Properties

Label 7728.2.a.cb.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.07830 q^{11} +5.16394 q^{13} -1.08564 q^{15} -2.56511 q^{17} -4.07830 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.82139 q^{25} +1.00000 q^{27} -10.6326 q^{29} -6.55044 q^{31} +4.07830 q^{33} +1.08564 q^{35} -3.90703 q^{37} +5.16394 q^{39} -1.43489 q^{41} -12.8484 q^{43} -1.08564 q^{45} +9.98533 q^{47} +1.00000 q^{49} -2.56511 q^{51} +9.15309 q^{53} -4.42756 q^{55} -4.07830 q^{57} -3.63989 q^{59} +1.90414 q^{61} -1.00000 q^{63} -5.60617 q^{65} +8.28331 q^{67} +1.00000 q^{69} -14.1903 q^{71} -4.64723 q^{73} -3.82139 q^{75} -4.07830 q^{77} +6.55044 q^{79} +1.00000 q^{81} -8.86597 q^{83} +2.78478 q^{85} -10.6326 q^{87} -10.6501 q^{89} -5.16394 q^{91} -6.55044 q^{93} +4.42756 q^{95} -4.64723 q^{97} +4.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.08564 −0.485512 −0.242756 0.970087i \(-0.578051\pi\)
−0.242756 + 0.970087i \(0.578051\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.07830 1.22965 0.614827 0.788662i \(-0.289226\pi\)
0.614827 + 0.788662i \(0.289226\pi\)
\(12\) 0 0
\(13\) 5.16394 1.43222 0.716110 0.697988i \(-0.245921\pi\)
0.716110 + 0.697988i \(0.245921\pi\)
\(14\) 0 0
\(15\) −1.08564 −0.280310
\(16\) 0 0
\(17\) −2.56511 −0.622130 −0.311065 0.950389i \(-0.600686\pi\)
−0.311065 + 0.950389i \(0.600686\pi\)
\(18\) 0 0
\(19\) −4.07830 −0.935627 −0.467814 0.883827i \(-0.654958\pi\)
−0.467814 + 0.883827i \(0.654958\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.82139 −0.764278
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.6326 −1.97442 −0.987208 0.159435i \(-0.949033\pi\)
−0.987208 + 0.159435i \(0.949033\pi\)
\(30\) 0 0
\(31\) −6.55044 −1.17649 −0.588247 0.808681i \(-0.700182\pi\)
−0.588247 + 0.808681i \(0.700182\pi\)
\(32\) 0 0
\(33\) 4.07830 0.709942
\(34\) 0 0
\(35\) 1.08564 0.183506
\(36\) 0 0
\(37\) −3.90703 −0.642312 −0.321156 0.947026i \(-0.604071\pi\)
−0.321156 + 0.947026i \(0.604071\pi\)
\(38\) 0 0
\(39\) 5.16394 0.826892
\(40\) 0 0
\(41\) −1.43489 −0.224093 −0.112046 0.993703i \(-0.535740\pi\)
−0.112046 + 0.993703i \(0.535740\pi\)
\(42\) 0 0
\(43\) −12.8484 −1.95936 −0.979682 0.200556i \(-0.935725\pi\)
−0.979682 + 0.200556i \(0.935725\pi\)
\(44\) 0 0
\(45\) −1.08564 −0.161837
\(46\) 0 0
\(47\) 9.98533 1.45651 0.728255 0.685306i \(-0.240332\pi\)
0.728255 + 0.685306i \(0.240332\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.56511 −0.359187
\(52\) 0 0
\(53\) 9.15309 1.25727 0.628637 0.777699i \(-0.283613\pi\)
0.628637 + 0.777699i \(0.283613\pi\)
\(54\) 0 0
\(55\) −4.42756 −0.597012
\(56\) 0 0
\(57\) −4.07830 −0.540185
\(58\) 0 0
\(59\) −3.63989 −0.473874 −0.236937 0.971525i \(-0.576143\pi\)
−0.236937 + 0.971525i \(0.576143\pi\)
\(60\) 0 0
\(61\) 1.90414 0.243800 0.121900 0.992542i \(-0.461101\pi\)
0.121900 + 0.992542i \(0.461101\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −5.60617 −0.695360
\(66\) 0 0
\(67\) 8.28331 1.01197 0.505983 0.862543i \(-0.331130\pi\)
0.505983 + 0.862543i \(0.331130\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.1903 −1.68408 −0.842041 0.539413i \(-0.818646\pi\)
−0.842041 + 0.539413i \(0.818646\pi\)
\(72\) 0 0
\(73\) −4.64723 −0.543917 −0.271958 0.962309i \(-0.587671\pi\)
−0.271958 + 0.962309i \(0.587671\pi\)
\(74\) 0 0
\(75\) −3.82139 −0.441256
\(76\) 0 0
\(77\) −4.07830 −0.464766
\(78\) 0 0
\(79\) 6.55044 0.736982 0.368491 0.929631i \(-0.379875\pi\)
0.368491 + 0.929631i \(0.379875\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.86597 −0.973166 −0.486583 0.873634i \(-0.661757\pi\)
−0.486583 + 0.873634i \(0.661757\pi\)
\(84\) 0 0
\(85\) 2.78478 0.302051
\(86\) 0 0
\(87\) −10.6326 −1.13993
\(88\) 0 0
\(89\) −10.6501 −1.12891 −0.564455 0.825464i \(-0.690914\pi\)
−0.564455 + 0.825464i \(0.690914\pi\)
\(90\) 0 0
\(91\) −5.16394 −0.541328
\(92\) 0 0
\(93\) −6.55044 −0.679249
\(94\) 0 0
\(95\) 4.42756 0.454258
\(96\) 0 0
\(97\) −4.64723 −0.471855 −0.235927 0.971771i \(-0.575813\pi\)
−0.235927 + 0.971771i \(0.575813\pi\)
