Properties

Label 8007.2.a.f.1.21
Level 8007
Weight 2
Character 8007.1
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 48
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.21
Character \(\chi\) = 8007.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +1.14691 q^{10} -3.76470 q^{11} +1.73563 q^{12} -1.73173 q^{13} -0.611367 q^{14} +2.23060 q^{15} +2.48367 q^{16} -1.00000 q^{17} -0.514170 q^{18} +1.01761 q^{19} +3.87149 q^{20} -1.18904 q^{21} +1.93569 q^{22} +2.15719 q^{23} -1.92075 q^{24} -0.0244263 q^{25} +0.890405 q^{26} -1.00000 q^{27} -2.06373 q^{28} -1.45443 q^{29} -1.14691 q^{30} -4.31473 q^{31} -5.11852 q^{32} +3.76470 q^{33} +0.514170 q^{34} -2.65226 q^{35} -1.73563 q^{36} -1.11433 q^{37} -0.523226 q^{38} +1.73173 q^{39} -4.28442 q^{40} +2.21959 q^{41} +0.611367 q^{42} +9.50603 q^{43} +6.53412 q^{44} -2.23060 q^{45} -1.10916 q^{46} +5.01983 q^{47} -2.48367 q^{48} -5.58619 q^{49} +0.0125592 q^{50} +1.00000 q^{51} +3.00565 q^{52} +13.2050 q^{53} +0.514170 q^{54} +8.39753 q^{55} +2.28384 q^{56} -1.01761 q^{57} +0.747824 q^{58} +3.08150 q^{59} -3.87149 q^{60} -9.73285 q^{61} +2.21851 q^{62} +1.18904 q^{63} -2.33555 q^{64} +3.86280 q^{65} -1.93569 q^{66} -1.07604 q^{67} +1.73563 q^{68} -2.15719 q^{69} +1.36371 q^{70} -13.7666 q^{71} +1.92075 q^{72} +5.31205 q^{73} +0.572955 q^{74} +0.0244263 q^{75} -1.76620 q^{76} -4.47636 q^{77} -0.890405 q^{78} -9.97851 q^{79} -5.54007 q^{80} +1.00000 q^{81} -1.14124 q^{82} +14.0231 q^{83} +2.06373 q^{84} +2.23060 q^{85} -4.88771 q^{86} +1.45443 q^{87} -7.23103 q^{88} +7.10670 q^{89} +1.14691 q^{90} -2.05909 q^{91} -3.74409 q^{92} +4.31473 q^{93} -2.58104 q^{94} -2.26989 q^{95} +5.11852 q^{96} +1.44167 q^{97} +2.87225 q^{98} -3.76470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} + O(q^{10}) \) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} - 20q^{10} + 5q^{11} - 45q^{12} - 8q^{13} + 4q^{14} - q^{15} + 39q^{16} - 48q^{17} - q^{18} - 6q^{19} + 6q^{20} + 13q^{21} - 35q^{22} - 8q^{23} + 6q^{24} + 13q^{25} + 17q^{26} - 48q^{27} - 38q^{28} + q^{29} + 20q^{30} - 21q^{31} - 3q^{32} - 5q^{33} + q^{34} + 19q^{35} + 45q^{36} - 58q^{37} - 14q^{38} + 8q^{39} - 54q^{40} - 3q^{41} - 4q^{42} - 33q^{43} + 2q^{44} + q^{45} - 26q^{46} + 9q^{47} - 39q^{48} + 11q^{49} + 4q^{50} + 48q^{51} - 31q^{52} - 33q^{53} + q^{54} - 21q^{55} + 6q^{57} - 55q^{58} + 77q^{59} - 6q^{60} - 29q^{61} - 46q^{62} - 13q^{63} + 24q^{64} - 49q^{65} + 35q^{66} - 44q^{67} - 45q^{68} + 8q^{69} + 4q^{70} + 22q^{71} - 6q^{72} - 63q^{73} - 16q^{74} - 13q^{75} - 46q^{76} - 30q^{77} - 17q^{78} - 46q^{79} - 14q^{80} + 48q^{81} - 75q^{82} + 11q^{83} + 38q^{84} - q^{85} + 8q^{86} - q^{87} - 116q^{88} + 10q^{89} - 20q^{90} - 67q^{91} - 64q^{92} + 21q^{93} - 16q^{94} - 8q^{95} + 3q^{96} - 96q^{97} - 46q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.514170 −0.363573 −0.181786 0.983338i \(-0.558188\pi\)
−0.181786 + 0.983338i \(0.558188\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.73563 −0.867815
\(5\) −2.23060 −0.997554 −0.498777 0.866730i \(-0.666217\pi\)
−0.498777 + 0.866730i \(0.666217\pi\)
\(6\) 0.514170 0.209909
\(7\) 1.18904 0.449414 0.224707 0.974426i \(-0.427858\pi\)
0.224707 + 0.974426i \(0.427858\pi\)
\(8\) 1.92075 0.679087
\(9\) 1.00000 0.333333
\(10\) 1.14691 0.362684
\(11\) −3.76470 −1.13510 −0.567549 0.823339i \(-0.692108\pi\)
−0.567549 + 0.823339i \(0.692108\pi\)
\(12\) 1.73563 0.501033
\(13\) −1.73173 −0.480296 −0.240148 0.970736i \(-0.577196\pi\)
−0.240148 + 0.970736i \(0.577196\pi\)
\(14\) −0.611367 −0.163395
\(15\) 2.23060 0.575938
\(16\) 2.48367 0.620917
\(17\) −1.00000 −0.242536
\(18\) −0.514170 −0.121191
\(19\) 1.01761 0.233456 0.116728 0.993164i \(-0.462759\pi\)
0.116728 + 0.993164i \(0.462759\pi\)
\(20\) 3.87149 0.865692
\(21\) −1.18904 −0.259469
\(22\) 1.93569 0.412691
\(23\) 2.15719 0.449806 0.224903 0.974381i \(-0.427793\pi\)
0.224903 + 0.974381i \(0.427793\pi\)
\(24\) −1.92075 −0.392071
\(25\) −0.0244263 −0.00488525
\(26\) 0.890405 0.174623
\(27\) −1.00000 −0.192450
\(28\) −2.06373 −0.390008
\(29\) −1.45443 −0.270081 −0.135040 0.990840i \(-0.543116\pi\)
−0.135040 + 0.990840i \(0.543116\pi\)
\(30\) −1.14691 −0.209396
\(31\) −4.31473 −0.774949 −0.387475 0.921880i \(-0.626652\pi\)
−0.387475 + 0.921880i \(0.626652\pi\)
\(32\) −5.11852 −0.904836
\(33\) 3.76470 0.655349
\(34\) 0.514170 0.0881794
\(35\) −2.65226 −0.448314
\(36\) −1.73563 −0.289272
\(37\) −1.11433 −0.183195 −0.0915974 0.995796i \(-0.529197\pi\)
−0.0915974 + 0.995796i \(0.529197\pi\)
\(38\) −0.523226 −0.0848784
\(39\) 1.73173 0.277299
\(40\) −4.28442 −0.677426
\(41\) 2.21959 0.346641 0.173321 0.984865i \(-0.444550\pi\)
0.173321 + 0.984865i \(0.444550\pi\)
\(42\) 0.611367 0.0943359
\(43\) 9.50603 1.44966 0.724828 0.688930i \(-0.241919\pi\)
0.724828 + 0.688930i \(0.241919\pi\)
\(44\) 6.53412 0.985055
\(45\) −2.23060 −0.332518
\(46\) −1.10916 −0.163537
\(47\) 5.01983 0.732217 0.366109 0.930572i \(-0.380690\pi\)
0.366109 + 0.930572i \(0.380690\pi\)
\(48\) −2.48367 −0.358487
\(49\) −5.58619 −0.798028
\(50\) 0.0125592 0.00177615
\(51\) 1.00000 0.140028
\(52\) 3.00565 0.416808
\(53\) 13.2050 1.81384 0.906921 0.421301i \(-0.138426\pi\)
0.906921 + 0.421301i \(0.138426\pi\)
