Properties

Label 1849.4.a.j.1.19
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 1.19
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.67189 q^{2} -3.03153 q^{3} -5.20480 q^{4} +19.6184 q^{5} +5.06836 q^{6} -28.4637 q^{7} +22.0769 q^{8} -17.8099 q^{9} +O(q^{10})\) \(q-1.67189 q^{2} -3.03153 q^{3} -5.20480 q^{4} +19.6184 q^{5} +5.06836 q^{6} -28.4637 q^{7} +22.0769 q^{8} -17.8099 q^{9} -32.7997 q^{10} -28.1590 q^{11} +15.7785 q^{12} +20.8592 q^{13} +47.5880 q^{14} -59.4736 q^{15} +4.72834 q^{16} +24.2325 q^{17} +29.7760 q^{18} -118.903 q^{19} -102.110 q^{20} +86.2883 q^{21} +47.0785 q^{22} +61.3604 q^{23} -66.9267 q^{24} +259.881 q^{25} -34.8741 q^{26} +135.842 q^{27} +148.148 q^{28} -70.4167 q^{29} +99.4331 q^{30} +263.721 q^{31} -184.521 q^{32} +85.3646 q^{33} -40.5139 q^{34} -558.411 q^{35} +92.6967 q^{36} +360.375 q^{37} +198.791 q^{38} -63.2350 q^{39} +433.113 q^{40} +67.4488 q^{41} -144.264 q^{42} +146.562 q^{44} -349.401 q^{45} -102.588 q^{46} -457.632 q^{47} -14.3341 q^{48} +467.180 q^{49} -434.491 q^{50} -73.4614 q^{51} -108.568 q^{52} -148.507 q^{53} -227.113 q^{54} -552.433 q^{55} -628.390 q^{56} +360.456 q^{57} +117.729 q^{58} -182.072 q^{59} +309.548 q^{60} -215.513 q^{61} -440.912 q^{62} +506.934 q^{63} +270.670 q^{64} +409.223 q^{65} -142.720 q^{66} -631.508 q^{67} -126.125 q^{68} -186.016 q^{69} +933.599 q^{70} +575.441 q^{71} -393.187 q^{72} +1111.81 q^{73} -602.505 q^{74} -787.836 q^{75} +618.864 q^{76} +801.507 q^{77} +105.722 q^{78} -622.227 q^{79} +92.7624 q^{80} +69.0570 q^{81} -112.767 q^{82} -475.871 q^{83} -449.113 q^{84} +475.402 q^{85} +213.470 q^{87} -621.663 q^{88} +1393.87 q^{89} +584.158 q^{90} -593.728 q^{91} -319.369 q^{92} -799.478 q^{93} +765.109 q^{94} -2332.68 q^{95} +559.379 q^{96} +1066.32 q^{97} -781.071 q^{98} +501.507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.67189 −0.591101 −0.295550 0.955327i \(-0.595503\pi\)
−0.295550 + 0.955327i \(0.595503\pi\)
\(3\) −3.03153 −0.583417 −0.291709 0.956507i \(-0.594224\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(4\) −5.20480 −0.650600
\(5\) 19.6184 1.75472 0.877361 0.479831i \(-0.159302\pi\)
0.877361 + 0.479831i \(0.159302\pi\)
\(6\) 5.06836 0.344858
\(7\) −28.4637 −1.53689 −0.768447 0.639914i \(-0.778970\pi\)
−0.768447 + 0.639914i \(0.778970\pi\)
\(8\) 22.0769 0.975671
\(9\) −17.8099 −0.659624
\(10\) −32.7997 −1.03722
\(11\) −28.1590 −0.771840 −0.385920 0.922532i \(-0.626116\pi\)
−0.385920 + 0.922532i \(0.626116\pi\)
\(12\) 15.7785 0.379571
\(13\) 20.8592 0.445022 0.222511 0.974930i \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(14\) 47.5880 0.908459
\(15\) −59.4736 −1.02374
\(16\) 4.72834 0.0738803
\(17\) 24.2325 0.345720 0.172860 0.984946i \(-0.444699\pi\)
0.172860 + 0.984946i \(0.444699\pi\)
\(18\) 29.7760 0.389904
\(19\) −118.903 −1.43569 −0.717845 0.696203i \(-0.754872\pi\)
−0.717845 + 0.696203i \(0.754872\pi\)
\(20\) −102.110 −1.14162
\(21\) 86.2883 0.896650
\(22\) 47.0785 0.456235
\(23\) 61.3604 0.556284 0.278142 0.960540i \(-0.410281\pi\)
0.278142 + 0.960540i \(0.410281\pi\)
\(24\) −66.9267 −0.569223
\(25\) 259.881 2.07905
\(26\) −34.8741 −0.263053
\(27\) 135.842 0.968254
\(28\) 148.148 0.999903
\(29\) −70.4167 −0.450898 −0.225449 0.974255i \(-0.572385\pi\)
−0.225449 + 0.974255i \(0.572385\pi\)
\(30\) 99.4331 0.605131
\(31\) 263.721 1.52793 0.763964 0.645259i \(-0.223251\pi\)
0.763964 + 0.645259i \(0.223251\pi\)
\(32\) −184.521 −1.01934
\(33\) 85.3646 0.450305
\(34\) −40.5139 −0.204355
\(35\) −558.411 −2.69682
\(36\) 92.6967 0.429152
\(37\) 360.375 1.60122 0.800611 0.599185i \(-0.204508\pi\)
0.800611 + 0.599185i \(0.204508\pi\)
\(38\) 198.791 0.848638
\(39\) −63.2350 −0.259634
\(40\) 433.113 1.71203
\(41\) 67.4488 0.256920 0.128460 0.991715i \(-0.458997\pi\)
0.128460 + 0.991715i \(0.458997\pi\)
\(42\) −144.264 −0.530011
\(43\) 0 0
\(44\) 146.562 0.502159
\(45\) −349.401 −1.15746
\(46\) −102.588 −0.328820
\(47\) −457.632 −1.42027 −0.710134 0.704067i \(-0.751365\pi\)
−0.710134 + 0.704067i \(0.751365\pi\)
\(48\) −14.3341 −0.0431030
\(49\) 467.180 1.36204
\(50\) −434.491 −1.22893
\(51\) −73.4614 −0.201699
\(52\) −108.568 −0.289531
\(53\) −148.507 −0.384887 −0.192443 0.981308i \(-0.561641\pi\)
−0.192443 + 0.981308i \(0.561641\pi\)
\(54\) −227.113 −0.572335
\(55\) −552.433 −1.35437
\(56\) −628.390 −1.49950
\(57\) 360.456 0.837607
\(58\) 117.729 0.266526
\(59\) −182.072 −0.401758 −0.200879 0.979616i \(-0.564380\pi\)
−0.200879 + 0.979616i \(0.564380\pi\)
\(60\) 309.548 0.666042
\(61\) −215.513 −0.452354 −0.226177 0.974086i \(-0.572623\pi\)
−0.226177 + 0.974086i \(0.572623\pi\)
\(62\) −440.912 −0.903159
\(63\) 506.934 1.01377
\(64\) 270.670 0.528653
\(65\) 409.223 0.780890
\(66\) −142.720 −0.266176
\(67\) −631.508 −1.15151 −0.575754 0.817623i \(-0.695291\pi\)
−0.575754 + 0.817623i \(0.695291\pi\)
\(68\) −126.125 −0.224925
\(69\) −186.016 −0.324546
\(70\) 933.599 1.59409
\(71\) 575.441 0.961863 0.480931 0.876758i \(-0.340299\pi\)
0.480931 + 0.876758i \(0.340299\pi\)
