Properties

Label 2475.2.a.bb.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} +O(q^{10})\) \(q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} -1.00000 q^{11} -2.96239 q^{13} +0.649738 q^{14} +3.77575 q^{16} -4.57452 q^{17} -4.31265 q^{19} +0.193937 q^{22} -6.70052 q^{23} +0.574515 q^{26} +6.57452 q^{28} +3.61213 q^{29} +9.92478 q^{31} -2.26916 q^{32} +0.887166 q^{34} +2.00000 q^{37} +0.836381 q^{38} +4.38787 q^{41} +9.27504 q^{43} +1.96239 q^{44} +1.29948 q^{46} -9.92478 q^{47} +4.22425 q^{49} +5.81336 q^{52} +4.70052 q^{53} -2.57452 q^{56} -0.700523 q^{58} -10.7005 q^{59} -8.70052 q^{61} -1.92478 q^{62} -7.11142 q^{64} -5.92478 q^{67} +8.97698 q^{68} -9.92478 q^{71} +7.73813 q^{73} -0.387873 q^{74} +8.46310 q^{76} +3.35026 q^{77} +11.5369 q^{79} -0.850969 q^{82} +10.8872 q^{83} -1.79877 q^{86} -0.768452 q^{88} +2.77575 q^{89} +9.92478 q^{91} +13.1490 q^{92} +1.92478 q^{94} -0.0752228 q^{97} -0.819237 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 9 q^{8} + O(q^{10}) \) \( 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41} - 4 q^{43} - 5 q^{44} + 24 q^{46} - 8 q^{47} + 11 q^{49} + 30 q^{52} - 6 q^{53} + 4 q^{56} + 18 q^{58} - 12 q^{59} - 6 q^{61} + 16 q^{62} + 13 q^{64} + 4 q^{67} + 42 q^{68} - 8 q^{71} + 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 26 q^{82} + 8 q^{86} + 9 q^{88} + 10 q^{89} + 8 q^{91} + 16 q^{92} - 16 q^{94} - 22 q^{97} + 39 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.193937 −0.137134 −0.0685669 0.997647i \(-0.521843\pi\)
−0.0685669 + 0.997647i \(0.521843\pi\)
\(3\) 0 0
\(4\) −1.96239 −0.981194
\(5\) 0 0
\(6\) 0 0
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 0.768452 0.271689
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.96239 −0.821619 −0.410809 0.911721i \(-0.634754\pi\)
−0.410809 + 0.911721i \(0.634754\pi\)
\(14\) 0.649738 0.173650
\(15\) 0 0
\(16\) 3.77575 0.943937
\(17\) −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 \(-0.664720\pi\)
−0.494695 + 0.869067i \(0.664720\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.193937 0.0413474
\(23\) −6.70052 −1.39716 −0.698578 0.715534i \(-0.746183\pi\)
−0.698578 + 0.715534i \(0.746183\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.574515 0.112672
\(27\) 0 0
\(28\) 6.57452 1.24247
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) 9.92478 1.78254 0.891271 0.453470i \(-0.149814\pi\)
0.891271 + 0.453470i \(0.149814\pi\)
\(32\) −2.26916 −0.401134
\(33\) 0 0
\(34\) 0.887166 0.152148
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0.836381 0.135679
\(39\) 0 0
\(40\) 0 0
\(41\) 4.38787 0.685271 0.342635 0.939468i \(-0.388680\pi\)
0.342635 + 0.939468i \(0.388680\pi\)
\(42\) 0 0
\(43\) 9.27504 1.41443 0.707215 0.706998i \(-0.249951\pi\)
0.707215 + 0.706998i \(0.249951\pi\)
\(44\) 1.96239 0.295841
\(45\) 0 0
\(46\) 1.29948 0.191597
\(47\) −9.92478 −1.44768 −0.723839 0.689969i \(-0.757624\pi\)
−0.723839 + 0.689969i \(0.757624\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) 0 0
\(52\) 5.81336 0.806168
\(53\) 4.70052 0.645667 0.322833 0.946456i \(-0.395365\pi\)
0.322833 + 0.946456i \(0.395365\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.57452 −0.344034
\(57\) 0 0
\(58\) −0.700523 −0.0919832
\(59\) −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 \(0.688045\pi\)
\(62\) −1.92478 −0.244447
\(63\) 0 0
\(64\) −7.11142 −0.888927
\(65\) 0 0
\(66\) 0 0
\(67\) −5.92478 −0.723827 −0.361913 0.932212i \(-0.617876\pi\)
−0.361913 + 0.932212i \(0.617876\pi\)
\(68\) 8.97698 1.08862
\(69\) 0 0
\(70\) 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 \(-0.350411\pi\)
0.452840 + 0.891592i \(0.350411\pi\)
\(74\) −0.387873 −0.0450893
\(75\) 0 0
\(76\) 8.46310 0.970784
\(77\) 3.35026 0.381798
\(78\) 0 0
\(79\) 11.5369 1.29800 0.649002 0.760787i \(-0.275187\pi\)
