Properties

Label 2640.2.a.be.1.1
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.35026 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.35026 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.96239 q^{13} -1.00000 q^{15} -4.57452 q^{17} +4.31265 q^{19} +3.35026 q^{21} +6.70052 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.61213 q^{29} -9.92478 q^{31} +1.00000 q^{33} -3.35026 q^{35} -2.00000 q^{37} -2.96239 q^{39} -4.38787 q^{41} +9.27504 q^{43} +1.00000 q^{45} +9.92478 q^{47} +4.22425 q^{49} +4.57452 q^{51} +4.70052 q^{53} -1.00000 q^{55} -4.31265 q^{57} -10.7005 q^{59} -8.70052 q^{61} -3.35026 q^{63} +2.96239 q^{65} -5.92478 q^{67} -6.70052 q^{69} -9.92478 q^{71} -7.73813 q^{73} -1.00000 q^{75} +3.35026 q^{77} -11.5369 q^{79} +1.00000 q^{81} -10.8872 q^{83} -4.57452 q^{85} +3.61213 q^{87} -2.77575 q^{89} -9.92478 q^{91} +9.92478 q^{93} +4.31265 q^{95} +0.0752228 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.96239 0.821619 0.410809 0.911721i \(-0.365246\pi\)
0.410809 + 0.911721i \(0.365246\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.57452 −1.10948 −0.554741 0.832023i \(-0.687183\pi\)
−0.554741 + 0.832023i \(0.687183\pi\)
\(18\) 0 0
\(19\) 4.31265 0.989390 0.494695 0.869067i \(-0.335280\pi\)
0.494695 + 0.869067i \(0.335280\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) 0 0
\(23\) 6.70052 1.39716 0.698578 0.715534i \(-0.253817\pi\)
0.698578 + 0.715534i \(0.253817\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) 0 0
\(31\) −9.92478 −1.78254 −0.891271 0.453470i \(-0.850186\pi\)
−0.891271 + 0.453470i \(0.850186\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −3.35026 −0.566298
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −2.96239 −0.474362
\(40\) 0 0
\(41\) −4.38787 −0.685271 −0.342635 0.939468i \(-0.611320\pi\)
−0.342635 + 0.939468i \(0.611320\pi\)
\(42\) 0 0
\(43\) 9.27504 1.41443 0.707215 0.706998i \(-0.249951\pi\)
0.707215 + 0.706998i \(0.249951\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 9.92478 1.44768 0.723839 0.689969i \(-0.242376\pi\)
0.723839 + 0.689969i \(0.242376\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) 4.57452 0.640560
\(52\) 0 0
\(53\) 4.70052 0.645667 0.322833 0.946456i \(-0.395365\pi\)
0.322833 + 0.946456i \(0.395365\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −4.31265 −0.571224
\(58\) 0 0
\(59\) −10.7005 −1.39309 −0.696545 0.717513i \(-0.745280\pi\)
−0.696545 + 0.717513i \(0.745280\pi\)
\(60\) 0 0
\(61\) −8.70052 −1.11399 −0.556994 0.830517i \(-0.688045\pi\)
−0.556994 + 0.830517i \(0.688045\pi\)
\(62\) 0 0
\(63\) −3.35026 −0.422093
\(64\) 0 0
\(65\) 2.96239 0.367439
\(66\) 0 0
\(67\) −5.92478 −0.723827 −0.361913 0.932212i \(-0.617876\pi\)
−0.361913 + 0.932212i \(0.617876\pi\)
\(68\) 0 0
\(69\) −6.70052 −0.806648
\(70\) 0 0
\(71\) −9.92478 −1.17785 −0.588927 0.808186i \(-0.700450\pi\)
−0.588927 + 0.808186i \(0.700450\pi\)
\(72\) 0 0
\(73\) −7.73813 −0.905680 −0.452840 0.891592i \(-0.649589\pi\)
−0.452840 + 0.891592i \(0.649589\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.35026 0.381798
\(78\) 0 0
\(79\) −11.5369 −1.29800 −0.649002 0.760787i \(-0.724813\pi\)
−0.649002 + 0.760787i \(0.724813\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.8872 −1.19502 −0.597511 0.801861i \(-0.703844\pi\)
−0.597511 + 0.801861i \(0.703844\pi\)
\(84\) 0 0
\(85\) −4.57452 −0.496176
\(86\) 0 0
\(87\) 3.61213 0.387261
\(88\) 0 0
\(89\) −2.77575 −0.294229 −0.147114 0.989120i \(-0.546999\pi\)
−0.147114 + 0.989120i \(0.546999\pi\)
\(90\) 0 0
\(91\) −9.92478 −1.04040
\(92\) 0 0
\(93\) 9.92478 1.02915
\(94\) 0 0
\(95\) 4.31265 0.442469
\(96\) 0 0
\(97\) 0.0752228 0.00763772 0.00381886 0.999993i \(-0.498784\pi\)
0.00381886 + 0.999993i \(0.498784\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −15.0884 −1.50135 −0.750676 0.660671i \(-0.770272\pi\)
−0.750676 + 0.660671i \(0.770272\pi\)
\(102\) 0 0
\(103\) 3.22425 0.317695 0.158848 0.987303i \(-0.449222\pi\)
