Properties

Label 1050.2.a.f.1.1
Level 1050
Weight 2
Character 1050.1
Self dual yes
Analytic conductor 8.384
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 1050.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -7.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{21} +2.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} +7.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -7.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} -7.00000 q^{39} +11.0000 q^{41} +1.00000 q^{42} -3.00000 q^{43} -2.00000 q^{44} -5.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +7.00000 q^{51} -7.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} -9.00000 q^{58} +7.00000 q^{59} -5.00000 q^{61} -1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +12.0000 q^{67} +7.00000 q^{68} +5.00000 q^{69} -4.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} +8.00000 q^{76} +2.00000 q^{77} +7.00000 q^{78} -6.00000 q^{79} +1.00000 q^{81} -11.0000 q^{82} +9.00000 q^{83} -1.00000 q^{84} +3.00000 q^{86} +9.00000 q^{87} +2.00000 q^{88} -10.0000 q^{89} +7.00000 q^{91} +5.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.00000 0.426401
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 7.00000 1.37281
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −8.00000 −1.29777
\(39\) −7.00000 −1.12090
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 1.00000 0.154303
\(43\) −3.00000 −0.457496 −0.228748 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) −7.00000 −0.970725
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) −9.00000 −1.18176
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −1.00000 −0.127000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 7.00000 0.848875
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 2.00000 0.227921
\(78\) 7.00000 0.792594
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −11.0000 −1.21475
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 3.00000 0.323498
\(87\) 9.00000 0.964901
\(88\) 2.00000 0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) 5.00000 0.521286
\(93\) 1.00000 0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −7.00000 −0.693103
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 7.00000 0.686406
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −7.00000 −0.647150
\(118\) −7.00000 −0.644402
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000 0.452679
\(123\) 11.0000 0.991837
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) −8.00000 −0.693688
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −5.00000 −0.425628
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 4.00000 0.335673
\(143\) 14.0000 1.17074
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) −8.00000 −0.648886
\(153\) 7.00000 0.565916
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −7.00000 −0.560449
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 6.00000 0.477334
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) −1.00000 −0.0785674
\(163\) 15.0000 1.17489 0.587445 0.809264i \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 1.00000 0.0771517
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −3.00000 −0.228748
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 7.00000 0.526152
\(178\) 10.0000 0.749532
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −7.00000 −0.518875
\(183\) −5.00000 −0.369611
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −14.0000 −1.02378
\(188\) −4.00000 −0.291730
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) 2.00000 0.142134
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) 7.00000 0.490098
\(205\) 0 0
\(206\) 5.00000 0.348367
\(207\) 5.00000 0.347524
\(208\) −7.00000 −0.485363
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −3.00000 −0.206041
\(213\) −4.00000 −0.274075
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −1.00000 −0.0678844
\(218\) −4.00000 −0.270914
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −49.0000 −3.29610
\(222\) −2.00000 −0.134231
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 8.00000 0.529813
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −9.00000 −0.590879
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 7.00000 0.457604
\(235\) 0 0
\(236\) 7.00000 0.455661
\(237\) −6.00000 −0.389742
\(238\) 7.00000 0.453743
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) −11.0000 −0.701334
\(247\) −56.0000 −3.56319
\(248\) −1.00000 −0.0635001
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −10.0000 −0.628695
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.00000 −0.311891 −0.155946 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 3.00000 0.186772
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 7.00000 0.424437
\(273\) 7.00000 0.423659
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 5.00000 0.300965
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −8.00000 −0.479808
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 4.00000 0.238197
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −14.0000 −0.827837
\(287\) −11.0000 −0.649309
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −2.00000 −0.116052
\(298\) 3.00000 0.173785
\(299\) −35.0000 −2.02410
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 22.0000 1.26596
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −7.00000 −0.400163
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 2.00000 0.113961
\(309\) −5.00000 −0.284440
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 7.00000 0.396297
