Properties

Label 950.2.a.l.1.1
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.77339\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.77339 q^{3} +1.00000 q^{4} -1.77339 q^{6} -2.69168 q^{7} +1.00000 q^{8} +0.144903 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.77339 q^{3} +1.00000 q^{4} -1.77339 q^{6} -2.69168 q^{7} +1.00000 q^{8} +0.144903 q^{9} +5.54677 q^{11} -1.77339 q^{12} -2.91829 q^{13} -2.69168 q^{14} +1.00000 q^{16} +4.91829 q^{17} +0.144903 q^{18} +1.00000 q^{19} +4.77339 q^{21} +5.54677 q^{22} +3.60997 q^{23} -1.77339 q^{24} -2.91829 q^{26} +5.06319 q^{27} -2.69168 q^{28} +1.08171 q^{29} +7.54677 q^{31} +1.00000 q^{32} -9.83658 q^{33} +4.91829 q^{34} +0.144903 q^{36} +4.54677 q^{37} +1.00000 q^{38} +5.17526 q^{39} +4.77339 q^{42} -9.54677 q^{43} +5.54677 q^{44} +3.60997 q^{46} -0.836581 q^{47} -1.77339 q^{48} +0.245129 q^{49} -8.72203 q^{51} -2.91829 q^{52} +9.78523 q^{53} +5.06319 q^{54} -2.69168 q^{56} -1.77339 q^{57} +1.08171 q^{58} +12.9933 q^{59} -7.38336 q^{61} +7.54677 q^{62} -0.390032 q^{63} +1.00000 q^{64} -9.83658 q^{66} +2.85510 q^{67} +4.91829 q^{68} -6.40187 q^{69} +14.4769 q^{71} +0.144903 q^{72} +5.15674 q^{73} +4.54677 q^{74} +1.00000 q^{76} -14.9301 q^{77} +5.17526 q^{78} -3.09355 q^{79} -9.41371 q^{81} +1.71019 q^{83} +4.77339 q^{84} -9.54677 q^{86} -1.91829 q^{87} +5.54677 q^{88} -5.09355 q^{89} +7.85510 q^{91} +3.60997 q^{92} -13.3834 q^{93} -0.836581 q^{94} -1.77339 q^{96} -17.2570 q^{97} +0.245129 q^{98} +0.803744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + 5q^{9} + 2q^{11} + 2q^{12} - 6q^{13} + 2q^{14} + 3q^{16} + 12q^{17} + 5q^{18} + 3q^{19} + 7q^{21} + 2q^{22} - 2q^{23} + 2q^{24} - 6q^{26} + 17q^{27} + 2q^{28} + 6q^{29} + 8q^{31} + 3q^{32} - 24q^{33} + 12q^{34} + 5q^{36} - q^{37} + 3q^{38} - 11q^{39} + 7q^{42} - 14q^{43} + 2q^{44} - 2q^{46} + 3q^{47} + 2q^{48} + 9q^{49} + 15q^{51} - 6q^{52} - 10q^{53} + 17q^{54} + 2q^{56} + 2q^{57} + 6q^{58} + 6q^{59} - 2q^{61} + 8q^{62} - 14q^{63} + 3q^{64} - 24q^{66} + 4q^{67} + 12q^{68} - 6q^{71} + 5q^{72} - 12q^{73} - q^{74} + 3q^{76} - 10q^{77} - 11q^{78} + 20q^{79} + 23q^{81} - 4q^{83} + 7q^{84} - 14q^{86} - 3q^{87} + 2q^{88} + 14q^{89} + 19q^{91} - 2q^{92} - 20q^{93} + 3q^{94} + 2q^{96} - 28q^{97} + 9q^{98} - 36q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.77339 −1.02387 −0.511933 0.859025i \(-0.671070\pi\)
−0.511933 + 0.859025i \(0.671070\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.77339 −0.723982
\(7\) −2.69168 −1.01736 −0.508679 0.860956i \(-0.669866\pi\)
−0.508679 + 0.860956i \(0.669866\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.144903 0.0483010
\(10\) 0 0
\(11\) 5.54677 1.67242 0.836208 0.548413i \(-0.184768\pi\)
0.836208 + 0.548413i \(0.184768\pi\)
\(12\) −1.77339 −0.511933
\(13\) −2.91829 −0.809388 −0.404694 0.914452i \(-0.632622\pi\)
−0.404694 + 0.914452i \(0.632622\pi\)
\(14\) −2.69168 −0.719381
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.91829 1.19286 0.596430 0.802665i \(-0.296585\pi\)
0.596430 + 0.802665i \(0.296585\pi\)
\(18\) 0.144903 0.0341539
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.77339 1.04164
\(22\) 5.54677 1.18258
\(23\) 3.60997 0.752730 0.376365 0.926471i \(-0.377174\pi\)
0.376365 + 0.926471i \(0.377174\pi\)
\(24\) −1.77339 −0.361991
\(25\) 0 0
\(26\) −2.91829 −0.572324
\(27\) 5.06319 0.974412
\(28\) −2.69168 −0.508679
\(29\) 1.08171 0.200868 0.100434 0.994944i \(-0.467977\pi\)
0.100434 + 0.994944i \(0.467977\pi\)
\(30\) 0 0
\(31\) 7.54677 1.35544 0.677720 0.735320i \(-0.262968\pi\)
0.677720 + 0.735320i \(0.262968\pi\)
\(32\) 1.00000 0.176777
\(33\) −9.83658 −1.71233
\(34\) 4.91829 0.843480
\(35\) 0 0
\(36\) 0.144903 0.0241505
\(37\) 4.54677 0.747485 0.373743 0.927532i \(-0.378074\pi\)
0.373743 + 0.927532i \(0.378074\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.17526 0.828705
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 4.77339 0.736550
\(43\) −9.54677 −1.45587 −0.727935 0.685646i \(-0.759520\pi\)
−0.727935 + 0.685646i \(0.759520\pi\)
\(44\) 5.54677 0.836208
\(45\) 0 0
\(46\) 3.60997 0.532261
\(47\) −0.836581 −0.122028 −0.0610139 0.998137i \(-0.519433\pi\)
−0.0610139 + 0.998137i \(0.519433\pi\)
\(48\) −1.77339 −0.255966
\(49\) 0.245129 0.0350184
\(50\) 0 0
\(51\) −8.72203 −1.22133
\(52\) −2.91829 −0.404694
\(53\) 9.78523 1.34410 0.672052 0.740504i \(-0.265413\pi\)
0.672052 + 0.740504i \(0.265413\pi\)
\(54\) 5.06319 0.689013
\(55\) 0 0
\(56\) −2.69168 −0.359691
\(57\) −1.77339 −0.234891
\(58\) 1.08171 0.142035
\(59\) 12.9933 1.69159 0.845793 0.533511i \(-0.179128\pi\)
0.845793 + 0.533511i \(0.179128\pi\)
\(60\) 0 0
\(61\) −7.38336 −0.945342 −0.472671 0.881239i \(-0.656710\pi\)
−0.472671 + 0.881239i \(0.656710\pi\)
\(62\) 7.54677 0.958441
