Properties

Label 8001.2.a.z.1.27
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 1.27
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} +O(q^{10})\) \(q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} -4.62544 q^{10} +2.89015 q^{11} +6.85121 q^{13} -1.96743 q^{14} -4.24174 q^{16} -2.39968 q^{17} -4.06451 q^{19} -4.39822 q^{20} +5.68618 q^{22} +2.32861 q^{23} +0.527232 q^{25} +13.4793 q^{26} -1.87078 q^{28} -3.14236 q^{29} -6.35057 q^{31} -7.83687 q^{32} -4.72121 q^{34} +2.35101 q^{35} +5.72445 q^{37} -7.99665 q^{38} +0.597690 q^{40} -5.44826 q^{41} -4.92110 q^{43} +5.40685 q^{44} +4.58138 q^{46} +6.42833 q^{47} +1.00000 q^{49} +1.03729 q^{50} +12.8171 q^{52} +13.2746 q^{53} -6.79477 q^{55} +0.254227 q^{56} -6.18238 q^{58} +4.10912 q^{59} +10.0147 q^{61} -12.4943 q^{62} -6.93502 q^{64} -16.1072 q^{65} -9.78035 q^{67} -4.48929 q^{68} +4.62544 q^{70} +0.218239 q^{71} -12.3655 q^{73} +11.2625 q^{74} -7.60382 q^{76} -2.89015 q^{77} -2.97850 q^{79} +9.97236 q^{80} -10.7191 q^{82} -13.0162 q^{83} +5.64167 q^{85} -9.68193 q^{86} -0.734756 q^{88} -7.20192 q^{89} -6.85121 q^{91} +4.35632 q^{92} +12.6473 q^{94} +9.55570 q^{95} -18.5452 q^{97} +1.96743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.96743 1.39118 0.695592 0.718437i \(-0.255142\pi\)
0.695592 + 0.718437i \(0.255142\pi\)
\(3\) 0 0
\(4\) 1.87078 0.935391
\(5\) −2.35101 −1.05140 −0.525701 0.850669i \(-0.676197\pi\)
−0.525701 + 0.850669i \(0.676197\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.254227 −0.0898829
\(9\) 0 0
\(10\) −4.62544 −1.46269
\(11\) 2.89015 0.871414 0.435707 0.900088i \(-0.356498\pi\)
0.435707 + 0.900088i \(0.356498\pi\)
\(12\) 0 0
\(13\) 6.85121 1.90018 0.950092 0.311970i \(-0.100989\pi\)
0.950092 + 0.311970i \(0.100989\pi\)
\(14\) −1.96743 −0.525818
\(15\) 0 0
\(16\) −4.24174 −1.06043
\(17\) −2.39968 −0.582009 −0.291005 0.956722i \(-0.593989\pi\)
−0.291005 + 0.956722i \(0.593989\pi\)
\(18\) 0 0
\(19\) −4.06451 −0.932464 −0.466232 0.884663i \(-0.654389\pi\)
−0.466232 + 0.884663i \(0.654389\pi\)
\(20\) −4.39822 −0.983472
\(21\) 0 0
\(22\) 5.68618 1.21230
\(23\) 2.32861 0.485549 0.242775 0.970083i \(-0.421942\pi\)
0.242775 + 0.970083i \(0.421942\pi\)
\(24\) 0 0
\(25\) 0.527232 0.105446
\(26\) 13.4793 2.64350
\(27\) 0 0
\(28\) −1.87078 −0.353545
\(29\) −3.14236 −0.583522 −0.291761 0.956491i \(-0.594241\pi\)
−0.291761 + 0.956491i \(0.594241\pi\)
\(30\) 0 0
\(31\) −6.35057 −1.14060 −0.570298 0.821438i \(-0.693172\pi\)
−0.570298 + 0.821438i \(0.693172\pi\)
\(32\) −7.83687 −1.38538
\(33\) 0 0
\(34\) −4.72121 −0.809681
\(35\) 2.35101 0.397393
\(36\) 0 0
\(37\) 5.72445 0.941095 0.470547 0.882375i \(-0.344057\pi\)
0.470547 + 0.882375i \(0.344057\pi\)
\(38\) −7.99665 −1.29723
\(39\) 0 0
\(40\) 0.597690 0.0945031
\(41\) −5.44826 −0.850875 −0.425438 0.904988i \(-0.639880\pi\)
−0.425438 + 0.904988i \(0.639880\pi\)
\(42\) 0 0
\(43\) −4.92110 −0.750461 −0.375231 0.926931i \(-0.622436\pi\)
−0.375231 + 0.926931i \(0.622436\pi\)
\(44\) 5.40685 0.815113
\(45\) 0 0
\(46\) 4.58138 0.675488
\(47\) 6.42833 0.937668 0.468834 0.883286i \(-0.344674\pi\)
0.468834 + 0.883286i \(0.344674\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.03729 0.146695
\(51\) 0 0
\(52\) 12.8171 1.77742
\(53\) 13.2746 1.82340 0.911702 0.410853i \(-0.134769\pi\)
0.911702 + 0.410853i \(0.134769\pi\)
\(54\) 0 0
\(55\) −6.79477 −0.916207
\(56\) 0.254227 0.0339726
\(57\) 0 0
\(58\) −6.18238 −0.811787
\(59\) 4.10912 0.534962 0.267481 0.963563i \(-0.413809\pi\)
0.267481 + 0.963563i \(0.413809\pi\)
\(60\) 0 0
\(61\) 10.0147 1.28225 0.641126 0.767436i \(-0.278468\pi\)
0.641126 + 0.767436i \(0.278468\pi\)
\(62\) −12.4943 −1.58678
\(63\) 0 0
\(64\) −6.93502 −0.866877
\(65\) −16.1072 −1.99786
\(66\) 0 0
\(67\) −9.78035 −1.19486 −0.597430 0.801921i \(-0.703811\pi\)
−0.597430 + 0.801921i \(0.703811\pi\)
\(68\) −4.48929 −0.544406
\(69\) 0 0
\(70\) 4.62544 0.552846
\(71\) 0.218239 0.0259003 0.0129501 0.999916i \(-0.495878\pi\)
0.0129501 + 0.999916i \(0.495878\pi\)
\(72\) 0 0
\(73\) −12.3655 −1.44727 −0.723635 0.690183i \(-0.757530\pi\)
−0.723635 + 0.690183i \(0.757530\pi\)
\(74\) 11.2625 1.30924
\(75\) 0 0
\(76\) −7.60382 −0.872218
\(77\) −2.89015 −0.329364
\(78\) 0 0
\(79\) −2.97850 −0.335107 −0.167553 0.985863i \(-0.553587\pi\)
−0.167553 + 0.985863i \(0.553587\pi\)
\(80\) 9.97236 1.11494
\(81\) 0 0
\(82\) −10.7191 −1.18372
