Properties

Label 10000.2.a.c.1.2
Level $10000$
Weight $2$
Character 10000.1
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.8504020213\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 10000.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.61803 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.61803 q^{7} -2.00000 q^{9} +0.763932 q^{11} -4.85410 q^{13} -0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{21} -8.23607 q^{23} +5.00000 q^{27} -1.38197 q^{29} +3.00000 q^{31} -0.763932 q^{33} +4.23607 q^{37} +4.85410 q^{39} -5.23607 q^{41} -1.85410 q^{43} +1.61803 q^{47} -4.38197 q^{49} +0.763932 q^{51} +5.47214 q^{53} -5.85410 q^{57} +4.14590 q^{59} -4.70820 q^{61} -3.23607 q^{63} -9.23607 q^{67} +8.23607 q^{69} +4.38197 q^{71} -9.00000 q^{73} +1.23607 q^{77} -3.09017 q^{79} +1.00000 q^{81} +1.76393 q^{83} +1.38197 q^{87} +8.94427 q^{89} -7.85410 q^{91} -3.00000 q^{93} +2.85410 q^{97} -1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + q^{7} - 4 q^{9} + 6 q^{11} - 3 q^{13} - 6 q^{17} + 5 q^{19} - q^{21} - 12 q^{23} + 10 q^{27} - 5 q^{29} + 6 q^{31} - 6 q^{33} + 4 q^{37} + 3 q^{39} - 6 q^{41} + 3 q^{43} + q^{47} - 11 q^{49} + 6 q^{51} + 2 q^{53} - 5 q^{57} + 15 q^{59} + 4 q^{61} - 2 q^{63} - 14 q^{67} + 12 q^{69} + 11 q^{71} - 18 q^{73} - 2 q^{77} + 5 q^{79} + 2 q^{81} + 8 q^{83} + 5 q^{87} - 9 q^{91} - 6 q^{93} - q^{97} - 12 q^{99} + O(q^{100}) \)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) −4.85410 −1.34629 −0.673143 0.739512i \(-0.735056\pi\)
−0.673143 + 0.739512i \(0.735056\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) 5.85410 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) 0 0
\(21\) −1.61803 −0.353084
\(22\) 0 0
\(23\) −8.23607 −1.71734 −0.858669 0.512530i \(-0.828708\pi\)
−0.858669 + 0.512530i \(0.828708\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −1.38197 −0.256625 −0.128312 0.991734i \(-0.540956\pi\)
−0.128312 + 0.991734i \(0.540956\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −0.763932 −0.132983
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.23607 0.696405 0.348203 0.937419i \(-0.386792\pi\)
0.348203 + 0.937419i \(0.386792\pi\)
\(38\) 0 0
\(39\) 4.85410 0.777278
\(40\) 0 0
\(41\) −5.23607 −0.817736 −0.408868 0.912593i \(-0.634076\pi\)
−0.408868 + 0.912593i \(0.634076\pi\)
\(42\) 0 0
\(43\) −1.85410 −0.282748 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.61803 0.236015 0.118007 0.993013i \(-0.462349\pi\)
0.118007 + 0.993013i \(0.462349\pi\)
\(48\) 0 0
\(49\) −4.38197 −0.625995
\(50\) 0 0
\(51\) 0.763932 0.106972
\(52\) 0 0
\(53\) 5.47214 0.751656 0.375828 0.926690i \(-0.377358\pi\)
0.375828 + 0.926690i \(0.377358\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.85410 −0.775395
\(58\) 0 0
\(59\) 4.14590 0.539750 0.269875 0.962895i \(-0.413018\pi\)
0.269875 + 0.962895i \(0.413018\pi\)
\(60\) 0 0
\(61\) −4.70820 −0.602824 −0.301412 0.953494i \(-0.597458\pi\)
−0.301412 + 0.953494i \(0.597458\pi\)
\(62\) 0 0
\(63\) −3.23607 −0.407706
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.23607 −1.12837 −0.564183 0.825650i \(-0.690809\pi\)
−0.564183 + 0.825650i \(0.690809\pi\)
\(68\) 0 0
\(69\) 8.23607 0.991506
\(70\) 0 0
\(71\) 4.38197 0.520044 0.260022 0.965603i \(-0.416270\pi\)
0.260022 + 0.965603i \(0.416270\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.23607 0.140863
\(78\) 0 0
\(79\) −3.09017 −0.347671 −0.173836 0.984775i \(-0.555616\pi\)
−0.173836 + 0.984775i \(0.555616\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.76393 0.193617 0.0968083 0.995303i \(-0.469137\pi\)
0.0968083 + 0.995303i \(0.469137\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.38197 0.148162
\(88\) 0 0
\(89\) 8.94427 0.948091 0.474045 0.880500i \(-0.342793\pi\)
0.474045 + 0.880500i \(0.342793\pi\)
\(90\) 0 0
\(91\) −7.85410 −0.823334
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.85410 0.289790 0.144895 0.989447i \(-0.453716\pi\)
0.144895 + 0.989447i \(0.453716\pi\)
\(98\) 0 0
\(99\) −1.52786 −0.153556
