Properties

Label 625.2.a.c.1.2
Level $625$
Weight $2$
Character 625.1
Self dual yes
Analytic conductor $4.991$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.99065012633\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\chi\) \(=\) 625.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} -5.23607 q^{11} -0.618034 q^{12} -1.85410 q^{13} -1.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} -3.23607 q^{18} +0.854102 q^{19} +0.618034 q^{21} -8.47214 q^{22} -3.76393 q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000 q^{27} -0.381966 q^{28} -3.61803 q^{29} -3.00000 q^{31} -3.38197 q^{32} +5.23607 q^{33} +8.47214 q^{34} -1.23607 q^{36} +0.236068 q^{37} +1.38197 q^{38} +1.85410 q^{39} -0.763932 q^{41} +1.00000 q^{42} +4.85410 q^{43} -3.23607 q^{44} -6.09017 q^{46} -0.618034 q^{47} +4.85410 q^{48} -6.61803 q^{49} -5.23607 q^{51} -1.14590 q^{52} +3.47214 q^{53} +8.09017 q^{54} +1.38197 q^{56} -0.854102 q^{57} -5.85410 q^{58} -10.8541 q^{59} +8.70820 q^{61} -4.85410 q^{62} +1.23607 q^{63} +4.23607 q^{64} +8.47214 q^{66} -4.76393 q^{67} +3.23607 q^{68} +3.76393 q^{69} -6.61803 q^{71} +4.47214 q^{72} +9.00000 q^{73} +0.381966 q^{74} +0.527864 q^{76} +3.23607 q^{77} +3.00000 q^{78} -8.09017 q^{79} +1.00000 q^{81} -1.23607 q^{82} +6.23607 q^{83} +0.381966 q^{84} +7.85410 q^{86} +3.61803 q^{87} +11.7082 q^{88} -8.94427 q^{89} +1.14590 q^{91} -2.32624 q^{92} +3.00000 q^{93} -1.00000 q^{94} +3.38197 q^{96} +3.85410 q^{97} -10.7082 q^{98} +10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} - 6 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} - 8 q^{22} - 12 q^{23} - 6 q^{26} + 10 q^{27} - 3 q^{28} - 5 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 2 q^{42} + 3 q^{43} - 2 q^{44} - q^{46} + q^{47} + 3 q^{48} - 11 q^{49} - 6 q^{51} - 9 q^{52} - 2 q^{53} + 5 q^{54} + 5 q^{56} + 5 q^{57} - 5 q^{58} - 15 q^{59} + 4 q^{61} - 3 q^{62} - 2 q^{63} + 4 q^{64} + 8 q^{66} - 14 q^{67} + 2 q^{68} + 12 q^{69} - 11 q^{71} + 18 q^{73} + 3 q^{74} + 10 q^{76} + 2 q^{77} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 2 q^{82} + 8 q^{83} + 3 q^{84} + 9 q^{86} + 5 q^{87} + 10 q^{88} + 9 q^{91} + 11 q^{92} + 6 q^{93} - 2 q^{94} + 9 q^{96} + q^{97} - 8 q^{98} + 12 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) −0.618034 −0.233595 −0.116797 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) −2.23607 −0.790569
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) −0.618034 −0.178411
\(13\) −1.85410 −0.514235 −0.257118 0.966380i \(-0.582773\pi\)
−0.257118 + 0.966380i \(0.582773\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) −3.23607 −0.762749
\(19\) 0.854102 0.195944 0.0979722 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) 0 0
\(21\) 0.618034 0.134866
\(22\) −8.47214 −1.80627
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 5.00000 0.962250
\(28\) −0.381966 −0.0721848
\(29\) −3.61803 −0.671852 −0.335926 0.941888i \(-0.609049\pi\)
−0.335926 + 0.941888i \(0.609049\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −3.38197 −0.597853
\(33\) 5.23607 0.911482
\(34\) 8.47214 1.45296
\(35\) 0 0
\(36\) −1.23607 −0.206011
\(37\) 0.236068 0.0388093 0.0194047 0.999812i \(-0.493823\pi\)
0.0194047 + 0.999812i \(0.493823\pi\)
\(38\) 1.38197 0.224184
\(39\) 1.85410 0.296894
\(40\) 0 0
\(41\) −0.763932 −0.119306 −0.0596531 0.998219i \(-0.518999\pi\)
−0.0596531 + 0.998219i \(0.518999\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.85410 0.740244 0.370122 0.928983i \(-0.379316\pi\)
0.370122 + 0.928983i \(0.379316\pi\)
\(44\) −3.23607 −0.487856
\(45\) 0 0
\(46\) −6.09017 −0.897947
\(47\) −0.618034 −0.0901495 −0.0450748 0.998984i \(-0.514353\pi\)
−0.0450748 + 0.998984i \(0.514353\pi\)
\(48\) 4.85410 0.700629
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) −5.23607 −0.733196
\(52\) −1.14590 −0.158907
\(53\) 3.47214 0.476935 0.238467 0.971151i \(-0.423355\pi\)
0.238467 + 0.971151i \(0.423355\pi\)
\(54\) 8.09017 1.10093
\(55\) 0 0
\(56\) 1.38197 0.184673
\(57\) −0.854102 −0.113129
\(58\) −5.85410 −0.768681
\(59\) −10.8541 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(60\) 0 0
\(61\) 8.70820 1.11497 0.557486 0.830187i \(-0.311766\pi\)
0.557486 + 0.830187i \(0.311766\pi\)
\(62\) −4.85410 −0.616472
\(63\) 1.23607 0.155730
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 8.47214 1.04285
\(67\) −4.76393 −0.582007 −0.291003 0.956722i \(-0.593989\pi\)
−0.291003 + 0.956722i \(0.593989\pi\)
\(68\) 3.23607 0.392431
\(69\) 3.76393 0.453124
\(70\) 0 0
\(71\) −6.61803 −0.785416 −0.392708 0.919663i \(-0.628462\pi\)
−0.392708 + 0.919663i \(0.628462\pi\)
\(72\) 4.47214 0.527046
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0.381966 0.0444026
\(75\) 0 0
\(76\) 0.527864 0.0605502
\(77\) 3.23607 0.368784
\(78\) 3.00000 0.339683
\(79\) −8.09017 −0.910215 −0.455108 0.890436i \(-0.650399\pi\)
