Properties

Label 5520.2.a.bu.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -0.732051 q^{11} -4.19615 q^{13} +1.00000 q^{15} -5.73205 q^{17} +6.73205 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.267949 q^{29} -3.92820 q^{31} -0.732051 q^{33} +3.00000 q^{35} +11.9282 q^{37} -4.19615 q^{39} -0.267949 q^{41} +4.53590 q^{43} +1.00000 q^{45} +9.66025 q^{47} +2.00000 q^{49} -5.73205 q^{51} -2.26795 q^{53} -0.732051 q^{55} +6.73205 q^{57} +3.19615 q^{59} +13.1244 q^{61} +3.00000 q^{63} -4.19615 q^{65} -0.464102 q^{67} -1.00000 q^{69} +0.267949 q^{71} +9.66025 q^{73} +1.00000 q^{75} -2.19615 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.2679 q^{83} -5.73205 q^{85} -0.267949 q^{87} +9.46410 q^{89} -12.5885 q^{91} -3.92820 q^{93} +6.73205 q^{95} -4.00000 q^{97} -0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 8 q^{17} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} + 2 q^{39} - 4 q^{41} + 16 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{49} - 8 q^{51} - 8 q^{53} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 2 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} + 6 q^{91} + 6 q^{93} + 10 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.732051 −0.220722 −0.110361 0.993892i \(-0.535201\pi\)
−0.110361 + 0.993892i \(0.535201\pi\)
\(12\) 0 0
\(13\) −4.19615 −1.16380 −0.581902 0.813259i \(-0.697691\pi\)
−0.581902 + 0.813259i \(0.697691\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −5.73205 −1.39023 −0.695113 0.718900i \(-0.744646\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) 6.73205 1.54444 0.772219 0.635356i \(-0.219147\pi\)
0.772219 + 0.635356i \(0.219147\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.267949 −0.0497569 −0.0248785 0.999690i \(-0.507920\pi\)
−0.0248785 + 0.999690i \(0.507920\pi\)
\(30\) 0 0
\(31\) −3.92820 −0.705526 −0.352763 0.935713i \(-0.614758\pi\)
−0.352763 + 0.935713i \(0.614758\pi\)
\(32\) 0 0
\(33\) −0.732051 −0.127434
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 11.9282 1.96098 0.980492 0.196558i \(-0.0629763\pi\)
0.980492 + 0.196558i \(0.0629763\pi\)
\(38\) 0 0
\(39\) −4.19615 −0.671922
\(40\) 0 0
\(41\) −0.267949 −0.0418466 −0.0209233 0.999781i \(-0.506661\pi\)
−0.0209233 + 0.999781i \(0.506661\pi\)
\(42\) 0 0
\(43\) 4.53590 0.691718 0.345859 0.938286i \(-0.387588\pi\)
0.345859 + 0.938286i \(0.387588\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 9.66025 1.40909 0.704546 0.709658i \(-0.251150\pi\)
0.704546 + 0.709658i \(0.251150\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −5.73205 −0.802648
\(52\) 0 0
\(53\) −2.26795 −0.311527 −0.155763 0.987794i \(-0.549784\pi\)
−0.155763 + 0.987794i \(0.549784\pi\)
\(54\) 0 0
\(55\) −0.732051 −0.0987097
\(56\) 0 0
\(57\) 6.73205 0.891682
\(58\) 0 0
\(59\) 3.19615 0.416104 0.208052 0.978118i \(-0.433288\pi\)
0.208052 + 0.978118i \(0.433288\pi\)
\(60\) 0 0
\(61\) 13.1244 1.68040 0.840201 0.542275i \(-0.182437\pi\)
0.840201 + 0.542275i \(0.182437\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −4.19615 −0.520469
\(66\) 0 0
\(67\) −0.464102 −0.0566990 −0.0283495 0.999598i \(-0.509025\pi\)
−0.0283495 + 0.999598i \(0.509025\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0.267949 0.0317997 0.0158999 0.999874i \(-0.494939\pi\)
0.0158999 + 0.999874i \(0.494939\pi\)
\(72\) 0 0
\(73\) 9.66025 1.13065 0.565324 0.824869i \(-0.308751\pi\)
0.565324 + 0.824869i \(0.308751\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.19615 −0.250275
\(78\) 0 0
\(79\) 6.92820 0.779484 0.389742 0.920924i \(-0.372564\pi\)
0.389742 + 0.920924i \(0.372564\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.2679 1.12705 0.563527 0.826098i \(-0.309444\pi\)
0.563527 + 0.826098i \(0.309444\pi\)
\(84\) 0 0
\(85\) −5.73205 −0.621728
\(86\) 0 0
\(87\) −0.267949 −0.0287272
\(88\) 0 0
\(89\) 9.46410 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(90\) 0 0
\(91\) −12.5885 −1.31963
