Properties

Label 4029.2.a.k.1.13
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 1.13
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.626356 q^{2} +1.00000 q^{3} -1.60768 q^{4} -1.73582 q^{5} -0.626356 q^{6} -3.52892 q^{7} +2.25969 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.626356 q^{2} +1.00000 q^{3} -1.60768 q^{4} -1.73582 q^{5} -0.626356 q^{6} -3.52892 q^{7} +2.25969 q^{8} +1.00000 q^{9} +1.08724 q^{10} -5.10748 q^{11} -1.60768 q^{12} -6.25342 q^{13} +2.21036 q^{14} -1.73582 q^{15} +1.79999 q^{16} +1.00000 q^{17} -0.626356 q^{18} -1.32233 q^{19} +2.79065 q^{20} -3.52892 q^{21} +3.19910 q^{22} -4.67299 q^{23} +2.25969 q^{24} -1.98692 q^{25} +3.91686 q^{26} +1.00000 q^{27} +5.67338 q^{28} -1.94307 q^{29} +1.08724 q^{30} -2.74559 q^{31} -5.64681 q^{32} -5.10748 q^{33} -0.626356 q^{34} +6.12559 q^{35} -1.60768 q^{36} +2.56583 q^{37} +0.828249 q^{38} -6.25342 q^{39} -3.92242 q^{40} -5.78701 q^{41} +2.21036 q^{42} +2.16341 q^{43} +8.21118 q^{44} -1.73582 q^{45} +2.92695 q^{46} -12.8066 q^{47} +1.79999 q^{48} +5.45331 q^{49} +1.24452 q^{50} +1.00000 q^{51} +10.0535 q^{52} -4.09434 q^{53} -0.626356 q^{54} +8.86568 q^{55} -7.97428 q^{56} -1.32233 q^{57} +1.21705 q^{58} -5.16861 q^{59} +2.79065 q^{60} -9.98052 q^{61} +1.71971 q^{62} -3.52892 q^{63} -0.0630639 q^{64} +10.8548 q^{65} +3.19910 q^{66} -2.06118 q^{67} -1.60768 q^{68} -4.67299 q^{69} -3.83680 q^{70} +7.20147 q^{71} +2.25969 q^{72} +10.0866 q^{73} -1.60712 q^{74} -1.98692 q^{75} +2.12588 q^{76} +18.0239 q^{77} +3.91686 q^{78} +1.00000 q^{79} -3.12446 q^{80} +1.00000 q^{81} +3.62473 q^{82} +16.6953 q^{83} +5.67338 q^{84} -1.73582 q^{85} -1.35506 q^{86} -1.94307 q^{87} -11.5413 q^{88} -14.4469 q^{89} +1.08724 q^{90} +22.0679 q^{91} +7.51267 q^{92} -2.74559 q^{93} +8.02149 q^{94} +2.29533 q^{95} -5.64681 q^{96} -4.30641 q^{97} -3.41571 q^{98} -5.10748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.626356 −0.442900 −0.221450 0.975172i \(-0.571079\pi\)
−0.221450 + 0.975172i \(0.571079\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.60768 −0.803839
\(5\) −1.73582 −0.776284 −0.388142 0.921600i \(-0.626883\pi\)
−0.388142 + 0.921600i \(0.626883\pi\)
\(6\) −0.626356 −0.255709
\(7\) −3.52892 −1.33381 −0.666904 0.745144i \(-0.732381\pi\)
−0.666904 + 0.745144i \(0.732381\pi\)
\(8\) 2.25969 0.798921
\(9\) 1.00000 0.333333
\(10\) 1.08724 0.343816
\(11\) −5.10748 −1.53996 −0.769981 0.638067i \(-0.779734\pi\)
−0.769981 + 0.638067i \(0.779734\pi\)
\(12\) −1.60768 −0.464097
\(13\) −6.25342 −1.73439 −0.867193 0.497972i \(-0.834078\pi\)
−0.867193 + 0.497972i \(0.834078\pi\)
\(14\) 2.21036 0.590744
\(15\) −1.73582 −0.448188
\(16\) 1.79999 0.449997
\(17\) 1.00000 0.242536
\(18\) −0.626356 −0.147633
\(19\) −1.32233 −0.303363 −0.151682 0.988429i \(-0.548469\pi\)
−0.151682 + 0.988429i \(0.548469\pi\)
\(20\) 2.79065 0.624007
\(21\) −3.52892 −0.770075
\(22\) 3.19910 0.682050
\(23\) −4.67299 −0.974386 −0.487193 0.873294i \(-0.661979\pi\)
−0.487193 + 0.873294i \(0.661979\pi\)
\(24\) 2.25969 0.461257
\(25\) −1.98692 −0.397383
\(26\) 3.91686 0.768160
\(27\) 1.00000 0.192450
\(28\) 5.67338 1.07217
\(29\) −1.94307 −0.360819 −0.180410 0.983592i \(-0.557742\pi\)
−0.180410 + 0.983592i \(0.557742\pi\)
\(30\) 1.08724 0.198502
\(31\) −2.74559 −0.493122 −0.246561 0.969127i \(-0.579301\pi\)
−0.246561 + 0.969127i \(0.579301\pi\)
\(32\) −5.64681 −0.998225
\(33\) −5.10748 −0.889098
\(34\) −0.626356 −0.107419
\(35\) 6.12559 1.03541
\(36\) −1.60768 −0.267946
\(37\) 2.56583 0.421820 0.210910 0.977505i \(-0.432357\pi\)
0.210910 + 0.977505i \(0.432357\pi\)
\(38\) 0.828249 0.134360
\(39\) −6.25342 −1.00135
\(40\) −3.92242 −0.620189
\(41\) −5.78701 −0.903780 −0.451890 0.892074i \(-0.649250\pi\)
−0.451890 + 0.892074i \(0.649250\pi\)
\(42\) 2.21036 0.341066
\(43\) 2.16341 0.329917 0.164959 0.986301i \(-0.447251\pi\)
0.164959 + 0.986301i \(0.447251\pi\)
\(44\) 8.21118 1.23788
\(45\) −1.73582 −0.258761
\(46\) 2.92695 0.431556
\(47\) −12.8066 −1.86803 −0.934017 0.357228i \(-0.883722\pi\)
−0.934017 + 0.357228i \(0.883722\pi\)
\(48\) 1.79999 0.259806
\(49\) 5.45331 0.779044
\(50\) 1.24452 0.176001
\(51\) 1.00000 0.140028
\(52\) 10.0535 1.39417
\(53\) −4.09434 −0.562401 −0.281201 0.959649i \(-0.590733\pi\)
−0.281201 + 0.959649i \(0.590733\pi\)
\(54\) −0.626356 −0.0852362
\(55\) 8.86568 1.19545
\(56\) −7.97428 −1.06561
\(57\) −1.32233 −0.175147
\(58\) 1.21705 0.159807
\(59\) −5.16861 −0.672896 −0.336448 0.941702i \(-0.609226\pi\)
−0.336448 + 0.941702i \(0.609226\pi\)
\(60\) 2.79065 0.360271
\(61\) −9.98052 −1.27788 −0.638938 0.769259i \(-0.720626\pi\)
−0.638938 + 0.769259i \(0.720626\pi\)
\(62\) 1.71971 0.218404
\(63\) −3.52892 −0.444603
\(64\) −0.0630639 −0.00788298
\(65\) 10.8548 1.34638
\(66\) 3.19910 0.393782
\(67\) −2.06118 −0.251813 −0.125907 0.992042i \(-0.540184\pi\)
−0.125907 + 0.992042i \(0.540184\pi\)
\(68\) −1.60768 −0.194960
\(69\) −4.67299 −0.562562
\(70\) −3.83680 −0.458585
\(71\) 7.20147 0.854657 0.427329 0.904096i \(-0.359455\pi\)
0.427329 + 0.904096i \(0.359455\pi\)
