Properties

Label 1850.2.a.y.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
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 1.1
Root \(3.13264\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} -4.13264 q^{7} -1.00000 q^{8} +6.81342 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} -4.13264 q^{7} -1.00000 q^{8} +6.81342 q^{9} -3.68078 q^{11} -3.13264 q^{12} +0.132637 q^{13} +4.13264 q^{14} +1.00000 q^{16} +5.13264 q^{17} -6.81342 q^{18} +0.451857 q^{19} +12.9461 q^{21} +3.68078 q^{22} +4.26527 q^{23} +3.13264 q^{24} -0.132637 q^{26} -11.9461 q^{27} -4.13264 q^{28} -10.2653 q^{29} +5.58449 q^{31} -1.00000 q^{32} +11.5305 q^{33} -5.13264 q^{34} +6.81342 q^{36} -1.00000 q^{37} -0.451857 q^{38} -0.415505 q^{39} +10.3616 q^{41} -12.9461 q^{42} +5.07869 q^{43} -3.68078 q^{44} -4.26527 q^{46} -3.13264 q^{48} +10.0787 q^{49} -16.0787 q^{51} +0.132637 q^{52} +11.6268 q^{53} +11.9461 q^{54} +4.13264 q^{56} -1.41551 q^{57} +10.2653 q^{58} -13.0787 q^{59} -4.39791 q^{61} -5.58449 q^{62} -28.1574 q^{63} +1.00000 q^{64} -11.5305 q^{66} +0.228923 q^{67} +5.13264 q^{68} -13.3616 q^{69} +9.49420 q^{71} -6.81342 q^{72} -3.54814 q^{73} +1.00000 q^{74} +0.451857 q^{76} +15.2113 q^{77} +0.415505 q^{78} -0.903715 q^{79} +16.9824 q^{81} -10.3616 q^{82} +13.9461 q^{83} +12.9461 q^{84} -5.07869 q^{86} +32.1574 q^{87} +3.68078 q^{88} +0.777066 q^{89} -0.548143 q^{91} +4.26527 q^{92} -17.4942 q^{93} +3.13264 q^{96} -0.638440 q^{97} -10.0787 q^{98} -25.0787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 5 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 7 q^{17} - 6 q^{18} - q^{19} + 16 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} + 8 q^{26} - 13 q^{27} - 4 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} - 3 q^{37} + q^{38} - 12 q^{39} + 19 q^{41} - 16 q^{42} - 16 q^{43} - 5 q^{44} + 4 q^{46} - q^{48} - q^{49} - 17 q^{51} - 8 q^{52} + 6 q^{53} + 13 q^{54} + 4 q^{56} - 15 q^{57} + 14 q^{58} - 8 q^{59} + 12 q^{61} - 6 q^{62} - 22 q^{63} + 3 q^{64} - q^{66} - 3 q^{67} + 7 q^{68} - 28 q^{69} + 8 q^{71} - 6 q^{72} - 13 q^{73} + 3 q^{74} - q^{76} + 6 q^{77} + 12 q^{78} + 2 q^{79} + 15 q^{81} - 19 q^{82} + 19 q^{83} + 16 q^{84} + 16 q^{86} + 34 q^{87} + 5 q^{88} + q^{89} - 4 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 14 q^{97} + q^{98} - 44 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −3.13264 −1.80863 −0.904315 0.426867i \(-0.859617\pi\)
−0.904315 + 0.426867i \(0.859617\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 3.13264 1.27889
\(7\) −4.13264 −1.56199 −0.780995 0.624537i \(-0.785288\pi\)
−0.780995 + 0.624537i \(0.785288\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.81342 2.27114
\(10\) 0 0
\(11\) −3.68078 −1.10980 −0.554898 0.831918i \(-0.687243\pi\)
−0.554898 + 0.831918i \(0.687243\pi\)
\(12\) −3.13264 −0.904315
\(13\) 0.132637 0.0367870 0.0183935 0.999831i \(-0.494145\pi\)
0.0183935 + 0.999831i \(0.494145\pi\)
\(14\) 4.13264 1.10449
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.13264 1.24485 0.622424 0.782680i \(-0.286148\pi\)
0.622424 + 0.782680i \(0.286148\pi\)
\(18\) −6.81342 −1.60594
\(19\) 0.451857 0.103663 0.0518316 0.998656i \(-0.483494\pi\)
0.0518316 + 0.998656i \(0.483494\pi\)
\(20\) 0 0
\(21\) 12.9461 2.82506
\(22\) 3.68078 0.784745
\(23\) 4.26527 0.889371 0.444686 0.895687i \(-0.353315\pi\)
0.444686 + 0.895687i \(0.353315\pi\)
\(24\) 3.13264 0.639447
\(25\) 0 0
\(26\) −0.132637 −0.0260124
\(27\) −11.9461 −2.29902
\(28\) −4.13264 −0.780995
\(29\) −10.2653 −1.90621 −0.953107 0.302634i \(-0.902134\pi\)
−0.953107 + 0.302634i \(0.902134\pi\)
\(30\) 0 0
\(31\) 5.58449 1.00300 0.501502 0.865156i \(-0.332781\pi\)
0.501502 + 0.865156i \(0.332781\pi\)
\(32\) −1.00000 −0.176777
\(33\) 11.5305 2.00721
\(34\) −5.13264 −0.880240
\(35\) 0 0
\(36\) 6.81342 1.13557
\(37\) −1.00000 −0.164399
\(38\) −0.451857 −0.0733009
\(39\) −0.415505 −0.0665341
\(40\) 0 0
\(41\) 10.3616 1.61820 0.809102 0.587668i \(-0.199954\pi\)
0.809102 + 0.587668i \(0.199954\pi\)
\(42\) −12.9461 −1.99762
\(43\) 5.07869 0.774493 0.387247 0.921976i \(-0.373426\pi\)
0.387247 + 0.921976i \(0.373426\pi\)
\(44\) −3.68078 −0.554898
\(45\) 0 0
\(46\) −4.26527 −0.628880
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −3.13264 −0.452157
\(49\) 10.0787 1.43981
\(50\) 0 0
\(51\) −16.0787 −2.25147
\(52\) 0.132637 0.0183935
\(53\) 11.6268 1.59707 0.798534 0.601949i \(-0.205609\pi\)
0.798534 + 0.601949i \(0.205609\pi\)
\(54\) 11.9461 1.62565
\(55\) 0 0
\(56\) 4.13264 0.552247
\(57\) −1.41551 −0.187488
\(58\) 10.2653 1.34790
\(59\) −13.0787 −1.70270 −0.851350 0.524597i \(-0.824216\pi\)
−0.851350 + 0.524597i \(0.824216\pi\)
\(60\) 0 0
\(61\) −4.39791 −0.563095 −0.281547 0.959547i \(-0.590848\pi\)
