Properties

Label 105.4.a.g.1.1
Level 105
Weight 4
Character 105.1
Self dual yes
Analytic conductor 6.195
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of \(x^{2} - x - 10\)
Character \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.70156 q^{2} +3.00000 q^{3} -5.10469 q^{4} -5.00000 q^{5} -5.10469 q^{6} +7.00000 q^{7} +22.2984 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.70156 q^{2} +3.00000 q^{3} -5.10469 q^{4} -5.00000 q^{5} -5.10469 q^{6} +7.00000 q^{7} +22.2984 q^{8} +9.00000 q^{9} +8.50781 q^{10} +37.4031 q^{11} -15.3141 q^{12} +29.0156 q^{13} -11.9109 q^{14} -15.0000 q^{15} +2.89531 q^{16} +58.4187 q^{17} -15.3141 q^{18} -54.5969 q^{19} +25.5234 q^{20} +21.0000 q^{21} -63.6437 q^{22} +161.675 q^{23} +66.8953 q^{24} +25.0000 q^{25} -49.3719 q^{26} +27.0000 q^{27} -35.7328 q^{28} +137.581 q^{29} +25.5234 q^{30} +154.659 q^{31} -183.314 q^{32} +112.209 q^{33} -99.4031 q^{34} -35.0000 q^{35} -45.9422 q^{36} -350.125 q^{37} +92.9000 q^{38} +87.0469 q^{39} -111.492 q^{40} +353.769 q^{41} -35.7328 q^{42} -518.156 q^{43} -190.931 q^{44} -45.0000 q^{45} -275.100 q^{46} -542.219 q^{47} +8.68594 q^{48} +49.0000 q^{49} -42.5391 q^{50} +175.256 q^{51} -148.116 q^{52} +305.309 q^{53} -45.9422 q^{54} -187.016 q^{55} +156.089 q^{56} -163.791 q^{57} -234.103 q^{58} +14.6813 q^{59} +76.5703 q^{60} -171.069 q^{61} -263.163 q^{62} +63.0000 q^{63} +288.758 q^{64} -145.078 q^{65} -190.931 q^{66} +551.956 q^{67} -298.209 q^{68} +485.025 q^{69} +59.5547 q^{70} -120.334 q^{71} +200.686 q^{72} +284.659 q^{73} +595.759 q^{74} +75.0000 q^{75} +278.700 q^{76} +261.822 q^{77} -148.116 q^{78} +941.612 q^{79} -14.4766 q^{80} +81.0000 q^{81} -601.959 q^{82} +377.150 q^{83} -107.198 q^{84} -292.094 q^{85} +881.675 q^{86} +412.744 q^{87} +834.031 q^{88} -677.725 q^{89} +76.5703 q^{90} +203.109 q^{91} -825.300 q^{92} +463.978 q^{93} +922.619 q^{94} +272.984 q^{95} -549.942 q^{96} -1225.03 q^{97} -83.3765 q^{98} +336.628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + 51q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + 51q^{8} + 18q^{9} - 15q^{10} + 62q^{11} + 27q^{12} - 6q^{13} + 21q^{14} - 30q^{15} + 25q^{16} + 40q^{17} + 27q^{18} - 122q^{19} - 45q^{20} + 42q^{21} + 52q^{22} + 16q^{23} + 153q^{24} + 50q^{25} - 214q^{26} + 54q^{27} + 63q^{28} + 352q^{29} - 45q^{30} + 66q^{31} - 309q^{32} + 186q^{33} - 186q^{34} - 70q^{35} + 81q^{36} - 188q^{37} - 224q^{38} - 18q^{39} - 255q^{40} + 16q^{41} + 63q^{42} - 396q^{43} + 156q^{44} - 90q^{45} - 960q^{46} - 188q^{47} + 75q^{48} + 98q^{49} + 75q^{50} + 120q^{51} - 642q^{52} + 982q^{53} + 81q^{54} - 310q^{55} + 357q^{56} - 366q^{57} + 774q^{58} + 516q^{59} - 135q^{60} - 880q^{61} - 680q^{62} + 126q^{63} - 479q^{64} + 30q^{65} + 156q^{66} - 356q^{67} - 558q^{68} + 48q^{69} - 105q^{70} + 310q^{71} + 459q^{72} + 326q^{73} + 1358q^{74} + 150q^{75} - 672q^{76} + 434q^{77} - 642q^{78} + 1832q^{79} - 125q^{80} + 162q^{81} - 2190q^{82} - 680q^{83} + 189q^{84} - 200q^{85} + 1456q^{86} + 1056q^{87} + 1540q^{88} + 796q^{89} - 135q^{90} - 42q^{91} - 2880q^{92} + 198q^{93} + 2588q^{94} + 610q^{95} - 927q^{96} - 670q^{97} + 147q^{98} + 558q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.70156 −0.601593 −0.300797 0.953688i \(-0.597253\pi\)
−0.300797 + 0.953688i \(0.597253\pi\)
\(3\) 3.00000 0.577350
\(4\) −5.10469 −0.638086
\(5\) −5.00000 −0.447214
\(6\) −5.10469 −0.347330
\(7\) 7.00000 0.377964
\(8\) 22.2984 0.985461
\(9\) 9.00000 0.333333
\(10\) 8.50781 0.269041
\(11\) 37.4031 1.02522 0.512612 0.858620i \(-0.328678\pi\)
0.512612 + 0.858620i \(0.328678\pi\)
\(12\) −15.3141 −0.368399
\(13\) 29.0156 0.619037 0.309519 0.950893i \(-0.399832\pi\)
0.309519 + 0.950893i \(0.399832\pi\)
\(14\) −11.9109 −0.227381
\(15\) −15.0000 −0.258199
\(16\) 2.89531 0.0452393
\(17\) 58.4187 0.833449 0.416724 0.909033i \(-0.363178\pi\)
0.416724 + 0.909033i \(0.363178\pi\)
\(18\) −15.3141 −0.200531
\(19\) −54.5969 −0.659231 −0.329615 0.944115i \(-0.606919\pi\)
−0.329615 + 0.944115i \(0.606919\pi\)
\(20\) 25.5234 0.285361
\(21\) 21.0000 0.218218
\(22\) −63.6437 −0.616768
\(23\) 161.675 1.46572 0.732860 0.680379i \(-0.238185\pi\)
0.732860 + 0.680379i \(0.238185\pi\)
\(24\) 66.8953 0.568956
\(25\) 25.0000 0.200000
\(26\) −49.3719 −0.372409
\(27\) 27.0000 0.192450
\(28\) −35.7328 −0.241174
\(29\) 137.581 0.880972 0.440486 0.897759i \(-0.354806\pi\)
0.440486 + 0.897759i \(0.354806\pi\)
\(30\) 25.5234 0.155331
\(31\) 154.659 0.896053 0.448026 0.894020i \(-0.352127\pi\)
0.448026 + 0.894020i \(0.352127\pi\)
\(32\) −183.314 −1.01268
\(33\) 112.209 0.591913
\(34\) −99.4031 −0.501397
\(35\) −35.0000 −0.169031
\(36\) −45.9422 −0.212695
\(37\) −350.125 −1.55568 −0.777840 0.628462i \(-0.783685\pi\)
−0.777840 + 0.628462i \(0.783685\pi\)
\(38\) 92.9000 0.396589
\(39\) 87.0469 0.357401
\(40\) −111.492 −0.440712
\(41\) 353.769 1.34755 0.673773 0.738938i \(-0.264673\pi\)
0.673773 + 0.738938i \(0.264673\pi\)
\(42\) −35.7328 −0.131278
\(43\) −518.156 −1.83763 −0.918815 0.394689i \(-0.870852\pi\)
−0.918815 + 0.394689i \(0.870852\pi\)
\(44\) −190.931 −0.654181
\(45\) −45.0000 −0.149071
\(46\) −275.100 −0.881767
\(47\) −542.219 −1.68278 −0.841391 0.540427i \(-0.818263\pi\)
−0.841391 + 0.540427i \(0.818263\pi\)
\(48\) 8.68594 0.0261189