\(63\) −0.390032 −0.0491394
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −9.83658 −1.21080
\(67\) 2.85510 0.348806 0.174403 0.984674i \(-0.444201\pi\)
0.174403 + 0.984674i \(0.444201\pi\)
\(68\) 4.91829 0.596430
\(69\) −6.40187 −0.770695
\(70\) 0 0
\(71\) 14.4769 1.71809 0.859046 0.511898i \(-0.171057\pi\)
0.859046 + 0.511898i \(0.171057\pi\)
\(72\) 0.144903 0.0170770
\(73\) 5.15674 0.603551 0.301776 0.953379i \(-0.402421\pi\)
0.301776 + 0.953379i \(0.402421\pi\)
\(74\) 4.54677 0.528552
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −14.9301 −1.70145
\(78\) 5.17526 0.585983
\(79\) −3.09355 −0.348052 −0.174026 0.984741i \(-0.555678\pi\)
−0.174026 + 0.984741i \(0.555678\pi\)
\(80\) 0 0
\(81\) −9.41371 −1.04597
\(82\) 0 0
\(83\) 1.71019 0.187718 0.0938591 0.995585i \(-0.470080\pi\)
0.0938591 + 0.995585i \(0.470080\pi\)
\(84\) 4.77339 0.520819
\(85\) 0 0
\(86\) −9.54677 −1.02946
\(87\) −1.91829 −0.205662
\(88\) 5.54677 0.591288
\(89\) −5.09355 −0.539915 −0.269958 0.962872i \(-0.587010\pi\)
−0.269958 + 0.962872i \(0.587010\pi\)
\(90\) 0 0
\(91\) 7.85510 0.823438
\(92\) 3.60997 0.376365
\(93\) −13.3834 −1.38779
\(94\) −0.836581 −0.0862867
\(95\) 0 0
\(96\) −1.77339 −0.180996
\(97\) −17.2570 −1.75218 −0.876090 0.482148i \(-0.839857\pi\)
−0.876090 + 0.482148i \(0.839857\pi\)
\(98\) 0.245129 0.0247618
\(99\) 0.803744 0.0807793
\(100\) 0 0
\(101\) −9.38336 −0.933679 −0.466839 0.884342i \(-0.654607\pi\)
−0.466839 + 0.884342i \(0.654607\pi\)
\(102\) −8.72203 −0.863610
\(103\) −12.9301 −1.27404 −0.637022 0.770846i \(-0.719834\pi\)
−0.637022 + 0.770846i \(0.719834\pi\)
\(104\) −2.91829 −0.286162
\(105\) 0 0
\(106\) 9.78523 0.950425
\(107\) 18.9420 1.83119 0.915595 0.402102i \(-0.131720\pi\)
0.915595 + 0.402102i \(0.131720\pi\)
\(108\) 5.06319 0.487206
\(109\) −2.69168 −0.257816 −0.128908 0.991657i \(-0.541147\pi\)
−0.128908 + 0.991657i \(0.541147\pi\)
\(110\) 0 0
\(111\) −8.06319 −0.765324
\(112\) −2.69168 −0.254340
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −1.77339 −0.166093
\(115\) 0 0
\(116\) 1.08171 0.100434
\(117\) −0.422869 −0.0390942
\(118\) 12.9933 1.19613
\(119\) −13.2385 −1.21357
\(120\) 0 0
\(121\) 19.7667 1.79697
\(122\) −7.38336 −0.668458
\(123\) 0 0
\(124\) 7.54677 0.677720
\(125\) 0 0
\(126\) −0.390032 −0.0347468
\(127\) −1.87361 −0.166256 −0.0831282 0.996539i \(-0.526491\pi\)
−0.0831282 + 0.996539i \(0.526491\pi\)
\(128\) 1.00000 0.0883883
\(129\) 16.9301 1.49061
\(130\) 0 0
\(131\) −11.5468 −1.00885 −0.504423 0.863457i \(-0.668295\pi\)
−0.504423 + 0.863457i \(0.668295\pi\)
\(132\) −9.83658 −0.856164
\(133\) −2.69168 −0.233398
\(134\) 2.85510 0.246643
\(135\) 0 0
\(136\) 4.91829 0.421740
\(137\) 4.23845 0.362115 0.181058 0.983472i \(-0.442048\pi\)
0.181058 + 0.983472i \(0.442048\pi\)
\(138\) −6.40187 −0.544964
\(139\) 9.21994 0.782025 0.391012 0.920385i \(-0.372125\pi\)
0.391012 + 0.920385i \(0.372125\pi\)
\(140\) 0 0
\(141\) 1.48358 0.124940
\(142\) 14.4769 1.21487
\(143\) −16.1871 −1.35363
\(144\) 0.144903 0.0120752
\(145\) 0 0
\(146\) 5.15674 0.426775
\(147\) −0.434709 −0.0358542
\(148\) 4.54677 0.373743
\(149\) −7.25697 −0.594514 −0.297257 0.954797i \(-0.596072\pi\)
−0.297257 + 0.954797i \(0.596072\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0.712675 0.0576163
\(154\) −14.9301 −1.20310
\(155\) 0 0
\(156\) 5.17526 0.414352
\(157\) −9.54677 −0.761916 −0.380958 0.924592i \(-0.624406\pi\)
−0.380958 + 0.924592i \(0.624406\pi\)
\(158\) −3.09355 −0.246110
\(159\) −17.3530 −1.37618
\(160\) 0 0
\(161\) −9.71687 −0.765797
\(162\) −9.41371 −0.739611
\(163\) 15.7102 1.23052 0.615259 0.788325i \(-0.289052\pi\)
0.615259 + 0.788325i \(0.289052\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.71019 0.132737
\(167\) 6.45323 0.499366 0.249683 0.968328i \(-0.419674\pi\)
0.249683 + 0.968328i \(0.419674\pi\)
\(168\) 4.77339 0.368275
\(169\) −4.48358 −0.344891
\(170\) 0 0
\(171\) 0.144903 0.0110810
\(172\) −9.54677 −0.727935
\(173\) −0.873614 −0.0664196 −0.0332098 0.999448i \(-0.510573\pi\)
−0.0332098 + 0.999448i \(0.510573\pi\)
\(174\) −1.91829 −0.145425
\(175\) 0 0
\(176\) 5.54677 0.418104
\(177\) −23.0422 −1.73196
\(178\) −5.09355 −0.381778
\(179\) −12.3834 −0.925575 −0.462788 0.886469i \(-0.653151\pi\)
−0.462788 + 0.886469i \(0.653151\pi\)
\(180\) 0 0
\(181\) −17.6403 −1.31119 −0.655597 0.755111i \(-0.727583\pi\)
−0.655597 + 0.755111i \(0.727583\pi\)
\(182\) 7.85510 0.582259
\(183\) 13.0935 0.967903
\(184\) 3.60997 0.266130
\(185\) 0 0
\(186\) −13.3834 −0.981315
\(187\) 27.2806 1.99496
\(188\) −0.836581 −0.0610139
\(189\) −13.6285 −0.991326
\(190\) 0 0
\(191\) −15.1567 −1.09670 −0.548352 0.836248i \(-0.684744\pi\)
−0.548352 + 0.836248i \(0.684744\pi\)
\(192\) −1.77339 −0.127983
\(193\) 23.3834 1.68317 0.841585 0.540124i \(-0.181623\pi\)
