Properties

Label 625.2.b.a.624.1
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: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
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 624.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 625.624
Dual form 625.2.b.a.624.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} +0.618034i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} +0.618034i q^{7} -2.23607i q^{8} +2.00000 q^{9} -5.23607 q^{11} +0.618034i q^{12} -1.85410i q^{13} +1.00000 q^{14} -4.85410 q^{16} -5.23607i q^{17} -3.23607i q^{18} -0.854102 q^{19} +0.618034 q^{21} +8.47214i q^{22} -3.76393i q^{23} -2.23607 q^{24} -3.00000 q^{26} -5.00000i q^{27} -0.381966i q^{28} +3.61803 q^{29} -3.00000 q^{31} +3.38197i q^{32} +5.23607i q^{33} -8.47214 q^{34} -1.23607 q^{36} -0.236068i q^{37} +1.38197i q^{38} -1.85410 q^{39} -0.763932 q^{41} -1.00000i q^{42} +4.85410i q^{43} +3.23607 q^{44} -6.09017 q^{46} +0.618034i q^{47} +4.85410i q^{48} +6.61803 q^{49} -5.23607 q^{51} +1.14590i q^{52} +3.47214i q^{53} -8.09017 q^{54} +1.38197 q^{56} +0.854102i q^{57} -5.85410i q^{58} +10.8541 q^{59} +8.70820 q^{61} +4.85410i q^{62} +1.23607i q^{63} -4.23607 q^{64} +8.47214 q^{66} +4.76393i q^{67} +3.23607i q^{68} -3.76393 q^{69} -6.61803 q^{71} -4.47214i q^{72} +9.00000i q^{73} -0.381966 q^{74} +0.527864 q^{76} -3.23607i q^{77} +3.00000i q^{78} +8.09017 q^{79} +1.00000 q^{81} +1.23607i q^{82} +6.23607i q^{83} -0.381966 q^{84} +7.85410 q^{86} -3.61803i q^{87} +11.7082i q^{88} +8.94427 q^{89} +1.14590 q^{91} +2.32624i q^{92} +3.00000i q^{93} +1.00000 q^{94} +3.38197 q^{96} -3.85410i q^{97} -10.7082i q^{98} -10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.61803i − 1.14412i −0.820211 0.572061i \(-0.806144\pi\)
0.820211 0.572061i \(-0.193856\pi\)
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) −0.618034 −0.309017
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) 0.618034i 0.233595i 0.993156 + 0.116797i \(0.0372628\pi\)
−0.993156 + 0.116797i \(0.962737\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 0.618034i 0.178411i
\(13\) − 1.85410i − 0.514235i −0.966380 0.257118i \(-0.917227\pi\)
0.966380 0.257118i \(-0.0827728\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) − 5.23607i − 1.26993i −0.772540 0.634967i \(-0.781014\pi\)
0.772540 0.634967i \(-0.218986\pi\)
\(18\) − 3.23607i − 0.762749i
\(19\) −0.854102 −0.195944 −0.0979722 0.995189i \(-0.531236\pi\)
−0.0979722 + 0.995189i \(0.531236\pi\)
\(20\) 0 0
\(21\) 0.618034 0.134866
\(22\) 8.47214i 1.80627i
\(23\) − 3.76393i − 0.784834i −0.919787 0.392417i \(-0.871639\pi\)
0.919787 0.392417i \(-0.128361\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) − 5.00000i − 0.962250i
\(28\) − 0.381966i − 0.0721848i
\(29\) 3.61803 0.671852 0.335926 0.941888i \(-0.390951\pi\)
0.335926 + 0.941888i \(0.390951\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 3.38197i 0.597853i
\(33\) 5.23607i 0.911482i
\(34\) −8.47214 −1.45296
\(35\) 0 0
\(36\) −1.23607 −0.206011
\(37\) − 0.236068i − 0.0388093i −0.999812 0.0194047i \(-0.993823\pi\)
0.999812 0.0194047i \(-0.00617709\pi\)
\(38\) 1.38197i 0.224184i
\(39\) −1.85410 −0.296894
\(40\) 0 0
\(41\) −0.763932 −0.119306 −0.0596531 0.998219i \(-0.518999\pi\)
−0.0596531 + 0.998219i \(0.518999\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 4.85410i 0.740244i 0.928983 + 0.370122i \(0.120684\pi\)
−0.928983 + 0.370122i \(0.879316\pi\)
\(44\) 3.23607 0.487856
\(45\) 0 0
\(46\) −6.09017 −0.897947
\(47\) 0.618034i 0.0901495i 0.998984 + 0.0450748i \(0.0143526\pi\)
−0.998984 + 0.0450748i \(0.985647\pi\)
\(48\) 4.85410i 0.700629i
\(49\) 6.61803 0.945433
\(50\) 0 0
\(51\) −5.23607 −0.733196
\(52\) 1.14590i 0.158907i
\(53\) 3.47214i 0.476935i 0.971151 + 0.238467i \(0.0766450\pi\)
−0.971151 + 0.238467i \(0.923355\pi\)
\(54\) −8.09017 −1.10093
\(55\) 0 0
\(56\) 1.38197 0.184673
\(57\) 0.854102i 0.113129i
\(58\) − 5.85410i − 0.768681i
\(59\) 10.8541 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(60\) 0 0
\(61\) 8.70820 1.11497 0.557486 0.830187i \(-0.311766\pi\)
0.557486 + 0.830187i \(0.311766\pi\)
\(62\) 4.85410i 0.616472i
\(63\) 1.23607i 0.155730i
\(64\) −4.23607 −0.529508
\(65\) 0 0
\(66\) 8.47214 1.04285
\(67\) 4.76393i 0.582007i 0.956722 + 0.291003i \(0.0939891\pi\)
−0.956722 + 0.291003i \(0.906011\pi\)
\(68\) 3.23607i 0.392431i
\(69\) −3.76393 −0.453124
\(70\) 0 0
\(71\) −6.61803 −0.785416 −0.392708 0.919663i \(-0.628462\pi\)
−0.392708 + 0.919663i \(0.628462\pi\)
\(72\) − 4.47214i − 0.527046i
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) −0.381966 −0.0444026
\(75\) 0 0
\(76\) 0.527864 0.0605502
\(77\) − 3.23607i − 0.368784i
\(78\) 3.00000i 0.339683i
\(79\) 8.09017 0.910215 0.455108 0.890436i \(-0.349601\pi\)
0.455108 + 0.890436i \(0.349601\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.23607i 0.136501i
\(83\) 6.23607i 0.684497i 0.939609 + 0.342249i \(0.111189\pi\)
−0.939609 + 0.342249i \(0.888811\pi\)
\(84\) −0.381966 −0.0416759
\(85\) 0 0