0.649002 + 0.760787i \(0.275187\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.850969 −0.0939738
\(83\) 10.8872 1.19502 0.597511 0.801861i \(-0.296156\pi\)
0.597511 + 0.801861i \(0.296156\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.79877 −0.193966
\(87\) 0 0
\(88\) −0.768452 −0.0819173
\(89\) 2.77575 0.294229 0.147114 0.989120i \(-0.453001\pi\)
0.147114 + 0.989120i \(0.453001\pi\)
\(90\) 0 0
\(91\) 9.92478 1.04040
\(92\) 13.1490 1.37088
\(93\) 0 0
\(94\) 1.92478 0.198526
\(95\) 0 0
\(96\) 0 0
\(97\) −0.0752228 −0.00763772 −0.00381886 0.999993i \(-0.501216\pi\)
−0.00381886 + 0.999993i \(0.501216\pi\)
\(98\) −0.819237 −0.0827555
\(99\) 0 0
\(100\) 0 0
\(101\) 15.0884 1.50135 0.750676 0.660671i \(-0.229728\pi\)
0.750676 + 0.660671i \(0.229728\pi\)
\(102\) 0 0
\(103\) 3.22425 0.317695 0.158848 0.987303i \(-0.449222\pi\)
0.158848 + 0.987303i \(0.449222\pi\)
\(104\) −2.27645 −0.223225
\(105\) 0 0
\(106\) −0.911603 −0.0885427
\(107\) −0.962389 −0.0930376 −0.0465188 0.998917i \(-0.514813\pi\)
−0.0465188 + 0.998917i \(0.514813\pi\)
\(108\) 0 0
\(109\) 11.4010 1.09202 0.546011 0.837778i \(-0.316146\pi\)
0.546011 + 0.837778i \(0.316146\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −12.6497 −1.19529
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.08840 −0.658141
\(117\) 0 0
\(118\) 2.07522 0.191040
\(119\) 15.3258 1.40492
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.68735 0.152765
\(123\) 0 0
\(124\) −19.4763 −1.74902
\(125\) 0 0
\(126\) 0 0
\(127\) 14.5745 1.29328 0.646640 0.762796i \(-0.276174\pi\)
0.646640 + 0.762796i \(0.276174\pi\)
\(128\) 5.91748 0.523037
\(129\) 0 0
\(130\) 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
\(134\) 1.14903 0.0992612
\(135\) 0 0
\(136\) −3.51530 −0.301434
\(137\) 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 \(-0.304113\pi\)
0.577283 + 0.816544i \(0.304113\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.92478 0.161524
\(143\) 2.96239 0.247727
\(144\) 0 0
\(145\) 0 0
\(146\) −1.50071 −0.124199
\(147\) 0 0
\(148\) −3.92478 −0.322615
\(149\) −1.53690 −0.125908 −0.0629540 0.998016i \(-0.520052\pi\)
−0.0629540 + 0.998016i \(0.520052\pi\)
\(150\) 0 0
\(151\) −6.76116 −0.550215 −0.275108 0.961413i \(-0.588713\pi\)
−0.275108 + 0.961413i \(0.588713\pi\)
\(152\) −3.31406 −0.268806
\(153\) 0 0
\(154\) −0.649738 −0.0523574
\(155\) 0 0
\(156\) 0 0
\(157\) 5.47627 0.437054 0.218527 0.975831i \(-0.429875\pi\)
0.218527 + 0.975831i \(0.429875\pi\)
\(158\) −2.23743 −0.178000
\(159\) 0 0
\(160\) 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 \(0.664629\pi\)
\(164\) −8.61071 −0.672384
\(165\) 0 0
\(166\) −2.11142 −0.163878
\(167\) 18.3634 1.42101 0.710503 0.703695i \(-0.248468\pi\)
0.710503 + 0.703695i \(0.248468\pi\)
\(168\) 0 0
\(169\) −4.22425 −0.324943
\(170\) 0 0
\(171\) 0 0
\(172\) −18.2012 −1.38783
\(173\) −8.57452 −0.651908 −0.325954 0.945386i \(-0.605686\pi\)
−0.325954 + 0.945386i \(0.605686\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.77575 −0.284608
\(177\) 0 0
\(178\) −0.538319 −0.0403487
\(179\) −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 \(0.562197\pi\)
\(182\) −1.92478 −0.142674
\(183\) 0 0
\(184\) −5.14903 −0.379592
\(185\) 0 0
\(186\) 0 0
\(187\) 4.57452 0.334522
\(188\) 19.4763 1.42045
\(189\) 0 0
\(190\) 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 \(-0.299548\pi\)
0.588933 + 0.808182i \(0.299548\pi\)
\(194\) 0.0145884 0.00104739
\(195\) 0 0
\(196\) −8.28963 −0.592116
\(197\) −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 \(-0.598896\pi\)
−0.305716 + 0.952123i \(0.598896\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.92619 −0.205886
\(203\) −12.1016 −0.849364
\(204\) 0 0
\(205\) 0 0
\(206\) −0.625301 −0.0435668
\(207\) 0 0
\(208\) −11.1852 −0.775556
\(209\) 4.31265 0.298312
\(210\) 0 0
\(211\) 9.08840 0.625671 0.312836 0.949807i \(-0.398721\pi\)