\(49\) 49.0000 0.142857
\(50\) −42.5391 −0.120319
\(51\) 175.256 0.481192
\(52\) −148.116 −0.394999
\(53\) 305.309 0.791273 0.395637 0.918407i \(-0.370524\pi\)
0.395637 + 0.918407i \(0.370524\pi\)
\(54\) −45.9422 −0.115777
\(55\) −187.016 −0.458494
\(56\) 156.089 0.372469
\(57\) −163.791 −0.380607
\(58\) −234.103 −0.529987
\(59\) 14.6813 0.0323956 0.0161978 0.999869i \(-0.494844\pi\)
0.0161978 + 0.999869i \(0.494844\pi\)
\(60\) 76.5703 0.164753
\(61\) −171.069 −0.359067 −0.179534 0.983752i \(-0.557459\pi\)
−0.179534 + 0.983752i \(0.557459\pi\)
\(62\) −263.163 −0.539059
\(63\) 63.0000 0.125988
\(64\) 288.758 0.563980
\(65\) −145.078 −0.276842
\(66\) −190.931 −0.356091
\(67\) 551.956 1.00645 0.503225 0.864155i \(-0.332147\pi\)
0.503225 + 0.864155i \(0.332147\pi\)
\(68\) −298.209 −0.531812
\(69\) 485.025 0.846234
\(70\) 59.5547 0.101688
\(71\) −120.334 −0.201142 −0.100571 0.994930i \(-0.532067\pi\)
−0.100571 + 0.994930i \(0.532067\pi\)
\(72\) 200.686 0.328487
\(73\) 284.659 0.456395 0.228198 0.973615i \(-0.426717\pi\)
0.228198 + 0.973615i \(0.426717\pi\)
\(74\) 595.759 0.935887
\(75\) 75.0000 0.115470
\(76\) 278.700 0.420646
\(77\) 261.822 0.387498
\(78\) −148.116 −0.215010
\(79\) 941.612 1.34101 0.670504 0.741906i \(-0.266078\pi\)
0.670504 + 0.741906i \(0.266078\pi\)
\(80\) −14.4766 −0.0202316
\(81\) 81.0000 0.111111
\(82\) −601.959 −0.810674
\(83\) 377.150 0.498766 0.249383 0.968405i \(-0.419772\pi\)
0.249383 + 0.968405i \(0.419772\pi\)
\(84\) −107.198 −0.139242
\(85\) −292.094 −0.372730
\(86\) 881.675 1.10551
\(87\) 412.744 0.508630
\(88\) 834.031 1.01032
\(89\) −677.725 −0.807176 −0.403588 0.914941i \(-0.632237\pi\)
−0.403588 + 0.914941i \(0.632237\pi\)
\(90\) 76.5703 0.0896802
\(91\) 203.109 0.233974
\(92\) −825.300 −0.935255
\(93\) 463.978 0.517336
\(94\) 922.619 1.01235
\(95\) 272.984 0.294817
\(96\) −549.942 −0.584669
\(97\) −1225.03 −1.28230 −0.641151 0.767414i \(-0.721543\pi\)
−0.641151 + 0.767414i \(0.721543\pi\)
\(98\) −83.3765 −0.0859419
\(99\) 336.628 0.341741
\(100\) −127.617 −0.127617
\(101\) −338.144 −0.333134 −0.166567 0.986030i \(-0.553268\pi\)
−0.166567 + 0.986030i \(0.553268\pi\)
\(102\) −298.209 −0.289482
\(103\) −566.700 −0.542122 −0.271061 0.962562i \(-0.587375\pi\)
−0.271061 + 0.962562i \(0.587375\pi\)
\(104\) 647.003 0.610037
\(105\) −105.000 −0.0975900
\(106\) −519.503 −0.476024
\(107\) −562.531 −0.508242 −0.254121 0.967172i \(-0.581786\pi\)
−0.254121 + 0.967172i \(0.581786\pi\)
\(108\) −137.827 −0.122800
\(109\) 1830.79 1.60879 0.804396 0.594094i \(-0.202489\pi\)
0.804396 + 0.594094i \(0.202489\pi\)
\(110\) 318.219 0.275827
\(111\) −1050.37 −0.898173
\(112\) 20.2672 0.0170988
\(113\) −31.8032 −0.0264761 −0.0132380 0.999912i \(-0.504214\pi\)
−0.0132380 + 0.999912i \(0.504214\pi\)
\(114\) 278.700 0.228971
\(115\) −808.375 −0.655490
\(116\) −702.309 −0.562136
\(117\) 261.141 0.206346
\(118\) −24.9811 −0.0194890
\(119\) 408.931 0.315014
\(120\) −334.477 −0.254445
\(121\) 67.9937 0.0510847
\(122\) 291.084 0.216012
\(123\) 1061.31 0.778006
\(124\) −789.488 −0.571759
\(125\) −125.000 −0.0894427
\(126\) −107.198 −0.0757936
\(127\) 2220.81 1.55169 0.775847 0.630921i \(-0.217323\pi\)
0.775847 + 0.630921i \(0.217323\pi\)
\(128\) 975.173 0.673390
\(129\) −1554.47 −1.06096
\(130\) 246.859 0.166546
\(131\) 646.512 0.431191 0.215596 0.976483i \(-0.430831\pi\)
0.215596 + 0.976483i \(0.430831\pi\)
\(132\) −572.794 −0.377692
\(133\) −382.178 −0.249166
\(134\) −939.188 −0.605474
\(135\) −135.000 −0.0860663
\(136\) 1302.65 0.821331
\(137\) −896.009 −0.558768 −0.279384 0.960179i \(-0.590130\pi\)
−0.279384 + 0.960179i \(0.590130\pi\)
\(138\) −825.300 −0.509088
\(139\) −2313.61 −1.41178 −0.705891 0.708320i \(-0.749453\pi\)
−0.705891 + 0.708320i \(0.749453\pi\)
\(140\) 178.664 0.107856
\(141\) −1626.66 −0.971554
\(142\) 204.756 0.121005
\(143\) 1085.27 0.634652
\(144\) 26.0578 0.0150798
\(145\) −687.906 −0.393983
\(146\) −484.366 −0.274564
\(147\) 147.000 0.0824786
\(148\) 1787.28 0.992658
\(149\) 819.337 0.450488 0.225244 0.974302i \(-0.427682\pi\)
0.225244 + 0.974302i \(0.427682\pi\)
\(150\) −127.617 −0.0694660
\(151\) 534.744 0.288191 0.144095 0.989564i \(-0.453973\pi\)
0.144095 + 0.989564i \(0.453973\pi\)
\(152\) −1217.43 −0.649646
\(153\) 525.769 0.277816
\(154\) −445.506 −0.233116
\(155\) −773.297 −0.400727
\(156\) −444.347 −0.228053
\(157\) 1564.76 0.795423 0.397711 0.917511i \(-0.369805\pi\)
0.397711 + 0.917511i \(0.369805\pi\)
\(158\) −1602.21 −0.806741
\(159\) 915.928 0.456842
\(160\) 916.570 0.452883
\(161\) 1131.72 0.553990
\(162\) −137.827 −0.0668437
\(163\) −1114.31 −0.535455 −0.267728 0.963495i \(-0.586273\pi\)
−0.267728 + 0.963495i \(0.586273\pi\)
\(164\) −1805.88 −0.859850
\(165\) −561.047 −0.264712
\(166\) −641.744 −0.300054
\(167\) −1774.47 −0.822231 −0.411115 0.911583i \(-0.634861\pi\)
−0.411115 + 0.911583i \(0.634861\pi\)
\(168\) 468.267 0.215045
\(169\) −1355.09 −0.616793
\(170\) 497.016 0.224232
\(171\) −491.372 −0.219744
\(172\) 2645.02 1.17257
\(173\) −4215.88 −1.85276 −0.926380 0.376590i \(-0.877097\pi\)
−0.926380 + 0.376590i \(0.877097\pi\)
\(174\) −702.309 −0.305988
\(175\) 175.000 0.0755929
\(176\) 108.294 0.0463804
\(177\) 44.0438 0.0187036
\(178\) 1153.19 0.485592
\(179\) −2430.70 −1.01497 −0.507483 0.861662i \(-0.669424\pi\)
