Properties

Label 4029.2.a.k.1.8
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.49453 q^{2} +1.00000 q^{3} +0.233609 q^{4} -1.69557 q^{5} -1.49453 q^{6} +0.268778 q^{7} +2.63992 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.49453 q^{2} +1.00000 q^{3} +0.233609 q^{4} -1.69557 q^{5} -1.49453 q^{6} +0.268778 q^{7} +2.63992 q^{8} +1.00000 q^{9} +2.53407 q^{10} +0.913698 q^{11} +0.233609 q^{12} +6.63232 q^{13} -0.401696 q^{14} -1.69557 q^{15} -4.41265 q^{16} +1.00000 q^{17} -1.49453 q^{18} +0.913394 q^{19} -0.396101 q^{20} +0.268778 q^{21} -1.36555 q^{22} +4.49061 q^{23} +2.63992 q^{24} -2.12505 q^{25} -9.91218 q^{26} +1.00000 q^{27} +0.0627891 q^{28} +5.79951 q^{29} +2.53407 q^{30} -6.13148 q^{31} +1.31498 q^{32} +0.913698 q^{33} -1.49453 q^{34} -0.455732 q^{35} +0.233609 q^{36} +0.597523 q^{37} -1.36509 q^{38} +6.63232 q^{39} -4.47616 q^{40} +6.73930 q^{41} -0.401696 q^{42} +6.30611 q^{43} +0.213448 q^{44} -1.69557 q^{45} -6.71133 q^{46} +8.20679 q^{47} -4.41265 q^{48} -6.92776 q^{49} +3.17594 q^{50} +1.00000 q^{51} +1.54937 q^{52} +7.64323 q^{53} -1.49453 q^{54} -1.54924 q^{55} +0.709552 q^{56} +0.913394 q^{57} -8.66753 q^{58} -13.4445 q^{59} -0.396101 q^{60} +0.904939 q^{61} +9.16365 q^{62} +0.268778 q^{63} +6.86002 q^{64} -11.2456 q^{65} -1.36555 q^{66} -5.93614 q^{67} +0.233609 q^{68} +4.49061 q^{69} +0.681103 q^{70} +3.32023 q^{71} +2.63992 q^{72} -0.542859 q^{73} -0.893014 q^{74} -2.12505 q^{75} +0.213377 q^{76} +0.245582 q^{77} -9.91218 q^{78} +1.00000 q^{79} +7.48194 q^{80} +1.00000 q^{81} -10.0721 q^{82} -3.99615 q^{83} +0.0627891 q^{84} -1.69557 q^{85} -9.42465 q^{86} +5.79951 q^{87} +2.41209 q^{88} -11.2379 q^{89} +2.53407 q^{90} +1.78262 q^{91} +1.04905 q^{92} -6.13148 q^{93} -12.2653 q^{94} -1.54872 q^{95} +1.31498 q^{96} -16.5060 q^{97} +10.3537 q^{98} +0.913698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.49453 −1.05679 −0.528395 0.848999i \(-0.677206\pi\)
−0.528395 + 0.848999i \(0.677206\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.233609 0.116805
\(5\) −1.69557 −0.758281 −0.379141 0.925339i \(-0.623780\pi\)
−0.379141 + 0.925339i \(0.623780\pi\)
\(6\) −1.49453 −0.610138
\(7\) 0.268778 0.101589 0.0507943 0.998709i \(-0.483825\pi\)
0.0507943 + 0.998709i \(0.483825\pi\)
\(8\) 2.63992 0.933352
\(9\) 1.00000 0.333333
\(10\) 2.53407 0.801344
\(11\) 0.913698 0.275490 0.137745 0.990468i \(-0.456014\pi\)
0.137745 + 0.990468i \(0.456014\pi\)
\(12\) 0.233609 0.0674372
\(13\) 6.63232 1.83947 0.919737 0.392535i \(-0.128402\pi\)
0.919737 + 0.392535i \(0.128402\pi\)
\(14\) −0.401696 −0.107358
\(15\) −1.69557 −0.437794
\(16\) −4.41265 −1.10316
\(17\) 1.00000 0.242536
\(18\) −1.49453 −0.352263
\(19\) 0.913394 0.209547 0.104773 0.994496i \(-0.466588\pi\)
0.104773 + 0.994496i \(0.466588\pi\)
\(20\) −0.396101 −0.0885708
\(21\) 0.268778 0.0586522
\(22\) −1.36555 −0.291135
\(23\) 4.49061 0.936356 0.468178 0.883634i \(-0.344911\pi\)
0.468178 + 0.883634i \(0.344911\pi\)
\(24\) 2.63992 0.538871
\(25\) −2.12505 −0.425009
\(26\) −9.91218 −1.94394
\(27\) 1.00000 0.192450
\(28\) 0.0627891 0.0118660
\(29\) 5.79951 1.07694 0.538471 0.842644i \(-0.319002\pi\)
0.538471 + 0.842644i \(0.319002\pi\)
\(30\) 2.53407 0.462656
\(31\) −6.13148 −1.10125 −0.550623 0.834754i \(-0.685610\pi\)
−0.550623 + 0.834754i \(0.685610\pi\)
\(32\) 1.31498 0.232458
\(33\) 0.913698 0.159054
\(34\) −1.49453 −0.256309
\(35\) −0.455732 −0.0770327
\(36\) 0.233609 0.0389349
\(37\) 0.597523 0.0982322 0.0491161 0.998793i \(-0.484360\pi\)
0.0491161 + 0.998793i \(0.484360\pi\)
\(38\) −1.36509 −0.221447
\(39\) 6.63232 1.06202
\(40\) −4.47616 −0.707743
\(41\) 6.73930 1.05250 0.526251 0.850329i \(-0.323597\pi\)
0.526251 + 0.850329i \(0.323597\pi\)
\(42\) −0.401696 −0.0619830
\(43\) 6.30611 0.961673 0.480836 0.876810i \(-0.340333\pi\)
0.480836 + 0.876810i \(0.340333\pi\)
\(44\) 0.213448 0.0321786
\(45\) −1.69557 −0.252760
\(46\) −6.71133 −0.989531
\(47\) 8.20679 1.19708 0.598542 0.801092i \(-0.295747\pi\)
0.598542 + 0.801092i \(0.295747\pi\)
\(48\) −4.41265 −0.636911
\(49\) −6.92776 −0.989680
\(50\) 3.17594 0.449146
\(51\) 1.00000 0.140028
\(52\) 1.54937 0.214859
\(53\) 7.64323 1.04988 0.524939 0.851140i \(-0.324088\pi\)
0.524939 + 0.851140i \(0.324088\pi\)
\(54\) −1.49453 −0.203379
\(55\) −1.54924 −0.208899
\(56\) 0.709552 0.0948179
\(57\) 0.913394 0.120982
\(58\) −8.66753 −1.13810
\(59\) −13.4445 −1.75033 −0.875164 0.483827i \(-0.839247\pi\)
−0.875164 + 0.483827i \(0.839247\pi\)
\(60\) −0.396101 −0.0511364
\(61\) 0.904939 0.115866 0.0579328 0.998320i \(-0.481549\pi\)
0.0579328 + 0.998320i \(0.481549\pi\)
\(62\) 9.16365 1.16379
\(63\) 0.268778 0.0338629
\(64\) 6.86002 0.857502
\(65\) −11.2456 −1.39484
\(66\) −1.36555 −0.168087
\(67\) −5.93614 −0.725215 −0.362608 0.931942i \(-0.618113\pi\)
−0.362608 + 0.931942i \(0.618113\pi\)
\(68\) 0.233609 0.0283293
\(69\) 4.49061 0.540605
\(70\) 0.681103 0.0814074
\(71\) 3.32023 0.394039 0.197019 0.980400i \(-0.436874\pi\)
0.197019 + 0.980400i \(0.436874\pi\)
\(72\) 2.63992 0.311117
\(73\) −0.542859 −0.0635369 −0.0317684 0.999495i \(-0.510114\pi\)
