Properties

Label 4598.2.a.bu.1.4
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.33452.1
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 7x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.94749\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} +2.94749 q^{6} -1.66073 q^{7} +1.00000 q^{8} +5.68769 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} +2.94749 q^{6} -1.66073 q^{7} +1.00000 q^{8} +5.68769 q^{9} +2.36624 q^{10} +2.94749 q^{12} -0.633765 q^{13} -1.66073 q^{14} +6.97445 q^{15} +1.00000 q^{16} +0.581254 q^{17} +5.68769 q^{18} -1.00000 q^{19} +2.36624 q^{20} -4.89498 q^{21} +3.60822 q^{23} +2.94749 q^{24} +0.599069 q^{25} -0.633765 q^{26} +7.92194 q^{27} -1.66073 q^{28} +9.13696 q^{29} +6.97445 q^{30} +1.02555 q^{31} +1.00000 q^{32} +0.581254 q^{34} -3.92967 q^{35} +5.68769 q^{36} +2.15895 q^{37} -1.00000 q^{38} -1.86801 q^{39} +2.36624 q^{40} -4.40093 q^{41} -4.89498 q^{42} -9.58267 q^{43} +13.4584 q^{45} +3.60822 q^{46} +8.73247 q^{47} +2.94749 q^{48} -4.24198 q^{49} +0.599069 q^{50} +1.71324 q^{51} -0.633765 q^{52} -12.5302 q^{53} +7.92194 q^{54} -1.66073 q^{56} -2.94749 q^{57} +9.13696 q^{58} +6.47623 q^{59} +6.97445 q^{60} -1.32146 q^{61} +1.02555 q^{62} -9.44571 q^{63} +1.00000 q^{64} -1.49964 q^{65} -0.393199 q^{67} +0.581254 q^{68} +10.6352 q^{69} -3.92967 q^{70} -0.312308 q^{71} +5.68769 q^{72} +9.31372 q^{73} +2.15895 q^{74} +1.76575 q^{75} -1.00000 q^{76} -1.86801 q^{78} -11.2704 q^{79} +2.36624 q^{80} +6.28676 q^{81} -4.40093 q^{82} +6.90413 q^{83} -4.89498 q^{84} +1.37538 q^{85} -9.58267 q^{86} +26.9311 q^{87} +0.519595 q^{89} +13.4584 q^{90} +1.05251 q^{91} +3.60822 q^{92} +3.02279 q^{93} +8.73247 q^{94} -2.36624 q^{95} +2.94749 q^{96} -8.37894 q^{97} -4.24198 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9} + 3 q^{10} + q^{12} - 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} - 2 q^{17} + 5 q^{18} - 4 q^{19} + 3 q^{20} + 2 q^{21} - 2 q^{23} + q^{24} + 15 q^{25} - 9 q^{26} - 2 q^{27} - q^{28} + 5 q^{29} + 5 q^{30} + 27 q^{31} + 4 q^{32} - 2 q^{34} + 12 q^{35} + 5 q^{36} + 6 q^{37} - 4 q^{38} + 2 q^{39} + 3 q^{40} - 5 q^{41} + 2 q^{42} + q^{43} + 11 q^{45} - 2 q^{46} + 22 q^{47} + q^{48} - 7 q^{49} + 15 q^{50} + 12 q^{51} - 9 q^{52} - 2 q^{54} - q^{56} - q^{57} + 5 q^{58} + 5 q^{60} + 6 q^{61} + 27 q^{62} - 30 q^{63} + 4 q^{64} + 26 q^{65} + 17 q^{67} - 2 q^{68} + 14 q^{69} + 12 q^{70} - 19 q^{71} + 5 q^{72} + 20 q^{73} + 6 q^{74} + 23 q^{75} - 4 q^{76} + 2 q^{78} + 12 q^{79} + 3 q^{80} + 20 q^{81} - 5 q^{82} - 23 q^{83} + 2 q^{84} - 30 q^{85} + q^{86} + 45 q^{87} + 16 q^{89} + 11 q^{90} + 15 q^{91} - 2 q^{92} - 12 q^{93} + 22 q^{94} - 3 q^{95} + q^{96} + 8 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.94749 1.70173 0.850867 0.525381i \(-0.176077\pi\)
0.850867 + 0.525381i \(0.176077\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.36624 1.05821 0.529106 0.848556i \(-0.322527\pi\)
0.529106 + 0.848556i \(0.322527\pi\)
\(6\) 2.94749 1.20331
\(7\) −1.66073 −0.627696 −0.313848 0.949473i \(-0.601618\pi\)
−0.313848 + 0.949473i \(0.601618\pi\)
\(8\) 1.00000 0.353553
\(9\) 5.68769 1.89590
\(10\) 2.36624 0.748269
\(11\) 0 0
\(12\) 2.94749 0.850867
\(13\) −0.633765 −0.175775 −0.0878874 0.996130i \(-0.528012\pi\)
−0.0878874 + 0.996130i \(0.528012\pi\)
\(14\) −1.66073 −0.443848
\(15\) 6.97445 1.80080
\(16\) 1.00000 0.250000
\(17\) 0.581254 0.140975 0.0704874 0.997513i \(-0.477545\pi\)
0.0704874 + 0.997513i \(0.477545\pi\)
\(18\) 5.68769 1.34060
\(19\) −1.00000 −0.229416
\(20\) 2.36624 0.529106
\(21\) −4.89498 −1.06817
\(22\) 0 0
\(23\) 3.60822 0.752365 0.376183 0.926546i \(-0.377237\pi\)
0.376183 + 0.926546i \(0.377237\pi\)
\(24\) 2.94749 0.601654
\(25\) 0.599069 0.119814
\(26\) −0.633765 −0.124291
\(27\) 7.92194 1.52458
\(28\) −1.66073 −0.313848
\(29\) 9.13696 1.69669 0.848345 0.529443i \(-0.177599\pi\)
0.848345 + 0.529443i \(0.177599\pi\)
\(30\) 6.97445 1.27335
\(31\) 1.02555 0.184194 0.0920969 0.995750i \(-0.470643\pi\)
0.0920969 + 0.995750i \(0.470643\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.581254 0.0996842
\(35\) −3.92967 −0.664236
\(36\) 5.68769 0.947949
\(37\) 2.15895 0.354929 0.177464 0.984127i \(-0.443211\pi\)
0.177464 + 0.984127i \(0.443211\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.86801 −0.299122
\(40\) 2.36624 0.374135
\(41\) −4.40093 −0.687310 −0.343655 0.939096i \(-0.611665\pi\)
−0.343655 + 0.939096i \(0.611665\pi\)
\(42\) −4.89498 −0.755312
\(43\) −9.58267 −1.46134 −0.730672 0.682729i \(-0.760793\pi\)
−0.730672 + 0.682729i \(0.760793\pi\)
\(44\) 0 0
\(45\) 13.4584 2.00626
\(46\) 3.60822 0.532003
\(47\) 8.73247 1.27376 0.636881 0.770962i \(-0.280224\pi\)
0.636881 + 0.770962i \(0.280224\pi\)
\(48\) 2.94749 0.425433
\(49\) −4.24198 −0.605997
\(50\) 0.599069 0.0847212
\(51\) 1.71324 0.239901
\(52\) −0.633765 −0.0878874
\(53\) −12.5302 −1.72115 −0.860575 0.509324i \(-0.829895\pi\)
−0.860575 + 0.509324i \(0.829895\pi\)
\(54\) 7.92194 1.07804
\(55\) 0 0
\(56\) −1.66073 −0.221924
