Properties

Label 4027.2.a.c.1.12
Level 4027
Weight 2
Character 4027.1
Self dual yes
Analytic conductor 32.156
Analytic rank 0
Dimension 174
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4027 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4027.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1557568940\)
Analytic rank: \(0\)
Dimension: \(174\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) = 4027.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.50780 q^{2} +2.09202 q^{3} +4.28906 q^{4} -3.15733 q^{5} -5.24636 q^{6} -2.77421 q^{7} -5.74049 q^{8} +1.37654 q^{9} +O(q^{10})\) \(q-2.50780 q^{2} +2.09202 q^{3} +4.28906 q^{4} -3.15733 q^{5} -5.24636 q^{6} -2.77421 q^{7} -5.74049 q^{8} +1.37654 q^{9} +7.91796 q^{10} +2.37154 q^{11} +8.97279 q^{12} +1.38971 q^{13} +6.95717 q^{14} -6.60520 q^{15} +5.81789 q^{16} -2.15703 q^{17} -3.45210 q^{18} -2.24645 q^{19} -13.5420 q^{20} -5.80370 q^{21} -5.94735 q^{22} +5.80938 q^{23} -12.0092 q^{24} +4.96876 q^{25} -3.48512 q^{26} -3.39630 q^{27} -11.8987 q^{28} -4.81200 q^{29} +16.5645 q^{30} -5.81211 q^{31} -3.10911 q^{32} +4.96131 q^{33} +5.40941 q^{34} +8.75911 q^{35} +5.90408 q^{36} -9.46646 q^{37} +5.63364 q^{38} +2.90730 q^{39} +18.1247 q^{40} +8.95246 q^{41} +14.5545 q^{42} +0.699540 q^{43} +10.1717 q^{44} -4.34621 q^{45} -14.5687 q^{46} +11.6297 q^{47} +12.1711 q^{48} +0.696251 q^{49} -12.4607 q^{50} -4.51256 q^{51} +5.96055 q^{52} +5.27972 q^{53} +8.51724 q^{54} -7.48775 q^{55} +15.9253 q^{56} -4.69961 q^{57} +12.0675 q^{58} -8.25919 q^{59} -28.3301 q^{60} -0.889360 q^{61} +14.5756 q^{62} -3.81883 q^{63} -3.83875 q^{64} -4.38778 q^{65} -12.4420 q^{66} +8.64543 q^{67} -9.25164 q^{68} +12.1533 q^{69} -21.9661 q^{70} -5.37954 q^{71} -7.90204 q^{72} -15.2353 q^{73} +23.7400 q^{74} +10.3947 q^{75} -9.63514 q^{76} -6.57916 q^{77} -7.29093 q^{78} +13.3694 q^{79} -18.3690 q^{80} -11.2348 q^{81} -22.4510 q^{82} -10.6453 q^{83} -24.8924 q^{84} +6.81048 q^{85} -1.75431 q^{86} -10.0668 q^{87} -13.6138 q^{88} +3.24014 q^{89} +10.8994 q^{90} -3.85535 q^{91} +24.9167 q^{92} -12.1590 q^{93} -29.1649 q^{94} +7.09279 q^{95} -6.50432 q^{96} -8.37419 q^{97} -1.74606 q^{98} +3.26453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 174q + 21q^{2} + 17q^{3} + 187q^{4} + 72q^{5} + 21q^{6} + 24q^{7} + 54q^{8} + 197q^{9} + O(q^{10}) \) \( 174q + 21q^{2} + 17q^{3} + 187q^{4} + 72q^{5} + 21q^{6} + 24q^{7} + 54q^{8} + 197q^{9} + 20q^{10} + 35q^{11} + 23q^{12} + 91q^{13} + 18q^{14} + 16q^{15} + 201q^{16} + 148q^{17} + 39q^{18} + 36q^{19} + 128q^{20} + 57q^{21} + 17q^{22} + 96q^{23} + 24q^{24} + 226q^{25} + 44q^{26} + 62q^{27} + 32q^{28} + 122q^{29} + 25q^{30} + 23q^{31} + 104q^{32} + 91q^{33} + 6q^{34} + 80q^{35} + 222q^{36} + 71q^{37} + 125q^{38} + 16q^{39} + 53q^{40} + 97q^{41} + 14q^{42} + 38q^{43} + 70q^{44} + 185q^{45} - 23q^{46} + 110q^{47} + 36q^{48} + 210q^{49} + 51q^{50} + 33q^{51} + 118q^{52} + 214q^{53} + 8q^{54} + 37q^{55} + 41q^{56} + 76q^{57} + 2q^{58} + 66q^{59} - 12q^{60} + 114q^{61} + 175q^{62} + 62q^{63} + 190q^{64} + 128q^{65} + 12q^{66} - 6q^{67} + 348q^{68} + 115q^{69} - 38q^{70} + 54q^{71} + 101q^{72} + 107q^{73} + 71q^{74} - q^{75} + 31q^{76} + 368q^{77} - 14q^{78} - 14q^{79} + 205q^{80} + 222q^{81} + 26q^{82} + 246q^{83} + 41q^{84} + 87q^{85} + 33q^{86} + 100q^{87} - 6q^{88} + 147q^{89} + 50q^{90} - 23q^{91} + 189q^{92} + 117q^{93} + 23q^{94} + 42q^{95} + 38q^{96} + 52q^{97} + 148q^{98} + 38q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50780 −1.77328 −0.886641 0.462459i \(-0.846967\pi\)
−0.886641 + 0.462459i \(0.846967\pi\)
\(3\) 2.09202 1.20783 0.603914 0.797050i \(-0.293607\pi\)
0.603914 + 0.797050i \(0.293607\pi\)
\(4\) 4.28906 2.14453
\(5\) −3.15733 −1.41200 −0.706001 0.708210i \(-0.749503\pi\)
−0.706001 + 0.708210i \(0.749503\pi\)
\(6\) −5.24636 −2.14182
\(7\) −2.77421 −1.04855 −0.524277 0.851548i \(-0.675664\pi\)
−0.524277 + 0.851548i \(0.675664\pi\)
\(8\) −5.74049 −2.02957
\(9\) 1.37654 0.458848
\(10\) 7.91796 2.50388
\(11\) 2.37154 0.715046 0.357523 0.933904i \(-0.383621\pi\)
0.357523 + 0.933904i \(0.383621\pi\)
\(12\) 8.97279 2.59022
\(13\) 1.38971 0.385437 0.192718 0.981254i \(-0.438270\pi\)
0.192718 + 0.981254i \(0.438270\pi\)
\(14\) 6.95717 1.85938
\(15\) −6.60520 −1.70546
\(16\) 5.81789 1.45447
\(17\) −2.15703 −0.523158 −0.261579 0.965182i \(-0.584243\pi\)
−0.261579 + 0.965182i \(0.584243\pi\)
\(18\) −3.45210 −0.813667
\(19\) −2.24645 −0.515371 −0.257685 0.966229i \(-0.582960\pi\)
−0.257685 + 0.966229i \(0.582960\pi\)
\(20\) −13.5420 −3.02808
\(21\) −5.80370 −1.26647
\(22\) −5.94735 −1.26798
\(23\) 5.80938 1.21134 0.605669 0.795716i \(-0.292905\pi\)
0.605669 + 0.795716i \(0.292905\pi\)
\(24\) −12.0092 −2.45137
\(25\) 4.96876 0.993752
\(26\) −3.48512 −0.683488
\(27\) −3.39630 −0.653618
\(28\) −11.8987 −2.24865
\(29\) −4.81200 −0.893566 −0.446783 0.894642i \(-0.647430\pi\)
−0.446783 + 0.894642i \(0.647430\pi\)
\(30\) 16.5645 3.02425
\(31\) −5.81211 −1.04389 −0.521943 0.852981i \(-0.674793\pi\)
−0.521943 + 0.852981i \(0.674793\pi\)
\(32\) −3.10911 −0.549618
\(33\) 4.96131 0.863653
\(34\) 5.40941 0.927706
\(35\) 8.75911 1.48056
\(36\) 5.90408 0.984013
\(37\) −9.46646 −1.55628 −0.778138 0.628093i \(-0.783836\pi\)