0.841585 + 0.540124i \(0.181623\pi\)
\(194\) −17.2570 −1.23898
\(195\) 0 0
\(196\) 0.245129 0.0175092
\(197\) −23.0935 −1.64535 −0.822674 0.568514i \(-0.807519\pi\)
−0.822674 + 0.568514i \(0.807519\pi\)
\(198\) 0.803744 0.0571196
\(199\) −25.7852 −1.82787 −0.913933 0.405865i \(-0.866970\pi\)
−0.913933 + 0.405865i \(0.866970\pi\)
\(200\) 0 0
\(201\) −5.06319 −0.357130
\(202\) −9.38336 −0.660211
\(203\) −2.91161 −0.204355
\(204\) −8.72203 −0.610665
\(205\) 0 0
\(206\) −12.9301 −0.900885
\(207\) 0.523095 0.0363576
\(208\) −2.91829 −0.202347
\(209\) 5.54677 0.383678
\(210\) 0 0
\(211\) 14.2266 0.979400 0.489700 0.871891i \(-0.337106\pi\)
0.489700 + 0.871891i \(0.337106\pi\)
\(212\) 9.78523 0.672052
\(213\) −25.6732 −1.75910
\(214\) 18.9420 1.29485
\(215\) 0 0
\(216\) 5.06319 0.344507
\(217\) −20.3135 −1.37897
\(218\) −2.69168 −0.182303
\(219\) −9.14490 −0.617955
\(220\) 0 0
\(221\) −14.3530 −0.965487
\(222\) −8.06319 −0.541166
\(223\) −4.45323 −0.298210 −0.149105 0.988821i \(-0.547639\pi\)
−0.149105 + 0.988821i \(0.547639\pi\)
\(224\) −2.69168 −0.179845
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 5.94865 0.394826 0.197413 0.980320i \(-0.436746\pi\)
0.197413 + 0.980320i \(0.436746\pi\)
\(228\) −1.77339 −0.117445
\(229\) 1.09355 0.0722638 0.0361319 0.999347i \(-0.488496\pi\)
0.0361319 + 0.999347i \(0.488496\pi\)
\(230\) 0 0
\(231\) 26.4769 1.74205
\(232\) 1.08171 0.0710177
\(233\) 14.0935 0.923299 0.461650 0.887062i \(-0.347258\pi\)
0.461650 + 0.887062i \(0.347258\pi\)
\(234\) −0.422869 −0.0276438
\(235\) 0 0
\(236\) 12.9933 0.845793
\(237\) 5.48606 0.356358
\(238\) −13.2385 −0.858121
\(239\) 15.6100 1.00972 0.504862 0.863200i \(-0.331543\pi\)
0.504862 + 0.863200i \(0.331543\pi\)
\(240\) 0 0
\(241\) −5.21994 −0.336246 −0.168123 0.985766i \(-0.553771\pi\)
−0.168123 + 0.985766i \(0.553771\pi\)
\(242\) 19.7667 1.27065
\(243\) 1.50458 0.0965188
\(244\) −7.38336 −0.472671
\(245\) 0 0
\(246\) 0 0
\(247\) −2.91829 −0.185686
\(248\) 7.54677 0.479221
\(249\) −3.03284 −0.192198
\(250\) 0 0
\(251\) −28.6403 −1.80776 −0.903881 0.427785i \(-0.859294\pi\)
−0.903881 + 0.427785i \(0.859294\pi\)
\(252\) −0.390032 −0.0245697
\(253\) 20.0237 1.25888
\(254\) −1.87361 −0.117561
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.45323 0.402541 0.201271 0.979536i \(-0.435493\pi\)
0.201271 + 0.979536i \(0.435493\pi\)
\(258\) 16.9301 1.05402
\(259\) −12.2385 −0.760460
\(260\) 0 0
\(261\) 0.156743 0.00970214
\(262\) −11.5468 −0.713362
\(263\) 22.1871 1.36812 0.684058 0.729428i \(-0.260214\pi\)
0.684058 + 0.729428i \(0.260214\pi\)
\(264\) −9.83658 −0.605400
\(265\) 0 0
\(266\) −2.69168 −0.165037
\(267\) 9.03284 0.552801
\(268\) 2.85510 0.174403
\(269\) −20.4070 −1.24424 −0.622119 0.782922i \(-0.713728\pi\)
−0.622119 + 0.782922i \(0.713728\pi\)
\(270\) 0 0
\(271\) 15.9368 0.968092 0.484046 0.875043i \(-0.339167\pi\)
0.484046 + 0.875043i \(0.339167\pi\)
\(272\) 4.91829 0.298215
\(273\) −13.9301 −0.843090
\(274\) 4.23845 0.256054
\(275\) 0 0
\(276\) −6.40187 −0.385347
\(277\) 4.02368 0.241759 0.120880 0.992667i \(-0.461428\pi\)
0.120880 + 0.992667i \(0.461428\pi\)
\(278\) 9.21994 0.552975
\(279\) 1.09355 0.0654691
\(280\) 0 0
\(281\) 1.67316 0.0998124 0.0499062 0.998754i \(-0.484108\pi\)
0.0499062 + 0.998754i \(0.484108\pi\)
\(282\) 1.48358 0.0883460
\(283\) 16.8931 1.00419 0.502095 0.864812i \(-0.332563\pi\)
0.502095 + 0.864812i \(0.332563\pi\)
\(284\) 14.4769 0.859046
\(285\) 0 0
\(286\) −16.1871 −0.957163
\(287\) 0 0
\(288\) 0.144903 0.00853849
\(289\) 7.18958 0.422916
\(290\) 0 0
\(291\) 30.6033 1.79400
\(292\) 5.15674 0.301776
\(293\) −1.08171 −0.0631942 −0.0315971 0.999501i \(-0.510059\pi\)
−0.0315971 + 0.999501i \(0.510059\pi\)
\(294\) −0.434709 −0.0253527
\(295\) 0 0
\(296\) 4.54677 0.264276
\(297\) 28.0844 1.62962
\(298\) −7.25697 −0.420385
\(299\) −10.5349 −0.609251
\(300\) 0 0
\(301\) 25.6968 1.48114
\(302\) 20.0000 1.15087
\(303\) 16.6403 0.955962
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0.712675 0.0407409
\(307\) 2.38336 0.136025 0.0680126 0.997684i \(-0.478334\pi\)
0.0680126 + 0.997684i \(0.478334\pi\)
\(308\) −14.9301 −0.850723
\(309\) 22.9301 1.30445
\(310\) 0 0
\(311\) −29.4584 −1.67043 −0.835216 0.549922i \(-0.814657\pi\)
−0.835216 + 0.549922i \(0.814657\pi\)
\(312\) 5.17526 0.292991
\(313\) 32.0355 1.81075 0.905377 0.424608i \(-0.139588\pi\)
0.905377 + 0.424608i \(0.139588\pi\)
\(314\) −9.54677 −0.538756
\(315\) 0 0
\(316\) −3.09355 −0.174026
\(317\) −26.6665 −1.49774 −0.748870 0.662717i \(-0.769403\pi\)
−0.748870 + 0.662717i \(0.769403\pi\)
\(318\) −17.3530 −0.973108
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) −33.5915 −1.87489
\(322\) −9.71687 −0.541500
\(323\) 4.91829 0.273661
\(324\) −9.41371 −0.522984
