Properties

Label 1849.4.a.j.1.15
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 1.15
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +O(q^{10})\) \(q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +43.8722 q^{10} +5.04244 q^{11} -6.07358 q^{12} -48.6308 q^{13} +1.20206 q^{14} -45.4401 q^{15} -37.7429 q^{16} -6.35958 q^{17} +49.4825 q^{18} -123.762 q^{19} +46.7805 q^{20} -1.24502 q^{21} -11.8250 q^{22} +141.010 q^{23} +59.8111 q^{24} +224.993 q^{25} +114.044 q^{26} -116.831 q^{27} +1.28175 q^{28} +206.672 q^{29} +106.561 q^{30} +164.801 q^{31} -108.487 q^{32} +12.2476 q^{33} +14.9138 q^{34} +9.58954 q^{35} +52.7627 q^{36} +365.186 q^{37} +290.233 q^{38} -118.119 q^{39} -460.682 q^{40} -175.812 q^{41} +2.91969 q^{42} -12.6089 q^{44} +394.749 q^{45} -330.680 q^{46} +219.602 q^{47} -91.6737 q^{48} -342.737 q^{49} -527.630 q^{50} -15.4468 q^{51} +121.604 q^{52} +34.4084 q^{53} +273.980 q^{54} -94.3346 q^{55} -12.6223 q^{56} -300.605 q^{57} -484.665 q^{58} -771.285 q^{59} +113.625 q^{60} +432.289 q^{61} -386.473 q^{62} +10.8158 q^{63} +556.356 q^{64} +909.791 q^{65} -28.7217 q^{66} -644.829 q^{67} +15.9024 q^{68} +342.498 q^{69} -22.4883 q^{70} +919.149 q^{71} -519.593 q^{72} -98.7029 q^{73} -856.394 q^{74} +546.486 q^{75} +309.472 q^{76} -2.58469 q^{77} +277.001 q^{78} +411.930 q^{79} +706.098 q^{80} +285.940 q^{81} +412.295 q^{82} +271.229 q^{83} +3.11324 q^{84} +118.976 q^{85} +501.986 q^{87} +124.169 q^{88} -115.517 q^{89} -925.723 q^{90} +24.9276 q^{91} -352.601 q^{92} +400.285 q^{93} -514.986 q^{94} +2315.35 q^{95} -263.505 q^{96} +623.381 q^{97} +803.750 q^{98} -106.398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.34509 −0.829115 −0.414557 0.910023i \(-0.636064\pi\)
−0.414557 + 0.910023i \(0.636064\pi\)
\(3\) 2.42890 0.467442 0.233721 0.972304i \(-0.424910\pi\)
0.233721 + 0.972304i \(0.424910\pi\)
\(4\) −2.50055 −0.312569
\(5\) −18.7081 −1.67330 −0.836652 0.547735i \(-0.815490\pi\)
−0.836652 + 0.547735i \(0.815490\pi\)
\(6\) −5.69599 −0.387563
\(7\) −0.512587 −0.0276771 −0.0138386 0.999904i \(-0.504405\pi\)
−0.0138386 + 0.999904i \(0.504405\pi\)
\(8\) 24.6247 1.08827
\(9\) −21.1004 −0.781498
\(10\) 43.8722 1.38736
\(11\) 5.04244 0.138214 0.0691070 0.997609i \(-0.477985\pi\)
0.0691070 + 0.997609i \(0.477985\pi\)
\(12\) −6.07358 −0.146108
\(13\) −48.6308 −1.03752 −0.518760 0.854920i \(-0.673606\pi\)
−0.518760 + 0.854920i \(0.673606\pi\)
\(14\) 1.20206 0.0229475
\(15\) −45.4401 −0.782173
\(16\) −37.7429 −0.589732
\(17\) −6.35958 −0.0907308 −0.0453654 0.998970i \(-0.514445\pi\)
−0.0453654 + 0.998970i \(0.514445\pi\)
\(18\) 49.4825 0.647951
\(19\) −123.762 −1.49436 −0.747182 0.664620i \(-0.768594\pi\)
−0.747182 + 0.664620i \(0.768594\pi\)
\(20\) 46.7805 0.523022
\(21\) −1.24502 −0.0129375
\(22\) −11.8250 −0.114595
\(23\) 141.010 1.27837 0.639185 0.769053i \(-0.279272\pi\)
0.639185 + 0.769053i \(0.279272\pi\)
\(24\) 59.8111 0.508703
\(25\) 224.993 1.79995
\(26\) 114.044 0.860224
\(27\) −116.831 −0.832747
\(28\) 1.28175 0.00865100
\(29\) 206.672 1.32338 0.661690 0.749777i \(-0.269839\pi\)
0.661690 + 0.749777i \(0.269839\pi\)
\(30\) 106.561 0.648511
\(31\) 164.801 0.954809 0.477405 0.878684i \(-0.341578\pi\)
0.477405 + 0.878684i \(0.341578\pi\)
\(32\) −108.487 −0.599314
\(33\) 12.2476 0.0646071
\(34\) 14.9138 0.0752263
\(35\) 9.58954 0.0463122
\(36\) 52.7627 0.244272
\(37\) 365.186 1.62260 0.811300 0.584630i \(-0.198760\pi\)
0.811300 + 0.584630i \(0.198760\pi\)
\(38\) 290.233 1.23900
\(39\) −118.119 −0.484981
\(40\) −460.682 −1.82101
\(41\) −175.812 −0.669689 −0.334844 0.942273i \(-0.608684\pi\)
−0.334844 + 0.942273i \(0.608684\pi\)
\(42\) 2.91969 0.0107266
\(43\) 0 0
\(44\) −12.6089 −0.0432013
\(45\) 394.749 1.30768
\(46\) −330.680 −1.05992
\(47\) 219.602 0.681537 0.340768 0.940147i \(-0.389313\pi\)
0.340768 + 0.940147i \(0.389313\pi\)
\(48\) −91.6737 −0.275666
\(49\) −342.737 −0.999234
\(50\) −527.630 −1.49236
\(51\) −15.4468 −0.0424114
\(52\) 121.604 0.324296
\(53\) 34.4084 0.0891765 0.0445883 0.999005i \(-0.485802\pi\)
0.0445883 + 0.999005i \(0.485802\pi\)
\(54\) 273.980 0.690443
\(55\) −94.3346 −0.231274
\(56\) −12.6223 −0.0301202
\(57\) −300.605 −0.698529
\(58\) −484.665 −1.09723
\(59\) −771.285 −1.70191 −0.850956 0.525238i \(-0.823976\pi\)
−0.850956 + 0.525238i \(0.823976\pi\)
\(60\) 113.625 0.244483
\(61\) 432.289 0.907360 0.453680 0.891165i \(-0.350111\pi\)
0.453680 + 0.891165i \(0.350111\pi\)
\(62\) −386.473 −0.791647
\(63\) 10.8158 0.0216296
\(64\) 556.356 1.08663
\(65\) 909.791 1.73609
\(66\) −28.7217 −0.0535667
\(67\) −644.829 −1.17580 −0.587899 0.808935i \(-0.700045\pi\)
−0.587899 + 0.808935i \(0.700045\pi\)
\(68\) 15.9024 0.0283596
\(69\) 342.498 0.597564
\(70\) −22.4883 −0.0383982
\(71\) 919.149 1.53638 0.768190 0.640222i \(-0.221158\pi\)
0.768190 + 0.640222i \(0.221158\pi\)
\(72\) −519.593 −0.850481
\(73\) −98.7029 −0.158251 −0.0791253 0.996865i \(-0.525213\pi\)
−0.0791253 + 0.996865i \(0.525213\pi\)
\(74\) −856.394 −1.34532
\(75\) 546.486 0.841371
