Properties

Label 1859.2.a.g.1.3
Level $1859$
Weight $2$
Character 1859.1
Self dual yes
Analytic conductor $14.844$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.361.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.22188\) of defining polynomial
Character \(\chi\) \(=\) 1859.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.28514 q^{2} +1.22188 q^{3} +3.22188 q^{4} -2.50702 q^{5} +2.79216 q^{6} -4.50702 q^{7} +2.79216 q^{8} -1.50702 q^{9} +O(q^{10})\) \(q+2.28514 q^{2} +1.22188 q^{3} +3.22188 q^{4} -2.50702 q^{5} +2.79216 q^{6} -4.50702 q^{7} +2.79216 q^{8} -1.50702 q^{9} -5.72889 q^{10} -1.00000 q^{11} +3.93673 q^{12} -10.2992 q^{14} -3.06327 q^{15} -0.0632663 q^{16} -3.22188 q^{17} -3.44375 q^{18} +2.34841 q^{19} -8.07730 q^{20} -5.50702 q^{21} -2.28514 q^{22} +5.79216 q^{23} +3.41168 q^{24} +1.28514 q^{25} -5.50702 q^{27} -14.5211 q^{28} -0.619514 q^{29} -7.00000 q^{30} -6.01404 q^{31} -5.72889 q^{32} -1.22188 q^{33} -7.36245 q^{34} +11.2992 q^{35} -4.85543 q^{36} -5.06327 q^{37} +5.36645 q^{38} -7.00000 q^{40} +8.36245 q^{41} -12.5843 q^{42} +11.0913 q^{43} -3.22188 q^{44} +3.77812 q^{45} +13.2359 q^{46} -6.74293 q^{47} -0.0773036 q^{48} +13.3132 q^{49} +2.93673 q^{50} -3.93673 q^{51} -4.06327 q^{53} -12.5843 q^{54} +2.50702 q^{55} -12.5843 q^{56} +2.86946 q^{57} -1.41568 q^{58} -11.0633 q^{59} -9.86946 q^{60} -0.144573 q^{61} -13.7429 q^{62} +6.79216 q^{63} -12.9648 q^{64} -2.79216 q^{66} -8.72889 q^{67} -10.3805 q^{68} +7.07730 q^{69} +25.8202 q^{70} +1.71486 q^{71} -4.20784 q^{72} +2.00000 q^{73} -11.5703 q^{74} +1.57028 q^{75} +7.56628 q^{76} +4.50702 q^{77} +1.87347 q^{79} +0.158610 q^{80} -2.20784 q^{81} +19.1094 q^{82} +5.38049 q^{83} -17.7429 q^{84} +8.07730 q^{85} +25.3453 q^{86} -0.756969 q^{87} -2.79216 q^{88} +8.50702 q^{89} +8.63355 q^{90} +18.6616 q^{92} -7.34841 q^{93} -15.4086 q^{94} -5.88750 q^{95} -7.00000 q^{96} -0.112495 q^{97} +30.4226 q^{98} +1.50702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 3 q^{11} + 15 q^{12} - 8 q^{14} - 6 q^{15} + 3 q^{16} - 7 q^{17} - 5 q^{18} - 2 q^{19} - 4 q^{20} - 8 q^{21} - q^{22} + 3 q^{23} - 2 q^{24} - 2 q^{25} - 8 q^{27} - 18 q^{28} - 4 q^{29} - 21 q^{30} - q^{31} - 6 q^{32} - q^{33} + 4 q^{34} + 11 q^{35} + 3 q^{36} - 12 q^{37} + 31 q^{38} - 21 q^{40} - q^{41} - 9 q^{42} - 4 q^{43} - 7 q^{44} + 14 q^{45} + 20 q^{46} + 8 q^{47} + 20 q^{48} + 12 q^{50} - 15 q^{51} - 9 q^{53} - 9 q^{54} - q^{55} - 9 q^{56} - 26 q^{57} - 33 q^{58} - 30 q^{59} + 5 q^{60} - 18 q^{61} - 13 q^{62} + 6 q^{63} - 8 q^{64} + 6 q^{66} - 15 q^{67} - 29 q^{68} + q^{69} + 29 q^{70} + 11 q^{71} - 27 q^{72} + 6 q^{73} - 23 q^{74} - 7 q^{75} - 30 q^{76} + 5 q^{77} + 12 q^{79} + q^{80} - 21 q^{81} + 44 q^{82} + 14 q^{83} - 25 q^{84} + 4 q^{85} + 43 q^{86} + 43 q^{87} + 6 q^{88} + 17 q^{89} + 11 q^{90} + 7 q^{92} - 13 q^{93} - 10 q^{94} - 7 q^{95} - 21 q^{96} - 11 q^{97} + 38 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.28514 1.61584 0.807920 0.589292i \(-0.200593\pi\)
0.807920 + 0.589292i \(0.200593\pi\)
\(3\) 1.22188 0.705451 0.352725 0.935727i \(-0.385255\pi\)
0.352725 + 0.935727i \(0.385255\pi\)
\(4\) 3.22188 1.61094
\(5\) −2.50702 −1.12117 −0.560586 0.828096i \(-0.689424\pi\)
−0.560586 + 0.828096i \(0.689424\pi\)
\(6\) 2.79216 1.13990
\(7\) −4.50702 −1.70349 −0.851746 0.523954i \(-0.824456\pi\)
−0.851746 + 0.523954i \(0.824456\pi\)
\(8\) 2.79216 0.987178
\(9\) −1.50702 −0.502340
\(10\) −5.72889 −1.81164
\(11\) −1.00000 −0.301511
\(12\) 3.93673 1.13644
\(13\) 0 0
\(14\) −10.2992 −2.75257
\(15\) −3.06327 −0.790932
\(16\) −0.0632663 −0.0158166
\(17\) −3.22188 −0.781420 −0.390710 0.920514i \(-0.627770\pi\)
−0.390710 + 0.920514i \(0.627770\pi\)
\(18\) −3.44375 −0.811700
\(19\) 2.34841 0.538762 0.269381 0.963034i \(-0.413181\pi\)
0.269381 + 0.963034i \(0.413181\pi\)
\(20\) −8.07730 −1.80614
\(21\) −5.50702 −1.20173
\(22\) −2.28514 −0.487194
\(23\) 5.79216 1.20775 0.603875 0.797079i \(-0.293623\pi\)
0.603875 + 0.797079i \(0.293623\pi\)
\(24\) 3.41168 0.696405
\(25\) 1.28514 0.257028
\(26\) 0 0
\(27\) −5.50702 −1.05983
\(28\) −14.5211 −2.74422
\(29\) −0.619514 −0.115041 −0.0575204 0.998344i \(-0.518319\pi\)
−0.0575204 + 0.998344i \(0.518319\pi\)
\(30\) −7.00000 −1.27802
\(31\) −6.01404 −1.08015 −0.540076 0.841616i \(-0.681605\pi\)
−0.540076 + 0.841616i \(0.681605\pi\)
\(32\) −5.72889 −1.01274
\(33\) −1.22188 −0.212701
\(34\) −7.36245 −1.26265
\(35\) 11.2992 1.90991
\(36\) −4.85543 −0.809238
\(37\) −5.06327 −0.832396 −0.416198 0.909274i \(-0.636638\pi\)
−0.416198 + 0.909274i \(0.636638\pi\)
\(38\) 5.36645 0.870553
\(39\) 0 0
\(40\) −7.00000 −1.10680
\(41\) 8.36245 1.30599 0.652997 0.757360i \(-0.273511\pi\)
0.652997 + 0.757360i \(0.273511\pi\)
\(42\) −12.5843 −1.94180
\(43\) 11.0913 1.69141 0.845707 0.533648i \(-0.179179\pi\)
0.845707 + 0.533648i \(0.179179\pi\)
\(44\) −3.22188 −0.485716
\(45\) 3.77812 0.563209
\(46\) 13.2359 1.95153
\(47\) −6.74293 −0.983558 −0.491779 0.870720i \(-0.663653\pi\)
−0.491779 + 0.870720i \(0.663653\pi\)
\(48\) −0.0773036 −0.0111578