\(325\) 0 0
\(326\) 15.7102 0.870107
\(327\) 4.77339 0.263969
\(328\) 0 0
\(329\) 2.25181 0.124146
\(330\) 0 0
\(331\) −19.8721 −1.09227 −0.546135 0.837697i \(-0.683901\pi\)
−0.546135 + 0.837697i \(0.683901\pi\)
\(332\) 1.71019 0.0938591
\(333\) 0.658841 0.0361043
\(334\) 6.45323 0.353105
\(335\) 0 0
\(336\) 4.77339 0.260410
\(337\) 27.5705 1.50186 0.750929 0.660383i \(-0.229606\pi\)
0.750929 + 0.660383i \(0.229606\pi\)
\(338\) −4.48358 −0.243875
\(339\) 10.6403 0.577903
\(340\) 0 0
\(341\) 41.8603 2.26686
\(342\) 0.144903 0.00783545
\(343\) 18.1819 0.981732
\(344\) −9.54677 −0.514728
\(345\) 0 0
\(346\) −0.873614 −0.0469658
\(347\) −27.2806 −1.46450 −0.732251 0.681035i \(-0.761530\pi\)
−0.732251 + 0.681035i \(0.761530\pi\)
\(348\) −1.91829 −0.102831
\(349\) 30.9538 1.65692 0.828460 0.560049i \(-0.189218\pi\)
0.828460 + 0.560049i \(0.189218\pi\)
\(350\) 0 0
\(351\) −14.7759 −0.788677
\(352\) 5.54677 0.295644
\(353\) 11.1819 0.595154 0.297577 0.954698i \(-0.403821\pi\)
0.297577 + 0.954698i \(0.403821\pi\)
\(354\) −23.0422 −1.22468
\(355\) 0 0
\(356\) −5.09355 −0.269958
\(357\) 23.4769 1.24253
\(358\) −12.3834 −0.654481
\(359\) 20.3387 1.07343 0.536717 0.843762i \(-0.319664\pi\)
0.536717 + 0.843762i \(0.319664\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −17.6403 −0.927155
\(363\) −35.0540 −1.83986
\(364\) 7.85510 0.411719
\(365\) 0 0
\(366\) 13.0935 0.684411
\(367\) −32.9538 −1.72017 −0.860087 0.510147i \(-0.829591\pi\)
−0.860087 + 0.510147i \(0.829591\pi\)
\(368\) 3.60997 0.188183
\(369\) 0 0
\(370\) 0 0
\(371\) −26.3387 −1.36744
\(372\) −13.3834 −0.693895
\(373\) −10.0869 −0.522278 −0.261139 0.965301i \(-0.584098\pi\)
−0.261139 + 0.965301i \(0.584098\pi\)
\(374\) 27.2806 1.41065
\(375\) 0 0
\(376\) −0.836581 −0.0431434
\(377\) −3.15674 −0.162581
\(378\) −13.6285 −0.700974
\(379\) −18.4256 −0.946457 −0.473229 0.880940i \(-0.656912\pi\)
−0.473229 + 0.880940i \(0.656912\pi\)
\(380\) 0 0
\(381\) 3.32264 0.170224
\(382\) −15.1567 −0.775486
\(383\) 28.1871 1.44029 0.720147 0.693822i \(-0.244074\pi\)
0.720147 + 0.693822i \(0.244074\pi\)
\(384\) −1.77339 −0.0904978
\(385\) 0 0
\(386\) 23.3834 1.19018
\(387\) −1.38336 −0.0703199
\(388\) −17.2570 −0.876090
\(389\) 20.0237 1.01524 0.507620 0.861581i \(-0.330525\pi\)
0.507620 + 0.861581i \(0.330525\pi\)
\(390\) 0 0
\(391\) 17.7549 0.897902
\(392\) 0.245129 0.0123809
\(393\) 20.4769 1.03292
\(394\) −23.0935 −1.16344
\(395\) 0 0
\(396\) 0.803744 0.0403896
\(397\) −8.32684 −0.417912 −0.208956 0.977925i \(-0.567007\pi\)
−0.208956 + 0.977925i \(0.567007\pi\)
\(398\) −25.7852 −1.29250
\(399\) 4.77339 0.238968
\(400\) 0 0
\(401\) −22.1501 −1.10612 −0.553061 0.833141i \(-0.686540\pi\)
−0.553061 + 0.833141i \(0.686540\pi\)
\(402\) −5.06319 −0.252529
\(403\) −22.0237 −1.09708
\(404\) −9.38336 −0.466839
\(405\) 0 0
\(406\) −2.91161 −0.144501
\(407\) 25.2199 1.25011
\(408\) −8.72203 −0.431805
\(409\) 34.0237 1.68236 0.841181 0.540753i \(-0.181861\pi\)
0.841181 + 0.540753i \(0.181861\pi\)
\(410\) 0 0
\(411\) −7.51642 −0.370758
\(412\) −12.9301 −0.637022
\(413\) −34.9738 −1.72095
\(414\) 0.523095 0.0257087
\(415\) 0 0
\(416\) −2.91829 −0.143081
\(417\) −16.3505 −0.800688
\(418\) 5.54677 0.271302
\(419\) 37.5334 1.83363 0.916814 0.399315i \(-0.130752\pi\)
0.916814 + 0.399315i \(0.130752\pi\)
\(420\) 0 0
\(421\) 33.8341 1.64897 0.824487 0.565882i \(-0.191464\pi\)
0.824487 + 0.565882i \(0.191464\pi\)
\(422\) 14.2266 0.692541
\(423\) −0.121223 −0.00589406
\(424\) 9.78523 0.475213
\(425\) 0 0
\(426\) −25.6732 −1.24387
\(427\) 19.8736 0.961752
\(428\) 18.9420 0.915595
\(429\) 28.7060 1.38594
\(430\) 0 0
\(431\) 24.8037 1.19475 0.597377 0.801960i \(-0.296210\pi\)
0.597377 + 0.801960i \(0.296210\pi\)
\(432\) 5.06319 0.243603
\(433\) 1.68651 0.0810487 0.0405244 0.999179i \(-0.487097\pi\)
0.0405244 + 0.999179i \(0.487097\pi\)
\(434\) −20.3135 −0.975079
\(435\) 0 0
\(436\) −2.69168 −0.128908
\(437\) 3.60997 0.172688
\(438\) −9.14490 −0.436960
\(439\) 0.326839 0.0155992 0.00779958 0.999970i \(-0.497517\pi\)
0.00779958 + 0.999970i \(0.497517\pi\)
\(440\) 0 0
\(441\) 0.0355199 0.00169142
\(442\) −14.3530 −0.682703
\(443\) 0.490258 0.0232929 0.0116464 0.999932i \(-0.496293\pi\)
0.0116464 + 0.999932i \(0.496293\pi\)
\(444\) −8.06319 −0.382662
\(445\) 0 0
\(446\) −4.45323 −0.210866
\(447\) 12.8694 0.608703
\(448\) −2.69168 −0.127170
\(449\) 23.5468 1.11124 0.555621 0.831436i \(-0.312481\pi\)
0.555621 + 0.831436i \(0.312481\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −35.4677 −1.66642
\(454\) 5.94865 0.279184
\(455\) 0 0
\(456\) −1.77339 −0.0830465
\(457\) −8.01184 −0.374778 −0.187389 0.982286i \(-0.560003\pi\)
−0.187389 + 0.982286i \(0.560003\pi\)
\(458\) 1.09355 0.0510982
\(459\) 24.9023 1.16234
\(460\) 0 0