−0.455108 + 0.890436i \(0.650399\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.23607 −0.136501
\(83\) 6.23607 0.684497 0.342249 0.939609i \(-0.388811\pi\)
0.342249 + 0.939609i \(0.388811\pi\)
\(84\) 0.381966 0.0416759
\(85\) 0 0
\(86\) 7.85410 0.846930
\(87\) 3.61803 0.387894
\(88\) 11.7082 1.24810
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 0 0
\(91\) 1.14590 0.120123
\(92\) −2.32624 −0.242527
\(93\) 3.00000 0.311086
\(94\) −1.00000 −0.103142
\(95\) 0 0
\(96\) 3.38197 0.345170
\(97\) 3.85410 0.391325 0.195662 0.980671i \(-0.437314\pi\)
0.195662 + 0.980671i \(0.437314\pi\)
\(98\) −10.7082 −1.08169
\(99\) 10.4721 1.05249
\(100\) 0 0
\(101\) 1.47214 0.146483 0.0732415 0.997314i \(-0.476666\pi\)
0.0732415 + 0.997314i \(0.476666\pi\)
\(102\) −8.47214 −0.838866
\(103\) −8.56231 −0.843669 −0.421835 0.906673i \(-0.638614\pi\)
−0.421835 + 0.906673i \(0.638614\pi\)
\(104\) 4.14590 0.406539
\(105\) 0 0
\(106\) 5.61803 0.545672
\(107\) 16.4164 1.58703 0.793517 0.608548i \(-0.208248\pi\)
0.793517 + 0.608548i \(0.208248\pi\)
\(108\) 3.09017 0.297352
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −0.236068 −0.0224066
\(112\) 3.00000 0.283473
\(113\) −16.8541 −1.58550 −0.792750 0.609547i \(-0.791352\pi\)
−0.792750 + 0.609547i \(0.791352\pi\)
\(114\) −1.38197 −0.129433
\(115\) 0 0
\(116\) −2.23607 −0.207614
\(117\) 3.70820 0.342824
\(118\) −17.5623 −1.61674
\(119\) −3.23607 −0.296650
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 14.0902 1.27566
\(123\) 0.763932 0.0688814
\(124\) −1.85410 −0.166503
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −19.8885 −1.76482 −0.882411 0.470479i \(-0.844081\pi\)
−0.882411 + 0.470479i \(0.844081\pi\)
\(128\) 13.6180 1.20368
\(129\) −4.85410 −0.427380
\(130\) 0 0
\(131\) 6.79837 0.593977 0.296988 0.954881i \(-0.404018\pi\)
0.296988 + 0.954881i \(0.404018\pi\)
\(132\) 3.23607 0.281664
\(133\) −0.527864 −0.0457716
\(134\) −7.70820 −0.665887
\(135\) 0 0
\(136\) −11.7082 −1.00397
\(137\) 11.9443 1.02047 0.510234 0.860036i \(-0.329559\pi\)
0.510234 + 0.860036i \(0.329559\pi\)
\(138\) 6.09017 0.518430
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0.618034 0.0520479
\(142\) −10.7082 −0.898613
\(143\) 9.70820 0.811841
\(144\) 9.70820 0.809017
\(145\) 0 0
\(146\) 14.5623 1.20519
\(147\) 6.61803 0.545846
\(148\) 0.145898 0.0119927
\(149\) −3.94427 −0.323127 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(150\) 0 0
\(151\) 14.5623 1.18506 0.592532 0.805547i \(-0.298128\pi\)
0.592532 + 0.805547i \(0.298128\pi\)
\(152\) −1.90983 −0.154908
\(153\) −10.4721 −0.846622
\(154\) 5.23607 0.421934
\(155\) 0 0
\(156\) 1.14590 0.0917453
\(157\) −13.1803 −1.05191 −0.525953 0.850514i \(-0.676291\pi\)
−0.525953 + 0.850514i \(0.676291\pi\)
\(158\) −13.0902 −1.04140
\(159\) −3.47214 −0.275358
\(160\) 0 0
\(161\) 2.32624 0.183333
\(162\) 1.61803 0.127125
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −0.472136 −0.0368676
\(165\) 0 0
\(166\) 10.0902 0.783149
\(167\) −14.5623 −1.12687 −0.563433 0.826162i \(-0.690520\pi\)
−0.563433 + 0.826162i \(0.690520\pi\)
\(168\) −1.38197 −0.106621
\(169\) −9.56231 −0.735562
\(170\) 0 0
\(171\) −1.70820 −0.130630
\(172\) 3.00000 0.228748
\(173\) −18.8885 −1.43607 −0.718035 0.696007i \(-0.754958\pi\)
−0.718035 + 0.696007i \(0.754958\pi\)
\(174\) 5.85410 0.443798
\(175\) 0 0
\(176\) 25.4164 1.91583
\(177\) 10.8541 0.815844
\(178\) −14.4721 −1.08473
\(179\) −0.527864 −0.0394544 −0.0197272 0.999805i \(-0.506280\pi\)
−0.0197272 + 0.999805i \(0.506280\pi\)
\(180\) 0 0
\(181\) 0.291796 0.0216890 0.0108445 0.999941i \(-0.496548\pi\)
0.0108445 + 0.999941i \(0.496548\pi\)
\(182\) 1.85410 0.137435
\(183\) −8.70820 −0.643729
\(184\) 8.41641 0.620466
\(185\) 0 0
\(186\) 4.85410 0.355920
\(187\) −27.4164 −2.00489
\(188\) −0.381966 −0.0278577
\(189\) −3.09017 −0.224777
\(190\) 0 0
\(191\) −1.81966 −0.131666 −0.0658330 0.997831i \(-0.520970\pi\)
−0.0658330 + 0.997831i \(0.520970\pi\)
\(192\) −4.23607 −0.305712
\(193\) −7.70820 −0.554849 −0.277424 0.960747i \(-0.589481\pi\)
−0.277424 + 0.960747i \(0.589481\pi\)
\(194\) 6.23607 0.447724
\(195\) 0 0
\(196\) −4.09017 −0.292155
\(197\) −3.70820 −0.264199 −0.132099 0.991236i \(-0.542172\pi\)
−0.132099 + 0.991236i \(0.542172\pi\)
\(198\) 16.9443 1.20418
\(199\) −17.5623 −1.24496 −0.622479 0.782636i \(-0.713875\pi\)
−0.622479 + 0.782636i \(0.713875\pi\)
\(200\) 0 0
\(201\) 4.76393 0.336022
\(202\) 2.38197 0.167595
\(203\) 2.23607 0.156941
\(204\) −3.23607 −0.226570
\(205\) 0 0
\(206\) −13.8541 −0.965261
\(207\) 7.52786 0.523223
\(208\) 9.00000 0.624038
\(209\) −4.47214 −0.309344
\(210\) 0 0
\(211\) −9.18034 −0.632001 −0.316000 0.948759i \(-0.602340\pi\)
−0.316000 + 0.948759i \(0.602340\pi\)
\(212\) 2.14590 0.147381
\(213\) 6.61803 0.453460
\(214\) 26.5623 1.81576
\(215\) 0 0
\(216\) −11.1803 −0.760726
