Properties

Label 3150.2.g.a.2899.1
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(1.00000i\)
Character \(\chi\) = 3150.2899
Dual form 3150.2.g.a.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} +6.00000i q^{22} +3.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +10.0000i q^{37} -4.00000i q^{38} -9.00000 q^{41} -1.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{49} +1.00000i q^{52} -9.00000i q^{53} +1.00000 q^{56} -3.00000i q^{58} +9.00000 q^{59} +11.0000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +9.00000i q^{82} -9.00000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -14.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 12q^{11} - 2q^{14} + 2q^{16} + 8q^{19} - 2q^{26} + 6q^{29} + 10q^{31} + 6q^{34} - 18q^{41} + 12q^{44} + 6q^{46} - 2q^{49} + 2q^{56} + 18q^{59} + 22q^{61} - 2q^{64} + 24q^{71} + 20q^{74} - 8q^{76} + 20q^{79} - 2q^{86} - 12q^{89} - 2q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 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 \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) − 3.00000i − 0.393919i
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) − 5.00000i − 0.635001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.00000i 0.993884i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) − 3.00000i − 0.312772i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 17.0000i 1.67506i 0.546392 + 0.837530i \(0.316001\pi\)
−0.546392 + 0.837530i \(0.683999\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) − 9.00000i − 0.828517i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) − 11.0000i − 0.995893i
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.0000i − 1.00702i
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) − 10.0000i − 0.821995i
\(149\) −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 \(0.474068\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000i 0.0762493i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 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 \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) − 18.0000i − 1.31629i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) − 22.0000i − 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 12.0000i − 0.844317i
\(203\) − 3.00000i − 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) − 5.00000i − 0.339422i
\(218\) 8.00000i 0.541828i
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 3.00000i 0.199117i 0.995032 + 0.0995585i \(0.0317430\pi\)
−0.995032 + 0.0995585i \(0.968257\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000i 0.196960i
\(233\) − 12.0000i − 0.786146i −0.919507 0.393073i \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 0 0
\(238\) − 3.00000i − 0.194461i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) 0 0
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.00000i − 0.254514i
\(248\) 5.00000i 0.317500i
\(249\) 0 0
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) − 18.0000i − 1.13165i
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000i 0.935674i 0.883815 + 0.467837i \(0.154967\pi\)
−0.883815 + 0.467837i \(0.845033\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) − 4.00000i − 0.244339i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 9.00000i 0.531253i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000i 0.585206i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 9.00000i 0.521356i
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) − 2.00000i − 0.115087i
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) − 27.0000i − 1.51647i −0.651981 0.758236i \(-0.726062\pi\)
0.651981 0.758236i \(-0.273938\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) − 3.00000i − 0.167183i
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 0 0
\(326\) 5.00000 0.276924
\(327\) 0 0
\(328\) − 9.00000i − 0.496942i
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 9.00000i 0.493939i
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.00000i 0.319801i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) − 18.0000i − 0.951330i
\(359\) −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
\(362\) − 2.00000i − 0.105118i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 37.0000i 1.93138i 0.259690 + 0.965692i \(0.416380\pi\)
−0.259690 + 0.965692i \(0.583620\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 8.00000i 0.414224i 0.978317 + 0.207112i \(0.0664065\pi\)
−0.978317 + 0.207112i \(0.933593\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.00000i − 0.154508i
\(378\) 0 0
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 15.0000i − 0.767467i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 14.0000i 0.710742i
\(389\) −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
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 0 0
\(397\) − 29.0000i − 1.45547i −0.685859 0.727734i \(-0.740573\pi\)
