Properties

Label 2205.2.a.z.1.2
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: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} +O(q^{10})\) \(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} -2.73205 q^{10} -0.732051 q^{11} +2.26795 q^{13} +14.9282 q^{16} -3.26795 q^{17} +4.46410 q^{19} -5.46410 q^{20} -2.00000 q^{22} +4.73205 q^{23} +1.00000 q^{25} +6.19615 q^{26} +4.19615 q^{29} -0.464102 q^{31} +21.8564 q^{32} -8.92820 q^{34} -3.19615 q^{37} +12.1962 q^{38} -9.46410 q^{40} +0.732051 q^{41} +3.19615 q^{43} -4.00000 q^{44} +12.9282 q^{46} -2.00000 q^{47} +2.73205 q^{50} +12.3923 q^{52} -12.3923 q^{53} +0.732051 q^{55} +11.4641 q^{58} +0.196152 q^{59} +4.00000 q^{61} -1.26795 q^{62} +29.8564 q^{64} -2.26795 q^{65} -14.6603 q^{67} -17.8564 q^{68} -6.19615 q^{71} +12.6603 q^{73} -8.73205 q^{74} +24.3923 q^{76} -7.39230 q^{79} -14.9282 q^{80} +2.00000 q^{82} -15.1244 q^{83} +3.26795 q^{85} +8.73205 q^{86} -6.92820 q^{88} -15.1244 q^{89} +25.8564 q^{92} -5.46410 q^{94} -4.46410 q^{95} +14.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 4q^{4} - 2q^{5} + 12q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 4q^{4} - 2q^{5} + 12q^{8} - 2q^{10} + 2q^{11} + 8q^{13} + 16q^{16} - 10q^{17} + 2q^{19} - 4q^{20} - 4q^{22} + 6q^{23} + 2q^{25} + 2q^{26} - 2q^{29} + 6q^{31} + 16q^{32} - 4q^{34} + 4q^{37} + 14q^{38} - 12q^{40} - 2q^{41} - 4q^{43} - 8q^{44} + 12q^{46} - 4q^{47} + 2q^{50} + 4q^{52} - 4q^{53} - 2q^{55} + 16q^{58} - 10q^{59} + 8q^{61} - 6q^{62} + 32q^{64} - 8q^{65} - 12q^{67} - 8q^{68} - 2q^{71} + 8q^{73} - 14q^{74} + 28q^{76} + 6q^{79} - 16q^{80} + 4q^{82} - 6q^{83} + 10q^{85} + 14q^{86} - 6q^{89} + 24q^{92} - 4q^{94} - 2q^{95} + 16q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73205 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(3\) 0 0
\(4\) 5.46410 2.73205
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 9.46410 3.34607
\(9\) 0 0
\(10\) −2.73205 −0.863950
\(11\) −0.732051 −0.220722 −0.110361 0.993892i \(-0.535201\pi\)
−0.110361 + 0.993892i \(0.535201\pi\)
\(12\) 0 0
\(13\) 2.26795 0.629016 0.314508 0.949255i \(-0.398160\pi\)
0.314508 + 0.949255i \(0.398160\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 14.9282 3.73205
\(17\) −3.26795 −0.792594 −0.396297 0.918122i \(-0.629705\pi\)
−0.396297 + 0.918122i \(0.629705\pi\)
\(18\) 0 0
\(19\) 4.46410 1.02414 0.512068 0.858945i \(-0.328880\pi\)
0.512068 + 0.858945i \(0.328880\pi\)
\(20\) −5.46410 −1.22181
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 4.73205 0.986701 0.493350 0.869831i \(-0.335772\pi\)
0.493350 + 0.869831i \(0.335772\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.19615 1.21517
\(27\) 0 0
\(28\) 0 0
\(29\) 4.19615 0.779206 0.389603 0.920983i \(-0.372612\pi\)
0.389603 + 0.920983i \(0.372612\pi\)
\(30\) 0 0
\(31\) −0.464102 −0.0833551 −0.0416776 0.999131i \(-0.513270\pi\)
−0.0416776 + 0.999131i \(0.513270\pi\)
\(32\) 21.8564 3.86370
\(33\) 0 0
\(34\) −8.92820 −1.53117
\(35\) 0 0
\(36\) 0 0
\(37\) −3.19615 −0.525444 −0.262722 0.964872i \(-0.584620\pi\)
−0.262722 + 0.964872i \(0.584620\pi\)
\(38\) 12.1962 1.97848
\(39\) 0 0
\(40\) −9.46410 −1.49641
\(41\) 0.732051 0.114327 0.0571636 0.998365i \(-0.481794\pi\)
0.0571636 + 0.998365i \(0.481794\pi\)
\(42\) 0 0
\(43\) 3.19615 0.487409 0.243704 0.969850i \(-0.421637\pi\)
0.243704 + 0.969850i \(0.421637\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 12.9282 1.90616
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.73205 0.386370
\(51\) 0 0
\(52\) 12.3923 1.71850
\(53\) −12.3923 −1.70221 −0.851107 0.524992i \(-0.824068\pi\)
−0.851107 + 0.524992i \(0.824068\pi\)
\(54\) 0 0
\(55\) 0.732051 0.0987097
\(56\) 0 0
\(57\) 0 0
\(58\) 11.4641 1.50531
\(59\) 0.196152 0.0255369 0.0127684 0.999918i \(-0.495936\pi\)
0.0127684 + 0.999918i \(0.495936\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −1.26795 −0.161030
\(63\) 0 0
\(64\) 29.8564 3.73205
\(65\) −2.26795 −0.281304
\(66\) 0 0
\(67\) −14.6603 −1.79104 −0.895518 0.445026i \(-0.853194\pi\)
−0.895518 + 0.445026i \(0.853194\pi\)
\(68\) −17.8564 −2.16541
\(69\) 0 0
\(70\) 0 0
\(71\) −6.19615 −0.735348 −0.367674 0.929955i \(-0.619846\pi\)
−0.367674 + 0.929955i \(0.619846\pi\)
\(72\) 0 0
\(73\) 12.6603 1.48177 0.740885 0.671632i \(-0.234406\pi\)
0.740885 + 0.671632i \(0.234406\pi\)
\(74\) −8.73205 −1.01508
\(75\) 0 0
\(76\) 24.3923 2.79799
