Properties

Label 625.2.b.a.624.2
Level $625$
Weight $2$
Character 625.624
Analytic conductor $4.991$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 624.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 625.624
Dual form 625.2.b.a.624.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +1.61803i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +1.61803i q^{7} -2.23607i q^{8} +2.00000 q^{9} -0.763932 q^{11} +1.61803i q^{12} -4.85410i q^{13} +1.00000 q^{14} +1.85410 q^{16} +0.763932i q^{17} -1.23607i q^{18} +5.85410 q^{19} -1.61803 q^{21} +0.472136i q^{22} +8.23607i q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000i q^{27} +2.61803i q^{28} +1.38197 q^{29} -3.00000 q^{31} -5.61803i q^{32} -0.763932i q^{33} +0.472136 q^{34} +3.23607 q^{36} -4.23607i q^{37} -3.61803i q^{38} +4.85410 q^{39} -5.23607 q^{41} +1.00000i q^{42} +1.85410i q^{43} -1.23607 q^{44} +5.09017 q^{46} +1.61803i q^{47} +1.85410i q^{48} +4.38197 q^{49} -0.763932 q^{51} -7.85410i q^{52} +5.47214i q^{53} +3.09017 q^{54} +3.61803 q^{56} +5.85410i q^{57} -0.854102i q^{58} +4.14590 q^{59} -4.70820 q^{61} +1.85410i q^{62} +3.23607i q^{63} +0.236068 q^{64} -0.472136 q^{66} -9.23607i q^{67} +1.23607i q^{68} -8.23607 q^{69} -4.38197 q^{71} -4.47214i q^{72} -9.00000i q^{73} -2.61803 q^{74} +9.47214 q^{76} -1.23607i q^{77} -3.00000i q^{78} -3.09017 q^{79} +1.00000 q^{81} +3.23607i q^{82} -1.76393i q^{83} -2.61803 q^{84} +1.14590 q^{86} +1.38197i q^{87} +1.70820i q^{88} -8.94427 q^{89} +7.85410 q^{91} +13.3262i q^{92} -3.00000i q^{93} +1.00000 q^{94} +5.61803 q^{96} -2.85410i q^{97} -2.70820i q^{98} -1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.618034i − 0.437016i −0.975835 0.218508i \(-0.929881\pi\)
0.975835 0.218508i \(-0.0701190\pi\)
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) 1.61803i 0.611559i 0.952102 + 0.305780i \(0.0989171\pi\)
−0.952102 + 0.305780i \(0.901083\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −0.763932 −0.230334 −0.115167 0.993346i \(-0.536740\pi\)
−0.115167 + 0.993346i \(0.536740\pi\)
\(12\) 1.61803i 0.467086i
\(13\) − 4.85410i − 1.34629i −0.739512 0.673143i \(-0.764944\pi\)
0.739512 0.673143i \(-0.235056\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 0.763932i 0.185281i 0.995700 + 0.0926404i \(0.0295307\pi\)
−0.995700 + 0.0926404i \(0.970469\pi\)
\(18\) − 1.23607i − 0.291344i
\(19\) 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
\(22\) 0.472136i 0.100660i
\(23\) 8.23607i 1.71734i 0.512530 + 0.858669i \(0.328708\pi\)
−0.512530 + 0.858669i \(0.671292\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 5.00000i 0.962250i
\(28\) 2.61803i 0.494762i
\(29\) 1.38197 0.256625 0.128312 0.991734i \(-0.459044\pi\)
0.128312 + 0.991734i \(0.459044\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) − 5.61803i − 0.993137i
\(33\) − 0.763932i − 0.132983i
\(34\) 0.472136 0.0809706
\(35\) 0 0
\(36\) 3.23607 0.539345
\(37\) − 4.23607i − 0.696405i −0.937419 0.348203i \(-0.886792\pi\)
0.937419 0.348203i \(-0.113208\pi\)
\(38\) − 3.61803i − 0.586923i
\(39\) 4.85410 0.777278
\(40\) 0 0
\(41\) −5.23607 −0.817736 −0.408868 0.912593i \(-0.634076\pi\)
−0.408868 + 0.912593i \(0.634076\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 1.85410i 0.282748i 0.989956 + 0.141374i \(0.0451520\pi\)
−0.989956 + 0.141374i \(0.954848\pi\)
\(44\) −1.23607 −0.186344
\(45\) 0 0
\(46\) 5.09017 0.750505
\(47\) 1.61803i 0.236015i 0.993013 + 0.118007i \(0.0376506\pi\)
−0.993013 + 0.118007i \(0.962349\pi\)
\(48\) 1.85410i 0.267617i
\(49\) 4.38197 0.625995
\(50\) 0 0
\(51\) −0.763932 −0.106972
\(52\) − 7.85410i − 1.08917i
\(53\) 5.47214i 0.751656i 0.926690 + 0.375828i \(0.122642\pi\)
−0.926690 + 0.375828i \(0.877358\pi\)
\(54\) 3.09017 0.420519
\(55\) 0 0
\(56\) 3.61803 0.483480
\(57\) 5.85410i 0.775395i
\(58\) − 0.854102i − 0.112149i
\(59\) 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 \(0.597458\pi\)
\(62\) 1.85410i 0.235471i
\(63\) 3.23607i 0.407706i
\(64\) 0.236068 0.0295085
\(65\) 0 0
\(66\) −0.472136 −0.0581159
\(67\) − 9.23607i − 1.12837i −0.825650 0.564183i \(-0.809191\pi\)
0.825650 0.564183i \(-0.190809\pi\)
\(68\) 1.23607i 0.149895i
\(69\) −8.23607 −0.991506
\(70\) 0 0
\(71\) −4.38197 −0.520044 −0.260022 0.965603i \(-0.583730\pi\)
−0.260022 + 0.965603i \(0.583730\pi\)
\(72\) − 4.47214i − 0.527046i
\(73\) − 9.00000i − 1.05337i −0.850060 0.526685i \(-0.823435\pi\)
0.850060 0.526685i \(-0.176565\pi\)
\(74\) −2.61803 −0.304340
\(75\) 0 0
\(76\) 9.47214 1.08653
\(77\) − 1.23607i − 0.140863i
\(78\) − 3.00000i − 0.339683i
\(79\) −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
\(82\) 3.23607i 0.357364i
\(83\) − 1.76393i − 0.193617i −0.995303 0.0968083i \(-0.969137\pi\)
0.995303 0.0968083i \(-0.0308634\pi\)
\(84\) −2.61803 −0.285651
