Properties

Label 7623.2.a.ci.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.737640\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.45589 q^{2} +0.119606 q^{4} -2.38705 q^{5} -1.00000 q^{7} +2.73764 q^{8} +O(q^{10})\) \(q-1.45589 q^{2} +0.119606 q^{4} -2.38705 q^{5} -1.00000 q^{7} +2.73764 q^{8} +3.47528 q^{10} -0.911774 q^{13} +1.45589 q^{14} -4.22491 q^{16} +2.18663 q^{17} +0.711349 q^{19} -0.285507 q^{20} +7.80466 q^{23} +0.698028 q^{25} +1.32744 q^{26} -0.119606 q^{28} -8.86742 q^{29} -6.62627 q^{31} +0.675706 q^{32} -3.18348 q^{34} +2.38705 q^{35} -4.20551 q^{37} -1.03564 q^{38} -6.53490 q^{40} +11.9576 q^{41} +7.51601 q^{43} -11.3627 q^{46} -6.27171 q^{47} +1.00000 q^{49} -1.01625 q^{50} -0.109054 q^{52} +6.56545 q^{53} -2.73764 q^{56} +12.9100 q^{58} +9.62312 q^{59} -0.0627598 q^{61} +9.64709 q^{62} +7.46606 q^{64} +2.17645 q^{65} -11.3294 q^{67} +0.261535 q^{68} -3.47528 q^{70} +5.96945 q^{71} -4.38705 q^{73} +6.12275 q^{74} +0.0850818 q^{76} -4.85725 q^{79} +10.0851 q^{80} -17.4089 q^{82} -10.0490 q^{83} -5.21960 q^{85} -10.9425 q^{86} +0.261535 q^{89} +0.911774 q^{91} +0.933487 q^{92} +9.13090 q^{94} -1.69803 q^{95} +9.88466 q^{97} -1.45589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + O(q^{10}) \) \( 4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + 10q^{10} + 6q^{13} + q^{14} - 3q^{16} - 8q^{17} - 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} - 3q^{28} - 18q^{31} + 2q^{32} + 18q^{34} - 4q^{35} - 2q^{37} + 8q^{38} + 6q^{40} - 10q^{41} - 4q^{43} + 11q^{46} - 4q^{47} + 4q^{49} + 9q^{50} + 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} + 14q^{61} - 11q^{64} + 28q^{65} - 28q^{67} + 16q^{68} - 10q^{70} + 18q^{71} - 4q^{73} + 41q^{74} - 4q^{76} - 20q^{79} + 36q^{80} - 24q^{82} - 6q^{83} - 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} - 16q^{94} - 16q^{95} + 32q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.45589 −1.02947 −0.514734 0.857350i \(-0.672109\pi\)
−0.514734 + 0.857350i \(0.672109\pi\)
\(3\) 0 0
\(4\) 0.119606 0.0598032
\(5\) −2.38705 −1.06752 −0.533762 0.845635i \(-0.679222\pi\)
−0.533762 + 0.845635i \(0.679222\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.73764 0.967902
\(9\) 0 0
\(10\) 3.47528 1.09898
\(11\) 0 0
\(12\) 0 0
\(13\) −0.911774 −0.252880 −0.126440 0.991974i \(-0.540355\pi\)
−0.126440 + 0.991974i \(0.540355\pi\)
\(14\) 1.45589 0.389102
\(15\) 0 0
\(16\) −4.22491 −1.05623
\(17\) 2.18663 0.530335 0.265168 0.964202i \(-0.414573\pi\)
0.265168 + 0.964202i \(0.414573\pi\)
\(18\) 0 0
\(19\) 0.711349 0.163195 0.0815973 0.996665i \(-0.473998\pi\)
0.0815973 + 0.996665i \(0.473998\pi\)
\(20\) −0.285507 −0.0638413
\(21\) 0 0
\(22\) 0 0
\(23\) 7.80466 1.62738 0.813692 0.581296i \(-0.197454\pi\)
0.813692 + 0.581296i \(0.197454\pi\)
\(24\) 0 0
\(25\) 0.698028 0.139606
\(26\) 1.32744 0.260332
\(27\) 0 0
\(28\) −0.119606 −0.0226035
\(29\) −8.86742 −1.64664 −0.823320 0.567578i \(-0.807880\pi\)
−0.823320 + 0.567578i \(0.807880\pi\)
\(30\) 0 0
\(31\) −6.62627 −1.19011 −0.595056 0.803684i \(-0.702870\pi\)
−0.595056 + 0.803684i \(0.702870\pi\)
\(32\) 0.675706 0.119449
\(33\) 0 0
\(34\) −3.18348 −0.545963
\(35\) 2.38705 0.403486
\(36\) 0 0
\(37\) −4.20551 −0.691382 −0.345691 0.938348i \(-0.612355\pi\)
−0.345691 + 0.938348i \(0.612355\pi\)
\(38\) −1.03564 −0.168003
\(39\) 0 0
\(40\) −6.53490 −1.03326
\(41\) 11.9576 1.86746 0.933731 0.357975i \(-0.116533\pi\)
0.933731 + 0.357975i \(0.116533\pi\)
\(42\) 0 0
\(43\) 7.51601 1.14618 0.573091 0.819492i \(-0.305744\pi\)
0.573091 + 0.819492i \(0.305744\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −11.3627 −1.67534
\(47\) −6.27171 −0.914823 −0.457412 0.889255i \(-0.651223\pi\)
−0.457412 + 0.889255i \(0.651223\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.01625 −0.143719
\(51\) 0 0
\(52\) −0.109054 −0.0151231
\(53\) 6.56545 0.901834 0.450917 0.892566i \(-0.351097\pi\)
0.450917 + 0.892566i \(0.351097\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.73764 −0.365833
\(57\) 0 0
\(58\) 12.9100 1.69516
\(59\) 9.62312 1.25282 0.626412 0.779492i \(-0.284523\pi\)
0.626412 + 0.779492i \(0.284523\pi\)
\(60\) 0 0
\(61\) −0.0627598 −0.00803556 −0.00401778 0.999992i \(-0.501279\pi\)
−0.00401778 + 0.999992i \(0.501279\pi\)
\(62\) 9.64709 1.22518
\(63\) 0 0
\(64\) 7.46606 0.933258
\(65\) 2.17645 0.269956
\(66\) 0 0
\(67\) −11.3294 −1.38410 −0.692052 0.721847i \(-0.743293\pi\)
−0.692052 + 0.721847i \(0.743293\pi\)
\(68\) 0.261535 0.0317157