0.312836 + 0.949807i \(0.398721\pi\)
\(212\) −9.22425 −0.633524
\(213\) 0 0
\(214\) 0.186642 0.0127586
\(215\) 0 0
\(216\) 0 0
\(217\) −33.2506 −2.25720
\(218\) −2.21108 −0.149753
\(219\) 0 0
\(220\) 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 \(0.427974\pi\)
\(224\) 7.60228 0.507949
\(225\) 0 0
\(226\) 1.16362 0.0774028
\(227\) 16.9624 1.12583 0.562917 0.826514i \(-0.309679\pi\)
0.562917 + 0.826514i \(0.309679\pi\)
\(228\) 0 0
\(229\) 25.8496 1.70819 0.854093 0.520120i \(-0.174113\pi\)
0.854093 + 0.520120i \(0.174113\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.77575 0.182237
\(233\) −19.2750 −1.26275 −0.631375 0.775478i \(-0.717509\pi\)
−0.631375 + 0.775478i \(0.717509\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 20.9986 1.36689
\(237\) 0 0
\(238\) −2.97224 −0.192662
\(239\) −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 \(0.128565\pi\)
\(242\) −0.193937 −0.0124667
\(243\) 0 0
\(244\) 17.0738 1.09304
\(245\) 0 0
\(246\) 0 0
\(247\) 12.7757 0.812901
\(248\) 7.62672 0.484297
\(249\) 0 0
\(250\) 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
\(254\) −2.82653 −0.177352
\(255\) 0 0
\(256\) 13.0752 0.817201
\(257\) 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
\(261\) 0 0
\(262\) −1.14903 −0.0709874
\(263\) 12.2882 0.757724 0.378862 0.925453i \(-0.376316\pi\)
0.378862 + 0.925453i \(0.376316\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.80209 −0.171807
\(267\) 0 0
\(268\) 11.6267 0.710215
\(269\) 5.84955 0.356654 0.178327 0.983971i \(-0.442932\pi\)
0.178327 + 0.983971i \(0.442932\pi\)
\(270\) 0 0
\(271\) −5.08840 −0.309098 −0.154549 0.987985i \(-0.549392\pi\)
−0.154549 + 0.987985i \(0.549392\pi\)
\(272\) −17.2722 −1.04728
\(273\) 0 0
\(274\) −2.68594 −0.162263
\(275\) 0 0
\(276\) 0 0
\(277\) −1.41090 −0.0847725 −0.0423863 0.999101i \(-0.513496\pi\)
−0.0423863 + 0.999101i \(0.513496\pi\)
\(278\) −2.63989 −0.158330
\(279\) 0 0
\(280\) 0 0
\(281\) 4.38787 0.261759 0.130879 0.991398i \(-0.458220\pi\)
0.130879 + 0.991398i \(0.458220\pi\)
\(282\) 0 0
\(283\) −26.5745 −1.57969 −0.789845 0.613306i \(-0.789839\pi\)
−0.789845 + 0.613306i \(0.789839\pi\)
\(284\) 19.4763 1.15570
\(285\) 0 0
\(286\) −0.574515 −0.0339718
\(287\) −14.7005 −0.867744
\(288\) 0 0
\(289\) 3.92619 0.230952
\(290\) 0 0
\(291\) 0 0
\(292\) −15.1852 −0.888648
\(293\) −3.42548 −0.200119 −0.100059 0.994981i \(-0.531903\pi\)
−0.100059 + 0.994981i \(0.531903\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.53690 0.0893307
\(297\) 0 0
\(298\) 0.298062 0.0172663
\(299\) 19.8496 1.14793
\(300\) 0 0
\(301\) −31.0738 −1.79106
\(302\) 1.31124 0.0754531
\(303\) 0 0
\(304\) −16.2835 −0.933921
\(305\) 0 0
\(306\) 0 0
\(307\) 16.6497 0.950251 0.475125 0.879918i \(-0.342403\pi\)
0.475125 + 0.879918i \(0.342403\pi\)
\(308\) −6.57452 −0.374618
\(309\) 0 0
\(310\) 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 \(-0.643343\pi\)
−0.435259 + 0.900305i \(0.643343\pi\)
\(314\) −1.06205 −0.0599349
\(315\) 0 0
\(316\) −22.6399 −1.27359
\(317\) 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 0
\(322\) −4.35359 −0.242616
\(323\) 19.7283 1.09771
\(324\) 0 0
\(325\) 0 0
\(326\) 2.44851 0.135610
\(327\) 0 0
\(328\) 3.37187 0.186180
\(329\) 33.2506 1.83316
\(330\) 0 0
\(331\) −14.5501 −0.799745 −0.399872 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(332\) −21.3649 −1.17255
\(333\) 0 0
\(334\) −3.56134 −0.194868
\(335\) 0 0
\(336\) 0 0
\(337\) −16.2619 −0.885840 −0.442920 0.896561i \(-0.646057\pi\)
−0.442920 + 0.896561i \(0.646057\pi\)
\(338\) 0.819237 0.0445606
\(339\) 0 0
\(340\) 0 0
\(341\) −9.92478 −0.537457
\(342\) 0 0
\(343\) 9.29948 0.502125
\(344\) 7.12742 0.384285
\(345\) 0 0
\(346\) 1.66291 0.0893987
\(347\) −0.962389 −0.0516637 −0.0258319 0.999666i \(-0.508223\pi\)
−0.0258319 + 0.999666i \(0.508223\pi\)
\(348\) 0 0