\(49\) 13.3132 1.90189
\(50\) 2.93673 0.415317
\(51\) −3.93673 −0.551253
\(52\) 0 0
\(53\) −4.06327 −0.558133 −0.279066 0.960272i \(-0.590025\pi\)
−0.279066 + 0.960272i \(0.590025\pi\)
\(54\) −12.5843 −1.71251
\(55\) 2.50702 0.338046
\(56\) −12.5843 −1.68165
\(57\) 2.86946 0.380070
\(58\) −1.41568 −0.185888
\(59\) −11.0633 −1.44031 −0.720157 0.693811i \(-0.755930\pi\)
−0.720157 + 0.693811i \(0.755930\pi\)
\(60\) −9.86946 −1.27414
\(61\) −0.144573 −0.0185106 −0.00925531 0.999957i \(-0.502946\pi\)
−0.00925531 + 0.999957i \(0.502946\pi\)
\(62\) −13.7429 −1.74535
\(63\) 6.79216 0.855732
\(64\) −12.9648 −1.62060
\(65\) 0 0
\(66\) −2.79216 −0.343691
\(67\) −8.72889 −1.06640 −0.533202 0.845988i \(-0.679011\pi\)
−0.533202 + 0.845988i \(0.679011\pi\)
\(68\) −10.3805 −1.25882
\(69\) 7.07730 0.852007
\(70\) 25.8202 3.08611
\(71\) 1.71486 0.203516 0.101758 0.994809i \(-0.467553\pi\)
0.101758 + 0.994809i \(0.467553\pi\)
\(72\) −4.20784 −0.495899
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −11.5703 −1.34502
\(75\) 1.57028 0.181321
\(76\) 7.56628 0.867912
\(77\) 4.50702 0.513622
\(78\) 0 0
\(79\) 1.87347 0.210782 0.105391 0.994431i \(-0.466391\pi\)
0.105391 + 0.994431i \(0.466391\pi\)
\(80\) 0.158610 0.0177331
\(81\) −2.20784 −0.245315
\(82\) 19.1094 2.11028
\(83\) 5.38049 0.590585 0.295293 0.955407i \(-0.404583\pi\)
0.295293 + 0.955407i \(0.404583\pi\)
\(84\) −17.7429 −1.93591
\(85\) 8.07730 0.876107
\(86\) 25.3453 2.73305
\(87\) −0.756969 −0.0811556
\(88\) −2.79216 −0.297645
\(89\) 8.50702 0.901742 0.450871 0.892589i \(-0.351113\pi\)
0.450871 + 0.892589i \(0.351113\pi\)
\(90\) 8.63355 0.910056
\(91\) 0 0
\(92\) 18.6616 1.94561
\(93\) −7.34841 −0.761994
\(94\) −15.4086 −1.58927
\(95\) −5.88750 −0.604045
\(96\) −7.00000 −0.714435
\(97\) −0.112495 −0.0114222 −0.00571109 0.999984i \(-0.501818\pi\)
−0.00571109 + 0.999984i \(0.501818\pi\)
\(98\) 30.4226 3.07315
\(99\) 1.50702 0.151461
\(100\) 4.14057 0.414057
\(101\) −12.7570 −1.26937 −0.634683 0.772773i \(-0.718869\pi\)
−0.634683 + 0.772773i \(0.718869\pi\)
\(102\) −8.99600 −0.890736
\(103\) 15.6304 1.54011 0.770056 0.637976i \(-0.220228\pi\)
0.770056 + 0.637976i \(0.220228\pi\)
\(104\) 0 0
\(105\) 13.8062 1.34735
\(106\) −9.28514 −0.901853
\(107\) −5.61951 −0.543259 −0.271629 0.962402i \(-0.587563\pi\)
−0.271629 + 0.962402i \(0.587563\pi\)
\(108\) −17.7429 −1.70731
\(109\) −9.90466 −0.948694 −0.474347 0.880338i \(-0.657316\pi\)
−0.474347 + 0.880338i \(0.657316\pi\)
\(110\) 5.72889 0.546229
\(111\) −6.18668 −0.587214
\(112\) 0.285142 0.0269434
\(113\) 8.24995 0.776090 0.388045 0.921640i \(-0.373150\pi\)
0.388045 + 0.921640i \(0.373150\pi\)
\(114\) 6.55714 0.614132
\(115\) −14.5211 −1.35410
\(116\) −1.99600 −0.185324
\(117\) 0 0
\(118\) −25.2811 −2.32732
\(119\) 14.5211 1.33114
\(120\) −8.55313 −0.780791
\(121\) 1.00000 0.0909091
\(122\) −0.330369 −0.0299102
\(123\) 10.2179 0.921315
\(124\) −19.3765 −1.74006
\(125\) 9.31322 0.832999
\(126\) 15.5211 1.38273
\(127\) −6.07730 −0.539273 −0.269637 0.962962i \(-0.586904\pi\)
−0.269637 + 0.962962i \(0.586904\pi\)
\(128\) −18.1686 −1.60590
\(129\) 13.5522 1.19321
\(130\) 0 0
\(131\) −11.9508 −1.04414 −0.522072 0.852902i \(-0.674841\pi\)
−0.522072 + 0.852902i \(0.674841\pi\)
\(132\) −3.93673 −0.342649
\(133\) −10.5843 −0.917777
\(134\) −19.9468 −1.72314
\(135\) 13.8062 1.18825
\(136\) −8.99600 −0.771400
\(137\) 2.27111 0.194034 0.0970168 0.995283i \(-0.469070\pi\)
0.0970168 + 0.995283i \(0.469070\pi\)
\(138\) 16.1726 1.37671
\(139\) −10.1546 −0.861303 −0.430651 0.902518i \(-0.641716\pi\)
−0.430651 + 0.902518i \(0.641716\pi\)
\(140\) 36.4046 3.07675
\(141\) −8.23903 −0.693851
\(142\) 3.91869 0.328849
\(143\) 0 0
\(144\) 0.0953435 0.00794529
\(145\) 1.55313 0.128981
\(146\) 4.57028 0.378239
\(147\) 16.2671 1.34169
\(148\) −16.3132 −1.34094
\(149\) 10.9648 0.898272 0.449136 0.893463i \(-0.351732\pi\)
0.449136 + 0.893463i \(0.351732\pi\)
\(150\) 3.58832 0.292985
\(151\) 10.8835 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(152\) 6.55714 0.531854
\(153\) 4.85543 0.392538
\(154\) 10.2992 0.829932
\(155\) 15.0773 1.21104
\(156\) 0 0
\(157\) 13.9820 1.11588 0.557941 0.829881i \(-0.311592\pi\)
0.557941 + 0.829881i \(0.311592\pi\)
\(158\) 4.28114 0.340589
\(159\) −4.96481 −0.393735
\(160\) 14.3624 1.13545
\(161\) −26.1054 −2.05739
\(162\) −5.04523 −0.396390
\(163\) −1.39452 −0.109227 −0.0546137 0.998508i \(-0.517393\pi\)
−0.0546137 + 0.998508i \(0.517393\pi\)
\(164\) 26.9428 2.10388
\(165\) 3.06327 0.238475
\(166\) 12.2952 0.954291
\(167\) −23.8343 −1.84435 −0.922176 0.386771i \(-0.873590\pi\)
−0.922176 + 0.386771i \(0.873590\pi\)
\(168\) −15.3765 −1.18632
\(169\) 0 0
\(170\) 18.4578 1.41565
\(171\) −3.53910 −0.270641
\(172\) 35.7349 2.72476
\(173\) −7.39764 −0.562432 −0.281216 0.959644i \(-0.590738\pi\)
−0.281216 + 0.959644i \(0.590738\pi\)
\(174\) −1.72978 −0.131135
\(175\) −5.79216 −0.437846
\(176\) 0.0632663 0.00476888
\(177\) −13.5179 −1.01607
\(178\) 19.4397 1.45707
\(179\) 18.5211 1.38433 0.692164 0.721740i \(-0.256657\pi\)
0.692164 + 0.721740i \(0.256657\pi\)
