Properties

Label 6864.2.a.bn.1.2
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +2.39593 q^{15} -5.96633 q^{17} +5.05137 q^{19} +1.05137 q^{21} -9.15412 q^{23} +0.740489 q^{25} -1.00000 q^{27} -2.65544 q^{29} +9.70682 q^{31} +1.00000 q^{33} +2.51902 q^{35} -4.51902 q^{37} -1.00000 q^{39} +3.57040 q^{41} -7.96633 q^{43} -2.39593 q^{45} +1.20814 q^{47} -5.89461 q^{49} +5.96633 q^{51} -0.689115 q^{53} +2.39593 q^{55} -5.05137 q^{57} -11.4136 q^{59} -6.89461 q^{61} -1.05137 q^{63} -2.39593 q^{65} +14.2258 q^{67} +9.15412 q^{69} +8.10275 q^{71} +2.25951 q^{73} -0.740489 q^{75} +1.05137 q^{77} +11.4473 q^{79} +1.00000 q^{81} -15.4136 q^{83} +14.2949 q^{85} +2.65544 q^{87} -11.8096 q^{89} -1.05137 q^{91} -9.70682 q^{93} -12.1027 q^{95} -14.1027 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} - 3q^{11} + 3q^{13} + 4q^{15} + 6q^{19} - 6q^{21} + 5q^{25} - 3q^{27} - 2q^{29} + 14q^{31} + 3q^{33} + 2q^{35} - 8q^{37} - 3q^{39} - 4q^{41} - 6q^{43} - 4q^{45} + 10q^{47} + 7q^{49} - 14q^{53} + 4q^{55} - 6q^{57} - 4q^{59} + 4q^{61} + 6q^{63} - 4q^{65} + 22q^{67} + 6q^{71} + 4q^{73} - 5q^{75} - 6q^{77} + 22q^{79} + 3q^{81} - 16q^{83} - 14q^{85} + 2q^{87} - 2q^{89} + 6q^{91} - 14q^{93} - 18q^{95} - 24q^{97} - 3q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.39593 −1.07149 −0.535747 0.844379i \(-0.679970\pi\)
−0.535747 + 0.844379i \(0.679970\pi\)
\(6\) 0 0
\(7\) −1.05137 −0.397382 −0.198691 0.980062i \(-0.563669\pi\)
−0.198691 + 0.980062i \(0.563669\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.39593 0.618627
\(16\) 0 0
\(17\) −5.96633 −1.44705 −0.723523 0.690300i \(-0.757479\pi\)
−0.723523 + 0.690300i \(0.757479\pi\)
\(18\) 0 0
\(19\) 5.05137 1.15886 0.579432 0.815020i \(-0.303274\pi\)
0.579432 + 0.815020i \(0.303274\pi\)
\(20\) 0 0
\(21\) 1.05137 0.229429
\(22\) 0 0
\(23\) −9.15412 −1.90877 −0.954383 0.298584i \(-0.903486\pi\)
−0.954383 + 0.298584i \(0.903486\pi\)
\(24\) 0 0
\(25\) 0.740489 0.148098
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.65544 −0.493103 −0.246552 0.969130i \(-0.579297\pi\)
−0.246552 + 0.969130i \(0.579297\pi\)
\(30\) 0 0
\(31\) 9.70682 1.74340 0.871698 0.490044i \(-0.163019\pi\)
0.871698 + 0.490044i \(0.163019\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 2.51902 0.425792
\(36\) 0 0
\(37\) −4.51902 −0.742922 −0.371461 0.928448i \(-0.621143\pi\)
−0.371461 + 0.928448i \(0.621143\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 3.57040 0.557602 0.278801 0.960349i \(-0.410063\pi\)
0.278801 + 0.960349i \(0.410063\pi\)
\(42\) 0 0
\(43\) −7.96633 −1.21485 −0.607427 0.794376i \(-0.707798\pi\)
−0.607427 + 0.794376i \(0.707798\pi\)
\(44\) 0 0
\(45\) −2.39593 −0.357164
\(46\) 0 0
\(47\) 1.20814 0.176225 0.0881124 0.996111i \(-0.471917\pi\)
0.0881124 + 0.996111i \(0.471917\pi\)
\(48\) 0 0
\(49\) −5.89461 −0.842087
\(50\) 0 0
\(51\) 5.96633 0.835453
\(52\) 0 0
\(53\) −0.689115 −0.0946573 −0.0473286 0.998879i \(-0.515071\pi\)
−0.0473286 + 0.998879i \(0.515071\pi\)
\(54\) 0 0
\(55\) 2.39593 0.323067
\(56\) 0 0
\(57\) −5.05137 −0.669071
\(58\) 0 0
\(59\) −11.4136 −1.48593 −0.742964 0.669331i \(-0.766581\pi\)
−0.742964 + 0.669331i \(0.766581\pi\)
\(60\) 0 0
\(61\) −6.89461 −0.882765 −0.441382 0.897319i \(-0.645512\pi\)
−0.441382 + 0.897319i \(0.645512\pi\)
\(62\) 0 0
\(63\) −1.05137 −0.132461
\(64\) 0 0
\(65\) −2.39593 −0.297179
\(66\) 0 0
\(67\) 14.2258 1.73796 0.868981 0.494845i \(-0.164775\pi\)
0.868981 + 0.494845i \(0.164775\pi\)
\(68\) 0 0
\(69\) 9.15412 1.10203
\(70\) 0 0
\(71\) 8.10275 0.961619 0.480810 0.876825i \(-0.340343\pi\)
0.480810 + 0.876825i \(0.340343\pi\)
\(72\) 0 0
\(73\) 2.25951 0.264456 0.132228 0.991219i \(-0.457787\pi\)
0.132228 + 0.991219i \(0.457787\pi\)
\(74\) 0 0
\(75\) −0.740489 −0.0855044
\(76\) 0 0
\(77\) 1.05137 0.119815
\(78\) 0 0
\(79\) 11.4473 1.28792 0.643961 0.765058i \(-0.277290\pi\)
0.643961 + 0.765058i \(0.277290\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.4136 −1.69187 −0.845933 0.533289i \(-0.820956\pi\)
−0.845933 + 0.533289i \(0.820956\pi\)
\(84\) 0 0
\(85\) 14.2949 1.55050