−0.778138 + 0.628093i \(0.783836\pi\)
\(38\) 5.63364 0.913897
\(39\) 2.90730 0.465541
\(40\) 18.1247 2.86576
\(41\) 8.95246 1.39814 0.699069 0.715054i \(-0.253598\pi\)
0.699069 + 0.715054i \(0.253598\pi\)
\(42\) 14.5545 2.24581
\(43\) 0.699540 0.106679 0.0533394 0.998576i \(-0.483013\pi\)
0.0533394 + 0.998576i \(0.483013\pi\)
\(44\) 10.1717 1.53344
\(45\) −4.34621 −0.647895
\(46\) −14.5687 −2.14804
\(47\) 11.6297 1.69637 0.848183 0.529703i \(-0.177697\pi\)
0.848183 + 0.529703i \(0.177697\pi\)
\(48\) 12.1711 1.75675
\(49\) 0.696251 0.0994644
\(50\) −12.4607 −1.76220
\(51\) −4.51256 −0.631885
\(52\) 5.96055 0.826579
\(53\) 5.27972 0.725225 0.362612 0.931940i \(-0.381885\pi\)
0.362612 + 0.931940i \(0.381885\pi\)
\(54\) 8.51724 1.15905
\(55\) −7.48775 −1.00965
\(56\) 15.9253 2.12811
\(57\) −4.69961 −0.622479
\(58\) 12.0675 1.58454
\(59\) −8.25919 −1.07525 −0.537627 0.843183i \(-0.680679\pi\)
−0.537627 + 0.843183i \(0.680679\pi\)
\(60\) −28.3301 −3.65740
\(61\) −0.889360 −0.113871 −0.0569355 0.998378i \(-0.518133\pi\)
−0.0569355 + 0.998378i \(0.518133\pi\)
\(62\) 14.5756 1.85110
\(63\) −3.81883 −0.481127
\(64\) −3.83875 −0.479844
\(65\) −4.38778 −0.544238
\(66\) −12.4420 −1.53150
\(67\) 8.64543 1.05621 0.528103 0.849180i \(-0.322903\pi\)
0.528103 + 0.849180i \(0.322903\pi\)
\(68\) −9.25164 −1.12193
\(69\) 12.1533 1.46309
\(70\) −21.9661 −2.62545
\(71\) −5.37954 −0.638433 −0.319217 0.947682i \(-0.603420\pi\)
−0.319217 + 0.947682i \(0.603420\pi\)
\(72\) −7.90204 −0.931265
\(73\) −15.2353 −1.78316 −0.891579 0.452866i \(-0.850402\pi\)
−0.891579 + 0.452866i \(0.850402\pi\)
\(74\) 23.7400 2.75972
\(75\) 10.3947 1.20028
\(76\) −9.63514 −1.10523
\(77\) −6.57916 −0.749764
\(78\) −7.29093 −0.825535
\(79\) 13.3694 1.50417 0.752086 0.659065i \(-0.229048\pi\)
0.752086 + 0.659065i \(0.229048\pi\)
\(80\) −18.3690 −2.05372
\(81\) −11.2348 −1.24831
\(82\) −22.4510 −2.47929
\(83\) −10.6453 −1.16847 −0.584237 0.811583i \(-0.698606\pi\)
−0.584237 + 0.811583i \(0.698606\pi\)
\(84\) −24.8924 −2.71598
\(85\) 6.81048 0.738700
\(86\) −1.75431 −0.189172
\(87\) −10.0668 −1.07927
\(88\) −13.6138 −1.45124
\(89\) 3.24014 0.343454 0.171727 0.985145i \(-0.445065\pi\)
0.171727 + 0.985145i \(0.445065\pi\)
\(90\) 10.8994 1.14890
\(91\) −3.85535 −0.404151
\(92\) 24.9167 2.59775
\(93\) −12.1590 −1.26083
\(94\) −29.1649 −3.00813
\(95\) 7.09279 0.727705
\(96\) −6.50432 −0.663844
\(97\) −8.37419 −0.850270 −0.425135 0.905130i \(-0.639773\pi\)
−0.425135 + 0.905130i \(0.639773\pi\)
\(98\) −1.74606 −0.176378
\(99\) 3.26453 0.328098
\(100\) 21.3113 2.13113
\(101\) 11.1897 1.11341 0.556707 0.830709i \(-0.312064\pi\)
0.556707 + 0.830709i \(0.312064\pi\)
\(102\) 11.3166 1.12051
\(103\) 16.5627 1.63198 0.815988 0.578069i \(-0.196194\pi\)
0.815988 + 0.578069i \(0.196194\pi\)
\(104\) −7.97763 −0.782271
\(105\) 18.3242 1.78826
\(106\) −13.2405 −1.28603
\(107\) −16.9325 −1.63692 −0.818461 0.574562i \(-0.805173\pi\)
−0.818461 + 0.574562i \(0.805173\pi\)
\(108\) −14.5669 −1.40170
\(109\) 3.76552 0.360671 0.180336 0.983605i \(-0.442282\pi\)
0.180336 + 0.983605i \(0.442282\pi\)
\(110\) 18.7778 1.79039
\(111\) −19.8040 −1.87971
\(112\) −16.1401 −1.52509
\(113\) 8.84687 0.832244 0.416122 0.909309i \(-0.363389\pi\)
0.416122 + 0.909309i \(0.363389\pi\)
\(114\) 11.7857 1.10383
\(115\) −18.3421 −1.71041
\(116\) −20.6389 −1.91628
\(117\) 1.91300 0.176857
\(118\) 20.7124 1.90673
\(119\) 5.98407 0.548559
\(120\) 37.9171 3.46134
\(121\) −5.37579 −0.488709
\(122\) 2.23034 0.201925
\(123\) 18.7287 1.68871
\(124\) −24.9285 −2.23864
\(125\) 0.0986302 0.00882176
\(126\) 9.57685 0.853173
\(127\) 10.3795 0.921036 0.460518 0.887650i \(-0.347664\pi\)
0.460518 + 0.887650i \(0.347664\pi\)
\(128\) 15.8450 1.40052
\(129\) 1.46345 0.128850
\(130\) 11.0037 0.965086
\(131\) 12.5787 1.09901 0.549505 0.835490i \(-0.314816\pi\)
0.549505 + 0.835490i \(0.314816\pi\)
\(132\) 21.2793 1.85213
\(133\) 6.23212 0.540394
\(134\) −21.6810 −1.87295
\(135\) 10.7233 0.922911
\(136\) 12.3824 1.06179
\(137\) −18.2136 −1.55609 −0.778046 0.628208i \(-0.783789\pi\)
−0.778046 + 0.628208i \(0.783789\pi\)
\(138\) −30.4781 −2.59447
\(139\) 10.0012 0.848291 0.424146 0.905594i \(-0.360574\pi\)
0.424146 + 0.905594i \(0.360574\pi\)
\(140\) 37.5683 3.17510
\(141\) 24.3296 2.04892
\(142\) 13.4908 1.13212
\(143\) 3.29576 0.275605
\(144\) 8.00858 0.667382
\(145\) 15.1931 1.26172
\(146\) 38.2071 3.16204
\(147\) 1.45657 0.120136
\(148\) −40.6022 −3.33748
\(149\) 23.0912 1.89170 0.945851 0.324600i \(-0.105230\pi\)
0.945851 + 0.324600i \(0.105230\pi\)
\(150\) −26.0679 −2.12844
\(151\) −10.4148 −0.847548 −0.423774 0.905768i \(-0.639295\pi\)
−0.423774 + 0.905768i \(0.639295\pi\)
\(152\) 12.8957 1.04598
\(153\) −2.96925 −0.240050
\(154\) 16.4992 1.32954
\(155\) 18.3508 1.47397
\(156\) 12.4696 0.998366
\(157\) −4.71576 −0.376358 −0.188179 0.982135i \(-0.560259\pi\)
−0.188179 + 0.982135i \(0.560259\pi\)
\(158\) −33.5277 −2.66732
\(159\) 11.0453 0.875947
\(160\) 9.81650 0.776062
\(161\) −16.1164 −1.27015
\(162\) 28.1745 2.21360
\(163\) 1.42115 0.111313 0.0556564 0.998450i \(-0.482275\pi\)
0.0556564 + 0.998450i \(0.482275\pi\)
\(164\) 38.3976 2.99835
\(165\) −15.6645 −1.21948
\(166\) 26.6963 2.07203