\(349\) 20.7005 1.10807 0.554037 0.832492i \(-0.313087\pi\)
0.554037 + 0.832492i \(0.313087\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.26916 0.120947
\(353\) 20.5501 1.09377 0.546885 0.837208i \(-0.315813\pi\)
0.546885 + 0.837208i \(0.315813\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −5.44709 −0.288695
\(357\) 0 0
\(358\) 2.74940 0.145310
\(359\) −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
\(362\) 1.01317 0.0532512
\(363\) 0 0
\(364\) −19.4763 −1.02083
\(365\) 0 0
\(366\) 0 0
\(367\) 29.6531 1.54788 0.773939 0.633261i \(-0.218284\pi\)
0.773939 + 0.633261i \(0.218284\pi\)
\(368\) −25.2995 −1.31883
\(369\) 0 0
\(370\) 0 0
\(371\) −15.7480 −0.817595
\(372\) 0 0
\(373\) 9.13918 0.473209 0.236604 0.971606i \(-0.423965\pi\)
0.236604 + 0.971606i \(0.423965\pi\)
\(374\) −0.887166 −0.0458743
\(375\) 0 0
\(376\) −7.62672 −0.393318
\(377\) −10.7005 −0.551105
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.22425 −0.164967
\(383\) −34.9234 −1.78450 −0.892250 0.451541i \(-0.850874\pi\)
−0.892250 + 0.451541i \(0.850874\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.17347 −0.161525
\(387\) 0 0
\(388\) 0.147616 0.00749408
\(389\) −2.77575 −0.140736 −0.0703680 0.997521i \(-0.522417\pi\)
−0.0703680 + 0.997521i \(0.522417\pi\)
\(390\) 0 0
\(391\) 30.6516 1.55012
\(392\) 3.24614 0.163955
\(393\) 0 0
\(394\) 3.96097 0.199551
\(395\) 0 0
\(396\) 0 0
\(397\) 19.9248 0.999996 0.499998 0.866027i \(-0.333334\pi\)
0.499998 + 0.866027i \(0.333334\pi\)
\(398\) 1.67276 0.0838479
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −29.4010 −1.46457
\(404\) −29.6093 −1.47312
\(405\) 0 0
\(406\) 2.34694 0.116477
\(407\) −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
\(411\) 0 0
\(412\) −6.32724 −0.311721
\(413\) 35.8496 1.76404
\(414\) 0 0
\(415\) 0 0
\(416\) 6.72213 0.329580
\(417\) 0 0
\(418\) −0.836381 −0.0409087
\(419\) −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 \(0.231832\pi\)
\(422\) −1.76257 −0.0858007
\(423\) 0 0
\(424\) 3.61213 0.175420
\(425\) 0 0
\(426\) 0 0
\(427\) 29.1490 1.41062
\(428\) 1.88858 0.0912880
\(429\) 0 0
\(430\) 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 \(-0.426879\pi\)
0.227700 + 0.973731i \(0.426879\pi\)
\(434\) 6.44851 0.309538
\(435\) 0 0
\(436\) −22.3733 −1.07149
\(437\) 28.8970 1.38233
\(438\) 0 0
\(439\) −29.4617 −1.40613 −0.703065 0.711126i \(-0.748186\pi\)
−0.703065 + 0.711126i \(0.748186\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.62813 −0.125007
\(443\) −19.0738 −0.906224 −0.453112 0.891454i \(-0.649686\pi\)
−0.453112 + 0.891454i \(0.649686\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.29948 −0.0615320
\(447\) 0 0
\(448\) 23.8251 1.12563
\(449\) −35.8759 −1.69309 −0.846544 0.532318i \(-0.821321\pi\)
−0.846544 + 0.532318i \(0.821321\pi\)
\(450\) 0 0
\(451\) −4.38787 −0.206617
\(452\) 11.7743 0.553818
\(453\) 0 0
\(454\) −3.28963 −0.154390
\(455\) 0 0
\(456\) 0 0
\(457\) −5.28963 −0.247438 −0.123719 0.992317i \(-0.539482\pi\)
−0.123719 + 0.992317i \(0.539482\pi\)
\(458\) −5.01317 −0.234250
\(459\) 0 0
\(460\) 0 0
\(461\) −36.3390 −1.69248 −0.846238 0.532805i \(-0.821138\pi\)
−0.846238 + 0.532805i \(0.821138\pi\)
\(462\) 0 0
\(463\) −10.5501 −0.490304 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(464\) 13.6385 0.633150
\(465\) 0 0
\(466\) 3.73813 0.173166
\(467\) 18.7005 0.865357 0.432679 0.901548i \(-0.357569\pi\)
0.432679 + 0.901548i \(0.357569\pi\)
\(468\) 0 0
\(469\) 19.8496 0.916567
\(470\) 0 0
\(471\) 0 0
\(472\) −8.22284 −0.378487
\(473\) −9.27504 −0.426467
\(474\) 0 0
\(475\) 0 0
\(476\) −30.0752 −1.37850
\(477\) 0 0
\(478\) 5.14903 0.235511
\(479\) 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
\(482\) −5.53690 −0.252199
\(483\) 0 0
\(484\) −1.96239 −0.0891995
\(485\) 0 0
\(486\) 0 0
\(487\) 35.4763 1.60758 0.803792 0.594911i \(-0.202813\pi\)