\(86\) 7.85410 0.846930
\(87\) − 3.61803i − 0.387894i
\(88\) 11.7082i 1.24810i
\(89\) 8.94427 0.948091 0.474045 0.880500i \(-0.342793\pi\)
0.474045 + 0.880500i \(0.342793\pi\)
\(90\) 0 0
\(91\) 1.14590 0.120123
\(92\) 2.32624i 0.242527i
\(93\) 3.00000i 0.311086i
\(94\) 1.00000 0.103142
\(95\) 0 0
\(96\) 3.38197 0.345170
\(97\) − 3.85410i − 0.391325i −0.980671 0.195662i \(-0.937314\pi\)
0.980671 0.195662i \(-0.0626857\pi\)
\(98\) − 10.7082i − 1.08169i
\(99\) −10.4721 −1.05249
\(100\) 0 0
\(101\) 1.47214 0.146483 0.0732415 0.997314i \(-0.476666\pi\)
0.0732415 + 0.997314i \(0.476666\pi\)
\(102\) 8.47214i 0.838866i
\(103\) − 8.56231i − 0.843669i −0.906673 0.421835i \(-0.861386\pi\)
0.906673 0.421835i \(-0.138614\pi\)
\(104\) −4.14590 −0.406539
\(105\) 0 0
\(106\) 5.61803 0.545672
\(107\) − 16.4164i − 1.58703i −0.608548 0.793517i \(-0.708248\pi\)
0.608548 0.793517i \(-0.291752\pi\)
\(108\) 3.09017i 0.297352i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −0.236068 −0.0224066
\(112\) − 3.00000i − 0.283473i
\(113\) − 16.8541i − 1.58550i −0.609547 0.792750i \(-0.708648\pi\)
0.609547 0.792750i \(-0.291352\pi\)
\(114\) 1.38197 0.129433
\(115\) 0 0
\(116\) −2.23607 −0.207614
\(117\) − 3.70820i − 0.342824i
\(118\) − 17.5623i − 1.61674i
\(119\) 3.23607 0.296650
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) − 14.0902i − 1.27566i
\(123\) 0.763932i 0.0688814i
\(124\) 1.85410 0.166503
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 19.8885i 1.76482i 0.470479 + 0.882411i \(0.344081\pi\)
−0.470479 + 0.882411i \(0.655919\pi\)
\(128\) 13.6180i 1.20368i
\(129\) 4.85410 0.427380
\(130\) 0 0
\(131\) 6.79837 0.593977 0.296988 0.954881i \(-0.404018\pi\)
0.296988 + 0.954881i \(0.404018\pi\)
\(132\) − 3.23607i − 0.281664i
\(133\) − 0.527864i − 0.0457716i
\(134\) 7.70820 0.665887
\(135\) 0 0
\(136\) −11.7082 −1.00397
\(137\) − 11.9443i − 1.02047i −0.860036 0.510234i \(-0.829559\pi\)
0.860036 0.510234i \(-0.170441\pi\)
\(138\) 6.09017i 0.518430i
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 0.618034 0.0520479
\(142\) 10.7082i 0.898613i
\(143\) 9.70820i 0.811841i
\(144\) −9.70820 −0.809017
\(145\) 0 0
\(146\) 14.5623 1.20519
\(147\) − 6.61803i − 0.545846i
\(148\) 0.145898i 0.0119927i
\(149\) 3.94427 0.323127 0.161564 0.986862i \(-0.448346\pi\)
0.161564 + 0.986862i \(0.448346\pi\)
\(150\) 0 0
\(151\) 14.5623 1.18506 0.592532 0.805547i \(-0.298128\pi\)
0.592532 + 0.805547i \(0.298128\pi\)
\(152\) 1.90983i 0.154908i
\(153\) − 10.4721i − 0.846622i
\(154\) −5.23607 −0.421934
\(155\) 0 0
\(156\) 1.14590 0.0917453
\(157\) 13.1803i 1.05191i 0.850514 + 0.525953i \(0.176291\pi\)
−0.850514 + 0.525953i \(0.823709\pi\)
\(158\) − 13.0902i − 1.04140i
\(159\) 3.47214 0.275358
\(160\) 0 0
\(161\) 2.32624 0.183333
\(162\) − 1.61803i − 0.127125i
\(163\) − 11.0000i − 0.861586i −0.902451 0.430793i \(-0.858234\pi\)
0.902451 0.430793i \(-0.141766\pi\)
\(164\) 0.472136 0.0368676
\(165\) 0 0
\(166\) 10.0902 0.783149
\(167\) 14.5623i 1.12687i 0.826162 + 0.563433i \(0.190520\pi\)
−0.826162 + 0.563433i \(0.809480\pi\)
\(168\) − 1.38197i − 0.106621i
\(169\) 9.56231 0.735562
\(170\) 0 0
\(171\) −1.70820 −0.130630
\(172\) − 3.00000i − 0.228748i
\(173\) − 18.8885i − 1.43607i −0.696007 0.718035i \(-0.745042\pi\)
0.696007 0.718035i \(-0.254958\pi\)
\(174\) −5.85410 −0.443798
\(175\) 0 0
\(176\) 25.4164 1.91583
\(177\) − 10.8541i − 0.815844i
\(178\) − 14.4721i − 1.08473i
\(179\) 0.527864 0.0394544 0.0197272 0.999805i \(-0.493720\pi\)
0.0197272 + 0.999805i \(0.493720\pi\)
\(180\) 0 0
\(181\) 0.291796 0.0216890 0.0108445 0.999941i \(-0.496548\pi\)
0.0108445 + 0.999941i \(0.496548\pi\)
\(182\) − 1.85410i − 0.137435i
\(183\) − 8.70820i − 0.643729i
\(184\) −8.41641 −0.620466
\(185\) 0 0
\(186\) 4.85410 0.355920
\(187\) 27.4164i 2.00489i
\(188\) − 0.381966i − 0.0278577i
\(189\) 3.09017 0.224777
\(190\) 0 0
\(191\) −1.81966 −0.131666 −0.0658330 0.997831i \(-0.520970\pi\)
−0.0658330 + 0.997831i \(0.520970\pi\)
\(192\) 4.23607i 0.305712i
\(193\) − 7.70820i − 0.554849i −0.960747 0.277424i \(-0.910519\pi\)
0.960747 0.277424i \(-0.0894808\pi\)
\(194\) −6.23607 −0.447724
\(195\) 0 0
\(196\) −4.09017 −0.292155
\(197\) 3.70820i 0.264199i 0.991236 + 0.132099i \(0.0421718\pi\)
−0.991236 + 0.132099i \(0.957828\pi\)
\(198\) 16.9443i 1.20418i
\(199\) 17.5623 1.24496 0.622479 0.782636i \(-0.286125\pi\)
0.622479 + 0.782636i \(0.286125\pi\)
\(200\) 0 0
\(201\) 4.76393 0.336022
\(202\) − 2.38197i − 0.167595i
\(203\) 2.23607i 0.156941i
\(204\) 3.23607 0.226570
\(205\) 0 0
\(206\) −13.8541 −0.965261
\(207\) − 7.52786i − 0.523223i
\(208\) 9.00000i 0.624038i
\(209\) 4.47214 0.309344
\(210\) 0 0
\(211\) −9.18034 −0.632001 −0.316000 0.948759i \(-0.602340\pi\)
−0.316000 + 0.948759i \(0.602340\pi\)
\(212\) − 2.14590i − 0.147381i
\(213\) 6.61803i 0.453460i
\(214\) −26.5623 −1.81576
\(215\) 0 0
\(216\) −11.1803 −0.760726
\(217\) − 1.85410i − 0.125865i
\(218\) 16.1803i 1.09587i
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −9.70820 −0.653044