\(217\) 1.85410 0.125865
\(218\) 16.1803 1.09587
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −9.70820 −0.653044
\(222\) −0.381966 −0.0256359
\(223\) 0.180340 0.0120765 0.00603823 0.999982i \(-0.498078\pi\)
0.00603823 + 0.999982i \(0.498078\pi\)
\(224\) 2.09017 0.139655
\(225\) 0 0
\(226\) −27.2705 −1.81401
\(227\) −14.7639 −0.979917 −0.489958 0.871746i \(-0.662988\pi\)
−0.489958 + 0.871746i \(0.662988\pi\)
\(228\) −0.527864 −0.0349587
\(229\) 21.7082 1.43452 0.717259 0.696806i \(-0.245396\pi\)
0.717259 + 0.696806i \(0.245396\pi\)
\(230\) 0 0
\(231\) −3.23607 −0.212918
\(232\) 8.09017 0.531146
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −6.70820 −0.436667
\(237\) 8.09017 0.525513
\(238\) −5.23607 −0.339404
\(239\) −20.5279 −1.32784 −0.663919 0.747805i \(-0.731108\pi\)
−0.663919 + 0.747805i \(0.731108\pi\)
\(240\) 0 0
\(241\) 2.52786 0.162834 0.0814170 0.996680i \(-0.474055\pi\)
0.0814170 + 0.996680i \(0.474055\pi\)
\(242\) 26.5623 1.70749
\(243\) −16.0000 −1.02640
\(244\) 5.38197 0.344545
\(245\) 0 0
\(246\) 1.23607 0.0788088
\(247\) −1.58359 −0.100762
\(248\) 6.70820 0.425971
\(249\) −6.23607 −0.395195
\(250\) 0 0
\(251\) −29.1803 −1.84185 −0.920923 0.389744i \(-0.872564\pi\)
−0.920923 + 0.389744i \(0.872564\pi\)
\(252\) 0.763932 0.0481232
\(253\) 19.7082 1.23904
\(254\) −32.1803 −2.01917
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −22.8541 −1.42560 −0.712800 0.701367i \(-0.752573\pi\)
−0.712800 + 0.701367i \(0.752573\pi\)
\(258\) −7.85410 −0.488975
\(259\) −0.145898 −0.00906566
\(260\) 0 0
\(261\) 7.23607 0.447901
\(262\) 11.0000 0.679582
\(263\) 10.9098 0.672729 0.336364 0.941732i \(-0.390803\pi\)
0.336364 + 0.941732i \(0.390803\pi\)
\(264\) −11.7082 −0.720590
\(265\) 0 0
\(266\) −0.854102 −0.0523684
\(267\) 8.94427 0.547381
\(268\) −2.94427 −0.179850
\(269\) −12.7639 −0.778231 −0.389115 0.921189i \(-0.627219\pi\)
−0.389115 + 0.921189i \(0.627219\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −25.4164 −1.54110
\(273\) −1.14590 −0.0693529
\(274\) 19.3262 1.16754
\(275\) 0 0
\(276\) 2.32624 0.140023
\(277\) 24.7082 1.48457 0.742286 0.670083i \(-0.233742\pi\)
0.742286 + 0.670083i \(0.233742\pi\)
\(278\) 8.09017 0.485216
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 10.0902 0.601929 0.300965 0.953635i \(-0.402691\pi\)
0.300965 + 0.953635i \(0.402691\pi\)
\(282\) 1.00000 0.0595491
\(283\) 29.8541 1.77464 0.887321 0.461152i \(-0.152564\pi\)
0.887321 + 0.461152i \(0.152564\pi\)
\(284\) −4.09017 −0.242707
\(285\) 0 0
\(286\) 15.7082 0.928846
\(287\) 0.472136 0.0278693
\(288\) 6.76393 0.398569
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) −3.85410 −0.225931
\(292\) 5.56231 0.325509
\(293\) 19.5279 1.14083 0.570415 0.821357i \(-0.306782\pi\)
0.570415 + 0.821357i \(0.306782\pi\)
\(294\) 10.7082 0.624515
\(295\) 0 0
\(296\) −0.527864 −0.0306815
\(297\) −26.1803 −1.51914
\(298\) −6.38197 −0.369697
\(299\) 6.97871 0.403589
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 23.5623 1.35586
\(303\) −1.47214 −0.0845720
\(304\) −4.14590 −0.237784
\(305\) 0 0
\(306\) −16.9443 −0.968640
\(307\) −9.23607 −0.527130 −0.263565 0.964642i \(-0.584898\pi\)
−0.263565 + 0.964642i \(0.584898\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.56231 0.487093
\(310\) 0 0
\(311\) 8.50658 0.482364 0.241182 0.970480i \(-0.422465\pi\)
0.241182 + 0.970480i \(0.422465\pi\)
\(312\) −4.14590 −0.234715
\(313\) 16.7639 0.947553 0.473777 0.880645i \(-0.342890\pi\)
0.473777 + 0.880645i \(0.342890\pi\)
\(314\) −21.3262 −1.20351
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −7.65248 −0.429806 −0.214903 0.976635i \(-0.568944\pi\)
−0.214903 + 0.976635i \(0.568944\pi\)
\(318\) −5.61803 −0.315044
\(319\) 18.9443 1.06068
\(320\) 0 0
\(321\) −16.4164 −0.916275
\(322\) 3.76393 0.209756
\(323\) 4.47214 0.248836
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) −17.7984 −0.985761
\(327\) −10.0000 −0.553001
\(328\) 1.70820 0.0943198
\(329\) 0.381966 0.0210585
\(330\) 0 0
\(331\) −23.1246 −1.27104 −0.635522 0.772083i \(-0.719215\pi\)
−0.635522 + 0.772083i \(0.719215\pi\)
\(332\) 3.85410 0.211521
\(333\) −0.472136 −0.0258729
\(334\) −23.5623 −1.28927
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −7.85410 −0.427840 −0.213920 0.976851i \(-0.568623\pi\)
−0.213920 + 0.976851i \(0.568623\pi\)
\(338\) −15.4721 −0.841573
\(339\) 16.8541 0.915389
\(340\) 0 0
\(341\) 15.7082 0.850647
\(342\) −2.76393 −0.149456
\(343\) 8.41641 0.454443
\(344\) −10.8541 −0.585214
\(345\) 0 0
\(346\) −30.5623 −1.64304
\(347\) 19.9098 1.06882 0.534408 0.845227i \(-0.320535\pi\)
0.534408 + 0.845227i \(0.320535\pi\)
\(348\) 2.23607 0.119866
\(349\) 21.7082 1.16201 0.581007 0.813899i \(-0.302659\pi\)
0.581007 + 0.813899i \(0.302659\pi\)
\(350\) 0 0
\(351\) −9.27051 −0.494823
\(352\) 17.7082 0.943850