\(98\) 0 0
\(99\) 4.07830 0.409885
\(100\) 0 0
\(101\) −11.9003 −1.18413 −0.592063 0.805892i \(-0.701686\pi\)
−0.592063 + 0.805892i \(0.701686\pi\)
\(102\) 0 0
\(103\) 11.3689 1.12022 0.560108 0.828420i \(-0.310760\pi\)
0.560108 + 0.828420i \(0.310760\pi\)
\(104\) 0 0
\(105\) 1.08564 0.105947
\(106\) 0 0
\(107\) 15.7857 1.52606 0.763028 0.646365i \(-0.223712\pi\)
0.763028 + 0.646365i \(0.223712\pi\)
\(108\) 0 0
\(109\) −1.33903 −0.128256 −0.0641281 0.997942i \(-0.520427\pi\)
−0.0641281 + 0.997942i \(0.520427\pi\)
\(110\) 0 0
\(111\) −3.90703 −0.370839
\(112\) 0 0
\(113\) 6.58417 0.619386 0.309693 0.950837i \(-0.399774\pi\)
0.309693 + 0.950837i \(0.399774\pi\)
\(114\) 0 0
\(115\) −1.08564 −0.101236
\(116\) 0 0
\(117\) 5.16394 0.477407
\(118\) 0 0
\(119\) 2.56511 0.235143
\(120\) 0 0
\(121\) 5.63256 0.512051
\(122\) 0 0
\(123\) −1.43489 −0.129380
\(124\) 0 0
\(125\) 9.57683 0.856578
\(126\) 0 0
\(127\) 12.3616 1.09692 0.548458 0.836178i \(-0.315215\pi\)
0.548458 + 0.836178i \(0.315215\pi\)
\(128\) 0 0
\(129\) −12.8484 −1.13124
\(130\) 0 0
\(131\) −7.59150 −0.663272 −0.331636 0.943407i \(-0.607601\pi\)
−0.331636 + 0.943407i \(0.607601\pi\)
\(132\) 0 0
\(133\) 4.07830 0.353634
\(134\) 0 0
\(135\) −1.08564 −0.0934368
\(136\) 0 0
\(137\) −10.8185 −0.924287 −0.462144 0.886805i \(-0.652920\pi\)
−0.462144 + 0.886805i \(0.652920\pi\)
\(138\) 0 0
\(139\) 12.6733 1.07494 0.537468 0.843284i \(-0.319381\pi\)
0.537468 + 0.843284i \(0.319381\pi\)
\(140\) 0 0
\(141\) 9.98533 0.840917
\(142\) 0 0
\(143\) 21.0601 1.76114
\(144\) 0 0
\(145\) 11.5431 0.958603
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −12.5613 −1.02906 −0.514530 0.857472i \(-0.672034\pi\)
−0.514530 + 0.857472i \(0.672034\pi\)
\(150\) 0 0
\(151\) 21.3543 1.73779 0.868893 0.495000i \(-0.164832\pi\)
0.868893 + 0.495000i \(0.164832\pi\)
\(152\) 0 0
\(153\) −2.56511 −0.207377
\(154\) 0 0
\(155\) 7.11140 0.571202
\(156\) 0 0
\(157\) −11.4311 −0.912299 −0.456150 0.889903i \(-0.650772\pi\)
−0.456150 + 0.889903i \(0.650772\pi\)
\(158\) 0 0
\(159\) 9.15309 0.725887
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −16.8376 −1.31882 −0.659410 0.751784i \(-0.729194\pi\)
−0.659410 + 0.751784i \(0.729194\pi\)
\(164\) 0 0
\(165\) −4.42756 −0.344685
\(166\) 0 0
\(167\) 6.82554 0.528176 0.264088 0.964499i \(-0.414929\pi\)
0.264088 + 0.964499i \(0.414929\pi\)
\(168\) 0 0
\(169\) 13.6663 1.05125
\(170\) 0 0
\(171\) −4.07830 −0.311876
\(172\) 0 0
\(173\) −24.2753 −1.84562 −0.922810 0.385255i \(-0.874113\pi\)
−0.922810 + 0.385255i \(0.874113\pi\)
\(174\) 0 0
\(175\) 3.82139 0.288870
\(176\) 0 0
\(177\) −3.63989 −0.273591
\(178\) 0 0
\(179\) −22.4803 −1.68026 −0.840130 0.542385i \(-0.817521\pi\)
−0.840130 + 0.542385i \(0.817521\pi\)
\(180\) 0 0
\(181\) 9.19385 0.683374 0.341687 0.939814i \(-0.389002\pi\)
0.341687 + 0.939814i \(0.389002\pi\)
\(182\) 0 0
\(183\) 1.90414 0.140758
\(184\) 0 0
\(185\) 4.24162 0.311850
\(186\) 0 0
\(187\) −10.4613 −0.765005
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −21.3543 −1.54514 −0.772571 0.634929i \(-0.781029\pi\)
−0.772571 + 0.634929i \(0.781029\pi\)
\(192\) 0 0
\(193\) −18.2458 −1.31336 −0.656679 0.754170i \(-0.728039\pi\)
−0.656679 + 0.754170i \(0.728039\pi\)
\(194\) 0 0
\(195\) −5.60617 −0.401466
\(196\) 0 0
\(197\) 15.1493 1.07934 0.539671 0.841876i \(-0.318549\pi\)
0.539671 + 0.841876i \(0.318549\pi\)
\(198\) 0 0
\(199\) 6.53138 0.462997 0.231499 0.972835i \(-0.425637\pi\)
0.231499 + 0.972835i \(0.425637\pi\)
\(200\) 0 0
\(201\) 8.28331 0.584259
\(202\) 0 0
\(203\) 10.6326 0.746259
\(204\) 0 0
\(205\) 1.55777 0.108800
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −16.6326 −1.15050
\(210\) 0 0
\(211\) 2.64723 0.182243 0.0911214 0.995840i \(-0.470955\pi\)
0.0911214 + 0.995840i \(0.470955\pi\)
\(212\) 0 0
\(213\) −14.1903 −0.972306
\(214\) 0 0
\(215\) 13.9487 0.951295
\(216\) 0 0
\(217\) 6.55044 0.444673
\(218\) 0 0
\(219\) −4.64723 −0.314031
\(220\) 0 0
\(221\) −13.2461 −0.891027
\(222\) 0 0