\(57\) −2.94749 −0.390404
\(58\) 9.13696 1.19974
\(59\) 6.47623 0.843134 0.421567 0.906797i \(-0.361480\pi\)
0.421567 + 0.906797i \(0.361480\pi\)
\(60\) 6.97445 0.900398
\(61\) −1.32146 −0.169195 −0.0845976 0.996415i \(-0.526960\pi\)
−0.0845976 + 0.996415i \(0.526960\pi\)
\(62\) 1.02555 0.130245
\(63\) −9.44571 −1.19005
\(64\) 1.00000 0.125000
\(65\) −1.49964 −0.186007
\(66\) 0 0
\(67\) −0.393199 −0.0480369 −0.0240184 0.999712i \(-0.507646\pi\)
−0.0240184 + 0.999712i \(0.507646\pi\)
\(68\) 0.581254 0.0704874
\(69\) 10.6352 1.28033
\(70\) −3.92967 −0.469686
\(71\) −0.312308 −0.0370642 −0.0185321 0.999828i \(-0.505899\pi\)
−0.0185321 + 0.999828i \(0.505899\pi\)
\(72\) 5.68769 0.670301
\(73\) 9.31372 1.09009 0.545044 0.838407i \(-0.316513\pi\)
0.545044 + 0.838407i \(0.316513\pi\)
\(74\) 2.15895 0.250973
\(75\) 1.76575 0.203891
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −1.86801 −0.211511
\(79\) −11.2704 −1.26801 −0.634007 0.773327i \(-0.718591\pi\)
−0.634007 + 0.773327i \(0.718591\pi\)
\(80\) 2.36624 0.264553
\(81\) 6.28676 0.698529
\(82\) −4.40093 −0.486001
\(83\) 6.90413 0.757826 0.378913 0.925432i \(-0.376298\pi\)
0.378913 + 0.925432i \(0.376298\pi\)
\(84\) −4.89498 −0.534086
\(85\) 1.37538 0.149181
\(86\) −9.58267 −1.03333
\(87\) 26.9311 2.88732
\(88\) 0 0
\(89\) 0.519595 0.0550769 0.0275385 0.999621i \(-0.491233\pi\)
0.0275385 + 0.999621i \(0.491233\pi\)
\(90\) 13.4584 1.41864
\(91\) 1.05251 0.110333
\(92\) 3.60822 0.376183
\(93\) 3.02279 0.313449
\(94\) 8.73247 0.900686
\(95\) −2.36624 −0.242771
\(96\) 2.94749 0.300827
\(97\) −8.37894 −0.850753 −0.425376 0.905017i \(-0.639858\pi\)
−0.425376 + 0.905017i \(0.639858\pi\)
\(98\) −4.24198 −0.428505
\(99\) 0 0
\(100\) 0.599069 0.0599069
\(101\) 16.5379 1.64558 0.822791 0.568344i \(-0.192416\pi\)
0.822791 + 0.568344i \(0.192416\pi\)
\(102\) 1.71324 0.169636
\(103\) 2.79627 0.275525 0.137762 0.990465i \(-0.456009\pi\)
0.137762 + 0.990465i \(0.456009\pi\)
\(104\) −0.633765 −0.0621457
\(105\) −11.5827 −1.13035
\(106\) −12.5302 −1.21704
\(107\) −18.9681 −1.83372 −0.916859 0.399210i \(-0.869284\pi\)
−0.916859 + 0.399210i \(0.869284\pi\)
\(108\) 7.92194 0.762289
\(109\) −17.0732 −1.63531 −0.817656 0.575707i \(-0.804727\pi\)
−0.817656 + 0.575707i \(0.804727\pi\)
\(110\) 0 0
\(111\) 6.36348 0.603995
\(112\) −1.66073 −0.156924
\(113\) 21.1114 1.98599 0.992997 0.118137i \(-0.0376922\pi\)
0.992997 + 0.118137i \(0.0376922\pi\)
\(114\) −2.94749 −0.276058
\(115\) 8.53789 0.796162
\(116\) 9.13696 0.848345
\(117\) −3.60466 −0.333251
\(118\) 6.47623 0.596185
\(119\) −0.965305 −0.0884893
\(120\) 6.97445 0.636677
\(121\) 0 0
\(122\) −1.32146 −0.119639
\(123\) −12.9717 −1.16962
\(124\) 1.02555 0.0920969
\(125\) −10.4136 −0.931424
\(126\) −9.44571 −0.841491
\(127\) −4.62745 −0.410620 −0.205310 0.978697i \(-0.565820\pi\)
−0.205310 + 0.978697i \(0.565820\pi\)
\(128\) 1.00000 0.0883883
\(129\) −28.2448 −2.48682
\(130\) −1.49964 −0.131527
\(131\) 15.7763 1.37838 0.689191 0.724579i \(-0.257966\pi\)
0.689191 + 0.724579i \(0.257966\pi\)
\(132\) 0 0
\(133\) 1.66073 0.144003
\(134\) −0.393199 −0.0339672
\(135\) 18.7452 1.61333
\(136\) 0.581254 0.0498421
\(137\) −20.6366 −1.76310 −0.881552 0.472087i \(-0.843501\pi\)
−0.881552 + 0.472087i \(0.843501\pi\)
\(138\) 10.6352 0.905327
\(139\) 9.56721 0.811480 0.405740 0.913989i \(-0.367014\pi\)
0.405740 + 0.913989i \(0.367014\pi\)
\(140\) −3.92967 −0.332118
\(141\) 25.7389 2.16760
\(142\) −0.312308 −0.0262083
\(143\) 0 0
\(144\) 5.68769 0.473974
\(145\) 21.6202 1.79546
\(146\) 9.31372 0.770809
\(147\) −12.5032 −1.03125
\(148\) 2.15895 0.177464
\(149\) −18.3096 −1.49998 −0.749988 0.661451i \(-0.769941\pi\)
−0.749988 + 0.661451i \(0.769941\pi\)
\(150\) 1.76575 0.144173
\(151\) −6.25562 −0.509075 −0.254538 0.967063i \(-0.581923\pi\)
−0.254538 + 0.967063i \(0.581923\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 3.30599 0.267274
\(154\) 0 0
\(155\) 2.42669 0.194916
\(156\) −1.86801 −0.149561
\(157\) −6.15619 −0.491318 −0.245659 0.969356i \(-0.579004\pi\)
−0.245659 + 0.969356i \(0.579004\pi\)
\(158\) −11.2704 −0.896622
\(159\) −36.9325 −2.92894
\(160\) 2.36624 0.187067
\(161\) −5.99227 −0.472257
\(162\) 6.28676 0.493935
\(163\) −11.8439 −0.927685 −0.463842 0.885918i \(-0.653530\pi\)
−0.463842 + 0.885918i \(0.653530\pi\)
\(164\) −4.40093 −0.343655
\(165\) 0 0
\(166\) 6.90413 0.535864
\(167\) 4.89142 0.378509 0.189255 0.981928i \(-0.439393\pi\)
0.189255 + 0.981928i \(0.439393\pi\)
\(168\) −4.89498 −0.377656
\(169\) −12.5983 −0.969103
\(170\) 1.37538 0.105487
\(171\) −5.68769 −0.434949
\(172\) −9.58267 −0.730672
\(173\) 18.1206 1.37768 0.688840 0.724913i \(-0.258120\pi\)
0.688840 + 0.724913i \(0.258120\pi\)
\(174\) 26.9311 2.04164
\(175\) −0.994891 −0.0752067
\(176\) 0 0
\(177\) 19.0886 1.43479
\(178\) 0.519595 0.0389453
\(179\) 3.07947 0.230171 0.115085 0.993356i \(-0.463286\pi\)
0.115085 + 0.993356i \(0.463286\pi\)
\(180\) 13.4584 1.00313
\(181\) 0.0693910 0.00515779 0.00257889 0.999997i \(-0.499179\pi\)