\(72\) −393.187 −0.643576
\(73\) 1111.81 1.78256 0.891281 0.453452i \(-0.149808\pi\)
0.891281 + 0.453452i \(0.149808\pi\)
\(74\) −602.505 −0.946483
\(75\) −787.836 −1.21295
\(76\) 618.864 0.934060
\(77\) 801.507 1.18624
\(78\) 105.722 0.153470
\(79\) −622.227 −0.886152 −0.443076 0.896484i \(-0.646113\pi\)
−0.443076 + 0.896484i \(0.646113\pi\)
\(80\) 92.7624 0.129639
\(81\) 69.0570 0.0947284
\(82\) −112.767 −0.151866
\(83\) −475.871 −0.629320 −0.314660 0.949204i \(-0.601891\pi\)
−0.314660 + 0.949204i \(0.601891\pi\)
\(84\) −449.113 −0.583361
\(85\) 475.402 0.606642
\(86\) 0 0
\(87\) 213.470 0.263062
\(88\) −621.663 −0.753062
\(89\) 1393.87 1.66011 0.830053 0.557684i \(-0.188310\pi\)
0.830053 + 0.557684i \(0.188310\pi\)
\(90\) 584.158 0.684174
\(91\) −593.728 −0.683951
\(92\) −319.369 −0.361918
\(93\) −799.478 −0.891419
\(94\) 765.109 0.839521
\(95\) −2332.68 −2.51924
\(96\) 559.379 0.594701
\(97\) 1066.32 1.11617 0.558084 0.829785i \(-0.311537\pi\)
0.558084 + 0.829785i \(0.311537\pi\)
\(98\) −781.071 −0.805103
\(99\) 501.507 0.509125
\(100\) −1352.63 −1.35263
\(101\) 1127.76 1.11105 0.555527 0.831498i \(-0.312516\pi\)
0.555527 + 0.831498i \(0.312516\pi\)
\(102\) 122.819 0.119224
\(103\) −450.354 −0.430822 −0.215411 0.976523i \(-0.569109\pi\)
−0.215411 + 0.976523i \(0.569109\pi\)
\(104\) 460.506 0.434195
\(105\) 1692.84 1.57337
\(106\) 248.286 0.227507
\(107\) 512.081 0.462661 0.231331 0.972875i \(-0.425692\pi\)
0.231331 + 0.972875i \(0.425692\pi\)
\(108\) −707.031 −0.629946
\(109\) 535.897 0.470914 0.235457 0.971885i \(-0.424341\pi\)
0.235457 + 0.971885i \(0.424341\pi\)
\(110\) 923.605 0.800566
\(111\) −1092.48 −0.934180
\(112\) −134.586 −0.113546
\(113\) −2126.44 −1.77025 −0.885125 0.465353i \(-0.845927\pi\)
−0.885125 + 0.465353i \(0.845927\pi\)
\(114\) −602.641 −0.495110
\(115\) 1203.79 0.976124
\(116\) 366.505 0.293354
\(117\) −371.498 −0.293547
\(118\) 304.403 0.237479
\(119\) −689.745 −0.531335
\(120\) −1312.99 −0.998828
\(121\) −538.074 −0.404263
\(122\) 360.313 0.267387
\(123\) −204.473 −0.149892
\(124\) −1372.62 −0.994070
\(125\) 2646.15 1.89343
\(126\) −847.535 −0.599241
\(127\) −778.542 −0.543972 −0.271986 0.962301i \(-0.587680\pi\)
−0.271986 + 0.962301i \(0.587680\pi\)
\(128\) 1023.63 0.706854
\(129\) 0 0
\(130\) −684.174 −0.461585
\(131\) 39.6709 0.0264585 0.0132292 0.999912i \(-0.495789\pi\)
0.0132292 + 0.999912i \(0.495789\pi\)
\(132\) −444.305 −0.292968
\(133\) 3384.40 2.20650
\(134\) 1055.81 0.680657
\(135\) 2665.01 1.69902
\(136\) 534.978 0.337309
\(137\) −137.298 −0.0856219 −0.0428109 0.999083i \(-0.513631\pi\)
−0.0428109 + 0.999083i \(0.513631\pi\)
\(138\) 310.997 0.191839
\(139\) 975.906 0.595505 0.297753 0.954643i \(-0.403763\pi\)
0.297753 + 0.954643i \(0.403763\pi\)
\(140\) 2906.42 1.75455
\(141\) 1387.32 0.828608
\(142\) −962.071 −0.568558
\(143\) −587.372 −0.343486
\(144\) −84.2110 −0.0487332
\(145\) −1381.46 −0.791201
\(146\) −1858.81 −1.05367
\(147\) −1416.27 −0.794638
\(148\) −1875.68 −1.04175
\(149\) −2381.63 −1.30947 −0.654734 0.755860i \(-0.727219\pi\)
−0.654734 + 0.755860i \(0.727219\pi\)
\(150\) 1317.17 0.716978
\(151\) 62.4879 0.0336768 0.0168384 0.999858i \(-0.494640\pi\)
0.0168384 + 0.999858i \(0.494640\pi\)
\(152\) −2625.00 −1.40076
\(153\) −431.577 −0.228045
\(154\) −1340.03 −0.701185
\(155\) 5173.79 2.68109
\(156\) 329.126 0.168918
\(157\) −2186.50 −1.11148 −0.555738 0.831358i \(-0.687564\pi\)
−0.555738 + 0.831358i \(0.687564\pi\)
\(158\) 1040.29 0.523805
\(159\) 450.202 0.224549
\(160\) −3620.00 −1.78866
\(161\) −1746.54 −0.854949
\(162\) −115.455 −0.0559940
\(163\) 2358.30 1.13323 0.566614 0.823984i \(-0.308253\pi\)
0.566614 + 0.823984i \(0.308253\pi\)
\(164\) −351.057 −0.167152
\(165\) 1674.72 0.790160
\(166\) 795.601 0.371992
\(167\) −25.7856 −0.0119482 −0.00597410 0.999982i \(-0.501902\pi\)
−0.00597410 + 0.999982i \(0.501902\pi\)
\(168\) 1904.98 0.874835
\(169\) −1761.90 −0.801955
\(170\) −794.818 −0.358587
\(171\) 2117.64 0.947016
\(172\) 0 0
\(173\) −1222.65 −0.537318 −0.268659 0.963235i \(-0.586581\pi\)
−0.268659 + 0.963235i \(0.586581\pi\)
\(174\) −356.897 −0.155496
\(175\) −7397.17 −3.19528
\(176\) −133.145 −0.0570238
\(177\) 551.955 0.234393
\(178\) −2330.38 −0.981290
\(179\) 1466.11 0.612190 0.306095 0.952001i \(-0.400977\pi\)
0.306095 + 0.952001i \(0.400977\pi\)
\(180\) 1818.56 0.753042
\(181\) −288.210 −0.118356 −0.0591782 0.998247i \(-0.518848\pi\)
−0.0591782 + 0.998247i \(0.518848\pi\)
\(182\) 992.645 0.404284
\(183\) 653.333 0.263911
\(184\) 1354.65 0.542750
\(185\) 7069.97 2.80970
\(186\) 1336.64 0.526919
\(187\) −682.361 −0.266841
\(188\) 2381.89 0.924026
\(189\) −3866.57 −1.48810
\(190\) 3899.97 1.48912
\(191\) 4516.92 1.71117 0.855584 0.517664i \(-0.173198\pi\)
0.855584 + 0.517664i \(0.173198\pi\)
\(192\) −820.544 −0.308425
\(193\) 333.088 0.124229 0.0621145 0.998069i \(-0.480216\pi\)
0.0621145 + 0.998069i \(0.480216\pi\)
\(194\) −1782.76 −0.659767
\(195\) −1240.57 −0.455585
\(196\) −2431.58 −0.886144
