Properties

Label 8034.2.a.bc.1.8
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.113262\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.113262 q^{5} -1.00000 q^{6} +1.75034 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.113262 q^{5} -1.00000 q^{6} +1.75034 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.113262 q^{10} +3.65617 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.75034 q^{14} +0.113262 q^{15} +1.00000 q^{16} +6.05562 q^{17} +1.00000 q^{18} -7.50794 q^{19} -0.113262 q^{20} -1.75034 q^{21} +3.65617 q^{22} +5.43702 q^{23} -1.00000 q^{24} -4.98717 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.75034 q^{28} +9.14306 q^{29} +0.113262 q^{30} +4.35795 q^{31} +1.00000 q^{32} -3.65617 q^{33} +6.05562 q^{34} -0.198247 q^{35} +1.00000 q^{36} +0.0304744 q^{37} -7.50794 q^{38} +1.00000 q^{39} -0.113262 q^{40} +5.21159 q^{41} -1.75034 q^{42} +6.58092 q^{43} +3.65617 q^{44} -0.113262 q^{45} +5.43702 q^{46} -11.8201 q^{47} -1.00000 q^{48} -3.93630 q^{49} -4.98717 q^{50} -6.05562 q^{51} -1.00000 q^{52} -3.10712 q^{53} -1.00000 q^{54} -0.414104 q^{55} +1.75034 q^{56} +7.50794 q^{57} +9.14306 q^{58} +6.97905 q^{59} +0.113262 q^{60} +10.2132 q^{61} +4.35795 q^{62} +1.75034 q^{63} +1.00000 q^{64} +0.113262 q^{65} -3.65617 q^{66} -12.1648 q^{67} +6.05562 q^{68} -5.43702 q^{69} -0.198247 q^{70} -5.75511 q^{71} +1.00000 q^{72} -6.79650 q^{73} +0.0304744 q^{74} +4.98717 q^{75} -7.50794 q^{76} +6.39954 q^{77} +1.00000 q^{78} -5.30030 q^{79} -0.113262 q^{80} +1.00000 q^{81} +5.21159 q^{82} +11.0499 q^{83} -1.75034 q^{84} -0.685871 q^{85} +6.58092 q^{86} -9.14306 q^{87} +3.65617 q^{88} +10.4079 q^{89} -0.113262 q^{90} -1.75034 q^{91} +5.43702 q^{92} -4.35795 q^{93} -11.8201 q^{94} +0.850363 q^{95} -1.00000 q^{96} +10.1090 q^{97} -3.93630 q^{98} +3.65617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.113262 −0.0506522 −0.0253261 0.999679i \(-0.508062\pi\)
−0.0253261 + 0.999679i \(0.508062\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.75034 0.661568 0.330784 0.943707i \(-0.392687\pi\)
0.330784 + 0.943707i \(0.392687\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.113262 −0.0358165
\(11\) 3.65617 1.10238 0.551188 0.834381i \(-0.314175\pi\)
0.551188 + 0.834381i \(0.314175\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.75034 0.467799
\(15\) 0.113262 0.0292441
\(16\) 1.00000 0.250000
\(17\) 6.05562 1.46870 0.734352 0.678769i \(-0.237486\pi\)
0.734352 + 0.678769i \(0.237486\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.50794 −1.72244 −0.861220 0.508232i \(-0.830299\pi\)
−0.861220 + 0.508232i \(0.830299\pi\)
\(20\) −0.113262 −0.0253261
\(21\) −1.75034 −0.381956
\(22\) 3.65617 0.779497
\(23\) 5.43702 1.13370 0.566848 0.823822i \(-0.308163\pi\)
0.566848 + 0.823822i \(0.308163\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.98717 −0.997434
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.75034 0.330784
\(29\) 9.14306 1.69782 0.848912 0.528534i \(-0.177258\pi\)
0.848912 + 0.528534i \(0.177258\pi\)
\(30\) 0.113262 0.0206787
\(31\) 4.35795 0.782710 0.391355 0.920240i \(-0.372006\pi\)
0.391355 + 0.920240i \(0.372006\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.65617 −0.636457
\(34\) 6.05562 1.03853
\(35\) −0.198247 −0.0335099
\(36\) 1.00000 0.166667
\(37\) 0.0304744 0.00500996 0.00250498 0.999997i \(-0.499203\pi\)
0.00250498 + 0.999997i \(0.499203\pi\)
\(38\) −7.50794 −1.21795
\(39\) 1.00000 0.160128
\(40\) −0.113262 −0.0179083
\(41\) 5.21159 0.813914 0.406957 0.913447i \(-0.366590\pi\)
0.406957 + 0.913447i \(0.366590\pi\)
\(42\) −1.75034 −0.270084
\(43\) 6.58092 1.00358 0.501791 0.864989i \(-0.332675\pi\)
0.501791 + 0.864989i \(0.332675\pi\)
\(44\) 3.65617 0.551188
\(45\) −0.113262 −0.0168841
\(46\) 5.43702 0.801644
\(47\) −11.8201 −1.72414 −0.862068 0.506792i \(-0.830831\pi\)
−0.862068 + 0.506792i \(0.830831\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.93630 −0.562328
\(50\) −4.98717 −0.705293
\(51\) −6.05562 −0.847957
\(52\) −1.00000 −0.138675
\(53\) −3.10712 −0.426796 −0.213398 0.976965i \(-0.568453\pi\)
−0.213398 + 0.976965i \(0.568453\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.414104 −0.0558378
\(56\) 1.75034 0.233899
\(57\) 7.50794 0.994451
\(58\) 9.14306 1.20054
\(59\) 6.97905 0.908595 0.454298 0.890850i \(-0.349890\pi\)
0.454298 + 0.890850i \(0.349890\pi\)
\(60\) 0.113262 0.0146220
\(61\) 10.2132 1.30767 0.653833 0.756639i \(-0.273160\pi\)
0.653833 + 0.756639i \(0.273160\pi\)
\(62\) 4.35795 0.553460
\(63\) 1.75034 0.220523
\(64\) 1.00000 0.125000
\(65\) 0.113262 0.0140484
\(66\) −3.65617 −0.450043
\(67\) −12.1648 −1.48617 −0.743085 0.669197i \(-0.766638\pi\)
−0.743085 + 0.669197i \(0.766638\pi\)
\(68\) 6.05562 0.734352
\(69\) −5.43702 −0.654540
\(70\) −0.198247 −0.0236951
\(71\) −5.75511 −0.683006 −0.341503 0.939881i \(-0.610936\pi\)
−0.341503 + 0.939881i \(0.610936\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.79650 −0.795470 −0.397735 0.917500i \(-0.630204\pi\)
−0.397735 + 0.917500i \(0.630204\pi\)
\(74\) 0.0304744 0.00354257
\(75\) 4.98717 0.575869
\(76\) −7.50794 −0.861220