\(100\) 0 0
\(101\) −7.47214 −0.743505 −0.371753 0.928332i \(-0.621243\pi\)
−0.371753 + 0.928332i \(0.621243\pi\)
\(102\) 0 0
\(103\) 11.5623 1.13927 0.569634 0.821899i \(-0.307085\pi\)
0.569634 + 0.821899i \(0.307085\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.4164 −1.00699 −0.503496 0.863998i \(-0.667953\pi\)
−0.503496 + 0.863998i \(0.667953\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −4.23607 −0.402070
\(112\) 0 0
\(113\) 10.1459 0.954446 0.477223 0.878782i \(-0.341643\pi\)
0.477223 + 0.878782i \(0.341643\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.70820 0.897524
\(118\) 0 0
\(119\) −1.23607 −0.113310
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) 5.23607 0.472120
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8885 1.40988 0.704940 0.709267i \(-0.250974\pi\)
0.704940 + 0.709267i \(0.250974\pi\)
\(128\) 0 0
\(129\) 1.85410 0.163245
\(130\) 0 0
\(131\) 17.7984 1.55505 0.777526 0.628851i \(-0.216475\pi\)
0.777526 + 0.628851i \(0.216475\pi\)
\(132\) 0 0
\(133\) 9.47214 0.821338
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.94427 0.507853 0.253927 0.967223i \(-0.418278\pi\)
0.253927 + 0.967223i \(0.418278\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −1.61803 −0.136263
\(142\) 0 0
\(143\) −3.70820 −0.310096
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.38197 0.361418
\(148\) 0 0
\(149\) 13.9443 1.14236 0.571180 0.820825i \(-0.306486\pi\)
0.571180 + 0.820825i \(0.306486\pi\)
\(150\) 0 0
\(151\) 5.56231 0.452654 0.226327 0.974051i \(-0.427328\pi\)
0.226327 + 0.974051i \(0.427328\pi\)
\(152\) 0 0
\(153\) 1.52786 0.123520
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.18034 −0.732671 −0.366335 0.930483i \(-0.619388\pi\)
−0.366335 + 0.930483i \(0.619388\pi\)
\(158\) 0 0
\(159\) −5.47214 −0.433969
\(160\) 0 0
\(161\) −13.3262 −1.05025
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.56231 0.430424 0.215212 0.976567i \(-0.430956\pi\)
0.215212 + 0.976567i \(0.430956\pi\)
\(168\) 0 0
\(169\) 10.5623 0.812485
\(170\) 0 0
\(171\) −11.7082 −0.895349
\(172\) 0 0
\(173\) −16.8885 −1.28401 −0.642006 0.766700i \(-0.721898\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.14590 −0.311625
\(178\) 0 0
\(179\) 9.47214 0.707981 0.353990 0.935249i \(-0.384825\pi\)
0.353990 + 0.935249i \(0.384825\pi\)
\(180\) 0 0
\(181\) 13.7082 1.01892 0.509461 0.860494i \(-0.329845\pi\)
0.509461 + 0.860494i \(0.329845\pi\)
\(182\) 0 0
\(183\) 4.70820 0.348040
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.583592 −0.0426765
\(188\) 0 0
\(189\) 8.09017 0.588473
\(190\) 0 0
\(191\) 24.1803 1.74963 0.874814 0.484459i \(-0.160984\pi\)
0.874814 + 0.484459i \(0.160984\pi\)
\(192\) 0 0
\(193\) −5.70820 −0.410886 −0.205443 0.978669i \(-0.565863\pi\)
−0.205443 + 0.978669i \(0.565863\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.70820 −0.691681 −0.345840 0.938293i \(-0.612406\pi\)
−0.345840 + 0.938293i \(0.612406\pi\)
\(198\) 0 0
\(199\) −2.56231 −0.181637 −0.0908185 0.995867i \(-0.528948\pi\)
−0.0908185 + 0.995867i \(0.528948\pi\)
\(200\) 0 0
\(201\) 9.23607 0.651462
\(202\) 0 0
\(203\) −2.23607 −0.156941
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.4721 1.14489
\(208\) 0 0
\(209\) 4.47214 0.309344
\(210\) 0 0
\(211\) −13.1803 −0.907372 −0.453686 0.891162i \(-0.649891\pi\)
−0.453686 + 0.891162i \(0.649891\pi\)
\(212\) 0 0
\(213\) −4.38197 −0.300247
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.85410 0.329518
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 3.70820 0.249441
\(222\) 0 0
\(223\) −22.1803 −1.48531 −0.742653 0.669677i \(-0.766433\pi\)
−0.742653 + 0.669677i \(0.766433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.2361 −1.27674 −0.638371 0.769729i \(-0.720392\pi\)
−0.638371 + 0.769729i \(0.720392\pi\)
\(228\) 0 0
\(229\) 8.29180 0.547937 0.273969 0.961739i \(-0.411664\pi\)
0.273969 + 0.961739i \(0.411664\pi\)
\(230\) 0 0
\(231\) −1.23607 −0.0813273
\(232\) 0 0
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.09017 0.200728