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −23.0000 −1.29181 −0.645904 0.763418i \(-0.723520\pi\)
−0.645904 + 0.763418i \(0.723520\pi\)
\(318\) 3.00000 0.168232
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 5.00000 0.278639
\(323\) 56.0000 3.11592
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −15.0000 −0.830773
\(327\) 4.00000 0.221201
\(328\) −11.0000 −0.607373
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 9.00000 0.493939
\(333\) 2.00000 0.109599
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) −36.0000 −1.95814
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) −8.00000 −0.432590
\(343\) −1.00000 −0.0539949
\(344\) 3.00000 0.161749
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 9.00000 0.482451
\(349\) −7.00000 −0.374701 −0.187351 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) 0 0
\(351\) −7.00000 −0.373632
\(352\) 2.00000 0.106600
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −7.00000 −0.372046
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −7.00000 −0.370479
\(358\) 2.00000 0.105703
\(359\) 19.0000 1.00278 0.501391 0.865221i \(-0.332822\pi\)
0.501391 + 0.865221i \(0.332822\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) −7.00000 −0.367405
\(364\) 7.00000 0.366900
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 5.00000 0.260643
\(369\) 11.0000 0.572637
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 1.00000 0.0518476
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 14.0000 0.723923
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −63.0000 −3.24467
\(378\) 1.00000 0.0514344
\(379\) 7.00000 0.359566 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 1.00000 0.0511645
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −3.00000 −0.152499
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 35.0000 1.77003
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −21.0000 −1.05796
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −37.0000 −1.85698 −0.928488 0.371361i \(-0.878891\pi\)
−0.928488 + 0.371361i \(0.878891\pi\)
\(398\) 16.0000 0.802008
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −12.0000 −0.598506
\(403\) −7.00000 −0.348695
\(404\) 0 0
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −4.00000 −0.198273
\(408\) −7.00000 −0.346552
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) −5.00000 −0.246332
\(413\) −7.00000 −0.344447
\(414\) −5.00000 −0.245737
\(415\) 0 0
\(416\) 7.00000 0.343203
\(417\) 8.00000 0.391762
\(418\) 16.0000 0.782586
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 13.0000 0.632830
\(423\) −4.00000 −0.194487
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 5.00000 0.241967
\(428\) 2.00000 0.0966736
\(429\) 14.0000 0.675926
\(430\) 0 0
\(431\) 39.0000 1.87856 0.939282 0.343146i \(-0.111493\pi\)
0.939282 + 0.343146i \(0.111493\pi\)
\(432\) 1.00000 0.0481125
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 40.0000 1.91346
\(438\) 10.0000 0.477818
\(439\) −19.0000 −0.906821 −0.453410 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 49.0000 2.33069
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) −3.00000 −0.141895
\(448\) −1.00000 −0.0472456
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −22.0000 −1.03594
\(452\) 14.0000 0.658505
\(453\) −22.0000 −1.03365
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 14.0000 0.654177
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) 29.0000 1.34196 0.670980 0.741475i \(-0.265874\pi\)
0.670980 + 0.741475i \(0.265874\pi\)
\(468\) −7.00000 −0.323575
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −7.00000 −0.322201
\(473\) 6.00000 0.275880
\(474\) 6.00000 0.275589
\(475\) 0 0
\(476\) −7.00000 −0.320844
\(477\) −3.00000 −0.137361
\(478\) −8.00000 −0.365911
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 30.0000 1.36646
\(483\) −5.00000 −0.227508
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −6.00000 −0.271886 −0.135943 0.990717i \(-0.543406\pi\)
−0.135943 + 0.990717i \(0.543406\pi\)
\(488\) 5.00000 0.226339
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 11.0000 0.495918
\(493\) 63.0000 2.83738
\(494\) 56.0000 2.51956
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 4.00000 0.179425
\(498\) −9.00000 −0.403300
\(499\) −23.0000 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 5.00000 0.223161
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 10.0000 0.444554
\(507\) 36.0000 1.59882
\(508\) 18.0000 0.798621
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 5.00000 0.220541
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) 8.00000 0.351840
\(518\) 2.00000 0.0878750
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) −9.00000 −0.393919
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 7.00000 0.304925
\(528\) −2.00000 −0.0870388
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 7.00000 0.303774
\(532\) −8.00000 −0.346844
\(533\) −77.0000 −3.33524
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −2.00000 −0.0863064
\(538\) 6.00000 0.258678
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −12.0000 −0.515444
\(543\) 2.00000 0.0858282
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) −7.00000 −0.299572
\(547\) 35.0000 1.49649 0.748246 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) −16.0000 −0.683486
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 72.0000 3.06730
\(552\) −5.00000 −0.212814
\(553\) 6.00000 0.255146
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) −14.0000 −0.591080
\(562\) 8.00000 0.337460