\(180\) 12.1726 0.907296
\(181\) −18.0561 −1.34210 −0.671051 0.741411i \(-0.734157\pi\)
−0.671051 + 0.741411i \(0.734157\pi\)
\(182\) 0 0
\(183\) −0.176650 −0.0130583
\(184\) 16.1726 1.19226
\(185\) 12.6937 0.933260
\(186\) −16.7922 −1.23126
\(187\) 3.22188 0.235607
\(188\) −21.7249 −1.58445
\(189\) 24.8202 1.80541
\(190\) −13.4538 −0.976040
\(191\) −3.36245 −0.243298 −0.121649 0.992573i \(-0.538818\pi\)
−0.121649 + 0.992573i \(0.538818\pi\)
\(192\) −15.8414 −1.14325
\(193\) −15.5984 −1.12279 −0.561397 0.827546i \(-0.689736\pi\)
−0.561397 + 0.827546i \(0.689736\pi\)
\(194\) −0.257068 −0.0184564
\(195\) 0 0
\(196\) 42.8935 3.06382
\(197\) 21.0281 1.49819 0.749094 0.662464i \(-0.230489\pi\)
0.749094 + 0.662464i \(0.230489\pi\)
\(198\) 3.44375 0.244737
\(199\) −22.6124 −1.60295 −0.801475 0.598028i \(-0.795951\pi\)
−0.801475 + 0.598028i \(0.795951\pi\)
\(200\) 3.58832 0.253733
\(201\) −10.6656 −0.752295
\(202\) −29.1515 −2.05109
\(203\) 2.79216 0.195971
\(204\) −12.6837 −0.888034
\(205\) −20.9648 −1.46425
\(206\) 35.7178 2.48858
\(207\) −8.72889 −0.606700
\(208\) 0 0
\(209\) −2.34841 −0.162443
\(210\) 31.5491 2.17710
\(211\) 20.3765 1.40277 0.701387 0.712780i \(-0.252564\pi\)
0.701387 + 0.712780i \(0.252564\pi\)
\(212\) −13.0913 −0.899117
\(213\) 2.09534 0.143571
\(214\) −12.8414 −0.877819
\(215\) −27.8062 −1.89637
\(216\) −15.3765 −1.04624
\(217\) 27.1054 1.84003
\(218\) −22.6336 −1.53294
\(219\) 2.44375 0.165133
\(220\) 8.07730 0.544572
\(221\) 0 0
\(222\) −14.1375 −0.948844
\(223\) 17.7741 1.19024 0.595122 0.803636i \(-0.297104\pi\)
0.595122 + 0.803636i \(0.297104\pi\)
\(224\) 25.8202 1.72519
\(225\) −1.93673 −0.129116
\(226\) 18.8523 1.25404
\(227\) −5.82024 −0.386303 −0.193151 0.981169i \(-0.561871\pi\)
−0.193151 + 0.981169i \(0.561871\pi\)
\(228\) 9.24506 0.612269
\(229\) −23.6656 −1.56387 −0.781934 0.623361i \(-0.785767\pi\)
−0.781934 + 0.623361i \(0.785767\pi\)
\(230\) −33.1827 −2.18800
\(231\) 5.50702 0.362335
\(232\) −1.72978 −0.113566
\(233\) −22.8875 −1.49941 −0.749705 0.661772i \(-0.769805\pi\)
−0.749705 + 0.661772i \(0.769805\pi\)
\(234\) 0 0
\(235\) 16.9047 1.10274
\(236\) −35.6445 −2.32026
\(237\) 2.28915 0.148696
\(238\) 33.1827 2.15091
\(239\) −16.2007 −1.04794 −0.523969 0.851737i \(-0.675549\pi\)
−0.523969 + 0.851737i \(0.675549\pi\)
\(240\) 0.193802 0.0125098
\(241\) −0.193802 −0.0124839 −0.00624193 0.999981i \(-0.501987\pi\)
−0.00624193 + 0.999981i \(0.501987\pi\)
\(242\) 2.28514 0.146895
\(243\) 13.8234 0.886768
\(244\) −0.465795 −0.0298195
\(245\) −33.3765 −2.13235
\(246\) 23.3493 1.48870
\(247\) 0 0
\(248\) −16.7922 −1.06630
\(249\) 6.57429 0.416629
\(250\) 21.2820 1.34599
\(251\) 8.14057 0.513828 0.256914 0.966434i \(-0.417294\pi\)
0.256914 + 0.966434i \(0.417294\pi\)
\(252\) 21.8835 1.37853
\(253\) −5.79216 −0.364150
\(254\) −13.8875 −0.871380
\(255\) 9.86946 0.618050
\(256\) −15.5883 −0.974270
\(257\) −17.8975 −1.11642 −0.558209 0.829700i \(-0.688511\pi\)
−0.558209 + 0.829700i \(0.688511\pi\)
\(258\) 30.9688 1.92803
\(259\) 22.8202 1.41798
\(260\) 0 0
\(261\) 0.933619 0.0577896
\(262\) −27.3092 −1.68717
\(263\) −15.5672 −0.959913 −0.479956 0.877292i \(-0.659348\pi\)
−0.479956 + 0.877292i \(0.659348\pi\)
\(264\) −3.41168 −0.209974
\(265\) 10.1867 0.625763
\(266\) −24.1867 −1.48298
\(267\) 10.3945 0.636134
\(268\) −28.1234 −1.71791
\(269\) −1.49298 −0.0910287 −0.0455143 0.998964i \(-0.514493\pi\)
−0.0455143 + 0.998964i \(0.514493\pi\)
\(270\) 31.5491 1.92002
\(271\) −23.2711 −1.41362 −0.706809 0.707404i \(-0.749866\pi\)
−0.706809 + 0.707404i \(0.749866\pi\)
\(272\) 0.203836 0.0123594
\(273\) 0 0
\(274\) 5.18980 0.313527
\(275\) −1.28514 −0.0774970
\(276\) 22.8022 1.37253
\(277\) −14.0602 −0.844793 −0.422396 0.906411i \(-0.638811\pi\)
−0.422396 + 0.906411i \(0.638811\pi\)
\(278\) −23.2047 −1.39173
\(279\) 9.06327 0.542604
\(280\) 31.5491 1.88542
\(281\) −11.6797 −0.696750 −0.348375 0.937355i \(-0.613266\pi\)
−0.348375 + 0.937355i \(0.613266\pi\)
\(282\) −18.8274 −1.12115
\(283\) −3.47183 −0.206379 −0.103189 0.994662i \(-0.532905\pi\)
−0.103189 + 0.994662i \(0.532905\pi\)
\(284\) 5.52506 0.327852
\(285\) −7.19380 −0.426124
\(286\) 0 0
\(287\) −37.6897 −2.22475
\(288\) 8.63355 0.508737
\(289\) −6.61951 −0.389383
\(290\) 3.54913 0.208412
\(291\) −0.137455 −0.00805778
\(292\) 6.44375 0.377092
\(293\) −6.37960 −0.372700 −0.186350 0.982483i \(-0.559666\pi\)
−0.186350 + 0.982483i \(0.559666\pi\)
\(294\) 37.1726 2.16795
\(295\) 27.7358 1.61484
\(296\) −14.1375 −0.821723
\(297\) 5.50702 0.319550
\(298\) 25.0561 1.45146
\(299\) 0 0
\(300\) 5.05926 0.292097
\(301\) −49.9889 −2.88131
\(302\) 24.8704 1.43113
\(303\) −15.5874 −0.895475
\(304\) −0.148575 −0.00852137
\(305\) 0.362446 0.0207536
\(306\) 11.0953 0.634279
\(307\) −5.06327 −0.288976 −0.144488 0.989507i \(-0.546153\pi\)
−0.144488 + 0.989507i \(0.546153\pi\)
\(308\) 14.5211 0.827414
\(309\) 19.0985 1.08647
\(310\) 34.4538 1.95684
\(311\) 2.91958 0.165554 0.0827771 0.996568i \(-0.473621\pi\)
0.0827771 + 0.996568i \(0.473621\pi\)
\(312\) 0 0