\(77\) 6.39954 0.729296
\(78\) 1.00000 0.113228
\(79\) −5.30030 −0.596330 −0.298165 0.954514i \(-0.596375\pi\)
−0.298165 + 0.954514i \(0.596375\pi\)
\(80\) −0.113262 −0.0126631
\(81\) 1.00000 0.111111
\(82\) 5.21159 0.575524
\(83\) 11.0499 1.21288 0.606442 0.795128i \(-0.292596\pi\)
0.606442 + 0.795128i \(0.292596\pi\)
\(84\) −1.75034 −0.190978
\(85\) −0.685871 −0.0743932
\(86\) 6.58092 0.709639
\(87\) −9.14306 −0.980239
\(88\) 3.65617 0.389749
\(89\) 10.4079 1.10324 0.551619 0.834096i \(-0.314010\pi\)
0.551619 + 0.834096i \(0.314010\pi\)
\(90\) −0.113262 −0.0119388
\(91\) −1.75034 −0.183486
\(92\) 5.43702 0.566848
\(93\) −4.35795 −0.451898
\(94\) −11.8201 −1.21915
\(95\) 0.850363 0.0872454
\(96\) −1.00000 −0.102062
\(97\) 10.1090 1.02641 0.513207 0.858265i \(-0.328457\pi\)
0.513207 + 0.858265i \(0.328457\pi\)
\(98\) −3.93630 −0.397626
\(99\) 3.65617 0.367459
\(100\) −4.98717 −0.498717
\(101\) −1.08143 −0.107606 −0.0538030 0.998552i \(-0.517134\pi\)
−0.0538030 + 0.998552i \(0.517134\pi\)
\(102\) −6.05562 −0.599596
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 0.198247 0.0193469
\(106\) −3.10712 −0.301790
\(107\) 2.13304 0.206209 0.103105 0.994671i \(-0.467122\pi\)
0.103105 + 0.994671i \(0.467122\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.39240 −0.229150 −0.114575 0.993415i \(-0.536551\pi\)
−0.114575 + 0.993415i \(0.536551\pi\)
\(110\) −0.414104 −0.0394833
\(111\) −0.0304744 −0.00289250
\(112\) 1.75034 0.165392
\(113\) 1.47065 0.138347 0.0691736 0.997605i \(-0.477964\pi\)
0.0691736 + 0.997605i \(0.477964\pi\)
\(114\) 7.50794 0.703183
\(115\) −0.615806 −0.0574242
\(116\) 9.14306 0.848912
\(117\) −1.00000 −0.0924500
\(118\) 6.97905 0.642474
\(119\) 10.5994 0.971647
\(120\) 0.113262 0.0103393
\(121\) 2.36755 0.215232
\(122\) 10.2132 0.924659
\(123\) −5.21159 −0.469913
\(124\) 4.35795 0.391355
\(125\) 1.13117 0.101174
\(126\) 1.75034 0.155933
\(127\) −5.04339 −0.447528 −0.223764 0.974643i \(-0.571835\pi\)
−0.223764 + 0.974643i \(0.571835\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.58092 −0.579418
\(130\) 0.113262 0.00993372
\(131\) −4.59698 −0.401640 −0.200820 0.979628i \(-0.564361\pi\)
−0.200820 + 0.979628i \(0.564361\pi\)
\(132\) −3.65617 −0.318228
\(133\) −13.1415 −1.13951
\(134\) −12.1648 −1.05088
\(135\) 0.113262 0.00974803
\(136\) 6.05562 0.519265
\(137\) 15.0484 1.28567 0.642836 0.766004i \(-0.277758\pi\)
0.642836 + 0.766004i \(0.277758\pi\)
\(138\) −5.43702 −0.462830
\(139\) 5.89502 0.500009 0.250004 0.968245i \(-0.419568\pi\)
0.250004 + 0.968245i \(0.419568\pi\)
\(140\) −0.198247 −0.0167549
\(141\) 11.8201 0.995431
\(142\) −5.75511 −0.482958
\(143\) −3.65617 −0.305744
\(144\) 1.00000 0.0833333
\(145\) −1.03556 −0.0859986
\(146\) −6.79650 −0.562483
\(147\) 3.93630 0.324660
\(148\) 0.0304744 0.00250498
\(149\) 0.0191072 0.00156532 0.000782660 1.00000i \(-0.499751\pi\)
0.000782660 1.00000i \(0.499751\pi\)
\(150\) 4.98717 0.407201
\(151\) −2.37037 −0.192898 −0.0964488 0.995338i \(-0.530748\pi\)
−0.0964488 + 0.995338i \(0.530748\pi\)
\(152\) −7.50794 −0.608975
\(153\) 6.05562 0.489568
\(154\) 6.39954 0.515690
\(155\) −0.493589 −0.0396460
\(156\) 1.00000 0.0800641
\(157\) −18.0051 −1.43697 −0.718483 0.695544i \(-0.755163\pi\)
−0.718483 + 0.695544i \(0.755163\pi\)
\(158\) −5.30030 −0.421669
\(159\) 3.10712 0.246411
\(160\) −0.113262 −0.00895413
\(161\) 9.51665 0.750017
\(162\) 1.00000 0.0785674
\(163\) 17.8749 1.40007 0.700034 0.714109i \(-0.253168\pi\)
0.700034 + 0.714109i \(0.253168\pi\)
\(164\) 5.21159 0.406957
\(165\) 0.414104 0.0322380
\(166\) 11.0499 0.857638
\(167\) 4.42869 0.342702 0.171351 0.985210i \(-0.445187\pi\)
0.171351 + 0.985210i \(0.445187\pi\)
\(168\) −1.75034 −0.135042
\(169\) 1.00000 0.0769231
\(170\) −0.685871 −0.0526039
\(171\) −7.50794 −0.574147
\(172\) 6.58092 0.501791
\(173\) 9.46262 0.719430 0.359715 0.933062i \(-0.382874\pi\)
0.359715 + 0.933062i \(0.382874\pi\)
\(174\) −9.14306 −0.693134
\(175\) −8.72926 −0.659870
\(176\) 3.65617 0.275594
\(177\) −6.97905 −0.524578
\(178\) 10.4079 0.780107
\(179\) 11.2831 0.843339 0.421669 0.906750i \(-0.361444\pi\)
0.421669 + 0.906750i \(0.361444\pi\)
\(180\) −0.113262 −0.00844204
\(181\) −17.1528 −1.27496 −0.637480 0.770467i \(-0.720023\pi\)
−0.637480 + 0.770467i \(0.720023\pi\)
\(182\) −1.75034 −0.129744
\(183\) −10.2132 −0.754981
\(184\) 5.43702 0.400822
\(185\) −0.00345158 −0.000253765 0
\(186\) −4.35795 −0.319540
\(187\) 22.1404 1.61906
\(188\) −11.8201 −0.862068
\(189\) −1.75034 −0.127319
\(190\) 0.850363 0.0616918
\(191\) 3.75731 0.271870 0.135935 0.990718i \(-0.456596\pi\)
0.135935 + 0.990718i \(0.456596\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.89362 −0.352250 −0.176125 0.984368i \(-0.556356\pi\)
−0.176125 + 0.984368i \(0.556356\pi\)
\(194\) 10.1090 0.725785
\(195\) −0.113262 −0.00811085
\(196\) −3.93630 −0.281164
\(197\) −24.9189 −1.77540 −0.887698 0.460426i \(-0.847697\pi\)
−0.887698 + 0.460426i \(0.847697\pi\)
\(198\) 3.65617 0.259832
\(199\) 12.8498 0.910900 0.455450 0.890261i \(-0.349478\pi\)
0.455450 + 0.890261i \(0.349478\pi\)
\(200\) −4.98717 −0.352646
