Properties

Label 8880.2.a.cg.1.1
Level $8880$
Weight $2$
Character 8880.1
Self dual yes
Analytic conductor $70.907$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.9071569949\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.67673\) of defining polynomial
Character \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -5.13277 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -5.13277 q^{7} +1.00000 q^{9} -2.22069 q^{11} -6.60629 q^{13} -1.00000 q^{15} +5.63841 q^{17} +7.51836 q^{19} +5.13277 q^{21} +6.10064 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.29767 q^{29} +0.967873 q^{31} +2.22069 q^{33} -5.13277 q^{35} -1.00000 q^{37} +6.60629 q^{39} -1.03213 q^{41} -5.29767 q^{43} +1.00000 q^{45} +5.85911 q^{47} +19.3453 q^{49} -5.63841 q^{51} -9.63841 q^{53} -2.22069 q^{55} -7.51836 q^{57} +10.2655 q^{59} +6.32134 q^{61} -5.13277 q^{63} -6.60629 q^{65} -1.49436 q^{67} -6.10064 q^{69} -0.329796 q^{71} +12.8718 q^{73} -1.00000 q^{75} +11.3983 q^{77} -4.00000 q^{79} +1.00000 q^{81} -9.04767 q^{83} +5.63841 q^{85} +7.29767 q^{87} -2.27649 q^{89} +33.9086 q^{91} -0.967873 q^{93} +7.51836 q^{95} +17.5982 q^{97} -2.22069 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} - 2q^{11} - 3q^{13} - 4q^{15} + 6q^{17} - 3q^{19} + 4q^{21} + q^{23} + 4q^{25} - 4q^{27} - 3q^{29} - 3q^{31} + 2q^{33} - 4q^{35} - 4q^{37} + 3q^{39} - 11q^{41} + 5q^{43} + 4q^{45} + 14q^{49} - 6q^{51} - 22q^{53} - 2q^{55} + 3q^{57} + 8q^{59} - 5q^{61} - 4q^{63} - 3q^{65} - 6q^{67} - q^{69} + 18q^{71} - 5q^{73} - 4q^{75} - 4q^{77} - 16q^{79} + 4q^{81} + q^{83} + 6q^{85} + 3q^{87} - 5q^{89} + 13q^{91} + 3q^{93} - 3q^{95} + 7q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.13277 −1.94001 −0.970003 0.243094i \(-0.921838\pi\)
−0.970003 + 0.243094i \(0.921838\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.22069 −0.669564 −0.334782 0.942296i \(-0.608663\pi\)
−0.334782 + 0.942296i \(0.608663\pi\)
\(12\) 0 0
\(13\) −6.60629 −1.83225 −0.916127 0.400888i \(-0.868702\pi\)
−0.916127 + 0.400888i \(0.868702\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.63841 1.36752 0.683758 0.729709i \(-0.260344\pi\)
0.683758 + 0.729709i \(0.260344\pi\)
\(18\) 0 0
\(19\) 7.51836 1.72483 0.862415 0.506201i \(-0.168951\pi\)
0.862415 + 0.506201i \(0.168951\pi\)
\(20\) 0 0
\(21\) 5.13277 1.12006
\(22\) 0 0
\(23\) 6.10064 1.27207 0.636036 0.771659i \(-0.280573\pi\)
0.636036 + 0.771659i \(0.280573\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.29767 −1.35514 −0.677572 0.735457i \(-0.736968\pi\)
−0.677572 + 0.735457i \(0.736968\pi\)
\(30\) 0 0
\(31\) 0.967873 0.173835 0.0869176 0.996216i \(-0.472298\pi\)
0.0869176 + 0.996216i \(0.472298\pi\)
\(32\) 0 0
\(33\) 2.22069 0.386573
\(34\) 0 0
\(35\) −5.13277 −0.867597
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 6.60629 1.05785
\(40\) 0 0
\(41\) −1.03213 −0.161191 −0.0805955 0.996747i \(-0.525682\pi\)
−0.0805955 + 0.996747i \(0.525682\pi\)
\(42\) 0 0
\(43\) −5.29767 −0.807887 −0.403944 0.914784i \(-0.632361\pi\)
−0.403944 + 0.914784i \(0.632361\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 5.85911 0.854638 0.427319 0.904101i \(-0.359458\pi\)
0.427319 + 0.904101i \(0.359458\pi\)
\(48\) 0 0
\(49\) 19.3453 2.76362
\(50\) 0 0
\(51\) −5.63841 −0.789536
\(52\) 0 0
\(53\) −9.63841 −1.32394 −0.661969 0.749531i \(-0.730279\pi\)
−0.661969 + 0.749531i \(0.730279\pi\)
\(54\) 0 0
\(55\) −2.22069 −0.299438
\(56\) 0 0
\(57\) −7.51836 −0.995832
\(58\) 0 0
\(59\) 10.2655 1.33646 0.668230 0.743955i \(-0.267052\pi\)
0.668230 + 0.743955i \(0.267052\pi\)
\(60\) 0 0
\(61\) 6.32134 0.809365 0.404682 0.914457i \(-0.367382\pi\)
0.404682 + 0.914457i \(0.367382\pi\)
\(62\) 0 0
\(63\) −5.13277 −0.646668
\(64\) 0 0
\(65\) −6.60629 −0.819409
\(66\) 0 0
\(67\) −1.49436 −0.182565 −0.0912825 0.995825i \(-0.529097\pi\)
−0.0912825 + 0.995825i \(0.529097\pi\)
\(68\) 0 0
\(69\) −6.10064 −0.734431
\(70\) 0 0
\(71\) −0.329796 −0.0391396 −0.0195698 0.999808i \(-0.506230\pi\)
−0.0195698 + 0.999808i \(0.506230\pi\)
\(72\) 0 0
\(73\) 12.8718 1.50653 0.753267 0.657715i \(-0.228477\pi\)