−0.281547 + 0.959547i \(0.590848\pi\)
\(62\) −5.58449 −0.709232
\(63\) −28.1574 −3.54750
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −11.5305 −1.41931
\(67\) 0.228923 0.0279674 0.0139837 0.999902i \(-0.495549\pi\)
0.0139837 + 0.999902i \(0.495549\pi\)
\(68\) 5.13264 0.622424
\(69\) −13.3616 −1.60854
\(70\) 0 0
\(71\) 9.49420 1.12675 0.563377 0.826200i \(-0.309502\pi\)
0.563377 + 0.826200i \(0.309502\pi\)
\(72\) −6.81342 −0.802969
\(73\) −3.54814 −0.415279 −0.207639 0.978205i \(-0.566578\pi\)
−0.207639 + 0.978205i \(0.566578\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 0.451857 0.0518316
\(77\) 15.2113 1.73349
\(78\) 0.415505 0.0470467
\(79\) −0.903715 −0.101676 −0.0508379 0.998707i \(-0.516189\pi\)
−0.0508379 + 0.998707i \(0.516189\pi\)
\(80\) 0 0
\(81\) 16.9824 1.88693
\(82\) −10.3616 −1.14424
\(83\) 13.9461 1.53078 0.765389 0.643568i \(-0.222546\pi\)
0.765389 + 0.643568i \(0.222546\pi\)
\(84\) 12.9461 1.41253
\(85\) 0 0
\(86\) −5.07869 −0.547650
\(87\) 32.1574 3.44763
\(88\) 3.68078 0.392372
\(89\) 0.777066 0.0823688 0.0411844 0.999152i \(-0.486887\pi\)
0.0411844 + 0.999152i \(0.486887\pi\)
\(90\) 0 0
\(91\) −0.548143 −0.0574610
\(92\) 4.26527 0.444686
\(93\) −17.4942 −1.81406
\(94\) 0 0
\(95\) 0 0
\(96\) 3.13264 0.319723
\(97\) −0.638440 −0.0648237 −0.0324119 0.999475i \(-0.510319\pi\)
−0.0324119 + 0.999475i \(0.510319\pi\)
\(98\) −10.0787 −1.01810
\(99\) −25.0787 −2.52050
\(100\) 0 0
\(101\) −5.62684 −0.559891 −0.279946 0.960016i \(-0.590316\pi\)
−0.279946 + 0.960016i \(0.590316\pi\)
\(102\) 16.0787 1.59203
\(103\) −10.0423 −0.989501 −0.494751 0.869035i \(-0.664741\pi\)
−0.494751 + 0.869035i \(0.664741\pi\)
\(104\) −0.132637 −0.0130062
\(105\) 0 0
\(106\) −11.6268 −1.12930
\(107\) −19.6632 −1.90091 −0.950456 0.310859i \(-0.899383\pi\)
−0.950456 + 0.310859i \(0.899383\pi\)
\(108\) −11.9461 −1.14951
\(109\) 13.6268 1.30521 0.652607 0.757697i \(-0.273675\pi\)
0.652607 + 0.757697i \(0.273675\pi\)
\(110\) 0 0
\(111\) 3.13264 0.297337
\(112\) −4.13264 −0.390498
\(113\) −7.94606 −0.747502 −0.373751 0.927529i \(-0.621929\pi\)
−0.373751 + 0.927529i \(0.621929\pi\)
\(114\) 1.41551 0.132574
\(115\) 0 0
\(116\) −10.2653 −0.953107
\(117\) 0.903715 0.0835484
\(118\) 13.0787 1.20399
\(119\) −21.2113 −1.94444
\(120\) 0 0
\(121\) 2.54814 0.231649
\(122\) 4.39791 0.398168
\(123\) −32.4590 −2.92673
\(124\) 5.58449 0.501502
\(125\) 0 0
\(126\) 28.1574 2.50846
\(127\) −17.4942 −1.55236 −0.776180 0.630512i \(-0.782845\pi\)
−0.776180 + 0.630512i \(0.782845\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.9097 −1.40077
\(130\) 0 0
\(131\) 8.81342 0.770032 0.385016 0.922910i \(-0.374196\pi\)
0.385016 + 0.922910i \(0.374196\pi\)
\(132\) 11.5305 1.00361
\(133\) −1.86736 −0.161921
\(134\) −0.228923 −0.0197759
\(135\) 0 0
\(136\) −5.13264 −0.440120
\(137\) −18.8921 −1.61406 −0.807031 0.590509i \(-0.798927\pi\)
−0.807031 + 0.590509i \(0.798927\pi\)
\(138\) 13.3616 1.13741
\(139\) −14.8498 −1.25954 −0.629771 0.776781i \(-0.716851\pi\)
−0.629771 + 0.776781i \(0.716851\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −9.49420 −0.796735
\(143\) −0.488209 −0.0408261
\(144\) 6.81342 0.567785
\(145\) 0 0
\(146\) 3.54814 0.293646
\(147\) −31.5729 −2.60409
\(148\) −1.00000 −0.0821995
\(149\) −2.41551 −0.197886 −0.0989429 0.995093i \(-0.531546\pi\)
−0.0989429 + 0.995093i \(0.531546\pi\)
\(150\) 0 0
\(151\) 10.7231 0.872635 0.436318 0.899793i \(-0.356282\pi\)
0.436318 + 0.899793i \(0.356282\pi\)
\(152\) −0.451857 −0.0366505
\(153\) 34.9708 2.82722
\(154\) −15.2113 −1.22576
\(155\) 0 0
\(156\) −0.415505 −0.0332670
\(157\) 3.31922 0.264903 0.132451 0.991190i \(-0.457715\pi\)
0.132451 + 0.991190i \(0.457715\pi\)
\(158\) 0.903715 0.0718957
\(159\) −36.4227 −2.88850
\(160\) 0 0
\(161\) −17.6268 −1.38919
\(162\) −16.9824 −1.33426
\(163\) −15.5481 −1.21782 −0.608912 0.793238i \(-0.708394\pi\)
−0.608912 + 0.793238i \(0.708394\pi\)
\(164\) 10.3616 0.809102
\(165\) 0 0
\(166\) −13.9461 −1.08242
\(167\) 3.77707 0.292278 0.146139 0.989264i \(-0.453315\pi\)
0.146139 + 0.989264i \(0.453315\pi\)
\(168\) −12.9461 −0.998810
\(169\) −12.9824 −0.998647
\(170\) 0 0
\(171\) 3.07869 0.235434
\(172\) 5.07869 0.387247
\(173\) −20.1574 −1.53254 −0.766269 0.642520i \(-0.777889\pi\)
−0.766269 + 0.642520i \(0.777889\pi\)
\(174\) −32.1574 −2.43785
\(175\) 0 0
\(176\) −3.68078 −0.277449
\(177\) 40.9708 3.07955
\(178\) −0.777066 −0.0582435
\(179\) 20.2537 1.51383 0.756915 0.653513i \(-0.226706\pi\)
0.756915 + 0.653513i \(0.226706\pi\)
\(180\) 0 0
\(181\) −17.1690 −1.27616 −0.638080 0.769970i \(-0.720271\pi\)
−0.638080 + 0.769970i \(0.720271\pi\)
\(182\) 0.548143 0.0406310
\(183\) 13.7771 1.01843
\(184\) −4.26527 −0.314440
\(185\) 0 0