0.00257889 + 0.999997i \(0.499179\pi\)
\(182\) 1.05251 0.0780173
\(183\) −3.89498 −0.287925
\(184\) 3.60822 0.266001
\(185\) 5.10858 0.375590
\(186\) 3.02279 0.221642
\(187\) 0 0
\(188\) 8.73247 0.636881
\(189\) −13.1562 −0.956972
\(190\) −2.36624 −0.171665
\(191\) 13.6776 0.989677 0.494838 0.868985i \(-0.335227\pi\)
0.494838 + 0.868985i \(0.335227\pi\)
\(192\) 2.94749 0.212717
\(193\) −5.61830 −0.404414 −0.202207 0.979343i \(-0.564811\pi\)
−0.202207 + 0.979343i \(0.564811\pi\)
\(194\) −8.37894 −0.601573
\(195\) −4.42016 −0.316534
\(196\) −4.24198 −0.302999
\(197\) −5.46494 −0.389361 −0.194680 0.980867i \(-0.562367\pi\)
−0.194680 + 0.980867i \(0.562367\pi\)
\(198\) 0 0
\(199\) −13.5571 −0.961039 −0.480519 0.876984i \(-0.659552\pi\)
−0.480519 + 0.876984i \(0.659552\pi\)
\(200\) 0.599069 0.0423606
\(201\) −1.15895 −0.0817459
\(202\) 16.5379 1.16360
\(203\) −15.1740 −1.06501
\(204\) 1.71324 0.119951
\(205\) −10.4136 −0.727320
\(206\) 2.79627 0.194826
\(207\) 20.5224 1.42641
\(208\) −0.633765 −0.0439437
\(209\) 0 0
\(210\) −11.5827 −0.799280
\(211\) −14.3562 −0.988318 −0.494159 0.869371i \(-0.664524\pi\)
−0.494159 + 0.869371i \(0.664524\pi\)
\(212\) −12.5302 −0.860575
\(213\) −0.920526 −0.0630734
\(214\) −18.9681 −1.29664
\(215\) −22.6749 −1.54641
\(216\) 7.92194 0.539020
\(217\) −1.70316 −0.115618
\(218\) −17.0732 −1.15634
\(219\) 27.4521 1.85504
\(220\) 0 0
\(221\) −0.368378 −0.0247798
\(222\) 6.36348 0.427089
\(223\) −5.16251 −0.345707 −0.172854 0.984948i \(-0.555299\pi\)
−0.172854 + 0.984948i \(0.555299\pi\)
\(224\) −1.66073 −0.110962
\(225\) 3.40732 0.227155
\(226\) 21.1114 1.40431
\(227\) 18.3435 1.21750 0.608751 0.793361i \(-0.291671\pi\)
0.608751 + 0.793361i \(0.291671\pi\)
\(228\) −2.94749 −0.195202
\(229\) 18.3344 1.21157 0.605785 0.795629i \(-0.292859\pi\)
0.605785 + 0.795629i \(0.292859\pi\)
\(230\) 8.53789 0.562972
\(231\) 0 0
\(232\) 9.13696 0.599871
\(233\) 14.6429 0.959289 0.479645 0.877463i \(-0.340766\pi\)
0.479645 + 0.877463i \(0.340766\pi\)
\(234\) −3.60466 −0.235644
\(235\) 20.6631 1.34791
\(236\) 6.47623 0.421567
\(237\) −33.2193 −2.15782
\(238\) −0.965305 −0.0625714
\(239\) 12.5868 0.814175 0.407088 0.913389i \(-0.366544\pi\)
0.407088 + 0.913389i \(0.366544\pi\)
\(240\) 6.97445 0.450199
\(241\) 2.79627 0.180124 0.0900619 0.995936i \(-0.471294\pi\)
0.0900619 + 0.995936i \(0.471294\pi\)
\(242\) 0 0
\(243\) −5.23567 −0.335868
\(244\) −1.32146 −0.0845976
\(245\) −10.0375 −0.641274
\(246\) −12.9717 −0.827045
\(247\) 0.633765 0.0403255
\(248\) 1.02555 0.0651223
\(249\) 20.3498 1.28962
\(250\) −10.4136 −0.658616
\(251\) −1.72891 −0.109128 −0.0545640 0.998510i \(-0.517377\pi\)
−0.0545640 + 0.998510i \(0.517377\pi\)
\(252\) −9.44571 −0.595024
\(253\) 0 0
\(254\) −4.62745 −0.290352
\(255\) 4.05393 0.253867
\(256\) 1.00000 0.0625000
\(257\) 21.5414 1.34372 0.671859 0.740679i \(-0.265496\pi\)
0.671859 + 0.740679i \(0.265496\pi\)
\(258\) −28.2448 −1.75845
\(259\) −3.58543 −0.222788
\(260\) −1.49964 −0.0930035
\(261\) 51.9682 3.21675
\(262\) 15.7763 0.974664
\(263\) 22.7920 1.40541 0.702707 0.711479i \(-0.251974\pi\)
0.702707 + 0.711479i \(0.251974\pi\)
\(264\) 0 0
\(265\) −29.6493 −1.82134
\(266\) 1.66073 0.101826
\(267\) 1.53150 0.0937263
\(268\) −0.393199 −0.0240184
\(269\) −18.6429 −1.13668 −0.568339 0.822794i \(-0.692414\pi\)
−0.568339 + 0.822794i \(0.692414\pi\)
\(270\) 18.7452 1.14080
\(271\) 3.33571 0.202630 0.101315 0.994854i \(-0.467695\pi\)
0.101315 + 0.994854i \(0.467695\pi\)
\(272\) 0.581254 0.0352437
\(273\) 3.10226 0.187758
\(274\) −20.6366 −1.24670
\(275\) 0 0
\(276\) 10.6352 0.640163
\(277\) −23.5883 −1.41728 −0.708641 0.705570i \(-0.750691\pi\)
−0.708641 + 0.705570i \(0.750691\pi\)
\(278\) 9.56721 0.573803
\(279\) 5.83300 0.349212
\(280\) −3.92967 −0.234843
\(281\) −31.6366 −1.88728 −0.943641 0.330972i \(-0.892623\pi\)
−0.943641 + 0.330972i \(0.892623\pi\)
\(282\) 25.7389 1.53273
\(283\) −6.06757 −0.360680 −0.180340 0.983604i \(-0.557720\pi\)
−0.180340 + 0.983604i \(0.557720\pi\)
\(284\) −0.312308 −0.0185321
\(285\) −6.97445 −0.413131
\(286\) 0 0
\(287\) 7.30875 0.431422
\(288\) 5.68769 0.335150
\(289\) −16.6621 −0.980126
\(290\) 21.6202 1.26958
\(291\) −24.6968 −1.44775
\(292\) 9.31372 0.545044
\(293\) −16.8312 −0.983288 −0.491644 0.870796i \(-0.663604\pi\)
−0.491644 + 0.870796i \(0.663604\pi\)
\(294\) −12.5032 −0.729201
\(295\) 15.3243 0.892215
\(296\) 2.15895 0.125486
\(297\) 0 0
\(298\) −18.3096 −1.06064
\(299\) −2.28676 −0.132247
\(300\) 1.76575 0.101946
\(301\) 15.9142 0.917280
\(302\) −6.25562 −0.359971
\(303\) 48.7453 2.80034
\(304\) −1.00000 −0.0573539
\(305\) −3.12688 −0.179044
\(306\) 3.30599 0.188991
\(307\) 5.10858 0.291562 0.145781 0.989317i \(-0.453430\pi\)
0.145781 + 0.989317i \(0.453430\pi\)
\(308\) 0 0
\(309\) 8.24198 0.468870
\(310\) 2.42669 0.137827
\(311\) −24.5635 −1.39287 −0.696435 0.717620i \(-0.745231\pi\)
−0.696435 + 0.717620i \(0.745231\pi\)
\(312\) −1.86801 −0.105756
\(313\) −24.2868 −1.37277 −0.686387 0.727237i \(-0.740804\pi\)