\(238\) 0 0
\(239\) 29.4721 1.90639 0.953197 0.302350i \(-0.0977711\pi\)
0.953197 + 0.302350i \(0.0977711\pi\)
\(240\) 0 0
\(241\) 11.4721 0.738985 0.369493 0.929234i \(-0.379532\pi\)
0.369493 + 0.929234i \(0.379532\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −28.4164 −1.80809
\(248\) 0 0
\(249\) −1.76393 −0.111785
\(250\) 0 0
\(251\) 6.81966 0.430453 0.215227 0.976564i \(-0.430951\pi\)
0.215227 + 0.976564i \(0.430951\pi\)
\(252\) 0 0
\(253\) −6.29180 −0.395562
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.1459 1.00715 0.503577 0.863951i \(-0.332017\pi\)
0.503577 + 0.863951i \(0.332017\pi\)
\(258\) 0 0
\(259\) 6.85410 0.425893
\(260\) 0 0
\(261\) 2.76393 0.171083
\(262\) 0 0
\(263\) 22.0902 1.36214 0.681069 0.732219i \(-0.261515\pi\)
0.681069 + 0.732219i \(0.261515\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8.94427 −0.547381
\(268\) 0 0
\(269\) −17.2361 −1.05090 −0.525451 0.850824i \(-0.676103\pi\)
−0.525451 + 0.850824i \(0.676103\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 7.85410 0.475352
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.2918 −0.678458 −0.339229 0.940704i \(-0.610166\pi\)
−0.339229 + 0.940704i \(0.610166\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −1.09017 −0.0650341 −0.0325170 0.999471i \(-0.510352\pi\)
−0.0325170 + 0.999471i \(0.510352\pi\)
\(282\) 0 0
\(283\) 23.1459 1.37588 0.687940 0.725767i \(-0.258515\pi\)
0.687940 + 0.725767i \(0.258515\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.47214 −0.500094
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) −2.85410 −0.167310
\(292\) 0 0
\(293\) −28.4721 −1.66336 −0.831680 0.555255i \(-0.812621\pi\)
−0.831680 + 0.555255i \(0.812621\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.81966 0.221639
\(298\) 0 0
\(299\) 39.9787 2.31203
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 7.47214 0.429263
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.76393 −0.271892 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(308\) 0 0
\(309\) −11.5623 −0.657757
\(310\) 0 0
\(311\) 29.5066 1.67316 0.836582 0.547841i \(-0.184550\pi\)
0.836582 + 0.547841i \(0.184550\pi\)
\(312\) 0 0
\(313\) −21.2361 −1.20033 −0.600167 0.799875i \(-0.704899\pi\)
−0.600167 + 0.799875i \(0.704899\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −23.6525 −1.32846 −0.664228 0.747530i \(-0.731239\pi\)
−0.664228 + 0.747530i \(0.731239\pi\)
\(318\) 0 0
\(319\) −1.05573 −0.0591094
\(320\) 0 0
\(321\) 10.4164 0.581387
\(322\) 0 0
\(323\) −4.47214 −0.248836
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 2.61803 0.144337
\(330\) 0 0
\(331\) −17.1246 −0.941254 −0.470627 0.882332i \(-0.655972\pi\)
−0.470627 + 0.882332i \(0.655972\pi\)
\(332\) 0 0
\(333\) −8.47214 −0.464270
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.14590 0.0624210 0.0312105 0.999513i \(-0.490064\pi\)
0.0312105 + 0.999513i \(0.490064\pi\)
\(338\) 0 0
\(339\) −10.1459 −0.551050
\(340\) 0 0
\(341\) 2.29180 0.124108
\(342\) 0 0
\(343\) −18.4164 −0.994393
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 31.0902 1.66901 0.834504 0.551002i \(-0.185754\pi\)
0.834504 + 0.551002i \(0.185754\pi\)
\(348\) 0 0
\(349\) 8.29180 0.443850 0.221925 0.975064i \(-0.428766\pi\)
0.221925 + 0.975064i \(0.428766\pi\)
\(350\) 0 0
\(351\) −24.2705 −1.29546
\(352\) 0 0
\(353\) 24.0902 1.28219 0.641095 0.767461i \(-0.278480\pi\)
0.641095 + 0.767461i \(0.278480\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.23607 0.0654197
\(358\) 0 0
\(359\) 28.7426 1.51698 0.758489 0.651685i \(-0.225938\pi\)
0.758489 + 0.651685i \(0.225938\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) 0 0
\(363\) 10.4164 0.546720
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.43769 0.283845 0.141923 0.989878i \(-0.454672\pi\)
0.141923 + 0.989878i \(0.454672\pi\)
\(368\) 0 0
\(369\) 10.4721 0.545158
\(370\) 0 0
\(371\) 8.85410 0.459682
\(372\) 0 0
\(373\) 5.27051 0.272897 0.136448 0.990647i \(-0.456431\pi\)