\(83\) −13.0162 −1.42871 −0.714355 0.699783i \(-0.753280\pi\)
−0.714355 + 0.699783i \(0.753280\pi\)
\(84\) 0 0
\(85\) 5.64167 0.611926
\(86\) −9.68193 −1.04403
\(87\) 0 0
\(88\) −0.734756 −0.0783253
\(89\) −7.20192 −0.763402 −0.381701 0.924286i \(-0.624662\pi\)
−0.381701 + 0.924286i \(0.624662\pi\)
\(90\) 0 0
\(91\) −6.85121 −0.718202
\(92\) 4.35632 0.454178
\(93\) 0 0
\(94\) 12.6473 1.30447
\(95\) 9.55570 0.980394
\(96\) 0 0
\(97\) −18.5452 −1.88298 −0.941491 0.337039i \(-0.890574\pi\)
−0.941491 + 0.337039i \(0.890574\pi\)
\(98\) 1.96743 0.198740
\(99\) 0 0
\(100\) 0.986336 0.0986336
\(101\) 4.67088 0.464770 0.232385 0.972624i \(-0.425347\pi\)
0.232385 + 0.972624i \(0.425347\pi\)
\(102\) 0 0
\(103\) −5.40554 −0.532624 −0.266312 0.963887i \(-0.585805\pi\)
−0.266312 + 0.963887i \(0.585805\pi\)
\(104\) −1.74176 −0.170794
\(105\) 0 0
\(106\) 26.1168 2.53669
\(107\) −12.5665 −1.21485 −0.607425 0.794377i \(-0.707797\pi\)
−0.607425 + 0.794377i \(0.707797\pi\)
\(108\) 0 0
\(109\) −16.6939 −1.59898 −0.799492 0.600676i \(-0.794898\pi\)
−0.799492 + 0.600676i \(0.794898\pi\)
\(110\) −13.3682 −1.27461
\(111\) 0 0
\(112\) 4.24174 0.400807
\(113\) −4.88624 −0.459658 −0.229829 0.973231i \(-0.573817\pi\)
−0.229829 + 0.973231i \(0.573817\pi\)
\(114\) 0 0
\(115\) −5.47458 −0.510507
\(116\) −5.87868 −0.545822
\(117\) 0 0
\(118\) 8.08440 0.744230
\(119\) 2.39968 0.219979
\(120\) 0 0
\(121\) −2.64701 −0.240637
\(122\) 19.7032 1.78385
\(123\) 0 0
\(124\) −11.8805 −1.06690
\(125\) 10.5155 0.940536
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 2.02957 0.179391
\(129\) 0 0
\(130\) −31.6899 −2.77939
\(131\) −21.4230 −1.87173 −0.935866 0.352356i \(-0.885381\pi\)
−0.935866 + 0.352356i \(0.885381\pi\)
\(132\) 0 0
\(133\) 4.06451 0.352438
\(134\) −19.2422 −1.66227
\(135\) 0 0
\(136\) 0.610065 0.0523127
\(137\) 8.73744 0.746490 0.373245 0.927733i \(-0.378245\pi\)
0.373245 + 0.927733i \(0.378245\pi\)
\(138\) 0 0
\(139\) −7.92793 −0.672438 −0.336219 0.941784i \(-0.609148\pi\)
−0.336219 + 0.941784i \(0.609148\pi\)
\(140\) 4.39822 0.371718
\(141\) 0 0
\(142\) 0.429371 0.0360320
\(143\) 19.8011 1.65585
\(144\) 0 0
\(145\) 7.38772 0.613517
\(146\) −24.3282 −2.01342
\(147\) 0 0
\(148\) 10.7092 0.880291
\(149\) −0.536374 −0.0439415 −0.0219707 0.999759i \(-0.506994\pi\)
−0.0219707 + 0.999759i \(0.506994\pi\)
\(150\) 0 0
\(151\) 12.9566 1.05439 0.527196 0.849744i \(-0.323243\pi\)
0.527196 + 0.849744i \(0.323243\pi\)
\(152\) 1.03331 0.0838125
\(153\) 0 0
\(154\) −5.68618 −0.458205
\(155\) 14.9302 1.19923
\(156\) 0 0
\(157\) −1.78766 −0.142671 −0.0713353 0.997452i \(-0.522726\pi\)
−0.0713353 + 0.997452i \(0.522726\pi\)
\(158\) −5.85998 −0.466195
\(159\) 0 0
\(160\) 18.4245 1.45659
\(161\) −2.32861 −0.183520
\(162\) 0 0
\(163\) −20.0026 −1.56672 −0.783362 0.621566i \(-0.786497\pi\)
−0.783362 + 0.621566i \(0.786497\pi\)
\(164\) −10.1925 −0.795901
\(165\) 0 0
\(166\) −25.6084 −1.98760
\(167\) 8.46145 0.654767 0.327383 0.944892i \(-0.393833\pi\)
0.327383 + 0.944892i \(0.393833\pi\)
\(168\) 0 0
\(169\) 33.9391 2.61070
\(170\) 11.0996 0.851301
\(171\) 0 0
\(172\) −9.20631 −0.701975
\(173\) −17.8933 −1.36040 −0.680202 0.733024i \(-0.738108\pi\)
−0.680202 + 0.733024i \(0.738108\pi\)
\(174\) 0 0
\(175\) −0.527232 −0.0398550
\(176\) −12.2593 −0.924078
\(177\) 0 0
\(178\) −14.1693 −1.06203
\(179\) 2.60034 0.194359 0.0971793 0.995267i \(-0.469018\pi\)
0.0971793 + 0.995267i \(0.469018\pi\)
\(180\) 0 0
\(181\) 2.34080 0.173990 0.0869952 0.996209i \(-0.472274\pi\)
0.0869952 + 0.996209i \(0.472274\pi\)
\(182\) −13.4793 −0.999151
\(183\) 0 0
\(184\) −0.591997 −0.0436426
\(185\) −13.4582 −0.989469
\(186\) 0 0
\(187\) −6.93546 −0.507171
\(188\) 12.0260 0.877087
\(189\) 0 0
\(190\) 18.8002 1.36391
\(191\) 14.5839 1.05525 0.527626 0.849477i \(-0.323082\pi\)
0.527626 + 0.849477i \(0.323082\pi\)
\(192\) 0 0
\(193\) 4.35927 0.313787 0.156894 0.987615i \(-0.449852\pi\)
0.156894 + 0.987615i \(0.449852\pi\)
\(194\) −36.4864 −2.61957
\(195\) 0 0
\(196\) 1.87078 0.133627
\(197\) 21.7166 1.54724 0.773620 0.633649i \(-0.218444\pi\)
0.773620 + 0.633649i \(0.218444\pi\)
\(198\) 0 0
\(199\) −12.8349 −0.909842 −0.454921 0.890532i \(-0.650332\pi\)
−0.454921 + 0.890532i \(0.650332\pi\)
\(200\) −0.134037 −0.00947783
\(201\) 0 0
\(202\) 9.18963 0.646580
\(203\) 3.14236 0.220551
\(204\) 0 0
\(205\) 12.8089 0.894612