\(197\) −2433.98 −0.880273 −0.440136 0.897931i \(-0.645070\pi\)
−0.440136 + 0.897931i \(0.645070\pi\)
\(198\) −838.462 −0.300944
\(199\) 450.298 0.160406 0.0802030 0.996779i \(-0.474443\pi\)
0.0802030 + 0.996779i \(0.474443\pi\)
\(200\) 5737.37 2.02847
\(201\) 1914.43 0.671809
\(202\) −1885.49 −0.656745
\(203\) 2004.32 0.692982
\(204\) 382.352 0.131225
\(205\) 1323.24 0.450824
\(206\) 752.940 0.254659
\(207\) −1092.82 −0.366938
\(208\) 98.6291 0.0328784
\(209\) 3348.17 1.10812
\(210\) −2830.23 −0.930021
\(211\) 2584.53 0.843253 0.421626 0.906770i \(-0.361459\pi\)
0.421626 + 0.906770i \(0.361459\pi\)
\(212\) 772.949 0.250407
\(213\) −1744.46 −0.561167
\(214\) −856.141 −0.273479
\(215\) 0 0
\(216\) 2998.98 0.944697
\(217\) −7506.48 −2.34826
\(218\) −895.958 −0.278357
\(219\) −3370.47 −1.03998
\(220\) 2875.30 0.881150
\(221\) 505.469 0.153853
\(222\) 1826.51 0.552195
\(223\) −1716.32 −0.515395 −0.257698 0.966226i \(-0.582964\pi\)
−0.257698 + 0.966226i \(0.582964\pi\)
\(224\) 5252.13 1.56662
\(225\) −4628.45 −1.37139
\(226\) 3555.16 1.04640
\(227\) −4106.23 −1.20062 −0.600308 0.799769i \(-0.704955\pi\)
−0.600308 + 0.799769i \(0.704955\pi\)
\(228\) −1876.10 −0.544947
\(229\) 3533.42 1.01963 0.509815 0.860284i \(-0.329714\pi\)
0.509815 + 0.860284i \(0.329714\pi\)
\(230\) −2012.60 −0.576987
\(231\) −2429.79 −0.692071
\(232\) −1554.58 −0.439928
\(233\) 4199.28 1.18070 0.590351 0.807146i \(-0.298989\pi\)
0.590351 + 0.807146i \(0.298989\pi\)
\(234\) 621.103 0.173516
\(235\) −8978.01 −2.49217
\(236\) 947.647 0.261384
\(237\) 1886.30 0.516996
\(238\) 1153.17 0.314072
\(239\) −2813.11 −0.761358 −0.380679 0.924707i \(-0.624310\pi\)
−0.380679 + 0.924707i \(0.624310\pi\)
\(240\) −281.212 −0.0756339
\(241\) −62.8282 −0.0167930 −0.00839652 0.999965i \(-0.502673\pi\)
−0.00839652 + 0.999965i \(0.502673\pi\)
\(242\) 899.597 0.238960
\(243\) −3877.09 −1.02352
\(244\) 1121.70 0.294302
\(245\) 9165.32 2.39000
\(246\) 341.855 0.0886011
\(247\) −2480.21 −0.638914
\(248\) 5822.15 1.49075
\(249\) 1442.61 0.367156
\(250\) −4424.06 −1.11921
\(251\) −4502.21 −1.13218 −0.566090 0.824344i \(-0.691544\pi\)
−0.566090 + 0.824344i \(0.691544\pi\)
\(252\) −2638.49 −0.659560
\(253\) −1727.84 −0.429362
\(254\) 1301.63 0.321542
\(255\) −1441.19 −0.353926
\(256\) −3876.76 −0.946475
\(257\) 2165.60 0.525629 0.262815 0.964846i \(-0.415349\pi\)
0.262815 + 0.964846i \(0.415349\pi\)
\(258\) 0 0
\(259\) −10257.6 −2.46091
\(260\) −2129.92 −0.508047
\(261\) 1254.11 0.297423
\(262\) −66.3251 −0.0156396
\(263\) −3071.42 −0.720122 −0.360061 0.932929i \(-0.617244\pi\)
−0.360061 + 0.932929i \(0.617244\pi\)
\(264\) 1884.59 0.439349
\(265\) −2913.47 −0.675369
\(266\) −5658.33 −1.30427
\(267\) −4225.54 −0.968535
\(268\) 3286.87 0.749171
\(269\) −4903.70 −1.11146 −0.555732 0.831362i \(-0.687562\pi\)
−0.555732 + 0.831362i \(0.687562\pi\)
\(270\) −4455.58 −1.00429
\(271\) −7712.31 −1.72874 −0.864371 0.502854i \(-0.832283\pi\)
−0.864371 + 0.502854i \(0.832283\pi\)
\(272\) 114.579 0.0255419
\(273\) 1799.90 0.399029
\(274\) 229.547 0.0506112
\(275\) −7317.98 −1.60469
\(276\) 968.174 0.211149
\(277\) −4685.10 −1.01625 −0.508124 0.861284i \(-0.669661\pi\)
−0.508124 + 0.861284i \(0.669661\pi\)
\(278\) −1631.60 −0.352004
\(279\) −4696.84 −1.00786
\(280\) −12328.0 −2.63121
\(281\) 1467.16 0.311471 0.155736 0.987799i \(-0.450225\pi\)
0.155736 + 0.987799i \(0.450225\pi\)
\(282\) −2319.45 −0.489791
\(283\) −2041.78 −0.428873 −0.214437 0.976738i \(-0.568792\pi\)
−0.214437 + 0.976738i \(0.568792\pi\)
\(284\) −2995.05 −0.625788
\(285\) 7071.57 1.46977
\(286\) 982.018 0.203035
\(287\) −1919.84 −0.394859
\(288\) 3286.28 0.672382
\(289\) −4325.79 −0.880478
\(290\) 2309.65 0.467680
\(291\) −3232.57 −0.651191
\(292\) −5786.72 −1.15973
\(293\) −2842.26 −0.566712 −0.283356 0.959015i \(-0.591448\pi\)
−0.283356 + 0.959015i \(0.591448\pi\)
\(294\) 2367.84 0.469711
\(295\) −3571.95 −0.704974
\(296\) 7955.96 1.56227
\(297\) −3825.17 −0.747337
\(298\) 3981.81 0.774027
\(299\) 1279.93 0.247559
\(300\) 4100.53 0.789147
\(301\) 0 0
\(302\) −104.473 −0.0199064
\(303\) −3418.84 −0.648209
\(304\) −562.212 −0.106069
\(305\) −4228.02 −0.793756
\(306\) 721.547 0.134798
\(307\) 3037.25 0.564641 0.282321 0.959320i \(-0.408896\pi\)
0.282321 + 0.959320i \(0.408896\pi\)
\(308\) −4171.68 −0.771765
\(309\) 1365.26 0.251349
\(310\) −8649.98 −1.58479
\(311\) −1850.61 −0.337423 −0.168712 0.985665i \(-0.553961\pi\)
−0.168712 + 0.985665i \(0.553961\pi\)
\(312\) −1396.03 −0.253317
\(313\) −5020.17 −0.906570 −0.453285 0.891366i \(-0.649748\pi\)
−0.453285 + 0.891366i \(0.649748\pi\)
\(314\) 3655.58 0.656994
\(315\) 9945.22 1.77889
\(316\) 3238.57 0.576530
\(317\) −5164.62 −0.915060 −0.457530 0.889194i \(-0.651266\pi\)
−0.457530 + 0.889194i \(0.651266\pi\)
\(318\) −752.687 −0.132731
\(319\) 1982.86 0.348021
\(320\) 5310.12 0.927639
\(321\) −1552.39 −0.269925
\(322\) 2920.02 0.505361
\(323\) −2881.30 −0.496347
\(324\) −359.428 −0.0616303
\(325\) 5420.90 0.925223
\(326\) −3942.80 −0.669851
\(327\) −1624.58 −0.274739
\(328\) 1489.06 0.250670