−0.507483 + 0.861662i \(0.669424\pi\)
\(180\) 229.711 0.0951202
\(181\) −2700.91 −1.10916 −0.554578 0.832132i \(-0.687120\pi\)
−0.554578 + 0.832132i \(0.687120\pi\)
\(182\) −345.603 −0.140757
\(183\) −513.206 −0.207308
\(184\) 3605.10 1.44441
\(185\) 1750.62 0.695722
\(186\) −789.488 −0.311226
\(187\) 2185.04 0.854472
\(188\) 2767.86 1.07376
\(189\) 189.000 0.0727393
\(190\) −464.500 −0.177360
\(191\) 3611.10 1.36801 0.684005 0.729478i \(-0.260237\pi\)
0.684005 + 0.729478i \(0.260237\pi\)
\(192\) 866.273 0.325614
\(193\) −4468.33 −1.66651 −0.833257 0.552886i \(-0.813526\pi\)
−0.833257 + 0.552886i \(0.813526\pi\)
\(194\) 2084.47 0.771425
\(195\) −435.234 −0.159835
\(196\) −250.130 −0.0911551
\(197\) −434.422 −0.157113 −0.0785566 0.996910i \(-0.525031\pi\)
−0.0785566 + 0.996910i \(0.525031\pi\)
\(198\) −572.794 −0.205589
\(199\) −468.915 −0.167038 −0.0835189 0.996506i \(-0.526616\pi\)
−0.0835189 + 0.996506i \(0.526616\pi\)
\(200\) 557.461 0.197092
\(201\) 1655.87 0.581074
\(202\) 575.372 0.200411
\(203\) 963.069 0.332976
\(204\) −894.628 −0.307042
\(205\) −1768.84 −0.602641
\(206\) 964.275 0.326137
\(207\) 1455.07 0.488573
\(208\) 84.0093 0.0280048
\(209\) −2042.09 −0.675859
\(210\) 178.664 0.0587095
\(211\) 3735.51 1.21878 0.609392 0.792869i \(-0.291414\pi\)
0.609392 + 0.792869i \(0.291414\pi\)
\(212\) −1558.51 −0.504900
\(213\) −361.003 −0.116129
\(214\) 957.182 0.305755
\(215\) 2590.78 0.821813
\(216\) 602.058 0.189652
\(217\) 1082.62 0.338676
\(218\) −3115.21 −0.967838
\(219\) 853.978 0.263500
\(220\) 954.656 0.292559
\(221\) 1695.06 0.515936
\(222\) 1787.28 0.540334
\(223\) 842.806 0.253087 0.126544 0.991961i \(-0.459612\pi\)
0.126544 + 0.991961i \(0.459612\pi\)
\(224\) −1283.20 −0.382756
\(225\) 225.000 0.0666667
\(226\) 54.1152 0.0159278
\(227\) 992.150 0.290094 0.145047 0.989425i \(-0.453667\pi\)
0.145047 + 0.989425i \(0.453667\pi\)
\(228\) 836.100 0.242860
\(229\) −6411.39 −1.85012 −0.925059 0.379825i \(-0.875984\pi\)
−0.925059 + 0.379825i \(0.875984\pi\)
\(230\) 1375.50 0.394338
\(231\) 785.466 0.223722
\(232\) 3067.85 0.868164
\(233\) −2274.35 −0.639476 −0.319738 0.947506i \(-0.603595\pi\)
−0.319738 + 0.947506i \(0.603595\pi\)
\(234\) −444.347 −0.124136
\(235\) 2711.09 0.752563
\(236\) −74.9433 −0.0206712
\(237\) 2824.84 0.774232
\(238\) −695.822 −0.189510
\(239\) 2863.12 0.774893 0.387447 0.921892i \(-0.373357\pi\)
0.387447 + 0.921892i \(0.373357\pi\)
\(240\) −43.4297 −0.0116807
\(241\) −5364.23 −1.43378 −0.716889 0.697187i \(-0.754435\pi\)
−0.716889 + 0.697187i \(0.754435\pi\)
\(242\) −115.696 −0.0307322
\(243\) 243.000 0.0641500
\(244\) 873.252 0.229116
\(245\) −245.000 −0.0638877
\(246\) −1805.88 −0.468043
\(247\) −1584.16 −0.408088
\(248\) 3448.66 0.883025
\(249\) 1131.45 0.287963
\(250\) 212.695 0.0538081
\(251\) 5569.81 1.40065 0.700325 0.713824i \(-0.253039\pi\)
0.700325 + 0.713824i \(0.253039\pi\)
\(252\) −321.595 −0.0803913
\(253\) 6047.15 1.50269
\(254\) −3778.85 −0.933489
\(255\) −876.281 −0.215196
\(256\) −3969.38 −0.969087
\(257\) 2095.36 0.508580 0.254290 0.967128i \(-0.418158\pi\)
0.254290 + 0.967128i \(0.418158\pi\)
\(258\) 2645.02 0.638264
\(259\) −2450.87 −0.587992
\(260\) 740.578 0.176649
\(261\) 1238.23 0.293657
\(262\) −1100.08 −0.259402
\(263\) −7465.88 −1.75044 −0.875220 0.483724i \(-0.839284\pi\)
−0.875220 + 0.483724i \(0.839284\pi\)
\(264\) 2502.09 0.583308
\(265\) −1526.55 −0.353868
\(266\) 650.300 0.149896
\(267\) −2033.17 −0.466023
\(268\) −2817.56 −0.642202
\(269\) −6521.38 −1.47812 −0.739062 0.673637i \(-0.764731\pi\)
−0.739062 + 0.673637i \(0.764731\pi\)
\(270\) 229.711 0.0517769
\(271\) 2409.70 0.540144 0.270072 0.962840i \(-0.412952\pi\)
0.270072 + 0.962840i \(0.412952\pi\)
\(272\) 169.141 0.0377046
\(273\) 609.328 0.135085
\(274\) 1524.62 0.336151
\(275\) 935.078 0.205045
\(276\) −2475.90 −0.539970
\(277\) −2219.83 −0.481503 −0.240752 0.970587i \(-0.577394\pi\)
−0.240752 + 0.970587i \(0.577394\pi\)
\(278\) 3936.75 0.849319
\(279\) 1391.93 0.298684
\(280\) −780.445 −0.166573
\(281\) 5838.56 1.23950 0.619749 0.784800i \(-0.287234\pi\)
0.619749 + 0.784800i \(0.287234\pi\)
\(282\) 2767.86 0.584480
\(283\) −3645.04 −0.765636 −0.382818 0.923824i \(-0.625046\pi\)
−0.382818 + 0.923824i \(0.625046\pi\)
\(284\) 614.269 0.128346
\(285\) 818.953 0.170213
\(286\) −1846.66 −0.381802
\(287\) 2476.38 0.509325
\(288\) −1649.83 −0.337559
\(289\) −1500.25 −0.305363
\(290\) 1170.52 0.237017
\(291\) −3675.10 −0.740338
\(292\) −1453.10 −0.291219
\(293\) 3777.91 0.753268 0.376634 0.926362i \(-0.377081\pi\)
0.376634 + 0.926362i \(0.377081\pi\)
\(294\) −250.130 −0.0496186
\(295\) −73.4064 −0.0144877
\(296\) −7807.24 −1.53306
\(297\) 1009.88 0.197304
\(298\) −1394.15 −0.271011
\(299\) 4691.10 0.907336
\(300\) −382.851 −0.0736798
\(301\) −3627.09 −0.694559
\(302\) −909.900 −0.173374
\(303\) −1014.43 −0.192335
\(304\) −158.075 −0.0298231
\(305\) 855.344 0.160580
\(306\) −894.628 −0.167132
\(307\) −4799.64 −0.892281 −0.446140 0.894963i \(-0.647202\pi\)
−0.446140 + 0.894963i \(0.647202\pi\)
\(308\) −1336.52 −0.247257
\(309\) −1700.10 −0.312994
\(310\) 1315.81 0.241075
\(311\) 580.113 0.105772 0.0528861 0.998601i \(-0.483158\pi\)
0.0528861 + 0.998601i \(0.483158\pi\)
\(312\) 1941.01 0.352205