\(92\) 0 0
\(93\) −3.92820 −0.407336
\(94\) 0 0
\(95\) 6.73205 0.690694
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) −0.732051 −0.0735739
\(100\) 0 0
\(101\) −5.19615 −0.517036 −0.258518 0.966006i \(-0.583234\pi\)
−0.258518 + 0.966006i \(0.583234\pi\)
\(102\) 0 0
\(103\) 11.8564 1.16825 0.584123 0.811665i \(-0.301438\pi\)
0.584123 + 0.811665i \(0.301438\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 18.1244 1.75215 0.876074 0.482177i \(-0.160154\pi\)
0.876074 + 0.482177i \(0.160154\pi\)
\(108\) 0 0
\(109\) 6.73205 0.644814 0.322407 0.946601i \(-0.395508\pi\)
0.322407 + 0.946601i \(0.395508\pi\)
\(110\) 0 0
\(111\) 11.9282 1.13217
\(112\) 0 0
\(113\) −9.19615 −0.865101 −0.432551 0.901610i \(-0.642386\pi\)
−0.432551 + 0.901610i \(0.642386\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −4.19615 −0.387934
\(118\) 0 0
\(119\) −17.1962 −1.57637
\(120\) 0 0
\(121\) −10.4641 −0.951282
\(122\) 0 0
\(123\) −0.267949 −0.0241602
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.80385 −0.160066 −0.0800328 0.996792i \(-0.525503\pi\)
−0.0800328 + 0.996792i \(0.525503\pi\)
\(128\) 0 0
\(129\) 4.53590 0.399364
\(130\) 0 0
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) 20.1962 1.75123
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 20.7846 1.77575 0.887875 0.460086i \(-0.152181\pi\)
0.887875 + 0.460086i \(0.152181\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 9.66025 0.813540
\(142\) 0 0
\(143\) 3.07180 0.256877
\(144\) 0 0
\(145\) −0.267949 −0.0222520
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −16.1962 −1.32684 −0.663420 0.748247i \(-0.730896\pi\)
−0.663420 + 0.748247i \(0.730896\pi\)
\(150\) 0 0
\(151\) 18.3923 1.49674 0.748372 0.663279i \(-0.230836\pi\)
0.748372 + 0.663279i \(0.230836\pi\)
\(152\) 0 0
\(153\) −5.73205 −0.463409
\(154\) 0 0
\(155\) −3.92820 −0.315521
\(156\) 0 0
\(157\) −13.9282 −1.11159 −0.555796 0.831319i \(-0.687586\pi\)
−0.555796 + 0.831319i \(0.687586\pi\)
\(158\) 0 0
\(159\) −2.26795 −0.179860
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) −0.732051 −0.0569901
\(166\) 0 0
\(167\) 6.33975 0.490584 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(168\) 0 0
\(169\) 4.60770 0.354438
\(170\) 0 0
\(171\) 6.73205 0.514813
\(172\) 0 0
\(173\) −25.8564 −1.96583 −0.982913 0.184070i \(-0.941073\pi\)
−0.982913 + 0.184070i \(0.941073\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) 3.19615 0.240238
\(178\) 0 0
\(179\) −11.4641 −0.856867 −0.428434 0.903573i \(-0.640934\pi\)
−0.428434 + 0.903573i \(0.640934\pi\)
\(180\) 0 0
\(181\) −8.39230 −0.623795 −0.311898 0.950116i \(-0.600965\pi\)
−0.311898 + 0.950116i \(0.600965\pi\)
\(182\) 0 0
\(183\) 13.1244 0.970180
\(184\) 0 0
\(185\) 11.9282 0.876979
\(186\) 0 0
\(187\) 4.19615 0.306853
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 13.6603 0.988421 0.494211 0.869342i \(-0.335457\pi\)
0.494211 + 0.869342i \(0.335457\pi\)
\(192\) 0 0
\(193\) 17.4641 1.25709 0.628547 0.777772i \(-0.283650\pi\)
0.628547 + 0.777772i \(0.283650\pi\)
\(194\) 0 0
\(195\) −4.19615 −0.300493
\(196\) 0 0
\(197\) −22.3923 −1.59539 −0.797693 0.603064i \(-0.793946\pi\)
−0.797693 + 0.603064i \(0.793946\pi\)
\(198\) 0 0
\(199\) 4.53590 0.321541 0.160771 0.986992i \(-0.448602\pi\)
0.160771 + 0.986992i \(0.448602\pi\)
\(200\) 0 0
\(201\) −0.464102 −0.0327352
\(202\) 0 0
\(203\) −0.803848 −0.0564190
\(204\) 0 0
\(205\) −0.267949 −0.0187144
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −4.92820 −0.340891
\(210\) 0 0
\(211\) −19.7846 −1.36203 −0.681014 0.732270i \(-0.738461\pi\)
−0.681014 + 0.732270i \(0.738461\pi\)
\(212\) 0 0
\(213\) 0.267949 0.0183596
\(214\) 0 0
\(215\) 4.53590 0.309346
\(216\) 0 0
\(217\) −11.7846 −0.799991
\(218\) 0 0
\(219\) 9.66025 0.652779
\(220\) 0 0
\(221\) 24.0526 1.61795
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.07180 −0.602116 −0.301058 0.953606i \(-0.597340\pi\)