\(77\) 0 0
\(78\) 0 0
\(79\) −7.39230 −0.831699 −0.415850 0.909433i \(-0.636516\pi\)
−0.415850 + 0.909433i \(0.636516\pi\)
\(80\) −14.9282 −1.66902
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −15.1244 −1.66011 −0.830057 0.557679i \(-0.811692\pi\)
−0.830057 + 0.557679i \(0.811692\pi\)
\(84\) 0 0
\(85\) 3.26795 0.354459
\(86\) 8.73205 0.941601
\(87\) 0 0
\(88\) −6.92820 −0.738549
\(89\) −15.1244 −1.60318 −0.801589 0.597875i \(-0.796012\pi\)
−0.801589 + 0.597875i \(0.796012\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 25.8564 2.69572
\(93\) 0 0
\(94\) −5.46410 −0.563579
\(95\) −4.46410 −0.458007
\(96\) 0 0
\(97\) 14.9282 1.51573 0.757865 0.652412i \(-0.226243\pi\)
0.757865 + 0.652412i \(0.226243\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.46410 0.546410
\(101\) 7.26795 0.723188 0.361594 0.932336i \(-0.382233\pi\)
0.361594 + 0.932336i \(0.382233\pi\)
\(102\) 0 0
\(103\) −9.19615 −0.906124 −0.453062 0.891479i \(-0.649668\pi\)
−0.453062 + 0.891479i \(0.649668\pi\)
\(104\) 21.4641 2.10473
\(105\) 0 0
\(106\) −33.8564 −3.28842
\(107\) −2.19615 −0.212310 −0.106155 0.994350i \(-0.533854\pi\)
−0.106155 + 0.994350i \(0.533854\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −8.92820 −0.839895 −0.419947 0.907548i \(-0.637951\pi\)
−0.419947 + 0.907548i \(0.637951\pi\)
\(114\) 0 0
\(115\) −4.73205 −0.441266
\(116\) 22.9282 2.12883
\(117\) 0 0
\(118\) 0.535898 0.0493334
\(119\) 0 0
\(120\) 0 0
\(121\) −10.4641 −0.951282
\(122\) 10.9282 0.989393
\(123\) 0 0
\(124\) −2.53590 −0.227730
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.80385 0.426273 0.213136 0.977022i \(-0.431632\pi\)
0.213136 + 0.977022i \(0.431632\pi\)
\(128\) 37.8564 3.34607
\(129\) 0 0
\(130\) −6.19615 −0.543439
\(131\) −15.4641 −1.35110 −0.675552 0.737312i \(-0.736095\pi\)
−0.675552 + 0.737312i \(0.736095\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −40.0526 −3.46001
\(135\) 0 0
\(136\) −30.9282 −2.65207
\(137\) −2.19615 −0.187630 −0.0938150 0.995590i \(-0.529906\pi\)
−0.0938150 + 0.995590i \(0.529906\pi\)
\(138\) 0 0
\(139\) 5.92820 0.502824 0.251412 0.967880i \(-0.419105\pi\)
0.251412 + 0.967880i \(0.419105\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.9282 −1.42058
\(143\) −1.66025 −0.138837
\(144\) 0 0
\(145\) −4.19615 −0.348471
\(146\) 34.5885 2.86256
\(147\) 0 0
\(148\) −17.4641 −1.43554
\(149\) −5.85641 −0.479776 −0.239888 0.970801i \(-0.577111\pi\)
−0.239888 + 0.970801i \(0.577111\pi\)
\(150\) 0 0
\(151\) −8.92820 −0.726567 −0.363283 0.931679i \(-0.618344\pi\)
−0.363283 + 0.931679i \(0.618344\pi\)
\(152\) 42.2487 3.42682
\(153\) 0 0
\(154\) 0 0
\(155\) 0.464102 0.0372775
\(156\) 0 0
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) −20.1962 −1.60672
\(159\) 0 0
\(160\) −21.8564 −1.72790
\(161\) 0 0
\(162\) 0 0
\(163\) 21.8564 1.71193 0.855963 0.517037i \(-0.172965\pi\)
0.855963 + 0.517037i \(0.172965\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −41.3205 −3.20709
\(167\) 17.6603 1.36659 0.683296 0.730142i \(-0.260546\pi\)
0.683296 + 0.730142i \(0.260546\pi\)
\(168\) 0 0
\(169\) −7.85641 −0.604339
\(170\) 8.92820 0.684762
\(171\) 0 0
\(172\) 17.4641 1.33163
\(173\) −14.5359 −1.10514 −0.552572 0.833465i \(-0.686354\pi\)
−0.552572 + 0.833465i \(0.686354\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −10.9282 −0.823744
\(177\) 0 0
\(178\) −41.3205 −3.09710
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −24.3205 −1.80773 −0.903865 0.427819i \(-0.859282\pi\)
−0.903865 + 0.427819i \(0.859282\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 44.7846 3.30157
\(185\) 3.19615 0.234986
\(186\) 0 0
\(187\) 2.39230 0.174943
\(188\) −10.9282 −0.797021
\(189\) 0 0
\(190\) −12.1962 −0.884802
\(191\) 8.92820 0.646022 0.323011 0.946395i \(-0.395305\pi\)
0.323011 + 0.946395i \(0.395305\pi\)
\(192\) 0 0
\(193\) 1.19615 0.0861009 0.0430505 0.999073i \(-0.486292\pi\)
0.0430505 + 0.999073i \(0.486292\pi\)
\(194\) 40.7846 2.92816
\(195\) 0 0
\(196\) 0 0
\(197\) 0.339746 0.0242059 0.0121029 0.999927i \(-0.496147\pi\)
0.0121029 + 0.999927i \(0.496147\pi\)
\(198\) 0 0
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 9.46410 0.669213
\(201\) 0 0
\(202\) 19.8564 1.39709
\(203\) 0 0
\(204\) 0 0
\(205\) −0.732051 −0.0511286
\(206\) −25.1244 −1.75050
\(207\) 0 0
\(208\) 33.8564 2.34752
\(209\) −3.26795 −0.226049
\(210\) 0 0