\(86\) 0 0
\(87\) 2.65544 0.284693
\(88\) 0 0
\(89\) −11.8096 −1.25181 −0.625906 0.779899i \(-0.715271\pi\)
−0.625906 + 0.779899i \(0.715271\pi\)
\(90\) 0 0
\(91\) −1.05137 −0.110214
\(92\) 0 0
\(93\) −9.70682 −1.00655
\(94\) 0 0
\(95\) −12.1027 −1.24172
\(96\) 0 0
\(97\) −14.1027 −1.43192 −0.715959 0.698143i \(-0.754010\pi\)
−0.715959 + 0.698143i \(0.754010\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −9.17446 −0.912893 −0.456447 0.889751i \(-0.650878\pi\)
−0.456447 + 0.889751i \(0.650878\pi\)
\(102\) 0 0
\(103\) −2.62177 −0.258331 −0.129165 0.991623i \(-0.541230\pi\)
−0.129165 + 0.991623i \(0.541230\pi\)
\(104\) 0 0
\(105\) −2.51902 −0.245831
\(106\) 0 0
\(107\) −6.37559 −0.616352 −0.308176 0.951329i \(-0.599719\pi\)
−0.308176 + 0.951329i \(0.599719\pi\)
\(108\) 0 0
\(109\) −1.22147 −0.116995 −0.0584977 0.998288i \(-0.518631\pi\)
−0.0584977 + 0.998288i \(0.518631\pi\)
\(110\) 0 0
\(111\) 4.51902 0.428926
\(112\) 0 0
\(113\) 13.4136 1.26185 0.630924 0.775844i \(-0.282676\pi\)
0.630924 + 0.775844i \(0.282676\pi\)
\(114\) 0 0
\(115\) 21.9327 2.04523
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 6.27284 0.575031
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.57040 −0.321932
\(124\) 0 0
\(125\) 10.2055 0.912807
\(126\) 0 0
\(127\) −8.06908 −0.716015 −0.358007 0.933719i \(-0.616544\pi\)
−0.358007 + 0.933719i \(0.616544\pi\)
\(128\) 0 0
\(129\) 7.96633 0.701396
\(130\) 0 0
\(131\) 18.3082 1.59960 0.799799 0.600267i \(-0.204939\pi\)
0.799799 + 0.600267i \(0.204939\pi\)
\(132\) 0 0
\(133\) −5.31088 −0.460512
\(134\) 0 0
\(135\) 2.39593 0.206209
\(136\) 0 0
\(137\) −18.2258 −1.55714 −0.778569 0.627559i \(-0.784054\pi\)
−0.778569 + 0.627559i \(0.784054\pi\)
\(138\) 0 0
\(139\) 4.65544 0.394869 0.197435 0.980316i \(-0.436739\pi\)
0.197435 + 0.980316i \(0.436739\pi\)
\(140\) 0 0
\(141\) −1.20814 −0.101743
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 6.36226 0.528357
\(146\) 0 0
\(147\) 5.89461 0.486179
\(148\) 0 0
\(149\) 11.7759 0.964719 0.482359 0.875973i \(-0.339780\pi\)
0.482359 + 0.875973i \(0.339780\pi\)
\(150\) 0 0
\(151\) 22.6351 1.84202 0.921009 0.389541i \(-0.127366\pi\)
0.921009 + 0.389541i \(0.127366\pi\)
\(152\) 0 0
\(153\) −5.96633 −0.482349
\(154\) 0 0
\(155\) −23.2569 −1.86804
\(156\) 0 0
\(157\) 0.259511 0.0207112 0.0103556 0.999946i \(-0.496704\pi\)
0.0103556 + 0.999946i \(0.496704\pi\)
\(158\) 0 0
\(159\) 0.689115 0.0546504
\(160\) 0 0
\(161\) 9.62441 0.758510
\(162\) 0 0
\(163\) −2.60143 −0.203760 −0.101880 0.994797i \(-0.532486\pi\)
−0.101880 + 0.994797i \(0.532486\pi\)
\(164\) 0 0
\(165\) −2.39593 −0.186523
\(166\) 0 0
\(167\) 9.06471 0.701448 0.350724 0.936479i \(-0.385936\pi\)
0.350724 + 0.936479i \(0.385936\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.05137 0.386288
\(172\) 0 0
\(173\) −17.8990 −1.36083 −0.680417 0.732825i \(-0.738201\pi\)
−0.680417 + 0.732825i \(0.738201\pi\)
\(174\) 0 0
\(175\) −0.778532 −0.0588515
\(176\) 0 0
\(177\) 11.4136 0.857901
\(178\) 0 0
\(179\) 12.8813 0.962792 0.481396 0.876503i \(-0.340130\pi\)
0.481396 + 0.876503i \(0.340130\pi\)
\(180\) 0 0
\(181\) −5.84324 −0.434324 −0.217162 0.976136i \(-0.569680\pi\)
−0.217162 + 0.976136i \(0.569680\pi\)
\(182\) 0 0
\(183\) 6.89461 0.509664
\(184\) 0 0
\(185\) 10.8273 0.796036
\(186\) 0 0
\(187\) 5.96633 0.436301
\(188\) 0 0
\(189\) 1.05137 0.0764762
\(190\) 0 0
\(191\) 9.82991 0.711267 0.355634 0.934625i \(-0.384265\pi\)
0.355634 + 0.934625i \(0.384265\pi\)
\(192\) 0 0
\(193\) −17.0868 −1.22993 −0.614967 0.788553i \(-0.710831\pi\)
−0.614967 + 0.788553i \(0.710831\pi\)
\(194\) 0 0
\(195\) 2.39593 0.171576
\(196\) 0 0
\(197\) −5.74049 −0.408993 −0.204496 0.978867i \(-0.565556\pi\)
−0.204496 + 0.978867i \(0.565556\pi\)
\(198\) 0 0
\(199\) −16.2055 −1.14878 −0.574389 0.818583i \(-0.694760\pi\)
−0.574389 + 0.818583i \(0.694760\pi\)
\(200\) 0 0
\(201\) −14.2258 −1.00341
\(202\) 0 0
\(203\) 2.79186 0.195950
\(204\) 0 0
\(205\) −8.55442 −0.597467
\(206\) 0 0
\(207\) −9.15412 −0.636256
\(208\) 0 0
\(209\) −5.05137 −0.349411
\(210\) 0 0
