Properties

Label 8001.2.a.z.1.31
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.31
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.49464 q^{2} +4.22324 q^{4} +0.373762 q^{5} -1.00000 q^{7} +5.54618 q^{8} +O(q^{10})\) \(q+2.49464 q^{2} +4.22324 q^{4} +0.373762 q^{5} -1.00000 q^{7} +5.54618 q^{8} +0.932401 q^{10} -6.40807 q^{11} +2.54237 q^{13} -2.49464 q^{14} +5.38926 q^{16} +4.88897 q^{17} -5.83201 q^{19} +1.57848 q^{20} -15.9858 q^{22} -8.35556 q^{23} -4.86030 q^{25} +6.34231 q^{26} -4.22324 q^{28} -1.63505 q^{29} -6.96821 q^{31} +2.35191 q^{32} +12.1962 q^{34} -0.373762 q^{35} +7.46339 q^{37} -14.5488 q^{38} +2.07295 q^{40} -9.02251 q^{41} -7.64100 q^{43} -27.0628 q^{44} -20.8441 q^{46} +6.14673 q^{47} +1.00000 q^{49} -12.1247 q^{50} +10.7370 q^{52} -2.53581 q^{53} -2.39509 q^{55} -5.54618 q^{56} -4.07885 q^{58} +1.55774 q^{59} -11.0709 q^{61} -17.3832 q^{62} -4.91135 q^{64} +0.950241 q^{65} +11.4944 q^{67} +20.6473 q^{68} -0.932401 q^{70} +7.04746 q^{71} +5.49704 q^{73} +18.6185 q^{74} -24.6300 q^{76} +6.40807 q^{77} -7.21739 q^{79} +2.01430 q^{80} -22.5079 q^{82} +4.98747 q^{83} +1.82731 q^{85} -19.0616 q^{86} -35.5403 q^{88} +6.29176 q^{89} -2.54237 q^{91} -35.2875 q^{92} +15.3339 q^{94} -2.17978 q^{95} -15.3484 q^{97} +2.49464 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.49464 1.76398 0.881989 0.471270i \(-0.156204\pi\)
0.881989 + 0.471270i \(0.156204\pi\)
\(3\) 0 0
\(4\) 4.22324 2.11162
\(5\) 0.373762 0.167151 0.0835757 0.996501i \(-0.473366\pi\)
0.0835757 + 0.996501i \(0.473366\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 5.54618 1.96087
\(9\) 0 0
\(10\) 0.932401 0.294851
\(11\) −6.40807 −1.93210 −0.966052 0.258346i \(-0.916823\pi\)
−0.966052 + 0.258346i \(0.916823\pi\)
\(12\) 0 0
\(13\) 2.54237 0.705127 0.352564 0.935788i \(-0.385310\pi\)
0.352564 + 0.935788i \(0.385310\pi\)
\(14\) −2.49464 −0.666721
\(15\) 0 0
\(16\) 5.38926 1.34731
\(17\) 4.88897 1.18575 0.592875 0.805295i \(-0.297993\pi\)
0.592875 + 0.805295i \(0.297993\pi\)
\(18\) 0 0
\(19\) −5.83201 −1.33796 −0.668978 0.743283i \(-0.733268\pi\)
−0.668978 + 0.743283i \(0.733268\pi\)
\(20\) 1.57848 0.352960
\(21\) 0 0
\(22\) −15.9858 −3.40819
\(23\) −8.35556 −1.74225 −0.871127 0.491058i \(-0.836610\pi\)
−0.871127 + 0.491058i \(0.836610\pi\)
\(24\) 0 0
\(25\) −4.86030 −0.972060
\(26\) 6.34231 1.24383
\(27\) 0 0
\(28\) −4.22324 −0.798117
\(29\) −1.63505 −0.303620 −0.151810 0.988410i \(-0.548510\pi\)
−0.151810 + 0.988410i \(0.548510\pi\)
\(30\) 0 0
\(31\) −6.96821 −1.25153 −0.625764 0.780013i \(-0.715213\pi\)
−0.625764 + 0.780013i \(0.715213\pi\)
\(32\) 2.35191 0.415763
\(33\) 0 0
\(34\) 12.1962 2.09164
\(35\) −0.373762 −0.0631773
\(36\) 0 0
\(37\) 7.46339 1.22697 0.613487 0.789705i \(-0.289766\pi\)
0.613487 + 0.789705i \(0.289766\pi\)
\(38\) −14.5488 −2.36012
\(39\) 0 0
\(40\) 2.07295 0.327762
\(41\) −9.02251 −1.40908 −0.704540 0.709664i \(-0.748847\pi\)
−0.704540 + 0.709664i \(0.748847\pi\)
\(42\) 0 0
\(43\) −7.64100 −1.16524 −0.582621 0.812744i \(-0.697973\pi\)
−0.582621 + 0.812744i \(0.697973\pi\)
\(44\) −27.0628 −4.07987
\(45\) 0 0
\(46\) −20.8441 −3.07330
\(47\) 6.14673 0.896592 0.448296 0.893885i \(-0.352031\pi\)
0.448296 + 0.893885i \(0.352031\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −12.1247 −1.71469
\(51\) 0 0
\(52\) 10.7370 1.48896
\(53\) −2.53581 −0.348321 −0.174160 0.984717i \(-0.555721\pi\)
−0.174160 + 0.984717i \(0.555721\pi\)
\(54\) 0 0
\(55\) −2.39509 −0.322954
\(56\) −5.54618 −0.741140
\(57\) 0 0
\(58\) −4.07885 −0.535580
\(59\) 1.55774 0.202800 0.101400 0.994846i \(-0.467668\pi\)
0.101400 + 0.994846i \(0.467668\pi\)
\(60\) 0 0
\(61\) −11.0709 −1.41749 −0.708745 0.705465i \(-0.750738\pi\)
−0.708745 + 0.705465i \(0.750738\pi\)
\(62\) −17.3832 −2.20767
\(63\) 0 0
\(64\) −4.91135 −0.613918
\(65\) 0.950241 0.117863
\(66\) 0 0
\(67\) 11.4944 1.40427 0.702134 0.712045i \(-0.252231\pi\)
0.702134 + 0.712045i \(0.252231\pi\)
\(68\) 20.6473 2.50385
\(69\) 0 0
\(70\) −0.932401 −0.111443
\(71\) 7.04746 0.836380 0.418190 0.908360i \(-0.362665\pi\)
0.418190 + 0.908360i \(0.362665\pi\)
\(72\) 0 0
\(73\) 5.49704 0.643380 0.321690 0.946845i \(-0.395749\pi\)
0.321690 + 0.946845i \(0.395749\pi\)
\(74\) 18.6185 2.16436
\(75\) 0 0
\(76\) −24.6300 −2.82525
\(77\) 6.40807 0.730267
\(78\) 0 0
\(79\) −7.21739 −0.812020 −0.406010 0.913869i \(-0.633080\pi\)