\(201\) 12.1648 0.858041
\(202\) −1.08143 −0.0760889
\(203\) 16.0035 1.12323
\(204\) −6.05562 −0.423978
\(205\) −0.590274 −0.0412266
\(206\) −1.00000 −0.0696733
\(207\) 5.43702 0.377899
\(208\) −1.00000 −0.0693375
\(209\) −27.4503 −1.89878
\(210\) 0.198247 0.0136803
\(211\) 16.3252 1.12388 0.561938 0.827180i \(-0.310056\pi\)
0.561938 + 0.827180i \(0.310056\pi\)
\(212\) −3.10712 −0.213398
\(213\) 5.75511 0.394333
\(214\) 2.13304 0.145812
\(215\) −0.745368 −0.0508336
\(216\) −1.00000 −0.0680414
\(217\) 7.62790 0.517816
\(218\) −2.39240 −0.162033
\(219\) 6.79650 0.459265
\(220\) −0.414104 −0.0279189
\(221\) −6.05562 −0.407345
\(222\) −0.0304744 −0.00204531
\(223\) −14.1757 −0.949273 −0.474637 0.880182i \(-0.657420\pi\)
−0.474637 + 0.880182i \(0.657420\pi\)
\(224\) 1.75034 0.116950
\(225\) −4.98717 −0.332478
\(226\) 1.47065 0.0978263
\(227\) 7.24312 0.480743 0.240371 0.970681i \(-0.422731\pi\)
0.240371 + 0.970681i \(0.422731\pi\)
\(228\) 7.50794 0.497226
\(229\) −1.26769 −0.0837713 −0.0418857 0.999122i \(-0.513337\pi\)
−0.0418857 + 0.999122i \(0.513337\pi\)
\(230\) −0.615806 −0.0406051
\(231\) −6.39954 −0.421059
\(232\) 9.14306 0.600272
\(233\) −8.23879 −0.539741 −0.269871 0.962897i \(-0.586981\pi\)
−0.269871 + 0.962897i \(0.586981\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 1.33876 0.0873314
\(236\) 6.97905 0.454298
\(237\) 5.30030 0.344291
\(238\) 10.5994 0.687058
\(239\) −2.25279 −0.145721 −0.0728606 0.997342i \(-0.523213\pi\)
−0.0728606 + 0.997342i \(0.523213\pi\)
\(240\) 0.113262 0.00731102
\(241\) 13.8985 0.895284 0.447642 0.894213i \(-0.352264\pi\)
0.447642 + 0.894213i \(0.352264\pi\)
\(242\) 2.36755 0.152192
\(243\) −1.00000 −0.0641500
\(244\) 10.2132 0.653833
\(245\) 0.445832 0.0284832
\(246\) −5.21159 −0.332279
\(247\) 7.50794 0.477719
\(248\) 4.35795 0.276730
\(249\) −11.0499 −0.700259
\(250\) 1.13117 0.0715412
\(251\) −28.5671 −1.80314 −0.901568 0.432637i \(-0.857583\pi\)
−0.901568 + 0.432637i \(0.857583\pi\)
\(252\) 1.75034 0.110261
\(253\) 19.8786 1.24976
\(254\) −5.04339 −0.316450
\(255\) 0.685871 0.0429509
\(256\) 1.00000 0.0625000
\(257\) −17.5467 −1.09453 −0.547265 0.836959i \(-0.684331\pi\)
−0.547265 + 0.836959i \(0.684331\pi\)
\(258\) −6.58092 −0.409710
\(259\) 0.0533406 0.00331442
\(260\) 0.113262 0.00702420
\(261\) 9.14306 0.565942
\(262\) −4.59698 −0.284002
\(263\) −9.97158 −0.614874 −0.307437 0.951568i \(-0.599471\pi\)
−0.307437 + 0.951568i \(0.599471\pi\)
\(264\) −3.65617 −0.225021
\(265\) 0.351918 0.0216182
\(266\) −13.1415 −0.805756
\(267\) −10.4079 −0.636955
\(268\) −12.1648 −0.743085
\(269\) −5.46710 −0.333335 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(270\) 0.113262 0.00689289
\(271\) 22.8053 1.38533 0.692663 0.721262i \(-0.256438\pi\)
0.692663 + 0.721262i \(0.256438\pi\)
\(272\) 6.05562 0.367176
\(273\) 1.75034 0.105936
\(274\) 15.0484 0.909107
\(275\) −18.2339 −1.09955
\(276\) −5.43702 −0.327270
\(277\) 24.2026 1.45419 0.727096 0.686536i \(-0.240870\pi\)
0.727096 + 0.686536i \(0.240870\pi\)
\(278\) 5.89502 0.353560
\(279\) 4.35795 0.260903
\(280\) −0.198247 −0.0118475
\(281\) −4.92762 −0.293957 −0.146979 0.989140i \(-0.546955\pi\)
−0.146979 + 0.989140i \(0.546955\pi\)
\(282\) 11.8201 0.703876
\(283\) −10.6051 −0.630409 −0.315205 0.949024i \(-0.602073\pi\)
−0.315205 + 0.949024i \(0.602073\pi\)
\(284\) −5.75511 −0.341503
\(285\) −0.850363 −0.0503712
\(286\) −3.65617 −0.216194
\(287\) 9.12208 0.538459
\(288\) 1.00000 0.0589256
\(289\) 19.6706 1.15709
\(290\) −1.03556 −0.0608102
\(291\) −10.1090 −0.592601
\(292\) −6.79650 −0.397735
\(293\) 20.3878 1.19107 0.595533 0.803331i \(-0.296941\pi\)
0.595533 + 0.803331i \(0.296941\pi\)
\(294\) 3.93630 0.229570
\(295\) −0.790460 −0.0460224
\(296\) 0.0304744 0.00177129
\(297\) −3.65617 −0.212152
\(298\) 0.0191072 0.00110685
\(299\) −5.43702 −0.314431
\(300\) 4.98717 0.287934
\(301\) 11.5189 0.663937
\(302\) −2.37037 −0.136399
\(303\) 1.08143 0.0621263
\(304\) −7.50794 −0.430610
\(305\) −1.15676 −0.0662362
\(306\) 6.05562 0.346177
\(307\) 5.60226 0.319738 0.159869 0.987138i \(-0.448893\pi\)
0.159869 + 0.987138i \(0.448893\pi\)
\(308\) 6.39954 0.364648
\(309\) 1.00000 0.0568880
\(310\) −0.493589 −0.0280340
\(311\) −18.0590 −1.02403 −0.512015 0.858976i \(-0.671101\pi\)
−0.512015 + 0.858976i \(0.671101\pi\)
\(312\) 1.00000 0.0566139
\(313\) −4.76033 −0.269070 −0.134535 0.990909i \(-0.542954\pi\)
−0.134535 + 0.990909i \(0.542954\pi\)
\(314\) −18.0051 −1.01609
\(315\) −0.198247 −0.0111700
\(316\) −5.30030 −0.298165
\(317\) 27.1745 1.52627 0.763136 0.646238i \(-0.223659\pi\)
0.763136 + 0.646238i \(0.223659\pi\)
\(318\) 3.10712 0.174239
\(319\) 33.4286 1.87164
\(320\) −0.113262 −0.00633153
\(321\) −2.13304 −0.119055
\(322\) 9.51665 0.530342
\(323\) −45.4653 −2.52976
\(324\) 1.00000 0.0555556
\(325\) 4.98717 0.276639
\(326\) 17.8749 0.989998
\(327\) 2.39240 0.132300
\(328\) 5.21159 0.287762
\(329\) −20.6892 −1.14063
\(330\) 0.414104 0.0227957
\(331\) 28.1489 1.54720 0.773601 0.633673i \(-0.218453\pi\)
0.773601 + 0.633673i \(0.218453\pi\)
\(332\) 11.0499 0.606442
\(333\) 0.0304744 0.00166999