\(211\) −5.27721 −0.363298 −0.181649 0.983363i \(-0.558144\pi\)
−0.181649 + 0.983363i \(0.558144\pi\)
\(212\) 0 0
\(213\) −8.10275 −0.555191
\(214\) 0 0
\(215\) 19.0868 1.30171
\(216\) 0 0
\(217\) −10.2055 −0.692794
\(218\) 0 0
\(219\) −2.25951 −0.152684
\(220\) 0 0
\(221\) −5.96633 −0.401339
\(222\) 0 0
\(223\) 20.0824 1.34482 0.672409 0.740180i \(-0.265260\pi\)
0.672409 + 0.740180i \(0.265260\pi\)
\(224\) 0 0
\(225\) 0.740489 0.0493660
\(226\) 0 0
\(227\) 5.25687 0.348911 0.174455 0.984665i \(-0.444184\pi\)
0.174455 + 0.984665i \(0.444184\pi\)
\(228\) 0 0
\(229\) −16.7919 −1.10964 −0.554819 0.831971i \(-0.687212\pi\)
−0.554819 + 0.831971i \(0.687212\pi\)
\(230\) 0 0
\(231\) −1.05137 −0.0691753
\(232\) 0 0
\(233\) 5.93092 0.388548 0.194274 0.980947i \(-0.437765\pi\)
0.194274 + 0.980947i \(0.437765\pi\)
\(234\) 0 0
\(235\) −2.89461 −0.188824
\(236\) 0 0
\(237\) −11.4473 −0.743582
\(238\) 0 0
\(239\) 20.3977 1.31942 0.659708 0.751522i \(-0.270680\pi\)
0.659708 + 0.751522i \(0.270680\pi\)
\(240\) 0 0
\(241\) 17.1001 1.10151 0.550757 0.834665i \(-0.314339\pi\)
0.550757 + 0.834665i \(0.314339\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 14.1231 0.902291
\(246\) 0 0
\(247\) 5.05137 0.321411
\(248\) 0 0
\(249\) 15.4136 0.976799
\(250\) 0 0
\(251\) 0.997361 0.0629528 0.0314764 0.999504i \(-0.489979\pi\)
0.0314764 + 0.999504i \(0.489979\pi\)
\(252\) 0 0
\(253\) 9.15412 0.575515
\(254\) 0 0
\(255\) −14.2949 −0.895182
\(256\) 0 0
\(257\) 29.3463 1.83057 0.915286 0.402806i \(-0.131965\pi\)
0.915286 + 0.402806i \(0.131965\pi\)
\(258\) 0 0
\(259\) 4.75118 0.295224
\(260\) 0 0
\(261\) −2.65544 −0.164368
\(262\) 0 0
\(263\) 25.2029 1.55407 0.777037 0.629454i \(-0.216722\pi\)
0.777037 + 0.629454i \(0.216722\pi\)
\(264\) 0 0
\(265\) 1.65107 0.101425
\(266\) 0 0
\(267\) 11.8096 0.722734
\(268\) 0 0
\(269\) 4.68912 0.285900 0.142950 0.989730i \(-0.454341\pi\)
0.142950 + 0.989730i \(0.454341\pi\)
\(270\) 0 0
\(271\) −0.465007 −0.0282472 −0.0141236 0.999900i \(-0.504496\pi\)
−0.0141236 + 0.999900i \(0.504496\pi\)
\(272\) 0 0
\(273\) 1.05137 0.0636321
\(274\) 0 0
\(275\) −0.740489 −0.0446532
\(276\) 0 0
\(277\) 20.3082 1.22020 0.610102 0.792323i \(-0.291128\pi\)
0.610102 + 0.792323i \(0.291128\pi\)
\(278\) 0 0
\(279\) 9.70682 0.581132
\(280\) 0 0
\(281\) 17.4003 1.03801 0.519007 0.854770i \(-0.326302\pi\)
0.519007 + 0.854770i \(0.326302\pi\)
\(282\) 0 0
\(283\) −31.1692 −1.85282 −0.926408 0.376522i \(-0.877120\pi\)
−0.926408 + 0.376522i \(0.877120\pi\)
\(284\) 0 0
\(285\) 12.1027 0.716905
\(286\) 0 0
\(287\) −3.75382 −0.221581
\(288\) 0 0
\(289\) 18.5971 1.09394
\(290\) 0 0
\(291\) 14.1027 0.826718
\(292\) 0 0
\(293\) 19.8432 1.15925 0.579627 0.814882i \(-0.303198\pi\)
0.579627 + 0.814882i \(0.303198\pi\)
\(294\) 0 0
\(295\) 27.3463 1.59216
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −9.15412 −0.529397
\(300\) 0 0
\(301\) 8.37559 0.482761
\(302\) 0 0
\(303\) 9.17446 0.527059
\(304\) 0 0
\(305\) 16.5190 0.945876
\(306\) 0 0
\(307\) 13.5704 0.774503 0.387252 0.921974i \(-0.373424\pi\)
0.387252 + 0.921974i \(0.373424\pi\)
\(308\) 0 0
\(309\) 2.62177 0.149147
\(310\) 0 0
\(311\) 24.3216 1.37915 0.689575 0.724214i \(-0.257797\pi\)
0.689575 + 0.724214i \(0.257797\pi\)
\(312\) 0 0
\(313\) 18.6084 1.05181 0.525906 0.850543i \(-0.323727\pi\)
0.525906 + 0.850543i \(0.323727\pi\)
\(314\) 0 0
\(315\) 2.51902 0.141931
\(316\) 0 0
\(317\) 25.6395 1.44006 0.720028 0.693945i \(-0.244129\pi\)
0.720028 + 0.693945i \(0.244129\pi\)
\(318\) 0 0
\(319\) 2.65544 0.148676
\(320\) 0 0
\(321\) 6.37559 0.355851
\(322\) 0 0
\(323\) −30.1382 −1.67693
\(324\) 0 0
\(325\) 0.740489 0.0410750
\(326\) 0 0
\(327\) 1.22147 0.0675474
\(328\) 0 0
\(329\) −1.27020 −0.0700286
\(330\) 0 0
\(331\) −33.7422 −1.85464 −0.927320 0.374269i \(-0.877894\pi\)
−0.927320 + 0.374269i \(0.877894\pi\)
\(332\) 0 0
\(333\) −4.51902 −0.247641
\(334\) 0 0
\(335\) −34.0841 −1.86222
\(336\) 0 0
\(337\) 12.1701 0.662947 0.331474 0.943464i \(-0.392454\pi\)
0.331474 + 0.943464i \(0.392454\pi\)
\(338\) 0 0
\(339\) −13.4136 −0.728529