−0.406010 + 0.913869i \(0.633080\pi\)
\(80\) 2.01430 0.225205
\(81\) 0 0
\(82\) −22.5079 −2.48559
\(83\) 4.98747 0.547445 0.273723 0.961809i \(-0.411745\pi\)
0.273723 + 0.961809i \(0.411745\pi\)
\(84\) 0 0
\(85\) 1.82731 0.198200
\(86\) −19.0616 −2.05546
\(87\) 0 0
\(88\) −35.5403 −3.78861
\(89\) 6.29176 0.666925 0.333463 0.942763i \(-0.391783\pi\)
0.333463 + 0.942763i \(0.391783\pi\)
\(90\) 0 0
\(91\) −2.54237 −0.266513
\(92\) −35.2875 −3.67898
\(93\) 0 0
\(94\) 15.3339 1.58157
\(95\) −2.17978 −0.223641
\(96\) 0 0
\(97\) −15.3484 −1.55840 −0.779199 0.626776i \(-0.784374\pi\)
−0.779199 + 0.626776i \(0.784374\pi\)
\(98\) 2.49464 0.251997
\(99\) 0 0
\(100\) −20.5262 −2.05262
\(101\) 14.8823 1.48085 0.740424 0.672140i \(-0.234625\pi\)
0.740424 + 0.672140i \(0.234625\pi\)
\(102\) 0 0
\(103\) 10.5754 1.04203 0.521013 0.853548i \(-0.325554\pi\)
0.521013 + 0.853548i \(0.325554\pi\)
\(104\) 14.1005 1.38266
\(105\) 0 0
\(106\) −6.32594 −0.614430
\(107\) 14.1331 1.36630 0.683150 0.730278i \(-0.260609\pi\)
0.683150 + 0.730278i \(0.260609\pi\)
\(108\) 0 0
\(109\) 7.01182 0.671610 0.335805 0.941932i \(-0.390992\pi\)
0.335805 + 0.941932i \(0.390992\pi\)
\(110\) −5.97489 −0.569684
\(111\) 0 0
\(112\) −5.38926 −0.509237
\(113\) 14.2300 1.33865 0.669325 0.742970i \(-0.266583\pi\)
0.669325 + 0.742970i \(0.266583\pi\)
\(114\) 0 0
\(115\) −3.12299 −0.291220
\(116\) −6.90519 −0.641130
\(117\) 0 0
\(118\) 3.88600 0.357735
\(119\) −4.88897 −0.448171
\(120\) 0 0
\(121\) 30.0633 2.73303
\(122\) −27.6181 −2.50042
\(123\) 0 0
\(124\) −29.4284 −2.64275
\(125\) −3.68540 −0.329632
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −16.9559 −1.49870
\(129\) 0 0
\(130\) 2.37051 0.207908
\(131\) −19.2456 −1.68150 −0.840750 0.541424i \(-0.817885\pi\)
−0.840750 + 0.541424i \(0.817885\pi\)
\(132\) 0 0
\(133\) 5.83201 0.505700
\(134\) 28.6745 2.47710
\(135\) 0 0
\(136\) 27.1151 2.32510
\(137\) −0.417270 −0.0356498 −0.0178249 0.999841i \(-0.505674\pi\)
−0.0178249 + 0.999841i \(0.505674\pi\)
\(138\) 0 0
\(139\) −7.80248 −0.661798 −0.330899 0.943666i \(-0.607352\pi\)
−0.330899 + 0.943666i \(0.607352\pi\)
\(140\) −1.57848 −0.133406
\(141\) 0 0
\(142\) 17.5809 1.47536
\(143\) −16.2917 −1.36238
\(144\) 0 0
\(145\) −0.611117 −0.0507505
\(146\) 13.7132 1.13491
\(147\) 0 0
\(148\) 31.5197 2.59090
\(149\) 0.799730 0.0655165 0.0327582 0.999463i \(-0.489571\pi\)
0.0327582 + 0.999463i \(0.489571\pi\)
\(150\) 0 0
\(151\) −17.4297 −1.41841 −0.709204 0.705003i \(-0.750946\pi\)
−0.709204 + 0.705003i \(0.750946\pi\)
\(152\) −32.3454 −2.62356
\(153\) 0 0
\(154\) 15.9858 1.28817
\(155\) −2.60445 −0.209194
\(156\) 0 0
\(157\) −22.3819 −1.78627 −0.893133 0.449792i \(-0.851498\pi\)
−0.893133 + 0.449792i \(0.851498\pi\)
\(158\) −18.0048 −1.43238
\(159\) 0 0
\(160\) 0.879054 0.0694953
\(161\) 8.35556 0.658510
\(162\) 0 0
\(163\) 17.1243 1.34128 0.670638 0.741785i \(-0.266021\pi\)
0.670638 + 0.741785i \(0.266021\pi\)
\(164\) −38.1042 −2.97544
\(165\) 0 0
\(166\) 12.4419 0.965682
\(167\) 8.64194 0.668734 0.334367 0.942443i \(-0.391478\pi\)
0.334367 + 0.942443i \(0.391478\pi\)
\(168\) 0 0
\(169\) −6.53634 −0.502796
\(170\) 4.55848 0.349620
\(171\) 0 0
\(172\) −32.2697 −2.46055
\(173\) −25.8065 −1.96203 −0.981016 0.193925i \(-0.937878\pi\)
−0.981016 + 0.193925i \(0.937878\pi\)
\(174\) 0 0
\(175\) 4.86030 0.367404
\(176\) −34.5347 −2.60315
\(177\) 0 0
\(178\) 15.6957 1.17644
\(179\) 14.0778 1.05222 0.526111 0.850416i \(-0.323649\pi\)
0.526111 + 0.850416i \(0.323649\pi\)
\(180\) 0 0
\(181\) −16.7679 −1.24635 −0.623174 0.782083i \(-0.714157\pi\)
−0.623174 + 0.782083i \(0.714157\pi\)
\(182\) −6.34231 −0.470123
\(183\) 0 0
\(184\) −46.3414 −3.41634
\(185\) 2.78953 0.205090
\(186\) 0 0
\(187\) −31.3289 −2.29099
\(188\) 25.9591 1.89326
\(189\) 0 0
\(190\) −5.43778 −0.394498
\(191\) −22.6982 −1.64238 −0.821191 0.570653i \(-0.806690\pi\)
−0.821191 + 0.570653i \(0.806690\pi\)
\(192\) 0 0
\(193\) 1.65402 0.119059 0.0595296 0.998227i \(-0.481040\pi\)
0.0595296 + 0.998227i \(0.481040\pi\)
\(194\) −38.2889 −2.74898
\(195\) 0 0
\(196\) 4.22324 0.301660
\(197\) −12.5827 −0.896481 −0.448241 0.893913i \(-0.647949\pi\)
−0.448241 + 0.893913i \(0.647949\pi\)
\(198\) 0 0
\(199\) 10.6610 0.755737 0.377869 0.925859i \(-0.376657\pi\)
0.377869 + 0.925859i \(0.376657\pi\)
\(200\) −26.9561 −1.90609
\(201\) 0 0
\(202\) 37.1261 2.61218
\(203\) 1.63505 0.114758
