Properties

Label 2205.2.a.x.1.1
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} +2.43845 q^{8} -1.56155 q^{10} -2.56155 q^{11} -4.56155 q^{13} -4.68466 q^{16} -4.56155 q^{17} -1.12311 q^{19} +0.438447 q^{20} +4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} +7.12311 q^{26} +5.68466 q^{29} +2.43845 q^{32} +7.12311 q^{34} +6.00000 q^{37} +1.75379 q^{38} +2.43845 q^{40} -3.12311 q^{41} +9.12311 q^{43} -1.12311 q^{44} -8.00000 q^{46} +3.68466 q^{47} -1.56155 q^{50} -2.00000 q^{52} -3.12311 q^{53} -2.56155 q^{55} -8.87689 q^{58} -4.00000 q^{59} +9.36932 q^{61} +5.56155 q^{64} -4.56155 q^{65} -6.24621 q^{67} -2.00000 q^{68} -8.00000 q^{71} -4.24621 q^{73} -9.36932 q^{74} -0.492423 q^{76} -6.56155 q^{79} -4.68466 q^{80} +4.87689 q^{82} +4.00000 q^{83} -4.56155 q^{85} -14.2462 q^{86} -6.24621 q^{88} +7.12311 q^{89} +2.24621 q^{92} -5.75379 q^{94} -1.12311 q^{95} +14.8078 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + q^{10} - q^{11} - 5q^{13} + 3q^{16} - 5q^{17} + 6q^{19} + 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} + 6q^{26} - q^{29} + 9q^{32} + 6q^{34} + 12q^{37} + 20q^{38} + 9q^{40} + 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} - 5q^{47} + q^{50} - 4q^{52} + 2q^{53} - q^{55} - 26q^{58} - 8q^{59} - 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} - 4q^{68} - 16q^{71} + 8q^{73} + 6q^{74} + 32q^{76} - 9q^{79} + 3q^{80} + 18q^{82} + 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} + 6q^{89} - 12q^{92} - 28q^{94} + 6q^{95} + 9q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) −1.56155 −0.493806
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) −4.56155 −1.26515 −0.632574 0.774500i \(-0.718001\pi\)
−0.632574 + 0.774500i \(0.718001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) −4.56155 −1.10634 −0.553170 0.833069i \(-0.686582\pi\)
−0.553170 + 0.833069i \(0.686582\pi\)
\(18\) 0 0
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 0.438447 0.0980398
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 5.12311 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 7.12311 1.39696
\(27\) 0 0
\(28\) 0 0
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) 7.12311 1.22160
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 1.75379 0.284502
\(39\) 0 0
\(40\) 2.43845 0.385552
\(41\) −3.12311 −0.487747 −0.243874 0.969807i \(-0.578418\pi\)
−0.243874 + 0.969807i \(0.578418\pi\)
\(42\) 0 0
\(43\) 9.12311 1.39126 0.695630 0.718400i \(-0.255125\pi\)
0.695630 + 0.718400i \(0.255125\pi\)
\(44\) −1.12311 −0.169315
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.68466 0.537463 0.268731 0.963215i \(-0.413396\pi\)
0.268731 + 0.963215i \(0.413396\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.56155 −0.220837
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −3.12311 −0.428992 −0.214496 0.976725i \(-0.568811\pi\)
−0.214496 + 0.976725i \(0.568811\pi\)
\(54\) 0 0
\(55\) −2.56155 −0.345400
\(56\) 0 0
\(57\) 0 0
\(58\) −8.87689 −1.16559
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 9.36932 1.19962 0.599809 0.800143i \(-0.295243\pi\)
0.599809 + 0.800143i \(0.295243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) −4.56155 −0.565791
\(66\) 0 0
\(67\) −6.24621 −0.763096 −0.381548 0.924349i \(-0.624609\pi\)
−0.381548 + 0.924349i \(0.624609\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) −9.36932 −1.08916
\(75\) 0 0
\(76\) −0.492423 −0.0564847
\(77\) 0 0
\(78\) 0 0
\(79\) −6.56155 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(80\) −4.68466 −0.523761
\(81\) 0 0
\(82\) 4.87689 0.538563
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) −14.2462 −1.53621
\(87\) 0 0
\(88\) −6.24621 −0.665848
\(89\) 7.12311 0.755048 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.24621 0.234184
\(93\) 0 0
\(94\) −5.75379 −0.593458
\(95\) −1.12311 −0.115228
\(96\) 0 0
\(97\) 14.8078 1.50350 0.751750 0.659448i \(-0.229210\pi\)
0.751750 + 0.659448i \(0.229210\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.438447 0.0438447
