Properties

Label 1380.2.a.j.1.3
Level $1380$
Weight $2$
Character 1380.1
Self dual yes
Analytic conductor $11.019$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 1380.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +4.20905 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +4.20905 q^{7} +1.00000 q^{9} +2.75353 q^{11} -0.753525 q^{13} -1.00000 q^{15} -4.96257 q^{17} +4.75353 q^{19} +4.20905 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.96257 q^{29} -0.209050 q^{31} +2.75353 q^{33} -4.20905 q^{35} -5.71610 q^{37} -0.753525 q^{39} -9.38067 q^{41} +12.4181 q^{43} -1.00000 q^{45} +7.17162 q^{47} +10.7161 q^{49} -4.96257 q^{51} -9.38067 q^{53} -2.75353 q^{55} +4.75353 q^{57} -4.96257 q^{59} -5.17162 q^{61} +4.20905 q^{63} +0.753525 q^{65} -0.209050 q^{67} +1.00000 q^{69} +9.38067 q^{71} +10.2606 q^{73} +1.00000 q^{75} +11.5897 q^{77} -2.41810 q^{79} +1.00000 q^{81} +5.45552 q^{83} +4.96257 q^{85} +4.96257 q^{87} +4.91105 q^{89} -3.17162 q^{91} -0.209050 q^{93} -4.75353 q^{95} +10.3432 q^{97} +2.75353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{15} + 10 q^{19} + 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} + 8 q^{41} + 16 q^{43} - 3 q^{45} - 4 q^{47} + 13 q^{49} + 8 q^{53} - 4 q^{55} + 10 q^{57} + 10 q^{61} + 2 q^{63} - 2 q^{65} + 10 q^{67} + 3 q^{69} - 8 q^{71} + 18 q^{73} + 3 q^{75} - 12 q^{77} + 14 q^{79} + 3 q^{81} + 10 q^{83} + 2 q^{89} + 16 q^{91} + 10 q^{93} - 10 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.20905 1.59087 0.795436 0.606038i \(-0.207242\pi\)
0.795436 + 0.606038i \(0.207242\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.75353 0.830219 0.415110 0.909771i \(-0.363743\pi\)
0.415110 + 0.909771i \(0.363743\pi\)
\(12\) 0 0
\(13\) −0.753525 −0.208990 −0.104495 0.994525i \(-0.533323\pi\)
−0.104495 + 0.994525i \(0.533323\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.96257 −1.20360 −0.601801 0.798646i \(-0.705550\pi\)
−0.601801 + 0.798646i \(0.705550\pi\)
\(18\) 0 0
\(19\) 4.75353 1.09053 0.545267 0.838263i \(-0.316428\pi\)
0.545267 + 0.838263i \(0.316428\pi\)
\(20\) 0 0
\(21\) 4.20905 0.918490
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.96257 0.921527 0.460764 0.887523i \(-0.347576\pi\)
0.460764 + 0.887523i \(0.347576\pi\)
\(30\) 0 0
\(31\) −0.209050 −0.0375464 −0.0187732 0.999824i \(-0.505976\pi\)
−0.0187732 + 0.999824i \(0.505976\pi\)
\(32\) 0 0
\(33\) 2.75353 0.479327
\(34\) 0 0
\(35\) −4.20905 −0.711459
\(36\) 0 0
\(37\) −5.71610 −0.939721 −0.469861 0.882741i \(-0.655696\pi\)
−0.469861 + 0.882741i \(0.655696\pi\)
\(38\) 0 0
\(39\) −0.753525 −0.120661
\(40\) 0 0
\(41\) −9.38067 −1.46502 −0.732508 0.680759i \(-0.761650\pi\)
−0.732508 + 0.680759i \(0.761650\pi\)
\(42\) 0 0
\(43\) 12.4181 1.89374 0.946871 0.321613i \(-0.104225\pi\)
0.946871 + 0.321613i \(0.104225\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 7.17162 1.04609 0.523044 0.852305i \(-0.324796\pi\)
0.523044 + 0.852305i \(0.324796\pi\)
\(48\) 0 0
\(49\) 10.7161 1.53087
\(50\) 0 0
\(51\) −4.96257 −0.694899
\(52\) 0 0
\(53\) −9.38067 −1.28853 −0.644267 0.764800i \(-0.722838\pi\)
−0.644267 + 0.764800i \(0.722838\pi\)
\(54\) 0 0
\(55\) −2.75353 −0.371285
\(56\) 0 0
\(57\) 4.75353 0.629620
\(58\) 0 0
\(59\) −4.96257 −0.646072 −0.323036 0.946387i \(-0.604704\pi\)
−0.323036 + 0.946387i \(0.604704\pi\)
\(60\) 0 0
\(61\) −5.17162 −0.662159 −0.331079 0.943603i \(-0.607413\pi\)
−0.331079 + 0.943603i \(0.607413\pi\)
\(62\) 0 0
\(63\) 4.20905 0.530290
\(64\) 0 0
\(65\) 0.753525 0.0934633
\(66\) 0 0
\(67\) −0.209050 −0.0255395 −0.0127697 0.999918i \(-0.504065\pi\)
−0.0127697 + 0.999918i \(0.504065\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 9.38067 1.11328 0.556641 0.830753i \(-0.312090\pi\)
0.556641 + 0.830753i \(0.312090\pi\)
\(72\) 0 0
\(73\) 10.2606 1.20091 0.600455 0.799659i \(-0.294986\pi\)
0.600455 + 0.799659i \(0.294986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 11.5897 1.32077
\(78\) 0 0
\(79\) −2.41810 −0.272057 −0.136029 0.990705i \(-0.543434\pi\)
−0.136029 + 0.990705i \(0.543434\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.45552 0.598822 0.299411 0.954124i \(-0.403210\pi\)