\(329\) 13025.9 2.18280
\(330\) −2799.93 −0.467064
\(331\) 848.187 0.140848 0.0704238 0.997517i \(-0.477565\pi\)
0.0704238 + 0.997517i \(0.477565\pi\)
\(332\) 2476.81 0.409436
\(333\) −6418.22 −1.05620
\(334\) 43.1105 0.00706259
\(335\) −12389.2 −2.02058
\(336\) 408.000 0.0662448
\(337\) −10239.4 −1.65512 −0.827560 0.561377i \(-0.810272\pi\)
−0.827560 + 0.561377i \(0.810272\pi\)
\(338\) 2945.69 0.474036
\(339\) 6446.35 1.03279
\(340\) −2474.37 −0.394682
\(341\) −7426.12 −1.17932
\(342\) −3540.45 −0.559782
\(343\) −3534.62 −0.556418
\(344\) 0 0
\(345\) −3649.33 −0.569487
\(346\) 2044.12 0.317609
\(347\) 5710.49 0.883445 0.441722 0.897152i \(-0.354368\pi\)
0.441722 + 0.897152i \(0.354368\pi\)
\(348\) −1111.07 −0.171148
\(349\) −11866.8 −1.82009 −0.910047 0.414504i \(-0.863955\pi\)
−0.910047 + 0.414504i \(0.863955\pi\)
\(350\) 12367.2 1.88873
\(351\) 2833.55 0.430894
\(352\) 5195.90 0.786769
\(353\) −9689.28 −1.46093 −0.730465 0.682950i \(-0.760697\pi\)
−0.730465 + 0.682950i \(0.760697\pi\)
\(354\) −922.805 −0.138550
\(355\) 11289.2 1.68780
\(356\) −7254.79 −1.08007
\(357\) 2090.98 0.309990
\(358\) −2451.16 −0.361866
\(359\) −1852.48 −0.272340 −0.136170 0.990685i \(-0.543479\pi\)
−0.136170 + 0.990685i \(0.543479\pi\)
\(360\) −7713.69 −1.12930
\(361\) 7278.82 1.06121
\(362\) 481.855 0.0699606
\(363\) 1631.18 0.235854
\(364\) 3090.23 0.444979
\(365\) 21811.8 3.12790
\(366\) −1092.30 −0.155998
\(367\) −10639.3 −1.51326 −0.756631 0.653842i \(-0.773156\pi\)
−0.756631 + 0.653842i \(0.773156\pi\)
\(368\) 290.133 0.0410984
\(369\) −1201.25 −0.169471
\(370\) −11820.2 −1.66081
\(371\) 4227.05 0.591530
\(372\) 4161.12 0.579957
\(373\) 6298.76 0.874363 0.437182 0.899373i \(-0.355977\pi\)
0.437182 + 0.899373i \(0.355977\pi\)
\(374\) 1140.83 0.157730
\(375\) −8021.87 −1.10466
\(376\) −10103.1 −1.38571
\(377\) −1468.83 −0.200660
\(378\) 6464.46 0.879618
\(379\) 9123.57 1.23653 0.618267 0.785968i \(-0.287835\pi\)
0.618267 + 0.785968i \(0.287835\pi\)
\(380\) 12141.1 1.63902
\(381\) 2360.17 0.317363
\(382\) −7551.78 −1.01147
\(383\) −1733.50 −0.231274 −0.115637 0.993292i \(-0.536891\pi\)
−0.115637 + 0.993292i \(0.536891\pi\)
\(384\) −3103.17 −0.412391
\(385\) 15724.3 2.08151
\(386\) −556.885 −0.0734318
\(387\) 0 0
\(388\) −5549.97 −0.726179
\(389\) 7013.26 0.914104 0.457052 0.889440i \(-0.348905\pi\)
0.457052 + 0.889440i \(0.348905\pi\)
\(390\) 2074.09 0.269296
\(391\) 1486.91 0.192318
\(392\) 10313.9 1.32890
\(393\) −120.263 −0.0154363
\(394\) 4069.33 0.520330
\(395\) −12207.1 −1.55495
\(396\) −2610.24 −0.331236
\(397\) 2694.24 0.340605 0.170303 0.985392i \(-0.445525\pi\)
0.170303 + 0.985392i \(0.445525\pi\)
\(398\) −752.847 −0.0948161
\(399\) −10259.9 −1.28731
\(400\) 1228.81 0.153601
\(401\) −8286.12 −1.03189 −0.515946 0.856621i \(-0.672560\pi\)
−0.515946 + 0.856621i \(0.672560\pi\)
\(402\) −3200.71 −0.397107
\(403\) 5501.00 0.679962
\(404\) −5869.78 −0.722852
\(405\) 1354.79 0.166222
\(406\) −3350.99 −0.409622
\(407\) −10147.8 −1.23589
\(408\) −1621.80 −0.196792
\(409\) −15422.0 −1.86447 −0.932233 0.361858i \(-0.882142\pi\)
−0.932233 + 0.361858i \(0.882142\pi\)
\(410\) −2212.30 −0.266482
\(411\) 416.224 0.0499533
\(412\) 2344.00 0.280293
\(413\) 5182.43 0.617459
\(414\) 1827.07 0.216898
\(415\) −9335.81 −1.10428
\(416\) −3848.94 −0.453629
\(417\) −2958.48 −0.347428
\(418\) −5597.76 −0.655013
\(419\) 1966.87 0.229327 0.114663 0.993404i \(-0.463421\pi\)
0.114663 + 0.993404i \(0.463421\pi\)
\(420\) −8810.88 −1.02364
\(421\) −4738.17 −0.548514 −0.274257 0.961656i \(-0.588432\pi\)
−0.274257 + 0.961656i \(0.588432\pi\)
\(422\) −4321.04 −0.498447
\(423\) 8150.37 0.936843
\(424\) −3278.57 −0.375523
\(425\) 6297.56 0.718769
\(426\) 2916.54 0.331706
\(427\) 6134.29 0.695220
\(428\) −2665.28 −0.301007
\(429\) 1780.63 0.200396
\(430\) 0 0
\(431\) −6730.57 −0.752205 −0.376102 0.926578i \(-0.622736\pi\)
−0.376102 + 0.926578i \(0.622736\pi\)
\(432\) 642.308 0.0715349
\(433\) −7458.24 −0.827760 −0.413880 0.910331i \(-0.635827\pi\)
−0.413880 + 0.910331i \(0.635827\pi\)
\(434\) 12550.0 1.38806
\(435\) 4187.94 0.461600
\(436\) −2789.23 −0.306376
\(437\) −7295.91 −0.798652
\(438\) 5635.03 0.614731
\(439\) 11055.7 1.20196 0.600979 0.799265i \(-0.294778\pi\)
0.600979 + 0.799265i \(0.294778\pi\)
\(440\) −12196.0 −1.32141
\(441\) −8320.41 −0.898435
\(442\) −845.086 −0.0909426
\(443\) 9314.23 0.998944 0.499472 0.866330i \(-0.333527\pi\)
0.499472 + 0.866330i \(0.333527\pi\)
\(444\) 5686.16 0.607778
\(445\) 27345.4 2.91303
\(446\) 2869.49 0.304650
\(447\) 7219.97 0.763966
\(448\) −7704.27 −0.812483
\(449\) −8831.64 −0.928265 −0.464132 0.885766i \(-0.653634\pi\)
−0.464132 + 0.885766i \(0.653634\pi\)
\(450\) 7738.23 0.810630
\(451\) −1899.29 −0.198301
\(452\) 11067.7 1.15172
\(453\) −189.434 −0.0196476
\(454\) 6865.14 0.709685
\(455\) −11648.0 −1.20014
\(456\) 7957.76 0.817228
\(457\) 4617.15 0.472607 0.236303 0.971679i \(-0.424064\pi\)
0.236303 + 0.971679i \(0.424064\pi\)
\(458\) −5907.48 −0.602704
\(459\) 3291.79 0.334745