\(461\) 3.83658 0.178687 0.0893437 0.996001i \(-0.471523\pi\)
0.0893437 + 0.996001i \(0.471523\pi\)
\(462\) 26.4769 1.23182
\(463\) −6.58381 −0.305975 −0.152988 0.988228i \(-0.548890\pi\)
−0.152988 + 0.988228i \(0.548890\pi\)
\(464\) 1.08171 0.0502171
\(465\) 0 0
\(466\) 14.0935 0.652871
\(467\) 22.6033 1.04596 0.522978 0.852346i \(-0.324821\pi\)
0.522978 + 0.852346i \(0.324821\pi\)
\(468\) −0.422869 −0.0195471
\(469\) −7.68500 −0.354860
\(470\) 0 0
\(471\) 16.9301 0.780099
\(472\) 12.9933 0.598066
\(473\) −52.9538 −2.43482
\(474\) 5.48606 0.251983
\(475\) 0 0
\(476\) −13.2385 −0.606783
\(477\) 1.41791 0.0649215
\(478\) 15.6100 0.713983
\(479\) −35.7904 −1.63530 −0.817652 0.575712i \(-0.804725\pi\)
−0.817652 + 0.575712i \(0.804725\pi\)
\(480\) 0 0
\(481\) −13.2688 −0.605006
\(482\) −5.21994 −0.237762
\(483\) 17.2318 0.784073
\(484\) 19.7667 0.898487
\(485\) 0 0
\(486\) 1.50458 0.0682491
\(487\) −21.4070 −0.970045 −0.485023 0.874502i \(-0.661189\pi\)
−0.485023 + 0.874502i \(0.661189\pi\)
\(488\) −7.38336 −0.334229
\(489\) −27.8603 −1.25988
\(490\) 0 0
\(491\) −4.32684 −0.195267 −0.0976337 0.995222i \(-0.531127\pi\)
−0.0976337 + 0.995222i \(0.531127\pi\)
\(492\) 0 0
\(493\) 5.32016 0.239608
\(494\) −2.91829 −0.131300
\(495\) 0 0
\(496\) 7.54677 0.338860
\(497\) −38.9672 −1.74792
\(498\) −3.03284 −0.135905
\(499\) −37.6968 −1.68754 −0.843771 0.536703i \(-0.819670\pi\)
−0.843771 + 0.536703i \(0.819670\pi\)
\(500\) 0 0
\(501\) −11.4441 −0.511283
\(502\) −28.6403 −1.27828
\(503\) −16.9183 −0.754349 −0.377175 0.926142i \(-0.623104\pi\)
−0.377175 + 0.926142i \(0.623104\pi\)
\(504\) −0.390032 −0.0173734
\(505\) 0 0
\(506\) 20.0237 0.890161
\(507\) 7.95113 0.353122
\(508\) −1.87361 −0.0831282
\(509\) −6.79955 −0.301385 −0.150692 0.988581i \(-0.548150\pi\)
−0.150692 + 0.988581i \(0.548150\pi\)
\(510\) 0 0
\(511\) −13.8803 −0.614028
\(512\) 1.00000 0.0441942
\(513\) 5.06319 0.223545
\(514\) 6.45323 0.284640
\(515\) 0 0
\(516\) 16.9301 0.745307
\(517\) −4.64032 −0.204081
\(518\) −12.2385 −0.537727
\(519\) 1.54926 0.0680048
\(520\) 0 0
\(521\) −1.35968 −0.0595685 −0.0297842 0.999556i \(-0.509482\pi\)
−0.0297842 + 0.999556i \(0.509482\pi\)
\(522\) 0.156743 0.00686045
\(523\) −21.9486 −0.959747 −0.479874 0.877338i \(-0.659318\pi\)
−0.479874 + 0.877338i \(0.659318\pi\)
\(524\) −11.5468 −0.504423
\(525\) 0 0
\(526\) 22.1871 0.967404
\(527\) 37.1172 1.61685
\(528\) −9.83658 −0.428082
\(529\) −9.96813 −0.433397
\(530\) 0 0
\(531\) 1.88277 0.0817053
\(532\) −2.69168 −0.116699
\(533\) 0 0
\(534\) 9.03284 0.390889
\(535\) 0 0
\(536\) 2.85510 0.123321
\(537\) 21.9605 0.947665
\(538\) −20.4070 −0.879810
\(539\) 1.35968 0.0585654
\(540\) 0 0
\(541\) −39.8973 −1.71532 −0.857659 0.514218i \(-0.828082\pi\)
−0.857659 + 0.514218i \(0.828082\pi\)
\(542\) 15.9368 0.684544
\(543\) 31.2831 1.34249
\(544\) 4.91829 0.210870
\(545\) 0 0
\(546\) −13.9301 −0.596155
\(547\) −7.34632 −0.314106 −0.157053 0.987590i \(-0.550199\pi\)
−0.157053 + 0.987590i \(0.550199\pi\)
\(548\) 4.23845 0.181058
\(549\) −1.06987 −0.0456609
\(550\) 0 0
\(551\) 1.08171 0.0460824
\(552\) −6.40187 −0.272482
\(553\) 8.32684 0.354093
\(554\) 4.02368 0.170950
\(555\) 0 0
\(556\) 9.21994 0.391012
\(557\) −26.9672 −1.14264 −0.571318 0.820729i \(-0.693568\pi\)
−0.571318 + 0.820729i \(0.693568\pi\)
\(558\) 1.09355 0.0462936
\(559\) 27.8603 1.17836
\(560\) 0 0
\(561\) −48.3792 −2.04257
\(562\) 1.67316 0.0705780
\(563\) −44.8973 −1.89220 −0.946098 0.323881i \(-0.895012\pi\)
−0.946098 + 0.323881i \(0.895012\pi\)
\(564\) 1.48358 0.0624701
\(565\) 0 0
\(566\) 16.8931 0.710070
\(567\) 25.3387 1.06412
\(568\) 14.4769 0.607437
\(569\) 31.1172 1.30450 0.652251 0.758003i \(-0.273825\pi\)
0.652251 + 0.758003i \(0.273825\pi\)
\(570\) 0 0
\(571\) 33.2570 1.39176 0.695880 0.718158i \(-0.255014\pi\)
0.695880 + 0.718158i \(0.255014\pi\)
\(572\) −16.1871 −0.676817
\(573\) 26.8788 1.12288
\(574\) 0 0
\(575\) 0 0
\(576\) 0.144903 0.00603762
\(577\) 23.4702 0.977078 0.488539 0.872542i \(-0.337530\pi\)
0.488539 + 0.872542i \(0.337530\pi\)
\(578\) 7.18958 0.299047
\(579\) −41.4677 −1.72334
\(580\) 0 0
\(581\) −4.60329 −0.190977
\(582\) 30.6033 1.26855
\(583\) 54.2765 2.24790
\(584\) 5.15674 0.213388
\(585\) 0 0
\(586\) −1.08171 −0.0446850
\(587\) −10.1501 −0.418938 −0.209469 0.977815i \(-0.567174\pi\)
−0.209469 + 0.977815i \(0.567174\pi\)
\(588\) −0.434709 −0.0179271
\(589\) 7.54677 0.310959
\(590\) 0 0
\(591\) 40.9538 1.68461
\(592\) 4.54677 0.186871
\(593\) −16.6732 −0.684685 −0.342342 0.939575i \(-0.611220\pi\)
−0.342342 + 0.939575i \(0.611220\pi\)
\(594\) 28.0844 1.15232
\(595\) 0 0
\(596\) −7.25697 −0.297257
\(597\) 45.7272 1.87149
\(598\) −10.5349 −0.430806
\(599\) 18.2765 0.746756 0.373378 0.927679i \(-0.378200\pi\)
0.373378 + 0.927679i \(0.378200\pi\)