\(211\) 7.07180 0.486843 0.243421 0.969921i \(-0.421730\pi\)
0.243421 + 0.969921i \(0.421730\pi\)
\(212\) −67.7128 −4.65054
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) −3.19615 −0.217976
\(216\) 0 0
\(217\) 0 0
\(218\) 30.0526 2.03542
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −7.41154 −0.498554
\(222\) 0 0
\(223\) −20.3923 −1.36557 −0.682785 0.730619i \(-0.739231\pi\)
−0.682785 + 0.730619i \(0.739231\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −24.3923 −1.62255
\(227\) 1.66025 0.110195 0.0550975 0.998481i \(-0.482453\pi\)
0.0550975 + 0.998481i \(0.482453\pi\)
\(228\) 0 0
\(229\) −3.00000 −0.198246 −0.0991228 0.995075i \(-0.531604\pi\)
−0.0991228 + 0.995075i \(0.531604\pi\)
\(230\) −12.9282 −0.852460
\(231\) 0 0
\(232\) 39.7128 2.60727
\(233\) −17.3205 −1.13470 −0.567352 0.823475i \(-0.692032\pi\)
−0.567352 + 0.823475i \(0.692032\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 1.07180 0.0697680
\(237\) 0 0
\(238\) 0 0
\(239\) −7.07180 −0.457437 −0.228718 0.973493i \(-0.573453\pi\)
−0.228718 + 0.973493i \(0.573453\pi\)
\(240\) 0 0
\(241\) 13.4641 0.867299 0.433650 0.901082i \(-0.357226\pi\)
0.433650 + 0.901082i \(0.357226\pi\)
\(242\) −28.5885 −1.83774
\(243\) 0 0
\(244\) 21.8564 1.39921
\(245\) 0 0
\(246\) 0 0
\(247\) 10.1244 0.644197
\(248\) −4.39230 −0.278912
\(249\) 0 0
\(250\) −2.73205 −0.172790
\(251\) 24.5885 1.55201 0.776005 0.630727i \(-0.217243\pi\)
0.776005 + 0.630727i \(0.217243\pi\)
\(252\) 0 0
\(253\) −3.46410 −0.217786
\(254\) 13.1244 0.823495
\(255\) 0 0
\(256\) 43.7128 2.73205
\(257\) 5.66025 0.353077 0.176538 0.984294i \(-0.443510\pi\)
0.176538 + 0.984294i \(0.443510\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −12.3923 −0.768538
\(261\) 0 0
\(262\) −42.2487 −2.61013
\(263\) −8.39230 −0.517492 −0.258746 0.965945i \(-0.583309\pi\)
−0.258746 + 0.965945i \(0.583309\pi\)
\(264\) 0 0
\(265\) 12.3923 0.761253
\(266\) 0 0
\(267\) 0 0
\(268\) −80.1051 −4.89320
\(269\) 12.5359 0.764327 0.382164 0.924095i \(-0.375179\pi\)
0.382164 + 0.924095i \(0.375179\pi\)
\(270\) 0 0
\(271\) 3.07180 0.186598 0.0932992 0.995638i \(-0.470259\pi\)
0.0932992 + 0.995638i \(0.470259\pi\)
\(272\) −48.7846 −2.95800
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −0.732051 −0.0441443
\(276\) 0 0
\(277\) −14.6603 −0.880849 −0.440425 0.897790i \(-0.645172\pi\)
−0.440425 + 0.897790i \(0.645172\pi\)
\(278\) 16.1962 0.971381
\(279\) 0 0
\(280\) 0 0
\(281\) −13.8564 −0.826604 −0.413302 0.910594i \(-0.635625\pi\)
−0.413302 + 0.910594i \(0.635625\pi\)
\(282\) 0 0
\(283\) −24.1244 −1.43404 −0.717022 0.697050i \(-0.754495\pi\)
−0.717022 + 0.697050i \(0.754495\pi\)
\(284\) −33.8564 −2.00901
\(285\) 0 0
\(286\) −4.53590 −0.268213
\(287\) 0 0
\(288\) 0 0
\(289\) −6.32051 −0.371795
\(290\) −11.4641 −0.673195
\(291\) 0 0
\(292\) 69.1769 4.04827
\(293\) −18.9282 −1.10580 −0.552899 0.833248i \(-0.686478\pi\)
−0.552899 + 0.833248i \(0.686478\pi\)
\(294\) 0 0
\(295\) −0.196152 −0.0114204
\(296\) −30.2487 −1.75817
\(297\) 0 0
\(298\) −16.0000 −0.926855
\(299\) 10.7321 0.620651
\(300\) 0 0
\(301\) 0 0
\(302\) −24.3923 −1.40362
\(303\) 0 0
\(304\) 66.6410 3.82212
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −32.1244 −1.83343 −0.916717 0.399537i \(-0.869171\pi\)
−0.916717 + 0.399537i \(0.869171\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.26795 0.0720147
\(311\) −9.12436 −0.517395 −0.258697 0.965958i \(-0.583293\pi\)
−0.258697 + 0.965958i \(0.583293\pi\)
\(312\) 0 0
\(313\) −12.6603 −0.715600 −0.357800 0.933798i \(-0.616473\pi\)
−0.357800 + 0.933798i \(0.616473\pi\)
\(314\) 17.4641 0.985556
\(315\) 0 0
\(316\) −40.3923 −2.27224
\(317\) 28.4449 1.59762 0.798811 0.601582i \(-0.205463\pi\)
0.798811 + 0.601582i \(0.205463\pi\)
\(318\) 0 0
\(319\) −3.07180 −0.171988
\(320\) −29.8564 −1.66902
\(321\) 0 0
\(322\) 0 0
\(323\) −14.5885 −0.811723
\(324\) 0 0
\(325\) 2.26795 0.125803
\(326\) 59.7128 3.30719
\(327\) 0 0
\(328\) 6.92820 0.382546
\(329\) 0 0
\(330\) 0 0
\(331\) 8.07180 0.443666 0.221833 0.975085i \(-0.428796\pi\)
0.221833 + 0.975085i \(0.428796\pi\)
\(332\) −82.6410 −4.53551
\(333\) 0 0
\(334\) 48.2487 2.64005
\(335\) 14.6603 0.800975
\(336\) 0 0
\(337\) 17.9808 0.979475 0.489737 0.871870i \(-0.337093\pi\)
0.489737 + 0.871870i \(0.337093\pi\)
\(338\) −21.4641 −1.16749
\(339\) 0 0
\(340\) 17.8564 0.968400
\(341\) 0.339746 0.0183983