\(223\) 17.7818 1.19076 0.595380 0.803444i \(-0.297002\pi\)
0.595380 + 0.803444i \(0.297002\pi\)
\(224\) 0 0
\(225\) −3.82139 −0.254759
\(226\) 0 0
\(227\) 13.7695 0.913913 0.456956 0.889489i \(-0.348940\pi\)
0.456956 + 0.889489i \(0.348940\pi\)
\(228\) 0 0
\(229\) −20.6112 −1.36203 −0.681013 0.732272i \(-0.738460\pi\)
−0.681013 + 0.732272i \(0.738460\pi\)
\(230\) 0 0
\(231\) −4.07830 −0.268333
\(232\) 0 0
\(233\) 3.37183 0.220896 0.110448 0.993882i \(-0.464771\pi\)
0.110448 + 0.993882i \(0.464771\pi\)
\(234\) 0 0
\(235\) −10.8405 −0.707153
\(236\) 0 0
\(237\) 6.55044 0.425497
\(238\) 0 0
\(239\) −17.7818 −1.15021 −0.575106 0.818079i \(-0.695039\pi\)
−0.575106 + 0.818079i \(0.695039\pi\)
\(240\) 0 0
\(241\) −25.0630 −1.61445 −0.807225 0.590244i \(-0.799032\pi\)
−0.807225 + 0.590244i \(0.799032\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.08564 −0.0693588
\(246\) 0 0
\(247\) −21.0601 −1.34002
\(248\) 0 0
\(249\) −8.86597 −0.561858
\(250\) 0 0
\(251\) −0.465103 −0.0293571 −0.0146785 0.999892i \(-0.504672\pi\)
−0.0146785 + 0.999892i \(0.504672\pi\)
\(252\) 0 0
\(253\) 4.07830 0.256401
\(254\) 0 0
\(255\) 2.78478 0.174389
\(256\) 0 0
\(257\) 17.5930 1.09742 0.548711 0.836012i \(-0.315119\pi\)
0.548711 + 0.836012i \(0.315119\pi\)
\(258\) 0 0
\(259\) 3.90703 0.242771
\(260\) 0 0
\(261\) −10.6326 −0.658139
\(262\) 0 0
\(263\) −5.69532 −0.351189 −0.175594 0.984463i \(-0.556185\pi\)
−0.175594 + 0.984463i \(0.556185\pi\)
\(264\) 0 0
\(265\) −9.93694 −0.610421
\(266\) 0 0
\(267\) −10.6501 −0.651777
\(268\) 0 0
\(269\) 8.27949 0.504809 0.252405 0.967622i \(-0.418779\pi\)
0.252405 + 0.967622i \(0.418779\pi\)
\(270\) 0 0
\(271\) 12.0366 0.731172 0.365586 0.930778i \(-0.380869\pi\)
0.365586 + 0.930778i \(0.380869\pi\)
\(272\) 0 0
\(273\) −5.16394 −0.312536
\(274\) 0 0
\(275\) −15.5848 −0.939798
\(276\) 0 0
\(277\) −12.5329 −0.753028 −0.376514 0.926411i \(-0.622877\pi\)
−0.376514 + 0.926411i \(0.622877\pi\)
\(278\) 0 0
\(279\) −6.55044 −0.392165
\(280\) 0 0
\(281\) −13.2181 −0.788526 −0.394263 0.918998i \(-0.629000\pi\)
−0.394263 + 0.918998i \(0.629000\pi\)
\(282\) 0 0
\(283\) −15.3314 −0.911357 −0.455679 0.890144i \(-0.650603\pi\)
−0.455679 + 0.890144i \(0.650603\pi\)
\(284\) 0 0
\(285\) 4.42756 0.262266
\(286\) 0 0
\(287\) 1.43489 0.0846990
\(288\) 0 0
\(289\) −10.4202 −0.612954
\(290\) 0 0
\(291\) −4.64723 −0.272425
\(292\) 0 0
\(293\) −11.9977 −0.700912 −0.350456 0.936579i \(-0.613973\pi\)
−0.350456 + 0.936579i \(0.613973\pi\)
\(294\) 0 0
\(295\) 3.95161 0.230071
\(296\) 0 0
\(297\) 4.07830 0.236647
\(298\) 0 0
\(299\) 5.16394 0.298638
\(300\) 0 0
\(301\) 12.8484 0.740570
\(302\) 0 0
\(303\) −11.9003 −0.683656
\(304\) 0 0
\(305\) −2.06721 −0.118368
\(306\) 0 0
\(307\) 14.7147 0.839811 0.419906 0.907568i \(-0.362063\pi\)
0.419906 + 0.907568i \(0.362063\pi\)
\(308\) 0 0
\(309\) 11.3689 0.646757
\(310\) 0 0
\(311\) 31.0455 1.76043 0.880213 0.474579i \(-0.157400\pi\)
0.880213 + 0.474579i \(0.157400\pi\)
\(312\) 0 0
\(313\) −23.9502 −1.35375 −0.676873 0.736100i \(-0.736665\pi\)
−0.676873 + 0.736100i \(0.736665\pi\)
\(314\) 0 0
\(315\) 1.08564 0.0611687
\(316\) 0 0
\(317\) −9.79650 −0.550226 −0.275113 0.961412i \(-0.588715\pi\)
−0.275113 + 0.961412i \(0.588715\pi\)
\(318\) 0 0
\(319\) −43.3628 −2.42785
\(320\) 0 0
\(321\) 15.7857 0.881069
\(322\) 0 0
\(323\) 10.4613 0.582082
\(324\) 0 0
\(325\) −19.7334 −1.09461
\(326\) 0 0
\(327\) −1.33903 −0.0740487
\(328\) 0 0
\(329\) −9.98533 −0.550509
\(330\) 0 0
\(331\) 3.37658 0.185593 0.0927967 0.995685i \(-0.470419\pi\)
0.0927967 + 0.995685i \(0.470419\pi\)
\(332\) 0 0
\(333\) −3.90703 −0.214104
\(334\) 0 0
\(335\) −8.99267 −0.491322
\(336\) 0 0
\(337\) −23.1142 −1.25911 −0.629554 0.776956i \(-0.716762\pi\)
−0.629554 + 0.776956i \(0.716762\pi\)
\(338\) 0 0
\(339\) 6.58417 0.357603
\(340\) 0 0
\(341\) −26.7147 −1.44668
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.08564 −0.0584488
\(346\) 0 0
\(347\) 21.6443 1.16193 0.580963 0.813930i \(-0.302676\pi\)
0.580963 + 0.813930i \(0.302676\pi\)