−0.686387 + 0.727237i \(0.740804\pi\)
\(314\) −6.15619 −0.347414
\(315\) −22.3508 −1.25932
\(316\) −11.2704 −0.634007
\(317\) −28.6225 −1.60760 −0.803801 0.594898i \(-0.797192\pi\)
−0.803801 + 0.594898i \(0.797192\pi\)
\(318\) −36.9325 −2.07107
\(319\) 0 0
\(320\) 2.36624 0.132277
\(321\) −55.9084 −3.12050
\(322\) −5.99227 −0.333936
\(323\) −0.581254 −0.0323418
\(324\) 6.28676 0.349264
\(325\) −0.379669 −0.0210602
\(326\) −11.8439 −0.655972
\(327\) −50.3229 −2.78287
\(328\) −4.40093 −0.243001
\(329\) −14.5023 −0.799535
\(330\) 0 0
\(331\) −2.40960 −0.132444 −0.0662218 0.997805i \(-0.521094\pi\)
−0.0662218 + 0.997805i \(0.521094\pi\)
\(332\) 6.90413 0.378913
\(333\) 12.2794 0.672909
\(334\) 4.89142 0.267647
\(335\) −0.930401 −0.0508332
\(336\) −4.89498 −0.267043
\(337\) 17.5195 0.954346 0.477173 0.878809i \(-0.341662\pi\)
0.477173 + 0.878809i \(0.341662\pi\)
\(338\) −12.5983 −0.685259
\(339\) 62.2257 3.37963
\(340\) 1.37538 0.0745906
\(341\) 0 0
\(342\) −5.68769 −0.307555
\(343\) 18.6699 1.00808
\(344\) −9.58267 −0.516663
\(345\) 25.1653 1.35486
\(346\) 18.1206 0.974167
\(347\) 32.1534 1.72609 0.863043 0.505130i \(-0.168555\pi\)
0.863043 + 0.505130i \(0.168555\pi\)
\(348\) 26.9311 1.44366
\(349\) 0.407455 0.0218106 0.0109053 0.999941i \(-0.496529\pi\)
0.0109053 + 0.999941i \(0.496529\pi\)
\(350\) −0.994891 −0.0531792
\(351\) −5.02065 −0.267982
\(352\) 0 0
\(353\) 3.94158 0.209789 0.104895 0.994483i \(-0.466549\pi\)
0.104895 + 0.994483i \(0.466549\pi\)
\(354\) 19.0886 1.01455
\(355\) −0.738995 −0.0392218
\(356\) 0.519595 0.0275385
\(357\) −2.84522 −0.150585
\(358\) 3.07947 0.162755
\(359\) −20.0781 −1.05968 −0.529842 0.848097i \(-0.677749\pi\)
−0.529842 + 0.848097i \(0.677749\pi\)
\(360\) 13.4584 0.709321
\(361\) 1.00000 0.0526316
\(362\) 0.0693910 0.00364711
\(363\) 0 0
\(364\) 1.05251 0.0551666
\(365\) 22.0385 1.15355
\(366\) −3.89498 −0.203594
\(367\) −20.5918 −1.07488 −0.537442 0.843301i \(-0.680609\pi\)
−0.537442 + 0.843301i \(0.680609\pi\)
\(368\) 3.60822 0.188091
\(369\) −25.0311 −1.30307
\(370\) 5.10858 0.265582
\(371\) 20.8092 1.08036
\(372\) 3.02279 0.156724
\(373\) −20.8813 −1.08119 −0.540597 0.841282i \(-0.681802\pi\)
−0.540597 + 0.841282i \(0.681802\pi\)
\(374\) 0 0
\(375\) −30.6941 −1.58504
\(376\) 8.73247 0.450343
\(377\) −5.79068 −0.298235
\(378\) −13.1562 −0.676681
\(379\) −0.753845 −0.0387224 −0.0193612 0.999813i \(-0.506163\pi\)
−0.0193612 + 0.999813i \(0.506163\pi\)
\(380\) −2.36624 −0.121385
\(381\) −13.6394 −0.698765
\(382\) 13.6776 0.699807
\(383\) −18.3142 −0.935812 −0.467906 0.883778i \(-0.654991\pi\)
−0.467906 + 0.883778i \(0.654991\pi\)
\(384\) 2.94749 0.150413
\(385\) 0 0
\(386\) −5.61830 −0.285964
\(387\) −54.5033 −2.77056
\(388\) −8.37894 −0.425376
\(389\) −26.1206 −1.32436 −0.662182 0.749343i \(-0.730370\pi\)
−0.662182 + 0.749343i \(0.730370\pi\)
\(390\) −4.42016 −0.223824
\(391\) 2.09729 0.106065
\(392\) −4.24198 −0.214252
\(393\) 46.5005 2.34564
\(394\) −5.46494 −0.275320
\(395\) −26.6683 −1.34183
\(396\) 0 0
\(397\) −34.9079 −1.75198 −0.875988 0.482332i \(-0.839790\pi\)
−0.875988 + 0.482332i \(0.839790\pi\)
\(398\) −13.5571 −0.679557
\(399\) 4.89498 0.245055
\(400\) 0.599069 0.0299535
\(401\) 6.35353 0.317280 0.158640 0.987336i \(-0.449289\pi\)
0.158640 + 0.987336i \(0.449289\pi\)
\(402\) −1.15895 −0.0578031
\(403\) −0.649956 −0.0323766
\(404\) 16.5379 0.822791
\(405\) 14.8760 0.739192
\(406\) −15.1740 −0.753073
\(407\) 0 0
\(408\) 1.71324 0.0848180
\(409\) 26.1816 1.29460 0.647299 0.762237i \(-0.275899\pi\)
0.647299 + 0.762237i \(0.275899\pi\)
\(410\) −10.4136 −0.514293
\(411\) −60.8261 −3.00033
\(412\) 2.79627 0.137762
\(413\) −10.7553 −0.529232
\(414\) 20.5224 1.00862
\(415\) 16.3368 0.801941
\(416\) −0.633765 −0.0310729
\(417\) 28.1992 1.38092
\(418\) 0 0
\(419\) 8.15060 0.398183 0.199091 0.979981i \(-0.436201\pi\)
0.199091 + 0.979981i \(0.436201\pi\)
\(420\) −11.5827 −0.565176
\(421\) 10.9531 0.533820 0.266910 0.963721i \(-0.413997\pi\)
0.266910 + 0.963721i \(0.413997\pi\)
\(422\) −14.3562 −0.698847
\(423\) 49.6676 2.41492
\(424\) −12.5302 −0.608518
\(425\) 0.348211 0.0168907
\(426\) −0.920526 −0.0445996
\(427\) 2.19458 0.106203
\(428\) −18.9681 −0.916859
\(429\) 0 0
\(430\) −22.6749 −1.09348
\(431\) 17.8510 0.859852 0.429926 0.902864i \(-0.358540\pi\)
0.429926 + 0.902864i \(0.358540\pi\)
\(432\) 7.92194 0.381145
\(433\) 39.4138 1.89411 0.947054 0.321074i \(-0.104044\pi\)
0.947054 + 0.321074i \(0.104044\pi\)
\(434\) −1.70316 −0.0817541
\(435\) 63.7253 3.05539
\(436\) −17.0732 −0.817656
\(437\) −3.60822 −0.172604
\(438\) 27.4521 1.31171
\(439\) 16.9993 0.811331 0.405666 0.914022i \(-0.367040\pi\)
0.405666 + 0.914022i \(0.367040\pi\)
\(440\) 0 0
\(441\) −24.1271 −1.14891
\(442\) −0.368378 −0.0175220
\(443\) 16.2668 0.772859 0.386430 0.922319i \(-0.373708\pi\)
0.386430 + 0.922319i \(0.373708\pi\)
\(444\) 6.36348 0.301997
\(445\) 1.22948 0.0582831
\(446\) −5.16251 −0.244452
\(447\) −53.9672 −2.55256
\(448\) −1.66073 −0.0784620
\(449\) 16.5534 0.781201 0.390601 0.920560i \(-0.372267\pi\)