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) −1.43845 −0.141734 −0.0708672 0.997486i \(-0.522577\pi\)
−0.0708672 + 0.997486i \(0.522577\pi\)
\(104\) −11.1231 −1.09071
\(105\) 0 0
\(106\) 4.87689 0.473686
\(107\) 11.3693 1.09911 0.549557 0.835456i \(-0.314797\pi\)
0.549557 + 0.835456i \(0.314797\pi\)
\(108\) 0 0
\(109\) 17.6847 1.69388 0.846942 0.531686i \(-0.178441\pi\)
0.846942 + 0.531686i \(0.178441\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 5.12311 0.477732
\(116\) 2.49242 0.231416
\(117\) 0 0
\(118\) 6.24621 0.575010
\(119\) 0 0
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) −14.6307 −1.32460
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) 7.12311 0.624738
\(131\) −9.12311 −0.797089 −0.398545 0.917149i \(-0.630485\pi\)
−0.398545 + 0.917149i \(0.630485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 9.75379 0.842599
\(135\) 0 0
\(136\) −11.1231 −0.953798
\(137\) 8.87689 0.758404 0.379202 0.925314i \(-0.376199\pi\)
0.379202 + 0.925314i \(0.376199\pi\)
\(138\) 0 0
\(139\) 6.87689 0.583291 0.291645 0.956527i \(-0.405797\pi\)
0.291645 + 0.956527i \(0.405797\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.4924 1.04834
\(143\) 11.6847 0.977120
\(144\) 0 0
\(145\) 5.68466 0.472085
\(146\) 6.63068 0.548759
\(147\) 0 0
\(148\) 2.63068 0.216241
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) 21.9309 1.78471 0.892354 0.451335i \(-0.149052\pi\)
0.892354 + 0.451335i \(0.149052\pi\)
\(152\) −2.73863 −0.222133
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.75379 −0.299585 −0.149792 0.988717i \(-0.547861\pi\)
−0.149792 + 0.988717i \(0.547861\pi\)
\(158\) 10.2462 0.815145
\(159\) 0 0
\(160\) 2.43845 0.192776
\(161\) 0 0
\(162\) 0 0
\(163\) 1.12311 0.0879684 0.0439842 0.999032i \(-0.485995\pi\)
0.0439842 + 0.999032i \(0.485995\pi\)
\(164\) −1.36932 −0.106926
\(165\) 0 0
\(166\) −6.24621 −0.484800
\(167\) 21.9309 1.69706 0.848531 0.529146i \(-0.177488\pi\)
0.848531 + 0.529146i \(0.177488\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) 7.12311 0.546317
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −8.56155 −0.650923 −0.325461 0.945555i \(-0.605520\pi\)
−0.325461 + 0.945555i \(0.605520\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) −11.1231 −0.833712
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −23.6155 −1.75533 −0.877664 0.479276i \(-0.840899\pi\)
−0.877664 + 0.479276i \(0.840899\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.4924 0.920954
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 11.6847 0.854467
\(188\) 1.61553 0.117824
\(189\) 0 0
\(190\) 1.75379 0.127233
\(191\) 9.43845 0.682942 0.341471 0.939892i \(-0.389075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(192\) 0 0
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) −23.1231 −1.66014
\(195\) 0 0
\(196\) 0 0
\(197\) 7.12311 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(198\) 0 0
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) 2.43845 0.172424
\(201\) 0 0
\(202\) −0.384472 −0.0270513
\(203\) 0 0
\(204\) 0 0
\(205\) −3.12311 −0.218127
\(206\) 2.24621 0.156501
\(207\) 0 0
\(208\) 21.3693 1.48170
\(209\) 2.87689 0.198999
\(210\) 0 0
\(211\) −23.0540 −1.58710 −0.793551 0.608504i \(-0.791770\pi\)
−0.793551 + 0.608504i \(0.791770\pi\)
\(212\) −1.36932 −0.0940451
\(213\) 0 0
\(214\) −17.7538 −1.21362
\(215\) 9.12311 0.622191
\(216\) 0 0
\(217\) 0 0
\(218\) −27.6155 −1.87036
\(219\) 0 0
\(220\) −1.12311 −0.0757198
\(221\) 20.8078 1.39968
\(222\) 0 0
\(223\) 6.56155 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −21.8617 −1.45422
\(227\) 23.6847 1.57201 0.786003 0.618223i \(-0.212147\pi\)
0.786003 + 0.618223i \(0.212147\pi\)
\(228\) 0 0
\(229\) −19.1231 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 13.8617 0.910068
\(233\) 3.12311 0.204601 0.102301 0.994754i \(-0.467380\pi\)
0.102301 + 0.994754i \(0.467380\pi\)
\(234\) 0 0
\(235\) 3.68466 0.240361