\(69\) 0 0
\(70\) −3.47528 −0.415375
\(71\) 5.96945 0.708443 0.354221 0.935162i \(-0.384746\pi\)
0.354221 + 0.935162i \(0.384746\pi\)
\(72\) 0 0
\(73\) −4.38705 −0.513466 −0.256733 0.966482i \(-0.582646\pi\)
−0.256733 + 0.966482i \(0.582646\pi\)
\(74\) 6.12275 0.711755
\(75\) 0 0
\(76\) 0.0850818 0.00975955
\(77\) 0 0
\(78\) 0 0
\(79\) −4.85725 −0.546483 −0.273241 0.961945i \(-0.588096\pi\)
−0.273241 + 0.961945i \(0.588096\pi\)
\(80\) 10.0851 1.12755
\(81\) 0 0
\(82\) −17.4089 −1.92249
\(83\) −10.0490 −1.10302 −0.551509 0.834169i \(-0.685948\pi\)
−0.551509 + 0.834169i \(0.685948\pi\)
\(84\) 0 0
\(85\) −5.21960 −0.566145
\(86\) −10.9425 −1.17996
\(87\) 0 0
\(88\) 0 0
\(89\) 0.261535 0.0277226 0.0138613 0.999904i \(-0.495588\pi\)
0.0138613 + 0.999904i \(0.495588\pi\)
\(90\) 0 0
\(91\) 0.911774 0.0955798
\(92\) 0.933487 0.0973227
\(93\) 0 0
\(94\) 9.13090 0.941781
\(95\) −1.69803 −0.174214
\(96\) 0 0
\(97\) 9.88466 1.00363 0.501817 0.864974i \(-0.332665\pi\)
0.501817 + 0.864974i \(0.332665\pi\)
\(98\) −1.45589 −0.147067
\(99\) 0 0
\(100\) 0.0834885 0.00834885
\(101\) −9.94742 −0.989805 −0.494902 0.868949i \(-0.664796\pi\)
−0.494902 + 0.868949i \(0.664796\pi\)
\(102\) 0 0
\(103\) −0.658765 −0.0649101 −0.0324550 0.999473i \(-0.510333\pi\)
−0.0324550 + 0.999473i \(0.510333\pi\)
\(104\) −2.49611 −0.244764
\(105\) 0 0
\(106\) −9.55855 −0.928409
\(107\) −0.283088 −0.0273672 −0.0136836 0.999906i \(-0.504356\pi\)
−0.0136836 + 0.999906i \(0.504356\pi\)
\(108\) 0 0
\(109\) −8.97962 −0.860092 −0.430046 0.902807i \(-0.641503\pi\)
−0.430046 + 0.902807i \(0.641503\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.22491 0.399216
\(113\) −1.32938 −0.125058 −0.0625289 0.998043i \(-0.519917\pi\)
−0.0625289 + 0.998043i \(0.519917\pi\)
\(114\) 0 0
\(115\) −18.6302 −1.73727
\(116\) −1.06060 −0.0984742
\(117\) 0 0
\(118\) −14.0102 −1.28974
\(119\) −2.18663 −0.200448
\(120\) 0 0
\(121\) 0 0
\(122\) 0.0913711 0.00827235
\(123\) 0 0
\(124\) −0.792543 −0.0711725
\(125\) 10.2690 0.918491
\(126\) 0 0
\(127\) −21.8372 −1.93773 −0.968867 0.247580i \(-0.920365\pi\)
−0.968867 + 0.247580i \(0.920365\pi\)
\(128\) −12.2212 −1.08021
\(129\) 0 0
\(130\) −3.16867 −0.277911
\(131\) 9.22227 0.805754 0.402877 0.915254i \(-0.368010\pi\)
0.402877 + 0.915254i \(0.368010\pi\)
\(132\) 0 0
\(133\) −0.711349 −0.0616817
\(134\) 16.4943 1.42489
\(135\) 0 0
\(136\) 5.98620 0.513313
\(137\) 2.04750 0.174929 0.0874647 0.996168i \(-0.472124\pi\)
0.0874647 + 0.996168i \(0.472124\pi\)
\(138\) 0 0
\(139\) −12.8454 −1.08953 −0.544766 0.838588i \(-0.683382\pi\)
−0.544766 + 0.838588i \(0.683382\pi\)
\(140\) 0.285507 0.0241297
\(141\) 0 0
\(142\) −8.69084 −0.729319
\(143\) 0 0
\(144\) 0 0
\(145\) 21.1670 1.75783
\(146\) 6.38705 0.528596
\(147\) 0 0
\(148\) −0.503006 −0.0413468
\(149\) −15.3451 −1.25712 −0.628561 0.777761i \(-0.716356\pi\)
−0.628561 + 0.777761i \(0.716356\pi\)
\(150\) 0 0
\(151\) −6.90936 −0.562275 −0.281138 0.959667i \(-0.590712\pi\)
−0.281138 + 0.959667i \(0.590712\pi\)
\(152\) 1.94742 0.157956
\(153\) 0 0
\(154\) 0 0
\(155\) 15.8173 1.27047
\(156\) 0 0
\(157\) −9.33762 −0.745223 −0.372611 0.927987i \(-0.621538\pi\)
−0.372611 + 0.927987i \(0.621538\pi\)
\(158\) 7.07160 0.562586
\(159\) 0 0
\(160\) −1.61295 −0.127515
\(161\) −7.80466 −0.615094
\(162\) 0 0
\(163\) 10.4309 0.817013 0.408507 0.912755i \(-0.366050\pi\)
0.408507 + 0.912755i \(0.366050\pi\)
\(164\) 1.43020 0.111680
\(165\) 0 0
\(166\) 14.6302 1.13552
\(167\) −1.35141 −0.104575 −0.0522877 0.998632i \(-0.516651\pi\)
−0.0522877 + 0.998632i \(0.516651\pi\)
\(168\) 0 0
\(169\) −12.1687 −0.936051
\(170\) 7.59915 0.582828
\(171\) 0 0
\(172\) 0.898962 0.0685452
\(173\) −19.4855 −1.48145 −0.740726 0.671807i \(-0.765518\pi\)
−0.740726 + 0.671807i \(0.765518\pi\)
\(174\) 0 0
\(175\) −0.698028 −0.0527659
\(176\) 0 0
\(177\) 0 0
\(178\) −0.380765 −0.0285395
\(179\) 12.8541 0.960761 0.480380 0.877060i \(-0.340499\pi\)
0.480380 + 0.877060i \(0.340499\pi\)
\(180\) 0 0
\(181\) 14.3749 1.06848 0.534239 0.845333i \(-0.320598\pi\)
0.534239 + 0.845333i \(0.320598\pi\)
\(182\) −1.32744 −0.0983963
\(183\) 0 0
\(184\) 21.3664 1.57515
\(185\) 10.0388 0.738066
\(186\) 0 0
\(187\) 0 0
\(188\) −0.750136 −0.0547093
\(189\) 0 0
\(190\) 2.47214 0.179348
\(191\) 23.9520 1.73311 0.866554 0.499083i \(-0.166330\pi\)
0.866554 + 0.499083i \(0.166330\pi\)
\(192\) 0 0
\(193\) 0.139609 0.0100493 0.00502463 0.999987i \(-0.498401\pi\)
0.00502463 + 0.999987i \(0.498401\pi\)