\(206\) −10.6350 −0.740977
\(207\) 0 0
\(208\) −29.0610 −2.01502
\(209\) −11.7471 −0.812562
\(210\) 0 0
\(211\) −16.0684 −1.10619 −0.553097 0.833117i \(-0.686554\pi\)
−0.553097 + 0.833117i \(0.686554\pi\)
\(212\) 24.8338 1.70560
\(213\) 0 0
\(214\) −24.7237 −1.69008
\(215\) 11.5695 0.789037
\(216\) 0 0
\(217\) 6.35057 0.431105
\(218\) −32.8441 −2.22448
\(219\) 0 0
\(220\) −12.7115 −0.857012
\(221\) −16.4407 −1.10592
\(222\) 0 0
\(223\) −21.4097 −1.43370 −0.716848 0.697229i \(-0.754416\pi\)
−0.716848 + 0.697229i \(0.754416\pi\)
\(224\) 7.83687 0.523623
\(225\) 0 0
\(226\) −9.61333 −0.639469
\(227\) −21.6474 −1.43679 −0.718395 0.695635i \(-0.755123\pi\)
−0.718395 + 0.695635i \(0.755123\pi\)
\(228\) 0 0
\(229\) −24.4516 −1.61580 −0.807902 0.589316i \(-0.799397\pi\)
−0.807902 + 0.589316i \(0.799397\pi\)
\(230\) −10.7709 −0.710209
\(231\) 0 0
\(232\) 0.798875 0.0524487
\(233\) −5.42084 −0.355131 −0.177566 0.984109i \(-0.556822\pi\)
−0.177566 + 0.984109i \(0.556822\pi\)
\(234\) 0 0
\(235\) −15.1130 −0.985867
\(236\) 7.68727 0.500398
\(237\) 0 0
\(238\) 4.72121 0.306031
\(239\) 13.9893 0.904895 0.452448 0.891791i \(-0.350551\pi\)
0.452448 + 0.891791i \(0.350551\pi\)
\(240\) 0 0
\(241\) 9.01626 0.580788 0.290394 0.956907i \(-0.406214\pi\)
0.290394 + 0.956907i \(0.406214\pi\)
\(242\) −5.20780 −0.334770
\(243\) 0 0
\(244\) 18.7353 1.19941
\(245\) −2.35101 −0.150200
\(246\) 0 0
\(247\) −27.8468 −1.77185
\(248\) 1.61449 0.102520
\(249\) 0 0
\(250\) 20.6885 1.30846
\(251\) 18.3298 1.15697 0.578485 0.815693i \(-0.303644\pi\)
0.578485 + 0.815693i \(0.303644\pi\)
\(252\) 0 0
\(253\) 6.73005 0.423114
\(254\) −1.96743 −0.123448
\(255\) 0 0
\(256\) 17.8631 1.11644
\(257\) 1.18621 0.0739939 0.0369970 0.999315i \(-0.488221\pi\)
0.0369970 + 0.999315i \(0.488221\pi\)
\(258\) 0 0
\(259\) −5.72445 −0.355700
\(260\) −30.1331 −1.86878
\(261\) 0 0
\(262\) −42.1482 −2.60392
\(263\) −13.8991 −0.857054 −0.428527 0.903529i \(-0.640967\pi\)
−0.428527 + 0.903529i \(0.640967\pi\)
\(264\) 0 0
\(265\) −31.2086 −1.91713
\(266\) 7.99665 0.490306
\(267\) 0 0
\(268\) −18.2969 −1.11766
\(269\) −24.4471 −1.49057 −0.745283 0.666748i \(-0.767686\pi\)
−0.745283 + 0.666748i \(0.767686\pi\)
\(270\) 0 0
\(271\) 5.36960 0.326180 0.163090 0.986611i \(-0.447854\pi\)
0.163090 + 0.986611i \(0.447854\pi\)
\(272\) 10.1788 0.617183
\(273\) 0 0
\(274\) 17.1903 1.03850
\(275\) 1.52378 0.0918875
\(276\) 0 0
\(277\) −10.2489 −0.615799 −0.307899 0.951419i \(-0.599626\pi\)
−0.307899 + 0.951419i \(0.599626\pi\)
\(278\) −15.5976 −0.935485
\(279\) 0 0
\(280\) −0.597690 −0.0357188
\(281\) −22.3072 −1.33074 −0.665369 0.746515i \(-0.731726\pi\)
−0.665369 + 0.746515i \(0.731726\pi\)
\(282\) 0 0
\(283\) 21.7199 1.29111 0.645557 0.763712i \(-0.276625\pi\)
0.645557 + 0.763712i \(0.276625\pi\)
\(284\) 0.408278 0.0242269
\(285\) 0 0
\(286\) 38.9572 2.30359
\(287\) 5.44826 0.321601
\(288\) 0 0
\(289\) −11.2415 −0.661266
\(290\) 14.5348 0.853514
\(291\) 0 0
\(292\) −23.1331 −1.35376
\(293\) 10.2743 0.600233 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(294\) 0 0
\(295\) −9.66057 −0.562460
\(296\) −1.45531 −0.0845883
\(297\) 0 0
\(298\) −1.05528 −0.0611306
\(299\) 15.9538 0.922633
\(300\) 0 0
\(301\) 4.92110 0.283648
\(302\) 25.4912 1.46685
\(303\) 0 0
\(304\) 17.2406 0.988817
\(305\) −23.5446 −1.34816
\(306\) 0 0
\(307\) −15.5926 −0.889915 −0.444957 0.895552i \(-0.646781\pi\)
−0.444957 + 0.895552i \(0.646781\pi\)
\(308\) −5.40685 −0.308084
\(309\) 0 0
\(310\) 29.3742 1.66834
\(311\) 7.50153 0.425373 0.212686 0.977121i \(-0.431779\pi\)
0.212686 + 0.977121i \(0.431779\pi\)
\(312\) 0 0
\(313\) 28.8638 1.63148 0.815739 0.578420i \(-0.196330\pi\)
0.815739 + 0.578420i \(0.196330\pi\)
\(314\) −3.51709 −0.198481
\(315\) 0 0
\(316\) −5.57212 −0.313456
\(317\) 20.0007 1.12335 0.561675 0.827358i \(-0.310157\pi\)
0.561675 + 0.827358i \(0.310157\pi\)
\(318\) 0 0
\(319\) −9.08192 −0.508490
\(320\) 16.3043 0.911437
\(321\) 0 0
\(322\) −4.58138 −0.255310
\(323\) 9.75355 0.542702
\(324\) 0 0
\(325\) 3.61218 0.200368
\(326\) −39.3537 −2.17960
\(327\) 0 0
\(328\) 1.38510 0.0764792
\(329\) −6.42833 −0.354405
\(330\) 0 0
\(331\) −11.3898 −0.626039 −0.313020 0.949747i \(-0.601341\pi\)
−0.313020 + 0.949747i \(0.601341\pi\)
\(332\) −24.3504 −1.33640
\(333\) 0 0
\(334\) 16.6473 0.910901
\(335\) 22.9937 1.25628
\(336\) 0 0
\(337\) 22.2950 1.21448 0.607242 0.794517i \(-0.292276\pi\)