\(76\) 309.472 0.467091
\(77\) −2.58469 −0.00382536
\(78\) 277.001 0.402105
\(79\) 411.930 0.586655 0.293327 0.956012i \(-0.405237\pi\)
0.293327 + 0.956012i \(0.405237\pi\)
\(80\) 706.098 0.986802
\(81\) 285.940 0.392237
\(82\) 412.295 0.555249
\(83\) 271.229 0.358689 0.179345 0.983786i \(-0.442602\pi\)
0.179345 + 0.983786i \(0.442602\pi\)
\(84\) 3.11324 0.00404384
\(85\) 118.976 0.151820
\(86\) 0 0
\(87\) 501.986 0.618604
\(88\) 124.169 0.150414
\(89\) −115.517 −0.137582 −0.0687908 0.997631i \(-0.521914\pi\)
−0.0687908 + 0.997631i \(0.521914\pi\)
\(90\) −925.723 −1.08422
\(91\) 24.9276 0.0287156
\(92\) −352.601 −0.399578
\(93\) 400.285 0.446318
\(94\) −514.986 −0.565072
\(95\) 2315.35 2.50053
\(96\) −263.505 −0.280145
\(97\) 623.381 0.652523 0.326261 0.945280i \(-0.394211\pi\)
0.326261 + 0.945280i \(0.394211\pi\)
\(98\) 803.750 0.828480
\(99\) −106.398 −0.108014
\(100\) −562.606 −0.562606
\(101\) 134.852 0.132854 0.0664270 0.997791i \(-0.478840\pi\)
0.0664270 + 0.997791i \(0.478840\pi\)
\(102\) 36.2241 0.0351639
\(103\) −246.042 −0.235371 −0.117686 0.993051i \(-0.537547\pi\)
−0.117686 + 0.993051i \(0.537547\pi\)
\(104\) −1197.52 −1.12910
\(105\) 23.2920 0.0216483
\(106\) −80.6908 −0.0739376
\(107\) −615.063 −0.555704 −0.277852 0.960624i \(-0.589623\pi\)
−0.277852 + 0.960624i \(0.589623\pi\)
\(108\) 292.142 0.260291
\(109\) 1860.95 1.63529 0.817646 0.575721i \(-0.195279\pi\)
0.817646 + 0.575721i \(0.195279\pi\)
\(110\) 221.223 0.191753
\(111\) 887.001 0.758472
\(112\) 19.3465 0.0163221
\(113\) 2228.96 1.85560 0.927802 0.373072i \(-0.121696\pi\)
0.927802 + 0.373072i \(0.121696\pi\)
\(114\) 704.947 0.579161
\(115\) −2638.02 −2.13910
\(116\) −516.793 −0.413647
\(117\) 1026.13 0.810820
\(118\) 1808.73 1.41108
\(119\) 3.25984 0.00251117
\(120\) −1118.95 −0.851215
\(121\) −1305.57 −0.980897
\(122\) −1013.76 −0.752306
\(123\) −427.030 −0.313041
\(124\) −412.092 −0.298443
\(125\) −1870.68 −1.33855
\(126\) −25.3641 −0.0179334
\(127\) 453.295 0.316720 0.158360 0.987381i \(-0.449379\pi\)
0.158360 + 0.987381i \(0.449379\pi\)
\(128\) −436.806 −0.301629
\(129\) 0 0
\(130\) −2133.54 −1.43942
\(131\) 995.763 0.664124 0.332062 0.943258i \(-0.392256\pi\)
0.332062 + 0.943258i \(0.392256\pi\)
\(132\) −30.6257 −0.0201941
\(133\) 63.4388 0.0413597
\(134\) 1512.18 0.974871
\(135\) 2185.69 1.39344
\(136\) −156.603 −0.0987397
\(137\) 2491.26 1.55359 0.776797 0.629751i \(-0.216843\pi\)
0.776797 + 0.629751i \(0.216843\pi\)
\(138\) −803.189 −0.495449
\(139\) −1951.81 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(140\) −23.9791 −0.0144757
\(141\) 533.391 0.318579
\(142\) −2155.49 −1.27384
\(143\) −245.218 −0.143400
\(144\) 796.391 0.460875
\(145\) −3866.44 −2.21442
\(146\) 231.467 0.131208
\(147\) −832.475 −0.467084
\(148\) −913.165 −0.507174
\(149\) 2006.42 1.10317 0.551585 0.834119i \(-0.314023\pi\)
0.551585 + 0.834119i \(0.314023\pi\)
\(150\) −1281.56 −0.697593
\(151\) −3455.23 −1.86213 −0.931067 0.364847i \(-0.881121\pi\)
−0.931067 + 0.364847i \(0.881121\pi\)
\(152\) −3047.60 −1.62627
\(153\) 134.190 0.0709059
\(154\) 6.06134 0.00317167
\(155\) −3083.11 −1.59769
\(156\) 295.363 0.151590
\(157\) 380.367 0.193354 0.0966771 0.995316i \(-0.469179\pi\)
0.0966771 + 0.995316i \(0.469179\pi\)
\(158\) −966.013 −0.486404
\(159\) 83.5746 0.0416849
\(160\) 2029.59 1.00283
\(161\) −72.2797 −0.0353816
\(162\) −670.556 −0.325209
\(163\) 1067.78 0.513096 0.256548 0.966532i \(-0.417415\pi\)
0.256548 + 0.966532i \(0.417415\pi\)
\(164\) 439.627 0.209324
\(165\) −229.129 −0.108107
\(166\) −636.056 −0.297395
\(167\) −4232.47 −1.96119 −0.980593 0.196055i \(-0.937187\pi\)
−0.980593 + 0.196055i \(0.937187\pi\)
\(168\) −30.6584 −0.0140794
\(169\) 167.959 0.0764490
\(170\) −279.009 −0.125876
\(171\) 2611.43 1.16784
\(172\) 0 0
\(173\) −3617.39 −1.58974 −0.794869 0.606781i \(-0.792461\pi\)
−0.794869 + 0.606781i \(0.792461\pi\)
\(174\) −1177.20 −0.512894
\(175\) −115.329 −0.0498173
\(176\) −190.316 −0.0815093
\(177\) −1873.37 −0.795545
\(178\) 270.897 0.114071
\(179\) −643.127 −0.268545 −0.134272 0.990944i \(-0.542870\pi\)
−0.134272 + 0.990944i \(0.542870\pi\)
\(180\) −987.090 −0.408741
\(181\) 3174.93 1.30382 0.651909 0.758297i \(-0.273968\pi\)
0.651909 + 0.758297i \(0.273968\pi\)
\(182\) −58.4574 −0.0238085
\(183\) 1049.99 0.424138
\(184\) 3472.32 1.39121
\(185\) −6831.94 −2.71510
\(186\) −938.704 −0.370049
\(187\) −32.0678 −0.0125403
\(188\) −549.125 −0.213027
\(189\) 59.8862 0.0230480
\(190\) −5429.71 −2.07322
\(191\) 464.296 0.175892 0.0879458 0.996125i \(-0.471970\pi\)
0.0879458 + 0.996125i \(0.471970\pi\)
\(192\) 1351.33 0.507938
\(193\) −3898.46 −1.45397 −0.726987 0.686651i \(-0.759080\pi\)
−0.726987 + 0.686651i \(0.759080\pi\)
\(194\) −1461.88 −0.541017
\(195\) 2209.79 0.811521
\(196\) 857.031 0.312329
\(197\) 2153.97 0.779005 0.389503 0.921025i \(-0.372647\pi\)
0.389503 + 0.921025i \(0.372647\pi\)
\(198\) 249.512 0.0895560
\(199\) −3038.98 −1.08255 −0.541275 0.840845i \(-0.682058\pi\)
−0.541275 + 0.840845i \(0.682058\pi\)