0.299411 + 0.954124i \(0.403210\pi\)
\(84\) 0 0
\(85\) 4.96257 0.538267
\(86\) 0 0
\(87\) 4.96257 0.532044
\(88\) 0 0
\(89\) 4.91105 0.520570 0.260285 0.965532i \(-0.416183\pi\)
0.260285 + 0.965532i \(0.416183\pi\)
\(90\) 0 0
\(91\) −3.17162 −0.332477
\(92\) 0 0
\(93\) −0.209050 −0.0216775
\(94\) 0 0
\(95\) −4.75353 −0.487701
\(96\) 0 0
\(97\) 10.3432 1.05020 0.525099 0.851041i \(-0.324028\pi\)
0.525099 + 0.851041i \(0.324028\pi\)
\(98\) 0 0
\(99\) 2.75353 0.276740
\(100\) 0 0
\(101\) 14.8877 1.48138 0.740692 0.671845i \(-0.234498\pi\)
0.740692 + 0.671845i \(0.234498\pi\)
\(102\) 0 0
\(103\) 17.9251 1.76622 0.883109 0.469168i \(-0.155446\pi\)
0.883109 + 0.469168i \(0.155446\pi\)
\(104\) 0 0
\(105\) −4.20905 −0.410761
\(106\) 0 0
\(107\) −16.4696 −1.59218 −0.796089 0.605179i \(-0.793102\pi\)
−0.796089 + 0.605179i \(0.793102\pi\)
\(108\) 0 0
\(109\) −6.26058 −0.599654 −0.299827 0.953994i \(-0.596929\pi\)
−0.299827 + 0.953994i \(0.596929\pi\)
\(110\) 0 0
\(111\) −5.71610 −0.542548
\(112\) 0 0
\(113\) −9.38067 −0.882460 −0.441230 0.897394i \(-0.645458\pi\)
−0.441230 + 0.897394i \(0.645458\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −0.753525 −0.0696634
\(118\) 0 0
\(119\) −20.8877 −1.91477
\(120\) 0 0
\(121\) −3.41810 −0.310736
\(122\) 0 0
\(123\) −9.38067 −0.845827
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −22.0827 −1.95952 −0.979760 0.200175i \(-0.935849\pi\)
−0.979760 + 0.200175i \(0.935849\pi\)
\(128\) 0 0
\(129\) 12.4181 1.09335
\(130\) 0 0
\(131\) −1.58190 −0.138211 −0.0691056 0.997609i \(-0.522015\pi\)
−0.0691056 + 0.997609i \(0.522015\pi\)
\(132\) 0 0
\(133\) 20.0078 1.73490
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −4.91105 −0.419579 −0.209790 0.977747i \(-0.567278\pi\)
−0.209790 + 0.977747i \(0.567278\pi\)
\(138\) 0 0
\(139\) 14.1342 1.19885 0.599424 0.800432i \(-0.295397\pi\)
0.599424 + 0.800432i \(0.295397\pi\)
\(140\) 0 0
\(141\) 7.17162 0.603960
\(142\) 0 0
\(143\) −2.07485 −0.173508
\(144\) 0 0
\(145\) −4.96257 −0.412119
\(146\) 0 0
\(147\) 10.7161 0.883849
\(148\) 0 0
\(149\) 3.24647 0.265962 0.132981 0.991119i \(-0.457545\pi\)
0.132981 + 0.991119i \(0.457545\pi\)
\(150\) 0 0
\(151\) 0.418100 0.0340245 0.0170122 0.999855i \(-0.494585\pi\)
0.0170122 + 0.999855i \(0.494585\pi\)
\(152\) 0 0
\(153\) −4.96257 −0.401200
\(154\) 0 0
\(155\) 0.209050 0.0167913
\(156\) 0 0
\(157\) −21.0452 −1.67959 −0.839797 0.542901i \(-0.817326\pi\)
−0.839797 + 0.542901i \(0.817326\pi\)
\(158\) 0 0
\(159\) −9.38067 −0.743936
\(160\) 0 0
\(161\) 4.20905 0.331720
\(162\) 0 0
\(163\) −7.43220 −0.582135 −0.291067 0.956703i \(-0.594010\pi\)
−0.291067 + 0.956703i \(0.594010\pi\)
\(164\) 0 0
\(165\) −2.75353 −0.214362
\(166\) 0 0
\(167\) −19.1716 −1.48354 −0.741772 0.670652i \(-0.766015\pi\)
−0.741772 + 0.670652i \(0.766015\pi\)
\(168\) 0 0
\(169\) −12.4322 −0.956323
\(170\) 0 0
\(171\) 4.75353 0.363511
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 4.20905 0.318174
\(176\) 0 0
\(177\) −4.96257 −0.373010
\(178\) 0 0
\(179\) 13.8503 1.03522 0.517610 0.855617i \(-0.326822\pi\)
0.517610 + 0.855617i \(0.326822\pi\)
\(180\) 0 0
\(181\) 12.9110 0.959671 0.479835 0.877359i \(-0.340696\pi\)
0.479835 + 0.877359i \(0.340696\pi\)
\(182\) 0 0
\(183\) −5.17162 −0.382297
\(184\) 0 0
\(185\) 5.71610 0.420256
\(186\) 0 0
\(187\) −13.6646 −0.999253
\(188\) 0 0
\(189\) 4.20905 0.306163
\(190\) 0 0
\(191\) −8.15752 −0.590258 −0.295129 0.955457i \(-0.595363\pi\)
−0.295129 + 0.955457i \(0.595363\pi\)
\(192\) 0 0
\(193\) −16.7613 −1.20651 −0.603254 0.797549i \(-0.706130\pi\)
−0.603254 + 0.797549i \(0.706130\pi\)
\(194\) 0 0
\(195\) 0.753525 0.0539611
\(196\) 0 0
\(197\) −10.9110 −0.777380 −0.388690 0.921369i \(-0.627072\pi\)
−0.388690 + 0.921369i \(0.627072\pi\)
\(198\) 0 0
\(199\) −4.49295 −0.318497 −0.159248 0.987239i \(-0.550907\pi\)
−0.159248 + 0.987239i \(0.550907\pi\)
\(200\) 0 0
\(201\) −0.209050 −0.0147452
\(202\) 0 0
\(203\) 20.8877 1.46603
\(204\) 0 0
\(205\) 9.38067 0.655175