0.753267 + 0.657715i \(0.228477\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 11.3983 1.29896
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.04767 −0.993111 −0.496556 0.868005i \(-0.665402\pi\)
−0.496556 + 0.868005i \(0.665402\pi\)
\(84\) 0 0
\(85\) 5.63841 0.611572
\(86\) 0 0
\(87\) 7.29767 0.782392
\(88\) 0 0
\(89\) −2.27649 −0.241307 −0.120654 0.992695i \(-0.538499\pi\)
−0.120654 + 0.992695i \(0.538499\pi\)
\(90\) 0 0
\(91\) 33.9086 3.55458
\(92\) 0 0
\(93\) −0.967873 −0.100364
\(94\) 0 0
\(95\) 7.51836 0.771368
\(96\) 0 0
\(97\) 17.5982 1.78682 0.893411 0.449239i \(-0.148305\pi\)
0.893411 + 0.449239i \(0.148305\pi\)
\(98\) 0 0
\(99\) −2.22069 −0.223188
\(100\) 0 0
\(101\) 7.61901 0.758120 0.379060 0.925372i \(-0.376247\pi\)
0.379060 + 0.925372i \(0.376247\pi\)
\(102\) 0 0
\(103\) −5.85911 −0.577315 −0.288657 0.957432i \(-0.593209\pi\)
−0.288657 + 0.957432i \(0.593209\pi\)
\(104\) 0 0
\(105\) 5.13277 0.500907
\(106\) 0 0
\(107\) −5.65926 −0.547101 −0.273551 0.961858i \(-0.588198\pi\)
−0.273551 + 0.961858i \(0.588198\pi\)
\(108\) 0 0
\(109\) −3.16772 −0.303413 −0.151706 0.988426i \(-0.548477\pi\)
−0.151706 + 0.988426i \(0.548477\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −0.702331 −0.0660697 −0.0330348 0.999454i \(-0.510517\pi\)
−0.0330348 + 0.999454i \(0.510517\pi\)
\(114\) 0 0
\(115\) 6.10064 0.568888
\(116\) 0 0
\(117\) −6.60629 −0.610751
\(118\) 0 0
\(119\) −28.9407 −2.65299
\(120\) 0 0
\(121\) −6.06852 −0.551683
\(122\) 0 0
\(123\) 1.03213 0.0930637
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.3132 1.71377 0.856885 0.515507i \(-0.172396\pi\)
0.856885 + 0.515507i \(0.172396\pi\)
\(128\) 0 0
\(129\) 5.29767 0.466434
\(130\) 0 0
\(131\) 18.5953 1.62468 0.812341 0.583183i \(-0.198193\pi\)
0.812341 + 0.583183i \(0.198193\pi\)
\(132\) 0 0
\(133\) −38.5900 −3.34618
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −19.5074 −1.66663 −0.833316 0.552798i \(-0.813560\pi\)
−0.833316 + 0.552798i \(0.813560\pi\)
\(138\) 0 0
\(139\) −10.9036 −0.924833 −0.462417 0.886663i \(-0.653018\pi\)
−0.462417 + 0.886663i \(0.653018\pi\)
\(140\) 0 0
\(141\) −5.85911 −0.493426
\(142\) 0 0
\(143\) 14.6705 1.22681
\(144\) 0 0
\(145\) −7.29767 −0.606038
\(146\) 0 0
\(147\) −19.3453 −1.59558
\(148\) 0 0
\(149\) −12.6720 −1.03813 −0.519065 0.854735i \(-0.673720\pi\)
−0.519065 + 0.854735i \(0.673720\pi\)
\(150\) 0 0
\(151\) −10.5420 −0.857898 −0.428949 0.903329i \(-0.641116\pi\)
−0.428949 + 0.903329i \(0.641116\pi\)
\(152\) 0 0
\(153\) 5.63841 0.455839
\(154\) 0 0
\(155\) 0.967873 0.0777415
\(156\) 0 0
\(157\) −15.4990 −1.23695 −0.618476 0.785804i \(-0.712250\pi\)
−0.618476 + 0.785804i \(0.712250\pi\)
\(158\) 0 0
\(159\) 9.63841 0.764376
\(160\) 0 0
\(161\) −31.3132 −2.46783
\(162\) 0 0
\(163\) 4.12149 0.322820 0.161410 0.986887i \(-0.448396\pi\)
0.161410 + 0.986887i \(0.448396\pi\)
\(164\) 0 0
\(165\) 2.22069 0.172881
\(166\) 0 0
\(167\) −21.9474 −1.69834 −0.849169 0.528121i \(-0.822897\pi\)
−0.849169 + 0.528121i \(0.822897\pi\)
\(168\) 0 0
\(169\) 30.6430 2.35715
\(170\) 0 0
\(171\) 7.51836 0.574944
\(172\) 0 0
\(173\) −0.425841 −0.0323761 −0.0161880 0.999869i \(-0.505153\pi\)
−0.0161880 + 0.999869i \(0.505153\pi\)
\(174\) 0 0
\(175\) −5.13277 −0.388001
\(176\) 0 0
\(177\) −10.2655 −0.771605
\(178\) 0 0
\(179\) 17.5424 1.31118 0.655589 0.755118i \(-0.272420\pi\)
0.655589 + 0.755118i \(0.272420\pi\)
\(180\) 0 0
\(181\) 18.5311 1.37740 0.688702 0.725044i \(-0.258181\pi\)
0.688702 + 0.725044i \(0.258181\pi\)
\(182\) 0 0
\(183\) −6.32134 −0.467287
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −12.5212 −0.915640
\(188\) 0 0
\(189\) 5.13277 0.373354
\(190\) 0 0
\(191\) −16.1695 −1.16998 −0.584992 0.811039i \(-0.698902\pi\)
−0.584992 + 0.811039i \(0.698902\pi\)
\(192\) 0 0
\(193\) −11.8591 −0.853637 −0.426819 0.904337i \(-0.640366\pi\)
−0.426819 + 0.904337i \(0.640366\pi\)
\(194\) 0 0
\(195\) 6.60629 0.473086
\(196\) 0 0
\(197\) 0.100645 0.00717066 0.00358533 0.999994i \(-0.498859\pi\)
0.00358533 + 0.999994i \(0.498859\pi\)
\(198\) 0 0
\(199\) 2.70693 0.191889 0.0959446 0.995387i \(-0.469413\pi\)