\(200\) 5540.40 1.95883
\(201\) −1566.23 −0.549617
\(202\) −316.240 −0.110151
\(203\) −105.937 −0.0366274
\(204\) 38.6254 0.0132565
\(205\) 3289.11 1.12059
\(206\) 576.990 0.195150
\(207\) −2975.36 −0.999043
\(208\) 1835.47 0.611860
\(209\) −624.062 −0.206542
\(210\) −54.6220 −0.0179489
\(211\) 969.498 0.316318 0.158159 0.987414i \(-0.449444\pi\)
0.158159 + 0.987414i \(0.449444\pi\)
\(212\) −86.0398 −0.0278738
\(213\) 2232.52 0.718169
\(214\) 1442.38 0.460743
\(215\) 0 0
\(216\) −2876.94 −0.906254
\(217\) −84.4748 −0.0264264
\(218\) −4364.10 −1.35584
\(219\) −239.740 −0.0739730
\(220\) 235.888 0.0722890
\(221\) 309.272 0.0941351
\(222\) −2080.10 −0.628860
\(223\) 446.780 0.134164 0.0670821 0.997747i \(-0.478631\pi\)
0.0670821 + 0.997747i \(0.478631\pi\)
\(224\) 55.6093 0.0165873
\(225\) −4747.46 −1.40665
\(226\) −5227.13 −1.53851
\(227\) −552.010 −0.161402 −0.0807009 0.996738i \(-0.525716\pi\)
−0.0807009 + 0.996738i \(0.525716\pi\)
\(228\) 751.678 0.218338
\(229\) −393.795 −0.113636 −0.0568181 0.998385i \(-0.518096\pi\)
−0.0568181 + 0.998385i \(0.518096\pi\)
\(230\) 6186.40 1.77356
\(231\) −6.27796 −0.00178814
\(232\) 5089.25 1.44020
\(233\) 335.082 0.0942143 0.0471071 0.998890i \(-0.485000\pi\)
0.0471071 + 0.998890i \(0.485000\pi\)
\(234\) −2406.37 −0.672263
\(235\) −4108.34 −1.14042
\(236\) 1928.64 0.531964
\(237\) 1000.54 0.274227
\(238\) −7.64462 −0.00208205
\(239\) −5612.62 −1.51904 −0.759520 0.650485i \(-0.774566\pi\)
−0.759520 + 0.650485i \(0.774566\pi\)
\(240\) 1715.04 0.461273
\(241\) 2947.42 0.787800 0.393900 0.919153i \(-0.371126\pi\)
0.393900 + 0.919153i \(0.371126\pi\)
\(242\) 3061.69 0.813276
\(243\) 3848.96 1.01610
\(244\) −1080.96 −0.283612
\(245\) 6411.97 1.67202
\(246\) 1001.42 0.259547
\(247\) 6018.64 1.55043
\(248\) 4058.18 1.03909
\(249\) 658.788 0.167667
\(250\) 4386.93 1.10981
\(251\) 1443.00 0.362874 0.181437 0.983403i \(-0.441925\pi\)
0.181437 + 0.983403i \(0.441925\pi\)
\(252\) −27.0455 −0.00676073
\(253\) 711.032 0.176689
\(254\) −1063.02 −0.262597
\(255\) 288.980 0.0709672
\(256\) −3426.50 −0.836547
\(257\) −7394.73 −1.79483 −0.897414 0.441189i \(-0.854557\pi\)
−0.897414 + 0.441189i \(0.854557\pi\)
\(258\) 0 0
\(259\) −187.190 −0.0449089
\(260\) −2274.98 −0.542646
\(261\) −4360.87 −1.03422
\(262\) −2335.16 −0.550635
\(263\) −7763.59 −1.82024 −0.910121 0.414343i \(-0.864012\pi\)
−0.910121 + 0.414343i \(0.864012\pi\)
\(264\) 301.594 0.0703099
\(265\) −643.716 −0.149219
\(266\) −148.770 −0.0342919
\(267\) −280.579 −0.0643114
\(268\) 1612.43 0.367517
\(269\) 6410.90 1.45308 0.726541 0.687123i \(-0.241127\pi\)
0.726541 + 0.687123i \(0.241127\pi\)
\(270\) −5125.64 −1.15532
\(271\) 1239.58 0.277857 0.138929 0.990302i \(-0.455634\pi\)
0.138929 + 0.990302i \(0.455634\pi\)
\(272\) 240.029 0.0535069
\(273\) 60.5466 0.0134229
\(274\) −5842.22 −1.28811
\(275\) 1134.52 0.248778
\(276\) −856.433 −0.186780
\(277\) 1265.98 0.274605 0.137302 0.990529i \(-0.456157\pi\)
0.137302 + 0.990529i \(0.456157\pi\)
\(278\) 4577.18 0.987485
\(279\) −3477.37 −0.746181
\(280\) 236.140 0.0504002
\(281\) −2547.49 −0.540821 −0.270410 0.962745i \(-0.587159\pi\)
−0.270410 + 0.962745i \(0.587159\pi\)
\(282\) −1250.85 −0.264139
\(283\) −2198.22 −0.461733 −0.230866 0.972985i \(-0.574156\pi\)
−0.230866 + 0.972985i \(0.574156\pi\)
\(284\) −2298.38 −0.480224
\(285\) 5623.76 1.16885
\(286\) 575.059 0.118895
\(287\) 90.1191 0.0185351
\(288\) 2289.13 0.468363
\(289\) −4872.56 −0.991768
\(290\) 9067.16 1.83601
\(291\) 1514.13 0.305017
\(292\) 246.811 0.0494642
\(293\) 3476.05 0.693082 0.346541 0.938035i \(-0.387356\pi\)
0.346541 + 0.938035i \(0.387356\pi\)
\(294\) 1952.23 0.387266
\(295\) 14429.3 2.84781
\(296\) 8992.61 1.76583
\(297\) −589.115 −0.115097
\(298\) −4705.24 −0.914655
\(299\) −6857.41 −1.32634
\(300\) −1366.52 −0.262986
\(301\) 0 0
\(302\) 8102.82 1.54392
\(303\) 327.542 0.0621015
\(304\) 4671.13 0.881275
\(305\) −8087.31 −1.51829
\(306\) −314.688 −0.0587892
\(307\) 1861.37 0.346039 0.173019 0.984918i \(-0.444648\pi\)
0.173019 + 0.984918i \(0.444648\pi\)
\(308\) 6.46315 0.00119569
\(309\) −597.611 −0.110022
\(310\) 7230.17 1.32467
\(311\) −7750.28 −1.41311 −0.706556 0.707657i \(-0.749752\pi\)
−0.706556 + 0.707657i \(0.749752\pi\)
\(312\) −2908.66 −0.527790
\(313\) −1674.83 −0.302451 −0.151225 0.988499i \(-0.548322\pi\)
−0.151225 + 0.988499i \(0.548322\pi\)
\(314\) −891.996 −0.160313
\(315\) −202.344 −0.0361929
\(316\) −1030.05 −0.183370
\(317\) −4814.34 −0.852998 −0.426499 0.904488i \(-0.640253\pi\)
−0.426499 + 0.904488i \(0.640253\pi\)
\(318\) −195.990 −0.0345616
\(319\) 1042.13 0.182910
\(320\) −10408.4 −1.81827
\(321\) −1493.93 −0.259760
\(322\) 169.502 0.0293354
\(323\) 787.073 0.135585
\(324\) −715.008 −0.122601
\(325\) −10941.6 −1.86748
\(326\) −2504.03 −0.425415
\(327\) 4520.07 0.764404
\(328\) −4329.33 −0.728802
\(329\) −112.565 −0.0188630
\(330\) 537.329 0.0896333
\(331\) 10848.9 1.80154 0.900768 0.434301i \(-0.143005\pi\)
0.900768 + 0.434301i \(0.143005\pi\)
\(332\) −678.220 −0.112115