0.158848 + 0.987303i \(0.449222\pi\)
\(104\) 0 0
\(105\) 3.35026 0.326952
\(106\) 0 0
\(107\) 0.962389 0.0930376 0.0465188 0.998917i \(-0.485187\pi\)
0.0465188 + 0.998917i \(0.485187\pi\)
\(108\) 0 0
\(109\) 11.4010 1.09202 0.546011 0.837778i \(-0.316146\pi\)
0.546011 + 0.837778i \(0.316146\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 6.70052 0.624827
\(116\) 0 0
\(117\) 2.96239 0.273873
\(118\) 0 0
\(119\) 15.3258 1.40492
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 4.38787 0.395641
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.5745 1.29328 0.646640 0.762796i \(-0.276174\pi\)
0.646640 + 0.762796i \(0.276174\pi\)
\(128\) 0 0
\(129\) −9.27504 −0.816622
\(130\) 0 0
\(131\) 5.92478 0.517650 0.258825 0.965924i \(-0.416665\pi\)
0.258825 + 0.965924i \(0.416665\pi\)
\(132\) 0 0
\(133\) −14.4485 −1.25284
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 13.8496 1.18325 0.591624 0.806214i \(-0.298487\pi\)
0.591624 + 0.806214i \(0.298487\pi\)
\(138\) 0 0
\(139\) −13.6121 −1.15457 −0.577283 0.816544i \(-0.695887\pi\)
−0.577283 + 0.816544i \(0.695887\pi\)
\(140\) 0 0
\(141\) −9.92478 −0.835817
\(142\) 0 0
\(143\) −2.96239 −0.247727
\(144\) 0 0
\(145\) −3.61213 −0.299971
\(146\) 0 0
\(147\) −4.22425 −0.348411
\(148\) 0 0
\(149\) 1.53690 0.125908 0.0629540 0.998016i \(-0.479948\pi\)
0.0629540 + 0.998016i \(0.479948\pi\)
\(150\) 0 0
\(151\) 6.76116 0.550215 0.275108 0.961413i \(-0.411287\pi\)
0.275108 + 0.961413i \(0.411287\pi\)
\(152\) 0 0
\(153\) −4.57452 −0.369828
\(154\) 0 0
\(155\) −9.92478 −0.797177
\(156\) 0 0
\(157\) −5.47627 −0.437054 −0.218527 0.975831i \(-0.570125\pi\)
−0.218527 + 0.975831i \(0.570125\pi\)
\(158\) 0 0
\(159\) −4.70052 −0.372776
\(160\) 0 0
\(161\) −22.4485 −1.76919
\(162\) 0 0
\(163\) −12.6253 −0.988890 −0.494445 0.869209i \(-0.664629\pi\)
−0.494445 + 0.869209i \(0.664629\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −18.3634 −1.42101 −0.710503 0.703695i \(-0.751532\pi\)
−0.710503 + 0.703695i \(0.751532\pi\)
\(168\) 0 0
\(169\) −4.22425 −0.324943
\(170\) 0 0
\(171\) 4.31265 0.329797
\(172\) 0 0
\(173\) −8.57452 −0.651908 −0.325954 0.945386i \(-0.605686\pi\)
−0.325954 + 0.945386i \(0.605686\pi\)
\(174\) 0 0
\(175\) −3.35026 −0.253256
\(176\) 0 0
\(177\) 10.7005 0.804301
\(178\) 0 0
\(179\) −14.1768 −1.05962 −0.529812 0.848115i \(-0.677737\pi\)
−0.529812 + 0.848115i \(0.677737\pi\)
\(180\) 0 0
\(181\) −5.22425 −0.388316 −0.194158 0.980970i \(-0.562197\pi\)
−0.194158 + 0.980970i \(0.562197\pi\)
\(182\) 0 0
\(183\) 8.70052 0.643161
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 4.57452 0.334522
\(188\) 0 0
\(189\) 3.35026 0.243696
\(190\) 0 0
\(191\) 16.6253 1.20296 0.601482 0.798886i \(-0.294577\pi\)
0.601482 + 0.798886i \(0.294577\pi\)
\(192\) 0 0
\(193\) −16.3634 −1.17787 −0.588933 0.808182i \(-0.700452\pi\)
−0.588933 + 0.808182i \(0.700452\pi\)
\(194\) 0 0
\(195\) −2.96239 −0.212141
\(196\) 0 0
\(197\) −20.4241 −1.45515 −0.727577 0.686026i \(-0.759354\pi\)
−0.727577 + 0.686026i \(0.759354\pi\)
\(198\) 0 0
\(199\) 8.62530 0.611431 0.305716 0.952123i \(-0.401104\pi\)
0.305716 + 0.952123i \(0.401104\pi\)
\(200\) 0 0
\(201\) 5.92478 0.417902
\(202\) 0 0
\(203\) 12.1016 0.849364
\(204\) 0 0
\(205\) −4.38787 −0.306462
\(206\) 0 0
\(207\) 6.70052 0.465719
\(208\) 0 0
\(209\) −4.31265 −0.298312
\(210\) 0 0
\(211\) −9.08840 −0.625671 −0.312836 0.949807i \(-0.601279\pi\)
−0.312836 + 0.949807i \(0.601279\pi\)
\(212\) 0 0
\(213\) 9.92478 0.680035
\(214\) 0 0
\(215\) 9.27504 0.632552
\(216\) 0 0
\(217\) 33.2506 2.25720
\(218\) 0 0
\(219\) 7.73813 0.522895
\(220\) 0 0
\(221\) −13.5515 −0.911572
\(222\) 0 0
\(223\) 6.70052 0.448700 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −16.9624 −1.12583 −0.562917 0.826514i \(-0.690321\pi\)
−0.562917 + 0.826514i \(0.690321\pi\)
\(228\) 0 0
\(229\) 25.8496 1.70819 0.854093 0.520120i \(-0.174113\pi\)
0.854093 + 0.520120i \(0.174113\pi\)
\(230\) 0 0
\(231\) −3.35026 −0.220431
\(232\) 0 0