0.390601 + 0.920560i \(0.372267\pi\)
\(450\) 3.40732 0.160623
\(451\) 0 0
\(452\) 21.1114 0.992997
\(453\) −18.4384 −0.866311
\(454\) 18.3435 0.860904
\(455\) 2.49049 0.116756
\(456\) −2.94749 −0.138029
\(457\) −11.5805 −0.541714 −0.270857 0.962620i \(-0.587307\pi\)
−0.270857 + 0.962620i \(0.587307\pi\)
\(458\) 18.3344 0.856709
\(459\) 4.60466 0.214927
\(460\) 8.53789 0.398081
\(461\) −5.42648 −0.252736 −0.126368 0.991983i \(-0.540332\pi\)
−0.126368 + 0.991983i \(0.540332\pi\)
\(462\) 0 0
\(463\) 26.9168 1.25093 0.625466 0.780252i \(-0.284909\pi\)
0.625466 + 0.780252i \(0.284909\pi\)
\(464\) 9.13696 0.424173
\(465\) 7.15263 0.331695
\(466\) 14.6429 0.678320
\(467\) 12.5581 0.581118 0.290559 0.956857i \(-0.406159\pi\)
0.290559 + 0.956857i \(0.406159\pi\)
\(468\) −3.60466 −0.166625
\(469\) 0.652996 0.0301526
\(470\) 20.6631 0.953117
\(471\) −18.1453 −0.836092
\(472\) 6.47623 0.298093
\(473\) 0 0
\(474\) −33.2193 −1.52581
\(475\) −0.599069 −0.0274872
\(476\) −0.965305 −0.0442447
\(477\) −71.2677 −3.26312
\(478\) 12.5868 0.575709
\(479\) 40.2322 1.83826 0.919128 0.393960i \(-0.128895\pi\)
0.919128 + 0.393960i \(0.128895\pi\)
\(480\) 6.97445 0.318339
\(481\) −1.36827 −0.0623875
\(482\) 2.79627 0.127367
\(483\) −17.6621 −0.803655
\(484\) 0 0
\(485\) −19.8265 −0.900277
\(486\) −5.23567 −0.237495
\(487\) −14.2795 −0.647066 −0.323533 0.946217i \(-0.604871\pi\)
−0.323533 + 0.946217i \(0.604871\pi\)
\(488\) −1.32146 −0.0598195
\(489\) −34.9097 −1.57867
\(490\) −10.0375 −0.453449
\(491\) 14.0094 0.632233 0.316117 0.948720i \(-0.397621\pi\)
0.316117 + 0.948720i \(0.397621\pi\)
\(492\) −12.9717 −0.584809
\(493\) 5.31089 0.239191
\(494\) 0.633765 0.0285144
\(495\) 0 0
\(496\) 1.02555 0.0460484
\(497\) 0.518659 0.0232651
\(498\) 20.3498 0.911898
\(499\) 18.1506 0.812533 0.406266 0.913755i \(-0.366831\pi\)
0.406266 + 0.913755i \(0.366831\pi\)
\(500\) −10.4136 −0.465712
\(501\) 14.4174 0.644122
\(502\) −1.72891 −0.0771651
\(503\) −29.8672 −1.33171 −0.665855 0.746081i \(-0.731933\pi\)
−0.665855 + 0.746081i \(0.731933\pi\)
\(504\) −9.44571 −0.420745
\(505\) 39.1325 1.74138
\(506\) 0 0
\(507\) −37.1335 −1.64916
\(508\) −4.62745 −0.205310
\(509\) 24.2045 1.07285 0.536423 0.843949i \(-0.319775\pi\)
0.536423 + 0.843949i \(0.319775\pi\)
\(510\) 4.05393 0.179511
\(511\) −15.4676 −0.684245
\(512\) 1.00000 0.0441942
\(513\) −7.92194 −0.349762
\(514\) 21.5414 0.950153
\(515\) 6.61664 0.291564
\(516\) −28.2448 −1.24341
\(517\) 0 0
\(518\) −3.58543 −0.157535
\(519\) 53.4102 2.34445
\(520\) −1.49964 −0.0657634
\(521\) −6.36899 −0.279031 −0.139515 0.990220i \(-0.544554\pi\)
−0.139515 + 0.990220i \(0.544554\pi\)
\(522\) 51.9682 2.27459
\(523\) −21.2749 −0.930284 −0.465142 0.885236i \(-0.653997\pi\)
−0.465142 + 0.885236i \(0.653997\pi\)
\(524\) 15.7763 0.689191
\(525\) −2.93243 −0.127982
\(526\) 22.7920 0.993778
\(527\) 0.596103 0.0259667
\(528\) 0 0
\(529\) −9.98077 −0.433946
\(530\) −29.6493 −1.28788
\(531\) 36.8348 1.59849
\(532\) 1.66073 0.0720017
\(533\) 2.78915 0.120812
\(534\) 1.53150 0.0662745
\(535\) −44.8831 −1.94046
\(536\) −0.393199 −0.0169836
\(537\) 9.07672 0.391689
\(538\) −18.6429 −0.803753
\(539\) 0 0
\(540\) 18.7452 0.806664
\(541\) −34.2668 −1.47324 −0.736622 0.676304i \(-0.763580\pi\)
−0.736622 + 0.676304i \(0.763580\pi\)
\(542\) 3.33571 0.143281
\(543\) 0.204529 0.00877718
\(544\) 0.581254 0.0249211
\(545\) −40.3991 −1.73051
\(546\) 3.10226 0.132765
\(547\) −31.2732 −1.33715 −0.668573 0.743647i \(-0.733094\pi\)
−0.668573 + 0.743647i \(0.733094\pi\)
\(548\) −20.6366 −0.881552
\(549\) −7.51604 −0.320777
\(550\) 0 0
\(551\) −9.13696 −0.389248
\(552\) 10.6352 0.452663
\(553\) 18.7170 0.795928
\(554\) −23.5883 −1.00217
\(555\) 15.0575 0.639155
\(556\) 9.56721 0.405740
\(557\) 24.7198 1.04741 0.523707 0.851899i \(-0.324549\pi\)
0.523707 + 0.851899i \(0.324549\pi\)
\(558\) 5.83300 0.246931
\(559\) 6.07316 0.256867
\(560\) −3.92967 −0.166059
\(561\) 0 0
\(562\) −31.6366 −1.33451
\(563\) 8.94251 0.376882 0.188441 0.982085i \(-0.439657\pi\)
0.188441 + 0.982085i \(0.439657\pi\)
\(564\) 25.7389 1.08380
\(565\) 49.9546 2.10160
\(566\) −6.06757 −0.255039
\(567\) −10.4406 −0.438464
\(568\) −0.312308 −0.0131042
\(569\) −0.764409 −0.0320457 −0.0160228 0.999872i \(-0.505100\pi\)
−0.0160228 + 0.999872i \(0.505100\pi\)
\(570\) −6.97445 −0.292128
\(571\) 4.11866 0.172361 0.0861804 0.996280i \(-0.472534\pi\)
0.0861804 + 0.996280i \(0.472534\pi\)
\(572\) 0 0
\(573\) 40.3146 1.68417
\(574\) 7.30875 0.305061
\(575\) 2.16157 0.0901438
\(576\) 5.68769 0.236987
\(577\) 26.8538 1.11794 0.558968 0.829189i \(-0.311197\pi\)
0.558968 + 0.829189i \(0.311197\pi\)
\(578\) −16.6621 −0.693054
\(579\) −16.5599 −0.688205
\(580\) 21.6202 0.897730
\(581\) −11.4659 −0.475685
\(582\) −24.6968 −1.02372
\(583\) 0 0
\(584\) 9.31372 0.385405
\(585\) −8.52947 −0.352650
\(586\) −16.8312 −0.695289
\(587\) 11.4832 0.473964 0.236982 0.971514i \(-0.423842\pi\)
0.236982 + 0.971514i \(0.423842\pi\)
\(588\) −12.5032 −0.515623