\(186\) 17.4942 1.28274
\(187\) −18.8921 −1.38153
\(188\) 0 0
\(189\) 49.3687 3.59105
\(190\) 0 0
\(191\) −0.192571 −0.0139339 −0.00696696 0.999976i \(-0.502218\pi\)
−0.00696696 + 0.999976i \(0.502218\pi\)
\(192\) −3.13264 −0.226079
\(193\) 14.6692 1.05591 0.527955 0.849272i \(-0.322959\pi\)
0.527955 + 0.849272i \(0.322959\pi\)
\(194\) 0.638440 0.0458373
\(195\) 0 0
\(196\) 10.0787 0.719907
\(197\) 7.84977 0.559273 0.279636 0.960106i \(-0.409786\pi\)
0.279636 + 0.960106i \(0.409786\pi\)
\(198\) 25.0787 1.78227
\(199\) −11.3616 −0.805400 −0.402700 0.915332i \(-0.631928\pi\)
−0.402700 + 0.915332i \(0.631928\pi\)
\(200\) 0 0
\(201\) −0.717132 −0.0505826
\(202\) 5.62684 0.395903
\(203\) 42.4227 2.97749
\(204\) −16.0787 −1.12573
\(205\) 0 0
\(206\) 10.0423 0.699683
\(207\) 29.0611 2.01989
\(208\) 0.132637 0.00919676
\(209\) −1.66319 −0.115045
\(210\) 0 0
\(211\) −12.2289 −0.841874 −0.420937 0.907090i \(-0.638299\pi\)
−0.420937 + 0.907090i \(0.638299\pi\)
\(212\) 11.6268 0.798534
\(213\) −29.7419 −2.03788
\(214\) 19.6632 1.34415
\(215\) 0 0
\(216\) 11.9461 0.812826
\(217\) −23.0787 −1.56668
\(218\) −13.6268 −0.922926
\(219\) 11.1150 0.751085
\(220\) 0 0
\(221\) 0.680780 0.0457942
\(222\) −3.13264 −0.210249
\(223\) −28.1574 −1.88556 −0.942779 0.333418i \(-0.891798\pi\)
−0.942779 + 0.333418i \(0.891798\pi\)
\(224\) 4.13264 0.276123
\(225\) 0 0
\(226\) 7.94606 0.528564
\(227\) −0.282868 −0.0187746 −0.00938729 0.999956i \(-0.502988\pi\)
−0.00938729 + 0.999956i \(0.502988\pi\)
\(228\) −1.41551 −0.0937441
\(229\) −8.45785 −0.558910 −0.279455 0.960159i \(-0.590154\pi\)
−0.279455 + 0.960159i \(0.590154\pi\)
\(230\) 0 0
\(231\) −47.6516 −3.13524
\(232\) 10.2653 0.673948
\(233\) 14.2477 0.933397 0.466698 0.884417i \(-0.345443\pi\)
0.466698 + 0.884417i \(0.345443\pi\)
\(234\) −0.903715 −0.0590777
\(235\) 0 0
\(236\) −13.0787 −0.851350
\(237\) 2.83101 0.183894
\(238\) 21.2113 1.37493
\(239\) 21.8921 1.41608 0.708041 0.706171i \(-0.249579\pi\)
0.708041 + 0.706171i \(0.249579\pi\)
\(240\) 0 0
\(241\) −9.13264 −0.588285 −0.294142 0.955762i \(-0.595034\pi\)
−0.294142 + 0.955762i \(0.595034\pi\)
\(242\) −2.54814 −0.163801
\(243\) −17.3616 −1.11374
\(244\) −4.39791 −0.281547
\(245\) 0 0
\(246\) 32.4590 2.06951
\(247\) 0.0599332 0.00381346
\(248\) −5.58449 −0.354616
\(249\) −43.6879 −2.76861
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) −28.1574 −1.77375
\(253\) −15.6995 −0.987022
\(254\) 17.4942 1.09768
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.90371 0.555398 0.277699 0.960668i \(-0.410428\pi\)
0.277699 + 0.960668i \(0.410428\pi\)
\(258\) 15.9097 0.990495
\(259\) 4.13264 0.256790
\(260\) 0 0
\(261\) −69.9416 −4.32928
\(262\) −8.81342 −0.544495
\(263\) 14.9037 0.919002 0.459501 0.888177i \(-0.348028\pi\)
0.459501 + 0.888177i \(0.348028\pi\)
\(264\) −11.5305 −0.709656
\(265\) 0 0
\(266\) 1.86736 0.114495
\(267\) −2.43426 −0.148975
\(268\) 0.228923 0.0139837
\(269\) −1.55413 −0.0947570 −0.0473785 0.998877i \(-0.515087\pi\)
−0.0473785 + 0.998877i \(0.515087\pi\)
\(270\) 0 0
\(271\) 5.68677 0.345447 0.172723 0.984970i \(-0.444743\pi\)
0.172723 + 0.984970i \(0.444743\pi\)
\(272\) 5.13264 0.311212
\(273\) 1.71713 0.103926
\(274\) 18.8921 1.14131
\(275\) 0 0
\(276\) −13.3616 −0.804271
\(277\) 12.9037 0.775309 0.387655 0.921805i \(-0.373285\pi\)
0.387655 + 0.921805i \(0.373285\pi\)
\(278\) 14.8498 0.890630
\(279\) 38.0495 2.27796
\(280\) 0 0
\(281\) −12.2829 −0.732734 −0.366367 0.930470i \(-0.619399\pi\)
−0.366367 + 0.930470i \(0.619399\pi\)
\(282\) 0 0
\(283\) 26.8745 1.59752 0.798762 0.601647i \(-0.205489\pi\)
0.798762 + 0.601647i \(0.205489\pi\)
\(284\) 9.49420 0.563377
\(285\) 0 0
\(286\) 0.488209 0.0288684
\(287\) −42.8206 −2.52762
\(288\) −6.81342 −0.401484
\(289\) 9.34397 0.549645
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −3.54814 −0.207639
\(293\) 16.7535 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(294\) 31.5729 1.84137
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 43.9708 2.55144
\(298\) 2.41551 0.139926
\(299\) 0.565735 0.0327173
\(300\) 0 0
\(301\) −20.9884 −1.20975
\(302\) −10.7231 −0.617046
\(303\) 17.6268 1.01264
\(304\) 0.451857 0.0259158
\(305\) 0 0
\(306\) −34.9708 −1.99915
\(307\) 8.47661 0.483785 0.241893 0.970303i \(-0.422232\pi\)
0.241893 + 0.970303i \(0.422232\pi\)
\(308\) 15.2113 0.866746
\(309\) 31.4590 1.78964
\(310\) 0 0
\(311\) 32.4650 1.84092 0.920461 0.390835i \(-0.127814\pi\)
0.920461 + 0.390835i \(0.127814\pi\)
\(312\) 0.415505 0.0235233
\(313\) −28.8134 −1.62863 −0.814315 0.580423i \(-0.802887\pi\)
−0.814315 + 0.580423i \(0.802887\pi\)
\(314\) −3.31922 −0.187314
\(315\) 0 0
\(316\) −0.903715 −0.0508379
\(317\) 6.87335 0.386046 0.193023 0.981194i \(-0.438171\pi\)