\(233\) −19.2750 −1.26275 −0.631375 0.775478i \(-0.717509\pi\)
−0.631375 + 0.775478i \(0.717509\pi\)
\(234\) 0 0
\(235\) 9.92478 0.647421
\(236\) 0 0
\(237\) 11.5369 0.749402
\(238\) 0 0
\(239\) −26.5501 −1.71738 −0.858691 0.512494i \(-0.828722\pi\)
−0.858691 + 0.512494i \(0.828722\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.22425 0.269878
\(246\) 0 0
\(247\) 12.7757 0.812901
\(248\) 0 0
\(249\) 10.8872 0.689946
\(250\) 0 0
\(251\) −29.9248 −1.88884 −0.944418 0.328748i \(-0.893373\pi\)
−0.944418 + 0.328748i \(0.893373\pi\)
\(252\) 0 0
\(253\) −6.70052 −0.421258
\(254\) 0 0
\(255\) 4.57452 0.286467
\(256\) 0 0
\(257\) 8.70052 0.542724 0.271362 0.962477i \(-0.412526\pi\)
0.271362 + 0.962477i \(0.412526\pi\)
\(258\) 0 0
\(259\) 6.70052 0.416350
\(260\) 0 0
\(261\) −3.61213 −0.223585
\(262\) 0 0
\(263\) −12.2882 −0.757724 −0.378862 0.925453i \(-0.623684\pi\)
−0.378862 + 0.925453i \(0.623684\pi\)
\(264\) 0 0
\(265\) 4.70052 0.288751
\(266\) 0 0
\(267\) 2.77575 0.169873
\(268\) 0 0
\(269\) −5.84955 −0.356654 −0.178327 0.983971i \(-0.557068\pi\)
−0.178327 + 0.983971i \(0.557068\pi\)
\(270\) 0 0
\(271\) 5.08840 0.309098 0.154549 0.987985i \(-0.450608\pi\)
0.154549 + 0.987985i \(0.450608\pi\)
\(272\) 0 0
\(273\) 9.92478 0.600675
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 1.41090 0.0847725 0.0423863 0.999101i \(-0.486504\pi\)
0.0423863 + 0.999101i \(0.486504\pi\)
\(278\) 0 0
\(279\) −9.92478 −0.594181
\(280\) 0 0
\(281\) −4.38787 −0.261759 −0.130879 0.991398i \(-0.541780\pi\)
−0.130879 + 0.991398i \(0.541780\pi\)
\(282\) 0 0
\(283\) −26.5745 −1.57969 −0.789845 0.613306i \(-0.789839\pi\)
−0.789845 + 0.613306i \(0.789839\pi\)
\(284\) 0 0
\(285\) −4.31265 −0.255459
\(286\) 0 0
\(287\) 14.7005 0.867744
\(288\) 0 0
\(289\) 3.92619 0.230952
\(290\) 0 0
\(291\) −0.0752228 −0.00440964
\(292\) 0 0
\(293\) −3.42548 −0.200119 −0.100059 0.994981i \(-0.531903\pi\)
−0.100059 + 0.994981i \(0.531903\pi\)
\(294\) 0 0
\(295\) −10.7005 −0.623009
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 19.8496 1.14793
\(300\) 0 0
\(301\) −31.0738 −1.79106
\(302\) 0 0
\(303\) 15.0884 0.866806
\(304\) 0 0
\(305\) −8.70052 −0.498191
\(306\) 0 0
\(307\) 16.6497 0.950251 0.475125 0.879918i \(-0.342403\pi\)
0.475125 + 0.879918i \(0.342403\pi\)
\(308\) 0 0
\(309\) −3.22425 −0.183421
\(310\) 0 0
\(311\) −32.9986 −1.87118 −0.935589 0.353091i \(-0.885131\pi\)
−0.935589 + 0.353091i \(0.885131\pi\)
\(312\) 0 0
\(313\) 15.4010 0.870519 0.435259 0.900305i \(-0.356657\pi\)
0.435259 + 0.900305i \(0.356657\pi\)
\(314\) 0 0
\(315\) −3.35026 −0.188766
\(316\) 0 0
\(317\) 2.15045 0.120781 0.0603905 0.998175i \(-0.480765\pi\)
0.0603905 + 0.998175i \(0.480765\pi\)
\(318\) 0 0
\(319\) 3.61213 0.202240
\(320\) 0 0
\(321\) −0.962389 −0.0537153
\(322\) 0 0
\(323\) −19.7283 −1.09771
\(324\) 0 0
\(325\) 2.96239 0.164324
\(326\) 0 0
\(327\) −11.4010 −0.630479
\(328\) 0 0
\(329\) −33.2506 −1.83316
\(330\) 0 0
\(331\) 14.5501 0.799745 0.399872 0.916571i \(-0.369054\pi\)
0.399872 + 0.916571i \(0.369054\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −5.92478 −0.323705
\(336\) 0 0
\(337\) 16.2619 0.885840 0.442920 0.896561i \(-0.353943\pi\)
0.442920 + 0.896561i \(0.353943\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 9.92478 0.537457
\(342\) 0 0
\(343\) 9.29948 0.502125
\(344\) 0 0
\(345\) −6.70052 −0.360744
\(346\) 0 0
\(347\) 0.962389 0.0516637 0.0258319 0.999666i \(-0.491777\pi\)
0.0258319 + 0.999666i \(0.491777\pi\)
\(348\) 0 0
\(349\) 20.7005 1.10807 0.554037 0.832492i \(-0.313087\pi\)
0.554037 + 0.832492i \(0.313087\pi\)
\(350\) 0 0
\(351\) −2.96239 −0.158121
\(352\) 0 0
\(353\) 20.5501 1.09377 0.546885 0.837208i \(-0.315813\pi\)
0.546885 + 0.837208i \(0.315813\pi\)
\(354\) 0 0
\(355\) −9.92478 −0.526752
\(356\) 0 0
\(357\) −15.3258 −0.811129
\(358\) 0 0
\(359\) −17.9248 −0.946034 −0.473017 0.881053i \(-0.656835\pi\)
−0.473017 + 0.881053i \(0.656835\pi\)
\(360\) 0 0
\(361\) −0.401047 −0.0211077