\(353\) −12.9098 −0.687121 −0.343560 0.939131i \(-0.611633\pi\)
−0.343560 + 0.939131i \(0.611633\pi\)
\(354\) 17.5623 0.933426
\(355\) 0 0
\(356\) −5.52786 −0.292976
\(357\) 3.23607 0.171271
\(358\) −0.854102 −0.0451407
\(359\) 13.7426 0.725309 0.362655 0.931924i \(-0.381870\pi\)
0.362655 + 0.931924i \(0.381870\pi\)
\(360\) 0 0
\(361\) −18.2705 −0.961606
\(362\) 0.472136 0.0248149
\(363\) −16.4164 −0.861638
\(364\) 0.708204 0.0371200
\(365\) 0 0
\(366\) −14.0902 −0.736505
\(367\) 25.5623 1.33434 0.667171 0.744905i \(-0.267505\pi\)
0.667171 + 0.744905i \(0.267505\pi\)
\(368\) 18.2705 0.952416
\(369\) 1.52786 0.0795374
\(370\) 0 0
\(371\) −2.14590 −0.111409
\(372\) 1.85410 0.0961307
\(373\) 28.2705 1.46379 0.731896 0.681417i \(-0.238636\pi\)
0.731896 + 0.681417i \(0.238636\pi\)
\(374\) −44.3607 −2.29384
\(375\) 0 0
\(376\) 1.38197 0.0712695
\(377\) 6.70820 0.345490
\(378\) −5.00000 −0.257172
\(379\) 14.5967 0.749785 0.374892 0.927068i \(-0.377680\pi\)
0.374892 + 0.927068i \(0.377680\pi\)
\(380\) 0 0
\(381\) 19.8885 1.01892
\(382\) −2.94427 −0.150642
\(383\) −33.3607 −1.70465 −0.852326 0.523012i \(-0.824808\pi\)
−0.852326 + 0.523012i \(0.824808\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) −12.4721 −0.634815
\(387\) −9.70820 −0.493496
\(388\) 2.38197 0.120926
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) −19.7082 −0.996687
\(392\) 14.7984 0.747431
\(393\) −6.79837 −0.342933
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 6.47214 0.325237
\(397\) −29.0344 −1.45720 −0.728598 0.684941i \(-0.759828\pi\)
−0.728598 + 0.684941i \(0.759828\pi\)
\(398\) −28.4164 −1.42439
\(399\) 0.527864 0.0264263
\(400\) 0 0
\(401\) 26.5967 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(402\) 7.70820 0.384450
\(403\) 5.56231 0.277078
\(404\) 0.909830 0.0452657
\(405\) 0 0
\(406\) 3.61803 0.179560
\(407\) −1.23607 −0.0612696
\(408\) 11.7082 0.579642
\(409\) 1.58359 0.0783036 0.0391518 0.999233i \(-0.487534\pi\)
0.0391518 + 0.999233i \(0.487534\pi\)
\(410\) 0 0
\(411\) −11.9443 −0.589167
\(412\) −5.29180 −0.260708
\(413\) 6.70820 0.330089
\(414\) 12.1803 0.598631
\(415\) 0 0
\(416\) 6.27051 0.307437
\(417\) −5.00000 −0.244851
\(418\) −7.23607 −0.353928
\(419\) −9.47214 −0.462744 −0.231372 0.972865i \(-0.574321\pi\)
−0.231372 + 0.972865i \(0.574321\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) −14.8541 −0.723086
\(423\) 1.23607 0.0600997
\(424\) −7.76393 −0.377050
\(425\) 0 0
\(426\) 10.7082 0.518814
\(427\) −5.38197 −0.260452
\(428\) 10.1459 0.490420
\(429\) −9.70820 −0.468717
\(430\) 0 0
\(431\) −29.8328 −1.43700 −0.718498 0.695529i \(-0.755170\pi\)
−0.718498 + 0.695529i \(0.755170\pi\)
\(432\) −24.2705 −1.16772
\(433\) −26.8541 −1.29053 −0.645263 0.763961i \(-0.723252\pi\)
−0.645263 + 0.763961i \(0.723252\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) 6.18034 0.295985
\(437\) −3.21478 −0.153784
\(438\) −14.5623 −0.695814
\(439\) −40.9787 −1.95581 −0.977904 0.209056i \(-0.932961\pi\)
−0.977904 + 0.209056i \(0.932961\pi\)
\(440\) 0 0
\(441\) 13.2361 0.630289
\(442\) −15.7082 −0.747163
\(443\) −29.9443 −1.42270 −0.711348 0.702840i \(-0.751915\pi\)
−0.711348 + 0.702840i \(0.751915\pi\)
\(444\) −0.145898 −0.00692401
\(445\) 0 0
\(446\) 0.291796 0.0138169
\(447\) 3.94427 0.186558
\(448\) −2.61803 −0.123690
\(449\) 4.67376 0.220568 0.110284 0.993900i \(-0.464824\pi\)
0.110284 + 0.993900i \(0.464824\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −10.4164 −0.489947
\(453\) −14.5623 −0.684197
\(454\) −23.8885 −1.12114
\(455\) 0 0
\(456\) 1.90983 0.0894360
\(457\) 21.4164 1.00182 0.500909 0.865500i \(-0.332999\pi\)
0.500909 + 0.865500i \(0.332999\pi\)
\(458\) 35.1246 1.64127
\(459\) 26.1803 1.22199
\(460\) 0 0
\(461\) 0.819660 0.0381754 0.0190877 0.999818i \(-0.493924\pi\)
0.0190877 + 0.999818i \(0.493924\pi\)
\(462\) −5.23607 −0.243604
\(463\) 24.1246 1.12117 0.560583 0.828098i \(-0.310577\pi\)
0.560583 + 0.828098i \(0.310577\pi\)
\(464\) 17.5623 0.815310
\(465\) 0 0
\(466\) 4.76393 0.220685
\(467\) −27.4508 −1.27027 −0.635137 0.772400i \(-0.719056\pi\)
−0.635137 + 0.772400i \(0.719056\pi\)
\(468\) 2.29180 0.105938
\(469\) 2.94427 0.135954
\(470\) 0 0
\(471\) 13.1803 0.607318
\(472\) 24.2705 1.11714
\(473\) −25.4164 −1.16865
\(474\) 13.0902 0.601251
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) −6.94427 −0.317956
\(478\) −33.2148 −1.51921
\(479\) −10.8541 −0.495937 −0.247968 0.968768i \(-0.579763\pi\)
−0.247968 + 0.968768i \(0.579763\pi\)
\(480\) 0 0
\(481\) −0.437694 −0.0199571
\(482\) 4.09017 0.186302
\(483\) −2.32624 −0.105847
\(484\) 10.1459 0.461177
\(485\) 0 0
\(486\) −25.8885 −1.17433
\(487\) 36.4164 1.65018 0.825092 0.564998i \(-0.191123\pi\)
0.825092 + 0.564998i \(0.191123\pi\)
\(488\) −19.4721 −0.881462
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) −43.2492 −1.95181 −0.975905 0.218196i \(-0.929983\pi\)