\(340\) 0 0
\(341\) −9.70682 −0.525654
\(342\) 0 0
\(343\) 13.5571 0.732013
\(344\) 0 0
\(345\) −21.9327 −1.18081
\(346\) 0 0
\(347\) −23.4490 −1.25881 −0.629405 0.777077i \(-0.716701\pi\)
−0.629405 + 0.777077i \(0.716701\pi\)
\(348\) 0 0
\(349\) 20.8946 1.11846 0.559231 0.829012i \(-0.311096\pi\)
0.559231 + 0.829012i \(0.311096\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 3.97966 0.211816 0.105908 0.994376i \(-0.466225\pi\)
0.105908 + 0.994376i \(0.466225\pi\)
\(354\) 0 0
\(355\) −19.4136 −1.03037
\(356\) 0 0
\(357\) −6.27284 −0.331994
\(358\) 0 0
\(359\) −36.9433 −1.94980 −0.974898 0.222654i \(-0.928528\pi\)
−0.974898 + 0.222654i \(0.928528\pi\)
\(360\) 0 0
\(361\) 6.51638 0.342967
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −5.41363 −0.283363
\(366\) 0 0
\(367\) 14.8946 0.777492 0.388746 0.921345i \(-0.372908\pi\)
0.388746 + 0.921345i \(0.372908\pi\)
\(368\) 0 0
\(369\) 3.57040 0.185867
\(370\) 0 0
\(371\) 0.724518 0.0376151
\(372\) 0 0
\(373\) −10.2462 −0.530527 −0.265264 0.964176i \(-0.585459\pi\)
−0.265264 + 0.964176i \(0.585459\pi\)
\(374\) 0 0
\(375\) −10.2055 −0.527010
\(376\) 0 0
\(377\) −2.65544 −0.136762
\(378\) 0 0
\(379\) 35.6395 1.83068 0.915338 0.402686i \(-0.131923\pi\)
0.915338 + 0.402686i \(0.131923\pi\)
\(380\) 0 0
\(381\) 8.06908 0.413391
\(382\) 0 0
\(383\) 23.9327 1.22290 0.611451 0.791282i \(-0.290586\pi\)
0.611451 + 0.791282i \(0.290586\pi\)
\(384\) 0 0
\(385\) −2.51902 −0.128381
\(386\) 0 0
\(387\) −7.96633 −0.404951
\(388\) 0 0
\(389\) −6.45168 −0.327113 −0.163556 0.986534i \(-0.552297\pi\)
−0.163556 + 0.986534i \(0.552297\pi\)
\(390\) 0 0
\(391\) 54.6165 2.76207
\(392\) 0 0
\(393\) −18.3082 −0.923529
\(394\) 0 0
\(395\) −27.4270 −1.38000
\(396\) 0 0
\(397\) −4.51902 −0.226803 −0.113402 0.993549i \(-0.536175\pi\)
−0.113402 + 0.993549i \(0.536175\pi\)
\(398\) 0 0
\(399\) 5.31088 0.265877
\(400\) 0 0
\(401\) 1.33123 0.0664782 0.0332391 0.999447i \(-0.489418\pi\)
0.0332391 + 0.999447i \(0.489418\pi\)
\(402\) 0 0
\(403\) 9.70682 0.483531
\(404\) 0 0
\(405\) −2.39593 −0.119055
\(406\) 0 0
\(407\) 4.51902 0.224000
\(408\) 0 0
\(409\) 14.4650 0.715249 0.357624 0.933866i \(-0.383587\pi\)
0.357624 + 0.933866i \(0.383587\pi\)
\(410\) 0 0
\(411\) 18.2258 0.899014
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 36.9300 1.81282
\(416\) 0 0
\(417\) −4.65544 −0.227978
\(418\) 0 0
\(419\) −0.156762 −0.00765833 −0.00382916 0.999993i \(-0.501219\pi\)
−0.00382916 + 0.999993i \(0.501219\pi\)
\(420\) 0 0
\(421\) 15.8973 0.774785 0.387392 0.921915i \(-0.373376\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(422\) 0 0
\(423\) 1.20814 0.0587416
\(424\) 0 0
\(425\) −4.41800 −0.214305
\(426\) 0 0
\(427\) 7.24882 0.350795
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 1.69175 0.0814889 0.0407445 0.999170i \(-0.487027\pi\)
0.0407445 + 0.999170i \(0.487027\pi\)
\(432\) 0 0
\(433\) −9.16745 −0.440560 −0.220280 0.975437i \(-0.570697\pi\)
−0.220280 + 0.975437i \(0.570697\pi\)
\(434\) 0 0
\(435\) −6.36226 −0.305047
\(436\) 0 0
\(437\) −46.2409 −2.21200
\(438\) 0 0
\(439\) 18.5208 0.883947 0.441974 0.897028i \(-0.354278\pi\)
0.441974 + 0.897028i \(0.354278\pi\)
\(440\) 0 0
\(441\) −5.89461 −0.280696
\(442\) 0 0
\(443\) −5.03804 −0.239365 −0.119682 0.992812i \(-0.538188\pi\)
−0.119682 + 0.992812i \(0.538188\pi\)
\(444\) 0 0
\(445\) 28.2949 1.34131
\(446\) 0 0
\(447\) −11.7759 −0.556981
\(448\) 0 0
\(449\) −2.70946 −0.127867 −0.0639336 0.997954i \(-0.520365\pi\)
−0.0639336 + 0.997954i \(0.520365\pi\)
\(450\) 0 0
\(451\) −3.57040 −0.168123
\(452\) 0 0
\(453\) −22.6351 −1.06349
\(454\) 0 0
\(455\) 2.51902 0.118094
\(456\) 0 0
\(457\) 19.8973 0.930754 0.465377 0.885113i \(-0.345919\pi\)
0.465377 + 0.885113i \(0.345919\pi\)
\(458\) 0 0
\(459\) 5.96633 0.278484
\(460\) 0 0
\(461\) −27.4624 −1.27905 −0.639525 0.768770i \(-0.720869\pi\)
−0.639525 + 0.768770i \(0.720869\pi\)
\(462\) 0 0
\(463\) 0.0470050 0.00218451 0.00109225 0.999999i \(-0.499652\pi\)
0.00109225 + 0.999999i \(0.499652\pi\)
\(464\) 0 0
\(465\) 23.2569 1.07851