0.136448 + 0.990647i \(0.456431\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.70820 0.345490
\(378\) 0 0
\(379\) 34.5967 1.77712 0.888558 0.458765i \(-0.151708\pi\)
0.888558 + 0.458765i \(0.151708\pi\)
\(380\) 0 0
\(381\) −15.8885 −0.813995
\(382\) 0 0
\(383\) 11.3607 0.580504 0.290252 0.956950i \(-0.406261\pi\)
0.290252 + 0.956950i \(0.406261\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.70820 0.188499
\(388\) 0 0
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 6.29180 0.318190
\(392\) 0 0
\(393\) −17.7984 −0.897809
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.0344419 −0.00172859 −0.000864294 1.00000i \(-0.500275\pi\)
−0.000864294 1.00000i \(0.500275\pi\)
\(398\) 0 0
\(399\) −9.47214 −0.474200
\(400\) 0 0
\(401\) −22.5967 −1.12843 −0.564214 0.825629i \(-0.690821\pi\)
−0.564214 + 0.825629i \(0.690821\pi\)
\(402\) 0 0
\(403\) −14.5623 −0.725400
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.23607 0.160406
\(408\) 0 0
\(409\) 28.4164 1.40510 0.702550 0.711634i \(-0.252045\pi\)
0.702550 + 0.711634i \(0.252045\pi\)
\(410\) 0 0
\(411\) −5.94427 −0.293209
\(412\) 0 0
\(413\) 6.70820 0.330089
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 0.527864 0.0257878 0.0128939 0.999917i \(-0.495896\pi\)
0.0128939 + 0.999917i \(0.495896\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 0 0
\(423\) −3.23607 −0.157343
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.61803 −0.368663
\(428\) 0 0
\(429\) 3.70820 0.179034
\(430\) 0 0
\(431\) −23.8328 −1.14799 −0.573993 0.818860i \(-0.694606\pi\)
−0.573993 + 0.818860i \(0.694606\pi\)
\(432\) 0 0
\(433\) 20.1459 0.968150 0.484075 0.875026i \(-0.339156\pi\)
0.484075 + 0.875026i \(0.339156\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −48.2148 −2.30643
\(438\) 0 0
\(439\) −5.97871 −0.285348 −0.142674 0.989770i \(-0.545570\pi\)
−0.142674 + 0.989770i \(0.545570\pi\)
\(440\) 0 0
\(441\) 8.76393 0.417330
\(442\) 0 0
\(443\) −12.0557 −0.572785 −0.286392 0.958112i \(-0.592456\pi\)
−0.286392 + 0.958112i \(0.592456\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.9443 −0.659541
\(448\) 0 0
\(449\) 20.3262 0.959254 0.479627 0.877472i \(-0.340772\pi\)
0.479627 + 0.877472i \(0.340772\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) −5.56231 −0.261340
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.41641 0.253369 0.126684 0.991943i \(-0.459566\pi\)
0.126684 + 0.991943i \(0.459566\pi\)
\(458\) 0 0
\(459\) −3.81966 −0.178286
\(460\) 0 0
\(461\) 23.1803 1.07962 0.539808 0.841788i \(-0.318497\pi\)
0.539808 + 0.841788i \(0.318497\pi\)
\(462\) 0 0
\(463\) −16.1246 −0.749374 −0.374687 0.927151i \(-0.622250\pi\)
−0.374687 + 0.927151i \(0.622250\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.4508 1.31655 0.658274 0.752778i \(-0.271287\pi\)
0.658274 + 0.752778i \(0.271287\pi\)
\(468\) 0 0
\(469\) −14.9443 −0.690062
\(470\) 0 0
\(471\) 9.18034 0.423008
\(472\) 0 0
\(473\) −1.41641 −0.0651265
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.9443 −0.501104
\(478\) 0 0
\(479\) 4.14590 0.189431 0.0947155 0.995504i \(-0.469806\pi\)
0.0947155 + 0.995504i \(0.469806\pi\)
\(480\) 0 0
\(481\) −20.5623 −0.937560
\(482\) 0 0
\(483\) 13.3262 0.606365
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.58359 0.434274 0.217137 0.976141i \(-0.430328\pi\)
0.217137 + 0.976141i \(0.430328\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) −37.2492 −1.68103 −0.840517 0.541785i \(-0.817749\pi\)
−0.840517 + 0.541785i \(0.817749\pi\)
\(492\) 0 0
\(493\) 1.05573 0.0475476
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.09017 0.318038
\(498\) 0 0
\(499\) 12.5623 0.562366 0.281183 0.959654i \(-0.409273\pi\)
0.281183 + 0.959654i \(0.409273\pi\)
\(500\) 0 0
\(501\) −5.56231 −0.248506
\(502\) 0 0
\(503\) 10.5836 0.471899 0.235950 0.971765i \(-0.424180\pi\)
0.235950 + 0.971765i \(0.424180\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.5623 −0.469088
\(508\) 0 0
\(509\) −4.67376 −0.207161 −0.103580 0.994621i \(-0.533030\pi\)