0.0959446 + 0.995387i \(0.469413\pi\)
\(200\) 0 0
\(201\) 1.49436 0.105404
\(202\) 0 0
\(203\) 37.4573 2.62898
\(204\) 0 0
\(205\) −1.03213 −0.0720868
\(206\) 0 0
\(207\) 6.10064 0.424024
\(208\) 0 0
\(209\) −16.6960 −1.15489
\(210\) 0 0
\(211\) −11.8033 −0.812573 −0.406287 0.913746i \(-0.633177\pi\)
−0.406287 + 0.913746i \(0.633177\pi\)
\(212\) 0 0
\(213\) 0.329796 0.0225972
\(214\) 0 0
\(215\) −5.29767 −0.361298
\(216\) 0 0
\(217\) −4.96787 −0.337241
\(218\) 0 0
\(219\) −12.8718 −0.869798
\(220\) 0 0
\(221\) −37.2490 −2.50564
\(222\) 0 0
\(223\) −13.1692 −0.881872 −0.440936 0.897538i \(-0.645354\pi\)
−0.440936 + 0.897538i \(0.645354\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −5.57946 −0.370322 −0.185161 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(228\) 0 0
\(229\) −1.75990 −0.116298 −0.0581488 0.998308i \(-0.518520\pi\)
−0.0581488 + 0.998308i \(0.518520\pi\)
\(230\) 0 0
\(231\) −11.3983 −0.749954
\(232\) 0 0
\(233\) −6.07664 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(234\) 0 0
\(235\) 5.85911 0.382206
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −13.9403 −0.901726 −0.450863 0.892593i \(-0.648884\pi\)
−0.450863 + 0.892593i \(0.648884\pi\)
\(240\) 0 0
\(241\) −3.89406 −0.250838 −0.125419 0.992104i \(-0.540028\pi\)
−0.125419 + 0.992104i \(0.540028\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 19.3453 1.23593
\(246\) 0 0
\(247\) −49.6685 −3.16033
\(248\) 0 0
\(249\) 9.04767 0.573373
\(250\) 0 0
\(251\) −20.9308 −1.32114 −0.660570 0.750765i \(-0.729685\pi\)
−0.660570 + 0.750765i \(0.729685\pi\)
\(252\) 0 0
\(253\) −13.5477 −0.851734
\(254\) 0 0
\(255\) −5.63841 −0.353091
\(256\) 0 0
\(257\) −4.43044 −0.276363 −0.138182 0.990407i \(-0.544126\pi\)
−0.138182 + 0.990407i \(0.544126\pi\)
\(258\) 0 0
\(259\) 5.13277 0.318935
\(260\) 0 0
\(261\) −7.29767 −0.451714
\(262\) 0 0
\(263\) −4.20975 −0.259584 −0.129792 0.991541i \(-0.541431\pi\)
−0.129792 + 0.991541i \(0.541431\pi\)
\(264\) 0 0
\(265\) −9.63841 −0.592083
\(266\) 0 0
\(267\) 2.27649 0.139319
\(268\) 0 0
\(269\) 12.5071 0.762570 0.381285 0.924457i \(-0.375482\pi\)
0.381285 + 0.924457i \(0.375482\pi\)
\(270\) 0 0
\(271\) 6.70693 0.407417 0.203709 0.979032i \(-0.434701\pi\)
0.203709 + 0.979032i \(0.434701\pi\)
\(272\) 0 0
\(273\) −33.9086 −2.05224
\(274\) 0 0
\(275\) −2.22069 −0.133913
\(276\) 0 0
\(277\) −27.0477 −1.62514 −0.812569 0.582866i \(-0.801931\pi\)
−0.812569 + 0.582866i \(0.801931\pi\)
\(278\) 0 0
\(279\) 0.967873 0.0579451
\(280\) 0 0
\(281\) −1.39371 −0.0831420 −0.0415710 0.999136i \(-0.513236\pi\)
−0.0415710 + 0.999136i \(0.513236\pi\)
\(282\) 0 0
\(283\) 15.3132 0.910276 0.455138 0.890421i \(-0.349590\pi\)
0.455138 + 0.890421i \(0.349590\pi\)
\(284\) 0 0
\(285\) −7.51836 −0.445349
\(286\) 0 0
\(287\) 5.29767 0.312712
\(288\) 0 0
\(289\) 14.7917 0.870100
\(290\) 0 0
\(291\) −17.5982 −1.03162
\(292\) 0 0
\(293\) −12.8672 −0.751712 −0.375856 0.926678i \(-0.622651\pi\)
−0.375856 + 0.926678i \(0.622651\pi\)
\(294\) 0 0
\(295\) 10.2655 0.597683
\(296\) 0 0
\(297\) 2.22069 0.128858
\(298\) 0 0
\(299\) −40.3026 −2.33076
\(300\) 0 0
\(301\) 27.1917 1.56731
\(302\) 0 0
\(303\) −7.61901 −0.437701
\(304\) 0 0
\(305\) 6.32134 0.361959
\(306\) 0 0
\(307\) 4.08970 0.233411 0.116706 0.993167i \(-0.462767\pi\)
0.116706 + 0.993167i \(0.462767\pi\)
\(308\) 0 0
\(309\) 5.85911 0.333313
\(310\) 0 0
\(311\) 10.3090 0.584567 0.292283 0.956332i \(-0.405585\pi\)
0.292283 + 0.956332i \(0.405585\pi\)
\(312\) 0 0
\(313\) −2.64653 −0.149591 −0.0747955 0.997199i \(-0.523830\pi\)
−0.0747955 + 0.997199i \(0.523830\pi\)
\(314\) 0 0
\(315\) −5.13277 −0.289199
\(316\) 0 0
\(317\) −8.06885 −0.453192 −0.226596 0.973989i \(-0.572760\pi\)
−0.226596 + 0.973989i \(0.572760\pi\)
\(318\) 0 0
\(319\) 16.2059 0.907356
\(320\) 0 0
\(321\) 5.65926 0.315869
\(322\) 0 0
\(323\) 42.3916 2.35873
\(324\) 0 0
\(325\) −6.60629 −0.366451
\(326\) 0 0
\(327\) 3.16772 0.175175
\(328\) 0 0
\(329\) −30.0735 −1.65800
\(330\) 0 0
\(331\) −19.6833 −1.08189 −0.540945 0.841058i \(-0.681933\pi\)
−0.540945 + 0.841058i \(0.681933\pi\)