0.607242 + 0.794517i \(0.292276\pi\)
\(338\) 66.7728 3.63196
\(339\) 0 0
\(340\) 10.5543 0.572390
\(341\) −18.3541 −0.993932
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.25108 0.0674537
\(345\) 0 0
\(346\) −35.2039 −1.89257
\(347\) 15.8395 0.850306 0.425153 0.905121i \(-0.360220\pi\)
0.425153 + 0.905121i \(0.360220\pi\)
\(348\) 0 0
\(349\) 17.2082 0.921133 0.460567 0.887625i \(-0.347646\pi\)
0.460567 + 0.887625i \(0.347646\pi\)
\(350\) −1.03729 −0.0554456
\(351\) 0 0
\(352\) −22.6498 −1.20724
\(353\) −7.97989 −0.424727 −0.212363 0.977191i \(-0.568116\pi\)
−0.212363 + 0.977191i \(0.568116\pi\)
\(354\) 0 0
\(355\) −0.513082 −0.0272316
\(356\) −13.4732 −0.714079
\(357\) 0 0
\(358\) 5.11599 0.270388
\(359\) −20.0243 −1.05684 −0.528420 0.848983i \(-0.677216\pi\)
−0.528420 + 0.848983i \(0.677216\pi\)
\(360\) 0 0
\(361\) −2.47972 −0.130512
\(362\) 4.60536 0.242053
\(363\) 0 0
\(364\) −12.8171 −0.671800
\(365\) 29.0713 1.52166
\(366\) 0 0
\(367\) 27.9152 1.45716 0.728581 0.684960i \(-0.240180\pi\)
0.728581 + 0.684960i \(0.240180\pi\)
\(368\) −9.87736 −0.514893
\(369\) 0 0
\(370\) −26.4781 −1.37653
\(371\) −13.2746 −0.689182
\(372\) 0 0
\(373\) 8.01565 0.415035 0.207517 0.978231i \(-0.433462\pi\)
0.207517 + 0.978231i \(0.433462\pi\)
\(374\) −13.6450 −0.705568
\(375\) 0 0
\(376\) −1.63426 −0.0842804
\(377\) −21.5290 −1.10880
\(378\) 0 0
\(379\) −2.39197 −0.122867 −0.0614336 0.998111i \(-0.519567\pi\)
−0.0614336 + 0.998111i \(0.519567\pi\)
\(380\) 17.8766 0.917052
\(381\) 0 0
\(382\) 28.6927 1.46805
\(383\) −19.5791 −1.00045 −0.500223 0.865896i \(-0.666749\pi\)
−0.500223 + 0.865896i \(0.666749\pi\)
\(384\) 0 0
\(385\) 6.79477 0.346294
\(386\) 8.57657 0.436536
\(387\) 0 0
\(388\) −34.6941 −1.76132
\(389\) 3.39595 0.172181 0.0860907 0.996287i \(-0.472563\pi\)
0.0860907 + 0.996287i \(0.472563\pi\)
\(390\) 0 0
\(391\) −5.58793 −0.282594
\(392\) −0.254227 −0.0128404
\(393\) 0 0
\(394\) 42.7258 2.15250
\(395\) 7.00246 0.352332
\(396\) 0 0
\(397\) −6.78359 −0.340459 −0.170229 0.985404i \(-0.554451\pi\)
−0.170229 + 0.985404i \(0.554451\pi\)
\(398\) −25.2518 −1.26576
\(399\) 0 0
\(400\) −2.23638 −0.111819
\(401\) 33.3109 1.66347 0.831734 0.555174i \(-0.187349\pi\)
0.831734 + 0.555174i \(0.187349\pi\)
\(402\) 0 0
\(403\) −43.5091 −2.16734
\(404\) 8.73820 0.434742
\(405\) 0 0
\(406\) 6.18238 0.306827
\(407\) 16.5446 0.820083
\(408\) 0 0
\(409\) 20.2438 1.00099 0.500497 0.865738i \(-0.333151\pi\)
0.500497 + 0.865738i \(0.333151\pi\)
\(410\) 25.2006 1.24457
\(411\) 0 0
\(412\) −10.1126 −0.498211
\(413\) −4.10912 −0.202197
\(414\) 0 0
\(415\) 30.6011 1.50215
\(416\) −53.6921 −2.63247
\(417\) 0 0
\(418\) −23.1115 −1.13042
\(419\) −28.0474 −1.37020 −0.685102 0.728447i \(-0.740242\pi\)
−0.685102 + 0.728447i \(0.740242\pi\)
\(420\) 0 0
\(421\) 20.8208 1.01474 0.507372 0.861727i \(-0.330617\pi\)
0.507372 + 0.861727i \(0.330617\pi\)
\(422\) −31.6134 −1.53892
\(423\) 0 0
\(424\) −3.37476 −0.163893
\(425\) −1.26519 −0.0613708
\(426\) 0 0
\(427\) −10.0147 −0.484646
\(428\) −23.5092 −1.13636
\(429\) 0 0
\(430\) 22.7623 1.09769
\(431\) 8.67253 0.417741 0.208870 0.977943i \(-0.433021\pi\)
0.208870 + 0.977943i \(0.433021\pi\)
\(432\) 0 0
\(433\) −3.89886 −0.187367 −0.0936837 0.995602i \(-0.529864\pi\)
−0.0936837 + 0.995602i \(0.529864\pi\)
\(434\) 12.4943 0.599746
\(435\) 0 0
\(436\) −31.2306 −1.49568
\(437\) −9.46467 −0.452757
\(438\) 0 0
\(439\) 8.02937 0.383221 0.191610 0.981471i \(-0.438629\pi\)
0.191610 + 0.981471i \(0.438629\pi\)
\(440\) 1.72742 0.0823514
\(441\) 0 0
\(442\) −32.3460 −1.53854
\(443\) −12.5240 −0.595036 −0.297518 0.954716i \(-0.596159\pi\)
−0.297518 + 0.954716i \(0.596159\pi\)
\(444\) 0 0
\(445\) 16.9318 0.802643
\(446\) −42.1220 −1.99453
\(447\) 0 0
\(448\) 6.93502 0.327649
\(449\) −8.48558 −0.400459 −0.200230 0.979749i \(-0.564169\pi\)
−0.200230 + 0.979749i \(0.564169\pi\)
\(450\) 0 0
\(451\) −15.7463 −0.741465
\(452\) −9.14108 −0.429960
\(453\) 0 0
\(454\) −42.5898 −1.99884
\(455\) 16.1072 0.755119
\(456\) 0 0
\(457\) −24.8695 −1.16335 −0.581673 0.813423i \(-0.697602\pi\)
−0.581673 + 0.813423i \(0.697602\pi\)
\(458\) −48.1067 −2.24788
\(459\) 0 0
\(460\) −10.2417 −0.477524
\(461\) −16.4708 −0.767121 −0.383560 0.923516i \(-0.625302\pi\)
−0.383560 + 0.923516i \(0.625302\pi\)
\(462\) 0 0
\(463\) 40.8908 1.90035 0.950177 0.311710i \(-0.100902\pi\)