\(342\) 0 0
\(343\) 0 0
\(344\) 30.2487 1.63090
\(345\) 0 0
\(346\) −39.7128 −2.13497
\(347\) 21.0718 1.13119 0.565597 0.824682i \(-0.308646\pi\)
0.565597 + 0.824682i \(0.308646\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −16.0000 −0.852803
\(353\) 3.12436 0.166293 0.0831463 0.996537i \(-0.473503\pi\)
0.0831463 + 0.996537i \(0.473503\pi\)
\(354\) 0 0
\(355\) 6.19615 0.328858
\(356\) −82.6410 −4.37997
\(357\) 0 0
\(358\) 27.3205 1.44393
\(359\) 1.26795 0.0669198 0.0334599 0.999440i \(-0.489347\pi\)
0.0334599 + 0.999440i \(0.489347\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) −66.4449 −3.49226
\(363\) 0 0
\(364\) 0 0
\(365\) −12.6603 −0.662668
\(366\) 0 0
\(367\) 11.1962 0.584434 0.292217 0.956352i \(-0.405607\pi\)
0.292217 + 0.956352i \(0.405607\pi\)
\(368\) 70.6410 3.68242
\(369\) 0 0
\(370\) 8.73205 0.453958
\(371\) 0 0
\(372\) 0 0
\(373\) 26.5167 1.37298 0.686490 0.727139i \(-0.259150\pi\)
0.686490 + 0.727139i \(0.259150\pi\)
\(374\) 6.53590 0.337963
\(375\) 0 0
\(376\) −18.9282 −0.976148
\(377\) 9.51666 0.490133
\(378\) 0 0
\(379\) 6.32051 0.324663 0.162331 0.986736i \(-0.448099\pi\)
0.162331 + 0.986736i \(0.448099\pi\)
\(380\) −24.3923 −1.25130
\(381\) 0 0
\(382\) 24.3923 1.24802
\(383\) 23.3205 1.19162 0.595811 0.803125i \(-0.296831\pi\)
0.595811 + 0.803125i \(0.296831\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.26795 0.166334
\(387\) 0 0
\(388\) 81.5692 4.14105
\(389\) −5.41154 −0.274376 −0.137188 0.990545i \(-0.543806\pi\)
−0.137188 + 0.990545i \(0.543806\pi\)
\(390\) 0 0
\(391\) −15.4641 −0.782053
\(392\) 0 0
\(393\) 0 0
\(394\) 0.928203 0.0467622
\(395\) 7.39230 0.371947
\(396\) 0 0
\(397\) −31.1962 −1.56569 −0.782845 0.622217i \(-0.786232\pi\)
−0.782845 + 0.622217i \(0.786232\pi\)
\(398\) 60.1051 3.01280
\(399\) 0 0
\(400\) 14.9282 0.746410
\(401\) 16.3923 0.818593 0.409296 0.912402i \(-0.365774\pi\)
0.409296 + 0.912402i \(0.365774\pi\)
\(402\) 0 0
\(403\) −1.05256 −0.0524317
\(404\) 39.7128 1.97579
\(405\) 0 0
\(406\) 0 0
\(407\) 2.33975 0.115977
\(408\) 0 0
\(409\) 3.14359 0.155441 0.0777203 0.996975i \(-0.475236\pi\)
0.0777203 + 0.996975i \(0.475236\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −50.2487 −2.47558
\(413\) 0 0
\(414\) 0 0
\(415\) 15.1244 0.742425
\(416\) 49.5692 2.43033
\(417\) 0 0
\(418\) −8.92820 −0.436693
\(419\) 35.4641 1.73253 0.866267 0.499581i \(-0.166513\pi\)
0.866267 + 0.499581i \(0.166513\pi\)
\(420\) 0 0
\(421\) 0.0717968 0.00349916 0.00174958 0.999998i \(-0.499443\pi\)
0.00174958 + 0.999998i \(0.499443\pi\)
\(422\) 19.3205 0.940508
\(423\) 0 0
\(424\) −117.282 −5.69572
\(425\) −3.26795 −0.158519
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.73205 −0.421097
\(431\) −17.3205 −0.834300 −0.417150 0.908838i \(-0.636971\pi\)
−0.417150 + 0.908838i \(0.636971\pi\)
\(432\) 0 0
\(433\) 15.1962 0.730280 0.365140 0.930953i \(-0.381021\pi\)
0.365140 + 0.930953i \(0.381021\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 60.1051 2.87851
\(437\) 21.1244 1.01051
\(438\) 0 0
\(439\) 0.535898 0.0255770 0.0127885 0.999918i \(-0.495929\pi\)
0.0127885 + 0.999918i \(0.495929\pi\)
\(440\) 6.92820 0.330289
\(441\) 0 0
\(442\) −20.2487 −0.963133
\(443\) −9.46410 −0.449653 −0.224827 0.974399i \(-0.572182\pi\)
−0.224827 + 0.974399i \(0.572182\pi\)
\(444\) 0 0
\(445\) 15.1244 0.716963
\(446\) −55.7128 −2.63808
\(447\) 0 0
\(448\) 0 0
\(449\) 35.8564 1.69217 0.846084 0.533049i \(-0.178954\pi\)
0.846084 + 0.533049i \(0.178954\pi\)
\(450\) 0 0
\(451\) −0.535898 −0.0252345
\(452\) −48.7846 −2.29464
\(453\) 0 0
\(454\) 4.53590 0.212880
\(455\) 0 0
\(456\) 0 0
\(457\) −16.6603 −0.779334 −0.389667 0.920956i \(-0.627410\pi\)
−0.389667 + 0.920956i \(0.627410\pi\)
\(458\) −8.19615 −0.382981
\(459\) 0 0
\(460\) −25.8564 −1.20556
\(461\) −16.9808 −0.790873 −0.395436 0.918493i \(-0.629407\pi\)
−0.395436 + 0.918493i \(0.629407\pi\)
\(462\) 0 0
\(463\) 25.7321 1.19587 0.597935 0.801545i \(-0.295988\pi\)
0.597935 + 0.801545i \(0.295988\pi\)
\(464\) 62.6410 2.90804
\(465\) 0 0
\(466\) −47.3205 −2.19208
\(467\) 0.143594 0.00664472 0.00332236 0.999994i \(-0.498942\pi\)
0.00332236 + 0.999994i \(0.498942\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5.46410 0.252040
\(471\) 0 0
\(472\) 1.85641 0.0854480
\(473\) −2.33975 −0.107582
\(474\) 0 0
\(475\) 4.46410 0.204827
\(476\) 0 0
\(477\) 0 0
\(478\) −19.3205 −0.883699