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 13.0890 0.905382
\(210\) 0 0
\(211\) 9.71610 0.668884 0.334442 0.942416i \(-0.391452\pi\)
0.334442 + 0.942416i \(0.391452\pi\)
\(212\) 0 0
\(213\) 9.38067 0.642753
\(214\) 0 0
\(215\) −12.4181 −0.846907
\(216\) 0 0
\(217\) −0.879901 −0.0597316
\(218\) 0 0
\(219\) 10.2606 0.693345
\(220\) 0 0
\(221\) 3.73942 0.251541
\(222\) 0 0
\(223\) −19.4322 −1.30128 −0.650638 0.759388i \(-0.725499\pi\)
−0.650638 + 0.759388i \(0.725499\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.985900 −0.0654365 −0.0327182 0.999465i \(-0.510416\pi\)
−0.0327182 + 0.999465i \(0.510416\pi\)
\(228\) 0 0
\(229\) 3.08895 0.204124 0.102062 0.994778i \(-0.467456\pi\)
0.102062 + 0.994778i \(0.467456\pi\)
\(230\) 0 0
\(231\) 11.5897 0.762548
\(232\) 0 0
\(233\) 8.83620 0.578879 0.289439 0.957196i \(-0.406531\pi\)
0.289439 + 0.957196i \(0.406531\pi\)
\(234\) 0 0
\(235\) −7.17162 −0.467825
\(236\) 0 0
\(237\) −2.41810 −0.157072
\(238\) 0 0
\(239\) −21.3807 −1.38300 −0.691500 0.722376i \(-0.743050\pi\)
−0.691500 + 0.722376i \(0.743050\pi\)
\(240\) 0 0
\(241\) 22.2606 1.43393 0.716965 0.697109i \(-0.245531\pi\)
0.716965 + 0.697109i \(0.245531\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −10.7161 −0.684627
\(246\) 0 0
\(247\) −3.58190 −0.227911
\(248\) 0 0
\(249\) 5.45552 0.345730
\(250\) 0 0
\(251\) −6.59600 −0.416336 −0.208168 0.978093i \(-0.566750\pi\)
−0.208168 + 0.978093i \(0.566750\pi\)
\(252\) 0 0
\(253\) 2.75353 0.173113
\(254\) 0 0
\(255\) 4.96257 0.310768
\(256\) 0 0
\(257\) −18.1857 −1.13439 −0.567197 0.823582i \(-0.691972\pi\)
−0.567197 + 0.823582i \(0.691972\pi\)
\(258\) 0 0
\(259\) −24.0593 −1.49498
\(260\) 0 0
\(261\) 4.96257 0.307176
\(262\) 0 0
\(263\) −15.3807 −0.948413 −0.474207 0.880414i \(-0.657265\pi\)
−0.474207 + 0.880414i \(0.657265\pi\)
\(264\) 0 0
\(265\) 9.38067 0.576250
\(266\) 0 0
\(267\) 4.91105 0.300551
\(268\) 0 0
\(269\) 0.544475 0.0331972 0.0165986 0.999862i \(-0.494716\pi\)
0.0165986 + 0.999862i \(0.494716\pi\)
\(270\) 0 0
\(271\) −12.2090 −0.741647 −0.370823 0.928703i \(-0.620925\pi\)
−0.370823 + 0.928703i \(0.620925\pi\)
\(272\) 0 0
\(273\) −3.17162 −0.191955
\(274\) 0 0
\(275\) 2.75353 0.166044
\(276\) 0 0
\(277\) −21.0141 −1.26261 −0.631307 0.775533i \(-0.717481\pi\)
−0.631307 + 0.775533i \(0.717481\pi\)
\(278\) 0 0
\(279\) −0.209050 −0.0125155
\(280\) 0 0
\(281\) −24.1857 −1.44280 −0.721400 0.692519i \(-0.756501\pi\)
−0.721400 + 0.692519i \(0.756501\pi\)
\(282\) 0 0
\(283\) 15.1201 0.898797 0.449398 0.893331i \(-0.351638\pi\)
0.449398 + 0.893331i \(0.351638\pi\)
\(284\) 0 0
\(285\) −4.75353 −0.281575
\(286\) 0 0
\(287\) −39.4837 −2.33065
\(288\) 0 0
\(289\) 7.62715 0.448656
\(290\) 0 0
\(291\) 10.3432 0.606332
\(292\) 0 0
\(293\) 10.4696 0.611642 0.305821 0.952089i \(-0.401069\pi\)
0.305821 + 0.952089i \(0.401069\pi\)
\(294\) 0 0
\(295\) 4.96257 0.288932
\(296\) 0 0
\(297\) 2.75353 0.159776
\(298\) 0 0
\(299\) −0.753525 −0.0435775
\(300\) 0 0
\(301\) 52.2684 3.01270
\(302\) 0 0
\(303\) 14.8877 0.855277
\(304\) 0 0
\(305\) 5.17162 0.296126
\(306\) 0 0
\(307\) −12.1575 −0.693867 −0.346933 0.937890i \(-0.612777\pi\)
−0.346933 + 0.937890i \(0.612777\pi\)
\(308\) 0 0
\(309\) 17.9251 1.01973
\(310\) 0 0
\(311\) −34.4181 −1.95167 −0.975836 0.218506i \(-0.929882\pi\)
−0.975836 + 0.218506i \(0.929882\pi\)
\(312\) 0 0
\(313\) 29.5664 1.67119 0.835596 0.549345i \(-0.185123\pi\)
0.835596 + 0.549345i \(0.185123\pi\)
\(314\) 0 0
\(315\) −4.20905 −0.237153
\(316\) 0 0
\(317\) 1.66457 0.0934918 0.0467459 0.998907i \(-0.485115\pi\)
0.0467459 + 0.998907i \(0.485115\pi\)
\(318\) 0 0
\(319\) 13.6646 0.765069
\(320\) 0 0
\(321\) −16.4696 −0.919245
\(322\) 0 0
\(323\) −23.5897 −1.31257
\(324\) 0 0
\(325\) −0.753525 −0.0417981
\(326\) 0 0
\(327\) −6.26058 −0.346211
\(328\) 0 0
\(329\) 30.1857 1.66419
\(330\) 0 0
\(331\) 3.12010 0.171496 0.0857481 0.996317i \(-0.472672\pi\)
0.0857481 + 0.996317i \(0.472672\pi\)
\(332\) 0 0
\(333\) −5.71610 −0.313240
\(334\) 0 0
\(335\) 0.209050 0.0114216
\(336\) 0 0
\(337\) −11.5819 −0.630906 −0.315453 0.948941i \(-0.602157\pi\)