\(332\) 0 0
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) −1.49436 −0.0816456
\(336\) 0 0
\(337\) −23.4891 −1.27953 −0.639765 0.768570i \(-0.720968\pi\)
−0.639765 + 0.768570i \(0.720968\pi\)
\(338\) 0 0
\(339\) 0.702331 0.0381454
\(340\) 0 0
\(341\) −2.14935 −0.116394
\(342\) 0 0
\(343\) −63.3658 −3.42143
\(344\) 0 0
\(345\) −6.10064 −0.328448
\(346\) 0 0
\(347\) −18.7069 −1.00424 −0.502120 0.864798i \(-0.667447\pi\)
−0.502120 + 0.864798i \(0.667447\pi\)
\(348\) 0 0
\(349\) 16.1123 0.862470 0.431235 0.902240i \(-0.358078\pi\)
0.431235 + 0.902240i \(0.358078\pi\)
\(350\) 0 0
\(351\) 6.60629 0.352617
\(352\) 0 0
\(353\) 2.37320 0.126313 0.0631565 0.998004i \(-0.479883\pi\)
0.0631565 + 0.998004i \(0.479883\pi\)
\(354\) 0 0
\(355\) −0.329796 −0.0175038
\(356\) 0 0
\(357\) 28.9407 1.53170
\(358\) 0 0
\(359\) 18.5953 0.981424 0.490712 0.871322i \(-0.336737\pi\)
0.490712 + 0.871322i \(0.336737\pi\)
\(360\) 0 0
\(361\) 37.5258 1.97504
\(362\) 0 0
\(363\) 6.06852 0.318515
\(364\) 0 0
\(365\) 12.8718 0.673742
\(366\) 0 0
\(367\) 29.9520 1.56348 0.781740 0.623605i \(-0.214332\pi\)
0.781740 + 0.623605i \(0.214332\pi\)
\(368\) 0 0
\(369\) −1.03213 −0.0537304
\(370\) 0 0
\(371\) 49.4718 2.56845
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 48.2105 2.48297
\(378\) 0 0
\(379\) −23.3185 −1.19779 −0.598896 0.800827i \(-0.704394\pi\)
−0.598896 + 0.800827i \(0.704394\pi\)
\(380\) 0 0
\(381\) −19.3132 −0.989446
\(382\) 0 0
\(383\) 16.6063 0.848542 0.424271 0.905535i \(-0.360530\pi\)
0.424271 + 0.905535i \(0.360530\pi\)
\(384\) 0 0
\(385\) 11.3983 0.580912
\(386\) 0 0
\(387\) −5.29767 −0.269296
\(388\) 0 0
\(389\) 15.5795 0.789910 0.394955 0.918701i \(-0.370760\pi\)
0.394955 + 0.918701i \(0.370760\pi\)
\(390\) 0 0
\(391\) 34.3980 1.73958
\(392\) 0 0
\(393\) −18.5953 −0.938011
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −23.2126 −1.16501 −0.582503 0.812829i \(-0.697926\pi\)
−0.582503 + 0.812829i \(0.697926\pi\)
\(398\) 0 0
\(399\) 38.5900 1.93192
\(400\) 0 0
\(401\) 5.72351 0.285818 0.142909 0.989736i \(-0.454354\pi\)
0.142909 + 0.989736i \(0.454354\pi\)
\(402\) 0 0
\(403\) −6.39405 −0.318510
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.22069 0.110076
\(408\) 0 0
\(409\) 28.0254 1.38577 0.692885 0.721049i \(-0.256340\pi\)
0.692885 + 0.721049i \(0.256340\pi\)
\(410\) 0 0
\(411\) 19.5074 0.962230
\(412\) 0 0
\(413\) −52.6907 −2.59274
\(414\) 0 0
\(415\) −9.04767 −0.444133
\(416\) 0 0
\(417\) 10.9036 0.533953
\(418\) 0 0
\(419\) −16.2415 −0.793451 −0.396726 0.917937i \(-0.629854\pi\)
−0.396726 + 0.917937i \(0.629854\pi\)
\(420\) 0 0
\(421\) 6.03495 0.294126 0.147063 0.989127i \(-0.453018\pi\)
0.147063 + 0.989127i \(0.453018\pi\)
\(422\) 0 0
\(423\) 5.85911 0.284879
\(424\) 0 0
\(425\) 5.63841 0.273503
\(426\) 0 0
\(427\) −32.4460 −1.57017
\(428\) 0 0
\(429\) −14.6705 −0.708300
\(430\) 0 0
\(431\) 33.2924 1.60364 0.801819 0.597568i \(-0.203866\pi\)
0.801819 + 0.597568i \(0.203866\pi\)
\(432\) 0 0
\(433\) 8.67054 0.416680 0.208340 0.978057i \(-0.433194\pi\)
0.208340 + 0.978057i \(0.433194\pi\)
\(434\) 0 0
\(435\) 7.29767 0.349896
\(436\) 0 0
\(437\) 45.8669 2.19411
\(438\) 0 0
\(439\) −32.8874 −1.56963 −0.784814 0.619731i \(-0.787242\pi\)
−0.784814 + 0.619731i \(0.787242\pi\)
\(440\) 0 0
\(441\) 19.3453 0.921207
\(442\) 0 0
\(443\) −14.8185 −0.704049 −0.352025 0.935991i \(-0.614507\pi\)
−0.352025 + 0.935991i \(0.614507\pi\)
\(444\) 0 0
\(445\) −2.27649 −0.107916
\(446\) 0 0
\(447\) 12.6720 0.599364
\(448\) 0 0
\(449\) 9.22882 0.435535 0.217767 0.976001i \(-0.430123\pi\)
0.217767 + 0.976001i \(0.430123\pi\)
\(450\) 0 0
\(451\) 2.29204 0.107928
\(452\) 0 0
\(453\) 10.5420 0.495308
\(454\) 0 0
\(455\) 33.9086 1.58966
\(456\) 0 0
\(457\) −36.4753 −1.70624 −0.853121 0.521713i \(-0.825293\pi\)
−0.853121 + 0.521713i \(0.825293\pi\)
\(458\) 0 0
\(459\) −5.63841 −0.263179
\(460\) 0 0
\(461\) 26.6642 1.24188 0.620938 0.783860i \(-0.286752\pi\)
0.620938 + 0.783860i \(0.286752\pi\)
\(462\) 0 0
\(463\) −6.25249 −0.290578 −0.145289 0.989389i \(-0.546411\pi\)