\(348\) 0 0
\(349\) 2.42756 0.129944 0.0649721 0.997887i \(-0.479304\pi\)
0.0649721 + 0.997887i \(0.479304\pi\)
\(350\) 0 0
\(351\) 5.16394 0.275631
\(352\) 0 0
\(353\) −15.3528 −0.817146 −0.408573 0.912726i \(-0.633973\pi\)
−0.408573 + 0.912726i \(0.633973\pi\)
\(354\) 0 0
\(355\) 15.4056 0.817642
\(356\) 0 0
\(357\) 2.56511 0.135760
\(358\) 0 0
\(359\) −14.0646 −0.742299 −0.371150 0.928573i \(-0.621036\pi\)
−0.371150 + 0.928573i \(0.621036\pi\)
\(360\) 0 0
\(361\) −2.36744 −0.124602
\(362\) 0 0
\(363\) 5.63256 0.295633
\(364\) 0 0
\(365\) 5.04520 0.264078
\(366\) 0 0
\(367\) −21.9018 −1.14327 −0.571633 0.820509i \(-0.693690\pi\)
−0.571633 + 0.820509i \(0.693690\pi\)
\(368\) 0 0
\(369\) −1.43489 −0.0746975
\(370\) 0 0
\(371\) −9.15309 −0.475205
\(372\) 0 0
\(373\) 26.1458 1.35378 0.676888 0.736086i \(-0.263328\pi\)
0.676888 + 0.736086i \(0.263328\pi\)
\(374\) 0 0
\(375\) 9.57683 0.494546
\(376\) 0 0
\(377\) −54.9059 −2.82780
\(378\) 0 0
\(379\) −20.2657 −1.04098 −0.520491 0.853867i \(-0.674251\pi\)
−0.520491 + 0.853867i \(0.674251\pi\)
\(380\) 0 0
\(381\) 12.3616 0.633304
\(382\) 0 0
\(383\) −7.05723 −0.360608 −0.180304 0.983611i \(-0.557708\pi\)
−0.180304 + 0.983611i \(0.557708\pi\)
\(384\) 0 0
\(385\) 4.42756 0.225649
\(386\) 0 0
\(387\) −12.8484 −0.653122
\(388\) 0 0
\(389\) −29.7956 −1.51070 −0.755348 0.655324i \(-0.772532\pi\)
−0.755348 + 0.655324i \(0.772532\pi\)
\(390\) 0 0
\(391\) −2.56511 −0.129723
\(392\) 0 0
\(393\) −7.59150 −0.382941
\(394\) 0 0
\(395\) −7.11140 −0.357813
\(396\) 0 0
\(397\) 33.9356 1.70318 0.851588 0.524211i \(-0.175640\pi\)
0.851588 + 0.524211i \(0.175640\pi\)
\(398\) 0 0
\(399\) 4.07830 0.204171
\(400\) 0 0
\(401\) −1.27829 −0.0638345 −0.0319173 0.999491i \(-0.510161\pi\)
−0.0319173 + 0.999491i \(0.510161\pi\)
\(402\) 0 0
\(403\) −33.8261 −1.68500
\(404\) 0 0
\(405\) −1.08564 −0.0539458
\(406\) 0 0
\(407\) −15.9340 −0.789822
\(408\) 0 0
\(409\) −2.53000 −0.125100 −0.0625501 0.998042i \(-0.519923\pi\)
−0.0625501 + 0.998042i \(0.519923\pi\)
\(410\) 0 0
\(411\) −10.8185 −0.533637
\(412\) 0 0
\(413\) 3.63989 0.179107
\(414\) 0 0
\(415\) 9.62523 0.472484
\(416\) 0 0
\(417\) 12.6733 0.620615
\(418\) 0 0
\(419\) 8.79010 0.429424 0.214712 0.976677i \(-0.431119\pi\)
0.214712 + 0.976677i \(0.431119\pi\)
\(420\) 0 0
\(421\) −3.70353 −0.180499 −0.0902495 0.995919i \(-0.528766\pi\)
−0.0902495 + 0.995919i \(0.528766\pi\)
\(422\) 0 0
\(423\) 9.98533 0.485503
\(424\) 0 0
\(425\) 9.80228 0.475480
\(426\) 0 0
\(427\) −1.90414 −0.0921478
\(428\) 0 0
\(429\) 21.0601 1.01679
\(430\) 0 0
\(431\) −18.8080 −0.905949 −0.452974 0.891524i \(-0.649637\pi\)
−0.452974 + 0.891524i \(0.649637\pi\)
\(432\) 0 0
\(433\) 3.80702 0.182954 0.0914770 0.995807i \(-0.470841\pi\)
0.0914770 + 0.995807i \(0.470841\pi\)
\(434\) 0 0
\(435\) 11.5431 0.553450
\(436\) 0 0
\(437\) −4.07830 −0.195092
\(438\) 0 0
\(439\) 37.6385 1.79639 0.898194 0.439599i \(-0.144880\pi\)
0.898194 + 0.439599i \(0.144880\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 1.62937 0.0774138 0.0387069 0.999251i \(-0.487676\pi\)
0.0387069 + 0.999251i \(0.487676\pi\)
\(444\) 0 0
\(445\) 11.5622 0.548099
\(446\) 0 0
\(447\) −12.5613 −0.594129
\(448\) 0 0
\(449\) −22.4816 −1.06097 −0.530486 0.847694i \(-0.677991\pi\)
−0.530486 + 0.847694i \(0.677991\pi\)
\(450\) 0 0
\(451\) −5.85193 −0.275557
\(452\) 0 0
\(453\) 21.3543 1.00331
\(454\) 0 0
\(455\) 5.60617 0.262821
\(456\) 0 0
\(457\) 20.7554 0.970899 0.485449 0.874265i \(-0.338656\pi\)
0.485449 + 0.874265i \(0.338656\pi\)
\(458\) 0 0
\(459\) −2.56511 −0.119729
\(460\) 0 0
\(461\) 35.4393 1.65057 0.825286 0.564715i \(-0.191014\pi\)
0.825286 + 0.564715i \(0.191014\pi\)
\(462\) 0 0
\(463\) 37.6015 1.74749 0.873746 0.486383i \(-0.161684\pi\)
0.873746 + 0.486383i \(0.161684\pi\)
\(464\) 0 0
\(465\) 7.11140 0.329783
\(466\) 0 0
\(467\) −32.0490 −1.48305 −0.741525 0.670926i \(-0.765897\pi\)
−0.741525 + 0.670926i \(0.765897\pi\)
\(468\) 0 0
\(469\) −8.28331 −0.382488
\(470\) 0 0