\(194\) −14.3909 −1.03321
\(195\) 0 0
\(196\) 0.119606 0.00854331
\(197\) −14.6113 −1.04101 −0.520505 0.853859i \(-0.674256\pi\)
−0.520505 + 0.853859i \(0.674256\pi\)
\(198\) 0 0
\(199\) −26.4702 −1.87642 −0.938210 0.346066i \(-0.887517\pi\)
−0.938210 + 0.346066i \(0.887517\pi\)
\(200\) 1.91095 0.135124
\(201\) 0 0
\(202\) 14.4823 1.01897
\(203\) 8.86742 0.622371
\(204\) 0 0
\(205\) −28.5434 −1.99356
\(206\) 0.959087 0.0668228
\(207\) 0 0
\(208\) 3.85216 0.267099
\(209\) 0 0
\(210\) 0 0
\(211\) 17.5618 1.20901 0.604503 0.796603i \(-0.293372\pi\)
0.604503 + 0.796603i \(0.293372\pi\)
\(212\) 0.785269 0.0539325
\(213\) 0 0
\(214\) 0.412145 0.0281736
\(215\) −17.9411 −1.22357
\(216\) 0 0
\(217\) 6.62627 0.449820
\(218\) 13.0733 0.885436
\(219\) 0 0
\(220\) 0 0
\(221\) −1.99371 −0.134111
\(222\) 0 0
\(223\) 22.5839 1.51233 0.756164 0.654383i \(-0.227071\pi\)
0.756164 + 0.654383i \(0.227071\pi\)
\(224\) −0.675706 −0.0451475
\(225\) 0 0
\(226\) 1.93543 0.128743
\(227\) −21.9714 −1.45829 −0.729146 0.684358i \(-0.760083\pi\)
−0.729146 + 0.684358i \(0.760083\pi\)
\(228\) 0 0
\(229\) 23.3803 1.54501 0.772507 0.635007i \(-0.219003\pi\)
0.772507 + 0.635007i \(0.219003\pi\)
\(230\) 27.1234 1.78846
\(231\) 0 0
\(232\) −24.2758 −1.59379
\(233\) 1.04315 0.0683390 0.0341695 0.999416i \(-0.489121\pi\)
0.0341695 + 0.999416i \(0.489121\pi\)
\(234\) 0 0
\(235\) 14.9709 0.976595
\(236\) 1.15099 0.0749228
\(237\) 0 0
\(238\) 3.18348 0.206355
\(239\) 12.4140 0.802994 0.401497 0.915860i \(-0.368490\pi\)
0.401497 + 0.915860i \(0.368490\pi\)
\(240\) 0 0
\(241\) 9.42435 0.607076 0.303538 0.952819i \(-0.401832\pi\)
0.303538 + 0.952819i \(0.401832\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −0.00750646 −0.000480552 0
\(245\) −2.38705 −0.152503
\(246\) 0 0
\(247\) −0.648589 −0.0412687
\(248\) −18.1403 −1.15191
\(249\) 0 0
\(250\) −14.9506 −0.945557
\(251\) 3.07219 0.193915 0.0969576 0.995289i \(-0.469089\pi\)
0.0969576 + 0.995289i \(0.469089\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 31.7924 1.99483
\(255\) 0 0
\(256\) 2.86049 0.178781
\(257\) −0.626267 −0.0390654 −0.0195327 0.999809i \(-0.506218\pi\)
−0.0195327 + 0.999809i \(0.506218\pi\)
\(258\) 0 0
\(259\) 4.20551 0.261318
\(260\) 0.260318 0.0161442
\(261\) 0 0
\(262\) −13.4266 −0.829497
\(263\) −1.02517 −0.0632149 −0.0316074 0.999500i \(-0.510063\pi\)
−0.0316074 + 0.999500i \(0.510063\pi\)
\(264\) 0 0
\(265\) −15.6721 −0.962729
\(266\) 1.03564 0.0634993
\(267\) 0 0
\(268\) −1.35507 −0.0827738
\(269\) −32.0066 −1.95147 −0.975737 0.218945i \(-0.929738\pi\)
−0.975737 + 0.218945i \(0.929738\pi\)
\(270\) 0 0
\(271\) 5.95536 0.361762 0.180881 0.983505i \(-0.442105\pi\)
0.180881 + 0.983505i \(0.442105\pi\)
\(272\) −9.23830 −0.560155
\(273\) 0 0
\(274\) −2.98092 −0.180084
\(275\) 0 0
\(276\) 0 0
\(277\) −1.27771 −0.0767700 −0.0383850 0.999263i \(-0.512221\pi\)
−0.0383850 + 0.999263i \(0.512221\pi\)
\(278\) 18.7014 1.12164
\(279\) 0 0
\(280\) 6.53490 0.390535
\(281\) 4.45011 0.265471 0.132736 0.991151i \(-0.457624\pi\)
0.132736 + 0.991151i \(0.457624\pi\)
\(282\) 0 0
\(283\) 22.7964 1.35511 0.677554 0.735473i \(-0.263040\pi\)
0.677554 + 0.735473i \(0.263040\pi\)
\(284\) 0.713983 0.0423671
\(285\) 0 0
\(286\) 0 0
\(287\) −11.9576 −0.705834
\(288\) 0 0
\(289\) −12.2187 −0.718744
\(290\) −30.8168 −1.80962
\(291\) 0 0
\(292\) −0.524719 −0.0307069
\(293\) 8.39334 0.490344 0.245172 0.969480i \(-0.421156\pi\)
0.245172 + 0.969480i \(0.421156\pi\)
\(294\) 0 0
\(295\) −22.9709 −1.33742
\(296\) −11.5132 −0.669190
\(297\) 0 0
\(298\) 22.3408 1.29417
\(299\) −7.11609 −0.411534
\(300\) 0 0
\(301\) −7.51601 −0.433216
\(302\) 10.0592 0.578844
\(303\) 0 0
\(304\) −3.00538 −0.172370
\(305\) 0.149811 0.00857815
\(306\) 0 0
\(307\) 11.0560 0.630999 0.315500 0.948926i \(-0.397828\pi\)
0.315500 + 0.948926i \(0.397828\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −23.0281 −1.30791
\(311\) 22.5914 1.28104 0.640519 0.767942i \(-0.278719\pi\)
0.640519 + 0.767942i \(0.278719\pi\)
\(312\) 0 0
\(313\) 2.54869 0.144061 0.0720303 0.997402i \(-0.477052\pi\)
0.0720303 + 0.997402i \(0.477052\pi\)
\(314\) 13.5945 0.767183
\(315\) 0 0
\(316\) −0.580957 −0.0326814
\(317\) −31.3434 −1.76042 −0.880212 0.474581i \(-0.842599\pi\)
−0.880212 + 0.474581i \(0.842599\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −17.8219 −0.996274
\(321\) 0 0
\(322\) 11.3627 0.633219
\(323\) 1.55546 0.0865479
\(324\) 0 0
\(325\) −0.636443 −0.0353035