−0.145289 + 0.989389i \(0.546411\pi\)
\(464\) 0 0
\(465\) −0.967873 −0.0448841
\(466\) 0 0
\(467\) −16.1748 −0.748480 −0.374240 0.927332i \(-0.622096\pi\)
−0.374240 + 0.927332i \(0.622096\pi\)
\(468\) 0 0
\(469\) 7.67020 0.354177
\(470\) 0 0
\(471\) 15.4990 0.714154
\(472\) 0 0
\(473\) 11.7645 0.540932
\(474\) 0 0
\(475\) 7.51836 0.344966
\(476\) 0 0
\(477\) −9.63841 −0.441313
\(478\) 0 0
\(479\) −7.72351 −0.352896 −0.176448 0.984310i \(-0.556461\pi\)
−0.176448 + 0.984310i \(0.556461\pi\)
\(480\) 0 0
\(481\) 6.60629 0.301221
\(482\) 0 0
\(483\) 31.3132 1.42480
\(484\) 0 0
\(485\) 17.5982 0.799091
\(486\) 0 0
\(487\) −35.8429 −1.62420 −0.812098 0.583522i \(-0.801674\pi\)
−0.812098 + 0.583522i \(0.801674\pi\)
\(488\) 0 0
\(489\) −4.12149 −0.186380
\(490\) 0 0
\(491\) −2.74718 −0.123978 −0.0619892 0.998077i \(-0.519744\pi\)
−0.0619892 + 0.998077i \(0.519744\pi\)
\(492\) 0 0
\(493\) −41.1473 −1.85318
\(494\) 0 0
\(495\) −2.22069 −0.0998128
\(496\) 0 0
\(497\) 1.69277 0.0759310
\(498\) 0 0
\(499\) 3.07698 0.137744 0.0688722 0.997625i \(-0.478060\pi\)
0.0688722 + 0.997625i \(0.478060\pi\)
\(500\) 0 0
\(501\) 21.9474 0.980536
\(502\) 0 0
\(503\) −8.88278 −0.396063 −0.198032 0.980196i \(-0.563455\pi\)
−0.198032 + 0.980196i \(0.563455\pi\)
\(504\) 0 0
\(505\) 7.61901 0.339041
\(506\) 0 0
\(507\) −30.6430 −1.36090
\(508\) 0 0
\(509\) 2.32836 0.103203 0.0516013 0.998668i \(-0.483568\pi\)
0.0516013 + 0.998668i \(0.483568\pi\)
\(510\) 0 0
\(511\) −66.0682 −2.92268
\(512\) 0 0
\(513\) −7.51836 −0.331944
\(514\) 0 0
\(515\) −5.85911 −0.258183
\(516\) 0 0
\(517\) −13.0113 −0.572236
\(518\) 0 0
\(519\) 0.425841 0.0186923
\(520\) 0 0
\(521\) −26.3344 −1.15373 −0.576865 0.816839i \(-0.695724\pi\)
−0.576865 + 0.816839i \(0.695724\pi\)
\(522\) 0 0
\(523\) −10.9887 −0.480503 −0.240252 0.970711i \(-0.577230\pi\)
−0.240252 + 0.970711i \(0.577230\pi\)
\(524\) 0 0
\(525\) 5.13277 0.224013
\(526\) 0 0
\(527\) 5.45727 0.237722
\(528\) 0 0
\(529\) 14.2179 0.618168
\(530\) 0 0
\(531\) 10.2655 0.445487
\(532\) 0 0
\(533\) 6.81852 0.295343
\(534\) 0 0
\(535\) −5.65926 −0.244671
\(536\) 0 0
\(537\) −17.5424 −0.757009
\(538\) 0 0
\(539\) −42.9601 −1.85042
\(540\) 0 0
\(541\) −6.61934 −0.284588 −0.142294 0.989824i \(-0.545448\pi\)
−0.142294 + 0.989824i \(0.545448\pi\)
\(542\) 0 0
\(543\) −18.5311 −0.795245
\(544\) 0 0
\(545\) −3.16772 −0.135690
\(546\) 0 0
\(547\) 23.1801 0.991110 0.495555 0.868577i \(-0.334965\pi\)
0.495555 + 0.868577i \(0.334965\pi\)
\(548\) 0 0
\(549\) 6.32134 0.269788
\(550\) 0 0
\(551\) −54.8665 −2.33739
\(552\) 0 0
\(553\) 20.5311 0.873071
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) 0 0
\(557\) 11.3411 0.480537 0.240268 0.970706i \(-0.422765\pi\)
0.240268 + 0.970706i \(0.422765\pi\)
\(558\) 0 0
\(559\) 34.9979 1.48025
\(560\) 0 0
\(561\) 12.5212 0.528645
\(562\) 0 0
\(563\) −33.3705 −1.40640 −0.703198 0.710994i \(-0.748245\pi\)
−0.703198 + 0.710994i \(0.748245\pi\)
\(564\) 0 0
\(565\) −0.702331 −0.0295473
\(566\) 0 0
\(567\) −5.13277 −0.215556
\(568\) 0 0
\(569\) −0.500343 −0.0209755 −0.0104877 0.999945i \(-0.503338\pi\)
−0.0104877 + 0.999945i \(0.503338\pi\)
\(570\) 0 0
\(571\) −27.5463 −1.15278 −0.576388 0.817176i \(-0.695538\pi\)
−0.576388 + 0.817176i \(0.695538\pi\)
\(572\) 0 0
\(573\) 16.1695 0.675490
\(574\) 0 0
\(575\) 6.10064 0.254414
\(576\) 0 0
\(577\) 12.3902 0.515810 0.257905 0.966170i \(-0.416968\pi\)
0.257905 + 0.966170i \(0.416968\pi\)
\(578\) 0 0
\(579\) 11.8591 0.492848
\(580\) 0 0
\(581\) 46.4396 1.92664
\(582\) 0 0
\(583\) 21.4040 0.886462
\(584\) 0 0
\(585\) −6.60629 −0.273136
\(586\) 0 0
\(587\) −0.808273 −0.0333610 −0.0166805 0.999861i \(-0.505310\pi\)
−0.0166805 + 0.999861i \(0.505310\pi\)
\(588\) 0 0
\(589\) 7.27682 0.299836
\(590\) 0 0
\(591\) −0.100645 −0.00413998
\(592\) 0 0
\(593\) −24.4121 −1.00248 −0.501242 0.865307i \(-0.667123\pi\)
−0.501242 + 0.865307i \(0.667123\pi\)
\(594\) 0 0
\(595\) −28.9407 −1.18645
\(596\) 0 0
\(597\) −2.70693 −0.110787
\(598\) 0 0
\(599\) 29.3440 1.19896 0.599481 0.800389i \(-0.295374\pi\)
0.599481 + 0.800389i \(0.295374\pi\)
\(600\) 0 0