\(460\) −6265.50 −0.635066
\(461\) 3614.97 0.365219 0.182610 0.983186i \(-0.441546\pi\)
0.182610 + 0.983186i \(0.441546\pi\)
\(462\) 4062.33 0.409083
\(463\) 7007.76 0.703409 0.351704 0.936111i \(-0.385602\pi\)
0.351704 + 0.936111i \(0.385602\pi\)
\(464\) −332.954 −0.0333125
\(465\) −15684.5 −1.56419
\(466\) −7020.71 −0.697914
\(467\) −15127.4 −1.49896 −0.749479 0.662028i \(-0.769696\pi\)
−0.749479 + 0.662028i \(0.769696\pi\)
\(468\) 1933.57 0.190982
\(469\) 17975.0 1.76974
\(470\) 15010.2 1.47313
\(471\) 6628.43 0.648454
\(472\) −4019.58 −0.391984
\(473\) 0 0
\(474\) −3153.67 −0.305597
\(475\) −30900.5 −2.98487
\(476\) 3589.98 0.345686
\(477\) 2644.89 0.253881
\(478\) 4703.19 0.450039
\(479\) −18354.2 −1.75078 −0.875392 0.483415i \(-0.839396\pi\)
−0.875392 + 0.483415i \(0.839396\pi\)
\(480\) 10974.1 1.04354
\(481\) 7517.11 0.712579
\(482\) 105.042 0.00992638
\(483\) 5294.69 0.498792
\(484\) 2800.56 0.263013
\(485\) 20919.5 1.95856
\(486\) 6482.05 0.605003
\(487\) −4112.61 −0.382669 −0.191335 0.981525i \(-0.561282\pi\)
−0.191335 + 0.981525i \(0.561282\pi\)
\(488\) −4757.86 −0.441349
\(489\) −7149.23 −0.661144
\(490\) −15323.4 −1.41273
\(491\) 7080.85 0.650824 0.325412 0.945572i \(-0.394497\pi\)
0.325412 + 0.945572i \(0.394497\pi\)
\(492\) 1064.24 0.0975195
\(493\) −1706.37 −0.155885
\(494\) 4146.62 0.377663
\(495\) 9838.76 0.893372
\(496\) 1246.96 0.112884
\(497\) −16379.2 −1.47828
\(498\) −2411.88 −0.217026
\(499\) −2894.70 −0.259688 −0.129844 0.991534i \(-0.541448\pi\)
−0.129844 + 0.991534i \(0.541448\pi\)
\(500\) −13772.7 −1.23187
\(501\) 78.1696 0.00697078
\(502\) 7527.18 0.669232
\(503\) 933.370 0.0827374 0.0413687 0.999144i \(-0.486828\pi\)
0.0413687 + 0.999144i \(0.486828\pi\)
\(504\) 11191.5 0.989108
\(505\) 22124.9 1.94959
\(506\) 2888.76 0.253796
\(507\) 5341.23 0.467875
\(508\) 4052.16 0.353908
\(509\) 14759.5 1.28527 0.642636 0.766171i \(-0.277841\pi\)
0.642636 + 0.766171i \(0.277841\pi\)
\(510\) 2409.51 0.209206
\(511\) −31646.1 −2.73961
\(512\) −1707.57 −0.147392
\(513\) −16152.0 −1.39011
\(514\) −3620.64 −0.310700
\(515\) −8835.22 −0.755973
\(516\) 0 0
\(517\) 12886.4 1.09622
\(518\) 17149.5 1.45464
\(519\) 3706.48 0.313481
\(520\) 9034.38 0.761892
\(521\) −532.873 −0.0448092 −0.0224046 0.999749i \(-0.507132\pi\)
−0.0224046 + 0.999749i \(0.507132\pi\)
\(522\) −2096.73 −0.175807
\(523\) −4181.75 −0.349627 −0.174814 0.984602i \(-0.555932\pi\)
−0.174814 + 0.984602i \(0.555932\pi\)
\(524\) −206.479 −0.0172139
\(525\) 22424.7 1.86418
\(526\) 5135.07 0.425665
\(527\) 6390.62 0.528235
\(528\) 403.633 0.0332687
\(529\) −8401.90 −0.690548
\(530\) 4870.98 0.399211
\(531\) 3242.67 0.265009
\(532\) −17615.1 −1.43555
\(533\) 1406.92 0.114335
\(534\) 7064.62 0.572502
\(535\) 10046.2 0.811842
\(536\) −13941.7 −1.12349
\(537\) −4444.54 −0.357162
\(538\) 8198.42 0.656987
\(539\) −13155.3 −1.05128
\(540\) −13870.8 −1.10538
\(541\) −2004.57 −0.159304 −0.0796518 0.996823i \(-0.525381\pi\)
−0.0796518 + 0.996823i \(0.525381\pi\)
\(542\) 12894.1 1.02186
\(543\) 873.717 0.0690512
\(544\) −4471.39 −0.352407
\(545\) 10513.4 0.826323
\(546\) −3009.23 −0.235866
\(547\) 540.046 0.0422134 0.0211067 0.999777i \(-0.493281\pi\)
0.0211067 + 0.999777i \(0.493281\pi\)
\(548\) 714.611 0.0557056
\(549\) 3838.26 0.298384
\(550\) 12234.8 0.948536
\(551\) 8372.73 0.647350
\(552\) −4106.65 −0.316650
\(553\) 17710.8 1.36192
\(554\) 7832.95 0.600704
\(555\) −21432.8 −1.63923
\(556\) −5079.39 −0.387436
\(557\) −10875.1 −0.827280 −0.413640 0.910441i \(-0.635743\pi\)
−0.413640 + 0.910441i \(0.635743\pi\)
\(558\) 7852.58 0.595746
\(559\) 0 0
\(560\) −2640.36 −0.199242
\(561\) 2068.60 0.155679
\(562\) −2452.92 −0.184111
\(563\) −15197.2 −1.13763 −0.568814 0.822466i \(-0.692597\pi\)
−0.568814 + 0.822466i \(0.692597\pi\)
\(564\) −7220.74 −0.539093
\(565\) −41717.3 −3.10630
\(566\) 3413.62 0.253507
\(567\) −1965.62 −0.145587
\(568\) 12704.0 0.938462
\(569\) −24697.5 −1.81963 −0.909817 0.415010i \(-0.863778\pi\)
−0.909817 + 0.415010i \(0.863778\pi\)
\(570\) −11822.9 −0.868780
\(571\) 22999.3 1.68562 0.842810 0.538210i \(-0.180899\pi\)
0.842810 + 0.538210i \(0.180899\pi\)
\(572\) 3057.15 0.223472
\(573\) −13693.2 −0.998325
\(574\) 3209.75 0.233401
\(575\) 15946.4 1.15654
\(576\) −4820.60 −0.348712
\(577\) −9086.76 −0.655610 −0.327805 0.944745i \(-0.606309\pi\)
−0.327805 + 0.944745i \(0.606309\pi\)
\(578\) 7232.22 0.520451
\(579\) −1009.76 −0.0724773
\(580\) 7190.23 0.514755
\(581\) 13545.0 0.967198
\(582\) 5404.49 0.384920
\(583\) 4181.80 0.297071
\(584\) 24545.2 1.73919
\(585\) −7288.20 −0.515094
\(586\) 4751.93 0.334984
\(587\) 9602.21 0.675171 0.337586 0.941295i \(-0.390390\pi\)
0.337586 + 0.941295i \(0.390390\pi\)
\(588\) 7371.39 0.516992
\(589\) −31357.1 −2.19363
\(590\) 5971.90 0.416710
\(591\) 7378.67 0.513566
\(592\) 1703.97 0.118299
\(593\) 20476.0 1.41796 0.708980 0.705229i \(-0.249156\pi\)
0.708980 + 0.705229i \(0.249156\pi\)
\(594\) 6395.25 0.441751
\(595\) −13531.7 −0.932345
\(596\) 12395.9 0.851940