−0.0317684 + 0.999495i \(0.510114\pi\)
\(74\) −0.893014 −0.103811
\(75\) −2.12505 −0.245379
\(76\) 0.213377 0.0244761
\(77\) 0.245582 0.0279867
\(78\) −9.91218 −1.12233
\(79\) 1.00000 0.112509
\(80\) 7.48194 0.836507
\(81\) 1.00000 0.111111
\(82\) −10.0721 −1.11227
\(83\) −3.99615 −0.438635 −0.219317 0.975654i \(-0.570383\pi\)
−0.219317 + 0.975654i \(0.570383\pi\)
\(84\) 0.0627891 0.00685085
\(85\) −1.69557 −0.183910
\(86\) −9.42465 −1.01629
\(87\) 5.79951 0.621773
\(88\) 2.41209 0.257129
\(89\) −11.2379 −1.19122 −0.595608 0.803276i \(-0.703089\pi\)
−0.595608 + 0.803276i \(0.703089\pi\)
\(90\) 2.53407 0.267115
\(91\) 1.78262 0.186870
\(92\) 1.04905 0.109371
\(93\) −6.13148 −0.635805
\(94\) −12.2653 −1.26507
\(95\) −1.54872 −0.158895
\(96\) 1.31498 0.134210
\(97\) −16.5060 −1.67593 −0.837965 0.545724i \(-0.816255\pi\)
−0.837965 + 0.545724i \(0.816255\pi\)
\(98\) 10.3537 1.04588
\(99\) 0.913698 0.0918301
\(100\) −0.496431 −0.0496431
\(101\) 4.16692 0.414624 0.207312 0.978275i \(-0.433529\pi\)
0.207312 + 0.978275i \(0.433529\pi\)
\(102\) −1.49453 −0.147980
\(103\) 8.38683 0.826378 0.413189 0.910645i \(-0.364415\pi\)
0.413189 + 0.910645i \(0.364415\pi\)
\(104\) 17.5088 1.71688
\(105\) −0.455732 −0.0444749
\(106\) −11.4230 −1.10950
\(107\) 9.23784 0.893056 0.446528 0.894770i \(-0.352660\pi\)
0.446528 + 0.894770i \(0.352660\pi\)
\(108\) 0.233609 0.0224791
\(109\) −14.0431 −1.34508 −0.672542 0.740059i \(-0.734797\pi\)
−0.672542 + 0.740059i \(0.734797\pi\)
\(110\) 2.31538 0.220763
\(111\) 0.597523 0.0567144
\(112\) −1.18602 −0.112069
\(113\) 13.5140 1.27129 0.635644 0.771983i \(-0.280735\pi\)
0.635644 + 0.771983i \(0.280735\pi\)
\(114\) −1.36509 −0.127852
\(115\) −7.61413 −0.710021
\(116\) 1.35482 0.125792
\(117\) 6.63232 0.613158
\(118\) 20.0932 1.84973
\(119\) 0.268778 0.0246388
\(120\) −4.47616 −0.408616
\(121\) −10.1652 −0.924105
\(122\) −1.35246 −0.122446
\(123\) 6.73930 0.607662
\(124\) −1.43237 −0.128631
\(125\) 12.0810 1.08056
\(126\) −0.401696 −0.0357859
\(127\) −2.43833 −0.216367 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(128\) −12.8824 −1.13866
\(129\) 6.30611 0.555222
\(130\) 16.8068 1.47405
\(131\) −13.8564 −1.21064 −0.605321 0.795981i \(-0.706955\pi\)
−0.605321 + 0.795981i \(0.706955\pi\)
\(132\) 0.213448 0.0185783
\(133\) 0.245500 0.0212876
\(134\) 8.87172 0.766400
\(135\) −1.69557 −0.145931
\(136\) 2.63992 0.226371
\(137\) −7.56539 −0.646355 −0.323178 0.946338i \(-0.604751\pi\)
−0.323178 + 0.946338i \(0.604751\pi\)
\(138\) −6.71133 −0.571306
\(139\) −0.216038 −0.0183241 −0.00916207 0.999958i \(-0.502916\pi\)
−0.00916207 + 0.999958i \(0.502916\pi\)
\(140\) −0.106463 −0.00899778
\(141\) 8.20679 0.691137
\(142\) −4.96217 −0.416416
\(143\) 6.05994 0.506757
\(144\) −4.41265 −0.367720
\(145\) −9.83347 −0.816626
\(146\) 0.811318 0.0671451
\(147\) −6.92776 −0.571392
\(148\) 0.139587 0.0114740
\(149\) −18.1225 −1.48465 −0.742326 0.670039i \(-0.766277\pi\)
−0.742326 + 0.670039i \(0.766277\pi\)
\(150\) 3.17594 0.259314
\(151\) 2.30948 0.187943 0.0939713 0.995575i \(-0.470044\pi\)
0.0939713 + 0.995575i \(0.470044\pi\)
\(152\) 2.41128 0.195581
\(153\) 1.00000 0.0808452
\(154\) −0.367029 −0.0295760
\(155\) 10.3963 0.835054
\(156\) 1.54937 0.124049
\(157\) 24.6989 1.97119 0.985593 0.169134i \(-0.0540972\pi\)
0.985593 + 0.169134i \(0.0540972\pi\)
\(158\) −1.49453 −0.118898
\(159\) 7.64323 0.606147
\(160\) −2.22964 −0.176268
\(161\) 1.20698 0.0951231
\(162\) −1.49453 −0.117421
\(163\) 12.9693 1.01584 0.507918 0.861405i \(-0.330415\pi\)
0.507918 + 0.861405i \(0.330415\pi\)
\(164\) 1.57436 0.122937
\(165\) −1.54924 −0.120608
\(166\) 5.97236 0.463545
\(167\) 13.3746 1.03496 0.517480 0.855695i \(-0.326870\pi\)
0.517480 + 0.855695i \(0.326870\pi\)
\(168\) 0.709552 0.0547431
\(169\) 30.9877 2.38367
\(170\) 2.53407 0.194354
\(171\) 0.913394 0.0698490
\(172\) 1.47317 0.112328
\(173\) 2.47979 0.188535 0.0942675 0.995547i \(-0.469949\pi\)
0.0942675 + 0.995547i \(0.469949\pi\)
\(174\) −8.66753 −0.657084
\(175\) −0.571166 −0.0431761
\(176\) −4.03183 −0.303910
\(177\) −13.4445 −1.01055
\(178\) 16.7953 1.25886
\(179\) 21.9042 1.63720 0.818599 0.574366i \(-0.194751\pi\)
0.818599 + 0.574366i \(0.194751\pi\)
\(180\) −0.396101 −0.0295236
\(181\) −12.8695 −0.956585 −0.478293 0.878201i \(-0.658744\pi\)
−0.478293 + 0.878201i \(0.658744\pi\)
\(182\) −2.66418 −0.197482
\(183\) 0.904939 0.0668950
\(184\) 11.8548 0.873950
\(185\) −1.01314 −0.0744876
\(186\) 9.16365 0.671912
\(187\) 0.913698 0.0668162
\(188\) 1.91718 0.139825
\(189\) 0.268778 0.0195507
\(190\) 2.31461 0.167919
\(191\) 19.1035 1.38228 0.691142 0.722719i \(-0.257108\pi\)
0.691142 + 0.722719i \(0.257108\pi\)
\(192\) 6.86002 0.495079
\(193\) 23.4384 1.68713 0.843566 0.537026i \(-0.180452\pi\)
0.843566 + 0.537026i \(0.180452\pi\)
\(194\) 24.6686 1.77111
\(195\) −11.2456 −0.805311
\(196\) −1.61839 −0.115599
\(197\) −21.0035 −1.49644 −0.748219 0.663452i \(-0.769091\pi\)
−0.748219 + 0.663452i \(0.769091\pi\)
\(198\) −1.36555 −0.0970451
\(199\) 0.224447 0.0159106 0.00795532 0.999968i \(-0.497468\pi\)