\(600\) 0 0
\(601\) 7.96297 0.324816 0.162408 0.986724i \(-0.448074\pi\)
0.162408 + 0.986724i \(0.448074\pi\)
\(602\) 25.6968 1.04733
\(603\) 0.413712 0.0168477
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 16.6403 0.675967
\(607\) −6.93013 −0.281285 −0.140643 0.990060i \(-0.544917\pi\)
−0.140643 + 0.990060i \(0.544917\pi\)
\(608\) 1.00000 0.0405554
\(609\) 5.16342 0.209232
\(610\) 0 0
\(611\) 2.44139 0.0987679
\(612\) 0.712675 0.0288082
\(613\) −34.2765 −1.38441 −0.692206 0.721700i \(-0.743361\pi\)
−0.692206 + 0.721700i \(0.743361\pi\)
\(614\) 2.38336 0.0961844
\(615\) 0 0
\(616\) −14.9301 −0.601552
\(617\) 35.5139 1.42974 0.714869 0.699259i \(-0.246486\pi\)
0.714869 + 0.699259i \(0.246486\pi\)
\(618\) 22.9301 0.922385
\(619\) −29.5334 −1.18705 −0.593524 0.804816i \(-0.702264\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(620\) 0 0
\(621\) 18.2780 0.733470
\(622\) −29.4584 −1.18117
\(623\) 13.7102 0.549287
\(624\) 5.17526 0.207176
\(625\) 0 0
\(626\) 32.0355 1.28040
\(627\) −9.83658 −0.392835
\(628\) −9.54677 −0.380958
\(629\) 22.3624 0.891646
\(630\) 0 0
\(631\) −15.1634 −0.603646 −0.301823 0.953364i \(-0.597595\pi\)
−0.301823 + 0.953364i \(0.597595\pi\)
\(632\) −3.09355 −0.123055
\(633\) −25.2293 −1.00277
\(634\) −26.6665 −1.05906
\(635\) 0 0
\(636\) −17.3530 −0.688091
\(637\) −0.715358 −0.0283435
\(638\) 6.00000 0.237542
\(639\) 2.09775 0.0829855
\(640\) 0 0
\(641\) −25.0802 −0.990608 −0.495304 0.868720i \(-0.664943\pi\)
−0.495304 + 0.868720i \(0.664943\pi\)
\(642\) −33.5915 −1.32575
\(643\) 19.1306 0.754437 0.377218 0.926124i \(-0.376881\pi\)
0.377218 + 0.926124i \(0.376881\pi\)
\(644\) −9.71687 −0.382898
\(645\) 0 0
\(646\) 4.91829 0.193508
\(647\) 12.4019 0.487568 0.243784 0.969830i \(-0.421611\pi\)
0.243784 + 0.969830i \(0.421611\pi\)
\(648\) −9.41371 −0.369806
\(649\) 72.0710 2.82904
\(650\) 0 0
\(651\) 36.0237 1.41188
\(652\) 15.7102 0.615259
\(653\) −10.7801 −0.421856 −0.210928 0.977502i \(-0.567649\pi\)
−0.210928 + 0.977502i \(0.567649\pi\)
\(654\) 4.77339 0.186654
\(655\) 0 0
\(656\) 0 0
\(657\) 0.747227 0.0291521
\(658\) 2.25181 0.0877845
\(659\) 8.56529 0.333656 0.166828 0.985986i \(-0.446647\pi\)
0.166828 + 0.985986i \(0.446647\pi\)
\(660\) 0 0
\(661\) −26.2883 −1.02250 −0.511248 0.859433i \(-0.670817\pi\)
−0.511248 + 0.859433i \(0.670817\pi\)
\(662\) −19.8721 −0.772351
\(663\) 25.4534 0.988529
\(664\) 1.71019 0.0663684
\(665\) 0 0
\(666\) 0.658841 0.0255296
\(667\) 3.90494 0.151200
\(668\) 6.45323 0.249683
\(669\) 7.89729 0.305327
\(670\) 0 0
\(671\) −40.9538 −1.58100
\(672\) 4.77339 0.184137
\(673\) −12.9301 −0.498420 −0.249210 0.968449i \(-0.580171\pi\)
−0.249210 + 0.968449i \(0.580171\pi\)
\(674\) 27.5705 1.06197
\(675\) 0 0
\(676\) −4.48358 −0.172445
\(677\) −15.9250 −0.612046 −0.306023 0.952024i \(-0.598998\pi\)
−0.306023 + 0.952024i \(0.598998\pi\)
\(678\) 10.6403 0.408639
\(679\) 46.4502 1.78260
\(680\) 0 0
\(681\) −10.5493 −0.404248
\(682\) 41.8603 1.60291
\(683\) 13.2898 0.508520 0.254260 0.967136i \(-0.418168\pi\)
0.254260 + 0.967136i \(0.418168\pi\)
\(684\) 0.144903 0.00554050
\(685\) 0 0
\(686\) 18.1819 0.694190
\(687\) −1.93929 −0.0739884
\(688\) −9.54677 −0.363967
\(689\) −28.5561 −1.08790
\(690\) 0 0
\(691\) 27.2570 1.03690 0.518452 0.855107i \(-0.326508\pi\)
0.518452 + 0.855107i \(0.326508\pi\)
\(692\) −0.873614 −0.0332098
\(693\) −2.16342 −0.0821815
\(694\) −27.2806 −1.03556
\(695\) 0 0
\(696\) −1.91829 −0.0727126
\(697\) 0 0
\(698\) 30.9538 1.17162
\(699\) −24.9933 −0.945334
\(700\) 0 0
\(701\) −41.4441 −1.56532 −0.782660 0.622449i \(-0.786138\pi\)
−0.782660 + 0.622449i \(0.786138\pi\)
\(702\) −14.7759 −0.557679
\(703\) 4.54677 0.171485
\(704\) 5.54677 0.209052
\(705\) 0 0
\(706\) 11.1819 0.420838
\(707\) 25.2570 0.949886
\(708\) −23.0422 −0.865979
\(709\) −36.7904 −1.38169 −0.690846 0.723002i \(-0.742762\pi\)
−0.690846 + 0.723002i \(0.742762\pi\)
\(710\) 0 0
\(711\) −0.448264 −0.0168112
\(712\) −5.09355 −0.190889
\(713\) 27.2436 1.02028
\(714\) 23.4769 0.878601
\(715\) 0 0
\(716\) −12.3834 −0.462788
\(717\) −27.6825 −1.03382
\(718\) 20.3387 0.759033
\(719\) −4.13726 −0.154294 −0.0771469 0.997020i \(-0.524581\pi\)
−0.0771469 + 0.997020i \(0.524581\pi\)
\(720\) 0 0
\(721\) 34.8037 1.29616
\(722\) 1.00000 0.0372161
\(723\) 9.25697 0.344270
\(724\) −17.6403 −0.655597
\(725\) 0 0
\(726\) −35.0540 −1.30098
\(727\) −40.4374 −1.49974 −0.749870 0.661585i \(-0.769884\pi\)
−0.749870 + 0.661585i \(0.769884\pi\)
\(728\) 7.85510 0.291129
\(729\) 25.5729 0.947146
\(730\) 0 0
\(731\) −46.9538 −1.73665
\(732\) 13.0935 0.483952
\(733\) 28.1264 1.03887 0.519436 0.854509i \(-0.326142\pi\)
0.519436 + 0.854509i \(0.326142\pi\)
\(734\) −32.9538 −1.21635
\(735\) 0 0
\(736\) 3.60997 0.133065
\(737\) 15.8366 0.583348