\(334\) 4.42869 0.242327
\(335\) 1.37781 0.0752778
\(336\) −1.75034 −0.0954890
\(337\) −9.70607 −0.528723 −0.264362 0.964424i \(-0.585161\pi\)
−0.264362 + 0.964424i \(0.585161\pi\)
\(338\) 1.00000 0.0543928
\(339\) −1.47065 −0.0798748
\(340\) −0.685871 −0.0371966
\(341\) 15.9334 0.862841
\(342\) −7.50794 −0.405983
\(343\) −19.1423 −1.03359
\(344\) 6.58092 0.354820
\(345\) 0.615806 0.0331539
\(346\) 9.46262 0.508714
\(347\) 0.347167 0.0186369 0.00931844 0.999957i \(-0.497034\pi\)
0.00931844 + 0.999957i \(0.497034\pi\)
\(348\) −9.14306 −0.490120
\(349\) 15.1633 0.811672 0.405836 0.913946i \(-0.366980\pi\)
0.405836 + 0.913946i \(0.366980\pi\)
\(350\) −8.72926 −0.466599
\(351\) 1.00000 0.0533761
\(352\) 3.65617 0.194874
\(353\) 23.8881 1.27144 0.635718 0.771921i \(-0.280704\pi\)
0.635718 + 0.771921i \(0.280704\pi\)
\(354\) −6.97905 −0.370932
\(355\) 0.651834 0.0345958
\(356\) 10.4079 0.551619
\(357\) −10.5994 −0.560981
\(358\) 11.2831 0.596331
\(359\) 21.9488 1.15841 0.579206 0.815181i \(-0.303363\pi\)
0.579206 + 0.815181i \(0.303363\pi\)
\(360\) −0.113262 −0.00596942
\(361\) 37.3692 1.96680
\(362\) −17.1528 −0.901533
\(363\) −2.36755 −0.124264
\(364\) −1.75034 −0.0917429
\(365\) 0.769784 0.0402923
\(366\) −10.2132 −0.533852
\(367\) 9.65935 0.504214 0.252107 0.967699i \(-0.418876\pi\)
0.252107 + 0.967699i \(0.418876\pi\)
\(368\) 5.43702 0.283424
\(369\) 5.21159 0.271305
\(370\) −0.00345158 −0.000179439 0
\(371\) −5.43853 −0.282354
\(372\) −4.35795 −0.225949
\(373\) 33.1269 1.71525 0.857623 0.514279i \(-0.171940\pi\)
0.857623 + 0.514279i \(0.171940\pi\)
\(374\) 22.1404 1.14485
\(375\) −1.13117 −0.0584131
\(376\) −11.8201 −0.609574
\(377\) −9.14306 −0.470892
\(378\) −1.75034 −0.0900279
\(379\) −13.3594 −0.686225 −0.343113 0.939294i \(-0.611481\pi\)
−0.343113 + 0.939294i \(0.611481\pi\)
\(380\) 0.850363 0.0436227
\(381\) 5.04339 0.258381
\(382\) 3.75731 0.192241
\(383\) 19.2044 0.981297 0.490649 0.871357i \(-0.336760\pi\)
0.490649 + 0.871357i \(0.336760\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.724824 −0.0369405
\(386\) −4.89362 −0.249079
\(387\) 6.58092 0.334527
\(388\) 10.1090 0.513207
\(389\) 27.5591 1.39730 0.698651 0.715463i \(-0.253784\pi\)
0.698651 + 0.715463i \(0.253784\pi\)
\(390\) −0.113262 −0.00573524
\(391\) 32.9245 1.66507
\(392\) −3.93630 −0.198813
\(393\) 4.59698 0.231887
\(394\) −24.9189 −1.25539
\(395\) 0.600321 0.0302054
\(396\) 3.65617 0.183729
\(397\) 12.6689 0.635834 0.317917 0.948119i \(-0.397017\pi\)
0.317917 + 0.948119i \(0.397017\pi\)
\(398\) 12.8498 0.644104
\(399\) 13.1415 0.657897
\(400\) −4.98717 −0.249359
\(401\) 5.67991 0.283641 0.141821 0.989892i \(-0.454704\pi\)
0.141821 + 0.989892i \(0.454704\pi\)
\(402\) 12.1648 0.606726
\(403\) −4.35795 −0.217085
\(404\) −1.08143 −0.0538030
\(405\) −0.113262 −0.00562803
\(406\) 16.0035 0.794240
\(407\) 0.111419 0.00552285
\(408\) −6.05562 −0.299798
\(409\) −26.1681 −1.29393 −0.646965 0.762520i \(-0.723962\pi\)
−0.646965 + 0.762520i \(0.723962\pi\)
\(410\) −0.590274 −0.0291516
\(411\) −15.0484 −0.742283
\(412\) −1.00000 −0.0492665
\(413\) 12.2157 0.601097
\(414\) 5.43702 0.267215
\(415\) −1.25153 −0.0614353
\(416\) −1.00000 −0.0490290
\(417\) −5.89502 −0.288680
\(418\) −27.4503 −1.34264
\(419\) −28.2916 −1.38214 −0.691068 0.722790i \(-0.742859\pi\)
−0.691068 + 0.722790i \(0.742859\pi\)
\(420\) 0.198247 0.00967347
\(421\) 10.1314 0.493776 0.246888 0.969044i \(-0.420592\pi\)
0.246888 + 0.969044i \(0.420592\pi\)
\(422\) 16.3252 0.794700
\(423\) −11.8201 −0.574712
\(424\) −3.10712 −0.150895
\(425\) −30.2004 −1.46494
\(426\) 5.75511 0.278836
\(427\) 17.8766 0.865109
\(428\) 2.13304 0.103105
\(429\) 3.65617 0.176521
\(430\) −0.745368 −0.0359448
\(431\) 38.4390 1.85154 0.925771 0.378084i \(-0.123417\pi\)
0.925771 + 0.378084i \(0.123417\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −23.8800 −1.14760 −0.573800 0.818995i \(-0.694531\pi\)
−0.573800 + 0.818995i \(0.694531\pi\)
\(434\) 7.62790 0.366151
\(435\) 1.03556 0.0496513
\(436\) −2.39240 −0.114575
\(437\) −40.8208 −1.95272
\(438\) 6.79650 0.324749
\(439\) −36.5348 −1.74371 −0.871856 0.489763i \(-0.837083\pi\)
−0.871856 + 0.489763i \(0.837083\pi\)
\(440\) −0.414104 −0.0197416
\(441\) −3.93630 −0.187443
\(442\) −6.05562 −0.288037
\(443\) 39.2216 1.86347 0.931737 0.363134i \(-0.118293\pi\)
0.931737 + 0.363134i \(0.118293\pi\)
\(444\) −0.0304744 −0.00144625
\(445\) −1.17882 −0.0558815
\(446\) −14.1757 −0.671237
\(447\) −0.0191072 −0.000903738 0
\(448\) 1.75034 0.0826959
\(449\) 7.10810 0.335452 0.167726 0.985834i \(-0.446358\pi\)
0.167726 + 0.985834i \(0.446358\pi\)
\(450\) −4.98717 −0.235098
\(451\) 19.0544 0.897239
\(452\) 1.47065 0.0691736
\(453\) 2.37037 0.111369
\(454\) 7.24312 0.339936
\(455\) 0.198247 0.00929396
\(456\) 7.50794 0.351592
\(457\) 3.49794 0.163627 0.0818133 0.996648i \(-0.473929\pi\)
0.0818133 + 0.996648i \(0.473929\pi\)
\(458\) −1.26769 −0.0592353
\(459\) −6.05562 −0.282652
\(460\) −0.615806 −0.0287121
\(461\) −16.7541 −0.780317 −0.390159 0.920748i \(-0.627580\pi\)
−0.390159 + 0.920748i \(0.627580\pi\)
\(462\) −6.39954 −0.297734