0.00795532 + 0.999968i \(0.497468\pi\)
\(200\) −5.60995 −0.396683
\(201\) −5.93614 −0.418703
\(202\) −6.22757 −0.438170
\(203\) 1.55878 0.109405
\(204\) 0.233609 0.0163559
\(205\) −11.4269 −0.798093
\(206\) −12.5343 −0.873308
\(207\) 4.49061 0.312119
\(208\) −29.2661 −2.02924
\(209\) 0.834566 0.0577282
\(210\) 0.681103 0.0470006
\(211\) −22.7743 −1.56785 −0.783923 0.620858i \(-0.786784\pi\)
−0.783923 + 0.620858i \(0.786784\pi\)
\(212\) 1.78553 0.122631
\(213\) 3.32023 0.227498
\(214\) −13.8062 −0.943772
\(215\) −10.6924 −0.729219
\(216\) 2.63992 0.179624
\(217\) −1.64801 −0.111874
\(218\) 20.9878 1.42147
\(219\) −0.542859 −0.0366830
\(220\) −0.361916 −0.0244004
\(221\) 6.63232 0.446138
\(222\) −0.893014 −0.0599352
\(223\) 10.1612 0.680446 0.340223 0.940345i \(-0.389497\pi\)
0.340223 + 0.940345i \(0.389497\pi\)
\(224\) 0.353438 0.0236151
\(225\) −2.12505 −0.141670
\(226\) −20.1970 −1.34348
\(227\) 20.4863 1.35972 0.679862 0.733340i \(-0.262040\pi\)
0.679862 + 0.733340i \(0.262040\pi\)
\(228\) 0.213377 0.0141313
\(229\) 17.0913 1.12942 0.564711 0.825289i \(-0.308988\pi\)
0.564711 + 0.825289i \(0.308988\pi\)
\(230\) 11.3795 0.750343
\(231\) 0.245582 0.0161581
\(232\) 15.3102 1.00517
\(233\) 25.7969 1.69001 0.845005 0.534758i \(-0.179597\pi\)
0.845005 + 0.534758i \(0.179597\pi\)
\(234\) −9.91218 −0.647979
\(235\) −13.9152 −0.907726
\(236\) −3.14077 −0.204446
\(237\) 1.00000 0.0649570
\(238\) −0.401696 −0.0260381
\(239\) 10.4203 0.674035 0.337017 0.941498i \(-0.390582\pi\)
0.337017 + 0.941498i \(0.390582\pi\)
\(240\) 7.48194 0.482957
\(241\) 19.5047 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(242\) 15.1921 0.976585
\(243\) 1.00000 0.0641500
\(244\) 0.211402 0.0135336
\(245\) 11.7465 0.750456
\(246\) −10.0721 −0.642171
\(247\) 6.05792 0.385456
\(248\) −16.1866 −1.02785
\(249\) −3.99615 −0.253246
\(250\) −18.0554 −1.14192
\(251\) 6.65199 0.419870 0.209935 0.977715i \(-0.432675\pi\)
0.209935 + 0.977715i \(0.432675\pi\)
\(252\) 0.0627891 0.00395534
\(253\) 4.10306 0.257957
\(254\) 3.64415 0.228654
\(255\) −1.69557 −0.106181
\(256\) 5.53311 0.345819
\(257\) 11.9295 0.744144 0.372072 0.928204i \(-0.378647\pi\)
0.372072 + 0.928204i \(0.378647\pi\)
\(258\) −9.42465 −0.586753
\(259\) 0.160601 0.00997926
\(260\) −2.62707 −0.162924
\(261\) 5.79951 0.358981
\(262\) 20.7088 1.27939
\(263\) 16.0526 0.989848 0.494924 0.868936i \(-0.335196\pi\)
0.494924 + 0.868936i \(0.335196\pi\)
\(264\) 2.41209 0.148454
\(265\) −12.9596 −0.796103
\(266\) −0.366907 −0.0224965
\(267\) −11.2379 −0.687748
\(268\) −1.38674 −0.0847085
\(269\) 10.2575 0.625411 0.312705 0.949850i \(-0.398765\pi\)
0.312705 + 0.949850i \(0.398765\pi\)
\(270\) 2.53407 0.154219
\(271\) −5.11607 −0.310779 −0.155389 0.987853i \(-0.549663\pi\)
−0.155389 + 0.987853i \(0.549663\pi\)
\(272\) −4.41265 −0.267556
\(273\) 1.78262 0.107889
\(274\) 11.3067 0.683061
\(275\) −1.94165 −0.117086
\(276\) 1.04905 0.0631452
\(277\) −30.1878 −1.81381 −0.906905 0.421335i \(-0.861562\pi\)
−0.906905 + 0.421335i \(0.861562\pi\)
\(278\) 0.322875 0.0193648
\(279\) −6.13148 −0.367082
\(280\) −1.20309 −0.0718986
\(281\) −23.7660 −1.41776 −0.708882 0.705328i \(-0.750800\pi\)
−0.708882 + 0.705328i \(0.750800\pi\)
\(282\) −12.2653 −0.730386
\(283\) −8.04642 −0.478310 −0.239155 0.970981i \(-0.576870\pi\)
−0.239155 + 0.970981i \(0.576870\pi\)
\(284\) 0.775637 0.0460256
\(285\) −1.54872 −0.0917384
\(286\) −9.05674 −0.535536
\(287\) 1.81138 0.106922
\(288\) 1.31498 0.0774860
\(289\) 1.00000 0.0588235
\(290\) 14.6964 0.863002
\(291\) −16.5060 −0.967599
\(292\) −0.126817 −0.00742140
\(293\) 31.4243 1.83583 0.917915 0.396778i \(-0.129872\pi\)
0.917915 + 0.396778i \(0.129872\pi\)
\(294\) 10.3537 0.603841
\(295\) 22.7961 1.32724
\(296\) 1.57741 0.0916852
\(297\) 0.913698 0.0530181
\(298\) 27.0845 1.56897
\(299\) 29.7831 1.72240
\(300\) −0.496431 −0.0286615
\(301\) 1.69494 0.0976950
\(302\) −3.45158 −0.198616
\(303\) 4.16692 0.239383
\(304\) −4.03048 −0.231164
\(305\) −1.53439 −0.0878587
\(306\) −1.49453 −0.0854364
\(307\) 4.02189 0.229541 0.114771 0.993392i \(-0.463387\pi\)
0.114771 + 0.993392i \(0.463387\pi\)
\(308\) 0.0573703 0.00326897
\(309\) 8.38683 0.477110
\(310\) −15.5376 −0.882477
\(311\) −15.6167 −0.885540 −0.442770 0.896635i \(-0.646004\pi\)
−0.442770 + 0.896635i \(0.646004\pi\)
\(312\) 17.5088 0.991239
\(313\) −0.155288 −0.00877739 −0.00438870 0.999990i \(-0.501397\pi\)
−0.00438870 + 0.999990i \(0.501397\pi\)
\(314\) −36.9131 −2.08313
\(315\) −0.455732 −0.0256776
\(316\) 0.233609 0.0131416
\(317\) 4.16068 0.233687 0.116843 0.993150i \(-0.462722\pi\)
0.116843 + 0.993150i \(0.462722\pi\)
\(318\) −11.4230 −0.640570
\(319\) 5.29901 0.296687
\(320\) −11.6316 −0.650228
\(321\) 9.23784 0.515606
\(322\) −1.80386 −0.100525
\(323\) 0.913394 0.0508226
\(324\) 0.233609 0.0129783
\(325\) −14.0940 −0.781794
\(326\) −19.3830 −1.07353
\(327\) −14.0431 −0.776584
\(328\) 17.7912 0.982355
\(329\) 2.20581 0.121610
\(330\) 2.31538 0.127457
\(331\) −11.9884 −0.658942 −0.329471 0.944166i \(-0.606870\pi\)
−0.329471 + 0.944166i \(0.606870\pi\)