\(54\) 0.514170 0.0699696
\(55\) 8.39753 1.13232
\(56\) 2.28384 0.305191
\(57\) −1.01761 −0.134786
\(58\) 0.747824 0.0981941
\(59\) 3.08150 0.401177 0.200589 0.979676i \(-0.435715\pi\)
0.200589 + 0.979676i \(0.435715\pi\)
\(60\) −3.87149 −0.499808
\(61\) −9.73285 −1.24616 −0.623082 0.782156i \(-0.714120\pi\)
−0.623082 + 0.782156i \(0.714120\pi\)
\(62\) 2.21851 0.281751
\(63\) 1.18904 0.149805
\(64\) −2.33555 −0.291943
\(65\) 3.86280 0.479122
\(66\) −1.93569 −0.238267
\(67\) −1.07604 −0.131459 −0.0657294 0.997837i \(-0.520937\pi\)
−0.0657294 + 0.997837i \(0.520937\pi\)
\(68\) 1.73563 0.210476
\(69\) −2.15719 −0.259696
\(70\) 1.36371 0.162995
\(71\) −13.7666 −1.63380 −0.816898 0.576782i \(-0.804308\pi\)
−0.816898 + 0.576782i \(0.804308\pi\)
\(72\) 1.92075 0.226362
\(73\) 5.31205 0.621729 0.310864 0.950454i \(-0.399381\pi\)
0.310864 + 0.950454i \(0.399381\pi\)
\(74\) 0.572955 0.0666047
\(75\) 0.0244263 0.00282050
\(76\) −1.76620 −0.202597
\(77\) −4.47636 −0.510129
\(78\) −0.890405 −0.100819
\(79\) −9.97851 −1.12267 −0.561335 0.827589i \(-0.689712\pi\)
−0.561335 + 0.827589i \(0.689712\pi\)
\(80\) −5.54007 −0.619399
\(81\) 1.00000 0.111111
\(82\) −1.14124 −0.126029
\(83\) 14.0231 1.53923 0.769617 0.638506i \(-0.220447\pi\)
0.769617 + 0.638506i \(0.220447\pi\)
\(84\) 2.06373 0.225171
\(85\) 2.23060 0.241942
\(86\) −4.88771 −0.527056
\(87\) 1.45443 0.155931
\(88\) −7.23103 −0.770831
\(89\) 7.10670 0.753308 0.376654 0.926354i \(-0.377075\pi\)
0.376654 + 0.926354i \(0.377075\pi\)
\(90\) 1.14691 0.120895
\(91\) −2.05909 −0.215852
\(92\) −3.74409 −0.390348
\(93\) 4.31473 0.447417
\(94\) −2.58104 −0.266214
\(95\) −2.26989 −0.232885
\(96\) 5.11852 0.522407
\(97\) 1.44167 0.146380 0.0731899 0.997318i \(-0.476682\pi\)
0.0731899 + 0.997318i \(0.476682\pi\)
\(98\) 2.87225 0.290141
\(99\) −3.76470 −0.378366
\(100\) 0.0423949 0.00423949
\(101\) 10.1000 1.00499 0.502495 0.864580i \(-0.332416\pi\)
0.502495 + 0.864580i \(0.332416\pi\)
\(102\) −0.514170 −0.0509104
\(103\) 10.6269 1.04710 0.523549 0.851996i \(-0.324608\pi\)
0.523549 + 0.851996i \(0.324608\pi\)
\(104\) −3.32622 −0.326163
\(105\) 2.65226 0.258834
\(106\) −6.78960 −0.659464
\(107\) −10.1754 −0.983689 −0.491845 0.870683i \(-0.663677\pi\)
−0.491845 + 0.870683i \(0.663677\pi\)
\(108\) 1.73563 0.167011
\(109\) −1.10382 −0.105727 −0.0528636 0.998602i \(-0.516835\pi\)
−0.0528636 + 0.998602i \(0.516835\pi\)
\(110\) −4.31776 −0.411682
\(111\) 1.11433 0.105768
\(112\) 2.95317 0.279049
\(113\) −0.726543 −0.0683475 −0.0341737 0.999416i \(-0.510880\pi\)
−0.0341737 + 0.999416i \(0.510880\pi\)
\(114\) 0.523226 0.0490046
\(115\) −4.81184 −0.448706
\(116\) 2.52435 0.234380
\(117\) −1.73173 −0.160099
\(118\) −1.58442 −0.145857
\(119\) −1.18904 −0.108999
\(120\) 4.28442 0.391112
\(121\) 3.17293 0.288449
\(122\) 5.00434 0.453072
\(123\) −2.21959 −0.200133
\(124\) 7.48878 0.672512
\(125\) 11.2075 1.00243
\(126\) −0.611367 −0.0544649
\(127\) 16.5326 1.46703 0.733517 0.679671i \(-0.237877\pi\)
0.733517 + 0.679671i \(0.237877\pi\)
\(128\) 11.4379 1.01098
\(129\) −9.50603 −0.836959
\(130\) −1.98614 −0.174196
\(131\) 5.81654 0.508194 0.254097 0.967179i \(-0.418222\pi\)
0.254097 + 0.967179i \(0.418222\pi\)
\(132\) −6.53412 −0.568722
\(133\) 1.20998 0.104918
\(134\) 0.553265 0.0477948
\(135\) 2.23060 0.191979
\(136\) −1.92075 −0.164703
\(137\) 21.1478 1.80678 0.903389 0.428821i \(-0.141071\pi\)
0.903389 + 0.428821i \(0.141071\pi\)
\(138\) 1.10916 0.0944183
\(139\) −15.9119 −1.34963 −0.674814 0.737988i \(-0.735776\pi\)
−0.674814 + 0.737988i \(0.735776\pi\)
\(140\) 4.60335 0.389054
\(141\) −5.01983 −0.422746
\(142\) 7.07838 0.594004
\(143\) 6.51945 0.545184
\(144\) 2.48367 0.206972
\(145\) 3.24425 0.269420
\(146\) −2.73130 −0.226044
\(147\) 5.58619 0.460741
\(148\) 1.93407 0.158979
\(149\) 20.2298 1.65729 0.828644 0.559776i \(-0.189113\pi\)
0.828644 + 0.559776i \(0.189113\pi\)
\(150\) −0.0125592 −0.00102546
\(151\) 10.1850 0.828841 0.414421 0.910085i \(-0.363984\pi\)
0.414421 + 0.910085i \(0.363984\pi\)
\(152\) 1.95458 0.158537
\(153\) −1.00000 −0.0808452
\(154\) 2.30161 0.185469
\(155\) 9.62444 0.773054
\(156\) −3.00565 −0.240644
\(157\) −1.00000 −0.0798087
\(158\) 5.13065 0.408172
\(159\) −13.2050 −1.04722
\(160\) 11.4174 0.902623
\(161\) 2.56498 0.202149
\(162\) −0.514170 −0.0403970
\(163\) −7.74482 −0.606621 −0.303311 0.952892i \(-0.598092\pi\)
−0.303311 + 0.952892i \(0.598092\pi\)
\(164\) −3.85238 −0.300820
\(165\) −8.39753 −0.653747
\(166\) −7.21024 −0.559624
\(167\) 2.88188 0.223007 0.111503 0.993764i \(-0.464433\pi\)
0.111503 + 0.993764i \(0.464433\pi\)
\(168\) −2.28384 −0.176202
\(169\) −10.0011 −0.769315
\(170\) −1.14691 −0.0879637
\(171\) 1.01761 0.0778188
\(172\) −16.4989 −1.25803
\(173\) −11.1660 −0.848938 −0.424469 0.905443i \(-0.639539\pi\)
−0.424469 + 0.905443i \(0.639539\pi\)
\(174\) −0.747824 −0.0566924
\(175\) −0.0290437 −0.00219550
\(176\) −9.35026 −0.704802
\(177\) −3.08150 −0.231620
\(178\) −3.65405 −0.273883
\(179\) 8.15506 0.609538 0.304769 0.952426i \(-0.401421\pi\)
0.304769 + 0.952426i \(0.401421\pi\)
\(180\) 3.87149 0.288564
\(181\) 6.14306 0.456610 0.228305 0.973590i \(-0.426682\pi\)
0.228305 + 0.973590i \(0.426682\pi\)