\(85\) 0 0
\(86\) 1.14590 0.123565
\(87\) 1.38197i 0.148162i
\(88\) 1.70820i 0.182095i
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 0 0
\(91\) 7.85410 0.823334
\(92\) 13.3262i 1.38936i
\(93\) − 3.00000i − 0.311086i
\(94\) 1.00000 0.103142
\(95\) 0 0
\(96\) 5.61803 0.573388
\(97\) − 2.85410i − 0.289790i −0.989447 0.144895i \(-0.953716\pi\)
0.989447 0.144895i \(-0.0462845\pi\)
\(98\) − 2.70820i − 0.273570i
\(99\) −1.52786 −0.153556
\(100\) 0 0
\(101\) −7.47214 −0.743505 −0.371753 0.928332i \(-0.621243\pi\)
−0.371753 + 0.928332i \(0.621243\pi\)
\(102\) 0.472136i 0.0467484i
\(103\) − 11.5623i − 1.13927i −0.821899 0.569634i \(-0.807085\pi\)
0.821899 0.569634i \(-0.192915\pi\)
\(104\) −10.8541 −1.06433
\(105\) 0 0
\(106\) 3.38197 0.328486
\(107\) − 10.4164i − 1.00699i −0.863998 0.503496i \(-0.832047\pi\)
0.863998 0.503496i \(-0.167953\pi\)
\(108\) 8.09017i 0.778477i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 4.23607 0.402070
\(112\) 3.00000i 0.283473i
\(113\) 10.1459i 0.954446i 0.878782 + 0.477223i \(0.158357\pi\)
−0.878782 + 0.477223i \(0.841643\pi\)
\(114\) 3.61803 0.338860
\(115\) 0 0
\(116\) 2.23607 0.207614
\(117\) − 9.70820i − 0.897524i
\(118\) − 2.56231i − 0.235879i
\(119\) −1.23607 −0.113310
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 2.90983i 0.263444i
\(123\) − 5.23607i − 0.472120i
\(124\) −4.85410 −0.435911
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 15.8885i 1.40988i 0.709267 + 0.704940i \(0.249026\pi\)
−0.709267 + 0.704940i \(0.750974\pi\)
\(128\) − 11.3820i − 1.00603i
\(129\) −1.85410 −0.163245
\(130\) 0 0
\(131\) −17.7984 −1.55505 −0.777526 0.628851i \(-0.783525\pi\)
−0.777526 + 0.628851i \(0.783525\pi\)
\(132\) − 1.23607i − 0.107586i
\(133\) 9.47214i 0.821338i
\(134\) −5.70820 −0.493114
\(135\) 0 0
\(136\) 1.70820 0.146477
\(137\) − 5.94427i − 0.507853i −0.967223 0.253927i \(-0.918278\pi\)
0.967223 0.253927i \(-0.0817222\pi\)
\(138\) 5.09017i 0.433304i
\(139\) −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
\(142\) 2.70820i 0.227267i
\(143\) 3.70820i 0.310096i
\(144\) 3.70820 0.309017
\(145\) 0 0
\(146\) −5.56231 −0.460340
\(147\) 4.38197i 0.361418i
\(148\) − 6.85410i − 0.563404i
\(149\) −13.9443 −1.14236 −0.571180 0.820825i \(-0.693514\pi\)
−0.571180 + 0.820825i \(0.693514\pi\)
\(150\) 0 0
\(151\) −5.56231 −0.452654 −0.226327 0.974051i \(-0.572672\pi\)
−0.226327 + 0.974051i \(0.572672\pi\)
\(152\) − 13.0902i − 1.06175i
\(153\) 1.52786i 0.123520i
\(154\) −0.763932 −0.0615594
\(155\) 0 0
\(156\) 7.85410 0.628831
\(157\) 9.18034i 0.732671i 0.930483 + 0.366335i \(0.119388\pi\)
−0.930483 + 0.366335i \(0.880612\pi\)
\(158\) 1.90983i 0.151938i
\(159\) −5.47214 −0.433969
\(160\) 0 0
\(161\) −13.3262 −1.05025
\(162\) − 0.618034i − 0.0485573i
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) −8.47214 −0.661563
\(165\) 0 0
\(166\) −1.09017 −0.0846136
\(167\) 5.56231i 0.430424i 0.976567 + 0.215212i \(0.0690443\pi\)
−0.976567 + 0.215212i \(0.930956\pi\)
\(168\) 3.61803i 0.279137i
\(169\) −10.5623 −0.812485
\(170\) 0 0
\(171\) 11.7082 0.895349
\(172\) 3.00000i 0.228748i
\(173\) − 16.8885i − 1.28401i −0.766700 0.642006i \(-0.778102\pi\)
0.766700 0.642006i \(-0.221898\pi\)
\(174\) 0.854102 0.0647493
\(175\) 0 0
\(176\) −1.41641 −0.106766
\(177\) 4.14590i 0.311625i
\(178\) 5.52786i 0.414331i
\(179\) 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 \(0.329845\pi\)
\(182\) − 4.85410i − 0.359810i
\(183\) − 4.70820i − 0.348040i
\(184\) 18.4164 1.35768
\(185\) 0 0
\(186\) −1.85410 −0.135949
\(187\) − 0.583592i − 0.0426765i
\(188\) 2.61803i 0.190940i
\(189\) −8.09017 −0.588473
\(190\) 0 0
\(191\) −24.1803 −1.74963 −0.874814 0.484459i \(-0.839016\pi\)
−0.874814 + 0.484459i \(0.839016\pi\)
\(192\) 0.236068i 0.0170367i
\(193\) − 5.70820i − 0.410886i −0.978669 0.205443i \(-0.934137\pi\)
0.978669 0.205443i \(-0.0658634\pi\)
\(194\) −1.76393 −0.126643
\(195\) 0 0
\(196\) 7.09017 0.506441
\(197\) 9.70820i 0.691681i 0.938293 + 0.345840i \(0.112406\pi\)
−0.938293 + 0.345840i \(0.887594\pi\)
\(198\) 0.944272i 0.0671065i
\(199\) −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
\(202\) 4.61803i 0.324924i
\(203\) 2.23607i 0.156941i
\(204\) −1.23607 −0.0865421
\(205\) 0 0
\(206\) −7.14590 −0.497878
\(207\) 16.4721i 1.14489i
\(208\) − 9.00000i − 0.624038i
\(209\) −4.47214 −0.309344
\(210\) 0 0
\(211\) 13.1803 0.907372 0.453686 0.891162i \(-0.350109\pi\)
0.453686 + 0.891162i \(0.350109\pi\)
\(212\) 8.85410i 0.608102i
\(213\) − 4.38197i − 0.300247i
\(214\) −6.43769 −0.440072
\(215\) 0 0
\(216\) 11.1803 0.760726
\(217\) − 4.85410i − 0.329518i
\(218\) 6.18034i 0.418585i
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 3.70820 0.249441
\(222\) − 2.61803i − 0.175711i
\(223\) 22.1803i 1.48531i 0.669677 + 0.742653i \(0.266433\pi\)