\(222\) 0.381966i 0.0256359i
\(223\) 0.180340i 0.0120765i 0.999982 + 0.00603823i \(0.00192204\pi\)
−0.999982 + 0.00603823i \(0.998078\pi\)
\(224\) −2.09017 −0.139655
\(225\) 0 0
\(226\) −27.2705 −1.81401
\(227\) 14.7639i 0.979917i 0.871746 + 0.489958i \(0.162988\pi\)
−0.871746 + 0.489958i \(0.837012\pi\)
\(228\) − 0.527864i − 0.0349587i
\(229\) −21.7082 −1.43452 −0.717259 0.696806i \(-0.754604\pi\)
−0.717259 + 0.696806i \(0.754604\pi\)
\(230\) 0 0
\(231\) −3.23607 −0.212918
\(232\) − 8.09017i − 0.531146i
\(233\) 2.94427i 0.192886i 0.995339 + 0.0964428i \(0.0307465\pi\)
−0.995339 + 0.0964428i \(0.969254\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −6.70820 −0.436667
\(237\) − 8.09017i − 0.525513i
\(238\) − 5.23607i − 0.339404i
\(239\) 20.5279 1.32784 0.663919 0.747805i \(-0.268892\pi\)
0.663919 + 0.747805i \(0.268892\pi\)
\(240\) 0 0
\(241\) 2.52786 0.162834 0.0814170 0.996680i \(-0.474055\pi\)
0.0814170 + 0.996680i \(0.474055\pi\)
\(242\) − 26.5623i − 1.70749i
\(243\) − 16.0000i − 1.02640i
\(244\) −5.38197 −0.344545
\(245\) 0 0
\(246\) 1.23607 0.0788088
\(247\) 1.58359i 0.100762i
\(248\) 6.70820i 0.425971i
\(249\) 6.23607 0.395195
\(250\) 0 0
\(251\) −29.1803 −1.84185 −0.920923 0.389744i \(-0.872564\pi\)
−0.920923 + 0.389744i \(0.872564\pi\)
\(252\) − 0.763932i − 0.0481232i
\(253\) 19.7082i 1.23904i
\(254\) 32.1803 2.01917
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 22.8541i 1.42560i 0.701367 + 0.712800i \(0.252573\pi\)
−0.701367 + 0.712800i \(0.747427\pi\)
\(258\) − 7.85410i − 0.488975i
\(259\) 0.145898 0.00906566
\(260\) 0 0
\(261\) 7.23607 0.447901
\(262\) − 11.0000i − 0.679582i
\(263\) 10.9098i 0.672729i 0.941732 + 0.336364i \(0.109197\pi\)
−0.941732 + 0.336364i \(0.890803\pi\)
\(264\) 11.7082 0.720590
\(265\) 0 0
\(266\) −0.854102 −0.0523684
\(267\) − 8.94427i − 0.547381i
\(268\) − 2.94427i − 0.179850i
\(269\) 12.7639 0.778231 0.389115 0.921189i \(-0.372781\pi\)
0.389115 + 0.921189i \(0.372781\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 25.4164i 1.54110i
\(273\) − 1.14590i − 0.0693529i
\(274\) −19.3262 −1.16754
\(275\) 0 0
\(276\) 2.32624 0.140023
\(277\) − 24.7082i − 1.48457i −0.670083 0.742286i \(-0.733742\pi\)
0.670083 0.742286i \(-0.266258\pi\)
\(278\) 8.09017i 0.485216i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 10.0902 0.601929 0.300965 0.953635i \(-0.402691\pi\)
0.300965 + 0.953635i \(0.402691\pi\)
\(282\) − 1.00000i − 0.0595491i
\(283\) 29.8541i 1.77464i 0.461152 + 0.887321i \(0.347436\pi\)
−0.461152 + 0.887321i \(0.652564\pi\)
\(284\) 4.09017 0.242707
\(285\) 0 0
\(286\) 15.7082 0.928846
\(287\) − 0.472136i − 0.0278693i
\(288\) 6.76393i 0.398569i
\(289\) −10.4164 −0.612730
\(290\) 0 0
\(291\) −3.85410 −0.225931
\(292\) − 5.56231i − 0.325509i
\(293\) 19.5279i 1.14083i 0.821357 + 0.570415i \(0.193218\pi\)
−0.821357 + 0.570415i \(0.806782\pi\)
\(294\) −10.7082 −0.624515
\(295\) 0 0
\(296\) −0.527864 −0.0306815
\(297\) 26.1803i 1.51914i
\(298\) − 6.38197i − 0.369697i
\(299\) −6.97871 −0.403589
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) − 23.5623i − 1.35586i
\(303\) − 1.47214i − 0.0845720i
\(304\) 4.14590 0.237784
\(305\) 0 0
\(306\) −16.9443 −0.968640
\(307\) 9.23607i 0.527130i 0.964642 + 0.263565i \(0.0848984\pi\)
−0.964642 + 0.263565i \(0.915102\pi\)
\(308\) 2.00000i 0.113961i
\(309\) −8.56231 −0.487093
\(310\) 0 0
\(311\) 8.50658 0.482364 0.241182 0.970480i \(-0.422465\pi\)
0.241182 + 0.970480i \(0.422465\pi\)
\(312\) 4.14590i 0.234715i
\(313\) 16.7639i 0.947553i 0.880645 + 0.473777i \(0.157110\pi\)
−0.880645 + 0.473777i \(0.842890\pi\)
\(314\) 21.3262 1.20351
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 7.65248i 0.429806i 0.976635 + 0.214903i \(0.0689435\pi\)
−0.976635 + 0.214903i \(0.931056\pi\)
\(318\) − 5.61803i − 0.315044i
\(319\) −18.9443 −1.06068
\(320\) 0 0
\(321\) −16.4164 −0.916275
\(322\) − 3.76393i − 0.209756i
\(323\) 4.47214i 0.248836i
\(324\) −0.618034 −0.0343352
\(325\) 0 0
\(326\) −17.7984 −0.985761
\(327\) 10.0000i 0.553001i
\(328\) 1.70820i 0.0943198i
\(329\) −0.381966 −0.0210585
\(330\) 0 0
\(331\) −23.1246 −1.27104 −0.635522 0.772083i \(-0.719215\pi\)
−0.635522 + 0.772083i \(0.719215\pi\)
\(332\) − 3.85410i − 0.211521i
\(333\) − 0.472136i − 0.0258729i
\(334\) 23.5623 1.28927
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 7.85410i 0.427840i 0.976851 + 0.213920i \(0.0686232\pi\)
−0.976851 + 0.213920i \(0.931377\pi\)
\(338\) − 15.4721i − 0.841573i
\(339\) −16.8541 −0.915389
\(340\) 0 0
\(341\) 15.7082 0.850647
\(342\) 2.76393i 0.149456i
\(343\) 8.41641i 0.454443i
\(344\) 10.8541 0.585214
\(345\) 0 0
\(346\) −30.5623 −1.64304
\(347\) − 19.9098i − 1.06882i −0.845227 0.534408i \(-0.820535\pi\)
0.845227 0.534408i \(-0.179465\pi\)
\(348\) 2.23607i 0.119866i
\(349\) −21.7082 −1.16201 −0.581007 0.813899i \(-0.697341\pi\)
−0.581007 + 0.813899i \(0.697341\pi\)
\(350\) 0 0
\(351\) −9.27051 −0.494823
\(352\) − 17.7082i − 0.943850i
\(353\) − 12.9098i − 0.687121i −0.939131 0.343560i \(-0.888367\pi\)