−0.301058 + 0.953606i \(0.597340\pi\)
\(228\) 0 0
\(229\) −11.3205 −0.748080 −0.374040 0.927413i \(-0.622028\pi\)
−0.374040 + 0.927413i \(0.622028\pi\)
\(230\) 0 0
\(231\) −2.19615 −0.144496
\(232\) 0 0
\(233\) 25.8564 1.69391 0.846955 0.531665i \(-0.178433\pi\)
0.846955 + 0.531665i \(0.178433\pi\)
\(234\) 0 0
\(235\) 9.66025 0.630165
\(236\) 0 0
\(237\) 6.92820 0.450035
\(238\) 0 0
\(239\) −21.0526 −1.36178 −0.680888 0.732387i \(-0.738406\pi\)
−0.680888 + 0.732387i \(0.738406\pi\)
\(240\) 0 0
\(241\) −30.7321 −1.97963 −0.989813 0.142376i \(-0.954526\pi\)
−0.989813 + 0.142376i \(0.954526\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −28.2487 −1.79742
\(248\) 0 0
\(249\) 10.2679 0.650705
\(250\) 0 0
\(251\) −19.8564 −1.25333 −0.626663 0.779291i \(-0.715580\pi\)
−0.626663 + 0.779291i \(0.715580\pi\)
\(252\) 0 0
\(253\) 0.732051 0.0460236
\(254\) 0 0
\(255\) −5.73205 −0.358955
\(256\) 0 0
\(257\) −5.66025 −0.353077 −0.176538 0.984294i \(-0.556490\pi\)
−0.176538 + 0.984294i \(0.556490\pi\)
\(258\) 0 0
\(259\) 35.7846 2.22355
\(260\) 0 0
\(261\) −0.267949 −0.0165856
\(262\) 0 0
\(263\) −8.26795 −0.509824 −0.254912 0.966964i \(-0.582046\pi\)
−0.254912 + 0.966964i \(0.582046\pi\)
\(264\) 0 0
\(265\) −2.26795 −0.139319
\(266\) 0 0
\(267\) 9.46410 0.579194
\(268\) 0 0
\(269\) 28.6603 1.74745 0.873723 0.486423i \(-0.161699\pi\)
0.873723 + 0.486423i \(0.161699\pi\)
\(270\) 0 0
\(271\) −5.39230 −0.327559 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(272\) 0 0
\(273\) −12.5885 −0.761888
\(274\) 0 0
\(275\) −0.732051 −0.0441443
\(276\) 0 0
\(277\) 14.7846 0.888321 0.444161 0.895947i \(-0.353502\pi\)
0.444161 + 0.895947i \(0.353502\pi\)
\(278\) 0 0
\(279\) −3.92820 −0.235175
\(280\) 0 0
\(281\) −16.0526 −0.957615 −0.478808 0.877920i \(-0.658931\pi\)
−0.478808 + 0.877920i \(0.658931\pi\)
\(282\) 0 0
\(283\) 6.46410 0.384251 0.192125 0.981370i \(-0.438462\pi\)
0.192125 + 0.981370i \(0.438462\pi\)
\(284\) 0 0
\(285\) 6.73205 0.398772
\(286\) 0 0
\(287\) −0.803848 −0.0474496
\(288\) 0 0
\(289\) 15.8564 0.932730
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 0 0
\(293\) −7.73205 −0.451711 −0.225856 0.974161i \(-0.572518\pi\)
−0.225856 + 0.974161i \(0.572518\pi\)
\(294\) 0 0
\(295\) 3.19615 0.186087
\(296\) 0 0
\(297\) −0.732051 −0.0424779
\(298\) 0 0
\(299\) 4.19615 0.242670
\(300\) 0 0
\(301\) 13.6077 0.784335
\(302\) 0 0
\(303\) −5.19615 −0.298511
\(304\) 0 0
\(305\) 13.1244 0.751498
\(306\) 0 0
\(307\) −24.1962 −1.38095 −0.690474 0.723358i \(-0.742598\pi\)
−0.690474 + 0.723358i \(0.742598\pi\)
\(308\) 0 0
\(309\) 11.8564 0.674487
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 5.92820 0.335082 0.167541 0.985865i \(-0.446417\pi\)
0.167541 + 0.985865i \(0.446417\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) −27.5167 −1.54549 −0.772745 0.634717i \(-0.781117\pi\)
−0.772745 + 0.634717i \(0.781117\pi\)
\(318\) 0 0
\(319\) 0.196152 0.0109824
\(320\) 0 0
\(321\) 18.1244 1.01160
\(322\) 0 0
\(323\) −38.5885 −2.14712
\(324\) 0 0
\(325\) −4.19615 −0.232761
\(326\) 0 0
\(327\) 6.73205 0.372283
\(328\) 0 0
\(329\) 28.9808 1.59776
\(330\) 0 0
\(331\) 14.3205 0.787126 0.393563 0.919298i \(-0.371242\pi\)
0.393563 + 0.919298i \(0.371242\pi\)
\(332\) 0 0
\(333\) 11.9282 0.653662
\(334\) 0 0
\(335\) −0.464102 −0.0253566
\(336\) 0 0
\(337\) −22.7846 −1.24116 −0.620578 0.784144i \(-0.713102\pi\)
−0.620578 + 0.784144i \(0.713102\pi\)
\(338\) 0 0
\(339\) −9.19615 −0.499466
\(340\) 0 0
\(341\) 2.87564 0.155725
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −7.32051 −0.392985 −0.196493 0.980505i \(-0.562955\pi\)
−0.196493 + 0.980505i \(0.562955\pi\)
\(348\) 0 0
\(349\) 30.3205 1.62302 0.811510 0.584339i \(-0.198646\pi\)
0.811510 + 0.584339i \(0.198646\pi\)
\(350\) 0 0
\(351\) −4.19615 −0.223974
\(352\) 0 0
\(353\) 1.26795 0.0674861 0.0337431 0.999431i \(-0.489257\pi\)
0.0337431 + 0.999431i \(0.489257\pi\)
\(354\) 0 0
\(355\) 0.267949 0.0142213