\(72\) 2.25969 0.266307
\(73\) 10.0866 1.18055 0.590276 0.807202i \(-0.299019\pi\)
0.590276 + 0.807202i \(0.299019\pi\)
\(74\) −1.60712 −0.186824
\(75\) −1.98692 −0.229429
\(76\) 2.12588 0.243855
\(77\) 18.0239 2.05401
\(78\) 3.91686 0.443498
\(79\) 1.00000 0.112509
\(80\) −3.12446 −0.349325
\(81\) 1.00000 0.111111
\(82\) 3.62473 0.400284
\(83\) 16.6953 1.83255 0.916274 0.400551i \(-0.131181\pi\)
0.916274 + 0.400551i \(0.131181\pi\)
\(84\) 5.67338 0.619016
\(85\) −1.73582 −0.188276
\(86\) −1.35506 −0.146120
\(87\) −1.94307 −0.208319
\(88\) −11.5413 −1.23031
\(89\) −14.4469 −1.53136 −0.765682 0.643219i \(-0.777598\pi\)
−0.765682 + 0.643219i \(0.777598\pi\)
\(90\) 1.08724 0.114605
\(91\) 22.0679 2.31334
\(92\) 7.51267 0.783250
\(93\) −2.74559 −0.284704
\(94\) 8.02149 0.827353
\(95\) 2.29533 0.235496
\(96\) −5.64681 −0.576325
\(97\) −4.30641 −0.437250 −0.218625 0.975809i \(-0.570157\pi\)
−0.218625 + 0.975809i \(0.570157\pi\)
\(98\) −3.41571 −0.345039
\(99\) −5.10748 −0.513321
\(100\) 3.19432 0.319432
\(101\) 10.3955 1.03439 0.517194 0.855868i \(-0.326977\pi\)
0.517194 + 0.855868i \(0.326977\pi\)
\(102\) −0.626356 −0.0620184
\(103\) −3.10646 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(104\) −14.1308 −1.38564
\(105\) 6.12559 0.597796
\(106\) 2.56452 0.249088
\(107\) 10.2602 0.991889 0.495944 0.868354i \(-0.334822\pi\)
0.495944 + 0.868354i \(0.334822\pi\)
\(108\) −1.60768 −0.154699
\(109\) −2.89139 −0.276945 −0.138473 0.990366i \(-0.544219\pi\)
−0.138473 + 0.990366i \(0.544219\pi\)
\(110\) −5.55307 −0.529464
\(111\) 2.56583 0.243538
\(112\) −6.35202 −0.600210
\(113\) 1.87221 0.176123 0.0880614 0.996115i \(-0.471933\pi\)
0.0880614 + 0.996115i \(0.471933\pi\)
\(114\) 0.828249 0.0775726
\(115\) 8.11149 0.756400
\(116\) 3.12383 0.290041
\(117\) −6.25342 −0.578129
\(118\) 3.23739 0.298026
\(119\) −3.52892 −0.323496
\(120\) −3.92242 −0.358067
\(121\) 15.0863 1.37148
\(122\) 6.25136 0.565971
\(123\) −5.78701 −0.521797
\(124\) 4.41402 0.396391
\(125\) 12.1281 1.08477
\(126\) 2.21036 0.196915
\(127\) −16.9024 −1.49984 −0.749921 0.661528i \(-0.769908\pi\)
−0.749921 + 0.661528i \(0.769908\pi\)
\(128\) 11.3331 1.00172
\(129\) 2.16341 0.190478
\(130\) −6.79899 −0.596310
\(131\) −2.88794 −0.252321 −0.126160 0.992010i \(-0.540265\pi\)
−0.126160 + 0.992010i \(0.540265\pi\)
\(132\) 8.21118 0.714692
\(133\) 4.66640 0.404628
\(134\) 1.29103 0.111528
\(135\) −1.73582 −0.149396
\(136\) 2.25969 0.193767
\(137\) 4.43192 0.378645 0.189322 0.981915i \(-0.439371\pi\)
0.189322 + 0.981915i \(0.439371\pi\)
\(138\) 2.92695 0.249159
\(139\) 2.19056 0.185801 0.0929005 0.995675i \(-0.470386\pi\)
0.0929005 + 0.995675i \(0.470386\pi\)
\(140\) −9.84798 −0.832306
\(141\) −12.8066 −1.07851
\(142\) −4.51068 −0.378528
\(143\) 31.9392 2.67089
\(144\) 1.79999 0.149999
\(145\) 3.37283 0.280098
\(146\) −6.31782 −0.522866
\(147\) 5.45331 0.449782
\(148\) −4.12503 −0.339076
\(149\) −9.39545 −0.769705 −0.384853 0.922978i \(-0.625748\pi\)
−0.384853 + 0.922978i \(0.625748\pi\)
\(150\) 1.24452 0.101614
\(151\) −14.3366 −1.16669 −0.583347 0.812223i \(-0.698257\pi\)
−0.583347 + 0.812223i \(0.698257\pi\)
\(152\) −2.98806 −0.242363
\(153\) 1.00000 0.0808452
\(154\) −11.2894 −0.909724
\(155\) 4.76585 0.382802
\(156\) 10.0535 0.804923
\(157\) −15.6020 −1.24517 −0.622586 0.782551i \(-0.713918\pi\)
−0.622586 + 0.782551i \(0.713918\pi\)
\(158\) −0.626356 −0.0498302
\(159\) −4.09434 −0.324703
\(160\) 9.80187 0.774906
\(161\) 16.4906 1.29964
\(162\) −0.626356 −0.0492111
\(163\) −3.58037 −0.280436 −0.140218 0.990121i \(-0.544780\pi\)
−0.140218 + 0.990121i \(0.544780\pi\)
\(164\) 9.30366 0.726494
\(165\) 8.86568 0.690192
\(166\) −10.4572 −0.811636
\(167\) 11.7391 0.908397 0.454198 0.890901i \(-0.349926\pi\)
0.454198 + 0.890901i \(0.349926\pi\)
\(168\) −7.97428 −0.615229
\(169\) 26.1053 2.00810
\(170\) 1.08724 0.0833877
\(171\) −1.32233 −0.101121
\(172\) −3.47807 −0.265200
\(173\) −6.72410 −0.511224 −0.255612 0.966779i \(-0.582277\pi\)
−0.255612 + 0.966779i \(0.582277\pi\)
\(174\) 1.21705 0.0922646
\(175\) 7.01168 0.530033
\(176\) −9.19340 −0.692979
\(177\) −5.16861 −0.388497
\(178\) 9.04887 0.678242
\(179\) 23.0226 1.72079 0.860394 0.509629i \(-0.170217\pi\)
0.860394 + 0.509629i \(0.170217\pi\)
\(180\) 2.79065 0.208002
\(181\) 0.0754824 0.00561057 0.00280528 0.999996i \(-0.499107\pi\)
0.00280528 + 0.999996i \(0.499107\pi\)
\(182\) −13.8223 −1.02458
\(183\) −9.98052 −0.737782
\(184\) −10.5595 −0.778458
\(185\) −4.45383 −0.327452
\(186\) 1.71971 0.126095
\(187\) −5.10748 −0.373496
\(188\) 20.5889 1.50160
\(189\) −3.52892 −0.256692
\(190\) −1.43769 −0.104301
\(191\) −16.2498 −1.17579 −0.587897 0.808936i \(-0.700044\pi\)
−0.587897 + 0.808936i \(0.700044\pi\)
\(192\) −0.0630639 −0.00455124
\(193\) −16.3940 −1.18007 −0.590035 0.807378i \(-0.700886\pi\)
−0.590035 + 0.807378i \(0.700886\pi\)
\(194\) 2.69735 0.193658
\(195\) 10.8548 0.777331
\(196\) −8.76717 −0.626227
\(197\) −22.5649 −1.60769 −0.803843 0.594842i \(-0.797215\pi\)
−0.803843 + 0.594842i \(0.797215\pi\)
\(198\) 3.19910 0.227350
