Properties

Label 7350.2.a.r.1.1
Level 7350
Weight 2
Character 7350.1
Self dual yes
Analytic conductor 58.690
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 7350.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -6.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -9.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{48} +3.00000 q^{51} +1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} -3.00000 q^{58} -9.00000 q^{59} -11.0000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -4.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} +4.00000 q^{76} +1.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -9.00000 q^{83} +1.00000 q^{86} -3.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -3.00000 q^{92} +5.00000 q^{93} +1.00000 q^{96} -14.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −3.00000 −0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) −3.00000 −0.321634
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −3.00000 −0.297044
\(103\) −17.0000 −1.67506 −0.837530 0.546392i \(-0.816001\pi\)
−0.837530 + 0.546392i \(0.816001\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 1.00000 0.0924500
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 11.0000 0.995893
\(123\) 9.00000 0.811503
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −3.00000 −0.255377
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −4.00000 −0.324443
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 10.0000 0.795557
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −1.00000 −0.0762493
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 9.00000 0.676481
\(178\) −6.00000 −0.449719
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) −5.00000 −0.366618
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) −6.00000 −0.426401
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) −3.00000 −0.208514
\(208\) 1.00000 0.0693375
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 9.00000 0.618123
\(213\) 12.0000 0.822226
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) −10.0000 −0.671156
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −4.00000 −0.264906
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 4.00000 0.254514
\(248\) 5.00000 0.317500
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −16.0000 −0.959616
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 10.0000 0.585206
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) −6.00000 −0.348155
\(298\) 9.00000 0.521356
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) −2.00000 −0.115087
\(303\) −12.0000 −0.689382
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 17.0000 0.967096
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 1.00000 0.0566139
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 9.00000 0.504695
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) −8.00000 −0.442401
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) −9.00000 −0.493939
\(333\) −10.0000 −0.547997
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 12.0000 0.652714
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) −3.00000 −0.160817
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −6.00000 −0.319801
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) −11.0000 −0.574979
\(367\) 37.0000 1.93138 0.965692 0.259690i \(-0.0836203\pi\)
0.965692 + 0.259690i \(0.0836203\pi\)
\(368\) −3.00000 −0.156386
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 0 0
\(372\) 5.00000 0.259238
\(373\) 8.00000 0.414224 0.207112 0.978317i \(-0.433593\pi\)
0.207112 + 0.978317i \(0.433593\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 35.0000 1.79783 0.898915 0.438124i \(-0.144357\pi\)
0.898915 + 0.438124i \(0.144357\pi\)
\(380\) 0 0
\(381\) 10.0000 0.512316
\(382\) 15.0000 0.767467
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −1.00000 −0.0508329
\(388\) −14.0000 −0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 0 0
\(394\) 15.0000 0.755689
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −4.00000 −0.199502
\(403\) −5.00000 −0.249068
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −60.0000 −2.97409
\(408\) −3.00000 −0.148522
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −17.0000 −0.837530
\(413\) 0 0
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) −16.0000 −0.783523
\(418\) −24.0000 −1.17388
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 25.0000 1.21698
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 18.0000 0.870063
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −12.0000 −0.574038
\(438\) 10.0000 0.477818
\(439\) −41.0000 −1.95682 −0.978412 0.206666i \(-0.933739\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −19.0000 −0.899676
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) −54.0000 −2.54276
\(452\) −6.00000 −0.282216
\(453\) −2.00000 −0.0939682
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −25.0000 −1.16945 −0.584725 0.811231i \(-0.698798\pi\)
−0.584725 + 0.811231i \(0.698798\pi\)
\(458\) −22.0000 −1.02799
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 9.00000 0.414259
\(473\) −6.00000 −0.275880
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 11.0000 0.497947
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) 9.00000 0.405751
\(493\) −9.00000 −0.405340
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) −19.0000 −0.850557 −0.425278 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) −27.0000 −1.20507
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) 12.0000 0.532939
\(508\) −10.0000 −0.443678
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) −3.00000 −0.131306
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 15.0000 0.653410
\(528\) −6.00000 −0.261116
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) −9.00000 −0.389833
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −18.0000 −0.776757
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 20.0000 0.859074
\(543\) 2.00000 0.0858282
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 0 0
\(549\) −11.0000 −0.469469
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) −3.00000 −0.127688
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 5.00000 0.211667
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 24.0000 1.01238
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 6.00000 0.250873
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 8.00000 0.332756
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 54.0000 2.23645
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 0 0
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 15.0000 0.617018
\(592\) −10.0000 −0.410997
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −9.00000 −0.368654
\(597\) 8.00000 0.327418