\(313\) 15.7882 0.892399 0.446200 0.894933i \(-0.352777\pi\)
0.446200 + 0.894933i \(0.352777\pi\)
\(314\) 31.9508 1.80309
\(315\) −17.0281 −0.959423
\(316\) 6.03608 0.339556
\(317\) 20.4397 1.14801 0.574005 0.818852i \(-0.305389\pi\)
0.574005 + 0.818852i \(0.305389\pi\)
\(318\) −11.3453 −0.636213
\(319\) 0.619514 0.0346861
\(320\) 32.5030 1.81697
\(321\) −6.86635 −0.383242
\(322\) −59.6545 −3.32442
\(323\) −7.56628 −0.420999
\(324\) −7.11338 −0.395188
\(325\) 0 0
\(326\) −3.18668 −0.176494
\(327\) −12.1023 −0.669257
\(328\) 23.3493 1.28925
\(329\) 30.3905 1.67548
\(330\) 7.00000 0.385337
\(331\) 28.2811 1.55447 0.777236 0.629209i \(-0.216621\pi\)
0.777236 + 0.629209i \(0.216621\pi\)
\(332\) 17.3353 0.951396
\(333\) 7.63044 0.418145
\(334\) −54.4647 −2.98018
\(335\) 21.8835 1.19562
\(336\) 0.348409 0.0190073
\(337\) −31.9256 −1.73910 −0.869550 0.493846i \(-0.835591\pi\)
−0.869550 + 0.493846i \(0.835591\pi\)
\(338\) 0 0
\(339\) 10.0804 0.547493
\(340\) 26.0241 1.41135
\(341\) 6.01404 0.325678
\(342\) −8.08734 −0.437313
\(343\) −28.4538 −1.53636
\(344\) 30.9688 1.66973
\(345\) −17.7429 −0.955247
\(346\) −16.9047 −0.908800
\(347\) −1.91869 −0.103001 −0.0515004 0.998673i \(-0.516400\pi\)
−0.0515004 + 0.998673i \(0.516400\pi\)
\(348\) −2.43886 −0.130737
\(349\) 0.189799 0.0101597 0.00507985 0.999987i \(-0.498383\pi\)
0.00507985 + 0.999987i \(0.498383\pi\)
\(350\) −13.2359 −0.707489
\(351\) 0 0
\(352\) 5.72889 0.305351
\(353\) 10.9499 0.582803 0.291402 0.956601i \(-0.405878\pi\)
0.291402 + 0.956601i \(0.405878\pi\)
\(354\) −30.8904 −1.64181
\(355\) −4.29918 −0.228177
\(356\) 27.4086 1.45265
\(357\) 17.7429 0.939056
\(358\) 42.3233 2.23685
\(359\) −3.79216 −0.200143 −0.100071 0.994980i \(-0.531907\pi\)
−0.100071 + 0.994980i \(0.531907\pi\)
\(360\) 10.5491 0.555988
\(361\) −13.4850 −0.709736
\(362\) −41.2609 −2.16862
\(363\) 1.22188 0.0641319
\(364\) 0 0
\(365\) −5.01404 −0.262447
\(366\) −0.403670 −0.0211002
\(367\) −25.0069 −1.30535 −0.652675 0.757638i \(-0.726353\pi\)
−0.652675 + 0.757638i \(0.726353\pi\)
\(368\) −0.366449 −0.0191025
\(369\) −12.6024 −0.656053
\(370\) 29.0069 1.50800
\(371\) 18.3132 0.950775
\(372\) −23.6757 −1.22753
\(373\) −8.67967 −0.449416 −0.224708 0.974426i \(-0.572143\pi\)
−0.224708 + 0.974426i \(0.572143\pi\)
\(374\) 7.36245 0.380703
\(375\) 11.3796 0.587640
\(376\) −18.8274 −0.970947
\(377\) 0 0
\(378\) 56.7178 2.91725
\(379\) −14.7149 −0.755851 −0.377926 0.925836i \(-0.623363\pi\)
−0.377926 + 0.925836i \(0.623363\pi\)
\(380\) −18.9688 −0.973079
\(381\) −7.42571 −0.380431
\(382\) −7.68367 −0.393131
\(383\) 5.39052 0.275443 0.137721 0.990471i \(-0.456022\pi\)
0.137721 + 0.990471i \(0.456022\pi\)
\(384\) −22.1998 −1.13288
\(385\) −11.2992 −0.575860
\(386\) −35.6445 −1.81426
\(387\) −16.7149 −0.849664
\(388\) −0.362446 −0.0184004
\(389\) −21.5531 −1.09279 −0.546394 0.837529i \(-0.684000\pi\)
−0.546394 + 0.837529i \(0.684000\pi\)
\(390\) 0 0
\(391\) −18.6616 −0.943759
\(392\) 37.1726 1.87750
\(393\) −14.6024 −0.736592
\(394\) 48.0521 2.42083
\(395\) −4.69682 −0.236323
\(396\) 4.85543 0.243994
\(397\) 17.5311 0.879860 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(398\) −51.6725 −2.59011
\(399\) −12.9327 −0.647446
\(400\) −0.0813062 −0.00406531
\(401\) 36.4366 1.81956 0.909779 0.415092i \(-0.136251\pi\)
0.909779 + 0.415092i \(0.136251\pi\)
\(402\) −24.3725 −1.21559
\(403\) 0 0
\(404\) −41.1014 −2.04487
\(405\) 5.53509 0.275041
\(406\) 6.38049 0.316658
\(407\) 5.06327 0.250977
\(408\) −10.9920 −0.544185
\(409\) −28.8664 −1.42735 −0.713675 0.700477i \(-0.752970\pi\)
−0.713675 + 0.700477i \(0.752970\pi\)
\(410\) −47.9076 −2.36599
\(411\) 2.77501 0.136881
\(412\) 50.3593 2.48103
\(413\) 49.8623 2.45357
\(414\) −19.9468 −0.980330
\(415\) −13.4890 −0.662148
\(416\) 0 0
\(417\) −12.4077 −0.607606
\(418\) −5.36645 −0.262482
\(419\) −4.20472 −0.205414 −0.102707 0.994712i \(-0.532750\pi\)
−0.102707 + 0.994712i \(0.532750\pi\)
\(420\) 44.4819 2.17049
\(421\) −13.5843 −0.662059 −0.331030 0.943620i \(-0.607396\pi\)
−0.331030 + 0.943620i \(0.607396\pi\)
\(422\) 46.5632 2.26666
\(423\) 10.1617 0.494080
\(424\) −11.3453 −0.550976
\(425\) −4.14057 −0.200847
\(426\) 4.78816 0.231987
\(427\) 0.651591 0.0315327
\(428\) −18.1054 −0.875156
\(429\) 0 0
\(430\) −63.5411 −3.06423
\(431\) 30.0381 1.44688 0.723442 0.690385i \(-0.242559\pi\)
0.723442 + 0.690385i \(0.242559\pi\)
\(432\) 0.348409 0.0167628
\(433\) 4.25707 0.204582 0.102291 0.994755i \(-0.467383\pi\)
0.102291 + 0.994755i \(0.467383\pi\)
\(434\) 61.9397 2.97320
\(435\) 1.89774 0.0909895
\(436\) −31.9116 −1.52829
\(437\) 13.6024 0.650689
\(438\) 5.58432 0.266829
\(439\) −5.00311 −0.238786 −0.119393 0.992847i \(-0.538095\pi\)
−0.119393 + 0.992847i \(0.538095\pi\)
\(440\) 7.00000 0.333712
\(441\) −20.0633 −0.955394
\(442\) 0 0
\(443\) 28.3101 1.34505 0.672527 0.740073i \(-0.265209\pi\)
0.672527 + 0.740073i \(0.265209\pi\)
\(444\) −19.9327 −0.945966
\(445\) −21.3273 −1.01101
\(446\) 40.6164 1.92324
\(447\) 13.3976 0.633687
\(448\) 58.4326 2.76068