\(167\) 17.8972 1.38493 0.692464 0.721452i \(-0.256525\pi\)
0.692464 + 0.721452i \(0.256525\pi\)
\(168\) 33.3161 2.57039
\(169\) −11.0687 −0.851439
\(170\) −17.0793 −1.30992
\(171\) −3.09234 −0.236477
\(172\) 3.00037 0.228776
\(173\) 19.0733 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(174\) 25.2455 1.91386
\(175\) −13.7844 −1.04200
\(176\) 13.7974 1.04001
\(177\) −17.2784 −1.29872
\(178\) −8.12562 −0.609041
\(179\) 14.9133 1.11467 0.557337 0.830286i \(-0.311823\pi\)
0.557337 + 0.830286i \(0.311823\pi\)
\(180\) −18.6411 −1.38943
\(181\) −3.64367 −0.270832 −0.135416 0.990789i \(-0.543237\pi\)
−0.135416 + 0.990789i \(0.543237\pi\)
\(182\) 9.66845 0.716673
\(183\) −1.86056 −0.137536
\(184\) −33.3487 −2.45850
\(185\) 29.8888 2.19747
\(186\) 30.4924 2.23581
\(187\) −5.11550 −0.374082
\(188\) 49.8804 3.63790
\(189\) 9.42205 0.685354
\(190\) −17.7873 −1.29043
\(191\) 8.60635 0.622734 0.311367 0.950290i \(-0.399213\pi\)
0.311367 + 0.950290i \(0.399213\pi\)
\(192\) −8.03075 −0.579569
\(193\) 20.6861 1.48902 0.744509 0.667612i \(-0.232684\pi\)
0.744509 + 0.667612i \(0.232684\pi\)
\(194\) 21.0008 1.50777
\(195\) −9.17933 −0.657345
\(196\) 2.98626 0.213304
\(197\) 19.4813 1.38798 0.693992 0.719983i \(-0.255850\pi\)
0.693992 + 0.719983i \(0.255850\pi\)
\(198\) −8.18679 −0.581810
\(199\) −3.70541 −0.262670 −0.131335 0.991338i \(-0.541926\pi\)
−0.131335 + 0.991338i \(0.541926\pi\)
\(200\) −28.5231 −2.01689
\(201\) 18.0864 1.27572
\(202\) −28.0615 −1.97440
\(203\) 13.3495 0.936952
\(204\) −19.3546 −1.35509
\(205\) −28.2659 −1.97418
\(206\) −41.5360 −2.89395
\(207\) 7.99687 0.555821
\(208\) 8.08518 0.560607
\(209\) −5.32754 −0.368514
\(210\) −45.9535 −3.17109
\(211\) −5.77506 −0.397571 −0.198786 0.980043i \(-0.563700\pi\)
−0.198786 + 0.980043i \(0.563700\pi\)
\(212\) 22.6450 1.55526
\(213\) −11.2541 −0.771118
\(214\) 42.4632 2.90273
\(215\) −2.20868 −0.150631
\(216\) 19.4964 1.32656
\(217\) 16.1240 1.09457
\(218\) −9.44316 −0.639572
\(219\) −31.8725 −2.15375
\(220\) −32.1154 −2.16522
\(221\) −2.99766 −0.201644
\(222\) 49.6645 3.33326
\(223\) 3.59749 0.240906 0.120453 0.992719i \(-0.461565\pi\)
0.120453 + 0.992719i \(0.461565\pi\)
\(224\) 8.62533 0.576304
\(225\) 6.83972 0.455981
\(226\) −22.1862 −1.47580
\(227\) 8.91046 0.591408 0.295704 0.955280i \(-0.404446\pi\)
0.295704 + 0.955280i \(0.404446\pi\)
\(228\) −20.1569 −1.33492
\(229\) −18.7226 −1.23723 −0.618613 0.785696i \(-0.712305\pi\)
−0.618613 + 0.785696i \(0.712305\pi\)
\(230\) 45.9984 3.03305
\(231\) −13.7637 −0.905586
\(232\) 27.6233 1.81356
\(233\) 13.3786 0.876463 0.438231 0.898862i \(-0.355605\pi\)
0.438231 + 0.898862i \(0.355605\pi\)
\(234\) −4.79742 −0.313617
\(235\) −36.7188 −2.39527
\(236\) −35.4241 −2.30591
\(237\) 27.9690 1.81678
\(238\) −15.0068 −0.972750
\(239\) −16.4951 −1.06698 −0.533491 0.845806i \(-0.679120\pi\)
−0.533491 + 0.845806i \(0.679120\pi\)
\(240\) −38.4283 −2.48054
\(241\) −0.552647 −0.0355991 −0.0177996 0.999842i \(-0.505666\pi\)
−0.0177996 + 0.999842i \(0.505666\pi\)
\(242\) 13.4814 0.866618
\(243\) −13.3144 −0.854121
\(244\) −3.81452 −0.244199
\(245\) −2.19830 −0.140444
\(246\) −46.9678 −2.99456
\(247\) −3.12191 −0.198643
\(248\) 33.3644 2.11864
\(249\) −22.2702 −1.41132
\(250\) −0.247345 −0.0156435
\(251\) −0.564031 −0.0356013 −0.0178007 0.999842i \(-0.505666\pi\)
−0.0178007 + 0.999842i \(0.505666\pi\)
\(252\) −16.3792 −1.03179
\(253\) 13.7772 0.866163
\(254\) −26.0298 −1.63326
\(255\) 14.2477 0.892223
\(256\) −32.0587 −2.00367
\(257\) −20.0823 −1.25270 −0.626350 0.779542i \(-0.715452\pi\)
−0.626350 + 0.779542i \(0.715452\pi\)
\(258\) −3.67004 −0.228487
\(259\) 26.2620 1.63184
\(260\) −18.8194 −1.16713
\(261\) −6.62393 −0.410011
\(262\) −31.5450 −1.94885
\(263\) −6.87332 −0.423827 −0.211914 0.977288i \(-0.567970\pi\)
−0.211914 + 0.977288i \(0.567970\pi\)
\(264\) −28.4804 −1.75284
\(265\) −16.6698 −1.02402
\(266\) −15.6289 −0.958270
\(267\) 6.77844 0.414834
\(268\) 37.0807 2.26507
\(269\) −0.918070 −0.0559757 −0.0279879 0.999608i \(-0.508910\pi\)
−0.0279879 + 0.999608i \(0.508910\pi\)
\(270\) −26.8918 −1.63658
\(271\) 24.9286 1.51431 0.757153 0.653237i \(-0.226590\pi\)
0.757153 + 0.653237i \(0.226590\pi\)
\(272\) −12.5494 −0.760918
\(273\) −8.06547 −0.488145
\(274\) 45.6760 2.75939
\(275\) 11.7836 0.710579
\(276\) 52.1263 3.13763
\(277\) 16.5526 0.994548 0.497274 0.867594i \(-0.334334\pi\)
0.497274 + 0.867594i \(0.334334\pi\)
\(278\) −25.0810 −1.50426
\(279\) −8.00063 −0.478985
\(280\) −50.2816 −3.00490
\(281\) −3.90216 −0.232784 −0.116392 0.993203i \(-0.537133\pi\)
−0.116392 + 0.993203i \(0.537133\pi\)
\(282\) −61.0136 −3.63331
\(283\) 12.3796 0.735889 0.367945 0.929848i \(-0.380062\pi\)
0.367945 + 0.929848i \(0.380062\pi\)
\(284\) −23.0731 −1.36914
\(285\) 14.8383 0.878942
\(286\) −8.26509 −0.488725
\(287\) −24.8360 −1.46602
\(288\) −4.27983 −0.252191
\(289\) −12.3472 −0.726306
\(290\) −38.1012 −2.23738
\(291\) −17.5190 −1.02698
\(292\) −65.3451 −3.82403
\(293\) 13.0448 0.762087 0.381044 0.924557i \(-0.375565\pi\)
0.381044 + 0.924557i \(0.375565\pi\)
\(294\) −3.65278 −0.213035
\(295\) 26.0770 1.51826
\(296\) 54.3421 3.15857
\(297\) −8.05446 −0.467367
\(298\) −57.9080 −3.35452
\(299\) 8.07336 0.466894