\(199\) 16.8334 1.19329 0.596643 0.802506i \(-0.296501\pi\)
0.596643 + 0.802506i \(0.296501\pi\)
\(200\) −4.48982 −0.317478
\(201\) −2.06118 −0.145385
\(202\) −6.51126 −0.458131
\(203\) 6.85695 0.481264
\(204\) −1.60768 −0.112560
\(205\) 10.0452 0.701590
\(206\) 1.94575 0.135567
\(207\) −4.67299 −0.324795
\(208\) −11.2561 −0.780469
\(209\) 6.75377 0.467168
\(210\) −3.83680 −0.264764
\(211\) −0.565736 −0.0389469 −0.0194734 0.999810i \(-0.506199\pi\)
−0.0194734 + 0.999810i \(0.506199\pi\)
\(212\) 6.58239 0.452080
\(213\) 7.20147 0.493437
\(214\) −6.42652 −0.439308
\(215\) −3.75530 −0.256109
\(216\) 2.25969 0.153752
\(217\) 9.68897 0.657730
\(218\) 1.81104 0.122659
\(219\) 10.0866 0.681592
\(220\) −14.2532 −0.960948
\(221\) −6.25342 −0.420651
\(222\) −1.60712 −0.107863
\(223\) 3.71956 0.249080 0.124540 0.992215i \(-0.460254\pi\)
0.124540 + 0.992215i \(0.460254\pi\)
\(224\) 19.9272 1.33144
\(225\) −1.98692 −0.132461
\(226\) −1.17267 −0.0780048
\(227\) −12.1820 −0.808550 −0.404275 0.914638i \(-0.632476\pi\)
−0.404275 + 0.914638i \(0.632476\pi\)
\(228\) 2.12588 0.140790
\(229\) 8.01590 0.529705 0.264853 0.964289i \(-0.414677\pi\)
0.264853 + 0.964289i \(0.414677\pi\)
\(230\) −5.08068 −0.335010
\(231\) 18.0239 1.18589
\(232\) −4.39074 −0.288266
\(233\) 6.77495 0.443842 0.221921 0.975065i \(-0.428767\pi\)
0.221921 + 0.975065i \(0.428767\pi\)
\(234\) 3.91686 0.256053
\(235\) 22.2300 1.45013
\(236\) 8.30947 0.540900
\(237\) 1.00000 0.0649570
\(238\) 2.21036 0.143276
\(239\) 19.1256 1.23713 0.618566 0.785733i \(-0.287714\pi\)
0.618566 + 0.785733i \(0.287714\pi\)
\(240\) −3.12446 −0.201683
\(241\) 11.5938 0.746822 0.373411 0.927666i \(-0.378188\pi\)
0.373411 + 0.927666i \(0.378188\pi\)
\(242\) −9.44941 −0.607431
\(243\) 1.00000 0.0641500
\(244\) 16.0455 1.02721
\(245\) −9.46599 −0.604760
\(246\) 3.62473 0.231104
\(247\) 8.26908 0.526149
\(248\) −6.20417 −0.393965
\(249\) 16.6953 1.05802
\(250\) −7.59647 −0.480443
\(251\) 4.85890 0.306691 0.153346 0.988173i \(-0.450995\pi\)
0.153346 + 0.988173i \(0.450995\pi\)
\(252\) 5.67338 0.357389
\(253\) 23.8672 1.50052
\(254\) 10.5869 0.664280
\(255\) −1.73582 −0.108701
\(256\) −6.97244 −0.435777
\(257\) 21.2911 1.32810 0.664051 0.747687i \(-0.268836\pi\)
0.664051 + 0.747687i \(0.268836\pi\)
\(258\) −1.35506 −0.0843626
\(259\) −9.05463 −0.562627
\(260\) −17.4511 −1.08227
\(261\) −1.94307 −0.120273
\(262\) 1.80888 0.111753
\(263\) 15.1049 0.931405 0.465703 0.884941i \(-0.345802\pi\)
0.465703 + 0.884941i \(0.345802\pi\)
\(264\) −11.5413 −0.710319
\(265\) 7.10706 0.436583
\(266\) −2.92283 −0.179210
\(267\) −14.4469 −0.884134
\(268\) 3.31372 0.202417
\(269\) −8.22219 −0.501316 −0.250658 0.968076i \(-0.580647\pi\)
−0.250658 + 0.968076i \(0.580647\pi\)
\(270\) 1.08724 0.0661675
\(271\) −9.72698 −0.590872 −0.295436 0.955362i \(-0.595465\pi\)
−0.295436 + 0.955362i \(0.595465\pi\)
\(272\) 1.79999 0.109140
\(273\) 22.0679 1.33561
\(274\) −2.77596 −0.167702
\(275\) 10.1481 0.611956
\(276\) 7.51267 0.452210
\(277\) 7.03106 0.422455 0.211228 0.977437i \(-0.432254\pi\)
0.211228 + 0.977437i \(0.432254\pi\)
\(278\) −1.37207 −0.0822913
\(279\) −2.74559 −0.164374
\(280\) 13.8419 0.827214
\(281\) −18.2953 −1.09141 −0.545703 0.837979i \(-0.683737\pi\)
−0.545703 + 0.837979i \(0.683737\pi\)
\(282\) 8.02149 0.477673
\(283\) −24.7438 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(284\) −11.5777 −0.687007
\(285\) 2.29533 0.135964
\(286\) −20.0053 −1.18294
\(287\) 20.4219 1.20547
\(288\) −5.64681 −0.332742
\(289\) 1.00000 0.0588235
\(290\) −2.11259 −0.124056
\(291\) −4.30641 −0.252446
\(292\) −16.2161 −0.948973
\(293\) −5.77026 −0.337102 −0.168551 0.985693i \(-0.553909\pi\)
−0.168551 + 0.985693i \(0.553909\pi\)
\(294\) −3.41571 −0.199208
\(295\) 8.97180 0.522358
\(296\) 5.79799 0.337001
\(297\) −5.10748 −0.296366
\(298\) 5.88489 0.340903
\(299\) 29.2222 1.68996
\(300\) 3.19432 0.184424
\(301\) −7.63451 −0.440046
\(302\) 8.97979 0.516729
\(303\) 10.3955 0.597204
\(304\) −2.38018 −0.136513
\(305\) 17.3244 0.991994
\(306\) −0.626356 −0.0358064
\(307\) −26.8692 −1.53351 −0.766753 0.641942i \(-0.778129\pi\)
−0.766753 + 0.641942i \(0.778129\pi\)
\(308\) −28.9766 −1.65110
\(309\) −3.10646 −0.176720
\(310\) −2.98512 −0.169543
\(311\) 6.33580 0.359270 0.179635 0.983733i \(-0.442508\pi\)
0.179635 + 0.983733i \(0.442508\pi\)
\(312\) −14.1308 −0.799998
\(313\) −8.48733 −0.479733 −0.239866 0.970806i \(-0.577104\pi\)
−0.239866 + 0.970806i \(0.577104\pi\)
\(314\) 9.77238 0.551487
\(315\) 6.12559 0.345138
\(316\) −1.60768 −0.0904390
\(317\) −16.2865 −0.914742 −0.457371 0.889276i \(-0.651209\pi\)
−0.457371 + 0.889276i \(0.651209\pi\)
\(318\) 2.56452 0.143811
\(319\) 9.92419 0.555648
\(320\) 0.109468 0.00611943
\(321\) 10.2602 0.572667
\(322\) −10.3290 −0.575613
\(323\) −1.32233 −0.0735764
\(324\) −1.60768 −0.0893155
\(325\) 12.4250 0.689217
\(326\) 2.24259 0.124205
\(327\) −2.89139 −0.159894
\(328\) −13.0769 −0.722049
\(329\) 45.1935 2.49160
\(330\) −5.55307 −0.305686
\(331\) −1.73976 −0.0956260 −0.0478130 0.998856i \(-0.515225\pi\)