\(601\) −9.55791 −0.389875 −0.194938 0.980816i \(-0.562450\pi\)
−0.194938 + 0.980816i \(0.562450\pi\)
\(602\) 0 0
\(603\) −1.49436 −0.0608550
\(604\) 0 0
\(605\) −6.06852 −0.246720
\(606\) 0 0
\(607\) 31.4912 1.27819 0.639094 0.769129i \(-0.279310\pi\)
0.639094 + 0.769129i \(0.279310\pi\)
\(608\) 0 0
\(609\) −37.4573 −1.51785
\(610\) 0 0
\(611\) −38.7069 −1.56591
\(612\) 0 0
\(613\) 6.41489 0.259095 0.129548 0.991573i \(-0.458648\pi\)
0.129548 + 0.991573i \(0.458648\pi\)
\(614\) 0 0
\(615\) 1.03213 0.0416194
\(616\) 0 0
\(617\) −38.8153 −1.56265 −0.781323 0.624127i \(-0.785455\pi\)
−0.781323 + 0.624127i \(0.785455\pi\)
\(618\) 0 0
\(619\) 0.00460032 0.000184902 0 9.24512e−5 1.00000i \(-0.499971\pi\)
9.24512e−5 1.00000i \(0.499971\pi\)
\(620\) 0 0
\(621\) −6.10064 −0.244810
\(622\) 0 0
\(623\) 11.6847 0.468138
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 16.6960 0.666773
\(628\) 0 0
\(629\) −5.63841 −0.224818
\(630\) 0 0
\(631\) −34.5999 −1.37740 −0.688701 0.725046i \(-0.741819\pi\)
−0.688701 + 0.725046i \(0.741819\pi\)
\(632\) 0 0
\(633\) 11.8033 0.469139
\(634\) 0 0
\(635\) 19.3132 0.766422
\(636\) 0 0
\(637\) −127.801 −5.06365
\(638\) 0 0
\(639\) −0.329796 −0.0130465
\(640\) 0 0
\(641\) −40.8711 −1.61431 −0.807156 0.590338i \(-0.798995\pi\)
−0.807156 + 0.590338i \(0.798995\pi\)
\(642\) 0 0
\(643\) −11.4788 −0.452680 −0.226340 0.974048i \(-0.572676\pi\)
−0.226340 + 0.974048i \(0.572676\pi\)
\(644\) 0 0
\(645\) 5.29767 0.208596
\(646\) 0 0
\(647\) 17.7490 0.697783 0.348892 0.937163i \(-0.386558\pi\)
0.348892 + 0.937163i \(0.386558\pi\)
\(648\) 0 0
\(649\) −22.7966 −0.894846
\(650\) 0 0
\(651\) 4.96787 0.194706
\(652\) 0 0
\(653\) 21.2768 0.832626 0.416313 0.909221i \(-0.363322\pi\)
0.416313 + 0.909221i \(0.363322\pi\)
\(654\) 0 0
\(655\) 18.5953 0.726580
\(656\) 0 0
\(657\) 12.8718 0.502178
\(658\) 0 0
\(659\) −23.5548 −0.917563 −0.458781 0.888549i \(-0.651714\pi\)
−0.458781 + 0.888549i \(0.651714\pi\)
\(660\) 0 0
\(661\) 1.34357 0.0522587 0.0261294 0.999659i \(-0.491682\pi\)
0.0261294 + 0.999659i \(0.491682\pi\)
\(662\) 0 0
\(663\) 37.2490 1.44663
\(664\) 0 0
\(665\) −38.5900 −1.49646
\(666\) 0 0
\(667\) −44.5205 −1.72384
\(668\) 0 0
\(669\) 13.1692 0.509149
\(670\) 0 0
\(671\) −14.0378 −0.541922
\(672\) 0 0
\(673\) 17.5562 0.676742 0.338371 0.941013i \(-0.390124\pi\)
0.338371 + 0.941013i \(0.390124\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −19.2073 −0.738196 −0.369098 0.929391i \(-0.620333\pi\)
−0.369098 + 0.929391i \(0.620333\pi\)
\(678\) 0 0
\(679\) −90.3274 −3.46645
\(680\) 0 0
\(681\) 5.57946 0.213805
\(682\) 0 0
\(683\) 34.7532 1.32980 0.664898 0.746935i \(-0.268475\pi\)
0.664898 + 0.746935i \(0.268475\pi\)
\(684\) 0 0
\(685\) −19.5074 −0.745340
\(686\) 0 0
\(687\) 1.75990 0.0671445
\(688\) 0 0
\(689\) 63.6741 2.42579
\(690\) 0 0
\(691\) −24.3344 −0.925724 −0.462862 0.886430i \(-0.653177\pi\)
−0.462862 + 0.886430i \(0.653177\pi\)
\(692\) 0 0
\(693\) 11.3983 0.432986
\(694\) 0 0
\(695\) −10.9036 −0.413598
\(696\) 0 0
\(697\) −5.81955 −0.220431
\(698\) 0 0
\(699\) 6.07664 0.229840
\(700\) 0 0
\(701\) −8.81354 −0.332883 −0.166441 0.986051i \(-0.553228\pi\)
−0.166441 + 0.986051i \(0.553228\pi\)
\(702\) 0 0
\(703\) −7.51836 −0.283560
\(704\) 0 0
\(705\) −5.85911 −0.220667
\(706\) 0 0
\(707\) −39.1066 −1.47076
\(708\) 0 0
\(709\) −9.07238 −0.340720 −0.170360 0.985382i \(-0.554493\pi\)
−0.170360 + 0.985382i \(0.554493\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 5.90465 0.221131
\(714\) 0 0
\(715\) 14.6705 0.548647
\(716\) 0 0
\(717\) 13.9403 0.520612
\(718\) 0 0
\(719\) −22.8665 −0.852778 −0.426389 0.904540i \(-0.640214\pi\)
−0.426389 + 0.904540i \(0.640214\pi\)
\(720\) 0 0
\(721\) 30.0735 1.11999
\(722\) 0 0
\(723\) 3.89406 0.144822
\(724\) 0 0
\(725\) −7.29767 −0.271029
\(726\) 0 0
\(727\) 2.86407 0.106222 0.0531112 0.998589i \(-0.483086\pi\)
0.0531112 + 0.998589i \(0.483086\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.8704 −1.10480
\(732\) 0 0
\(733\) −21.6360 −0.799144 −0.399572 0.916702i \(-0.630841\pi\)
−0.399572 + 0.916702i \(0.630841\pi\)