\(332\) −0.933539 −0.0512346
\(333\) 0.597523 0.0327441
\(334\) −19.9888 −1.09374
\(335\) 10.0651 0.549917
\(336\) −1.18602 −0.0647028
\(337\) −23.5136 −1.28087 −0.640433 0.768014i \(-0.721245\pi\)
−0.640433 + 0.768014i \(0.721245\pi\)
\(338\) −46.3119 −2.51903
\(339\) 13.5140 0.733978
\(340\) −0.396101 −0.0214816
\(341\) −5.60232 −0.303383
\(342\) −1.36509 −0.0738157
\(343\) −3.74348 −0.202129
\(344\) 16.6476 0.897579
\(345\) −7.61413 −0.409931
\(346\) −3.70611 −0.199242
\(347\) −12.5334 −0.672828 −0.336414 0.941714i \(-0.609214\pi\)
−0.336414 + 0.941714i \(0.609214\pi\)
\(348\) 1.35482 0.0726260
\(349\) 6.40596 0.342903 0.171452 0.985193i \(-0.445154\pi\)
0.171452 + 0.985193i \(0.445154\pi\)
\(350\) 0.853623 0.0456281
\(351\) 6.63232 0.354007
\(352\) 1.20150 0.0640399
\(353\) 22.7658 1.21170 0.605849 0.795579i \(-0.292833\pi\)
0.605849 + 0.795579i \(0.292833\pi\)
\(354\) 20.0932 1.06794
\(355\) −5.62968 −0.298792
\(356\) −2.62528 −0.139140
\(357\) 0.268778 0.0142252
\(358\) −32.7364 −1.73017
\(359\) −15.2071 −0.802599 −0.401300 0.915947i \(-0.631441\pi\)
−0.401300 + 0.915947i \(0.631441\pi\)
\(360\) −4.47616 −0.235914
\(361\) −18.1657 −0.956090
\(362\) 19.2339 1.01091
\(363\) −10.1652 −0.533532
\(364\) 0.416437 0.0218272
\(365\) 0.920455 0.0481788
\(366\) −1.35246 −0.0706940
\(367\) 11.7881 0.615336 0.307668 0.951494i \(-0.400451\pi\)
0.307668 + 0.951494i \(0.400451\pi\)
\(368\) −19.8154 −1.03295
\(369\) 6.73930 0.350834
\(370\) 1.51417 0.0787177
\(371\) 2.05433 0.106656
\(372\) −1.43237 −0.0742649
\(373\) 16.1504 0.836238 0.418119 0.908392i \(-0.362690\pi\)
0.418119 + 0.908392i \(0.362690\pi\)
\(374\) −1.36555 −0.0706107
\(375\) 12.0810 0.623860
\(376\) 21.6653 1.11730
\(377\) 38.4642 1.98101
\(378\) −0.401696 −0.0206610
\(379\) 23.7553 1.22023 0.610113 0.792314i \(-0.291124\pi\)
0.610113 + 0.792314i \(0.291124\pi\)
\(380\) −0.361796 −0.0185597
\(381\) −2.43833 −0.124919
\(382\) −28.5507 −1.46078
\(383\) −8.29335 −0.423770 −0.211885 0.977295i \(-0.567960\pi\)
−0.211885 + 0.977295i \(0.567960\pi\)
\(384\) −12.8824 −0.657404
\(385\) −0.416401 −0.0212218
\(386\) −35.0293 −1.78294
\(387\) 6.30611 0.320558
\(388\) −3.85596 −0.195756
\(389\) −16.4233 −0.832692 −0.416346 0.909206i \(-0.636690\pi\)
−0.416346 + 0.909206i \(0.636690\pi\)
\(390\) 16.8068 0.851044
\(391\) 4.49061 0.227100
\(392\) −18.2887 −0.923719
\(393\) −13.8564 −0.698964
\(394\) 31.3903 1.58142
\(395\) −1.69557 −0.0853133
\(396\) 0.213448 0.0107262
\(397\) −26.5201 −1.33101 −0.665503 0.746395i \(-0.731783\pi\)
−0.665503 + 0.746395i \(0.731783\pi\)
\(398\) −0.335442 −0.0168142
\(399\) 0.245500 0.0122904
\(400\) 9.37708 0.468854
\(401\) −4.81158 −0.240279 −0.120139 0.992757i \(-0.538334\pi\)
−0.120139 + 0.992757i \(0.538334\pi\)
\(402\) 8.87172 0.442481
\(403\) −40.6659 −2.02571
\(404\) 0.973431 0.0484300
\(405\) −1.69557 −0.0842535
\(406\) −2.32964 −0.115618
\(407\) 0.545956 0.0270620
\(408\) 2.63992 0.130695
\(409\) −36.5884 −1.80918 −0.904590 0.426284i \(-0.859823\pi\)
−0.904590 + 0.426284i \(0.859823\pi\)
\(410\) 17.0779 0.843416
\(411\) −7.56539 −0.373173
\(412\) 1.95924 0.0965249
\(413\) −3.61359 −0.177813
\(414\) −6.71133 −0.329844
\(415\) 6.77575 0.332609
\(416\) 8.72137 0.427600
\(417\) −0.216038 −0.0105794
\(418\) −1.24728 −0.0610065
\(419\) 25.8357 1.26216 0.631079 0.775719i \(-0.282612\pi\)
0.631079 + 0.775719i \(0.282612\pi\)
\(420\) −0.106463 −0.00519487
\(421\) −31.2098 −1.52107 −0.760537 0.649295i \(-0.775064\pi\)
−0.760537 + 0.649295i \(0.775064\pi\)
\(422\) 34.0368 1.65688
\(423\) 8.20679 0.399028
\(424\) 20.1775 0.979905
\(425\) −2.12505 −0.103080
\(426\) −4.96217 −0.240418
\(427\) 0.243228 0.0117706
\(428\) 2.15805 0.104313
\(429\) 6.05994 0.292577
\(430\) 15.9801 0.770631
\(431\) −19.7100 −0.949398 −0.474699 0.880148i \(-0.657443\pi\)
−0.474699 + 0.880148i \(0.657443\pi\)
\(432\) −4.41265 −0.212304
\(433\) 5.24733 0.252171 0.126085 0.992019i \(-0.459759\pi\)
0.126085 + 0.992019i \(0.459759\pi\)
\(434\) 2.46299 0.118227
\(435\) −9.83347 −0.471479
\(436\) −3.28060 −0.157112
\(437\) 4.10169 0.196210
\(438\) 0.811318 0.0387663
\(439\) 13.1009 0.625271 0.312636 0.949873i \(-0.398788\pi\)
0.312636 + 0.949873i \(0.398788\pi\)
\(440\) −4.08986 −0.194976
\(441\) −6.92776 −0.329893
\(442\) −9.91218 −0.471474
\(443\) 13.0342 0.619275 0.309638 0.950855i \(-0.399792\pi\)
0.309638 + 0.950855i \(0.399792\pi\)
\(444\) 0.139587 0.00662450
\(445\) 19.0546 0.903276
\(446\) −15.1862 −0.719088
\(447\) −18.1225 −0.857164
\(448\) 1.84382 0.0871124
\(449\) 38.2315 1.80426 0.902128 0.431470i \(-0.142005\pi\)
0.902128 + 0.431470i \(0.142005\pi\)
\(450\) 3.17594 0.149715
\(451\) 6.15769 0.289954
\(452\) 3.15699 0.148492
\(453\) 2.30948 0.108509
\(454\) −30.6173 −1.43694
\(455\) −3.02256 −0.141700
\(456\) 2.41128 0.112919
\(457\) 13.7193 0.641762 0.320881 0.947120i \(-0.396021\pi\)
0.320881 + 0.947120i \(0.396021\pi\)
\(458\) −25.5433 −1.19356
\(459\) 1.00000 0.0466760
\(460\) −1.77873 −0.0829338
\(461\) 41.3034 1.92369 0.961846 0.273592i \(-0.0882116\pi\)