−0.315453 + 0.948941i \(0.602157\pi\)
\(338\) 0 0
\(339\) −9.38067 −0.509488
\(340\) 0 0
\(341\) −0.575624 −0.0311718
\(342\) 0 0
\(343\) 15.6412 0.844548
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 26.3432 1.41418 0.707090 0.707124i \(-0.250008\pi\)
0.707090 + 0.707124i \(0.250008\pi\)
\(348\) 0 0
\(349\) 9.22315 0.493704 0.246852 0.969053i \(-0.420604\pi\)
0.246852 + 0.969053i \(0.420604\pi\)
\(350\) 0 0
\(351\) −0.753525 −0.0402202
\(352\) 0 0
\(353\) −18.1857 −0.967928 −0.483964 0.875088i \(-0.660804\pi\)
−0.483964 + 0.875088i \(0.660804\pi\)
\(354\) 0 0
\(355\) −9.38067 −0.497875
\(356\) 0 0
\(357\) −20.8877 −1.10550
\(358\) 0 0
\(359\) 7.17162 0.378504 0.189252 0.981929i \(-0.439394\pi\)
0.189252 + 0.981929i \(0.439394\pi\)
\(360\) 0 0
\(361\) 3.59600 0.189263
\(362\) 0 0
\(363\) −3.41810 −0.179404
\(364\) 0 0
\(365\) −10.2606 −0.537063
\(366\) 0 0
\(367\) 17.5664 0.916959 0.458479 0.888705i \(-0.348394\pi\)
0.458479 + 0.888705i \(0.348394\pi\)
\(368\) 0 0
\(369\) −9.38067 −0.488338
\(370\) 0 0
\(371\) −39.4837 −2.04989
\(372\) 0 0
\(373\) −33.6724 −1.74349 −0.871745 0.489959i \(-0.837012\pi\)
−0.871745 + 0.489959i \(0.837012\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −3.73942 −0.192590
\(378\) 0 0
\(379\) 18.4181 0.946074 0.473037 0.881043i \(-0.343158\pi\)
0.473037 + 0.881043i \(0.343158\pi\)
\(380\) 0 0
\(381\) −22.0827 −1.13133
\(382\) 0 0
\(383\) −20.7847 −1.06205 −0.531024 0.847357i \(-0.678192\pi\)
−0.531024 + 0.847357i \(0.678192\pi\)
\(384\) 0 0
\(385\) −11.5897 −0.590667
\(386\) 0 0
\(387\) 12.4181 0.631247
\(388\) 0 0
\(389\) 22.5212 1.14187 0.570934 0.820996i \(-0.306581\pi\)
0.570934 + 0.820996i \(0.306581\pi\)
\(390\) 0 0
\(391\) −4.96257 −0.250968
\(392\) 0 0
\(393\) −1.58190 −0.0797963
\(394\) 0 0
\(395\) 2.41810 0.121668
\(396\) 0 0
\(397\) −15.4040 −0.773105 −0.386552 0.922267i \(-0.626334\pi\)
−0.386552 + 0.922267i \(0.626334\pi\)
\(398\) 0 0
\(399\) 20.0078 1.00164
\(400\) 0 0
\(401\) −10.5212 −0.525401 −0.262701 0.964877i \(-0.584613\pi\)
−0.262701 + 0.964877i \(0.584613\pi\)
\(402\) 0 0
\(403\) 0.157524 0.00784684
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −15.7394 −0.780174
\(408\) 0 0
\(409\) −26.0593 −1.28855 −0.644276 0.764793i \(-0.722841\pi\)
−0.644276 + 0.764793i \(0.722841\pi\)
\(410\) 0 0
\(411\) −4.91105 −0.242244
\(412\) 0 0
\(413\) −20.8877 −1.02782
\(414\) 0 0
\(415\) −5.45552 −0.267801
\(416\) 0 0
\(417\) 14.1342 0.692155
\(418\) 0 0
\(419\) −3.73942 −0.182683 −0.0913414 0.995820i \(-0.529115\pi\)
−0.0913414 + 0.995820i \(0.529115\pi\)
\(420\) 0 0
\(421\) −3.09677 −0.150928 −0.0754638 0.997149i \(-0.524044\pi\)
−0.0754638 + 0.997149i \(0.524044\pi\)
\(422\) 0 0
\(423\) 7.17162 0.348696
\(424\) 0 0
\(425\) −4.96257 −0.240720
\(426\) 0 0
\(427\) −21.7676 −1.05341
\(428\) 0 0
\(429\) −2.07485 −0.100175
\(430\) 0 0
\(431\) −36.1653 −1.74202 −0.871012 0.491262i \(-0.836536\pi\)
−0.871012 + 0.491262i \(0.836536\pi\)
\(432\) 0 0
\(433\) −16.6271 −0.799050 −0.399525 0.916722i \(-0.630825\pi\)
−0.399525 + 0.916722i \(0.630825\pi\)
\(434\) 0 0
\(435\) −4.96257 −0.237937
\(436\) 0 0
\(437\) 4.75353 0.227392
\(438\) 0 0
\(439\) 38.7613 1.84998 0.924989 0.379994i \(-0.124074\pi\)
0.924989 + 0.379994i \(0.124074\pi\)
\(440\) 0 0
\(441\) 10.7161 0.510290
\(442\) 0 0
\(443\) 33.5149 1.59234 0.796170 0.605073i \(-0.206856\pi\)
0.796170 + 0.605073i \(0.206856\pi\)
\(444\) 0 0
\(445\) −4.91105 −0.232806
\(446\) 0 0
\(447\) 3.24647 0.153553
\(448\) 0 0
\(449\) −13.9018 −0.656068 −0.328034 0.944666i \(-0.606386\pi\)
−0.328034 + 0.944666i \(0.606386\pi\)
\(450\) 0 0
\(451\) −25.8299 −1.21628
\(452\) 0 0
\(453\) 0.418100 0.0196440
\(454\) 0 0
\(455\) 3.17162 0.148688
\(456\) 0 0
\(457\) −2.28390 −0.106836 −0.0534182 0.998572i \(-0.517012\pi\)
−0.0534182 + 0.998572i \(0.517012\pi\)
\(458\) 0 0
\(459\) −4.96257 −0.231633
\(460\) 0 0
\(461\) 8.34325 0.388584 0.194292 0.980944i \(-0.437759\pi\)
0.194292 + 0.980944i \(0.437759\pi\)
\(462\) 0 0
\(463\) −27.6928 −1.28699 −0.643496 0.765449i \(-0.722517\pi\)
−0.643496 + 0.765449i \(0.722517\pi\)
\(464\) 0 0