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −7.73813 −0.405032
\(366\) 0 0
\(367\) 29.6531 1.54788 0.773939 0.633261i \(-0.218284\pi\)
0.773939 + 0.633261i \(0.218284\pi\)
\(368\) 0 0
\(369\) −4.38787 −0.228424
\(370\) 0 0
\(371\) −15.7480 −0.817595
\(372\) 0 0
\(373\) −9.13918 −0.473209 −0.236604 0.971606i \(-0.576035\pi\)
−0.236604 + 0.971606i \(0.576035\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −10.7005 −0.551105
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −14.5745 −0.746675
\(382\) 0 0
\(383\) 34.9234 1.78450 0.892250 0.451541i \(-0.149126\pi\)
0.892250 + 0.451541i \(0.149126\pi\)
\(384\) 0 0
\(385\) 3.35026 0.170745
\(386\) 0 0
\(387\) 9.27504 0.471477
\(388\) 0 0
\(389\) 2.77575 0.140736 0.0703680 0.997521i \(-0.477583\pi\)
0.0703680 + 0.997521i \(0.477583\pi\)
\(390\) 0 0
\(391\) −30.6516 −1.55012
\(392\) 0 0
\(393\) −5.92478 −0.298865
\(394\) 0 0
\(395\) −11.5369 −0.580485
\(396\) 0 0
\(397\) −19.9248 −0.999996 −0.499998 0.866027i \(-0.666666\pi\)
−0.499998 + 0.866027i \(0.666666\pi\)
\(398\) 0 0
\(399\) 14.4485 0.723330
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −29.4010 −1.46457
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −13.0738 −0.646458 −0.323229 0.946321i \(-0.604768\pi\)
−0.323229 + 0.946321i \(0.604768\pi\)
\(410\) 0 0
\(411\) −13.8496 −0.683148
\(412\) 0 0
\(413\) 35.8496 1.76404
\(414\) 0 0
\(415\) −10.8872 −0.534430
\(416\) 0 0
\(417\) 13.6121 0.666589
\(418\) 0 0
\(419\) −7.22425 −0.352928 −0.176464 0.984307i \(-0.556466\pi\)
−0.176464 + 0.984307i \(0.556466\pi\)
\(420\) 0 0
\(421\) 30.6253 1.49259 0.746293 0.665618i \(-0.231832\pi\)
0.746293 + 0.665618i \(0.231832\pi\)
\(422\) 0 0
\(423\) 9.92478 0.482559
\(424\) 0 0
\(425\) −4.57452 −0.221897
\(426\) 0 0
\(427\) 29.1490 1.41062
\(428\) 0 0
\(429\) 2.96239 0.143025
\(430\) 0 0
\(431\) 33.8759 1.63174 0.815872 0.578232i \(-0.196257\pi\)
0.815872 + 0.578232i \(0.196257\pi\)
\(432\) 0 0
\(433\) −9.47627 −0.455400 −0.227700 0.973731i \(-0.573121\pi\)
−0.227700 + 0.973731i \(0.573121\pi\)
\(434\) 0 0
\(435\) 3.61213 0.173188
\(436\) 0 0
\(437\) 28.8970 1.38233
\(438\) 0 0
\(439\) 29.4617 1.40613 0.703065 0.711126i \(-0.251814\pi\)
0.703065 + 0.711126i \(0.251814\pi\)
\(440\) 0 0
\(441\) 4.22425 0.201155
\(442\) 0 0
\(443\) 19.0738 0.906224 0.453112 0.891454i \(-0.350314\pi\)
0.453112 + 0.891454i \(0.350314\pi\)
\(444\) 0 0
\(445\) −2.77575 −0.131583
\(446\) 0 0
\(447\) −1.53690 −0.0726931
\(448\) 0 0
\(449\) 35.8759 1.69309 0.846544 0.532318i \(-0.178679\pi\)
0.846544 + 0.532318i \(0.178679\pi\)
\(450\) 0 0
\(451\) 4.38787 0.206617
\(452\) 0 0
\(453\) −6.76116 −0.317667
\(454\) 0 0
\(455\) −9.92478 −0.465281
\(456\) 0 0
\(457\) 5.28963 0.247438 0.123719 0.992317i \(-0.460518\pi\)
0.123719 + 0.992317i \(0.460518\pi\)
\(458\) 0 0
\(459\) 4.57452 0.213520
\(460\) 0 0
\(461\) 36.3390 1.69248 0.846238 0.532805i \(-0.178862\pi\)
0.846238 + 0.532805i \(0.178862\pi\)
\(462\) 0 0
\(463\) −10.5501 −0.490304 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(464\) 0 0
\(465\) 9.92478 0.460251
\(466\) 0 0
\(467\) −18.7005 −0.865357 −0.432679 0.901548i \(-0.642431\pi\)
−0.432679 + 0.901548i \(0.642431\pi\)
\(468\) 0 0
\(469\) 19.8496 0.916567
\(470\) 0 0
\(471\) 5.47627 0.252333
\(472\) 0 0
\(473\) −9.27504 −0.426467
\(474\) 0 0
\(475\) 4.31265 0.197878
\(476\) 0 0
\(477\) 4.70052 0.215222
\(478\) 0 0
\(479\) 9.29948 0.424904 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(480\) 0 0
\(481\) −5.92478 −0.270147
\(482\) 0 0
\(483\) 22.4485 1.02144
\(484\) 0 0
\(485\) 0.0752228 0.00341569
\(486\) 0 0
\(487\) 35.4763 1.60758 0.803792 0.594911i \(-0.202813\pi\)
0.803792 + 0.594911i \(0.202813\pi\)
\(488\) 0 0
\(489\) 12.6253 0.570936
\(490\) 0 0
\(491\) −24.7757 −1.11811 −0.559057 0.829129i \(-0.688837\pi\)
−0.559057 + 0.829129i \(0.688837\pi\)
\(492\) 0 0
\(493\) 16.5237 0.744191
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 33.2506 1.49149
\(498\) 0 0