\(333\) −7705.58 −1.26806
\(334\) 9925.52 1.62605
\(335\) 12063.5 1.96747
\(336\) 46.9908 0.00762964
\(337\) −8725.22 −1.41037 −0.705183 0.709026i \(-0.749135\pi\)
−0.705183 + 0.709026i \(0.749135\pi\)
\(338\) −393.878 −0.0633850
\(339\) 5413.93 0.867388
\(340\) −297.504 −0.0474542
\(341\) 830.999 0.131968
\(342\) −6124.04 −0.968275
\(343\) 351.500 0.0553330
\(344\) 0 0
\(345\) −6407.49 −0.999907
\(346\) 8483.10 1.31808
\(347\) −5950.59 −0.920589 −0.460295 0.887766i \(-0.652256\pi\)
−0.460295 + 0.887766i \(0.652256\pi\)
\(348\) −1255.24 −0.193356
\(349\) −5162.99 −0.791887 −0.395944 0.918275i \(-0.629582\pi\)
−0.395944 + 0.918275i \(0.629582\pi\)
\(350\) 270.456 0.0413043
\(351\) 5681.60 0.863992
\(352\) −547.042 −0.0828336
\(353\) −6312.98 −0.951859 −0.475929 0.879484i \(-0.657888\pi\)
−0.475929 + 0.879484i \(0.657888\pi\)
\(354\) 4393.23 0.659598
\(355\) −17195.5 −2.57083
\(356\) 288.855 0.0430037
\(357\) 7.91783 0.00117383
\(358\) 1508.19 0.222655
\(359\) −2094.51 −0.307921 −0.153961 0.988077i \(-0.549203\pi\)
−0.153961 + 0.988077i \(0.549203\pi\)
\(360\) 9720.60 1.42311
\(361\) 8457.99 1.23312
\(362\) −7445.51 −1.08101
\(363\) −3171.11 −0.458513
\(364\) −62.3325 −0.00897559
\(365\) 1846.54 0.264801
\(366\) −2462.32 −0.351659
\(367\) 11071.4 1.57473 0.787363 0.616489i \(-0.211446\pi\)
0.787363 + 0.616489i \(0.211446\pi\)
\(368\) −5322.10 −0.753896
\(369\) 3709.71 0.523360
\(370\) 16021.5 2.25113
\(371\) −17.6373 −0.00246815
\(372\) −1000.93 −0.139505
\(373\) 1517.88 0.210705 0.105353 0.994435i \(-0.466403\pi\)
0.105353 + 0.994435i \(0.466403\pi\)
\(374\) 75.2019 0.0103973
\(375\) −4543.71 −0.625696
\(376\) 5407.64 0.741696
\(377\) −10050.6 −1.37303
\(378\) −140.439 −0.0191095
\(379\) 13541.0 1.83524 0.917618 0.397463i \(-0.130109\pi\)
0.917618 + 0.397463i \(0.130109\pi\)
\(380\) −5789.64 −0.781585
\(381\) 1101.01 0.148048
\(382\) −1088.82 −0.145834
\(383\) 1160.65 0.154847 0.0774235 0.996998i \(-0.475331\pi\)
0.0774235 + 0.996998i \(0.475331\pi\)
\(384\) −1060.96 −0.140994
\(385\) 48.3547 0.00640100
\(386\) 9142.24 1.20551
\(387\) 0 0
\(388\) −1558.79 −0.203958
\(389\) −2125.41 −0.277024 −0.138512 0.990361i \(-0.544232\pi\)
−0.138512 + 0.990361i \(0.544232\pi\)
\(390\) −5182.16 −0.672844
\(391\) −896.761 −0.115988
\(392\) −8439.82 −1.08744
\(393\) 2418.61 0.310440
\(394\) −5051.26 −0.645885
\(395\) −7706.43 −0.981652
\(396\) 266.053 0.0337618
\(397\) 1017.45 0.128626 0.0643128 0.997930i \(-0.479514\pi\)
0.0643128 + 0.997930i \(0.479514\pi\)
\(398\) 7126.69 0.897559
\(399\) 154.086 0.0193333
\(400\) −8491.89 −1.06149
\(401\) 4536.33 0.564922 0.282461 0.959279i \(-0.408849\pi\)
0.282461 + 0.959279i \(0.408849\pi\)
\(402\) 3672.94 0.455696
\(403\) −8014.40 −0.990634
\(404\) −337.203 −0.0415260
\(405\) −5349.40 −0.656331
\(406\) 248.433 0.0303683
\(407\) 1841.43 0.224266
\(408\) −380.373 −0.0461551
\(409\) −12621.0 −1.52584 −0.762918 0.646495i \(-0.776234\pi\)
−0.762918 + 0.646495i \(0.776234\pi\)
\(410\) −7713.27 −0.929100
\(411\) 6051.01 0.726215
\(412\) 615.239 0.0735696
\(413\) 395.351 0.0471040
\(414\) 6977.50 0.828322
\(415\) −5074.18 −0.600196
\(416\) 5275.84 0.621801
\(417\) −4740.76 −0.556729
\(418\) 1463.48 0.171247
\(419\) −7940.92 −0.925869 −0.462934 0.886392i \(-0.653203\pi\)
−0.462934 + 0.886392i \(0.653203\pi\)
\(420\) −58.2429 −0.00676657
\(421\) 3246.56 0.375838 0.187919 0.982185i \(-0.439826\pi\)
0.187919 + 0.982185i \(0.439826\pi\)
\(422\) −2273.56 −0.262264
\(423\) −4633.70 −0.532619
\(424\) 847.298 0.0970482
\(425\) −1430.86 −0.163311
\(426\) −5235.47 −0.595444
\(427\) −221.586 −0.0251131
\(428\) 1537.99 0.173696
\(429\) −595.611 −0.0670312
\(430\) 0 0
\(431\) −10024.0 −1.12028 −0.560141 0.828398i \(-0.689253\pi\)
−0.560141 + 0.828398i \(0.689253\pi\)
\(432\) 4409.55 0.491098
\(433\) 9624.64 1.06820 0.534100 0.845421i \(-0.320651\pi\)
0.534100 + 0.845421i \(0.320651\pi\)
\(434\) 198.101 0.0219105
\(435\) −9391.21 −1.03511
\(436\) −4653.40 −0.511141
\(437\) −17451.6 −1.91035
\(438\) 562.211 0.0613321
\(439\) −7330.44 −0.796954 −0.398477 0.917178i \(-0.630461\pi\)
−0.398477 + 0.917178i \(0.630461\pi\)
\(440\) −2322.96 −0.251689
\(441\) 7231.91 0.780899
\(442\) −725.270 −0.0780488
\(443\) −1082.51 −0.116098 −0.0580492 0.998314i \(-0.518488\pi\)
−0.0580492 + 0.998314i \(0.518488\pi\)
\(444\) −2217.99 −0.237074
\(445\) 2161.10 0.230216
\(446\) −1047.74 −0.111238
\(447\) 4873.39 0.515668
\(448\) −285.181 −0.0300749
\(449\) 5924.49 0.622703 0.311352 0.950295i \(-0.399218\pi\)
0.311352 + 0.950295i \(0.399218\pi\)
\(450\) 11133.2 1.16628
\(451\) −886.523 −0.0925604
\(452\) −5573.63 −0.580004
\(453\) −8392.40 −0.870440
\(454\) 1294.51 0.133821
\(455\) −466.347 −0.0480499
\(456\) −7402.33 −0.760188
\(457\) −9192.85 −0.940970 −0.470485 0.882408i \(-0.655921\pi\)
−0.470485 + 0.882408i \(0.655921\pi\)
\(458\) 923.485 0.0942175
\(459\) 742.997 0.0755559
\(460\) 6596.50 0.668616
\(461\) −1014.04 −0.102448 −0.0512242 0.998687i \(-0.516312\pi\)
−0.0512242 + 0.998687i \(0.516312\pi\)