\(465\) 0.209050 0.00969445
\(466\) 0 0
\(467\) 7.53037 0.348464 0.174232 0.984705i \(-0.444256\pi\)
0.174232 + 0.984705i \(0.444256\pi\)
\(468\) 0 0
\(469\) −0.879901 −0.0406300
\(470\) 0 0
\(471\) −21.0452 −0.969714
\(472\) 0 0
\(473\) 34.1935 1.57222
\(474\) 0 0
\(475\) 4.75353 0.218107
\(476\) 0 0
\(477\) −9.38067 −0.429512
\(478\) 0 0
\(479\) −18.1857 −0.830927 −0.415463 0.909610i \(-0.636381\pi\)
−0.415463 + 0.909610i \(0.636381\pi\)
\(480\) 0 0
\(481\) 4.30722 0.196393
\(482\) 0 0
\(483\) 4.20905 0.191518
\(484\) 0 0
\(485\) −10.3432 −0.469663
\(486\) 0 0
\(487\) −24.2606 −1.09935 −0.549676 0.835378i \(-0.685249\pi\)
−0.549676 + 0.835378i \(0.685249\pi\)
\(488\) 0 0
\(489\) −7.43220 −0.336096
\(490\) 0 0
\(491\) 14.8877 0.671874 0.335937 0.941885i \(-0.390947\pi\)
0.335937 + 0.941885i \(0.390947\pi\)
\(492\) 0 0
\(493\) −24.6271 −1.10915
\(494\) 0 0
\(495\) −2.75353 −0.123762
\(496\) 0 0
\(497\) 39.4837 1.77109
\(498\) 0 0
\(499\) −0.312101 −0.0139716 −0.00698578 0.999976i \(-0.502224\pi\)
−0.00698578 + 0.999976i \(0.502224\pi\)
\(500\) 0 0
\(501\) −19.1716 −0.856525
\(502\) 0 0
\(503\) 20.8877 0.931338 0.465669 0.884959i \(-0.345814\pi\)
0.465669 + 0.884959i \(0.345814\pi\)
\(504\) 0 0
\(505\) −14.8877 −0.662495
\(506\) 0 0
\(507\) −12.4322 −0.552133
\(508\) 0 0
\(509\) 27.9251 1.23776 0.618880 0.785485i \(-0.287587\pi\)
0.618880 + 0.785485i \(0.287587\pi\)
\(510\) 0 0
\(511\) 43.1873 1.91049
\(512\) 0 0
\(513\) 4.75353 0.209873
\(514\) 0 0
\(515\) −17.9251 −0.789876
\(516\) 0 0
\(517\) 19.7472 0.868483
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.0360 1.40352 0.701762 0.712412i \(-0.252397\pi\)
0.701762 + 0.712412i \(0.252397\pi\)
\(522\) 0 0
\(523\) 38.8644 1.69942 0.849711 0.527249i \(-0.176777\pi\)
0.849711 + 0.527249i \(0.176777\pi\)
\(524\) 0 0
\(525\) 4.20905 0.183698
\(526\) 0 0
\(527\) 1.03743 0.0451909
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.96257 −0.215357
\(532\) 0 0
\(533\) 7.06857 0.306174
\(534\) 0 0
\(535\) 16.4696 0.712044
\(536\) 0 0
\(537\) 13.8503 0.597685
\(538\) 0 0
\(539\) 29.5071 1.27096
\(540\) 0 0
\(541\) 4.07485 0.175191 0.0875957 0.996156i \(-0.472082\pi\)
0.0875957 + 0.996156i \(0.472082\pi\)
\(542\) 0 0
\(543\) 12.9110 0.554066
\(544\) 0 0
\(545\) 6.26058 0.268174
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −5.17162 −0.220720
\(550\) 0 0
\(551\) 23.5897 1.00496
\(552\) 0 0
\(553\) −10.1779 −0.432808
\(554\) 0 0
\(555\) 5.71610 0.242635
\(556\) 0 0
\(557\) 1.63343 0.0692105 0.0346052 0.999401i \(-0.488983\pi\)
0.0346052 + 0.999401i \(0.488983\pi\)
\(558\) 0 0
\(559\) −9.35735 −0.395774
\(560\) 0 0
\(561\) −13.6646 −0.576919
\(562\) 0 0
\(563\) −23.2310 −0.979069 −0.489534 0.871984i \(-0.662833\pi\)
−0.489534 + 0.871984i \(0.662833\pi\)
\(564\) 0 0
\(565\) 9.38067 0.394648
\(566\) 0 0
\(567\) 4.20905 0.176763
\(568\) 0 0
\(569\) 21.3291 0.894164 0.447082 0.894493i \(-0.352463\pi\)
0.447082 + 0.894493i \(0.352463\pi\)
\(570\) 0 0
\(571\) −4.18573 −0.175167 −0.0875836 0.996157i \(-0.527914\pi\)
−0.0875836 + 0.996157i \(0.527914\pi\)
\(572\) 0 0
\(573\) −8.15752 −0.340785
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 10.5678 0.439943 0.219972 0.975506i \(-0.429404\pi\)
0.219972 + 0.975506i \(0.429404\pi\)
\(578\) 0 0
\(579\) −16.7613 −0.696578
\(580\) 0 0
\(581\) 22.9626 0.952648
\(582\) 0 0
\(583\) −25.8299 −1.06977
\(584\) 0 0
\(585\) 0.753525 0.0311544
\(586\) 0 0
\(587\) 28.4181 1.17294 0.586470 0.809971i \(-0.300517\pi\)
0.586470 + 0.809971i \(0.300517\pi\)
\(588\) 0 0
\(589\) −0.993723 −0.0409457
\(590\) 0 0
\(591\) −10.9110 −0.448821
\(592\) 0 0
\(593\) 28.9937 1.19063 0.595315 0.803493i \(-0.297027\pi\)
0.595315 + 0.803493i \(0.297027\pi\)
\(594\) 0 0
\(595\) 20.8877 0.856313
\(596\) 0 0
\(597\) −4.49295 −0.183884
\(598\) 0 0
\(599\) −6.10305 −0.249364 −0.124682 0.992197i \(-0.539791\pi\)
−0.124682 + 0.992197i \(0.539791\pi\)
\(600\) 0 0
\(601\) −29.3885 −1.19878 −0.599391 0.800456i \(-0.704590\pi\)
−0.599391 + 0.800456i \(0.704590\pi\)
\(602\) 0 0
\(603\) −0.209050 −0.00851316
\(604\) 0 0