0.961846 + 0.273592i \(0.0882116\pi\)
\(462\) −0.367029 −0.0170757
\(463\) −8.14549 −0.378553 −0.189277 0.981924i \(-0.560614\pi\)
−0.189277 + 0.981924i \(0.560614\pi\)
\(464\) −25.5912 −1.18804
\(465\) 10.3963 0.482119
\(466\) −38.5541 −1.78599
\(467\) 11.3426 0.524871 0.262435 0.964950i \(-0.415474\pi\)
0.262435 + 0.964950i \(0.415474\pi\)
\(468\) 1.54937 0.0716197
\(469\) −1.59550 −0.0736736
\(470\) 20.7966 0.959276
\(471\) 24.6989 1.13806
\(472\) −35.4924 −1.63367
\(473\) 5.76188 0.264932
\(474\) −1.49453 −0.0686459
\(475\) −1.94100 −0.0890594
\(476\) 0.0627891 0.00287793
\(477\) 7.64323 0.349959
\(478\) −15.5735 −0.712313
\(479\) 14.7144 0.672316 0.336158 0.941806i \(-0.390872\pi\)
0.336158 + 0.941806i \(0.390872\pi\)
\(480\) −2.22964 −0.101769
\(481\) 3.96296 0.180696
\(482\) −29.1503 −1.32776
\(483\) 1.20698 0.0549193
\(484\) −2.37468 −0.107940
\(485\) 27.9870 1.27083
\(486\) −1.49453 −0.0677931
\(487\) 35.1004 1.59055 0.795275 0.606249i \(-0.207326\pi\)
0.795275 + 0.606249i \(0.207326\pi\)
\(488\) 2.38897 0.108143
\(489\) 12.9693 0.586494
\(490\) −17.5554 −0.793074
\(491\) −19.0577 −0.860062 −0.430031 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(492\) 1.57436 0.0709778
\(493\) 5.79951 0.261197
\(494\) −9.05372 −0.407346
\(495\) −1.54924 −0.0696331
\(496\) 27.0560 1.21485
\(497\) 0.892405 0.0400298
\(498\) 5.97236 0.267628
\(499\) 2.43422 0.108970 0.0544852 0.998515i \(-0.482648\pi\)
0.0544852 + 0.998515i \(0.482648\pi\)
\(500\) 2.82224 0.126214
\(501\) 13.3746 0.597535
\(502\) −9.94157 −0.443714
\(503\) 10.6480 0.474770 0.237385 0.971416i \(-0.423710\pi\)
0.237385 + 0.971416i \(0.423710\pi\)
\(504\) 0.709552 0.0316060
\(505\) −7.06529 −0.314401
\(506\) −6.13213 −0.272606
\(507\) 30.9877 1.37621
\(508\) −0.569616 −0.0252726
\(509\) −22.6656 −1.00463 −0.502317 0.864683i \(-0.667519\pi\)
−0.502317 + 0.864683i \(0.667519\pi\)
\(510\) 2.53407 0.112211
\(511\) −0.145909 −0.00645462
\(512\) 17.4955 0.773199
\(513\) 0.913394 0.0403273
\(514\) −17.8290 −0.786404
\(515\) −14.2204 −0.626627
\(516\) 1.47317 0.0648525
\(517\) 7.49853 0.329785
\(518\) −0.240023 −0.0105460
\(519\) 2.47979 0.108851
\(520\) −29.6873 −1.30188
\(521\) 11.9235 0.522379 0.261189 0.965288i \(-0.415885\pi\)
0.261189 + 0.965288i \(0.415885\pi\)
\(522\) −8.66753 −0.379367
\(523\) 15.3283 0.670262 0.335131 0.942172i \(-0.391219\pi\)
0.335131 + 0.942172i \(0.391219\pi\)
\(524\) −3.23699 −0.141409
\(525\) −0.571166 −0.0249277
\(526\) −23.9911 −1.04606
\(527\) −6.13148 −0.267091
\(528\) −4.03183 −0.175463
\(529\) −2.83446 −0.123238
\(530\) 19.3685 0.841313
\(531\) −13.4445 −0.583443
\(532\) 0.0573512 0.00248649
\(533\) 44.6972 1.93605
\(534\) 16.7953 0.726805
\(535\) −15.6634 −0.677188
\(536\) −15.6709 −0.676881
\(537\) 21.9042 0.945237
\(538\) −15.3301 −0.660928
\(539\) −6.32988 −0.272647
\(540\) −0.396101 −0.0170455
\(541\) −28.7904 −1.23780 −0.618898 0.785471i \(-0.712421\pi\)
−0.618898 + 0.785471i \(0.712421\pi\)
\(542\) 7.64610 0.328428
\(543\) −12.8695 −0.552285
\(544\) 1.31498 0.0563793
\(545\) 23.8110 1.01995
\(546\) −2.66418 −0.114016
\(547\) 38.4943 1.64590 0.822949 0.568115i \(-0.192327\pi\)
0.822949 + 0.568115i \(0.192327\pi\)
\(548\) −1.76735 −0.0754973
\(549\) 0.904939 0.0386219
\(550\) 2.90185 0.123735
\(551\) 5.29724 0.225670
\(552\) 11.8548 0.504575
\(553\) 0.268778 0.0114296
\(554\) 45.1165 1.91682
\(555\) −1.01314 −0.0430054
\(556\) −0.0504686 −0.00214035
\(557\) −29.4924 −1.24963 −0.624816 0.780772i \(-0.714826\pi\)
−0.624816 + 0.780772i \(0.714826\pi\)
\(558\) 9.16365 0.387928
\(559\) 41.8241 1.76897
\(560\) 2.01098 0.0849795
\(561\) 0.913698 0.0385764
\(562\) 35.5190 1.49828
\(563\) −32.4871 −1.36917 −0.684584 0.728934i \(-0.740016\pi\)
−0.684584 + 0.728934i \(0.740016\pi\)
\(564\) 1.91718 0.0807280
\(565\) −22.9139 −0.963993
\(566\) 12.0256 0.505473
\(567\) 0.268778 0.0112876
\(568\) 8.76513 0.367777
\(569\) 0.738394 0.0309551 0.0154776 0.999880i \(-0.495073\pi\)
0.0154776 + 0.999880i \(0.495073\pi\)
\(570\) 2.31461 0.0969482
\(571\) 22.8366 0.955684 0.477842 0.878446i \(-0.341419\pi\)
0.477842 + 0.878446i \(0.341419\pi\)
\(572\) 1.41566 0.0591916
\(573\) 19.1035 0.798062
\(574\) −2.70715 −0.112994
\(575\) −9.54275 −0.397960
\(576\) 6.86002 0.285834
\(577\) 39.4170 1.64095 0.820475 0.571682i \(-0.193709\pi\)
0.820475 + 0.571682i \(0.193709\pi\)
\(578\) −1.49453 −0.0621641
\(579\) 23.4384 0.974066
\(580\) −2.29719 −0.0953857
\(581\) −1.07408 −0.0445603
\(582\) 24.6686 1.02255
\(583\) 6.98360 0.289231
\(584\) −1.43310 −0.0593023
\(585\) −11.2456 −0.464946
\(586\) −46.9645 −1.94009
\(587\) 20.9512 0.864747 0.432374 0.901695i \(-0.357676\pi\)
0.432374 + 0.901695i \(0.357676\pi\)
\(588\) −1.61839 −0.0667412
\(589\) −5.60045 −0.230763
\(590\) −34.0694 −1.40261
\(591\) −21.0035 −0.863969
\(592\) −2.63666 −0.108366
\(593\) 17.6487 0.724744 0.362372 0.932033i \(-0.381967\pi\)
0.362372 + 0.932033i \(0.381967\pi\)
\(594\) −1.36555 −0.0560290
\(595\) −0.455732 −0.0186832
\(596\) −4.23358 −0.173414
\(597\) 0.224447 0.00918601
\(598\) −44.5117 −1.82022