0.685859 0.727734i \(-0.259427\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) − 5.00000i − 0.249068i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) − 60.0000i − 2.97409i
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 17.0000i − 0.837530i
\(413\) − 9.00000i − 0.442861i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 24.0000i 1.17388i
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 25.0000i 1.21698i
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) − 11.0000i − 0.532327i
\(428\) − 18.0000i − 0.870063i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 0 0
\(433\) 38.0000i 1.82616i 0.407777 + 0.913082i \(0.366304\pi\)
−0.407777 + 0.913082i \(0.633696\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) −41.0000 −1.95682 −0.978412 0.206666i \(-0.933739\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.00000i − 0.142695i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −19.0000 −0.899676
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 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
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0000i 1.16945i 0.811231 + 0.584725i \(0.198798\pi\)
−0.811231 + 0.584725i \(0.801202\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 9.00000i 0.414259i
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) − 2.00000i − 0.0910975i
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 11.0000i 0.497947i
\(489\) 0 0
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 9.00000i 0.405340i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 27.0000i − 1.20507i
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) − 10.0000i − 0.443678i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) − 10.0000i − 0.439375i
\(519\) 0 0
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 15.0000i 0.653410i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) 9.00000i 0.389833i
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 18.0000i 0.776035i
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) − 17.0000i − 0.726868i −0.931620 0.363434i \(-0.881604\pi\)
0.931620 0.363434i \(-0.118396\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) − 10.0000i − 0.425243i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) − 24.0000i − 1.01238i
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) −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 \(0.587691\pi\)
\(572\) − 6.00000i − 0.250873i
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) 54.0000i 2.23645i
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000i 1.36206i 0.732257 + 0.681028i \(0.238467\pi\)
−0.732257 + 0.681028i \(0.761533\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000i 0.410997i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) − 3.00000i − 0.122679i
\(599\) 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 \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 18.0000i − 0.721734i
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 0 0
\(634\) −27.0000 −1.07231
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 18.0000i 0.712627i
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 0 0
\(649\) −54.0000 −2.11969
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.00000i − 0.195815i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 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 \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 19.0000i 0.738456i
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) − 18.0000i − 0.696441i
\(669\) 0 0
\(670\) 0 0
\(671\) −66.0000 −2.54790
\(672\) 0 0
\(673\) − 37.0000i − 1.42625i −0.701039 0.713123i \(-0.747280\pi\)
0.701039 0.713123i \(-0.252720\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 12.0000i − 0.461197i −0.973049 0.230599i \(-0.925932\pi\)
0.973049 0.230599i \(-0.0740685\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 30.0000i 1.14876i
\(683\) − 18.0000i − 0.688751i −0.938832 0.344375i \(-0.888091\pi\)
0.938832 0.344375i \(-0.111909\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) − 1.00000i − 0.0381246i
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) 0 0
\(697\) − 27.0000i − 1.02270i
\(698\) 17.0000i 0.643459i
\(699\) 0 0
\(700\) 0 0
\(701\) −3.00000 −0.113308 −0.0566542 0.998394i \(-0.518043\pi\)
−0.0566542 + 0.998394i \(0.518043\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) 15.0000i 0.561754i
\(714\) 0 0
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) 3.00000i 0.111959i
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) 3.00000i 0.111648i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 7.00000i 0.259616i 0.991539 + 0.129808i \(0.0414360\pi\)
−0.991539 + 0.129808i \(0.958564\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 29.0000i 1.07114i 0.844491 + 0.535570i \(0.179903\pi\)
−0.844491 + 0.535570i \(0.820097\pi\)