\(182\) 1.05872 0.0784778
\(183\) 9.73285 0.719473
\(184\) 4.14343 0.305457
\(185\) 2.48563 0.182747
\(186\) −2.21851 −0.162669
\(187\) 3.76470 0.275302
\(188\) −8.71256 −0.635429
\(189\) −1.18904 −0.0864897
\(190\) 1.16711 0.0846708
\(191\) −6.14219 −0.444433 −0.222217 0.974997i \(-0.571329\pi\)
−0.222217 + 0.974997i \(0.571329\pi\)
\(192\) 2.33555 0.168554
\(193\) −1.60323 −0.115403 −0.0577015 0.998334i \(-0.518377\pi\)
−0.0577015 + 0.998334i \(0.518377\pi\)
\(194\) −0.741265 −0.0532197
\(195\) −3.86280 −0.276621
\(196\) 9.69556 0.692540
\(197\) 5.34589 0.380879 0.190440 0.981699i \(-0.439009\pi\)
0.190440 + 0.981699i \(0.439009\pi\)
\(198\) 1.93569 0.137564
\(199\) −4.94252 −0.350366 −0.175183 0.984536i \(-0.556052\pi\)
−0.175183 + 0.984536i \(0.556052\pi\)
\(200\) −0.0469167 −0.00331751
\(201\) 1.07604 0.0758977
\(202\) −5.19313 −0.365387
\(203\) −1.72937 −0.121378
\(204\) −1.73563 −0.121518
\(205\) −4.95101 −0.345794
\(206\) −5.46402 −0.380697
\(207\) 2.15719 0.149935
\(208\) −4.30105 −0.298224
\(209\) −3.83100 −0.264996
\(210\) −1.36371 −0.0941052
\(211\) 5.63736 0.388092 0.194046 0.980992i \(-0.437839\pi\)
0.194046 + 0.980992i \(0.437839\pi\)
\(212\) −22.9189 −1.57408
\(213\) 13.7666 0.943273
\(214\) 5.23186 0.357643
\(215\) −21.2041 −1.44611
\(216\) −1.92075 −0.130690
\(217\) −5.13038 −0.348273
\(218\) 0.567553 0.0384395
\(219\) −5.31205 −0.358955
\(220\) −14.5750 −0.982646
\(221\) 1.73173 0.116489
\(222\) −0.572955 −0.0384542
\(223\) −2.41484 −0.161710 −0.0808550 0.996726i \(-0.525765\pi\)
−0.0808550 + 0.996726i \(0.525765\pi\)
\(224\) −6.08611 −0.406645
\(225\) −0.0244263 −0.00162842
\(226\) 0.373567 0.0248493
\(227\) 22.6950 1.50632 0.753162 0.657836i \(-0.228528\pi\)
0.753162 + 0.657836i \(0.228528\pi\)
\(228\) 1.76620 0.116969
\(229\) 2.88354 0.190550 0.0952750 0.995451i \(-0.469627\pi\)
0.0952750 + 0.995451i \(0.469627\pi\)
\(230\) 2.47410 0.163137
\(231\) 4.47636 0.294523
\(232\) −2.79359 −0.183408
\(233\) 17.2482 1.12997 0.564983 0.825102i \(-0.308883\pi\)
0.564983 + 0.825102i \(0.308883\pi\)
\(234\) 0.890405 0.0582076
\(235\) −11.1972 −0.730426
\(236\) −5.34835 −0.348148
\(237\) 9.97851 0.648174
\(238\) 0.611367 0.0396290
\(239\) −27.3485 −1.76903 −0.884515 0.466512i \(-0.845511\pi\)
−0.884515 + 0.466512i \(0.845511\pi\)
\(240\) 5.54007 0.357610
\(241\) 2.84765 0.183433 0.0917165 0.995785i \(-0.470765\pi\)
0.0917165 + 0.995785i \(0.470765\pi\)
\(242\) −1.63143 −0.104872
\(243\) −1.00000 −0.0641500
\(244\) 16.8926 1.08144
\(245\) 12.4606 0.796076
\(246\) 1.14124 0.0727631
\(247\) −1.76223 −0.112128
\(248\) −8.28752 −0.526258
\(249\) −14.0231 −0.888677
\(250\) −5.76255 −0.364456
\(251\) −24.5200 −1.54769 −0.773845 0.633375i \(-0.781669\pi\)
−0.773845 + 0.633375i \(0.781669\pi\)
\(252\) −2.06373 −0.130003
\(253\) −8.12118 −0.510574
\(254\) −8.50058 −0.533374
\(255\) −2.23060 −0.139686
\(256\) −1.20994 −0.0756210
\(257\) −28.6769 −1.78881 −0.894406 0.447255i \(-0.852402\pi\)
−0.894406 + 0.447255i \(0.852402\pi\)
\(258\) 4.88771 0.304296
\(259\) −1.32498 −0.0823302
\(260\) −6.70440 −0.415789
\(261\) −1.45443 −0.0900269
\(262\) −2.99069 −0.184765
\(263\) −2.63094 −0.162231 −0.0811154 0.996705i \(-0.525848\pi\)
−0.0811154 + 0.996705i \(0.525848\pi\)
\(264\) 7.23103 0.445039
\(265\) −29.4550 −1.80941
\(266\) −0.622134 −0.0381455
\(267\) −7.10670 −0.434923
\(268\) 1.86760 0.114082
\(269\) 8.89967 0.542622 0.271311 0.962492i \(-0.412543\pi\)
0.271311 + 0.962492i \(0.412543\pi\)
\(270\) −1.14691 −0.0697985
\(271\) −5.43976 −0.330442 −0.165221 0.986257i \(-0.552834\pi\)
−0.165221 + 0.986257i \(0.552834\pi\)
\(272\) −2.48367 −0.150595
\(273\) 2.05909 0.124622
\(274\) −10.8736 −0.656896
\(275\) 0.0919574 0.00554524
\(276\) 3.74409 0.225368
\(277\) 6.91104 0.415244 0.207622 0.978209i \(-0.433428\pi\)
0.207622 + 0.978209i \(0.433428\pi\)
\(278\) 8.18141 0.490688
\(279\) −4.31473 −0.258316
\(280\) −5.09433 −0.304444
\(281\) −27.8390 −1.66073 −0.830367 0.557216i \(-0.811869\pi\)
−0.830367 + 0.557216i \(0.811869\pi\)
\(282\) 2.58104 0.153699
\(283\) 12.7768 0.759504 0.379752 0.925088i \(-0.376009\pi\)
0.379752 + 0.925088i \(0.376009\pi\)
\(284\) 23.8937 1.41783
\(285\) 2.26989 0.134456
\(286\) −3.35210 −0.198214
\(287\) 2.63917 0.155785
\(288\) −5.11852 −0.301612
\(289\) 1.00000 0.0588235
\(290\) −1.66810 −0.0979539
\(291\) −1.44167 −0.0845124
\(292\) −9.21976 −0.539545
\(293\) −14.6412 −0.855346 −0.427673 0.903934i \(-0.640666\pi\)
−0.427673 + 0.903934i \(0.640666\pi\)
\(294\) −2.87225 −0.167513
\(295\) −6.87360 −0.400196
\(296\) −2.14035 −0.124405
\(297\) 3.76470 0.218450
\(298\) −10.4015 −0.602545
\(299\) −3.73569 −0.216040
\(300\) −0.0423949 −0.00244767
\(301\) 11.3030 0.651495
\(302\) −5.23680 −0.301344
\(303\) −10.1000 −0.580231
\(304\) 2.52741 0.144957
\(305\) 21.7101 1.24312
\(306\) 0.514170 0.0293931
\(307\) −31.3082 −1.78685 −0.893426 0.449209i \(-0.851706\pi\)
−0.893426 + 0.449209i \(0.851706\pi\)
\(308\) 7.76930 0.442697
\(309\) −10.6269 −0.604542
\(310\) −4.94860 −0.281061
\(311\) 9.90880 0.561876 0.280938 0.959726i \(-0.409354\pi\)
0.280938 + 0.959726i \(0.409354\pi\)
\(312\) 3.32622 0.188310
\(313\) 24.9421 1.40981 0.704907 0.709300i \(-0.250989\pi\)