\(313\) −6114.78 −1.10424 −0.552121 0.833764i \(-0.686182\pi\)
−0.552121 + 0.833764i \(0.686182\pi\)
\(314\) −2662.53 −0.478521
\(315\) −315.000 −0.0563436
\(316\) −4806.64 −0.855679
\(317\) −4300.63 −0.761979 −0.380989 0.924579i \(-0.624417\pi\)
−0.380989 + 0.924579i \(0.624417\pi\)
\(318\) −1558.51 −0.274833
\(319\) 5145.97 0.903194
\(320\) −1443.79 −0.252220
\(321\) −1687.59 −0.293434
\(322\) −1925.70 −0.333277
\(323\) −3189.48 −0.549435
\(324\) −413.480 −0.0708984
\(325\) 725.391 0.123807
\(326\) 1896.06 0.322126
\(327\) 5492.38 0.928836
\(328\) 7888.49 1.32795
\(329\) −3795.53 −0.636032
\(330\) 954.656 0.159249
\(331\) −6687.54 −1.11051 −0.555257 0.831679i \(-0.687380\pi\)
−0.555257 + 0.831679i \(0.687380\pi\)
\(332\) −1925.23 −0.318256
\(333\) −3151.12 −0.518560
\(334\) 3019.37 0.494648
\(335\) −2759.78 −0.450098
\(336\) 60.8016 0.00987202
\(337\) 5869.28 0.948723 0.474362 0.880330i \(-0.342679\pi\)
0.474362 + 0.880330i \(0.342679\pi\)
\(338\) 2305.78 0.371058
\(339\) −95.4097 −0.0152860
\(340\) 1491.05 0.237833
\(341\) 5784.74 0.918655
\(342\) 836.100 0.132196
\(343\) 343.000 0.0539949
\(344\) −11554.1 −1.81091
\(345\) −2425.12 −0.378447
\(346\) 7173.58 1.11461
\(347\) 1937.22 0.299699 0.149850 0.988709i \(-0.452121\pi\)
0.149850 + 0.988709i \(0.452121\pi\)
\(348\) −2106.93 −0.324549
\(349\) −9748.82 −1.49525 −0.747625 0.664121i \(-0.768806\pi\)
−0.747625 + 0.664121i \(0.768806\pi\)
\(350\) −297.773 −0.0454762
\(351\) 783.422 0.119134
\(352\) −6856.52 −1.03822
\(353\) −4576.61 −0.690052 −0.345026 0.938593i \(-0.612130\pi\)
−0.345026 + 0.938593i \(0.612130\pi\)
\(354\) −74.9433 −0.0112520
\(355\) 601.672 0.0899533
\(356\) 3459.57 0.515048
\(357\) 1226.79 0.181873
\(358\) 4135.98 0.610596
\(359\) 10849.9 1.59509 0.797546 0.603258i \(-0.206131\pi\)
0.797546 + 0.603258i \(0.206131\pi\)
\(360\) −1003.43 −0.146904
\(361\) −3878.18 −0.565415
\(362\) 4595.77 0.667261
\(363\) 203.981 0.0294938
\(364\) −1036.81 −0.149296
\(365\) −1423.30 −0.204106
\(366\) 873.252 0.124715
\(367\) 11467.7 1.63108 0.815541 0.578699i \(-0.196439\pi\)
0.815541 + 0.578699i \(0.196439\pi\)
\(368\) 468.100 0.0663081
\(369\) 3183.92 0.449182
\(370\) −2978.80 −0.418541
\(371\) 2137.17 0.299073
\(372\) −2368.46 −0.330105
\(373\) 539.982 0.0749576 0.0374788 0.999297i \(-0.488067\pi\)
0.0374788 + 0.999297i \(0.488067\pi\)
\(374\) −3717.99 −0.514044
\(375\) −375.000 −0.0516398
\(376\) −12090.6 −1.65832
\(377\) 3992.01 0.545355
\(378\) −321.595 −0.0437595
\(379\) 8577.57 1.16253 0.581267 0.813713i \(-0.302557\pi\)
0.581267 + 0.813713i \(0.302557\pi\)
\(380\) −1393.50 −0.188118
\(381\) 6662.44 0.895871
\(382\) −6144.51 −0.822985
\(383\) 8627.96 1.15109 0.575546 0.817770i \(-0.304790\pi\)
0.575546 + 0.817770i \(0.304790\pi\)
\(384\) 2925.52 0.388782
\(385\) −1309.11 −0.173295
\(386\) 7603.13 1.00256
\(387\) −4663.41 −0.612543
\(388\) 6253.42 0.818219
\(389\) 9234.06 1.20356 0.601781 0.798661i \(-0.294458\pi\)
0.601781 + 0.798661i \(0.294458\pi\)
\(390\) 740.578 0.0961555
\(391\) 9444.85 1.22160
\(392\) 1092.62 0.140780
\(393\) 1939.54 0.248948
\(394\) 739.196 0.0945182
\(395\) −4708.06 −0.599717
\(396\) −1718.38 −0.218060
\(397\) 11618.0 1.46874 0.734372 0.678747i \(-0.237477\pi\)
0.734372 + 0.678747i \(0.237477\pi\)
\(398\) 797.889 0.100489
\(399\) −1146.53 −0.143856
\(400\) 72.3828 0.00904786
\(401\) 11157.1 1.38942 0.694711 0.719289i \(-0.255532\pi\)
0.694711 + 0.719289i \(0.255532\pi\)
\(402\) −2817.56 −0.349570
\(403\) 4487.54 0.554690
\(404\) 1726.12 0.212568
\(405\) −405.000 −0.0496904
\(406\) −1638.72 −0.200316
\(407\) −13095.8 −1.59492
\(408\) 3907.94 0.474196
\(409\) −7428.08 −0.898031 −0.449015 0.893524i \(-0.648225\pi\)
−0.449015 + 0.893524i \(0.648225\pi\)
\(410\) 3009.80 0.362545
\(411\) −2688.03 −0.322605
\(412\) 2892.83 0.345921
\(413\) 102.769 0.0122444
\(414\) −2475.90 −0.293922
\(415\) −1885.75 −0.223055
\(416\) −5318.97 −0.626885
\(417\) −6940.83 −0.815093
\(418\) 3474.75 0.406592
\(419\) 9644.74 1.12453 0.562263 0.826959i \(-0.309931\pi\)
0.562263 + 0.826959i \(0.309931\pi\)
\(420\) 535.992 0.0622708
\(421\) 9918.18 1.14818 0.574088 0.818793i \(-0.305357\pi\)
0.574088 + 0.818793i \(0.305357\pi\)
\(422\) −6356.21 −0.733212
\(423\) −4879.97 −0.560927
\(424\) 6807.92 0.779769
\(425\) 1460.47 0.166690
\(426\) 614.269 0.0698625
\(427\) −1197.48 −0.135715
\(428\) 2871.54 0.324302
\(429\) 3255.82 0.366417
\(430\) −4408.37 −0.494397
\(431\) −16324.1 −1.82437 −0.912185 0.409779i \(-0.865606\pi\)
−0.912185 + 0.409779i \(0.865606\pi\)
\(432\) 78.1735 0.00870630
\(433\) −5168.75 −0.573659 −0.286829 0.957982i \(-0.592601\pi\)
−0.286829 + 0.957982i \(0.592601\pi\)
\(434\) −1842.14 −0.203745
\(435\) −2063.72 −0.227466
\(436\) −9345.63 −1.02655
\(437\) −8826.95 −0.966248
\(438\) −1453.10 −0.158520
\(439\) 18339.0 1.99378 0.996892 0.0787782i \(-0.0251019\pi\)
0.996892 + 0.0787782i \(0.0251019\pi\)
\(440\) −4170.16 −0.451828
\(441\) 441.000 0.0476190
\(442\) −2884.24 −0.310383
\(443\) −1613.28 −0.173023 −0.0865113 0.996251i \(-0.527572\pi\)
−0.0865113 + 0.996251i \(0.527572\pi\)
\(444\) 5361.83 0.573111
\(445\) 3388.62 0.360980
\(446\) −1434.09 −0.152256
\(447\) 2458.01 0.260089
\(448\) 2021.30 0.213164
\(449\) −886.750 −0.0932034 −0.0466017 0.998914i \(-0.514839\pi\)