\(463\) 34.6202 1.60894 0.804469 0.593995i \(-0.202450\pi\)
0.804469 + 0.593995i \(0.202450\pi\)
\(464\) 9.14306 0.424456
\(465\) 0.493589 0.0228896
\(466\) −8.23879 −0.381655
\(467\) 27.0423 1.25137 0.625685 0.780076i \(-0.284820\pi\)
0.625685 + 0.780076i \(0.284820\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −21.2926 −0.983202
\(470\) 1.33876 0.0617526
\(471\) 18.0051 0.829633
\(472\) 6.97905 0.321237
\(473\) 24.0610 1.10632
\(474\) 5.30030 0.243451
\(475\) 37.4434 1.71802
\(476\) 10.5994 0.485824
\(477\) −3.10712 −0.142265
\(478\) −2.25279 −0.103040
\(479\) −3.48706 −0.159328 −0.0796640 0.996822i \(-0.525385\pi\)
−0.0796640 + 0.996822i \(0.525385\pi\)
\(480\) 0.113262 0.00516967
\(481\) −0.0304744 −0.00138951
\(482\) 13.8985 0.633061
\(483\) −9.51665 −0.433022
\(484\) 2.36755 0.107616
\(485\) −1.14496 −0.0519902
\(486\) −1.00000 −0.0453609
\(487\) −25.2054 −1.14217 −0.571084 0.820892i \(-0.693477\pi\)
−0.571084 + 0.820892i \(0.693477\pi\)
\(488\) 10.2132 0.462330
\(489\) −17.8749 −0.808330
\(490\) 0.445832 0.0201407
\(491\) −24.3399 −1.09844 −0.549221 0.835677i \(-0.685076\pi\)
−0.549221 + 0.835677i \(0.685076\pi\)
\(492\) −5.21159 −0.234957
\(493\) 55.3670 2.49360
\(494\) 7.50794 0.337798
\(495\) −0.414104 −0.0186126
\(496\) 4.35795 0.195678
\(497\) −10.0734 −0.451854
\(498\) −11.0499 −0.495158
\(499\) 4.90624 0.219633 0.109817 0.993952i \(-0.464974\pi\)
0.109817 + 0.993952i \(0.464974\pi\)
\(500\) 1.13117 0.0505872
\(501\) −4.42869 −0.197859
\(502\) −28.5671 −1.27501
\(503\) −11.7912 −0.525745 −0.262872 0.964831i \(-0.584670\pi\)
−0.262872 + 0.964831i \(0.584670\pi\)
\(504\) 1.75034 0.0779665
\(505\) 0.122484 0.00545048
\(506\) 19.8786 0.883713
\(507\) −1.00000 −0.0444116
\(508\) −5.04339 −0.223764
\(509\) 9.02388 0.399976 0.199988 0.979798i \(-0.435910\pi\)
0.199988 + 0.979798i \(0.435910\pi\)
\(510\) 0.685871 0.0303709
\(511\) −11.8962 −0.526257
\(512\) 1.00000 0.0441942
\(513\) 7.50794 0.331484
\(514\) −17.5467 −0.773950
\(515\) 0.113262 0.00499091
\(516\) −6.58092 −0.289709
\(517\) −43.2162 −1.90065
\(518\) 0.0533406 0.00234365
\(519\) −9.46262 −0.415363
\(520\) 0.113262 0.00496686
\(521\) −19.5572 −0.856818 −0.428409 0.903585i \(-0.640926\pi\)
−0.428409 + 0.903585i \(0.640926\pi\)
\(522\) 9.14306 0.400181
\(523\) −13.6811 −0.598232 −0.299116 0.954217i \(-0.596692\pi\)
−0.299116 + 0.954217i \(0.596692\pi\)
\(524\) −4.59698 −0.200820
\(525\) 8.72926 0.380976
\(526\) −9.97158 −0.434782
\(527\) 26.3901 1.14957
\(528\) −3.65617 −0.159114
\(529\) 6.56116 0.285268
\(530\) 0.351918 0.0152863
\(531\) 6.97905 0.302865
\(532\) −13.1415 −0.569755
\(533\) −5.21159 −0.225739
\(534\) −10.4079 −0.450395
\(535\) −0.241593 −0.0104450
\(536\) −12.1648 −0.525441
\(537\) −11.2831 −0.486902
\(538\) −5.46710 −0.235704
\(539\) −14.3918 −0.619897
\(540\) 0.113262 0.00487401
\(541\) 28.2190 1.21323 0.606615 0.794996i \(-0.292527\pi\)
0.606615 + 0.794996i \(0.292527\pi\)
\(542\) 22.8053 0.979573
\(543\) 17.1528 0.736099
\(544\) 6.05562 0.259633
\(545\) 0.270967 0.0116070
\(546\) 1.75034 0.0749078
\(547\) 23.1882 0.991457 0.495729 0.868477i \(-0.334901\pi\)
0.495729 + 0.868477i \(0.334901\pi\)
\(548\) 15.0484 0.642836
\(549\) 10.2132 0.435889
\(550\) −18.2339 −0.777497
\(551\) −68.6456 −2.92440
\(552\) −5.43702 −0.231415
\(553\) −9.27734 −0.394513
\(554\) 24.2026 1.02827
\(555\) 0.00345158 0.000146512 0
\(556\) 5.89502 0.250004
\(557\) 31.6205 1.33981 0.669903 0.742449i \(-0.266336\pi\)
0.669903 + 0.742449i \(0.266336\pi\)
\(558\) 4.35795 0.184487
\(559\) −6.58092 −0.278343
\(560\) −0.198247 −0.00837747
\(561\) −22.1404 −0.934767
\(562\) −4.92762 −0.207859
\(563\) −43.7292 −1.84297 −0.921483 0.388419i \(-0.873021\pi\)
−0.921483 + 0.388419i \(0.873021\pi\)
\(564\) 11.8201 0.497715
\(565\) −0.166569 −0.00700760
\(566\) −10.6051 −0.445767
\(567\) 1.75034 0.0735075
\(568\) −5.75511 −0.241479
\(569\) 47.0223 1.97128 0.985639 0.168867i \(-0.0540109\pi\)
0.985639 + 0.168867i \(0.0540109\pi\)
\(570\) −0.850363 −0.0356178
\(571\) −28.0637 −1.17443 −0.587214 0.809432i \(-0.699775\pi\)
−0.587214 + 0.809432i \(0.699775\pi\)
\(572\) −3.65617 −0.152872
\(573\) −3.75731 −0.156964
\(574\) 9.12208 0.380748
\(575\) −27.1153 −1.13079
\(576\) 1.00000 0.0416667
\(577\) −8.71369 −0.362756 −0.181378 0.983413i \(-0.558056\pi\)
−0.181378 + 0.983413i \(0.558056\pi\)
\(578\) 19.6706 0.818188
\(579\) 4.89362 0.203372
\(580\) −1.03556 −0.0429993
\(581\) 19.3411 0.802404
\(582\) −10.1090 −0.419032
\(583\) −11.3602 −0.470489
\(584\) −6.79650 −0.281241
\(585\) 0.113262 0.00468280
\(586\) 20.3878 0.842211
\(587\) −29.4117 −1.21395 −0.606976 0.794721i \(-0.707617\pi\)
−0.606976 + 0.794721i \(0.707617\pi\)
\(588\) 3.93630 0.162330
\(589\) −32.7192 −1.34817
\(590\) −0.790460 −0.0325427
\(591\) 24.9189 1.02503
\(592\) 0.0304744 0.00125249
\(593\) −31.1387 −1.27871 −0.639357 0.768910i \(-0.720800\pi\)
−0.639357 + 0.768910i \(0.720800\pi\)
\(594\) −3.65617 −0.150014
\(595\) −1.20051 −0.0492161
\(596\) 0.0191072 0.000782660 0
\(597\) −12.8498 −0.525908
\(598\) −5.43702 −0.222336
\(599\) 3.34070 0.136497 0.0682486 0.997668i \(-0.478259\pi\)