0.704907 + 0.709300i \(0.250989\pi\)
\(314\) 0.514170 0.0290163
\(315\) −2.65226 −0.149438
\(316\) 17.3190 0.974269
\(317\) 6.05204 0.339917 0.169958 0.985451i \(-0.445637\pi\)
0.169958 + 0.985451i \(0.445637\pi\)
\(318\) 6.78960 0.380742
\(319\) 5.47549 0.306568
\(320\) 5.20967 0.291229
\(321\) 10.1754 0.567933
\(322\) −1.31884 −0.0734959
\(323\) −1.01761 −0.0566215
\(324\) −1.73563 −0.0964239
\(325\) 0.0422998 0.00234637
\(326\) 3.98215 0.220551
\(327\) 1.10382 0.0610416
\(328\) 4.26327 0.235400
\(329\) 5.96876 0.329068
\(330\) 4.31776 0.237685
\(331\) 20.4687 1.12506 0.562532 0.826776i \(-0.309827\pi\)
0.562532 + 0.826776i \(0.309827\pi\)
\(332\) −24.3389 −1.33577
\(333\) −1.11433 −0.0610650
\(334\) −1.48178 −0.0810791
\(335\) 2.40021 0.131137
\(336\) −2.95317 −0.161109
\(337\) −26.6402 −1.45119 −0.725593 0.688125i \(-0.758434\pi\)
−0.725593 + 0.688125i \(0.758434\pi\)
\(338\) 5.14226 0.279702
\(339\) 0.726543 0.0394604
\(340\) −3.87149 −0.209961
\(341\) 16.2437 0.879643
\(342\) −0.523226 −0.0282928
\(343\) −14.9654 −0.808058
\(344\) 18.2587 0.984442
\(345\) 4.81184 0.259061
\(346\) 5.74124 0.308651
\(347\) 15.8160 0.849048 0.424524 0.905417i \(-0.360441\pi\)
0.424524 + 0.905417i \(0.360441\pi\)
\(348\) −2.52435 −0.135319
\(349\) −29.5438 −1.58144 −0.790721 0.612177i \(-0.790294\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(350\) 0.0149334 0.000798224 0
\(351\) 1.73173 0.0924331
\(352\) 19.2697 1.02708
\(353\) −11.7051 −0.623002 −0.311501 0.950246i \(-0.600832\pi\)
−0.311501 + 0.950246i \(0.600832\pi\)
\(354\) 1.58442 0.0842107
\(355\) 30.7078 1.62980
\(356\) −12.3346 −0.653732
\(357\) 1.18904 0.0629305
\(358\) −4.19309 −0.221611
\(359\) 20.6109 1.08780 0.543902 0.839149i \(-0.316946\pi\)
0.543902 + 0.839149i \(0.316946\pi\)
\(360\) −4.28442 −0.225809
\(361\) −17.9645 −0.945498
\(362\) −3.15857 −0.166011
\(363\) −3.17293 −0.166536
\(364\) 3.57382 0.187319
\(365\) −11.8491 −0.620208
\(366\) −5.00434 −0.261581
\(367\) −17.4966 −0.913313 −0.456656 0.889643i \(-0.650953\pi\)
−0.456656 + 0.889643i \(0.650953\pi\)
\(368\) 5.35775 0.279292
\(369\) 2.21959 0.115547
\(370\) −1.27803 −0.0664418
\(371\) 15.7012 0.815165
\(372\) −7.48878 −0.388275
\(373\) 6.81511 0.352873 0.176436 0.984312i \(-0.443543\pi\)
0.176436 + 0.984312i \(0.443543\pi\)
\(374\) −1.93569 −0.100092
\(375\) −11.2075 −0.578752
\(376\) 9.64182 0.497239
\(377\) 2.51869 0.129719
\(378\) 0.611367 0.0314453
\(379\) 16.5901 0.852175 0.426088 0.904682i \(-0.359891\pi\)
0.426088 + 0.904682i \(0.359891\pi\)
\(380\) 3.93968 0.202101
\(381\) −16.5326 −0.846993
\(382\) 3.15813 0.161584
\(383\) −14.5530 −0.743623 −0.371812 0.928308i \(-0.621263\pi\)
−0.371812 + 0.928308i \(0.621263\pi\)
\(384\) −11.4379 −0.583689
\(385\) 9.98497 0.508881
\(386\) 0.824332 0.0419574
\(387\) 9.50603 0.483219
\(388\) −2.50221 −0.127030
\(389\) 28.2560 1.43264 0.716318 0.697774i \(-0.245826\pi\)
0.716318 + 0.697774i \(0.245826\pi\)
\(390\) 1.98614 0.100572
\(391\) −2.15719 −0.109094
\(392\) −10.7297 −0.541930
\(393\) −5.81654 −0.293406
\(394\) −2.74870 −0.138477
\(395\) 22.2581 1.11992
\(396\) 6.53412 0.328352
\(397\) −17.0174 −0.854080 −0.427040 0.904233i \(-0.640444\pi\)
−0.427040 + 0.904233i \(0.640444\pi\)
\(398\) 2.54129 0.127383
\(399\) −1.20998 −0.0605747
\(400\) −0.0606667 −0.00303334
\(401\) −20.1447 −1.00598 −0.502990 0.864292i \(-0.667767\pi\)
−0.502990 + 0.864292i \(0.667767\pi\)
\(402\) −0.553265 −0.0275944
\(403\) 7.47197 0.372205
\(404\) −17.5299 −0.872145
\(405\) −2.23060 −0.110839
\(406\) 0.889190 0.0441298
\(407\) 4.19512 0.207944
\(408\) 1.92075 0.0950912
\(409\) −17.6560 −0.873032 −0.436516 0.899696i \(-0.643788\pi\)
−0.436516 + 0.899696i \(0.643788\pi\)
\(410\) 2.54566 0.125721
\(411\) −21.1478 −1.04314
\(412\) −18.4443 −0.908687
\(413\) 3.66402 0.180295
\(414\) −1.10916 −0.0545124
\(415\) −31.2799 −1.53547
\(416\) 8.86392 0.434589
\(417\) 15.9119 0.779208
\(418\) 1.96979 0.0963454
\(419\) −26.2708 −1.28341 −0.641706 0.766951i \(-0.721773\pi\)
−0.641706 + 0.766951i \(0.721773\pi\)
\(420\) −4.60335 −0.224620
\(421\) 0.854664 0.0416538 0.0208269 0.999783i \(-0.493370\pi\)
0.0208269 + 0.999783i \(0.493370\pi\)
\(422\) −2.89856 −0.141100
\(423\) 5.01983 0.244072
\(424\) 25.3634 1.23176
\(425\) 0.0244263 0.00118485
\(426\) −7.07838 −0.342949
\(427\) −11.5727 −0.560043
\(428\) 17.6607 0.853660
\(429\) −6.51945 −0.314762
\(430\) 10.9025 0.525767
\(431\) −38.6065 −1.85961 −0.929805 0.368053i \(-0.880024\pi\)
−0.929805 + 0.368053i \(0.880024\pi\)
\(432\) −2.48367 −0.119496
\(433\) −2.39340 −0.115019 −0.0575096 0.998345i \(-0.518316\pi\)
−0.0575096 + 0.998345i \(0.518316\pi\)
\(434\) 2.63788 0.126622
\(435\) −3.24425 −0.155550
\(436\) 1.91583 0.0917516
\(437\) 2.19519 0.105010
\(438\) 2.73130 0.130506
\(439\) −20.9687 −1.00078 −0.500391 0.865800i \(-0.666810\pi\)
−0.500391 + 0.865800i \(0.666810\pi\)
\(440\) 16.1295 0.768945
\(441\) −5.58619 −0.266009
\(442\) −0.890405 −0.0423523
\(443\) −13.0580 −0.620403 −0.310201 0.950671i \(-0.600397\pi\)
−0.310201 + 0.950671i \(0.600397\pi\)
\(444\) −1.93407 −0.0917867
\(445\) −15.8522 −0.751466
\(446\) 1.24164 0.0587934
\(447\) −20.2298 −0.956836
\(448\) −2.77705 −0.131203