\(204\) 0 0
\(205\) −3.37227 −0.235529
\(206\) 26.3819 1.83811
\(207\) 0 0
\(208\) 13.7015 0.950028
\(209\) 37.3719 2.58507
\(210\) 0 0
\(211\) −8.06656 −0.555325 −0.277662 0.960679i \(-0.589560\pi\)
−0.277662 + 0.960679i \(0.589560\pi\)
\(212\) −10.7093 −0.735520
\(213\) 0 0
\(214\) 35.2571 2.41012
\(215\) −2.85591 −0.194772
\(216\) 0 0
\(217\) 6.96821 0.473033
\(218\) 17.4920 1.18471
\(219\) 0 0
\(220\) −10.1150 −0.681955
\(221\) 12.4296 0.836104
\(222\) 0 0
\(223\) −3.52281 −0.235905 −0.117952 0.993019i \(-0.537633\pi\)
−0.117952 + 0.993019i \(0.537633\pi\)
\(224\) −2.35191 −0.157144
\(225\) 0 0
\(226\) 35.4989 2.36135
\(227\) 11.6455 0.772938 0.386469 0.922302i \(-0.373695\pi\)
0.386469 + 0.922302i \(0.373695\pi\)
\(228\) 0 0
\(229\) 21.3408 1.41024 0.705120 0.709088i \(-0.250893\pi\)
0.705120 + 0.709088i \(0.250893\pi\)
\(230\) −7.79073 −0.513706
\(231\) 0 0
\(232\) −9.06826 −0.595360
\(233\) 3.63820 0.238346 0.119173 0.992873i \(-0.461976\pi\)
0.119173 + 0.992873i \(0.461976\pi\)
\(234\) 0 0
\(235\) 2.29741 0.149867
\(236\) 6.57870 0.428237
\(237\) 0 0
\(238\) −12.1962 −0.790564
\(239\) 3.73257 0.241440 0.120720 0.992687i \(-0.461480\pi\)
0.120720 + 0.992687i \(0.461480\pi\)
\(240\) 0 0
\(241\) 5.73060 0.369141 0.184570 0.982819i \(-0.440911\pi\)
0.184570 + 0.982819i \(0.440911\pi\)
\(242\) 74.9972 4.82100
\(243\) 0 0
\(244\) −46.7552 −2.99320
\(245\) 0.373762 0.0238788
\(246\) 0 0
\(247\) −14.8271 −0.943429
\(248\) −38.6470 −2.45408
\(249\) 0 0
\(250\) −9.19376 −0.581464
\(251\) 11.4934 0.725458 0.362729 0.931895i \(-0.381845\pi\)
0.362729 + 0.931895i \(0.381845\pi\)
\(252\) 0 0
\(253\) 53.5430 3.36622
\(254\) −2.49464 −0.156528
\(255\) 0 0
\(256\) −32.4761 −2.02976
\(257\) −24.9218 −1.55458 −0.777290 0.629143i \(-0.783406\pi\)
−0.777290 + 0.629143i \(0.783406\pi\)
\(258\) 0 0
\(259\) −7.46339 −0.463753
\(260\) 4.01309 0.248882
\(261\) 0 0
\(262\) −48.0110 −2.96613
\(263\) −22.7571 −1.40326 −0.701632 0.712539i \(-0.747545\pi\)
−0.701632 + 0.712539i \(0.747545\pi\)
\(264\) 0 0
\(265\) −0.947789 −0.0582222
\(266\) 14.5488 0.892043
\(267\) 0 0
\(268\) 48.5437 2.96528
\(269\) 6.75574 0.411905 0.205952 0.978562i \(-0.433971\pi\)
0.205952 + 0.978562i \(0.433971\pi\)
\(270\) 0 0
\(271\) −20.8092 −1.26407 −0.632036 0.774939i \(-0.717780\pi\)
−0.632036 + 0.774939i \(0.717780\pi\)
\(272\) 26.3479 1.59758
\(273\) 0 0
\(274\) −1.04094 −0.0628854
\(275\) 31.1451 1.87812
\(276\) 0 0
\(277\) 16.4321 0.987311 0.493655 0.869658i \(-0.335660\pi\)
0.493655 + 0.869658i \(0.335660\pi\)
\(278\) −19.4644 −1.16740
\(279\) 0 0
\(280\) −2.07295 −0.123882
\(281\) 17.3979 1.03787 0.518935 0.854814i \(-0.326329\pi\)
0.518935 + 0.854814i \(0.326329\pi\)
\(282\) 0 0
\(283\) −20.8735 −1.24080 −0.620400 0.784285i \(-0.713030\pi\)
−0.620400 + 0.784285i \(0.713030\pi\)
\(284\) 29.7631 1.76612
\(285\) 0 0
\(286\) −40.6419 −2.40321
\(287\) 9.02251 0.532582
\(288\) 0 0
\(289\) 6.90205 0.406003
\(290\) −1.52452 −0.0895228
\(291\) 0 0
\(292\) 23.2153 1.35857
\(293\) 2.66238 0.155538 0.0777690 0.996971i \(-0.475220\pi\)
0.0777690 + 0.996971i \(0.475220\pi\)
\(294\) 0 0
\(295\) 0.582223 0.0338984
\(296\) 41.3933 2.40594
\(297\) 0 0
\(298\) 1.99504 0.115570
\(299\) −21.2429 −1.22851
\(300\) 0 0
\(301\) 7.64100 0.440420
\(302\) −43.4808 −2.50204
\(303\) 0 0
\(304\) −31.4302 −1.80265
\(305\) −4.13790 −0.236935
\(306\) 0 0
\(307\) −19.1541 −1.09318 −0.546592 0.837399i \(-0.684075\pi\)
−0.546592 + 0.837399i \(0.684075\pi\)
\(308\) 27.0628 1.54205
\(309\) 0 0
\(310\) −6.49717 −0.369014
\(311\) 17.6665 1.00177 0.500887 0.865513i \(-0.333007\pi\)
0.500887 + 0.865513i \(0.333007\pi\)
\(312\) 0 0
\(313\) −21.4016 −1.20969 −0.604846 0.796343i \(-0.706765\pi\)
−0.604846 + 0.796343i \(0.706765\pi\)
\(314\) −55.8347 −3.15094
\(315\) 0 0
\(316\) −30.4807 −1.71468
\(317\) −11.4317 −0.642070 −0.321035 0.947067i \(-0.604031\pi\)
−0.321035 + 0.947067i \(0.604031\pi\)
\(318\) 0 0
\(319\) 10.4775 0.586626
\(320\) −1.83567 −0.102617
\(321\) 0 0
\(322\) 20.8441 1.16160
\(323\) −28.5125 −1.58648
\(324\) 0 0
\(325\) −12.3567 −0.685426
\(326\) 42.7189 2.36598
\(327\) 0 0
\(328\) −50.0405 −2.76302
\(329\) −6.14673 −0.338880
\(330\) 0 0
\(331\) 25.5607 1.40494 0.702472 0.711711i \(-0.252079\pi\)
0.702472 + 0.711711i \(0.252079\pi\)
\(332\) 21.0633 1.15600
\(333\) 0 0
\(334\) 21.5586 1.17963
\(335\) 4.29618 0.234725
\(336\) 0 0