\(598\) 3.00000 0.122679
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −17.0000 −0.683840
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) −24.0000 −0.958468
\(628\) 10.0000 0.399043
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 10.0000 0.397779
\(633\) 25.0000 0.993661
\(634\) 27.0000 1.07231
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 18.0000 0.710403
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −54.0000 −2.11969
\(650\) 0 0
\(651\) 0 0
\(652\) 5.00000 0.195815
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 42.0000 1.63609 0.818044 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 19.0000 0.738456
\(663\) 3.00000 0.116510
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) −9.00000 −0.348481
\(668\) −18.0000 −0.696441
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) −66.0000 −2.54790
\(672\) 0 0
\(673\) −37.0000 −1.42625 −0.713123 0.701039i \(-0.752720\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 30.0000 1.14876
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −1.00000 −0.0381246
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 34.0000 1.29342 0.646710 0.762736i \(-0.276144\pi\)
0.646710 + 0.762736i \(0.276144\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) 27.0000 1.02270
\(698\) 17.0000 0.643459
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 1.00000 0.0377426
\(703\) −40.0000 −1.50863
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 9.00000 0.338241
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) 0 0
\(718\) 3.00000 0.111959
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 11.0000 0.406572
\(733\) −29.0000 −1.07114 −0.535570 0.844491i \(-0.679903\pi\)
−0.535570 + 0.844491i \(0.679903\pi\)
\(734\) −37.0000 −1.36569
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −24.0000 −0.884051
\(738\) 9.00000 0.331295
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) −5.00000 −0.183309
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) −9.00000 −0.329293
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −27.0000 −0.983935
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) −35.0000 −1.27126
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −10.0000 −0.362262
\(763\) 0 0
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −9.00000 −0.324971
\(768\) −1.00000 −0.0360844
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) −22.0000 −0.791797
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) −9.00000 −0.321839
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.0000 −0.926800 −0.463400 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(788\) −15.0000 −0.534353
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) −11.0000 −0.390621
\(794\) 29.0000 1.02917
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 12.0000 0.423735
\(803\) 60.0000 2.11735
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 5.00000 0.176117
\(807\) −18.0000 −0.633630
\(808\) −12.0000 −0.422159
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 60.0000 2.10300
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −4.00000 −0.139942
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 17.0000 0.592223
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) −3.00000 −0.104257
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 5.00000 0.172825
\(838\) −21.0000 −0.725433
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000 1.17172
\(843\) 24.0000 0.826604
\(844\) −25.0000 −0.860535
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 12.0000 0.411113
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 6.00000 0.204837
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.0000 0.510902
\(863\) −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
\(866\) 38.0000 1.29129
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) −8.00000 −0.270914
\(873\) −14.0000 −0.473828
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 41.0000 1.38368
\(879\) 0 0
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 5.00000 0.168263 0.0841317 0.996455i \(-0.473188\pi\)
0.0841317 + 0.996455i \(0.473188\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −54.0000 −1.81314 −0.906571 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) −10.0000 −0.335578
\(889\) 0 0
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 19.0000 0.636167
\(893\) 0 0
\(894\) −9.00000 −0.301005
\(895\) 0 0
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) −24.0000 −0.800890
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 54.0000 1.79800
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 2.00000 0.0664455
\(907\) 53.0000 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) −4.00000 −0.132453
\(913\) −54.0000 −1.78714
\(914\) 25.0000 0.826927
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −3.00000 −0.0990148
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 12.0000 0.395199
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) −17.0000 −0.558353
\(928\) −3.00000 −0.0984798
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.0000 0.393073
\(933\) −18.0000 −0.589294
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 10.0000 0.325818
\(943\) 27.0000 0.879241
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 10.0000 0.324785
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) −18.0000 −0.581857
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 10.0000 0.322413
\(963\) 18.0000 0.580042
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −25.0000 −0.803530
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −11.0000 −0.352101
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 5.00000 0.159882
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) −42.0000 −1.34027
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 5.00000 0.158750
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 19.0000 0.601434
\(999\) 10.0000 0.316386
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 7350.2.a.r.1.1 1
5.4 even 2 7350.2.a.cz.1.1 1
7.6 odd 2 1050.2.a.j.1.1 1
21.20 even 2 3150.2.a.bg.1.1 1
28.27 even 2 8400.2.a.a.1.1 1
35.13 even 4 1050.2.g.e.799.2 2
35.27 even 4 1050.2.g.e.799.1 2
35.34 odd 2 1050.2.a.l.1.1 yes 1
105.62 odd 4 3150.2.g.a.2899.2 2
105.83 odd 4 3150.2.g.a.2899.1 2
105.104 even 2 3150.2.a.a.1.1 1
140.139 even 2 8400.2.a.ci.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.j.1.1 1 7.6 odd 2
1050.2.a.l.1.1 yes 1 35.34 odd 2
1050.2.g.e.799.1 2 35.27 even 4
1050.2.g.e.799.2 2 35.13 even 4
3150.2.a.a.1.1 1 105.104 even 2
3150.2.a.bg.1.1 1 21.20 even 2
3150.2.g.a.2899.1 2 105.83 odd 4
3150.2.g.a.2899.2 2 105.62 odd 4
7350.2.a.r.1.1 1 1.1 even 1 trivial
7350.2.a.cz.1.1 1 5.4 even 2
8400.2.a.a.1.1 1 28.27 even 2
8400.2.a.ci.1.1 1 140.139 even 2