\(300\) 44.5836 2.57404
\(301\) −1.94067 −0.111858
\(302\) 26.1183 1.50294
\(303\) 23.4090 1.34481
\(304\) −13.0696 −0.749592
\(305\) 2.80801 0.160786
\(306\) 7.44629 0.425676
\(307\) 7.91892 0.451957 0.225978 0.974132i \(-0.427442\pi\)
0.225978 + 0.974132i \(0.427442\pi\)
\(308\) −28.2184 −1.60789
\(309\) 34.6496 1.97115
\(310\) −46.0200 −2.61376
\(311\) 25.7388 1.45951 0.729757 0.683707i \(-0.239633\pi\)
0.729757 + 0.683707i \(0.239633\pi\)
\(312\) −16.6893 −0.944848
\(313\) 32.2311 1.82181 0.910904 0.412618i \(-0.135385\pi\)
0.910904 + 0.412618i \(0.135385\pi\)
\(314\) 11.8262 0.667390
\(315\) 12.0573 0.679353
\(316\) 57.3420 3.22574
\(317\) −7.07935 −0.397616 −0.198808 0.980038i \(-0.563707\pi\)
−0.198808 + 0.980038i \(0.563707\pi\)
\(318\) −27.6993 −1.55330
\(319\) −11.4119 −0.638941
\(320\) 12.1202 0.677541
\(321\) −35.4230 −1.97712
\(322\) 40.4168 2.25234
\(323\) 4.84567 0.269620
\(324\) −48.1865 −2.67703
\(325\) 6.90514 0.383028
\(326\) −3.56395 −0.197389
\(327\) 7.87754 0.435629
\(328\) −51.3915 −2.83762
\(329\) −32.2632 −1.77873
\(330\) 39.2834 2.16248
\(331\) 17.3309 0.952593 0.476296 0.879285i \(-0.341979\pi\)
0.476296 + 0.879285i \(0.341979\pi\)
\(332\) −45.6583 −2.50583
\(333\) −13.0310 −0.714095
\(334\) −44.8826 −2.45587
\(335\) −27.2965 −1.49137
\(336\) −33.7653 −1.84205
\(337\) 26.9371 1.46736 0.733680 0.679495i \(-0.237801\pi\)
0.733680 + 0.679495i \(0.237801\pi\)
\(338\) 27.7581 1.50984
\(339\) 18.5078 1.00521
\(340\) 29.2105 1.58416
\(341\) −13.7837 −0.746427
\(342\) 7.75496 0.419340
\(343\) 17.4879 0.944260
\(344\) −4.01570 −0.216512
\(345\) −38.3721 −2.06589
\(346\) −47.8321 −2.57147
\(347\) −31.5841 −1.69552 −0.847762 0.530377i \(-0.822050\pi\)
−0.847762 + 0.530377i \(0.822050\pi\)
\(348\) −43.1771 −2.31453
\(349\) −28.6483 −1.53351 −0.766755 0.641940i \(-0.778130\pi\)
−0.766755 + 0.641940i \(0.778130\pi\)
\(350\) 34.5685 1.84776
\(351\) −4.71988 −0.251928
\(352\) −7.37338 −0.393002
\(353\) −2.01142 −0.107057 −0.0535284 0.998566i \(-0.517047\pi\)
−0.0535284 + 0.998566i \(0.517047\pi\)
\(354\) 43.3307 2.30300
\(355\) 16.9850 0.901470
\(356\) 13.8971 0.736547
\(357\) 12.5188 0.662565
\(358\) −37.3996 −1.97663
\(359\) −2.60987 −0.137744 −0.0688719 0.997626i \(-0.521940\pi\)
−0.0688719 + 0.997626i \(0.521940\pi\)
\(360\) 24.9494 1.31495
\(361\) −13.9535 −0.734393
\(362\) 9.13759 0.480261
\(363\) −11.2463 −0.590276
\(364\) −16.5358 −0.866713
\(365\) 48.1029 2.51782
\(366\) 4.66591 0.243891
\(367\) 11.7850 0.615170 0.307585 0.951521i \(-0.400479\pi\)
0.307585 + 0.951521i \(0.400479\pi\)
\(368\) 33.7983 1.76186
\(369\) 12.3235 0.641534
\(370\) −74.9550 −3.89673
\(371\) −14.6470 −0.760437
\(372\) −52.1508 −2.70389
\(373\) 7.35336 0.380743 0.190371 0.981712i \(-0.439031\pi\)
0.190371 + 0.981712i \(0.439031\pi\)
\(374\) 12.8286 0.663353
\(375\) 0.206336 0.0106552
\(376\) −66.7602 −3.44289
\(377\) −6.68729 −0.344413
\(378\) −23.6286 −1.21533
\(379\) 32.8343 1.68658 0.843292 0.537455i \(-0.180614\pi\)
0.843292 + 0.537455i \(0.180614\pi\)
\(380\) 30.4214 1.56058
\(381\) 21.7142 1.11245
\(382\) −21.5830 −1.10428
\(383\) 20.5396 1.04952 0.524761 0.851249i \(-0.324155\pi\)
0.524761 + 0.851249i \(0.324155\pi\)
\(384\) 33.1481 1.69158
\(385\) 20.7726 1.05867
\(386\) −51.8766 −2.64045
\(387\) 0.962948 0.0489494
\(388\) −35.9174 −1.82343
\(389\) 24.5090 1.24266 0.621329 0.783550i \(-0.286593\pi\)
0.621329 + 0.783550i \(0.286593\pi\)
\(390\) 23.0199 1.16566
\(391\) −12.5310 −0.633721
\(392\) −3.99682 −0.201870
\(393\) 26.3150 1.32742
\(394\) −48.8551 −2.46129
\(395\) −42.2116 −2.12390
\(396\) 14.0018 0.703615
\(397\) 21.6119 1.08467 0.542334 0.840163i \(-0.317541\pi\)
0.542334 + 0.840163i \(0.317541\pi\)
\(398\) 9.29243 0.465787
\(399\) 13.0377 0.652702
\(400\) 28.9077 1.44538
\(401\) −19.5420 −0.975881 −0.487940 0.872877i \(-0.662252\pi\)
−0.487940 + 0.872877i \(0.662252\pi\)
\(402\) −45.3571 −2.26220
\(403\) −8.07715 −0.402352
\(404\) 47.9932 2.38775
\(405\) 35.4719 1.76261
\(406\) −33.4779 −1.66148
\(407\) −22.4501 −1.11281
\(408\) 25.9043 1.28245
\(409\) 5.64928 0.279339 0.139670 0.990198i \(-0.455396\pi\)
0.139670 + 0.990198i \(0.455396\pi\)
\(410\) 70.8852 3.50077
\(411\) −38.1032 −1.87949
\(412\) 71.0385 3.49982
\(413\) 22.9127 1.12746
\(414\) −20.0545 −0.985627
\(415\) 33.6108 1.64989
\(416\) −4.32076 −0.211843
\(417\) 20.9227 1.02459
\(418\) 13.3604 0.653479
\(419\) 16.0956 0.786321 0.393160 0.919470i \(-0.371382\pi\)
0.393160 + 0.919470i \(0.371382\pi\)
\(420\) 78.5937 3.83498
\(421\) 17.2086 0.838698 0.419349 0.907825i \(-0.362258\pi\)
0.419349 + 0.907825i \(0.362258\pi\)
\(422\) 14.4827 0.705006
\(423\) 16.0088 0.778374
\(424\) −30.3082 −1.47189
\(425\) −10.7178 −0.519889
\(426\) 28.2230 1.36741
\(427\) 2.46727 0.119400
\(428\) −72.6243 −3.51043
\(429\) 6.89479 0.332883
\(430\) 5.53893 0.267111
\(431\) −18.2825 −0.880635 −0.440318 0.897842i \(-0.645134\pi\)
−0.440318 + 0.897842i \(0.645134\pi\)
\(432\) −19.7593 −0.950669
\(433\) 5.90112 0.283590 0.141795 0.989896i \(-0.454713\pi\)
0.141795 + 0.989896i \(0.454713\pi\)
\(434\) −40.4358 −1.94098
\(435\) 31.7843 1.52394
\(436\) 16.1505 0.773470
\(437\) −13.0505 −0.624288
\(438\) 79.9299 3.81920
\(439\) −13.0195 −0.621387 −0.310693 0.950510i \(-0.600561\pi\)