−0.0466017 + 0.998914i \(0.514839\pi\)
\(450\) −382.851 −0.0401062
\(451\) 13232.1 1.38154
\(452\) 162.345 0.0168940
\(453\) 1604.23 0.166387
\(454\) −1688.21 −0.174518
\(455\) −1015.55 −0.104636
\(456\) −3652.28 −0.375073
\(457\) 7391.22 0.756557 0.378279 0.925692i \(-0.376516\pi\)
0.378279 + 0.925692i \(0.376516\pi\)
\(458\) 10909.4 1.11302
\(459\) 1577.31 0.160397
\(460\) 4126.50 0.418259
\(461\) −7133.35 −0.720679 −0.360340 0.932821i \(-0.617339\pi\)
−0.360340 + 0.932821i \(0.617339\pi\)
\(462\) −1336.52 −0.134590
\(463\) 14461.8 1.45162 0.725808 0.687897i \(-0.241466\pi\)
0.725808 + 0.687897i \(0.241466\pi\)
\(464\) 398.341 0.0398546
\(465\) −2319.89 −0.231360
\(466\) 3869.95 0.384704
\(467\) −16393.5 −1.62441 −0.812206 0.583370i \(-0.801734\pi\)
−0.812206 + 0.583370i \(0.801734\pi\)
\(468\) −1333.04 −0.131666
\(469\) 3863.69 0.380403
\(470\) −4613.09 −0.452737
\(471\) 4694.28 0.459238
\(472\) 327.370 0.0319246
\(473\) −19380.7 −1.88398
\(474\) −4806.64 −0.465772
\(475\) −1364.92 −0.131846
\(476\) −2087.47 −0.201006
\(477\) 2747.78 0.263758
\(478\) −4871.77 −0.466171
\(479\) −12991.4 −1.23923 −0.619617 0.784904i \(-0.712712\pi\)
−0.619617 + 0.784904i \(0.712712\pi\)
\(480\) 2749.71 0.261472
\(481\) −10159.1 −0.963025
\(482\) 9127.57 0.862551
\(483\) 3395.17 0.319846
\(484\) −347.087 −0.0325964
\(485\) 6125.17 0.573463
\(486\) −413.480 −0.0385922
\(487\) −12863.5 −1.19692 −0.598461 0.801152i \(-0.704221\pi\)
−0.598461 + 0.801152i \(0.704221\pi\)
\(488\) −3814.57 −0.353847
\(489\) −3342.92 −0.309145
\(490\) 416.883 0.0384344
\(491\) −4898.10 −0.450200 −0.225100 0.974336i \(-0.572271\pi\)
−0.225100 + 0.974336i \(0.572271\pi\)
\(492\) −5417.63 −0.496435
\(493\) 8037.32 0.734245
\(494\) 2695.55 0.245503
\(495\) −1683.14 −0.152831
\(496\) 447.787 0.0405368
\(497\) −842.340 −0.0760244
\(498\) −1925.23 −0.173236
\(499\) 10308.0 0.924746 0.462373 0.886686i \(-0.346998\pi\)
0.462373 + 0.886686i \(0.346998\pi\)
\(500\) 638.086 0.0570721
\(501\) −5323.41 −0.474715
\(502\) −9477.37 −0.842621
\(503\) −15119.6 −1.34026 −0.670130 0.742244i \(-0.733762\pi\)
−0.670130 + 0.742244i \(0.733762\pi\)
\(504\) 1404.80 0.124156
\(505\) 1690.72 0.148982
\(506\) −10289.6 −0.904009
\(507\) −4065.28 −0.356105
\(508\) −11336.6 −0.990114
\(509\) 14183.8 1.23514 0.617571 0.786515i \(-0.288117\pi\)
0.617571 + 0.786515i \(0.288117\pi\)
\(510\) 1491.05 0.129460
\(511\) 1992.62 0.172501
\(512\) −1047.24 −0.0903943
\(513\) −1474.12 −0.126869
\(514\) −3565.39 −0.305958
\(515\) 2833.50 0.242444
\(516\) 7935.07 0.676981
\(517\) −20280.7 −1.72523
\(518\) 4170.32 0.353732
\(519\) −12647.6 −1.06969
\(520\) −3235.02 −0.272817
\(521\) −7464.08 −0.627653 −0.313827 0.949480i \(-0.601611\pi\)
−0.313827 + 0.949480i \(0.601611\pi\)
\(522\) −2106.93 −0.176662
\(523\) −16642.9 −1.39148 −0.695739 0.718295i \(-0.744923\pi\)
−0.695739 + 0.718295i \(0.744923\pi\)
\(524\) −3300.24 −0.275137
\(525\) 525.000 0.0436436
\(526\) 12703.7 1.05305
\(527\) 9035.01 0.746814
\(528\) 324.881 0.0267777
\(529\) 13971.8 1.14834
\(530\) 2597.51 0.212885
\(531\) 132.132 0.0107985
\(532\) 1950.90 0.158989
\(533\) 10264.8 0.834181
\(534\) 3459.57 0.280356
\(535\) 2812.66 0.227293
\(536\) 12307.8 0.991818
\(537\) −7292.09 −0.585991
\(538\) 11096.5 0.889230
\(539\) 1832.75 0.146461
\(540\) 689.133 0.0549177
\(541\) 67.8755 0.00539408 0.00269704 0.999996i \(-0.499142\pi\)
0.00269704 + 0.999996i \(0.499142\pi\)
\(542\) −4100.26 −0.324947
\(543\) −8102.74 −0.640372
\(544\) −10709.0 −0.844014
\(545\) −9153.97 −0.719473
\(546\) −1036.81 −0.0812662
\(547\) −9212.91 −0.720138 −0.360069 0.932926i \(-0.617247\pi\)
−0.360069 + 0.932926i \(0.617247\pi\)
\(548\) 4573.85 0.356542
\(549\) −1539.62 −0.119689
\(550\) −1591.09 −0.123354
\(551\) −7511.51 −0.580764
\(552\) 10815.3 0.833931
\(553\) 6591.29 0.506854
\(554\) 3777.17 0.289669
\(555\) 5251.87 0.401675
\(556\) 11810.2 0.900838
\(557\) 16699.6 1.27035 0.635173 0.772370i \(-0.280929\pi\)
0.635173 + 0.772370i \(0.280929\pi\)
\(558\) −2368.46 −0.179686
\(559\) −15034.6 −1.13756
\(560\) −101.336 −0.00764683
\(561\) 6555.13 0.493329
\(562\) −9934.67 −0.745674
\(563\) 14772.8 1.10586 0.552931 0.833227i \(-0.313509\pi\)
0.552931 + 0.833227i \(0.313509\pi\)
\(564\) 8303.57 0.619935
\(565\) 159.016 0.0118405
\(566\) 6202.26 0.460601
\(567\) 567.000 0.0419961
\(568\) −2683.27 −0.198217
\(569\) 5663.76 0.417289 0.208644 0.977992i \(-0.433095\pi\)
0.208644 + 0.977992i \(0.433095\pi\)
\(570\) −1393.50 −0.102399
\(571\) 5579.58 0.408929 0.204464 0.978874i \(-0.434455\pi\)
0.204464 + 0.978874i \(0.434455\pi\)
\(572\) −5539.99 −0.404962
\(573\) 10833.3 0.789821
\(574\) −4213.72 −0.306406
\(575\) 4041.87 0.293144
\(576\) 2598.82 0.187993
\(577\) −2301.23 −0.166034 −0.0830170 0.996548i \(-0.526456\pi\)
−0.0830170 + 0.996548i \(0.526456\pi\)
\(578\) 2552.77 0.183704
\(579\) −13405.0 −0.962162
\(580\) 3511.55 0.251395
\(581\) 2640.05 0.188516
\(582\) 6253.42 0.445382
\(583\) 11419.5 0.811232
\(584\) 6347.46 0.449760
\(585\) −1305.70 −0.0922806
\(586\) −6428.34 −0.453161
\(587\) −16470.4 −1.15810 −0.579052 0.815291i \(-0.696577\pi\)
−0.579052 + 0.815291i \(0.696577\pi\)
\(588\) −750.389 −0.0526284
\(589\) −8443.92 −0.590706
\(590\) 124.906 0.00871573
\(591\) −1303.27 −0.0907093