\(466\) 0 0
\(467\) 26.2188 1.21326 0.606631 0.794983i \(-0.292520\pi\)
0.606631 + 0.794983i \(0.292520\pi\)
\(468\) 0 0
\(469\) −14.9567 −0.690635
\(470\) 0 0
\(471\) −0.259511 −0.0119576
\(472\) 0 0
\(473\) 7.96633 0.366292
\(474\) 0 0
\(475\) 3.74049 0.171625
\(476\) 0 0
\(477\) −0.689115 −0.0315524
\(478\) 0 0
\(479\) 30.6351 1.39975 0.699877 0.714264i \(-0.253238\pi\)
0.699877 + 0.714264i \(0.253238\pi\)
\(480\) 0 0
\(481\) −4.51902 −0.206050
\(482\) 0 0
\(483\) −9.62441 −0.437926
\(484\) 0 0
\(485\) 33.7892 1.53429
\(486\) 0 0
\(487\) −10.3553 −0.469241 −0.234621 0.972087i \(-0.575385\pi\)
−0.234621 + 0.972087i \(0.575385\pi\)
\(488\) 0 0
\(489\) 2.60143 0.117641
\(490\) 0 0
\(491\) −9.48098 −0.427871 −0.213935 0.976848i \(-0.568628\pi\)
−0.213935 + 0.976848i \(0.568628\pi\)
\(492\) 0 0
\(493\) 15.8432 0.713544
\(494\) 0 0
\(495\) 2.39593 0.107689
\(496\) 0 0
\(497\) −8.51902 −0.382130
\(498\) 0 0
\(499\) −44.2879 −1.98260 −0.991299 0.131626i \(-0.957980\pi\)
−0.991299 + 0.131626i \(0.957980\pi\)
\(500\) 0 0
\(501\) −9.06471 −0.404981
\(502\) 0 0
\(503\) −36.1115 −1.61013 −0.805066 0.593186i \(-0.797870\pi\)
−0.805066 + 0.593186i \(0.797870\pi\)
\(504\) 0 0
\(505\) 21.9814 0.978159
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 7.80957 0.346153 0.173076 0.984908i \(-0.444629\pi\)
0.173076 + 0.984908i \(0.444629\pi\)
\(510\) 0 0
\(511\) −2.37559 −0.105090
\(512\) 0 0
\(513\) −5.05137 −0.223024
\(514\) 0 0
\(515\) 6.28158 0.276800
\(516\) 0 0
\(517\) −1.20814 −0.0531338
\(518\) 0 0
\(519\) 17.8990 0.785678
\(520\) 0 0
\(521\) −18.7512 −0.821504 −0.410752 0.911747i \(-0.634734\pi\)
−0.410752 + 0.911747i \(0.634734\pi\)
\(522\) 0 0
\(523\) 2.92829 0.128045 0.0640225 0.997948i \(-0.479607\pi\)
0.0640225 + 0.997948i \(0.479607\pi\)
\(524\) 0 0
\(525\) 0.778532 0.0339779
\(526\) 0 0
\(527\) −57.9140 −2.52278
\(528\) 0 0
\(529\) 60.7980 2.64339
\(530\) 0 0
\(531\) −11.4136 −0.495309
\(532\) 0 0
\(533\) 3.57040 0.154651
\(534\) 0 0
\(535\) 15.2755 0.660417
\(536\) 0 0
\(537\) −12.8813 −0.555868
\(538\) 0 0
\(539\) 5.89461 0.253899
\(540\) 0 0
\(541\) −27.0380 −1.16246 −0.581228 0.813741i \(-0.697428\pi\)
−0.581228 + 0.813741i \(0.697428\pi\)
\(542\) 0 0
\(543\) 5.84324 0.250757
\(544\) 0 0
\(545\) 2.92656 0.125360
\(546\) 0 0
\(547\) −37.5588 −1.60590 −0.802949 0.596048i \(-0.796737\pi\)
−0.802949 + 0.596048i \(0.796737\pi\)
\(548\) 0 0
\(549\) −6.89461 −0.294255
\(550\) 0 0
\(551\) −13.4136 −0.571440
\(552\) 0 0
\(553\) −12.0354 −0.511797
\(554\) 0 0
\(555\) −10.8273 −0.459592
\(556\) 0 0
\(557\) 16.1161 0.682860 0.341430 0.939907i \(-0.389089\pi\)
0.341430 + 0.939907i \(0.389089\pi\)
\(558\) 0 0
\(559\) −7.96633 −0.336940
\(560\) 0 0
\(561\) −5.96633 −0.251899
\(562\) 0 0
\(563\) 2.10275 0.0886203 0.0443101 0.999018i \(-0.485891\pi\)
0.0443101 + 0.999018i \(0.485891\pi\)
\(564\) 0 0
\(565\) −32.1382 −1.35206
\(566\) 0 0
\(567\) −1.05137 −0.0441536
\(568\) 0 0
\(569\) 25.9663 1.08857 0.544283 0.838902i \(-0.316802\pi\)
0.544283 + 0.838902i \(0.316802\pi\)
\(570\) 0 0
\(571\) −6.54742 −0.274001 −0.137000 0.990571i \(-0.543746\pi\)
−0.137000 + 0.990571i \(0.543746\pi\)
\(572\) 0 0
\(573\) −9.82991 −0.410650
\(574\) 0 0
\(575\) −6.77853 −0.282684
\(576\) 0 0
\(577\) −24.3082 −1.01197 −0.505983 0.862544i \(-0.668870\pi\)
−0.505983 + 0.862544i \(0.668870\pi\)
\(578\) 0 0
\(579\) 17.0868 0.710102
\(580\) 0 0
\(581\) 16.2055 0.672317
\(582\) 0 0
\(583\) 0.689115 0.0285402
\(584\) 0 0
\(585\) −2.39593 −0.0990596
\(586\) 0 0
\(587\) 3.00264 0.123932 0.0619661 0.998078i \(-0.480263\pi\)
0.0619661 + 0.998078i \(0.480263\pi\)
\(588\) 0 0
\(589\) 49.0328 2.02036
\(590\) 0 0
\(591\) 5.74049 0.236132
\(592\) 0 0
\(593\) −32.7733 −1.34584 −0.672918 0.739717i \(-0.734959\pi\)
−0.672918 + 0.739717i \(0.734959\pi\)
\(594\) 0 0
\(595\) −15.0293 −0.616141
\(596\) 0 0
\(597\) 16.2055 0.663247
\(598\) 0 0
\(599\) −14.0354 −0.573471 −0.286736 0.958010i \(-0.592570\pi\)
−0.286736 + 0.958010i \(0.592570\pi\)
\(600\) 0 0
\(601\) 25.3730 1.03498 0.517492 0.855688i \(-0.326866\pi\)