\(605\) 3.41810 0.138966
\(606\) 0 0
\(607\) −3.58972 −0.145702 −0.0728512 0.997343i \(-0.523210\pi\)
−0.0728512 + 0.997343i \(0.523210\pi\)
\(608\) 0 0
\(609\) 20.8877 0.846413
\(610\) 0 0
\(611\) −5.40400 −0.218622
\(612\) 0 0
\(613\) −21.4040 −0.864499 −0.432250 0.901754i \(-0.642280\pi\)
−0.432250 + 0.901754i \(0.642280\pi\)
\(614\) 0 0
\(615\) 9.38067 0.378265
\(616\) 0 0
\(617\) 32.2917 1.30002 0.650008 0.759927i \(-0.274766\pi\)
0.650008 + 0.759927i \(0.274766\pi\)
\(618\) 0 0
\(619\) 47.1046 1.89329 0.946647 0.322273i \(-0.104447\pi\)
0.946647 + 0.322273i \(0.104447\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 20.6709 0.828160
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 13.0890 0.522722
\(628\) 0 0
\(629\) 28.3666 1.13105
\(630\) 0 0
\(631\) 23.5149 0.936112 0.468056 0.883699i \(-0.344954\pi\)
0.468056 + 0.883699i \(0.344954\pi\)
\(632\) 0 0
\(633\) 9.71610 0.386180
\(634\) 0 0
\(635\) 22.0827 0.876324
\(636\) 0 0
\(637\) −8.07485 −0.319937
\(638\) 0 0
\(639\) 9.38067 0.371094
\(640\) 0 0
\(641\) −6.67867 −0.263792 −0.131896 0.991264i \(-0.542106\pi\)
−0.131896 + 0.991264i \(0.542106\pi\)
\(642\) 0 0
\(643\) 27.1201 1.06951 0.534756 0.845006i \(-0.320403\pi\)
0.534756 + 0.845006i \(0.320403\pi\)
\(644\) 0 0
\(645\) −12.4181 −0.488962
\(646\) 0 0
\(647\) −27.7394 −1.09055 −0.545275 0.838257i \(-0.683575\pi\)
−0.545275 + 0.838257i \(0.683575\pi\)
\(648\) 0 0
\(649\) −13.6646 −0.536381
\(650\) 0 0
\(651\) −0.879901 −0.0344860
\(652\) 0 0
\(653\) −2.75353 −0.107754 −0.0538769 0.998548i \(-0.517158\pi\)
−0.0538769 + 0.998548i \(0.517158\pi\)
\(654\) 0 0
\(655\) 1.58190 0.0618100
\(656\) 0 0
\(657\) 10.2606 0.400303
\(658\) 0 0
\(659\) 11.5897 0.451472 0.225736 0.974189i \(-0.427521\pi\)
0.225736 + 0.974189i \(0.427521\pi\)
\(660\) 0 0
\(661\) 28.3432 1.10242 0.551212 0.834365i \(-0.314165\pi\)
0.551212 + 0.834365i \(0.314165\pi\)
\(662\) 0 0
\(663\) 3.73942 0.145227
\(664\) 0 0
\(665\) −20.0078 −0.775870
\(666\) 0 0
\(667\) 4.96257 0.192152
\(668\) 0 0
\(669\) −19.4322 −0.751292
\(670\) 0 0
\(671\) −14.2402 −0.549737
\(672\) 0 0
\(673\) −48.0360 −1.85165 −0.925826 0.377949i \(-0.876629\pi\)
−0.925826 + 0.377949i \(0.876629\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 2.51627 0.0967083 0.0483541 0.998830i \(-0.484602\pi\)
0.0483541 + 0.998830i \(0.484602\pi\)
\(678\) 0 0
\(679\) 43.5353 1.67073
\(680\) 0 0
\(681\) −0.985900 −0.0377798
\(682\) 0 0
\(683\) −3.84248 −0.147028 −0.0735141 0.997294i \(-0.523421\pi\)
−0.0735141 + 0.997294i \(0.523421\pi\)
\(684\) 0 0
\(685\) 4.91105 0.187642
\(686\) 0 0
\(687\) 3.08895 0.117851
\(688\) 0 0
\(689\) 7.06857 0.269291
\(690\) 0 0
\(691\) 2.59600 0.0987565 0.0493783 0.998780i \(-0.484276\pi\)
0.0493783 + 0.998780i \(0.484276\pi\)
\(692\) 0 0
\(693\) 11.5897 0.440257
\(694\) 0 0
\(695\) −14.1342 −0.536141
\(696\) 0 0
\(697\) 46.5523 1.76329
\(698\) 0 0
\(699\) 8.83620 0.334216
\(700\) 0 0
\(701\) 41.5897 1.57082 0.785411 0.618974i \(-0.212452\pi\)
0.785411 + 0.618974i \(0.212452\pi\)
\(702\) 0 0
\(703\) −27.1716 −1.02480
\(704\) 0 0
\(705\) −7.17162 −0.270099
\(706\) 0 0
\(707\) 62.6632 2.35669
\(708\) 0 0
\(709\) −53.1716 −1.99690 −0.998451 0.0556356i \(-0.982281\pi\)
−0.998451 + 0.0556356i \(0.982281\pi\)
\(710\) 0 0
\(711\) −2.41810 −0.0906858
\(712\) 0 0
\(713\) −0.209050 −0.00782898
\(714\) 0 0
\(715\) 2.07485 0.0775950
\(716\) 0 0
\(717\) −21.3807 −0.798476
\(718\) 0 0
\(719\) 45.3807 1.69241 0.846207 0.532855i \(-0.178881\pi\)
0.846207 + 0.532855i \(0.178881\pi\)
\(720\) 0 0
\(721\) 75.4478 2.80982
\(722\) 0 0
\(723\) 22.2606 0.827880
\(724\) 0 0
\(725\) 4.96257 0.184305
\(726\) 0 0
\(727\) −46.4026 −1.72098 −0.860489 0.509470i \(-0.829842\pi\)
−0.860489 + 0.509470i \(0.829842\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −61.6257 −2.27931
\(732\) 0 0
\(733\) 22.8051 0.842324 0.421162 0.906985i \(-0.361622\pi\)
0.421162 + 0.906985i \(0.361622\pi\)
\(734\) 0 0
\(735\) −10.7161 −0.395269
\(736\) 0 0
\(737\) −0.575624 −0.0212034
\(738\) 0 0
\(739\) −16.5241 −0.607849 −0.303924 0.952696i \(-0.598297\pi\)