\(462\) 14.7224 0.00148257
\(463\) 11579.2 1.16227 0.581134 0.813808i \(-0.302609\pi\)
0.581134 + 0.813808i \(0.302609\pi\)
\(464\) −7800.40 −0.780441
\(465\) −7488.57 −0.746826
\(466\) −785.797 −0.0781145
\(467\) −11479.0 −1.13744 −0.568721 0.822530i \(-0.692562\pi\)
−0.568721 + 0.822530i \(0.692562\pi\)
\(468\) −2565.89 −0.253437
\(469\) 330.531 0.0325427
\(470\) 9634.42 0.945538
\(471\) 923.875 0.0903820
\(472\) −18992.7 −1.85214
\(473\) 0 0
\(474\) −2346.35 −0.227366
\(475\) −27845.6 −2.68977
\(476\) −8.15139 −0.000784912 0
\(477\) −726.032 −0.0696913
\(478\) 13162.1 1.25946
\(479\) 11310.1 1.07886 0.539429 0.842031i \(-0.318640\pi\)
0.539429 + 0.842031i \(0.318640\pi\)
\(480\) 4929.68 0.468767
\(481\) −17759.3 −1.68348
\(482\) −6911.96 −0.653177
\(483\) −175.560 −0.0165389
\(484\) 3264.65 0.306597
\(485\) −11662.3 −1.09187
\(486\) −9026.17 −0.842460
\(487\) 13942.4 1.29731 0.648657 0.761081i \(-0.275331\pi\)
0.648657 + 0.761081i \(0.275331\pi\)
\(488\) 10645.0 0.987453
\(489\) 2593.52 0.239843
\(490\) −15036.6 −1.38630
\(491\) −6614.78 −0.607986 −0.303993 0.952674i \(-0.598320\pi\)
−0.303993 + 0.952674i \(0.598320\pi\)
\(492\) 1067.81 0.0978467
\(493\) −1314.35 −0.120071
\(494\) −14114.3 −1.28549
\(495\) 1990.50 0.180740
\(496\) −6220.06 −0.563082
\(497\) −471.144 −0.0425226
\(498\) −1544.92 −0.139015
\(499\) −1025.76 −0.0920229 −0.0460114 0.998941i \(-0.514651\pi\)
−0.0460114 + 0.998941i \(0.514651\pi\)
\(500\) 4677.74 0.418389
\(501\) −10280.2 −0.916741
\(502\) −3383.96 −0.300864
\(503\) −3411.78 −0.302433 −0.151217 0.988501i \(-0.548319\pi\)
−0.151217 + 0.988501i \(0.548319\pi\)
\(504\) 266.337 0.0235389
\(505\) −2522.82 −0.222305
\(506\) −1667.44 −0.146495
\(507\) 407.955 0.0357355
\(508\) −1133.49 −0.0989967
\(509\) −334.899 −0.0291634 −0.0145817 0.999894i \(-0.504642\pi\)
−0.0145817 + 0.999894i \(0.504642\pi\)
\(510\) −677.685 −0.0588400
\(511\) 50.5938 0.00437992
\(512\) 11529.9 0.995223
\(513\) 14459.2 1.24443
\(514\) 17341.3 1.48812
\(515\) 4602.98 0.393847
\(516\) 0 0
\(517\) 1107.33 0.0941979
\(518\) 438.977 0.0372346
\(519\) −8786.27 −0.743111
\(520\) 22403.4 1.88933
\(521\) 15973.0 1.34317 0.671585 0.740928i \(-0.265614\pi\)
0.671585 + 0.740928i \(0.265614\pi\)
\(522\) 10226.6 0.857486
\(523\) 4623.45 0.386557 0.193279 0.981144i \(-0.438088\pi\)
0.193279 + 0.981144i \(0.438088\pi\)
\(524\) −2489.95 −0.207584
\(525\) −280.122 −0.0232867
\(526\) 18206.3 1.50919
\(527\) −1048.06 −0.0866307
\(528\) −462.260 −0.0381009
\(529\) 7716.68 0.634230
\(530\) 1509.57 0.123720
\(531\) 16274.5 1.33004
\(532\) −158.632 −0.0129277
\(533\) 8549.89 0.694816
\(534\) 657.983 0.0533216
\(535\) 11506.7 0.929862
\(536\) −15878.8 −1.27959
\(537\) −1562.09 −0.125529
\(538\) −15034.1 −1.20477
\(539\) −1728.23 −0.138108
\(540\) −5465.42 −0.435545
\(541\) 5776.76 0.459080 0.229540 0.973299i \(-0.426278\pi\)
0.229540 + 0.973299i \(0.426278\pi\)
\(542\) −2906.93 −0.230376
\(543\) 7711.60 0.609459
\(544\) 689.934 0.0543763
\(545\) −34814.9 −2.73634
\(546\) −141.987 −0.0111291
\(547\) 20861.6 1.63067 0.815334 0.578990i \(-0.196553\pi\)
0.815334 + 0.578990i \(0.196553\pi\)
\(548\) −6229.51 −0.485605
\(549\) −9121.49 −0.709100
\(550\) −2660.54 −0.206265
\(551\) −25578.1 −1.97761
\(552\) 8433.93 0.650311
\(553\) −211.150 −0.0162369
\(554\) −2968.85 −0.227679
\(555\) −16594.1 −1.26915
\(556\) 4880.60 0.372273
\(557\) −4932.51 −0.375219 −0.187610 0.982244i \(-0.560074\pi\)
−0.187610 + 0.982244i \(0.560074\pi\)
\(558\) 8154.75 0.618670
\(559\) 0 0
\(560\) −361.937 −0.0273118
\(561\) −77.8895 −0.00586185
\(562\) 5974.10 0.448403
\(563\) −14078.4 −1.05388 −0.526939 0.849903i \(-0.676660\pi\)
−0.526939 + 0.849903i \(0.676660\pi\)
\(564\) −1333.77 −0.0995778
\(565\) −41699.7 −3.10499
\(566\) 5155.02 0.382830
\(567\) −146.569 −0.0108560
\(568\) 22633.8 1.67200
\(569\) −7165.14 −0.527905 −0.263953 0.964536i \(-0.585026\pi\)
−0.263953 + 0.964536i \(0.585026\pi\)
\(570\) −13188.2 −0.969112
\(571\) 15291.1 1.12069 0.560344 0.828260i \(-0.310669\pi\)
0.560344 + 0.828260i \(0.310669\pi\)
\(572\) 613.180 0.0448223
\(573\) 1127.73 0.0822192
\(574\) −211.337 −0.0153677
\(575\) 31726.2 2.30100
\(576\) −11739.4 −0.849201
\(577\) 10271.5 0.741089 0.370545 0.928815i \(-0.379171\pi\)
0.370545 + 0.928815i \(0.379171\pi\)
\(578\) 11426.6 0.822290
\(579\) −9468.97 −0.679649
\(580\) 9668.23 0.692157
\(581\) −139.028 −0.00992749
\(582\) −3550.77 −0.252894
\(583\) 173.502 0.0123254
\(584\) −2430.53 −0.172219
\(585\) −19197.0 −1.35675
\(586\) −8151.66 −0.574645
\(587\) 5869.33 0.412697 0.206349 0.978479i \(-0.433842\pi\)
0.206349 + 0.978479i \(0.433842\pi\)
\(588\) 2081.64 0.145996
\(589\) −20396.0 −1.42683
\(590\) −33838.0 −2.36117
\(591\) 5231.78 0.364140
\(592\) −13783.2 −0.956900
\(593\) −13637.1 −0.944367 −0.472184 0.881500i \(-0.656534\pi\)
−0.472184 + 0.881500i \(0.656534\pi\)
\(594\) 1381.53 0.0954289
\(595\) −60.9854 −0.00420195
\(596\) −5017.15 −0.344816
\(597\) −7381.38 −0.506030
\(598\) 16081.3 1.09968