0.517492 + 0.855688i \(0.326866\pi\)
\(602\) 0 0
\(603\) 14.2258 0.579321
\(604\) 0 0
\(605\) −2.39593 −0.0974085
\(606\) 0 0
\(607\) 45.5855 1.85026 0.925128 0.379655i \(-0.123957\pi\)
0.925128 + 0.379655i \(0.123957\pi\)
\(608\) 0 0
\(609\) −2.79186 −0.113132
\(610\) 0 0
\(611\) 1.20814 0.0488760
\(612\) 0 0
\(613\) 5.71383 0.230779 0.115390 0.993320i \(-0.463188\pi\)
0.115390 + 0.993320i \(0.463188\pi\)
\(614\) 0 0
\(615\) 8.55442 0.344948
\(616\) 0 0
\(617\) −19.4606 −0.783456 −0.391728 0.920081i \(-0.628123\pi\)
−0.391728 + 0.920081i \(0.628123\pi\)
\(618\) 0 0
\(619\) 11.3259 0.455228 0.227614 0.973751i \(-0.426908\pi\)
0.227614 + 0.973751i \(0.426908\pi\)
\(620\) 0 0
\(621\) 9.15412 0.367342
\(622\) 0 0
\(623\) 12.4163 0.497447
\(624\) 0 0
\(625\) −28.1541 −1.12616
\(626\) 0 0
\(627\) 5.05137 0.201732
\(628\) 0 0
\(629\) 26.9620 1.07504
\(630\) 0 0
\(631\) 11.7068 0.466041 0.233021 0.972472i \(-0.425139\pi\)
0.233021 + 0.972472i \(0.425139\pi\)
\(632\) 0 0
\(633\) 5.27721 0.209750
\(634\) 0 0
\(635\) 19.3330 0.767205
\(636\) 0 0
\(637\) −5.89461 −0.233553
\(638\) 0 0
\(639\) 8.10275 0.320540
\(640\) 0 0
\(641\) 22.7245 0.897564 0.448782 0.893641i \(-0.351858\pi\)
0.448782 + 0.893641i \(0.351858\pi\)
\(642\) 0 0
\(643\) 29.4340 1.16076 0.580381 0.814345i \(-0.302904\pi\)
0.580381 + 0.814345i \(0.302904\pi\)
\(644\) 0 0
\(645\) −19.0868 −0.751541
\(646\) 0 0
\(647\) −44.3170 −1.74228 −0.871140 0.491034i \(-0.836619\pi\)
−0.871140 + 0.491034i \(0.836619\pi\)
\(648\) 0 0
\(649\) 11.4136 0.448024
\(650\) 0 0
\(651\) 10.2055 0.399985
\(652\) 0 0
\(653\) −18.7245 −0.732747 −0.366374 0.930468i \(-0.619401\pi\)
−0.366374 + 0.930468i \(0.619401\pi\)
\(654\) 0 0
\(655\) −43.8653 −1.71396
\(656\) 0 0
\(657\) 2.25951 0.0881519
\(658\) 0 0
\(659\) −9.72716 −0.378916 −0.189458 0.981889i \(-0.560673\pi\)
−0.189458 + 0.981889i \(0.560673\pi\)
\(660\) 0 0
\(661\) 49.7980 1.93692 0.968458 0.249176i \(-0.0801599\pi\)
0.968458 + 0.249176i \(0.0801599\pi\)
\(662\) 0 0
\(663\) 5.96633 0.231713
\(664\) 0 0
\(665\) 12.7245 0.493436
\(666\) 0 0
\(667\) 24.3082 0.941219
\(668\) 0 0
\(669\) −20.0824 −0.776431
\(670\) 0 0
\(671\) 6.89461 0.266164
\(672\) 0 0
\(673\) 6.41627 0.247329 0.123665 0.992324i \(-0.460535\pi\)
0.123665 + 0.992324i \(0.460535\pi\)
\(674\) 0 0
\(675\) −0.740489 −0.0285015
\(676\) 0 0
\(677\) 20.7315 0.796777 0.398389 0.917217i \(-0.369570\pi\)
0.398389 + 0.917217i \(0.369570\pi\)
\(678\) 0 0
\(679\) 14.8273 0.569018
\(680\) 0 0
\(681\) −5.25687 −0.201444
\(682\) 0 0
\(683\) −6.85921 −0.262460 −0.131230 0.991352i \(-0.541893\pi\)
−0.131230 + 0.991352i \(0.541893\pi\)
\(684\) 0 0
\(685\) 43.6679 1.66846
\(686\) 0 0
\(687\) 16.7919 0.640650
\(688\) 0 0
\(689\) −0.689115 −0.0262532
\(690\) 0 0
\(691\) −39.8183 −1.51476 −0.757380 0.652975i \(-0.773521\pi\)
−0.757380 + 0.652975i \(0.773521\pi\)
\(692\) 0 0
\(693\) 1.05137 0.0399384
\(694\) 0 0
\(695\) −11.1541 −0.423100
\(696\) 0 0
\(697\) −21.3021 −0.806876
\(698\) 0 0
\(699\) −5.93092 −0.224328
\(700\) 0 0
\(701\) 30.1985 1.14058 0.570291 0.821443i \(-0.306831\pi\)
0.570291 + 0.821443i \(0.306831\pi\)
\(702\) 0 0
\(703\) −22.8273 −0.860947
\(704\) 0 0
\(705\) 2.89461 0.109017
\(706\) 0 0
\(707\) 9.64579 0.362767
\(708\) 0 0
\(709\) −29.7219 −1.11623 −0.558114 0.829764i \(-0.688475\pi\)
−0.558114 + 0.829764i \(0.688475\pi\)
\(710\) 0 0
\(711\) 11.4473 0.429308
\(712\) 0 0
\(713\) −88.8574 −3.32774
\(714\) 0 0
\(715\) 2.39593 0.0896028
\(716\) 0 0
\(717\) −20.3977 −0.761765
\(718\) 0 0
\(719\) 9.44904 0.352390 0.176195 0.984355i \(-0.443621\pi\)
0.176195 + 0.984355i \(0.443621\pi\)
\(720\) 0 0
\(721\) 2.75646 0.102656
\(722\) 0 0
\(723\) −17.1001 −0.635960
\(724\) 0 0
\(725\) −1.96633 −0.0730276
\(726\) 0 0
\(727\) 44.3436 1.64461 0.822307 0.569043i \(-0.192686\pi\)
0.822307 + 0.569043i \(0.192686\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 47.5297 1.75795
\(732\) 0 0
\(733\) −8.77325 −0.324047 −0.162024 0.986787i \(-0.551802\pi\)
−0.162024 + 0.986787i \(0.551802\pi\)
\(734\) 0 0
\(735\) −14.1231 −0.520938