\(471\) −11.4311 −0.526716
\(472\) 0 0
\(473\) −52.3997 −2.40934
\(474\) 0 0
\(475\) 15.5848 0.715079
\(476\) 0 0
\(477\) 9.15309 0.419091
\(478\) 0 0
\(479\) −29.0132 −1.32565 −0.662824 0.748775i \(-0.730642\pi\)
−0.662824 + 0.748775i \(0.730642\pi\)
\(480\) 0 0
\(481\) −20.1757 −0.919931
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 5.04520 0.229091
\(486\) 0 0
\(487\) −27.0279 −1.22475 −0.612375 0.790567i \(-0.709786\pi\)
−0.612375 + 0.790567i \(0.709786\pi\)
\(488\) 0 0
\(489\) −16.8376 −0.761421
\(490\) 0 0
\(491\) 4.85776 0.219228 0.109614 0.993974i \(-0.465039\pi\)
0.109614 + 0.993974i \(0.465039\pi\)
\(492\) 0 0
\(493\) 27.2737 1.22834
\(494\) 0 0
\(495\) −4.42756 −0.199004
\(496\) 0 0
\(497\) 14.1903 0.636523
\(498\) 0 0
\(499\) −5.49032 −0.245780 −0.122890 0.992420i \(-0.539216\pi\)
−0.122890 + 0.992420i \(0.539216\pi\)
\(500\) 0 0
\(501\) 6.82554 0.304942
\(502\) 0 0
\(503\) −4.70266 −0.209681 −0.104841 0.994489i \(-0.533433\pi\)
−0.104841 + 0.994489i \(0.533433\pi\)
\(504\) 0 0
\(505\) 12.9194 0.574907
\(506\) 0 0
\(507\) 13.6663 0.606941
\(508\) 0 0
\(509\) −13.1155 −0.581336 −0.290668 0.956824i \(-0.593878\pi\)
−0.290668 + 0.956824i \(0.593878\pi\)
\(510\) 0 0
\(511\) 4.64723 0.205581
\(512\) 0 0
\(513\) −4.07830 −0.180062
\(514\) 0 0
\(515\) −12.3425 −0.543878
\(516\) 0 0
\(517\) 40.7232 1.79101
\(518\) 0 0
\(519\) −24.2753 −1.06557
\(520\) 0 0
\(521\) −22.5068 −0.986040 −0.493020 0.870018i \(-0.664107\pi\)
−0.493020 + 0.870018i \(0.664107\pi\)
\(522\) 0 0
\(523\) 1.34192 0.0586781 0.0293391 0.999570i \(-0.490660\pi\)
0.0293391 + 0.999570i \(0.490660\pi\)
\(524\) 0 0
\(525\) 3.82139 0.166779
\(526\) 0 0
\(527\) 16.8026 0.731932
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.63989 −0.157958
\(532\) 0 0
\(533\) −7.40970 −0.320950
\(534\) 0 0
\(535\) −17.1375 −0.740918
\(536\) 0 0
\(537\) −22.4803 −0.970098
\(538\) 0 0
\(539\) 4.07830 0.175665
\(540\) 0 0
\(541\) 44.2909 1.90422 0.952108 0.305763i \(-0.0989117\pi\)
0.952108 + 0.305763i \(0.0989117\pi\)
\(542\) 0 0
\(543\) 9.19385 0.394546
\(544\) 0 0
\(545\) 1.45370 0.0622699
\(546\) 0 0
\(547\) 8.00414 0.342233 0.171116 0.985251i \(-0.445263\pi\)
0.171116 + 0.985251i \(0.445263\pi\)
\(548\) 0 0
\(549\) 1.90414 0.0812667
\(550\) 0 0
\(551\) 43.3628 1.84732
\(552\) 0 0
\(553\) −6.55044 −0.278553
\(554\) 0 0
\(555\) 4.24162 0.180047
\(556\) 0 0
\(557\) 32.9362 1.39555 0.697775 0.716317i \(-0.254173\pi\)
0.697775 + 0.716317i \(0.254173\pi\)
\(558\) 0 0
\(559\) −66.3484 −2.80624
\(560\) 0 0
\(561\) −10.4613 −0.441676
\(562\) 0 0
\(563\) −20.4033 −0.859896 −0.429948 0.902854i \(-0.641468\pi\)
−0.429948 + 0.902854i \(0.641468\pi\)
\(564\) 0 0
\(565\) −7.14802 −0.300719
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 5.50084 0.230607 0.115304 0.993330i \(-0.463216\pi\)
0.115304 + 0.993330i \(0.463216\pi\)
\(570\) 0 0
\(571\) 12.4285 0.520116 0.260058 0.965593i \(-0.416258\pi\)
0.260058 + 0.965593i \(0.416258\pi\)
\(572\) 0 0
\(573\) −21.3543 −0.892088
\(574\) 0 0
\(575\) −3.82139 −0.159363
\(576\) 0 0
\(577\) −9.83576 −0.409468 −0.204734 0.978818i \(-0.565633\pi\)
−0.204734 + 0.978818i \(0.565633\pi\)
\(578\) 0 0
\(579\) −18.2458 −0.758268
\(580\) 0 0
\(581\) 8.86597 0.367822
\(582\) 0 0
\(583\) 37.3291 1.54601
\(584\) 0 0
\(585\) −5.60617 −0.231787
\(586\) 0 0
\(587\) 9.20332 0.379861 0.189931 0.981797i \(-0.439174\pi\)
0.189931 + 0.981797i \(0.439174\pi\)
\(588\) 0 0
\(589\) 26.7147 1.10076
\(590\) 0 0
\(591\) 15.1493 0.623158
\(592\) 0 0
\(593\) 13.1772 0.541124 0.270562 0.962703i \(-0.412791\pi\)
0.270562 + 0.962703i \(0.412791\pi\)
\(594\) 0 0
\(595\) −2.78478 −0.114165
\(596\) 0 0
\(597\) 6.53138 0.267312
\(598\) 0 0
\(599\) −28.9499 −1.18286 −0.591431 0.806356i \(-0.701437\pi\)
−0.591431 + 0.806356i \(0.701437\pi\)
\(600\) 0 0
\(601\) 38.7542 1.58082 0.790408 0.612581i \(-0.209869\pi\)
0.790408 + 0.612581i \(0.209869\pi\)
\(602\) 0 0
\(603\) 8.28331 0.337322
\(604\) 0 0
\(605\) −6.11492 −0.248607
\(606\) 0 0