\(599\) 2989.04 0.203888 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(600\) 13457.1 0.915639
\(601\) 3752.57 0.254693 0.127347 0.991858i \(-0.459354\pi\)
0.127347 + 0.991858i \(0.459354\pi\)
\(602\) 0 0
\(603\) 13606.2 0.918883
\(604\) 8639.96 0.582045
\(605\) 24424.8 1.64134
\(606\) −768.115 −0.0514893
\(607\) −9165.95 −0.612906 −0.306453 0.951886i \(-0.599142\pi\)
−0.306453 + 0.951886i \(0.599142\pi\)
\(608\) 13426.6 0.895593
\(609\) −257.312 −0.0171212
\(610\) 18965.5 1.25884
\(611\) −10679.4 −0.707108
\(612\) −335.548 −0.0221630
\(613\) 8403.46 0.553691 0.276846 0.960914i \(-0.410711\pi\)
0.276846 + 0.960914i \(0.410711\pi\)
\(614\) −4365.08 −0.286906
\(615\) 7988.93 0.523812
\(616\) −63.6474 −0.00416303
\(617\) −14381.6 −0.938380 −0.469190 0.883097i \(-0.655454\pi\)
−0.469190 + 0.883097i \(0.655454\pi\)
\(618\) 1401.45 0.0912212
\(619\) 20695.9 1.34384 0.671921 0.740623i \(-0.265470\pi\)
0.671921 + 0.740623i \(0.265470\pi\)
\(620\) 7709.47 0.499386
\(621\) −16474.3 −1.06456
\(622\) 18175.1 1.17163
\(623\) 59.2125 0.00380786
\(624\) 4458.17 0.286009
\(625\) 6872.81 0.439860
\(626\) 3927.64 0.250767
\(627\) −1515.79 −0.0965465
\(628\) −951.127 −0.0604365
\(629\) −2322.43 −0.147220
\(630\) 474.514 0.0300081
\(631\) 18686.0 1.17889 0.589444 0.807809i \(-0.299347\pi\)
0.589444 + 0.807809i \(0.299347\pi\)
\(632\) 10143.7 0.638439
\(633\) 2354.81 0.147860
\(634\) 11290.1 0.707233
\(635\) −8480.29 −0.529969
\(636\) −208.982 −0.0130294
\(637\) 16667.6 1.03673
\(638\) −2443.89 −0.151653
\(639\) −19394.5 −1.20068
\(640\) 8171.81 0.504717
\(641\) 2741.94 0.168955 0.0844774 0.996425i \(-0.473078\pi\)
0.0844774 + 0.996425i \(0.473078\pi\)
\(642\) 3503.39 0.215371
\(643\) −9202.37 −0.564395 −0.282198 0.959356i \(-0.591063\pi\)
−0.282198 + 0.959356i \(0.591063\pi\)
\(644\) 180.739 0.0110592
\(645\) 0 0
\(646\) −1845.76 −0.112415
\(647\) −8101.38 −0.492269 −0.246135 0.969236i \(-0.579161\pi\)
−0.246135 + 0.969236i \(0.579161\pi\)
\(648\) 7041.21 0.426859
\(649\) −3889.16 −0.235228
\(650\) 25659.1 1.54836
\(651\) −205.181 −0.0123528
\(652\) −2670.02 −0.160378
\(653\) −2590.36 −0.155235 −0.0776175 0.996983i \(-0.524731\pi\)
−0.0776175 + 0.996983i \(0.524731\pi\)
\(654\) −10600.0 −0.633779
\(655\) −18628.8 −1.11128
\(656\) 6635.66 0.394937
\(657\) 2082.67 0.123673
\(658\) 263.976 0.0156396
\(659\) −5952.63 −0.351869 −0.175934 0.984402i \(-0.556295\pi\)
−0.175934 + 0.984402i \(0.556295\pi\)
\(660\) 572.949 0.0337909
\(661\) −23326.7 −1.37262 −0.686311 0.727309i \(-0.740771\pi\)
−0.686311 + 0.727309i \(0.740771\pi\)
\(662\) −25441.6 −1.49368
\(663\) 751.190 0.0440027
\(664\) 6678.94 0.390351
\(665\) −1186.82 −0.0692073
\(666\) 18070.3 1.05137
\(667\) 29142.7 1.69177
\(668\) 10583.5 0.613005
\(669\) 1085.19 0.0627140
\(670\) −28290.1 −1.63126
\(671\) 2179.79 0.125410
\(672\) 135.069 0.00775360
\(673\) −20507.9 −1.17462 −0.587311 0.809362i \(-0.699813\pi\)
−0.587311 + 0.809362i \(0.699813\pi\)
\(674\) 20461.4 1.16935
\(675\) −26286.2 −1.49890
\(676\) −419.988 −0.0238956
\(677\) −29831.9 −1.69355 −0.846774 0.531953i \(-0.821458\pi\)
−0.846774 + 0.531953i \(0.821458\pi\)
\(678\) −12696.2 −0.719164
\(679\) −319.537 −0.0180600
\(680\) 2929.74 0.165221
\(681\) −1340.78 −0.0754460
\(682\) −1948.77 −0.109417
\(683\) −6248.79 −0.350078 −0.175039 0.984562i \(-0.556005\pi\)
−0.175039 + 0.984562i \(0.556005\pi\)
\(684\) −6530.00 −0.365031
\(685\) −46606.7 −2.59964
\(686\) −824.300 −0.0458774
\(687\) −956.489 −0.0531184
\(688\) 0 0
\(689\) −1673.31 −0.0925225
\(690\) 15026.2 0.829037
\(691\) 17494.1 0.963107 0.481554 0.876417i \(-0.340073\pi\)
0.481554 + 0.876417i \(0.340073\pi\)
\(692\) 9045.45 0.496902
\(693\) 54.5382 0.00298951
\(694\) 13954.7 0.763274
\(695\) 36514.7 1.99292
\(696\) 12361.3 0.673208
\(697\) 1118.09 0.0607614
\(698\) 12107.7 0.656566
\(699\) 813.880 0.0440397
\(700\) 288.385 0.0155713
\(701\) 25115.2 1.35319 0.676596 0.736354i \(-0.263454\pi\)
0.676596 + 0.736354i \(0.263454\pi\)
\(702\) −13323.9 −0.716349
\(703\) −45196.1 −2.42476
\(704\) 2805.39 0.150188
\(705\) −9978.74 −0.533080
\(706\) 14804.5 0.789200
\(707\) −69.1233 −0.00367701
\(708\) 4684.46 0.248662
\(709\) 18394.0 0.974331 0.487165 0.873310i \(-0.338031\pi\)
0.487165 + 0.873310i \(0.338031\pi\)
\(710\) 40325.1 2.13151
\(711\) −8691.90 −0.458469
\(712\) −2844.57 −0.149726
\(713\) 23238.5 1.22060
\(714\) −18.5680 −0.000973237 0
\(715\) 4587.57 0.239952
\(716\) 1608.17 0.0839387
\(717\) −13632.5 −0.710063
\(718\) 4911.81 0.255302
\(719\) 30052.0 1.55876 0.779382 0.626549i \(-0.215533\pi\)
0.779382 + 0.626549i \(0.215533\pi\)
\(720\) −14899.0 −0.771183
\(721\) 126.118 0.00651439
\(722\) −19834.8 −1.02240
\(723\) 7158.99 0.368251
\(724\) −7939.07 −0.407532
\(725\) 46499.8 2.38201
\(726\) 7436.54 0.380160
\(727\) 4166.51 0.212555 0.106277 0.994337i \(-0.466107\pi\)
0.106277 + 0.994337i \(0.466107\pi\)
\(728\) 613.835 0.0312503
\(729\) 1628.36 0.0827293
\(730\) −4330.31 −0.219551
\(731\) 0 0
\(732\) −2625.54 −0.132572