0.803792 + 0.594911i \(0.202813\pi\)
\(488\) −6.68594 −0.302658
\(489\) 0 0
\(490\) 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
\(494\) −2.47768 −0.111476
\(495\) 0 0
\(496\) 37.4734 1.68261
\(497\) 33.2506 1.49149
\(498\) 0 0
\(499\) 14.1768 0.634640 0.317320 0.948318i \(-0.397217\pi\)
0.317320 + 0.948318i \(0.397217\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.80351 0.259023
\(503\) −8.43866 −0.376261 −0.188131 0.982144i \(-0.560243\pi\)
−0.188131 + 0.982144i \(0.560243\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.29948 −0.0577688
\(507\) 0 0
\(508\) −28.6009 −1.26896
\(509\) −1.10299 −0.0488890 −0.0244445 0.999701i \(-0.507782\pi\)
−0.0244445 + 0.999701i \(0.507782\pi\)
\(510\) 0 0
\(511\) −25.9248 −1.14684
\(512\) −14.3707 −0.635103
\(513\) 0 0
\(514\) −1.68735 −0.0744258
\(515\) 0 0
\(516\) 0 0
\(517\) 9.92478 0.436491
\(518\) 1.29948 0.0570957
\(519\) 0 0
\(520\) 0 0
\(521\) 12.4485 0.545379 0.272690 0.962102i \(-0.412087\pi\)
0.272690 + 0.962102i \(0.412087\pi\)
\(522\) 0 0
\(523\) −30.0508 −1.31403 −0.657015 0.753878i \(-0.728181\pi\)
−0.657015 + 0.753878i \(0.728181\pi\)
\(524\) −11.6267 −0.507915
\(525\) 0 0
\(526\) −2.38313 −0.103910
\(527\) −45.4010 −1.97770
\(528\) 0 0
\(529\) 21.8970 0.952044
\(530\) 0 0
\(531\) 0 0
\(532\) −28.3536 −1.22928
\(533\) −12.9986 −0.563031
\(534\) 0 0
\(535\) 0 0
\(536\) −4.55291 −0.196656
\(537\) 0 0
\(538\) −1.13444 −0.0489093
\(539\) −4.22425 −0.181951
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0.986826 0.0423878
\(543\) 0 0
\(544\) 10.3803 0.445052
\(545\) 0 0
\(546\) 0 0
\(547\) 14.3028 0.611544 0.305772 0.952105i \(-0.401086\pi\)
0.305772 + 0.952105i \(0.401086\pi\)
\(548\) −27.1782 −1.16100
\(549\) 0 0
\(550\) 0 0
\(551\) −15.5778 −0.663638
\(552\) 0 0
\(553\) −38.6516 −1.64364
\(554\) 0.273624 0.0116252
\(555\) 0 0
\(556\) −26.7123 −1.13285
\(557\) −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
\(561\) 0 0
\(562\) −0.850969 −0.0358960
\(563\) 30.4847 1.28478 0.642389 0.766379i \(-0.277944\pi\)
0.642389 + 0.766379i \(0.277944\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.15377 0.216629
\(567\) 0 0
\(568\) −7.62672 −0.320010
\(569\) 27.0884 1.13560 0.567802 0.823165i \(-0.307794\pi\)
0.567802 + 0.823165i \(0.307794\pi\)
\(570\) 0 0
\(571\) 7.28489 0.304863 0.152432 0.988314i \(-0.451290\pi\)
0.152432 + 0.988314i \(0.451290\pi\)
\(572\) −5.81336 −0.243069
\(573\) 0 0
\(574\) 2.85097 0.118997
\(575\) 0 0
\(576\) 0 0
\(577\) 31.6239 1.31652 0.658260 0.752791i \(-0.271293\pi\)
0.658260 + 0.752791i \(0.271293\pi\)
\(578\) −0.761432 −0.0316714
\(579\) 0 0
\(580\) 0 0
\(581\) −36.4749 −1.51323
\(582\) 0 0
\(583\) −4.70052 −0.194676
\(584\) 5.94639 0.246063
\(585\) 0 0
\(586\) 0.664327 0.0274431
\(587\) 33.1490 1.36821 0.684103 0.729385i \(-0.260194\pi\)
0.684103 + 0.729385i \(0.260194\pi\)
\(588\) 0 0
\(589\) −42.8021 −1.76363
\(590\) 0 0
\(591\) 0 0
\(592\) 7.55149 0.310364
\(593\) 34.4993 1.41672 0.708358 0.705853i \(-0.249436\pi\)
0.708358 + 0.705853i \(0.249436\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.01600 0.123540
\(597\) 0 0
\(598\) −3.84955 −0.157420
\(599\) 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 \(0.605295\pi\)
\(602\) 6.02635 0.245616
\(603\) 0 0
\(604\) 13.2680 0.539868
\(605\) 0 0
\(606\) 0 0
\(607\) 14.5745 0.591561 0.295781 0.955256i \(-0.404420\pi\)
0.295781 + 0.955256i \(0.404420\pi\)
\(608\) 9.78609 0.396878
\(609\) 0 0
\(610\) 0 0
\(611\) 29.4010 1.18944
\(612\) 0 0
\(613\) −16.4123 −0.662887 −0.331443 0.943475i \(-0.607536\pi\)
−0.331443 + 0.943475i \(0.607536\pi\)
\(614\) −3.22899 −0.130312
\(615\) 0 0
\(616\) 2.57452 0.103730
\(617\) −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 \(-0.502575\pi\)
−0.00808815 + 0.999967i \(0.502575\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.39963 0.256602