\(236\) −1.75379 −0.114162
\(237\) 0 0
\(238\) 0 0
\(239\) 0.807764 0.0522499 0.0261250 0.999659i \(-0.491683\pi\)
0.0261250 + 0.999659i \(0.491683\pi\)
\(240\) 0 0
\(241\) −12.2462 −0.788848 −0.394424 0.918929i \(-0.629056\pi\)
−0.394424 + 0.918929i \(0.629056\pi\)
\(242\) 6.93087 0.445533
\(243\) 0 0
\(244\) 4.10795 0.262985
\(245\) 0 0
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) 0 0
\(250\) −1.56155 −0.0987613
\(251\) −17.1231 −1.08080 −0.540400 0.841408i \(-0.681727\pi\)
−0.540400 + 0.841408i \(0.681727\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) 22.4924 1.40304 0.701519 0.712650i \(-0.252505\pi\)
0.701519 + 0.712650i \(0.252505\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 14.2462 0.880134
\(263\) 21.1231 1.30251 0.651253 0.758860i \(-0.274244\pi\)
0.651253 + 0.758860i \(0.274244\pi\)
\(264\) 0 0
\(265\) −3.12311 −0.191851
\(266\) 0 0
\(267\) 0 0
\(268\) −2.73863 −0.167289
\(269\) 28.7386 1.75223 0.876113 0.482106i \(-0.160128\pi\)
0.876113 + 0.482106i \(0.160128\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 21.3693 1.29571
\(273\) 0 0
\(274\) −13.8617 −0.837418
\(275\) −2.56155 −0.154467
\(276\) 0 0
\(277\) 16.2462 0.976140 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(278\) −10.7386 −0.644060
\(279\) 0 0
\(280\) 0 0
\(281\) −16.5616 −0.987979 −0.493990 0.869468i \(-0.664462\pi\)
−0.493990 + 0.869468i \(0.664462\pi\)
\(282\) 0 0
\(283\) 23.6847 1.40791 0.703953 0.710246i \(-0.251416\pi\)
0.703953 + 0.710246i \(0.251416\pi\)
\(284\) −3.50758 −0.208136
\(285\) 0 0
\(286\) −18.2462 −1.07892
\(287\) 0 0
\(288\) 0 0
\(289\) 3.80776 0.223986
\(290\) −8.87689 −0.521269
\(291\) 0 0
\(292\) −1.86174 −0.108950
\(293\) 9.68466 0.565784 0.282892 0.959152i \(-0.408706\pi\)
0.282892 + 0.959152i \(0.408706\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 14.6307 0.850391
\(297\) 0 0
\(298\) −6.63068 −0.384105
\(299\) −23.3693 −1.35148
\(300\) 0 0
\(301\) 0 0
\(302\) −34.2462 −1.97065
\(303\) 0 0
\(304\) 5.26137 0.301760
\(305\) 9.36932 0.536486
\(306\) 0 0
\(307\) 31.6847 1.80834 0.904169 0.427174i \(-0.140491\pi\)
0.904169 + 0.427174i \(0.140491\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.61553 −0.545247 −0.272623 0.962121i \(-0.587891\pi\)
−0.272623 + 0.962121i \(0.587891\pi\)
\(312\) 0 0
\(313\) −31.3002 −1.76919 −0.884596 0.466359i \(-0.845566\pi\)
−0.884596 + 0.466359i \(0.845566\pi\)
\(314\) 5.86174 0.330797
\(315\) 0 0
\(316\) −2.87689 −0.161838
\(317\) 22.4924 1.26330 0.631650 0.775254i \(-0.282378\pi\)
0.631650 + 0.775254i \(0.282378\pi\)
\(318\) 0 0
\(319\) −14.5616 −0.815290
\(320\) 5.56155 0.310900
\(321\) 0 0
\(322\) 0 0
\(323\) 5.12311 0.285057
\(324\) 0 0
\(325\) −4.56155 −0.253029
\(326\) −1.75379 −0.0971334
\(327\) 0 0
\(328\) −7.61553 −0.420497
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 1.75379 0.0962517
\(333\) 0 0
\(334\) −34.2462 −1.87387
\(335\) −6.24621 −0.341267
\(336\) 0 0
\(337\) −34.4924 −1.87892 −0.939461 0.342656i \(-0.888674\pi\)
−0.939461 + 0.342656i \(0.888674\pi\)
\(338\) −12.1922 −0.663170
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 22.2462 1.19944
\(345\) 0 0
\(346\) 13.3693 0.718739
\(347\) 1.12311 0.0602915 0.0301457 0.999546i \(-0.490403\pi\)
0.0301457 + 0.999546i \(0.490403\pi\)
\(348\) 0 0
\(349\) 22.4924 1.20399 0.601996 0.798499i \(-0.294372\pi\)
0.601996 + 0.798499i \(0.294372\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.24621 −0.332924
\(353\) −14.8078 −0.788138 −0.394069 0.919081i \(-0.628933\pi\)
−0.394069 + 0.919081i \(0.628933\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 3.12311 0.165524
\(357\) 0 0
\(358\) 31.2311 1.65061
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 36.8769 1.93821
\(363\) 0 0
\(364\) 0 0
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) −3.68466 −0.192338 −0.0961688 0.995365i \(-0.530659\pi\)
−0.0961688 + 0.995365i \(0.530659\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) −9.36932 −0.487088