\(607\) −2.54630 −0.103351 −0.0516755 0.998664i \(-0.516456\pi\)
−0.0516755 + 0.998664i \(0.516456\pi\)
\(608\) 0 0
\(609\) 10.6326 0.430853
\(610\) 0 0
\(611\) 51.5637 2.08604
\(612\) 0 0
\(613\) 15.1251 0.610899 0.305449 0.952208i \(-0.401193\pi\)
0.305449 + 0.952208i \(0.401193\pi\)
\(614\) 0 0
\(615\) 1.55777 0.0628155
\(616\) 0 0
\(617\) 3.59439 0.144705 0.0723523 0.997379i \(-0.476949\pi\)
0.0723523 + 0.997379i \(0.476949\pi\)
\(618\) 0 0
\(619\) 28.2231 1.13438 0.567192 0.823586i \(-0.308030\pi\)
0.567192 + 0.823586i \(0.308030\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 10.6501 0.426688
\(624\) 0 0
\(625\) 8.70999 0.348400
\(626\) 0 0
\(627\) −16.6326 −0.664240
\(628\) 0 0
\(629\) 10.0219 0.399601
\(630\) 0 0
\(631\) 35.4596 1.41162 0.705812 0.708399i \(-0.250582\pi\)
0.705812 + 0.708399i \(0.250582\pi\)
\(632\) 0 0
\(633\) 2.64723 0.105218
\(634\) 0 0
\(635\) −13.4202 −0.532565
\(636\) 0 0
\(637\) 5.16394 0.204603
\(638\) 0 0
\(639\) −14.1903 −0.561361
\(640\) 0 0
\(641\) 28.5710 1.12849 0.564243 0.825609i \(-0.309168\pi\)
0.564243 + 0.825609i \(0.309168\pi\)
\(642\) 0 0
\(643\) −26.1429 −1.03097 −0.515487 0.856897i \(-0.672389\pi\)
−0.515487 + 0.856897i \(0.672389\pi\)
\(644\) 0 0
\(645\) 13.9487 0.549230
\(646\) 0 0
\(647\) 42.6235 1.67570 0.837852 0.545897i \(-0.183811\pi\)
0.837852 + 0.545897i \(0.183811\pi\)
\(648\) 0 0
\(649\) −14.8446 −0.582701
\(650\) 0 0
\(651\) 6.55044 0.256732
\(652\) 0 0
\(653\) −13.8857 −0.543388 −0.271694 0.962384i \(-0.587584\pi\)
−0.271694 + 0.962384i \(0.587584\pi\)
\(654\) 0 0
\(655\) 8.24162 0.322027
\(656\) 0 0
\(657\) −4.64723 −0.181306
\(658\) 0 0
\(659\) −37.1498 −1.44715 −0.723576 0.690244i \(-0.757503\pi\)
−0.723576 + 0.690244i \(0.757503\pi\)
\(660\) 0 0
\(661\) 16.5088 0.642116 0.321058 0.947060i \(-0.395962\pi\)
0.321058 + 0.947060i \(0.395962\pi\)
\(662\) 0 0
\(663\) −13.2461 −0.514434
\(664\) 0 0
\(665\) −4.42756 −0.171693
\(666\) 0 0
\(667\) −10.6326 −0.411694
\(668\) 0 0
\(669\) 17.7818 0.687485
\(670\) 0 0
\(671\) 7.76566 0.299790
\(672\) 0 0
\(673\) −0.536374 −0.0206757 −0.0103379 0.999947i \(-0.503291\pi\)
−0.0103379 + 0.999947i \(0.503291\pi\)
\(674\) 0 0
\(675\) −3.82139 −0.147085
\(676\) 0 0
\(677\) 10.9376 0.420365 0.210182 0.977662i \(-0.432594\pi\)
0.210182 + 0.977662i \(0.432594\pi\)
\(678\) 0 0
\(679\) 4.64723 0.178344
\(680\) 0 0
\(681\) 13.7695 0.527648
\(682\) 0 0
\(683\) −18.6472 −0.713516 −0.356758 0.934197i \(-0.616118\pi\)
−0.356758 + 0.934197i \(0.616118\pi\)
\(684\) 0 0
\(685\) 11.7450 0.448752
\(686\) 0 0
\(687\) −20.6112 −0.786366
\(688\) 0 0
\(689\) 47.2660 1.80069
\(690\) 0 0
\(691\) −42.6385 −1.62204 −0.811022 0.585016i \(-0.801088\pi\)
−0.811022 + 0.585016i \(0.801088\pi\)
\(692\) 0 0
\(693\) −4.07830 −0.154922
\(694\) 0 0
\(695\) −13.7586 −0.521895
\(696\) 0 0
\(697\) 3.68065 0.139415
\(698\) 0 0
\(699\) 3.37183 0.127534
\(700\) 0 0
\(701\) −43.8910 −1.65774 −0.828870 0.559442i \(-0.811016\pi\)
−0.828870 + 0.559442i \(0.811016\pi\)
\(702\) 0 0
\(703\) 15.9340 0.600964
\(704\) 0 0
\(705\) −10.8405 −0.408275
\(706\) 0 0
\(707\) 11.9003 0.447558
\(708\) 0 0
\(709\) 0.123938 0.00465461 0.00232730 0.999997i \(-0.499259\pi\)
0.00232730 + 0.999997i \(0.499259\pi\)
\(710\) 0 0
\(711\) 6.55044 0.245661
\(712\) 0 0
\(713\) −6.55044 −0.245316
\(714\) 0 0
\(715\) −22.8637 −0.855052
\(716\) 0 0
\(717\) −17.7818 −0.664075
\(718\) 0 0
\(719\) 44.8215 1.67156 0.835780 0.549064i \(-0.185016\pi\)
0.835780 + 0.549064i \(0.185016\pi\)
\(720\) 0 0
\(721\) −11.3689 −0.423402
\(722\) 0 0
\(723\) −25.0630 −0.932103
\(724\) 0 0
\(725\) 40.6312 1.50900
\(726\) 0 0
\(727\) −15.5023 −0.574950 −0.287475 0.957788i \(-0.592816\pi\)
−0.287475 + 0.957788i \(0.592816\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.9576 1.21898
\(732\) 0 0
\(733\) −0.458725 −0.0169434 −0.00847169 0.999964i \(-0.502697\pi\)
−0.00847169 + 0.999964i \(0.502697\pi\)
\(734\) 0 0
\(735\) −1.08564 −0.0400443
\(736\) 0 0
\(737\) 33.7818 1.24437
\(738\) 0 0