\(326\) −15.1863 −0.841089
\(327\) 0 0
\(328\) 32.7356 1.80752
\(329\) 6.27171 0.345771
\(330\) 0 0
\(331\) −4.82789 −0.265365 −0.132682 0.991159i \(-0.542359\pi\)
−0.132682 + 0.991159i \(0.542359\pi\)
\(332\) −1.20192 −0.0659639
\(333\) 0 0
\(334\) 1.96750 0.107657
\(335\) 27.0438 1.47756
\(336\) 0 0
\(337\) 21.1153 1.15022 0.575112 0.818074i \(-0.304958\pi\)
0.575112 + 0.818074i \(0.304958\pi\)
\(338\) 17.7162 0.963634
\(339\) 0 0
\(340\) −0.624297 −0.0338573
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 20.5761 1.10939
\(345\) 0 0
\(346\) 28.3686 1.52511
\(347\) 12.4251 0.667015 0.333508 0.942747i \(-0.391768\pi\)
0.333508 + 0.942747i \(0.391768\pi\)
\(348\) 0 0
\(349\) 16.9682 0.908289 0.454145 0.890928i \(-0.349945\pi\)
0.454145 + 0.890928i \(0.349945\pi\)
\(350\) 1.01625 0.0543208
\(351\) 0 0
\(352\) 0 0
\(353\) 4.57101 0.243291 0.121645 0.992574i \(-0.461183\pi\)
0.121645 + 0.992574i \(0.461183\pi\)
\(354\) 0 0
\(355\) −14.2494 −0.756279
\(356\) 0.0312812 0.00165790
\(357\) 0 0
\(358\) −18.7141 −0.989072
\(359\) 30.9733 1.63471 0.817355 0.576134i \(-0.195440\pi\)
0.817355 + 0.576134i \(0.195440\pi\)
\(360\) 0 0
\(361\) −18.4940 −0.973368
\(362\) −20.9282 −1.09996
\(363\) 0 0
\(364\) 0.109054 0.00571598
\(365\) 10.4721 0.548137
\(366\) 0 0
\(367\) 14.7422 0.769536 0.384768 0.923013i \(-0.374281\pi\)
0.384768 + 0.923013i \(0.374281\pi\)
\(368\) −32.9740 −1.71889
\(369\) 0 0
\(370\) −14.6153 −0.759815
\(371\) −6.56545 −0.340861
\(372\) 0 0
\(373\) −12.5701 −0.650853 −0.325427 0.945567i \(-0.605508\pi\)
−0.325427 + 0.945567i \(0.605508\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −17.1697 −0.885459
\(377\) 8.08508 0.416403
\(378\) 0 0
\(379\) 16.0340 0.823610 0.411805 0.911272i \(-0.364899\pi\)
0.411805 + 0.911272i \(0.364899\pi\)
\(380\) −0.203095 −0.0104185
\(381\) 0 0
\(382\) −34.8714 −1.78418
\(383\) 2.81023 0.143596 0.0717979 0.997419i \(-0.477126\pi\)
0.0717979 + 0.997419i \(0.477126\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.203254 −0.0103454
\(387\) 0 0
\(388\) 1.18227 0.0600205
\(389\) 20.6050 1.04471 0.522357 0.852727i \(-0.325053\pi\)
0.522357 + 0.852727i \(0.325053\pi\)
\(390\) 0 0
\(391\) 17.0659 0.863060
\(392\) 2.73764 0.138272
\(393\) 0 0
\(394\) 21.2724 1.07169
\(395\) 11.5945 0.583383
\(396\) 0 0
\(397\) 28.6588 1.43834 0.719171 0.694833i \(-0.244522\pi\)
0.719171 + 0.694833i \(0.244522\pi\)
\(398\) 38.5376 1.93171
\(399\) 0 0
\(400\) −2.94910 −0.147455
\(401\) 22.3156 1.11439 0.557194 0.830383i \(-0.311878\pi\)
0.557194 + 0.830383i \(0.311878\pi\)
\(402\) 0 0
\(403\) 6.04165 0.300956
\(404\) −1.18977 −0.0591935
\(405\) 0 0
\(406\) −12.9100 −0.640711
\(407\) 0 0
\(408\) 0 0
\(409\) 16.4360 0.812709 0.406354 0.913716i \(-0.366800\pi\)
0.406354 + 0.913716i \(0.366800\pi\)
\(410\) 41.5560 2.05230
\(411\) 0 0
\(412\) −0.0787925 −0.00388183
\(413\) −9.62312 −0.473523
\(414\) 0 0
\(415\) 23.9874 1.17750
\(416\) −0.616090 −0.0302063
\(417\) 0 0
\(418\) 0 0
\(419\) 12.7725 0.623975 0.311988 0.950086i \(-0.399005\pi\)
0.311988 + 0.950086i \(0.399005\pi\)
\(420\) 0 0
\(421\) 10.8032 0.526514 0.263257 0.964726i \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(422\) −25.5680 −1.24463
\(423\) 0 0
\(424\) 17.9738 0.872887
\(425\) 1.52633 0.0740378
\(426\) 0 0
\(427\) 0.0627598 0.00303716
\(428\) −0.0338592 −0.00163664
\(429\) 0 0
\(430\) 26.1202 1.25963
\(431\) −27.5395 −1.32653 −0.663266 0.748384i \(-0.730830\pi\)
−0.663266 + 0.748384i \(0.730830\pi\)
\(432\) 0 0
\(433\) 15.2600 0.733351 0.366675 0.930349i \(-0.380496\pi\)
0.366675 + 0.930349i \(0.380496\pi\)
\(434\) −9.64709 −0.463075
\(435\) 0 0
\(436\) −1.07402 −0.0514362
\(437\) 5.55184 0.265580
\(438\) 0 0
\(439\) −28.5500 −1.36262 −0.681308 0.731997i \(-0.738589\pi\)
−0.681308 + 0.731997i \(0.738589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.90262 0.138063
\(443\) −11.3310 −0.538354 −0.269177 0.963091i \(-0.586752\pi\)
−0.269177 + 0.963091i \(0.586752\pi\)
\(444\) 0 0
\(445\) −0.624297 −0.0295945
\(446\) −32.8795 −1.55689
\(447\) 0 0
\(448\) −7.46606 −0.352738
\(449\) 26.8968 1.26934 0.634669 0.772784i \(-0.281137\pi\)
0.634669 + 0.772784i \(0.281137\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.159003 −0.00747885
\(453\) 0 0
\(454\) 31.9879 1.50126
\(455\) −2.17645 −0.102034
\(456\) 0 0
\(457\) −28.9694 −1.35513 −0.677567 0.735461i \(-0.736966\pi\)
−0.677567 + 0.735461i \(0.736966\pi\)
\(458\) −34.0391 −1.59054
\(459\) 0 0
\(460\) −2.22828 −0.103894