\(499\) −14.1768 −0.634640 −0.317320 0.948318i \(-0.602783\pi\)
−0.317320 + 0.948318i \(0.602783\pi\)
\(500\) 0 0
\(501\) 18.3634 0.820418
\(502\) 0 0
\(503\) 8.43866 0.376261 0.188131 0.982144i \(-0.439757\pi\)
0.188131 + 0.982144i \(0.439757\pi\)
\(504\) 0 0
\(505\) −15.0884 −0.671425
\(506\) 0 0
\(507\) 4.22425 0.187606
\(508\) 0 0
\(509\) 1.10299 0.0488890 0.0244445 0.999701i \(-0.492218\pi\)
0.0244445 + 0.999701i \(0.492218\pi\)
\(510\) 0 0
\(511\) 25.9248 1.14684
\(512\) 0 0
\(513\) −4.31265 −0.190408
\(514\) 0 0
\(515\) 3.22425 0.142078
\(516\) 0 0
\(517\) −9.92478 −0.436491
\(518\) 0 0
\(519\) 8.57452 0.376379
\(520\) 0 0
\(521\) −12.4485 −0.545379 −0.272690 0.962102i \(-0.587913\pi\)
−0.272690 + 0.962102i \(0.587913\pi\)
\(522\) 0 0
\(523\) −30.0508 −1.31403 −0.657015 0.753878i \(-0.728181\pi\)
−0.657015 + 0.753878i \(0.728181\pi\)
\(524\) 0 0
\(525\) 3.35026 0.146217
\(526\) 0 0
\(527\) 45.4010 1.97770
\(528\) 0 0
\(529\) 21.8970 0.952044
\(530\) 0 0
\(531\) −10.7005 −0.464363
\(532\) 0 0
\(533\) −12.9986 −0.563031
\(534\) 0 0
\(535\) 0.962389 0.0416077
\(536\) 0 0
\(537\) 14.1768 0.611774
\(538\) 0 0
\(539\) −4.22425 −0.181951
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 5.22425 0.224194
\(544\) 0 0
\(545\) 11.4010 0.488367
\(546\) 0 0
\(547\) 14.3028 0.611544 0.305772 0.952105i \(-0.401086\pi\)
0.305772 + 0.952105i \(0.401086\pi\)
\(548\) 0 0
\(549\) −8.70052 −0.371329
\(550\) 0 0
\(551\) −15.5778 −0.663638
\(552\) 0 0
\(553\) 38.6516 1.64364
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −11.7988 −0.499930 −0.249965 0.968255i \(-0.580419\pi\)
−0.249965 + 0.968255i \(0.580419\pi\)
\(558\) 0 0
\(559\) 27.4763 1.16212
\(560\) 0 0
\(561\) −4.57452 −0.193136
\(562\) 0 0
\(563\) −30.4847 −1.28478 −0.642389 0.766379i \(-0.722056\pi\)
−0.642389 + 0.766379i \(0.722056\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −3.35026 −0.140698
\(568\) 0 0
\(569\) −27.0884 −1.13560 −0.567802 0.823165i \(-0.692206\pi\)
−0.567802 + 0.823165i \(0.692206\pi\)
\(570\) 0 0
\(571\) −7.28489 −0.304863 −0.152432 0.988314i \(-0.548710\pi\)
−0.152432 + 0.988314i \(0.548710\pi\)
\(572\) 0 0
\(573\) −16.6253 −0.694532
\(574\) 0 0
\(575\) 6.70052 0.279431
\(576\) 0 0
\(577\) −31.6239 −1.31652 −0.658260 0.752791i \(-0.728707\pi\)
−0.658260 + 0.752791i \(0.728707\pi\)
\(578\) 0 0
\(579\) 16.3634 0.680041
\(580\) 0 0
\(581\) 36.4749 1.51323
\(582\) 0 0
\(583\) −4.70052 −0.194676
\(584\) 0 0
\(585\) 2.96239 0.122480
\(586\) 0 0
\(587\) −33.1490 −1.36821 −0.684103 0.729385i \(-0.739806\pi\)
−0.684103 + 0.729385i \(0.739806\pi\)
\(588\) 0 0
\(589\) −42.8021 −1.76363
\(590\) 0 0
\(591\) 20.4241 0.840134
\(592\) 0 0
\(593\) 34.4993 1.41672 0.708358 0.705853i \(-0.249436\pi\)
0.708358 + 0.705853i \(0.249436\pi\)
\(594\) 0 0
\(595\) 15.3258 0.628298
\(596\) 0 0
\(597\) −8.62530 −0.353010
\(598\) 0 0
\(599\) 14.4485 0.590350 0.295175 0.955443i \(-0.404622\pi\)
0.295175 + 0.955443i \(0.404622\pi\)
\(600\) 0 0
\(601\) −15.9248 −0.649585 −0.324793 0.945785i \(-0.605295\pi\)
−0.324793 + 0.945785i \(0.605295\pi\)
\(602\) 0 0
\(603\) −5.92478 −0.241276
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 14.5745 0.591561 0.295781 0.955256i \(-0.404420\pi\)
0.295781 + 0.955256i \(0.404420\pi\)
\(608\) 0 0
\(609\) −12.1016 −0.490380
\(610\) 0 0
\(611\) 29.4010 1.18944
\(612\) 0 0
\(613\) 16.4123 0.662887 0.331443 0.943475i \(-0.392464\pi\)
0.331443 + 0.943475i \(0.392464\pi\)
\(614\) 0 0
\(615\) 4.38787 0.176936
\(616\) 0 0
\(617\) −17.8496 −0.718596 −0.359298 0.933223i \(-0.616984\pi\)
−0.359298 + 0.933223i \(0.616984\pi\)
\(618\) 0 0
\(619\) 0.402462 0.0161763 0.00808815 0.999967i \(-0.497425\pi\)
0.00808815 + 0.999967i \(0.497425\pi\)
\(620\) 0 0
\(621\) −6.70052 −0.268883
\(622\) 0 0
\(623\) 9.29948 0.372576
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.31265 0.172231
\(628\) 0 0
\(629\) 9.14903 0.364796
\(630\) 0 0
\(631\) 38.0263 1.51380 0.756902 0.653528i \(-0.226712\pi\)