−0.310693 + 0.950510i \(0.600561\pi\)
\(440\) 42.9834 2.04915
\(441\) 0.958420 0.0456390
\(442\) 7.51752 0.357572
\(443\) 13.8135 0.656301 0.328151 0.944625i \(-0.393575\pi\)
0.328151 + 0.944625i \(0.393575\pi\)
\(444\) −84.9405 −4.03110
\(445\) −10.2302 −0.484959
\(446\) −9.02178 −0.427194
\(447\) 48.3072 2.28485
\(448\) 10.6495 0.503142
\(449\) −6.45993 −0.304863 −0.152431 0.988314i \(-0.548710\pi\)
−0.152431 + 0.988314i \(0.548710\pi\)
\(450\) −17.1526 −0.808584
\(451\) 21.2311 0.999734
\(452\) 37.9447 1.78477
\(453\) −21.7881 −1.02369
\(454\) −22.3456 −1.04873
\(455\) 12.1726 0.570662
\(456\) 26.9781 1.26336
\(457\) 22.0973 1.03367 0.516835 0.856085i \(-0.327110\pi\)
0.516835 + 0.856085i \(0.327110\pi\)
\(458\) 46.9526 2.19395
\(459\) 7.32594 0.341945
\(460\) −78.6705 −3.66803
\(461\) −6.55784 −0.305429 −0.152714 0.988270i \(-0.548801\pi\)
−0.152714 + 0.988270i \(0.548801\pi\)
\(462\) 34.5166 1.60586
\(463\) −18.2710 −0.849124 −0.424562 0.905399i \(-0.639572\pi\)
−0.424562 + 0.905399i \(0.639572\pi\)
\(464\) −27.9957 −1.29967
\(465\) 38.3902 1.78030
\(466\) −33.5509 −1.55422
\(467\) 24.9818 1.15602 0.578010 0.816030i \(-0.303830\pi\)
0.578010 + 0.816030i \(0.303830\pi\)
\(468\) 8.20496 0.379274
\(469\) −23.9842 −1.10749
\(470\) 92.0835 4.24749
\(471\) −9.86546 −0.454576
\(472\) 47.4118 2.18230
\(473\) 1.65899 0.0762803
\(474\) −70.1406 −3.22167
\(475\) −11.1621 −0.512151
\(476\) 25.6660 1.17640
\(477\) 7.26776 0.332768
\(478\) 41.3665 1.89206
\(479\) −20.5003 −0.936683 −0.468341 0.883548i \(-0.655148\pi\)
−0.468341 + 0.883548i \(0.655148\pi\)
\(480\) 20.5363 0.937350
\(481\) −13.1556 −0.599846
\(482\) 1.38593 0.0631273
\(483\) −33.7159 −1.53413
\(484\) −23.0571 −1.04805
\(485\) 26.4401 1.20058
\(486\) 33.3899 1.51460
\(487\) −7.00703 −0.317519 −0.158759 0.987317i \(-0.550749\pi\)
−0.158759 + 0.987317i \(0.550749\pi\)
\(488\) 5.10536 0.231109
\(489\) 2.97307 0.134447
\(490\) 5.51288 0.249047
\(491\) −8.72994 −0.393977 −0.196988 0.980406i \(-0.563116\pi\)
−0.196988 + 0.980406i \(0.563116\pi\)
\(492\) 80.3285 3.62149
\(493\) 10.3797 0.467476
\(494\) 7.82913 0.352249
\(495\) −10.3072 −0.463275
\(496\) −33.8142 −1.51830
\(497\) 14.9240 0.669432
\(498\) 55.8491 2.50266
\(499\) 13.3535 0.597784 0.298892 0.954287i \(-0.403383\pi\)
0.298892 + 0.954287i \(0.403383\pi\)
\(500\) 0.423031 0.0189185
\(501\) 37.4413 1.67276
\(502\) 1.41448 0.0631312
\(503\) 32.4448 1.44664 0.723321 0.690511i \(-0.242614\pi\)
0.723321 + 0.690511i \(0.242614\pi\)
\(504\) 21.9219 0.976481
\(505\) −35.3296 −1.57214
\(506\) −34.5504 −1.53595
\(507\) −23.1559 −1.02839
\(508\) 44.5184 1.97519
\(509\) −25.9994 −1.15240 −0.576201 0.817308i \(-0.695466\pi\)
−0.576201 + 0.817308i \(0.695466\pi\)
\(510\) −35.7303 −1.58216
\(511\) 42.2660 1.86974
\(512\) 48.7066 2.15255
\(513\) 7.62961 0.336856
\(514\) 50.3624 2.22139
\(515\) −52.2941 −2.30435
\(516\) 6.27682 0.276322
\(517\) 27.5803 1.21298
\(518\) −65.8597 −2.89371
\(519\) 39.9018 1.75149
\(520\) 25.1880 1.10457
\(521\) −13.9201 −0.609851 −0.304926 0.952376i \(-0.598632\pi\)
−0.304926 + 0.952376i \(0.598632\pi\)
\(522\) 16.6115 0.727065
\(523\) −40.3251 −1.76329 −0.881646 0.471912i \(-0.843564\pi\)
−0.881646 + 0.471912i \(0.843564\pi\)
\(524\) 53.9509 2.35686
\(525\) −28.8372 −1.25856
\(526\) 17.2369 0.751565
\(527\) 12.5369 0.546117
\(528\) 28.8643 1.25616
\(529\) 10.7489 0.467342
\(530\) 41.8046 1.81587
\(531\) −11.3691 −0.493379
\(532\) 26.7299 1.15889
\(533\) 12.4413 0.538894
\(534\) −16.9990 −0.735617
\(535\) 53.4614 2.31134
\(536\) −49.6290 −2.14365
\(537\) 31.1990 1.34633
\(538\) 2.30233 0.0992607
\(539\) 1.65119 0.0711216
\(540\) 45.9926 1.97921
\(541\) −12.4075 −0.533438 −0.266719 0.963774i \(-0.585940\pi\)
−0.266719 + 0.963774i \(0.585940\pi\)
\(542\) −62.5160 −2.68529
\(543\) −7.62262 −0.327118
\(544\) 6.70646 0.287537
\(545\) −11.8890 −0.509269
\(546\) 20.2266 0.865618
\(547\) 7.89112 0.337400 0.168700 0.985667i \(-0.446043\pi\)
0.168700 + 0.985667i \(0.446043\pi\)
\(548\) −78.1191 −3.33708
\(549\) −1.22424 −0.0522495
\(550\) −29.5510 −1.26006
\(551\) 10.8099 0.460518
\(552\) −69.7661 −2.96944
\(553\) −37.0895 −1.57721
\(554\) −41.5105 −1.76361
\(555\) 62.5279 2.65416
\(556\) 42.8957 1.81918
\(557\) 36.0448 1.52727 0.763633 0.645651i \(-0.223414\pi\)
0.763633 + 0.645651i \(0.223414\pi\)
\(558\) 20.0640 0.849375
\(559\) 0.972159 0.0411179
\(560\) 50.9595 2.15343
\(561\) −10.7017 −0.451827
\(562\) 9.78584 0.412791
\(563\) 2.45029 0.103268 0.0516338 0.998666i \(-0.483557\pi\)
0.0516338 + 0.998666i \(0.483557\pi\)
\(564\) 104.351 4.39396
\(565\) −27.9325 −1.17513
\(566\) −31.0455 −1.30494
\(567\) 31.1676 1.30892
\(568\) 30.8812 1.29575
\(569\) 5.27470 0.221127 0.110563 0.993869i \(-0.464734\pi\)
0.110563 + 0.993869i \(0.464734\pi\)
\(570\) −37.2114 −1.55861
\(571\) −23.9847 −1.00373 −0.501863 0.864947i \(-0.667352\pi\)
−0.501863 + 0.864947i \(0.667352\pi\)
\(572\) 14.1357 0.591043
\(573\) 18.0046 0.752155
\(574\) 62.2837 2.59967
\(575\) 28.8654 1.20377
\(576\) −5.28422 −0.220176
\(577\) −0.512046 −0.0213168 −0.0106584 0.999943i \(-0.503393\pi\)
−0.0106584 + 0.999943i \(0.503393\pi\)
\(578\) 30.9643 1.28794
\(579\) 43.2757 1.79848