0.950177 + 0.311710i \(0.100902\pi\)
\(464\) 13.3291 0.618787
\(465\) 0 0
\(466\) −10.6651 −0.494053
\(467\) −24.6291 −1.13970 −0.569850 0.821749i \(-0.692999\pi\)
−0.569850 + 0.821749i \(0.692999\pi\)
\(468\) 0 0
\(469\) 9.78035 0.451615
\(470\) −29.7339 −1.37152
\(471\) 0 0
\(472\) −1.04465 −0.0480839
\(473\) −14.2227 −0.653963
\(474\) 0 0
\(475\) −2.14294 −0.0983249
\(476\) 4.48929 0.205766
\(477\) 0 0
\(478\) 27.5231 1.25888
\(479\) −15.8536 −0.724369 −0.362184 0.932106i \(-0.617969\pi\)
−0.362184 + 0.932106i \(0.617969\pi\)
\(480\) 0 0
\(481\) 39.2194 1.78825
\(482\) 17.7389 0.807983
\(483\) 0 0
\(484\) −4.95197 −0.225090
\(485\) 43.5999 1.97977
\(486\) 0 0
\(487\) 10.3699 0.469904 0.234952 0.972007i \(-0.424507\pi\)
0.234952 + 0.972007i \(0.424507\pi\)
\(488\) −2.54601 −0.115253
\(489\) 0 0
\(490\) −4.62544 −0.208956
\(491\) 38.5101 1.73794 0.868968 0.494869i \(-0.164784\pi\)
0.868968 + 0.494869i \(0.164784\pi\)
\(492\) 0 0
\(493\) 7.54068 0.339615
\(494\) −54.7867 −2.46497
\(495\) 0 0
\(496\) 26.9375 1.20953
\(497\) −0.218239 −0.00978937
\(498\) 0 0
\(499\) −19.6920 −0.881536 −0.440768 0.897621i \(-0.645294\pi\)
−0.440768 + 0.897621i \(0.645294\pi\)
\(500\) 19.6722 0.879768
\(501\) 0 0
\(502\) 36.0627 1.60956
\(503\) −10.8754 −0.484909 −0.242454 0.970163i \(-0.577953\pi\)
−0.242454 + 0.970163i \(0.577953\pi\)
\(504\) 0 0
\(505\) −10.9813 −0.488660
\(506\) 13.2409 0.588630
\(507\) 0 0
\(508\) −1.87078 −0.0830025
\(509\) −36.5494 −1.62002 −0.810011 0.586415i \(-0.800539\pi\)
−0.810011 + 0.586415i \(0.800539\pi\)
\(510\) 0 0
\(511\) 12.3655 0.547016
\(512\) 31.0852 1.37379
\(513\) 0 0
\(514\) 2.33379 0.102939
\(515\) 12.7085 0.560002
\(516\) 0 0
\(517\) 18.5789 0.817098
\(518\) −11.2625 −0.494844
\(519\) 0 0
\(520\) 4.09490 0.179573
\(521\) 30.1034 1.31885 0.659426 0.751769i \(-0.270799\pi\)
0.659426 + 0.751769i \(0.270799\pi\)
\(522\) 0 0
\(523\) −33.0254 −1.44410 −0.722049 0.691842i \(-0.756800\pi\)
−0.722049 + 0.691842i \(0.756800\pi\)
\(524\) −40.0777 −1.75080
\(525\) 0 0
\(526\) −27.3455 −1.19232
\(527\) 15.2394 0.663837
\(528\) 0 0
\(529\) −17.5776 −0.764242
\(530\) −61.4008 −2.66708
\(531\) 0 0
\(532\) 7.60382 0.329667
\(533\) −37.3272 −1.61682
\(534\) 0 0
\(535\) 29.5439 1.27730
\(536\) 2.48643 0.107397
\(537\) 0 0
\(538\) −48.0980 −2.07365
\(539\) 2.89015 0.124488
\(540\) 0 0
\(541\) −3.86545 −0.166189 −0.0830943 0.996542i \(-0.526480\pi\)
−0.0830943 + 0.996542i \(0.526480\pi\)
\(542\) 10.5643 0.453776
\(543\) 0 0
\(544\) 18.8060 0.806301
\(545\) 39.2474 1.68118
\(546\) 0 0
\(547\) −4.24929 −0.181687 −0.0908433 0.995865i \(-0.528956\pi\)
−0.0908433 + 0.995865i \(0.528956\pi\)
\(548\) 16.3458 0.698260
\(549\) 0 0
\(550\) 2.99794 0.127832
\(551\) 12.7722 0.544113
\(552\) 0 0
\(553\) 2.97850 0.126659
\(554\) −20.1641 −0.856689
\(555\) 0 0
\(556\) −14.8314 −0.628993
\(557\) 42.3091 1.79269 0.896347 0.443352i \(-0.146211\pi\)
0.896347 + 0.443352i \(0.146211\pi\)
\(558\) 0 0
\(559\) −33.7155 −1.42601
\(560\) −9.97236 −0.421409
\(561\) 0 0
\(562\) −43.8879 −1.85130
\(563\) −32.4926 −1.36940 −0.684701 0.728825i \(-0.740067\pi\)
−0.684701 + 0.728825i \(0.740067\pi\)
\(564\) 0 0
\(565\) 11.4876 0.483286
\(566\) 42.7324 1.79618
\(567\) 0 0
\(568\) −0.0554824 −0.00232799
\(569\) 13.2130 0.553917 0.276959 0.960882i \(-0.410673\pi\)
0.276959 + 0.960882i \(0.410673\pi\)
\(570\) 0 0
\(571\) 11.5633 0.483907 0.241954 0.970288i \(-0.422212\pi\)
0.241954 + 0.970288i \(0.422212\pi\)
\(572\) 37.0435 1.54887
\(573\) 0 0
\(574\) 10.7191 0.447405
\(575\) 1.22772 0.0511994
\(576\) 0 0
\(577\) −31.1420 −1.29646 −0.648230 0.761445i \(-0.724490\pi\)
−0.648230 + 0.761445i \(0.724490\pi\)
\(578\) −22.1169 −0.919942
\(579\) 0 0
\(580\) 13.8208 0.573878
\(581\) 13.0162 0.540002
\(582\) 0 0
\(583\) 38.3656 1.58894
\(584\) 3.14364 0.130085
\(585\) 0 0
\(586\) 20.2141 0.835035
\(587\) 35.5404 1.46691 0.733455 0.679738i \(-0.237907\pi\)
0.733455 + 0.679738i \(0.237907\pi\)
\(588\) 0 0
\(589\) 25.8120 1.06356
\(590\) −19.0065 −0.782485
\(591\) 0 0
\(592\) −24.2816 −0.997969
\(593\) 47.0247 1.93107 0.965536 0.260271i \(-0.0838119\pi\)
0.965536 + 0.260271i \(0.0838119\pi\)
\(594\) 0 0
\(595\) −5.64167 −0.231286
\(596\) −1.00344 −0.0411025
\(597\) 0 0
\(598\) 31.3880 1.28355
\(599\) 24.5946 1.00491 0.502453 0.864604i \(-0.332431\pi\)
0.502453 + 0.864604i \(0.332431\pi\)
\(600\) 0 0