\(623\) −9.29948 −0.372576
\(624\) 0 0
\(625\) 0 0
\(626\) 2.98683 0.119378
\(627\) 0 0
\(628\) −10.7466 −0.428835
\(629\) −9.14903 −0.364796
\(630\) 0 0
\(631\) −38.0263 −1.51380 −0.756902 0.653528i \(-0.773288\pi\)
−0.756902 + 0.653528i \(0.773288\pi\)
\(632\) 8.86556 0.352653
\(633\) 0 0
\(634\) −0.417050 −0.0165632
\(635\) 0 0
\(636\) 0 0
\(637\) −12.5139 −0.495818
\(638\) 0.700523 0.0277340
\(639\) 0 0
\(640\) 0 0
\(641\) 28.0263 1.10697 0.553487 0.832858i \(-0.313297\pi\)
0.553487 + 0.832858i \(0.313297\pi\)
\(642\) 0 0
\(643\) 4.62530 0.182404 0.0912020 0.995832i \(-0.470929\pi\)
0.0912020 + 0.995832i \(0.470929\pi\)
\(644\) −44.0527 −1.73592
\(645\) 0 0
\(646\) −3.82604 −0.150533
\(647\) 23.5778 0.926941 0.463470 0.886112i \(-0.346604\pi\)
0.463470 + 0.886112i \(0.346604\pi\)
\(648\) 0 0
\(649\) 10.7005 0.420032
\(650\) 0 0
\(651\) 0 0
\(652\) 24.7757 0.970293
\(653\) −2.25202 −0.0881282 −0.0440641 0.999029i \(-0.514031\pi\)
−0.0440641 + 0.999029i \(0.514031\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 16.5675 0.646852
\(657\) 0 0
\(658\) −6.44851 −0.251389
\(659\) 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 \(0.478931\pi\)
\(662\) 2.82179 0.109672
\(663\) 0 0
\(664\) 8.36626 0.324674
\(665\) 0 0
\(666\) 0 0
\(667\) −24.2031 −0.937149
\(668\) −36.0362 −1.39428
\(669\) 0 0
\(670\) 0 0
\(671\) 8.70052 0.335880
\(672\) 0 0
\(673\) −0.887166 −0.0341977 −0.0170989 0.999854i \(-0.505443\pi\)
−0.0170989 + 0.999854i \(0.505443\pi\)
\(674\) 3.15377 0.121479
\(675\) 0 0
\(676\) 8.28963 0.318832
\(677\) 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
\(681\) 0 0
\(682\) 1.92478 0.0737035
\(683\) −20.8773 −0.798848 −0.399424 0.916766i \(-0.630790\pi\)
−0.399424 + 0.916766i \(0.630790\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.80351 −0.0688583
\(687\) 0 0
\(688\) 35.0202 1.33513
\(689\) −13.9248 −0.530492
\(690\) 0 0
\(691\) −2.44851 −0.0931456 −0.0465728 0.998915i \(-0.514830\pi\)
−0.0465728 + 0.998915i \(0.514830\pi\)
\(692\) 16.8265 0.639649
\(693\) 0 0
\(694\) 0.186642 0.00708485
\(695\) 0 0
\(696\) 0 0
\(697\) −20.0724 −0.760296
\(698\) −4.01459 −0.151954
\(699\) 0 0
\(700\) 0 0
\(701\) 2.98683 0.112811 0.0564054 0.998408i \(-0.482036\pi\)
0.0564054 + 0.998408i \(0.482036\pi\)
\(702\) 0 0
\(703\) −8.62530 −0.325309
\(704\) 7.11142 0.268022
\(705\) 0 0
\(706\) −3.98541 −0.149993
\(707\) −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
\(711\) 0 0
\(712\) 2.13303 0.0799386
\(713\) −66.5012 −2.49049
\(714\) 0 0
\(715\) 0 0
\(716\) 27.8204 1.03970
\(717\) 0 0
\(718\) 3.47627 0.129733
\(719\) 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
\(722\) 0.0777777 0.00289459
\(723\) 0 0
\(724\) 10.2520 0.381013
\(725\) 0 0
\(726\) 0 0
\(727\) −14.9525 −0.554559 −0.277279 0.960789i \(-0.589433\pi\)
−0.277279 + 0.960789i \(0.589433\pi\)
\(728\) 7.62672 0.282665
\(729\) 0 0
\(730\) 0 0
\(731\) −42.4288 −1.56929
\(732\) 0 0
\(733\) 19.1128 0.705949 0.352974 0.935633i \(-0.385170\pi\)
0.352974 + 0.935633i \(0.385170\pi\)
\(734\) −5.75081 −0.212266
\(735\) 0 0
\(736\) 15.2046 0.560447
\(737\) 5.92478 0.218242
\(738\) 0 0
\(739\) −3.31406 −0.121910 −0.0609549 0.998141i \(-0.519415\pi\)
−0.0609549 + 0.998141i \(0.519415\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.05411 0.112120
\(743\) −34.9887 −1.28361 −0.641806 0.766867i \(-0.721815\pi\)
−0.641806 + 0.766867i \(0.721815\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.77242 −0.0648930
\(747\) 0 0
\(748\) −8.97698 −0.328231
\(749\) 3.22425 0.117812
\(750\) 0 0
\(751\) −26.9234 −0.982447 −0.491224 0.871033i \(-0.663450\pi\)
−0.491224 + 0.871033i \(0.663450\pi\)
\(752\) −37.4734 −1.36652
\(753\) 0 0
\(754\) 2.07522 0.0755752
\(755\) 0 0
\(756\) 0 0
\(757\) −15.9248 −0.578796 −0.289398 0.957209i \(-0.593455\pi\)
−0.289398 + 0.957209i \(0.593455\pi\)
\(758\) 3.87873 0.140882