\(449\) −19.2732 −0.909559 −0.454780 0.890604i \(-0.650282\pi\)
−0.454780 + 0.890604i \(0.650282\pi\)
\(450\) 0.0125592 0.000592048 0
\(451\) −8.35607 −0.393472
\(452\) 1.26101 0.0593129
\(453\) −10.1850 −0.478532
\(454\) −11.6691 −0.547658
\(455\) 4.59301 0.215324
\(456\) −1.95458 −0.0915315
\(457\) 12.6683 0.592598 0.296299 0.955095i \(-0.404248\pi\)
0.296299 + 0.955095i \(0.404248\pi\)
\(458\) −1.48263 −0.0692788
\(459\) 1.00000 0.0466760
\(460\) 8.35156 0.389394
\(461\) −3.79484 −0.176743 −0.0883716 0.996088i \(-0.528166\pi\)
−0.0883716 + 0.996088i \(0.528166\pi\)
\(462\) −2.30161 −0.107081
\(463\) −29.9544 −1.39210 −0.696048 0.717995i \(-0.745060\pi\)
−0.696048 + 0.717995i \(0.745060\pi\)
\(464\) −3.61232 −0.167698
\(465\) −9.62444 −0.446323
\(466\) −8.86850 −0.410825
\(467\) 20.5814 0.952391 0.476196 0.879339i \(-0.342015\pi\)
0.476196 + 0.879339i \(0.342015\pi\)
\(468\) 3.00565 0.138936
\(469\) −1.27945 −0.0590793
\(470\) 5.75727 0.265563
\(471\) 1.00000 0.0460776
\(472\) 5.91879 0.272434
\(473\) −35.7873 −1.64550
\(474\) −5.13065 −0.235658
\(475\) −0.0248565 −0.00114049
\(476\) 2.06373 0.0945907
\(477\) 13.2050 0.604614
\(478\) 14.0618 0.643172
\(479\) −30.3123 −1.38500 −0.692502 0.721416i \(-0.743491\pi\)
−0.692502 + 0.721416i \(0.743491\pi\)
\(480\) −11.4174 −0.521129
\(481\) 1.92972 0.0879878
\(482\) −1.46417 −0.0666913
\(483\) −2.56498 −0.116711
\(484\) −5.50704 −0.250320
\(485\) −3.21580 −0.146022
\(486\) 0.514170 0.0233232
\(487\) 9.18808 0.416352 0.208176 0.978091i \(-0.433247\pi\)
0.208176 + 0.978091i \(0.433247\pi\)
\(488\) −18.6944 −0.846254
\(489\) 7.74482 0.350233
\(490\) −6.40684 −0.289432
\(491\) −21.0280 −0.948981 −0.474491 0.880260i \(-0.657368\pi\)
−0.474491 + 0.880260i \(0.657368\pi\)
\(492\) 3.85238 0.173679
\(493\) 1.45443 0.0655042
\(494\) 0.906088 0.0407668
\(495\) 8.39753 0.377441
\(496\) −10.7164 −0.481179
\(497\) −16.3690 −0.734250
\(498\) 7.21024 0.323099
\(499\) 18.9397 0.847859 0.423930 0.905695i \(-0.360650\pi\)
0.423930 + 0.905695i \(0.360650\pi\)
\(500\) −19.4520 −0.869921
\(501\) −2.88188 −0.128753
\(502\) 12.6075 0.562698
\(503\) 17.8046 0.793870 0.396935 0.917847i \(-0.370074\pi\)
0.396935 + 0.917847i \(0.370074\pi\)
\(504\) 2.28384 0.101730
\(505\) −22.5291 −1.00253
\(506\) 4.17566 0.185631
\(507\) 10.0011 0.444164
\(508\) −28.6945 −1.27311
\(509\) 24.6273 1.09158 0.545792 0.837921i \(-0.316229\pi\)
0.545792 + 0.837921i \(0.316229\pi\)
\(510\) 1.14691 0.0507859
\(511\) 6.31623 0.279413
\(512\) −22.2537 −0.983485
\(513\) −1.01761 −0.0449287
\(514\) 14.7448 0.650364
\(515\) −23.7043 −1.04454
\(516\) 16.4989 0.726325
\(517\) −18.8981 −0.831138
\(518\) 0.681265 0.0299330
\(519\) 11.1660 0.490134
\(520\) 7.41947 0.325365
\(521\) 22.4184 0.982169 0.491084 0.871112i \(-0.336601\pi\)
0.491084 + 0.871112i \(0.336601\pi\)
\(522\) 0.747824 0.0327314
\(523\) −42.0016 −1.83660 −0.918300 0.395884i \(-0.870438\pi\)
−0.918300 + 0.395884i \(0.870438\pi\)
\(524\) −10.0954 −0.441018
\(525\) 0.0290437 0.00126757
\(526\) 1.35275 0.0589827
\(527\) 4.31473 0.187953
\(528\) 9.35026 0.406918
\(529\) −18.3465 −0.797675
\(530\) 15.1449 0.657851
\(531\) 3.08150 0.133726
\(532\) −2.10007 −0.0910498
\(533\) −3.84373 −0.166491
\(534\) 3.65405 0.158126
\(535\) 22.6971 0.981283
\(536\) −2.06679 −0.0892719
\(537\) −8.15506 −0.351917
\(538\) −4.57594 −0.197283
\(539\) 21.0303 0.905840
\(540\) −3.87149 −0.166603
\(541\) −39.4376 −1.69556 −0.847778 0.530352i \(-0.822060\pi\)
−0.847778 + 0.530352i \(0.822060\pi\)
\(542\) 2.79696 0.120140
\(543\) −6.14306 −0.263624
\(544\) 5.11852 0.219455
\(545\) 2.46219 0.105469
\(546\) −1.05872 −0.0453092
\(547\) 10.3572 0.442842 0.221421 0.975178i \(-0.428931\pi\)
0.221421 + 0.975178i \(0.428931\pi\)
\(548\) −36.7047 −1.56795
\(549\) −9.73285 −0.415388
\(550\) −0.0472817 −0.00201610
\(551\) −1.48005 −0.0630521
\(552\) −4.14343 −0.176356
\(553\) −11.8648 −0.504543
\(554\) −3.55345 −0.150972
\(555\) −2.48563 −0.105509
\(556\) 27.6171 1.17123
\(557\) 34.5040 1.46198 0.730992 0.682386i \(-0.239058\pi\)
0.730992 + 0.682386i \(0.239058\pi\)
\(558\) 2.21851 0.0939168
\(559\) −16.4619 −0.696265
\(560\) −6.58734 −0.278366
\(561\) −3.76470 −0.158946
\(562\) 14.3140 0.603798
\(563\) −34.1976 −1.44126 −0.720629 0.693321i \(-0.756147\pi\)
−0.720629 + 0.693321i \(0.756147\pi\)
\(564\) 8.71256 0.366865
\(565\) 1.62063 0.0681803
\(566\) −6.56946 −0.276135
\(567\) 1.18904 0.0499348
\(568\) −26.4422 −1.10949
\(569\) −22.0129 −0.922828 −0.461414 0.887185i \(-0.652658\pi\)
−0.461414 + 0.887185i \(0.652658\pi\)
\(570\) −1.16711 −0.0488847
\(571\) −15.6963 −0.656870 −0.328435 0.944527i \(-0.606521\pi\)
−0.328435 + 0.944527i \(0.606521\pi\)
\(572\) −11.3153 −0.473119
\(573\) 6.14219 0.256594
\(574\) −1.35698 −0.0566393
\(575\) −0.0526922 −0.00219742
\(576\) −2.33555 −0.0973144
\(577\) −32.1369 −1.33788 −0.668938 0.743319i \(-0.733251\pi\)
−0.668938 + 0.743319i \(0.733251\pi\)
\(578\) −0.514170 −0.0213866
\(579\) 1.60323 0.0666279
\(580\) −5.63082 −0.233807
\(581\) 16.6740 0.691752
\(582\) 0.741265 0.0307264
\(583\) −49.7127 −2.05889
\(584\) 10.2031 0.422208
\(585\) 3.86280 0.159707
\(586\) 7.52804 0.310981
\(587\) 30.0234 1.23920 0.619600 0.784918i \(-0.287295\pi\)