\(597\) −1365.09 −0.0935837
\(598\) −2139.89 −0.146332
\(599\) 11861.1 0.809070 0.404535 0.914523i \(-0.367433\pi\)
0.404535 + 0.914523i \(0.367433\pi\)
\(600\) −17393.0 −1.18344
\(601\) 11364.0 0.771296 0.385648 0.922646i \(-0.373978\pi\)
0.385648 + 0.922646i \(0.373978\pi\)
\(602\) 0 0
\(603\) 11247.1 0.759562
\(604\) −325.237 −0.0219101
\(605\) −10556.1 −0.709368
\(606\) 5715.91 0.383157
\(607\) 15781.3 1.05526 0.527631 0.849474i \(-0.323080\pi\)
0.527631 + 0.849474i \(0.323080\pi\)
\(608\) 21940.0 1.46346
\(609\) −6076.14 −0.404298
\(610\) 7068.76 0.469190
\(611\) −9545.82 −0.632050
\(612\) 2246.27 0.148366
\(613\) 9869.44 0.650282 0.325141 0.945666i \(-0.394588\pi\)
0.325141 + 0.945666i \(0.394588\pi\)
\(614\) −5077.93 −0.333760
\(615\) −4011.42 −0.263018
\(616\) 17694.8 1.15738
\(617\) −1089.90 −0.0711148 −0.0355574 0.999368i \(-0.511321\pi\)
−0.0355574 + 0.999368i \(0.511321\pi\)
\(618\) −2282.56 −0.148573
\(619\) 5880.92 0.381865 0.190932 0.981603i \(-0.438849\pi\)
0.190932 + 0.981603i \(0.438849\pi\)
\(620\) −26928.5 −1.74432
\(621\) 8335.33 0.538624
\(622\) 3094.01 0.199451
\(623\) −39674.5 −2.55141
\(624\) −298.997 −0.0191818
\(625\) 19428.1 1.24340
\(626\) 8393.14 0.535874
\(627\) −10150.1 −0.646499
\(628\) 11380.3 0.723126
\(629\) 8732.77 0.553574
\(630\) −16627.3 −1.05150
\(631\) −26475.3 −1.67031 −0.835155 0.550014i \(-0.814622\pi\)
−0.835155 + 0.550014i \(0.814622\pi\)
\(632\) −13736.8 −0.864592
\(633\) −7835.07 −0.491968
\(634\) 8634.66 0.540893
\(635\) −15273.7 −0.954520
\(636\) −2343.21 −0.146092
\(637\) 9744.98 0.606138
\(638\) −3315.11 −0.205716
\(639\) −10248.5 −0.634468
\(640\) 20082.1 1.24033
\(641\) 31481.3 1.93984 0.969918 0.243431i \(-0.0782731\pi\)
0.969918 + 0.243431i \(0.0782731\pi\)
\(642\) 2595.41 0.159553
\(643\) 3775.28 0.231543 0.115772 0.993276i \(-0.463066\pi\)
0.115772 + 0.993276i \(0.463066\pi\)
\(644\) 9090.40 0.556230
\(645\) 0 0
\(646\) 4817.21 0.293391
\(647\) 20064.1 1.21917 0.609583 0.792722i \(-0.291337\pi\)
0.609583 + 0.792722i \(0.291337\pi\)
\(648\) 1524.57 0.0924237
\(649\) 5126.95 0.310093
\(650\) −9063.12 −0.546900
\(651\) 22756.1 1.37002
\(652\) −12274.5 −0.737277
\(653\) −4142.49 −0.248252 −0.124126 0.992266i \(-0.539613\pi\)
−0.124126 + 0.992266i \(0.539613\pi\)
\(654\) 2716.12 0.162399
\(655\) 778.278 0.0464272
\(656\) 318.921 0.0189813
\(657\) −19801.1 −1.17582
\(658\) −21777.8 −1.29025
\(659\) −5352.37 −0.316386 −0.158193 0.987408i \(-0.550567\pi\)
−0.158193 + 0.987408i \(0.550567\pi\)
\(660\) −8716.56 −0.514078
\(661\) 32014.0 1.88381 0.941906 0.335876i \(-0.109032\pi\)
0.941906 + 0.335876i \(0.109032\pi\)
\(662\) −1418.07 −0.0832552
\(663\) −1532.34 −0.0897605
\(664\) −10505.8 −0.614009
\(665\) 66396.5 3.87180
\(666\) 10730.5 0.624323
\(667\) −4320.80 −0.250827
\(668\) 134.209 0.00777349
\(669\) 5203.06 0.300690
\(670\) 20713.3 1.19436
\(671\) 6068.62 0.349145
\(672\) −15922.0 −0.913993
\(673\) 19539.2 1.11914 0.559568 0.828784i \(-0.310967\pi\)
0.559568 + 0.828784i \(0.310967\pi\)
\(674\) 17119.1 0.978343
\(675\) 35302.8 2.01305
\(676\) 9170.32 0.521752
\(677\) −17239.8 −0.978701 −0.489350 0.872087i \(-0.662766\pi\)
−0.489350 + 0.872087i \(0.662766\pi\)
\(678\) −10777.5 −0.610486
\(679\) −30351.3 −1.71543
\(680\) 10495.4 0.591883
\(681\) 12448.1 0.700460
\(682\) 12415.6 0.697094
\(683\) −1098.55 −0.0615447 −0.0307723 0.999526i \(-0.509797\pi\)
−0.0307723 + 0.999526i \(0.509797\pi\)
\(684\) −11021.9 −0.616129
\(685\) −2693.57 −0.150243
\(686\) 5909.48 0.328899
\(687\) −10711.7 −0.594870
\(688\) 0 0
\(689\) −3097.73 −0.171283
\(690\) 6101.26 0.336624
\(691\) −329.691 −0.0181506 −0.00907528 0.999959i \(-0.502889\pi\)
−0.00907528 + 0.999959i \(0.502889\pi\)
\(692\) 6363.63 0.349579
\(693\) −14274.7 −0.782470
\(694\) −9547.29 −0.522205
\(695\) 19145.7 1.04495
\(696\) 4712.76 0.256662
\(697\) 1634.45 0.0888224
\(698\) 19839.9 1.07586
\(699\) −12730.2 −0.688843
\(700\) 38500.8 2.07885
\(701\) −14077.1 −0.758467 −0.379233 0.925301i \(-0.623812\pi\)
−0.379233 + 0.925301i \(0.623812\pi\)
\(702\) −4737.38 −0.254702
\(703\) −42849.5 −2.29886
\(704\) −7621.79 −0.408036
\(705\) 27217.1 1.45398
\(706\) 16199.4 0.863557
\(707\) −32100.2 −1.70757
\(708\) −2872.82 −0.152496
\(709\) −18090.1 −0.958235 −0.479118 0.877751i \(-0.659043\pi\)
−0.479118 + 0.877751i \(0.659043\pi\)
\(710\) −18874.3 −0.997661
\(711\) 11081.8 0.584527
\(712\) 30772.3 1.61972
\(713\) 16182.1 0.849962
\(714\) −3495.88 −0.183235
\(715\) −11523.3 −0.602722
\(716\) −7630.79 −0.398291
\(717\) 8528.00 0.444190
\(718\) 3097.13 0.160980
\(719\) 23754.1 1.23210 0.616050 0.787707i \(-0.288732\pi\)
0.616050 + 0.787707i \(0.288732\pi\)
\(720\) −1652.08 −0.0855133
\(721\) 12818.7 0.662128
\(722\) −12169.4 −0.627281
\(723\) 190.465 0.00979735
\(724\) 1500.08 0.0770027
\(725\) −18300.0 −0.937440
\(726\) −2727.15 −0.139413
\(727\) −31123.3 −1.58776 −0.793879 0.608076i \(-0.791942\pi\)
−0.793879 + 0.608076i \(0.791942\pi\)
\(728\) −13107.7 −0.667311
\(729\) 9888.95 0.502411
\(730\) −36466.9 −1.84890