−0.669677 + 0.742653i \(0.733567\pi\)
\(224\) 9.09017 0.607363
\(225\) 0 0
\(226\) 6.27051 0.417108
\(227\) − 19.2361i − 1.27674i −0.769729 0.638371i \(-0.779608\pi\)
0.769729 0.638371i \(-0.220392\pi\)
\(228\) 9.47214i 0.627308i
\(229\) −8.29180 −0.547937 −0.273969 0.961739i \(-0.588336\pi\)
−0.273969 + 0.961739i \(0.588336\pi\)
\(230\) 0 0
\(231\) 1.23607 0.0813273
\(232\) − 3.09017i − 0.202880i
\(233\) 14.9443i 0.979032i 0.871994 + 0.489516i \(0.162826\pi\)
−0.871994 + 0.489516i \(0.837174\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 6.70820 0.436667
\(237\) − 3.09017i − 0.200728i
\(238\) 0.763932i 0.0495184i
\(239\) 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 \(0.379532\pi\)
\(242\) 6.43769i 0.413831i
\(243\) 16.0000i 1.02640i
\(244\) −7.61803 −0.487695
\(245\) 0 0
\(246\) −3.23607 −0.206324
\(247\) − 28.4164i − 1.80809i
\(248\) 6.70820i 0.425971i
\(249\) 1.76393 0.111785
\(250\) 0 0
\(251\) −6.81966 −0.430453 −0.215227 0.976564i \(-0.569049\pi\)
−0.215227 + 0.976564i \(0.569049\pi\)
\(252\) 5.23607i 0.329841i
\(253\) − 6.29180i − 0.395562i
\(254\) 9.81966 0.616140
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) − 16.1459i − 1.00715i −0.863951 0.503577i \(-0.832017\pi\)
0.863951 0.503577i \(-0.167983\pi\)
\(258\) 1.14590i 0.0713405i
\(259\) 6.85410 0.425893
\(260\) 0 0
\(261\) 2.76393 0.171083
\(262\) 11.0000i 0.679582i
\(263\) − 22.0902i − 1.36214i −0.732219 0.681069i \(-0.761515\pi\)
0.732219 0.681069i \(-0.238485\pi\)
\(264\) −1.70820 −0.105133
\(265\) 0 0
\(266\) 5.85410 0.358938
\(267\) − 8.94427i − 0.547381i
\(268\) − 14.9443i − 0.912867i
\(269\) 17.2361 1.05090 0.525451 0.850824i \(-0.323897\pi\)
0.525451 + 0.850824i \(0.323897\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 1.41641i 0.0858823i
\(273\) 7.85410i 0.475352i
\(274\) −3.67376 −0.221940
\(275\) 0 0
\(276\) −13.3262 −0.802145
\(277\) 11.2918i 0.678458i 0.940704 + 0.339229i \(0.110166\pi\)
−0.940704 + 0.339229i \(0.889834\pi\)
\(278\) 3.09017i 0.185336i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −1.09017 −0.0650341 −0.0325170 0.999471i \(-0.510352\pi\)
−0.0325170 + 0.999471i \(0.510352\pi\)
\(282\) 1.00000i 0.0595491i
\(283\) − 23.1459i − 1.37588i −0.725767 0.687940i \(-0.758515\pi\)
0.725767 0.687940i \(-0.241485\pi\)
\(284\) −7.09017 −0.420724
\(285\) 0 0
\(286\) 2.29180 0.135517
\(287\) − 8.47214i − 0.500094i
\(288\) − 11.2361i − 0.662092i
\(289\) 16.4164 0.965671
\(290\) 0 0
\(291\) 2.85410 0.167310
\(292\) − 14.5623i − 0.852194i
\(293\) − 28.4721i − 1.66336i −0.555255 0.831680i \(-0.687379\pi\)
0.555255 0.831680i \(-0.312621\pi\)
\(294\) 2.70820 0.157946
\(295\) 0 0
\(296\) −9.47214 −0.550557
\(297\) − 3.81966i − 0.221639i
\(298\) 8.61803i 0.499229i
\(299\) 39.9787 2.31203
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 3.43769i 0.197817i
\(303\) − 7.47214i − 0.429263i
\(304\) 10.8541 0.622525
\(305\) 0 0
\(306\) 0.944272 0.0539804
\(307\) − 4.76393i − 0.271892i −0.990716 0.135946i \(-0.956593\pi\)
0.990716 0.135946i \(-0.0434074\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) 11.5623 0.657757
\(310\) 0 0
\(311\) −29.5066 −1.67316 −0.836582 0.547841i \(-0.815450\pi\)
−0.836582 + 0.547841i \(0.815450\pi\)
\(312\) − 10.8541i − 0.614493i
\(313\) − 21.2361i − 1.20033i −0.799875 0.600167i \(-0.795101\pi\)
0.799875 0.600167i \(-0.204899\pi\)
\(314\) 5.67376 0.320189
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 23.6525i 1.32846i 0.747530 + 0.664228i \(0.231239\pi\)
−0.747530 + 0.664228i \(0.768761\pi\)
\(318\) 3.38197i 0.189651i
\(319\) −1.05573 −0.0591094
\(320\) 0 0
\(321\) 10.4164 0.581387
\(322\) 8.23607i 0.458978i
\(323\) 4.47214i 0.248836i
\(324\) 1.61803 0.0898908
\(325\) 0 0
\(326\) 6.79837 0.376527
\(327\) − 10.0000i − 0.553001i
\(328\) 11.7082i 0.646477i
\(329\) −2.61803 −0.144337
\(330\) 0 0
\(331\) 17.1246 0.941254 0.470627 0.882332i \(-0.344028\pi\)
0.470627 + 0.882332i \(0.344028\pi\)
\(332\) − 2.85410i − 0.156639i
\(333\) − 8.47214i − 0.464270i
\(334\) 3.43769 0.188102
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) − 1.14590i − 0.0624210i −0.999513 0.0312105i \(-0.990064\pi\)
0.999513 0.0312105i \(-0.00993623\pi\)
\(338\) 6.52786i 0.355069i
\(339\) −10.1459 −0.551050
\(340\) 0 0
\(341\) 2.29180 0.124108
\(342\) − 7.23607i − 0.391282i
\(343\) 18.4164i 0.994393i
\(344\) 4.14590 0.223532
\(345\) 0 0
\(346\) −10.4377 −0.561134
\(347\) 31.0902i 1.66901i 0.551002 + 0.834504i \(0.314246\pi\)
−0.551002 + 0.834504i \(0.685754\pi\)
\(348\) 2.23607i 0.119866i
\(349\) −8.29180 −0.443850 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(350\) 0 0
\(351\) 24.2705 1.29546
\(352\) 4.29180i 0.228753i
\(353\) 24.0902i 1.28219i 0.767461 + 0.641095i \(0.221520\pi\)