0.193023 + 0.981194i \(0.438171\pi\)
\(318\) 36.4227 2.04248
\(319\) 37.7842 2.11551
\(320\) 0 0
\(321\) 61.5976 3.43804
\(322\) 17.6268 0.982305
\(323\) 2.31922 0.129045
\(324\) 16.9824 0.943467
\(325\) 0 0
\(326\) 15.5481 0.861132
\(327\) −42.6879 −2.36065
\(328\) −10.3616 −0.572121
\(329\) 0 0
\(330\) 0 0
\(331\) −1.65603 −0.0910238 −0.0455119 0.998964i \(-0.514492\pi\)
−0.0455119 + 0.998964i \(0.514492\pi\)
\(332\) 13.9461 0.765389
\(333\) −6.81342 −0.373373
\(334\) −3.77707 −0.206672
\(335\) 0 0
\(336\) 12.9461 0.706265
\(337\) −10.2537 −0.558553 −0.279277 0.960211i \(-0.590095\pi\)
−0.279277 + 0.960211i \(0.590095\pi\)
\(338\) 12.9824 0.706150
\(339\) 24.8921 1.35195
\(340\) 0 0
\(341\) −20.5553 −1.11313
\(342\) −3.07869 −0.166477
\(343\) −12.7231 −0.686984
\(344\) −5.07869 −0.273825
\(345\) 0 0
\(346\) 20.1574 1.08367
\(347\) −25.8805 −1.38934 −0.694669 0.719329i \(-0.744449\pi\)
−0.694669 + 0.719329i \(0.744449\pi\)
\(348\) 32.1574 1.72382
\(349\) 12.7535 0.682678 0.341339 0.939940i \(-0.389120\pi\)
0.341339 + 0.939940i \(0.389120\pi\)
\(350\) 0 0
\(351\) −1.58449 −0.0845741
\(352\) 3.68078 0.196186
\(353\) −8.72312 −0.464285 −0.232142 0.972682i \(-0.574574\pi\)
−0.232142 + 0.972682i \(0.574574\pi\)
\(354\) −40.9708 −2.17757
\(355\) 0 0
\(356\) 0.777066 0.0411844
\(357\) 66.4474 3.51677
\(358\) −20.2537 −1.07044
\(359\) 14.9037 0.786588 0.393294 0.919413i \(-0.371335\pi\)
0.393294 + 0.919413i \(0.371335\pi\)
\(360\) 0 0
\(361\) −18.7958 −0.989254
\(362\) 17.1690 0.902382
\(363\) −7.98241 −0.418968
\(364\) −0.548143 −0.0287305
\(365\) 0 0
\(366\) −13.7771 −0.720139
\(367\) 14.8558 0.775464 0.387732 0.921772i \(-0.373259\pi\)
0.387732 + 0.921772i \(0.373259\pi\)
\(368\) 4.26527 0.222343
\(369\) 70.5976 3.67517
\(370\) 0 0
\(371\) −48.0495 −2.49461
\(372\) −17.4942 −0.907032
\(373\) 1.77707 0.0920130 0.0460065 0.998941i \(-0.485351\pi\)
0.0460065 + 0.998941i \(0.485351\pi\)
\(374\) 18.8921 0.976888
\(375\) 0 0
\(376\) 0 0
\(377\) −1.36156 −0.0701239
\(378\) −49.3687 −2.53925
\(379\) −26.3863 −1.35537 −0.677687 0.735351i \(-0.737017\pi\)
−0.677687 + 0.735351i \(0.737017\pi\)
\(380\) 0 0
\(381\) 54.8030 2.80764
\(382\) 0.192571 0.00985277
\(383\) −4.57289 −0.233664 −0.116832 0.993152i \(-0.537274\pi\)
−0.116832 + 0.993152i \(0.537274\pi\)
\(384\) 3.13264 0.159862
\(385\) 0 0
\(386\) −14.6692 −0.746641
\(387\) 34.6033 1.75898
\(388\) −0.638440 −0.0324119
\(389\) 5.03635 0.255353 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(390\) 0 0
\(391\) 21.8921 1.10713
\(392\) −10.0787 −0.509051
\(393\) −27.6092 −1.39270
\(394\) −7.84977 −0.395466
\(395\) 0 0
\(396\) −25.0787 −1.26025
\(397\) −34.4650 −1.72975 −0.864874 0.501988i \(-0.832602\pi\)
−0.864874 + 0.501988i \(0.832602\pi\)
\(398\) 11.3616 0.569504
\(399\) 5.84977 0.292855
\(400\) 0 0
\(401\) −4.47661 −0.223551 −0.111775 0.993733i \(-0.535654\pi\)
−0.111775 + 0.993733i \(0.535654\pi\)
\(402\) 0.717132 0.0357673
\(403\) 0.740713 0.0368976
\(404\) −5.62684 −0.279946
\(405\) 0 0
\(406\) −42.4227 −2.10540
\(407\) 3.68078 0.182449
\(408\) 16.0787 0.796014
\(409\) −34.7243 −1.71701 −0.858503 0.512809i \(-0.828605\pi\)
−0.858503 + 0.512809i \(0.828605\pi\)
\(410\) 0 0
\(411\) 59.1821 2.91924
\(412\) −10.0423 −0.494751
\(413\) 54.0495 2.65960
\(414\) −29.0611 −1.42828
\(415\) 0 0
\(416\) −0.132637 −0.00650309
\(417\) 46.5189 2.27804
\(418\) 1.66319 0.0813492
\(419\) −7.15023 −0.349312 −0.174656 0.984630i \(-0.555881\pi\)
−0.174656 + 0.984630i \(0.555881\pi\)
\(420\) 0 0
\(421\) 26.6033 1.29656 0.648282 0.761401i \(-0.275488\pi\)
0.648282 + 0.761401i \(0.275488\pi\)
\(422\) 12.2289 0.595295
\(423\) 0 0
\(424\) −11.6268 −0.564649
\(425\) 0 0
\(426\) 29.7419 1.44100
\(427\) 18.1750 0.879549
\(428\) −19.6632 −0.950456
\(429\) 1.52938 0.0738393
\(430\) 0 0
\(431\) −1.16899 −0.0563082 −0.0281541 0.999604i \(-0.508963\pi\)
−0.0281541 + 0.999604i \(0.508963\pi\)
\(432\) −11.9461 −0.574755
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 23.0787 1.10781
\(435\) 0 0
\(436\) 13.6268 0.652607
\(437\) 1.92730 0.0921951
\(438\) −11.1150 −0.531097
\(439\) −32.3148 −1.54230 −0.771150 0.636654i \(-0.780318\pi\)
−0.771150 + 0.636654i \(0.780318\pi\)
\(440\) 0 0
\(441\) 68.6703 3.27002
\(442\) −0.680780 −0.0323814
\(443\) 17.4155 0.827436 0.413718 0.910405i \(-0.364230\pi\)
0.413718 + 0.910405i \(0.364230\pi\)
\(444\) 3.13264 0.148668
\(445\) 0 0
\(446\) 28.1574 1.33329
\(447\) 7.56690 0.357902
\(448\) −4.13264 −0.195249
\(449\) −7.38032 −0.348299 −0.174149 0.984719i \(-0.555718\pi\)
−0.174149 + 0.984719i \(0.555718\pi\)
\(450\) 0 0
\(451\) −38.1386 −1.79588
\(452\) −7.94606 −0.373751
\(453\) −33.5916 −1.57827
\(454\) 0.282868 0.0132756
\(455\) 0 0
\(456\) 1.41551 0.0662871