0.939131 0.343560i \(-0.111633\pi\)
\(354\) −17.5623 −0.933426
\(355\) 0 0
\(356\) −5.52786 −0.292976
\(357\) − 3.23607i − 0.171271i
\(358\) − 0.854102i − 0.0451407i
\(359\) −13.7426 −0.725309 −0.362655 0.931924i \(-0.618130\pi\)
−0.362655 + 0.931924i \(0.618130\pi\)
\(360\) 0 0
\(361\) −18.2705 −0.961606
\(362\) − 0.472136i − 0.0248149i
\(363\) − 16.4164i − 0.861638i
\(364\) −0.708204 −0.0371200
\(365\) 0 0
\(366\) −14.0902 −0.736505
\(367\) − 25.5623i − 1.33434i −0.744905 0.667171i \(-0.767505\pi\)
0.744905 0.667171i \(-0.232495\pi\)
\(368\) 18.2705i 0.952416i
\(369\) −1.52786 −0.0795374
\(370\) 0 0
\(371\) −2.14590 −0.111409
\(372\) − 1.85410i − 0.0961307i
\(373\) 28.2705i 1.46379i 0.681417 + 0.731896i \(0.261364\pi\)
−0.681417 + 0.731896i \(0.738636\pi\)
\(374\) 44.3607 2.29384
\(375\) 0 0
\(376\) 1.38197 0.0712695
\(377\) − 6.70820i − 0.345490i
\(378\) − 5.00000i − 0.257172i
\(379\) −14.5967 −0.749785 −0.374892 0.927068i \(-0.622320\pi\)
−0.374892 + 0.927068i \(0.622320\pi\)
\(380\) 0 0
\(381\) 19.8885 1.01892
\(382\) 2.94427i 0.150642i
\(383\) − 33.3607i − 1.70465i −0.523012 0.852326i \(-0.675192\pi\)
0.523012 0.852326i \(-0.324808\pi\)
\(384\) 13.6180 0.694942
\(385\) 0 0
\(386\) −12.4721 −0.634815
\(387\) 9.70820i 0.493496i
\(388\) 2.38197i 0.120926i
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −19.7082 −0.996687
\(392\) − 14.7984i − 0.747431i
\(393\) − 6.79837i − 0.342933i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 6.47214 0.325237
\(397\) 29.0344i 1.45720i 0.684941 + 0.728598i \(0.259828\pi\)
−0.684941 + 0.728598i \(0.740172\pi\)
\(398\) − 28.4164i − 1.42439i
\(399\) −0.527864 −0.0264263
\(400\) 0 0
\(401\) 26.5967 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(402\) − 7.70820i − 0.384450i
\(403\) 5.56231i 0.277078i
\(404\) −0.909830 −0.0452657
\(405\) 0 0
\(406\) 3.61803 0.179560
\(407\) 1.23607i 0.0612696i
\(408\) 11.7082i 0.579642i
\(409\) −1.58359 −0.0783036 −0.0391518 0.999233i \(-0.512466\pi\)
−0.0391518 + 0.999233i \(0.512466\pi\)
\(410\) 0 0
\(411\) −11.9443 −0.589167
\(412\) 5.29180i 0.260708i
\(413\) 6.70820i 0.330089i
\(414\) −12.1803 −0.598631
\(415\) 0 0
\(416\) 6.27051 0.307437
\(417\) 5.00000i 0.244851i
\(418\) − 7.23607i − 0.353928i
\(419\) 9.47214 0.462744 0.231372 0.972865i \(-0.425679\pi\)
0.231372 + 0.972865i \(0.425679\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 14.8541i 0.723086i
\(423\) 1.23607i 0.0600997i
\(424\) 7.76393 0.377050
\(425\) 0 0
\(426\) 10.7082 0.518814
\(427\) 5.38197i 0.260452i
\(428\) 10.1459i 0.490420i
\(429\) 9.70820 0.468717
\(430\) 0 0
\(431\) −29.8328 −1.43700 −0.718498 0.695529i \(-0.755170\pi\)
−0.718498 + 0.695529i \(0.755170\pi\)
\(432\) 24.2705i 1.16772i
\(433\) − 26.8541i − 1.29053i −0.763961 0.645263i \(-0.776748\pi\)
0.763961 0.645263i \(-0.223252\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 6.18034 0.295985
\(437\) 3.21478i 0.153784i
\(438\) − 14.5623i − 0.695814i
\(439\) 40.9787 1.95581 0.977904 0.209056i \(-0.0670391\pi\)
0.977904 + 0.209056i \(0.0670391\pi\)
\(440\) 0 0
\(441\) 13.2361 0.630289
\(442\) 15.7082i 0.747163i
\(443\) − 29.9443i − 1.42270i −0.702840 0.711348i \(-0.748085\pi\)
0.702840 0.711348i \(-0.251915\pi\)
\(444\) 0.145898 0.00692401
\(445\) 0 0
\(446\) 0.291796 0.0138169
\(447\) − 3.94427i − 0.186558i
\(448\) − 2.61803i − 0.123690i
\(449\) −4.67376 −0.220568 −0.110284 0.993900i \(-0.535176\pi\)
−0.110284 + 0.993900i \(0.535176\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 10.4164i 0.489947i
\(453\) − 14.5623i − 0.684197i
\(454\) 23.8885 1.12114
\(455\) 0 0
\(456\) 1.90983 0.0894360
\(457\) − 21.4164i − 1.00182i −0.865500 0.500909i \(-0.832999\pi\)
0.865500 0.500909i \(-0.167001\pi\)
\(458\) 35.1246i 1.64127i
\(459\) −26.1803 −1.22199
\(460\) 0 0
\(461\) 0.819660 0.0381754 0.0190877 0.999818i \(-0.493924\pi\)
0.0190877 + 0.999818i \(0.493924\pi\)
\(462\) 5.23607i 0.243604i
\(463\) 24.1246i 1.12117i 0.828098 + 0.560583i \(0.189423\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(464\) −17.5623 −0.815310
\(465\) 0 0
\(466\) 4.76393 0.220685
\(467\) 27.4508i 1.27027i 0.772400 + 0.635137i \(0.219056\pi\)
−0.772400 + 0.635137i \(0.780944\pi\)
\(468\) 2.29180i 0.105938i
\(469\) −2.94427 −0.135954
\(470\) 0 0
\(471\) 13.1803 0.607318
\(472\) − 24.2705i − 1.11714i
\(473\) − 25.4164i − 1.16865i
\(474\) −13.0902 −0.601251
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 6.94427i 0.317956i
\(478\) − 33.2148i − 1.51921i
\(479\) 10.8541 0.495937 0.247968 0.968768i \(-0.420237\pi\)
0.247968 + 0.968768i \(0.420237\pi\)
\(480\) 0 0
\(481\) −0.437694 −0.0199571
\(482\) − 4.09017i − 0.186302i
\(483\) − 2.32624i − 0.105847i
\(484\) −10.1459 −0.461177
\(485\) 0 0
\(486\) −25.8885 −1.17433
\(487\) − 36.4164i − 1.65018i −0.564998 0.825092i \(-0.691123\pi\)
0.564998 0.825092i \(-0.308877\pi\)
\(488\) − 19.4721i − 0.881462i
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) −43.2492 −1.95181 −0.975905 0.218196i \(-0.929983\pi\)