\(592\) −1013.72 −0.0703779
\(593\) −13570.0 −0.939715 −0.469858 0.882742i \(-0.655695\pi\)
−0.469858 + 0.882742i \(0.655695\pi\)
\(594\) −1718.38 −0.118697
\(595\) −2044.66 −0.140879
\(596\) −4182.46 −0.287450
\(597\) −1406.75 −0.0964393
\(598\) −7982.20 −0.545847
\(599\) 27814.1 1.89725 0.948625 0.316403i \(-0.102475\pi\)
0.948625 + 0.316403i \(0.102475\pi\)
\(600\) 1672.38 0.113791
\(601\) 20646.1 1.40128 0.700641 0.713514i \(-0.252898\pi\)
0.700641 + 0.713514i \(0.252898\pi\)
\(602\) 6171.72 0.417842
\(603\) 4967.61 0.335483
\(604\) −2729.70 −0.183891
\(605\) −339.969 −0.0228458
\(606\) 1726.12 0.115707
\(607\) 3315.28 0.221686 0.110843 0.993838i \(-0.464645\pi\)
0.110843 + 0.993838i \(0.464645\pi\)
\(608\) 10008.4 0.667588
\(609\) 2889.21 0.192244
\(610\) −1455.42 −0.0966037
\(611\) −15732.8 −1.04170
\(612\) −2683.88 −0.177271
\(613\) −11113.9 −0.732278 −0.366139 0.930560i \(-0.619320\pi\)
−0.366139 + 0.930560i \(0.619320\pi\)
\(614\) 8166.89 0.536790
\(615\) −5306.53 −0.347935
\(616\) 5838.22 0.381865
\(617\) 7871.34 0.513595 0.256797 0.966465i \(-0.417333\pi\)
0.256797 + 0.966465i \(0.417333\pi\)
\(618\) 2892.83 0.188295
\(619\) −19107.1 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(620\) 3947.44 0.255698
\(621\) 4365.22 0.282078
\(622\) −987.098 −0.0636319
\(623\) −4744.07 −0.305084
\(624\) 252.028 0.0161686
\(625\) 625.000 0.0400000
\(626\) 10404.7 0.664305
\(627\) −6126.28 −0.390208
\(628\) −7987.60 −0.507548
\(629\) −20453.9 −1.29658
\(630\) 535.992 0.0338959
\(631\) −25769.9 −1.62580 −0.812902 0.582401i \(-0.802113\pi\)
−0.812902 + 0.582401i \(0.802113\pi\)
\(632\) 20996.5 1.32151
\(633\) 11206.5 0.703665
\(634\) 7317.79 0.458401
\(635\) −11104.1 −0.693939
\(636\) −4675.53 −0.291504
\(637\) 1421.77 0.0884339
\(638\) −8756.19 −0.543355
\(639\) −1083.01 −0.0670472
\(640\) −4875.87 −0.301149
\(641\) −1954.61 −0.120440 −0.0602202 0.998185i \(-0.519180\pi\)
−0.0602202 + 0.998185i \(0.519180\pi\)
\(642\) 2871.54 0.176528
\(643\) −19396.5 −1.18961 −0.594807 0.803868i \(-0.702772\pi\)
−0.594807 + 0.803868i \(0.702772\pi\)
\(644\) −5777.10 −0.353493
\(645\) 7772.34 0.474474
\(646\) 5427.10 0.330536
\(647\) 31264.3 1.89973 0.949865 0.312661i \(-0.101220\pi\)
0.949865 + 0.312661i \(0.101220\pi\)
\(648\) 1806.17 0.109496
\(649\) 549.126 0.0332127
\(650\) −1234.30 −0.0744817
\(651\) 3247.85 0.195535
\(652\) 5688.18 0.341666
\(653\) 6442.75 0.386102 0.193051 0.981189i \(-0.438162\pi\)
0.193051 + 0.981189i \(0.438162\pi\)
\(654\) −9345.63 −0.558781
\(655\) −3232.56 −0.192835
\(656\) 1024.27 0.0609620
\(657\) 2561.93 0.152132
\(658\) 6458.33 0.382632
\(659\) −3584.70 −0.211897 −0.105949 0.994372i \(-0.533788\pi\)
−0.105949 + 0.994372i \(0.533788\pi\)
\(660\) 2863.97 0.168909
\(661\) 6294.43 0.370386 0.185193 0.982702i \(-0.440709\pi\)
0.185193 + 0.982702i \(0.440709\pi\)
\(662\) 11379.3 0.668078
\(663\) 5085.17 0.297876
\(664\) 8409.85 0.491514
\(665\) 1910.89 0.111430
\(666\) 5361.83 0.311962
\(667\) 22243.4 1.29126
\(668\) 9058.11 0.524654
\(669\) 2528.42 0.146120
\(670\) 4695.94 0.270776
\(671\) −6398.51 −0.368125
\(672\) −3849.60 −0.220984
\(673\) −10233.9 −0.586162 −0.293081 0.956088i \(-0.594681\pi\)
−0.293081 + 0.956088i \(0.594681\pi\)
\(674\) −9986.94 −0.570745
\(675\) 675.000 0.0384900
\(676\) 6917.33 0.393567
\(677\) −7100.75 −0.403108 −0.201554 0.979477i \(-0.564599\pi\)
−0.201554 + 0.979477i \(0.564599\pi\)
\(678\) 162.345 0.00919593
\(679\) −8575.24 −0.484665
\(680\) −6513.23 −0.367310
\(681\) 2976.45 0.167486
\(682\) −9843.10 −0.552657
\(683\) −35274.6 −1.97620 −0.988100 0.153813i \(-0.950845\pi\)
−0.988100 + 0.153813i \(0.950845\pi\)
\(684\) 2508.30 0.140215
\(685\) 4480.05 0.249889
\(686\) −583.636 −0.0324830
\(687\) −19234.2 −1.06817
\(688\) −1500.22 −0.0831330
\(689\) 8858.74 0.489828
\(690\) 4126.50 0.227671
\(691\) 4945.12 0.272245 0.136122 0.990692i \(-0.456536\pi\)
0.136122 + 0.990692i \(0.456536\pi\)
\(692\) 21520.8 1.18222
\(693\) 2356.40 0.129166
\(694\) −3296.31 −0.180297
\(695\) 11568.0 0.631368
\(696\) 9203.54 0.501235
\(697\) 20666.7 1.12311
\(698\) 16588.2 0.899532
\(699\) −6823.06 −0.369201
\(700\) −893.320 −0.0482348
\(701\) −15300.4 −0.824379 −0.412190 0.911098i \(-0.635236\pi\)
−0.412190 + 0.911098i \(0.635236\pi\)
\(702\) −1333.04 −0.0716701
\(703\) 19115.7 1.02555
\(704\) 10800.4 0.578206
\(705\) 8133.28 0.434492
\(706\) 7787.38 0.415130
\(707\) −2367.01 −0.125913
\(708\) −224.830 −0.0119345
\(709\) −28297.4 −1.49892 −0.749458 0.662052i \(-0.769686\pi\)
−0.749458 + 0.662052i \(0.769686\pi\)
\(710\) −1023.78 −0.0541153
\(711\) 8474.51 0.447003
\(712\) −15112.2 −0.795441
\(713\) 25004.5 1.31336
\(714\) −2087.47 −0.109414
\(715\) −5426.37 −0.283825
\(716\) 12407.9 0.647635
\(717\) 8589.35 0.447385
\(718\) −18461.9 −0.959597
\(719\) 8548.96 0.443425 0.221712 0.975112i \(-0.428835\pi\)
0.221712 + 0.975112i \(0.428835\pi\)
\(720\) −130.289 −0.00674387
\(721\) −3966.90 −0.204903
\(722\) 6598.97 0.340150
\(723\) −16092.7 −0.827792
\(724\) 13787.3 0.707737
\(725\) 3439.53 0.176194
\(726\) −347.087 −0.0177432
\(727\) 14345.3 0.731827 0.365913 0.930649i \(-0.380757\pi\)
0.365913 + 0.930649i \(0.380757\pi\)
\(728\) 4529.02 0.230572
\(729\) 729.000 0.0370370
\(730\) 2421.83 0.122789
\(731\) −30270.0 −1.53157