\(563\) −27.0000 −1.13791 −0.568957 0.822367i \(-0.692653\pi\)
−0.568957 + 0.822367i \(0.692653\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) −1.00000 −0.0419961
\(568\) 4.00000 0.167836
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 14.0000 0.585369
\(573\) −1.00000 −0.0417756
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −32.0000 −1.33102
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 10.0000 0.414513
\(583\) 6.00000 0.248495
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −39.0000 −1.60970 −0.804851 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 1.00000 0.0412393
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) 2.00000 0.0821995
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −3.00000 −0.122885
\(597\) −16.0000 −0.654836
\(598\) 35.0000 1.43126
\(599\) −1.00000 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −3.00000 −0.122271
\(603\) 12.0000 0.488678
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) 36.0000 1.46119 0.730597 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) −8.00000 −0.324443
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 28.0000 1.13276
\(612\) 7.00000 0.282958
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 5.00000 0.201129
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) −2.00000 −0.0801927
\(623\) 10.0000 0.400642
\(624\) −7.00000 −0.280224
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) −16.0000 −0.638978
\(628\) 10.0000 0.399043
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) 42.0000 1.67199 0.835997 0.548734i \(-0.184890\pi\)
0.835997 + 0.548734i \(0.184890\pi\)
\(632\) 6.00000 0.238667
\(633\) −13.0000 −0.516704
\(634\) 23.0000 0.913447
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) −7.00000 −0.277350
\(638\) 18.0000 0.712627
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) −56.0000 −2.20329
\(647\) −46.0000 −1.80845 −0.904223 0.427060i \(-0.859549\pi\)
−0.904223 + 0.427060i \(0.859549\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −14.0000 −0.549548
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 15.0000 0.587445
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) −10.0000 −0.390137
\(658\) −4.00000 −0.155936
\(659\) 34.0000 1.32445 0.662226 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −25.0000 −0.971653
\(663\) −49.0000 −1.90300
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 45.0000 1.74241
\(668\) −2.00000 −0.0773823
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 1.00000 0.0385758
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −14.0000 −0.537667
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 2.00000 0.0765840
\(683\) −38.0000 −1.45403 −0.727015 0.686622i \(-0.759093\pi\)
−0.727015 + 0.686622i \(0.759093\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −14.0000 −0.534133
\(688\) −3.00000 −0.114374
\(689\) 21.0000 0.800036
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 16.0000 0.608229
\(693\) 2.00000 0.0759737
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 77.0000 2.91658
\(698\) 7.00000 0.264954
\(699\) −16.0000 −0.605176
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 7.00000 0.264198
\(703\) 16.0000 0.603451
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 7.00000 0.263076
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 10.0000 0.374766
\(713\) 5.00000 0.187251
\(714\) 7.00000 0.261968
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 8.00000 0.298765
\(718\) −19.0000 −0.709074
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) −45.0000 −1.67473
\(723\) −30.0000 −1.11571
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) −21.0000 −0.778847 −0.389423 0.921059i \(-0.627326\pi\)
−0.389423 + 0.921059i \(0.627326\pi\)
\(728\) −7.00000 −0.259437
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.0000 −0.776713
\(732\) −5.00000 −0.184805
\(733\) −45.0000 −1.66211 −0.831056 0.556188i \(-0.812263\pi\)
−0.831056 + 0.556188i \(0.812263\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −24.0000 −0.884051
\(738\) −11.0000 −0.404916
\(739\) 15.0000 0.551784 0.275892 0.961189i \(-0.411027\pi\)
0.275892 + 0.961189i \(0.411027\pi\)
\(740\) 0 0
\(741\) −56.0000 −2.05721
\(742\) −3.00000 −0.110133
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 9.00000 0.329293
\(748\) −14.0000 −0.511891
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −4.00000 −0.145865
\(753\) −5.00000 −0.182210
\(754\) 63.0000 2.29432
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −7.00000 −0.254251
\(759\) −10.0000 −0.362977
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) −18.0000 −0.652071
\(763\) −4.00000 −0.144810
\(764\) −1.00000 −0.0361787
\(765\) 0 0
\(766\) 0 0
\(767\) −49.0000 −1.76929
\(768\) 1.00000 0.0360844
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) −10.0000 −0.359908
\(773\) −28.0000 −1.00709 −0.503545 0.863969i \(-0.667971\pi\)
−0.503545 + 0.863969i \(0.667971\pi\)
\(774\) 3.00000 0.107833
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) −2.00000 −0.0717496
\(778\) −6.00000 −0.215110
\(779\) 88.0000 3.15293
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −35.0000 −1.25160
\(783\) 9.00000 0.321634
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −26.0000 −0.926800 −0.463400 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(788\) 21.0000 0.748094
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 2.00000 0.0710669
\(793\) 35.0000 1.24289
\(794\) 37.0000 1.31308
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 8.00000 0.283197