\(371\) 0 0
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) −18.2462 −0.943489
\(375\) 0 0
\(376\) 8.98485 0.463358
\(377\) −25.9309 −1.33551
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) −0.492423 −0.0252607
\(381\) 0 0
\(382\) −14.7386 −0.754094
\(383\) −10.2462 −0.523557 −0.261778 0.965128i \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.38447 0.426758
\(387\) 0 0
\(388\) 6.49242 0.329603
\(389\) −3.93087 −0.199303 −0.0996515 0.995022i \(-0.531773\pi\)
−0.0996515 + 0.995022i \(0.531773\pi\)
\(390\) 0 0
\(391\) −23.3693 −1.18184
\(392\) 0 0
\(393\) 0 0
\(394\) −11.1231 −0.560374
\(395\) −6.56155 −0.330148
\(396\) 0 0
\(397\) −23.4384 −1.17634 −0.588171 0.808737i \(-0.700152\pi\)
−0.588171 + 0.808737i \(0.700152\pi\)
\(398\) −28.4924 −1.42820
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) −27.4384 −1.37021 −0.685105 0.728444i \(-0.740244\pi\)
−0.685105 + 0.728444i \(0.740244\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.107951 0.00537074
\(405\) 0 0
\(406\) 0 0
\(407\) −15.3693 −0.761829
\(408\) 0 0
\(409\) 26.4924 1.30997 0.654983 0.755644i \(-0.272676\pi\)
0.654983 + 0.755644i \(0.272676\pi\)
\(410\) 4.87689 0.240853
\(411\) 0 0
\(412\) −0.630683 −0.0310715
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −11.1231 −0.545355
\(417\) 0 0
\(418\) −4.49242 −0.219732
\(419\) 9.75379 0.476504 0.238252 0.971203i \(-0.423426\pi\)
0.238252 + 0.971203i \(0.423426\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 36.0000 1.75245
\(423\) 0 0
\(424\) −7.61553 −0.369843
\(425\) −4.56155 −0.221268
\(426\) 0 0
\(427\) 0 0
\(428\) 4.98485 0.240952
\(429\) 0 0
\(430\) −14.2462 −0.687013
\(431\) −0.807764 −0.0389086 −0.0194543 0.999811i \(-0.506193\pi\)
−0.0194543 + 0.999811i \(0.506193\pi\)
\(432\) 0 0
\(433\) 8.24621 0.396288 0.198144 0.980173i \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.75379 0.371339
\(437\) −5.75379 −0.275241
\(438\) 0 0
\(439\) 15.3693 0.733537 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(440\) −6.24621 −0.297776
\(441\) 0 0
\(442\) −32.4924 −1.54551
\(443\) 27.3693 1.30036 0.650178 0.759782i \(-0.274694\pi\)
0.650178 + 0.759782i \(0.274694\pi\)
\(444\) 0 0
\(445\) 7.12311 0.337668
\(446\) −10.2462 −0.485172
\(447\) 0 0
\(448\) 0 0
\(449\) −18.8078 −0.887593 −0.443797 0.896128i \(-0.646369\pi\)
−0.443797 + 0.896128i \(0.646369\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 6.13826 0.288719
\(453\) 0 0
\(454\) −36.9848 −1.73578
\(455\) 0 0
\(456\) 0 0
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) 29.8617 1.39535
\(459\) 0 0
\(460\) 2.24621 0.104730
\(461\) −4.87689 −0.227140 −0.113570 0.993530i \(-0.536229\pi\)
−0.113570 + 0.993530i \(0.536229\pi\)
\(462\) 0 0
\(463\) −20.4924 −0.952364 −0.476182 0.879347i \(-0.657980\pi\)
−0.476182 + 0.879347i \(0.657980\pi\)
\(464\) −26.6307 −1.23630
\(465\) 0 0
\(466\) −4.87689 −0.225918
\(467\) 26.5616 1.22912 0.614561 0.788869i \(-0.289333\pi\)
0.614561 + 0.788869i \(0.289333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −5.75379 −0.265402
\(471\) 0 0
\(472\) −9.75379 −0.448955
\(473\) −23.3693 −1.07452
\(474\) 0 0
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) 0 0
\(478\) −1.26137 −0.0576935
\(479\) 13.1231 0.599610 0.299805 0.954001i \(-0.403078\pi\)
0.299805 + 0.954001i \(0.403078\pi\)
\(480\) 0 0
\(481\) −27.3693 −1.24793
\(482\) 19.1231 0.871034
\(483\) 0 0
\(484\) −1.94602 −0.0884557
\(485\) 14.8078 0.672386
\(486\) 0 0
\(487\) 5.12311 0.232150 0.116075 0.993240i \(-0.462969\pi\)
0.116075 + 0.993240i \(0.462969\pi\)
\(488\) 22.8466 1.03422
\(489\) 0 0
\(490\) 0 0
\(491\) −4.17708 −0.188509 −0.0942545 0.995548i \(-0.530047\pi\)
−0.0942545 + 0.995548i \(0.530047\pi\)
\(492\) 0 0
\(493\) −25.9309 −1.16787
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.17708 −0.186992 −0.0934959 0.995620i \(-0.529804\pi\)
−0.0934959 + 0.995620i \(0.529804\pi\)
\(500\) 0.438447 0.0196080
\(501\) 0 0
\(502\) 26.7386 1.19340