\(589\) −1.02555 −0.0422570
\(590\) 15.3243 0.630891
\(591\) −16.1079 −0.662589
\(592\) 2.15895 0.0887322
\(593\) 32.3508 1.32849 0.664245 0.747515i \(-0.268753\pi\)
0.664245 + 0.747515i \(0.268753\pi\)
\(594\) 0 0
\(595\) −2.28414 −0.0936405
\(596\) −18.3096 −0.749988
\(597\) −39.9595 −1.63543
\(598\) −2.28676 −0.0935126
\(599\) 6.06380 0.247760 0.123880 0.992297i \(-0.460466\pi\)
0.123880 + 0.992297i \(0.460466\pi\)
\(600\) 1.76575 0.0720864
\(601\) −23.0923 −0.941953 −0.470976 0.882146i \(-0.656098\pi\)
−0.470976 + 0.882146i \(0.656098\pi\)
\(602\) 15.9142 0.648615
\(603\) −2.23639 −0.0910730
\(604\) −6.25562 −0.254538
\(605\) 0 0
\(606\) 48.7453 1.98014
\(607\) −11.1808 −0.453815 −0.226907 0.973916i \(-0.572861\pi\)
−0.226907 + 0.973916i \(0.572861\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −44.7252 −1.81236
\(610\) −3.12688 −0.126604
\(611\) −5.53433 −0.223895
\(612\) 3.30599 0.133637
\(613\) −33.8355 −1.36660 −0.683302 0.730136i \(-0.739457\pi\)
−0.683302 + 0.730136i \(0.739457\pi\)
\(614\) 5.10858 0.206166
\(615\) −30.6941 −1.23770
\(616\) 0 0
\(617\) 24.9927 1.00617 0.503085 0.864237i \(-0.332198\pi\)
0.503085 + 0.864237i \(0.332198\pi\)
\(618\) 8.24198 0.331541
\(619\) 7.68493 0.308884 0.154442 0.988002i \(-0.450642\pi\)
0.154442 + 0.988002i \(0.450642\pi\)
\(620\) 2.42669 0.0974581
\(621\) 28.5841 1.14704
\(622\) −24.5635 −0.984907
\(623\) −0.862906 −0.0345716
\(624\) −1.86801 −0.0747804
\(625\) −27.6365 −1.10546
\(626\) −24.2868 −0.970697
\(627\) 0 0
\(628\) −6.15619 −0.245659
\(629\) 1.25490 0.0500360
\(630\) −22.3508 −0.890476
\(631\) 25.7004 1.02312 0.511558 0.859249i \(-0.329068\pi\)
0.511558 + 0.859249i \(0.329068\pi\)
\(632\) −11.2704 −0.448311
\(633\) −42.3146 −1.68185
\(634\) −28.6225 −1.13675
\(635\) −10.9496 −0.434523
\(636\) −36.9325 −1.46447
\(637\) 2.68842 0.106519
\(638\) 0 0
\(639\) −1.77631 −0.0702699
\(640\) 2.36624 0.0935337
\(641\) 4.03846 0.159510 0.0797548 0.996815i \(-0.474586\pi\)
0.0797548 + 0.996815i \(0.474586\pi\)
\(642\) −55.9084 −2.20653
\(643\) 5.44194 0.214609 0.107305 0.994226i \(-0.465778\pi\)
0.107305 + 0.994226i \(0.465778\pi\)
\(644\) −5.99227 −0.236128
\(645\) −66.8339 −2.63158
\(646\) −0.581254 −0.0228691
\(647\) −32.4046 −1.27395 −0.636977 0.770882i \(-0.719816\pi\)
−0.636977 + 0.770882i \(0.719816\pi\)
\(648\) 6.28676 0.246967
\(649\) 0 0
\(650\) −0.379669 −0.0148918
\(651\) −5.02003 −0.196751
\(652\) −11.8439 −0.463842
\(653\) 24.7672 0.969217 0.484609 0.874731i \(-0.338962\pi\)
0.484609 + 0.874731i \(0.338962\pi\)
\(654\) −50.3229 −1.96778
\(655\) 37.3305 1.45862
\(656\) −4.40093 −0.171827
\(657\) 52.9736 2.06670
\(658\) −14.5023 −0.565357
\(659\) 27.5600 1.07358 0.536792 0.843715i \(-0.319636\pi\)
0.536792 + 0.843715i \(0.319636\pi\)
\(660\) 0 0
\(661\) −7.53372 −0.293028 −0.146514 0.989209i \(-0.546805\pi\)
−0.146514 + 0.989209i \(0.546805\pi\)
\(662\) −2.40960 −0.0936517
\(663\) −1.08579 −0.0421686
\(664\) 6.90413 0.267932
\(665\) 3.92967 0.152386
\(666\) 12.2794 0.475818
\(667\) 32.9681 1.27653
\(668\) 4.89142 0.189255
\(669\) −15.2164 −0.588301
\(670\) −0.930401 −0.0359445
\(671\) 0 0
\(672\) −4.89498 −0.188828
\(673\) 8.79534 0.339035 0.169518 0.985527i \(-0.445779\pi\)
0.169518 + 0.985527i \(0.445779\pi\)
\(674\) 17.5195 0.674824
\(675\) 4.74579 0.182666
\(676\) −12.5983 −0.484552
\(677\) −15.8310 −0.608434 −0.304217 0.952603i \(-0.598395\pi\)
−0.304217 + 0.952603i \(0.598395\pi\)
\(678\) 62.2257 2.38976
\(679\) 13.9151 0.534014
\(680\) 1.37538 0.0527435
\(681\) 54.0673 2.07186
\(682\) 0 0
\(683\) 38.6640 1.47944 0.739719 0.672916i \(-0.234958\pi\)
0.739719 + 0.672916i \(0.234958\pi\)
\(684\) −5.68769 −0.217474
\(685\) −48.8310 −1.86574
\(686\) 18.6699 0.712819
\(687\) 54.0404 2.06177
\(688\) −9.58267 −0.365336
\(689\) 7.94117 0.302535
\(690\) 25.1653 0.958028
\(691\) −27.4678 −1.04492 −0.522462 0.852663i \(-0.674986\pi\)
−0.522462 + 0.852663i \(0.674986\pi\)
\(692\) 18.1206 0.688840
\(693\) 0 0
\(694\) 32.1534 1.22053
\(695\) 22.6383 0.858718
\(696\) 26.9311 1.02082
\(697\) −2.55806 −0.0968933
\(698\) 0.407455 0.0154224
\(699\) 43.1598 1.63245
\(700\) −0.994891 −0.0376034
\(701\) 1.50892 0.0569911 0.0284955 0.999594i \(-0.490928\pi\)
0.0284955 + 0.999594i \(0.490928\pi\)
\(702\) −5.02065 −0.189492
\(703\) −2.15895 −0.0814263
\(704\) 0 0
\(705\) 60.9042 2.29378
\(706\) 3.94158 0.148343
\(707\) −27.4649 −1.03293
\(708\) 19.0886 0.717394
\(709\) 16.2959 0.612006 0.306003 0.952031i \(-0.401008\pi\)
0.306003 + 0.952031i \(0.401008\pi\)
\(710\) −0.738995 −0.0277340
\(711\) −64.1023 −2.40403
\(712\) 0.519595 0.0194726
\(713\) 3.70040 0.138581
\(714\) −2.84522 −0.106480
\(715\) 0 0
\(716\) 3.07947 0.115085
\(717\) 37.0996 1.38551
\(718\) −20.0781 −0.749309
\(719\) 40.6785 1.51705 0.758526 0.651643i \(-0.225920\pi\)
0.758526 + 0.651643i \(0.225920\pi\)
\(720\) 13.4584 0.501566
\(721\) −4.64385 −0.172946
\(722\) 1.00000 0.0372161
\(723\) 8.24198 0.306523
\(724\) 0.0693910 0.00257889
\(725\) 5.47367 0.203287
\(726\) 0 0