\(461\) 41.9375 1.95322 0.976612 0.215008i \(-0.0689778\pi\)
0.976612 + 0.215008i \(0.0689778\pi\)
\(462\) 0 0
\(463\) −1.14904 −0.0534005 −0.0267003 0.999643i \(-0.508500\pi\)
−0.0267003 + 0.999643i \(0.508500\pi\)
\(464\) 37.4640 1.73922
\(465\) 0 0
\(466\) −1.51871 −0.0703528
\(467\) −29.5887 −1.36920 −0.684600 0.728919i \(-0.740023\pi\)
−0.684600 + 0.728919i \(0.740023\pi\)
\(468\) 0 0
\(469\) 11.3294 0.523142
\(470\) −21.7960 −1.00537
\(471\) 0 0
\(472\) 26.3446 1.21261
\(473\) 0 0
\(474\) 0 0
\(475\) 0.496541 0.0227829
\(476\) −0.261535 −0.0119874
\(477\) 0 0
\(478\) −18.0734 −0.826656
\(479\) 10.1728 0.464809 0.232404 0.972619i \(-0.425341\pi\)
0.232404 + 0.972619i \(0.425341\pi\)
\(480\) 0 0
\(481\) 3.83448 0.174837
\(482\) −13.7208 −0.624965
\(483\) 0 0
\(484\) 0 0
\(485\) −23.5952 −1.07140
\(486\) 0 0
\(487\) 43.8320 1.98622 0.993110 0.117187i \(-0.0373878\pi\)
0.993110 + 0.117187i \(0.0373878\pi\)
\(488\) −0.171814 −0.00777764
\(489\) 0 0
\(490\) 3.47528 0.156997
\(491\) −19.1207 −0.862906 −0.431453 0.902135i \(-0.641999\pi\)
−0.431453 + 0.902135i \(0.641999\pi\)
\(492\) 0 0
\(493\) −19.3898 −0.873271
\(494\) 0.944272 0.0424848
\(495\) 0 0
\(496\) 27.9954 1.25703
\(497\) −5.96945 −0.267766
\(498\) 0 0
\(499\) 20.4971 0.917576 0.458788 0.888546i \(-0.348284\pi\)
0.458788 + 0.888546i \(0.348284\pi\)
\(500\) 1.22824 0.0549287
\(501\) 0 0
\(502\) −4.47277 −0.199629
\(503\) −18.9794 −0.846251 −0.423126 0.906071i \(-0.639067\pi\)
−0.423126 + 0.906071i \(0.639067\pi\)
\(504\) 0 0
\(505\) 23.7450 1.05664
\(506\) 0 0
\(507\) 0 0
\(508\) −2.61186 −0.115883
\(509\) −23.5548 −1.04405 −0.522023 0.852931i \(-0.674823\pi\)
−0.522023 + 0.852931i \(0.674823\pi\)
\(510\) 0 0
\(511\) 4.38705 0.194072
\(512\) 20.2778 0.896159
\(513\) 0 0
\(514\) 0.911774 0.0402166
\(515\) 1.57251 0.0692930
\(516\) 0 0
\(517\) 0 0
\(518\) −6.12275 −0.269018
\(519\) 0 0
\(520\) 5.95835 0.261291
\(521\) 5.09046 0.223017 0.111509 0.993763i \(-0.464432\pi\)
0.111509 + 0.993763i \(0.464432\pi\)
\(522\) 0 0
\(523\) 22.7768 0.995959 0.497979 0.867189i \(-0.334075\pi\)
0.497979 + 0.867189i \(0.334075\pi\)
\(524\) 1.10304 0.0481866
\(525\) 0 0
\(526\) 1.49254 0.0650777
\(527\) −14.4892 −0.631159
\(528\) 0 0
\(529\) 37.9128 1.64838
\(530\) 22.8168 0.991098
\(531\) 0 0
\(532\) −0.0850818 −0.00368876
\(533\) −10.9026 −0.472245
\(534\) 0 0
\(535\) 0.675747 0.0292151
\(536\) −31.0158 −1.33968
\(537\) 0 0
\(538\) 46.5979 2.00898
\(539\) 0 0
\(540\) 0 0
\(541\) −11.4416 −0.491912 −0.245956 0.969281i \(-0.579102\pi\)
−0.245956 + 0.969281i \(0.579102\pi\)
\(542\) −8.67032 −0.372422
\(543\) 0 0
\(544\) 1.47752 0.0633480
\(545\) 21.4348 0.918168
\(546\) 0 0
\(547\) 3.79691 0.162344 0.0811720 0.996700i \(-0.474134\pi\)
0.0811720 + 0.996700i \(0.474134\pi\)
\(548\) 0.244893 0.0104613
\(549\) 0 0
\(550\) 0 0
\(551\) −6.30783 −0.268723
\(552\) 0 0
\(553\) 4.85725 0.206551
\(554\) 1.86020 0.0790322
\(555\) 0 0
\(556\) −1.53639 −0.0651575
\(557\) 3.48937 0.147849 0.0739247 0.997264i \(-0.476448\pi\)
0.0739247 + 0.997264i \(0.476448\pi\)
\(558\) 0 0
\(559\) −6.85290 −0.289847
\(560\) −10.0851 −0.426172
\(561\) 0 0
\(562\) −6.47885 −0.273294
\(563\) 28.1619 1.18688 0.593441 0.804877i \(-0.297769\pi\)
0.593441 + 0.804877i \(0.297769\pi\)
\(564\) 0 0
\(565\) 3.17331 0.133502
\(566\) −33.1890 −1.39504
\(567\) 0 0
\(568\) 16.3422 0.685703
\(569\) −21.5992 −0.905483 −0.452742 0.891642i \(-0.649554\pi\)
−0.452742 + 0.891642i \(0.649554\pi\)
\(570\) 0 0
\(571\) 14.4959 0.606636 0.303318 0.952889i \(-0.401906\pi\)
0.303318 + 0.952889i \(0.401906\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 17.4089 0.726634
\(575\) 5.44787 0.227192
\(576\) 0 0
\(577\) 5.07081 0.211101 0.105550 0.994414i \(-0.466340\pi\)
0.105550 + 0.994414i \(0.466340\pi\)
\(578\) 17.7890 0.739924
\(579\) 0 0
\(580\) 2.53171 0.105124
\(581\) 10.0490 0.416901
\(582\) 0 0
\(583\) 0 0
\(584\) −12.0102 −0.496985
\(585\) 0 0
\(586\) −12.2198 −0.504794
\(587\) 43.3905 1.79092 0.895458 0.445146i \(-0.146848\pi\)
0.895458 + 0.445146i \(0.146848\pi\)
\(588\) 0 0
\(589\) −4.71359 −0.194220
\(590\) 33.4430 1.37683
\(591\) 0 0
\(592\) 17.7679 0.730256
\(593\) 31.4532 1.29163 0.645815 0.763494i \(-0.276518\pi\)
0.645815 + 0.763494i \(0.276518\pi\)
\(594\) 0 0
\(595\) 5.21960 0.213983
\(596\) −1.83537 −0.0751798
\(597\) 0 0
\(598\) 10.3602 0.423661
\(599\) 22.1842 0.906422 0.453211 0.891403i \(-0.350278\pi\)
0.453211 + 0.891403i \(0.350278\pi\)