−0.975905 + 0.218196i \(0.929983\pi\)
\(492\) 0.472136 0.0212855
\(493\) −18.9443 −0.853207
\(494\) −2.56231 −0.115284
\(495\) 0 0
\(496\) 14.5623 0.653867
\(497\) 4.09017 0.183469
\(498\) −10.0902 −0.452151
\(499\) 7.56231 0.338535 0.169268 0.985570i \(-0.445860\pi\)
0.169268 + 0.985570i \(0.445860\pi\)
\(500\) 0 0
\(501\) 14.5623 0.650596
\(502\) −47.2148 −2.10730
\(503\) 37.4164 1.66832 0.834158 0.551526i \(-0.185954\pi\)
0.834158 + 0.551526i \(0.185954\pi\)
\(504\) −2.76393 −0.123115
\(505\) 0 0
\(506\) 31.8885 1.41762
\(507\) 9.56231 0.424677
\(508\) −12.2918 −0.545360
\(509\) −20.3262 −0.900945 −0.450472 0.892790i \(-0.648744\pi\)
−0.450472 + 0.892790i \(0.648744\pi\)
\(510\) 0 0
\(511\) −5.56231 −0.246062
\(512\) −5.29180 −0.233867
\(513\) 4.27051 0.188548
\(514\) −36.9787 −1.63106
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) 3.23607 0.142322
\(518\) −0.236068 −0.0103722
\(519\) 18.8885 0.829115
\(520\) 0 0
\(521\) 29.3607 1.28631 0.643157 0.765734i \(-0.277624\pi\)
0.643157 + 0.765734i \(0.277624\pi\)
\(522\) 11.7082 0.512454
\(523\) 13.1459 0.574830 0.287415 0.957806i \(-0.407204\pi\)
0.287415 + 0.957806i \(0.407204\pi\)
\(524\) 4.20163 0.183549
\(525\) 0 0
\(526\) 17.6525 0.769685
\(527\) −15.7082 −0.684260
\(528\) −25.4164 −1.10611
\(529\) −8.83282 −0.384035
\(530\) 0 0
\(531\) 21.7082 0.942056
\(532\) −0.326238 −0.0141442
\(533\) 1.41641 0.0613514
\(534\) 14.4721 0.626271
\(535\) 0 0
\(536\) 10.6525 0.460117
\(537\) 0.527864 0.0227790
\(538\) −20.6525 −0.890391
\(539\) 34.6525 1.49259
\(540\) 0 0
\(541\) 27.1246 1.16618 0.583089 0.812408i \(-0.301844\pi\)
0.583089 + 0.812408i \(0.301844\pi\)
\(542\) −12.9443 −0.556004
\(543\) −0.291796 −0.0125222
\(544\) −17.7082 −0.759233
\(545\) 0 0
\(546\) −1.85410 −0.0793482
\(547\) 21.2918 0.910371 0.455186 0.890397i \(-0.349573\pi\)
0.455186 + 0.890397i \(0.349573\pi\)
\(548\) 7.38197 0.315342
\(549\) −17.4164 −0.743314
\(550\) 0 0
\(551\) −3.09017 −0.131646
\(552\) −8.41641 −0.358226
\(553\) 5.00000 0.212622
\(554\) 39.9787 1.69853
\(555\) 0 0
\(556\) 3.09017 0.131052
\(557\) −4.76393 −0.201854 −0.100927 0.994894i \(-0.532181\pi\)
−0.100927 + 0.994894i \(0.532181\pi\)
\(558\) 9.70820 0.410981
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 27.4164 1.15752
\(562\) 16.3262 0.688681
\(563\) −7.38197 −0.311113 −0.155556 0.987827i \(-0.549717\pi\)
−0.155556 + 0.987827i \(0.549717\pi\)
\(564\) 0.381966 0.0160837
\(565\) 0 0
\(566\) 48.3050 2.03041
\(567\) −0.618034 −0.0259550
\(568\) 14.7984 0.620926
\(569\) 20.5279 0.860573 0.430286 0.902692i \(-0.358413\pi\)
0.430286 + 0.902692i \(0.358413\pi\)
\(570\) 0 0
\(571\) −8.12461 −0.340004 −0.170002 0.985444i \(-0.554377\pi\)
−0.170002 + 0.985444i \(0.554377\pi\)
\(572\) 6.00000 0.250873
\(573\) 1.81966 0.0760174
\(574\) 0.763932 0.0318859
\(575\) 0 0
\(576\) −8.47214 −0.353006
\(577\) 33.7771 1.40616 0.703079 0.711111i \(-0.251808\pi\)
0.703079 + 0.711111i \(0.251808\pi\)
\(578\) 16.8541 0.701038
\(579\) 7.70820 0.320342
\(580\) 0 0
\(581\) −3.85410 −0.159895
\(582\) −6.23607 −0.258493
\(583\) −18.1803 −0.752953
\(584\) −20.1246 −0.832762
\(585\) 0 0
\(586\) 31.5967 1.30525
\(587\) −5.29180 −0.218416 −0.109208 0.994019i \(-0.534831\pi\)
−0.109208 + 0.994019i \(0.534831\pi\)
\(588\) 4.09017 0.168676
\(589\) −2.56231 −0.105578
\(590\) 0 0
\(591\) 3.70820 0.152535
\(592\) −1.14590 −0.0470961
\(593\) 10.9098 0.448013 0.224007 0.974588i \(-0.428086\pi\)
0.224007 + 0.974588i \(0.428086\pi\)
\(594\) −42.3607 −1.73808
\(595\) 0 0
\(596\) −2.43769 −0.0998518
\(597\) 17.5623 0.718777
\(598\) 11.2918 0.461756
\(599\) −9.47214 −0.387021 −0.193510 0.981098i \(-0.561987\pi\)
−0.193510 + 0.981098i \(0.561987\pi\)
\(600\) 0 0
\(601\) 2.72949 0.111338 0.0556691 0.998449i \(-0.482271\pi\)
0.0556691 + 0.998449i \(0.482271\pi\)
\(602\) −4.85410 −0.197838
\(603\) 9.52786 0.388005
\(604\) 9.00000 0.366205
\(605\) 0 0
\(606\) −2.38197 −0.0967608
\(607\) 35.5623 1.44343 0.721715 0.692191i \(-0.243354\pi\)
0.721715 + 0.692191i \(0.243354\pi\)
\(608\) −2.88854 −0.117146
\(609\) −2.23607 −0.0906100
\(610\) 0 0
\(611\) 1.14590 0.0463581
\(612\) −6.47214 −0.261621
\(613\) 14.9787 0.604985 0.302492 0.953152i \(-0.402181\pi\)
0.302492 + 0.953152i \(0.402181\pi\)
\(614\) −14.9443 −0.603102
\(615\) 0 0
\(616\) −7.23607 −0.291549
\(617\) −14.2361 −0.573123 −0.286561 0.958062i \(-0.592512\pi\)
−0.286561 + 0.958062i \(0.592512\pi\)
\(618\) 13.8541 0.557294
\(619\) 30.5279 1.22702 0.613509 0.789688i \(-0.289757\pi\)
0.613509 + 0.789688i \(0.289757\pi\)
\(620\) 0 0
\(621\) −18.8197 −0.755207
\(622\) 13.7639 0.551883
\(623\) 5.52786 0.221469
\(624\) −9.00000 −0.360288
\(625\) 0 0
\(626\) 27.1246 1.08412
\(627\) 4.47214 0.178600
\(628\) −8.14590 −0.325057
\(629\) 1.23607 0.0492853
\(630\) 0 0
\(631\) −10.2361 −0.407491 −0.203746 0.979024i \(-0.565312\pi\)