\(759\) 0 0
\(760\) 0 0
\(761\) 30.9380 1.12150 0.560750 0.827985i \(-0.310513\pi\)
0.560750 + 0.827985i \(0.310513\pi\)
\(762\) 0 0
\(763\) −38.1965 −1.38281
\(764\) −32.6253 −1.18034
\(765\) 0 0
\(766\) 6.77292 0.244715
\(767\) 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
\(771\) 0 0
\(772\) −32.1114 −1.15572
\(773\) 44.7005 1.60777 0.803883 0.594787i \(-0.202764\pi\)
0.803883 + 0.594787i \(0.202764\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.0578051 −0.00207508
\(777\) 0 0
\(778\) 0.538319 0.0192997
\(779\) −18.9234 −0.678000
\(780\) 0 0
\(781\) 9.92478 0.355136
\(782\) −5.94448 −0.212574
\(783\) 0 0
\(784\) 15.9497 0.569633
\(785\) 0 0
\(786\) 0 0
\(787\) −21.6775 −0.772719 −0.386360 0.922348i \(-0.626268\pi\)
−0.386360 + 0.922348i \(0.626268\pi\)
\(788\) 40.0800 1.42779
\(789\) 0 0
\(790\) 0 0
\(791\) 20.1016 0.714730
\(792\) 0 0
\(793\) 25.7743 0.915273
\(794\) −3.86414 −0.137133
\(795\) 0 0
\(796\) 16.9262 0.599933
\(797\) 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
\(801\) 0 0
\(802\) 0.387873 0.0136963
\(803\) −7.73813 −0.273073
\(804\) 0 0
\(805\) 0 0
\(806\) 5.70194 0.200842
\(807\) 0 0
\(808\) 11.5947 0.407900
\(809\) 23.6121 0.830158 0.415079 0.909785i \(-0.363754\pi\)
0.415079 + 0.909785i \(0.363754\pi\)
\(810\) 0 0
\(811\) −26.0870 −0.916038 −0.458019 0.888942i \(-0.651441\pi\)
−0.458019 + 0.888942i \(0.651441\pi\)
\(812\) 23.7480 0.833391
\(813\) 0 0
\(814\) 0.387873 0.0135949
\(815\) 0 0
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) 2.53549 0.0886513
\(819\) 0 0
\(820\) 0 0
\(821\) 54.4142 1.89907 0.949535 0.313662i \(-0.101556\pi\)
0.949535 + 0.313662i \(0.101556\pi\)
\(822\) 0 0
\(823\) −0.121269 −0.00422716 −0.00211358 0.999998i \(-0.500673\pi\)
−0.00211358 + 0.999998i \(0.500673\pi\)
\(824\) 2.47768 0.0863142
\(825\) 0 0
\(826\) −6.95254 −0.241910
\(827\) −18.2130 −0.633328 −0.316664 0.948538i \(-0.602563\pi\)
−0.316664 + 0.948538i \(0.602563\pi\)
\(828\) 0 0
\(829\) 13.0738 0.454072 0.227036 0.973886i \(-0.427096\pi\)
0.227036 + 0.973886i \(0.427096\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 21.0668 0.730359
\(833\) −19.3239 −0.669534
\(834\) 0 0
\(835\) 0 0
\(836\) −8.46310 −0.292702
\(837\) 0 0
\(838\) 1.40105 0.0483984
\(839\) 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
\(842\) −5.93937 −0.204684
\(843\) 0 0
\(844\) −17.8350 −0.613905
\(845\) 0 0
\(846\) 0 0
\(847\) −3.35026 −0.115116
\(848\) 17.7480 0.609468
\(849\) 0 0
\(850\) 0 0
\(851\) −13.4010 −0.459382
\(852\) 0 0
\(853\) −40.6155 −1.39065 −0.695323 0.718697i \(-0.744739\pi\)
−0.695323 + 0.718697i \(0.744739\pi\)
\(854\) −5.65306 −0.193444
\(855\) 0 0
\(856\) −0.739549 −0.0252773
\(857\) 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 \(-0.378685\pi\)
0.371963 + 0.928248i \(0.378685\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.56978 −0.223767
\(863\) 35.4274 1.20596 0.602981 0.797755i \(-0.293979\pi\)
0.602981 + 0.797755i \(0.293979\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.83780 −0.0624508
\(867\) 0 0
\(868\) 65.2506 2.21475
\(869\) −11.5369 −0.391363
\(870\) 0 0
\(871\) 17.5515 0.594710
\(872\) 8.76116 0.296690
\(873\) 0 0
\(874\) −5.60419 −0.189564
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0362 0.473969 0.236984 0.971513i \(-0.423841\pi\)
0.236984 + 0.971513i \(0.423841\pi\)
\(878\) 5.71370 0.192828
\(879\) 0 0
\(880\) 0 0
\(881\) 21.0738 0.709995 0.354997 0.934867i \(-0.384482\pi\)
0.354997 + 0.934867i \(0.384482\pi\)
\(882\) 0 0
\(883\) 42.1476 1.41838 0.709190 0.705017i \(-0.249061\pi\)
0.709190 + 0.705017i \(0.249061\pi\)
\(884\) −26.5933 −0.894429
\(885\) 0 0
\(886\) 3.69911 0.124274
\(887\) 6.93604 0.232889 0.116445 0.993197i \(-0.462850\pi\)
0.116445 + 0.993197i \(0.462850\pi\)
\(888\) 0 0
\(889\) −48.8284 −1.63765
\(890\) 0 0
\(891\) 0 0
\(892\) −13.1490 −0.440262