0.756902 + 0.653528i \(0.226712\pi\)
\(632\) 0 0
\(633\) 9.08840 0.361231
\(634\) 0 0
\(635\) 14.5745 0.578372
\(636\) 0 0
\(637\) 12.5139 0.495818
\(638\) 0 0
\(639\) −9.92478 −0.392618
\(640\) 0 0
\(641\) −28.0263 −1.10697 −0.553487 0.832858i \(-0.686703\pi\)
−0.553487 + 0.832858i \(0.686703\pi\)
\(642\) 0 0
\(643\) 4.62530 0.182404 0.0912020 0.995832i \(-0.470929\pi\)
0.0912020 + 0.995832i \(0.470929\pi\)
\(644\) 0 0
\(645\) −9.27504 −0.365204
\(646\) 0 0
\(647\) −23.5778 −0.926941 −0.463470 0.886112i \(-0.653396\pi\)
−0.463470 + 0.886112i \(0.653396\pi\)
\(648\) 0 0
\(649\) 10.7005 0.420032
\(650\) 0 0
\(651\) −33.2506 −1.30319
\(652\) 0 0
\(653\) −2.25202 −0.0881282 −0.0440641 0.999029i \(-0.514031\pi\)
−0.0440641 + 0.999029i \(0.514031\pi\)
\(654\) 0 0
\(655\) 5.92478 0.231500
\(656\) 0 0
\(657\) −7.73813 −0.301893
\(658\) 0 0
\(659\) 41.4010 1.61276 0.806378 0.591401i \(-0.201425\pi\)
0.806378 + 0.591401i \(0.201425\pi\)
\(660\) 0 0
\(661\) 3.40105 0.132285 0.0661427 0.997810i \(-0.478931\pi\)
0.0661427 + 0.997810i \(0.478931\pi\)
\(662\) 0 0
\(663\) 13.5515 0.526296
\(664\) 0 0
\(665\) −14.4485 −0.560289
\(666\) 0 0
\(667\) −24.2031 −0.937149
\(668\) 0 0
\(669\) −6.70052 −0.259057
\(670\) 0 0
\(671\) 8.70052 0.335880
\(672\) 0 0
\(673\) 0.887166 0.0341977 0.0170989 0.999854i \(-0.494557\pi\)
0.0170989 + 0.999854i \(0.494557\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 18.9018 0.726453 0.363227 0.931701i \(-0.381675\pi\)
0.363227 + 0.931701i \(0.381675\pi\)
\(678\) 0 0
\(679\) −0.252016 −0.00967149
\(680\) 0 0
\(681\) 16.9624 0.650000
\(682\) 0 0
\(683\) 20.8773 0.798848 0.399424 0.916766i \(-0.369210\pi\)
0.399424 + 0.916766i \(0.369210\pi\)
\(684\) 0 0
\(685\) 13.8496 0.529164
\(686\) 0 0
\(687\) −25.8496 −0.986222
\(688\) 0 0
\(689\) 13.9248 0.530492
\(690\) 0 0
\(691\) 2.44851 0.0931456 0.0465728 0.998915i \(-0.485170\pi\)
0.0465728 + 0.998915i \(0.485170\pi\)
\(692\) 0 0
\(693\) 3.35026 0.127266
\(694\) 0 0
\(695\) −13.6121 −0.516337
\(696\) 0 0
\(697\) 20.0724 0.760296
\(698\) 0 0
\(699\) 19.2750 0.729049
\(700\) 0 0
\(701\) −2.98683 −0.112811 −0.0564054 0.998408i \(-0.517964\pi\)
−0.0564054 + 0.998408i \(0.517964\pi\)
\(702\) 0 0
\(703\) −8.62530 −0.325309
\(704\) 0 0
\(705\) −9.92478 −0.373789
\(706\) 0 0
\(707\) 50.5501 1.90113
\(708\) 0 0
\(709\) 24.1768 0.907979 0.453989 0.891007i \(-0.350000\pi\)
0.453989 + 0.891007i \(0.350000\pi\)
\(710\) 0 0
\(711\) −11.5369 −0.432668
\(712\) 0 0
\(713\) −66.5012 −2.49049
\(714\) 0 0
\(715\) −2.96239 −0.110787
\(716\) 0 0
\(717\) 26.5501 0.991531
\(718\) 0 0
\(719\) 30.0263 1.11979 0.559897 0.828562i \(-0.310841\pi\)
0.559897 + 0.828562i \(0.310841\pi\)
\(720\) 0 0
\(721\) −10.8021 −0.402291
\(722\) 0 0
\(723\) −28.5501 −1.06179
\(724\) 0 0
\(725\) −3.61213 −0.134151
\(726\) 0 0
\(727\) −14.9525 −0.554559 −0.277279 0.960789i \(-0.589433\pi\)
−0.277279 + 0.960789i \(0.589433\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −42.4288 −1.56929
\(732\) 0 0
\(733\) −19.1128 −0.705949 −0.352974 0.935633i \(-0.614830\pi\)
−0.352974 + 0.935633i \(0.614830\pi\)
\(734\) 0 0
\(735\) −4.22425 −0.155814
\(736\) 0 0
\(737\) 5.92478 0.218242
\(738\) 0 0
\(739\) 3.31406 0.121910 0.0609549 0.998141i \(-0.480585\pi\)
0.0609549 + 0.998141i \(0.480585\pi\)
\(740\) 0 0
\(741\) −12.7757 −0.469329
\(742\) 0 0
\(743\) 34.9887 1.28361 0.641806 0.766867i \(-0.278185\pi\)
0.641806 + 0.766867i \(0.278185\pi\)
\(744\) 0 0
\(745\) 1.53690 0.0563078
\(746\) 0 0
\(747\) −10.8872 −0.398341
\(748\) 0 0
\(749\) −3.22425 −0.117812
\(750\) 0 0
\(751\) 26.9234 0.982447 0.491224 0.871033i \(-0.336550\pi\)
0.491224 + 0.871033i \(0.336550\pi\)
\(752\) 0 0
\(753\) 29.9248 1.09052
\(754\) 0 0
\(755\) 6.76116 0.246064
\(756\) 0 0
\(757\) 15.9248 0.578796 0.289398 0.957209i \(-0.406545\pi\)
0.289398 + 0.957209i \(0.406545\pi\)
\(758\) 0 0
\(759\) 6.70052 0.243214
\(760\) 0 0
\(761\) −30.9380 −1.12150 −0.560750 0.827985i \(-0.689487\pi\)