\(600\) 0 0
\(601\) 32.2182 1.31421 0.657103 0.753800i \(-0.271781\pi\)
0.657103 + 0.753800i \(0.271781\pi\)
\(602\) 10.9425 0.445981
\(603\) 0 0
\(604\) −0.826403 −0.0336258
\(605\) 0 0
\(606\) 0 0
\(607\) 30.2533 1.22794 0.613971 0.789328i \(-0.289571\pi\)
0.613971 + 0.789328i \(0.289571\pi\)
\(608\) 0.480662 0.0194934
\(609\) 0 0
\(610\) −0.218108 −0.00883092
\(611\) 5.71838 0.231341
\(612\) 0 0
\(613\) −4.18959 −0.169216 −0.0846080 0.996414i \(-0.526964\pi\)
−0.0846080 + 0.996414i \(0.526964\pi\)
\(614\) −16.0963 −0.649593
\(615\) 0 0
\(616\) 0 0
\(617\) 31.1457 1.25388 0.626939 0.779068i \(-0.284307\pi\)
0.626939 + 0.779068i \(0.284307\pi\)
\(618\) 0 0
\(619\) −47.8439 −1.92301 −0.961504 0.274790i \(-0.911392\pi\)
−0.961504 + 0.274790i \(0.911392\pi\)
\(620\) 1.89184 0.0759783
\(621\) 0 0
\(622\) −32.8905 −1.31879
\(623\) −0.261535 −0.0104782
\(624\) 0 0
\(625\) −28.0029 −1.12012
\(626\) −3.71061 −0.148306
\(627\) 0 0
\(628\) −1.11684 −0.0445667
\(629\) −9.19590 −0.366664
\(630\) 0 0
\(631\) 4.01691 0.159911 0.0799554 0.996798i \(-0.474522\pi\)
0.0799554 + 0.996798i \(0.474522\pi\)
\(632\) −13.2974 −0.528942
\(633\) 0 0
\(634\) 45.6325 1.81230
\(635\) 52.1265 2.06858
\(636\) 0 0
\(637\) −0.911774 −0.0361258
\(638\) 0 0
\(639\) 0 0
\(640\) 29.1726 1.15315
\(641\) −27.6852 −1.09350 −0.546750 0.837296i \(-0.684135\pi\)
−0.546750 + 0.837296i \(0.684135\pi\)
\(642\) 0 0
\(643\) −4.25792 −0.167916 −0.0839579 0.996469i \(-0.526756\pi\)
−0.0839579 + 0.996469i \(0.526756\pi\)
\(644\) −0.933487 −0.0367845
\(645\) 0 0
\(646\) −2.26457 −0.0890982
\(647\) −5.24503 −0.206203 −0.103102 0.994671i \(-0.532877\pi\)
−0.103102 + 0.994671i \(0.532877\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.926589 0.0363438
\(651\) 0 0
\(652\) 1.24761 0.0488600
\(653\) 9.83716 0.384958 0.192479 0.981301i \(-0.438347\pi\)
0.192479 + 0.981301i \(0.438347\pi\)
\(654\) 0 0
\(655\) −22.0141 −0.860161
\(656\) −50.5197 −1.97246
\(657\) 0 0
\(658\) −9.13090 −0.355960
\(659\) −50.5575 −1.96944 −0.984720 0.174145i \(-0.944284\pi\)
−0.984720 + 0.174145i \(0.944284\pi\)
\(660\) 0 0
\(661\) 22.9751 0.893629 0.446814 0.894627i \(-0.352558\pi\)
0.446814 + 0.894627i \(0.352558\pi\)
\(662\) 7.02887 0.273185
\(663\) 0 0
\(664\) −27.5104 −1.06761
\(665\) 1.69803 0.0658467
\(666\) 0 0
\(667\) −69.2072 −2.67972
\(668\) −0.161637 −0.00625394
\(669\) 0 0
\(670\) −39.3728 −1.52110
\(671\) 0 0
\(672\) 0 0
\(673\) −21.8706 −0.843048 −0.421524 0.906817i \(-0.638505\pi\)
−0.421524 + 0.906817i \(0.638505\pi\)
\(674\) −30.7415 −1.18412
\(675\) 0 0
\(676\) −1.45545 −0.0559788
\(677\) 14.6716 0.563876 0.281938 0.959433i \(-0.409023\pi\)
0.281938 + 0.959433i \(0.409023\pi\)
\(678\) 0 0
\(679\) −9.88466 −0.379338
\(680\) −14.2894 −0.547973
\(681\) 0 0
\(682\) 0 0
\(683\) 9.70579 0.371382 0.185691 0.982608i \(-0.440548\pi\)
0.185691 + 0.982608i \(0.440548\pi\)
\(684\) 0 0
\(685\) −4.88748 −0.186741
\(686\) 1.45589 0.0555860
\(687\) 0 0
\(688\) −31.7545 −1.21063
\(689\) −5.98620 −0.228056
\(690\) 0 0
\(691\) −43.6828 −1.66177 −0.830887 0.556441i \(-0.812166\pi\)
−0.830887 + 0.556441i \(0.812166\pi\)
\(692\) −2.33058 −0.0885955
\(693\) 0 0
\(694\) −18.0896 −0.686670
\(695\) 30.6627 1.16310
\(696\) 0 0
\(697\) 26.1468 0.990381
\(698\) −24.7038 −0.935054
\(699\) 0 0
\(700\) −0.0834885 −0.00315557
\(701\) 20.7227 0.782687 0.391343 0.920245i \(-0.372011\pi\)
0.391343 + 0.920245i \(0.372011\pi\)
\(702\) 0 0
\(703\) −2.99159 −0.112830
\(704\) 0 0
\(705\) 0 0
\(706\) −6.65488 −0.250460
\(707\) 9.94742 0.374111
\(708\) 0 0
\(709\) −19.2327 −0.722301 −0.361150 0.932508i \(-0.617616\pi\)
−0.361150 + 0.932508i \(0.617616\pi\)
\(710\) 20.7455 0.778565
\(711\) 0 0
\(712\) 0.715988 0.0268328
\(713\) −51.7158 −1.93677
\(714\) 0 0
\(715\) 0 0
\(716\) 1.53743 0.0574565
\(717\) 0 0
\(718\) −45.0937 −1.68288
\(719\) 5.36985 0.200261 0.100131 0.994974i \(-0.468074\pi\)
0.100131 + 0.994974i \(0.468074\pi\)
\(720\) 0 0
\(721\) 0.658765 0.0245337
\(722\) 26.9251 1.00205
\(723\) 0 0
\(724\) 1.71933 0.0638984
\(725\) −6.18971 −0.229880
\(726\) 0 0
\(727\) 8.19052 0.303769 0.151885 0.988398i \(-0.451466\pi\)
0.151885 + 0.988398i \(0.451466\pi\)
\(728\) 2.49611 0.0925119
\(729\) 0 0
\(730\) −15.2462 −0.564289
\(731\) 16.4347 0.607860
\(732\) 0 0
\(733\) 24.3265 0.898521 0.449260 0.893401i \(-0.351688\pi\)
0.449260 + 0.893401i \(0.351688\pi\)
\(734\) −21.4630 −0.792213
\(735\) 0 0
\(736\) 5.27365 0.194389
\(737\) 0 0
\(738\) 0 0