\(739\) 46.2707 1.70210 0.851048 0.525088i \(-0.175968\pi\)
0.851048 + 0.525088i \(0.175968\pi\)
\(740\) 0 0
\(741\) −21.0601 −0.773663
\(742\) 0 0
\(743\) 13.9461 0.511632 0.255816 0.966726i \(-0.417656\pi\)
0.255816 + 0.966726i \(0.417656\pi\)
\(744\) 0 0
\(745\) 13.6370 0.499621
\(746\) 0 0
\(747\) −8.86597 −0.324389
\(748\) 0 0
\(749\) −15.7857 −0.576795
\(750\) 0 0
\(751\) −35.3355 −1.28941 −0.644705 0.764432i \(-0.723020\pi\)
−0.644705 + 0.764432i \(0.723020\pi\)
\(752\) 0 0
\(753\) −0.465103 −0.0169493
\(754\) 0 0
\(755\) −23.1830 −0.843716
\(756\) 0 0
\(757\) 21.8249 0.793240 0.396620 0.917983i \(-0.370183\pi\)
0.396620 + 0.917983i \(0.370183\pi\)
\(758\) 0 0
\(759\) 4.07830 0.148033
\(760\) 0 0
\(761\) 1.68979 0.0612549 0.0306275 0.999531i \(-0.490249\pi\)
0.0306275 + 0.999531i \(0.490249\pi\)
\(762\) 0 0
\(763\) 1.33903 0.0484763
\(764\) 0 0
\(765\) 2.78478 0.100684
\(766\) 0 0
\(767\) −18.7962 −0.678691
\(768\) 0 0
\(769\) −5.72490 −0.206445 −0.103223 0.994658i \(-0.532915\pi\)
−0.103223 + 0.994658i \(0.532915\pi\)
\(770\) 0 0
\(771\) 17.5930 0.633597
\(772\) 0 0
\(773\) 15.9451 0.573507 0.286754 0.958004i \(-0.407424\pi\)
0.286754 + 0.958004i \(0.407424\pi\)
\(774\) 0 0
\(775\) 25.0318 0.899169
\(776\) 0 0
\(777\) 3.90703 0.140164
\(778\) 0 0
\(779\) 5.85193 0.209667
\(780\) 0 0
\(781\) −57.8725 −2.07084
\(782\) 0 0
\(783\) −10.6326 −0.379977
\(784\) 0 0
\(785\) 12.4100 0.442932
\(786\) 0 0
\(787\) 32.2082 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(788\) 0 0
\(789\) −5.69532 −0.202759
\(790\) 0 0
\(791\) −6.58417 −0.234106
\(792\) 0 0
\(793\) 9.83287 0.349175
\(794\) 0 0
\(795\) −9.93694 −0.352427
\(796\) 0 0
\(797\) 20.1238 0.712822 0.356411 0.934329i \(-0.384000\pi\)
0.356411 + 0.934329i \(0.384000\pi\)
\(798\) 0 0
\(799\) −25.6134 −0.906139
\(800\) 0 0
\(801\) −10.6501 −0.376303
\(802\) 0 0
\(803\) −18.9528 −0.668830
\(804\) 0 0
\(805\) 1.08564 0.0382637
\(806\) 0 0
\(807\) 8.27949 0.291452
\(808\) 0 0
\(809\) −36.4437 −1.28129 −0.640647 0.767836i \(-0.721334\pi\)
−0.640647 + 0.767836i \(0.721334\pi\)
\(810\) 0 0
\(811\) 47.7264 1.67590 0.837950 0.545746i \(-0.183754\pi\)
0.837950 + 0.545746i \(0.183754\pi\)
\(812\) 0 0
\(813\) 12.0366 0.422143
\(814\) 0 0
\(815\) 18.2795 0.640303
\(816\) 0 0
\(817\) 52.3997 1.83323
\(818\) 0 0
\(819\) −5.16394 −0.180443
\(820\) 0 0
\(821\) 14.3883 0.502155 0.251078 0.967967i \(-0.419215\pi\)
0.251078 + 0.967967i \(0.419215\pi\)
\(822\) 0 0
\(823\) −32.9484 −1.14851 −0.574255 0.818677i \(-0.694708\pi\)
−0.574255 + 0.818677i \(0.694708\pi\)
\(824\) 0 0
\(825\) −15.5848 −0.542593
\(826\) 0 0
\(827\) −7.61162 −0.264682 −0.132341 0.991204i \(-0.542249\pi\)
−0.132341 + 0.991204i \(0.542249\pi\)
\(828\) 0 0
\(829\) −40.2033 −1.39632 −0.698159 0.715943i \(-0.745997\pi\)
−0.698159 + 0.715943i \(0.745997\pi\)
\(830\) 0 0
\(831\) −12.5329 −0.434761
\(832\) 0 0
\(833\) −2.56511 −0.0888757
\(834\) 0 0
\(835\) −7.41006 −0.256436
\(836\) 0 0
\(837\) −6.55044 −0.226416
\(838\) 0 0
\(839\) 4.18330 0.144424 0.0722118 0.997389i \(-0.476994\pi\)
0.0722118 + 0.997389i \(0.476994\pi\)
\(840\) 0 0
\(841\) 84.0513 2.89832
\(842\) 0 0
\(843\) −13.2181 −0.455256
\(844\) 0 0
\(845\) −14.8366 −0.510396
\(846\) 0 0
\(847\) −5.63256 −0.193537
\(848\) 0 0
\(849\) −15.3314 −0.526172
\(850\) 0 0
\(851\) −3.90703 −0.133931
\(852\) 0 0
\(853\) −1.72490 −0.0590596 −0.0295298 0.999564i \(-0.509401\pi\)
−0.0295298 + 0.999564i \(0.509401\pi\)
\(854\) 0 0
\(855\) 4.42756 0.151419
\(856\) 0 0
\(857\) −37.8445 −1.29274 −0.646372 0.763022i \(-0.723715\pi\)
−0.646372 + 0.763022i \(0.723715\pi\)
\(858\) 0 0
\(859\) 29.2651 0.998513 0.499257 0.866454i \(-0.333607\pi\)
0.499257 + 0.866454i \(0.333607\pi\)
\(860\) 0 0
\(861\) 1.43489 0.0489010
\(862\) 0 0
\(863\) −51.5637 −1.75525 −0.877624 0.479350i \(-0.840872\pi\)
−0.877624 + 0.479350i \(0.840872\pi\)
\(864\) 0 0
\(865\) 26.3542 0.896070
\(866\) 0 0
\(867\) −10.4202 −0.353889
\(868\) 0 0
\(869\) 26.7147 0.906234
\(870\) 0 0
\(871\) 42.7745 1.44936