\(356\) 0 0
\(357\) −17.1962 −0.910117
\(358\) 0 0
\(359\) 27.1244 1.43157 0.715784 0.698321i \(-0.246069\pi\)
0.715784 + 0.698321i \(0.246069\pi\)
\(360\) 0 0
\(361\) 26.3205 1.38529
\(362\) 0 0
\(363\) −10.4641 −0.549223
\(364\) 0 0
\(365\) 9.66025 0.505641
\(366\) 0 0
\(367\) 15.0000 0.782994 0.391497 0.920179i \(-0.371957\pi\)
0.391497 + 0.920179i \(0.371957\pi\)
\(368\) 0 0
\(369\) −0.267949 −0.0139489
\(370\) 0 0
\(371\) −6.80385 −0.353238
\(372\) 0 0
\(373\) 32.3923 1.67721 0.838605 0.544740i \(-0.183372\pi\)
0.838605 + 0.544740i \(0.183372\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.12436 0.0579073
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −1.80385 −0.0924139
\(382\) 0 0
\(383\) 2.41154 0.123224 0.0616120 0.998100i \(-0.480376\pi\)
0.0616120 + 0.998100i \(0.480376\pi\)
\(384\) 0 0
\(385\) −2.19615 −0.111926
\(386\) 0 0
\(387\) 4.53590 0.230573
\(388\) 0 0
\(389\) 3.46410 0.175637 0.0878185 0.996136i \(-0.472010\pi\)
0.0878185 + 0.996136i \(0.472010\pi\)
\(390\) 0 0
\(391\) 5.73205 0.289882
\(392\) 0 0
\(393\) −5.07180 −0.255838
\(394\) 0 0
\(395\) 6.92820 0.348596
\(396\) 0 0
\(397\) 24.7846 1.24390 0.621952 0.783055i \(-0.286340\pi\)
0.621952 + 0.783055i \(0.286340\pi\)
\(398\) 0 0
\(399\) 20.1962 1.01107
\(400\) 0 0
\(401\) 5.32051 0.265693 0.132847 0.991137i \(-0.457588\pi\)
0.132847 + 0.991137i \(0.457588\pi\)
\(402\) 0 0
\(403\) 16.4833 0.821094
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.73205 −0.432832
\(408\) 0 0
\(409\) −11.3923 −0.563313 −0.281657 0.959515i \(-0.590884\pi\)
−0.281657 + 0.959515i \(0.590884\pi\)
\(410\) 0 0
\(411\) 20.7846 1.02523
\(412\) 0 0
\(413\) 9.58846 0.471817
\(414\) 0 0
\(415\) 10.2679 0.504034
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −11.2679 −0.550475 −0.275238 0.961376i \(-0.588757\pi\)
−0.275238 + 0.961376i \(0.588757\pi\)
\(420\) 0 0
\(421\) 21.6603 1.05566 0.527828 0.849351i \(-0.323007\pi\)
0.527828 + 0.849351i \(0.323007\pi\)
\(422\) 0 0
\(423\) 9.66025 0.469698
\(424\) 0 0
\(425\) −5.73205 −0.278045
\(426\) 0 0
\(427\) 39.3731 1.90540
\(428\) 0 0
\(429\) 3.07180 0.148308
\(430\) 0 0
\(431\) 16.3923 0.789590 0.394795 0.918769i \(-0.370816\pi\)
0.394795 + 0.918769i \(0.370816\pi\)
\(432\) 0 0
\(433\) 24.0718 1.15682 0.578408 0.815747i \(-0.303674\pi\)
0.578408 + 0.815747i \(0.303674\pi\)
\(434\) 0 0
\(435\) −0.267949 −0.0128472
\(436\) 0 0
\(437\) −6.73205 −0.322038
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −19.5167 −0.927265 −0.463632 0.886028i \(-0.653454\pi\)
−0.463632 + 0.886028i \(0.653454\pi\)
\(444\) 0 0
\(445\) 9.46410 0.448641
\(446\) 0 0
\(447\) −16.1962 −0.766052
\(448\) 0 0
\(449\) −29.0526 −1.37108 −0.685538 0.728037i \(-0.740433\pi\)
−0.685538 + 0.728037i \(0.740433\pi\)
\(450\) 0 0
\(451\) 0.196152 0.00923646
\(452\) 0 0
\(453\) 18.3923 0.864146
\(454\) 0 0
\(455\) −12.5885 −0.590156
\(456\) 0 0
\(457\) −23.9282 −1.11931 −0.559657 0.828724i \(-0.689067\pi\)
−0.559657 + 0.828724i \(0.689067\pi\)
\(458\) 0 0
\(459\) −5.73205 −0.267549
\(460\) 0 0
\(461\) −27.4641 −1.27913 −0.639565 0.768737i \(-0.720886\pi\)
−0.639565 + 0.768737i \(0.720886\pi\)
\(462\) 0 0
\(463\) −17.6603 −0.820742 −0.410371 0.911919i \(-0.634601\pi\)
−0.410371 + 0.911919i \(0.634601\pi\)
\(464\) 0 0
\(465\) −3.92820 −0.182166
\(466\) 0 0
\(467\) 12.2679 0.567693 0.283846 0.958870i \(-0.408389\pi\)
0.283846 + 0.958870i \(0.408389\pi\)
\(468\) 0 0
\(469\) −1.39230 −0.0642907
\(470\) 0 0
\(471\) −13.9282 −0.641778
\(472\) 0 0
\(473\) −3.32051 −0.152677
\(474\) 0 0
\(475\) 6.73205 0.308888
\(476\) 0 0
\(477\) −2.26795 −0.103842
\(478\) 0 0
\(479\) 38.4449 1.75659 0.878295 0.478119i \(-0.158681\pi\)
0.878295 + 0.478119i \(0.158681\pi\)
\(480\) 0 0
\(481\) −50.0526 −2.28220
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −7.66025 −0.347119 −0.173560 0.984823i \(-0.555527\pi\)
−0.173560 + 0.984823i \(0.555527\pi\)