\(739\) −18.9692 −0.697795 −0.348897 0.937161i \(-0.613444\pi\)
−0.348897 + 0.937161i \(0.613444\pi\)
\(740\) 1.20070 0.0441387
\(741\) 0 0
\(742\) 9.55855 0.350906
\(743\) 16.8997 0.619988 0.309994 0.950738i \(-0.399673\pi\)
0.309994 + 0.950738i \(0.399673\pi\)
\(744\) 0 0
\(745\) 36.6296 1.34201
\(746\) 18.3006 0.670032
\(747\) 0 0
\(748\) 0 0
\(749\) 0.283088 0.0103438
\(750\) 0 0
\(751\) −22.5942 −0.824475 −0.412237 0.911076i \(-0.635253\pi\)
−0.412237 + 0.911076i \(0.635253\pi\)
\(752\) 26.4974 0.966261
\(753\) 0 0
\(754\) −11.7710 −0.428673
\(755\) 16.4930 0.600242
\(756\) 0 0
\(757\) 3.87744 0.140928 0.0704640 0.997514i \(-0.477552\pi\)
0.0704640 + 0.997514i \(0.477552\pi\)
\(758\) −23.3436 −0.847879
\(759\) 0 0
\(760\) −4.64859 −0.168622
\(761\) −37.4591 −1.35789 −0.678945 0.734189i \(-0.737563\pi\)
−0.678945 + 0.734189i \(0.737563\pi\)
\(762\) 0 0
\(763\) 8.97962 0.325084
\(764\) 2.86481 0.103645
\(765\) 0 0
\(766\) −4.09137 −0.147827
\(767\) −8.77411 −0.316815
\(768\) 0 0
\(769\) −20.6137 −0.743349 −0.371674 0.928363i \(-0.621216\pi\)
−0.371674 + 0.928363i \(0.621216\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.0166981 0.000600977 0
\(773\) −9.03686 −0.325033 −0.162517 0.986706i \(-0.551961\pi\)
−0.162517 + 0.986706i \(0.551961\pi\)
\(774\) 0 0
\(775\) −4.62532 −0.166146
\(776\) 27.0606 0.971420
\(777\) 0 0
\(778\) −29.9985 −1.07550
\(779\) 8.50602 0.304760
\(780\) 0 0
\(781\) 0 0
\(782\) −24.8460 −0.888492
\(783\) 0 0
\(784\) −4.22491 −0.150890
\(785\) 22.2894 0.795543
\(786\) 0 0
\(787\) −33.4585 −1.19267 −0.596333 0.802737i \(-0.703376\pi\)
−0.596333 + 0.802737i \(0.703376\pi\)
\(788\) −1.74760 −0.0622557
\(789\) 0 0
\(790\) −16.8803 −0.600574
\(791\) 1.32938 0.0472674
\(792\) 0 0
\(793\) 0.0572227 0.00203204
\(794\) −41.7239 −1.48073
\(795\) 0 0
\(796\) −3.16600 −0.112216
\(797\) −0.766169 −0.0271391 −0.0135695 0.999908i \(-0.504319\pi\)
−0.0135695 + 0.999908i \(0.504319\pi\)
\(798\) 0 0
\(799\) −13.7139 −0.485163
\(800\) 0.471661 0.0166757
\(801\) 0 0
\(802\) −32.4890 −1.14723
\(803\) 0 0
\(804\) 0 0
\(805\) 18.6302 0.656627
\(806\) −8.79597 −0.309825
\(807\) 0 0
\(808\) −27.2324 −0.958034
\(809\) 10.2311 0.359708 0.179854 0.983693i \(-0.442438\pi\)
0.179854 + 0.983693i \(0.442438\pi\)
\(810\) 0 0
\(811\) 10.4196 0.365880 0.182940 0.983124i \(-0.441439\pi\)
0.182940 + 0.983124i \(0.441439\pi\)
\(812\) 1.06060 0.0372198
\(813\) 0 0
\(814\) 0 0
\(815\) −24.8992 −0.872181
\(816\) 0 0
\(817\) 5.34650 0.187051
\(818\) −23.9290 −0.836657
\(819\) 0 0
\(820\) −3.41397 −0.119221
\(821\) −27.7513 −0.968527 −0.484264 0.874922i \(-0.660912\pi\)
−0.484264 + 0.874922i \(0.660912\pi\)
\(822\) 0 0
\(823\) −18.7419 −0.653302 −0.326651 0.945145i \(-0.605920\pi\)
−0.326651 + 0.945145i \(0.605920\pi\)
\(824\) −1.80346 −0.0628266
\(825\) 0 0
\(826\) 14.0102 0.487476
\(827\) 55.6432 1.93490 0.967451 0.253058i \(-0.0814364\pi\)
0.967451 + 0.253058i \(0.0814364\pi\)
\(828\) 0 0
\(829\) −37.5600 −1.30451 −0.652257 0.757998i \(-0.726178\pi\)
−0.652257 + 0.757998i \(0.726178\pi\)
\(830\) −34.9230 −1.21219
\(831\) 0 0
\(832\) −6.80736 −0.236003
\(833\) 2.18663 0.0757622
\(834\) 0 0
\(835\) 3.22589 0.111637
\(836\) 0 0
\(837\) 0 0
\(838\) −18.5953 −0.642362
\(839\) 44.6850 1.54270 0.771348 0.636413i \(-0.219583\pi\)
0.771348 + 0.636413i \(0.219583\pi\)
\(840\) 0 0
\(841\) 49.6312 1.71142
\(842\) −15.7282 −0.542029
\(843\) 0 0
\(844\) 2.10051 0.0723024
\(845\) 29.0473 0.999257
\(846\) 0 0
\(847\) 0 0
\(848\) −27.7384 −0.952541
\(849\) 0 0
\(850\) −2.22216 −0.0762195
\(851\) −32.8226 −1.12514
\(852\) 0 0
\(853\) 55.6861 1.90666 0.953329 0.301934i \(-0.0976323\pi\)
0.953329 + 0.301934i \(0.0976323\pi\)
\(854\) −0.0913711 −0.00312665
\(855\) 0 0
\(856\) −0.774994 −0.0264888
\(857\) 37.5988 1.28435 0.642176 0.766557i \(-0.278032\pi\)
0.642176 + 0.766557i \(0.278032\pi\)
\(858\) 0 0
\(859\) 44.1084 1.50496 0.752479 0.658616i \(-0.228858\pi\)
0.752479 + 0.658616i \(0.228858\pi\)
\(860\) −2.14587 −0.0731736
\(861\) 0 0
\(862\) 40.0944 1.36562
\(863\) 4.51049 0.153539 0.0767694 0.997049i \(-0.475539\pi\)
0.0767694 + 0.997049i \(0.475539\pi\)
\(864\) 0 0
\(865\) 46.5128 1.58148
\(866\) −22.2169 −0.754961
\(867\) 0 0
\(868\) 0.792543 0.0269007
\(869\) 0 0
\(870\) 0 0
\(871\) 10.3298 0.350013
\(872\) −24.5830 −0.832485
\(873\) 0 0
\(874\) −8.08284 −0.273406
\(875\) −10.2690 −0.347157
\(876\) 0 0
\(877\) −36.0877 −1.21860 −0.609298 0.792941i \(-0.708549\pi\)