\(599\) −23.0178 −0.940482 −0.470241 0.882538i \(-0.655833\pi\)
−0.470241 + 0.882538i \(0.655833\pi\)
\(600\) −5.60995 −0.229025
\(601\) 36.4769 1.48792 0.743962 0.668222i \(-0.232945\pi\)
0.743962 + 0.668222i \(0.232945\pi\)
\(602\) −2.53314 −0.103243
\(603\) −5.93614 −0.241738
\(604\) 0.539516 0.0219526
\(605\) 17.2357 0.700732
\(606\) −6.22757 −0.252978
\(607\) 10.7471 0.436213 0.218106 0.975925i \(-0.430012\pi\)
0.218106 + 0.975925i \(0.430012\pi\)
\(608\) 1.20109 0.0487108
\(609\) 1.55878 0.0631651
\(610\) 2.29318 0.0928482
\(611\) 54.4301 2.20200
\(612\) 0.233609 0.00944310
\(613\) 38.0515 1.53689 0.768443 0.639919i \(-0.221032\pi\)
0.768443 + 0.639919i \(0.221032\pi\)
\(614\) −6.01082 −0.242577
\(615\) −11.4269 −0.460779
\(616\) 0.648316 0.0261214
\(617\) −10.7668 −0.433456 −0.216728 0.976232i \(-0.569539\pi\)
−0.216728 + 0.976232i \(0.569539\pi\)
\(618\) −12.5343 −0.504205
\(619\) 13.9789 0.561860 0.280930 0.959728i \(-0.409357\pi\)
0.280930 + 0.959728i \(0.409357\pi\)
\(620\) 2.42868 0.0975382
\(621\) 4.49061 0.180202
\(622\) 23.3395 0.935830
\(623\) −3.02050 −0.121014
\(624\) −29.2661 −1.17158
\(625\) −9.85894 −0.394357
\(626\) 0.232082 0.00927586
\(627\) 0.834566 0.0333294
\(628\) 5.76989 0.230244
\(629\) 0.597523 0.0238248
\(630\) 0.681103 0.0271358
\(631\) −21.6632 −0.862398 −0.431199 0.902257i \(-0.641909\pi\)
−0.431199 + 0.902257i \(0.641909\pi\)
\(632\) 2.63992 0.105010
\(633\) −22.7743 −0.905196
\(634\) −6.21824 −0.246958
\(635\) 4.13435 0.164067
\(636\) 1.78553 0.0708008
\(637\) −45.9471 −1.82049
\(638\) −7.91950 −0.313536
\(639\) 3.32023 0.131346
\(640\) 21.8431 0.863423
\(641\) 5.84816 0.230989 0.115494 0.993308i \(-0.463155\pi\)
0.115494 + 0.993308i \(0.463155\pi\)
\(642\) −13.8062 −0.544887
\(643\) −41.6959 −1.64432 −0.822162 0.569253i \(-0.807232\pi\)
−0.822162 + 0.569253i \(0.807232\pi\)
\(644\) 0.281961 0.0111108
\(645\) −10.6924 −0.421015
\(646\) −1.36509 −0.0537088
\(647\) −28.8401 −1.13382 −0.566911 0.823779i \(-0.691862\pi\)
−0.566911 + 0.823779i \(0.691862\pi\)
\(648\) 2.63992 0.103706
\(649\) −12.2842 −0.482198
\(650\) 21.0638 0.826192
\(651\) −1.64801 −0.0645905
\(652\) 3.02976 0.118654
\(653\) −25.9219 −1.01440 −0.507201 0.861828i \(-0.669320\pi\)
−0.507201 + 0.861828i \(0.669320\pi\)
\(654\) 20.9878 0.820686
\(655\) 23.4945 0.918007
\(656\) −29.7381 −1.16108
\(657\) −0.542859 −0.0211790
\(658\) −3.29664 −0.128516
\(659\) −32.0745 −1.24944 −0.624722 0.780847i \(-0.714788\pi\)
−0.624722 + 0.780847i \(0.714788\pi\)
\(660\) −0.361916 −0.0140876
\(661\) −6.40457 −0.249109 −0.124554 0.992213i \(-0.539750\pi\)
−0.124554 + 0.992213i \(0.539750\pi\)
\(662\) 17.9170 0.696363
\(663\) 6.63232 0.257578
\(664\) −10.5495 −0.409401
\(665\) −0.416262 −0.0161420
\(666\) −0.893014 −0.0346036
\(667\) 26.0433 1.00840
\(668\) 3.12444 0.120888
\(669\) 10.1612 0.392856
\(670\) −15.0426 −0.581147
\(671\) 0.826841 0.0319199
\(672\) 0.353438 0.0136342
\(673\) −27.5993 −1.06387 −0.531937 0.846784i \(-0.678536\pi\)
−0.531937 + 0.846784i \(0.678536\pi\)
\(674\) 35.1417 1.35361
\(675\) −2.12505 −0.0817931
\(676\) 7.23901 0.278423
\(677\) 24.6468 0.947254 0.473627 0.880725i \(-0.342944\pi\)
0.473627 + 0.880725i \(0.342944\pi\)
\(678\) −20.1970 −0.775660
\(679\) −4.43645 −0.170255
\(680\) −4.47616 −0.171653
\(681\) 20.4863 0.785037
\(682\) 8.37281 0.320612
\(683\) 28.1993 1.07902 0.539509 0.841980i \(-0.318610\pi\)
0.539509 + 0.841980i \(0.318610\pi\)
\(684\) 0.213377 0.00815869
\(685\) 12.8276 0.490119
\(686\) 5.59472 0.213608
\(687\) 17.0913 0.652072
\(688\) −27.8266 −1.06088
\(689\) 50.6923 1.93122
\(690\) 11.3795 0.433211
\(691\) 34.9187 1.32837 0.664185 0.747569i \(-0.268779\pi\)
0.664185 + 0.747569i \(0.268779\pi\)
\(692\) 0.579302 0.0220218
\(693\) 0.245582 0.00932889
\(694\) 18.7315 0.711038
\(695\) 0.366308 0.0138949
\(696\) 15.3102 0.580333
\(697\) 6.73930 0.255269
\(698\) −9.57388 −0.362377
\(699\) 25.7969 0.975728
\(700\) −0.133430 −0.00504317
\(701\) −17.1683 −0.648436 −0.324218 0.945982i \(-0.605101\pi\)
−0.324218 + 0.945982i \(0.605101\pi\)
\(702\) −9.91218 −0.374111
\(703\) 0.545774 0.0205842
\(704\) 6.26799 0.236234
\(705\) −13.9152 −0.524076
\(706\) −34.0240 −1.28051
\(707\) 1.11998 0.0421210
\(708\) −3.14077 −0.118037
\(709\) −26.4785 −0.994420 −0.497210 0.867630i \(-0.665642\pi\)
−0.497210 + 0.867630i \(0.665642\pi\)
\(710\) 8.41370 0.315761
\(711\) 1.00000 0.0375029
\(712\) −29.6671 −1.11182
\(713\) −27.5340 −1.03116
\(714\) −0.401696 −0.0150331
\(715\) −10.2750 −0.384265
\(716\) 5.11703 0.191232
\(717\) 10.4203 0.389154
\(718\) 22.7274 0.848179
\(719\) −36.2797 −1.35300 −0.676502 0.736441i \(-0.736505\pi\)
−0.676502 + 0.736441i \(0.736505\pi\)
\(720\) 7.48194 0.278836
\(721\) 2.25420 0.0839506
\(722\) 27.1491 1.01039
\(723\) 19.5047 0.725388
\(724\) −3.00644 −0.111734
\(725\) −12.3242 −0.457711
\(726\) 15.1921 0.563831
\(727\) −37.9117 −1.40607 −0.703034 0.711157i \(-0.748172\pi\)
−0.703034 + 0.711157i \(0.748172\pi\)
\(728\) 4.70598 0.174415
\(729\) 1.00000 0.0370370
\(730\) −1.37564 −0.0509149
\(731\) 6.30611 0.233240