−0.975905 + 0.218196i \(0.929983\pi\)
\(492\) − 0.472136i − 0.0212855i
\(493\) − 18.9443i − 0.853207i
\(494\) 2.56231 0.115284
\(495\) 0 0
\(496\) 14.5623 0.653867
\(497\) − 4.09017i − 0.183469i
\(498\) − 10.0902i − 0.452151i
\(499\) −7.56231 −0.338535 −0.169268 0.985570i \(-0.554140\pi\)
−0.169268 + 0.985570i \(0.554140\pi\)
\(500\) 0 0
\(501\) 14.5623 0.650596
\(502\) 47.2148i 2.10730i
\(503\) 37.4164i 1.66832i 0.551526 + 0.834158i \(0.314046\pi\)
−0.551526 + 0.834158i \(0.685954\pi\)
\(504\) 2.76393 0.123115
\(505\) 0 0
\(506\) 31.8885 1.41762
\(507\) − 9.56231i − 0.424677i
\(508\) − 12.2918i − 0.545360i
\(509\) 20.3262 0.900945 0.450472 0.892790i \(-0.351256\pi\)
0.450472 + 0.892790i \(0.351256\pi\)
\(510\) 0 0
\(511\) −5.56231 −0.246062
\(512\) 5.29180i 0.233867i
\(513\) 4.27051i 0.188548i
\(514\) 36.9787 1.63106
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) − 3.23607i − 0.142322i
\(518\) − 0.236068i − 0.0103722i
\(519\) −18.8885 −0.829115
\(520\) 0 0
\(521\) 29.3607 1.28631 0.643157 0.765734i \(-0.277624\pi\)
0.643157 + 0.765734i \(0.277624\pi\)
\(522\) − 11.7082i − 0.512454i
\(523\) 13.1459i 0.574830i 0.957806 + 0.287415i \(0.0927959\pi\)
−0.957806 + 0.287415i \(0.907204\pi\)
\(524\) −4.20163 −0.183549
\(525\) 0 0
\(526\) 17.6525 0.769685
\(527\) 15.7082i 0.684260i
\(528\) − 25.4164i − 1.10611i
\(529\) 8.83282 0.384035
\(530\) 0 0
\(531\) 21.7082 0.942056
\(532\) 0.326238i 0.0141442i
\(533\) 1.41641i 0.0613514i
\(534\) −14.4721 −0.626271
\(535\) 0 0
\(536\) 10.6525 0.460117
\(537\) − 0.527864i − 0.0227790i
\(538\) − 20.6525i − 0.890391i
\(539\) −34.6525 −1.49259
\(540\) 0 0
\(541\) 27.1246 1.16618 0.583089 0.812408i \(-0.301844\pi\)
0.583089 + 0.812408i \(0.301844\pi\)
\(542\) 12.9443i 0.556004i
\(543\) − 0.291796i − 0.0125222i
\(544\) 17.7082 0.759233
\(545\) 0 0
\(546\) −1.85410 −0.0793482
\(547\) − 21.2918i − 0.910371i −0.890397 0.455186i \(-0.849573\pi\)
0.890397 0.455186i \(-0.150427\pi\)
\(548\) 7.38197i 0.315342i
\(549\) 17.4164 0.743314
\(550\) 0 0
\(551\) −3.09017 −0.131646
\(552\) 8.41641i 0.358226i
\(553\) 5.00000i 0.212622i
\(554\) −39.9787 −1.69853
\(555\) 0 0
\(556\) 3.09017 0.131052
\(557\) 4.76393i 0.201854i 0.994894 + 0.100927i \(0.0321809\pi\)
−0.994894 + 0.100927i \(0.967819\pi\)
\(558\) 9.70820i 0.410981i
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) 27.4164 1.15752
\(562\) − 16.3262i − 0.688681i
\(563\) − 7.38197i − 0.311113i −0.987827 0.155556i \(-0.950283\pi\)
0.987827 0.155556i \(-0.0497170\pi\)
\(564\) −0.381966 −0.0160837
\(565\) 0 0
\(566\) 48.3050 2.03041
\(567\) 0.618034i 0.0259550i
\(568\) 14.7984i 0.620926i
\(569\) −20.5279 −0.860573 −0.430286 0.902692i \(-0.641587\pi\)
−0.430286 + 0.902692i \(0.641587\pi\)
\(570\) 0 0
\(571\) −8.12461 −0.340004 −0.170002 0.985444i \(-0.554377\pi\)
−0.170002 + 0.985444i \(0.554377\pi\)
\(572\) − 6.00000i − 0.250873i
\(573\) 1.81966i 0.0760174i
\(574\) −0.763932 −0.0318859
\(575\) 0 0
\(576\) −8.47214 −0.353006
\(577\) − 33.7771i − 1.40616i −0.711111 0.703079i \(-0.751808\pi\)
0.711111 0.703079i \(-0.248192\pi\)
\(578\) 16.8541i 0.701038i
\(579\) −7.70820 −0.320342
\(580\) 0 0
\(581\) −3.85410 −0.159895
\(582\) 6.23607i 0.258493i
\(583\) − 18.1803i − 0.752953i
\(584\) 20.1246 0.832762
\(585\) 0 0
\(586\) 31.5967 1.30525
\(587\) 5.29180i 0.218416i 0.994019 + 0.109208i \(0.0348314\pi\)
−0.994019 + 0.109208i \(0.965169\pi\)
\(588\) 4.09017i 0.168676i
\(589\) 2.56231 0.105578
\(590\) 0 0
\(591\) 3.70820 0.152535
\(592\) 1.14590i 0.0470961i
\(593\) 10.9098i 0.448013i 0.974588 + 0.224007i \(0.0719137\pi\)
−0.974588 + 0.224007i \(0.928086\pi\)
\(594\) 42.3607 1.73808
\(595\) 0 0
\(596\) −2.43769 −0.0998518
\(597\) − 17.5623i − 0.718777i
\(598\) 11.2918i 0.461756i
\(599\) 9.47214 0.387021 0.193510 0.981098i \(-0.438013\pi\)
0.193510 + 0.981098i \(0.438013\pi\)
\(600\) 0 0
\(601\) 2.72949 0.111338 0.0556691 0.998449i \(-0.482271\pi\)
0.0556691 + 0.998449i \(0.482271\pi\)
\(602\) 4.85410i 0.197838i
\(603\) 9.52786i 0.388005i
\(604\) −9.00000 −0.366205
\(605\) 0 0
\(606\) −2.38197 −0.0967608
\(607\) − 35.5623i − 1.44343i −0.692191 0.721715i \(-0.743354\pi\)
0.692191 0.721715i \(-0.256646\pi\)
\(608\) − 2.88854i − 0.117146i
\(609\) 2.23607 0.0906100
\(610\) 0 0
\(611\) 1.14590 0.0463581
\(612\) 6.47214i 0.261621i
\(613\) 14.9787i 0.604985i 0.953152 + 0.302492i \(0.0978186\pi\)
−0.953152 + 0.302492i \(0.902181\pi\)
\(614\) 14.9443 0.603102
\(615\) 0 0
\(616\) −7.23607 −0.291549
\(617\) 14.2361i 0.573123i 0.958062 + 0.286561i \(0.0925122\pi\)
−0.958062 + 0.286561i \(0.907488\pi\)
\(618\) 13.8541i 0.557294i
\(619\) −30.5279 −1.22702 −0.613509 0.789688i \(-0.710243\pi\)
−0.613509 + 0.789688i \(0.710243\pi\)
\(620\) 0 0
\(621\) −18.8197 −0.755207
\(622\) − 13.7639i − 0.551883i
\(623\) 5.52786i 0.221469i
\(624\) 9.00000 0.360288
\(625\) 0 0
\(626\) 27.1246 1.08412
\(627\) − 4.47214i − 0.178600i
\(628\) − 8.14590i − 0.325057i
\(629\) −1.23607 −0.0492853
\(630\) 0 0