\(488\) 0 0
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) 25.4449 1.14831 0.574155 0.818746i \(-0.305331\pi\)
0.574155 + 0.818746i \(0.305331\pi\)
\(492\) 0 0
\(493\) 1.53590 0.0691734
\(494\) 0 0
\(495\) −0.732051 −0.0329032
\(496\) 0 0
\(497\) 0.803848 0.0360575
\(498\) 0 0
\(499\) −14.6077 −0.653930 −0.326965 0.945036i \(-0.606026\pi\)
−0.326965 + 0.945036i \(0.606026\pi\)
\(500\) 0 0
\(501\) 6.33975 0.283239
\(502\) 0 0
\(503\) 5.33975 0.238088 0.119044 0.992889i \(-0.462017\pi\)
0.119044 + 0.992889i \(0.462017\pi\)
\(504\) 0 0
\(505\) −5.19615 −0.231226
\(506\) 0 0
\(507\) 4.60770 0.204635
\(508\) 0 0
\(509\) −34.9282 −1.54817 −0.774083 0.633084i \(-0.781789\pi\)
−0.774083 + 0.633084i \(0.781789\pi\)
\(510\) 0 0
\(511\) 28.9808 1.28203
\(512\) 0 0
\(513\) 6.73205 0.297227
\(514\) 0 0
\(515\) 11.8564 0.522456
\(516\) 0 0
\(517\) −7.07180 −0.311017
\(518\) 0 0
\(519\) −25.8564 −1.13497
\(520\) 0 0
\(521\) −12.7321 −0.557801 −0.278901 0.960320i \(-0.589970\pi\)
−0.278901 + 0.960320i \(0.589970\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 22.5167 0.980841
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.19615 0.138701
\(532\) 0 0
\(533\) 1.12436 0.0487012
\(534\) 0 0
\(535\) 18.1244 0.783584
\(536\) 0 0
\(537\) −11.4641 −0.494713
\(538\) 0 0
\(539\) −1.46410 −0.0630633
\(540\) 0 0
\(541\) −1.85641 −0.0798131 −0.0399066 0.999203i \(-0.512706\pi\)
−0.0399066 + 0.999203i \(0.512706\pi\)
\(542\) 0 0
\(543\) −8.39230 −0.360148
\(544\) 0 0
\(545\) 6.73205 0.288369
\(546\) 0 0
\(547\) −34.9282 −1.49342 −0.746711 0.665149i \(-0.768368\pi\)
−0.746711 + 0.665149i \(0.768368\pi\)
\(548\) 0 0
\(549\) 13.1244 0.560134
\(550\) 0 0
\(551\) −1.80385 −0.0768465
\(552\) 0 0
\(553\) 20.7846 0.883852
\(554\) 0 0
\(555\) 11.9282 0.506324
\(556\) 0 0
\(557\) 3.58846 0.152048 0.0760239 0.997106i \(-0.475777\pi\)
0.0760239 + 0.997106i \(0.475777\pi\)
\(558\) 0 0
\(559\) −19.0333 −0.805024
\(560\) 0 0
\(561\) 4.19615 0.177162
\(562\) 0 0
\(563\) 4.51666 0.190355 0.0951773 0.995460i \(-0.469658\pi\)
0.0951773 + 0.995460i \(0.469658\pi\)
\(564\) 0 0
\(565\) −9.19615 −0.386885
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) −17.3205 −0.726113 −0.363057 0.931767i \(-0.618267\pi\)
−0.363057 + 0.931767i \(0.618267\pi\)
\(570\) 0 0
\(571\) −14.5885 −0.610508 −0.305254 0.952271i \(-0.598741\pi\)
−0.305254 + 0.952271i \(0.598741\pi\)
\(572\) 0 0
\(573\) 13.6603 0.570665
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −10.9282 −0.454947 −0.227474 0.973784i \(-0.573047\pi\)
−0.227474 + 0.973784i \(0.573047\pi\)
\(578\) 0 0
\(579\) 17.4641 0.725783
\(580\) 0 0
\(581\) 30.8038 1.27796
\(582\) 0 0
\(583\) 1.66025 0.0687607
\(584\) 0 0
\(585\) −4.19615 −0.173490
\(586\) 0 0
\(587\) −19.8564 −0.819562 −0.409781 0.912184i \(-0.634395\pi\)
−0.409781 + 0.912184i \(0.634395\pi\)
\(588\) 0 0
\(589\) −26.4449 −1.08964
\(590\) 0 0
\(591\) −22.3923 −0.921096
\(592\) 0 0
\(593\) 4.58846 0.188425 0.0942127 0.995552i \(-0.469967\pi\)
0.0942127 + 0.995552i \(0.469967\pi\)
\(594\) 0 0
\(595\) −17.1962 −0.704974
\(596\) 0 0
\(597\) 4.53590 0.185642
\(598\) 0 0
\(599\) −27.3205 −1.11629 −0.558143 0.829745i \(-0.688486\pi\)
−0.558143 + 0.829745i \(0.688486\pi\)
\(600\) 0 0
\(601\) 2.46410 0.100513 0.0502564 0.998736i \(-0.483996\pi\)
0.0502564 + 0.998736i \(0.483996\pi\)
\(602\) 0 0
\(603\) −0.464102 −0.0188997
\(604\) 0 0
\(605\) −10.4641 −0.425426
\(606\) 0 0
\(607\) −37.1244 −1.50683 −0.753416 0.657545i \(-0.771595\pi\)
−0.753416 + 0.657545i \(0.771595\pi\)
\(608\) 0 0
\(609\) −0.803848 −0.0325735
\(610\) 0 0
\(611\) −40.5359 −1.63991
\(612\) 0 0
\(613\) 20.1436 0.813592 0.406796 0.913519i \(-0.366646\pi\)
0.406796 + 0.913519i \(0.366646\pi\)
\(614\) 0 0
\(615\) −0.267949 −0.0108048
\(616\) 0 0
\(617\) −40.5167 −1.63114 −0.815570 0.578659i \(-0.803576\pi\)
−0.815570 + 0.578659i \(0.803576\pi\)
\(618\) 0 0
\(619\) −40.7846 −1.63927 −0.819636 0.572885i \(-0.805824\pi\)