−0.103580 + 0.994621i \(0.533030\pi\)
\(510\) 0 0
\(511\) −14.5623 −0.644198
\(512\) 0 0
\(513\) 29.2705 1.29232
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.23607 0.0543622
\(518\) 0 0
\(519\) 16.8885 0.741325
\(520\) 0 0
\(521\) −15.3607 −0.672964 −0.336482 0.941690i \(-0.609237\pi\)
−0.336482 + 0.941690i \(0.609237\pi\)
\(522\) 0 0
\(523\) 19.8541 0.868159 0.434080 0.900875i \(-0.357074\pi\)
0.434080 + 0.900875i \(0.357074\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.29180 −0.0998322
\(528\) 0 0
\(529\) 44.8328 1.94925
\(530\) 0 0
\(531\) −8.29180 −0.359833
\(532\) 0 0
\(533\) 25.4164 1.10091
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9.47214 −0.408753
\(538\) 0 0
\(539\) −3.34752 −0.144188
\(540\) 0 0
\(541\) −13.1246 −0.564271 −0.282136 0.959375i \(-0.591043\pi\)
−0.282136 + 0.959375i \(0.591043\pi\)
\(542\) 0 0
\(543\) −13.7082 −0.588275
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 34.7082 1.48402 0.742008 0.670391i \(-0.233874\pi\)
0.742008 + 0.670391i \(0.233874\pi\)
\(548\) 0 0
\(549\) 9.41641 0.401882
\(550\) 0 0
\(551\) −8.09017 −0.344653
\(552\) 0 0
\(553\) −5.00000 −0.212622
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.23607 0.391345 0.195672 0.980669i \(-0.437311\pi\)
0.195672 + 0.980669i \(0.437311\pi\)
\(558\) 0 0
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) 0.583592 0.0246393
\(562\) 0 0
\(563\) −9.61803 −0.405352 −0.202676 0.979246i \(-0.564964\pi\)
−0.202676 + 0.979246i \(0.564964\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.61803 0.0679510
\(568\) 0 0
\(569\) 29.4721 1.23554 0.617768 0.786360i \(-0.288037\pi\)
0.617768 + 0.786360i \(0.288037\pi\)
\(570\) 0 0
\(571\) −32.1246 −1.34437 −0.672187 0.740382i \(-0.734645\pi\)
−0.672187 + 0.740382i \(0.734645\pi\)
\(572\) 0 0
\(573\) −24.1803 −1.01015
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 37.7771 1.57268 0.786340 0.617794i \(-0.211973\pi\)
0.786340 + 0.617794i \(0.211973\pi\)
\(578\) 0 0
\(579\) 5.70820 0.237225
\(580\) 0 0
\(581\) 2.85410 0.118408
\(582\) 0 0
\(583\) 4.18034 0.173132
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.7082 −0.772170 −0.386085 0.922463i \(-0.626173\pi\)
−0.386085 + 0.922463i \(0.626173\pi\)
\(588\) 0 0
\(589\) 17.5623 0.723642
\(590\) 0 0
\(591\) 9.70820 0.399342
\(592\) 0 0
\(593\) −22.0902 −0.907135 −0.453567 0.891222i \(-0.649849\pi\)
−0.453567 + 0.891222i \(0.649849\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.56231 0.104868
\(598\) 0 0
\(599\) 0.527864 0.0215679 0.0107840 0.999942i \(-0.496567\pi\)
0.0107840 + 0.999942i \(0.496567\pi\)
\(600\) 0 0
\(601\) 36.2705 1.47950 0.739752 0.672879i \(-0.234943\pi\)
0.739752 + 0.672879i \(0.234943\pi\)
\(602\) 0 0
\(603\) 18.4721 0.752244
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.4377 0.626597 0.313298 0.949655i \(-0.398566\pi\)
0.313298 + 0.949655i \(0.398566\pi\)
\(608\) 0 0
\(609\) 2.23607 0.0906100
\(610\) 0 0
\(611\) −7.85410 −0.317743
\(612\) 0 0
\(613\) 31.9787 1.29161 0.645804 0.763503i \(-0.276522\pi\)
0.645804 + 0.763503i \(0.276522\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.76393 0.393081 0.196541 0.980496i \(-0.437029\pi\)
0.196541 + 0.980496i \(0.437029\pi\)
\(618\) 0 0
\(619\) −39.4721 −1.58652 −0.793260 0.608884i \(-0.791618\pi\)
−0.793260 + 0.608884i \(0.791618\pi\)
\(620\) 0 0
\(621\) −41.1803 −1.65251
\(622\) 0 0
\(623\) 14.4721 0.579814
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.47214 −0.178600
\(628\) 0 0
\(629\) −3.23607 −0.129030
\(630\) 0 0
\(631\) 5.76393 0.229459 0.114729 0.993397i \(-0.463400\pi\)
0.114729 + 0.993397i \(0.463400\pi\)
\(632\) 0 0
\(633\) 13.1803 0.523871
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 21.2705 0.842768
\(638\) 0 0
\(639\) −8.76393 −0.346696
\(640\) 0 0
\(641\) 10.0902 0.398538 0.199269 0.979945i \(-0.436143\pi\)
0.199269 + 0.979945i \(0.436143\pi\)
\(642\) 0 0
\(643\) −22.8328 −0.900438 −0.450219 0.892918i \(-0.648654\pi\)