\(631\) −10.2361 −0.407491 −0.203746 0.979024i \(-0.565312\pi\)
−0.203746 + 0.979024i \(0.565312\pi\)
\(632\) − 18.0902i − 0.719588i
\(633\) 9.18034i 0.364886i
\(634\) 12.3820 0.491751
\(635\) 0 0
\(636\) −2.14590 −0.0850904
\(637\) − 12.2705i − 0.486175i
\(638\) 30.6525i 1.21354i
\(639\) −13.2361 −0.523611
\(640\) 0 0
\(641\) −1.09017 −0.0430591 −0.0215296 0.999768i \(-0.506854\pi\)
−0.0215296 + 0.999768i \(0.506854\pi\)
\(642\) 26.5623i 1.04833i
\(643\) 30.8328i 1.21593i 0.793965 + 0.607964i \(0.208013\pi\)
−0.793965 + 0.607964i \(0.791987\pi\)
\(644\) −1.43769 −0.0566531
\(645\) 0 0
\(646\) 7.23607 0.284699
\(647\) − 36.5410i − 1.43658i −0.695746 0.718288i \(-0.744926\pi\)
0.695746 0.718288i \(-0.255074\pi\)
\(648\) − 2.23607i − 0.0878410i
\(649\) −56.8328 −2.23088
\(650\) 0 0
\(651\) −1.85410 −0.0726680
\(652\) 6.79837i 0.266245i
\(653\) − 19.0902i − 0.747056i −0.927619 0.373528i \(-0.878148\pi\)
0.927619 0.373528i \(-0.121852\pi\)
\(654\) 16.1803 0.632701
\(655\) 0 0
\(656\) 3.70820 0.144781
\(657\) 18.0000i 0.702247i
\(658\) 0.618034i 0.0240935i
\(659\) −15.5279 −0.604880 −0.302440 0.953168i \(-0.597801\pi\)
−0.302440 + 0.953168i \(0.597801\pi\)
\(660\) 0 0
\(661\) 19.6869 0.765732 0.382866 0.923804i \(-0.374937\pi\)
0.382866 + 0.923804i \(0.374937\pi\)
\(662\) 37.4164i 1.45423i
\(663\) 9.70820i 0.377035i
\(664\) 13.9443 0.541143
\(665\) 0 0
\(666\) −0.763932 −0.0296018
\(667\) − 13.6180i − 0.527292i
\(668\) − 9.00000i − 0.348220i
\(669\) 0.180340 0.00697234
\(670\) 0 0
\(671\) −45.5967 −1.76024
\(672\) 2.09017i 0.0806301i
\(673\) − 12.1803i − 0.469518i −0.972054 0.234759i \(-0.924570\pi\)
0.972054 0.234759i \(-0.0754300\pi\)
\(674\) 12.7082 0.489502
\(675\) 0 0
\(676\) −5.90983 −0.227301
\(677\) 10.6180i 0.408084i 0.978962 + 0.204042i \(0.0654079\pi\)
−0.978962 + 0.204042i \(0.934592\pi\)
\(678\) 27.2705i 1.04732i
\(679\) 2.38197 0.0914115
\(680\) 0 0
\(681\) 14.7639 0.565755
\(682\) − 25.4164i − 0.973245i
\(683\) 13.4721i 0.515497i 0.966212 + 0.257748i \(0.0829806\pi\)
−0.966212 + 0.257748i \(0.917019\pi\)
\(684\) 1.05573 0.0403668
\(685\) 0 0
\(686\) 13.6180 0.519939
\(687\) 21.7082i 0.828220i
\(688\) − 23.5623i − 0.898304i
\(689\) 6.43769 0.245257
\(690\) 0 0
\(691\) 36.2705 1.37980 0.689898 0.723907i \(-0.257656\pi\)
0.689898 + 0.723907i \(0.257656\pi\)
\(692\) 11.6738i 0.443770i
\(693\) − 6.47214i − 0.245856i
\(694\) −32.2148 −1.22286
\(695\) 0 0
\(696\) −8.09017 −0.306657
\(697\) 4.00000i 0.151511i
\(698\) 35.1246i 1.32949i
\(699\) 2.94427 0.111363
\(700\) 0 0
\(701\) −41.0132 −1.54905 −0.774523 0.632546i \(-0.782010\pi\)
−0.774523 + 0.632546i \(0.782010\pi\)
\(702\) 15.0000i 0.566139i
\(703\) 0.201626i 0.00760447i
\(704\) 22.1803 0.835953
\(705\) 0 0
\(706\) −20.8885 −0.786151
\(707\) 0.909830i 0.0342177i
\(708\) 6.70820i 0.252110i
\(709\) 33.5410 1.25966 0.629830 0.776733i \(-0.283125\pi\)
0.629830 + 0.776733i \(0.283125\pi\)
\(710\) 0 0
\(711\) 16.1803 0.606810
\(712\) − 20.0000i − 0.749532i
\(713\) 11.2918i 0.422881i
\(714\) −5.23607 −0.195955
\(715\) 0 0
\(716\) −0.326238 −0.0121921
\(717\) − 20.5279i − 0.766627i
\(718\) 22.2361i 0.829843i
\(719\) 23.2918 0.868637 0.434319 0.900759i \(-0.356989\pi\)
0.434319 + 0.900759i \(0.356989\pi\)
\(720\) 0 0
\(721\) 5.29180 0.197077
\(722\) 29.5623i 1.10020i
\(723\) − 2.52786i − 0.0940123i
\(724\) −0.180340 −0.00670228
\(725\) 0 0
\(726\) −26.5623 −0.985820
\(727\) 24.5623i 0.910966i 0.890245 + 0.455483i \(0.150533\pi\)
−0.890245 + 0.455483i \(0.849467\pi\)
\(728\) − 2.56231i − 0.0949654i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 25.4164 0.940060
\(732\) 5.38197i 0.198923i
\(733\) 19.9787i 0.737931i 0.929443 + 0.368965i \(0.120288\pi\)
−0.929443 + 0.368965i \(0.879712\pi\)
\(734\) −41.3607 −1.52665
\(735\) 0 0
\(736\) 12.7295 0.469215
\(737\) − 24.9443i − 0.918834i
\(738\) 2.47214i 0.0910006i
\(739\) −15.9787 −0.587786 −0.293893 0.955838i \(-0.594951\pi\)
−0.293893 + 0.955838i \(0.594951\pi\)
\(740\) 0 0
\(741\) 1.58359 0.0581747
\(742\) 3.47214i 0.127466i
\(743\) − 28.3607i − 1.04045i −0.854029 0.520226i \(-0.825848\pi\)
0.854029 0.520226i \(-0.174152\pi\)
\(744\) 6.70820 0.245935
\(745\) 0 0
\(746\) 45.7426 1.67476
\(747\) 12.4721i 0.456332i
\(748\) − 16.9443i − 0.619544i
\(749\) 10.1459 0.370723
\(750\) 0 0
\(751\) −5.11146 −0.186520 −0.0932598 0.995642i \(-0.529729\pi\)
−0.0932598 + 0.995642i \(0.529729\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) 29.1803i 1.06339i
\(754\) −10.8541 −0.395283
\(755\) 0 0
\(756\) −1.90983 −0.0694598
\(757\) 30.4164i 1.10550i 0.833346 + 0.552752i \(0.186422\pi\)
−0.833346 + 0.552752i \(0.813578\pi\)
\(758\) 23.6180i 0.857846i
\(759\) 19.7082 0.715362
\(760\) 0 0
\(761\) −18.4508 −0.668843 −0.334421 0.942424i \(-0.608541\pi\)
−0.334421 + 0.942424i \(0.608541\pi\)
\(762\) − 32.1803i − 1.16577i
\(763\) − 6.18034i − 0.223743i
\(764\) 1.12461 0.0406870
\(765\) 0 0
\(766\) −53.9787 −1.95033
\(767\) − 20.1246i − 0.726658i
\(768\) − 13.5623i − 0.489388i