\(727\) 27.7545 1.02936 0.514679 0.857383i \(-0.327911\pi\)
0.514679 + 0.857383i \(0.327911\pi\)
\(728\) 1.05251 0.0390087
\(729\) −34.2924 −1.27009
\(730\) 22.0385 0.815680
\(731\) −5.56996 −0.206013
\(732\) −3.89498 −0.143963
\(733\) 6.89854 0.254803 0.127402 0.991851i \(-0.459336\pi\)
0.127402 + 0.991851i \(0.459336\pi\)
\(734\) −20.5918 −0.760058
\(735\) −29.5855 −1.09128
\(736\) 3.60822 0.133001
\(737\) 0 0
\(738\) −25.0311 −0.921409
\(739\) −49.2039 −1.80999 −0.904997 0.425418i \(-0.860127\pi\)
−0.904997 + 0.425418i \(0.860127\pi\)
\(740\) 5.10858 0.187795
\(741\) 1.86801 0.0686232
\(742\) 20.8092 0.763929
\(743\) −10.4046 −0.381709 −0.190854 0.981618i \(-0.561126\pi\)
−0.190854 + 0.981618i \(0.561126\pi\)
\(744\) 3.02279 0.110821
\(745\) −43.3247 −1.58729
\(746\) −20.8813 −0.764520
\(747\) 39.2685 1.43676
\(748\) 0 0
\(749\) 31.5009 1.15102
\(750\) −30.6941 −1.12079
\(751\) 37.7103 1.37607 0.688035 0.725677i \(-0.258474\pi\)
0.688035 + 0.725677i \(0.258474\pi\)
\(752\) 8.73247 0.318440
\(753\) −5.09595 −0.185707
\(754\) −5.79068 −0.210884
\(755\) −14.8023 −0.538710
\(756\) −13.1562 −0.478486
\(757\) 14.3972 0.523274 0.261637 0.965166i \(-0.415738\pi\)
0.261637 + 0.965166i \(0.415738\pi\)
\(758\) −0.753845 −0.0273809
\(759\) 0 0
\(760\) −2.36624 −0.0858324
\(761\) 39.7311 1.44025 0.720126 0.693843i \(-0.244084\pi\)
0.720126 + 0.693843i \(0.244084\pi\)
\(762\) −13.6394 −0.494102
\(763\) 28.3539 1.02648
\(764\) 13.6776 0.494838
\(765\) 7.82276 0.282832
\(766\) −18.3142 −0.661719
\(767\) −4.10441 −0.148202
\(768\) 2.94749 0.106358
\(769\) 2.27170 0.0819197 0.0409598 0.999161i \(-0.486958\pi\)
0.0409598 + 0.999161i \(0.486958\pi\)
\(770\) 0 0
\(771\) 63.4932 2.28665
\(772\) −5.61830 −0.202207
\(773\) 3.64157 0.130978 0.0654891 0.997853i \(-0.479139\pi\)
0.0654891 + 0.997853i \(0.479139\pi\)
\(774\) −54.5033 −1.95908
\(775\) 0.614374 0.0220690
\(776\) −8.37894 −0.300786
\(777\) −10.5680 −0.379125
\(778\) −26.1206 −0.936467
\(779\) 4.40093 0.157680
\(780\) −4.42016 −0.158267
\(781\) 0 0
\(782\) 2.09729 0.0749989
\(783\) 72.3825 2.58674
\(784\) −4.24198 −0.151499
\(785\) −14.5670 −0.519918
\(786\) 46.5005 1.65862
\(787\) −0.279643 −0.00996821 −0.00498410 0.999988i \(-0.501586\pi\)
−0.00498410 + 0.999988i \(0.501586\pi\)
\(788\) −5.46494 −0.194680
\(789\) 67.1791 2.39164
\(790\) −26.6683 −0.948816
\(791\) −35.0603 −1.24660
\(792\) 0 0
\(793\) 0.837492 0.0297402
\(794\) −34.9079 −1.23883
\(795\) −87.3910 −3.09944
\(796\) −13.5571 −0.480519
\(797\) −3.66014 −0.129649 −0.0648243 0.997897i \(-0.520649\pi\)
−0.0648243 + 0.997897i \(0.520649\pi\)
\(798\) 4.89498 0.173280
\(799\) 5.07578 0.179568
\(800\) 0.599069 0.0211803
\(801\) 2.95530 0.104420
\(802\) 6.35353 0.224351
\(803\) 0 0
\(804\) −1.15895 −0.0408730
\(805\) −14.1791 −0.499748
\(806\) −0.649956 −0.0228937
\(807\) −54.9498 −1.93432
\(808\) 16.5379 0.581801
\(809\) 27.0990 0.952749 0.476375 0.879242i \(-0.341951\pi\)
0.476375 + 0.879242i \(0.341951\pi\)
\(810\) 14.8760 0.522688
\(811\) 31.4212 1.10335 0.551673 0.834060i \(-0.313990\pi\)
0.551673 + 0.834060i \(0.313990\pi\)
\(812\) −15.1740 −0.532503
\(813\) 9.83198 0.344823
\(814\) 0 0
\(815\) −28.0254 −0.981687
\(816\) 1.71324 0.0599754
\(817\) 9.58267 0.335255
\(818\) 26.1816 0.915418
\(819\) 5.98636 0.209180
\(820\) −10.4136 −0.363660
\(821\) −20.1304 −0.702557 −0.351279 0.936271i \(-0.614253\pi\)
−0.351279 + 0.936271i \(0.614253\pi\)
\(822\) −60.8261 −2.12156
\(823\) −7.23922 −0.252344 −0.126172 0.992008i \(-0.540269\pi\)
−0.126172 + 0.992008i \(0.540269\pi\)
\(824\) 2.79627 0.0974128
\(825\) 0 0
\(826\) −10.7553 −0.374223
\(827\) 23.1088 0.803571 0.401786 0.915734i \(-0.368390\pi\)
0.401786 + 0.915734i \(0.368390\pi\)
\(828\) 20.5224 0.713204
\(829\) 21.7081 0.753955 0.376977 0.926223i \(-0.376963\pi\)
0.376977 + 0.926223i \(0.376963\pi\)
\(830\) 16.3368 0.567058
\(831\) −69.5261 −2.41183
\(832\) −0.633765 −0.0219718
\(833\) −2.46567 −0.0854303
\(834\) 28.1992 0.976460
\(835\) 11.5742 0.400543
\(836\) 0 0
\(837\) 8.12433 0.280818
\(838\) 8.15060 0.281558
\(839\) −21.1935 −0.731681 −0.365841 0.930678i \(-0.619218\pi\)
−0.365841 + 0.930678i \(0.619218\pi\)
\(840\) −11.5827 −0.399640
\(841\) 54.4840 1.87876
\(842\) 10.9531 0.377468
\(843\) −93.2485 −3.21165
\(844\) −14.3562 −0.494159
\(845\) −29.8106 −1.02552
\(846\) 49.6676 1.70761
\(847\) 0 0
\(848\) −12.5302 −0.430287
\(849\) −17.8841 −0.613780
\(850\) 0.348211 0.0119435
\(851\) 7.78996 0.267036
\(852\) −0.920526 −0.0315367
\(853\) 30.5835 1.04716 0.523579 0.851977i \(-0.324596\pi\)
0.523579 + 0.851977i \(0.324596\pi\)
\(854\) 2.19458 0.0750970
\(855\) −13.4584 −0.460268
\(856\) −18.9681 −0.648318
\(857\) 48.3757 1.65248 0.826241 0.563317i \(-0.190475\pi\)
0.826241 + 0.563317i \(0.190475\pi\)
\(858\) 0 0
\(859\) −57.0941 −1.94802 −0.974012 0.226495i \(-0.927273\pi\)
−0.974012 + 0.226495i \(0.927273\pi\)
\(860\) −22.6749 −0.773206
\(861\) 21.5425 0.734165
\(862\) 17.8510 0.608007
\(863\) 3.31963 0.113002 0.0565008 0.998403i \(-0.482006\pi\)