\(734\) 37.0000 1.36569
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.00000i 0.330400i
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) 0 0
\(748\) 18.0000i 0.658145i
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.0000i − 0.508839i −0.967094 0.254419i \(-0.918116\pi\)
0.967094 0.254419i \(-0.0818843\pi\)
\(758\) 35.0000i 1.27126i
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 9.00000i − 0.324971i
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.0000i 0.791797i
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) 9.00000i 0.321839i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 26.0000i − 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) 15.0000i 0.534353i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) − 11.0000i − 0.390621i
\(794\) −29.0000 −1.02917
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 12.0000i − 0.423735i
\(803\) 60.0000i 2.11735i
\(804\) 0 0
\(805\) 0 0
\(806\) −5.00000 −0.176117
\(807\) 0 0
\(808\) 12.0000i 0.422159i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 3.00000i 0.105279i
\(813\) 0 0
\(814\) −60.0000 −2.10300
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.00000i − 0.139942i
\(818\) 38.0000i 1.32864i
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) − 34.0000i − 1.18517i −0.805510 0.592583i \(-0.798108\pi\)
0.805510 0.592583i \(-0.201892\pi\)
\(824\) −17.0000 −0.592223
\(825\) 0 0
\(826\) −9.00000 −0.313150
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) 0 0
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) − 3.00000i − 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 21.0000i 0.725433i
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000i 1.17172i
\(843\) 0 0
\(844\) 25.0000 0.860535
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.0000i − 0.859010i
\(848\) − 9.00000i − 0.309061i
\(849\) 0 0
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) −11.0000 −0.376412
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 15.0000i − 0.510902i
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 38.0000 1.29129
\(867\) 0 0
\(868\) 5.00000i 0.169711i
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) − 8.00000i − 0.270914i
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 41.0000i 1.38368i
\(879\) 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
\(883\) 5.00000i 0.168263i 0.996455 + 0.0841317i \(0.0268116\pi\)
−0.996455 + 0.0841317i \(0.973188\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 54.0000i 1.81314i 0.422053 + 0.906571i \(0.361310\pi\)
−0.422053 + 0.906571i \(0.638690\pi\)
\(888\) 0 0
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 0 0
\(892\) 19.0000i 0.636167i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) − 24.0000i − 0.800890i
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 27.0000 0.899500
\(902\) − 54.0000i − 1.79800i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) − 53.0000i − 1.75984i −0.475125 0.879918i \(-0.657597\pi\)
0.475125 0.879918i \(-0.342403\pi\)
\(908\) − 3.00000i − 0.0995585i
\(909\) 0 0
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 0 0
\(913\) 54.0000i 1.78714i
\(914\) 25.0000 0.826927
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12.0000i 0.395199i
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) − 3.00000i − 0.0984798i
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 12.0000i 0.393073i
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) − 27.0000i − 0.879241i
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 3.00000i 0.0972306i
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) − 10.0000i − 0.322413i
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) − 16.0000i − 0.512936i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) 42.0000i 1.34027i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 4.00000i 0.127257i
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 5.00000i − 0.158750i
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) − 19.0000i − 0.601434i
\(999\) 0 0
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 3150.2.g.a.2899.1 2
3.2 odd 2 1050.2.g.e.799.2 2
5.2 odd 4 3150.2.a.bg.1.1 1
5.3 odd 4 3150.2.a.a.1.1 1
5.4 even 2 inner 3150.2.g.a.2899.2 2
15.2 even 4 1050.2.a.j.1.1 1
15.8 even 4 1050.2.a.l.1.1 yes 1
15.14 odd 2 1050.2.g.e.799.1 2
60.23 odd 4 8400.2.a.ci.1.1 1
60.47 odd 4 8400.2.a.a.1.1 1
105.62 odd 4 7350.2.a.r.1.1 1
105.83 odd 4 7350.2.a.cz.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.j.1.1 1 15.2 even 4
1050.2.a.l.1.1 yes 1 15.8 even 4
1050.2.g.e.799.1 2 15.14 odd 2
1050.2.g.e.799.2 2 3.2 odd 2
3150.2.a.a.1.1 1 5.3 odd 4
3150.2.a.bg.1.1 1 5.2 odd 4
3150.2.g.a.2899.1 2 1.1 even 1 trivial
3150.2.g.a.2899.2 2 5.4 even 2 inner
7350.2.a.r.1.1 1 105.62 odd 4
7350.2.a.cz.1.1 1 105.83 odd 4
8400.2.a.a.1.1 1 60.47 odd 4
8400.2.a.ci.1.1 1 60.23 odd 4