\(734\) 0 0
\(735\) −19.3453 −0.713564
\(736\) 0 0
\(737\) 3.31851 0.122239
\(738\) 0 0
\(739\) −16.7278 −0.615341 −0.307671 0.951493i \(-0.599549\pi\)
−0.307671 + 0.951493i \(0.599549\pi\)
\(740\) 0 0
\(741\) 49.6685 1.82462
\(742\) 0 0
\(743\) 21.6624 0.794717 0.397358 0.917663i \(-0.369927\pi\)
0.397358 + 0.917663i \(0.369927\pi\)
\(744\) 0 0
\(745\) −12.6720 −0.464265
\(746\) 0 0
\(747\) −9.04767 −0.331037
\(748\) 0 0
\(749\) 29.0477 1.06138
\(750\) 0 0
\(751\) −24.8129 −0.905435 −0.452717 0.891654i \(-0.649545\pi\)
−0.452717 + 0.891654i \(0.649545\pi\)
\(752\) 0 0
\(753\) 20.9308 0.762760
\(754\) 0 0
\(755\) −10.5420 −0.383664
\(756\) 0 0
\(757\) 1.97281 0.0717030 0.0358515 0.999357i \(-0.488586\pi\)
0.0358515 + 0.999357i \(0.488586\pi\)
\(758\) 0 0
\(759\) 13.5477 0.491749
\(760\) 0 0
\(761\) 25.4735 0.923414 0.461707 0.887032i \(-0.347237\pi\)
0.461707 + 0.887032i \(0.347237\pi\)
\(762\) 0 0
\(763\) 16.2592 0.588622
\(764\) 0 0
\(765\) 5.63841 0.203857
\(766\) 0 0
\(767\) −67.8171 −2.44873
\(768\) 0 0
\(769\) −41.9675 −1.51339 −0.756694 0.653770i \(-0.773187\pi\)
−0.756694 + 0.653770i \(0.773187\pi\)
\(770\) 0 0
\(771\) 4.43044 0.159558
\(772\) 0 0
\(773\) 46.0410 1.65598 0.827990 0.560743i \(-0.189485\pi\)
0.827990 + 0.560743i \(0.189485\pi\)
\(774\) 0 0
\(775\) 0.967873 0.0347670
\(776\) 0 0
\(777\) −5.13277 −0.184137
\(778\) 0 0
\(779\) −7.75990 −0.278027
\(780\) 0 0
\(781\) 0.732376 0.0262065
\(782\) 0 0
\(783\) 7.29767 0.260797
\(784\) 0 0
\(785\) −15.4990 −0.553182
\(786\) 0 0
\(787\) 8.04801 0.286881 0.143440 0.989659i \(-0.454184\pi\)
0.143440 + 0.989659i \(0.454184\pi\)
\(788\) 0 0
\(789\) 4.20975 0.149871
\(790\) 0 0
\(791\) 3.60490 0.128176
\(792\) 0 0
\(793\) −41.7606 −1.48296
\(794\) 0 0
\(795\) 9.63841 0.341839
\(796\) 0 0
\(797\) −26.2662 −0.930397 −0.465198 0.885206i \(-0.654017\pi\)
−0.465198 + 0.885206i \(0.654017\pi\)
\(798\) 0 0
\(799\) 33.0361 1.16873
\(800\) 0 0
\(801\) −2.27649 −0.0804358
\(802\) 0 0
\(803\) −28.5844 −1.00872
\(804\) 0 0
\(805\) −31.3132 −1.10365
\(806\) 0 0
\(807\) −12.5071 −0.440270
\(808\) 0 0
\(809\) 13.3301 0.468662 0.234331 0.972157i \(-0.424710\pi\)
0.234331 + 0.972157i \(0.424710\pi\)
\(810\) 0 0
\(811\) 30.4251 1.06837 0.534186 0.845367i \(-0.320618\pi\)
0.534186 + 0.845367i \(0.320618\pi\)
\(812\) 0 0
\(813\) −6.70693 −0.235222
\(814\) 0 0
\(815\) 4.12149 0.144369
\(816\) 0 0
\(817\) −39.8298 −1.39347
\(818\) 0 0
\(819\) 33.9086 1.18486
\(820\) 0 0
\(821\) −26.2895 −0.917512 −0.458756 0.888562i \(-0.651705\pi\)
−0.458756 + 0.888562i \(0.651705\pi\)
\(822\) 0 0
\(823\) −24.4050 −0.850705 −0.425352 0.905028i \(-0.639850\pi\)
−0.425352 + 0.905028i \(0.639850\pi\)
\(824\) 0 0
\(825\) 2.22069 0.0773146
\(826\) 0 0
\(827\) 37.5887 1.30709 0.653543 0.756890i \(-0.273282\pi\)
0.653543 + 0.756890i \(0.273282\pi\)
\(828\) 0 0
\(829\) 36.4481 1.26589 0.632947 0.774195i \(-0.281845\pi\)
0.632947 + 0.774195i \(0.281845\pi\)
\(830\) 0 0
\(831\) 27.0477 0.938273
\(832\) 0 0
\(833\) 109.077 3.77929
\(834\) 0 0
\(835\) −21.9474 −0.759520
\(836\) 0 0
\(837\) −0.967873 −0.0334546
\(838\) 0 0
\(839\) 43.3277 1.49584 0.747919 0.663790i \(-0.231053\pi\)
0.747919 + 0.663790i \(0.231053\pi\)
\(840\) 0 0
\(841\) 24.2560 0.836413
\(842\) 0 0
\(843\) 1.39371 0.0480021
\(844\) 0 0
\(845\) 30.6430 1.05415
\(846\) 0 0
\(847\) 31.1483 1.07027
\(848\) 0 0
\(849\) −15.3132 −0.525548
\(850\) 0 0
\(851\) −6.10064 −0.209127
\(852\) 0 0
\(853\) 17.6430 0.604085 0.302043 0.953294i \(-0.402332\pi\)
0.302043 + 0.953294i \(0.402332\pi\)
\(854\) 0 0
\(855\) 7.51836 0.257123
\(856\) 0 0
\(857\) 35.6222 1.21683 0.608415 0.793619i \(-0.291806\pi\)
0.608415 + 0.793619i \(0.291806\pi\)
\(858\) 0 0
\(859\) 5.49292 0.187416 0.0937080 0.995600i \(-0.470128\pi\)
0.0937080 + 0.995600i \(0.470128\pi\)
\(860\) 0 0
\(861\) −5.29767 −0.180544
\(862\) 0 0
\(863\) 1.12501 0.0382958 0.0191479 0.999817i \(-0.493905\pi\)
0.0191479 + 0.999817i \(0.493905\pi\)
\(864\) 0 0
\(865\) −0.425841 −0.0144790
\(866\) 0 0
\(867\) −14.7917 −0.502352
\(868\) 0 0
\(869\) 8.88278 0.301328
\(870\) 0 0