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 12.0000 0.423735
\(803\) 20.0000 0.705785
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 7.00000 0.246564
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −44.0000 −1.54696 −0.773479 0.633822i \(-0.781485\pi\)
−0.773479 + 0.633822i \(0.781485\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) −9.00000 −0.315838
\(813\) 12.0000 0.420858
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) −24.0000 −0.839654
\(818\) −10.0000 −0.349642
\(819\) 7.00000 0.244600
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 16.0000 0.558064
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) 7.00000 0.243561
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 5.00000 0.173762
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −7.00000 −0.242681
\(833\) 7.00000 0.242536
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 1.00000 0.0345651
\(838\) 3.00000 0.103633
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −10.0000 −0.344623
\(843\) −8.00000 −0.275535
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 7.00000 0.240523
\(848\) −3.00000 −0.103020
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) −4.00000 −0.137038
\(853\) 35.0000 1.19838 0.599189 0.800608i \(-0.295490\pi\)
0.599189 + 0.800608i \(0.295490\pi\)
\(854\) −5.00000 −0.171096
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −14.0000 −0.477952
\(859\) −16.0000 −0.545913 −0.272956 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 0 0
\(861\) −11.0000 −0.374879
\(862\) −39.0000 −1.32835
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 6.00000 0.203888
\(867\) 32.0000 1.08678
\(868\) −1.00000 −0.0339422
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −84.0000 −2.84623
\(872\) −4.00000 −0.135457
\(873\) −10.0000 −0.338449
\(874\) −40.0000 −1.35302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 19.0000 0.641219
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 23.0000 0.774012 0.387006 0.922077i \(-0.373509\pi\)
0.387006 + 0.922077i \(0.373509\pi\)
\(884\) −49.0000 −1.64805
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 14.0000 0.470074 0.235037 0.971986i \(-0.424479\pi\)
0.235037 + 0.971986i \(0.424479\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −9.00000 −0.301342
\(893\) −32.0000 −1.07084
\(894\) 3.00000 0.100335
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −35.0000 −1.16862
\(898\) 20.0000 0.667409
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) 22.0000 0.732520
\(903\) 3.00000 0.0998337
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 39.0000 1.29497 0.647487 0.762077i \(-0.275820\pi\)
0.647487 + 0.762077i \(0.275820\pi\)
\(908\) 3.00000 0.0995585
\(909\) 0 0
\(910\) 0 0
\(911\) −39.0000 −1.29213 −0.646064 0.763283i \(-0.723586\pi\)
−0.646064 + 0.763283i \(0.723586\pi\)
\(912\) 8.00000 0.264906
\(913\) −18.0000 −0.595713
\(914\) 7.00000 0.231539
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −7.00000 −0.231034
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) −8.00000 −0.263466
\(923\) 28.0000 0.921631
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −5.00000 −0.164222
\(928\) −9.00000 −0.295439
\(929\) 7.00000 0.229663 0.114831 0.993385i \(-0.463367\pi\)
0.114831 + 0.993385i \(0.463367\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) −16.0000 −0.524097
\(933\) 2.00000 0.0654771
\(934\) −29.0000 −0.948909
\(935\) 0 0
\(936\) 7.00000 0.228802
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 12.0000 0.391814
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) −10.0000 −0.325818
\(943\) 55.0000 1.79105
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −6.00000 −0.194871
\(949\) 70.0000 2.27230
\(950\) 0 0
\(951\) −23.0000 −0.745826
\(952\) 7.00000 0.226871
\(953\) 48.0000 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) −18.0000 −0.581857
\(958\) 12.0000 0.387702
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 14.0000 0.451378
\(963\) 2.00000 0.0644491
\(964\) −30.0000 −0.966235
\(965\) 0 0
\(966\) 5.00000 0.160872
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 7.00000 0.224989
\(969\) 56.0000 1.79898
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) 6.00000 0.192252
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −15.0000 −0.479647
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 42.0000 1.34027
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −11.0000 −0.350667
\(985\) 0 0
\(986\) −63.0000 −2.00633
\(987\) 4.00000 0.127321
\(988\) −56.0000 −1.78160
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 25.0000 0.793351
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 23.0000 0.728052
\(999\) 2.00000 0.0632772
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 1050.2.a.f.1.1 1
3.2 odd 2 3150.2.a.bd.1.1 1
4.3 odd 2 8400.2.a.bb.1.1 1
5.2 odd 4 1050.2.g.b.799.1 2
5.3 odd 4 1050.2.g.b.799.2 2
5.4 even 2 1050.2.a.n.1.1 yes 1
7.6 odd 2 7350.2.a.i.1.1 1
15.2 even 4 3150.2.g.p.2899.2 2
15.8 even 4 3150.2.g.p.2899.1 2
15.14 odd 2 3150.2.a.r.1.1 1
20.19 odd 2 8400.2.a.cb.1.1 1
35.34 odd 2 7350.2.a.cm.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.f.1.1 1 1.1 even 1 trivial
1050.2.a.n.1.1 yes 1 5.4 even 2
1050.2.g.b.799.1 2 5.2 odd 4
1050.2.g.b.799.2 2 5.3 odd 4
3150.2.a.r.1.1 1 15.14 odd 2
3150.2.a.bd.1.1 1 3.2 odd 2
3150.2.g.p.2899.1 2 15.8 even 4
3150.2.g.p.2899.2 2 15.2 even 4
7350.2.a.i.1.1 1 7.6 odd 2
7350.2.a.cm.1.1 1 35.34 odd 2
8400.2.a.bb.1.1 1 4.3 odd 2
8400.2.a.cb.1.1 1 20.19 odd 2