Properties

Label 4410.2.a.c.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -5.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} +3.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{26} +6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +11.0000 q^{37} -1.00000 q^{38} +1.00000 q^{40} +3.00000 q^{41} -10.0000 q^{43} -3.00000 q^{44} +3.00000 q^{46} +3.00000 q^{47} -1.00000 q^{50} -5.00000 q^{52} -3.00000 q^{53} +3.00000 q^{55} -6.00000 q^{58} +4.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +5.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -12.0000 q^{71} +4.00000 q^{73} -11.0000 q^{74} +1.00000 q^{76} -10.0000 q^{79} -1.00000 q^{80} -3.00000 q^{82} -12.0000 q^{83} -6.00000 q^{85} +10.0000 q^{86} +3.00000 q^{88} +6.00000 q^{89} -3.00000 q^{92} -3.00000 q^{94} -1.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −11.0000 −1.27872
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 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
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −5.00000 −0.438529
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 11.0000 0.904194
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −11.0000 −0.808736
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) −3.00000 −0.206041
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) −30.0000 −2.01802
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 19.0000 1.19217
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 5.00000 0.310087
\(261\) 0 0
\(262\) −3.00000 −0.185341
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −26.0000 −1.54554 −0.772770 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −15.0000 −0.886969
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −27.0000 −1.57736 −0.788678 0.614806i \(-0.789234\pi\)
−0.788678 + 0.614806i \(0.789234\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −11.0000 −0.639362
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 15.0000 0.867472
\(300\) 0 0
\(301\) 0 0
\(302\) −14.0000 −0.805609
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 5.00000 0.282166
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −9.00000 −0.492458
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −3.00000 −0.161281
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −3.00000 −0.158555
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 11.0000 0.571863
\(371\) 0 0
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −30.0000 −1.54508
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) −3.00000 −0.151138
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −21.0000 −1.04869 −0.524345 0.851506i \(-0.675690\pi\)
−0.524345 + 0.851506i \(0.675690\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −33.0000 −1.63575
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 3.00000 0.148159
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 3.00000 0.146735
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 1.00000 0.0486792
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 30.0000 1.42695
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) 3.00000 0.141579 0.0707894 0.997491i \(-0.477448\pi\)
0.0707894 + 0.997491i \(0.477448\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.00000 0.138380
\(471\) 0 0
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −55.0000 −2.50778
\(482\) −25.0000 −1.13872
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −4.00000 −0.181071
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 15.0000 0.669483
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) −19.0000 −0.842989
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) −5.00000 −0.219265
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 3.00000 0.127920
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 27.0000 1.14403 0.572013 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) 0 0
\(559\) 50.0000 2.11477
\(560\) 0 0
\(561\) 0 0
\(562\) −3.00000 −0.126547
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 26.0000 1.09286
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 15.0000 0.627182
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 27.0000 1.11536
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 11.0000 0.452097
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) −15.0000 −0.613396
\(599\) −42.0000 −1.71607 −0.858037 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 19.0000 0.771186 0.385593 0.922669i \(-0.373997\pi\)
0.385593 + 0.922669i \(0.373997\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −15.0000 −0.606835
\(612\) 0 0
\(613\) 47.0000 1.89831 0.949156 0.314806i \(-0.101939\pi\)
0.949156 + 0.314806i \(0.101939\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −5.00000 −0.199522
\(629\) 66.0000 2.63159
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 19.0000 0.753992
\(636\) 0 0
\(637\) 0 0