\(738\) 0 0
\(739\) −38.1501 −1.40337 −0.701686 0.712486i \(-0.747569\pi\)
−0.701686 + 0.712486i \(0.747569\pi\)
\(740\) 0 0
\(741\) 5.17526 0.190118
\(742\) −26.3387 −0.966923
\(743\) −17.8232 −0.653871 −0.326935 0.945047i \(-0.606016\pi\)
−0.326935 + 0.945047i \(0.606016\pi\)
\(744\) −13.3834 −0.490658
\(745\) 0 0
\(746\) −10.0869 −0.369307
\(747\) 0.247812 0.00906697
\(748\) 27.2806 0.997479
\(749\) −50.9857 −1.86298
\(750\) 0 0
\(751\) 23.6968 0.864710 0.432355 0.901703i \(-0.357683\pi\)
0.432355 + 0.901703i \(0.357683\pi\)
\(752\) −0.836581 −0.0305070
\(753\) 50.7904 1.85090
\(754\) −3.15674 −0.114962
\(755\) 0 0
\(756\) −13.6285 −0.495663
\(757\) −6.51394 −0.236753 −0.118377 0.992969i \(-0.537769\pi\)
−0.118377 + 0.992969i \(0.537769\pi\)
\(758\) −18.4256 −0.669246
\(759\) −35.5097 −1.28892
\(760\) 0 0
\(761\) 34.7786 1.26072 0.630361 0.776302i \(-0.282907\pi\)
0.630361 + 0.776302i \(0.282907\pi\)
\(762\) 3.32264 0.120367
\(763\) 7.24513 0.262291
\(764\) −15.1567 −0.548352
\(765\) 0 0
\(766\) 28.1871 1.01844
\(767\) −37.9183 −1.36915
\(768\) −1.77339 −0.0639916
\(769\) −27.6850 −0.998347 −0.499173 0.866502i \(-0.666363\pi\)
−0.499173 + 0.866502i \(0.666363\pi\)
\(770\) 0 0
\(771\) −11.4441 −0.412148
\(772\) 23.3834 0.841585
\(773\) −19.9738 −0.718409 −0.359205 0.933259i \(-0.616952\pi\)
−0.359205 + 0.933259i \(0.616952\pi\)
\(774\) −1.38336 −0.0497237
\(775\) 0 0
\(776\) −17.2570 −0.619489
\(777\) 21.7035 0.778609
\(778\) 20.0237 0.717884
\(779\) 0 0
\(780\) 0 0
\(781\) 80.3001 2.87336
\(782\) 17.7549 0.634913
\(783\) 5.47691 0.195729
\(784\) 0.245129 0.00875461
\(785\) 0 0
\(786\) 20.4769 0.730387
\(787\) 42.0118 1.49756 0.748780 0.662818i \(-0.230640\pi\)
0.748780 + 0.662818i \(0.230640\pi\)
\(788\) −23.0935 −0.822674
\(789\) −39.3463 −1.40077
\(790\) 0 0
\(791\) 16.1501 0.574230
\(792\) 0.803744 0.0285598
\(793\) 21.5468 0.765148
\(794\) −8.32684 −0.295508
\(795\) 0 0
\(796\) −25.7852 −0.913933
\(797\) −45.1054 −1.59771 −0.798857 0.601520i \(-0.794562\pi\)
−0.798857 + 0.601520i \(0.794562\pi\)
\(798\) 4.77339 0.168976
\(799\) −4.11455 −0.145562
\(800\) 0 0
\(801\) −0.738070 −0.0260784
\(802\) −22.1501 −0.782146
\(803\) 28.6033 1.00939
\(804\) −5.06319 −0.178565
\(805\) 0 0
\(806\) −22.0237 −0.775751
\(807\) 36.1896 1.27393
\(808\) −9.38336 −0.330105
\(809\) 6.85510 0.241012 0.120506 0.992713i \(-0.461548\pi\)
0.120506 + 0.992713i \(0.461548\pi\)
\(810\) 0 0
\(811\) −15.1819 −0.533110 −0.266555 0.963820i \(-0.585885\pi\)
−0.266555 + 0.963820i \(0.585885\pi\)
\(812\) −2.91161 −0.102178
\(813\) −28.2621 −0.991196
\(814\) 25.2199 0.883958
\(815\) 0 0
\(816\) −8.72203 −0.305332
\(817\) −9.54677 −0.333999
\(818\) 34.0237 1.18961
\(819\) 1.13823 0.0397729
\(820\) 0 0
\(821\) 22.5006 0.785276 0.392638 0.919693i \(-0.371563\pi\)
0.392638 + 0.919693i \(0.371563\pi\)
\(822\) −7.51642 −0.262165
\(823\) −16.1896 −0.564333 −0.282167 0.959365i \(-0.591053\pi\)
−0.282167 + 0.959365i \(0.591053\pi\)
\(824\) −12.9301 −0.450442
\(825\) 0 0
\(826\) −34.9738 −1.21690
\(827\) −25.0817 −0.872177 −0.436088 0.899904i \(-0.643636\pi\)
−0.436088 + 0.899904i \(0.643636\pi\)
\(828\) 0.523095 0.0181788
\(829\) 21.3068 0.740016 0.370008 0.929029i \(-0.379355\pi\)
0.370008 + 0.929029i \(0.379355\pi\)
\(830\) 0 0
\(831\) −7.13554 −0.247529
\(832\) −2.91829 −0.101174
\(833\) 1.20562 0.0417721
\(834\) −16.3505 −0.566172
\(835\) 0 0
\(836\) 5.54677 0.191839
\(837\) 38.2108 1.32076
\(838\) 37.5334 1.29657
\(839\) −11.4727 −0.396082 −0.198041 0.980194i \(-0.563458\pi\)
−0.198041 + 0.980194i \(0.563458\pi\)
\(840\) 0 0
\(841\) −27.8299 −0.959652
\(842\) 33.8341 1.16600
\(843\) −2.96716 −0.102195
\(844\) 14.2266 0.489700
\(845\) 0 0
\(846\) −0.121223 −0.00416773
\(847\) −53.2056 −1.82817
\(848\) 9.78523 0.336026
\(849\) −29.9580 −1.02816
\(850\) 0 0
\(851\) 16.4137 0.562655
\(852\) −25.6732 −0.879548
\(853\) −38.9908 −1.33502 −0.667511 0.744600i \(-0.732640\pi\)
−0.667511 + 0.744600i \(0.732640\pi\)
\(854\) 19.8736 0.680061
\(855\) 0 0
\(856\) 18.9420 0.647423
\(857\) −29.8973 −1.02127 −0.510636 0.859797i \(-0.670590\pi\)
−0.510636 + 0.859797i \(0.670590\pi\)
\(858\) 28.7060 0.980007
\(859\) −6.47691 −0.220989 −0.110495 0.993877i \(-0.535243\pi\)
−0.110495 + 0.993877i \(0.535243\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.8037 0.844819
\(863\) −5.09355 −0.173386 −0.0866932 0.996235i \(-0.527630\pi\)
−0.0866932 + 0.996235i \(0.527630\pi\)
\(864\) 5.06319 0.172253
\(865\) 0 0
\(866\) 1.68651 0.0573101
\(867\) −12.7499 −0.433010
\(868\) −20.3135 −0.689485
\(869\) −17.1592 −0.582087
\(870\) 0 0
\(871\) −8.33200 −0.282319
\(872\) −2.69168 −0.0911517
\(873\) −2.50059 −0.0846320
\(874\) 3.60997 0.122109
\(875\) 0 0
\(876\) −9.14490 −0.308978
