.474134\pi\)
0.0811699 + 0.996700i \(0.474134\pi\)
\(930\) −6510.66 −0.229562
\(931\) 777.245 0.0273611
\(932\) −7157.81 −0.251568
\(933\) 31910.7 1.11973
\(934\) −1034.23 −0.0362323
\(935\) 399.700 0.0139803
\(936\) −7988.52 −0.278967
\(937\) −42751.3 −1.49053 −0.745264 0.666769i \(-0.767677\pi\)
−0.745264 + 0.666769i \(0.767677\pi\)
\(938\) 1046.91 0.0364424
\(939\) −31351.5 −1.08958
\(940\) 1654.72 0.0574161
\(941\) −30236.3 −1.04747 −0.523737 0.851880i \(-0.675463\pi\)
−0.523737 + 0.851880i \(0.675463\pi\)
\(942\) −3078.70 −0.106486
\(943\) −3726.27 −0.128679
\(944\) −5872.13 −0.202459
\(945\) 518.909 0.0178625
\(946\) 0 0
\(947\) 24630.0 0.845160 0.422580 0.906326i \(-0.361125\pi\)
0.422580 + 0.906326i \(0.361125\pi\)
\(948\) −1578.16 −0.0540678
\(949\) 19393.0 0.663356
\(950\) −534.813 −0.0182649
\(951\) −42989.4 −1.46585
\(952\) −2773.19 −0.0944112
\(953\) −25475.3 −0.865925 −0.432963 0.901412i \(-0.642532\pi\)
−0.432963 + 0.901412i \(0.642532\pi\)
\(954\) 10310.3 0.349905
\(955\) 3230.05 0.109447
\(956\) −7856.25 −0.265784
\(957\) 2534.23 0.0856007
\(958\) 7582.75 0.255728
\(959\) −448.950 −0.0151172
\(960\) 4621.69 0.155380
\(961\) 58076.9 1.94948
\(962\) −11083.2 −0.371451
\(963\) 11028.7 0.369050
\(964\) 21672.3 0.724085
\(965\) −1697.26 −0.0566183
\(966\) −352.184 −0.0117302
\(967\) −39005.4 −1.29714 −0.648568 0.761157i \(-0.724632\pi\)
−0.648568 + 0.761157i \(0.724632\pi\)
\(968\) −30736.9 −1.02058
\(969\) −604.889 −0.0200535
\(970\) −3649.35 −0.120798
\(971\) 30577.5 1.01059 0.505293 0.862948i \(-0.331384\pi\)
0.505293 + 0.862948i \(0.331384\pi\)
\(972\) 12647.6 0.417359
\(973\) 6376.66 0.210099
\(974\) 18928.0 0.622681
\(975\) 23383.5 0.768073
\(976\) 6327.97 0.207534
\(977\) −22122.7 −0.724428 −0.362214 0.932095i \(-0.617979\pi\)
−0.362214 + 0.932095i \(0.617979\pi\)
\(978\) 7036.27 0.230056
\(979\) −6636.53 −0.216654
\(980\) 2768.40 0.0902381
\(981\) −16420.7 −0.534428
\(982\) 29817.2 0.968947
\(983\) 2783.02 0.0902998 0.0451499 0.998980i \(-0.485623\pi\)
0.0451499 + 0.998980i \(0.485623\pi\)
\(984\) 49725.5 1.61097
\(985\) −4820.01 −0.155917
\(986\) −6619.85 −0.213812
\(987\) 3424.63 0.110443
\(988\) −318.643 −0.0102605
\(989\) 0 0
\(990\) −194.122 −0.00623191
\(991\) 55111.8 1.76658 0.883291 0.468825i \(-0.155323\pi\)
0.883291 + 0.468825i \(0.155323\pi\)
\(992\) −50431.6 −1.61412
\(993\) 16546.9 0.528803
\(994\) 4728.96 0.150899
\(995\) 7692.96 0.245109
\(996\) −5081.70 −0.161667
\(997\) −25817.4 −0.820104 −0.410052 0.912062i \(-0.634489\pi\)
−0.410052 + 0.912062i \(0.634489\pi\)
\(998\) −4257.54 −0.135040
\(999\) 18531.7 0.586902
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.1.17 yes 50
43.42 odd 2 1849.4.a.i.1.34 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.34 50 43.42 odd 2
1849.4.a.j.1.17 yes 50 1.1 even 1 trivial