Properties

Label 2205.2.a.ba.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 \(-0.601840\pi\)
−0.314508 + 0.949255i \(0.601840\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 14.9282 3.73205
\(17\) 3.26795 0.792594 0.396297 0.918122i \(-0.370295\pi\)
0.396297 + 0.918122i \(0.370295\pi\)
\(18\) 0 0
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) 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 \(-0.486730\pi\)
0.0416776 + 0.999131i \(0.486730\pi\)
\(32\) 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 \(-0.518206\pi\)
−0.0571636 + 0.998365i \(0.518206\pi\)
\(42\) 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 \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 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 \(-0.504064\pi\)
−0.0127684 + 0.999918i \(0.504064\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 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 \(-0.765594\pi\)
−0.740885 + 0.671632i \(0.765594\pi\)
\(74\) −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 \(-0.188308\pi\)
0.830057 + 0.557679i \(0.188308\pi\)
\(84\) 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 \(-0.203988\pi\)
0.801589 + 0.597875i \(0.203988\pi\)
\(90\) 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 \(-0.773757\pi\)
−0.757865 + 0.652412i \(0.773757\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.46410 0.546410
\(101\) −7.26795 −0.723188 −0.361594 0.932336i \(-0.617767\pi\)
−0.361594 + 0.932336i \(0.617767\pi\)
\(102\) 0 0
\(103\) 9.19615 0.906124 0.453062 0.891479i \(-0.350332\pi\)
0.453062 + 0.891479i \(0.350332\pi\)
\(104\) −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 \(-0.263905\pi\)
0.675552 + 0.737312i \(0.263905\pi\)
\(132\) 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 \(-0.580895\pi\)
−0.251412 + 0.967880i \(0.580895\pi\)
\(140\) 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 \(-0.582102\pi\)
−0.255081 + 0.966920i \(0.582102\pi\)
\(158\) −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 \(-0.739454\pi\)
−0.683296 + 0.730142i \(0.739454\pi\)
\(168\) 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 \(-0.313646\pi\)
0.552572 + 0.833465i \(0.313646\pi\)
\(174\) 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 \(-0.140718\pi\)
0.903865 + 0.427819i \(0.140718\pi\)
\(182\) 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 \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 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 \(-0.260769\pi\)
0.682785 + 0.730619i \(0.260769\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −24.3923 −1.62255
\(227\) −1.66025 −0.110195 −0.0550975 0.998481i \(-0.517547\pi\)
−0.0550975 + 0.998481i \(0.517547\pi\)
\(228\) 0 0
\(229\) 3.00000 0.198246 0.0991228 0.995075i \(-0.468396\pi\)
0.0991228 + 0.995075i \(0.468396\pi\)
\(230\) 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 \(-0.642774\pi\)
−0.433650 + 0.901082i \(0.642774\pi\)
\(242\) −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 \(-0.782757\pi\)
−0.776005 + 0.630727i \(0.782757\pi\)
\(252\) 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 \(-0.556490\pi\)
−0.176538 + 0.984294i \(0.556490\pi\)
\(258\) 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 \(-0.624821\pi\)
−0.382164 + 0.924095i \(0.624821\pi\)
\(270\) 0 0
\(271\) −3.07180 −0.186598 −0.0932992 0.995638i \(-0.529741\pi\)
−0.0932992 + 0.995638i \(0.529741\pi\)
\(272\) 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 \(-0.245505\pi\)
0.717022 + 0.697050i \(0.245505\pi\)
\(284\) −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 \(-0.313522\pi\)
0.552899 + 0.833248i \(0.313522\pi\)
\(294\) 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 \(-0.130829\pi\)
0.916717 + 0.399537i \(0.130829\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.26795 0.0720147