−0.0478130 + 0.998856i \(0.515225\pi\)
\(332\) −26.8407 −1.47307
\(333\) 2.56583 0.140607
\(334\) −7.35283 −0.402329
\(335\) 3.57785 0.195479
\(336\) −6.35202 −0.346531
\(337\) −25.4157 −1.38448 −0.692242 0.721666i \(-0.743377\pi\)
−0.692242 + 0.721666i \(0.743377\pi\)
\(338\) −16.3512 −0.889387
\(339\) 1.87221 0.101685
\(340\) 2.79065 0.151344
\(341\) 14.0230 0.759389
\(342\) 0.828249 0.0447866
\(343\) 5.45815 0.294712
\(344\) 4.88864 0.263578
\(345\) 8.11149 0.436708
\(346\) 4.21168 0.226421
\(347\) 8.73270 0.468796 0.234398 0.972141i \(-0.424688\pi\)
0.234398 + 0.972141i \(0.424688\pi\)
\(348\) 3.12383 0.167455
\(349\) −15.1012 −0.808349 −0.404175 0.914682i \(-0.632441\pi\)
−0.404175 + 0.914682i \(0.632441\pi\)
\(350\) −4.39181 −0.234752
\(351\) −6.25342 −0.333783
\(352\) 28.8410 1.53723
\(353\) 17.8898 0.952177 0.476088 0.879397i \(-0.342054\pi\)
0.476088 + 0.879397i \(0.342054\pi\)
\(354\) 3.23739 0.172065
\(355\) −12.5005 −0.663457
\(356\) 23.2259 1.23097
\(357\) −3.52892 −0.186771
\(358\) −14.4203 −0.762138
\(359\) −14.9200 −0.787446 −0.393723 0.919229i \(-0.628813\pi\)
−0.393723 + 0.919229i \(0.628813\pi\)
\(360\) −3.92242 −0.206730
\(361\) −17.2514 −0.907971
\(362\) −0.0472788 −0.00248492
\(363\) 15.0863 0.791827
\(364\) −35.4780 −1.85955
\(365\) −17.5086 −0.916443
\(366\) 6.25136 0.326764
\(367\) 21.7045 1.13297 0.566483 0.824073i \(-0.308304\pi\)
0.566483 + 0.824073i \(0.308304\pi\)
\(368\) −8.41133 −0.438471
\(369\) −5.78701 −0.301260
\(370\) 2.78968 0.145029
\(371\) 14.4486 0.750136
\(372\) 4.41402 0.228856
\(373\) −31.6257 −1.63752 −0.818758 0.574138i \(-0.805337\pi\)
−0.818758 + 0.574138i \(0.805337\pi\)
\(374\) 3.19910 0.165421
\(375\) 12.1281 0.626290
\(376\) −28.9389 −1.49241
\(377\) 12.1508 0.625800
\(378\) 2.21036 0.113689
\(379\) −0.419298 −0.0215379 −0.0107689 0.999942i \(-0.503428\pi\)
−0.0107689 + 0.999942i \(0.503428\pi\)
\(380\) −3.69015 −0.189301
\(381\) −16.9024 −0.865934
\(382\) 10.1781 0.520759
\(383\) 36.8028 1.88053 0.940267 0.340437i \(-0.110575\pi\)
0.940267 + 0.340437i \(0.110575\pi\)
\(384\) 11.3331 0.578341
\(385\) −31.2863 −1.59450
\(386\) 10.2685 0.522653
\(387\) 2.16341 0.109972
\(388\) 6.92333 0.351479
\(389\) −21.3161 −1.08077 −0.540385 0.841418i \(-0.681721\pi\)
−0.540385 + 0.841418i \(0.681721\pi\)
\(390\) −6.79899 −0.344280
\(391\) −4.67299 −0.236323
\(392\) 12.3228 0.622395
\(393\) −2.88794 −0.145677
\(394\) 14.1337 0.712045
\(395\) −1.73582 −0.0873388
\(396\) 8.21118 0.412627
\(397\) 37.9639 1.90535 0.952677 0.303986i \(-0.0983175\pi\)
0.952677 + 0.303986i \(0.0983175\pi\)
\(398\) −10.5437 −0.528507
\(399\) 4.66640 0.233612
\(400\) −3.57643 −0.178821
\(401\) −5.80297 −0.289787 −0.144893 0.989447i \(-0.546284\pi\)
−0.144893 + 0.989447i \(0.546284\pi\)
\(402\) 1.29103 0.0643908
\(403\) 17.1693 0.855264
\(404\) −16.7126 −0.831481
\(405\) −1.73582 −0.0862538
\(406\) −4.29489 −0.213152
\(407\) −13.1049 −0.649587
\(408\) 2.25969 0.111871
\(409\) −18.1294 −0.896441 −0.448220 0.893923i \(-0.647942\pi\)
−0.448220 + 0.893923i \(0.647942\pi\)
\(410\) −6.29189 −0.310734
\(411\) 4.43192 0.218611
\(412\) 4.99419 0.246046
\(413\) 18.2396 0.897514
\(414\) 2.92695 0.143852
\(415\) −28.9801 −1.42258
\(416\) 35.3119 1.73131
\(417\) 2.19056 0.107272
\(418\) −4.23026 −0.206909
\(419\) −14.0537 −0.686567 −0.343283 0.939232i \(-0.611539\pi\)
−0.343283 + 0.939232i \(0.611539\pi\)
\(420\) −9.84798 −0.480532
\(421\) −9.92808 −0.483865 −0.241932 0.970293i \(-0.577781\pi\)
−0.241932 + 0.970293i \(0.577781\pi\)
\(422\) 0.354352 0.0172496
\(423\) −12.8066 −0.622678
\(424\) −9.25195 −0.449314
\(425\) −1.98692 −0.0963796
\(426\) −4.51068 −0.218543
\(427\) 35.2205 1.70444
\(428\) −16.4951 −0.797319
\(429\) 31.9392 1.54204
\(430\) 2.35215 0.113431
\(431\) −23.4739 −1.13070 −0.565349 0.824852i \(-0.691258\pi\)
−0.565349 + 0.824852i \(0.691258\pi\)
\(432\) 1.79999 0.0866020
\(433\) −25.4109 −1.22117 −0.610584 0.791951i \(-0.709065\pi\)
−0.610584 + 0.791951i \(0.709065\pi\)
\(434\) −6.06874 −0.291309
\(435\) 3.37283 0.161715
\(436\) 4.64843 0.222619
\(437\) 6.17924 0.295593
\(438\) −6.31782 −0.301877
\(439\) 26.5277 1.26610 0.633048 0.774113i \(-0.281804\pi\)
0.633048 + 0.774113i \(0.281804\pi\)
\(440\) 20.0337 0.955068
\(441\) 5.45331 0.259681
\(442\) 3.91686 0.186306
\(443\) −34.5811 −1.64300 −0.821498 0.570211i \(-0.806861\pi\)
−0.821498 + 0.570211i \(0.806861\pi\)
\(444\) −4.12503 −0.195765
\(445\) 25.0772 1.18877
\(446\) −2.32977 −0.110318
\(447\) −9.39545 −0.444390
\(448\) 0.222548 0.0105144
\(449\) 25.5656 1.20652 0.603258 0.797546i \(-0.293869\pi\)
0.603258 + 0.797546i \(0.293869\pi\)
\(450\) 1.24452 0.0586671
\(451\) 29.5570 1.39179
\(452\) −3.00991 −0.141574
\(453\) −14.3366 −0.673591
\(454\) 7.63028 0.358107
\(455\) −38.3059 −1.79581
\(456\) −2.98806 −0.139929
\(457\) −29.1740 −1.36470 −0.682352 0.731024i \(-0.739043\pi\)
−0.682352 + 0.731024i \(0.739043\pi\)
\(458\) −5.02080 −0.234607
\(459\) 1.00000 0.0466760
\(460\) −13.0407 −0.608024
\(461\) −4.00888 −0.186712 −0.0933560 0.995633i \(-0.529759\pi\)