\(337\) −0.321760 −0.0175274 −0.00876370 0.999962i \(-0.502790\pi\)
−0.00876370 + 0.999962i \(0.502790\pi\)
\(338\) −16.3058 −0.886921
\(339\) 0 0
\(340\) 7.71717 0.418522
\(341\) 44.6528 2.41808
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −42.3784 −2.28489
\(345\) 0 0
\(346\) −64.3780 −3.46098
\(347\) −11.5029 −0.617507 −0.308753 0.951142i \(-0.599912\pi\)
−0.308753 + 0.951142i \(0.599912\pi\)
\(348\) 0 0
\(349\) 15.3049 0.819252 0.409626 0.912254i \(-0.365659\pi\)
0.409626 + 0.912254i \(0.365659\pi\)
\(350\) 12.1247 0.648093
\(351\) 0 0
\(352\) −15.0712 −0.803298
\(353\) −7.04701 −0.375075 −0.187537 0.982257i \(-0.560051\pi\)
−0.187537 + 0.982257i \(0.560051\pi\)
\(354\) 0 0
\(355\) 2.63407 0.139802
\(356\) 26.5716 1.40829
\(357\) 0 0
\(358\) 35.1190 1.85610
\(359\) 17.9118 0.945350 0.472675 0.881237i \(-0.343289\pi\)
0.472675 + 0.881237i \(0.343289\pi\)
\(360\) 0 0
\(361\) 15.0124 0.790124
\(362\) −41.8299 −2.19853
\(363\) 0 0
\(364\) −10.7370 −0.562774
\(365\) 2.05458 0.107542
\(366\) 0 0
\(367\) 2.13230 0.111305 0.0556527 0.998450i \(-0.482276\pi\)
0.0556527 + 0.998450i \(0.482276\pi\)
\(368\) −45.0303 −2.34736
\(369\) 0 0
\(370\) 6.95888 0.361775
\(371\) 2.53581 0.131653
\(372\) 0 0
\(373\) −29.0911 −1.50628 −0.753140 0.657861i \(-0.771462\pi\)
−0.753140 + 0.657861i \(0.771462\pi\)
\(374\) −78.1543 −4.04126
\(375\) 0 0
\(376\) 34.0909 1.75810
\(377\) −4.15689 −0.214091
\(378\) 0 0
\(379\) −19.7521 −1.01460 −0.507298 0.861771i \(-0.669356\pi\)
−0.507298 + 0.861771i \(0.669356\pi\)
\(380\) −9.20574 −0.472244
\(381\) 0 0
\(382\) −56.6238 −2.89713
\(383\) −13.0295 −0.665776 −0.332888 0.942966i \(-0.608023\pi\)
−0.332888 + 0.942966i \(0.608023\pi\)
\(384\) 0 0
\(385\) 2.39509 0.122065
\(386\) 4.12619 0.210018
\(387\) 0 0
\(388\) −64.8201 −3.29074
\(389\) 27.0519 1.37159 0.685794 0.727796i \(-0.259455\pi\)
0.685794 + 0.727796i \(0.259455\pi\)
\(390\) 0 0
\(391\) −40.8501 −2.06588
\(392\) 5.54618 0.280124
\(393\) 0 0
\(394\) −31.3894 −1.58137
\(395\) −2.69758 −0.135730
\(396\) 0 0
\(397\) −7.86212 −0.394589 −0.197294 0.980344i \(-0.563215\pi\)
−0.197294 + 0.980344i \(0.563215\pi\)
\(398\) 26.5953 1.33310
\(399\) 0 0
\(400\) −26.1934 −1.30967
\(401\) 7.53993 0.376526 0.188263 0.982119i \(-0.439714\pi\)
0.188263 + 0.982119i \(0.439714\pi\)
\(402\) 0 0
\(403\) −17.7158 −0.882486
\(404\) 62.8517 3.12699
\(405\) 0 0
\(406\) 4.07885 0.202430
\(407\) −47.8259 −2.37064
\(408\) 0 0
\(409\) −5.18427 −0.256346 −0.128173 0.991752i \(-0.540911\pi\)
−0.128173 + 0.991752i \(0.540911\pi\)
\(410\) −8.41260 −0.415469
\(411\) 0 0
\(412\) 44.6625 2.20036
\(413\) −1.55774 −0.0766514
\(414\) 0 0
\(415\) 1.86412 0.0915062
\(416\) 5.97943 0.293166
\(417\) 0 0
\(418\) 93.2296 4.56001
\(419\) −23.7502 −1.16028 −0.580138 0.814518i \(-0.697001\pi\)
−0.580138 + 0.814518i \(0.697001\pi\)
\(420\) 0 0
\(421\) 17.3466 0.845419 0.422710 0.906265i \(-0.361079\pi\)
0.422710 + 0.906265i \(0.361079\pi\)
\(422\) −20.1232 −0.979581
\(423\) 0 0
\(424\) −14.0641 −0.683012
\(425\) −23.7619 −1.15262
\(426\) 0 0
\(427\) 11.0709 0.535761
\(428\) 59.6875 2.88511
\(429\) 0 0
\(430\) −7.12448 −0.343573
\(431\) −18.6704 −0.899324 −0.449662 0.893199i \(-0.648456\pi\)
−0.449662 + 0.893199i \(0.648456\pi\)
\(432\) 0 0
\(433\) 13.6587 0.656398 0.328199 0.944609i \(-0.393558\pi\)
0.328199 + 0.944609i \(0.393558\pi\)
\(434\) 17.3832 0.834420
\(435\) 0 0
\(436\) 29.6126 1.41818
\(437\) 48.7297 2.33106
\(438\) 0 0
\(439\) 25.1317 1.19947 0.599736 0.800198i \(-0.295272\pi\)
0.599736 + 0.800198i \(0.295272\pi\)
\(440\) −13.2836 −0.633271
\(441\) 0 0
\(442\) 31.0074 1.47487
\(443\) −26.5215 −1.26008 −0.630038 0.776565i \(-0.716961\pi\)
−0.630038 + 0.776565i \(0.716961\pi\)
\(444\) 0 0
\(445\) 2.35162 0.111477
\(446\) −8.78815 −0.416131
\(447\) 0 0
\(448\) 4.91135 0.232039
\(449\) 16.1644 0.762843 0.381422 0.924401i \(-0.375435\pi\)
0.381422 + 0.924401i \(0.375435\pi\)
\(450\) 0 0
\(451\) 57.8169 2.72249
\(452\) 60.0968 2.82672
\(453\) 0 0
\(454\) 29.0513 1.36345
\(455\) −0.950241 −0.0445480
\(456\) 0 0
\(457\) −3.04113 −0.142258 −0.0711290 0.997467i \(-0.522660\pi\)
−0.0711290 + 0.997467i \(0.522660\pi\)
\(458\) 53.2377 2.48763
\(459\) 0 0
\(460\) −13.1891 −0.614946
\(461\) 21.0282 0.979379 0.489689 0.871897i \(-0.337110\pi\)
0.489689 + 0.871897i \(0.337110\pi\)
\(462\) 0 0
\(463\) 13.6609 0.634877 0.317439 0.948279i \(-0.397177\pi\)