−0.203746 + 0.979024i \(0.565312\pi\)
\(632\) 18.0902 0.719588
\(633\) 9.18034 0.364886
\(634\) −12.3820 −0.491751
\(635\) 0 0
\(636\) −2.14590 −0.0850904
\(637\) 12.2705 0.486175
\(638\) 30.6525 1.21354
\(639\) 13.2361 0.523611
\(640\) 0 0
\(641\) −1.09017 −0.0430591 −0.0215296 0.999768i \(-0.506854\pi\)
−0.0215296 + 0.999768i \(0.506854\pi\)
\(642\) −26.5623 −1.04833
\(643\) 30.8328 1.21593 0.607964 0.793965i \(-0.291987\pi\)
0.607964 + 0.793965i \(0.291987\pi\)
\(644\) 1.43769 0.0566531
\(645\) 0 0
\(646\) 7.23607 0.284699
\(647\) 36.5410 1.43658 0.718288 0.695746i \(-0.244926\pi\)
0.718288 + 0.695746i \(0.244926\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 56.8328 2.23088
\(650\) 0 0
\(651\) −1.85410 −0.0726680
\(652\) −6.79837 −0.266245
\(653\) −19.0902 −0.747056 −0.373528 0.927619i \(-0.621852\pi\)
−0.373528 + 0.927619i \(0.621852\pi\)
\(654\) −16.1803 −0.632701
\(655\) 0 0
\(656\) 3.70820 0.144781
\(657\) −18.0000 −0.702247
\(658\) 0.618034 0.0240935
\(659\) 15.5279 0.604880 0.302440 0.953168i \(-0.402199\pi\)
0.302440 + 0.953168i \(0.402199\pi\)
\(660\) 0 0
\(661\) 19.6869 0.765732 0.382866 0.923804i \(-0.374937\pi\)
0.382866 + 0.923804i \(0.374937\pi\)
\(662\) −37.4164 −1.45423
\(663\) 9.70820 0.377035
\(664\) −13.9443 −0.541143
\(665\) 0 0
\(666\) −0.763932 −0.0296018
\(667\) 13.6180 0.527292
\(668\) −9.00000 −0.348220
\(669\) −0.180340 −0.00697234
\(670\) 0 0
\(671\) −45.5967 −1.76024
\(672\) −2.09017 −0.0806301
\(673\) −12.1803 −0.469518 −0.234759 0.972054i \(-0.575430\pi\)
−0.234759 + 0.972054i \(0.575430\pi\)
\(674\) −12.7082 −0.489502
\(675\) 0 0
\(676\) −5.90983 −0.227301
\(677\) −10.6180 −0.408084 −0.204042 0.978962i \(-0.565408\pi\)
−0.204042 + 0.978962i \(0.565408\pi\)
\(678\) 27.2705 1.04732
\(679\) −2.38197 −0.0914115
\(680\) 0 0
\(681\) 14.7639 0.565755
\(682\) 25.4164 0.973245
\(683\) 13.4721 0.515497 0.257748 0.966212i \(-0.417019\pi\)
0.257748 + 0.966212i \(0.417019\pi\)
\(684\) −1.05573 −0.0403668
\(685\) 0 0
\(686\) 13.6180 0.519939
\(687\) −21.7082 −0.828220
\(688\) −23.5623 −0.898304
\(689\) −6.43769 −0.245257
\(690\) 0 0
\(691\) 36.2705 1.37980 0.689898 0.723907i \(-0.257656\pi\)
0.689898 + 0.723907i \(0.257656\pi\)
\(692\) −11.6738 −0.443770
\(693\) −6.47214 −0.245856
\(694\) 32.2148 1.22286
\(695\) 0 0
\(696\) −8.09017 −0.306657
\(697\) −4.00000 −0.151511
\(698\) 35.1246 1.32949
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) −41.0132 −1.54905 −0.774523 0.632546i \(-0.782010\pi\)
−0.774523 + 0.632546i \(0.782010\pi\)
\(702\) −15.0000 −0.566139
\(703\) 0.201626 0.00760447
\(704\) −22.1803 −0.835953
\(705\) 0 0
\(706\) −20.8885 −0.786151
\(707\) −0.909830 −0.0342177
\(708\) 6.70820 0.252110
\(709\) −33.5410 −1.25966 −0.629830 0.776733i \(-0.716875\pi\)
−0.629830 + 0.776733i \(0.716875\pi\)
\(710\) 0 0
\(711\) 16.1803 0.606810
\(712\) 20.0000 0.749532
\(713\) 11.2918 0.422881
\(714\) 5.23607 0.195955
\(715\) 0 0
\(716\) −0.326238 −0.0121921
\(717\) 20.5279 0.766627
\(718\) 22.2361 0.829843
\(719\) −23.2918 −0.868637 −0.434319 0.900759i \(-0.643011\pi\)
−0.434319 + 0.900759i \(0.643011\pi\)
\(720\) 0 0
\(721\) 5.29180 0.197077
\(722\) −29.5623 −1.10020
\(723\) −2.52786 −0.0940123
\(724\) 0.180340 0.00670228
\(725\) 0 0
\(726\) −26.5623 −0.985820
\(727\) −24.5623 −0.910966 −0.455483 0.890245i \(-0.650533\pi\)
−0.455483 + 0.890245i \(0.650533\pi\)
\(728\) −2.56231 −0.0949654
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 25.4164 0.940060
\(732\) −5.38197 −0.198923
\(733\) 19.9787 0.737931 0.368965 0.929443i \(-0.379712\pi\)
0.368965 + 0.929443i \(0.379712\pi\)
\(734\) 41.3607 1.52665
\(735\) 0 0
\(736\) 12.7295 0.469215
\(737\) 24.9443 0.918834
\(738\) 2.47214 0.0910006
\(739\) 15.9787 0.587786 0.293893 0.955838i \(-0.405049\pi\)
0.293893 + 0.955838i \(0.405049\pi\)
\(740\) 0 0
\(741\) 1.58359 0.0581747
\(742\) −3.47214 −0.127466
\(743\) −28.3607 −1.04045 −0.520226 0.854029i \(-0.674152\pi\)
−0.520226 + 0.854029i \(0.674152\pi\)
\(744\) −6.70820 −0.245935
\(745\) 0 0
\(746\) 45.7426 1.67476
\(747\) −12.4721 −0.456332
\(748\) −16.9443 −0.619544
\(749\) −10.1459 −0.370723
\(750\) 0 0
\(751\) −5.11146 −0.186520 −0.0932598 0.995642i \(-0.529729\pi\)
−0.0932598 + 0.995642i \(0.529729\pi\)
\(752\) 3.00000 0.109399
\(753\) 29.1803 1.06339
\(754\) 10.8541 0.395283
\(755\) 0 0
\(756\) −1.90983 −0.0694598
\(757\) −30.4164 −1.10550 −0.552752 0.833346i \(-0.686422\pi\)
−0.552752 + 0.833346i \(0.686422\pi\)
\(758\) 23.6180 0.857846
\(759\) −19.7082 −0.715362
\(760\) 0 0
\(761\) −18.4508 −0.668843 −0.334421 0.942424i \(-0.608541\pi\)
−0.334421 + 0.942424i \(0.608541\pi\)
\(762\) 32.1803 1.16577
\(763\) −6.18034 −0.223743
\(764\) −1.12461 −0.0406870
\(765\) 0 0
\(766\) −53.9787 −1.95033
\(767\) 20.1246 0.726658
\(768\) −13.5623 −0.489388