\(449\) 5.72578 0.270216 0.135108 0.990831i \(-0.456862\pi\)
0.135108 + 0.990831i \(0.456862\pi\)
\(450\) −4.42571 −0.208630
\(451\) −8.36245 −0.393772
\(452\) 26.5803 1.25023
\(453\) 13.2983 0.624808
\(454\) −13.3001 −0.624203
\(455\) 0 0
\(456\) 8.01201 0.375197
\(457\) 12.4899 0.584251 0.292126 0.956380i \(-0.405637\pi\)
0.292126 + 0.956380i \(0.405637\pi\)
\(458\) −54.0793 −2.52696
\(459\) 17.7429 0.828169
\(460\) −46.7850 −2.18136
\(461\) 13.1726 0.613511 0.306756 0.951788i \(-0.400757\pi\)
0.306756 + 0.951788i \(0.400757\pi\)
\(462\) 12.5843 0.585476
\(463\) 10.4086 0.483727 0.241863 0.970310i \(-0.422241\pi\)
0.241863 + 0.970310i \(0.422241\pi\)
\(464\) 0.0391944 0.00181955
\(465\) 18.4226 0.854328
\(466\) −52.3012 −2.42281
\(467\) 23.3553 1.08076 0.540378 0.841422i \(-0.318281\pi\)
0.540378 + 0.841422i \(0.318281\pi\)
\(468\) 0 0
\(469\) 39.3413 1.81661
\(470\) 38.6295 1.78185
\(471\) 17.0842 0.787199
\(472\) −30.8904 −1.42185
\(473\) −11.0913 −0.509980
\(474\) 5.23102 0.240269
\(475\) 3.01804 0.138477
\(476\) 46.7850 2.14439
\(477\) 6.12342 0.280372
\(478\) −37.0210 −1.69330
\(479\) −8.22188 −0.375667 −0.187834 0.982201i \(-0.560147\pi\)
−0.187834 + 0.982201i \(0.560147\pi\)
\(480\) 17.5491 0.801005
\(481\) 0 0
\(482\) −0.442864 −0.0201719
\(483\) −31.8975 −1.45139
\(484\) 3.22188 0.146449
\(485\) 0.282028 0.0128062
\(486\) 31.5883 1.43288
\(487\) −3.73893 −0.169427 −0.0847135 0.996405i \(-0.526998\pi\)
−0.0847135 + 0.996405i \(0.526998\pi\)
\(488\) −0.403670 −0.0182733
\(489\) −1.70393 −0.0770546
\(490\) −76.2700 −3.44553
\(491\) −8.17665 −0.369007 −0.184504 0.982832i \(-0.559068\pi\)
−0.184504 + 0.982832i \(0.559068\pi\)
\(492\) 32.9207 1.48418
\(493\) 1.99600 0.0898952
\(494\) 0 0
\(495\) −3.77812 −0.169814
\(496\) 0.380486 0.0170843
\(497\) −7.72889 −0.346688
\(498\) 15.0232 0.673205
\(499\) 0.918694 0.0411264 0.0205632 0.999789i \(-0.493454\pi\)
0.0205632 + 0.999789i \(0.493454\pi\)
\(500\) 30.0060 1.34191
\(501\) −29.1225 −1.30110
\(502\) 18.6024 0.830264
\(503\) −18.1085 −0.807418 −0.403709 0.914887i \(-0.632279\pi\)
−0.403709 + 0.914887i \(0.632279\pi\)
\(504\) 18.9648 0.844760
\(505\) 31.9820 1.42318
\(506\) −13.2359 −0.588408
\(507\) 0 0
\(508\) −19.5803 −0.868736
\(509\) −20.4086 −0.904594 −0.452297 0.891867i \(-0.649395\pi\)
−0.452297 + 0.891867i \(0.649395\pi\)
\(510\) 22.5531 0.998670
\(511\) −9.01404 −0.398758
\(512\) 0.715746 0.0316318
\(513\) −12.9327 −0.570994
\(514\) −40.8984 −1.80395
\(515\) −39.1858 −1.72673
\(516\) 43.6637 1.92219
\(517\) 6.74293 0.296554
\(518\) 52.1475 2.29123
\(519\) −9.03900 −0.396768
\(520\) 0 0
\(521\) −12.0069 −0.526033 −0.263016 0.964791i \(-0.584717\pi\)
−0.263016 + 0.964791i \(0.584717\pi\)
\(522\) 2.13345 0.0933787
\(523\) 17.8414 0.780150 0.390075 0.920783i \(-0.372449\pi\)
0.390075 + 0.920783i \(0.372449\pi\)
\(524\) −38.5039 −1.68205
\(525\) −7.07730 −0.308879
\(526\) −35.5732 −1.55107
\(527\) 19.3765 0.844053
\(528\) 0.0773036 0.00336421
\(529\) 10.5491 0.458658
\(530\) 23.2780 1.01113
\(531\) 16.6725 0.723527
\(532\) −34.1014 −1.47848
\(533\) 0 0
\(534\) 23.7530 1.02789
\(535\) 14.0882 0.609087
\(536\) −24.3725 −1.05273
\(537\) 22.6304 0.976575
\(538\) −3.41168 −0.147088
\(539\) −13.3132 −0.573441
\(540\) 44.4819 1.91419
\(541\) 19.1234 0.822180 0.411090 0.911595i \(-0.365148\pi\)
0.411090 + 0.911595i \(0.365148\pi\)
\(542\) −53.1778 −2.28418
\(543\) −22.0624 −0.946787
\(544\) 18.4578 0.791371
\(545\) 24.8312 1.06365
\(546\) 0 0
\(547\) −44.1225 −1.88654 −0.943272 0.332022i \(-0.892269\pi\)
−0.943272 + 0.332022i \(0.892269\pi\)
\(548\) 7.31722 0.312576
\(549\) 0.217874 0.00929862
\(550\) −2.93673 −0.125223
\(551\) −1.45487 −0.0619796
\(552\) 19.7610 0.841083
\(553\) −8.44375 −0.359065
\(554\) −32.1295 −1.36505
\(555\) 15.5101 0.658368
\(556\) −32.7169 −1.38751
\(557\) 21.3453 0.904429 0.452215 0.891909i \(-0.350634\pi\)
0.452215 + 0.891909i \(0.350634\pi\)
\(558\) 20.7109 0.876760
\(559\) 0 0
\(560\) −0.714858 −0.0302082
\(561\) 3.93673 0.166209
\(562\) −26.6897 −1.12584
\(563\) −19.9187 −0.839473 −0.419736 0.907646i \(-0.637877\pi\)
−0.419736 + 0.907646i \(0.637877\pi\)
\(564\) −26.5451 −1.11775
\(565\) −20.6828 −0.870131
\(566\) −7.93362 −0.333475
\(567\) 9.95077 0.417893
\(568\) 4.78816 0.200907
\(569\) 46.4217 1.94610 0.973050 0.230596i \(-0.0740675\pi\)
0.973050 + 0.230596i \(0.0740675\pi\)
\(570\) −16.4389 −0.688548
\(571\) −5.67878 −0.237649 −0.118825 0.992915i \(-0.537913\pi\)
−0.118825 + 0.992915i \(0.537913\pi\)
\(572\) 0 0
\(573\) −4.10849 −0.171635
\(574\) −86.1263 −3.59484
\(575\) 7.44375 0.310426
\(576\) 19.5382 0.814092
\(577\) 6.37648 0.265456 0.132728 0.991152i \(-0.457626\pi\)
0.132728 + 0.991152i \(0.457626\pi\)
\(578\) −15.1265 −0.629181
\(579\) −19.0593 −0.792076
\(580\) 5.00400 0.207780
\(581\) −24.2500 −1.00606
\(582\) −0.314105 −0.0130201
\(583\) 4.06327 0.168283
\(584\) 5.58432 0.231081
\(585\) 0 0
\(586\) −14.5783 −0.602224
\(587\) 0.873467 0.0360519 0.0180259 0.999838i \(-0.494262\pi\)
0.0180259 + 0.999838i \(0.494262\pi\)
\(588\) 52.4106 2.16138