0.619600 + 0.784918i \(0.287295\pi\)
\(588\) −9.69556 −0.399838
\(589\) −4.39073 −0.180917
\(590\) 3.53420 0.145501
\(591\) −5.34589 −0.219901
\(592\) −2.76763 −0.113749
\(593\) 8.34511 0.342692 0.171346 0.985211i \(-0.445188\pi\)
0.171346 + 0.985211i \(0.445188\pi\)
\(594\) −1.93569 −0.0794224
\(595\) 2.65226 0.108732
\(596\) −35.1114 −1.43822
\(597\) 4.94252 0.202284
\(598\) 1.92078 0.0785464
\(599\) −23.3385 −0.953585 −0.476793 0.879016i \(-0.658201\pi\)
−0.476793 + 0.879016i \(0.658201\pi\)
\(600\) 0.0469167 0.00191537
\(601\) 43.3537 1.76843 0.884216 0.467078i \(-0.154693\pi\)
0.884216 + 0.467078i \(0.154693\pi\)
\(602\) −5.81167 −0.236866
\(603\) −1.07604 −0.0438196
\(604\) −17.6773 −0.719281
\(605\) −7.07754 −0.287743
\(606\) 5.19313 0.210956
\(607\) 33.8996 1.37594 0.687970 0.725739i \(-0.258502\pi\)
0.687970 + 0.725739i \(0.258502\pi\)
\(608\) −5.20867 −0.211240
\(609\) 1.72937 0.0700776
\(610\) −11.1627 −0.451964
\(611\) −8.69300 −0.351681
\(612\) 1.73563 0.0701587
\(613\) −2.02264 −0.0816936 −0.0408468 0.999165i \(-0.513006\pi\)
−0.0408468 + 0.999165i \(0.513006\pi\)
\(614\) 16.0977 0.649651
\(615\) 4.95101 0.199644
\(616\) −8.59796 −0.346422
\(617\) −44.5900 −1.79513 −0.897563 0.440886i \(-0.854664\pi\)
−0.897563 + 0.440886i \(0.854664\pi\)
\(618\) 5.46402 0.219795
\(619\) −29.7208 −1.19458 −0.597290 0.802025i \(-0.703756\pi\)
−0.597290 + 0.802025i \(0.703756\pi\)
\(620\) −16.7045 −0.670867
\(621\) −2.15719 −0.0865652
\(622\) −5.09481 −0.204283
\(623\) 8.45012 0.338547
\(624\) 4.30105 0.172180
\(625\) −24.8773 −0.995091
\(626\) −12.8245 −0.512570
\(627\) 3.83100 0.152996
\(628\) 1.73563 0.0692592
\(629\) 1.11433 0.0444313
\(630\) 1.36371 0.0543317
\(631\) 44.1891 1.75914 0.879570 0.475770i \(-0.157830\pi\)
0.879570 + 0.475770i \(0.157830\pi\)
\(632\) −19.1662 −0.762390
\(633\) −5.63736 −0.224065
\(634\) −3.11178 −0.123584
\(635\) −36.8777 −1.46345
\(636\) 22.9189 0.908795
\(637\) 9.67380 0.383290
\(638\) −2.81533 −0.111460
\(639\) −13.7666 −0.544599
\(640\) −25.5134 −1.00851
\(641\) 19.6845 0.777489 0.388745 0.921346i \(-0.372909\pi\)
0.388745 + 0.921346i \(0.372909\pi\)
\(642\) −5.23186 −0.206485
\(643\) 3.57002 0.140788 0.0703939 0.997519i \(-0.477574\pi\)
0.0703939 + 0.997519i \(0.477574\pi\)
\(644\) −4.45186 −0.175428
\(645\) 21.2041 0.834912
\(646\) 0.523226 0.0205860
\(647\) 34.1186 1.34134 0.670670 0.741755i \(-0.266007\pi\)
0.670670 + 0.741755i \(0.266007\pi\)
\(648\) 1.92075 0.0754541
\(649\) −11.6009 −0.455376
\(650\) −0.0217493 −0.000853076 0
\(651\) 5.13038 0.201075
\(652\) 13.4421 0.526435
\(653\) −23.5821 −0.922838 −0.461419 0.887182i \(-0.652659\pi\)
−0.461419 + 0.887182i \(0.652659\pi\)
\(654\) −0.567553 −0.0221931
\(655\) −12.9744 −0.506951
\(656\) 5.51272 0.215235
\(657\) 5.31205 0.207243
\(658\) −3.06895 −0.119640
\(659\) 17.8392 0.694916 0.347458 0.937696i \(-0.387045\pi\)
0.347458 + 0.937696i \(0.387045\pi\)
\(660\) 14.5750 0.567331
\(661\) −33.9635 −1.32103 −0.660513 0.750815i \(-0.729661\pi\)
−0.660513 + 0.750815i \(0.729661\pi\)
\(662\) −10.5244 −0.409043
\(663\) −1.73173 −0.0672550
\(664\) 26.9348 1.04527
\(665\) −2.69898 −0.104662
\(666\) 0.572955 0.0222016
\(667\) −3.13749 −0.121484
\(668\) −5.00187 −0.193528
\(669\) 2.41484 0.0933633
\(670\) −1.23411 −0.0476780
\(671\) 36.6412 1.41452
\(672\) 6.08611 0.234777
\(673\) −7.16175 −0.276065 −0.138033 0.990428i \(-0.544078\pi\)
−0.138033 + 0.990428i \(0.544078\pi\)
\(674\) 13.6976 0.527612
\(675\) 0.0244263 0.000940167 0
\(676\) 17.3582 0.667623
\(677\) 38.5677 1.48228 0.741139 0.671352i \(-0.234286\pi\)
0.741139 + 0.671352i \(0.234286\pi\)
\(678\) −0.373567 −0.0143467
\(679\) 1.71420 0.0657850
\(680\) 4.28442 0.164300
\(681\) −22.6950 −0.869676
\(682\) −8.35200 −0.319815
\(683\) 4.09135 0.156551 0.0782756 0.996932i \(-0.475059\pi\)
0.0782756 + 0.996932i \(0.475059\pi\)
\(684\) −1.76620 −0.0675323
\(685\) −47.1723 −1.80236
\(686\) 7.69478 0.293788
\(687\) −2.88354 −0.110014
\(688\) 23.6098 0.900116
\(689\) −22.8675 −0.871182
\(690\) −2.47410 −0.0941874
\(691\) −40.2715 −1.53200 −0.766001 0.642840i \(-0.777756\pi\)
−0.766001 + 0.642840i \(0.777756\pi\)
\(692\) 19.3801 0.736721
\(693\) −4.47636 −0.170043
\(694\) −8.13211 −0.308691
\(695\) 35.4930 1.34633
\(696\) 2.79359 0.105891
\(697\) −2.21959 −0.0840729
\(698\) 15.1905 0.574970
\(699\) −17.2482 −0.652386
\(700\) 0.0504091 0.00190529
\(701\) −51.0135 −1.92675 −0.963377 0.268152i \(-0.913587\pi\)
−0.963377 + 0.268152i \(0.913587\pi\)
\(702\) −0.890405 −0.0336062
\(703\) −1.13396 −0.0427680
\(704\) 8.79262 0.331384
\(705\) 11.1972 0.421712
\(706\) 6.01843 0.226507
\(707\) 12.0093 0.451656
\(708\) 5.34835 0.201003
\(709\) 29.9318 1.12411 0.562056 0.827099i \(-0.310010\pi\)
0.562056 + 0.827099i \(0.310010\pi\)
\(710\) −15.7890 −0.592552
\(711\) −9.97851 −0.374223
\(712\) 13.6502 0.511562
\(713\) −9.30772 −0.348577
\(714\) −0.611367 −0.0228798
\(715\) −14.5423 −0.543850
\(716\) −14.1542 −0.528966
\(717\) 27.3485 1.02135
\(718\) −10.5975 −0.395496
\(719\) 16.1469 0.602177 0.301088 0.953596i \(-0.402650\pi\)
0.301088 + 0.953596i \(0.402650\pi\)
\(720\) −5.54007 −0.206466
\(721\) 12.6358 0.470580
\(722\) 9.23679 0.343758
\(723\) −2.84765 −0.105905
\(724\) −10.6621 −0.396253