\(733\) 23201.4 1.16912 0.584559 0.811351i \(-0.301267\pi\)
0.584559 + 0.811351i \(0.301267\pi\)
\(734\) −25963.5 −1.30563
\(735\) 15574.0 0.781574
\(736\) −15297.8 −0.766145
\(737\) −3251.51 −0.162512
\(738\) −8699.62 −0.433926
\(739\) 27055.3 1.34675 0.673373 0.739303i \(-0.264845\pi\)
0.673373 + 0.739303i \(0.264845\pi\)
\(740\) 17083.6 0.848656
\(741\) 14618.7 0.724738
\(742\) 41.3611 0.00204638
\(743\) 16610.7 0.820172 0.410086 0.912047i \(-0.365499\pi\)
0.410086 + 0.912047i \(0.365499\pi\)
\(744\) 9856.91 0.485715
\(745\) −37536.3 −1.84594
\(746\) −3559.58 −0.174699
\(747\) −5723.05 −0.280315
\(748\) 80.1871 0.00391969
\(749\) 315.273 0.0153803
\(750\) 10655.4 0.518774
\(751\) −5199.15 −0.252623 −0.126311 0.991991i \(-0.540314\pi\)
−0.126311 + 0.991991i \(0.540314\pi\)
\(752\) −8288.41 −0.401924
\(753\) 3504.90 0.169622
\(754\) 23569.7 1.13840
\(755\) 64640.7 3.11592
\(756\) −149.748 −0.00720409
\(757\) 30028.9 1.44177 0.720884 0.693056i \(-0.243736\pi\)
0.720884 + 0.693056i \(0.243736\pi\)
\(758\) −31754.9 −1.52162
\(759\) 1727.03 0.0825917
\(760\) 57014.9 2.72125
\(761\) −13045.8 −0.621433 −0.310716 0.950503i \(-0.600569\pi\)
−0.310716 + 0.950503i \(0.600569\pi\)
\(762\) −2581.97 −0.122749
\(763\) −953.900 −0.0452602
\(764\) −1161.00 −0.0549782
\(765\) −2510.44 −0.118647
\(766\) −2721.83 −0.128386
\(767\) 37508.2 1.76577
\(768\) −8322.63 −0.391038
\(769\) 18935.3 0.887938 0.443969 0.896042i \(-0.353570\pi\)
0.443969 + 0.896042i \(0.353570\pi\)
\(770\) −113.396 −0.00530716
\(771\) −17961.1 −0.838978
\(772\) 9748.28 0.454467
\(773\) 7052.59 0.328155 0.164078 0.986447i \(-0.447535\pi\)
0.164078 + 0.986447i \(0.447535\pi\)
\(774\) 0 0
\(775\) 37079.1 1.71861
\(776\) 15350.6 0.710121
\(777\) −454.665 −0.0209923
\(778\) 4984.28 0.229685
\(779\) 21758.8 1.00076
\(780\) −5525.69 −0.253656
\(781\) 4634.76 0.212349
\(782\) 2102.99 0.0961670
\(783\) −24145.7 −1.10204
\(784\) 12935.9 0.589281
\(785\) −7115.95 −0.323540
\(786\) −5671.86 −0.257390
\(787\) 31343.9 1.41968 0.709841 0.704362i \(-0.248767\pi\)
0.709841 + 0.704362i \(0.248767\pi\)
\(788\) −5386.11 −0.243492
\(789\) −18857.0 −0.850858
\(790\) 18072.3 0.813902
\(791\) −1142.54 −0.0513578
\(792\) −2620.02 −0.117548
\(793\) −21022.6 −0.941405
\(794\) −2386.01 −0.106645
\(795\) −1563.52 −0.0697515
\(796\) 7599.12 0.338371
\(797\) 35516.9 1.57851 0.789255 0.614066i \(-0.210467\pi\)
0.789255 + 0.614066i \(0.210467\pi\)
\(798\) −361.347 −0.0160295
\(799\) −1396.58 −0.0618364
\(800\) −24408.9 −1.07873
\(801\) 2437.46 0.107520
\(802\) −10638.1 −0.468385
\(803\) −497.704 −0.0218725
\(804\) 3916.42 0.171793
\(805\) 1352.22 0.0592042
\(806\) 18794.5 0.821350
\(807\) 15571.4 0.679232
\(808\) 3320.69 0.144581
\(809\) −18244.6 −0.792886 −0.396443 0.918059i \(-0.629756\pi\)
−0.396443 + 0.918059i \(0.629756\pi\)
\(810\) 12544.8 0.544174
\(811\) −26372.1 −1.14186 −0.570930 0.820999i \(-0.693417\pi\)
−0.570930 + 0.820999i \(0.693417\pi\)
\(812\) 264.902 0.0114486
\(813\) 3010.82 0.129882
\(814\) −4318.32 −0.185942
\(815\) −19976.1 −0.858565
\(816\) 583.006 0.0250114
\(817\) 0 0
\(818\) 29597.3 1.26509
\(819\) −525.982 −0.0224412
\(820\) −8224.58 −0.350262
\(821\) −3948.12 −0.167832 −0.0839162 0.996473i \(-0.526743\pi\)
−0.0839162 + 0.996473i \(0.526743\pi\)
\(822\) −14190.2 −0.602116
\(823\) −36600.5 −1.55020 −0.775100 0.631839i \(-0.782301\pi\)
−0.775100 + 0.631839i \(0.782301\pi\)
\(824\) −6058.71 −0.256147
\(825\) 2755.63 0.116289
\(826\) −927.134 −0.0390546
\(827\) 8511.44 0.357886 0.178943 0.983859i \(-0.442732\pi\)
0.178943 + 0.983859i \(0.442732\pi\)
\(828\) 7440.04 0.312270
\(829\) −20202.2 −0.846382 −0.423191 0.906041i \(-0.639090\pi\)
−0.423191 + 0.906041i \(0.639090\pi\)
\(830\) 11899.4 0.497632
\(831\) 3074.95 0.128362
\(832\) −27056.1 −1.12740
\(833\) 2179.66 0.0906613
\(834\) 11117.5 0.461592
\(835\) 79181.4 3.28166
\(836\) 1560.50 0.0645585
\(837\) −19253.9 −0.795115
\(838\) 18622.2 0.767652
\(839\) 5893.79 0.242522 0.121261 0.992621i \(-0.461306\pi\)
0.121261 + 0.992621i \(0.461306\pi\)
\(840\) 573.561 0.0235592
\(841\) 18324.3 0.751336
\(842\) −7613.48 −0.311613
\(843\) −6187.61 −0.252803
\(844\) −2424.28 −0.0988709
\(845\) −3142.19 −0.127922
\(846\) 10866.4 0.441603
\(847\) 669.221 0.0271484
\(848\) −1298.67 −0.0525903
\(849\) −5339.25 −0.215833
\(850\) 3355.50 0.135403
\(851\) 51494.7 2.07428
\(852\) −5582.53 −0.224477
\(853\) 29634.1 1.18951 0.594756 0.803906i \(-0.297249\pi\)
0.594756 + 0.803906i \(0.297249\pi\)
\(854\) 519.639 0.0208217
\(855\) −48854.9 −1.95415
\(856\) −15145.8 −0.604756
\(857\) 18851.6 0.751411 0.375705 0.926739i \(-0.377401\pi\)
0.375705 + 0.926739i \(0.377401\pi\)
\(858\) 1396.76 0.0555765
\(859\) 8874.89 0.352511 0.176256 0.984344i \(-0.443601\pi\)
0.176256 + 0.984344i \(0.443601\pi\)
\(860\) 0 0
\(861\) 218.890 0.00866407
\(862\) 23507.3 0.928842
\(863\) −34956.5 −1.37883 −0.689417 0.724365i \(-0.742133\pi\)
−0.689417 + 0.724365i \(0.742133\pi\)
\(864\) 12674.7 0.499077
\(865\) 67674.5 2.66012
\(866\) −22570.7 −0.885660