−0.767461 + 0.641095i \(0.778480\pi\)
\(354\) 2.56231 0.136185
\(355\) 0 0
\(356\) −14.4721 −0.767022
\(357\) − 1.23607i − 0.0654197i
\(358\) − 5.85410i − 0.309399i
\(359\) 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
\(362\) − 8.47214i − 0.445286i
\(363\) − 10.4164i − 0.546720i
\(364\) 12.7082 0.666091
\(365\) 0 0
\(366\) −2.90983 −0.152099
\(367\) 5.43769i 0.283845i 0.989878 + 0.141923i \(0.0453284\pi\)
−0.989878 + 0.141923i \(0.954672\pi\)
\(368\) 15.2705i 0.796030i
\(369\) −10.4721 −0.545158
\(370\) 0 0
\(371\) −8.85410 −0.459682
\(372\) − 4.85410i − 0.251673i
\(373\) 5.27051i 0.272897i 0.990647 + 0.136448i \(0.0435688\pi\)
−0.990647 + 0.136448i \(0.956431\pi\)
\(374\) −0.360680 −0.0186503
\(375\) 0 0
\(376\) 3.61803 0.186586
\(377\) − 6.70820i − 0.345490i
\(378\) 5.00000i 0.257172i
\(379\) 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
\(382\) 14.9443i 0.764615i
\(383\) − 11.3607i − 0.580504i −0.956950 0.290252i \(-0.906261\pi\)
0.956950 0.290252i \(-0.0937391\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) −3.52786 −0.179564
\(387\) 3.70820i 0.188499i
\(388\) − 4.61803i − 0.234445i
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −6.29180 −0.318190
\(392\) − 9.79837i − 0.494893i
\(393\) − 17.7984i − 0.897809i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −2.47214 −0.124230
\(397\) 0.0344419i 0.00172859i 1.00000 0.000864294i \(0.000275113\pi\)
−1.00000 0.000864294i \(0.999725\pi\)
\(398\) 1.58359i 0.0793783i
\(399\) −9.47214 −0.474200
\(400\) 0 0
\(401\) −22.5967 −1.12843 −0.564214 0.825629i \(-0.690821\pi\)
−0.564214 + 0.825629i \(0.690821\pi\)
\(402\) − 5.70820i − 0.284699i
\(403\) 14.5623i 0.725400i
\(404\) −12.0902 −0.601508
\(405\) 0 0
\(406\) 1.38197 0.0685858
\(407\) 3.23607i 0.160406i
\(408\) 1.70820i 0.0845687i
\(409\) −28.4164 −1.40510 −0.702550 0.711634i \(-0.747955\pi\)
−0.702550 + 0.711634i \(0.747955\pi\)
\(410\) 0 0
\(411\) 5.94427 0.293209
\(412\) − 18.7082i − 0.921687i
\(413\) 6.70820i 0.330089i
\(414\) 10.1803 0.500336
\(415\) 0 0
\(416\) −27.2705 −1.33705
\(417\) − 5.00000i − 0.244851i
\(418\) 2.76393i 0.135188i
\(419\) 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 \(0.215325\pi\)
\(422\) − 8.14590i − 0.396536i
\(423\) 3.23607i 0.157343i
\(424\) 12.2361 0.594236
\(425\) 0 0
\(426\) −2.70820 −0.131213
\(427\) − 7.61803i − 0.368663i
\(428\) − 16.8541i − 0.814674i
\(429\) −3.70820 −0.179034
\(430\) 0 0
\(431\) 23.8328 1.14799 0.573993 0.818860i \(-0.305394\pi\)
0.573993 + 0.818860i \(0.305394\pi\)
\(432\) 9.27051i 0.446028i
\(433\) 20.1459i 0.968150i 0.875026 + 0.484075i \(0.160844\pi\)
−0.875026 + 0.484075i \(0.839156\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −16.1803 −0.774898
\(437\) 48.2148i 2.30643i
\(438\) − 5.56231i − 0.265777i
\(439\) −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
\(442\) − 2.29180i − 0.109010i
\(443\) 12.0557i 0.572785i 0.958112 + 0.286392i \(0.0924561\pi\)
−0.958112 + 0.286392i \(0.907544\pi\)
\(444\) 6.85410 0.325281
\(445\) 0 0
\(446\) 13.7082 0.649102
\(447\) − 13.9443i − 0.659541i
\(448\) 0.381966i 0.0180462i
\(449\) −20.3262 −0.959254 −0.479627 0.877472i \(-0.659228\pi\)
−0.479627 + 0.877472i \(0.659228\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 16.4164i 0.772163i
\(453\) − 5.56231i − 0.261340i
\(454\) −11.8885 −0.557957
\(455\) 0 0
\(456\) 13.0902 0.613003
\(457\) − 5.41641i − 0.253369i −0.991943 0.126684i \(-0.959566\pi\)
0.991943 0.126684i \(-0.0404336\pi\)
\(458\) 5.12461i 0.239457i
\(459\) −3.81966 −0.178286
\(460\) 0 0
\(461\) 23.1803 1.07962 0.539808 0.841788i \(-0.318497\pi\)
0.539808 + 0.841788i \(0.318497\pi\)
\(462\) − 0.763932i − 0.0355413i
\(463\) 16.1246i 0.749374i 0.927151 + 0.374687i \(0.122250\pi\)
−0.927151 + 0.374687i \(0.877750\pi\)
\(464\) 2.56231 0.118952
\(465\) 0 0
\(466\) 9.23607 0.427853
\(467\) 28.4508i 1.31655i 0.752778 + 0.658274i \(0.228713\pi\)
−0.752778 + 0.658274i \(0.771287\pi\)
\(468\) − 15.7082i − 0.726112i
\(469\) 14.9443 0.690062
\(470\) 0 0
\(471\) −9.18034 −0.423008
\(472\) − 9.27051i − 0.426710i
\(473\) − 1.41641i − 0.0651265i
\(474\) −1.90983 −0.0877214
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 10.9443i 0.501104i
\(478\) − 18.2148i − 0.833125i
\(479\) 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
\(482\) − 7.09017i − 0.322948i
\(483\) − 13.3262i − 0.606365i
\(484\) −16.8541 −0.766096
\(485\) 0 0
\(486\) 9.88854 0.448553
\(487\) 9.58359i 0.434274i 0.976141 + 0.217137i \(0.0696718\pi\)
−0.976141 + 0.217137i \(0.930328\pi\)
\(488\) 10.5279i 0.476574i
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) 37.2492 1.68103 0.840517 0.541785i \(-0.182251\pi\)
0.840517 + 0.541785i \(0.182251\pi\)
\(492\) − 8.47214i − 0.381953i
\(493\) 1.05573i 0.0475476i
\(494\) −17.5623 −0.790165