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 0 0
\(481\) −7.24871 −0.330513
\(482\) 36.7846 1.67549
\(483\) 0 0
\(484\) −57.1769 −2.59895
\(485\) −14.9282 −0.677855
\(486\) 0 0
\(487\) −0.411543 −0.0186488 −0.00932439 0.999957i \(-0.502968\pi\)
−0.00932439 + 0.999957i \(0.502968\pi\)
\(488\) 37.8564 1.71368
\(489\) 0 0
\(490\) 0 0
\(491\) 38.2487 1.72614 0.863070 0.505084i \(-0.168539\pi\)
0.863070 + 0.505084i \(0.168539\pi\)
\(492\) 0 0
\(493\) −13.7128 −0.617594
\(494\) 27.6603 1.24449
\(495\) 0 0
\(496\) −6.92820 −0.311086
\(497\) 0 0
\(498\) 0 0
\(499\) −13.5359 −0.605950 −0.302975 0.952998i \(-0.597980\pi\)
−0.302975 + 0.952998i \(0.597980\pi\)
\(500\) −5.46410 −0.244362
\(501\) 0 0
\(502\) 67.1769 2.99825
\(503\) −14.3923 −0.641721 −0.320861 0.947126i \(-0.603972\pi\)
−0.320861 + 0.947126i \(0.603972\pi\)
\(504\) 0 0
\(505\) −7.26795 −0.323419
\(506\) −9.46410 −0.420731
\(507\) 0 0
\(508\) 26.2487 1.16460
\(509\) −4.53590 −0.201050 −0.100525 0.994935i \(-0.532052\pi\)
−0.100525 + 0.994935i \(0.532052\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 43.7128 1.93185
\(513\) 0 0
\(514\) 15.4641 0.682092
\(515\) 9.19615 0.405231
\(516\) 0 0
\(517\) 1.46410 0.0643911
\(518\) 0 0
\(519\) 0 0
\(520\) −21.4641 −0.941263
\(521\) 5.46410 0.239387 0.119693 0.992811i \(-0.461809\pi\)
0.119693 + 0.992811i \(0.461809\pi\)
\(522\) 0 0
\(523\) −27.7321 −1.21264 −0.606319 0.795222i \(-0.707355\pi\)
−0.606319 + 0.795222i \(0.707355\pi\)
\(524\) −84.4974 −3.69129
\(525\) 0 0
\(526\) −22.9282 −0.999717
\(527\) 1.51666 0.0660668
\(528\) 0 0
\(529\) −0.607695 −0.0264215
\(530\) 33.8564 1.47063
\(531\) 0 0
\(532\) 0 0
\(533\) 1.66025 0.0719136
\(534\) 0 0
\(535\) 2.19615 0.0949479
\(536\) −138.746 −5.99292
\(537\) 0 0
\(538\) 34.2487 1.47657
\(539\) 0 0
\(540\) 0 0
\(541\) 5.78461 0.248700 0.124350 0.992238i \(-0.460315\pi\)
0.124350 + 0.992238i \(0.460315\pi\)
\(542\) 8.39230 0.360480
\(543\) 0 0
\(544\) −71.4256 −3.06235
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) −26.2487 −1.12231 −0.561157 0.827709i \(-0.689644\pi\)
−0.561157 + 0.827709i \(0.689644\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 18.7321 0.798012
\(552\) 0 0
\(553\) 0 0
\(554\) −40.0526 −1.70167
\(555\) 0 0
\(556\) 32.3923 1.37374
\(557\) −14.7846 −0.626444 −0.313222 0.949680i \(-0.601408\pi\)
−0.313222 + 0.949680i \(0.601408\pi\)
\(558\) 0 0
\(559\) 7.24871 0.306588
\(560\) 0 0
\(561\) 0 0
\(562\) −37.8564 −1.59688
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 8.92820 0.375612
\(566\) −65.9090 −2.77036
\(567\) 0 0
\(568\) −58.6410 −2.46052
\(569\) −32.4449 −1.36016 −0.680080 0.733138i \(-0.738055\pi\)
−0.680080 + 0.733138i \(0.738055\pi\)
\(570\) 0 0
\(571\) 18.6077 0.778708 0.389354 0.921088i \(-0.372698\pi\)
0.389354 + 0.921088i \(0.372698\pi\)
\(572\) −9.07180 −0.379311
\(573\) 0 0
\(574\) 0 0
\(575\) 4.73205 0.197340
\(576\) 0 0
\(577\) 28.6603 1.19314 0.596571 0.802560i \(-0.296529\pi\)
0.596571 + 0.802560i \(0.296529\pi\)
\(578\) −17.2679 −0.718252
\(579\) 0 0
\(580\) −22.9282 −0.952042
\(581\) 0 0
\(582\) 0 0
\(583\) 9.07180 0.375715
\(584\) 119.818 4.95810
\(585\) 0 0
\(586\) −51.7128 −2.13624
\(587\) −40.7321 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(588\) 0 0
\(589\) −2.07180 −0.0853669
\(590\) −0.535898 −0.0220626
\(591\) 0 0
\(592\) −47.7128 −1.96098
\(593\) −27.9090 −1.14608 −0.573042 0.819526i \(-0.694237\pi\)
−0.573042 + 0.819526i \(0.694237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −32.0000 −1.31077
\(597\) 0 0
\(598\) 29.3205 1.19900
\(599\) 38.2487 1.56280 0.781400 0.624030i \(-0.214506\pi\)
0.781400 + 0.624030i \(0.214506\pi\)
\(600\) 0 0
\(601\) −0.0717968 −0.00292865 −0.00146433 0.999999i \(-0.500466\pi\)
−0.00146433 + 0.999999i \(0.500466\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −48.7846 −1.98502
\(605\) 10.4641 0.425426
\(606\) 0 0
\(607\) 3.19615 0.129728 0.0648639 0.997894i \(-0.479339\pi\)
0.0648639 + 0.997894i \(0.479339\pi\)
\(608\) 97.5692 3.95695
\(609\) 0 0
\(610\) −10.9282 −0.442470
\(611\) −4.53590 −0.183503
\(612\) 0 0
\(613\) −26.9282 −1.08762 −0.543810 0.839208i \(-0.683019\pi\)
−0.543810 + 0.839208i \(0.683019\pi\)
\(614\) −87.7654 −3.54192
\(615\) 0 0
\(616\) 0 0
\(617\) 36.2487 1.45932 0.729659 0.683811i \(-0.239679\pi\)
0.729659 + 0.683811i \(0.239679\pi\)
\(618\) 0 0