−0.450219 + 0.892918i \(0.648654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.5410 −1.20069 −0.600346 0.799741i \(-0.704970\pi\)
−0.600346 + 0.799741i \(0.704970\pi\)
\(648\) 0 0
\(649\) 3.16718 0.124323
\(650\) 0 0
\(651\) −4.85410 −0.190247
\(652\) 0 0
\(653\) 7.90983 0.309536 0.154768 0.987951i \(-0.450537\pi\)
0.154768 + 0.987951i \(0.450537\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) −24.4721 −0.953299 −0.476650 0.879093i \(-0.658149\pi\)
−0.476650 + 0.879093i \(0.658149\pi\)
\(660\) 0 0
\(661\) −40.6869 −1.58254 −0.791269 0.611468i \(-0.790579\pi\)
−0.791269 + 0.611468i \(0.790579\pi\)
\(662\) 0 0
\(663\) −3.70820 −0.144015
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.3820 0.440711
\(668\) 0 0
\(669\) 22.1803 0.857541
\(670\) 0 0
\(671\) −3.59675 −0.138851
\(672\) 0 0
\(673\) −10.1803 −0.392423 −0.196212 0.980562i \(-0.562864\pi\)
−0.196212 + 0.980562i \(0.562864\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.38197 0.322145 0.161073 0.986943i \(-0.448505\pi\)
0.161073 + 0.986943i \(0.448505\pi\)
\(678\) 0 0
\(679\) 4.61803 0.177224
\(680\) 0 0
\(681\) 19.2361 0.737128
\(682\) 0 0
\(683\) 4.52786 0.173254 0.0866270 0.996241i \(-0.472391\pi\)
0.0866270 + 0.996241i \(0.472391\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.29180 −0.316352
\(688\) 0 0
\(689\) −26.5623 −1.01194
\(690\) 0 0
\(691\) −2.72949 −0.103835 −0.0519173 0.998651i \(-0.516533\pi\)
−0.0519173 + 0.998651i \(0.516533\pi\)
\(692\) 0 0
\(693\) −2.47214 −0.0939087
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −14.9443 −0.565244
\(700\) 0 0
\(701\) 35.0132 1.32243 0.661214 0.750197i \(-0.270041\pi\)
0.661214 + 0.750197i \(0.270041\pi\)
\(702\) 0 0
\(703\) 24.7984 0.935288
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0902 −0.454698
\(708\) 0 0
\(709\) 33.5410 1.25966 0.629830 0.776733i \(-0.283125\pi\)
0.629830 + 0.776733i \(0.283125\pi\)
\(710\) 0 0
\(711\) 6.18034 0.231781
\(712\) 0 0
\(713\) −24.7082 −0.925330
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −29.4721 −1.10066
\(718\) 0 0
\(719\) 36.7082 1.36899 0.684493 0.729020i \(-0.260024\pi\)
0.684493 + 0.729020i \(0.260024\pi\)
\(720\) 0 0
\(721\) 18.7082 0.696730
\(722\) 0 0
\(723\) −11.4721 −0.426653
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −4.43769 −0.164585 −0.0822925 0.996608i \(-0.526224\pi\)
−0.0822925 + 0.996608i \(0.526224\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 1.41641 0.0523877
\(732\) 0 0
\(733\) 26.9787 0.996482 0.498241 0.867039i \(-0.333980\pi\)
0.498241 + 0.867039i \(0.333980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.05573 −0.259901
\(738\) 0 0
\(739\) 30.9787 1.13957 0.569785 0.821794i \(-0.307026\pi\)
0.569785 + 0.821794i \(0.307026\pi\)
\(740\) 0 0
\(741\) 28.4164 1.04390
\(742\) 0 0
\(743\) 16.3607 0.600215 0.300108 0.953905i \(-0.402977\pi\)
0.300108 + 0.953905i \(0.402977\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.52786 −0.129078
\(748\) 0 0
\(749\) −16.8541 −0.615835
\(750\) 0 0
\(751\) 40.8885 1.49204 0.746022 0.665921i \(-0.231961\pi\)
0.746022 + 0.665921i \(0.231961\pi\)
\(752\) 0 0
\(753\) −6.81966 −0.248522
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.58359 0.130248 0.0651239 0.997877i \(-0.479256\pi\)
0.0651239 + 0.997877i \(0.479256\pi\)
\(758\) 0 0
\(759\) 6.29180 0.228378
\(760\) 0 0
\(761\) 37.4508 1.35759 0.678796 0.734327i \(-0.262502\pi\)
0.678796 + 0.734327i \(0.262502\pi\)
\(762\) 0 0
\(763\) 16.1803 0.585768
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.1246 −0.726658
\(768\) 0 0
\(769\) −13.4164 −0.483808 −0.241904 0.970300i \(-0.577772\pi\)
−0.241904 + 0.970300i \(0.577772\pi\)
\(770\) 0 0
\(771\) −16.1459 −0.581480
\(772\) 0 0
\(773\) 33.1591 1.19265 0.596324 0.802744i \(-0.296627\pi\)
0.596324 + 0.802744i \(0.296627\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.85410 −0.245890
\(778\) 0 0
\(779\) −30.6525 −1.09824
\(780\) 0 0
\(781\) 3.34752 0.119784
\(782\) 0 0
\(783\) −6.90983 −0.246937