\(457\) −22.7419 −1.06382 −0.531910 0.846801i \(-0.678526\pi\)
−0.531910 + 0.846801i \(0.678526\pi\)
\(458\) 8.45785 0.395209
\(459\) −61.3148 −2.86193
\(460\) 0 0
\(461\) 39.7115 1.84955 0.924775 0.380515i \(-0.124253\pi\)
0.924775 + 0.380515i \(0.124253\pi\)
\(462\) 47.6516 2.21695
\(463\) 30.6456 1.42422 0.712111 0.702067i \(-0.247739\pi\)
0.712111 + 0.702067i \(0.247739\pi\)
\(464\) −10.2653 −0.476553
\(465\) 0 0
\(466\) −14.2477 −0.660011
\(467\) −24.7958 −1.14741 −0.573707 0.819061i \(-0.694495\pi\)
−0.573707 + 0.819061i \(0.694495\pi\)
\(468\) 0.903715 0.0417742
\(469\) −0.946055 −0.0436848
\(470\) 0 0
\(471\) −10.3979 −0.479111
\(472\) 13.0787 0.601996
\(473\) −18.6936 −0.859530
\(474\) −2.83101 −0.130033
\(475\) 0 0
\(476\) −21.2113 −0.972220
\(477\) 79.2185 3.62717
\(478\) −21.8921 −1.00132
\(479\) −15.5845 −0.712074 −0.356037 0.934472i \(-0.615872\pi\)
−0.356037 + 0.934472i \(0.615872\pi\)
\(480\) 0 0
\(481\) −0.132637 −0.00604775
\(482\) 9.13264 0.415980
\(483\) 55.2185 2.51253
\(484\) 2.54814 0.115825
\(485\) 0 0
\(486\) 17.3616 0.787536
\(487\) −3.65720 −0.165724 −0.0828618 0.996561i \(-0.526406\pi\)
−0.0828618 + 0.996561i \(0.526406\pi\)
\(488\) 4.39791 0.199084
\(489\) 48.7067 2.20259
\(490\) 0 0
\(491\) 33.3264 1.50400 0.751999 0.659164i \(-0.229090\pi\)
0.751999 + 0.659164i \(0.229090\pi\)
\(492\) −32.4590 −1.46337
\(493\) −52.6879 −2.37295
\(494\) −0.0599332 −0.00269652
\(495\) 0 0
\(496\) 5.58449 0.250751
\(497\) −39.2361 −1.75998
\(498\) 43.6879 1.95770
\(499\) 7.53654 0.337382 0.168691 0.985669i \(-0.446046\pi\)
0.168691 + 0.985669i \(0.446046\pi\)
\(500\) 0 0
\(501\) −11.8322 −0.528623
\(502\) 9.00000 0.401690
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 28.1574 1.25423
\(505\) 0 0
\(506\) 15.6995 0.697930
\(507\) 40.6692 1.80618
\(508\) −17.4942 −0.776180
\(509\) 24.1150 1.06888 0.534440 0.845206i \(-0.320522\pi\)
0.534440 + 0.845206i \(0.320522\pi\)
\(510\) 0 0
\(511\) 14.6632 0.648661
\(512\) −1.00000 −0.0441942
\(513\) −5.39791 −0.238324
\(514\) −8.90371 −0.392726
\(515\) 0 0
\(516\) −15.9097 −0.700386
\(517\) 0 0
\(518\) −4.13264 −0.181578
\(519\) 63.1458 2.77179
\(520\) 0 0
\(521\) −11.8134 −0.517555 −0.258778 0.965937i \(-0.583320\pi\)
−0.258778 + 0.965937i \(0.583320\pi\)
\(522\) 69.9416 3.06126
\(523\) 33.9532 1.48467 0.742335 0.670029i \(-0.233718\pi\)
0.742335 + 0.670029i \(0.233718\pi\)
\(524\) 8.81342 0.385016
\(525\) 0 0
\(526\) −14.9037 −0.649833
\(527\) 28.6632 1.24859
\(528\) 11.5305 0.501803
\(529\) −4.80743 −0.209019
\(530\) 0 0
\(531\) −89.1106 −3.86707
\(532\) −1.86736 −0.0809604
\(533\) 1.37433 0.0595289
\(534\) 2.43426 0.105341
\(535\) 0 0
\(536\) −0.228923 −0.00988796
\(537\) −63.4474 −2.73796
\(538\) 1.55413 0.0670033
\(539\) −37.0975 −1.59790
\(540\) 0 0
\(541\) −33.4590 −1.43852 −0.719258 0.694743i \(-0.755518\pi\)
−0.719258 + 0.694743i \(0.755518\pi\)
\(542\) −5.68677 −0.244268
\(543\) 53.7842 2.30810
\(544\) −5.13264 −0.220060
\(545\) 0 0
\(546\) −1.71713 −0.0734865
\(547\) 42.1514 1.80226 0.901132 0.433545i \(-0.142738\pi\)
0.901132 + 0.433545i \(0.142738\pi\)
\(548\) −18.8921 −0.807031
\(549\) −29.9648 −1.27887
\(550\) 0 0
\(551\) −4.63844 −0.197604
\(552\) 13.3616 0.568706
\(553\) 3.73473 0.158817
\(554\) −12.9037 −0.548226
\(555\) 0 0
\(556\) −14.8498 −0.629771
\(557\) −12.3732 −0.524268 −0.262134 0.965032i \(-0.584426\pi\)
−0.262134 + 0.965032i \(0.584426\pi\)
\(558\) −38.0495 −1.61076
\(559\) 0.673625 0.0284913
\(560\) 0 0
\(561\) 59.1821 2.49867
\(562\) 12.2829 0.518122
\(563\) −40.7782 −1.71860 −0.859299 0.511474i \(-0.829100\pi\)
−0.859299 + 0.511474i \(0.829100\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.8745 −1.12962
\(567\) −70.1821 −2.94737
\(568\) −9.49420 −0.398368
\(569\) −20.7419 −0.869545 −0.434772 0.900540i \(-0.643171\pi\)
−0.434772 + 0.900540i \(0.643171\pi\)
\(570\) 0 0
\(571\) −37.1458 −1.55450 −0.777251 0.629190i \(-0.783387\pi\)
−0.777251 + 0.629190i \(0.783387\pi\)
\(572\) −0.488209 −0.0204131
\(573\) 0.603254 0.0252013
\(574\) 42.8206 1.78730
\(575\) 0 0
\(576\) 6.81342 0.283892
\(577\) 23.0975 0.961560 0.480780 0.876841i \(-0.340354\pi\)
0.480780 + 0.876841i \(0.340354\pi\)
\(578\) −9.34397 −0.388658
\(579\) −45.9532 −1.90975
\(580\) 0 0
\(581\) −57.6340 −2.39106
\(582\) −2.00000 −0.0829027
\(583\) −42.7958 −1.77242
\(584\) 3.54814 0.146823
\(585\) 0 0
\(586\) −16.7535 −0.692080
\(587\) −11.0176 −0.454745 −0.227372 0.973808i \(-0.573013\pi\)
−0.227372 + 0.973808i \(0.573013\pi\)
\(588\) −31.5729 −1.30204
\(589\) 2.52339 0.103975
\(590\) 0 0
\(591\) −24.5905 −1.01152
\(592\) −1.00000 −0.0410997
\(593\) 0.469450 0.0192780 0.00963900 0.999954i \(-0.496932\pi\)
0.00963900 + 0.999954i \(0.496932\pi\)
\(594\) −43.9708 −1.80414