\(867\) −11835.0 −0.463594
\(868\) 211.233 0.00826005
\(869\) 2077.13 0.0810839
\(870\) 22023.2 0.858227
\(871\) 31358.6 1.21991
\(872\) 45825.4 1.77964
\(873\) −13153.6 −0.509945
\(874\) 40925.6 1.58390
\(875\) 958.889 0.0370473
\(876\) 599.480 0.0231216
\(877\) −21331.6 −0.821341 −0.410670 0.911784i \(-0.634705\pi\)
−0.410670 + 0.911784i \(0.634705\pi\)
\(878\) 17190.6 0.660767
\(879\) 8442.98 0.323976
\(880\) 3560.46 0.136390
\(881\) 28864.1 1.10381 0.551905 0.833907i \(-0.313901\pi\)
0.551905 + 0.833907i \(0.313901\pi\)
\(882\) −16959.5 −0.647455
\(883\) −14623.5 −0.557327 −0.278663 0.960389i \(-0.589891\pi\)
−0.278663 + 0.960389i \(0.589891\pi\)
\(884\) −773.349 −0.0294237
\(885\) 35047.3 1.33119
\(886\) 2538.58 0.0962589
\(887\) −31615.5 −1.19678 −0.598390 0.801205i \(-0.704193\pi\)
−0.598390 + 0.801205i \(0.704193\pi\)
\(888\) 21842.2 0.825422
\(889\) −232.353 −0.00876589
\(890\) −5067.98 −0.190875
\(891\) 1441.84 0.0542126
\(892\) −1117.20 −0.0419355
\(893\) −27178.3 −1.01846
\(894\) −11428.6 −0.427548
\(895\) 12031.7 0.449357
\(896\) 223.901 0.00834823
\(897\) −16656.0 −0.619985
\(898\) −13893.5 −0.516293
\(899\) 34059.7 1.26358
\(900\) 11871.2 0.439676
\(901\) −218.823 −0.00809106
\(902\) 2078.98 0.0767432
\(903\) 0 0
\(904\) 54887.7 2.01940
\(905\) −59397.0 −2.18168
\(906\) 19681.0 0.721695
\(907\) −32362.1 −1.18475 −0.592373 0.805664i \(-0.701809\pi\)
−0.592373 + 0.805664i \(0.701809\pi\)
\(908\) 1380.33 0.0504491
\(909\) −2845.43 −0.103825
\(910\) 1093.63 0.0398389
\(911\) 35154.1 1.27849 0.639247 0.769002i \(-0.279246\pi\)
0.639247 + 0.769002i \(0.279246\pi\)
\(912\) 11345.7 0.411945
\(913\) 1367.66 0.0495759
\(914\) 21558.1 0.780172
\(915\) −19643.3 −0.709712
\(916\) 984.703 0.0355191
\(917\) −510.416 −0.0183810
\(918\) −1742.40 −0.0626445
\(919\) −7359.98 −0.264182 −0.132091 0.991238i \(-0.542169\pi\)
−0.132091 + 0.991238i \(0.542169\pi\)
\(920\) −64960.6 −2.32792
\(921\) 4521.08 0.161753
\(922\) 2378.02 0.0849415
\(923\) −44699.0 −1.59403
\(924\) 15.6983 0.000558915 0
\(925\) 82164.4 2.92059
\(926\) −27154.2 −0.963654
\(927\) 5191.59 0.183942
\(928\) −22421.3 −0.793121
\(929\) −26410.1 −0.932708 −0.466354 0.884598i \(-0.654433\pi\)
−0.466354 + 0.884598i \(0.654433\pi\)
\(930\) 17561.4 0.619205
\(931\) 42417.8 1.49322
\(932\) −837.888 −0.0294484
\(933\) −18824.7 −0.660548
\(934\) 26919.3 0.943070
\(935\) 599.928 0.0209837
\(936\) 25268.2 0.882391
\(937\) −26123.6 −0.910800 −0.455400 0.890287i \(-0.650504\pi\)
−0.455400 + 0.890287i \(0.650504\pi\)
\(938\) −775.126 −0.0269816
\(939\) −4068.00 −0.141378
\(940\) 10273.1 0.356459
\(941\) 6602.83 0.228742 0.114371 0.993438i \(-0.463515\pi\)
0.114371 + 0.993438i \(0.463515\pi\)
\(942\) −2166.57 −0.0749370
\(943\) −24791.2 −0.856110
\(944\) 29110.5 1.00367
\(945\) −1120.36 −0.0385664
\(946\) 0 0
\(947\) −9913.13 −0.340162 −0.170081 0.985430i \(-0.554403\pi\)
−0.170081 + 0.985430i \(0.554403\pi\)
\(948\) −2501.89 −0.0857148
\(949\) 4800.00 0.164188
\(950\) 65300.4 2.23013
\(951\) −11693.6 −0.398727
\(952\) 80.2727 0.00273283
\(953\) 12334.8 0.419268 0.209634 0.977780i \(-0.432773\pi\)
0.209634 + 0.977780i \(0.432773\pi\)
\(954\) 1702.61 0.0577821
\(955\) −8686.10 −0.294320
\(956\) 14034.6 0.474804
\(957\) 2531.24 0.0854997
\(958\) −26523.3 −0.894497
\(959\) −1276.99 −0.0429990
\(960\) −25280.9 −0.849935
\(961\) −2631.71 −0.0883389
\(962\) 41647.2 1.39580
\(963\) 12978.1 0.434282
\(964\) −7370.16 −0.246242
\(965\) 72932.7 2.43294
\(966\) 411.705 0.0137126
\(967\) −43542.0 −1.44800 −0.724000 0.689800i \(-0.757699\pi\)
−0.724000 + 0.689800i \(0.757699\pi\)
\(968\) −32149.4 −1.06748
\(969\) 1911.72 0.0633781
\(970\) 27349.1 0.905285
\(971\) −47860.8 −1.58180 −0.790900 0.611946i \(-0.790387\pi\)
−0.790900 + 0.611946i \(0.790387\pi\)
\(972\) −9624.52 −0.317599
\(973\) 1000.47 0.0329638
\(974\) −32696.3 −1.07562
\(975\) −26576.1 −0.872940
\(976\) −16315.8 −0.535100
\(977\) 2411.64 0.0789716 0.0394858 0.999220i \(-0.487428\pi\)
0.0394858 + 0.999220i \(0.487428\pi\)
\(978\) −6082.04 −0.198857
\(979\) −582.487 −0.0190157
\(980\) −16033.4 −0.522621
\(981\) −39266.9 −1.27798
\(982\) 15512.3 0.504090
\(983\) −15548.6 −0.504500 −0.252250 0.967662i \(-0.581171\pi\)
−0.252250 + 0.967662i \(0.581171\pi\)
\(984\) −10515.5 −0.340673
\(985\) −40296.7 −1.30351
\(986\) 3082.26 0.0995530
\(987\) −273.410 −0.00881735
\(988\) −15049.9 −0.484617
\(989\) 0 0
\(990\) −4667.91 −0.149854
\(991\) −679.692 −0.0217872 −0.0108936 0.999941i \(-0.503468\pi\)
−0.0108936 + 0.999941i \(0.503468\pi\)
\(992\) −17878.8 −0.572231
\(993\) 26350.8 0.842114
\(994\) 1104.88 0.0352561
\(995\) 56853.6 1.81144
\(996\) −1647.33 −0.0524073
\(997\) −42549.5 −1.35161 −0.675806 0.737080i \(-0.736204\pi\)
−0.675806 + 0.737080i \(0.736204\pi\)
\(998\) 2405.51 0.0762976
\(999\) −42665.1 −1.35122
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.1.15 yes 50
43.42 odd 2 1849.4.a.i.1.36 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.36 50 43.42 odd 2
1849.4.a.j.1.15 yes 50 1.1 even 1 trivial