\(725\) 0.0355263 0.00131941
\(726\) 1.63143 0.0605479
\(727\) −1.25104 −0.0463986 −0.0231993 0.999731i \(-0.507385\pi\)
−0.0231993 + 0.999731i \(0.507385\pi\)
\(728\) −3.95500 −0.146582
\(729\) 1.00000 0.0370370
\(730\) 6.09243 0.225491
\(731\) −9.50603 −0.351593
\(732\) −16.8926 −0.624370
\(733\) 48.0206 1.77368 0.886841 0.462075i \(-0.152895\pi\)
0.886841 + 0.462075i \(0.152895\pi\)
\(734\) 8.99620 0.332056
\(735\) −12.4606 −0.459615
\(736\) −11.0416 −0.407000
\(737\) 4.05095 0.149219
\(738\) −1.14124 −0.0420098
\(739\) 3.29332 0.121147 0.0605733 0.998164i \(-0.480707\pi\)
0.0605733 + 0.998164i \(0.480707\pi\)
\(740\) −4.31412 −0.158590
\(741\) 1.76223 0.0647373
\(742\) −8.07308 −0.296372
\(743\) 32.2405 1.18279 0.591395 0.806382i \(-0.298577\pi\)
0.591395 + 0.806382i \(0.298577\pi\)
\(744\) 8.28752 0.303835
\(745\) −45.1245 −1.65323
\(746\) −3.50412 −0.128295
\(747\) 14.0231 0.513078
\(748\) −6.53412 −0.238911
\(749\) −12.0989 −0.442083
\(750\) 5.76255 0.210419
\(751\) 37.0044 1.35031 0.675155 0.737675i \(-0.264077\pi\)
0.675155 + 0.737675i \(0.264077\pi\)
\(752\) 12.4676 0.454646
\(753\) 24.5200 0.893559
\(754\) −1.29503 −0.0471623
\(755\) −22.7186 −0.826814
\(756\) 2.06373 0.0750570
\(757\) −48.5832 −1.76579 −0.882893 0.469574i \(-0.844407\pi\)
−0.882893 + 0.469574i \(0.844407\pi\)
\(758\) −8.53012 −0.309828
\(759\) 8.12118 0.294780
\(760\) −4.35988 −0.158149
\(761\) −39.8327 −1.44393 −0.721967 0.691928i \(-0.756762\pi\)
−0.721967 + 0.691928i \(0.756762\pi\)
\(762\) 8.50058 0.307944
\(763\) −1.31249 −0.0475152
\(764\) 10.6606 0.385686
\(765\) 2.23060 0.0806475
\(766\) 7.48271 0.270361
\(767\) −5.33634 −0.192684
\(768\) 1.20994 0.0436598
\(769\) 17.3906 0.627123 0.313561 0.949568i \(-0.398478\pi\)
0.313561 + 0.949568i \(0.398478\pi\)
\(770\) −5.13397 −0.185015
\(771\) 28.6769 1.03277
\(772\) 2.78261 0.100148
\(773\) −39.0106 −1.40311 −0.701556 0.712614i \(-0.747511\pi\)
−0.701556 + 0.712614i \(0.747511\pi\)
\(774\) −4.88771 −0.175685
\(775\) 0.105393 0.00378582
\(776\) 2.76909 0.0994046
\(777\) 1.32498 0.0475334
\(778\) −14.5284 −0.520867
\(779\) 2.25868 0.0809256
\(780\) 6.70440 0.240056
\(781\) 51.8271 1.85452
\(782\) 1.10916 0.0396636
\(783\) 1.45443 0.0519771
\(784\) −13.8742 −0.495509
\(785\) 2.23060 0.0796135
\(786\) 2.99069 0.106674
\(787\) 24.0086 0.855815 0.427907 0.903823i \(-0.359251\pi\)
0.427907 + 0.903823i \(0.359251\pi\)
\(788\) −9.27849 −0.330533
\(789\) 2.63094 0.0936640
\(790\) −11.4444 −0.407174
\(791\) −0.863887 −0.0307163
\(792\) −7.23103 −0.256944
\(793\) 16.8547 0.598528
\(794\) 8.74985 0.310521
\(795\) 29.4550 1.04466
\(796\) 8.57838 0.304052
\(797\) −39.0448 −1.38304 −0.691518 0.722359i \(-0.743058\pi\)
−0.691518 + 0.722359i \(0.743058\pi\)
\(798\) 0.622134 0.0220233
\(799\) −5.01983 −0.177589
\(800\) 0.125026 0.00442035
\(801\) 7.10670 0.251103
\(802\) 10.3578 0.365747
\(803\) −19.9983 −0.705724
\(804\) −1.86760 −0.0658652
\(805\) −5.72145 −0.201655
\(806\) −3.84186 −0.135324
\(807\) −8.89967 −0.313283
\(808\) 19.3996 0.682476
\(809\) −13.5099 −0.474982 −0.237491 0.971390i \(-0.576325\pi\)
−0.237491 + 0.971390i \(0.576325\pi\)
\(810\) 1.14691 0.0402982
\(811\) −24.7830 −0.870249 −0.435125 0.900370i \(-0.643296\pi\)
−0.435125 + 0.900370i \(0.643296\pi\)
\(812\) 3.00155 0.105334
\(813\) 5.43976 0.190781
\(814\) −2.15700 −0.0756029
\(815\) 17.2756 0.605138
\(816\) 2.48367 0.0869458
\(817\) 9.67346 0.338431
\(818\) 9.07818 0.317411
\(819\) −2.05909 −0.0719506
\(820\) 8.59312 0.300085
\(821\) 2.27618 0.0794391 0.0397196 0.999211i \(-0.487354\pi\)
0.0397196 + 0.999211i \(0.487354\pi\)
\(822\) 10.8736 0.379259
\(823\) 6.39602 0.222951 0.111476 0.993767i \(-0.464442\pi\)
0.111476 + 0.993767i \(0.464442\pi\)
\(824\) 20.4116 0.711071
\(825\) −0.0919574 −0.00320155
\(826\) −1.88393 −0.0655502
\(827\) 14.9349 0.519336 0.259668 0.965698i \(-0.416387\pi\)
0.259668 + 0.965698i \(0.416387\pi\)
\(828\) −3.74409 −0.130116
\(829\) 48.6569 1.68993 0.844963 0.534825i \(-0.179623\pi\)
0.844963 + 0.534825i \(0.179623\pi\)
\(830\) 16.0832 0.558255
\(831\) −6.91104 −0.239741
\(832\) 4.04454 0.140219
\(833\) 5.58619 0.193550
\(834\) −8.18141 −0.283299
\(835\) −6.42832 −0.222461
\(836\) 6.64920 0.229967
\(837\) 4.31473 0.149139
\(838\) 13.5076 0.466614
\(839\) 4.42332 0.152710 0.0763549 0.997081i \(-0.475672\pi\)
0.0763549 + 0.997081i \(0.475672\pi\)
\(840\) 5.09433 0.175771
\(841\) −26.8846 −0.927056
\(842\) −0.439442 −0.0151442
\(843\) 27.8390 0.958826
\(844\) −9.78436 −0.336792
\(845\) 22.3084 0.767434
\(846\) −2.58104 −0.0887381
\(847\) 3.77273 0.129633
\(848\) 32.7968 1.12625
\(849\) −12.7768 −0.438500
\(850\) −0.0125592 −0.000430778 0
\(851\) −2.40383 −0.0824021
\(852\) −23.8937 −0.818586
\(853\) −51.7029 −1.77027 −0.885136 0.465332i \(-0.845935\pi\)
−0.885136 + 0.465332i \(0.845935\pi\)
\(854\) 5.95034 0.203617
\(855\) −2.26989 −0.0776285
\(856\) −19.5443 −0.668010
\(857\) 48.2602 1.64854 0.824268 0.566199i \(-0.191587\pi\)
0.824268 + 0.566199i \(0.191587\pi\)
\(858\) 3.35210 0.114439
\(859\) 27.6188 0.942342 0.471171 0.882042i \(-0.343831\pi\)
0.471171 + 0.882042i \(0.343831\pi\)
\(860\) 36.8025 1.25496
\(861\) −2.63917 −0.0899427
\(862\) 19.8503 0.676104