−0.560750 + 0.827985i \(0.689487\pi\)
\(762\) 0 0
\(763\) −38.1965 −1.38281
\(764\) 0 0
\(765\) −4.57452 −0.165392
\(766\) 0 0
\(767\) −31.6991 −1.14459
\(768\) 0 0
\(769\) 9.32582 0.336298 0.168149 0.985762i \(-0.446221\pi\)
0.168149 + 0.985762i \(0.446221\pi\)
\(770\) 0 0
\(771\) −8.70052 −0.313342
\(772\) 0 0
\(773\) 44.7005 1.60777 0.803883 0.594787i \(-0.202764\pi\)
0.803883 + 0.594787i \(0.202764\pi\)
\(774\) 0 0
\(775\) −9.92478 −0.356509
\(776\) 0 0
\(777\) −6.70052 −0.240380
\(778\) 0 0
\(779\) −18.9234 −0.678000
\(780\) 0 0
\(781\) 9.92478 0.355136
\(782\) 0 0
\(783\) 3.61213 0.129087
\(784\) 0 0
\(785\) −5.47627 −0.195456
\(786\) 0 0
\(787\) −21.6775 −0.772719 −0.386360 0.922348i \(-0.626268\pi\)
−0.386360 + 0.922348i \(0.626268\pi\)
\(788\) 0 0
\(789\) 12.2882 0.437472
\(790\) 0 0
\(791\) 20.1016 0.714730
\(792\) 0 0
\(793\) −25.7743 −0.915273
\(794\) 0 0
\(795\) −4.70052 −0.166710
\(796\) 0 0
\(797\) 22.7466 0.805725 0.402862 0.915261i \(-0.368015\pi\)
0.402862 + 0.915261i \(0.368015\pi\)
\(798\) 0 0
\(799\) −45.4010 −1.60617
\(800\) 0 0
\(801\) −2.77575 −0.0980762
\(802\) 0 0
\(803\) 7.73813 0.273073
\(804\) 0 0
\(805\) −22.4485 −0.791206
\(806\) 0 0
\(807\) 5.84955 0.205914
\(808\) 0 0
\(809\) −23.6121 −0.830158 −0.415079 0.909785i \(-0.636246\pi\)
−0.415079 + 0.909785i \(0.636246\pi\)
\(810\) 0 0
\(811\) 26.0870 0.916038 0.458019 0.888942i \(-0.348559\pi\)
0.458019 + 0.888942i \(0.348559\pi\)
\(812\) 0 0
\(813\) −5.08840 −0.178458
\(814\) 0 0
\(815\) −12.6253 −0.442245
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) 0 0
\(819\) −9.92478 −0.346800
\(820\) 0 0
\(821\) −54.4142 −1.89907 −0.949535 0.313662i \(-0.898444\pi\)
−0.949535 + 0.313662i \(0.898444\pi\)
\(822\) 0 0
\(823\) −0.121269 −0.00422716 −0.00211358 0.999998i \(-0.500673\pi\)
−0.00211358 + 0.999998i \(0.500673\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 18.2130 0.633328 0.316664 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(828\) 0 0
\(829\) 13.0738 0.454072 0.227036 0.973886i \(-0.427096\pi\)
0.227036 + 0.973886i \(0.427096\pi\)
\(830\) 0 0
\(831\) −1.41090 −0.0489434
\(832\) 0 0
\(833\) −19.3239 −0.669534
\(834\) 0 0
\(835\) −18.3634 −0.635493
\(836\) 0 0
\(837\) 9.92478 0.343050
\(838\) 0 0
\(839\) 26.5501 0.916610 0.458305 0.888795i \(-0.348457\pi\)
0.458305 + 0.888795i \(0.348457\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) 0 0
\(843\) 4.38787 0.151126
\(844\) 0 0
\(845\) −4.22425 −0.145319
\(846\) 0 0
\(847\) −3.35026 −0.115116
\(848\) 0 0
\(849\) 26.5745 0.912035
\(850\) 0 0
\(851\) −13.4010 −0.459382
\(852\) 0 0
\(853\) 40.6155 1.39065 0.695323 0.718697i \(-0.255261\pi\)
0.695323 + 0.718697i \(0.255261\pi\)
\(854\) 0 0
\(855\) 4.31265 0.147490
\(856\) 0 0
\(857\) 20.1721 0.689064 0.344532 0.938775i \(-0.388038\pi\)
0.344532 + 0.938775i \(0.388038\pi\)
\(858\) 0 0
\(859\) −21.8035 −0.743926 −0.371963 0.928248i \(-0.621315\pi\)
−0.371963 + 0.928248i \(0.621315\pi\)
\(860\) 0 0
\(861\) −14.7005 −0.500993
\(862\) 0 0
\(863\) −35.4274 −1.20596 −0.602981 0.797755i \(-0.706021\pi\)
−0.602981 + 0.797755i \(0.706021\pi\)
\(864\) 0 0
\(865\) −8.57452 −0.291542
\(866\) 0 0
\(867\) −3.92619 −0.133340
\(868\) 0 0
\(869\) 11.5369 0.391363
\(870\) 0 0
\(871\) −17.5515 −0.594710
\(872\) 0 0
\(873\) 0.0752228 0.00254591
\(874\) 0 0
\(875\) −3.35026 −0.113260
\(876\) 0 0
\(877\) −14.0362 −0.473969 −0.236984 0.971513i \(-0.576159\pi\)
−0.236984 + 0.971513i \(0.576159\pi\)
\(878\) 0 0
\(879\) 3.42548 0.115539
\(880\) 0 0
\(881\) −21.0738 −0.709995 −0.354997 0.934867i \(-0.615518\pi\)
−0.354997 + 0.934867i \(0.615518\pi\)
\(882\) 0 0
\(883\) 42.1476 1.41838 0.709190 0.705017i \(-0.249061\pi\)
0.709190 + 0.705017i \(0.249061\pi\)
\(884\) 0 0
\(885\) 10.7005 0.359694
\(886\) 0 0
\(887\) −6.93604 −0.232889 −0.116445 0.993197i \(-0.537150\pi\)
−0.116445 + 0.993197i \(0.537150\pi\)
\(888\) 0 0
\(889\) −48.8284 −1.63765
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 42.8021 1.43232