\(595\) 0 0
\(596\) −2.41551 −0.0989429
\(597\) 35.5916 1.45667
\(598\) −0.565735 −0.0231346
\(599\) −24.7711 −1.01212 −0.506059 0.862499i \(-0.668898\pi\)
−0.506059 + 0.862499i \(0.668898\pi\)
\(600\) 0 0
\(601\) −23.4051 −0.954713 −0.477356 0.878710i \(-0.658405\pi\)
−0.477356 + 0.878710i \(0.658405\pi\)
\(602\) 20.9884 0.855423
\(603\) 1.55975 0.0635178
\(604\) 10.7231 0.436318
\(605\) 0 0
\(606\) −17.6268 −0.716041
\(607\) −0.342801 −0.0139139 −0.00695693 0.999976i \(-0.502214\pi\)
−0.00695693 + 0.999976i \(0.502214\pi\)
\(608\) −0.451857 −0.0183252
\(609\) −132.895 −5.38517
\(610\) 0 0
\(611\) 0 0
\(612\) 34.9708 1.41361
\(613\) −35.8921 −1.44967 −0.724834 0.688923i \(-0.758084\pi\)
−0.724834 + 0.688923i \(0.758084\pi\)
\(614\) −8.47661 −0.342088
\(615\) 0 0
\(616\) −15.2113 −0.612882
\(617\) −7.64443 −0.307753 −0.153877 0.988090i \(-0.549176\pi\)
−0.153877 + 0.988090i \(0.549176\pi\)
\(618\) −31.4590 −1.26547
\(619\) −10.4227 −0.418922 −0.209461 0.977817i \(-0.567171\pi\)
−0.209461 + 0.977817i \(0.567171\pi\)
\(620\) 0 0
\(621\) −50.9532 −2.04468
\(622\) −32.4650 −1.30173
\(623\) −3.21133 −0.128659
\(624\) −0.415505 −0.0166335
\(625\) 0 0
\(626\) 28.8134 1.15162
\(627\) 5.21016 0.208074
\(628\) 3.31922 0.132451
\(629\) −5.13264 −0.204652
\(630\) 0 0
\(631\) 26.4227 1.05187 0.525935 0.850525i \(-0.323716\pi\)
0.525935 + 0.850525i \(0.323716\pi\)
\(632\) 0.903715 0.0359478
\(633\) 38.3088 1.52264
\(634\) −6.87335 −0.272976
\(635\) 0 0
\(636\) −36.4227 −1.44425
\(637\) 1.33681 0.0529664
\(638\) −37.7842 −1.49589
\(639\) 64.6879 2.55902
\(640\) 0 0
\(641\) 11.8074 0.466365 0.233183 0.972433i \(-0.425086\pi\)
0.233183 + 0.972433i \(0.425086\pi\)
\(642\) −61.5976 −2.43106
\(643\) −18.3556 −0.723873 −0.361937 0.932203i \(-0.617884\pi\)
−0.361937 + 0.932203i \(0.617884\pi\)
\(644\) −17.6268 −0.694595
\(645\) 0 0
\(646\) −2.31922 −0.0912485
\(647\) −33.7771 −1.32791 −0.663957 0.747771i \(-0.731124\pi\)
−0.663957 + 0.747771i \(0.731124\pi\)
\(648\) −16.9824 −0.667132
\(649\) 48.1398 1.88965
\(650\) 0 0
\(651\) 72.2972 2.83355
\(652\) −15.5481 −0.608912
\(653\) −18.0247 −0.705363 −0.352681 0.935743i \(-0.614730\pi\)
−0.352681 + 0.935743i \(0.614730\pi\)
\(654\) 42.6879 1.66923
\(655\) 0 0
\(656\) 10.3616 0.404551
\(657\) −24.1750 −0.943156
\(658\) 0 0
\(659\) 16.5669 0.645355 0.322677 0.946509i \(-0.395417\pi\)
0.322677 + 0.946509i \(0.395417\pi\)
\(660\) 0 0
\(661\) −23.7842 −0.925099 −0.462549 0.886593i \(-0.653065\pi\)
−0.462549 + 0.886593i \(0.653065\pi\)
\(662\) 1.65603 0.0643635
\(663\) −2.13264 −0.0828248
\(664\) −13.9461 −0.541212
\(665\) 0 0
\(666\) 6.81342 0.264015
\(667\) −43.7842 −1.69533
\(668\) 3.77707 0.146139
\(669\) 88.2069 3.41028
\(670\) 0 0
\(671\) 16.1877 0.624921
\(672\) −12.9461 −0.499405
\(673\) 12.0551 0.464690 0.232345 0.972633i \(-0.425360\pi\)
0.232345 + 0.972633i \(0.425360\pi\)
\(674\) 10.2537 0.394957
\(675\) 0 0
\(676\) −12.9824 −0.499323
\(677\) 28.8805 1.10997 0.554984 0.831861i \(-0.312724\pi\)
0.554984 + 0.831861i \(0.312724\pi\)
\(678\) −24.8921 −0.955976
\(679\) 2.63844 0.101254
\(680\) 0 0
\(681\) 0.886122 0.0339563
\(682\) 20.5553 0.787103
\(683\) −17.6328 −0.674701 −0.337351 0.941379i \(-0.609531\pi\)
−0.337351 + 0.941379i \(0.609531\pi\)
\(684\) 3.07869 0.117717
\(685\) 0 0
\(686\) 12.7231 0.485771
\(687\) 26.4954 1.01086
\(688\) 5.07869 0.193623
\(689\) 1.54215 0.0587514
\(690\) 0 0
\(691\) −20.6572 −0.785837 −0.392918 0.919573i \(-0.628535\pi\)
−0.392918 + 0.919573i \(0.628535\pi\)
\(692\) −20.1574 −0.766269
\(693\) 103.641 3.93700
\(694\) 25.8805 0.982411
\(695\) 0 0
\(696\) −32.1574 −1.21892
\(697\) 53.1821 2.01442
\(698\) −12.7535 −0.482727
\(699\) −44.6328 −1.68817
\(700\) 0 0
\(701\) 49.2057 1.85847 0.929237 0.369484i \(-0.120466\pi\)
0.929237 + 0.369484i \(0.120466\pi\)
\(702\) 1.58449 0.0598029
\(703\) −0.451857 −0.0170421
\(704\) −3.68078 −0.138725
\(705\) 0 0
\(706\) 8.72312 0.328299
\(707\) 23.2537 0.874544
\(708\) 40.9708 1.53978
\(709\) 3.73473 0.140261 0.0701303 0.997538i \(-0.477659\pi\)
0.0701303 + 0.997538i \(0.477659\pi\)
\(710\) 0 0
\(711\) −6.15739 −0.230920
\(712\) −0.777066 −0.0291218
\(713\) 23.8194 0.892044
\(714\) −66.4474 −2.48673
\(715\) 0 0
\(716\) 20.2537 0.756915
\(717\) −68.5800 −2.56117
\(718\) −14.9037 −0.556202
\(719\) −47.0858 −1.75601 −0.878003 0.478655i \(-0.841124\pi\)
−0.878003 + 0.478655i \(0.841124\pi\)
\(720\) 0 0
\(721\) 41.5014 1.54559
\(722\) 18.7958 0.699508
\(723\) 28.6092 1.06399
\(724\) −17.1690 −0.638080
\(725\) 0 0
\(726\) 7.98241 0.296255
\(727\) −27.3616 −1.01478 −0.507392 0.861715i \(-0.669390\pi\)
−0.507392 + 0.861715i \(0.669390\pi\)
\(728\) 0.548143 0.0203155
\(729\) 3.44025 0.127417
\(730\) 0 0
\(731\) 26.0671 0.964126