\(619\) −30.0718 −1.20869 −0.604344 0.796724i \(-0.706565\pi\)
−0.604344 + 0.796724i \(0.706565\pi\)
\(620\) 2.53590 0.101844
\(621\) 0 0
\(622\) −24.9282 −0.999530
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.5885 −1.38243
\(627\) 0 0
\(628\) 34.9282 1.39379
\(629\) 10.4449 0.416464
\(630\) 0 0
\(631\) 48.7846 1.94208 0.971042 0.238908i \(-0.0767893\pi\)
0.971042 + 0.238908i \(0.0767893\pi\)
\(632\) −69.9615 −2.78292
\(633\) 0 0
\(634\) 77.7128 3.08637
\(635\) −4.80385 −0.190635
\(636\) 0 0
\(637\) 0 0
\(638\) −8.39230 −0.332255
\(639\) 0 0
\(640\) −37.8564 −1.49641
\(641\) −3.80385 −0.150243 −0.0751215 0.997174i \(-0.523934\pi\)
−0.0751215 + 0.997174i \(0.523934\pi\)
\(642\) 0 0
\(643\) 4.51666 0.178120 0.0890599 0.996026i \(-0.471614\pi\)
0.0890599 + 0.996026i \(0.471614\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −39.8564 −1.56813
\(647\) 27.9090 1.09721 0.548607 0.836080i \(-0.315158\pi\)
0.548607 + 0.836080i \(0.315158\pi\)
\(648\) 0 0
\(649\) −0.143594 −0.00563654
\(650\) 6.19615 0.243033
\(651\) 0 0
\(652\) 119.426 4.67707
\(653\) 44.5885 1.74488 0.872441 0.488720i \(-0.162536\pi\)
0.872441 + 0.488720i \(0.162536\pi\)
\(654\) 0 0
\(655\) 15.4641 0.604232
\(656\) 10.9282 0.426675
\(657\) 0 0
\(658\) 0 0
\(659\) −2.92820 −0.114067 −0.0570333 0.998372i \(-0.518164\pi\)
−0.0570333 + 0.998372i \(0.518164\pi\)
\(660\) 0 0
\(661\) 10.4641 0.407006 0.203503 0.979074i \(-0.434767\pi\)
0.203503 + 0.979074i \(0.434767\pi\)
\(662\) 22.0526 0.857097
\(663\) 0 0
\(664\) −143.138 −5.55485
\(665\) 0 0
\(666\) 0 0
\(667\) 19.8564 0.768843
\(668\) 96.4974 3.73360
\(669\) 0 0
\(670\) 40.0526 1.54737
\(671\) −2.92820 −0.113042
\(672\) 0 0
\(673\) −27.3397 −1.05387 −0.526935 0.849906i \(-0.676659\pi\)
−0.526935 + 0.849906i \(0.676659\pi\)
\(674\) 49.1244 1.89220
\(675\) 0 0
\(676\) −42.9282 −1.65108
\(677\) 33.1244 1.27307 0.636536 0.771247i \(-0.280367\pi\)
0.636536 + 0.771247i \(0.280367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 30.9282 1.18604
\(681\) 0 0
\(682\) 0.928203 0.0355427
\(683\) 28.0526 1.07340 0.536701 0.843773i \(-0.319670\pi\)
0.536701 + 0.843773i \(0.319670\pi\)
\(684\) 0 0
\(685\) 2.19615 0.0839107
\(686\) 0 0
\(687\) 0 0
\(688\) 47.7128 1.81903
\(689\) −28.1051 −1.07072
\(690\) 0 0
\(691\) 8.85641 0.336914 0.168457 0.985709i \(-0.446122\pi\)
0.168457 + 0.985709i \(0.446122\pi\)
\(692\) −79.4256 −3.01931
\(693\) 0 0
\(694\) 57.5692 2.18530
\(695\) −5.92820 −0.224870
\(696\) 0 0
\(697\) −2.39230 −0.0906150
\(698\) 60.1051 2.27501
\(699\) 0 0
\(700\) 0 0
\(701\) 8.58846 0.324382 0.162191 0.986759i \(-0.448144\pi\)
0.162191 + 0.986759i \(0.448144\pi\)
\(702\) 0 0
\(703\) −14.2679 −0.538126
\(704\) −21.8564 −0.823744
\(705\) 0 0
\(706\) 8.53590 0.321253
\(707\) 0 0
\(708\) 0 0
\(709\) −1.07180 −0.0402522 −0.0201261 0.999797i \(-0.506407\pi\)
−0.0201261 + 0.999797i \(0.506407\pi\)
\(710\) 16.9282 0.635304
\(711\) 0 0
\(712\) −143.138 −5.36434
\(713\) −2.19615 −0.0822466
\(714\) 0 0
\(715\) 1.66025 0.0620900
\(716\) 54.6410 2.04203
\(717\) 0 0
\(718\) 3.46410 0.129279
\(719\) 20.5359 0.765860 0.382930 0.923777i \(-0.374915\pi\)
0.382930 + 0.923777i \(0.374915\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.53590 0.0943764
\(723\) 0 0
\(724\) −132.890 −4.93881
\(725\) 4.19615 0.155841
\(726\) 0 0
\(727\) −13.3397 −0.494744 −0.247372 0.968921i \(-0.579567\pi\)
−0.247372 + 0.968921i \(0.579567\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −34.5885 −1.28018
\(731\) −10.4449 −0.386317
\(732\) 0 0
\(733\) −1.33975 −0.0494846 −0.0247423 0.999694i \(-0.507877\pi\)
−0.0247423 + 0.999694i \(0.507877\pi\)
\(734\) 30.5885 1.12904
\(735\) 0 0
\(736\) 103.426 3.81232
\(737\) 10.7321 0.395320
\(738\) 0 0
\(739\) 27.7846 1.02207 0.511037 0.859559i \(-0.329262\pi\)
0.511037 + 0.859559i \(0.329262\pi\)
\(740\) 17.4641 0.641993
\(741\) 0 0
\(742\) 0 0
\(743\) 15.9090 0.583643 0.291822 0.956473i \(-0.405739\pi\)
0.291822 + 0.956473i \(0.405739\pi\)
\(744\) 0 0
\(745\) 5.85641 0.214562
\(746\) 72.4449 2.65239
\(747\) 0 0
\(748\) 13.0718 0.477952
\(749\) 0 0
\(750\) 0 0
\(751\) 18.0718 0.659449 0.329725 0.944077i \(-0.393044\pi\)
0.329725 + 0.944077i \(0.393044\pi\)
\(752\) −29.8564 −1.08875
\(753\) 0 0
\(754\) 26.0000 0.946864
\(755\) 8.92820 0.324931
\(756\) 0 0
\(757\) −27.8564 −1.01246 −0.506229 0.862399i \(-0.668961\pi\)
−0.506229 + 0.862399i \(0.668961\pi\)