\(863\) 35.1708 1.19723 0.598613 0.801038i \(-0.295719\pi\)
0.598613 + 0.801038i \(0.295719\pi\)
\(864\) 5.11852 0.174136
\(865\) 24.9069 0.846861
\(866\) 1.23061 0.0418179
\(867\) −1.00000 −0.0339618
\(868\) 8.90443 0.302236
\(869\) 37.5660 1.27434
\(870\) 1.66810 0.0565537
\(871\) 1.86341 0.0631392
\(872\) −2.12017 −0.0717980
\(873\) 1.44167 0.0487932
\(874\) −1.12870 −0.0381788
\(875\) 13.3261 0.450505
\(876\) 9.21976 0.311507
\(877\) −54.2884 −1.83319 −0.916594 0.399820i \(-0.869073\pi\)
−0.916594 + 0.399820i \(0.869073\pi\)
\(878\) 10.7815 0.363857
\(879\) 14.6412 0.493834
\(880\) 20.8567 0.703078
\(881\) −7.11623 −0.239752 −0.119876 0.992789i \(-0.538250\pi\)
−0.119876 + 0.992789i \(0.538250\pi\)
\(882\) 2.87225 0.0967137
\(883\) −16.2851 −0.548037 −0.274019 0.961724i \(-0.588353\pi\)
−0.274019 + 0.961724i \(0.588353\pi\)
\(884\) −3.00565 −0.101091
\(885\) 6.87360 0.231053
\(886\) 6.71401 0.225562
\(887\) 1.96917 0.0661184 0.0330592 0.999453i \(-0.489475\pi\)
0.0330592 + 0.999453i \(0.489475\pi\)
\(888\) 2.14035 0.0718254
\(889\) 19.6579 0.659305
\(890\) 8.15072 0.273213
\(891\) −3.76470 −0.126122
\(892\) 4.19128 0.140334
\(893\) 5.10824 0.170941
\(894\) 10.4015 0.347880
\(895\) −18.1907 −0.608047
\(896\) 13.6001 0.454347
\(897\) 3.73569 0.124731
\(898\) 9.90971 0.330691
\(899\) 6.27548 0.209299
\(900\) 0.0423949 0.00141316
\(901\) −13.2050 −0.439921
\(902\) 4.29644 0.143056
\(903\) −11.3030 −0.376141
\(904\) −1.39551 −0.0464139
\(905\) −13.7027 −0.455493
\(906\) 5.23680 0.173981
\(907\) −6.69902 −0.222437 −0.111219 0.993796i \(-0.535475\pi\)
−0.111219 + 0.993796i \(0.535475\pi\)
\(908\) −39.3902 −1.30721
\(909\) 10.1000 0.334997
\(910\) −2.36159 −0.0782859
\(911\) −46.8026 −1.55064 −0.775319 0.631570i \(-0.782411\pi\)
−0.775319 + 0.631570i \(0.782411\pi\)
\(912\) −2.52741 −0.0836910
\(913\) −52.7926 −1.74718
\(914\) −6.51366 −0.215453
\(915\) −21.7101 −0.717714
\(916\) −5.00476 −0.165362
\(917\) 6.91608 0.228389
\(918\) −0.514170 −0.0169701
\(919\) 19.5255 0.644088 0.322044 0.946725i \(-0.395630\pi\)
0.322044 + 0.946725i \(0.395630\pi\)
\(920\) −9.24232 −0.304710
\(921\) 31.3082 1.03164
\(922\) 1.95119 0.0642590
\(923\) 23.8401 0.784707
\(924\) −7.76930 −0.255591
\(925\) 0.0272189 0.000894953 0
\(926\) 15.4016 0.506129
\(927\) 10.6269 0.349033
\(928\) 7.44453 0.244379
\(929\) −13.3463 −0.437877 −0.218939 0.975739i \(-0.570259\pi\)
−0.218939 + 0.975739i \(0.570259\pi\)
\(930\) 4.94860 0.162271
\(931\) −5.68458 −0.186305
\(932\) −29.9365 −0.980601
\(933\) −9.90880 −0.324400
\(934\) −10.5823 −0.346264
\(935\) −8.39753 −0.274629
\(936\) −3.32622 −0.108721
\(937\) 18.4543 0.602876 0.301438 0.953486i \(-0.402533\pi\)
0.301438 + 0.953486i \(0.402533\pi\)
\(938\) 0.657853 0.0214796
\(939\) −24.9421 −0.813956
\(940\) 19.4342 0.633875
\(941\) 17.5919 0.573480 0.286740 0.958008i \(-0.407428\pi\)
0.286740 + 0.958008i \(0.407428\pi\)
\(942\) −0.514170 −0.0167526
\(943\) 4.78808 0.155921
\(944\) 7.65343 0.249098
\(945\) 2.65226 0.0862781
\(946\) 18.4007 0.598260
\(947\) −21.4608 −0.697384 −0.348692 0.937237i \(-0.613374\pi\)
−0.348692 + 0.937237i \(0.613374\pi\)
\(948\) −17.3190 −0.562495
\(949\) −9.19906 −0.298614
\(950\) 0.0127804 0.000414652 0
\(951\) −6.05204 −0.196251
\(952\) −2.28384 −0.0740196
\(953\) −45.1751 −1.46336 −0.731682 0.681646i \(-0.761265\pi\)
−0.731682 + 0.681646i \(0.761265\pi\)
\(954\) −6.78960 −0.219821
\(955\) 13.7008 0.443346
\(956\) 47.4669 1.53519
\(957\) −5.47549 −0.176997
\(958\) 15.5857 0.503550
\(959\) 25.1455 0.811991
\(960\) −5.20967 −0.168141
\(961\) −12.3831 −0.399454
\(962\) −0.992206 −0.0319900
\(963\) −10.1754 −0.327896
\(964\) −4.94246 −0.159186
\(965\) 3.57616 0.115121
\(966\) 1.31884 0.0424329
\(967\) −6.97023 −0.224148 −0.112074 0.993700i \(-0.535749\pi\)
−0.112074 + 0.993700i \(0.535749\pi\)
\(968\) 6.09441 0.195882
\(969\) 1.01761 0.0326904
\(970\) 1.65346 0.0530896
\(971\) 54.9614 1.76380 0.881898 0.471441i \(-0.156266\pi\)
0.881898 + 0.471441i \(0.156266\pi\)
\(972\) 1.73563 0.0556703
\(973\) −18.9198 −0.606541
\(974\) −4.72423 −0.151374
\(975\) −0.0422998 −0.00135468
\(976\) −24.1732 −0.773765
\(977\) 54.5244 1.74439 0.872196 0.489157i \(-0.162695\pi\)
0.872196 + 0.489157i \(0.162695\pi\)
\(978\) −3.98215 −0.127335
\(979\) −26.7546 −0.855079
\(980\) −21.6269 −0.690846
\(981\) −1.10382 −0.0352424
\(982\) 10.8120 0.345024
\(983\) −32.0460 −1.02211 −0.511055 0.859548i \(-0.670745\pi\)
−0.511055 + 0.859548i \(0.670745\pi\)
\(984\) −4.26327 −0.135908
\(985\) −11.9245 −0.379948
\(986\) −0.747824 −0.0238156
\(987\) −5.96876 −0.189988
\(988\) 3.05859 0.0973066
\(989\) 20.5063 0.652064
\(990\) −4.31776 −0.137227
\(991\) −41.8283 −1.32872 −0.664360 0.747413i \(-0.731296\pi\)
−0.664360 + 0.747413i \(0.731296\pi\)
\(992\) 22.0851 0.701201
\(993\) −20.4687 −0.649556
\(994\) 8.41645 0.266954
\(995\) 11.0248 0.349509
\(996\) 24.3389 0.771207
\(997\) 46.0383 1.45805 0.729023 0.684489i \(-0.239975\pi\)
0.729023 + 0.684489i \(0.239975\pi\)
\(998\) −9.73824 −0.308259
\(999\) 1.11433 0.0352559
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 8007.2.a.f.1.21 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.f.1.21 48 1.1 even 1 trivial