\(503\) 10.0691 0.448960 0.224480 0.974479i \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 20.4924 0.910999
\(507\) 0 0
\(508\) 4.49242 0.199319
\(509\) −28.2462 −1.25199 −0.625996 0.779827i \(-0.715307\pi\)
−0.625996 + 0.779827i \(0.715307\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) −35.1231 −1.54921
\(515\) −1.43845 −0.0633856
\(516\) 0 0
\(517\) −9.43845 −0.415102
\(518\) 0 0
\(519\) 0 0
\(520\) −11.1231 −0.487780
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −7.50758 −0.328283 −0.164142 0.986437i \(-0.552485\pi\)
−0.164142 + 0.986437i \(0.552485\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −32.9848 −1.43821
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 4.87689 0.211839
\(531\) 0 0
\(532\) 0 0
\(533\) 14.2462 0.617072
\(534\) 0 0
\(535\) 11.3693 0.491538
\(536\) −15.2311 −0.657881
\(537\) 0 0
\(538\) −44.8769 −1.93478
\(539\) 0 0
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) −24.9848 −1.07319
\(543\) 0 0
\(544\) −11.1231 −0.476899
\(545\) 17.6847 0.757528
\(546\) 0 0
\(547\) 14.2462 0.609124 0.304562 0.952493i \(-0.401490\pi\)
0.304562 + 0.952493i \(0.401490\pi\)
\(548\) 3.89205 0.166260
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) −6.38447 −0.271988
\(552\) 0 0
\(553\) 0 0
\(554\) −25.3693 −1.07784
\(555\) 0 0
\(556\) 3.01515 0.127871
\(557\) 4.87689 0.206641 0.103320 0.994648i \(-0.467053\pi\)
0.103320 + 0.994648i \(0.467053\pi\)
\(558\) 0 0
\(559\) −41.6155 −1.76015
\(560\) 0 0
\(561\) 0 0
\(562\) 25.8617 1.09091
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) −36.9848 −1.55459
\(567\) 0 0
\(568\) −19.5076 −0.818520
\(569\) −34.9848 −1.46664 −0.733320 0.679883i \(-0.762031\pi\)
−0.733320 + 0.679883i \(0.762031\pi\)
\(570\) 0 0
\(571\) 7.50758 0.314182 0.157091 0.987584i \(-0.449788\pi\)
0.157091 + 0.987584i \(0.449788\pi\)
\(572\) 5.12311 0.214208
\(573\) 0 0
\(574\) 0 0
\(575\) 5.12311 0.213648
\(576\) 0 0
\(577\) −13.0540 −0.543444 −0.271722 0.962376i \(-0.587593\pi\)
−0.271722 + 0.962376i \(0.587593\pi\)
\(578\) −5.94602 −0.247322
\(579\) 0 0
\(580\) 2.49242 0.103492
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) −10.3542 −0.428458
\(585\) 0 0
\(586\) −15.1231 −0.624730
\(587\) −9.75379 −0.402582 −0.201291 0.979531i \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.24621 0.257152
\(591\) 0 0
\(592\) −28.1080 −1.15523
\(593\) −23.4384 −0.962502 −0.481251 0.876583i \(-0.659817\pi\)
−0.481251 + 0.876583i \(0.659817\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.86174 0.0762598
\(597\) 0 0
\(598\) 36.4924 1.49229
\(599\) −8.80776 −0.359875 −0.179938 0.983678i \(-0.557590\pi\)
−0.179938 + 0.983678i \(0.557590\pi\)
\(600\) 0 0
\(601\) 26.4924 1.08065 0.540324 0.841457i \(-0.318302\pi\)
0.540324 + 0.841457i \(0.318302\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 9.61553 0.391250
\(605\) −4.43845 −0.180449
\(606\) 0 0
\(607\) 4.94602 0.200753 0.100376 0.994950i \(-0.467995\pi\)
0.100376 + 0.994950i \(0.467995\pi\)
\(608\) −2.73863 −0.111066
\(609\) 0 0
\(610\) −14.6307 −0.592379
\(611\) −16.8078 −0.679969
\(612\) 0 0
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) −49.4773 −1.99674
\(615\) 0 0
\(616\) 0 0
\(617\) −15.7538 −0.634224 −0.317112 0.948388i \(-0.602713\pi\)
−0.317112 + 0.948388i \(0.602713\pi\)
\(618\) 0 0
\(619\) 42.1080 1.69246 0.846231 0.532817i \(-0.178866\pi\)
0.846231 + 0.532817i \(0.178866\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15.0152 0.602053
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 48.8769 1.95351
\(627\) 0 0
\(628\) −1.64584 −0.0656761
\(629\) −27.3693 −1.09129
\(630\) 0 0
\(631\) 8.80776 0.350632 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) −35.1231 −1.39492
\(635\) 10.2462 0.406608
\(636\) 0 0
\(637\) 0 0
\(638\) 22.7386 0.900231
\(639\) 0 0
\(640\) −13.5616 −0.536067
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 2.56155 0.101018 0.0505089 0.998724i \(-0.483916\pi\)