\(580\) 65.1640 2.70579
\(581\) 29.5323 1.22521
\(582\) 43.9340 1.82112
\(583\) 12.5211 0.518569
\(584\) 87.4581 3.61904
\(585\) −6.03998 −0.249722
\(586\) −32.7138 −1.35140
\(587\) 9.95942 0.411069 0.205535 0.978650i \(-0.434107\pi\)
0.205535 + 0.978650i \(0.434107\pi\)
\(588\) 6.24731 0.257635
\(589\) 13.0566 0.537988
\(590\) −65.3959 −2.69231
\(591\) 40.7552 1.67645
\(592\) −55.0748 −2.26356
\(593\) −13.1422 −0.539685 −0.269843 0.962904i \(-0.586972\pi\)
−0.269843 + 0.962904i \(0.586972\pi\)
\(594\) 20.1990 0.828774
\(595\) −18.8937 −0.774567
\(596\) 99.0393 4.05681
\(597\) −7.75179 −0.317260
\(598\) −20.2464 −0.827935
\(599\) 35.1687 1.43695 0.718477 0.695551i \(-0.244839\pi\)
0.718477 + 0.695551i \(0.244839\pi\)
\(600\) −59.6709 −2.43606
\(601\) 44.5769 1.81833 0.909165 0.416437i \(-0.136721\pi\)
0.909165 + 0.416437i \(0.136721\pi\)
\(602\) 4.86682 0.198357
\(603\) 11.9008 0.484639
\(604\) −44.6699 −1.81759
\(605\) 16.9732 0.690058
\(606\) −58.7051 −2.38473
\(607\) −5.26537 −0.213715 −0.106857 0.994274i \(-0.534079\pi\)
−0.106857 + 0.994274i \(0.534079\pi\)
\(608\) 6.98445 0.283257
\(609\) 27.9274 1.13168
\(610\) −7.04192 −0.285119
\(611\) 16.1619 0.653841
\(612\) −12.7353 −0.514794
\(613\) −20.1333 −0.813174 −0.406587 0.913612i \(-0.633281\pi\)
−0.406587 + 0.913612i \(0.633281\pi\)
\(614\) −19.8591 −0.801447
\(615\) −59.1328 −2.38447
\(616\) 37.7676 1.52170
\(617\) −41.5701 −1.67355 −0.836775 0.547547i \(-0.815562\pi\)
−0.836775 + 0.547547i \(0.815562\pi\)
\(618\) −86.8942 −3.49540
\(619\) 22.1384 0.889816 0.444908 0.895576i \(-0.353236\pi\)
0.444908 + 0.895576i \(0.353236\pi\)
\(620\) 78.7075 3.16097
\(621\) −19.7304 −0.791753
\(622\) −64.5477 −2.58813
\(623\) −8.98884 −0.360130
\(624\) 16.9144 0.677116
\(625\) −25.1552 −1.00621
\(626\) −80.8291 −3.23058
\(627\) −11.1453 −0.445101
\(628\) −20.2261 −0.807111
\(629\) 20.4195 0.814178
\(630\) −30.2373 −1.20468
\(631\) −9.18693 −0.365726 −0.182863 0.983138i \(-0.558536\pi\)
−0.182863 + 0.983138i \(0.558536\pi\)
\(632\) −76.7468 −3.05282
\(633\) −12.0815 −0.480198
\(634\) 17.7536 0.705085
\(635\) −32.7717 −1.30051
\(636\) 47.3738 1.87849
\(637\) 0.967587 0.0383372
\(638\) 28.6186 1.13302
\(639\) −7.40517 −0.292944
\(640\) −50.0281 −1.97753
\(641\) −43.6865 −1.72551 −0.862756 0.505620i \(-0.831264\pi\)
−0.862756 + 0.505620i \(0.831264\pi\)
\(642\) 88.8339 3.50599
\(643\) −3.46349 −0.136587 −0.0682933 0.997665i \(-0.521755\pi\)
−0.0682933 + 0.997665i \(0.521755\pi\)
\(644\) −69.1243 −2.72388
\(645\) −4.62060 −0.181936
\(646\) −12.1520 −0.478112
\(647\) −14.0554 −0.552573 −0.276286 0.961075i \(-0.589104\pi\)
−0.276286 + 0.961075i \(0.589104\pi\)
\(648\) 64.4930 2.53353
\(649\) −19.5870 −0.768857
\(650\) −17.3167 −0.679217
\(651\) 33.7318 1.32205
\(652\) 6.09538 0.238713
\(653\) 7.76287 0.303785 0.151892 0.988397i \(-0.451463\pi\)
0.151892 + 0.988397i \(0.451463\pi\)
\(654\) −19.7553 −0.772493
\(655\) −39.7153 −1.55181
\(656\) 52.0844 2.03355
\(657\) −20.9721 −0.818199
\(658\) 80.9097 3.15419
\(659\) 28.1715 1.09741 0.548703 0.836018i \(-0.315122\pi\)
0.548703 + 0.836018i \(0.315122\pi\)
\(660\) −67.1860 −2.61521
\(661\) −43.4156 −1.68867 −0.844336 0.535813i \(-0.820005\pi\)
−0.844336 + 0.535813i \(0.820005\pi\)
\(662\) −43.4624 −1.68922
\(663\) −6.27115 −0.243551
\(664\) 61.1093 2.37150
\(665\) −19.6769 −0.763037
\(666\) 32.6791 1.26629
\(667\) −27.9547 −1.08241
\(668\) 76.7622 2.97002
\(669\) 7.52602 0.290973
\(670\) 68.4541 2.64461
\(671\) −2.10915 −0.0814230
\(672\) 18.0444 0.696076
\(673\) −33.7929 −1.30262 −0.651311 0.758811i \(-0.725781\pi\)
−0.651311 + 0.758811i \(0.725781\pi\)
\(674\) −67.5529 −2.60204
\(675\) −16.8754 −0.649535
\(676\) −47.4743 −1.82593
\(677\) 1.09918 0.0422450 0.0211225 0.999777i \(-0.493276\pi\)
0.0211225 + 0.999777i \(0.493276\pi\)
\(678\) −46.4139 −1.78252
\(679\) 23.2318 0.891554
\(680\) −39.0955 −1.49924
\(681\) 18.6408 0.714319
\(682\) 34.5666 1.32362
\(683\) 38.5124 1.47364 0.736819 0.676090i \(-0.236327\pi\)
0.736819 + 0.676090i \(0.236327\pi\)
\(684\) −13.2632 −0.507131
\(685\) 57.5064 2.19721
\(686\) −43.8562 −1.67444
\(687\) −39.1681 −1.49436
\(688\) 4.06985 0.155161
\(689\) 7.33728 0.279528
\(690\) 96.2296 3.66340
\(691\) −41.0331 −1.56097 −0.780486 0.625173i \(-0.785028\pi\)
−0.780486 + 0.625173i \(0.785028\pi\)
\(692\) 81.8066 3.10982
\(693\) −9.05650 −0.344028
\(694\) 79.2066 3.00664
\(695\) −31.5772 −1.19779
\(696\) 57.7884 2.19046
\(697\) −19.3108 −0.731447
\(698\) 71.8443 2.71935
\(699\) 27.9883 1.05862
\(700\) −59.1220 −2.23460
\(701\) 2.98726 0.112827 0.0564136 0.998407i \(-0.482033\pi\)
0.0564136 + 0.998407i \(0.482033\pi\)
\(702\) 11.8365 0.446740
\(703\) 21.2659 0.802059
\(704\) −9.10376 −0.343111
\(705\) −76.8165 −2.89308
\(706\) 5.04423 0.189842
\(707\) −31.0425 −1.16747
\(708\) −74.1079 −2.78515
\(709\) −9.26225 −0.347851 −0.173926 0.984759i \(-0.555645\pi\)
−0.173926 + 0.984759i \(0.555645\pi\)
\(710\) −42.5949 −1.59856
\(711\) 18.4035 0.690187
\(712\) −18.6000 −0.697065
\(713\) −33.7647 −1.26450
\(714\) −31.3946 −1.17491
\(715\) −10.4058 −0.389155
\(716\) 63.9641 2.39045
\(717\) −34.5082 −1.28873
\(718\) 6.54503 0.244258
\(719\) 29.0064 1.08176 0.540878 0.841101i \(-0.318092\pi\)