0.0682486 + 0.997668i \(0.478259\pi\)
\(600\) 4.98717 0.203600
\(601\) 29.2746 1.19413 0.597067 0.802191i \(-0.296332\pi\)
0.597067 + 0.802191i \(0.296332\pi\)
\(602\) 11.5189 0.469474
\(603\) −12.1648 −0.495390
\(604\) −2.37037 −0.0964488
\(605\) −0.268153 −0.0109020
\(606\) 1.08143 0.0439300
\(607\) −38.4815 −1.56192 −0.780958 0.624584i \(-0.785269\pi\)
−0.780958 + 0.624584i \(0.785269\pi\)
\(608\) −7.50794 −0.304487
\(609\) −16.0035 −0.648495
\(610\) −1.15676 −0.0468360
\(611\) 11.8201 0.478190
\(612\) 6.05562 0.244784
\(613\) 25.6048 1.03417 0.517084 0.855935i \(-0.327017\pi\)
0.517084 + 0.855935i \(0.327017\pi\)
\(614\) 5.60226 0.226089
\(615\) 0.590274 0.0238022
\(616\) 6.39954 0.257845
\(617\) −5.82731 −0.234599 −0.117299 0.993097i \(-0.537424\pi\)
−0.117299 + 0.993097i \(0.537424\pi\)
\(618\) 1.00000 0.0402259
\(619\) −9.74174 −0.391554 −0.195777 0.980648i \(-0.562723\pi\)
−0.195777 + 0.980648i \(0.562723\pi\)
\(620\) −0.493589 −0.0198230
\(621\) −5.43702 −0.218180
\(622\) −18.0590 −0.724099
\(623\) 18.2174 0.729866
\(624\) 1.00000 0.0400320
\(625\) 24.8077 0.992310
\(626\) −4.76033 −0.190261
\(627\) 27.4503 1.09626
\(628\) −18.0051 −0.718483
\(629\) 0.184541 0.00735815
\(630\) −0.198247 −0.00789835
\(631\) 10.6825 0.425262 0.212631 0.977133i \(-0.431797\pi\)
0.212631 + 0.977133i \(0.431797\pi\)
\(632\) −5.30030 −0.210835
\(633\) −16.3252 −0.648870
\(634\) 27.1745 1.07924
\(635\) 0.571223 0.0226683
\(636\) 3.10712 0.123205
\(637\) 3.93630 0.155962
\(638\) 33.4286 1.32345
\(639\) −5.75511 −0.227669
\(640\) −0.113262 −0.00447707
\(641\) −6.41802 −0.253497 −0.126748 0.991935i \(-0.540454\pi\)
−0.126748 + 0.991935i \(0.540454\pi\)
\(642\) −2.13304 −0.0841846
\(643\) −20.5851 −0.811798 −0.405899 0.913918i \(-0.633041\pi\)
−0.405899 + 0.913918i \(0.633041\pi\)
\(644\) 9.51665 0.375008
\(645\) 0.745368 0.0293488
\(646\) −45.4653 −1.78881
\(647\) −3.21383 −0.126349 −0.0631744 0.998003i \(-0.520122\pi\)
−0.0631744 + 0.998003i \(0.520122\pi\)
\(648\) 1.00000 0.0392837
\(649\) 25.5166 1.00161
\(650\) 4.98717 0.195613
\(651\) −7.62790 −0.298961
\(652\) 17.8749 0.700034
\(653\) 8.29517 0.324615 0.162307 0.986740i \(-0.448106\pi\)
0.162307 + 0.986740i \(0.448106\pi\)
\(654\) 2.39240 0.0935501
\(655\) 0.520662 0.0203440
\(656\) 5.21159 0.203479
\(657\) −6.79650 −0.265157
\(658\) −20.6892 −0.806549
\(659\) −8.89655 −0.346560 −0.173280 0.984873i \(-0.555437\pi\)
−0.173280 + 0.984873i \(0.555437\pi\)
\(660\) 0.414104 0.0161190
\(661\) 18.4665 0.718264 0.359132 0.933287i \(-0.383073\pi\)
0.359132 + 0.933287i \(0.383073\pi\)
\(662\) 28.1489 1.09404
\(663\) 6.05562 0.235181
\(664\) 11.0499 0.428819
\(665\) 1.48843 0.0577187
\(666\) 0.0304744 0.00118086
\(667\) 49.7110 1.92482
\(668\) 4.42869 0.171351
\(669\) 14.1757 0.548063
\(670\) 1.37781 0.0532295
\(671\) 37.3411 1.44154
\(672\) −1.75034 −0.0675210
\(673\) −18.1098 −0.698082 −0.349041 0.937107i \(-0.613493\pi\)
−0.349041 + 0.937107i \(0.613493\pi\)
\(674\) −9.70607 −0.373864
\(675\) 4.98717 0.191956
\(676\) 1.00000 0.0384615
\(677\) 19.9368 0.766234 0.383117 0.923700i \(-0.374851\pi\)
0.383117 + 0.923700i \(0.374851\pi\)
\(678\) −1.47065 −0.0564800
\(679\) 17.6942 0.679042
\(680\) −0.685871 −0.0263020
\(681\) −7.24312 −0.277557
\(682\) 15.9334 0.610120
\(683\) −36.6572 −1.40265 −0.701324 0.712843i \(-0.747407\pi\)
−0.701324 + 0.712843i \(0.747407\pi\)
\(684\) −7.50794 −0.287073
\(685\) −1.70441 −0.0651221
\(686\) −19.1423 −0.730855
\(687\) 1.26769 0.0483654
\(688\) 6.58092 0.250895
\(689\) 3.10712 0.118372
\(690\) 0.615806 0.0234434
\(691\) 28.4139 1.08092 0.540459 0.841371i \(-0.318251\pi\)
0.540459 + 0.841371i \(0.318251\pi\)
\(692\) 9.46262 0.359715
\(693\) 6.39954 0.243099
\(694\) 0.347167 0.0131783
\(695\) −0.667680 −0.0253266
\(696\) −9.14306 −0.346567
\(697\) 31.5594 1.19540
\(698\) 15.1633 0.573939
\(699\) 8.23879 0.311620
\(700\) −8.72926 −0.329935
\(701\) −1.07339 −0.0405413 −0.0202706 0.999795i \(-0.506453\pi\)
−0.0202706 + 0.999795i \(0.506453\pi\)
\(702\) 1.00000 0.0377426
\(703\) −0.228800 −0.00862935
\(704\) 3.65617 0.137797
\(705\) −1.33876 −0.0504208
\(706\) 23.8881 0.899041
\(707\) −1.89287 −0.0711886
\(708\) −6.97905 −0.262289
\(709\) −38.6959 −1.45325 −0.726627 0.687032i \(-0.758913\pi\)
−0.726627 + 0.687032i \(0.758913\pi\)
\(710\) 0.651834 0.0244629
\(711\) −5.30030 −0.198777
\(712\) 10.4079 0.390054
\(713\) 23.6942 0.887356
\(714\) −10.5994 −0.396673
\(715\) 0.414104 0.0154866
\(716\) 11.2831 0.421669
\(717\) 2.25279 0.0841321
\(718\) 21.9488 0.819121
\(719\) −27.6041 −1.02946 −0.514730 0.857352i \(-0.672108\pi\)
−0.514730 + 0.857352i \(0.672108\pi\)
\(720\) −0.113262 −0.00422102
\(721\) −1.75034 −0.0651862
\(722\) 37.3692 1.39074
\(723\) −13.8985 −0.516892
\(724\) −17.1528 −0.637480
\(725\) −45.5980 −1.69347
\(726\) −2.36755 −0.0878680
\(727\) −28.5672 −1.05950 −0.529750 0.848154i \(-0.677714\pi\)
−0.529750 + 0.848154i \(0.677714\pi\)
\(728\) −1.75034 −0.0648720
\(729\) 1.00000 0.0370370
\(730\) 0.769784 0.0284910
\(731\) 39.8516 1.47396
\(732\) −10.2132 −0.377491
\(733\) 37.3573 1.37983 0.689913 0.723893i \(-0.257649\pi\)