\(758\) 17.2679 0.627200
\(759\) 0 0
\(760\) −42.2487 −1.53252
\(761\) −46.7321 −1.69404 −0.847018 0.531565i \(-0.821604\pi\)
−0.847018 + 0.531565i \(0.821604\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 48.7846 1.76497
\(765\) 0 0
\(766\) 63.7128 2.30204
\(767\) 0.444864 0.0160631
\(768\) 0 0
\(769\) 52.3205 1.88673 0.943363 0.331763i \(-0.107643\pi\)
0.943363 + 0.331763i \(0.107643\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.53590 0.235232
\(773\) −43.5167 −1.56519 −0.782593 0.622534i \(-0.786103\pi\)
−0.782593 + 0.622534i \(0.786103\pi\)
\(774\) 0 0
\(775\) −0.464102 −0.0166710
\(776\) 141.282 5.07173
\(777\) 0 0
\(778\) −14.7846 −0.530054
\(779\) 3.26795 0.117086
\(780\) 0 0
\(781\) 4.53590 0.162307
\(782\) −42.2487 −1.51081
\(783\) 0 0
\(784\) 0 0
\(785\) −6.39230 −0.228151
\(786\) 0 0
\(787\) 13.4641 0.479943 0.239972 0.970780i \(-0.422862\pi\)
0.239972 + 0.970780i \(0.422862\pi\)
\(788\) 1.85641 0.0661317
\(789\) 0 0
\(790\) 20.1962 0.718547
\(791\) 0 0
\(792\) 0 0
\(793\) 9.07180 0.322149
\(794\) −85.2295 −3.02468
\(795\) 0 0
\(796\) 120.210 4.26074
\(797\) 3.94744 0.139826 0.0699128 0.997553i \(-0.477728\pi\)
0.0699128 + 0.997553i \(0.477728\pi\)
\(798\) 0 0
\(799\) 6.53590 0.231223
\(800\) 21.8564 0.772741
\(801\) 0 0
\(802\) 44.7846 1.58140
\(803\) −9.26795 −0.327059
\(804\) 0 0
\(805\) 0 0
\(806\) −2.87564 −0.101290
\(807\) 0 0
\(808\) 68.7846 2.41983
\(809\) −25.7128 −0.904014 −0.452007 0.892014i \(-0.649292\pi\)
−0.452007 + 0.892014i \(0.649292\pi\)
\(810\) 0 0
\(811\) −3.46410 −0.121641 −0.0608205 0.998149i \(-0.519372\pi\)
−0.0608205 + 0.998149i \(0.519372\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.39230 0.224050
\(815\) −21.8564 −0.765597
\(816\) 0 0
\(817\) 14.2679 0.499172
\(818\) 8.58846 0.300288
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 25.5167 0.890538 0.445269 0.895397i \(-0.353108\pi\)
0.445269 + 0.895397i \(0.353108\pi\)
\(822\) 0 0
\(823\) 39.1769 1.36562 0.682811 0.730595i \(-0.260757\pi\)
0.682811 + 0.730595i \(0.260757\pi\)
\(824\) −87.0333 −3.03195
\(825\) 0 0
\(826\) 0 0
\(827\) 3.75129 0.130445 0.0652225 0.997871i \(-0.479224\pi\)
0.0652225 + 0.997871i \(0.479224\pi\)
\(828\) 0 0
\(829\) −4.60770 −0.160032 −0.0800159 0.996794i \(-0.525497\pi\)
−0.0800159 + 0.996794i \(0.525497\pi\)
\(830\) 41.3205 1.43426
\(831\) 0 0
\(832\) 67.7128 2.34752
\(833\) 0 0
\(834\) 0 0
\(835\) −17.6603 −0.611158
\(836\) −17.8564 −0.617577
\(837\) 0 0
\(838\) 96.8897 3.34700
\(839\) −18.4449 −0.636787 −0.318394 0.947959i \(-0.603143\pi\)
−0.318394 + 0.947959i \(0.603143\pi\)
\(840\) 0 0
\(841\) −11.3923 −0.392838
\(842\) 0.196152 0.00675986
\(843\) 0 0
\(844\) 38.6410 1.33008
\(845\) 7.85641 0.270269
\(846\) 0 0
\(847\) 0 0
\(848\) −184.995 −6.35275
\(849\) 0 0
\(850\) −8.92820 −0.306235
\(851\) −15.1244 −0.518456
\(852\) 0 0
\(853\) −31.9808 −1.09500 −0.547500 0.836806i \(-0.684420\pi\)
−0.547500 + 0.836806i \(0.684420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.7846 −0.710403
\(857\) −29.1244 −0.994869 −0.497435 0.867502i \(-0.665725\pi\)
−0.497435 + 0.867502i \(0.665725\pi\)
\(858\) 0 0
\(859\) 7.46410 0.254672 0.127336 0.991860i \(-0.459357\pi\)
0.127336 + 0.991860i \(0.459357\pi\)
\(860\) −17.4641 −0.595521
\(861\) 0 0
\(862\) −47.3205 −1.61174
\(863\) −14.3923 −0.489920 −0.244960 0.969533i \(-0.578775\pi\)
−0.244960 + 0.969533i \(0.578775\pi\)
\(864\) 0 0
\(865\) 14.5359 0.494235
\(866\) 41.5167 1.41079
\(867\) 0 0
\(868\) 0 0
\(869\) 5.41154 0.183574
\(870\) 0 0
\(871\) −33.2487 −1.12659
\(872\) 104.105 3.52544
\(873\) 0 0
\(874\) 57.7128 1.95217
\(875\) 0 0
\(876\) 0 0
\(877\) −4.14359 −0.139919 −0.0699596 0.997550i \(-0.522287\pi\)
−0.0699596 + 0.997550i \(0.522287\pi\)
\(878\) 1.46410 0.0494110
\(879\) 0 0
\(880\) 10.9282 0.368390
\(881\) −9.85641 −0.332071 −0.166035 0.986120i \(-0.553097\pi\)
−0.166035 + 0.986120i \(0.553097\pi\)
\(882\) 0 0
\(883\) 53.5885 1.80340 0.901698 0.432367i \(-0.142322\pi\)
0.901698 + 0.432367i \(0.142322\pi\)
\(884\) −40.4974 −1.36208
\(885\) 0 0
\(886\) −25.8564 −0.868663
\(887\) 25.2679 0.848415 0.424207 0.905565i \(-0.360553\pi\)
0.424207 + 0.905565i \(0.360553\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 41.3205 1.38507
\(891\) 0 0
\(892\) −111.426 −3.73081
\(893\) −8.92820 −0.298771
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) 0 0