\(894\) 0 0
\(895\) −14.1768 −0.473878
\(896\) 0 0
\(897\) −19.8496 −0.662757
\(898\) 0 0
\(899\) 35.8496 1.19565
\(900\) 0 0
\(901\) −21.5026 −0.716356
\(902\) 0 0
\(903\) 31.0738 1.03407
\(904\) 0 0
\(905\) −5.22425 −0.173660
\(906\) 0 0
\(907\) −53.2017 −1.76653 −0.883267 0.468870i \(-0.844661\pi\)
−0.883267 + 0.468870i \(0.844661\pi\)
\(908\) 0 0
\(909\) −15.0884 −0.500451
\(910\) 0 0
\(911\) −36.4749 −1.20847 −0.604233 0.796808i \(-0.706520\pi\)
−0.604233 + 0.796808i \(0.706520\pi\)
\(912\) 0 0
\(913\) 10.8872 0.360313
\(914\) 0 0
\(915\) 8.70052 0.287630
\(916\) 0 0
\(917\) −19.8496 −0.655490
\(918\) 0 0
\(919\) −9.73340 −0.321075 −0.160538 0.987030i \(-0.551323\pi\)
−0.160538 + 0.987030i \(0.551323\pi\)
\(920\) 0 0
\(921\) −16.6497 −0.548628
\(922\) 0 0
\(923\) −29.4010 −0.967747
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 3.22425 0.105898
\(928\) 0 0
\(929\) −24.1768 −0.793215 −0.396607 0.917988i \(-0.629813\pi\)
−0.396607 + 0.917988i \(0.629813\pi\)
\(930\) 0 0
\(931\) 18.2177 0.597062
\(932\) 0 0
\(933\) 32.9986 1.08033
\(934\) 0 0
\(935\) 4.57452 0.149603
\(936\) 0 0
\(937\) −7.48612 −0.244561 −0.122280 0.992496i \(-0.539021\pi\)
−0.122280 + 0.992496i \(0.539021\pi\)
\(938\) 0 0
\(939\) −15.4010 −0.502594
\(940\) 0 0
\(941\) −21.2360 −0.692274 −0.346137 0.938184i \(-0.612507\pi\)
−0.346137 + 0.938184i \(0.612507\pi\)
\(942\) 0 0
\(943\) −29.4010 −0.957430
\(944\) 0 0
\(945\) 3.35026 0.108984
\(946\) 0 0
\(947\) 15.4763 0.502911 0.251456 0.967869i \(-0.419091\pi\)
0.251456 + 0.967869i \(0.419091\pi\)
\(948\) 0 0
\(949\) −22.9234 −0.744124
\(950\) 0 0
\(951\) −2.15045 −0.0697330
\(952\) 0 0
\(953\) −32.0508 −1.03823 −0.519113 0.854705i \(-0.673738\pi\)
−0.519113 + 0.854705i \(0.673738\pi\)
\(954\) 0 0
\(955\) 16.6253 0.537982
\(956\) 0 0
\(957\) −3.61213 −0.116763
\(958\) 0 0
\(959\) −46.3996 −1.49832
\(960\) 0 0
\(961\) 67.5012 2.17746
\(962\) 0 0
\(963\) 0.962389 0.0310125
\(964\) 0 0
\(965\) −16.3634 −0.526758
\(966\) 0 0
\(967\) 17.3766 0.558794 0.279397 0.960176i \(-0.409865\pi\)
0.279397 + 0.960176i \(0.409865\pi\)
\(968\) 0 0
\(969\) 19.7283 0.633764
\(970\) 0 0
\(971\) −36.2031 −1.16181 −0.580907 0.813970i \(-0.697302\pi\)
−0.580907 + 0.813970i \(0.697302\pi\)
\(972\) 0 0
\(973\) 45.6042 1.46200
\(974\) 0 0
\(975\) −2.96239 −0.0948724
\(976\) 0 0
\(977\) −28.1476 −0.900522 −0.450261 0.892897i \(-0.648669\pi\)
−0.450261 + 0.892897i \(0.648669\pi\)
\(978\) 0 0
\(979\) 2.77575 0.0887132
\(980\) 0 0
\(981\) 11.4010 0.364007
\(982\) 0 0
\(983\) −7.07381 −0.225619 −0.112810 0.993617i \(-0.535985\pi\)
−0.112810 + 0.993617i \(0.535985\pi\)
\(984\) 0 0
\(985\) −20.4241 −0.650765
\(986\) 0 0
\(987\) 33.2506 1.05838
\(988\) 0 0
\(989\) 62.1476 1.97618
\(990\) 0 0
\(991\) −44.4260 −1.41124 −0.705619 0.708592i \(-0.749331\pi\)
−0.705619 + 0.708592i \(0.749331\pi\)
\(992\) 0 0
\(993\) −14.5501 −0.461733
\(994\) 0 0
\(995\) 8.62530 0.273440
\(996\) 0 0
\(997\) −28.4847 −0.902120 −0.451060 0.892494i \(-0.648954\pi\)
−0.451060 + 0.892494i \(0.648954\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 2640.2.a.be.1.1 3
3.2 odd 2 7920.2.a.cj.1.1 3
4.3 odd 2 165.2.a.c.1.2 3
12.11 even 2 495.2.a.e.1.2 3
20.3 even 4 825.2.c.g.199.4 6
20.7 even 4 825.2.c.g.199.3 6
20.19 odd 2 825.2.a.k.1.2 3
28.27 even 2 8085.2.a.bk.1.2 3
44.43 even 2 1815.2.a.m.1.2 3
60.23 odd 4 2475.2.c.r.199.3 6
60.47 odd 4 2475.2.c.r.199.4 6
60.59 even 2 2475.2.a.bb.1.2 3
132.131 odd 2 5445.2.a.z.1.2 3
220.219 even 2 9075.2.a.cf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 4.3 odd 2
495.2.a.e.1.2 3 12.11 even 2
825.2.a.k.1.2 3 20.19 odd 2
825.2.c.g.199.3 6 20.7 even 4
825.2.c.g.199.4 6 20.3 even 4
1815.2.a.m.1.2 3 44.43 even 2
2475.2.a.bb.1.2 3 60.59 even 2
2475.2.c.r.199.3 6 60.23 odd 4
2475.2.c.r.199.4 6 60.47 odd 4
2640.2.a.be.1.1 3 1.1 even 1 trivial
5445.2.a.z.1.2 3 132.131 odd 2
7920.2.a.cj.1.1 3 3.2 odd 2
8085.2.a.bk.1.2 3 28.27 even 2
9075.2.a.cf.1.2 3 220.219 even 2