0.540878 + 0.841101i \(0.318092\pi\)
\(720\) −25.2858 −0.942345
\(721\) −45.9485 −1.71121
\(722\) 34.9925 1.30229
\(723\) −1.15615 −0.0429976
\(724\) −15.6279 −0.580806
\(725\) −23.9097 −0.887983
\(726\) 28.2034 1.04673
\(727\) 30.2062 1.12029 0.560143 0.828396i \(-0.310746\pi\)
0.560143 + 0.828396i \(0.310746\pi\)
\(728\) 22.1316 0.820252
\(729\) 5.85023 0.216675
\(730\) −120.633 −4.46481
\(731\) −1.50893 −0.0558099
\(732\) −7.98004 −0.294951
\(733\) 49.4928 1.82806 0.914030 0.405647i \(-0.132954\pi\)
0.914030 + 0.405647i \(0.132954\pi\)
\(734\) −29.5543 −1.09087
\(735\) −4.59888 −0.169632
\(736\) −18.0620 −0.665774
\(737\) 20.5030 0.755237
\(738\) −30.9048 −1.13762
\(739\) 40.1038 1.47524 0.737620 0.675216i \(-0.235949\pi\)
0.737620 + 0.675216i \(0.235949\pi\)
\(740\) 128.195 4.71253
\(741\) −6.53110 −0.239926
\(742\) 36.7319 1.34847
\(743\) 1.42235 0.0521810 0.0260905 0.999660i \(-0.491694\pi\)
0.0260905 + 0.999660i \(0.491694\pi\)
\(744\) 69.7989 2.55895
\(745\) −72.9066 −2.67109
\(746\) −18.4407 −0.675164
\(747\) −14.6537 −0.536152
\(748\) −21.9406 −0.802230
\(749\) 46.9742 1.71640
\(750\) −0.517450 −0.0188946
\(751\) −17.5781 −0.641435 −0.320718 0.947175i \(-0.603924\pi\)
−0.320718 + 0.947175i \(0.603924\pi\)
\(752\) 67.6603 2.46732
\(753\) −1.17996 −0.0430003
\(754\) 16.7704 0.610741
\(755\) 32.8832 1.19674
\(756\) 40.4117 1.46976
\(757\) −14.6288 −0.531694 −0.265847 0.964015i \(-0.585652\pi\)
−0.265847 + 0.964015i \(0.585652\pi\)
\(758\) −82.3418 −2.99079
\(759\) 28.8221 1.04618
\(760\) −40.7161 −1.47693
\(761\) −30.4598 −1.10417 −0.552083 0.833789i \(-0.686167\pi\)
−0.552083 + 0.833789i \(0.686167\pi\)
\(762\) −54.4549 −1.97269
\(763\) −10.4463 −0.378183
\(764\) 36.9131 1.33547
\(765\) 9.37493 0.338951
\(766\) −51.5091 −1.86110
\(767\) −11.4779 −0.414442
\(768\) −67.0674 −2.42008
\(769\) 2.36118 0.0851465 0.0425732 0.999093i \(-0.486444\pi\)
0.0425732 + 0.999093i \(0.486444\pi\)
\(770\) −52.0935 −1.87732
\(771\) −42.0126 −1.51305
\(772\) 88.7238 3.19324
\(773\) −6.65925 −0.239516 −0.119758 0.992803i \(-0.538212\pi\)
−0.119758 + 0.992803i \(0.538212\pi\)
\(774\) −2.41488 −0.0868011
\(775\) −28.8790 −1.03736
\(776\) 48.0720 1.72568
\(777\) 54.9405 1.97098
\(778\) −61.4637 −2.20358
\(779\) −20.1112 −0.720560
\(780\) −39.3706 −1.40970
\(781\) −12.7578 −0.456510
\(782\) 31.4253 1.12377
\(783\) 16.3430 0.584051
\(784\) 4.05071 0.144668
\(785\) 14.8892 0.531419
\(786\) −65.9927 −2.35388
\(787\) 36.1520 1.28868 0.644340 0.764739i \(-0.277132\pi\)
0.644340 + 0.764739i \(0.277132\pi\)
\(788\) 83.5563 2.97657
\(789\) −14.3791 −0.511910
\(790\) 105.858 3.76627
\(791\) −24.5431 −0.872652
\(792\) −18.7400 −0.665897
\(793\) −1.23595 −0.0438900
\(794\) −54.1982 −1.92342
\(795\) −34.8736 −1.23684
\(796\) −15.8927 −0.563302
\(797\) 43.1287 1.52770 0.763848 0.645397i \(-0.223308\pi\)
0.763848 + 0.645397i \(0.223308\pi\)
\(798\) −32.6960 −1.15743
\(799\) −25.0857 −0.887467
\(800\) −15.4484 −0.546184
\(801\) 4.46020 0.157593
\(802\) 49.0074 1.73051
\(803\) −36.1311 −1.27504
\(804\) 77.5736 2.73581
\(805\) 50.8850 1.79346
\(806\) 20.2559 0.713483
\(807\) −1.92062 −0.0676090
\(808\) −64.2343 −2.25975
\(809\) 7.77831 0.273471 0.136735 0.990608i \(-0.456339\pi\)
0.136735 + 0.990608i \(0.456339\pi\)
\(810\) −88.9564 −3.12561
\(811\) −14.6808 −0.515514 −0.257757 0.966210i \(-0.582983\pi\)
−0.257757 + 0.966210i \(0.582983\pi\)
\(812\) 57.2568 2.00932
\(813\) 52.1512 1.82902
\(814\) 56.3003 1.97333
\(815\) −4.48704 −0.157174
\(816\) −26.2536 −0.919058
\(817\) −1.57148 −0.0549791
\(818\) −14.1673 −0.495347
\(819\) −5.30707 −0.185444
\(820\) −121.234 −4.23368
\(821\) −21.2187 −0.740538 −0.370269 0.928924i \(-0.620735\pi\)
−0.370269 + 0.928924i \(0.620735\pi\)
\(822\) 95.5551 3.33287
\(823\) −53.2220 −1.85520 −0.927601 0.373572i \(-0.878133\pi\)
−0.927601 + 0.373572i \(0.878133\pi\)
\(824\) −95.0783 −3.31221
\(825\) 24.6516 0.858257
\(826\) −57.4605 −1.99931
\(827\) −14.3032 −0.497372 −0.248686 0.968584i \(-0.579999\pi\)
−0.248686 + 0.968584i \(0.579999\pi\)
\(828\) 34.2990 1.19197
\(829\) 17.1601 0.595994 0.297997 0.954567i \(-0.403682\pi\)
0.297997 + 0.954567i \(0.403682\pi\)
\(830\) −84.2891 −2.92572
\(831\) 34.6283 1.20124
\(832\) −5.33476 −0.184950
\(833\) −1.50184 −0.0520356
\(834\) −52.4700 −1.81689
\(835\) −56.5075 −1.95552
\(836\) −22.8501 −0.790288
\(837\) 19.7397 0.682303
\(838\) −40.3645 −1.39437
\(839\) 17.0657 0.589173 0.294586 0.955625i \(-0.404818\pi\)
0.294586 + 0.955625i \(0.404818\pi\)
\(840\) −105.190 −3.62940
\(841\) −5.84465 −0.201540
\(842\) −43.1558 −1.48725
\(843\) −8.16340 −0.281163
\(844\) −24.7695 −0.852603
\(845\) 34.9476 1.20223
\(846\) −40.1469 −1.38028
\(847\) 14.9136 0.512437
\(848\) 30.7168 1.05482
\(849\) 25.8983 0.888827
\(850\) 26.8781 0.921910
\(851\) −54.9942 −1.88518
\(852\) −48.2694 −1.65368
\(853\) 15.9210 0.545124 0.272562 0.962138i \(-0.412129\pi\)
0.272562 + 0.962138i \(0.412129\pi\)
\(854\) −6.18743 −0.211729
\(855\) 9.76354 0.333906
\(856\) 97.2007 3.32225
\(857\) −3.76849 −0.128729 −0.0643645 0.997926i \(-0.520502\pi\)
−0.0643645 + 0.997926i \(0.520502\pi\)
\(858\) −17.2907 −0.590296
\(859\) −31.0748 −1.06026 −0.530129 0.847917i \(-0.677856\pi\)