0.689913 + 0.723893i \(0.257649\pi\)
\(734\) 9.65935 0.356533
\(735\) −0.445832 −0.0164448
\(736\) 5.43702 0.200411
\(737\) −44.4766 −1.63832
\(738\) 5.21159 0.191841
\(739\) 10.8084 0.397595 0.198798 0.980041i \(-0.436296\pi\)
0.198798 + 0.980041i \(0.436296\pi\)
\(740\) −0.00345158 −0.000126883 0
\(741\) −7.50794 −0.275811
\(742\) −5.43853 −0.199655
\(743\) −33.2240 −1.21887 −0.609435 0.792836i \(-0.708604\pi\)
−0.609435 + 0.792836i \(0.708604\pi\)
\(744\) −4.35795 −0.159770
\(745\) −0.00216411 −7.92870e−5 0
\(746\) 33.1269 1.21286
\(747\) 11.0499 0.404295
\(748\) 22.1404 0.809532
\(749\) 3.73356 0.136421
\(750\) −1.13117 −0.0413043
\(751\) −6.10923 −0.222929 −0.111465 0.993768i \(-0.535554\pi\)
−0.111465 + 0.993768i \(0.535554\pi\)
\(752\) −11.8201 −0.431034
\(753\) 28.5671 1.04104
\(754\) −9.14306 −0.332971
\(755\) 0.268472 0.00977069
\(756\) −1.75034 −0.0636594
\(757\) −22.2939 −0.810287 −0.405143 0.914253i \(-0.632778\pi\)
−0.405143 + 0.914253i \(0.632778\pi\)
\(758\) −13.3594 −0.485235
\(759\) −19.8786 −0.721549
\(760\) 0.850363 0.0308459
\(761\) −20.1551 −0.730622 −0.365311 0.930886i \(-0.619037\pi\)
−0.365311 + 0.930886i \(0.619037\pi\)
\(762\) 5.04339 0.182703
\(763\) −4.18751 −0.151598
\(764\) 3.75731 0.135935
\(765\) −0.685871 −0.0247977
\(766\) 19.2044 0.693882
\(767\) −6.97905 −0.251999
\(768\) −1.00000 −0.0360844
\(769\) 20.5590 0.741378 0.370689 0.928757i \(-0.379122\pi\)
0.370689 + 0.928757i \(0.379122\pi\)
\(770\) −0.724824 −0.0261208
\(771\) 17.5467 0.631927
\(772\) −4.89362 −0.176125
\(773\) 18.1145 0.651532 0.325766 0.945450i \(-0.394378\pi\)
0.325766 + 0.945450i \(0.394378\pi\)
\(774\) 6.58092 0.236546
\(775\) −21.7338 −0.780702
\(776\) 10.1090 0.362892
\(777\) −0.0533406 −0.00191358
\(778\) 27.5591 0.988042
\(779\) −39.1283 −1.40192
\(780\) −0.113262 −0.00405542
\(781\) −21.0416 −0.752929
\(782\) 32.9245 1.17738
\(783\) −9.14306 −0.326746
\(784\) −3.93630 −0.140582
\(785\) 2.03929 0.0727856
\(786\) 4.59698 0.163969
\(787\) −5.64316 −0.201157 −0.100579 0.994929i \(-0.532069\pi\)
−0.100579 + 0.994929i \(0.532069\pi\)
\(788\) −24.9189 −0.887698
\(789\) 9.97158 0.354998
\(790\) 0.600321 0.0213585
\(791\) 2.57414 0.0915260
\(792\) 3.65617 0.129916
\(793\) −10.2132 −0.362681
\(794\) 12.6689 0.449603
\(795\) −0.351918 −0.0124813
\(796\) 12.8498 0.455450
\(797\) 0.839598 0.0297401 0.0148700 0.999889i \(-0.495267\pi\)
0.0148700 + 0.999889i \(0.495267\pi\)
\(798\) 13.1415 0.465203
\(799\) −71.5780 −2.53225
\(800\) −4.98717 −0.176323
\(801\) 10.4079 0.367746
\(802\) 5.67991 0.200565
\(803\) −24.8491 −0.876907
\(804\) 12.1648 0.429020
\(805\) −1.07787 −0.0379900
\(806\) −4.35795 −0.153502
\(807\) 5.46710 0.192451
\(808\) −1.08143 −0.0380445
\(809\) −40.2280 −1.41434 −0.707171 0.707043i \(-0.750029\pi\)
−0.707171 + 0.707043i \(0.750029\pi\)
\(810\) −0.113262 −0.00397961
\(811\) −40.4036 −1.41876 −0.709381 0.704826i \(-0.751025\pi\)
−0.709381 + 0.704826i \(0.751025\pi\)
\(812\) 16.0035 0.561613
\(813\) −22.8053 −0.799818
\(814\) 0.111419 0.00390525
\(815\) −2.02454 −0.0709166
\(816\) −6.05562 −0.211989
\(817\) −49.4092 −1.72861
\(818\) −26.1681 −0.914946
\(819\) −1.75034 −0.0611619
\(820\) −0.590274 −0.0206133
\(821\) −21.7903 −0.760487 −0.380244 0.924886i \(-0.624160\pi\)
−0.380244 + 0.924886i \(0.624160\pi\)
\(822\) −15.0484 −0.524873
\(823\) −16.4108 −0.572044 −0.286022 0.958223i \(-0.592333\pi\)
−0.286022 + 0.958223i \(0.592333\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 18.2339 0.634824
\(826\) 12.2157 0.425040
\(827\) 17.7950 0.618794 0.309397 0.950933i \(-0.399873\pi\)
0.309397 + 0.950933i \(0.399873\pi\)
\(828\) 5.43702 0.188949
\(829\) −52.2100 −1.81333 −0.906665 0.421852i \(-0.861380\pi\)
−0.906665 + 0.421852i \(0.861380\pi\)
\(830\) −1.25153 −0.0434413
\(831\) −24.2026 −0.839578
\(832\) −1.00000 −0.0346688
\(833\) −23.8367 −0.825894
\(834\) −5.89502 −0.204128
\(835\) −0.501601 −0.0173586
\(836\) −27.4503 −0.949388
\(837\) −4.35795 −0.150633
\(838\) −28.2916 −0.977318
\(839\) 15.9315 0.550017 0.275009 0.961442i \(-0.411319\pi\)
0.275009 + 0.961442i \(0.411319\pi\)
\(840\) 0.198247 0.00684017
\(841\) 54.5956 1.88261
\(842\) 10.1314 0.349153
\(843\) 4.92762 0.169716
\(844\) 16.3252 0.561938
\(845\) −0.113262 −0.00389633
\(846\) −11.8201 −0.406383
\(847\) 4.14402 0.142390
\(848\) −3.10712 −0.106699
\(849\) 10.6051 0.363967
\(850\) −30.2004 −1.03587
\(851\) 0.165690 0.00567977
\(852\) 5.75511 0.197167
\(853\) −14.9322 −0.511269 −0.255634 0.966774i \(-0.582284\pi\)
−0.255634 + 0.966774i \(0.582284\pi\)
\(854\) 17.8766 0.611724
\(855\) 0.850363 0.0290818
\(856\) 2.13304 0.0729060
\(857\) −54.0921 −1.84775 −0.923876 0.382692i \(-0.874997\pi\)
−0.923876 + 0.382692i \(0.874997\pi\)
\(858\) 3.65617 0.124819
\(859\) −21.9404 −0.748596 −0.374298 0.927309i \(-0.622116\pi\)
−0.374298 + 0.927309i \(0.622116\pi\)
\(860\) −0.745368 −0.0254168
\(861\) −9.12208 −0.310880
\(862\) 38.4390 1.30924
\(863\) 5.05710 0.172146 0.0860729 0.996289i \(-0.472568\pi\)
0.0860729 + 0.996289i \(0.472568\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.07175 −0.0364407