\(311\) 9.12436 0.517395 0.258697 0.965958i \(-0.416707\pi\)
0.258697 + 0.965958i \(0.416707\pi\)
\(312\) 0 0
\(313\) 12.6603 0.715600 0.357800 0.933798i \(-0.383527\pi\)
0.357800 + 0.933798i \(0.383527\pi\)
\(314\) −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 \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −16.0000 −0.852803
\(353\) −3.12436 −0.166293 −0.0831463 0.996537i \(-0.526497\pi\)
−0.0831463 + 0.996537i \(0.526497\pi\)
\(354\) 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 \(-0.594393\pi\)
−0.292217 + 0.956352i \(0.594393\pi\)
\(368\) 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 \(-0.703169\pi\)
−0.595811 + 0.803125i \(0.703169\pi\)
\(384\) 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 \(-0.213768\pi\)
0.782845 + 0.622217i \(0.213768\pi\)
\(398\) −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 \(-0.524764\pi\)
−0.0777203 + 0.996975i \(0.524764\pi\)
\(410\) −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 \(-0.833487\pi\)
−0.866267 + 0.499581i \(0.833487\pi\)
\(420\) 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 \(-0.618979\pi\)
−0.365140 + 0.930953i \(0.618979\pi\)
\(434\) 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 \(-0.504071\pi\)
−0.0127885 + 0.999918i \(0.504071\pi\)
\(440\) −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 \(-0.370593\pi\)
0.395436 + 0.918493i \(0.370593\pi\)
\(462\) 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 \(-0.501058\pi\)
−0.00332236 + 0.999994i \(0.501058\pi\)
\(468\) 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 \(-0.435682\pi\)
0.200690 + 0.979655i \(0.435682\pi\)
\(480\) 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 \(-0.396028\pi\)
0.320861 + 0.947126i \(0.396028\pi\)
\(504\) 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 \(-0.467948\pi\)
0.100525 + 0.994935i \(0.467948\pi\)
\(510\) 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 \(-0.538191\pi\)
−0.119693 + 0.992811i \(0.538191\pi\)
\(522\) 0 0
\(523\) 27.7321 1.21264 0.606319 0.795222i \(-0.292645\pi\)
0.606319 + 0.795222i \(0.292645\pi\)
\(524\) 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 \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 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 \(-0.703471\pi\)
−0.596571 + 0.802560i \(0.703471\pi\)
\(578\) −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 \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(588\) 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 \(-0.305763\pi\)
0.573042 + 0.819526i \(0.305763\pi\)
\(594\) 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 \(-0.499534\pi\)
0.00146433 + 0.999999i \(0.499534\pi\)
\(602\) 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 \(-0.520661\pi\)
−0.0648639 + 0.997894i \(0.520661\pi\)
\(608\) −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 \(-0.293435\pi\)
0.604344 + 0.796724i \(0.293435\pi\)
\(620\) 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 \(-0.528386\pi\)
−0.0890599 + 0.996026i \(0.528386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −39.8564 −1.56813
\(647\) −27.9090 −1.09721 −0.548607 0.836080i \(-0.684842\pi\)
−0.548607 + 0.836080i \(0.684842\pi\)
\(648\) 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 \(-0.565233\pi\)
−0.203503 + 0.979074i \(0.565233\pi\)
\(662\) 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 \(-0.719633\pi\)
−0.636536 + 0.771247i \(0.719633\pi\)
\(678\) 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 \(-0.553878\pi\)
−0.168457 + 0.985709i \(0.553878\pi\)
\(692\) 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 \(-0.625085\pi\)
−0.382930 + 0.923777i \(0.625085\pi\)
\(720\) 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 \(-0.420433\pi\)
0.247372 + 0.968921i \(0.420433\pi\)