\(731\) 0 0
\(732\) −3400.47 −0.171701
\(733\) 1512.41 0.0762101 0.0381051 0.999274i \(-0.487868\pi\)
0.0381051 + 0.999274i \(0.487868\pi\)
\(734\) 17787.7 0.894490
\(735\) −27784.9 −1.39437
\(736\) −11322.3 −0.567043
\(737\) 17782.6 0.888780
\(738\) 2008.36 0.100174
\(739\) −30174.5 −1.50201 −0.751007 0.660295i \(-0.770431\pi\)
−0.751007 + 0.660295i \(0.770431\pi\)
\(740\) −36797.8 −1.82799
\(741\) 7518.81 0.372754
\(742\) −7067.14 −0.349654
\(743\) −14115.4 −0.696964 −0.348482 0.937315i \(-0.613303\pi\)
−0.348482 + 0.937315i \(0.613303\pi\)
\(744\) −17650.0 −0.869732
\(745\) −46723.7 −2.29775
\(746\) −10530.8 −0.516837
\(747\) 8475.19 0.415115
\(748\) 3551.55 0.173606
\(749\) −14575.7 −0.711061
\(750\) 13411.7 0.652966
\(751\) 2268.61 0.110230 0.0551151 0.998480i \(-0.482447\pi\)
0.0551151 + 0.998480i \(0.482447\pi\)
\(752\) −2163.84 −0.104930
\(753\) 13648.6 0.660533
\(754\) 2455.72 0.118610
\(755\) 1225.91 0.0590934
\(756\) 20124.7 0.968159
\(757\) 6032.16 0.289620 0.144810 0.989459i \(-0.453743\pi\)
0.144810 + 0.989459i \(0.453743\pi\)
\(758\) −15253.6 −0.730916
\(759\) 5238.01 0.250497
\(760\) −51498.3 −2.45795
\(761\) 23570.9 1.12279 0.561396 0.827547i \(-0.310265\pi\)
0.561396 + 0.827547i \(0.310265\pi\)
\(762\) −3945.93 −0.187593
\(763\) −15253.6 −0.723744
\(764\) −23509.7 −1.11329
\(765\) −8466.84 −0.400156
\(766\) 2898.22 0.136706
\(767\) −3797.86 −0.178791
\(768\) 11752.5 0.552190
\(769\) −5217.26 −0.244654 −0.122327 0.992490i \(-0.539036\pi\)
−0.122327 + 0.992490i \(0.539036\pi\)
\(770\) −26289.2 −1.23038
\(771\) −6565.08 −0.306661
\(772\) −1733.66 −0.0808234
\(773\) 21883.6 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(774\) 0 0
\(775\) 68536.2 3.17664
\(776\) 23541.0 1.08901
\(777\) 31096.1 1.43574
\(778\) −11725.4 −0.540327
\(779\) −8019.83 −0.368858
\(780\) 6456.92 0.296403
\(781\) −16203.8 −0.742405
\(782\) −2485.95 −0.113680
\(783\) −9565.56 −0.436584
\(784\) 2208.99 0.100628
\(785\) −42895.6 −1.95033
\(786\) 201.066 0.00912442
\(787\) 26159.7 1.18487 0.592435 0.805619i \(-0.298167\pi\)
0.592435 + 0.805619i \(0.298167\pi\)
\(788\) 12668.4 0.572706
\(789\) 9311.10 0.420132
\(790\) 20408.8 0.919132
\(791\) 60526.2 2.72069
\(792\) 11071.7 0.496738
\(793\) −4495.42 −0.201308
\(794\) −4504.47 −0.201332
\(795\) 8832.24 0.394022
\(796\) −2343.71 −0.104360
\(797\) 29914.6 1.32952 0.664761 0.747056i \(-0.268533\pi\)
0.664761 + 0.747056i \(0.268533\pi\)
\(798\) 17153.4 0.760931
\(799\) −11089.6 −0.491015
\(800\) −47953.4 −2.11926
\(801\) −24824.5 −1.09505
\(802\) 13853.4 0.609952
\(803\) −31307.3 −1.37585
\(804\) −9964.24 −0.437079
\(805\) −34264.3 −1.50020
\(806\) −9197.05 −0.401926
\(807\) 14865.7 0.648447
\(808\) 24897.5 1.08402
\(809\) −18721.9 −0.813630 −0.406815 0.913511i \(-0.633361\pi\)
−0.406815 + 0.913511i \(0.633361\pi\)
\(810\) −2265.05 −0.0982539
\(811\) −31993.9 −1.38527 −0.692637 0.721287i \(-0.743551\pi\)
−0.692637 + 0.721287i \(0.743551\pi\)
\(812\) −10432.1 −0.450854
\(813\) 23380.1 1.00858
\(814\) 16965.9 0.730534
\(815\) 46266.0 1.98850
\(816\) −347.350 −0.0149016
\(817\) 0 0
\(818\) 25783.7 1.10209
\(819\) 10574.2 0.451151
\(820\) −6887.18 −0.293306
\(821\) −10202.7 −0.433710 −0.216855 0.976204i \(-0.569580\pi\)
−0.216855 + 0.976204i \(0.569580\pi\)
\(822\) −695.878 −0.0295274
\(823\) 1991.87 0.0843647 0.0421824 0.999110i \(-0.486569\pi\)
0.0421824 + 0.999110i \(0.486569\pi\)
\(824\) −9942.42 −0.420341
\(825\) 22184.6 0.936206
\(826\) −8664.42 −0.364981
\(827\) 28028.9 1.17855 0.589274 0.807933i \(-0.299414\pi\)
0.589274 + 0.807933i \(0.299414\pi\)
\(828\) 5687.91 0.238730
\(829\) −7115.74 −0.298118 −0.149059 0.988828i \(-0.547624\pi\)
−0.149059 + 0.988828i \(0.547624\pi\)
\(830\) 15608.4 0.652742
\(831\) 14203.0 0.592896
\(832\) 5645.95 0.235262
\(833\) 11320.9 0.470885
\(834\) 4946.24 0.205365
\(835\) −505.872 −0.0209658
\(836\) −17426.6 −0.720945
\(837\) 35824.5 1.47942
\(838\) −3288.38 −0.135555
\(839\) 2743.35 0.112886 0.0564429 0.998406i \(-0.482024\pi\)
0.0564429 + 0.998406i \(0.482024\pi\)
\(840\) 37372.6 1.53509
\(841\) −19430.5 −0.796691
\(842\) 7921.67 0.324227
\(843\) −4447.73 −0.181718
\(844\) −13452.0 −0.548620
\(845\) −34565.6 −1.40721
\(846\) −13626.5 −0.553768
\(847\) 15315.5 0.621308
\(848\) −702.191 −0.0284355
\(849\) 6189.71 0.250212
\(850\) −10528.8 −0.424865
\(851\) 22112.7 0.890734
\(852\) 9079.58 0.365096
\(853\) −20954.5 −0.841111 −0.420555 0.907267i \(-0.638165\pi\)
−0.420555 + 0.907267i \(0.638165\pi\)
\(854\) −10255.8 −0.410945
\(855\) 41544.6 1.66175
\(856\) 11305.2 0.451405
\(857\) 11280.8 0.449644 0.224822 0.974400i \(-0.427820\pi\)
0.224822 + 0.974400i \(0.427820\pi\)
\(858\) −2977.01 −0.118454
\(859\) −5173.87 −0.205507 −0.102753 0.994707i \(-0.532765\pi\)
−0.102753 + 0.994707i \(0.532765\pi\)
\(860\) 0 0
\(861\) 5820.04 0.230368
\(862\) 11252.7 0.444629
\(863\) −8783.07 −0.346442 −0.173221 0.984883i \(-0.555417\pi\)
−0.173221 + 0.984883i \(0.555417\pi\)
\(864\) −25065.7 −0.986981
\(865\) −23986.3 −0.942844