\(769\) −13.4164 −0.483808 −0.241904 0.970300i \(-0.577772\pi\)
−0.241904 + 0.970300i \(0.577772\pi\)
\(770\) 0 0
\(771\) 22.8541 0.823070
\(772\) 4.76393i 0.171458i
\(773\) 36.1591i 1.30055i 0.759699 + 0.650275i \(0.225346\pi\)
−0.759699 + 0.650275i \(0.774654\pi\)
\(774\) 15.7082 0.564620
\(775\) 0 0
\(776\) −8.61803 −0.309369
\(777\) − 0.145898i − 0.00523406i
\(778\) 24.2705i 0.870140i
\(779\) 0.652476 0.0233774
\(780\) 0 0
\(781\) 34.6525 1.23996
\(782\) 31.8885i 1.14033i
\(783\) − 18.0902i − 0.646490i
\(784\) −32.1246 −1.14731
\(785\) 0 0
\(786\) −11.0000 −0.392357
\(787\) − 11.8197i − 0.421325i −0.977559 0.210663i \(-0.932438\pi\)
0.977559 0.210663i \(-0.0675622\pi\)
\(788\) − 2.29180i − 0.0816419i
\(789\) 10.9098 0.388400
\(790\) 0 0
\(791\) 10.4164 0.370365
\(792\) 23.4164i 0.832066i
\(793\) − 16.1459i − 0.573358i
\(794\) 46.9787 1.66721
\(795\) 0 0
\(796\) −10.8541 −0.384713
\(797\) 9.76393i 0.345856i 0.984934 + 0.172928i \(0.0553228\pi\)
−0.984934 + 0.172928i \(0.944677\pi\)
\(798\) 0.854102i 0.0302349i
\(799\) 3.23607 0.114484
\(800\) 0 0
\(801\) 17.8885 0.632061
\(802\) − 43.0344i − 1.51960i
\(803\) − 47.1246i − 1.66299i
\(804\) −2.94427 −0.103836
\(805\) 0 0
\(806\) 9.00000 0.317011
\(807\) − 12.7639i − 0.449312i
\(808\) − 3.29180i − 0.115805i
\(809\) −30.9787 −1.08915 −0.544577 0.838711i \(-0.683310\pi\)
−0.544577 + 0.838711i \(0.683310\pi\)
\(810\) 0 0
\(811\) −14.7082 −0.516475 −0.258237 0.966081i \(-0.583142\pi\)
−0.258237 + 0.966081i \(0.583142\pi\)
\(812\) − 1.38197i − 0.0484975i
\(813\) 8.00000i 0.280572i
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) 25.4164 0.889752
\(817\) − 4.14590i − 0.145047i
\(818\) 2.56231i 0.0895889i
\(819\) 2.29180 0.0800818
\(820\) 0 0
\(821\) −40.6869 −1.41998 −0.709992 0.704210i \(-0.751301\pi\)
−0.709992 + 0.704210i \(0.751301\pi\)
\(822\) 19.3262i 0.674080i
\(823\) − 47.7082i − 1.66300i −0.555522 0.831502i \(-0.687482\pi\)
0.555522 0.831502i \(-0.312518\pi\)
\(824\) −19.1459 −0.666979
\(825\) 0 0
\(826\) 10.8541 0.377663
\(827\) − 0.965558i − 0.0335757i −0.999859 0.0167879i \(-0.994656\pi\)
0.999859 0.0167879i \(-0.00534400\pi\)
\(828\) 4.65248i 0.161685i
\(829\) 35.8541 1.24526 0.622632 0.782515i \(-0.286063\pi\)
0.622632 + 0.782515i \(0.286063\pi\)
\(830\) 0 0
\(831\) −24.7082 −0.857118
\(832\) 7.85410i 0.272292i
\(833\) − 34.6525i − 1.20064i
\(834\) 8.09017 0.280140
\(835\) 0 0
\(836\) −2.76393 −0.0955926
\(837\) 15.0000i 0.518476i
\(838\) − 15.3262i − 0.529436i
\(839\) −10.8541 −0.374725 −0.187363 0.982291i \(-0.559994\pi\)
−0.187363 + 0.982291i \(0.559994\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) − 51.7771i − 1.78436i
\(843\) − 10.0902i − 0.347524i
\(844\) 5.67376 0.195299
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) 10.1459i 0.348617i
\(848\) − 16.8541i − 0.578772i
\(849\) 29.8541 1.02459
\(850\) 0 0
\(851\) −0.888544 −0.0304589
\(852\) − 4.09017i − 0.140127i
\(853\) 15.3050i 0.524032i 0.965064 + 0.262016i \(0.0843873\pi\)
−0.965064 + 0.262016i \(0.915613\pi\)
\(854\) 8.70820 0.297989
\(855\) 0 0
\(856\) −36.7082 −1.25466
\(857\) 19.6869i 0.672492i 0.941774 + 0.336246i \(0.109157\pi\)
−0.941774 + 0.336246i \(0.890843\pi\)
\(858\) − 15.7082i − 0.536269i
\(859\) 1.58359 0.0540315 0.0270157 0.999635i \(-0.491400\pi\)
0.0270157 + 0.999635i \(0.491400\pi\)
\(860\) 0 0
\(861\) −0.472136 −0.0160904
\(862\) 48.2705i 1.64410i
\(863\) 21.4377i 0.729748i 0.931057 + 0.364874i \(0.118888\pi\)
−0.931057 + 0.364874i \(0.881112\pi\)
\(864\) 16.9098 0.575284
\(865\) 0 0
\(866\) −43.4508 −1.47652
\(867\) 10.4164i 0.353760i
\(868\) 1.14590i 0.0388943i
\(869\) −42.3607 −1.43699
\(870\) 0 0
\(871\) 8.83282 0.299289
\(872\) 22.3607i 0.757228i
\(873\) − 7.70820i − 0.260883i
\(874\) 5.20163 0.175948
\(875\) 0 0
\(876\) −5.56231 −0.187933
\(877\) − 36.5410i − 1.23390i −0.787001 0.616951i \(-0.788368\pi\)
0.787001 0.616951i \(-0.211632\pi\)
\(878\) − 66.3050i − 2.23768i
\(879\) 19.5279 0.658659
\(880\) 0 0
\(881\) −40.3607 −1.35979 −0.679893 0.733311i \(-0.737974\pi\)
−0.679893 + 0.733311i \(0.737974\pi\)
\(882\) − 21.4164i − 0.721128i
\(883\) 20.5836i 0.692693i 0.938107 + 0.346347i \(0.112578\pi\)
−0.938107 + 0.346347i \(0.887422\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −48.4508 −1.62774
\(887\) 29.8885i 1.00356i 0.864996 + 0.501780i \(0.167321\pi\)
−0.864996 + 0.501780i \(0.832679\pi\)
\(888\) 0.527864i 0.0177140i
\(889\) −12.2918 −0.412254
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) − 0.111456i − 0.00373183i
\(893\) − 0.527864i − 0.0176643i
\(894\) −6.38197 −0.213445
\(895\) 0 0
\(896\) −8.41641 −0.281172
\(897\) 6.97871i 0.233012i
\(898\) 7.56231i 0.252357i
\(899\) −10.8541 −0.362005
\(900\) 0 0
\(901\) 18.1803 0.605675
\(902\) − 6.47214i − 0.215499i
\(903\) 3.00000i 0.0998337i
\(904\) −37.6869 −1.25345
\(905\) 0 0
\(906\) −23.5623 −0.782805
\(907\) − 33.2492i − 1.10402i −0.833837 0.552011i \(-0.813861\pi\)
0.833837 0.552011i \(-0.186139\pi\)
\(908\) − 9.12461i − 0.302811i
\(909\) 2.94427 0.0976553