\(728\) 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 \(-0.492123\pi\)
0.0247423 + 0.999694i \(0.492123\pi\)
\(734\) −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 \(-0.178396\pi\)
0.847018 + 0.531565i \(0.178396\pi\)
\(762\) 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 \(-0.892357\pi\)
−0.943363 + 0.331763i \(0.892357\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.53590 0.235232
\(773\) 43.5167 1.56519 0.782593 0.622534i \(-0.213897\pi\)
0.782593 + 0.622534i \(0.213897\pi\)
\(774\) 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 \(-0.577138\pi\)
−0.239972 + 0.970780i \(0.577138\pi\)
\(788\) 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 \(-0.522272\pi\)
−0.0699128 + 0.997553i \(0.522272\pi\)
\(798\) 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 \(-0.480628\pi\)
0.0608205 + 0.998149i \(0.480628\pi\)
\(812\) 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 \(-0.474503\pi\)
0.0800159 + 0.996794i \(0.474503\pi\)
\(830\) 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 \(-0.396857\pi\)
0.318394 + 0.947959i \(0.396857\pi\)
\(840\) 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 \(-0.315580\pi\)
0.547500 + 0.836806i \(0.315580\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.7846 −0.710403
\(857\) 29.1244 0.994869 0.497435 0.867502i \(-0.334275\pi\)
0.497435 + 0.867502i \(0.334275\pi\)
\(858\) 0 0
\(859\) −7.46410 −0.254672 −0.127336 0.991860i \(-0.540643\pi\)
−0.127336 + 0.991860i \(0.540643\pi\)
\(860\) 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 \(-0.446903\pi\)
0.166035 + 0.986120i \(0.446903\pi\)
\(882\) 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 \(-0.639447\pi\)
−0.424207 + 0.905565i \(0.639447\pi\)
\(888\) 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 \(-0.829743\pi\)
−0.860330 + 0.509737i \(0.829743\pi\)
\(930\) 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 \(-0.673443\pi\)
−0.518320 + 0.855186i \(0.673443\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10.9282 0.356439
\(941\) −30.0526 −0.979685 −0.489843 0.871811i \(-0.662946\pi\)
−0.489843 + 0.871811i \(0.662946\pi\)
\(942\) 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 \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 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 \(-0.450028\pi\)
0.156347 + 0.987702i \(0.450028\pi\)
\(984\) 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 \(-0.153154\pi\)
0.886464 + 0.462797i \(0.153154\pi\)
\(998\) −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.ba.1.2 2
3.2 odd 2 735.2.a.h.1.1 2
7.3 odd 6 315.2.j.c.226.1 4
7.5 odd 6 315.2.j.c.46.1 4
7.6 odd 2 2205.2.a.z.1.2 2
15.14 odd 2 3675.2.a.be.1.2 2
21.2 odd 6 735.2.i.l.361.2 4
21.5 even 6 105.2.i.d.46.2 yes 4
21.11 odd 6 735.2.i.l.226.2 4
21.17 even 6 105.2.i.d.16.2 4
21.20 even 2 735.2.a.g.1.1 2
84.47 odd 6 1680.2.bg.o.1201.1 4
84.59 odd 6 1680.2.bg.o.961.1 4
105.17 odd 12 525.2.r.f.499.2 4
105.38 odd 12 525.2.r.a.499.1 4
105.47 odd 12 525.2.r.a.424.1 4
105.59 even 6 525.2.i.f.226.1 4
105.68 odd 12 525.2.r.f.424.2 4
105.89 even 6 525.2.i.f.151.1 4
105.104 even 2 3675.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.d.16.2 4 21.17 even 6
105.2.i.d.46.2 yes 4 21.5 even 6
315.2.j.c.46.1 4 7.5 odd 6
315.2.j.c.226.1 4 7.3 odd 6
525.2.i.f.151.1 4 105.89 even 6
525.2.i.f.226.1 4 105.59 even 6
525.2.r.a.424.1 4 105.47 odd 12
525.2.r.a.499.1 4 105.38 odd 12
525.2.r.f.424.2 4 105.68 odd 12
525.2.r.f.499.2 4 105.17 odd 12
735.2.a.g.1.1 2 21.20 even 2
735.2.a.h.1.1 2 3.2 odd 2
735.2.i.l.226.2 4 21.11 odd 6
735.2.i.l.361.2 4 21.2 odd 6
1680.2.bg.o.961.1 4 84.59 odd 6
1680.2.bg.o.1201.1 4 84.47 odd 6
2205.2.a.z.1.2 2 7.6 odd 2
2205.2.a.ba.1.2 2 1.1 even 1 trivial
3675.2.a.be.1.2 2 15.14 odd 2
3675.2.a.bg.1.2 2 105.104 even 2