\(736\) 0 0
\(737\) −14.2258 −0.524015
\(738\) 0 0
\(739\) 31.3863 1.15456 0.577282 0.816545i \(-0.304114\pi\)
0.577282 + 0.816545i \(0.304114\pi\)
\(740\) 0 0
\(741\) −5.05137 −0.185567
\(742\) 0 0
\(743\) 17.5164 0.642614 0.321307 0.946975i \(-0.395878\pi\)
0.321307 + 0.946975i \(0.395878\pi\)
\(744\) 0 0
\(745\) −28.2142 −1.03369
\(746\) 0 0
\(747\) −15.4136 −0.563955
\(748\) 0 0
\(749\) 6.70313 0.244927
\(750\) 0 0
\(751\) −48.1382 −1.75659 −0.878293 0.478123i \(-0.841317\pi\)
−0.878293 + 0.478123i \(0.841317\pi\)
\(752\) 0 0
\(753\) −0.997361 −0.0363458
\(754\) 0 0
\(755\) −54.2322 −1.97371
\(756\) 0 0
\(757\) 19.0868 0.693721 0.346860 0.937917i \(-0.387248\pi\)
0.346860 + 0.937917i \(0.387248\pi\)
\(758\) 0 0
\(759\) −9.15412 −0.332274
\(760\) 0 0
\(761\) 35.1895 1.27562 0.637810 0.770194i \(-0.279841\pi\)
0.637810 + 0.770194i \(0.279841\pi\)
\(762\) 0 0
\(763\) 1.28422 0.0464919
\(764\) 0 0
\(765\) 14.2949 0.516834
\(766\) 0 0
\(767\) −11.4136 −0.412122
\(768\) 0 0
\(769\) −43.1088 −1.55454 −0.777272 0.629164i \(-0.783397\pi\)
−0.777272 + 0.629164i \(0.783397\pi\)
\(770\) 0 0
\(771\) −29.3463 −1.05688
\(772\) 0 0
\(773\) 12.8069 0.460633 0.230317 0.973116i \(-0.426024\pi\)
0.230317 + 0.973116i \(0.426024\pi\)
\(774\) 0 0
\(775\) 7.18780 0.258193
\(776\) 0 0
\(777\) −4.75118 −0.170448
\(778\) 0 0
\(779\) 18.0354 0.646185
\(780\) 0 0
\(781\) −8.10275 −0.289939
\(782\) 0 0
\(783\) 2.65544 0.0948978
\(784\) 0 0
\(785\) −0.621770 −0.0221919
\(786\) 0 0
\(787\) 8.50569 0.303195 0.151598 0.988442i \(-0.451558\pi\)
0.151598 + 0.988442i \(0.451558\pi\)
\(788\) 0 0
\(789\) −25.2029 −0.897245
\(790\) 0 0
\(791\) −14.1027 −0.501436
\(792\) 0 0
\(793\) −6.89461 −0.244835
\(794\) 0 0
\(795\) −1.65107 −0.0585575
\(796\) 0 0
\(797\) 0.129413 0.00458404 0.00229202 0.999997i \(-0.499270\pi\)
0.00229202 + 0.999997i \(0.499270\pi\)
\(798\) 0 0
\(799\) −7.20814 −0.255006
\(800\) 0 0
\(801\) −11.8096 −0.417270
\(802\) 0 0
\(803\) −2.25951 −0.0797364
\(804\) 0 0
\(805\) −23.0594 −0.812738
\(806\) 0 0
\(807\) −4.68912 −0.165065
\(808\) 0 0
\(809\) 31.9610 1.12369 0.561845 0.827242i \(-0.310092\pi\)
0.561845 + 0.827242i \(0.310092\pi\)
\(810\) 0 0
\(811\) −36.4243 −1.27903 −0.639516 0.768778i \(-0.720865\pi\)
−0.639516 + 0.768778i \(0.720865\pi\)
\(812\) 0 0
\(813\) 0.465007 0.0163085
\(814\) 0 0
\(815\) 6.23285 0.218327
\(816\) 0 0
\(817\) −40.2409 −1.40785
\(818\) 0 0
\(819\) −1.05137 −0.0367380
\(820\) 0 0
\(821\) −3.09206 −0.107913 −0.0539567 0.998543i \(-0.517183\pi\)
−0.0539567 + 0.998543i \(0.517183\pi\)
\(822\) 0 0
\(823\) 11.1001 0.386925 0.193463 0.981108i \(-0.438028\pi\)
0.193463 + 0.981108i \(0.438028\pi\)
\(824\) 0 0
\(825\) 0.740489 0.0257805
\(826\) 0 0
\(827\) −31.6998 −1.10231 −0.551155 0.834403i \(-0.685813\pi\)
−0.551155 + 0.834403i \(0.685813\pi\)
\(828\) 0 0
\(829\) −13.0328 −0.452647 −0.226323 0.974052i \(-0.572671\pi\)
−0.226323 + 0.974052i \(0.572671\pi\)
\(830\) 0 0
\(831\) −20.3082 −0.704485
\(832\) 0 0
\(833\) 35.1692 1.21854
\(834\) 0 0
\(835\) −21.7184 −0.751597
\(836\) 0 0
\(837\) −9.70682 −0.335517
\(838\) 0 0
\(839\) −2.93529 −0.101338 −0.0506688 0.998716i \(-0.516135\pi\)
−0.0506688 + 0.998716i \(0.516135\pi\)
\(840\) 0 0
\(841\) −21.9486 −0.756849
\(842\) 0 0
\(843\) −17.4003 −0.599298
\(844\) 0 0
\(845\) −2.39593 −0.0824226
\(846\) 0 0
\(847\) −1.05137 −0.0361256
\(848\) 0 0
\(849\) 31.1692 1.06972
\(850\) 0 0
\(851\) 41.3677 1.41807
\(852\) 0 0
\(853\) 39.4757 1.35162 0.675811 0.737075i \(-0.263794\pi\)
0.675811 + 0.737075i \(0.263794\pi\)
\(854\) 0 0
\(855\) −12.1027 −0.413905
\(856\) 0 0
\(857\) 42.4534 1.45018 0.725090 0.688654i \(-0.241798\pi\)
0.725090 + 0.688654i \(0.241798\pi\)
\(858\) 0 0
\(859\) 38.1329 1.30108 0.650538 0.759473i \(-0.274543\pi\)
0.650538 + 0.759473i \(0.274543\pi\)
\(860\) 0 0
\(861\) 3.75382 0.127930
\(862\) 0 0
\(863\) 18.3489 0.624605 0.312302 0.949983i \(-0.398900\pi\)
0.312302 + 0.949983i \(0.398900\pi\)
\(864\) 0 0
\(865\) 42.8847 1.45812
\(866\) 0 0
\(867\) −18.5971 −0.631589
\(868\) 0 0