−0.0933560 + 0.995633i \(0.529759\pi\)
\(462\) −11.2894 −0.525229
\(463\) −27.5624 −1.28093 −0.640466 0.767987i \(-0.721259\pi\)
−0.640466 + 0.767987i \(0.721259\pi\)
\(464\) −3.49750 −0.162368
\(465\) 4.76585 0.221011
\(466\) −4.24353 −0.196578
\(467\) 1.76806 0.0818161 0.0409080 0.999163i \(-0.486975\pi\)
0.0409080 + 0.999163i \(0.486975\pi\)
\(468\) 10.0535 0.464723
\(469\) 7.27375 0.335871
\(470\) −13.9239 −0.642261
\(471\) −15.6020 −0.718901
\(472\) −11.6795 −0.537591
\(473\) −11.0496 −0.508060
\(474\) −0.626356 −0.0287695
\(475\) 2.62736 0.120552
\(476\) 5.67338 0.260039
\(477\) −4.09434 −0.187467
\(478\) −11.9794 −0.547926
\(479\) 9.74005 0.445034 0.222517 0.974929i \(-0.428573\pi\)
0.222517 + 0.974929i \(0.428573\pi\)
\(480\) 9.80187 0.447392
\(481\) −16.0452 −0.731599
\(482\) −7.26184 −0.330768
\(483\) 16.4906 0.750350
\(484\) −24.2540 −1.10245
\(485\) 7.47517 0.339430
\(486\) −0.626356 −0.0284121
\(487\) −6.53289 −0.296034 −0.148017 0.988985i \(-0.547289\pi\)
−0.148017 + 0.988985i \(0.547289\pi\)
\(488\) −22.5529 −1.02092
\(489\) −3.58037 −0.161910
\(490\) 5.92907 0.267848
\(491\) 32.2958 1.45749 0.728745 0.684785i \(-0.240104\pi\)
0.728745 + 0.684785i \(0.240104\pi\)
\(492\) 9.30366 0.419441
\(493\) −1.94307 −0.0875115
\(494\) −5.17939 −0.233032
\(495\) 8.86568 0.398483
\(496\) −4.94202 −0.221903
\(497\) −25.4134 −1.13995
\(498\) −10.4572 −0.468598
\(499\) −20.5314 −0.919110 −0.459555 0.888149i \(-0.651991\pi\)
−0.459555 + 0.888149i \(0.651991\pi\)
\(500\) −19.4980 −0.871978
\(501\) 11.7391 0.524463
\(502\) −3.04340 −0.135834
\(503\) −26.7932 −1.19465 −0.597324 0.802000i \(-0.703769\pi\)
−0.597324 + 0.802000i \(0.703769\pi\)
\(504\) −7.97428 −0.355202
\(505\) −18.0447 −0.802978
\(506\) −14.9494 −0.664580
\(507\) 26.1053 1.15938
\(508\) 27.1736 1.20563
\(509\) −2.37616 −0.105322 −0.0526608 0.998612i \(-0.516770\pi\)
−0.0526608 + 0.998612i \(0.516770\pi\)
\(510\) 1.08724 0.0481439
\(511\) −35.5950 −1.57463
\(512\) −18.2990 −0.808710
\(513\) −1.32233 −0.0583823
\(514\) −13.3358 −0.588217
\(515\) 5.39227 0.237612
\(516\) −3.47807 −0.153113
\(517\) 65.4094 2.87670
\(518\) 5.67142 0.249188
\(519\) −6.72410 −0.295155
\(520\) 24.5286 1.07565
\(521\) 24.3131 1.06518 0.532588 0.846375i \(-0.321220\pi\)
0.532588 + 0.846375i \(0.321220\pi\)
\(522\) 1.21705 0.0532690
\(523\) −29.6613 −1.29700 −0.648499 0.761215i \(-0.724603\pi\)
−0.648499 + 0.761215i \(0.724603\pi\)
\(524\) 4.64288 0.202825
\(525\) 7.01168 0.306015
\(526\) −9.46101 −0.412520
\(527\) −2.74559 −0.119600
\(528\) −9.19340 −0.400091
\(529\) −1.16314 −0.0505713
\(530\) −4.45155 −0.193363
\(531\) −5.16861 −0.224299
\(532\) −7.50208 −0.325256
\(533\) 36.1886 1.56750
\(534\) 9.04887 0.391583
\(535\) −17.8099 −0.769987
\(536\) −4.65763 −0.201179
\(537\) 23.0226 0.993498
\(538\) 5.15002 0.222033
\(539\) −27.8527 −1.19970
\(540\) 2.79065 0.120090
\(541\) 7.81470 0.335980 0.167990 0.985789i \(-0.446272\pi\)
0.167990 + 0.985789i \(0.446272\pi\)
\(542\) 6.09255 0.261697
\(543\) 0.0754824 0.00323926
\(544\) −5.64681 −0.242105
\(545\) 5.01895 0.214988
\(546\) −13.8223 −0.591541
\(547\) 41.5044 1.77460 0.887301 0.461191i \(-0.152578\pi\)
0.887301 + 0.461191i \(0.152578\pi\)
\(548\) −7.12511 −0.304370
\(549\) −9.98052 −0.425958
\(550\) −6.35634 −0.271035
\(551\) 2.56938 0.109459
\(552\) −10.5595 −0.449443
\(553\) −3.52892 −0.150065
\(554\) −4.40394 −0.187106
\(555\) −4.45383 −0.189055
\(556\) −3.52172 −0.149354
\(557\) 20.4459 0.866322 0.433161 0.901316i \(-0.357398\pi\)
0.433161 + 0.901316i \(0.357398\pi\)
\(558\) 1.71971 0.0728013
\(559\) −13.5287 −0.572204
\(560\) 11.0260 0.465933
\(561\) −5.10748 −0.215638
\(562\) 11.4594 0.483384
\(563\) 9.16137 0.386106 0.193053 0.981188i \(-0.438161\pi\)
0.193053 + 0.981188i \(0.438161\pi\)
\(564\) 20.5889 0.866949
\(565\) −3.24983 −0.136721
\(566\) 15.4984 0.651446
\(567\) −3.52892 −0.148201
\(568\) 16.2731 0.682804
\(569\) 16.4516 0.689689 0.344844 0.938660i \(-0.387932\pi\)
0.344844 + 0.938660i \(0.387932\pi\)
\(570\) −1.43769 −0.0602184
\(571\) 35.0552 1.46702 0.733508 0.679681i \(-0.237882\pi\)
0.733508 + 0.679681i \(0.237882\pi\)
\(572\) −51.3480 −2.14697
\(573\) −16.2498 −0.678844
\(574\) −12.7914 −0.533902
\(575\) 9.28485 0.387205
\(576\) −0.0630639 −0.00262766
\(577\) −37.8797 −1.57695 −0.788476 0.615066i \(-0.789129\pi\)
−0.788476 + 0.615066i \(0.789129\pi\)
\(578\) −0.626356 −0.0260530
\(579\) −16.3940 −0.681313
\(580\) −5.42242 −0.225154
\(581\) −58.9165 −2.44427
\(582\) 2.69735 0.111809
\(583\) 20.9118 0.866077
\(584\) 22.7927 0.943167
\(585\) 10.8548 0.448792
\(586\) 3.61423 0.149303
\(587\) 9.23673 0.381241 0.190620 0.981664i \(-0.438950\pi\)
0.190620 + 0.981664i \(0.438950\pi\)
\(588\) −8.76717 −0.361552
\(589\) 3.63057 0.149595
\(590\) −5.61953 −0.231353
\(591\) −22.5649 −0.928198
\(592\) 4.61847 0.189818
\(593\) −25.1196 −1.03154 −0.515770 0.856727i \(-0.672494\pi\)
−0.515770 + 0.856727i \(0.672494\pi\)
\(594\) 3.19910 0.131261
\(595\) 6.12559 0.251125
\(596\) 15.1049 0.618719
\(597\) 16.8334 0.688944
\(598\) −18.3035 −0.748485