\(601\) −35.5297 −1.44929 −0.724643 0.689124i \(-0.757995\pi\)
−0.724643 + 0.689124i \(0.757995\pi\)
\(602\) 9.68193 0.394606
\(603\) 0 0
\(604\) 24.2390 0.986269
\(605\) 6.22313 0.253006
\(606\) 0 0
\(607\) 6.31589 0.256354 0.128177 0.991751i \(-0.459087\pi\)
0.128177 + 0.991751i \(0.459087\pi\)
\(608\) 31.8531 1.29181
\(609\) 0 0
\(610\) −46.3224 −1.87554
\(611\) 44.0419 1.78174
\(612\) 0 0
\(613\) −44.7009 −1.80545 −0.902727 0.430215i \(-0.858438\pi\)
−0.902727 + 0.430215i \(0.858438\pi\)
\(614\) −30.6773 −1.23803
\(615\) 0 0
\(616\) 0.734756 0.0296042
\(617\) 2.00783 0.0808322 0.0404161 0.999183i \(-0.487132\pi\)
0.0404161 + 0.999183i \(0.487132\pi\)
\(618\) 0 0
\(619\) 23.2979 0.936421 0.468211 0.883617i \(-0.344899\pi\)
0.468211 + 0.883617i \(0.344899\pi\)
\(620\) 27.9312 1.12174
\(621\) 0 0
\(622\) 14.7587 0.591771
\(623\) 7.20192 0.288539
\(624\) 0 0
\(625\) −27.3582 −1.09433
\(626\) 56.7875 2.26969
\(627\) 0 0
\(628\) −3.34432 −0.133453
\(629\) −13.7369 −0.547725
\(630\) 0 0
\(631\) 38.1985 1.52066 0.760329 0.649538i \(-0.225038\pi\)
0.760329 + 0.649538i \(0.225038\pi\)
\(632\) 0.757215 0.0301204
\(633\) 0 0
\(634\) 39.3499 1.56279
\(635\) 2.35101 0.0932969
\(636\) 0 0
\(637\) 6.85121 0.271455
\(638\) −17.8680 −0.707403
\(639\) 0 0
\(640\) −4.77154 −0.188612
\(641\) 48.1272 1.90091 0.950454 0.310864i \(-0.100618\pi\)
0.950454 + 0.310864i \(0.100618\pi\)
\(642\) 0 0
\(643\) 26.4929 1.04478 0.522390 0.852707i \(-0.325041\pi\)
0.522390 + 0.852707i \(0.325041\pi\)
\(644\) −4.35632 −0.171663
\(645\) 0 0
\(646\) 19.1894 0.754998
\(647\) 22.9476 0.902164 0.451082 0.892483i \(-0.351038\pi\)
0.451082 + 0.892483i \(0.351038\pi\)
\(648\) 0 0
\(649\) 11.8760 0.466173
\(650\) 7.10671 0.278748
\(651\) 0 0
\(652\) −37.4205 −1.46550
\(653\) 21.4210 0.838268 0.419134 0.907924i \(-0.362334\pi\)
0.419134 + 0.907924i \(0.362334\pi\)
\(654\) 0 0
\(655\) 50.3655 1.96794
\(656\) 23.1101 0.902298
\(657\) 0 0
\(658\) −12.6473 −0.493043
\(659\) 48.0153 1.87041 0.935205 0.354106i \(-0.115215\pi\)
0.935205 + 0.354106i \(0.115215\pi\)
\(660\) 0 0
\(661\) 13.4557 0.523367 0.261683 0.965154i \(-0.415722\pi\)
0.261683 + 0.965154i \(0.415722\pi\)
\(662\) −22.4086 −0.870935
\(663\) 0 0
\(664\) 3.30907 0.128417
\(665\) −9.55570 −0.370554
\(666\) 0 0
\(667\) −7.31735 −0.283329
\(668\) 15.8295 0.612463
\(669\) 0 0
\(670\) 45.2384 1.74771
\(671\) 28.9440 1.11737
\(672\) 0 0
\(673\) −31.6378 −1.21955 −0.609773 0.792576i \(-0.708739\pi\)
−0.609773 + 0.792576i \(0.708739\pi\)
\(674\) 43.8638 1.68957
\(675\) 0 0
\(676\) 63.4927 2.44203
\(677\) −9.63313 −0.370231 −0.185116 0.982717i \(-0.559266\pi\)
−0.185116 + 0.982717i \(0.559266\pi\)
\(678\) 0 0
\(679\) 18.5452 0.711700
\(680\) −1.43427 −0.0550017
\(681\) 0 0
\(682\) −36.1105 −1.38274
\(683\) −11.6247 −0.444808 −0.222404 0.974955i \(-0.571390\pi\)
−0.222404 + 0.974955i \(0.571390\pi\)
\(684\) 0 0
\(685\) −20.5418 −0.784861
\(686\) −1.96743 −0.0751168
\(687\) 0 0
\(688\) 20.8740 0.795815
\(689\) 90.9469 3.46480
\(690\) 0 0
\(691\) −12.6074 −0.479608 −0.239804 0.970821i \(-0.577083\pi\)
−0.239804 + 0.970821i \(0.577083\pi\)
\(692\) −33.4745 −1.27251
\(693\) 0 0
\(694\) 31.1630 1.18293
\(695\) 18.6386 0.707003
\(696\) 0 0
\(697\) 13.0741 0.495217
\(698\) 33.8559 1.28147
\(699\) 0 0
\(700\) −0.986336 −0.0372800
\(701\) 8.45875 0.319483 0.159741 0.987159i \(-0.448934\pi\)
0.159741 + 0.987159i \(0.448934\pi\)
\(702\) 0 0
\(703\) −23.2671 −0.877536
\(704\) −20.0433 −0.755409
\(705\) 0 0
\(706\) −15.6999 −0.590873
\(707\) −4.67088 −0.175667
\(708\) 0 0
\(709\) −28.2276 −1.06011 −0.530056 0.847963i \(-0.677829\pi\)
−0.530056 + 0.847963i \(0.677829\pi\)
\(710\) −1.00945 −0.0378841
\(711\) 0 0
\(712\) 1.83092 0.0686168
\(713\) −14.7880 −0.553815
\(714\) 0 0
\(715\) −46.5524 −1.74096
\(716\) 4.86467 0.181801
\(717\) 0 0
\(718\) −39.3964 −1.47026
\(719\) −20.8897 −0.779056 −0.389528 0.921015i \(-0.627362\pi\)
−0.389528 + 0.921015i \(0.627362\pi\)
\(720\) 0 0
\(721\) 5.40554 0.201313
\(722\) −4.87868 −0.181566
\(723\) 0 0
\(724\) 4.37913 0.162749
\(725\) −1.65676 −0.0615304
\(726\) 0 0
\(727\) −21.4574 −0.795811 −0.397906 0.917426i \(-0.630263\pi\)
−0.397906 + 0.917426i \(0.630263\pi\)
\(728\) 1.74176 0.0645541
\(729\) 0 0
\(730\) 57.1958 2.11691
\(731\) 11.8091 0.436775
\(732\) 0 0
\(733\) −48.1668 −1.77908 −0.889540 0.456857i \(-0.848975\pi\)