\(495\) 0 0
\(496\) −5.56231 −0.249755
\(497\) − 7.09017i − 0.318038i
\(498\) − 1.09017i − 0.0488517i
\(499\) 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
\(502\) 4.21478i 0.188115i
\(503\) − 10.5836i − 0.471899i −0.971765 0.235950i \(-0.924180\pi\)
0.971765 0.235950i \(-0.0758200\pi\)
\(504\) 7.23607 0.322320
\(505\) 0 0
\(506\) −3.88854 −0.172867
\(507\) − 10.5623i − 0.469088i
\(508\) 25.7082i 1.14062i
\(509\) 4.67376 0.207161 0.103580 0.994621i \(-0.466970\pi\)
0.103580 + 0.994621i \(0.466970\pi\)
\(510\) 0 0
\(511\) 14.5623 0.644198
\(512\) − 18.7082i − 0.826794i
\(513\) 29.2705i 1.29232i
\(514\) −9.97871 −0.440142
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) − 1.23607i − 0.0543622i
\(518\) − 4.23607i − 0.186122i
\(519\) 16.8885 0.741325
\(520\) 0 0
\(521\) −15.3607 −0.672964 −0.336482 0.941690i \(-0.609237\pi\)
−0.336482 + 0.941690i \(0.609237\pi\)
\(522\) − 1.70820i − 0.0747661i
\(523\) − 19.8541i − 0.868159i −0.900875 0.434080i \(-0.857074\pi\)
0.900875 0.434080i \(-0.142926\pi\)
\(524\) −28.7984 −1.25806
\(525\) 0 0
\(526\) −13.6525 −0.595276
\(527\) − 2.29180i − 0.0998322i
\(528\) − 1.41641i − 0.0616412i
\(529\) −44.8328 −1.94925
\(530\) 0 0
\(531\) 8.29180 0.359833
\(532\) 15.3262i 0.664477i
\(533\) 25.4164i 1.10091i
\(534\) −5.52786 −0.239214
\(535\) 0 0
\(536\) −20.6525 −0.892051
\(537\) 9.47214i 0.408753i
\(538\) − 10.6525i − 0.459261i
\(539\) −3.34752 −0.144188
\(540\) 0 0
\(541\) −13.1246 −0.564271 −0.282136 0.959375i \(-0.591043\pi\)
−0.282136 + 0.959375i \(0.591043\pi\)
\(542\) 4.94427i 0.212375i
\(543\) 13.7082i 0.588275i
\(544\) 4.29180 0.184009
\(545\) 0 0
\(546\) 4.85410 0.207736
\(547\) 34.7082i 1.48402i 0.670391 + 0.742008i \(0.266126\pi\)
−0.670391 + 0.742008i \(0.733874\pi\)
\(548\) − 9.61803i − 0.410862i
\(549\) −9.41641 −0.401882
\(550\) 0 0
\(551\) 8.09017 0.344653
\(552\) 18.4164i 0.783854i
\(553\) − 5.00000i − 0.212622i
\(554\) 6.97871 0.296497
\(555\) 0 0
\(556\) −8.09017 −0.343100
\(557\) − 9.23607i − 0.391345i −0.980669 0.195672i \(-0.937311\pi\)
0.980669 0.195672i \(-0.0626889\pi\)
\(558\) 3.70820i 0.156981i
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) 0.583592 0.0246393
\(562\) 0.673762i 0.0284209i
\(563\) 9.61803i 0.405352i 0.979246 + 0.202676i \(0.0649638\pi\)
−0.979246 + 0.202676i \(0.935036\pi\)
\(564\) −2.61803 −0.110239
\(565\) 0 0
\(566\) −14.3050 −0.601282
\(567\) 1.61803i 0.0679510i
\(568\) 9.79837i 0.411131i
\(569\) −29.4721 −1.23554 −0.617768 0.786360i \(-0.711963\pi\)
−0.617768 + 0.786360i \(0.711963\pi\)
\(570\) 0 0
\(571\) 32.1246 1.34437 0.672187 0.740382i \(-0.265355\pi\)
0.672187 + 0.740382i \(0.265355\pi\)
\(572\) 6.00000i 0.250873i
\(573\) − 24.1803i − 1.01015i
\(574\) −5.23607 −0.218549
\(575\) 0 0
\(576\) 0.472136 0.0196723
\(577\) − 37.7771i − 1.57268i −0.617794 0.786340i \(-0.711973\pi\)
0.617794 0.786340i \(-0.288027\pi\)
\(578\) − 10.1459i − 0.422014i
\(579\) 5.70820 0.237225
\(580\) 0 0
\(581\) 2.85410 0.118408
\(582\) − 1.76393i − 0.0731173i
\(583\) − 4.18034i − 0.173132i
\(584\) −20.1246 −0.832762
\(585\) 0 0
\(586\) −17.5967 −0.726915
\(587\) − 18.7082i − 0.772170i −0.922463 0.386085i \(-0.873827\pi\)
0.922463 0.386085i \(-0.126173\pi\)
\(588\) 7.09017i 0.292394i
\(589\) −17.5623 −0.723642
\(590\) 0 0
\(591\) −9.70820 −0.399342
\(592\) − 7.85410i − 0.322802i
\(593\) − 22.0902i − 0.907135i −0.891222 0.453567i \(-0.850151\pi\)
0.891222 0.453567i \(-0.149849\pi\)
\(594\) −2.36068 −0.0968599
\(595\) 0 0
\(596\) −22.5623 −0.924188
\(597\) − 2.56231i − 0.104868i
\(598\) − 24.7082i − 1.01039i
\(599\) 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 \(0.234943\pi\)
\(602\) 1.85410i 0.0755676i
\(603\) − 18.4721i − 0.752244i
\(604\) −9.00000 −0.366205
\(605\) 0 0
\(606\) −4.61803 −0.187595
\(607\) 15.4377i 0.626597i 0.949655 + 0.313298i \(0.101434\pi\)
−0.949655 + 0.313298i \(0.898566\pi\)
\(608\) − 32.8885i − 1.33381i
\(609\) −2.23607 −0.0906100
\(610\) 0 0
\(611\) 7.85410 0.317743
\(612\) 2.47214i 0.0999302i
\(613\) 31.9787i 1.29161i 0.763503 + 0.645804i \(0.223478\pi\)
−0.763503 + 0.645804i \(0.776522\pi\)
\(614\) −2.94427 −0.118821
\(615\) 0 0
\(616\) −2.76393 −0.111362
\(617\) − 9.76393i − 0.393081i −0.980496 0.196541i \(-0.937029\pi\)
0.980496 0.196541i \(-0.0629707\pi\)
\(618\) − 7.14590i − 0.287450i
\(619\) −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
\(622\) 18.2361i 0.731200i
\(623\) − 14.4721i − 0.579814i
\(624\) 9.00000 0.360288
\(625\) 0 0
\(626\) −13.1246 −0.524565
\(627\) − 4.47214i − 0.178600i
\(628\) 14.8541i 0.592743i
\(629\) 3.23607 0.129030
\(630\) 0 0
\(631\) −5.76393 −0.229459 −0.114729 0.993397i \(-0.536600\pi\)
−0.114729 + 0.993397i \(0.536600\pi\)
\(632\) 6.90983i 0.274858i
\(633\) 13.1803i 0.523871i