\(589\) −14.1234 −0.581945
\(590\) 63.3803 2.60933
\(591\) 25.6937 1.05690
\(592\) 0.320334 0.0131657
\(593\) −18.7850 −0.771409 −0.385705 0.922622i \(-0.626042\pi\)
−0.385705 + 0.922622i \(0.626042\pi\)
\(594\) 12.5843 0.516341
\(595\) −36.4046 −1.49244
\(596\) 35.3273 1.44706
\(597\) −27.6295 −1.13080
\(598\) 0 0
\(599\) 11.7601 0.480504 0.240252 0.970711i \(-0.422770\pi\)
0.240252 + 0.970711i \(0.422770\pi\)
\(600\) 4.38449 0.178996
\(601\) 28.5171 1.16323 0.581617 0.813462i \(-0.302420\pi\)
0.581617 + 0.813462i \(0.302420\pi\)
\(602\) −114.232 −4.65574
\(603\) 13.1546 0.535697
\(604\) 35.0653 1.42679
\(605\) −2.50702 −0.101925
\(606\) −35.6195 −1.44694
\(607\) 15.4718 0.627982 0.313991 0.949426i \(-0.398334\pi\)
0.313991 + 0.949426i \(0.398334\pi\)
\(608\) −13.4538 −0.545623
\(609\) 3.41168 0.138248
\(610\) 0.828241 0.0335345
\(611\) 0 0
\(612\) 15.6436 0.632354
\(613\) 19.5883 0.791165 0.395582 0.918430i \(-0.370543\pi\)
0.395582 + 0.918430i \(0.370543\pi\)
\(614\) −11.5703 −0.466939
\(615\) −25.6164 −1.03295
\(616\) 12.5843 0.507037
\(617\) 29.3101 1.17998 0.589990 0.807410i \(-0.299132\pi\)
0.589990 + 0.807410i \(0.299132\pi\)
\(618\) 43.6427 1.75557
\(619\) 3.74293 0.150441 0.0752206 0.997167i \(-0.476034\pi\)
0.0752206 + 0.997167i \(0.476034\pi\)
\(620\) 48.5772 1.95091
\(621\) −31.8975 −1.28000
\(622\) 6.67166 0.267509
\(623\) −38.3413 −1.53611
\(624\) 0 0
\(625\) −29.7741 −1.19096
\(626\) 36.0782 1.44197
\(627\) −2.86946 −0.114595
\(628\) 45.0481 1.79762
\(629\) 16.3132 0.650451
\(630\) −38.9116 −1.55027
\(631\) 22.5179 0.896425 0.448213 0.893927i \(-0.352061\pi\)
0.448213 + 0.893927i \(0.352061\pi\)
\(632\) 5.23102 0.208079
\(633\) 24.8975 0.989588
\(634\) 46.7077 1.85500
\(635\) 15.2359 0.604619
\(636\) −15.9960 −0.634283
\(637\) 0 0
\(638\) 1.41568 0.0560472
\(639\) −2.58432 −0.102234
\(640\) 45.5491 1.80049
\(641\) 26.5812 1.04989 0.524947 0.851135i \(-0.324085\pi\)
0.524947 + 0.851135i \(0.324085\pi\)
\(642\) −15.6906 −0.619258
\(643\) −1.28514 −0.0506811 −0.0253405 0.999679i \(-0.508067\pi\)
−0.0253405 + 0.999679i \(0.508067\pi\)
\(644\) −84.1083 −3.31433
\(645\) −33.9757 −1.33779
\(646\) −17.2900 −0.680267
\(647\) −13.7741 −0.541517 −0.270758 0.962647i \(-0.587274\pi\)
−0.270758 + 0.962647i \(0.587274\pi\)
\(648\) −6.16464 −0.242170
\(649\) 11.0633 0.434271
\(650\) 0 0
\(651\) 33.1194 1.29805
\(652\) −4.49298 −0.175959
\(653\) 15.5382 0.608057 0.304029 0.952663i \(-0.401668\pi\)
0.304029 + 0.952663i \(0.401668\pi\)
\(654\) −27.6554 −1.08141
\(655\) 29.9608 1.17067
\(656\) −0.529061 −0.0206564
\(657\) −3.01404 −0.117589
\(658\) 69.4467 2.70731
\(659\) 20.3734 0.793634 0.396817 0.917898i \(-0.370115\pi\)
0.396817 + 0.917898i \(0.370115\pi\)
\(660\) 9.86946 0.384168
\(661\) −0.989077 −0.0384706 −0.0192353 0.999815i \(-0.506123\pi\)
−0.0192353 + 0.999815i \(0.506123\pi\)
\(662\) 64.6264 2.51178
\(663\) 0 0
\(664\) 15.0232 0.583013
\(665\) 26.5351 1.02899
\(666\) 17.4366 0.675656
\(667\) −3.58832 −0.138940
\(668\) −76.7911 −2.97114
\(669\) 21.7178 0.839658
\(670\) 50.0069 1.93194
\(671\) 0.144573 0.00558116
\(672\) 31.5491 1.21703
\(673\) 30.2992 1.16795 0.583974 0.811773i \(-0.301497\pi\)
0.583974 + 0.811773i \(0.301497\pi\)
\(674\) −72.9546 −2.81011
\(675\) −7.07730 −0.272406
\(676\) 0 0
\(677\) 20.6788 0.794750 0.397375 0.917656i \(-0.369921\pi\)
0.397375 + 0.917656i \(0.369921\pi\)
\(678\) 23.0352 0.884661
\(679\) 0.507019 0.0194576
\(680\) 22.5531 0.864873
\(681\) −7.11161 −0.272517
\(682\) 13.7429 0.526244
\(683\) 0.439750 0.0168266 0.00841328 0.999965i \(-0.497322\pi\)
0.00841328 + 0.999965i \(0.497322\pi\)
\(684\) −11.4025 −0.435987
\(685\) −5.69370 −0.217545
\(686\) −65.0210 −2.48251
\(687\) −28.9165 −1.10323
\(688\) −0.701708 −0.0267524
\(689\) 0 0
\(690\) −40.5451 −1.54353
\(691\) −20.0421 −0.762438 −0.381219 0.924485i \(-0.624496\pi\)
−0.381219 + 0.924485i \(0.624496\pi\)
\(692\) −23.8343 −0.906043
\(693\) −6.79216 −0.258013
\(694\) −4.38449 −0.166433
\(695\) 25.4578 0.965669
\(696\) −2.11358 −0.0801151
\(697\) −26.9428 −1.02053
\(698\) 0.433718 0.0164165
\(699\) −27.9657 −1.05776
\(700\) −18.6616 −0.705343
\(701\) −36.9898 −1.39708 −0.698542 0.715569i \(-0.746168\pi\)
−0.698542 + 0.715569i \(0.746168\pi\)
\(702\) 0 0
\(703\) −11.8906 −0.448463
\(704\) 12.9648 0.488630
\(705\) 20.6554 0.777927
\(706\) 25.0220 0.941717
\(707\) 57.4959 2.16236
\(708\) −43.5531 −1.63683
\(709\) −33.8022 −1.26947 −0.634734 0.772731i \(-0.718890\pi\)
−0.634734 + 0.772731i \(0.718890\pi\)
\(710\) −9.82424 −0.368697
\(711\) −2.82335 −0.105884
\(712\) 23.7530 0.890180
\(713\) −34.8343 −1.30455
\(714\) 40.5451 1.51736
\(715\) 0 0
\(716\) 59.6725 2.23007
\(717\) −19.7953 −0.739268
\(718\) −8.66563 −0.323398
\(719\) 12.2210 0.455766 0.227883 0.973689i \(-0.426820\pi\)
0.227883 + 0.973689i \(0.426820\pi\)
\(720\) −0.239028 −0.00890805
\(721\) −70.4467 −2.62357
\(722\) −30.8151 −1.14682
\(723\) −0.236802 −0.00880674
\(724\) −58.1747 −2.16204
\(725\) −0.796164 −0.0295688
\(726\) 2.79216 0.103627
\(727\) 35.2771 1.30836 0.654178 0.756340i \(-0.273015\pi\)