\(897\) 0 0
\(898\) 97.9615 3.26902
\(899\) −1.94744 −0.0649508
\(900\) 0 0
\(901\) 40.4974 1.34916
\(902\) −1.46410 −0.0487493
\(903\) 0 0
\(904\) −84.4974 −2.81034
\(905\) 24.3205 0.808441
\(906\) 0 0
\(907\) 33.5885 1.11529 0.557643 0.830081i \(-0.311706\pi\)
0.557643 + 0.830081i \(0.311706\pi\)
\(908\) 9.07180 0.301058
\(909\) 0 0
\(910\) 0 0
\(911\) 14.7321 0.488095 0.244047 0.969763i \(-0.421525\pi\)
0.244047 + 0.969763i \(0.421525\pi\)
\(912\) 0 0
\(913\) 11.0718 0.366423
\(914\) −45.5167 −1.50556
\(915\) 0 0
\(916\) −16.3923 −0.541617
\(917\) 0 0
\(918\) 0 0
\(919\) −30.8564 −1.01786 −0.508929 0.860808i \(-0.669959\pi\)
−0.508929 + 0.860808i \(0.669959\pi\)
\(920\) −44.7846 −1.47650
\(921\) 0 0
\(922\) −46.3923 −1.52785
\(923\) −14.0526 −0.462546
\(924\) 0 0
\(925\) −3.19615 −0.105089
\(926\) 70.3013 2.31024
\(927\) 0 0
\(928\) 91.7128 3.01062
\(929\) 52.4449 1.72066 0.860330 0.509737i \(-0.170257\pi\)
0.860330 + 0.509737i \(0.170257\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −94.6410 −3.10007
\(933\) 0 0
\(934\) 0.392305 0.0128366
\(935\) −2.39230 −0.0782367
\(936\) 0 0
\(937\) 31.7321 1.03664 0.518320 0.855186i \(-0.326557\pi\)
0.518320 + 0.855186i \(0.326557\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10.9282 0.356439
\(941\) 30.0526 0.979685 0.489843 0.871811i \(-0.337054\pi\)
0.489843 + 0.871811i \(0.337054\pi\)
\(942\) 0 0
\(943\) 3.46410 0.112807
\(944\) 2.92820 0.0953049
\(945\) 0 0
\(946\) −6.39230 −0.207832
\(947\) 5.66025 0.183934 0.0919668 0.995762i \(-0.470685\pi\)
0.0919668 + 0.995762i \(0.470685\pi\)
\(948\) 0 0
\(949\) 28.7128 0.932057
\(950\) 12.1962 0.395695
\(951\) 0 0
\(952\) 0 0
\(953\) 36.1051 1.16956 0.584780 0.811192i \(-0.301181\pi\)
0.584780 + 0.811192i \(0.301181\pi\)
\(954\) 0 0
\(955\) −8.92820 −0.288910
\(956\) −38.6410 −1.24974
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −30.7846 −0.993052
\(962\) −19.8038 −0.638502
\(963\) 0 0
\(964\) 73.5692 2.36951
\(965\) −1.19615 −0.0385055
\(966\) 0 0
\(967\) −10.1244 −0.325577 −0.162789 0.986661i \(-0.552049\pi\)
−0.162789 + 0.986661i \(0.552049\pi\)
\(968\) −99.0333 −3.18305
\(969\) 0 0
\(970\) −40.7846 −1.30951
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1.12436 −0.0360267
\(975\) 0 0
\(976\) 59.7128 1.91136
\(977\) 16.5885 0.530712 0.265356 0.964151i \(-0.414511\pi\)
0.265356 + 0.964151i \(0.414511\pi\)
\(978\) 0 0
\(979\) 11.0718 0.353856
\(980\) 0 0
\(981\) 0 0
\(982\) 104.497 3.33465
\(983\) −9.80385 −0.312694 −0.156347 0.987702i \(-0.549972\pi\)
−0.156347 + 0.987702i \(0.549972\pi\)
\(984\) 0 0
\(985\) −0.339746 −0.0108252
\(986\) −37.4641 −1.19310
\(987\) 0 0
\(988\) 55.3205 1.75998
\(989\) 15.1244 0.480927
\(990\) 0 0
\(991\) −21.1051 −0.670426 −0.335213 0.942142i \(-0.608808\pi\)
−0.335213 + 0.942142i \(0.608808\pi\)
\(992\) −10.1436 −0.322059
\(993\) 0 0
\(994\) 0 0
\(995\) −22.0000 −0.697447
\(996\) 0 0
\(997\) −55.9808 −1.77293 −0.886464 0.462797i \(-0.846846\pi\)
−0.886464 + 0.462797i \(0.846846\pi\)
\(998\) −36.9808 −1.17061
\(999\) 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.z.1.2 2
3.2 odd 2 735.2.a.g.1.1 2
7.2 even 3 315.2.j.c.46.1 4
7.4 even 3 315.2.j.c.226.1 4
7.6 odd 2 2205.2.a.ba.1.2 2
15.14 odd 2 3675.2.a.bg.1.2 2
21.2 odd 6 105.2.i.d.46.2 yes 4
21.5 even 6 735.2.i.l.361.2 4
21.11 odd 6 105.2.i.d.16.2 4
21.17 even 6 735.2.i.l.226.2 4
21.20 even 2 735.2.a.h.1.1 2
84.11 even 6 1680.2.bg.o.961.1 4
84.23 even 6 1680.2.bg.o.1201.1 4
105.2 even 12 525.2.r.a.424.1 4
105.23 even 12 525.2.r.f.424.2 4
105.32 even 12 525.2.r.f.499.2 4
105.44 odd 6 525.2.i.f.151.1 4
105.53 even 12 525.2.r.a.499.1 4
105.74 odd 6 525.2.i.f.226.1 4
105.104 even 2 3675.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.d.16.2 4 21.11 odd 6
105.2.i.d.46.2 yes 4 21.2 odd 6
315.2.j.c.46.1 4 7.2 even 3
315.2.j.c.226.1 4 7.4 even 3
525.2.i.f.151.1 4 105.44 odd 6
525.2.i.f.226.1 4 105.74 odd 6
525.2.r.a.424.1 4 105.2 even 12
525.2.r.a.499.1 4 105.53 even 12
525.2.r.f.424.2 4 105.23 even 12
525.2.r.f.499.2 4 105.32 even 12
735.2.a.g.1.1 2 3.2 odd 2
735.2.a.h.1.1 2 21.20 even 2
735.2.i.l.226.2 4 21.17 even 6
735.2.i.l.361.2 4 21.5 even 6
1680.2.bg.o.961.1 4 84.11 even 6
1680.2.bg.o.1201.1 4 84.23 even 6
2205.2.a.z.1.2 2 1.1 even 1 trivial
2205.2.a.ba.1.2 2 7.6 odd 2
3675.2.a.be.1.2 2 105.104 even 2
3675.2.a.bg.1.2 2 15.14 odd 2