−0.530129 + 0.847917i \(0.677856\pi\)
\(860\) −9.47316 −0.323032
\(861\) −51.9574 −1.77070
\(862\) 45.8487 1.56161
\(863\) −47.5457 −1.61847 −0.809237 0.587482i \(-0.800119\pi\)
−0.809237 + 0.587482i \(0.800119\pi\)
\(864\) 10.5595 0.359240
\(865\) −60.2209 −2.04757
\(866\) −14.7988 −0.502885
\(867\) −25.8306 −0.877253
\(868\) 69.1568 2.34734
\(869\) 31.7060 1.07555
\(870\) −79.7085 −2.70237
\(871\) 12.0146 0.407101
\(872\) −21.6159 −0.732008
\(873\) −11.5274 −0.390145
\(874\) 32.7279 1.10704
\(875\) −0.273621 −0.00925008
\(876\) −136.703 −4.61877
\(877\) −25.7216 −0.868556 −0.434278 0.900779i \(-0.642997\pi\)
−0.434278 + 0.900779i \(0.642997\pi\)
\(878\) 32.6503 1.10189
\(879\) 27.2900 0.920470
\(880\) −43.5629 −1.46850
\(881\) −10.6114 −0.357506 −0.178753 0.983894i \(-0.557206\pi\)
−0.178753 + 0.983894i \(0.557206\pi\)
\(882\) −2.40352 −0.0809309
\(883\) 33.4492 1.12566 0.562828 0.826574i \(-0.309713\pi\)
0.562828 + 0.826574i \(0.309713\pi\)
\(884\) −12.8571 −0.432431
\(885\) 54.5536 1.83380
\(886\) −34.6416 −1.16381
\(887\) 20.5714 0.690720 0.345360 0.938470i \(-0.387757\pi\)
0.345360 + 0.938470i \(0.387757\pi\)
\(888\) 113.685 3.81501
\(889\) −28.7951 −0.965755
\(890\) 25.6553 0.859968
\(891\) −26.6437 −0.892597
\(892\) 15.4298 0.516629
\(893\) −26.1255 −0.874257
\(894\) −121.145 −4.05168
\(895\) −47.0863 −1.57392
\(896\) −43.9575 −1.46852
\(897\) 16.8896 0.563928
\(898\) 16.2002 0.540608
\(899\) 27.9679 0.932781
\(900\) 29.3359 0.977865
\(901\) −11.3885 −0.379407
\(902\) −53.2434 −1.77281
\(903\) −4.05992 −0.135106
\(904\) −50.7854 −1.68910
\(905\) 11.5043 0.382415
\(906\) 54.6401 1.81530
\(907\) 20.4540 0.679164 0.339582 0.940577i \(-0.389714\pi\)
0.339582 + 0.940577i \(0.389714\pi\)
\(908\) 38.2174 1.26829
\(909\) 15.4031 0.510888
\(910\) −30.5265 −1.01194
\(911\) −47.9509 −1.58868 −0.794342 0.607470i \(-0.792184\pi\)
−0.794342 + 0.607470i \(0.792184\pi\)
\(912\) −27.3418 −0.905378
\(913\) −25.2458 −0.835513
\(914\) −55.4157 −1.83299
\(915\) 5.87441 0.194202
\(916\) −80.3024 −2.65326
\(917\) −34.8961 −1.15237
\(918\) −18.3720 −0.606366
\(919\) 30.1172 0.993473 0.496737 0.867901i \(-0.334531\pi\)
0.496737 + 0.867901i \(0.334531\pi\)
\(920\) 105.293 3.47141
\(921\) 16.5665 0.545886
\(922\) 16.4457 0.541611
\(923\) −7.47600 −0.246076
\(924\) −59.0334 −1.94206
\(925\) −47.0366 −1.54655
\(926\) 45.8199 1.50574
\(927\) 22.7993 0.748829
\(928\) 14.9610 0.491120
\(929\) 16.9454 0.555960 0.277980 0.960587i \(-0.410335\pi\)
0.277980 + 0.960587i \(0.410335\pi\)
\(930\) −96.2748 −3.15698
\(931\) −1.56409 −0.0512610
\(932\) 57.3817 1.87960
\(933\) 53.8461 1.76284
\(934\) −62.6493 −2.04995
\(935\) 16.1513 0.528205
\(936\) −10.9816 −0.358943
\(937\) 14.3314 0.468188 0.234094 0.972214i \(-0.424788\pi\)
0.234094 + 0.972214i \(0.424788\pi\)
\(938\) 60.1477 1.96389
\(939\) 67.4280 2.20043
\(940\) −157.489 −5.13673
\(941\) −13.2249 −0.431121 −0.215560 0.976491i \(-0.569158\pi\)
−0.215560 + 0.976491i \(0.569158\pi\)
\(942\) 24.7406 0.806092
\(943\) 52.0082 1.69362
\(944\) −48.0510 −1.56393
\(945\) −29.7486 −0.967721
\(946\) −4.16041 −0.135267
\(947\) 20.4653 0.665031 0.332516 0.943098i \(-0.392103\pi\)
0.332516 + 0.943098i \(0.392103\pi\)
\(948\) 119.961 3.89614
\(949\) −21.1727 −0.687294
\(950\) 27.9922 0.908187
\(951\) −14.8101 −0.480252
\(952\) −34.3515 −1.11334
\(953\) 2.88689 0.0935154 0.0467577 0.998906i \(-0.485111\pi\)
0.0467577 + 0.998906i \(0.485111\pi\)
\(954\) −18.2261 −0.590091
\(955\) −27.1731 −0.879302
\(956\) −70.7486 −2.28817
\(957\) −23.8738 −0.771731
\(958\) 51.4106 1.66100
\(959\) 50.5283 1.63164
\(960\) 25.3558 0.818354
\(961\) 2.78060 0.0896968
\(962\) 32.9917 1.06370
\(963\) −23.3083 −0.751099
\(964\) −2.37034 −0.0763434
\(965\) −65.3129 −2.10250
\(966\) 84.5527 2.72044
\(967\) 6.95913 0.223791 0.111895 0.993720i \(-0.464308\pi\)
0.111895 + 0.993720i \(0.464308\pi\)
\(968\) 30.8597 0.991868
\(969\) 10.1372 0.325655
\(970\) −66.3065 −2.12897
\(971\) 30.7141 0.985663 0.492832 0.870125i \(-0.335962\pi\)
0.492832 + 0.870125i \(0.335962\pi\)
\(972\) −57.1063 −1.83169
\(973\) −27.7455 −0.889479
\(974\) 17.5722 0.563050
\(975\) 14.4457 0.462632
\(976\) −5.17420 −0.165622
\(977\) 7.14575 0.228613 0.114306 0.993446i \(-0.463535\pi\)
0.114306 + 0.993446i \(0.463535\pi\)
\(978\) −7.45585 −0.238412
\(979\) 7.68413 0.245586
\(980\) −9.42861 −0.301186
\(981\) 5.18340 0.165493
\(982\) 21.8929 0.698632
\(983\) 13.2992 0.424180 0.212090 0.977250i \(-0.431973\pi\)
0.212090 + 0.977250i \(0.431973\pi\)
\(984\) −107.512 −3.42736
\(985\) −61.5089 −1.95984
\(986\) −26.0301 −0.828967
\(987\) −67.4953 −2.14840
\(988\) −13.3901 −0.425995
\(989\) 4.06389 0.129224
\(990\) 25.8484 0.821517
\(991\) 25.1327 0.798366 0.399183 0.916871i \(-0.369294\pi\)
0.399183 + 0.916871i \(0.369294\pi\)
\(992\) 18.0705 0.573738
\(993\) 36.2566 1.15057
\(994\) −37.4263 −1.18709
\(995\) 11.6992 0.370890
\(996\) −95.5181 −3.02661
\(997\) 46.1536 1.46170 0.730850 0.682538i \(-0.239124\pi\)
0.730850 + 0.682538i \(0.239124\pi\)
\(998\) −33.4878 −1.06004
\(999\) 32.1509 1.01721
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 4027.2.a.c.1.12 174
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4027.2.a.c.1.12 174 1.1 even 1 trivial