−0.889540 + 0.456857i \(0.848975\pi\)
\(734\) 54.9212 2.02718
\(735\) 0 0
\(736\) −18.2490 −0.672668
\(737\) −28.2667 −1.04122
\(738\) 0 0
\(739\) 33.3207 1.22572 0.612861 0.790191i \(-0.290019\pi\)
0.612861 + 0.790191i \(0.290019\pi\)
\(740\) −25.1774 −0.925540
\(741\) 0 0
\(742\) −26.1168 −0.958778
\(743\) 16.0300 0.588085 0.294043 0.955792i \(-0.404999\pi\)
0.294043 + 0.955792i \(0.404999\pi\)
\(744\) 0 0
\(745\) 1.26102 0.0462002
\(746\) 15.7702 0.577389
\(747\) 0 0
\(748\) −12.9747 −0.474403
\(749\) 12.5665 0.459170
\(750\) 0 0
\(751\) −31.2938 −1.14193 −0.570964 0.820975i \(-0.693430\pi\)
−0.570964 + 0.820975i \(0.693430\pi\)
\(752\) −27.2673 −0.994336
\(753\) 0 0
\(754\) −42.3568 −1.54254
\(755\) −30.4610 −1.10859
\(756\) 0 0
\(757\) 6.16678 0.224136 0.112068 0.993701i \(-0.464253\pi\)
0.112068 + 0.993701i \(0.464253\pi\)
\(758\) −4.70604 −0.170931
\(759\) 0 0
\(760\) −2.42932 −0.0881207
\(761\) −42.7410 −1.54936 −0.774680 0.632353i \(-0.782089\pi\)
−0.774680 + 0.632353i \(0.782089\pi\)
\(762\) 0 0
\(763\) 16.6939 0.604359
\(764\) 27.2832 0.987072
\(765\) 0 0
\(766\) −38.5206 −1.39181
\(767\) 28.1524 1.01653
\(768\) 0 0
\(769\) 17.4128 0.627923 0.313961 0.949436i \(-0.398344\pi\)
0.313961 + 0.949436i \(0.398344\pi\)
\(770\) 13.3682 0.481758
\(771\) 0 0
\(772\) 8.15525 0.293514
\(773\) 29.8993 1.07540 0.537701 0.843135i \(-0.319293\pi\)
0.537701 + 0.843135i \(0.319293\pi\)
\(774\) 0 0
\(775\) −3.34823 −0.120272
\(776\) 4.71470 0.169248
\(777\) 0 0
\(778\) 6.68129 0.239536
\(779\) 22.1445 0.793410
\(780\) 0 0
\(781\) 0.630746 0.0225699
\(782\) −10.9939 −0.393140
\(783\) 0 0
\(784\) −4.24174 −0.151491
\(785\) 4.20279 0.150004
\(786\) 0 0
\(787\) −25.7598 −0.918238 −0.459119 0.888375i \(-0.651835\pi\)
−0.459119 + 0.888375i \(0.651835\pi\)
\(788\) 40.6269 1.44728
\(789\) 0 0
\(790\) 13.7769 0.490159
\(791\) 4.88624 0.173735
\(792\) 0 0
\(793\) 68.6129 2.43651
\(794\) −13.3462 −0.473640
\(795\) 0 0
\(796\) −24.0113 −0.851058
\(797\) 23.7025 0.839584 0.419792 0.907620i \(-0.362103\pi\)
0.419792 + 0.907620i \(0.362103\pi\)
\(798\) 0 0
\(799\) −15.4260 −0.545731
\(800\) −4.13185 −0.146083
\(801\) 0 0
\(802\) 65.5369 2.31419
\(803\) −35.7381 −1.26117
\(804\) 0 0
\(805\) 5.47458 0.192954
\(806\) −85.6011 −3.01517
\(807\) 0 0
\(808\) −1.18747 −0.0417749
\(809\) 46.8159 1.64596 0.822981 0.568070i \(-0.192310\pi\)
0.822981 + 0.568070i \(0.192310\pi\)
\(810\) 0 0
\(811\) −21.3447 −0.749515 −0.374758 0.927123i \(-0.622274\pi\)
−0.374758 + 0.927123i \(0.622274\pi\)
\(812\) 5.87868 0.206301
\(813\) 0 0
\(814\) 32.5503 1.14089
\(815\) 47.0262 1.64726
\(816\) 0 0
\(817\) 20.0019 0.699778
\(818\) 39.8283 1.39257
\(819\) 0 0
\(820\) 23.9627 0.836812
\(821\) −25.9217 −0.904673 −0.452337 0.891847i \(-0.649409\pi\)
−0.452337 + 0.891847i \(0.649409\pi\)
\(822\) 0 0
\(823\) 18.6224 0.649137 0.324568 0.945862i \(-0.394781\pi\)
0.324568 + 0.945862i \(0.394781\pi\)
\(824\) 1.37424 0.0478738
\(825\) 0 0
\(826\) −8.08440 −0.281292
\(827\) 3.32877 0.115753 0.0578763 0.998324i \(-0.481567\pi\)
0.0578763 + 0.998324i \(0.481567\pi\)
\(828\) 0 0
\(829\) 32.9044 1.14282 0.571409 0.820665i \(-0.306397\pi\)
0.571409 + 0.820665i \(0.306397\pi\)
\(830\) 60.2056 2.08977
\(831\) 0 0
\(832\) −47.5133 −1.64723
\(833\) −2.39968 −0.0831441
\(834\) 0 0
\(835\) −19.8929 −0.688423
\(836\) −21.9762 −0.760063
\(837\) 0 0
\(838\) −55.1813 −1.90620
\(839\) 32.9296 1.13686 0.568428 0.822733i \(-0.307552\pi\)
0.568428 + 0.822733i \(0.307552\pi\)
\(840\) 0 0
\(841\) −19.1255 −0.659502
\(842\) 40.9635 1.41170
\(843\) 0 0
\(844\) −30.0605 −1.03472
\(845\) −79.7910 −2.74490
\(846\) 0 0
\(847\) 2.64701 0.0909522
\(848\) −56.3073 −1.93360
\(849\) 0 0
\(850\) −2.48917 −0.0853780
\(851\) 13.3300 0.456948
\(852\) 0 0
\(853\) 19.4639 0.666431 0.333215 0.942851i \(-0.391866\pi\)
0.333215 + 0.942851i \(0.391866\pi\)
\(854\) −19.7032 −0.674231
\(855\) 0 0
\(856\) 3.19475 0.109194
\(857\) −27.8327 −0.950749 −0.475374 0.879784i \(-0.657687\pi\)
−0.475374 + 0.879784i \(0.657687\pi\)
\(858\) 0 0
\(859\) 50.4371 1.72089 0.860447 0.509541i \(-0.170185\pi\)
0.860447 + 0.509541i \(0.170185\pi\)
\(860\) 21.6441 0.738058
\(861\) 0 0
\(862\) 17.0626 0.581154
\(863\) −33.6526 −1.14555 −0.572774 0.819714i \(-0.694133\pi\)
−0.572774 + 0.819714i \(0.694133\pi\)
\(864\) 0 0
\(865\) 42.0673 1.43033
\(866\) −7.67074 −0.260662
\(867\) 0 0