\(634\) 14.6180 0.580556
\(635\) 0 0
\(636\) −8.85410 −0.351088
\(637\) − 21.2705i − 0.842768i
\(638\) 0.652476i 0.0258318i
\(639\) −8.76393 −0.346696
\(640\) 0 0
\(641\) 10.0902 0.398538 0.199269 0.979945i \(-0.436143\pi\)
0.199269 + 0.979945i \(0.436143\pi\)
\(642\) − 6.43769i − 0.254076i
\(643\) 22.8328i 0.900438i 0.892918 + 0.450219i \(0.148654\pi\)
−0.892918 + 0.450219i \(0.851346\pi\)
\(644\) −21.5623 −0.849674
\(645\) 0 0
\(646\) 2.76393 0.108745
\(647\) − 30.5410i − 1.20069i −0.799741 0.600346i \(-0.795030\pi\)
0.799741 0.600346i \(-0.204970\pi\)
\(648\) − 2.23607i − 0.0878410i
\(649\) −3.16718 −0.124323
\(650\) 0 0
\(651\) 4.85410 0.190247
\(652\) 17.7984i 0.697038i
\(653\) 7.90983i 0.309536i 0.987951 + 0.154768i \(0.0494629\pi\)
−0.987951 + 0.154768i \(0.950537\pi\)
\(654\) −6.18034 −0.241670
\(655\) 0 0
\(656\) −9.70820 −0.379042
\(657\) − 18.0000i − 0.702247i
\(658\) 1.61803i 0.0630775i
\(659\) −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 \(0.790579\pi\)
\(662\) − 10.5836i − 0.411343i
\(663\) 3.70820i 0.144015i
\(664\) −3.94427 −0.153067
\(665\) 0 0
\(666\) −5.23607 −0.202894
\(667\) 11.3820i 0.440711i
\(668\) 9.00000i 0.348220i
\(669\) −22.1803 −0.857541
\(670\) 0 0
\(671\) 3.59675 0.138851
\(672\) 9.09017i 0.350661i
\(673\) − 10.1803i − 0.392423i −0.980562 0.196212i \(-0.937136\pi\)
0.980562 0.196212i \(-0.0628640\pi\)
\(674\) −0.708204 −0.0272790
\(675\) 0 0
\(676\) −17.0902 −0.657314
\(677\) − 8.38197i − 0.322145i −0.986943 0.161073i \(-0.948505\pi\)
0.986943 0.161073i \(-0.0514953\pi\)
\(678\) 6.27051i 0.240817i
\(679\) 4.61803 0.177224
\(680\) 0 0
\(681\) 19.2361 0.737128
\(682\) − 1.41641i − 0.0542371i
\(683\) − 4.52786i − 0.173254i −0.996241 0.0866270i \(-0.972391\pi\)
0.996241 0.0866270i \(-0.0276088\pi\)
\(684\) 18.9443 0.724352
\(685\) 0 0
\(686\) 11.3820 0.434565
\(687\) − 8.29180i − 0.316352i
\(688\) 3.43769i 0.131061i
\(689\) 26.5623 1.01194
\(690\) 0 0
\(691\) 2.72949 0.103835 0.0519173 0.998651i \(-0.483467\pi\)
0.0519173 + 0.998651i \(0.483467\pi\)
\(692\) − 27.3262i − 1.03879i
\(693\) − 2.47214i − 0.0939087i
\(694\) 19.2148 0.729383
\(695\) 0 0
\(696\) 3.09017 0.117133
\(697\) − 4.00000i − 0.151511i
\(698\) 5.12461i 0.193969i
\(699\) −14.9443 −0.565244
\(700\) 0 0
\(701\) 35.0132 1.32243 0.661214 0.750197i \(-0.270041\pi\)
0.661214 + 0.750197i \(0.270041\pi\)
\(702\) − 15.0000i − 0.566139i
\(703\) − 24.7984i − 0.935288i
\(704\) −0.180340 −0.00679682
\(705\) 0 0
\(706\) 14.8885 0.560338
\(707\) − 12.0902i − 0.454698i
\(708\) 6.70820i 0.252110i
\(709\) −33.5410 −1.25966 −0.629830 0.776733i \(-0.716875\pi\)
−0.629830 + 0.776733i \(0.716875\pi\)
\(710\) 0 0
\(711\) −6.18034 −0.231781
\(712\) 20.0000i 0.749532i
\(713\) − 24.7082i − 0.925330i
\(714\) −0.763932 −0.0285894
\(715\) 0 0
\(716\) 15.3262 0.572768
\(717\) 29.4721i 1.10066i
\(718\) − 17.7639i − 0.662944i
\(719\) 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
\(722\) − 9.43769i − 0.351235i
\(723\) 11.4721i 0.426653i
\(724\) 22.1803 0.824326
\(725\) 0 0
\(726\) −6.43769 −0.238925
\(727\) − 4.43769i − 0.164585i −0.996608 0.0822925i \(-0.973776\pi\)
0.996608 0.0822925i \(-0.0262242\pi\)
\(728\) − 17.5623i − 0.650902i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −1.41641 −0.0523877
\(732\) − 7.61803i − 0.281571i
\(733\) 26.9787i 0.996482i 0.867039 + 0.498241i \(0.166020\pi\)
−0.867039 + 0.498241i \(0.833980\pi\)
\(734\) 3.36068 0.124045
\(735\) 0 0
\(736\) 46.2705 1.70555
\(737\) 7.05573i 0.259901i
\(738\) 6.47214i 0.238243i
\(739\) 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
\(742\) 5.47214i 0.200888i
\(743\) − 16.3607i − 0.600215i −0.953905 0.300108i \(-0.902977\pi\)
0.953905 0.300108i \(-0.0970226\pi\)
\(744\) −6.70820 −0.245935
\(745\) 0 0
\(746\) 3.25735 0.119260
\(747\) − 3.52786i − 0.129078i
\(748\) − 0.944272i − 0.0345260i
\(749\) 16.8541 0.615835
\(750\) 0 0
\(751\) −40.8885 −1.49204 −0.746022 0.665921i \(-0.768039\pi\)
−0.746022 + 0.665921i \(0.768039\pi\)
\(752\) 3.00000i 0.109399i
\(753\) − 6.81966i − 0.248522i
\(754\) −4.14590 −0.150985
\(755\) 0 0
\(756\) −13.0902 −0.476085
\(757\) − 3.58359i − 0.130248i −0.997877 0.0651239i \(-0.979256\pi\)
0.997877 0.0651239i \(-0.0207443\pi\)
\(758\) − 21.3820i − 0.776628i
\(759\) 6.29180 0.228378
\(760\) 0 0
\(761\) 37.4508 1.35759 0.678796 0.734327i \(-0.262502\pi\)
0.678796 + 0.734327i \(0.262502\pi\)
\(762\) 9.81966i 0.355729i
\(763\) − 16.1803i − 0.585768i
\(764\) −39.1246 −1.41548
\(765\) 0 0
\(766\) −7.02129 −0.253689
\(767\) − 20.1246i − 0.726658i
\(768\) − 6.56231i − 0.236797i
\(769\) 13.4164 0.483808 0.241904 0.970300i \(-0.422228\pi\)
0.241904 + 0.970300i \(0.422228\pi\)
\(770\) 0 0
\(771\) 16.1459 0.581480
\(772\) − 9.23607i − 0.332413i
\(773\) 33.1591i 1.19265i 0.802744 + 0.596324i \(0.203373\pi\)