\(732\) 0.211402 0.00781365
\(733\) −37.7122 −1.39293 −0.696467 0.717589i \(-0.745246\pi\)
−0.696467 + 0.717589i \(0.745246\pi\)
\(734\) −17.6177 −0.650281
\(735\) 11.7465 0.433276
\(736\) 5.90506 0.217663
\(737\) −5.42384 −0.199790
\(738\) −10.0721 −0.370758
\(739\) −48.3647 −1.77913 −0.889563 0.456813i \(-0.848991\pi\)
−0.889563 + 0.456813i \(0.848991\pi\)
\(740\) −0.236679 −0.00870050
\(741\) 6.05792 0.222543
\(742\) −3.07025 −0.112713
\(743\) 7.81976 0.286879 0.143440 0.989659i \(-0.454184\pi\)
0.143440 + 0.989659i \(0.454184\pi\)
\(744\) −16.1866 −0.593429
\(745\) 30.7279 1.12578
\(746\) −24.1373 −0.883728
\(747\) −3.99615 −0.146212
\(748\) 0.213448 0.00780445
\(749\) 2.48293 0.0907243
\(750\) −18.0554 −0.659289
\(751\) 40.6123 1.48196 0.740982 0.671524i \(-0.234360\pi\)
0.740982 + 0.671524i \(0.234360\pi\)
\(752\) −36.2137 −1.32058
\(753\) 6.65199 0.242412
\(754\) −57.4858 −2.09351
\(755\) −3.91588 −0.142513
\(756\) 0.0627891 0.00228362
\(757\) 31.5672 1.14733 0.573666 0.819090i \(-0.305521\pi\)
0.573666 + 0.819090i \(0.305521\pi\)
\(758\) −35.5029 −1.28952
\(759\) 4.10306 0.148932
\(760\) −4.08850 −0.148305
\(761\) −5.53161 −0.200521 −0.100260 0.994961i \(-0.531968\pi\)
−0.100260 + 0.994961i \(0.531968\pi\)
\(762\) 3.64415 0.132014
\(763\) −3.77447 −0.136645
\(764\) 4.46277 0.161457
\(765\) −1.69557 −0.0613034
\(766\) 12.3946 0.447836
\(767\) −89.1684 −3.21968
\(768\) 5.53311 0.199659
\(769\) 14.8677 0.536143 0.268071 0.963399i \(-0.413614\pi\)
0.268071 + 0.963399i \(0.413614\pi\)
\(770\) 0.622323 0.0224270
\(771\) 11.9295 0.429632
\(772\) 5.47543 0.197065
\(773\) −28.2841 −1.01731 −0.508655 0.860970i \(-0.669857\pi\)
−0.508655 + 0.860970i \(0.669857\pi\)
\(774\) −9.42465 −0.338762
\(775\) 13.0297 0.468040
\(776\) −43.5745 −1.56423
\(777\) 0.160601 0.00576153
\(778\) 24.5450 0.879981
\(779\) 6.15564 0.220549
\(780\) −2.62707 −0.0940641
\(781\) 3.03369 0.108554
\(782\) −6.71133 −0.239997
\(783\) 5.79951 0.207258
\(784\) 30.5697 1.09178
\(785\) −41.8787 −1.49471
\(786\) 20.7088 0.738659
\(787\) −52.6913 −1.87824 −0.939121 0.343586i \(-0.888358\pi\)
−0.939121 + 0.343586i \(0.888358\pi\)
\(788\) −4.90662 −0.174791
\(789\) 16.0526 0.571489
\(790\) 2.53407 0.0901582
\(791\) 3.63226 0.129148
\(792\) 2.41209 0.0857098
\(793\) 6.00185 0.213132
\(794\) 39.6350 1.40659
\(795\) −12.9596 −0.459630
\(796\) 0.0524330 0.00185844
\(797\) 1.06397 0.0376876 0.0188438 0.999822i \(-0.494001\pi\)
0.0188438 + 0.999822i \(0.494001\pi\)
\(798\) −0.366907 −0.0129884
\(799\) 8.20679 0.290335
\(800\) −2.79440 −0.0987968
\(801\) −11.2379 −0.397072
\(802\) 7.19104 0.253924
\(803\) −0.496010 −0.0175038
\(804\) −1.38674 −0.0489065
\(805\) −2.04651 −0.0721300
\(806\) 60.7763 2.14075
\(807\) 10.2575 0.361081
\(808\) 11.0003 0.386990
\(809\) 11.1172 0.390860 0.195430 0.980718i \(-0.437390\pi\)
0.195430 + 0.980718i \(0.437390\pi\)
\(810\) 2.53407 0.0890382
\(811\) 49.6601 1.74380 0.871902 0.489681i \(-0.162887\pi\)
0.871902 + 0.489681i \(0.162887\pi\)
\(812\) 0.364146 0.0127790
\(813\) −5.11607 −0.179428
\(814\) −0.815945 −0.0285989
\(815\) −21.9904 −0.770290
\(816\) −4.41265 −0.154473
\(817\) 5.75996 0.201516
\(818\) 54.6823 1.91192
\(819\) 1.78262 0.0622899
\(820\) −2.66944 −0.0932209
\(821\) 14.1357 0.493340 0.246670 0.969100i \(-0.420664\pi\)
0.246670 + 0.969100i \(0.420664\pi\)
\(822\) 11.3067 0.394366
\(823\) 15.4485 0.538502 0.269251 0.963070i \(-0.413224\pi\)
0.269251 + 0.963070i \(0.413224\pi\)
\(824\) 22.1405 0.771302
\(825\) −1.94165 −0.0675996
\(826\) 5.40061 0.187911
\(827\) 12.7533 0.443476 0.221738 0.975106i \(-0.428827\pi\)
0.221738 + 0.975106i \(0.428827\pi\)
\(828\) 1.04905 0.0364569
\(829\) −49.3228 −1.71305 −0.856527 0.516103i \(-0.827382\pi\)
−0.856527 + 0.516103i \(0.827382\pi\)
\(830\) −10.1265 −0.351497
\(831\) −30.1878 −1.04720
\(832\) 45.4978 1.57735
\(833\) −6.92776 −0.240033
\(834\) 0.322875 0.0111803
\(835\) −22.6776 −0.784792
\(836\) 0.194962 0.00674292
\(837\) −6.13148 −0.211935
\(838\) −38.6122 −1.33384
\(839\) −7.12653 −0.246035 −0.123018 0.992404i \(-0.539257\pi\)
−0.123018 + 0.992404i \(0.539257\pi\)
\(840\) −1.20309 −0.0415107
\(841\) 4.63436 0.159806
\(842\) 46.6439 1.60746
\(843\) −23.7660 −0.818546
\(844\) −5.32028 −0.183132
\(845\) −52.5417 −1.80749
\(846\) −12.2653 −0.421689
\(847\) −2.73217 −0.0938785
\(848\) −33.7269 −1.15818
\(849\) −8.04642 −0.276152
\(850\) 3.17594 0.108934
\(851\) 2.68324 0.0919803
\(852\) 0.775637 0.0265729
\(853\) −16.5825 −0.567775 −0.283887 0.958858i \(-0.591624\pi\)
−0.283887 + 0.958858i \(0.591624\pi\)
\(854\) −0.363511 −0.0124391
\(855\) −1.54872 −0.0529652
\(856\) 24.3871 0.833535
\(857\) −42.6112 −1.45557 −0.727786 0.685804i \(-0.759451\pi\)
−0.727786 + 0.685804i \(0.759451\pi\)
\(858\) −9.05674 −0.309192
\(859\) −52.4322 −1.78896 −0.894482 0.447104i \(-0.852455\pi\)
−0.894482 + 0.447104i \(0.852455\pi\)
\(860\) −2.49785 −0.0851761
\(861\) 1.81138 0.0617315
\(862\) 29.4571 1.00331
\(863\) −37.5798 −1.27923 −0.639615 0.768695i \(-0.720906\pi\)
−0.639615 + 0.768695i \(0.720906\pi\)
\(864\) 1.31498 0.0447365