\(910\) 0 0
\(911\) −40.2361 −1.33308 −0.666540 0.745469i \(-0.732226\pi\)
−0.666540 + 0.745469i \(0.732226\pi\)
\(912\) − 4.14590i − 0.137284i
\(913\) − 32.6525i − 1.08064i
\(914\) −34.6525 −1.14620
\(915\) 0 0
\(916\) 13.4164 0.443291
\(917\) 4.20163i 0.138750i
\(918\) 42.3607i 1.39811i
\(919\) 53.2148 1.75539 0.877697 0.479216i \(-0.159079\pi\)
0.877697 + 0.479216i \(0.159079\pi\)
\(920\) 0 0
\(921\) 9.23607 0.304339
\(922\) − 1.32624i − 0.0436773i
\(923\) 12.2705i 0.403889i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 39.0344 1.28275
\(927\) − 17.1246i − 0.562446i
\(928\) 12.2361i 0.401669i
\(929\) −41.6312 −1.36588 −0.682938 0.730477i \(-0.739298\pi\)
−0.682938 + 0.730477i \(0.739298\pi\)
\(930\) 0 0
\(931\) −5.65248 −0.185252
\(932\) − 1.81966i − 0.0596049i
\(933\) − 8.50658i − 0.278493i
\(934\) 44.4164 1.45335
\(935\) 0 0
\(936\) −8.29180 −0.271026
\(937\) 17.7295i 0.579197i 0.957148 + 0.289599i \(0.0935218\pi\)
−0.957148 + 0.289599i \(0.906478\pi\)
\(938\) 4.76393i 0.155548i
\(939\) 16.7639 0.547070
\(940\) 0 0
\(941\) −46.4164 −1.51313 −0.756566 0.653918i \(-0.773124\pi\)
−0.756566 + 0.653918i \(0.773124\pi\)
\(942\) − 21.3262i − 0.694846i
\(943\) 2.87539i 0.0936355i
\(944\) −52.6869 −1.71481
\(945\) 0 0
\(946\) −41.1246 −1.33708
\(947\) 2.65248i 0.0861939i 0.999071 + 0.0430969i \(0.0137224\pi\)
−0.999071 + 0.0430969i \(0.986278\pi\)
\(948\) 5.00000i 0.162392i
\(949\) 16.6869 0.541680
\(950\) 0 0
\(951\) 7.65248 0.248149
\(952\) − 7.23607i − 0.234522i
\(953\) 7.74265i 0.250809i 0.992106 + 0.125404i \(0.0400228\pi\)
−0.992106 + 0.125404i \(0.959977\pi\)
\(954\) 11.2361 0.363781
\(955\) 0 0
\(956\) −12.6869 −0.410324
\(957\) 18.9443i 0.612381i
\(958\) − 17.5623i − 0.567412i
\(959\) 7.38197 0.238376
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0.708204i 0.0228334i
\(963\) − 32.8328i − 1.05802i
\(964\) −1.56231 −0.0503185
\(965\) 0 0
\(966\) −3.76393 −0.121103
\(967\) 4.11146i 0.132216i 0.997812 + 0.0661078i \(0.0210581\pi\)
−0.997812 + 0.0661078i \(0.978942\pi\)
\(968\) − 36.7082i − 1.17985i
\(969\) 4.47214 0.143666
\(970\) 0 0
\(971\) 5.61803 0.180291 0.0901456 0.995929i \(-0.471267\pi\)
0.0901456 + 0.995929i \(0.471267\pi\)
\(972\) 9.88854i 0.317175i
\(973\) − 3.09017i − 0.0990663i
\(974\) −58.9230 −1.88801
\(975\) 0 0
\(976\) −42.2705 −1.35305
\(977\) − 2.34752i − 0.0751040i −0.999295 0.0375520i \(-0.988044\pi\)
0.999295 0.0375520i \(-0.0119560\pi\)
\(978\) 17.7984i 0.569129i
\(979\) −46.8328 −1.49678
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 69.9787i 2.23311i
\(983\) − 9.61803i − 0.306768i −0.988167 0.153384i \(-0.950983\pi\)
0.988167 0.153384i \(-0.0490171\pi\)
\(984\) 1.70820 0.0544556
\(985\) 0 0
\(986\) −30.6525 −0.976174
\(987\) 0.381966i 0.0121581i
\(988\) − 0.978714i − 0.0311370i
\(989\) 18.2705 0.580968
\(990\) 0 0
\(991\) −15.3607 −0.487948 −0.243974 0.969782i \(-0.578451\pi\)
−0.243974 + 0.969782i \(0.578451\pi\)
\(992\) − 10.1459i − 0.322133i
\(993\) 23.1246i 0.733837i
\(994\) −6.61803 −0.209911
\(995\) 0 0
\(996\) −3.85410 −0.122122
\(997\) 24.8885i 0.788228i 0.919062 + 0.394114i \(0.128949\pi\)
−0.919062 + 0.394114i \(0.871051\pi\)
\(998\) 12.2361i 0.387326i
\(999\) −1.18034 −0.0373443
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.624.1 4
5.2 odd 4 625.2.a.c.1.2 2
5.3 odd 4 625.2.a.b.1.1 2
5.4 even 2 inner 625.2.b.a.624.4 4
15.2 even 4 5625.2.a.d.1.1 2
15.8 even 4 5625.2.a.f.1.2 2
20.3 even 4 10000.2.a.c.1.1 2
20.7 even 4 10000.2.a.l.1.2 2
25.2 odd 20 125.2.d.a.101.1 4
25.3 odd 20 625.2.d.h.376.1 4
25.4 even 10 625.2.e.c.249.2 8
25.6 even 5 625.2.e.c.374.2 8
25.8 odd 20 625.2.d.h.251.1 4
25.9 even 10 125.2.e.a.99.1 8
25.11 even 5 125.2.e.a.24.1 8
25.12 odd 20 125.2.d.a.26.1 4
25.13 odd 20 25.2.d.a.6.1 4
25.14 even 10 125.2.e.a.24.2 8
25.16 even 5 125.2.e.a.99.2 8
25.17 odd 20 625.2.d.b.251.1 4
25.19 even 10 625.2.e.c.374.1 8
25.21 even 5 625.2.e.c.249.1 8
25.22 odd 20 625.2.d.b.376.1 4
25.23 odd 20 25.2.d.a.21.1 yes 4
75.23 even 20 225.2.h.b.46.1 4
75.38 even 20 225.2.h.b.181.1 4
100.23 even 20 400.2.u.b.321.1 4
100.63 even 20 400.2.u.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 25.13 odd 20
25.2.d.a.21.1 yes 4 25.23 odd 20
125.2.d.a.26.1 4 25.12 odd 20
125.2.d.a.101.1 4 25.2 odd 20
125.2.e.a.24.1 8 25.11 even 5
125.2.e.a.24.2 8 25.14 even 10
125.2.e.a.99.1 8 25.9 even 10
125.2.e.a.99.2 8 25.16 even 5
225.2.h.b.46.1 4 75.23 even 20
225.2.h.b.181.1 4 75.38 even 20
400.2.u.b.81.1 4 100.63 even 20
400.2.u.b.321.1 4 100.23 even 20
625.2.a.b.1.1 2 5.3 odd 4
625.2.a.c.1.2 2 5.2 odd 4
625.2.b.a.624.1 4 1.1 even 1 trivial
625.2.b.a.624.4 4 5.4 even 2 inner
625.2.d.b.251.1 4 25.17 odd 20
625.2.d.b.376.1 4 25.22 odd 20
625.2.d.h.251.1 4 25.8 odd 20
625.2.d.h.376.1 4 25.3 odd 20
625.2.e.c.249.1 8 25.21 even 5
625.2.e.c.249.2 8 25.4 even 10
625.2.e.c.374.1 8 25.19 even 10
625.2.e.c.374.2 8 25.6 even 5
5625.2.a.d.1.1 2 15.2 even 4
5625.2.a.f.1.2 2 15.8 even 4
10000.2.a.c.1.1 2 20.3 even 4
10000.2.a.l.1.2 2 20.7 even 4