−0.819636 + 0.572885i \(0.805824\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 28.3923 1.13751
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.92820 −0.196813
\(628\) 0 0
\(629\) −68.3731 −2.72621
\(630\) 0 0
\(631\) 17.1244 0.681710 0.340855 0.940116i \(-0.389284\pi\)
0.340855 + 0.940116i \(0.389284\pi\)
\(632\) 0 0
\(633\) −19.7846 −0.786368
\(634\) 0 0
\(635\) −1.80385 −0.0715835
\(636\) 0 0
\(637\) −8.39230 −0.332515
\(638\) 0 0
\(639\) 0.267949 0.0105999
\(640\) 0 0
\(641\) −25.5167 −1.00785 −0.503924 0.863748i \(-0.668111\pi\)
−0.503924 + 0.863748i \(0.668111\pi\)
\(642\) 0 0
\(643\) 43.6410 1.72103 0.860517 0.509422i \(-0.170141\pi\)
0.860517 + 0.509422i \(0.170141\pi\)
\(644\) 0 0
\(645\) 4.53590 0.178601
\(646\) 0 0
\(647\) 4.33975 0.170613 0.0853065 0.996355i \(-0.472813\pi\)
0.0853065 + 0.996355i \(0.472813\pi\)
\(648\) 0 0
\(649\) −2.33975 −0.0918431
\(650\) 0 0
\(651\) −11.7846 −0.461875
\(652\) 0 0
\(653\) 9.41154 0.368302 0.184151 0.982898i \(-0.441046\pi\)
0.184151 + 0.982898i \(0.441046\pi\)
\(654\) 0 0
\(655\) −5.07180 −0.198171
\(656\) 0 0
\(657\) 9.66025 0.376882
\(658\) 0 0
\(659\) −5.80385 −0.226086 −0.113043 0.993590i \(-0.536060\pi\)
−0.113043 + 0.993590i \(0.536060\pi\)
\(660\) 0 0
\(661\) 9.32051 0.362526 0.181263 0.983435i \(-0.441982\pi\)
0.181263 + 0.983435i \(0.441982\pi\)
\(662\) 0 0
\(663\) 24.0526 0.934124
\(664\) 0 0
\(665\) 20.1962 0.783173
\(666\) 0 0
\(667\) 0.267949 0.0103750
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −9.60770 −0.370901
\(672\) 0 0
\(673\) 23.6603 0.912036 0.456018 0.889971i \(-0.349275\pi\)
0.456018 + 0.889971i \(0.349275\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 39.8372 1.53107 0.765533 0.643396i \(-0.222475\pi\)
0.765533 + 0.643396i \(0.222475\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) −9.07180 −0.347632
\(682\) 0 0
\(683\) −10.0526 −0.384650 −0.192325 0.981331i \(-0.561603\pi\)
−0.192325 + 0.981331i \(0.561603\pi\)
\(684\) 0 0
\(685\) 20.7846 0.794139
\(686\) 0 0
\(687\) −11.3205 −0.431904
\(688\) 0 0
\(689\) 9.51666 0.362556
\(690\) 0 0
\(691\) 2.67949 0.101933 0.0509663 0.998700i \(-0.483770\pi\)
0.0509663 + 0.998700i \(0.483770\pi\)
\(692\) 0 0
\(693\) −2.19615 −0.0834249
\(694\) 0 0
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 1.53590 0.0581763
\(698\) 0 0
\(699\) 25.8564 0.977979
\(700\) 0 0
\(701\) −15.5167 −0.586056 −0.293028 0.956104i \(-0.594663\pi\)
−0.293028 + 0.956104i \(0.594663\pi\)
\(702\) 0 0
\(703\) 80.3013 3.02862
\(704\) 0 0
\(705\) 9.66025 0.363826
\(706\) 0 0
\(707\) −15.5885 −0.586264
\(708\) 0 0
\(709\) −7.66025 −0.287687 −0.143843 0.989600i \(-0.545946\pi\)
−0.143843 + 0.989600i \(0.545946\pi\)
\(710\) 0 0
\(711\) 6.92820 0.259828
\(712\) 0 0
\(713\) 3.92820 0.147112
\(714\) 0 0
\(715\) 3.07180 0.114879
\(716\) 0 0
\(717\) −21.0526 −0.786222
\(718\) 0 0
\(719\) −5.19615 −0.193784 −0.0968919 0.995295i \(-0.530890\pi\)
−0.0968919 + 0.995295i \(0.530890\pi\)
\(720\) 0 0
\(721\) 35.5692 1.32467
\(722\) 0 0
\(723\) −30.7321 −1.14294
\(724\) 0 0
\(725\) −0.267949 −0.00995138
\(726\) 0 0
\(727\) 50.1769 1.86096 0.930479 0.366344i \(-0.119391\pi\)
0.930479 + 0.366344i \(0.119391\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −26.0000 −0.961645
\(732\) 0 0
\(733\) −27.1051 −1.00115 −0.500575 0.865693i \(-0.666878\pi\)
−0.500575 + 0.865693i \(0.666878\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 0.339746 0.0125147
\(738\) 0 0
\(739\) −48.3205 −1.77750 −0.888749 0.458394i \(-0.848425\pi\)
−0.888749 + 0.458394i \(0.848425\pi\)
\(740\) 0 0
\(741\) −28.2487 −1.03774
\(742\) 0 0
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 0 0
\(745\) −16.1962 −0.593381
\(746\) 0 0
\(747\) 10.2679 0.375685
\(748\) 0 0
\(749\) 54.3731 1.98675
\(750\) 0 0
\(751\) −5.26795 −0.192230 −0.0961151 0.995370i \(-0.530642\pi\)
−0.0961151 + 0.995370i \(0.530642\pi\)