\(769\) 13.4164 0.483808 0.241904 0.970300i \(-0.422228\pi\)
0.241904 + 0.970300i \(0.422228\pi\)
\(770\) 0 0
\(771\) 22.8541 0.823070
\(772\) −4.76393 −0.171458
\(773\) 36.1591 1.30055 0.650275 0.759699i \(-0.274654\pi\)
0.650275 + 0.759699i \(0.274654\pi\)
\(774\) −15.7082 −0.564620
\(775\) 0 0
\(776\) −8.61803 −0.309369
\(777\) 0.145898 0.00523406
\(778\) 24.2705 0.870140
\(779\) −0.652476 −0.0233774
\(780\) 0 0
\(781\) 34.6525 1.23996
\(782\) −31.8885 −1.14033
\(783\) −18.0902 −0.646490
\(784\) 32.1246 1.14731
\(785\) 0 0
\(786\) −11.0000 −0.392357
\(787\) 11.8197 0.421325 0.210663 0.977559i \(-0.432438\pi\)
0.210663 + 0.977559i \(0.432438\pi\)
\(788\) −2.29180 −0.0816419
\(789\) −10.9098 −0.388400
\(790\) 0 0
\(791\) 10.4164 0.370365
\(792\) −23.4164 −0.832066
\(793\) −16.1459 −0.573358
\(794\) −46.9787 −1.66721
\(795\) 0 0
\(796\) −10.8541 −0.384713
\(797\) −9.76393 −0.345856 −0.172928 0.984934i \(-0.555323\pi\)
−0.172928 + 0.984934i \(0.555323\pi\)
\(798\) 0.854102 0.0302349
\(799\) −3.23607 −0.114484
\(800\) 0 0
\(801\) 17.8885 0.632061
\(802\) 43.0344 1.51960
\(803\) −47.1246 −1.66299
\(804\) 2.94427 0.103836
\(805\) 0 0
\(806\) 9.00000 0.317011
\(807\) 12.7639 0.449312
\(808\) −3.29180 −0.115805
\(809\) 30.9787 1.08915 0.544577 0.838711i \(-0.316690\pi\)
0.544577 + 0.838711i \(0.316690\pi\)
\(810\) 0 0
\(811\) −14.7082 −0.516475 −0.258237 0.966081i \(-0.583142\pi\)
−0.258237 + 0.966081i \(0.583142\pi\)
\(812\) 1.38197 0.0484975
\(813\) 8.00000 0.280572
\(814\) −2.00000 −0.0701000
\(815\) 0 0
\(816\) 25.4164 0.889752
\(817\) 4.14590 0.145047
\(818\) 2.56231 0.0895889
\(819\) −2.29180 −0.0800818
\(820\) 0 0
\(821\) −40.6869 −1.41998 −0.709992 0.704210i \(-0.751301\pi\)
−0.709992 + 0.704210i \(0.751301\pi\)
\(822\) −19.3262 −0.674080
\(823\) −47.7082 −1.66300 −0.831502 0.555522i \(-0.812518\pi\)
−0.831502 + 0.555522i \(0.812518\pi\)
\(824\) 19.1459 0.666979
\(825\) 0 0
\(826\) 10.8541 0.377663
\(827\) 0.965558 0.0335757 0.0167879 0.999859i \(-0.494656\pi\)
0.0167879 + 0.999859i \(0.494656\pi\)
\(828\) 4.65248 0.161685
\(829\) −35.8541 −1.24526 −0.622632 0.782515i \(-0.713937\pi\)
−0.622632 + 0.782515i \(0.713937\pi\)
\(830\) 0 0
\(831\) −24.7082 −0.857118
\(832\) −7.85410 −0.272292
\(833\) −34.6525 −1.20064
\(834\) −8.09017 −0.280140
\(835\) 0 0
\(836\) −2.76393 −0.0955926
\(837\) −15.0000 −0.518476
\(838\) −15.3262 −0.529436
\(839\) 10.8541 0.374725 0.187363 0.982291i \(-0.440006\pi\)
0.187363 + 0.982291i \(0.440006\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 51.7771 1.78436
\(843\) −10.0902 −0.347524
\(844\) −5.67376 −0.195299
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) −10.1459 −0.348617
\(848\) −16.8541 −0.578772
\(849\) −29.8541 −1.02459
\(850\) 0 0
\(851\) −0.888544 −0.0304589
\(852\) 4.09017 0.140127
\(853\) 15.3050 0.524032 0.262016 0.965064i \(-0.415613\pi\)
0.262016 + 0.965064i \(0.415613\pi\)
\(854\) −8.70820 −0.297989
\(855\) 0 0
\(856\) −36.7082 −1.25466
\(857\) −19.6869 −0.672492 −0.336246 0.941774i \(-0.609157\pi\)
−0.336246 + 0.941774i \(0.609157\pi\)
\(858\) −15.7082 −0.536269
\(859\) −1.58359 −0.0540315 −0.0270157 0.999635i \(-0.508600\pi\)
−0.0270157 + 0.999635i \(0.508600\pi\)
\(860\) 0 0
\(861\) −0.472136 −0.0160904
\(862\) −48.2705 −1.64410
\(863\) 21.4377 0.729748 0.364874 0.931057i \(-0.381112\pi\)
0.364874 + 0.931057i \(0.381112\pi\)
\(864\) −16.9098 −0.575284
\(865\) 0 0
\(866\) −43.4508 −1.47652
\(867\) −10.4164 −0.353760
\(868\) 1.14590 0.0388943
\(869\) 42.3607 1.43699
\(870\) 0 0
\(871\) 8.83282 0.299289
\(872\) −22.3607 −0.757228
\(873\) −7.70820 −0.260883
\(874\) −5.20163 −0.175948
\(875\) 0 0
\(876\) −5.56231 −0.187933
\(877\) 36.5410 1.23390 0.616951 0.787001i \(-0.288368\pi\)
0.616951 + 0.787001i \(0.288368\pi\)
\(878\) −66.3050 −2.23768
\(879\) −19.5279 −0.658659
\(880\) 0 0
\(881\) −40.3607 −1.35979 −0.679893 0.733311i \(-0.737974\pi\)
−0.679893 + 0.733311i \(0.737974\pi\)
\(882\) 21.4164 0.721128
\(883\) 20.5836 0.692693 0.346347 0.938107i \(-0.387422\pi\)
0.346347 + 0.938107i \(0.387422\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −48.4508 −1.62774
\(887\) −29.8885 −1.00356 −0.501780 0.864996i \(-0.667321\pi\)
−0.501780 + 0.864996i \(0.667321\pi\)
\(888\) 0.527864 0.0177140
\(889\) 12.2918 0.412254
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) 0.111456 0.00373183
\(893\) −0.527864 −0.0176643
\(894\) 6.38197 0.213445
\(895\) 0 0
\(896\) −8.41641 −0.281172
\(897\) −6.97871 −0.233012
\(898\) 7.56231 0.252357
\(899\) 10.8541 0.362005
\(900\) 0 0
\(901\) 18.1803 0.605675
\(902\) 6.47214 0.215499
\(903\) 3.00000 0.0998337
\(904\) 37.6869 1.25345
\(905\) 0 0
\(906\) −23.5623 −0.782805
\(907\) 33.2492 1.10402 0.552011 0.833837i \(-0.313861\pi\)
0.552011 + 0.833837i \(0.313861\pi\)
\(908\) −9.12461 −0.302811
\(909\) −2.94427 −0.0976553
\(910\) 0 0