\(877\) −29.8341 −1.00743 −0.503713 0.863871i \(-0.668033\pi\)
−0.503713 + 0.863871i \(0.668033\pi\)
\(878\) 0.326839 0.0110303
\(879\) 1.91829 0.0647023
\(880\) 0 0
\(881\) −11.5796 −0.390127 −0.195064 0.980791i \(-0.562491\pi\)
−0.195064 + 0.980791i \(0.562491\pi\)
\(882\) 0.0355199 0.00119602
\(883\) −48.0237 −1.61613 −0.808063 0.589096i \(-0.799484\pi\)
−0.808063 + 0.589096i \(0.799484\pi\)
\(884\) −14.3530 −0.482744
\(885\) 0 0
\(886\) 0.490258 0.0164705
\(887\) 9.38336 0.315062 0.157531 0.987514i \(-0.449647\pi\)
0.157531 + 0.987514i \(0.449647\pi\)
\(888\) −8.06319 −0.270583
\(889\) 5.04316 0.169142
\(890\) 0 0
\(891\) −52.2157 −1.74929
\(892\) −4.45323 −0.149105
\(893\) −0.836581 −0.0279951
\(894\) 12.8694 0.430418
\(895\) 0 0
\(896\) −2.69168 −0.0899226
\(897\) 18.6825 0.623791
\(898\) 23.5468 0.785766
\(899\) 8.16342 0.272265
\(900\) 0 0
\(901\) 48.1266 1.60333
\(902\) 0 0
\(903\) −45.5705 −1.51649
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −35.4677 −1.17834
\(907\) −19.4347 −0.645319 −0.322659 0.946515i \(-0.604577\pi\)
−0.322659 + 0.946515i \(0.604577\pi\)
\(908\) 5.94865 0.197413
\(909\) −1.35968 −0.0450976
\(910\) 0 0
\(911\) −19.9866 −0.662187 −0.331094 0.943598i \(-0.607418\pi\)
−0.331094 + 0.943598i \(0.607418\pi\)
\(912\) −1.77339 −0.0587227
\(913\) 9.48606 0.313943
\(914\) −8.01184 −0.265008
\(915\) 0 0
\(916\) 1.09355 0.0361319
\(917\) 31.0802 1.02636
\(918\) 24.9023 0.821897
\(919\) 6.15158 0.202922 0.101461 0.994840i \(-0.467648\pi\)
0.101461 + 0.994840i \(0.467648\pi\)
\(920\) 0 0
\(921\) −4.22661 −0.139272
\(922\) 3.83658 0.126351
\(923\) −42.2478 −1.39060
\(924\) 26.4769 0.871026
\(925\) 0 0
\(926\) −6.58381 −0.216357
\(927\) −1.87361 −0.0615375
\(928\) 1.08171 0.0355089
\(929\) −7.17010 −0.235243 −0.117622 0.993058i \(-0.537527\pi\)
−0.117622 + 0.993058i \(0.537527\pi\)
\(930\) 0 0
\(931\) 0.245129 0.00803378
\(932\) 14.0935 0.461650
\(933\) 52.2411 1.71030
\(934\) 22.6033 0.739602
\(935\) 0 0
\(936\) −0.422869 −0.0138219
\(937\) −24.2646 −0.792690 −0.396345 0.918102i \(-0.629722\pi\)
−0.396345 + 0.918102i \(0.629722\pi\)
\(938\) −7.68500 −0.250924
\(939\) −56.8114 −1.85397
\(940\) 0 0
\(941\) −7.44806 −0.242800 −0.121400 0.992604i \(-0.538738\pi\)
−0.121400 + 0.992604i \(0.538738\pi\)
\(942\) 16.9301 0.551613
\(943\) 0 0
\(944\) 12.9933 0.422897
\(945\) 0 0
\(946\) −52.9538 −1.72168
\(947\) −52.9538 −1.72077 −0.860384 0.509647i \(-0.829776\pi\)
−0.860384 + 0.509647i \(0.829776\pi\)
\(948\) 5.48606 0.178179
\(949\) −15.0489 −0.488507
\(950\) 0 0
\(951\) 47.2900 1.53348
\(952\) −13.2385 −0.429061
\(953\) −12.8694 −0.416881 −0.208441 0.978035i \(-0.566839\pi\)
−0.208441 + 0.978035i \(0.566839\pi\)
\(954\) 1.41791 0.0459065
\(955\) 0 0
\(956\) 15.6100 0.504862
\(957\) −10.6403 −0.343953
\(958\) −35.7904 −1.15634
\(959\) −11.4085 −0.368401
\(960\) 0 0
\(961\) 25.9538 0.837220
\(962\) −13.2688 −0.427804
\(963\) 2.74475 0.0884482
\(964\) −5.21994 −0.168123
\(965\) 0 0
\(966\) 17.2318 0.554423
\(967\) 25.8603 0.831610 0.415805 0.909454i \(-0.363500\pi\)
0.415805 + 0.909454i \(0.363500\pi\)
\(968\) 19.7667 0.635326
\(969\) −8.72203 −0.280192
\(970\) 0 0
\(971\) 35.0935 1.12621 0.563103 0.826387i \(-0.309608\pi\)
0.563103 + 0.826387i \(0.309608\pi\)
\(972\) 1.50458 0.0482594
\(973\) −24.8171 −0.795600
\(974\) −21.4070 −0.685926
\(975\) 0 0
\(976\) −7.38336 −0.236335
\(977\) −42.1737 −1.34926 −0.674629 0.738157i \(-0.735696\pi\)
−0.674629 + 0.738157i \(0.735696\pi\)
\(978\) −27.8603 −0.890873
\(979\) −28.2528 −0.902963
\(980\) 0 0
\(981\) −0.390032 −0.0124528
\(982\) −4.32684 −0.138075
\(983\) −42.6270 −1.35959 −0.679795 0.733403i \(-0.737931\pi\)
−0.679795 + 0.733403i \(0.737931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 5.32016 0.169428
\(987\) −3.99332 −0.127109
\(988\) −2.91829 −0.0928432
\(989\) −34.4636 −1.09588
\(990\) 0 0
\(991\) −34.1737 −1.08556 −0.542782 0.839873i \(-0.682629\pi\)
−0.542782 + 0.839873i \(0.682629\pi\)
\(992\) 7.54677 0.239610
\(993\) 35.2409 1.11834
\(994\) −38.9672 −1.23596
\(995\) 0 0
\(996\) −3.03284 −0.0960991
\(997\) 29.3834 0.930580 0.465290 0.885158i \(-0.345950\pi\)
0.465290 + 0.885158i \(0.345950\pi\)
\(998\) −37.6968 −1.19327
\(999\) 23.0212 0.728359
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 950.2.a.l.1.1 yes 3
3.2 odd 2 8550.2.a.ci.1.1 3
4.3 odd 2 7600.2.a.bk.1.3 3
5.2 odd 4 950.2.b.h.799.6 6
5.3 odd 4 950.2.b.h.799.1 6
5.4 even 2 950.2.a.j.1.3 3
15.14 odd 2 8550.2.a.cp.1.3 3
20.19 odd 2 7600.2.a.bz.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.3 3 5.4 even 2
950.2.a.l.1.1 yes 3 1.1 even 1 trivial
950.2.b.h.799.1 6 5.3 odd 4
950.2.b.h.799.6 6 5.2 odd 4
7600.2.a.bk.1.3 3 4.3 odd 2
7600.2.a.bz.1.1 3 20.19 odd 2
8550.2.a.ci.1.1 3 3.2 odd 2
8550.2.a.cp.1.3 3 15.14 odd 2