0.0505089 + 0.998724i \(0.483916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) 0 0
\(649\) 10.2462 0.402199
\(650\) 7.12311 0.279391
\(651\) 0 0
\(652\) 0.492423 0.0192848
\(653\) −49.2311 −1.92656 −0.963280 0.268499i \(-0.913473\pi\)
−0.963280 + 0.268499i \(0.913473\pi\)
\(654\) 0 0
\(655\) −9.12311 −0.356469
\(656\) 14.6307 0.571232
\(657\) 0 0
\(658\) 0 0
\(659\) 36.1771 1.40926 0.704629 0.709575i \(-0.251113\pi\)
0.704629 + 0.709575i \(0.251113\pi\)
\(660\) 0 0
\(661\) −3.12311 −0.121475 −0.0607374 0.998154i \(-0.519345\pi\)
−0.0607374 + 0.998154i \(0.519345\pi\)
\(662\) −18.7386 −0.728298
\(663\) 0 0
\(664\) 9.75379 0.378520
\(665\) 0 0
\(666\) 0 0
\(667\) 29.1231 1.12765
\(668\) 9.61553 0.372036
\(669\) 0 0
\(670\) 9.75379 0.376822
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −25.8617 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(674\) 53.8617 2.07468
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) −23.9309 −0.919738 −0.459869 0.887987i \(-0.652104\pi\)
−0.459869 + 0.887987i \(0.652104\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −11.1231 −0.426552
\(681\) 0 0
\(682\) 0 0
\(683\) −42.7386 −1.63535 −0.817674 0.575681i \(-0.804737\pi\)
−0.817674 + 0.575681i \(0.804737\pi\)
\(684\) 0 0
\(685\) 8.87689 0.339169
\(686\) 0 0
\(687\) 0 0
\(688\) −42.7386 −1.62940
\(689\) 14.2462 0.542737
\(690\) 0 0
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) −3.75379 −0.142698
\(693\) 0 0
\(694\) −1.75379 −0.0665729
\(695\) 6.87689 0.260855
\(696\) 0 0
\(697\) 14.2462 0.539614
\(698\) −35.1231 −1.32943
\(699\) 0 0
\(700\) 0 0
\(701\) −0.0691303 −0.00261102 −0.00130551 0.999999i \(-0.500416\pi\)
−0.00130551 + 0.999999i \(0.500416\pi\)
\(702\) 0 0
\(703\) −6.73863 −0.254152
\(704\) −14.2462 −0.536924
\(705\) 0 0
\(706\) 23.1231 0.870250
\(707\) 0 0
\(708\) 0 0
\(709\) −18.1771 −0.682655 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(710\) 12.4924 0.468832
\(711\) 0 0
\(712\) 17.3693 0.650943
\(713\) 0 0
\(714\) 0 0
\(715\) 11.6847 0.436981
\(716\) −8.76894 −0.327711
\(717\) 0 0
\(718\) 12.4924 0.466213
\(719\) 49.6155 1.85035 0.925173 0.379544i \(-0.123919\pi\)
0.925173 + 0.379544i \(0.123919\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 27.6998 1.03088
\(723\) 0 0
\(724\) −10.3542 −0.384809
\(725\) 5.68466 0.211123
\(726\) 0 0
\(727\) −19.5076 −0.723496 −0.361748 0.932276i \(-0.617820\pi\)
−0.361748 + 0.932276i \(0.617820\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.63068 0.245413
\(731\) −41.6155 −1.53921
\(732\) 0 0
\(733\) 5.68466 0.209968 0.104984 0.994474i \(-0.466521\pi\)
0.104984 + 0.994474i \(0.466521\pi\)
\(734\) 5.75379 0.212376
\(735\) 0 0
\(736\) 12.4924 0.460477
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 6.06913 0.223257 0.111628 0.993750i \(-0.464393\pi\)
0.111628 + 0.993750i \(0.464393\pi\)
\(740\) 2.63068 0.0967058
\(741\) 0 0
\(742\) 0 0
\(743\) −32.9848 −1.21010 −0.605048 0.796189i \(-0.706846\pi\)
−0.605048 + 0.796189i \(0.706846\pi\)
\(744\) 0 0
\(745\) 4.24621 0.155569
\(746\) −45.8617 −1.67912
\(747\) 0 0
\(748\) 5.12311 0.187319
\(749\) 0 0
\(750\) 0 0
\(751\) 45.9309 1.67604 0.838021 0.545639i \(-0.183713\pi\)
0.838021 + 0.545639i \(0.183713\pi\)
\(752\) −17.2614 −0.629457
\(753\) 0 0
\(754\) 40.4924 1.47465
\(755\) 21.9309 0.798146
\(756\) 0 0
\(757\) 14.6307 0.531761 0.265881 0.964006i \(-0.414337\pi\)
0.265881 + 0.964006i \(0.414337\pi\)
\(758\) −25.7538 −0.935420
\(759\) 0 0
\(760\) −2.73863 −0.0993407
\(761\) 31.7538 1.15107 0.575537 0.817776i \(-0.304793\pi\)
0.575537 + 0.817776i \(0.304793\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4.13826 0.149717
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 18.2462 0.658833
\(768\) 0 0
\(769\) 9.50758 0.342852 0.171426 0.985197i \(-0.445163\pi\)
0.171426 + 0.985197i \(0.445163\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.35416 −0.0847281
\(773\) 8.06913 0.290226 0.145113 0.989415i \(-0.453645\pi\)