\(752\) 0 0
\(753\) −19.8564 −0.723608
\(754\) 0 0
\(755\) 18.3923 0.669365
\(756\) 0 0
\(757\) −18.6077 −0.676308 −0.338154 0.941091i \(-0.609803\pi\)
−0.338154 + 0.941091i \(0.609803\pi\)
\(758\) 0 0
\(759\) 0.732051 0.0265718
\(760\) 0 0
\(761\) −3.33975 −0.121066 −0.0605328 0.998166i \(-0.519280\pi\)
−0.0605328 + 0.998166i \(0.519280\pi\)
\(762\) 0 0
\(763\) 20.1962 0.731150
\(764\) 0 0
\(765\) −5.73205 −0.207243
\(766\) 0 0
\(767\) −13.4115 −0.484263
\(768\) 0 0
\(769\) −1.80385 −0.0650484 −0.0325242 0.999471i \(-0.510355\pi\)
−0.0325242 + 0.999471i \(0.510355\pi\)
\(770\) 0 0
\(771\) −5.66025 −0.203849
\(772\) 0 0
\(773\) 15.4641 0.556205 0.278103 0.960551i \(-0.410294\pi\)
0.278103 + 0.960551i \(0.410294\pi\)
\(774\) 0 0
\(775\) −3.92820 −0.141105
\(776\) 0 0
\(777\) 35.7846 1.28377
\(778\) 0 0
\(779\) −1.80385 −0.0646295
\(780\) 0 0
\(781\) −0.196152 −0.00701889
\(782\) 0 0
\(783\) −0.267949 −0.00957572
\(784\) 0 0
\(785\) −13.9282 −0.497119
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 0 0
\(789\) −8.26795 −0.294347
\(790\) 0 0
\(791\) −27.5885 −0.980933
\(792\) 0 0
\(793\) −55.0718 −1.95566
\(794\) 0 0
\(795\) −2.26795 −0.0804359
\(796\) 0 0
\(797\) 29.9808 1.06197 0.530987 0.847380i \(-0.321821\pi\)
0.530987 + 0.847380i \(0.321821\pi\)
\(798\) 0 0
\(799\) −55.3731 −1.95896
\(800\) 0 0
\(801\) 9.46410 0.334398
\(802\) 0 0
\(803\) −7.07180 −0.249558
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 28.6603 1.00889
\(808\) 0 0
\(809\) 10.1244 0.355953 0.177977 0.984035i \(-0.443045\pi\)
0.177977 + 0.984035i \(0.443045\pi\)
\(810\) 0 0
\(811\) −3.00000 −0.105344 −0.0526721 0.998612i \(-0.516774\pi\)
−0.0526721 + 0.998612i \(0.516774\pi\)
\(812\) 0 0
\(813\) −5.39230 −0.189116
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) 30.5359 1.06832
\(818\) 0 0
\(819\) −12.5885 −0.439876
\(820\) 0 0
\(821\) −39.7128 −1.38599 −0.692993 0.720944i \(-0.743708\pi\)
−0.692993 + 0.720944i \(0.743708\pi\)
\(822\) 0 0
\(823\) 46.9282 1.63581 0.817907 0.575350i \(-0.195134\pi\)
0.817907 + 0.575350i \(0.195134\pi\)
\(824\) 0 0
\(825\) −0.732051 −0.0254867
\(826\) 0 0
\(827\) −32.3731 −1.12572 −0.562861 0.826552i \(-0.690299\pi\)
−0.562861 + 0.826552i \(0.690299\pi\)
\(828\) 0 0
\(829\) −27.9282 −0.969987 −0.484993 0.874518i \(-0.661178\pi\)
−0.484993 + 0.874518i \(0.661178\pi\)
\(830\) 0 0
\(831\) 14.7846 0.512872
\(832\) 0 0
\(833\) −11.4641 −0.397208
\(834\) 0 0
\(835\) 6.33975 0.219396
\(836\) 0 0
\(837\) −3.92820 −0.135779
\(838\) 0 0
\(839\) −11.8564 −0.409329 −0.204664 0.978832i \(-0.565610\pi\)
−0.204664 + 0.978832i \(0.565610\pi\)
\(840\) 0 0
\(841\) −28.9282 −0.997524
\(842\) 0 0
\(843\) −16.0526 −0.552879
\(844\) 0 0
\(845\) 4.60770 0.158510
\(846\) 0 0
\(847\) −31.3923 −1.07865
\(848\) 0 0
\(849\) 6.46410 0.221847
\(850\) 0 0
\(851\) −11.9282 −0.408894
\(852\) 0 0
\(853\) 15.4641 0.529481 0.264740 0.964320i \(-0.414714\pi\)
0.264740 + 0.964320i \(0.414714\pi\)
\(854\) 0 0
\(855\) 6.73205 0.230231
\(856\) 0 0
\(857\) −4.92820 −0.168344 −0.0841721 0.996451i \(-0.526825\pi\)
−0.0841721 + 0.996451i \(0.526825\pi\)
\(858\) 0 0
\(859\) −36.0718 −1.23075 −0.615377 0.788233i \(-0.710996\pi\)
−0.615377 + 0.788233i \(0.710996\pi\)
\(860\) 0 0
\(861\) −0.803848 −0.0273951
\(862\) 0 0
\(863\) 6.39230 0.217597 0.108798 0.994064i \(-0.465300\pi\)
0.108798 + 0.994064i \(0.465300\pi\)
\(864\) 0 0
\(865\) −25.8564 −0.879144
\(866\) 0 0
\(867\) 15.8564 0.538512
\(868\) 0 0
\(869\) −5.07180 −0.172049
\(870\) 0 0
\(871\) 1.94744 0.0659865
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 20.5359 0.693448 0.346724 0.937967i \(-0.387294\pi\)
0.346724 + 0.937967i \(0.387294\pi\)
\(878\) 0 0
\(879\) −7.73205 −0.260796
\(880\) 0 0
\(881\) −24.1962 −0.815189 −0.407595 0.913163i \(-0.633632\pi\)
−0.407595 + 0.913163i \(0.633632\pi\)
\(882\) 0 0
\(883\) 44.0526 1.48249 0.741243 0.671236i \(-0.234236\pi\)