\(866\) −23.8800 −0.811476
\(867\) −19.6706 −0.668048
\(868\) 7.62790 0.258908
\(869\) −19.3788 −0.657380
\(870\) 1.03556 0.0351088
\(871\) 12.1648 0.412189
\(872\) −2.39240 −0.0810167
\(873\) 10.1090 0.342138
\(874\) −40.8208 −1.38078
\(875\) 1.97993 0.0669338
\(876\) 6.79650 0.229633
\(877\) −6.50539 −0.219671 −0.109836 0.993950i \(-0.535032\pi\)
−0.109836 + 0.993950i \(0.535032\pi\)
\(878\) −36.5348 −1.23299
\(879\) −20.3878 −0.687663
\(880\) −0.414104 −0.0139594
\(881\) 52.8066 1.77910 0.889549 0.456839i \(-0.151019\pi\)
0.889549 + 0.456839i \(0.151019\pi\)
\(882\) −3.93630 −0.132542
\(883\) −10.8432 −0.364902 −0.182451 0.983215i \(-0.558403\pi\)
−0.182451 + 0.983215i \(0.558403\pi\)
\(884\) −6.05562 −0.203673
\(885\) 0.790460 0.0265710
\(886\) 39.2216 1.31768
\(887\) −13.2482 −0.444830 −0.222415 0.974952i \(-0.571394\pi\)
−0.222415 + 0.974952i \(0.571394\pi\)
\(888\) −0.0304744 −0.00102265
\(889\) −8.82766 −0.296070
\(890\) −1.17882 −0.0395142
\(891\) 3.65617 0.122486
\(892\) −14.1757 −0.474637
\(893\) 88.7446 2.96972
\(894\) −0.0191072 −0.000639039 0
\(895\) −1.27794 −0.0427170
\(896\) 1.75034 0.0584749
\(897\) 5.43702 0.181537
\(898\) 7.10810 0.237200
\(899\) 39.8450 1.32890
\(900\) −4.98717 −0.166239
\(901\) −18.8156 −0.626837
\(902\) 19.0544 0.634444
\(903\) −11.5189 −0.383324
\(904\) 1.47065 0.0489131
\(905\) 1.94276 0.0645796
\(906\) 2.37037 0.0787501
\(907\) 54.2853 1.80251 0.901257 0.433285i \(-0.142646\pi\)
0.901257 + 0.433285i \(0.142646\pi\)
\(908\) 7.24312 0.240371
\(909\) −1.08143 −0.0358687
\(910\) 0.198247 0.00657183
\(911\) −13.2915 −0.440366 −0.220183 0.975459i \(-0.570665\pi\)
−0.220183 + 0.975459i \(0.570665\pi\)
\(912\) 7.50794 0.248613
\(913\) 40.4002 1.33705
\(914\) 3.49794 0.115702
\(915\) 1.15676 0.0382415
\(916\) −1.26769 −0.0418857
\(917\) −8.04629 −0.265712
\(918\) −6.05562 −0.199865
\(919\) −33.7921 −1.11470 −0.557349 0.830278i \(-0.688182\pi\)
−0.557349 + 0.830278i \(0.688182\pi\)
\(920\) −0.615806 −0.0203025
\(921\) −5.60226 −0.184601
\(922\) −16.7541 −0.551768
\(923\) 5.75511 0.189432
\(924\) −6.39954 −0.210530
\(925\) −0.151981 −0.00499710
\(926\) 34.6202 1.13769
\(927\) −1.00000 −0.0328443
\(928\) 9.14306 0.300136
\(929\) −25.0860 −0.823045 −0.411523 0.911400i \(-0.635003\pi\)
−0.411523 + 0.911400i \(0.635003\pi\)
\(930\) 0.493589 0.0161854
\(931\) 29.5535 0.968577
\(932\) −8.23879 −0.269871
\(933\) 18.0590 0.591224
\(934\) 27.0423 0.884852
\(935\) −2.50766 −0.0820092
\(936\) −1.00000 −0.0326860
\(937\) −10.6702 −0.348579 −0.174289 0.984694i \(-0.555763\pi\)
−0.174289 + 0.984694i \(0.555763\pi\)
\(938\) −21.2926 −0.695229
\(939\) 4.76033 0.155348
\(940\) 1.33876 0.0436657
\(941\) −37.8436 −1.23367 −0.616833 0.787094i \(-0.711585\pi\)
−0.616833 + 0.787094i \(0.711585\pi\)
\(942\) 18.0051 0.586639
\(943\) 28.3355 0.922731
\(944\) 6.97905 0.227149
\(945\) 0.198247 0.00644898
\(946\) 24.0610 0.782289
\(947\) −21.8642 −0.710492 −0.355246 0.934773i \(-0.615603\pi\)
−0.355246 + 0.934773i \(0.615603\pi\)
\(948\) 5.30030 0.172146
\(949\) 6.79650 0.220624
\(950\) 37.4434 1.21482
\(951\) −27.1745 −0.881193
\(952\) 10.5994 0.343529
\(953\) −9.56399 −0.309808 −0.154904 0.987930i \(-0.549507\pi\)
−0.154904 + 0.987930i \(0.549507\pi\)
\(954\) −3.10712 −0.100597
\(955\) −0.425560 −0.0137708
\(956\) −2.25279 −0.0728606
\(957\) −33.4286 −1.08059
\(958\) −3.48706 −0.112662
\(959\) 26.3399 0.850559
\(960\) 0.113262 0.00365551
\(961\) −12.0083 −0.387365
\(962\) −0.0304744 −0.000982533 0
\(963\) 2.13304 0.0687364
\(964\) 13.8985 0.447642
\(965\) 0.554260 0.0178423
\(966\) −9.51665 −0.306193
\(967\) −43.1439 −1.38741 −0.693707 0.720257i \(-0.744024\pi\)
−0.693707 + 0.720257i \(0.744024\pi\)
\(968\) 2.36755 0.0760959
\(969\) 45.4653 1.46056
\(970\) −1.14496 −0.0367626
\(971\) 36.1333 1.15957 0.579786 0.814769i \(-0.303136\pi\)
0.579786 + 0.814769i \(0.303136\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 10.3183 0.330790
\(974\) −25.2054 −0.807634
\(975\) −4.98717 −0.159717
\(976\) 10.2132 0.326916
\(977\) −32.7260 −1.04700 −0.523499 0.852026i \(-0.675374\pi\)
−0.523499 + 0.852026i \(0.675374\pi\)
\(978\) −17.8749 −0.571575
\(979\) 38.0531 1.21618
\(980\) 0.445832 0.0142416
\(981\) −2.39240 −0.0763833
\(982\) −24.3399 −0.776716
\(983\) −0.489890 −0.0156251 −0.00781253 0.999969i \(-0.502487\pi\)
−0.00781253 + 0.999969i \(0.502487\pi\)
\(984\) −5.21159 −0.166140
\(985\) 2.82236 0.0899278
\(986\) 55.3670 1.76324
\(987\) 20.6892 0.658545
\(988\) 7.50794 0.238860
\(989\) 35.7806 1.13776
\(990\) −0.414104 −0.0131611
\(991\) 17.5448 0.557329 0.278665 0.960388i \(-0.410108\pi\)
0.278665 + 0.960388i \(0.410108\pi\)
\(992\) 4.35795 0.138365
\(993\) −28.1489 −0.893278
\(994\) −10.0734 −0.319509
\(995\) −1.45539 −0.0461391
\(996\) −11.0499 −0.350129
\(997\) −31.7460 −1.00541 −0.502703 0.864459i \(-0.667661\pi\)
−0.502703 + 0.864459i \(0.667661\pi\)
\(998\) 4.90624 0.155304
\(999\) −0.0304744 −0.000964167 0
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 8034.2.a.bc.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.8 15 1.1 even 1 trivial