0.145113 + 0.989415i \(0.453645\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 36.1080 1.29620
\(777\) 0 0
\(778\) 6.13826 0.220067
\(779\) 3.50758 0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 36.4924 1.30497
\(783\) 0 0
\(784\) 0 0
\(785\) −3.75379 −0.133978
\(786\) 0 0
\(787\) 3.82292 0.136272 0.0681362 0.997676i \(-0.478295\pi\)
0.0681362 + 0.997676i \(0.478295\pi\)
\(788\) 3.12311 0.111256
\(789\) 0 0
\(790\) 10.2462 0.364544
\(791\) 0 0
\(792\) 0 0
\(793\) −42.7386 −1.51769
\(794\) 36.6004 1.29890
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −13.0540 −0.462396 −0.231198 0.972907i \(-0.574264\pi\)
−0.231198 + 0.972907i \(0.574264\pi\)
\(798\) 0 0
\(799\) −16.8078 −0.594616
\(800\) 2.43845 0.0862121
\(801\) 0 0
\(802\) 42.8466 1.51297
\(803\) 10.8769 0.383837
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.600373 0.0211211
\(809\) 53.5464 1.88259 0.941296 0.337584i \(-0.109610\pi\)
0.941296 + 0.337584i \(0.109610\pi\)
\(810\) 0 0
\(811\) 21.6155 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) 1.12311 0.0393407
\(816\) 0 0
\(817\) −10.2462 −0.358470
\(818\) −41.3693 −1.44644
\(819\) 0 0
\(820\) −1.36932 −0.0478186
\(821\) −40.4233 −1.41078 −0.705391 0.708818i \(-0.749229\pi\)
−0.705391 + 0.708818i \(0.749229\pi\)
\(822\) 0 0
\(823\) −3.50758 −0.122266 −0.0611332 0.998130i \(-0.519471\pi\)
−0.0611332 + 0.998130i \(0.519471\pi\)
\(824\) −3.50758 −0.122192
\(825\) 0 0
\(826\) 0 0
\(827\) −19.3693 −0.673537 −0.336769 0.941587i \(-0.609334\pi\)
−0.336769 + 0.941587i \(0.609334\pi\)
\(828\) 0 0
\(829\) −43.1231 −1.49773 −0.748864 0.662724i \(-0.769400\pi\)
−0.748864 + 0.662724i \(0.769400\pi\)
\(830\) −6.24621 −0.216809
\(831\) 0 0
\(832\) −25.3693 −0.879523
\(833\) 0 0
\(834\) 0 0
\(835\) 21.9309 0.758949
\(836\) 1.26137 0.0436253
\(837\) 0 0
\(838\) −15.2311 −0.526148
\(839\) −37.1231 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) −15.1231 −0.521177
\(843\) 0 0
\(844\) −10.1080 −0.347930
\(845\) 7.80776 0.268595
\(846\) 0 0
\(847\) 0 0
\(848\) 14.6307 0.502420
\(849\) 0 0
\(850\) 7.12311 0.244321
\(851\) 30.7386 1.05371
\(852\) 0 0
\(853\) 56.7386 1.94269 0.971347 0.237666i \(-0.0763824\pi\)
0.971347 + 0.237666i \(0.0763824\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 27.7235 0.947569
\(857\) −32.2462 −1.10151 −0.550755 0.834667i \(-0.685660\pi\)
−0.550755 + 0.834667i \(0.685660\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 1.26137 0.0429623
\(863\) 42.2462 1.43808 0.719039 0.694970i \(-0.244582\pi\)
0.719039 + 0.694970i \(0.244582\pi\)
\(864\) 0 0
\(865\) −8.56155 −0.291102
\(866\) −12.8769 −0.437575
\(867\) 0 0
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 0 0
\(871\) 28.4924 0.965429
\(872\) 43.1231 1.46033
\(873\) 0 0
\(874\) 8.98485 0.303917
\(875\) 0 0
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) −24.0000 −0.809961
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) 45.8617 1.54512 0.772561 0.634941i \(-0.218976\pi\)
0.772561 + 0.634941i \(0.218976\pi\)
\(882\) 0 0
\(883\) 24.4924 0.824236 0.412118 0.911131i \(-0.364789\pi\)
0.412118 + 0.911131i \(0.364789\pi\)
\(884\) 9.12311 0.306843
\(885\) 0 0
\(886\) −42.7386 −1.43583
\(887\) −12.4924 −0.419454 −0.209727 0.977760i \(-0.567258\pi\)
−0.209727 + 0.977760i \(0.567258\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −11.1231 −0.372847
\(891\) 0 0
\(892\) 2.87689 0.0963255
\(893\) −4.13826 −0.138482
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 29.3693 0.980067
\(899\) 0 0
\(900\) 0 0
\(901\) 14.2462 0.474610
\(902\) −12.4924 −0.415952
\(903\) 0 0
\(904\) 34.1383 1.13542
\(905\) −23.6155 −0.785007
\(906\) 0 0
\(907\) 50.1080 1.66381 0.831904 0.554920i \(-0.187251\pi\)
0.831904 + 0.554920i \(0.187251\pi\)
\(908\) 10.3845 0.344621
\(909\) 0 0
\(910\) 0 0
\(911\) −4.49242 −0.148841 −0.0744203 0.997227i \(-0.523711\pi\)
−0.0744203 + 0.997227i \(0.523711\pi\)
\(912\) 0 0
\(913\) −10.2462 −0.339100
\(914\) 13.8617 0.458506
\(915\) 0 0