0.654178 + 0.756340i \(0.273015\pi\)
\(728\) 0 0
\(729\) 23.5139 0.870887
\(730\) −11.4578 −0.424072
\(731\) −35.7349 −1.32170
\(732\) −0.569144 −0.0210362
\(733\) −17.0953 −0.631431 −0.315715 0.948854i \(-0.602245\pi\)
−0.315715 + 0.948854i \(0.602245\pi\)
\(734\) −57.1444 −2.10924
\(735\) −40.7819 −1.50426
\(736\) −33.1827 −1.22313
\(737\) 8.72889 0.321533
\(738\) −28.7982 −1.06008
\(739\) 43.2279 1.59016 0.795082 0.606502i \(-0.207428\pi\)
0.795082 + 0.606502i \(0.207428\pi\)
\(740\) 40.8975 1.50342
\(741\) 0 0
\(742\) 41.8483 1.53630
\(743\) −1.45868 −0.0535137 −0.0267568 0.999642i \(-0.508518\pi\)
−0.0267568 + 0.999642i \(0.508518\pi\)
\(744\) −20.5179 −0.752224
\(745\) −27.4890 −1.00712
\(746\) −19.8343 −0.726184
\(747\) −8.10849 −0.296674
\(748\) 10.3805 0.379548
\(749\) 25.3273 0.925438
\(750\) 26.0040 0.949532
\(751\) 15.3593 0.560470 0.280235 0.959931i \(-0.409588\pi\)
0.280235 + 0.959931i \(0.409588\pi\)
\(752\) 0.426600 0.0155565
\(753\) 9.94677 0.362480
\(754\) 0 0
\(755\) −27.2851 −0.993008
\(756\) 79.9677 2.90840
\(757\) 34.3796 1.24955 0.624774 0.780806i \(-0.285191\pi\)
0.624774 + 0.780806i \(0.285191\pi\)
\(758\) −33.6255 −1.22133
\(759\) −7.07730 −0.256890
\(760\) −16.4389 −0.596300
\(761\) 3.88662 0.140890 0.0704449 0.997516i \(-0.477558\pi\)
0.0704449 + 0.997516i \(0.477558\pi\)
\(762\) −16.9688 −0.614715
\(763\) 44.6405 1.61609
\(764\) −10.8334 −0.391938
\(765\) −12.1726 −0.440103
\(766\) 12.3181 0.445071
\(767\) 0 0
\(768\) −19.0470 −0.687300
\(769\) −39.4257 −1.42173 −0.710864 0.703330i \(-0.751696\pi\)
−0.710864 + 0.703330i \(0.751696\pi\)
\(770\) −25.8202 −0.930497
\(771\) −21.8686 −0.787578
\(772\) −50.2560 −1.80875
\(773\) 24.4406 0.879069 0.439534 0.898226i \(-0.355143\pi\)
0.439534 + 0.898226i \(0.355143\pi\)
\(774\) −38.1958 −1.37292
\(775\) −7.72889 −0.277630
\(776\) −0.314105 −0.0112757
\(777\) 27.8835 1.00032
\(778\) −49.2520 −1.76577
\(779\) 19.6384 0.703620
\(780\) 0 0
\(781\) −1.71486 −0.0613624
\(782\) −42.6445 −1.52496
\(783\) 3.41168 0.121923
\(784\) −0.842278 −0.0300814
\(785\) −35.0530 −1.25110
\(786\) −33.3685 −1.19021
\(787\) 36.8171 1.31239 0.656194 0.754592i \(-0.272165\pi\)
0.656194 + 0.754592i \(0.272165\pi\)
\(788\) 67.7499 2.41349
\(789\) −19.0212 −0.677171
\(790\) −10.7329 −0.381859
\(791\) −37.1827 −1.32206
\(792\) 4.20784 0.149519
\(793\) 0 0
\(794\) 40.0610 1.42171
\(795\) 12.4469 0.441445
\(796\) −72.8543 −2.58225
\(797\) 11.5219 0.408128 0.204064 0.978958i \(-0.434585\pi\)
0.204064 + 0.978958i \(0.434585\pi\)
\(798\) −29.5531 −1.04617
\(799\) 21.7249 0.768571
\(800\) −7.36245 −0.260302
\(801\) −12.8202 −0.452981
\(802\) 83.2629 2.94012
\(803\) −2.00000 −0.0705785
\(804\) −34.3633 −1.21190
\(805\) 65.4467 2.30669
\(806\) 0 0
\(807\) −1.82424 −0.0642162
\(808\) −35.6195 −1.25309
\(809\) −24.4649 −0.860140 −0.430070 0.902795i \(-0.641511\pi\)
−0.430070 + 0.902795i \(0.641511\pi\)
\(810\) 12.6485 0.444422
\(811\) −9.19291 −0.322807 −0.161403 0.986889i \(-0.551602\pi\)
−0.161403 + 0.986889i \(0.551602\pi\)
\(812\) 8.99600 0.315698
\(813\) −28.4344 −0.997238
\(814\) 11.5703 0.405538
\(815\) 3.49610 0.122463
\(816\) 0.249063 0.00871894
\(817\) 26.0470 0.911269
\(818\) −65.9637 −2.30637
\(819\) 0 0
\(820\) −67.5460 −2.35881
\(821\) −30.5952 −1.06778 −0.533891 0.845553i \(-0.679271\pi\)
−0.533891 + 0.845553i \(0.679271\pi\)
\(822\) 6.34129 0.221178
\(823\) −19.2148 −0.669784 −0.334892 0.942256i \(-0.608700\pi\)
−0.334892 + 0.942256i \(0.608700\pi\)
\(824\) 43.6427 1.52037
\(825\) −1.57028 −0.0546703
\(826\) 113.943 3.96457
\(827\) −28.4990 −0.991008 −0.495504 0.868606i \(-0.665017\pi\)
−0.495504 + 0.868606i \(0.665017\pi\)
\(828\) −28.1234 −0.977356
\(829\) −6.68278 −0.232103 −0.116051 0.993243i \(-0.537024\pi\)
−0.116051 + 0.993243i \(0.537024\pi\)
\(830\) −30.8242 −1.06992
\(831\) −17.1798 −0.595959
\(832\) 0 0
\(833\) −42.8935 −1.48617
\(834\) −28.3533 −0.981794
\(835\) 59.7530 2.06784
\(836\) −7.56628 −0.261685
\(837\) 33.1194 1.14477
\(838\) −9.60839 −0.331916
\(839\) −8.63044 −0.297956 −0.148978 0.988841i \(-0.547598\pi\)
−0.148978 + 0.988841i \(0.547598\pi\)
\(840\) 38.5491 1.33007
\(841\) −28.6162 −0.986766
\(842\) −31.0421 −1.06978
\(843\) −14.2711 −0.491523
\(844\) 65.6505 2.25978
\(845\) 0 0
\(846\) 23.2210 0.798354
\(847\) −4.50702 −0.154863
\(848\) 0.257068 0.00882775
\(849\) −4.24214 −0.145590
\(850\) −9.46179 −0.324537
\(851\) −29.3273 −1.00533
\(852\) 6.75094 0.231283
\(853\) 18.2631 0.625317 0.312658 0.949866i \(-0.398781\pi\)
0.312658 + 0.949866i \(0.398781\pi\)
\(854\) 1.48898 0.0509518
\(855\) 8.87258 0.303436
\(856\) −15.6906 −0.536293
\(857\) 1.22899 0.0419816 0.0209908 0.999780i \(-0.493318\pi\)
0.0209908 + 0.999780i \(0.493318\pi\)
\(858\) 0 0
\(859\) 4.15772 0.141860 0.0709298 0.997481i \(-0.477403\pi\)
0.0709298 + 0.997481i \(0.477403\pi\)
\(860\) −89.5881 −3.05493
\(861\) −46.0521 −1.56945
\(862\) 68.6414 2.33793
\(863\) 30.0633 1.02337 0.511683 0.859174i \(-0.329022\pi\)
0.511683 + 0.859174i \(0.329022\pi\)
\(864\) 31.5491 1.07332