\(893\) 42.8021 1.43232
\(894\) 0 0
\(895\) 0 0
\(896\) −19.8251 −0.662311
\(897\) 0 0
\(898\) 6.95765 0.232180
\(899\) 35.8496 1.19565
\(900\) 0 0
\(901\) −21.5026 −0.716356
\(902\) 0.850969 0.0283342
\(903\) 0 0
\(904\) −4.61071 −0.153350
\(905\) 0 0
\(906\) 0 0
\(907\) −53.2017 −1.76653 −0.883267 0.468870i \(-0.844661\pi\)
−0.883267 + 0.468870i \(0.844661\pi\)
\(908\) −33.2868 −1.10466
\(909\) 0 0
\(910\) 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
\(914\) 1.02585 0.0339322
\(915\) 0 0
\(916\) −50.7269 −1.67606
\(917\) −19.8496 −0.655490
\(918\) 0 0
\(919\) 9.73340 0.321075 0.160538 0.987030i \(-0.448677\pi\)
0.160538 + 0.987030i \(0.448677\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 7.04746 0.232096
\(923\) 29.4010 0.967747
\(924\) 0 0
\(925\) 0 0
\(926\) 2.04605 0.0672372
\(927\) 0 0
\(928\) −8.19649 −0.269063
\(929\) 24.1768 0.793215 0.396607 0.917988i \(-0.370187\pi\)
0.396607 + 0.917988i \(0.370187\pi\)
\(930\) 0 0
\(931\) −18.2177 −0.597062
\(932\) 37.8251 1.23900
\(933\) 0 0
\(934\) −3.62672 −0.118670
\(935\) 0 0
\(936\) 0 0
\(937\) 7.48612 0.244561 0.122280 0.992496i \(-0.460979\pi\)
0.122280 + 0.992496i \(0.460979\pi\)
\(938\) −3.84955 −0.125692
\(939\) 0 0
\(940\) 0 0
\(941\) 21.2360 0.692274 0.346137 0.938184i \(-0.387493\pi\)
0.346137 + 0.938184i \(0.387493\pi\)
\(942\) 0 0
\(943\) −29.4010 −0.957430
\(944\) −40.4025 −1.31499
\(945\) 0 0
\(946\) 1.79877 0.0584830
\(947\) −15.4763 −0.502911 −0.251456 0.967869i \(-0.580909\pi\)
−0.251456 + 0.967869i \(0.580909\pi\)
\(948\) 0 0
\(949\) −22.9234 −0.744124
\(950\) 0 0
\(951\) 0 0
\(952\) 11.7772 0.381700
\(953\) −32.0508 −1.03823 −0.519113 0.854705i \(-0.673738\pi\)
−0.519113 + 0.854705i \(0.673738\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 52.1016 1.68509
\(957\) 0 0
\(958\) −1.80351 −0.0582687
\(959\) −46.3996 −1.49832
\(960\) 0 0
\(961\) 67.5012 2.17746
\(962\) 1.14903 0.0370462
\(963\) 0 0
\(964\) −56.0263 −1.80449
\(965\) 0 0
\(966\) 0 0
\(967\) 17.3766 0.558794 0.279397 0.960176i \(-0.409865\pi\)
0.279397 + 0.960176i \(0.409865\pi\)
\(968\) 0.768452 0.0246990
\(969\) 0 0
\(970\) 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
\(974\) −6.88015 −0.220454
\(975\) 0 0
\(976\) −32.8510 −1.05153
\(977\) −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
\(981\) 0 0
\(982\) 4.80492 0.153331
\(983\) 7.07381 0.225619 0.112810 0.993617i \(-0.464015\pi\)
0.112810 + 0.993617i \(0.464015\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3.20456 0.102054
\(987\) 0 0
\(988\) −25.0710 −0.797614
\(989\) −62.1476 −1.97618
\(990\) 0 0
\(991\) 44.4260 1.41124 0.705619 0.708592i \(-0.250669\pi\)
0.705619 + 0.708592i \(0.250669\pi\)
\(992\) −22.5209 −0.715039
\(993\) 0 0
\(994\) −6.44851 −0.204534
\(995\) 0 0
\(996\) 0 0
\(997\) 28.4847 0.902120 0.451060 0.892494i \(-0.351046\pi\)
0.451060 + 0.892494i \(0.351046\pi\)
\(998\) −2.74940 −0.0870307
\(999\) 0 0
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 2475.2.a.bb.1.2 3
3.2 odd 2 825.2.a.k.1.2 3
5.2 odd 4 2475.2.c.r.199.3 6
5.3 odd 4 2475.2.c.r.199.4 6
5.4 even 2 495.2.a.e.1.2 3
15.2 even 4 825.2.c.g.199.4 6
15.8 even 4 825.2.c.g.199.3 6
15.14 odd 2 165.2.a.c.1.2 3
20.19 odd 2 7920.2.a.cj.1.1 3
33.32 even 2 9075.2.a.cf.1.2 3
55.54 odd 2 5445.2.a.z.1.2 3
60.59 even 2 2640.2.a.be.1.1 3
105.104 even 2 8085.2.a.bk.1.2 3
165.164 even 2 1815.2.a.m.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 15.14 odd 2
495.2.a.e.1.2 3 5.4 even 2
825.2.a.k.1.2 3 3.2 odd 2
825.2.c.g.199.3 6 15.8 even 4
825.2.c.g.199.4 6 15.2 even 4
1815.2.a.m.1.2 3 165.164 even 2
2475.2.a.bb.1.2 3 1.1 even 1 trivial
2475.2.c.r.199.3 6 5.2 odd 4
2475.2.c.r.199.4 6 5.3 odd 4
2640.2.a.be.1.1 3 60.59 even 2
5445.2.a.z.1.2 3 55.54 odd 2
7920.2.a.cj.1.1 3 20.19 odd 2
8085.2.a.bk.1.2 3 105.104 even 2
9075.2.a.cf.1.2 3 33.32 even 2