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.1803 1.21840 0.609199 0.793018i \(-0.291491\pi\)
0.609199 + 0.793018i \(0.291491\pi\)
\(788\) 0 0
\(789\) −22.0902 −0.786431
\(790\) 0 0
\(791\) 16.4164 0.583700
\(792\) 0 0
\(793\) 22.8541 0.811573
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.2361 0.504267 0.252134 0.967692i \(-0.418868\pi\)
0.252134 + 0.967692i \(0.418868\pi\)
\(798\) 0 0
\(799\) −1.23607 −0.0437289
\(800\) 0 0
\(801\) −17.8885 −0.632061
\(802\) 0 0
\(803\) −6.87539 −0.242627
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17.2361 0.606738
\(808\) 0 0
\(809\) −15.9787 −0.561782 −0.280891 0.959740i \(-0.590630\pi\)
−0.280891 + 0.959740i \(0.590630\pi\)
\(810\) 0 0
\(811\) 1.29180 0.0453611 0.0226805 0.999743i \(-0.492780\pi\)
0.0226805 + 0.999743i \(0.492780\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.8541 −0.379737
\(818\) 0 0
\(819\) 15.7082 0.548889
\(820\) 0 0
\(821\) 19.6869 0.687078 0.343539 0.939138i \(-0.388374\pi\)
0.343539 + 0.939138i \(0.388374\pi\)
\(822\) 0 0
\(823\) −34.2918 −1.19534 −0.597668 0.801743i \(-0.703906\pi\)
−0.597668 + 0.801743i \(0.703906\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.0344 1.04440 0.522200 0.852823i \(-0.325111\pi\)
0.522200 + 0.852823i \(0.325111\pi\)
\(828\) 0 0
\(829\) −29.1459 −1.01228 −0.506139 0.862452i \(-0.668928\pi\)
−0.506139 + 0.862452i \(0.668928\pi\)
\(830\) 0 0
\(831\) 11.2918 0.391708
\(832\) 0 0
\(833\) 3.34752 0.115985
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 0 0
\(839\) −4.14590 −0.143132 −0.0715661 0.997436i \(-0.522800\pi\)
−0.0715661 + 0.997436i \(0.522800\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) 0 0
\(843\) 1.09017 0.0375474
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −16.8541 −0.579114
\(848\) 0 0
\(849\) −23.1459 −0.794365
\(850\) 0 0
\(851\) −34.8885 −1.19596
\(852\) 0 0
\(853\) 47.3050 1.61969 0.809845 0.586643i \(-0.199551\pi\)
0.809845 + 0.586643i \(0.199551\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.6869 −1.38984 −0.694919 0.719088i \(-0.744560\pi\)
−0.694919 + 0.719088i \(0.744560\pi\)
\(858\) 0 0
\(859\) 28.4164 0.969555 0.484778 0.874637i \(-0.338901\pi\)
0.484778 + 0.874637i \(0.338901\pi\)
\(860\) 0 0
\(861\) 8.47214 0.288730
\(862\) 0 0
\(863\) 41.5623 1.41480 0.707399 0.706815i \(-0.249869\pi\)
0.707399 + 0.706815i \(0.249869\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.4164 0.557530
\(868\) 0 0
\(869\) −2.36068 −0.0800806
\(870\) 0 0
\(871\) 44.8328 1.51910
\(872\) 0 0
\(873\) −5.70820 −0.193193
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.5410 1.03130 0.515648 0.856800i \(-0.327551\pi\)
0.515648 + 0.856800i \(0.327551\pi\)
\(878\) 0 0
\(879\) 28.4721 0.960341
\(880\) 0 0
\(881\) 4.36068 0.146915 0.0734575 0.997298i \(-0.476597\pi\)
0.0734575 + 0.997298i \(0.476597\pi\)
\(882\) 0 0
\(883\) 47.4164 1.59569 0.797845 0.602863i \(-0.205974\pi\)
0.797845 + 0.602863i \(0.205974\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.88854 0.197718 0.0988590 0.995101i \(-0.468481\pi\)
0.0988590 + 0.995101i \(0.468481\pi\)
\(888\) 0 0
\(889\) 25.7082 0.862225
\(890\) 0 0
\(891\) 0.763932 0.0255927
\(892\) 0 0
\(893\) 9.47214 0.316973
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −39.9787 −1.33485
\(898\) 0 0
\(899\) −4.14590 −0.138273
\(900\) 0 0
\(901\) −4.18034 −0.139267
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −47.2492 −1.56888 −0.784442 0.620202i \(-0.787051\pi\)
−0.784442 + 0.620202i \(0.787051\pi\)
\(908\) 0 0
\(909\) 14.9443 0.495670
\(910\) 0 0
\(911\) 35.7639 1.18491 0.592456 0.805603i \(-0.298158\pi\)
0.592456 + 0.805603i \(0.298158\pi\)
\(912\) 0 0
\(913\) 1.34752 0.0445965
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28.7984 0.951006
\(918\) 0 0
\(919\) 1.78522 0.0588889 0.0294445 0.999566i \(-0.490626\pi\)
0.0294445 + 0.999566i \(0.490626\pi\)
\(920\) 0 0
\(921\) 4.76393 0.156977
\(922\) 0 0
\(923\) −21.2705 −0.700127
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −23.1246 −0.759512