−0.303924 + 0.952696i \(0.598297\pi\)
\(740\) 0 0
\(741\) −3.58190 −0.131584
\(742\) 0 0
\(743\) 43.8503 1.60871 0.804356 0.594148i \(-0.202511\pi\)
0.804356 + 0.594148i \(0.202511\pi\)
\(744\) 0 0
\(745\) −3.24647 −0.118942
\(746\) 0 0
\(747\) 5.45552 0.199607
\(748\) 0 0
\(749\) −69.3215 −2.53295
\(750\) 0 0
\(751\) 5.73942 0.209435 0.104717 0.994502i \(-0.466606\pi\)
0.104717 + 0.994502i \(0.466606\pi\)
\(752\) 0 0
\(753\) −6.59600 −0.240372
\(754\) 0 0
\(755\) −0.418100 −0.0152162
\(756\) 0 0
\(757\) −9.86580 −0.358579 −0.179289 0.983796i \(-0.557380\pi\)
−0.179289 + 0.983796i \(0.557380\pi\)
\(758\) 0 0
\(759\) 2.75353 0.0999466
\(760\) 0 0
\(761\) 4.69418 0.170164 0.0850819 0.996374i \(-0.472885\pi\)
0.0850819 + 0.996374i \(0.472885\pi\)
\(762\) 0 0
\(763\) −26.3511 −0.953973
\(764\) 0 0
\(765\) 4.96257 0.179422
\(766\) 0 0
\(767\) 3.73942 0.135023
\(768\) 0 0
\(769\) 45.5431 1.64233 0.821163 0.570694i \(-0.193326\pi\)
0.821163 + 0.570694i \(0.193326\pi\)
\(770\) 0 0
\(771\) −18.1857 −0.654943
\(772\) 0 0
\(773\) 12.4929 0.449340 0.224670 0.974435i \(-0.427870\pi\)
0.224670 + 0.974435i \(0.427870\pi\)
\(774\) 0 0
\(775\) −0.209050 −0.00750929
\(776\) 0 0
\(777\) −24.0593 −0.863124
\(778\) 0 0
\(779\) −44.5913 −1.59765
\(780\) 0 0
\(781\) 25.8299 0.924267
\(782\) 0 0
\(783\) 4.96257 0.177348
\(784\) 0 0
\(785\) 21.0452 0.751137
\(786\) 0 0
\(787\) −5.71610 −0.203757 −0.101878 0.994797i \(-0.532485\pi\)
−0.101878 + 0.994797i \(0.532485\pi\)
\(788\) 0 0
\(789\) −15.3807 −0.547567
\(790\) 0 0
\(791\) −39.4837 −1.40388
\(792\) 0 0
\(793\) 3.89695 0.138385
\(794\) 0 0
\(795\) 9.38067 0.332698
\(796\) 0 0
\(797\) −9.48373 −0.335931 −0.167965 0.985793i \(-0.553720\pi\)
−0.167965 + 0.985793i \(0.553720\pi\)
\(798\) 0 0
\(799\) −35.5897 −1.25907
\(800\) 0 0
\(801\) 4.91105 0.173523
\(802\) 0 0
\(803\) 28.2528 0.997018
\(804\) 0 0
\(805\) −4.20905 −0.148350
\(806\) 0 0
\(807\) 0.544475 0.0191664
\(808\) 0 0
\(809\) 50.0672 1.76027 0.880134 0.474725i \(-0.157453\pi\)
0.880134 + 0.474725i \(0.157453\pi\)
\(810\) 0 0
\(811\) −9.14830 −0.321240 −0.160620 0.987016i \(-0.551349\pi\)
−0.160620 + 0.987016i \(0.551349\pi\)
\(812\) 0 0
\(813\) −12.2090 −0.428190
\(814\) 0 0
\(815\) 7.43220 0.260339
\(816\) 0 0
\(817\) 59.0297 2.06519
\(818\) 0 0
\(819\) −3.17162 −0.110826
\(820\) 0 0
\(821\) −4.91105 −0.171397 −0.0856984 0.996321i \(-0.527312\pi\)
−0.0856984 + 0.996321i \(0.527312\pi\)
\(822\) 0 0
\(823\) 23.7006 0.826151 0.413075 0.910697i \(-0.364455\pi\)
0.413075 + 0.910697i \(0.364455\pi\)
\(824\) 0 0
\(825\) 2.75353 0.0958654
\(826\) 0 0
\(827\) 13.1405 0.456939 0.228470 0.973551i \(-0.426628\pi\)
0.228470 + 0.973551i \(0.426628\pi\)
\(828\) 0 0
\(829\) −45.6412 −1.58519 −0.792593 0.609751i \(-0.791269\pi\)
−0.792593 + 0.609751i \(0.791269\pi\)
\(830\) 0 0
\(831\) −21.0141 −0.728971
\(832\) 0 0
\(833\) −53.1794 −1.84256
\(834\) 0 0
\(835\) 19.1716 0.663461
\(836\) 0 0
\(837\) −0.209050 −0.00722582
\(838\) 0 0
\(839\) −5.40400 −0.186567 −0.0932834 0.995640i \(-0.529736\pi\)
−0.0932834 + 0.995640i \(0.529736\pi\)
\(840\) 0 0
\(841\) −4.37285 −0.150788
\(842\) 0 0
\(843\) −24.1857 −0.833001
\(844\) 0 0
\(845\) 12.4322 0.427681
\(846\) 0 0
\(847\) −14.3870 −0.494341
\(848\) 0 0
\(849\) 15.1201 0.518920
\(850\) 0 0
\(851\) −5.71610 −0.195945
\(852\) 0 0
\(853\) −53.0297 −1.81570 −0.907852 0.419291i \(-0.862279\pi\)
−0.907852 + 0.419291i \(0.862279\pi\)
\(854\) 0 0
\(855\) −4.75353 −0.162567
\(856\) 0 0
\(857\) −14.4463 −0.493476 −0.246738 0.969082i \(-0.579359\pi\)
−0.246738 + 0.969082i \(0.579359\pi\)
\(858\) 0 0
\(859\) 13.8658 0.473095 0.236548 0.971620i \(-0.423984\pi\)
0.236548 + 0.971620i \(0.423984\pi\)
\(860\) 0 0
\(861\) −39.4837 −1.34560
\(862\) 0 0
\(863\) 52.5834 1.78996 0.894981 0.446105i \(-0.147189\pi\)
0.894981 + 0.446105i \(0.147189\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.62715 0.259032
\(868\) 0 0
\(869\) −6.65830 −0.225867
\(870\) 0 0
\(871\) 0.157524 0.00533751
\(872\) 0 0
\(873\) 10.3432 0.350066
\(874\) 0 0
\(875\) −4.20905 −0.142292
\(876\) 0 0