0.0565008 + 0.998403i \(0.482006\pi\)
\(864\) 7.92194 0.269510
\(865\) 42.8775 1.45788
\(866\) 39.4138 1.33934
\(867\) −49.1115 −1.66791
\(868\) −1.70316 −0.0578089
\(869\) 0 0
\(870\) 63.7253 2.16049
\(871\) 0.249195 0.00844367
\(872\) −17.0732 −0.578170
\(873\) −47.6568 −1.61294
\(874\) −3.60822 −0.122050
\(875\) 17.2942 0.584651
\(876\) 27.4521 0.927521
\(877\) 4.46676 0.150832 0.0754159 0.997152i \(-0.475972\pi\)
0.0754159 + 0.997152i \(0.475972\pi\)
\(878\) 16.9993 0.573698
\(879\) −49.6097 −1.67329
\(880\) 0 0
\(881\) 6.10320 0.205622 0.102811 0.994701i \(-0.467216\pi\)
0.102811 + 0.994701i \(0.467216\pi\)
\(882\) −24.1271 −0.812401
\(883\) 6.98526 0.235073 0.117536 0.993069i \(-0.462500\pi\)
0.117536 + 0.993069i \(0.462500\pi\)
\(884\) −0.368378 −0.0123899
\(885\) 45.1682 1.51831
\(886\) 16.2668 0.546494
\(887\) 39.3670 1.32182 0.660908 0.750467i \(-0.270171\pi\)
0.660908 + 0.750467i \(0.270171\pi\)
\(888\) 6.36348 0.213544
\(889\) 7.68493 0.257744
\(890\) 1.22948 0.0412124
\(891\) 0 0
\(892\) −5.16251 −0.172854
\(893\) −8.73247 −0.292221
\(894\) −53.9672 −1.80493
\(895\) 7.28676 0.243570
\(896\) −1.66073 −0.0554810
\(897\) −6.74020 −0.225049
\(898\) 16.5534 0.552393
\(899\) 9.37039 0.312520
\(900\) 3.40732 0.113577
\(901\) −7.28320 −0.242639
\(902\) 0 0
\(903\) 46.9070 1.56097
\(904\) 21.1114 0.702155
\(905\) 0.164195 0.00545804
\(906\) −18.4384 −0.612574
\(907\) 56.7300 1.88369 0.941844 0.336050i \(-0.109091\pi\)
0.941844 + 0.336050i \(0.109091\pi\)
\(908\) 18.3435 0.608751
\(909\) 94.0624 3.11985
\(910\) 2.49049 0.0825589
\(911\) −47.8355 −1.58486 −0.792431 0.609962i \(-0.791185\pi\)
−0.792431 + 0.609962i \(0.791185\pi\)
\(912\) −2.94749 −0.0976011
\(913\) 0 0
\(914\) −11.5805 −0.383050
\(915\) −9.21643 −0.304686
\(916\) 18.3344 0.605785
\(917\) −26.2002 −0.865206
\(918\) 4.60466 0.151976
\(919\) −0.295588 −0.00975054 −0.00487527 0.999988i \(-0.501552\pi\)
−0.00487527 + 0.999988i \(0.501552\pi\)
\(920\) 8.53789 0.281486
\(921\) 15.0575 0.496161
\(922\) −5.42648 −0.178712
\(923\) 0.197930 0.00651495
\(924\) 0 0
\(925\) 1.29336 0.0425254
\(926\) 26.9168 0.884542
\(927\) 15.9043 0.522367
\(928\) 9.13696 0.299935
\(929\) −18.6254 −0.611080 −0.305540 0.952179i \(-0.598837\pi\)
−0.305540 + 0.952179i \(0.598837\pi\)
\(930\) 7.15263 0.234544
\(931\) 4.24198 0.139025
\(932\) 14.6429 0.479645
\(933\) −72.4007 −2.37029
\(934\) 12.5581 0.410912
\(935\) 0 0
\(936\) −3.60466 −0.117822
\(937\) 50.5786 1.65233 0.826165 0.563428i \(-0.190518\pi\)
0.826165 + 0.563428i \(0.190518\pi\)
\(938\) 0.652996 0.0213211
\(939\) −71.5852 −2.33609
\(940\) 20.6631 0.673955
\(941\) −47.1169 −1.53597 −0.767983 0.640470i \(-0.778739\pi\)
−0.767983 + 0.640470i \(0.778739\pi\)
\(942\) −18.1453 −0.591206
\(943\) −15.8795 −0.517108
\(944\) 6.47623 0.210783
\(945\) −31.1306 −1.01268
\(946\) 0 0
\(947\) 8.28227 0.269138 0.134569 0.990904i \(-0.457035\pi\)
0.134569 + 0.990904i \(0.457035\pi\)
\(948\) −33.2193 −1.07891
\(949\) −5.90271 −0.191610
\(950\) −0.599069 −0.0194364
\(951\) −84.3646 −2.73571
\(952\) −0.965305 −0.0312857
\(953\) 10.0765 0.326410 0.163205 0.986592i \(-0.447817\pi\)
0.163205 + 0.986592i \(0.447817\pi\)
\(954\) −71.2677 −2.30738
\(955\) 32.3644 1.04729
\(956\) 12.5868 0.407088
\(957\) 0 0
\(958\) 40.2322 1.29984
\(959\) 34.2718 1.10669
\(960\) 6.97445 0.225099
\(961\) −29.9483 −0.966073
\(962\) −1.36827 −0.0441147
\(963\) −107.885 −3.47654
\(964\) 2.79627 0.0900619
\(965\) −13.2942 −0.427956
\(966\) −17.6621 −0.568270
\(967\) −48.6402 −1.56416 −0.782082 0.623175i \(-0.785842\pi\)
−0.782082 + 0.623175i \(0.785842\pi\)
\(968\) 0 0
\(969\) −1.71324 −0.0550372
\(970\) −19.8265 −0.636592
\(971\) −4.89578 −0.157113 −0.0785565 0.996910i \(-0.525031\pi\)
−0.0785565 + 0.996910i \(0.525031\pi\)
\(972\) −5.23567 −0.167934
\(973\) −15.8885 −0.509363
\(974\) −14.2795 −0.457545
\(975\) −1.11907 −0.0358389
\(976\) −1.32146 −0.0422988
\(977\) −36.1052 −1.15511 −0.577553 0.816353i \(-0.695992\pi\)
−0.577553 + 0.816353i \(0.695992\pi\)
\(978\) −34.9097 −1.11629
\(979\) 0 0
\(980\) −10.0375 −0.320637
\(981\) −97.1069 −3.10038
\(982\) 14.0094 0.447057
\(983\) −18.7371 −0.597621 −0.298811 0.954312i \(-0.596590\pi\)
−0.298811 + 0.954312i \(0.596590\pi\)
\(984\) −12.9717 −0.413523
\(985\) −12.9313 −0.412027
\(986\) 5.31089 0.169133
\(987\) −42.7453 −1.36060
\(988\) 0.633765 0.0201627
\(989\) −34.5764 −1.09946
\(990\) 0 0
\(991\) −8.60669 −0.273400 −0.136700 0.990612i \(-0.543650\pi\)
−0.136700 + 0.990612i \(0.543650\pi\)
\(992\) 1.02555 0.0325612
\(993\) −7.10226 −0.225384
\(994\) 0.518659 0.0164509
\(995\) −32.0793 −1.01698
\(996\) 20.3498 0.644809
\(997\) 13.3500 0.422798 0.211399 0.977400i \(-0.432198\pi\)
0.211399 + 0.977400i \(0.432198\pi\)
\(998\) 18.1506 0.574547
\(999\) 17.1031 0.541117
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 4598.2.a.bu.1.4 yes 4
11.10 odd 2 4598.2.a.br.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.br.1.4 4 11.10 odd 2
4598.2.a.bu.1.4 yes 4 1.1 even 1 trivial