\(868\) 11.8805 0.403252
\(869\) −8.60831 −0.292017
\(870\) 0 0
\(871\) −67.0072 −2.27045
\(872\) 4.24404 0.143721
\(873\) 0 0
\(874\) −18.6211 −0.629868
\(875\) −10.5155 −0.355489
\(876\) 0 0
\(877\) −45.8712 −1.54896 −0.774480 0.632598i \(-0.781989\pi\)
−0.774480 + 0.632598i \(0.781989\pi\)
\(878\) 15.7972 0.533131
\(879\) 0 0
\(880\) 28.8216 0.971578
\(881\) 46.2258 1.55739 0.778694 0.627404i \(-0.215882\pi\)
0.778694 + 0.627404i \(0.215882\pi\)
\(882\) 0 0
\(883\) 25.5211 0.858854 0.429427 0.903102i \(-0.358716\pi\)
0.429427 + 0.903102i \(0.358716\pi\)
\(884\) −30.7571 −1.03447
\(885\) 0 0
\(886\) −24.6402 −0.827803
\(887\) −54.5602 −1.83195 −0.915976 0.401232i \(-0.868582\pi\)
−0.915976 + 0.401232i \(0.868582\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 33.3121 1.11662
\(891\) 0 0
\(892\) −40.0528 −1.34107
\(893\) −26.1280 −0.874342
\(894\) 0 0
\(895\) −6.11342 −0.204349
\(896\) −2.02957 −0.0678033
\(897\) 0 0
\(898\) −16.6948 −0.557112
\(899\) 19.9558 0.665564
\(900\) 0 0
\(901\) −31.8548 −1.06124
\(902\) −30.9798 −1.03151
\(903\) 0 0
\(904\) 1.24221 0.0413154
\(905\) −5.50324 −0.182934
\(906\) 0 0
\(907\) 13.5809 0.450947 0.225474 0.974249i \(-0.427607\pi\)
0.225474 + 0.974249i \(0.427607\pi\)
\(908\) −40.4976 −1.34396
\(909\) 0 0
\(910\) 31.6899 1.05051
\(911\) 23.8052 0.788700 0.394350 0.918960i \(-0.370970\pi\)
0.394350 + 0.918960i \(0.370970\pi\)
\(912\) 0 0
\(913\) −37.6188 −1.24500
\(914\) −48.9290 −1.61843
\(915\) 0 0
\(916\) −45.7435 −1.51141
\(917\) 21.4230 0.707448
\(918\) 0 0
\(919\) −6.64193 −0.219097 −0.109549 0.993981i \(-0.534941\pi\)
−0.109549 + 0.993981i \(0.534941\pi\)
\(920\) 1.39179 0.0458859
\(921\) 0 0
\(922\) −32.4051 −1.06721
\(923\) 1.49520 0.0492152
\(924\) 0 0
\(925\) 3.01812 0.0992351
\(926\) 80.4497 2.64374
\(927\) 0 0
\(928\) 24.6263 0.808398
\(929\) 18.6804 0.612883 0.306441 0.951890i \(-0.400862\pi\)
0.306441 + 0.951890i \(0.400862\pi\)
\(930\) 0 0
\(931\) −4.06451 −0.133209
\(932\) −10.1412 −0.332187
\(933\) 0 0
\(934\) −48.4561 −1.58553
\(935\) 16.3053 0.533241
\(936\) 0 0
\(937\) 36.7818 1.20161 0.600805 0.799396i \(-0.294847\pi\)
0.600805 + 0.799396i \(0.294847\pi\)
\(938\) 19.2422 0.628279
\(939\) 0 0
\(940\) −28.2732 −0.922171
\(941\) 5.52071 0.179970 0.0899850 0.995943i \(-0.471318\pi\)
0.0899850 + 0.995943i \(0.471318\pi\)
\(942\) 0 0
\(943\) −12.6869 −0.413142
\(944\) −17.4298 −0.567292
\(945\) 0 0
\(946\) −27.9823 −0.909782
\(947\) −50.0953 −1.62788 −0.813940 0.580949i \(-0.802682\pi\)
−0.813940 + 0.580949i \(0.802682\pi\)
\(948\) 0 0
\(949\) −84.7185 −2.75008
\(950\) −4.21609 −0.136788
\(951\) 0 0
\(952\) −0.610065 −0.0197723
\(953\) 19.8040 0.641514 0.320757 0.947162i \(-0.396063\pi\)
0.320757 + 0.947162i \(0.396063\pi\)
\(954\) 0 0
\(955\) −34.2868 −1.10949
\(956\) 26.1710 0.846431
\(957\) 0 0
\(958\) −31.1908 −1.00773
\(959\) −8.73744 −0.282147
\(960\) 0 0
\(961\) 9.32975 0.300960
\(962\) 77.1615 2.48779
\(963\) 0 0
\(964\) 16.8675 0.543264
\(965\) −10.2487 −0.329917
\(966\) 0 0
\(967\) 20.7804 0.668253 0.334127 0.942528i \(-0.391559\pi\)
0.334127 + 0.942528i \(0.391559\pi\)
\(968\) 0.672941 0.0216292
\(969\) 0 0
\(970\) 85.7798 2.75422
\(971\) 32.0142 1.02739 0.513693 0.857974i \(-0.328277\pi\)
0.513693 + 0.857974i \(0.328277\pi\)
\(972\) 0 0
\(973\) 7.92793 0.254158
\(974\) 20.4020 0.653723
\(975\) 0 0
\(976\) −42.4798 −1.35974
\(977\) 30.7497 0.983772 0.491886 0.870660i \(-0.336308\pi\)
0.491886 + 0.870660i \(0.336308\pi\)
\(978\) 0 0
\(979\) −20.8147 −0.665240
\(980\) −4.39822 −0.140496
\(981\) 0 0
\(982\) 75.7659 2.41779
\(983\) 43.4959 1.38730 0.693651 0.720311i \(-0.256001\pi\)
0.693651 + 0.720311i \(0.256001\pi\)
\(984\) 0 0
\(985\) −51.0558 −1.62677
\(986\) 14.8358 0.472467
\(987\) 0 0
\(988\) −52.0954 −1.65737
\(989\) −11.4593 −0.364386
\(990\) 0 0
\(991\) −31.8498 −1.01174 −0.505871 0.862609i \(-0.668829\pi\)
−0.505871 + 0.862609i \(0.668829\pi\)
\(992\) 49.7686 1.58015
\(993\) 0 0
\(994\) −0.429371 −0.0136188
\(995\) 30.1749 0.956610
\(996\) 0 0
\(997\) 20.8478 0.660257 0.330129 0.943936i \(-0.392908\pi\)
0.330129 + 0.943936i \(0.392908\pi\)
\(998\) −38.7427 −1.22638
\(999\) 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.1.27 yes 32
3.2 odd 2 inner 8001.2.a.z.1.6 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.6 32 3.2 odd 2 inner
8001.2.a.z.1.27 yes 32 1.1 even 1 trivial