−0.609298 + 0.792941i \(0.708549\pi\)
\(878\) 41.5655 1.40277
\(879\) 0 0
\(880\) 0 0
\(881\) 12.8240 0.432052 0.216026 0.976388i \(-0.430690\pi\)
0.216026 + 0.976388i \(0.430690\pi\)
\(882\) 0 0
\(883\) −9.36217 −0.315062 −0.157531 0.987514i \(-0.550353\pi\)
−0.157531 + 0.987514i \(0.550353\pi\)
\(884\) −0.238460 −0.00802029
\(885\) 0 0
\(886\) 16.4967 0.554217
\(887\) 9.91449 0.332896 0.166448 0.986050i \(-0.446770\pi\)
0.166448 + 0.986050i \(0.446770\pi\)
\(888\) 0 0
\(889\) 21.8372 0.732395
\(890\) 0.908906 0.0304666
\(891\) 0 0
\(892\) 2.70117 0.0904419
\(893\) −4.46137 −0.149294
\(894\) 0 0
\(895\) −30.6834 −1.02563
\(896\) 12.2212 0.408280
\(897\) 0 0
\(898\) −39.1587 −1.30674
\(899\) 58.7579 1.95969
\(900\) 0 0
\(901\) 14.3562 0.478275
\(902\) 0 0
\(903\) 0 0
\(904\) −3.63937 −0.121044
\(905\) −34.3137 −1.14063
\(906\) 0 0
\(907\) −13.9881 −0.464469 −0.232234 0.972660i \(-0.574604\pi\)
−0.232234 + 0.972660i \(0.574604\pi\)
\(908\) −2.62792 −0.0872105
\(909\) 0 0
\(910\) 3.16867 0.105040
\(911\) 27.1168 0.898419 0.449210 0.893426i \(-0.351706\pi\)
0.449210 + 0.893426i \(0.351706\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 42.1762 1.39507
\(915\) 0 0
\(916\) 2.79643 0.0923967
\(917\) −9.22227 −0.304546
\(918\) 0 0
\(919\) 4.42881 0.146093 0.0730464 0.997329i \(-0.476728\pi\)
0.0730464 + 0.997329i \(0.476728\pi\)
\(920\) −51.0027 −1.68151
\(921\) 0 0
\(922\) −61.0563 −2.01078
\(923\) −5.44278 −0.179151
\(924\) 0 0
\(925\) −2.93556 −0.0965208
\(926\) 1.67288 0.0549741
\(927\) 0 0
\(928\) −5.99177 −0.196689
\(929\) 56.7744 1.86271 0.931353 0.364116i \(-0.118629\pi\)
0.931353 + 0.364116i \(0.118629\pi\)
\(930\) 0 0
\(931\) 0.711349 0.0233135
\(932\) 0.124767 0.00408689
\(933\) 0 0
\(934\) 43.0777 1.40955
\(935\) 0 0
\(936\) 0 0
\(937\) −2.85797 −0.0933659 −0.0466830 0.998910i \(-0.514865\pi\)
−0.0466830 + 0.998910i \(0.514865\pi\)
\(938\) −16.4943 −0.538558
\(939\) 0 0
\(940\) 1.79062 0.0584034
\(941\) −24.2516 −0.790580 −0.395290 0.918556i \(-0.629356\pi\)
−0.395290 + 0.918556i \(0.629356\pi\)
\(942\) 0 0
\(943\) 93.3250 3.03908
\(944\) −40.6568 −1.32327
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0901 1.46523 0.732616 0.680642i \(-0.238299\pi\)
0.732616 + 0.680642i \(0.238299\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −0.722907 −0.0234542
\(951\) 0 0
\(952\) −5.98620 −0.194014
\(953\) 37.7101 1.22155 0.610775 0.791804i \(-0.290858\pi\)
0.610775 + 0.791804i \(0.290858\pi\)
\(954\) 0 0
\(955\) −57.1748 −1.85013
\(956\) 1.48479 0.0480216
\(957\) 0 0
\(958\) −14.8105 −0.478505
\(959\) −2.04750 −0.0661171
\(960\) 0 0
\(961\) 12.9074 0.416368
\(962\) −5.58256 −0.179989
\(963\) 0 0
\(964\) 1.12721 0.0363050
\(965\) −0.333254 −0.0107278
\(966\) 0 0
\(967\) −60.7131 −1.95240 −0.976201 0.216870i \(-0.930415\pi\)
−0.976201 + 0.216870i \(0.930415\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 34.3520 1.10297
\(971\) 29.4859 0.946247 0.473124 0.880996i \(-0.343126\pi\)
0.473124 + 0.880996i \(0.343126\pi\)
\(972\) 0 0
\(973\) 12.8454 0.411804
\(974\) −63.8145 −2.04475
\(975\) 0 0
\(976\) 0.265154 0.00848738
\(977\) −4.55016 −0.145573 −0.0727863 0.997348i \(-0.523189\pi\)
−0.0727863 + 0.997348i \(0.523189\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.285507 −0.00912018
\(981\) 0 0
\(982\) 27.8376 0.888334
\(983\) 36.9736 1.17927 0.589637 0.807668i \(-0.299271\pi\)
0.589637 + 0.807668i \(0.299271\pi\)
\(984\) 0 0
\(985\) 34.8779 1.11130
\(986\) 28.2293 0.899004
\(987\) 0 0
\(988\) −0.0775753 −0.00246800
\(989\) 58.6599 1.86528
\(990\) 0 0
\(991\) 30.5701 0.971090 0.485545 0.874212i \(-0.338621\pi\)
0.485545 + 0.874212i \(0.338621\pi\)
\(992\) −4.47741 −0.142158
\(993\) 0 0
\(994\) 8.69084 0.275657
\(995\) 63.1857 2.00312
\(996\) 0 0
\(997\) −2.65445 −0.0840671 −0.0420336 0.999116i \(-0.513384\pi\)
−0.0420336 + 0.999116i \(0.513384\pi\)
\(998\) −29.8415 −0.944615
\(999\) 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 7623.2.a.ci.1.2 4
3.2 odd 2 2541.2.a.bn.1.3 4
11.3 even 5 693.2.m.f.64.1 8
11.4 even 5 693.2.m.f.379.1 8
11.10 odd 2 7623.2.a.cl.1.3 4
33.14 odd 10 231.2.j.f.64.2 8
33.26 odd 10 231.2.j.f.148.2 yes 8
33.32 even 2 2541.2.a.bm.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.64.2 8 33.14 odd 10
231.2.j.f.148.2 yes 8 33.26 odd 10
693.2.m.f.64.1 8 11.3 even 5
693.2.m.f.379.1 8 11.4 even 5
2541.2.a.bm.1.2 4 33.32 even 2
2541.2.a.bn.1.3 4 3.2 odd 2
7623.2.a.ci.1.2 4 1.1 even 1 trivial
7623.2.a.cl.1.3 4 11.10 odd 2