\(865\) 18.5460 0.630583
\(866\) 9.72801 0.330571
\(867\) −8.08823 −0.274691
\(868\) 87.3302 2.96418
\(869\) −1.87347 −0.0635530
\(870\) 4.33660 0.147024
\(871\) 0 0
\(872\) −27.6554 −0.936530
\(873\) 0.169533 0.00573781
\(874\) 31.0833 1.05141
\(875\) −41.9748 −1.41901
\(876\) 7.87347 0.266020
\(877\) −54.3092 −1.83389 −0.916946 0.399011i \(-0.869353\pi\)
−0.916946 + 0.399011i \(0.869353\pi\)
\(878\) −11.4328 −0.385839
\(879\) −7.79508 −0.262921
\(880\) −0.158610 −0.00534674
\(881\) 32.8162 1.10561 0.552803 0.833312i \(-0.313558\pi\)
0.552803 + 0.833312i \(0.313558\pi\)
\(882\) −45.8474 −1.54376
\(883\) −38.3444 −1.29039 −0.645196 0.764017i \(-0.723224\pi\)
−0.645196 + 0.764017i \(0.723224\pi\)
\(884\) 0 0
\(885\) 33.8897 1.13919
\(886\) 64.6926 2.17339
\(887\) 19.5382 0.656029 0.328014 0.944673i \(-0.393621\pi\)
0.328014 + 0.944673i \(0.393621\pi\)
\(888\) −17.2742 −0.579685
\(889\) 27.3905 0.918649
\(890\) −48.7358 −1.63363
\(891\) 2.20784 0.0739654
\(892\) 57.2660 1.91741
\(893\) −15.8352 −0.529903
\(894\) 30.6155 1.02394
\(895\) −46.4326 −1.55207
\(896\) 81.8864 2.73563
\(897\) 0 0
\(898\) 13.0842 0.436626
\(899\) 3.72578 0.124262
\(900\) −6.23992 −0.207997
\(901\) 13.0913 0.436136
\(902\) −19.1094 −0.636273
\(903\) −61.0802 −2.03262
\(904\) 23.0352 0.766139
\(905\) 45.2671 1.50473
\(906\) 30.3885 1.00959
\(907\) 39.2600 1.30361 0.651803 0.758388i \(-0.274013\pi\)
0.651803 + 0.758388i \(0.274013\pi\)
\(908\) −18.7521 −0.622310
\(909\) 19.2250 0.637653
\(910\) 0 0
\(911\) −11.1194 −0.368403 −0.184201 0.982889i \(-0.558970\pi\)
−0.184201 + 0.982889i \(0.558970\pi\)
\(912\) −0.181540 −0.00601141
\(913\) −5.38049 −0.178068
\(914\) 28.5411 0.944057
\(915\) 0.442864 0.0146406
\(916\) −76.2477 −2.51930
\(917\) 53.8623 1.77869
\(918\) 40.5451 1.33819
\(919\) 8.77501 0.289461 0.144730 0.989471i \(-0.453769\pi\)
0.144730 + 0.989471i \(0.453769\pi\)
\(920\) −40.5451 −1.33673
\(921\) −6.18668 −0.203858
\(922\) 30.1014 0.991336
\(923\) 0 0
\(924\) 17.7429 0.583700
\(925\) −6.50702 −0.213949
\(926\) 23.7850 0.781625
\(927\) −23.5554 −0.773660
\(928\) 3.54913 0.116506
\(929\) −25.2279 −0.827701 −0.413850 0.910345i \(-0.635816\pi\)
−0.413850 + 0.910345i \(0.635816\pi\)
\(930\) 42.0983 1.38046
\(931\) 31.2649 1.02466
\(932\) −73.7407 −2.41546
\(933\) 3.56737 0.116790
\(934\) 53.3703 1.74633
\(935\) −8.07730 −0.264156
\(936\) 0 0
\(937\) 5.68589 0.185750 0.0928750 0.995678i \(-0.470394\pi\)
0.0928750 + 0.995678i \(0.470394\pi\)
\(938\) 89.9005 2.93535
\(939\) 19.2912 0.629544
\(940\) 54.4647 1.77644
\(941\) −12.2468 −0.399235 −0.199618 0.979874i \(-0.563970\pi\)
−0.199618 + 0.979874i \(0.563970\pi\)
\(942\) 39.0399 1.27199
\(943\) 48.4366 1.57731
\(944\) 0.699932 0.0227809
\(945\) −62.2248 −2.02417
\(946\) −25.3453 −0.824047
\(947\) 27.5663 0.895784 0.447892 0.894088i \(-0.352175\pi\)
0.447892 + 0.894088i \(0.352175\pi\)
\(948\) 7.37534 0.239540
\(949\) 0 0
\(950\) 6.89665 0.223757
\(951\) 24.9748 0.809865
\(952\) 40.5451 1.31408
\(953\) 12.3444 0.399875 0.199937 0.979809i \(-0.435926\pi\)
0.199937 + 0.979809i \(0.435926\pi\)
\(954\) 13.9929 0.453036
\(955\) 8.42972 0.272779
\(956\) −52.1967 −1.68816
\(957\) 0.756969 0.0244693
\(958\) −18.7882 −0.607018
\(959\) −10.2359 −0.330535
\(960\) 39.7147 1.28179
\(961\) 5.16864 0.166730
\(962\) 0 0
\(963\) 8.46871 0.272900
\(964\) −0.624405 −0.0201107
\(965\) 39.1054 1.25885
\(966\) −72.8904 −2.34521
\(967\) −13.5451 −0.435582 −0.217791 0.975995i \(-0.569885\pi\)
−0.217791 + 0.975995i \(0.569885\pi\)
\(968\) 2.79216 0.0897435
\(969\) −9.24506 −0.296994
\(970\) 0.644474 0.0206928
\(971\) −1.48987 −0.0478121 −0.0239061 0.999714i \(-0.507610\pi\)
−0.0239061 + 0.999714i \(0.507610\pi\)
\(972\) 44.5371 1.42853
\(973\) 45.7670 1.46722
\(974\) −8.54399 −0.273767
\(975\) 0 0
\(976\) 0.00914657 0.000292775 0
\(977\) 40.1827 1.28556 0.642779 0.766052i \(-0.277781\pi\)
0.642779 + 0.766052i \(0.277781\pi\)
\(978\) −3.89373 −0.124508
\(979\) −8.50702 −0.271885
\(980\) −107.535 −3.43508
\(981\) 14.9265 0.476567
\(982\) −18.6848 −0.596256
\(983\) −26.9617 −0.859944 −0.429972 0.902842i \(-0.641477\pi\)
−0.429972 + 0.902842i \(0.641477\pi\)
\(984\) 28.5299 0.909502
\(985\) −52.7178 −1.67973
\(986\) 4.56114 0.145256
\(987\) 37.1335 1.18197
\(988\) 0 0
\(989\) 64.2428 2.04280
\(990\) −8.63355 −0.274392
\(991\) −41.9225 −1.33171 −0.665856 0.746080i \(-0.731934\pi\)
−0.665856 + 0.746080i \(0.731934\pi\)
\(992\) 34.4538 1.09391
\(993\) 34.5561 1.09660
\(994\) −17.6616 −0.560193
\(995\) 56.6897 1.79718
\(996\) 21.1815 0.671163
\(997\) −39.4037 −1.24793 −0.623963 0.781454i \(-0.714479\pi\)
−0.623963 + 0.781454i \(0.714479\pi\)
\(998\) 2.09935 0.0664536
\(999\) 27.8835 0.882195
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 1859.2.a.g.1.3 3
13.4 even 6 143.2.e.b.133.3 yes 6
13.10 even 6 143.2.e.b.100.3 6
13.12 even 2 1859.2.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
143.2.e.b.100.3 6 13.10 even 6
143.2.e.b.133.3 yes 6 13.4 even 6
1859.2.a.f.1.1 3 13.12 even 2
1859.2.a.g.1.3 3 1.1 even 1 trivial