\(599\) −25.5580 −1.04427 −0.522136 0.852862i \(-0.674865\pi\)
−0.522136 + 0.852862i \(0.674865\pi\)
\(600\) −4.48982 −0.183296
\(601\) −39.8088 −1.62384 −0.811918 0.583771i \(-0.801577\pi\)
−0.811918 + 0.583771i \(0.801577\pi\)
\(602\) 4.78192 0.194897
\(603\) −2.06118 −0.0839378
\(604\) 23.0486 0.937834
\(605\) −26.1872 −1.06466
\(606\) −6.51126 −0.264502
\(607\) 19.8694 0.806474 0.403237 0.915096i \(-0.367885\pi\)
0.403237 + 0.915096i \(0.367885\pi\)
\(608\) 7.46695 0.302825
\(609\) 6.85695 0.277858
\(610\) −10.8513 −0.439354
\(611\) 80.0851 3.23989
\(612\) −1.60768 −0.0649866
\(613\) −16.6254 −0.671494 −0.335747 0.941952i \(-0.608989\pi\)
−0.335747 + 0.941952i \(0.608989\pi\)
\(614\) 16.8297 0.679191
\(615\) 10.0452 0.405063
\(616\) 40.7284 1.64100
\(617\) 18.9369 0.762371 0.381186 0.924499i \(-0.375516\pi\)
0.381186 + 0.924499i \(0.375516\pi\)
\(618\) 1.94575 0.0782695
\(619\) −25.1054 −1.00907 −0.504535 0.863391i \(-0.668336\pi\)
−0.504535 + 0.863391i \(0.668336\pi\)
\(620\) −7.66196 −0.307712
\(621\) −4.67299 −0.187521
\(622\) −3.96846 −0.159121
\(623\) 50.9819 2.04255
\(624\) −11.2561 −0.450604
\(625\) −11.1176 −0.444703
\(626\) 5.31609 0.212474
\(627\) 6.75377 0.269720
\(628\) 25.0830 1.00092
\(629\) 2.56583 0.102306
\(630\) −3.83680 −0.152862
\(631\) −37.8868 −1.50825 −0.754125 0.656731i \(-0.771939\pi\)
−0.754125 + 0.656731i \(0.771939\pi\)
\(632\) 2.25969 0.0898856
\(633\) −0.565736 −0.0224860
\(634\) 10.2012 0.405140
\(635\) 29.3395 1.16430
\(636\) 6.58239 0.261009
\(637\) −34.1018 −1.35116
\(638\) −6.21607 −0.246097
\(639\) 7.20147 0.284886
\(640\) −19.6723 −0.777616
\(641\) 27.7198 1.09487 0.547434 0.836849i \(-0.315605\pi\)
0.547434 + 0.836849i \(0.315605\pi\)
\(642\) −6.42652 −0.253634
\(643\) −25.7562 −1.01572 −0.507862 0.861438i \(-0.669564\pi\)
−0.507862 + 0.861438i \(0.669564\pi\)
\(644\) −26.5116 −1.04471
\(645\) −3.75530 −0.147865
\(646\) 0.828249 0.0325870
\(647\) −0.609702 −0.0239698 −0.0119849 0.999928i \(-0.503815\pi\)
−0.0119849 + 0.999928i \(0.503815\pi\)
\(648\) 2.25969 0.0887690
\(649\) 26.3986 1.03623
\(650\) −7.78249 −0.305254
\(651\) 9.68897 0.379741
\(652\) 5.75609 0.225426
\(653\) −9.53439 −0.373109 −0.186555 0.982445i \(-0.559732\pi\)
−0.186555 + 0.982445i \(0.559732\pi\)
\(654\) 1.81104 0.0708173
\(655\) 5.01296 0.195873
\(656\) −10.4166 −0.406698
\(657\) 10.0866 0.393517
\(658\) −28.3072 −1.10353
\(659\) 32.3470 1.26006 0.630030 0.776571i \(-0.283043\pi\)
0.630030 + 0.776571i \(0.283043\pi\)
\(660\) −14.2532 −0.554804
\(661\) −0.160299 −0.00623489 −0.00311745 0.999995i \(-0.500992\pi\)
−0.00311745 + 0.999995i \(0.500992\pi\)
\(662\) 1.08971 0.0423528
\(663\) −6.25342 −0.242863
\(664\) 37.7262 1.46406
\(665\) −8.10005 −0.314107
\(666\) −1.60712 −0.0622748
\(667\) 9.07995 0.351577
\(668\) −18.8727 −0.730205
\(669\) 3.71956 0.143807
\(670\) −2.24100 −0.0865775
\(671\) 50.9753 1.96788
\(672\) 19.9272 0.768707
\(673\) −15.1200 −0.582835 −0.291417 0.956596i \(-0.594127\pi\)
−0.291417 + 0.956596i \(0.594127\pi\)
\(674\) 15.9193 0.613188
\(675\) −1.98692 −0.0764765
\(676\) −41.9689 −1.61419
\(677\) 2.69920 0.103739 0.0518694 0.998654i \(-0.483482\pi\)
0.0518694 + 0.998654i \(0.483482\pi\)
\(678\) −1.17267 −0.0450361
\(679\) 15.1970 0.583207
\(680\) −3.92242 −0.150418
\(681\) −12.1820 −0.466816
\(682\) −8.78340 −0.336334
\(683\) 20.0421 0.766890 0.383445 0.923564i \(-0.374738\pi\)
0.383445 + 0.923564i \(0.374738\pi\)
\(684\) 2.12588 0.0812851
\(685\) −7.69304 −0.293936
\(686\) −3.41874 −0.130528
\(687\) 8.01590 0.305826
\(688\) 3.89411 0.148462
\(689\) 25.6037 0.975421
\(690\) −5.08068 −0.193418
\(691\) 25.6849 0.977099 0.488550 0.872536i \(-0.337526\pi\)
0.488550 + 0.872536i \(0.337526\pi\)
\(692\) 10.8102 0.410942
\(693\) 18.0239 0.684672
\(694\) −5.46978 −0.207630
\(695\) −3.80243 −0.144234
\(696\) −4.39074 −0.166430
\(697\) −5.78701 −0.219199
\(698\) 9.45873 0.358018
\(699\) 6.77495 0.256252
\(700\) −11.2725 −0.426062
\(701\) 48.5394 1.83331 0.916654 0.399682i \(-0.130879\pi\)
0.916654 + 0.399682i \(0.130879\pi\)
\(702\) 3.91686 0.147833
\(703\) −3.39288 −0.127965
\(704\) 0.322097 0.0121395
\(705\) 22.2300 0.837230
\(706\) −11.2054 −0.421719
\(707\) −36.6848 −1.37967
\(708\) 8.30947 0.312289
\(709\) −8.73886 −0.328195 −0.164097 0.986444i \(-0.552471\pi\)
−0.164097 + 0.986444i \(0.552471\pi\)
\(710\) 7.82975 0.293845
\(711\) 1.00000 0.0375029
\(712\) −32.6454 −1.22344
\(713\) 12.8301 0.480491
\(714\) 2.21036 0.0827207
\(715\) −55.4408 −2.07337
\(716\) −37.0129 −1.38324
\(717\) 19.1256 0.714259
\(718\) 9.34521 0.348760
\(719\) −30.7778 −1.14782 −0.573910 0.818919i \(-0.694574\pi\)
−0.573910 + 0.818919i \(0.694574\pi\)
\(720\) −3.12446 −0.116442
\(721\) 10.9625 0.408263
\(722\) 10.8055 0.402140
\(723\) 11.5938 0.431178
\(724\) −0.121351 −0.00450999
\(725\) 3.86072 0.143384
\(726\) −9.44941 −0.350700
\(727\) 53.6437 1.98954 0.994768 0.102163i \(-0.0325763\pi\)
0.994768 + 0.102163i \(0.0325763\pi\)
\(728\) 49.8665 1.84818
\(729\) 1.00000 0.0370370
\(730\) 10.9666 0.405893
\(731\) 2.16341 0.0800166