0.317439 + 0.948279i \(0.397177\pi\)
\(464\) −8.81169 −0.409072
\(465\) 0 0
\(466\) 9.07601 0.420438
\(467\) −11.9422 −0.552619 −0.276310 0.961069i \(-0.589112\pi\)
−0.276310 + 0.961069i \(0.589112\pi\)
\(468\) 0 0
\(469\) −11.4944 −0.530763
\(470\) 5.73122 0.264361
\(471\) 0 0
\(472\) 8.63951 0.397665
\(473\) 48.9640 2.25137
\(474\) 0 0
\(475\) 28.3453 1.30057
\(476\) −20.6473 −0.946367
\(477\) 0 0
\(478\) 9.31144 0.425895
\(479\) 2.43999 0.111486 0.0557430 0.998445i \(-0.482247\pi\)
0.0557430 + 0.998445i \(0.482247\pi\)
\(480\) 0 0
\(481\) 18.9747 0.865173
\(482\) 14.2958 0.651156
\(483\) 0 0
\(484\) 126.965 5.77112
\(485\) −5.73666 −0.260488
\(486\) 0 0
\(487\) 33.2175 1.50523 0.752613 0.658463i \(-0.228793\pi\)
0.752613 + 0.658463i \(0.228793\pi\)
\(488\) −61.4015 −2.77951
\(489\) 0 0
\(490\) 0.932401 0.0421216
\(491\) −31.9094 −1.44005 −0.720025 0.693948i \(-0.755870\pi\)
−0.720025 + 0.693948i \(0.755870\pi\)
\(492\) 0 0
\(493\) −7.99369 −0.360018
\(494\) −36.9884 −1.66419
\(495\) 0 0
\(496\) −37.5535 −1.68620
\(497\) −7.04746 −0.316122
\(498\) 0 0
\(499\) 39.0736 1.74917 0.874586 0.484870i \(-0.161133\pi\)
0.874586 + 0.484870i \(0.161133\pi\)
\(500\) −15.5643 −0.696058
\(501\) 0 0
\(502\) 28.6719 1.27969
\(503\) 1.41048 0.0628903 0.0314451 0.999505i \(-0.489989\pi\)
0.0314451 + 0.999505i \(0.489989\pi\)
\(504\) 0 0
\(505\) 5.56245 0.247526
\(506\) 133.571 5.93793
\(507\) 0 0
\(508\) −4.22324 −0.187376
\(509\) 21.3887 0.948036 0.474018 0.880515i \(-0.342803\pi\)
0.474018 + 0.880515i \(0.342803\pi\)
\(510\) 0 0
\(511\) −5.49704 −0.243175
\(512\) −47.1046 −2.08175
\(513\) 0 0
\(514\) −62.1710 −2.74224
\(515\) 3.95269 0.174176
\(516\) 0 0
\(517\) −39.3886 −1.73231
\(518\) −18.6185 −0.818049
\(519\) 0 0
\(520\) 5.27021 0.231114
\(521\) 8.11999 0.355743 0.177872 0.984054i \(-0.443079\pi\)
0.177872 + 0.984054i \(0.443079\pi\)
\(522\) 0 0
\(523\) −19.7403 −0.863184 −0.431592 0.902069i \(-0.642048\pi\)
−0.431592 + 0.902069i \(0.642048\pi\)
\(524\) −81.2789 −3.55069
\(525\) 0 0
\(526\) −56.7709 −2.47533
\(527\) −34.0674 −1.48400
\(528\) 0 0
\(529\) 46.8153 2.03545
\(530\) −2.36440 −0.102703
\(531\) 0 0
\(532\) 24.6300 1.06784
\(533\) −22.9386 −0.993580
\(534\) 0 0
\(535\) 5.28242 0.228379
\(536\) 63.7502 2.75359
\(537\) 0 0
\(538\) 16.8532 0.726591
\(539\) −6.40807 −0.276015
\(540\) 0 0
\(541\) −7.35211 −0.316092 −0.158046 0.987432i \(-0.550519\pi\)
−0.158046 + 0.987432i \(0.550519\pi\)
\(542\) −51.9116 −2.22979
\(543\) 0 0
\(544\) 11.4984 0.492991
\(545\) 2.62075 0.112261
\(546\) 0 0
\(547\) −17.8879 −0.764831 −0.382416 0.923990i \(-0.624908\pi\)
−0.382416 + 0.923990i \(0.624908\pi\)
\(548\) −1.76223 −0.0752787
\(549\) 0 0
\(550\) 77.6960 3.31297
\(551\) 9.53561 0.406230
\(552\) 0 0
\(553\) 7.21739 0.306915
\(554\) 40.9923 1.74159
\(555\) 0 0
\(556\) −32.9517 −1.39746
\(557\) 11.6752 0.494693 0.247346 0.968927i \(-0.420441\pi\)
0.247346 + 0.968927i \(0.420441\pi\)
\(558\) 0 0
\(559\) −19.4263 −0.821643
\(560\) −2.01430 −0.0851197
\(561\) 0 0
\(562\) 43.4014 1.83078
\(563\) −23.7886 −1.00257 −0.501285 0.865282i \(-0.667139\pi\)
−0.501285 + 0.865282i \(0.667139\pi\)
\(564\) 0 0
\(565\) 5.31864 0.223757
\(566\) −52.0719 −2.18874
\(567\) 0 0
\(568\) 39.0865 1.64003
\(569\) −3.67726 −0.154159 −0.0770794 0.997025i \(-0.524560\pi\)
−0.0770794 + 0.997025i \(0.524560\pi\)
\(570\) 0 0
\(571\) −41.7874 −1.74875 −0.874375 0.485251i \(-0.838728\pi\)
−0.874375 + 0.485251i \(0.838728\pi\)
\(572\) −68.8037 −2.87683
\(573\) 0 0
\(574\) 22.5079 0.939463
\(575\) 40.6105 1.69358
\(576\) 0 0
\(577\) 35.4826 1.47716 0.738579 0.674167i \(-0.235497\pi\)
0.738579 + 0.674167i \(0.235497\pi\)
\(578\) 17.2181 0.716180
\(579\) 0 0
\(580\) −2.58089 −0.107166
\(581\) −4.98747 −0.206915
\(582\) 0 0
\(583\) 16.2497 0.672992
\(584\) 30.4876 1.26159
\(585\) 0 0
\(586\) 6.64169 0.274366
\(587\) 18.7445 0.773669 0.386834 0.922149i \(-0.373569\pi\)
0.386834 + 0.922149i \(0.373569\pi\)
\(588\) 0 0
\(589\) 40.6387 1.67449
\(590\) 1.45244 0.0597960
\(591\) 0 0
\(592\) 40.2222 1.65312
\(593\) −15.6452 −0.642470 −0.321235 0.947000i \(-0.604098\pi\)
−0.321235 + 0.947000i \(0.604098\pi\)
\(594\) 0 0
\(595\) −1.82731 −0.0749124
\(596\) 3.37745 0.138346
\(597\) 0 0
\(598\) −52.9935 −2.16707
\(599\) 21.2278 0.867343 0.433672 0.901071i \(-0.357218\pi\)
0.433672 + 0.901071i \(0.357218\pi\)