\(732\) 2619.76 0.132280
\(733\) −22624.0 −1.14002 −0.570012 0.821637i \(-0.693061\pi\)
−0.570012 + 0.821637i \(0.693061\pi\)
\(734\) −19512.9 −0.981248
\(735\) −735.000 −0.0368856
\(736\) −29637.3 −1.48430
\(737\) 20644.9 1.03184
\(738\) −5417.63 −0.270225
\(739\) 14837.3 0.738566 0.369283 0.929317i \(-0.379603\pi\)
0.369283 + 0.929317i \(0.379603\pi\)
\(740\) −8936.39 −0.443930
\(741\) −4752.49 −0.235610
\(742\) −3636.52 −0.179920
\(743\) 13073.1 0.645497 0.322749 0.946485i \(-0.395393\pi\)
0.322749 + 0.946485i \(0.395393\pi\)
\(744\) 10346.0 0.509815
\(745\) −4096.69 −0.201464
\(746\) −918.813 −0.0450940
\(747\) 3394.35 0.166255
\(748\) −11154.0 −0.545226
\(749\) −3937.72 −0.192098
\(750\) 638.086 0.0310661
\(751\) 16213.3 0.787791 0.393895 0.919155i \(-0.371127\pi\)
0.393895 + 0.919155i \(0.371127\pi\)
\(752\) −1569.89 −0.0761278
\(753\) 16709.4 0.808665
\(754\) −6792.65 −0.328082
\(755\) −2673.72 −0.128883
\(756\) −964.786 −0.0464139
\(757\) 19903.9 0.955642 0.477821 0.878457i \(-0.341427\pi\)
0.477821 + 0.878457i \(0.341427\pi\)
\(758\) −14595.3 −0.699372
\(759\) 18141.4 0.867580
\(760\) 6087.13 0.290531
\(761\) −30125.5 −1.43502 −0.717509 0.696549i \(-0.754718\pi\)
−0.717509 + 0.696549i \(0.754718\pi\)
\(762\) −11336.6 −0.538950
\(763\) 12815.6 0.608066
\(764\) −18433.5 −0.872907
\(765\) −2628.84 −0.124243
\(766\) −14681.0 −0.692489
\(767\) 425.986 0.0200541
\(768\) −11908.1 −0.559503
\(769\) −36049.1 −1.69046 −0.845230 0.534402i \(-0.820537\pi\)
−0.845230 + 0.534402i \(0.820537\pi\)
\(770\) 2227.53 0.104253
\(771\) 6286.09 0.293629
\(772\) 22809.4 1.06338
\(773\) −9644.77 −0.448769 −0.224384 0.974501i \(-0.572037\pi\)
−0.224384 + 0.974501i \(0.572037\pi\)
\(774\) 7935.07 0.368502
\(775\) 3866.48 0.179211
\(776\) −27316.4 −1.26366
\(777\) −7352.62 −0.339477
\(778\) −15712.3 −0.724054
\(779\) −19314.7 −0.888344
\(780\) 2221.73 0.101988
\(781\) −4500.88 −0.206215
\(782\) −16071.0 −0.734908
\(783\) 3714.69 0.169543
\(784\) 141.870 0.00646275
\(785\) −7823.80 −0.355724
\(786\) −3300.24 −0.149766
\(787\) −25218.6 −1.14224 −0.571122 0.820865i \(-0.693492\pi\)
−0.571122 + 0.820865i \(0.693492\pi\)
\(788\) 2217.59 0.100252
\(789\) −22397.6 −1.01062
\(790\) 8011.06 0.360786
\(791\) −222.623 −0.0100070
\(792\) 7506.28 0.336773
\(793\) −4963.67 −0.222276
\(794\) −19768.8 −0.883586
\(795\) −4579.64 −0.204306
\(796\) 2393.67 0.106584
\(797\) 32042.3 1.42409 0.712044 0.702135i \(-0.247770\pi\)
0.712044 + 0.702135i \(0.247770\pi\)
\(798\) 1950.90 0.0865427
\(799\) −31675.7 −1.40251
\(800\) −4582.85 −0.202535
\(801\) −6099.52 −0.269059
\(802\) −18984.5 −0.835867
\(803\) 10647.1 0.467908
\(804\) −8452.69 −0.370775
\(805\) −5658.62 −0.247752
\(806\) −7635.82 −0.333698
\(807\) −19564.1 −0.853396
\(808\) −7540.07 −0.328291
\(809\) 3427.90 0.148972 0.0744860 0.997222i \(-0.476268\pi\)
0.0744860 + 0.997222i \(0.476268\pi\)
\(810\) 689.133 0.0298934
\(811\) 23094.4 0.999943 0.499972 0.866042i \(-0.333344\pi\)
0.499972 + 0.866042i \(0.333344\pi\)
\(812\) −4916.16 −0.212467
\(813\) 7229.11 0.311852
\(814\) 22283.3 0.959494
\(815\) 5571.53 0.239463
\(816\) 507.422 0.0217688
\(817\) 28289.7 1.21142
\(818\) 12639.3 0.540249
\(819\) 1827.98 0.0779914
\(820\) 9029.39 0.384537
\(821\) 474.741 0.0201810 0.0100905 0.999949i \(-0.496788\pi\)
0.0100905 + 0.999949i \(0.496788\pi\)
\(822\) 4573.85 0.194077
\(823\) 24159.8 1.02328 0.511638 0.859201i \(-0.329039\pi\)
0.511638 + 0.859201i \(0.329039\pi\)
\(824\) −12636.5 −0.534240
\(825\) 2805.23 0.118383
\(826\) −174.868 −0.00736613
\(827\) −7566.35 −0.318147 −0.159074 0.987267i \(-0.550851\pi\)
−0.159074 + 0.987267i \(0.550851\pi\)
\(828\) −7427.70 −0.311752
\(829\) −10580.9 −0.443295 −0.221648 0.975127i \(-0.571143\pi\)
−0.221648 + 0.975127i \(0.571143\pi\)
\(830\) 3208.72 0.134188
\(831\) −6659.48 −0.277996
\(832\) 8378.49 0.349125
\(833\) 2862.52 0.119064
\(834\) 11810.2 0.490354
\(835\) 8872.34 0.367713
\(836\) 10424.2 0.431256
\(837\) 4175.80 0.172445
\(838\) −16411.1 −0.676507
\(839\) 15315.6 0.630218 0.315109 0.949055i \(-0.397959\pi\)
0.315109 + 0.949055i \(0.397959\pi\)
\(840\) −2341.34 −0.0961712
\(841\) −5460.40 −0.223888
\(842\) −16876.4 −0.690735
\(843\) 17515.7 0.715625
\(844\) −19068.6 −0.777688
\(845\) 6775.47 0.275838
\(846\) 8303.57 0.337450
\(847\) 475.956 0.0193082
\(848\) 883.966 0.0357966
\(849\) −10935.1 −0.442040
\(850\) −2485.08 −0.100279
\(851\) −56606.4 −2.28019
\(852\) 1842.81 0.0741004
\(853\) 18598.2 0.746528 0.373264 0.927725i \(-0.378239\pi\)
0.373264 + 0.927725i \(0.378239\pi\)
\(854\) 2037.59 0.0816450
\(855\) 2456.86 0.0982723
\(856\) −12543.6 −0.500853
\(857\) 41775.3 1.66513 0.832566 0.553926i \(-0.186871\pi\)
0.832566 + 0.553926i \(0.186871\pi\)
\(858\) −5539.99 −0.220434
\(859\) 32414.7 1.28752 0.643758 0.765229i \(-0.277374\pi\)
0.643758 + 0.765229i \(0.277374\pi\)
\(860\) −13225.1 −0.524387
\(861\) 7429.14 0.294059
\(862\) 27776.4 1.09753
\(863\) −31299.7 −1.23459 −0.617297 0.786730i \(-0.711772\pi\)
−0.617297 + 0.786730i \(0.711772\pi\)
\(864\) −4949.48 −0.194890
\(865\) 21079.4 0.828580
\(866\) 8794.94 0.345109
\(867\) −4500.75 −0.176302
\(868\) −5526.41 −0.216104
\(869\) 35219.2 1.37483
\(870\) 3511.55 0.136842
\(871\) 16015.4 0.623030
\(872\) 40823.8 1.58540