\(865\) −4.20465 −0.142963
\(866\) −7.84227 −0.266491
\(867\) 1.00000 0.0339618
\(868\) −0.384990 −0.0130674
\(869\) 0.913698 0.0309951
\(870\) 14.6964 0.498254
\(871\) −39.3704 −1.33401
\(872\) −37.0726 −1.25544
\(873\) −16.5060 −0.558643
\(874\) −6.13009 −0.207353
\(875\) 3.24711 0.109772
\(876\) −0.126817 −0.00428475
\(877\) 18.3324 0.619042 0.309521 0.950893i \(-0.399831\pi\)
0.309521 + 0.950893i \(0.399831\pi\)
\(878\) −19.5796 −0.660780
\(879\) 31.4243 1.05992
\(880\) 6.83624 0.230450
\(881\) −21.1097 −0.711204 −0.355602 0.934637i \(-0.615724\pi\)
−0.355602 + 0.934637i \(0.615724\pi\)
\(882\) 10.3537 0.348628
\(883\) 49.8242 1.67672 0.838359 0.545118i \(-0.183515\pi\)
0.838359 + 0.545118i \(0.183515\pi\)
\(884\) 1.54937 0.0521110
\(885\) 22.7961 0.766283
\(886\) −19.4800 −0.654444
\(887\) −39.3072 −1.31981 −0.659904 0.751350i \(-0.729403\pi\)
−0.659904 + 0.751350i \(0.729403\pi\)
\(888\) 1.57741 0.0529345
\(889\) −0.655369 −0.0219804
\(890\) −28.4777 −0.954573
\(891\) 0.913698 0.0306100
\(892\) 2.37376 0.0794793
\(893\) 7.49603 0.250845
\(894\) 27.0845 0.905842
\(895\) −37.1401 −1.24146
\(896\) −3.46252 −0.115675
\(897\) 29.7831 0.994430
\(898\) −57.1380 −1.90672
\(899\) −35.5596 −1.18598
\(900\) −0.496431 −0.0165477
\(901\) 7.64323 0.254633
\(902\) −9.20283 −0.306421
\(903\) 1.69494 0.0564042
\(904\) 35.6758 1.18656
\(905\) 21.8212 0.725361
\(906\) −3.45158 −0.114671
\(907\) 15.3365 0.509241 0.254621 0.967041i \(-0.418049\pi\)
0.254621 + 0.967041i \(0.418049\pi\)
\(908\) 4.78580 0.158822
\(909\) 4.16692 0.138208
\(910\) 4.51729 0.149747
\(911\) 57.1582 1.89374 0.946868 0.321624i \(-0.104229\pi\)
0.946868 + 0.321624i \(0.104229\pi\)
\(912\) −4.03048 −0.133463
\(913\) −3.65128 −0.120840
\(914\) −20.5039 −0.678207
\(915\) −1.53439 −0.0507253
\(916\) 3.99268 0.131922
\(917\) −3.72431 −0.122987
\(918\) −1.49453 −0.0493267
\(919\) −4.14263 −0.136653 −0.0683264 0.997663i \(-0.521766\pi\)
−0.0683264 + 0.997663i \(0.521766\pi\)
\(920\) −20.1007 −0.662700
\(921\) 4.02189 0.132526
\(922\) −61.7291 −2.03294
\(923\) 22.0208 0.724824
\(924\) 0.0573703 0.00188734
\(925\) −1.26976 −0.0417496
\(926\) 12.1737 0.400051
\(927\) 8.38683 0.275459
\(928\) 7.62625 0.250344
\(929\) 27.7073 0.909046 0.454523 0.890735i \(-0.349810\pi\)
0.454523 + 0.890735i \(0.349810\pi\)
\(930\) −15.5376 −0.509498
\(931\) −6.32777 −0.207384
\(932\) 6.02639 0.197401
\(933\) −15.6167 −0.511267
\(934\) −16.9517 −0.554678
\(935\) −1.54924 −0.0506655
\(936\) 17.5088 0.572292
\(937\) 44.8744 1.46598 0.732991 0.680238i \(-0.238124\pi\)
0.732991 + 0.680238i \(0.238124\pi\)
\(938\) 2.38452 0.0778575
\(939\) −0.155288 −0.00506763
\(940\) −3.25072 −0.106027
\(941\) 22.8211 0.743948 0.371974 0.928243i \(-0.378681\pi\)
0.371974 + 0.928243i \(0.378681\pi\)
\(942\) −36.9131 −1.20270
\(943\) 30.2635 0.985517
\(944\) 59.3259 1.93089
\(945\) −0.455732 −0.0148250
\(946\) −8.61129 −0.279977
\(947\) −32.5296 −1.05707 −0.528535 0.848912i \(-0.677258\pi\)
−0.528535 + 0.848912i \(0.677258\pi\)
\(948\) 0.233609 0.00758728
\(949\) −3.60042 −0.116874
\(950\) 2.90088 0.0941171
\(951\) 4.16068 0.134919
\(952\) 0.709552 0.0229967
\(953\) −19.5404 −0.632975 −0.316488 0.948597i \(-0.602504\pi\)
−0.316488 + 0.948597i \(0.602504\pi\)
\(954\) −11.4230 −0.369833
\(955\) −32.3914 −1.04816
\(956\) 2.43429 0.0787304
\(957\) 5.29901 0.171293
\(958\) −21.9910 −0.710497
\(959\) −2.03341 −0.0656623
\(960\) −11.6316 −0.375409
\(961\) 6.59501 0.212742
\(962\) −5.92275 −0.190957
\(963\) 9.23784 0.297685
\(964\) 4.55648 0.146754
\(965\) −39.7414 −1.27932
\(966\) −1.80386 −0.0580382
\(967\) 7.75381 0.249346 0.124673 0.992198i \(-0.460212\pi\)
0.124673 + 0.992198i \(0.460212\pi\)
\(968\) −26.8352 −0.862515
\(969\) 0.913394 0.0293424
\(970\) −41.8274 −1.34300
\(971\) 25.3639 0.813967 0.406983 0.913436i \(-0.366581\pi\)
0.406983 + 0.913436i \(0.366581\pi\)
\(972\) 0.233609 0.00749302
\(973\) −0.0580664 −0.00186152
\(974\) −52.4585 −1.68088
\(975\) −14.0940 −0.451369
\(976\) −3.99318 −0.127818
\(977\) 49.4803 1.58301 0.791507 0.611160i \(-0.209297\pi\)
0.791507 + 0.611160i \(0.209297\pi\)
\(978\) −19.3830 −0.619800
\(979\) −10.2681 −0.328168
\(980\) 2.74409 0.0876567
\(981\) −14.0431 −0.448361
\(982\) 28.4822 0.908905
\(983\) −42.2568 −1.34778 −0.673892 0.738830i \(-0.735379\pi\)
−0.673892 + 0.738830i \(0.735379\pi\)
\(984\) 17.7912 0.567163
\(985\) 35.6129 1.13472
\(986\) −8.66753 −0.276030
\(987\) 2.20581 0.0702116
\(988\) 1.41519 0.0450231
\(989\) 28.3183 0.900468
\(990\) 2.31538 0.0735875
\(991\) −3.25958 −0.103544 −0.0517720 0.998659i \(-0.516487\pi\)
−0.0517720 + 0.998659i \(0.516487\pi\)
\(992\) −8.06277 −0.255993
\(993\) −11.9884 −0.380440
\(994\) −1.33372 −0.0423031
\(995\) −0.380566 −0.0120647
\(996\) −0.933539 −0.0295803
\(997\) −11.0856 −0.351084 −0.175542 0.984472i \(-0.556168\pi\)
−0.175542 + 0.984472i \(0.556168\pi\)
\(998\) −3.63800 −0.115159
\(999\) 0.597523 0.0189048
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 4029.2.a.k.1.8 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.8 31 1.1 even 1 trivial