−0.802744 + 0.596324i \(0.796627\pi\)
\(774\) 2.29180 0.0823769
\(775\) 0 0
\(776\) −6.38197 −0.229099
\(777\) 6.85410i 0.245890i
\(778\) 9.27051i 0.332364i
\(779\) −30.6525 −1.09824
\(780\) 0 0
\(781\) 3.34752 0.119784
\(782\) 3.88854i 0.139054i
\(783\) 6.90983i 0.246937i
\(784\) 8.12461 0.290165
\(785\) 0 0
\(786\) −11.0000 −0.392357
\(787\) 34.1803i 1.21840i 0.793018 + 0.609199i \(0.208509\pi\)
−0.793018 + 0.609199i \(0.791491\pi\)
\(788\) 15.7082i 0.559582i
\(789\) 22.0902 0.786431
\(790\) 0 0
\(791\) −16.4164 −0.583700
\(792\) 3.41641i 0.121397i
\(793\) 22.8541i 0.811573i
\(794\) 0.0212862 0.000755420 0
\(795\) 0 0
\(796\) −4.14590 −0.146947
\(797\) − 14.2361i − 0.504267i −0.967692 0.252134i \(-0.918868\pi\)
0.967692 0.252134i \(-0.0811323\pi\)
\(798\) 5.85410i 0.207233i
\(799\) −1.23607 −0.0437289
\(800\) 0 0
\(801\) −17.8885 −0.632061
\(802\) 13.9656i 0.493141i
\(803\) 6.87539i 0.242627i
\(804\) 14.9443 0.527044
\(805\) 0 0
\(806\) 9.00000 0.317011
\(807\) 17.2361i 0.606738i
\(808\) 16.7082i 0.587793i
\(809\) 15.9787 0.561782 0.280891 0.959740i \(-0.409370\pi\)
0.280891 + 0.959740i \(0.409370\pi\)
\(810\) 0 0
\(811\) −1.29180 −0.0453611 −0.0226805 0.999743i \(-0.507220\pi\)
−0.0226805 + 0.999743i \(0.507220\pi\)
\(812\) 3.61803i 0.126968i
\(813\) − 8.00000i − 0.280572i
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) −1.41641 −0.0495842
\(817\) 10.8541i 0.379737i
\(818\) 17.5623i 0.614052i
\(819\) 15.7082 0.548889
\(820\) 0 0
\(821\) 19.6869 0.687078 0.343539 0.939138i \(-0.388374\pi\)
0.343539 + 0.939138i \(0.388374\pi\)
\(822\) − 3.67376i − 0.128137i
\(823\) 34.2918i 1.19534i 0.801743 + 0.597668i \(0.203906\pi\)
−0.801743 + 0.597668i \(0.796094\pi\)
\(824\) −25.8541 −0.900670
\(825\) 0 0
\(826\) 4.14590 0.144254
\(827\) 30.0344i 1.04440i 0.852823 + 0.522200i \(0.174889\pi\)
−0.852823 + 0.522200i \(0.825111\pi\)
\(828\) 26.6525i 0.926238i
\(829\) 29.1459 1.01228 0.506139 0.862452i \(-0.331072\pi\)
0.506139 + 0.862452i \(0.331072\pi\)
\(830\) 0 0
\(831\) −11.2918 −0.391708
\(832\) − 1.14590i − 0.0397269i
\(833\) 3.34752i 0.115985i
\(834\) −3.09017 −0.107004
\(835\) 0 0
\(836\) −7.23607 −0.250265
\(837\) − 15.0000i − 0.518476i
\(838\) − 0.326238i − 0.0112697i
\(839\) −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
\(842\) − 19.7771i − 0.681563i
\(843\) − 1.09017i − 0.0375474i
\(844\) 21.3262 0.734079
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) − 16.8541i − 0.579114i
\(848\) 10.1459i 0.348412i
\(849\) 23.1459 0.794365
\(850\) 0 0
\(851\) 34.8885 1.19596
\(852\) − 7.09017i − 0.242905i
\(853\) 47.3050i 1.61969i 0.586643 + 0.809845i \(0.300449\pi\)
−0.586643 + 0.809845i \(0.699551\pi\)
\(854\) −4.70820 −0.161111
\(855\) 0 0
\(856\) −23.2918 −0.796097
\(857\) 40.6869i 1.38984i 0.719088 + 0.694919i \(0.244560\pi\)
−0.719088 + 0.694919i \(0.755440\pi\)
\(858\) 2.29180i 0.0782406i
\(859\) 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
\(862\) − 14.7295i − 0.501688i
\(863\) − 41.5623i − 1.41480i −0.706815 0.707399i \(-0.749869\pi\)
0.706815 0.707399i \(-0.250131\pi\)
\(864\) 28.0902 0.955647
\(865\) 0 0
\(866\) 12.4508 0.423097
\(867\) 16.4164i 0.557530i
\(868\) − 7.85410i − 0.266586i
\(869\) 2.36068 0.0800806
\(870\) 0 0
\(871\) −44.8328 −1.51910
\(872\) 22.3607i 0.757228i
\(873\) − 5.70820i − 0.193193i
\(874\) 29.7984 1.00795
\(875\) 0 0
\(876\) 14.5623 0.492015
\(877\) − 30.5410i − 1.03130i −0.856800 0.515648i \(-0.827551\pi\)
0.856800 0.515648i \(-0.172449\pi\)
\(878\) 3.69505i 0.124702i
\(879\) 28.4721 0.960341
\(880\) 0 0
\(881\) 4.36068 0.146915 0.0734575 0.997298i \(-0.476597\pi\)
0.0734575 + 0.997298i \(0.476597\pi\)
\(882\) − 5.41641i − 0.182380i
\(883\) − 47.4164i − 1.59569i −0.602863 0.797845i \(-0.705974\pi\)
0.602863 0.797845i \(-0.294026\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 7.45085 0.250316
\(887\) 5.88854i 0.197718i 0.995101 + 0.0988590i \(0.0315193\pi\)
−0.995101 + 0.0988590i \(0.968481\pi\)
\(888\) − 9.47214i − 0.317864i
\(889\) −25.7082 −0.862225
\(890\) 0 0
\(891\) −0.763932 −0.0255927
\(892\) 35.8885i 1.20164i
\(893\) 9.47214i 0.316973i
\(894\) −8.61803 −0.288230
\(895\) 0 0
\(896\) 18.4164 0.615249
\(897\) 39.9787i 1.33485i
\(898\) 12.5623i 0.419210i
\(899\) −4.14590 −0.138273
\(900\) 0 0
\(901\) −4.18034 −0.139267
\(902\) − 2.47214i − 0.0823131i
\(903\) − 3.00000i − 0.0998337i
\(904\) 22.6869 0.754556
\(905\) 0 0
\(906\) −3.43769 −0.114210
\(907\) − 47.2492i − 1.56888i −0.620202 0.784442i \(-0.712949\pi\)
0.620202 0.784442i \(-0.287051\pi\)
\(908\) − 31.1246i − 1.03291i
\(909\) −14.9443 −0.495670
\(910\) 0 0
\(911\) −35.7639 −1.18491 −0.592456 0.805603i \(-0.701842\pi\)
−0.592456 + 0.805603i \(0.701842\pi\)
\(912\) 10.8541i 0.359415i
\(913\) 1.34752i 0.0445965i