\(732\) 13.7771 0.509215
\(733\) −47.5070 −1.75471 −0.877355 0.479842i \(-0.840694\pi\)
−0.877355 + 0.479842i \(0.840694\pi\)
\(734\) −14.8558 −0.548336
\(735\) 0 0
\(736\) −4.26527 −0.157220
\(737\) −0.842615 −0.0310381
\(738\) −70.5976 −2.59873
\(739\) 19.6149 0.721544 0.360772 0.932654i \(-0.382513\pi\)
0.360772 + 0.932654i \(0.382513\pi\)
\(740\) 0 0
\(741\) −0.187749 −0.00689713
\(742\) 48.0495 1.76395
\(743\) −22.3148 −0.818650 −0.409325 0.912389i \(-0.634236\pi\)
−0.409325 + 0.912389i \(0.634236\pi\)
\(744\) 17.4942 0.641368
\(745\) 0 0
\(746\) −1.77707 −0.0650630
\(747\) 95.0203 3.47661
\(748\) −18.8921 −0.690764
\(749\) 81.2608 2.96921
\(750\) 0 0
\(751\) 20.7958 0.758850 0.379425 0.925222i \(-0.376122\pi\)
0.379425 + 0.925222i \(0.376122\pi\)
\(752\) 0 0
\(753\) 28.1937 1.02744
\(754\) 1.36156 0.0495851
\(755\) 0 0
\(756\) 49.3687 1.79552
\(757\) 0.530550 0.0192832 0.00964158 0.999954i \(-0.496931\pi\)
0.00964158 + 0.999954i \(0.496931\pi\)
\(758\) 26.3863 0.958394
\(759\) 49.1810 1.78516
\(760\) 0 0
\(761\) 4.63245 0.167926 0.0839631 0.996469i \(-0.473242\pi\)
0.0839631 + 0.996469i \(0.473242\pi\)
\(762\) −54.8030 −1.98530
\(763\) −56.3148 −2.03873
\(764\) −0.192571 −0.00696696
\(765\) 0 0
\(766\) 4.57289 0.165225
\(767\) −1.73473 −0.0626373
\(768\) −3.13264 −0.113039
\(769\) 7.78867 0.280867 0.140433 0.990090i \(-0.455150\pi\)
0.140433 + 0.990090i \(0.455150\pi\)
\(770\) 0 0
\(771\) −27.8921 −1.00451
\(772\) 14.6692 0.527955
\(773\) 8.04234 0.289263 0.144631 0.989486i \(-0.453800\pi\)
0.144631 + 0.989486i \(0.453800\pi\)
\(774\) −34.6033 −1.24379
\(775\) 0 0
\(776\) 0.638440 0.0229186
\(777\) −12.9461 −0.464437
\(778\) −5.03635 −0.180562
\(779\) 4.68195 0.167748
\(780\) 0 0
\(781\) −34.9461 −1.25047
\(782\) −21.8921 −0.782860
\(783\) 122.630 4.38242
\(784\) 10.0787 0.359953
\(785\) 0 0
\(786\) 27.6092 0.984789
\(787\) −25.6268 −0.913498 −0.456749 0.889596i \(-0.650986\pi\)
−0.456749 + 0.889596i \(0.650986\pi\)
\(788\) 7.84977 0.279636
\(789\) −46.6879 −1.66213
\(790\) 0 0
\(791\) 32.8382 1.16759
\(792\) 25.0787 0.891133
\(793\) −0.583328 −0.0207146
\(794\) 34.4650 1.22312
\(795\) 0 0
\(796\) −11.3616 −0.402700
\(797\) −11.4095 −0.404146 −0.202073 0.979370i \(-0.564768\pi\)
−0.202073 + 0.979370i \(0.564768\pi\)
\(798\) −5.84977 −0.207080
\(799\) 0 0
\(800\) 0 0
\(801\) 5.29447 0.187071
\(802\) 4.47661 0.158074
\(803\) 13.0599 0.460875
\(804\) −0.717132 −0.0252913
\(805\) 0 0
\(806\) −0.740713 −0.0260905
\(807\) 4.86853 0.171380
\(808\) 5.62684 0.197951
\(809\) 21.5189 0.756566 0.378283 0.925690i \(-0.376515\pi\)
0.378283 + 0.925690i \(0.376515\pi\)
\(810\) 0 0
\(811\) −44.0847 −1.54802 −0.774011 0.633172i \(-0.781753\pi\)
−0.774011 + 0.633172i \(0.781753\pi\)
\(812\) 42.4227 1.48874
\(813\) −17.8146 −0.624785
\(814\) −3.68078 −0.129011
\(815\) 0 0
\(816\) −16.0787 −0.562867
\(817\) 2.29484 0.0802864
\(818\) 34.7243 1.21411
\(819\) −3.73473 −0.130502
\(820\) 0 0
\(821\) −9.95766 −0.347525 −0.173762 0.984788i \(-0.555592\pi\)
−0.173762 + 0.984788i \(0.555592\pi\)
\(822\) −59.1821 −2.06421
\(823\) −0.663187 −0.0231173 −0.0115586 0.999933i \(-0.503679\pi\)
−0.0115586 + 0.999933i \(0.503679\pi\)
\(824\) 10.0423 0.349842
\(825\) 0 0
\(826\) −54.0495 −1.88062
\(827\) 41.0671 1.42804 0.714021 0.700124i \(-0.246872\pi\)
0.714021 + 0.700124i \(0.246872\pi\)
\(828\) 29.0611 1.00994
\(829\) 6.48259 0.225150 0.112575 0.993643i \(-0.464090\pi\)
0.112575 + 0.993643i \(0.464090\pi\)
\(830\) 0 0
\(831\) −40.4227 −1.40225
\(832\) 0.132637 0.00459838
\(833\) 51.7303 1.79235
\(834\) −46.5189 −1.61082
\(835\) 0 0
\(836\) −1.66319 −0.0575225
\(837\) −66.7127 −2.30593
\(838\) 7.15023 0.247001
\(839\) 0.771077 0.0266205 0.0133103 0.999911i \(-0.495763\pi\)
0.0133103 + 0.999911i \(0.495763\pi\)
\(840\) 0 0
\(841\) 76.3759 2.63365
\(842\) −26.6033 −0.916809
\(843\) 38.4778 1.32524
\(844\) −12.2289 −0.420937
\(845\) 0 0
\(846\) 0 0
\(847\) −10.5305 −0.361834
\(848\) 11.6268 0.399267
\(849\) −84.1881 −2.88933
\(850\) 0 0
\(851\) −4.26527 −0.146212
\(852\) −29.7419 −1.01894
\(853\) −30.0975 −1.03052 −0.515259 0.857035i \(-0.672304\pi\)
−0.515259 + 0.857035i \(0.672304\pi\)
\(854\) −18.1750 −0.621935
\(855\) 0 0
\(856\) 19.6632 0.672074
\(857\) −8.22892 −0.281095 −0.140547 0.990074i \(-0.544886\pi\)
−0.140547 + 0.990074i \(0.544886\pi\)
\(858\) −1.52938 −0.0522123
\(859\) 14.4286 0.492299 0.246150 0.969232i \(-0.420835\pi\)
0.246150 + 0.969232i \(0.420835\pi\)
\(860\) 0 0
\(861\) 134.141 4.57152
\(862\) 1.16899 0.0398159
\(863\) −33.9049 −1.15414 −0.577068 0.816696i \(-0.695803\pi\)
−0.577068 + 0.816696i \(0.695803\pi\)
\(864\) 11.9461 0.406413
\(865\) 0 0
\(866\) 5.00000 0.169907
\(867\) −29.2713 −0.994104