\(732\) 16.0455 0.593058
\(733\) −36.9991 −1.36659 −0.683297 0.730141i \(-0.739454\pi\)
−0.683297 + 0.730141i \(0.739454\pi\)
\(734\) −13.5947 −0.501791
\(735\) −9.46599 −0.349158
\(736\) 26.3875 0.972657
\(737\) 10.5274 0.387783
\(738\) 3.62473 0.133428
\(739\) −31.6269 −1.16341 −0.581707 0.813399i \(-0.697615\pi\)
−0.581707 + 0.813399i \(0.697615\pi\)
\(740\) 7.16033 0.263219
\(741\) 8.26908 0.303772
\(742\) −9.04998 −0.332235
\(743\) 0.954026 0.0349998 0.0174999 0.999847i \(-0.494429\pi\)
0.0174999 + 0.999847i \(0.494429\pi\)
\(744\) −6.20417 −0.227456
\(745\) 16.3088 0.597510
\(746\) 19.8089 0.725257
\(747\) 16.6953 0.610850
\(748\) 8.21118 0.300231
\(749\) −36.2074 −1.32299
\(750\) −7.59647 −0.277384
\(751\) 18.0772 0.659647 0.329823 0.944043i \(-0.393011\pi\)
0.329823 + 0.944043i \(0.393011\pi\)
\(752\) −23.0517 −0.840610
\(753\) 4.85890 0.177068
\(754\) −7.61074 −0.277167
\(755\) 24.8858 0.905685
\(756\) 5.67338 0.206339
\(757\) −13.5096 −0.491016 −0.245508 0.969395i \(-0.578955\pi\)
−0.245508 + 0.969395i \(0.578955\pi\)
\(758\) 0.262630 0.00953914
\(759\) 23.8672 0.866325
\(760\) 5.18674 0.188143
\(761\) −37.8935 −1.37364 −0.686819 0.726829i \(-0.740993\pi\)
−0.686819 + 0.726829i \(0.740993\pi\)
\(762\) 10.5869 0.383522
\(763\) 10.2035 0.369392
\(764\) 26.1244 0.945149
\(765\) −1.73582 −0.0627588
\(766\) −23.0516 −0.832889
\(767\) 32.3215 1.16706
\(768\) −6.97244 −0.251596
\(769\) 39.1108 1.41037 0.705185 0.709023i \(-0.250864\pi\)
0.705185 + 0.709023i \(0.250864\pi\)
\(770\) 19.5964 0.706204
\(771\) 21.2911 0.766780
\(772\) 26.3564 0.948586
\(773\) −31.4211 −1.13014 −0.565070 0.825043i \(-0.691151\pi\)
−0.565070 + 0.825043i \(0.691151\pi\)
\(774\) −1.35506 −0.0487068
\(775\) 5.45525 0.195958
\(776\) −9.73115 −0.349328
\(777\) −9.05463 −0.324833
\(778\) 13.3515 0.478673
\(779\) 7.65234 0.274174
\(780\) −17.4511 −0.624849
\(781\) −36.7813 −1.31614
\(782\) 2.92695 0.104668
\(783\) −1.94307 −0.0694397
\(784\) 9.81590 0.350568
\(785\) 27.0823 0.966607
\(786\) 1.80888 0.0645206
\(787\) 37.7518 1.34571 0.672854 0.739776i \(-0.265068\pi\)
0.672854 + 0.739776i \(0.265068\pi\)
\(788\) 36.2772 1.29232
\(789\) 15.1049 0.537747
\(790\) 1.08724 0.0386824
\(791\) −6.60689 −0.234914
\(792\) −11.5413 −0.410103
\(793\) 62.4124 2.21633
\(794\) −23.7789 −0.843881
\(795\) 7.10706 0.252061
\(796\) −27.0627 −0.959211
\(797\) −34.8148 −1.23320 −0.616601 0.787276i \(-0.711491\pi\)
−0.616601 + 0.787276i \(0.711491\pi\)
\(798\) −2.92283 −0.103467
\(799\) −12.8066 −0.453065
\(800\) 11.2197 0.396678
\(801\) −14.4469 −0.510455
\(802\) 3.63472 0.128347
\(803\) −51.5173 −1.81800
\(804\) 3.31372 0.116866
\(805\) −28.6248 −1.00889
\(806\) −10.7541 −0.378797
\(807\) −8.22219 −0.289435
\(808\) 23.4905 0.826394
\(809\) 42.9762 1.51096 0.755481 0.655171i \(-0.227403\pi\)
0.755481 + 0.655171i \(0.227403\pi\)
\(810\) 1.08724 0.0382018
\(811\) −29.2375 −1.02667 −0.513334 0.858189i \(-0.671590\pi\)
−0.513334 + 0.858189i \(0.671590\pi\)
\(812\) −11.0238 −0.386859
\(813\) −9.72698 −0.341140
\(814\) 8.20835 0.287702
\(815\) 6.21490 0.217698
\(816\) 1.79999 0.0630122
\(817\) −2.86074 −0.100085
\(818\) 11.3554 0.397034
\(819\) 22.0679 0.771113
\(820\) −16.1495 −0.563965
\(821\) 8.94659 0.312238 0.156119 0.987738i \(-0.450102\pi\)
0.156119 + 0.987738i \(0.450102\pi\)
\(822\) −2.77596 −0.0968227
\(823\) −22.4428 −0.782308 −0.391154 0.920325i \(-0.627924\pi\)
−0.391154 + 0.920325i \(0.627924\pi\)
\(824\) −7.01964 −0.244541
\(825\) 10.1481 0.353313
\(826\) −11.4245 −0.397509
\(827\) −16.2648 −0.565582 −0.282791 0.959182i \(-0.591260\pi\)
−0.282791 + 0.959182i \(0.591260\pi\)
\(828\) 7.51267 0.261083
\(829\) 17.9406 0.623104 0.311552 0.950229i \(-0.399151\pi\)
0.311552 + 0.950229i \(0.399151\pi\)
\(830\) 18.1519 0.630060
\(831\) 7.03106 0.243905
\(832\) 0.394365 0.0136721
\(833\) 5.45331 0.188946
\(834\) −1.37207 −0.0475109
\(835\) −20.3770 −0.705174
\(836\) −10.8579 −0.375528
\(837\) −2.74559 −0.0949013
\(838\) 8.80260 0.304081
\(839\) −21.0534 −0.726844 −0.363422 0.931625i \(-0.618392\pi\)
−0.363422 + 0.931625i \(0.618392\pi\)
\(840\) 13.8419 0.477592
\(841\) −25.2245 −0.869810
\(842\) 6.21851 0.214304
\(843\) −18.2953 −0.630124
\(844\) 0.909522 0.0313070
\(845\) −45.3141 −1.55885
\(846\) 8.02149 0.275784
\(847\) −53.2385 −1.82930
\(848\) −7.36977 −0.253079
\(849\) −24.7438 −0.849204
\(850\) 1.24452 0.0426866
\(851\) −11.9901 −0.411016
\(852\) −11.5777 −0.396644
\(853\) −27.1058 −0.928086 −0.464043 0.885813i \(-0.653602\pi\)
−0.464043 + 0.885813i \(0.653602\pi\)
\(854\) −22.0606 −0.754897
\(855\) 2.29533 0.0784987
\(856\) 23.1848 0.792441
\(857\) −20.2780 −0.692683 −0.346341 0.938109i \(-0.612576\pi\)
−0.346341 + 0.938109i \(0.612576\pi\)
\(858\) −20.0053 −0.682970
\(859\) 46.5813 1.58933 0.794666 0.607047i \(-0.207646\pi\)
0.794666 + 0.607047i \(0.207646\pi\)
\(860\) 6.03731 0.205871
\(861\) 20.4219 0.695978
\(862\) 14.7030 0.500786
\(863\) 13.4056 0.456331 0.228166 0.973622i \(-0.426727\pi\)
0.228166 + 0.973622i \(0.426727\pi\)
\(864\) −5.64681 −0.192108