\(600\) 0 0
\(601\) −11.5015 −0.469154 −0.234577 0.972098i \(-0.575371\pi\)
−0.234577 + 0.972098i \(0.575371\pi\)
\(602\) 19.0616 0.776891
\(603\) 0 0
\(604\) −73.6097 −2.99514
\(605\) 11.2365 0.456829
\(606\) 0 0
\(607\) −29.5860 −1.20086 −0.600430 0.799677i \(-0.705004\pi\)
−0.600430 + 0.799677i \(0.705004\pi\)
\(608\) −13.7164 −0.556272
\(609\) 0 0
\(610\) −10.3226 −0.417949
\(611\) 15.6273 0.632212
\(612\) 0 0
\(613\) 16.0162 0.646888 0.323444 0.946247i \(-0.395159\pi\)
0.323444 + 0.946247i \(0.395159\pi\)
\(614\) −47.7827 −1.92835
\(615\) 0 0
\(616\) 35.5403 1.43196
\(617\) 32.3833 1.30370 0.651851 0.758347i \(-0.273993\pi\)
0.651851 + 0.758347i \(0.273993\pi\)
\(618\) 0 0
\(619\) −6.74750 −0.271205 −0.135602 0.990763i \(-0.543297\pi\)
−0.135602 + 0.990763i \(0.543297\pi\)
\(620\) −10.9992 −0.441739
\(621\) 0 0
\(622\) 44.0715 1.76711
\(623\) −6.29176 −0.252074
\(624\) 0 0
\(625\) 22.9240 0.916962
\(626\) −53.3894 −2.13387
\(627\) 0 0
\(628\) −94.5239 −3.77191
\(629\) 36.4883 1.45488
\(630\) 0 0
\(631\) −27.4499 −1.09276 −0.546381 0.837537i \(-0.683995\pi\)
−0.546381 + 0.837537i \(0.683995\pi\)
\(632\) −40.0289 −1.59227
\(633\) 0 0
\(634\) −28.5181 −1.13260
\(635\) −0.373762 −0.0148323
\(636\) 0 0
\(637\) 2.54237 0.100732
\(638\) 26.1376 1.03480
\(639\) 0 0
\(640\) −6.33745 −0.250510
\(641\) 25.6205 1.01195 0.505975 0.862548i \(-0.331133\pi\)
0.505975 + 0.862548i \(0.331133\pi\)
\(642\) 0 0
\(643\) −25.0718 −0.988735 −0.494368 0.869253i \(-0.664600\pi\)
−0.494368 + 0.869253i \(0.664600\pi\)
\(644\) 35.2875 1.39052
\(645\) 0 0
\(646\) −71.1286 −2.79852
\(647\) −26.5116 −1.04228 −0.521139 0.853472i \(-0.674493\pi\)
−0.521139 + 0.853472i \(0.674493\pi\)
\(648\) 0 0
\(649\) −9.98210 −0.391832
\(650\) −30.8255 −1.20908
\(651\) 0 0
\(652\) 72.3198 2.83226
\(653\) 0.745988 0.0291928 0.0145964 0.999893i \(-0.495354\pi\)
0.0145964 + 0.999893i \(0.495354\pi\)
\(654\) 0 0
\(655\) −7.19328 −0.281065
\(656\) −48.6247 −1.89847
\(657\) 0 0
\(658\) −15.3339 −0.597777
\(659\) 50.1415 1.95324 0.976619 0.214979i \(-0.0689685\pi\)
0.976619 + 0.214979i \(0.0689685\pi\)
\(660\) 0 0
\(661\) −11.0074 −0.428138 −0.214069 0.976819i \(-0.568672\pi\)
−0.214069 + 0.976819i \(0.568672\pi\)
\(662\) 63.7649 2.47829
\(663\) 0 0
\(664\) 27.6614 1.07347
\(665\) 2.17978 0.0845283
\(666\) 0 0
\(667\) 13.6617 0.528984
\(668\) 36.4970 1.41211
\(669\) 0 0
\(670\) 10.7174 0.414050
\(671\) 70.9434 2.73874
\(672\) 0 0
\(673\) −30.4045 −1.17201 −0.586003 0.810309i \(-0.699299\pi\)
−0.586003 + 0.810309i \(0.699299\pi\)
\(674\) −0.802676 −0.0309179
\(675\) 0 0
\(676\) −27.6045 −1.06171
\(677\) 27.3555 1.05136 0.525679 0.850683i \(-0.323811\pi\)
0.525679 + 0.850683i \(0.323811\pi\)
\(678\) 0 0
\(679\) 15.3484 0.589019
\(680\) 10.1346 0.388644
\(681\) 0 0
\(682\) 111.393 4.26544
\(683\) −17.0233 −0.651379 −0.325689 0.945477i \(-0.605596\pi\)
−0.325689 + 0.945477i \(0.605596\pi\)
\(684\) 0 0
\(685\) −0.155960 −0.00595891
\(686\) −2.49464 −0.0952459
\(687\) 0 0
\(688\) −41.1793 −1.56995
\(689\) −6.44698 −0.245610
\(690\) 0 0
\(691\) −48.1416 −1.83139 −0.915696 0.401871i \(-0.868360\pi\)
−0.915696 + 0.401871i \(0.868360\pi\)
\(692\) −108.987 −4.14307
\(693\) 0 0
\(694\) −28.6956 −1.08927
\(695\) −2.91627 −0.110620
\(696\) 0 0
\(697\) −44.1108 −1.67082
\(698\) 38.1802 1.44514
\(699\) 0 0
\(700\) 20.5262 0.775818
\(701\) 36.9208 1.39448 0.697240 0.716837i \(-0.254411\pi\)
0.697240 + 0.716837i \(0.254411\pi\)
\(702\) 0 0
\(703\) −43.5266 −1.64164
\(704\) 31.4722 1.18615
\(705\) 0 0
\(706\) −17.5798 −0.661623
\(707\) −14.8823 −0.559708
\(708\) 0 0
\(709\) −10.1340 −0.380592 −0.190296 0.981727i \(-0.560945\pi\)
−0.190296 + 0.981727i \(0.560945\pi\)
\(710\) 6.57107 0.246608
\(711\) 0 0
\(712\) 34.8952 1.30775
\(713\) 58.2233 2.18048
\(714\) 0 0
\(715\) −6.08921 −0.227724
\(716\) 59.4538 2.22189
\(717\) 0 0
\(718\) 44.6836 1.66758
\(719\) 6.11067 0.227890 0.113945 0.993487i \(-0.463651\pi\)
0.113945 + 0.993487i \(0.463651\pi\)
\(720\) 0 0
\(721\) −10.5754 −0.393849
\(722\) 37.4505 1.39376
\(723\) 0 0
\(724\) −70.8148 −2.63181
\(725\) 7.94682 0.295137
\(726\) 0 0
\(727\) 32.9229 1.22104 0.610521 0.792000i \(-0.290960\pi\)
0.610521 + 0.792000i \(0.290960\pi\)
\(728\) −14.1005 −0.522598
\(729\) 0 0
\(730\) 5.12545 0.189701
\(731\) −37.3566 −1.38168
\(732\) 0 0
\(733\) 11.5829 0.427823 0.213912 0.976853i \(-0.431380\pi\)