\(871\) 9.87216 0.334506
\(872\) 0 0
\(873\) 17.5982 0.595608
\(874\) 0 0
\(875\) −5.13277 −0.173519
\(876\) 0 0
\(877\) 31.6367 1.06829 0.534147 0.845392i \(-0.320633\pi\)
0.534147 + 0.845392i \(0.320633\pi\)
\(878\) 0 0
\(879\) 12.8672 0.434001
\(880\) 0 0
\(881\) −24.8012 −0.835575 −0.417787 0.908545i \(-0.637194\pi\)
−0.417787 + 0.908545i \(0.637194\pi\)
\(882\) 0 0
\(883\) 27.7980 0.935478 0.467739 0.883867i \(-0.345069\pi\)
0.467739 + 0.883867i \(0.345069\pi\)
\(884\) 0 0
\(885\) −10.2655 −0.345072
\(886\) 0 0
\(887\) 28.5226 0.957696 0.478848 0.877898i \(-0.341054\pi\)
0.478848 + 0.877898i \(0.341054\pi\)
\(888\) 0 0
\(889\) −99.1303 −3.32472
\(890\) 0 0
\(891\) −2.22069 −0.0743960
\(892\) 0 0
\(893\) 44.0509 1.47411
\(894\) 0 0
\(895\) 17.5424 0.586377
\(896\) 0 0
\(897\) 40.3026 1.34566
\(898\) 0 0
\(899\) −7.06322 −0.235572
\(900\) 0 0
\(901\) −54.3453 −1.81051
\(902\) 0 0
\(903\) −27.1917 −0.904884
\(904\) 0 0
\(905\) 18.5311 0.615994
\(906\) 0 0
\(907\) −41.1854 −1.36754 −0.683769 0.729698i \(-0.739660\pi\)
−0.683769 + 0.729698i \(0.739660\pi\)
\(908\) 0 0
\(909\) 7.61901 0.252707
\(910\) 0 0
\(911\) −21.3440 −0.707157 −0.353578 0.935405i \(-0.615035\pi\)
−0.353578 + 0.935405i \(0.615035\pi\)
\(912\) 0 0
\(913\) 20.0921 0.664952
\(914\) 0 0
\(915\) −6.32134 −0.208977
\(916\) 0 0
\(917\) −95.4456 −3.15189
\(918\) 0 0
\(919\) −36.9089 −1.21751 −0.608756 0.793358i \(-0.708331\pi\)
−0.608756 + 0.793358i \(0.708331\pi\)
\(920\) 0 0
\(921\) −4.08970 −0.134760
\(922\) 0 0
\(923\) 2.17873 0.0717137
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) −5.85911 −0.192438
\(928\) 0 0
\(929\) 21.0350 0.690136 0.345068 0.938578i \(-0.387856\pi\)
0.345068 + 0.938578i \(0.387856\pi\)
\(930\) 0 0
\(931\) 145.445 4.76678
\(932\) 0 0
\(933\) −10.3090 −0.337500
\(934\) 0 0
\(935\) −12.5212 −0.409487
\(936\) 0 0
\(937\) 9.67652 0.316118 0.158059 0.987430i \(-0.449476\pi\)
0.158059 + 0.987430i \(0.449476\pi\)
\(938\) 0 0
\(939\) 2.64653 0.0863664
\(940\) 0 0
\(941\) −48.1332 −1.56910 −0.784548 0.620068i \(-0.787105\pi\)
−0.784548 + 0.620068i \(0.787105\pi\)
\(942\) 0 0
\(943\) −6.29664 −0.205047
\(944\) 0 0
\(945\) 5.13277 0.166969
\(946\) 0 0
\(947\) 17.7871 0.578002 0.289001 0.957329i \(-0.406677\pi\)
0.289001 + 0.957329i \(0.406677\pi\)
\(948\) 0 0
\(949\) −85.0350 −2.76035
\(950\) 0 0
\(951\) 8.06885 0.261650
\(952\) 0 0
\(953\) 30.0442 0.973226 0.486613 0.873618i \(-0.338232\pi\)
0.486613 + 0.873618i \(0.338232\pi\)
\(954\) 0 0
\(955\) −16.1695 −0.523233
\(956\) 0 0
\(957\) −16.2059 −0.523862
\(958\) 0 0
\(959\) 100.127 3.23327
\(960\) 0 0
\(961\) −30.0632 −0.969781
\(962\) 0 0
\(963\) −5.65926 −0.182367
\(964\) 0 0
\(965\) −11.8591 −0.381758
\(966\) 0 0
\(967\) −15.7052 −0.505044 −0.252522 0.967591i \(-0.581260\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(968\) 0 0
\(969\) −42.3916 −1.36182
\(970\) 0 0
\(971\) −36.8693 −1.18319 −0.591597 0.806234i \(-0.701502\pi\)
−0.591597 + 0.806234i \(0.701502\pi\)
\(972\) 0 0
\(973\) 55.9658 1.79418
\(974\) 0 0
\(975\) 6.60629 0.211570
\(976\) 0 0
\(977\) 31.8404 1.01866 0.509332 0.860570i \(-0.329893\pi\)
0.509332 + 0.860570i \(0.329893\pi\)
\(978\) 0 0
\(979\) 5.05539 0.161571
\(980\) 0 0
\(981\) −3.16772 −0.101138
\(982\) 0 0
\(983\) −11.8708 −0.378618 −0.189309 0.981918i \(-0.560625\pi\)
−0.189309 + 0.981918i \(0.560625\pi\)
\(984\) 0 0
\(985\) 0.100645 0.00320682
\(986\) 0 0
\(987\) 30.0735 0.957249
\(988\) 0 0
\(989\) −32.3192 −1.02769
\(990\) 0 0
\(991\) 13.9990 0.444691 0.222346 0.974968i \(-0.428629\pi\)
0.222346 + 0.974968i \(0.428629\pi\)
\(992\) 0 0
\(993\) 19.6833 0.624629
\(994\) 0 0
\(995\) 2.70693 0.0858155
\(996\) 0 0
\(997\) −26.4050 −0.836255 −0.418127 0.908388i \(-0.637313\pi\)
−0.418127 + 0.908388i \(0.637313\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 8880.2.a.cg.1.1 4
4.3 odd 2 1110.2.a.s.1.4 4
12.11 even 2 3330.2.a.bj.1.4 4
20.19 odd 2 5550.2.a.cj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.4 4 4.3 odd 2
3330.2.a.bj.1.4 4 12.11 even 2
5550.2.a.cj.1.1 4 20.19 odd 2
8880.2.a.cg.1.1 4 1.1 even 1 trivial