\(869\) −11.4473 −0.388323
\(870\) 0 0
\(871\) 14.2258 0.482024
\(872\) 0 0
\(873\) −14.1027 −0.477306
\(874\) 0 0
\(875\) −10.7298 −0.362733
\(876\) 0 0
\(877\) −28.7599 −0.971154 −0.485577 0.874194i \(-0.661390\pi\)
−0.485577 + 0.874194i \(0.661390\pi\)
\(878\) 0 0
\(879\) −19.8432 −0.669296
\(880\) 0 0
\(881\) −41.5251 −1.39902 −0.699508 0.714624i \(-0.746598\pi\)
−0.699508 + 0.714624i \(0.746598\pi\)
\(882\) 0 0
\(883\) 1.03804 0.0349329 0.0174664 0.999847i \(-0.494440\pi\)
0.0174664 + 0.999847i \(0.494440\pi\)
\(884\) 0 0
\(885\) −27.3463 −0.919235
\(886\) 0 0
\(887\) −40.1329 −1.34753 −0.673765 0.738946i \(-0.735324\pi\)
−0.673765 + 0.738946i \(0.735324\pi\)
\(888\) 0 0
\(889\) 8.48362 0.284531
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 6.10275 0.204221
\(894\) 0 0
\(895\) −30.8627 −1.03163
\(896\) 0 0
\(897\) 9.15412 0.305647
\(898\) 0 0
\(899\) −25.7759 −0.859674
\(900\) 0 0
\(901\) 4.11149 0.136973
\(902\) 0 0
\(903\) −8.37559 −0.278722
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 18.2676 0.606565 0.303282 0.952901i \(-0.401917\pi\)
0.303282 + 0.952901i \(0.401917\pi\)
\(908\) 0 0
\(909\) −9.17446 −0.304298
\(910\) 0 0
\(911\) −33.5430 −1.11133 −0.555665 0.831406i \(-0.687536\pi\)
−0.555665 + 0.831406i \(0.687536\pi\)
\(912\) 0 0
\(913\) 15.4136 0.510117
\(914\) 0 0
\(915\) −16.5190 −0.546102
\(916\) 0 0
\(917\) −19.2488 −0.635652
\(918\) 0 0
\(919\) −14.6200 −0.482271 −0.241135 0.970492i \(-0.577520\pi\)
−0.241135 + 0.970492i \(0.577520\pi\)
\(920\) 0 0
\(921\) −13.5704 −0.447160
\(922\) 0 0
\(923\) 8.10275 0.266705
\(924\) 0 0
\(925\) −3.34629 −0.110025
\(926\) 0 0
\(927\) −2.62177 −0.0861102
\(928\) 0 0
\(929\) 12.2879 0.403153 0.201577 0.979473i \(-0.435394\pi\)
0.201577 + 0.979473i \(0.435394\pi\)
\(930\) 0 0
\(931\) −29.7759 −0.975865
\(932\) 0 0
\(933\) −24.3216 −0.796253
\(934\) 0 0
\(935\) −14.2949 −0.467494
\(936\) 0 0
\(937\) 0.930015 0.0303823 0.0151911 0.999885i \(-0.495164\pi\)
0.0151911 + 0.999885i \(0.495164\pi\)
\(938\) 0 0
\(939\) −18.6084 −0.607263
\(940\) 0 0
\(941\) −35.3277 −1.15165 −0.575825 0.817573i \(-0.695319\pi\)
−0.575825 + 0.817573i \(0.695319\pi\)
\(942\) 0 0
\(943\) −32.6838 −1.06433
\(944\) 0 0
\(945\) −2.51902 −0.0819438
\(946\) 0 0
\(947\) −7.65107 −0.248626 −0.124313 0.992243i \(-0.539673\pi\)
−0.124313 + 0.992243i \(0.539673\pi\)
\(948\) 0 0
\(949\) 2.25951 0.0733468
\(950\) 0 0
\(951\) −25.6395 −0.831417
\(952\) 0 0
\(953\) 56.3507 1.82538 0.912688 0.408656i \(-0.134002\pi\)
0.912688 + 0.408656i \(0.134002\pi\)
\(954\) 0 0
\(955\) −23.5518 −0.762118
\(956\) 0 0
\(957\) −2.65544 −0.0858383
\(958\) 0 0
\(959\) 19.1622 0.618779
\(960\) 0 0
\(961\) 63.2223 2.03943
\(962\) 0 0
\(963\) −6.37559 −0.205451
\(964\) 0 0
\(965\) 40.9388 1.31787
\(966\) 0 0
\(967\) −40.5625 −1.30440 −0.652201 0.758046i \(-0.726154\pi\)
−0.652201 + 0.758046i \(0.726154\pi\)
\(968\) 0 0
\(969\) 30.1382 0.968177
\(970\) 0 0
\(971\) −9.33296 −0.299509 −0.149754 0.988723i \(-0.547848\pi\)
−0.149754 + 0.988723i \(0.547848\pi\)
\(972\) 0 0
\(973\) −4.89461 −0.156914
\(974\) 0 0
\(975\) −0.740489 −0.0237146
\(976\) 0 0
\(977\) −51.5634 −1.64966 −0.824829 0.565382i \(-0.808729\pi\)
−0.824829 + 0.565382i \(0.808729\pi\)
\(978\) 0 0
\(979\) 11.8096 0.377435
\(980\) 0 0
\(981\) −1.22147 −0.0389985
\(982\) 0 0
\(983\) −27.3463 −0.872211 −0.436106 0.899896i \(-0.643643\pi\)
−0.436106 + 0.899896i \(0.643643\pi\)
\(984\) 0 0
\(985\) 13.7538 0.438233
\(986\) 0 0
\(987\) 1.27020 0.0404310
\(988\) 0 0
\(989\) 72.9247 2.31887
\(990\) 0 0
\(991\) 18.8679 0.599360 0.299680 0.954040i \(-0.403120\pi\)
0.299680 + 0.954040i \(0.403120\pi\)
\(992\) 0 0
\(993\) 33.7422 1.07078
\(994\) 0 0
\(995\) 38.8273 1.23091
\(996\) 0 0
\(997\) −3.41891 −0.108278 −0.0541390 0.998533i \(-0.517241\pi\)
−0.0541390 + 0.998533i \(0.517241\pi\)
\(998\) 0 0
\(999\) 4.51902 0.142975
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 6864.2.a.bn.1.2 3
4.3 odd 2 3432.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.n.1.2 3 4.3 odd 2
6864.2.a.bn.1.2 3 1.1 even 1 trivial