\(866\) 12469.3 0.489289
\(867\) 13113.7 0.513686
\(868\) 39069.7 1.52778
\(869\) 17521.2 0.683967
\(870\) −7001.75 −0.272852
\(871\) −13172.7 −0.512446
\(872\) 11830.9 0.459457
\(873\) −18991.0 −0.736251
\(874\) 12197.9 0.472084
\(875\) −75319.1 −2.91000
\(876\) 17542.6 0.676609
\(877\) −8379.48 −0.322640 −0.161320 0.986902i \(-0.551575\pi\)
−0.161320 + 0.986902i \(0.551575\pi\)
\(878\) −18483.9 −0.710478
\(879\) 8616.38 0.330629
\(880\) −2612.09 −0.100061
\(881\) −37719.4 −1.44245 −0.721225 0.692701i \(-0.756421\pi\)
−0.721225 + 0.692701i \(0.756421\pi\)
\(882\) 13910.8 0.531066
\(883\) −41618.6 −1.58616 −0.793079 0.609119i \(-0.791523\pi\)
−0.793079 + 0.609119i \(0.791523\pi\)
\(884\) −2630.86 −0.100097
\(885\) 10828.5 0.411294
\(886\) −15572.3 −0.590477
\(887\) 6659.01 0.252072 0.126036 0.992026i \(-0.459775\pi\)
0.126036 + 0.992026i \(0.459775\pi\)
\(888\) −24118.7 −0.911453
\(889\) 22160.2 0.836027
\(890\) −45718.4 −1.72189
\(891\) −1944.57 −0.0731152
\(892\) 8933.09 0.335316
\(893\) 54413.7 2.03906
\(894\) −12071.0 −0.451581
\(895\) 28762.7 1.07422
\(896\) −29136.4 −1.08636
\(897\) −3880.13 −0.144430
\(898\) 14765.5 0.548698
\(899\) −18570.4 −0.688940
\(900\) 24090.1 0.892227
\(901\) −3598.69 −0.133063
\(902\) 3175.39 0.117216
\(903\) 0 0
\(904\) −46945.1 −1.72718
\(905\) −5654.22 −0.207683
\(906\) 316.711 0.0116137
\(907\) −39091.2 −1.43109 −0.715547 0.698565i \(-0.753822\pi\)
−0.715547 + 0.698565i \(0.753822\pi\)
\(908\) 21372.1 0.781121
\(909\) −20085.3 −0.732879
\(910\) 19474.1 0.709406
\(911\) −27839.3 −1.01247 −0.506233 0.862397i \(-0.668962\pi\)
−0.506233 + 0.862397i \(0.668962\pi\)
\(912\) 1704.36 0.0618826
\(913\) 13400.0 0.485735
\(914\) −7719.35 −0.279358
\(915\) 12817.3 0.463091
\(916\) −18390.8 −0.663371
\(917\) −1129.18 −0.0406638
\(918\) −5503.50 −0.197868
\(919\) −33937.5 −1.21816 −0.609082 0.793107i \(-0.708462\pi\)
−0.609082 + 0.793107i \(0.708462\pi\)
\(920\) 26576.0 0.952375
\(921\) −9207.49 −0.329421
\(922\) −6043.82 −0.215881
\(923\) 12003.2 0.428050
\(924\) 12646.6 0.450261
\(925\) 93654.5 3.32902
\(926\) −11716.2 −0.415785
\(927\) 8020.74 0.284181
\(928\) 12993.3 0.459619
\(929\) −13108.9 −0.462958 −0.231479 0.972840i \(-0.574356\pi\)
−0.231479 + 0.972840i \(0.574356\pi\)
\(930\) 26222.6 0.924596
\(931\) −55548.9 −1.95547
\(932\) −21856.4 −0.768165
\(933\) 5610.18 0.196859
\(934\) 25291.3 0.886036
\(935\) −13386.8 −0.468231
\(936\) −8201.54 −0.286406
\(937\) 55740.1 1.94338 0.971691 0.236254i \(-0.0759198\pi\)
0.971691 + 0.236254i \(0.0759198\pi\)
\(938\) −30052.2 −1.04610
\(939\) 15218.8 0.528909
\(940\) 46728.7 1.62141
\(941\) −12532.4 −0.434159 −0.217080 0.976154i \(-0.569653\pi\)
−0.217080 + 0.976154i \(0.569653\pi\)
\(942\) −11082.0 −0.383302
\(943\) 4138.68 0.142921
\(944\) −860.897 −0.0296820
\(945\) −75855.8 −2.61121
\(946\) 0 0
\(947\) 3084.68 0.105849 0.0529243 0.998599i \(-0.483146\pi\)
0.0529243 + 0.998599i \(0.483146\pi\)
\(948\) −9817.79 −0.336358
\(949\) 23191.3 0.793279
\(950\) 51662.2 1.76436
\(951\) 15656.7 0.533862
\(952\) −15227.4 −0.518408
\(953\) −13984.3 −0.475337 −0.237669 0.971346i \(-0.576383\pi\)
−0.237669 + 0.971346i \(0.576383\pi\)
\(954\) −4421.95 −0.150069
\(955\) 88614.7 3.00262
\(956\) 14641.6 0.495340
\(957\) −6011.09 −0.203042
\(958\) 30686.1 1.03489
\(959\) 3908.02 0.131592
\(960\) −16097.8 −0.541201
\(961\) 39757.9 1.33456
\(962\) −12567.7 −0.421206
\(963\) −9120.09 −0.305183
\(964\) 327.008 0.0109255
\(965\) 6534.65 0.217987
\(966\) −8852.11 −0.294836
\(967\) −36409.1 −1.21079 −0.605396 0.795924i \(-0.706985\pi\)
−0.605396 + 0.795924i \(0.706985\pi\)
\(968\) −11879.0 −0.394427
\(969\) 8734.75 0.289577
\(970\) −34974.9 −1.15771
\(971\) −22165.7 −0.732577 −0.366289 0.930501i \(-0.619372\pi\)
−0.366289 + 0.930501i \(0.619372\pi\)
\(972\) 20179.5 0.665902
\(973\) −27777.9 −0.915228
\(974\) 6875.80 0.226196
\(975\) −16433.6 −0.539791
\(976\) −1019.02 −0.0334201
\(977\) 50362.6 1.64917 0.824587 0.565735i \(-0.191408\pi\)
0.824587 + 0.565735i \(0.191408\pi\)
\(978\) 11952.7 0.390803
\(979\) −39249.8 −1.28134
\(980\) −47703.6 −1.55494
\(981\) −9544.24 −0.310626
\(982\) −11838.4 −0.384702
\(983\) −5278.97 −0.171285 −0.0856424 0.996326i \(-0.527294\pi\)
−0.0856424 + 0.996326i \(0.527294\pi\)
\(984\) −4514.12 −0.146245
\(985\) −47750.7 −1.54463
\(986\) 2852.86 0.0921434
\(987\) −39488.3 −1.27348
\(988\) 12909.0 0.415677
\(989\) 0 0
\(990\) −16449.3 −0.528073
\(991\) 44303.9 1.42014 0.710070 0.704131i \(-0.248663\pi\)
0.710070 + 0.704131i \(0.248663\pi\)
\(992\) −48662.0 −1.55748
\(993\) −2571.30 −0.0821730
\(994\) 27384.1 0.873813
\(995\) 8834.13 0.281468
\(996\) −7508.52 −0.238872
\(997\) 21056.6 0.668874 0.334437 0.942418i \(-0.391454\pi\)
0.334437 + 0.942418i \(0.391454\pi\)
\(998\) 4839.60 0.153502
\(999\) 48954.1 1.55039
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.1.19 yes 50
43.42 odd 2 1849.4.a.i.1.32 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.32 50 43.42 odd 2
1849.4.a.j.1.19 yes 50 1.1 even 1 trivial