\(911\) −40.2361 −1.33308 −0.666540 0.745469i \(-0.732226\pi\)
−0.666540 + 0.745469i \(0.732226\pi\)
\(912\) 4.14590 0.137284
\(913\) −32.6525 −1.08064
\(914\) 34.6525 1.14620
\(915\) 0 0
\(916\) 13.4164 0.443291
\(917\) −4.20163 −0.138750
\(918\) 42.3607 1.39811
\(919\) −53.2148 −1.75539 −0.877697 0.479216i \(-0.840921\pi\)
−0.877697 + 0.479216i \(0.840921\pi\)
\(920\) 0 0
\(921\) 9.23607 0.304339
\(922\) 1.32624 0.0436773
\(923\) 12.2705 0.403889
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 39.0344 1.28275
\(927\) 17.1246 0.562446
\(928\) 12.2361 0.401669
\(929\) 41.6312 1.36588 0.682938 0.730477i \(-0.260702\pi\)
0.682938 + 0.730477i \(0.260702\pi\)
\(930\) 0 0
\(931\) −5.65248 −0.185252
\(932\) 1.81966 0.0596049
\(933\) −8.50658 −0.278493
\(934\) −44.4164 −1.45335
\(935\) 0 0
\(936\) −8.29180 −0.271026
\(937\) −17.7295 −0.579197 −0.289599 0.957148i \(-0.593522\pi\)
−0.289599 + 0.957148i \(0.593522\pi\)
\(938\) 4.76393 0.155548
\(939\) −16.7639 −0.547070
\(940\) 0 0
\(941\) −46.4164 −1.51313 −0.756566 0.653918i \(-0.773124\pi\)
−0.756566 + 0.653918i \(0.773124\pi\)
\(942\) 21.3262 0.694846
\(943\) 2.87539 0.0936355
\(944\) 52.6869 1.71481
\(945\) 0 0
\(946\) −41.1246 −1.33708
\(947\) −2.65248 −0.0861939 −0.0430969 0.999071i \(-0.513722\pi\)
−0.0430969 + 0.999071i \(0.513722\pi\)
\(948\) 5.00000 0.162392
\(949\) −16.6869 −0.541680
\(950\) 0 0
\(951\) 7.65248 0.248149
\(952\) 7.23607 0.234522
\(953\) 7.74265 0.250809 0.125404 0.992106i \(-0.459977\pi\)
0.125404 + 0.992106i \(0.459977\pi\)
\(954\) −11.2361 −0.363781
\(955\) 0 0
\(956\) −12.6869 −0.410324
\(957\) −18.9443 −0.612381
\(958\) −17.5623 −0.567412
\(959\) −7.38197 −0.238376
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −0.708204 −0.0228334
\(963\) −32.8328 −1.05802
\(964\) 1.56231 0.0503185
\(965\) 0 0
\(966\) −3.76393 −0.121103
\(967\) −4.11146 −0.132216 −0.0661078 0.997812i \(-0.521058\pi\)
−0.0661078 + 0.997812i \(0.521058\pi\)
\(968\) −36.7082 −1.17985
\(969\) −4.47214 −0.143666
\(970\) 0 0
\(971\) 5.61803 0.180291 0.0901456 0.995929i \(-0.471267\pi\)
0.0901456 + 0.995929i \(0.471267\pi\)
\(972\) −9.88854 −0.317175
\(973\) −3.09017 −0.0990663
\(974\) 58.9230 1.88801
\(975\) 0 0
\(976\) −42.2705 −1.35305
\(977\) 2.34752 0.0751040 0.0375520 0.999295i \(-0.488044\pi\)
0.0375520 + 0.999295i \(0.488044\pi\)
\(978\) 17.7984 0.569129
\(979\) 46.8328 1.49678
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) −69.9787 −2.23311
\(983\) −9.61803 −0.306768 −0.153384 0.988167i \(-0.549017\pi\)
−0.153384 + 0.988167i \(0.549017\pi\)
\(984\) −1.70820 −0.0544556
\(985\) 0 0
\(986\) −30.6525 −0.976174
\(987\) −0.381966 −0.0121581
\(988\) −0.978714 −0.0311370
\(989\) −18.2705 −0.580968
\(990\) 0 0
\(991\) −15.3607 −0.487948 −0.243974 0.969782i \(-0.578451\pi\)
−0.243974 + 0.969782i \(0.578451\pi\)
\(992\) 10.1459 0.322133
\(993\) 23.1246 0.733837
\(994\) 6.61803 0.209911
\(995\) 0 0
\(996\) −3.85410 −0.122122
\(997\) −24.8885 −0.788228 −0.394114 0.919062i \(-0.628949\pi\)
−0.394114 + 0.919062i \(0.628949\pi\)
\(998\) 12.2361 0.387326
\(999\) 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.a.c.1.2 2
3.2 odd 2 5625.2.a.d.1.1 2
4.3 odd 2 10000.2.a.l.1.2 2
5.2 odd 4 625.2.b.a.624.4 4
5.3 odd 4 625.2.b.a.624.1 4
5.4 even 2 625.2.a.b.1.1 2
15.14 odd 2 5625.2.a.f.1.2 2
20.19 odd 2 10000.2.a.c.1.1 2
25.2 odd 20 125.2.e.a.24.2 8
25.3 odd 20 625.2.e.c.249.1 8
25.4 even 10 625.2.d.h.376.1 4
25.6 even 5 625.2.d.b.251.1 4
25.8 odd 20 625.2.e.c.374.2 8
25.9 even 10 25.2.d.a.6.1 4
25.11 even 5 125.2.d.a.101.1 4
25.12 odd 20 125.2.e.a.99.1 8
25.13 odd 20 125.2.e.a.99.2 8
25.14 even 10 25.2.d.a.21.1 yes 4
25.16 even 5 125.2.d.a.26.1 4
25.17 odd 20 625.2.e.c.374.1 8
25.19 even 10 625.2.d.h.251.1 4
25.21 even 5 625.2.d.b.376.1 4
25.22 odd 20 625.2.e.c.249.2 8
25.23 odd 20 125.2.e.a.24.1 8
75.14 odd 10 225.2.h.b.46.1 4
75.59 odd 10 225.2.h.b.181.1 4
100.39 odd 10 400.2.u.b.321.1 4
100.59 odd 10 400.2.u.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 25.9 even 10
25.2.d.a.21.1 yes 4 25.14 even 10
125.2.d.a.26.1 4 25.16 even 5
125.2.d.a.101.1 4 25.11 even 5
125.2.e.a.24.1 8 25.23 odd 20
125.2.e.a.24.2 8 25.2 odd 20
125.2.e.a.99.1 8 25.12 odd 20
125.2.e.a.99.2 8 25.13 odd 20
225.2.h.b.46.1 4 75.14 odd 10
225.2.h.b.181.1 4 75.59 odd 10
400.2.u.b.81.1 4 100.59 odd 10
400.2.u.b.321.1 4 100.39 odd 10
625.2.a.b.1.1 2 5.4 even 2
625.2.a.c.1.2 2 1.1 even 1 trivial
625.2.b.a.624.1 4 5.3 odd 4
625.2.b.a.624.4 4 5.2 odd 4
625.2.d.b.251.1 4 25.6 even 5
625.2.d.b.376.1 4 25.21 even 5
625.2.d.h.251.1 4 25.19 even 10
625.2.d.h.376.1 4 25.4 even 10
625.2.e.c.249.1 8 25.3 odd 20
625.2.e.c.249.2 8 25.22 odd 20
625.2.e.c.374.1 8 25.17 odd 20
625.2.e.c.374.2 8 25.8 odd 20
5625.2.a.d.1.1 2 3.2 odd 2
5625.2.a.f.1.2 2 15.14 odd 2
10000.2.a.c.1.1 2 20.19 odd 2
10000.2.a.l.1.2 2 4.3 odd 2