\(873\) −11025.3 −0.427434
\(874\) 15019.6 0.581288
\(875\) −875.000 −0.0338062
\(876\) −4359.29 −0.168136
\(877\) −19973.0 −0.769031 −0.384515 0.923119i \(-0.625631\pi\)
−0.384515 + 0.923119i \(0.625631\pi\)
\(878\) −31204.9 −1.19945
\(879\) 11333.7 0.434900
\(880\) −541.469 −0.0207419
\(881\) 17367.9 0.664176 0.332088 0.943248i \(-0.392247\pi\)
0.332088 + 0.943248i \(0.392247\pi\)
\(882\) −750.389 −0.0286473
\(883\) −14364.2 −0.547446 −0.273723 0.961809i \(-0.588255\pi\)
−0.273723 + 0.961809i \(0.588255\pi\)
\(884\) −8652.73 −0.329211
\(885\) −220.219 −0.00836450
\(886\) 2745.09 0.104089
\(887\) −33738.1 −1.27713 −0.638564 0.769568i \(-0.720471\pi\)
−0.638564 + 0.769568i \(0.720471\pi\)
\(888\) −23421.7 −0.885114
\(889\) 15545.7 0.586485
\(890\) −5765.95 −0.217163
\(891\) 3029.65 0.113914
\(892\) −4302.26 −0.161491
\(893\) 29603.4 1.10934
\(894\) −4182.46 −0.156468
\(895\) 12153.5 0.453906
\(896\) 6826.21 0.254518
\(897\) 14073.3 0.523850
\(898\) 1508.86 0.0560705
\(899\) 21278.2 0.789398
\(900\) −1148.55 −0.0425391
\(901\) 17835.8 0.659485
\(902\) −22515.2 −0.831123
\(903\) −10881.3 −0.401004
\(904\) −709.162 −0.0260911
\(905\) 13504.6 0.496030
\(906\) −2729.70 −0.100097
\(907\) −32998.6 −1.20805 −0.604024 0.796966i \(-0.706437\pi\)
−0.604024 + 0.796966i \(0.706437\pi\)
\(908\) −5064.62 −0.185105
\(909\) −3043.29 −0.111045
\(910\) 1728.02 0.0629485
\(911\) 33446.3 1.21638 0.608192 0.793790i \(-0.291895\pi\)
0.608192 + 0.793790i \(0.291895\pi\)
\(912\) −474.225 −0.0172184
\(913\) 14106.6 0.511347
\(914\) −12576.6 −0.455139
\(915\) 2566.03 0.0927108
\(916\) 32728.2 1.18053
\(917\) 4525.59 0.162975
\(918\) −2683.88 −0.0964939
\(919\) 41708.7 1.49711 0.748554 0.663074i \(-0.230748\pi\)
0.748554 + 0.663074i \(0.230748\pi\)
\(920\) −18025.5 −0.645960
\(921\) −14398.9 −0.515158
\(922\) 12137.8 0.433556
\(923\) −3491.58 −0.124514
\(924\) −4009.56 −0.142754
\(925\) −8753.12 −0.311136
\(926\) −24607.7 −0.873283
\(927\) −5100.30 −0.180707
\(928\) −25220.6 −0.892140
\(929\) 49024.6 1.73137 0.865686 0.500587i \(-0.166883\pi\)
0.865686 + 0.500587i \(0.166883\pi\)
\(930\) 3947.44 0.139184
\(931\) −2675.25 −0.0941758
\(932\) 11609.9 0.408040
\(933\) 1740.34 0.0610677
\(934\) 27894.6 0.977235
\(935\) −10925.2 −0.382131
\(936\) 5823.03 0.203346
\(937\) −5447.58 −0.189930 −0.0949651 0.995481i \(-0.530274\pi\)
−0.0949651 + 0.995481i \(0.530274\pi\)
\(938\) −6574.31 −0.228848
\(939\) −18344.4 −0.637535
\(940\) −13839.3 −0.480200
\(941\) 4125.75 0.142928 0.0714642 0.997443i \(-0.477233\pi\)
0.0714642 + 0.997443i \(0.477233\pi\)
\(942\) −7987.60 −0.276274
\(943\) 57195.5 1.97513
\(944\) 42.5069 0.00146555
\(945\) −945.000 −0.0325300
\(946\) 32977.4 1.13339
\(947\) 17332.3 0.594747 0.297374 0.954761i \(-0.403889\pi\)
0.297374 + 0.954761i \(0.403889\pi\)
\(948\) −14419.9 −0.494026
\(949\) 8259.57 0.282526
\(950\) 2322.50 0.0793177
\(951\) −12901.9 −0.439929
\(952\) 9118.53 0.310434
\(953\) 56839.4 1.93201 0.966007 0.258517i \(-0.0832337\pi\)
0.966007 + 0.258517i \(0.0832337\pi\)
\(954\) −4675.53 −0.158675
\(955\) −18055.5 −0.611792
\(956\) −14615.3 −0.494449
\(957\) 15437.9 0.521459
\(958\) 22105.7 0.745515
\(959\) −6272.07 −0.211195
\(960\) −4331.37 −0.145619
\(961\) −5871.48 −0.197089
\(962\) 17286.3 0.579349
\(963\) −5062.78 −0.169414
\(964\) 27382.7 0.914873
\(965\) 22341.6 0.745287
\(966\) −5777.10 −0.192417
\(967\) −13284.2 −0.441769 −0.220884 0.975300i \(-0.570894\pi\)
−0.220884 + 0.975300i \(0.570894\pi\)
\(968\) 1516.15 0.0503420
\(969\) −9568.44 −0.317216
\(970\) −10422.4 −0.344992
\(971\) 12153.0 0.401658 0.200829 0.979626i \(-0.435636\pi\)
0.200829 + 0.979626i \(0.435636\pi\)
\(972\) −1240.44 −0.0409332
\(973\) −16195.3 −0.533604
\(974\) 21888.1 0.720060
\(975\) 2176.17 0.0714803
\(976\) −495.298 −0.0162440
\(977\) −37999.3 −1.24433 −0.622163 0.782888i \(-0.713746\pi\)
−0.622163 + 0.782888i \(0.713746\pi\)
\(978\) 5688.18 0.185980
\(979\) −25349.0 −0.827537
\(980\) 1250.65 0.0407658
\(981\) 16477.1 0.536264
\(982\) 8334.42 0.270837
\(983\) 22375.0 0.725993 0.362996 0.931791i \(-0.381754\pi\)
0.362996 + 0.931791i \(0.381754\pi\)
\(984\) 23665.5 0.766695
\(985\) 2172.11 0.0702631
\(986\) −13676.0 −0.441717
\(987\) −11386.6 −0.367213
\(988\) 8086.65 0.260395
\(989\) −83772.9 −2.69345
\(990\) 2863.97 0.0919423
\(991\) 18985.3 0.608564 0.304282 0.952582i \(-0.401583\pi\)
0.304282 + 0.952582i \(0.401583\pi\)
\(992\) −28351.2 −0.907412
\(993\) −20062.6 −0.641156
\(994\) 1433.29 0.0457358
\(995\) 2344.58 0.0747016
\(996\) −5775.70 −0.183745
\(997\) −56476.5 −1.79401 −0.897006 0.442019i \(-0.854262\pi\)
−0.897006 + 0.442019i \(0.854262\pi\)
\(998\) −17539.6 −0.556321
\(999\) −9453.37 −0.299391
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 105.4.a.g.1.1 2
3.2 odd 2 315.4.a.g.1.2 2
4.3 odd 2 1680.4.a.y.1.1 2
5.2 odd 4 525.4.d.j.274.2 4
5.3 odd 4 525.4.d.j.274.3 4
5.4 even 2 525.4.a.i.1.2 2
7.6 odd 2 735.4.a.q.1.1 2
15.14 odd 2 1575.4.a.y.1.1 2
21.20 even 2 2205.4.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.1 2 1.1 even 1 trivial
315.4.a.g.1.2 2 3.2 odd 2
525.4.a.i.1.2 2 5.4 even 2
525.4.d.j.274.2 4 5.2 odd 4
525.4.d.j.274.3 4 5.3 odd 4
735.4.a.q.1.1 2 7.6 odd 2
1575.4.a.y.1.1 2 15.14 odd 2
1680.4.a.y.1.1 2 4.3 odd 2
2205.4.a.v.1.2 2 21.20 even 2