\(865\) 11.6718 0.396855
\(866\) 15.9162 0.540856
\(867\) 1.00000 0.0339618
\(868\) −15.5767 −0.528709
\(869\) −5.10748 −0.173259
\(870\) −2.11259 −0.0716235
\(871\) 12.8894 0.436742
\(872\) −6.53365 −0.221257
\(873\) −4.30641 −0.145750
\(874\) −3.87040 −0.130918
\(875\) −42.7990 −1.44687
\(876\) −16.2161 −0.547890
\(877\) 51.3560 1.73417 0.867084 0.498162i \(-0.165991\pi\)
0.867084 + 0.498162i \(0.165991\pi\)
\(878\) −16.6157 −0.560754
\(879\) −5.77026 −0.194626
\(880\) 15.9581 0.537948
\(881\) −26.8889 −0.905911 −0.452955 0.891533i \(-0.649630\pi\)
−0.452955 + 0.891533i \(0.649630\pi\)
\(882\) −3.41571 −0.115013
\(883\) −18.6065 −0.626160 −0.313080 0.949727i \(-0.601361\pi\)
−0.313080 + 0.949727i \(0.601361\pi\)
\(884\) 10.0535 0.338135
\(885\) 8.97180 0.301584
\(886\) 21.6601 0.727684
\(887\) −27.9580 −0.938738 −0.469369 0.883002i \(-0.655519\pi\)
−0.469369 + 0.883002i \(0.655519\pi\)
\(888\) 5.79799 0.194568
\(889\) 59.6472 2.00050
\(890\) −15.7072 −0.526508
\(891\) −5.10748 −0.171107
\(892\) −5.97986 −0.200220
\(893\) 16.9346 0.566693
\(894\) 5.88489 0.196820
\(895\) −39.9631 −1.33582
\(896\) −39.9937 −1.33610
\(897\) 29.2222 0.975700
\(898\) −16.0132 −0.534367
\(899\) 5.33487 0.177928
\(900\) 3.19432 0.106477
\(901\) −4.09434 −0.136402
\(902\) −18.5132 −0.616423
\(903\) −7.63451 −0.254061
\(904\) 4.23062 0.140708
\(905\) −0.131024 −0.00435539
\(906\) 8.97979 0.298334
\(907\) −21.6435 −0.718660 −0.359330 0.933211i \(-0.616995\pi\)
−0.359330 + 0.933211i \(0.616995\pi\)
\(908\) 19.5848 0.649944
\(909\) 10.3955 0.344796
\(910\) 23.9931 0.795364
\(911\) −40.7545 −1.35026 −0.675128 0.737700i \(-0.735912\pi\)
−0.675128 + 0.737700i \(0.735912\pi\)
\(912\) −2.38018 −0.0788156
\(913\) −85.2709 −2.82206
\(914\) 18.2733 0.604428
\(915\) 17.3244 0.572728
\(916\) −12.8870 −0.425798
\(917\) 10.1913 0.336547
\(918\) −0.626356 −0.0206728
\(919\) −39.4323 −1.30075 −0.650376 0.759612i \(-0.725389\pi\)
−0.650376 + 0.759612i \(0.725389\pi\)
\(920\) 18.3294 0.604304
\(921\) −26.8692 −0.885371
\(922\) 2.51098 0.0826948
\(923\) −45.0338 −1.48231
\(924\) −28.9766 −0.953262
\(925\) −5.09810 −0.167624
\(926\) 17.2638 0.567325
\(927\) −3.10646 −0.102030
\(928\) 10.9722 0.360179
\(929\) −13.9715 −0.458389 −0.229195 0.973381i \(-0.573609\pi\)
−0.229195 + 0.973381i \(0.573609\pi\)
\(930\) −2.98512 −0.0978859
\(931\) −7.21108 −0.236333
\(932\) −10.8919 −0.356778
\(933\) 6.33580 0.207425
\(934\) −1.10743 −0.0362364
\(935\) 8.86568 0.289939
\(936\) −14.1308 −0.461879
\(937\) 30.7648 1.00504 0.502521 0.864565i \(-0.332406\pi\)
0.502521 + 0.864565i \(0.332406\pi\)
\(938\) −4.55596 −0.148757
\(939\) −8.48733 −0.276974
\(940\) −35.7387 −1.16567
\(941\) 14.9333 0.486810 0.243405 0.969925i \(-0.421736\pi\)
0.243405 + 0.969925i \(0.421736\pi\)
\(942\) 9.77238 0.318401
\(943\) 27.0427 0.880631
\(944\) −9.30344 −0.302801
\(945\) 6.12559 0.199265
\(946\) 6.92096 0.225020
\(947\) −31.9890 −1.03950 −0.519751 0.854318i \(-0.673975\pi\)
−0.519751 + 0.854318i \(0.673975\pi\)
\(948\) −1.60768 −0.0522150
\(949\) −63.0760 −2.04753
\(950\) −1.64566 −0.0533923
\(951\) −16.2865 −0.528127
\(952\) −7.97428 −0.258448
\(953\) −10.5114 −0.340499 −0.170250 0.985401i \(-0.554457\pi\)
−0.170250 + 0.985401i \(0.554457\pi\)
\(954\) 2.56452 0.0830292
\(955\) 28.2068 0.912749
\(956\) −30.7478 −0.994455
\(957\) 9.92419 0.320804
\(958\) −6.10073 −0.197106
\(959\) −15.6399 −0.505039
\(960\) 0.109468 0.00353306
\(961\) −23.4618 −0.756831
\(962\) 10.0500 0.324026
\(963\) 10.2602 0.330630
\(964\) −18.6391 −0.600325
\(965\) 28.4572 0.916069
\(966\) −10.3290 −0.332330
\(967\) 2.41004 0.0775016 0.0387508 0.999249i \(-0.487662\pi\)
0.0387508 + 0.999249i \(0.487662\pi\)
\(968\) 34.0904 1.09571
\(969\) −1.32233 −0.0424794
\(970\) −4.68211 −0.150334
\(971\) −54.5068 −1.74921 −0.874603 0.484839i \(-0.838878\pi\)
−0.874603 + 0.484839i \(0.838878\pi\)
\(972\) −1.60768 −0.0515663
\(973\) −7.73033 −0.247823
\(974\) 4.09191 0.131113
\(975\) 12.4250 0.397919
\(976\) −17.9648 −0.575040
\(977\) −55.2952 −1.76905 −0.884526 0.466492i \(-0.845518\pi\)
−0.884526 + 0.466492i \(0.845518\pi\)
\(978\) 2.24259 0.0717100
\(979\) 73.7870 2.35824
\(980\) 15.2183 0.486130
\(981\) −2.89139 −0.0923151
\(982\) −20.2287 −0.645523
\(983\) −15.3859 −0.490734 −0.245367 0.969430i \(-0.578908\pi\)
−0.245367 + 0.969430i \(0.578908\pi\)
\(984\) −13.0769 −0.416875
\(985\) 39.1688 1.24802
\(986\) 1.21705 0.0387589
\(987\) 45.1935 1.43853
\(988\) −13.2940 −0.422939
\(989\) −10.1096 −0.321467
\(990\) −5.55307 −0.176488
\(991\) −18.3831 −0.583958 −0.291979 0.956425i \(-0.594314\pi\)
−0.291979 + 0.956425i \(0.594314\pi\)
\(992\) 15.5038 0.492246
\(993\) −1.73976 −0.0552097
\(994\) 15.9179 0.504884
\(995\) −29.2198 −0.926329
\(996\) −26.8407 −0.850480
\(997\) −32.4101 −1.02644 −0.513220 0.858257i \(-0.671547\pi\)
−0.513220 + 0.858257i \(0.671547\pi\)
\(998\) 12.8599 0.407074
\(999\) 2.56583 0.0811794
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.1.13 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.13 31 1.1 even 1 trivial