\(916\) −8.38447 −0.277031
\(917\) 0 0
\(918\) 0 0
\(919\) −13.3002 −0.438733 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(920\) 12.4924 0.411863
\(921\) 0 0
\(922\) 7.61553 0.250804
\(923\) 36.4924 1.20116
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 13.8617 0.455034
\(929\) −52.1080 −1.70961 −0.854803 0.518952i \(-0.826322\pi\)
−0.854803 + 0.518952i \(0.826322\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.36932 0.0448535
\(933\) 0 0
\(934\) −41.4773 −1.35718
\(935\) 11.6847 0.382129
\(936\) 0 0
\(937\) −22.6695 −0.740580 −0.370290 0.928916i \(-0.620742\pi\)
−0.370290 + 0.928916i \(0.620742\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.61553 0.0526927
\(941\) −13.8617 −0.451880 −0.225940 0.974141i \(-0.572545\pi\)
−0.225940 + 0.974141i \(0.572545\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 18.7386 0.609891
\(945\) 0 0
\(946\) 36.4924 1.18647
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 19.3693 0.628755
\(950\) 1.75379 0.0569004
\(951\) 0 0
\(952\) 0 0
\(953\) 24.8769 0.805842 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(954\) 0 0
\(955\) 9.43845 0.305421
\(956\) 0.354162 0.0114544
\(957\) 0 0
\(958\) −20.4924 −0.662080
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 42.7386 1.37795
\(963\) 0 0
\(964\) −5.36932 −0.172934
\(965\) −5.36932 −0.172844
\(966\) 0 0
\(967\) −26.8769 −0.864303 −0.432151 0.901801i \(-0.642245\pi\)
−0.432151 + 0.901801i \(0.642245\pi\)
\(968\) −10.8229 −0.347862
\(969\) 0 0
\(970\) −23.1231 −0.742438
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −43.8920 −1.40495
\(977\) 49.2311 1.57504 0.787521 0.616288i \(-0.211364\pi\)
0.787521 + 0.616288i \(0.211364\pi\)
\(978\) 0 0
\(979\) −18.2462 −0.583151
\(980\) 0 0
\(981\) 0 0
\(982\) 6.52273 0.208149
\(983\) −10.4233 −0.332451 −0.166226 0.986088i \(-0.553158\pi\)
−0.166226 + 0.986088i \(0.553158\pi\)
\(984\) 0 0
\(985\) 7.12311 0.226961
\(986\) 40.4924 1.28954
\(987\) 0 0
\(988\) 2.24621 0.0714615
\(989\) 46.7386 1.48620
\(990\) 0 0
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.2462 0.578444
\(996\) 0 0
\(997\) −9.68466 −0.306716 −0.153358 0.988171i \(-0.549009\pi\)
−0.153358 + 0.988171i \(0.549009\pi\)
\(998\) 6.52273 0.206473
\(999\) 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 2205.2.a.x.1.1 2
3.2 odd 2 245.2.a.d.1.2 2
7.6 odd 2 315.2.a.e.1.1 2
12.11 even 2 3920.2.a.bs.1.1 2
15.2 even 4 1225.2.b.f.99.3 4
15.8 even 4 1225.2.b.f.99.2 4
15.14 odd 2 1225.2.a.s.1.1 2
21.2 odd 6 245.2.e.h.116.1 4
21.5 even 6 245.2.e.i.116.1 4
21.11 odd 6 245.2.e.h.226.1 4
21.17 even 6 245.2.e.i.226.1 4
21.20 even 2 35.2.a.b.1.2 2
28.27 even 2 5040.2.a.bt.1.2 2
35.13 even 4 1575.2.d.e.1324.3 4
35.27 even 4 1575.2.d.e.1324.2 4
35.34 odd 2 1575.2.a.p.1.2 2
84.83 odd 2 560.2.a.i.1.2 2
105.62 odd 4 175.2.b.b.99.3 4
105.83 odd 4 175.2.b.b.99.2 4
105.104 even 2 175.2.a.f.1.1 2
168.83 odd 2 2240.2.a.bd.1.1 2
168.125 even 2 2240.2.a.bh.1.2 2
231.230 odd 2 4235.2.a.m.1.1 2
273.272 even 2 5915.2.a.l.1.1 2
420.83 even 4 2800.2.g.t.449.4 4
420.167 even 4 2800.2.g.t.449.1 4
420.419 odd 2 2800.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 21.20 even 2
175.2.a.f.1.1 2 105.104 even 2
175.2.b.b.99.2 4 105.83 odd 4
175.2.b.b.99.3 4 105.62 odd 4
245.2.a.d.1.2 2 3.2 odd 2
245.2.e.h.116.1 4 21.2 odd 6
245.2.e.h.226.1 4 21.11 odd 6
245.2.e.i.116.1 4 21.5 even 6
245.2.e.i.226.1 4 21.17 even 6
315.2.a.e.1.1 2 7.6 odd 2
560.2.a.i.1.2 2 84.83 odd 2
1225.2.a.s.1.1 2 15.14 odd 2
1225.2.b.f.99.2 4 15.8 even 4
1225.2.b.f.99.3 4 15.2 even 4
1575.2.a.p.1.2 2 35.34 odd 2
1575.2.d.e.1324.2 4 35.27 even 4
1575.2.d.e.1324.3 4 35.13 even 4
2205.2.a.x.1.1 2 1.1 even 1 trivial
2240.2.a.bd.1.1 2 168.83 odd 2
2240.2.a.bh.1.2 2 168.125 even 2
2800.2.a.bi.1.1 2 420.419 odd 2
2800.2.g.t.449.1 4 420.167 even 4
2800.2.g.t.449.4 4 420.83 even 4
3920.2.a.bs.1.1 2 12.11 even 2
4235.2.a.m.1.1 2 231.230 odd 2
5040.2.a.bt.1.2 2 28.27 even 2
5915.2.a.l.1.1 2 273.272 even 2