\(928\) 0 0
\(929\) −36.6312 −1.20183 −0.600915 0.799313i \(-0.705197\pi\)
−0.600915 + 0.799313i \(0.705197\pi\)
\(930\) 0 0
\(931\) −25.6525 −0.840726
\(932\) 0 0
\(933\) −29.5066 −0.966002
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 51.2705 1.67493 0.837467 0.546487i \(-0.184035\pi\)
0.837467 + 0.546487i \(0.184035\pi\)
\(938\) 0 0
\(939\) 21.2361 0.693013
\(940\) 0 0
\(941\) −19.5836 −0.638407 −0.319203 0.947686i \(-0.603415\pi\)
−0.319203 + 0.947686i \(0.603415\pi\)
\(942\) 0 0
\(943\) 43.1246 1.40433
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.6525 0.931080 0.465540 0.885027i \(-0.345860\pi\)
0.465540 + 0.885027i \(0.345860\pi\)
\(948\) 0 0
\(949\) 43.6869 1.41814
\(950\) 0 0
\(951\) 23.6525 0.766984
\(952\) 0 0
\(953\) 34.7426 1.12542 0.562712 0.826653i \(-0.309758\pi\)
0.562712 + 0.826653i \(0.309758\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.05573 0.0341268
\(958\) 0 0
\(959\) 9.61803 0.310583
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 20.8328 0.671328
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −39.8885 −1.28273 −0.641365 0.767236i \(-0.721631\pi\)
−0.641365 + 0.767236i \(0.721631\pi\)
\(968\) 0 0
\(969\) 4.47214 0.143666
\(970\) 0 0
\(971\) −3.38197 −0.108532 −0.0542662 0.998527i \(-0.517282\pi\)
−0.0542662 + 0.998527i \(0.517282\pi\)
\(972\) 0 0
\(973\) −8.09017 −0.259359
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.6525 −1.07664 −0.538319 0.842741i \(-0.680940\pi\)
−0.538319 + 0.842741i \(0.680940\pi\)
\(978\) 0 0
\(979\) 6.83282 0.218378
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 0 0
\(983\) −7.38197 −0.235448 −0.117724 0.993046i \(-0.537560\pi\)
−0.117724 + 0.993046i \(0.537560\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.61803 −0.0833329
\(988\) 0 0
\(989\) 15.2705 0.485574
\(990\) 0 0
\(991\) −29.3607 −0.932673 −0.466336 0.884607i \(-0.654426\pi\)
−0.466336 + 0.884607i \(0.654426\pi\)
\(992\) 0 0
\(993\) 17.1246 0.543433
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −10.8885 −0.344844 −0.172422 0.985023i \(-0.555159\pi\)
−0.172422 + 0.985023i \(0.555159\pi\)
\(998\) 0 0
\(999\) 21.1803 0.670116
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 10000.2.a.c.1.2 2
4.3 odd 2 625.2.a.b.1.2 2
5.4 even 2 10000.2.a.l.1.1 2
12.11 even 2 5625.2.a.f.1.1 2
20.3 even 4 625.2.b.a.624.2 4
20.7 even 4 625.2.b.a.624.3 4
20.19 odd 2 625.2.a.c.1.1 2
25.6 even 5 400.2.u.b.161.1 4
25.21 even 5 400.2.u.b.241.1 4
60.59 even 2 5625.2.a.d.1.2 2
100.3 even 20 125.2.e.a.49.1 8
100.11 odd 10 625.2.d.h.501.1 4
100.19 odd 10 125.2.d.a.51.1 4
100.23 even 20 625.2.e.c.124.1 8
100.27 even 20 625.2.e.c.124.2 8
100.31 odd 10 25.2.d.a.11.1 4
100.39 odd 10 625.2.d.b.501.1 4
100.47 even 20 125.2.e.a.49.2 8
100.59 odd 10 625.2.d.b.126.1 4
100.63 even 20 625.2.e.c.499.2 8
100.67 even 20 125.2.e.a.74.1 8
100.71 odd 10 25.2.d.a.16.1 yes 4
100.79 odd 10 125.2.d.a.76.1 4
100.83 even 20 125.2.e.a.74.2 8
100.87 even 20 625.2.e.c.499.1 8
100.91 odd 10 625.2.d.h.126.1 4
300.71 even 10 225.2.h.b.91.1 4
300.131 even 10 225.2.h.b.136.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 4 100.31 odd 10
25.2.d.a.16.1 yes 4 100.71 odd 10
125.2.d.a.51.1 4 100.19 odd 10
125.2.d.a.76.1 4 100.79 odd 10
125.2.e.a.49.1 8 100.3 even 20
125.2.e.a.49.2 8 100.47 even 20
125.2.e.a.74.1 8 100.67 even 20
125.2.e.a.74.2 8 100.83 even 20
225.2.h.b.91.1 4 300.71 even 10
225.2.h.b.136.1 4 300.131 even 10
400.2.u.b.161.1 4 25.6 even 5
400.2.u.b.241.1 4 25.21 even 5
625.2.a.b.1.2 2 4.3 odd 2
625.2.a.c.1.1 2 20.19 odd 2
625.2.b.a.624.2 4 20.3 even 4
625.2.b.a.624.3 4 20.7 even 4
625.2.d.b.126.1 4 100.59 odd 10
625.2.d.b.501.1 4 100.39 odd 10
625.2.d.h.126.1 4 100.91 odd 10
625.2.d.h.501.1 4 100.11 odd 10
625.2.e.c.124.1 8 100.23 even 20
625.2.e.c.124.2 8 100.27 even 20
625.2.e.c.499.1 8 100.87 even 20
625.2.e.c.499.2 8 100.63 even 20
5625.2.a.d.1.2 2 60.59 even 2
5625.2.a.f.1.1 2 12.11 even 2
10000.2.a.c.1.2 2 1.1 even 1 trivial
10000.2.a.l.1.1 2 5.4 even 2