\(872\) 0 0
\(873\) −4.64723 −0.157285
\(874\) 0 0
\(875\) −9.57683 −0.323756
\(876\) 0 0
\(877\) 21.1583 0.714465 0.357232 0.934016i \(-0.383720\pi\)
0.357232 + 0.934016i \(0.383720\pi\)
\(878\) 0 0
\(879\) −11.9977 −0.404672
\(880\) 0 0
\(881\) 31.0258 1.04529 0.522643 0.852552i \(-0.324946\pi\)
0.522643 + 0.852552i \(0.324946\pi\)
\(882\) 0 0
\(883\) 48.1597 1.62070 0.810352 0.585943i \(-0.199276\pi\)
0.810352 + 0.585943i \(0.199276\pi\)
\(884\) 0 0
\(885\) 3.95161 0.132832
\(886\) 0 0
\(887\) −26.3176 −0.883659 −0.441829 0.897099i \(-0.645670\pi\)
−0.441829 + 0.897099i \(0.645670\pi\)
\(888\) 0 0
\(889\) −12.3616 −0.414595
\(890\) 0 0
\(891\) 4.07830 0.136628
\(892\) 0 0
\(893\) −40.7232 −1.36275
\(894\) 0 0
\(895\) 24.4055 0.815786
\(896\) 0 0
\(897\) 5.16394 0.172419
\(898\) 0 0
\(899\) 69.6479 2.32289
\(900\) 0 0
\(901\) −23.4787 −0.782188
\(902\) 0 0
\(903\) 12.8484 0.427568
\(904\) 0 0
\(905\) −9.98119 −0.331786
\(906\) 0 0
\(907\) 18.5303 0.615289 0.307645 0.951501i \(-0.400459\pi\)
0.307645 + 0.951501i \(0.400459\pi\)
\(908\) 0 0
\(909\) −11.9003 −0.394709
\(910\) 0 0
\(911\) 25.0117 0.828675 0.414338 0.910123i \(-0.364013\pi\)
0.414338 + 0.910123i \(0.364013\pi\)
\(912\) 0 0
\(913\) −36.1581 −1.19666
\(914\) 0 0
\(915\) −2.06721 −0.0683397
\(916\) 0 0
\(917\) 7.59150 0.250693
\(918\) 0 0
\(919\) −36.7434 −1.21205 −0.606027 0.795444i \(-0.707238\pi\)
−0.606027 + 0.795444i \(0.707238\pi\)
\(920\) 0 0
\(921\) 14.7147 0.484865
\(922\) 0 0
\(923\) −73.2780 −2.41198
\(924\) 0 0
\(925\) 14.9303 0.490905
\(926\) 0 0
\(927\) 11.3689 0.373405
\(928\) 0 0
\(929\) −57.9256 −1.90048 −0.950239 0.311521i \(-0.899161\pi\)
−0.950239 + 0.311521i \(0.899161\pi\)
\(930\) 0 0
\(931\) −4.07830 −0.133661
\(932\) 0 0
\(933\) 31.0455 1.01638
\(934\) 0 0
\(935\) 11.3572 0.371419
\(936\) 0 0
\(937\) 1.71762 0.0561123 0.0280562 0.999606i \(-0.491068\pi\)
0.0280562 + 0.999606i \(0.491068\pi\)
\(938\) 0 0
\(939\) −23.9502 −0.781586
\(940\) 0 0
\(941\) 29.2552 0.953691 0.476846 0.878987i \(-0.341780\pi\)
0.476846 + 0.878987i \(0.341780\pi\)
\(942\) 0 0
\(943\) −1.43489 −0.0467265
\(944\) 0 0
\(945\) 1.08564 0.0353158
\(946\) 0 0
\(947\) 11.2779 0.366484 0.183242 0.983068i \(-0.441341\pi\)
0.183242 + 0.983068i \(0.441341\pi\)
\(948\) 0 0
\(949\) −23.9980 −0.779008
\(950\) 0 0
\(951\) −9.79650 −0.317673
\(952\) 0 0
\(953\) −35.0449 −1.13521 −0.567607 0.823300i \(-0.692131\pi\)
−0.567607 + 0.823300i \(0.692131\pi\)
\(954\) 0 0
\(955\) 23.1830 0.750184
\(956\) 0 0
\(957\) −43.3628 −1.40172
\(958\) 0 0
\(959\) 10.8185 0.349348
\(960\) 0 0
\(961\) 11.9083 0.384137
\(962\) 0 0
\(963\) 15.7857 0.508686
\(964\) 0 0
\(965\) 19.8083 0.637651
\(966\) 0 0
\(967\) −43.4153 −1.39614 −0.698072 0.716028i \(-0.745958\pi\)
−0.698072 + 0.716028i \(0.745958\pi\)
\(968\) 0 0
\(969\) 10.4613 0.336065
\(970\) 0 0
\(971\) 19.2948 0.619199 0.309600 0.950867i \(-0.399805\pi\)
0.309600 + 0.950867i \(0.399805\pi\)
\(972\) 0 0
\(973\) −12.6733 −0.406288
\(974\) 0 0
\(975\) −19.7334 −0.631976
\(976\) 0 0
\(977\) −43.1065 −1.37910 −0.689550 0.724238i \(-0.742192\pi\)
−0.689550 + 0.724238i \(0.742192\pi\)
\(978\) 0 0
\(979\) −43.4344 −1.38817
\(980\) 0 0
\(981\) −1.33903 −0.0427520
\(982\) 0 0
\(983\) −30.7687 −0.981370 −0.490685 0.871337i \(-0.663253\pi\)
−0.490685 + 0.871337i \(0.663253\pi\)
\(984\) 0 0
\(985\) −16.4466 −0.524033
\(986\) 0 0
\(987\) −9.98533 −0.317837
\(988\) 0 0
\(989\) −12.8484 −0.408556
\(990\) 0 0
\(991\) −19.8948 −0.631979 −0.315989 0.948763i \(-0.602336\pi\)
−0.315989 + 0.948763i \(0.602336\pi\)
\(992\) 0 0
\(993\) 3.37658 0.107152
\(994\) 0 0
\(995\) −7.09071 −0.224791
\(996\) 0 0
\(997\) 18.2530 0.578080 0.289040 0.957317i \(-0.406664\pi\)
0.289040 + 0.957317i \(0.406664\pi\)
\(998\) 0 0
\(999\) −3.90703 −0.123613
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 7728.2.a.cb.1.2 4
4.3 odd 2 3864.2.a.r.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.r.1.2 4 4.3 odd 2
7728.2.a.cb.1.2 4 1.1 even 1 trivial