0.213912 + 0.976853i \(0.431380\pi\)
\(734\) 5.31933 0.196340
\(735\) 0 0
\(736\) −19.6515 −0.724365
\(737\) −73.6571 −2.71319
\(738\) 0 0
\(739\) −32.8534 −1.20853 −0.604267 0.796782i \(-0.706534\pi\)
−0.604267 + 0.796782i \(0.706534\pi\)
\(740\) 11.7808 0.433073
\(741\) 0 0
\(742\) 6.32594 0.232233
\(743\) 16.3217 0.598783 0.299392 0.954130i \(-0.403216\pi\)
0.299392 + 0.954130i \(0.403216\pi\)
\(744\) 0 0
\(745\) 0.298909 0.0109512
\(746\) −72.5718 −2.65704
\(747\) 0 0
\(748\) −132.309 −4.83770
\(749\) −14.1331 −0.516413
\(750\) 0 0
\(751\) −6.51082 −0.237583 −0.118792 0.992919i \(-0.537902\pi\)
−0.118792 + 0.992919i \(0.537902\pi\)
\(752\) 33.1263 1.20799
\(753\) 0 0
\(754\) −10.3700 −0.377652
\(755\) −6.51455 −0.237089
\(756\) 0 0
\(757\) 28.1014 1.02136 0.510682 0.859770i \(-0.329393\pi\)
0.510682 + 0.859770i \(0.329393\pi\)
\(758\) −49.2744 −1.78973
\(759\) 0 0
\(760\) −12.0895 −0.438531
\(761\) −17.9070 −0.649130 −0.324565 0.945863i \(-0.605218\pi\)
−0.324565 + 0.945863i \(0.605218\pi\)
\(762\) 0 0
\(763\) −7.01182 −0.253845
\(764\) −95.8598 −3.46809
\(765\) 0 0
\(766\) −32.5039 −1.17441
\(767\) 3.96035 0.143000
\(768\) 0 0
\(769\) −31.1782 −1.12432 −0.562158 0.827030i \(-0.690029\pi\)
−0.562158 + 0.827030i \(0.690029\pi\)
\(770\) 5.97489 0.215320
\(771\) 0 0
\(772\) 6.98533 0.251408
\(773\) 28.7394 1.03368 0.516842 0.856081i \(-0.327108\pi\)
0.516842 + 0.856081i \(0.327108\pi\)
\(774\) 0 0
\(775\) 33.8676 1.21656
\(776\) −85.1252 −3.05582
\(777\) 0 0
\(778\) 67.4849 2.41945
\(779\) 52.6194 1.88529
\(780\) 0 0
\(781\) −45.1606 −1.61597
\(782\) −101.906 −3.64416
\(783\) 0 0
\(784\) 5.38926 0.192474
\(785\) −8.36548 −0.298577
\(786\) 0 0
\(787\) 31.8508 1.13536 0.567680 0.823249i \(-0.307841\pi\)
0.567680 + 0.823249i \(0.307841\pi\)
\(788\) −53.1398 −1.89303
\(789\) 0 0
\(790\) −6.72950 −0.239425
\(791\) −14.2300 −0.505962
\(792\) 0 0
\(793\) −28.1465 −0.999511
\(794\) −19.6132 −0.696046
\(795\) 0 0
\(796\) 45.0239 1.59583
\(797\) −7.35905 −0.260671 −0.130335 0.991470i \(-0.541605\pi\)
−0.130335 + 0.991470i \(0.541605\pi\)
\(798\) 0 0
\(799\) 30.0512 1.06313
\(800\) −11.4310 −0.404147
\(801\) 0 0
\(802\) 18.8094 0.664184
\(803\) −35.2254 −1.24308
\(804\) 0 0
\(805\) 3.12299 0.110071
\(806\) −44.1945 −1.55669
\(807\) 0 0
\(808\) 82.5402 2.90375
\(809\) 37.0239 1.30169 0.650846 0.759210i \(-0.274414\pi\)
0.650846 + 0.759210i \(0.274414\pi\)
\(810\) 0 0
\(811\) −54.7240 −1.92162 −0.960809 0.277212i \(-0.910590\pi\)
−0.960809 + 0.277212i \(0.910590\pi\)
\(812\) 6.90519 0.242325
\(813\) 0 0
\(814\) −119.309 −4.18176
\(815\) 6.40039 0.224196
\(816\) 0 0
\(817\) 44.5624 1.55904
\(818\) −12.9329 −0.452188
\(819\) 0 0
\(820\) −14.2419 −0.497348
\(821\) 9.67818 0.337771 0.168885 0.985636i \(-0.445983\pi\)
0.168885 + 0.985636i \(0.445983\pi\)
\(822\) 0 0
\(823\) −11.7237 −0.408663 −0.204331 0.978902i \(-0.565502\pi\)
−0.204331 + 0.978902i \(0.565502\pi\)
\(824\) 58.6532 2.04328
\(825\) 0 0
\(826\) −3.88600 −0.135211
\(827\) −26.6769 −0.927646 −0.463823 0.885928i \(-0.653523\pi\)
−0.463823 + 0.885928i \(0.653523\pi\)
\(828\) 0 0
\(829\) 8.20281 0.284895 0.142448 0.989802i \(-0.454503\pi\)
0.142448 + 0.989802i \(0.454503\pi\)
\(830\) 4.65032 0.161415
\(831\) 0 0
\(832\) −12.4865 −0.432890
\(833\) 4.88897 0.169393
\(834\) 0 0
\(835\) 3.23003 0.111780
\(836\) 157.830 5.45868
\(837\) 0 0
\(838\) −59.2484 −2.04670
\(839\) −32.8245 −1.13323 −0.566614 0.823983i \(-0.691747\pi\)
−0.566614 + 0.823983i \(0.691747\pi\)
\(840\) 0 0
\(841\) −26.3266 −0.907815
\(842\) 43.2734 1.49130
\(843\) 0 0
\(844\) −34.0670 −1.17263
\(845\) −2.44303 −0.0840430
\(846\) 0 0
\(847\) −30.0633 −1.03299
\(848\) −13.6661 −0.469298
\(849\) 0 0
\(850\) −59.2774 −2.03320
\(851\) −62.3608 −2.13770
\(852\) 0 0
\(853\) −32.2937 −1.10571 −0.552857 0.833276i \(-0.686462\pi\)
−0.552857 + 0.833276i \(0.686462\pi\)
\(854\) 27.6181 0.945070
\(855\) 0 0
\(856\) 78.3849 2.67914
\(857\) 27.3034 0.932668 0.466334 0.884609i \(-0.345575\pi\)
0.466334 + 0.884609i \(0.345575\pi\)
\(858\) 0 0
\(859\) −51.1565 −1.74544 −0.872718 0.488225i \(-0.837645\pi\)
−0.872718 + 0.488225i \(0.837645\pi\)
\(860\) −12.0612 −0.411283
\(861\) 0 0
\(862\) −46.5761 −1.58639
\(863\) 38.4182 1.30777 0.653885 0.756594i \(-0.273138\pi\)
0.653885 + 0.756594i \(0.273138\pi\)
\(864\) 0 0
\(865\) −9.64549 −0.327956
\(866\) 34.0737 1.15787