\(868\) −23.0787 −0.783342
\(869\) 3.32637 0.112840
\(870\) 0 0
\(871\) 0.0303638 0.00102884
\(872\) −13.6268 −0.461463
\(873\) −4.34996 −0.147224
\(874\) −1.92730 −0.0651918
\(875\) 0 0
\(876\) 11.1150 0.375543
\(877\) −1.36872 −0.0462182 −0.0231091 0.999733i \(-0.507357\pi\)
−0.0231091 + 0.999733i \(0.507357\pi\)
\(878\) 32.3148 1.09057
\(879\) −52.4826 −1.77019
\(880\) 0 0
\(881\) 47.5741 1.60281 0.801405 0.598122i \(-0.204086\pi\)
0.801405 + 0.598122i \(0.204086\pi\)
\(882\) −68.6703 −2.31225
\(883\) −9.88051 −0.332506 −0.166253 0.986083i \(-0.553167\pi\)
−0.166253 + 0.986083i \(0.553167\pi\)
\(884\) 0.680780 0.0228971
\(885\) 0 0
\(886\) −17.4155 −0.585085
\(887\) −21.8801 −0.734663 −0.367331 0.930090i \(-0.619729\pi\)
−0.367331 + 0.930090i \(0.619729\pi\)
\(888\) −3.13264 −0.105124
\(889\) 72.2972 2.42477
\(890\) 0 0
\(891\) −62.5085 −2.09411
\(892\) −28.1574 −0.942779
\(893\) 0 0
\(894\) −7.56690 −0.253075
\(895\) 0 0
\(896\) 4.13264 0.138062
\(897\) −1.77224 −0.0591735
\(898\) 7.38032 0.246284
\(899\) −57.3264 −1.91194
\(900\) 0 0
\(901\) 59.6763 1.98811
\(902\) 38.1386 1.26988
\(903\) 65.7490 2.18799
\(904\) 7.94606 0.264282
\(905\) 0 0
\(906\) 33.5916 1.11601
\(907\) −8.46346 −0.281025 −0.140512 0.990079i \(-0.544875\pi\)
−0.140512 + 0.990079i \(0.544875\pi\)
\(908\) −0.282868 −0.00938729
\(909\) −38.3380 −1.27159
\(910\) 0 0
\(911\) −34.1997 −1.13309 −0.566544 0.824032i \(-0.691720\pi\)
−0.566544 + 0.824032i \(0.691720\pi\)
\(912\) −1.41551 −0.0468721
\(913\) −51.3324 −1.69885
\(914\) 22.7419 0.752235
\(915\) 0 0
\(916\) −8.45785 −0.279455
\(917\) −36.4227 −1.20278
\(918\) 61.3148 2.02369
\(919\) −0.723121 −0.0238536 −0.0119268 0.999929i \(-0.503797\pi\)
−0.0119268 + 0.999929i \(0.503797\pi\)
\(920\) 0 0
\(921\) −26.5541 −0.874988
\(922\) −39.7115 −1.30783
\(923\) 1.25929 0.0414499
\(924\) −47.6516 −1.56762
\(925\) 0 0
\(926\) −30.6456 −1.00708
\(927\) −68.4227 −2.24730
\(928\) 10.2653 0.336974
\(929\) −16.7902 −0.550869 −0.275434 0.961320i \(-0.588822\pi\)
−0.275434 + 0.961320i \(0.588822\pi\)
\(930\) 0 0
\(931\) 4.55413 0.149256
\(932\) 14.2477 0.466698
\(933\) −101.701 −3.32954
\(934\) 24.7958 0.811344
\(935\) 0 0
\(936\) −0.903715 −0.0295388
\(937\) −7.01759 −0.229255 −0.114627 0.993409i \(-0.536567\pi\)
−0.114627 + 0.993409i \(0.536567\pi\)
\(938\) 0.946055 0.0308898
\(939\) 90.2620 2.94559
\(940\) 0 0
\(941\) 30.7958 1.00392 0.501958 0.864892i \(-0.332613\pi\)
0.501958 + 0.864892i \(0.332613\pi\)
\(942\) 10.3979 0.338782
\(943\) 44.1949 1.43918
\(944\) −13.0787 −0.425675
\(945\) 0 0
\(946\) 18.6936 0.607780
\(947\) −6.19257 −0.201232 −0.100616 0.994925i \(-0.532081\pi\)
−0.100616 + 0.994925i \(0.532081\pi\)
\(948\) 2.83101 0.0919469
\(949\) −0.470617 −0.0152769
\(950\) 0 0
\(951\) −21.5317 −0.698214
\(952\) 21.2113 0.687463
\(953\) 37.1750 1.20422 0.602108 0.798415i \(-0.294328\pi\)
0.602108 + 0.798415i \(0.294328\pi\)
\(954\) −79.2185 −2.56479
\(955\) 0 0
\(956\) 21.8921 0.708041
\(957\) −118.364 −3.82617
\(958\) 15.5845 0.503512
\(959\) 78.0742 2.52115
\(960\) 0 0
\(961\) 0.186582 0.00601878
\(962\) 0.132637 0.00427640
\(963\) −133.974 −4.31724
\(964\) −9.13264 −0.294142
\(965\) 0 0
\(966\) −55.2185 −1.77663
\(967\) 34.4722 1.10855 0.554275 0.832334i \(-0.312996\pi\)
0.554275 + 0.832334i \(0.312996\pi\)
\(968\) −2.54814 −0.0819004
\(969\) −7.26527 −0.233394
\(970\) 0 0
\(971\) −54.2488 −1.74093 −0.870464 0.492232i \(-0.836181\pi\)
−0.870464 + 0.492232i \(0.836181\pi\)
\(972\) −17.3616 −0.556872
\(973\) 61.3687 1.96739
\(974\) 3.65720 0.117184
\(975\) 0 0
\(976\) −4.39791 −0.140774
\(977\) 55.4706 1.77466 0.887331 0.461133i \(-0.152557\pi\)
0.887331 + 0.461133i \(0.152557\pi\)
\(978\) −48.7067 −1.55747
\(979\) −2.86021 −0.0914126
\(980\) 0 0
\(981\) 92.8453 2.96432
\(982\) −33.3264 −1.06349
\(983\) 0.422660 0.0134808 0.00674038 0.999977i \(-0.497854\pi\)
0.00674038 + 0.999977i \(0.497854\pi\)
\(984\) 32.4590 1.03476
\(985\) 0 0
\(986\) 52.6879 1.67793
\(987\) 0 0
\(988\) 0.0599332 0.00190673
\(989\) 21.6620 0.688812
\(990\) 0 0
\(991\) 7.93445 0.252046 0.126023 0.992027i \(-0.459779\pi\)
0.126023 + 0.992027i \(0.459779\pi\)
\(992\) −5.58449 −0.177308
\(993\) 5.18775 0.164628
\(994\) 39.2361 1.24449
\(995\) 0 0
\(996\) −43.6879 −1.38431
\(997\) −48.8933 −1.54847 −0.774233 0.632901i \(-0.781864\pi\)
−0.774233 + 0.632901i \(0.781864\pi\)
\(998\) −7.53654 −0.238565
\(999\) 11.9461 0.377956
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 1850.2.a.y.1.1 3
5.2 odd 4 1850.2.b.p.149.3 6
5.3 odd 4 1850.2.b.p.149.4 6
5.4 even 2 1850.2.a.bc.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.1 3 1.1 even 1 trivial
1850.2.a.bc.1.3 yes 3 5.4 even 2
1850.2.b.p.149.3 6 5.2 odd 4
1850.2.b.p.149.4 6 5.3 odd 4