0.741243 + 0.671236i \(0.234236\pi\)
\(884\) 0 0
\(885\) 3.19615 0.107437
\(886\) 0 0
\(887\) 3.46410 0.116313 0.0581566 0.998307i \(-0.481478\pi\)
0.0581566 + 0.998307i \(0.481478\pi\)
\(888\) 0 0
\(889\) −5.41154 −0.181497
\(890\) 0 0
\(891\) −0.732051 −0.0245246
\(892\) 0 0
\(893\) 65.0333 2.17626
\(894\) 0 0
\(895\) −11.4641 −0.383203
\(896\) 0 0
\(897\) 4.19615 0.140105
\(898\) 0 0
\(899\) 1.05256 0.0351048
\(900\) 0 0
\(901\) 13.0000 0.433093
\(902\) 0 0
\(903\) 13.6077 0.452836
\(904\) 0 0
\(905\) −8.39230 −0.278970
\(906\) 0 0
\(907\) −40.1769 −1.33405 −0.667026 0.745034i \(-0.732433\pi\)
−0.667026 + 0.745034i \(0.732433\pi\)
\(908\) 0 0
\(909\) −5.19615 −0.172345
\(910\) 0 0
\(911\) 34.9282 1.15722 0.578612 0.815603i \(-0.303595\pi\)
0.578612 + 0.815603i \(0.303595\pi\)
\(912\) 0 0
\(913\) −7.51666 −0.248765
\(914\) 0 0
\(915\) 13.1244 0.433878
\(916\) 0 0
\(917\) −15.2154 −0.502456
\(918\) 0 0
\(919\) 33.8564 1.11682 0.558410 0.829565i \(-0.311412\pi\)
0.558410 + 0.829565i \(0.311412\pi\)
\(920\) 0 0
\(921\) −24.1962 −0.797290
\(922\) 0 0
\(923\) −1.12436 −0.0370086
\(924\) 0 0
\(925\) 11.9282 0.392197
\(926\) 0 0
\(927\) 11.8564 0.389415
\(928\) 0 0
\(929\) 29.9808 0.983637 0.491818 0.870698i \(-0.336332\pi\)
0.491818 + 0.870698i \(0.336332\pi\)
\(930\) 0 0
\(931\) 13.4641 0.441268
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 4.19615 0.137229
\(936\) 0 0
\(937\) 36.4974 1.19232 0.596159 0.802866i \(-0.296693\pi\)
0.596159 + 0.802866i \(0.296693\pi\)
\(938\) 0 0
\(939\) 5.92820 0.193460
\(940\) 0 0
\(941\) 48.6936 1.58737 0.793683 0.608332i \(-0.208161\pi\)
0.793683 + 0.608332i \(0.208161\pi\)
\(942\) 0 0
\(943\) 0.267949 0.00872563
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) −31.6077 −1.02711 −0.513556 0.858056i \(-0.671672\pi\)
−0.513556 + 0.858056i \(0.671672\pi\)
\(948\) 0 0
\(949\) −40.5359 −1.31585
\(950\) 0 0
\(951\) −27.5167 −0.892289
\(952\) 0 0
\(953\) 12.1436 0.393370 0.196685 0.980467i \(-0.436982\pi\)
0.196685 + 0.980467i \(0.436982\pi\)
\(954\) 0 0
\(955\) 13.6603 0.442035
\(956\) 0 0
\(957\) 0.196152 0.00634071
\(958\) 0 0
\(959\) 62.3538 2.01351
\(960\) 0 0
\(961\) −15.5692 −0.502233
\(962\) 0 0
\(963\) 18.1244 0.584049
\(964\) 0 0
\(965\) 17.4641 0.562189
\(966\) 0 0
\(967\) −5.80385 −0.186639 −0.0933196 0.995636i \(-0.529748\pi\)
−0.0933196 + 0.995636i \(0.529748\pi\)
\(968\) 0 0
\(969\) −38.5885 −1.23964
\(970\) 0 0
\(971\) −2.53590 −0.0813809 −0.0406904 0.999172i \(-0.512956\pi\)
−0.0406904 + 0.999172i \(0.512956\pi\)
\(972\) 0 0
\(973\) −15.0000 −0.480878
\(974\) 0 0
\(975\) −4.19615 −0.134384
\(976\) 0 0
\(977\) −16.1244 −0.515864 −0.257932 0.966163i \(-0.583041\pi\)
−0.257932 + 0.966163i \(0.583041\pi\)
\(978\) 0 0
\(979\) −6.92820 −0.221426
\(980\) 0 0
\(981\) 6.73205 0.214938
\(982\) 0 0
\(983\) 13.9808 0.445917 0.222959 0.974828i \(-0.428429\pi\)
0.222959 + 0.974828i \(0.428429\pi\)
\(984\) 0 0
\(985\) −22.3923 −0.713478
\(986\) 0 0
\(987\) 28.9808 0.922468
\(988\) 0 0
\(989\) −4.53590 −0.144233
\(990\) 0 0
\(991\) 25.0000 0.794151 0.397076 0.917786i \(-0.370025\pi\)
0.397076 + 0.917786i \(0.370025\pi\)
\(992\) 0 0
\(993\) 14.3205 0.454448
\(994\) 0 0
\(995\) 4.53590 0.143798
\(996\) 0 0
\(997\) −57.4641 −1.81991 −0.909953 0.414711i \(-0.863883\pi\)
−0.909953 + 0.414711i \(0.863883\pi\)
\(998\) 0 0
\(999\) 11.9282 0.377392
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 5520.2.a.bu.1.1 2
4.3 odd 2 345.2.a.g.1.2 2
12.11 even 2 1035.2.a.l.1.1 2
20.3 even 4 1725.2.b.p.1174.2 4
20.7 even 4 1725.2.b.p.1174.3 4
20.19 odd 2 1725.2.a.bd.1.1 2
60.59 even 2 5175.2.a.bd.1.2 2
92.91 even 2 7935.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.g.1.2 2 4.3 odd 2
1035.2.a.l.1.1 2 12.11 even 2
1725.2.a.bd.1.1 2 20.19 odd 2
1725.2.b.p.1174.2 4 20.3 even 4
1725.2.b.p.1174.3 4 20.7 even 4
5175.2.a.bd.1.2 2 60.59 even 2
5520.2.a.bu.1.1 2 1.1 even 1 trivial
7935.2.a.n.1.2 2 92.91 even 2