\(867\) 0 0
\(868\) 29.4284 0.998865
\(869\) 46.2495 1.56891
\(870\) 0 0
\(871\) 29.2231 0.990187
\(872\) 38.8888 1.31694
\(873\) 0 0
\(874\) 121.563 4.11194
\(875\) 3.68540 0.124589
\(876\) 0 0
\(877\) −53.3936 −1.80298 −0.901488 0.432805i \(-0.857524\pi\)
−0.901488 + 0.432805i \(0.857524\pi\)
\(878\) 62.6947 2.11584
\(879\) 0 0
\(880\) −12.9078 −0.435121
\(881\) −3.49548 −0.117766 −0.0588828 0.998265i \(-0.518754\pi\)
−0.0588828 + 0.998265i \(0.518754\pi\)
\(882\) 0 0
\(883\) −2.75544 −0.0927281 −0.0463640 0.998925i \(-0.514763\pi\)
−0.0463640 + 0.998925i \(0.514763\pi\)
\(884\) 52.4931 1.76553
\(885\) 0 0
\(886\) −66.1617 −2.22275
\(887\) 53.5674 1.79862 0.899308 0.437315i \(-0.144071\pi\)
0.899308 + 0.437315i \(0.144071\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 5.86644 0.196644
\(891\) 0 0
\(892\) −14.8777 −0.498141
\(893\) −35.8478 −1.19960
\(894\) 0 0
\(895\) 5.26174 0.175880
\(896\) 16.9559 0.566456
\(897\) 0 0
\(898\) 40.3243 1.34564
\(899\) 11.3933 0.379989
\(900\) 0 0
\(901\) −12.3975 −0.413021
\(902\) 144.232 4.80241
\(903\) 0 0
\(904\) 78.9224 2.62492
\(905\) −6.26720 −0.208329
\(906\) 0 0
\(907\) 11.2461 0.373422 0.186711 0.982415i \(-0.440217\pi\)
0.186711 + 0.982415i \(0.440217\pi\)
\(908\) 49.1816 1.63215
\(909\) 0 0
\(910\) −2.37051 −0.0785817
\(911\) −42.0363 −1.39272 −0.696362 0.717690i \(-0.745199\pi\)
−0.696362 + 0.717690i \(0.745199\pi\)
\(912\) 0 0
\(913\) −31.9600 −1.05772
\(914\) −7.58653 −0.250940
\(915\) 0 0
\(916\) 90.1273 2.97789
\(917\) 19.2456 0.635547
\(918\) 0 0
\(919\) −20.6524 −0.681261 −0.340630 0.940197i \(-0.610641\pi\)
−0.340630 + 0.940197i \(0.610641\pi\)
\(920\) −17.3207 −0.571045
\(921\) 0 0
\(922\) 52.4577 1.72760
\(923\) 17.9173 0.589754
\(924\) 0 0
\(925\) −36.2743 −1.19269
\(926\) 34.0791 1.11991
\(927\) 0 0
\(928\) −3.84548 −0.126234
\(929\) −8.98955 −0.294937 −0.147469 0.989067i \(-0.547113\pi\)
−0.147469 + 0.989067i \(0.547113\pi\)
\(930\) 0 0
\(931\) −5.83201 −0.191136
\(932\) 15.3650 0.503297
\(933\) 0 0
\(934\) −29.7915 −0.974808
\(935\) −11.7095 −0.382942
\(936\) 0 0
\(937\) −18.7903 −0.613853 −0.306927 0.951733i \(-0.599301\pi\)
−0.306927 + 0.951733i \(0.599301\pi\)
\(938\) −28.6745 −0.936255
\(939\) 0 0
\(940\) 9.70251 0.316461
\(941\) −34.8758 −1.13692 −0.568459 0.822712i \(-0.692460\pi\)
−0.568459 + 0.822712i \(0.692460\pi\)
\(942\) 0 0
\(943\) 75.3881 2.45497
\(944\) 8.39506 0.273236
\(945\) 0 0
\(946\) 122.148 3.97136
\(947\) 57.7138 1.87545 0.937723 0.347383i \(-0.112929\pi\)
0.937723 + 0.347383i \(0.112929\pi\)
\(948\) 0 0
\(949\) 13.9755 0.453665
\(950\) 70.7115 2.29418
\(951\) 0 0
\(952\) −27.1151 −0.878806
\(953\) −0.231947 −0.00751349 −0.00375675 0.999993i \(-0.501196\pi\)
−0.00375675 + 0.999993i \(0.501196\pi\)
\(954\) 0 0
\(955\) −8.48371 −0.274526
\(956\) 15.7635 0.509830
\(957\) 0 0
\(958\) 6.08690 0.196659
\(959\) 0.417270 0.0134744
\(960\) 0 0
\(961\) 17.5559 0.566321
\(962\) 47.3351 1.52615
\(963\) 0 0
\(964\) 24.2017 0.779484
\(965\) 0.618210 0.0199009
\(966\) 0 0
\(967\) −20.3332 −0.653871 −0.326935 0.945047i \(-0.606016\pi\)
−0.326935 + 0.945047i \(0.606016\pi\)
\(968\) 166.737 5.35912
\(969\) 0 0
\(970\) −14.3109 −0.459496
\(971\) 18.7408 0.601421 0.300711 0.953715i \(-0.402776\pi\)
0.300711 + 0.953715i \(0.402776\pi\)
\(972\) 0 0
\(973\) 7.80248 0.250136
\(974\) 82.8657 2.65519
\(975\) 0 0
\(976\) −59.6642 −1.90981
\(977\) −17.7823 −0.568906 −0.284453 0.958690i \(-0.591812\pi\)
−0.284453 + 0.958690i \(0.591812\pi\)
\(978\) 0 0
\(979\) −40.3180 −1.28857
\(980\) 1.57848 0.0504228
\(981\) 0 0
\(982\) −79.6025 −2.54022
\(983\) 54.0319 1.72335 0.861675 0.507460i \(-0.169416\pi\)
0.861675 + 0.507460i \(0.169416\pi\)
\(984\) 0 0
\(985\) −4.70294 −0.149848
\(986\) −19.9414 −0.635064
\(987\) 0 0
\(988\) −62.6185 −1.99216
\(989\) 63.8448 2.03015
\(990\) 0 0
\(991\) −12.9862 −0.412519 −0.206259 0.978497i \(-0.566129\pi\)
−0.206259 + 0.978497i \(0.566129\pi\)
\(992\) −16.3886 −0.520339
\(993\) 0 0
\(994\) −17.5809 −0.557632
\(995\) 3.98467 0.126323
\(996\) 0 0
\(997\) 9.65694 0.305838 0.152919 0.988239i \(-0.451133\pi\)
0.152919 + 0.988239i \(0.451133\pi\)
\(998\) 97.4745 3.08550
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.z.1.31 yes 32
3.2 odd 2 inner 8001.2.a.z.1.2 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.2 32 3.2 odd 2 inner
8001.2.a.z.1.31 yes 32 1.1 even 1 trivial