\(914\) −3.34752 −0.110726
\(915\) 0 0
\(916\) −13.4164 −0.443291
\(917\) − 28.7984i − 0.951006i
\(918\) 2.36068i 0.0779140i
\(919\) 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
\(922\) − 14.3262i − 0.471810i
\(923\) 21.2705i 0.700127i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 9.96556 0.327489
\(927\) − 23.1246i − 0.759512i
\(928\) − 7.76393i − 0.254864i
\(929\) 36.6312 1.20183 0.600915 0.799313i \(-0.294803\pi\)
0.600915 + 0.799313i \(0.294803\pi\)
\(930\) 0 0
\(931\) 25.6525 0.840726
\(932\) 24.1803i 0.792053i
\(933\) − 29.5066i − 0.966002i
\(934\) 17.5836 0.575353
\(935\) 0 0
\(936\) −21.7082 −0.709555
\(937\) − 51.2705i − 1.67493i −0.546487 0.837467i \(-0.684035\pi\)
0.546487 0.837467i \(-0.315965\pi\)
\(938\) − 9.23607i − 0.301568i
\(939\) 21.2361 0.693013
\(940\) 0 0
\(941\) −19.5836 −0.638407 −0.319203 0.947686i \(-0.603415\pi\)
−0.319203 + 0.947686i \(0.603415\pi\)
\(942\) 5.67376i 0.184861i
\(943\) − 43.1246i − 1.40433i
\(944\) 7.68692 0.250188
\(945\) 0 0
\(946\) −0.875388 −0.0284613
\(947\) 28.6525i 0.931080i 0.885027 + 0.465540i \(0.154140\pi\)
−0.885027 + 0.465540i \(0.845860\pi\)
\(948\) − 5.00000i − 0.162392i
\(949\) −43.6869 −1.41814
\(950\) 0 0
\(951\) −23.6525 −0.766984
\(952\) 2.76393i 0.0895796i
\(953\) 34.7426i 1.12542i 0.826653 + 0.562712i \(0.190242\pi\)
−0.826653 + 0.562712i \(0.809758\pi\)
\(954\) 6.76393 0.218990
\(955\) 0 0
\(956\) 47.6869 1.54231
\(957\) − 1.05573i − 0.0341268i
\(958\) − 2.56231i − 0.0827843i
\(959\) 9.61803 0.310583
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 12.7082i 0.409729i
\(963\) − 20.8328i − 0.671328i
\(964\) 18.5623 0.597852
\(965\) 0 0
\(966\) −8.23607 −0.264991
\(967\) − 39.8885i − 1.28273i −0.767236 0.641365i \(-0.778369\pi\)
0.767236 0.641365i \(-0.221631\pi\)
\(968\) 23.2918i 0.748627i
\(969\) −4.47214 −0.143666
\(970\) 0 0
\(971\) 3.38197 0.108532 0.0542662 0.998527i \(-0.482718\pi\)
0.0542662 + 0.998527i \(0.482718\pi\)
\(972\) 25.8885i 0.830375i
\(973\) − 8.09017i − 0.259359i
\(974\) 5.92299 0.189785
\(975\) 0 0
\(976\) −8.72949 −0.279424
\(977\) 33.6525i 1.07664i 0.842741 + 0.538319i \(0.180940\pi\)
−0.842741 + 0.538319i \(0.819060\pi\)
\(978\) 6.79837i 0.217388i
\(979\) 6.83282 0.218378
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) − 23.0213i − 0.734639i
\(983\) 7.38197i 0.235448i 0.993046 + 0.117724i \(0.0375598\pi\)
−0.993046 + 0.117724i \(0.962440\pi\)
\(984\) −11.7082 −0.373244
\(985\) 0 0
\(986\) 0.652476 0.0207791
\(987\) − 2.61803i − 0.0833329i
\(988\) − 45.9787i − 1.46278i
\(989\) −15.2705 −0.485574
\(990\) 0 0
\(991\) 29.3607 0.932673 0.466336 0.884607i \(-0.345574\pi\)
0.466336 + 0.884607i \(0.345574\pi\)
\(992\) 16.8541i 0.535118i
\(993\) 17.1246i 0.543433i
\(994\) −4.38197 −0.138988
\(995\) 0 0
\(996\) 2.85410 0.0904357
\(997\) 10.8885i 0.344844i 0.985023 + 0.172422i \(0.0551592\pi\)
−0.985023 + 0.172422i \(0.944841\pi\)
\(998\) − 7.76393i − 0.245763i
\(999\) 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 625.2.b.a.624.2 4
5.2 odd 4 625.2.a.b.1.2 2
5.3 odd 4 625.2.a.c.1.1 2
5.4 even 2 inner 625.2.b.a.624.3 4
15.2 even 4 5625.2.a.f.1.1 2
15.8 even 4 5625.2.a.d.1.2 2
20.3 even 4 10000.2.a.l.1.1 2
20.7 even 4 10000.2.a.c.1.2 2
25.2 odd 20 625.2.d.h.501.1 4
25.3 odd 20 125.2.d.a.76.1 4
25.4 even 10 125.2.e.a.49.2 8
25.6 even 5 125.2.e.a.74.2 8
25.8 odd 20 125.2.d.a.51.1 4
25.9 even 10 625.2.e.c.499.1 8
25.11 even 5 625.2.e.c.124.1 8
25.12 odd 20 625.2.d.h.126.1 4
25.13 odd 20 625.2.d.b.126.1 4
25.14 even 10 625.2.e.c.124.2 8
25.16 even 5 625.2.e.c.499.2 8
25.17 odd 20 25.2.d.a.11.1 4
25.19 even 10 125.2.e.a.74.1 8
25.21 even 5 125.2.e.a.49.1 8
25.22 odd 20 25.2.d.a.16.1 yes 4
25.23 odd 20 625.2.d.b.501.1 4
75.17 even 20 225.2.h.b.136.1 4
75.47 even 20 225.2.h.b.91.1 4
100.47 even 20 400.2.u.b.241.1 4
100.67 even 20 400.2.u.b.161.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 4 25.17 odd 20
25.2.d.a.16.1 yes 4 25.22 odd 20
125.2.d.a.51.1 4 25.8 odd 20
125.2.d.a.76.1 4 25.3 odd 20
125.2.e.a.49.1 8 25.21 even 5
125.2.e.a.49.2 8 25.4 even 10
125.2.e.a.74.1 8 25.19 even 10
125.2.e.a.74.2 8 25.6 even 5
225.2.h.b.91.1 4 75.47 even 20
225.2.h.b.136.1 4 75.17 even 20
400.2.u.b.161.1 4 100.67 even 20
400.2.u.b.241.1 4 100.47 even 20
625.2.a.b.1.2 2 5.2 odd 4
625.2.a.c.1.1 2 5.3 odd 4
625.2.b.a.624.2 4 1.1 even 1 trivial
625.2.b.a.624.3 4 5.4 even 2 inner
625.2.d.b.126.1 4 25.13 odd 20
625.2.d.b.501.1 4 25.23 odd 20
625.2.d.h.126.1 4 25.12 odd 20
625.2.d.h.501.1 4 25.2 odd 20
625.2.e.c.124.1 8 25.11 even 5
625.2.e.c.124.2 8 25.14 even 10
625.2.e.c.499.1 8 25.9 even 10
625.2.e.c.499.2 8 25.16 even 5
5625.2.a.d.1.2 2 15.8 even 4
5625.2.a.f.1.1 2 15.2 even 4
10000.2.a.c.1.2 2 20.7 even 4
10000.2.a.l.1.1 2 20.3 even 4