\(877\) −25.3291 −0.855305 −0.427652 0.903943i \(-0.640659\pi\)
−0.427652 + 0.903943i \(0.640659\pi\)
\(878\) 0 0
\(879\) 10.4696 0.353132
\(880\) 0 0
\(881\) 14.2606 0.480451 0.240225 0.970717i \(-0.422779\pi\)
0.240225 + 0.970717i \(0.422779\pi\)
\(882\) 0 0
\(883\) 40.6320 1.36738 0.683688 0.729774i \(-0.260375\pi\)
0.683688 + 0.729774i \(0.260375\pi\)
\(884\) 0 0
\(885\) 4.96257 0.166815
\(886\) 0 0
\(887\) −19.9534 −0.669968 −0.334984 0.942224i \(-0.608731\pi\)
−0.334984 + 0.942224i \(0.608731\pi\)
\(888\) 0 0
\(889\) −92.9471 −3.11734
\(890\) 0 0
\(891\) 2.75353 0.0922466
\(892\) 0 0
\(893\) 34.0905 1.14080
\(894\) 0 0
\(895\) −13.8503 −0.462964
\(896\) 0 0
\(897\) −0.753525 −0.0251595
\(898\) 0 0
\(899\) −1.03743 −0.0346001
\(900\) 0 0
\(901\) 46.5523 1.55088
\(902\) 0 0
\(903\) 52.2684 1.73938
\(904\) 0 0
\(905\) −12.9110 −0.429178
\(906\) 0 0
\(907\) −29.8814 −0.992197 −0.496099 0.868266i \(-0.665235\pi\)
−0.496099 + 0.868266i \(0.665235\pi\)
\(908\) 0 0
\(909\) 14.8877 0.493795
\(910\) 0 0
\(911\) 18.8644 0.625005 0.312503 0.949917i \(-0.398833\pi\)
0.312503 + 0.949917i \(0.398833\pi\)
\(912\) 0 0
\(913\) 15.0219 0.497153
\(914\) 0 0
\(915\) 5.17162 0.170969
\(916\) 0 0
\(917\) −6.65830 −0.219876
\(918\) 0 0
\(919\) 16.2402 0.535715 0.267857 0.963459i \(-0.413684\pi\)
0.267857 + 0.963459i \(0.413684\pi\)
\(920\) 0 0
\(921\) −12.1575 −0.400604
\(922\) 0 0
\(923\) −7.06857 −0.232665
\(924\) 0 0
\(925\) −5.71610 −0.187944
\(926\) 0 0
\(927\) 17.9251 0.588739
\(928\) 0 0
\(929\) 29.3340 0.962418 0.481209 0.876606i \(-0.340198\pi\)
0.481209 + 0.876606i \(0.340198\pi\)
\(930\) 0 0
\(931\) 50.9393 1.66947
\(932\) 0 0
\(933\) −34.4181 −1.12680
\(934\) 0 0
\(935\) 13.6646 0.446879
\(936\) 0 0
\(937\) −41.3573 −1.35109 −0.675543 0.737321i \(-0.736091\pi\)
−0.675543 + 0.737321i \(0.736091\pi\)
\(938\) 0 0
\(939\) 29.5664 0.964863
\(940\) 0 0
\(941\) 50.4259 1.64384 0.821919 0.569604i \(-0.192904\pi\)
0.821919 + 0.569604i \(0.192904\pi\)
\(942\) 0 0
\(943\) −9.38067 −0.305477
\(944\) 0 0
\(945\) −4.20905 −0.136920
\(946\) 0 0
\(947\) 5.50705 0.178955 0.0894775 0.995989i \(-0.471480\pi\)
0.0894775 + 0.995989i \(0.471480\pi\)
\(948\) 0 0
\(949\) −7.73160 −0.250978
\(950\) 0 0
\(951\) 1.66457 0.0539775
\(952\) 0 0
\(953\) 0.864400 0.0280007 0.0140003 0.999902i \(-0.495543\pi\)
0.0140003 + 0.999902i \(0.495543\pi\)
\(954\) 0 0
\(955\) 8.15752 0.263971
\(956\) 0 0
\(957\) 13.6646 0.441713
\(958\) 0 0
\(959\) −20.6709 −0.667497
\(960\) 0 0
\(961\) −30.9563 −0.998590
\(962\) 0 0
\(963\) −16.4696 −0.530726
\(964\) 0 0
\(965\) 16.7613 0.539567
\(966\) 0 0
\(967\) 10.7535 0.345810 0.172905 0.984939i \(-0.444685\pi\)
0.172905 + 0.984939i \(0.444685\pi\)
\(968\) 0 0
\(969\) −23.5897 −0.757811
\(970\) 0 0
\(971\) 35.1794 1.12896 0.564481 0.825446i \(-0.309076\pi\)
0.564481 + 0.825446i \(0.309076\pi\)
\(972\) 0 0
\(973\) 59.4915 1.90721
\(974\) 0 0
\(975\) −0.753525 −0.0241321
\(976\) 0 0
\(977\) 15.9767 0.511139 0.255570 0.966791i \(-0.417737\pi\)
0.255570 + 0.966791i \(0.417737\pi\)
\(978\) 0 0
\(979\) 13.5227 0.432187
\(980\) 0 0
\(981\) −6.26058 −0.199885
\(982\) 0 0
\(983\) −57.2592 −1.82628 −0.913142 0.407642i \(-0.866351\pi\)
−0.913142 + 0.407642i \(0.866351\pi\)
\(984\) 0 0
\(985\) 10.9110 0.347655
\(986\) 0 0
\(987\) 30.1857 0.960822
\(988\) 0 0
\(989\) 12.4181 0.394873
\(990\) 0 0
\(991\) 12.8799 0.409144 0.204572 0.978852i \(-0.434420\pi\)
0.204572 + 0.978852i \(0.434420\pi\)
\(992\) 0 0
\(993\) 3.12010 0.0990134
\(994\) 0 0
\(995\) 4.49295 0.142436
\(996\) 0 0
\(997\) −39.7754 −1.25970 −0.629851 0.776716i \(-0.716884\pi\)
−0.629851 + 0.776716i \(0.716884\pi\)
\(998\) 0 0
\(999\) −5.71610 −0.180849
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 1380.2.a.j.1.3 3
3.2 odd 2 4140.2.a.s.1.3 3
4.3 odd 2 5520.2.a.bv.1.1 3
5.2 odd 4 6900.2.f.r.6349.3 6
5.3 odd 4 6900.2.f.r.6349.4 6
5.4 even 2 6900.2.a.x.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.3 3 1.1 even 1 trivial
4140.2.a.s.1.3 3 3.2 odd 2
5520.2.a.bv.1.1 3 4.3 odd 2
6900.2.a.x.1.1 3 5.4 even 2
6900.2.f.r.6349.3 6 5.2 odd 4
6900.2.f.r.6349.4 6 5.3 odd 4