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 45.0000 1.77739 0.888697 0.458496i \(-0.151612\pi\)
0.888697 + 0.458496i \(0.151612\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 5.00000 0.196116
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −21.0000 −0.821794 −0.410897 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(654\) 0 0
\(655\) −3.00000 −0.117220
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −44.0000 −1.71140 −0.855701 0.517471i \(-0.826874\pi\)
−0.855701 + 0.517471i \(0.826874\pi\)
\(662\) 7.00000 0.272063
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 9.00000 0.348220
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 11.0000 0.414873
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 3.00000 0.112115
\(717\) 0 0
\(718\) 6.00000 0.223918
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −60.0000 −2.21918
\(732\) 0 0
\(733\) −47.0000 −1.73598 −0.867992 0.496578i \(-0.834590\pi\)
−0.867992 + 0.496578i \(0.834590\pi\)
\(734\) −1.00000 −0.0369107
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) −11.0000 −0.404368
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 34.0000 1.24483
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) 30.0000 1.09254
\(755\) −14.0000 −0.509512
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 25.0000 0.908041
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) −51.0000 −1.84875 −0.924374 0.381487i \(-0.875412\pi\)
−0.924374 + 0.381487i \(0.875412\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) 0 0
\(768\) 0 0
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 3.00000 0.107486
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 18.0000 0.643679
\(783\) 0 0
\(784\) 0 0
\(785\) 5.00000 0.178458
\(786\) 0 0
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) 3.00000 0.106871
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 21.0000 0.741536
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 33.0000 1.15665
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) −3.00000 −0.104765
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 54.0000 1.87776 0.938882 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) −3.00000 −0.103757
\(837\) 0 0
\(838\) −15.0000 −0.518166
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) −1.00000 −0.0344214
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) −33.0000 −1.13123
\(852\) 0 0
\(853\) 1.00000 0.0342393 0.0171197 0.999853i \(-0.494550\pi\)
0.0171197 + 0.999853i \(0.494550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 10.0000 0.340997
\(861\) 0 0
\(862\) 0 0
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) 0 0
\(865\) −3.00000 −0.102003
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) 0 0
\(877\) −7.00000 −0.236373 −0.118187 0.992991i \(-0.537708\pi\)
−0.118187 + 0.992991i \(0.537708\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 0 0
\(898\) −3.00000 −0.100111
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 60.0000 1.97492
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) 19.0000 0.624379
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 18.0000 0.588978
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.00000 −0.0978492
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) 0 0
\(945\) 0 0
\(946\) −30.0000 −0.975384
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 55.0000 1.77327
\(963\) 0 0
\(964\) 25.0000 0.805196
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 57.0000 1.81802 0.909009 0.416777i \(-0.136840\pi\)
0.909009 + 0.416777i \(0.136840\pi\)
\(984\) 0 0
\(985\) −3.00000 −0.0955879
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −5.00000 −0.159071
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 28.0000 0.886325
\(999\) 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 4410.2.a.c.1.1 1
3.2 odd 2 490.2.a.j.1.1 1
7.3 odd 6 630.2.k.e.541.1 2
7.5 odd 6 630.2.k.e.361.1 2
7.6 odd 2 4410.2.a.m.1.1 1
12.11 even 2 3920.2.a.g.1.1 1
15.2 even 4 2450.2.c.p.99.2 2
15.8 even 4 2450.2.c.p.99.1 2
15.14 odd 2 2450.2.a.f.1.1 1
21.2 odd 6 490.2.e.a.361.1 2
21.5 even 6 70.2.e.b.11.1 2
21.11 odd 6 490.2.e.a.471.1 2
21.17 even 6 70.2.e.b.51.1 yes 2
21.20 even 2 490.2.a.g.1.1 1
84.47 odd 6 560.2.q.d.81.1 2
84.59 odd 6 560.2.q.d.401.1 2
84.83 odd 2 3920.2.a.be.1.1 1
105.17 odd 12 350.2.j.a.149.1 4
105.38 odd 12 350.2.j.a.149.2 4
105.47 odd 12 350.2.j.a.249.2 4
105.59 even 6 350.2.e.h.51.1 2
105.62 odd 4 2450.2.c.f.99.2 2
105.68 odd 12 350.2.j.a.249.1 4
105.83 odd 4 2450.2.c.f.99.1 2
105.89 even 6 350.2.e.h.151.1 2
105.104 even 2 2450.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.b.11.1 2 21.5 even 6
70.2.e.b.51.1 yes 2 21.17 even 6
350.2.e.h.51.1 2 105.59 even 6
350.2.e.h.151.1 2 105.89 even 6
350.2.j.a.149.1 4 105.17 odd 12
350.2.j.a.149.2 4 105.38 odd 12
350.2.j.a.249.1 4 105.68 odd 12
350.2.j.a.249.2 4 105.47 odd 12
490.2.a.g.1.1 1 21.20 even 2
490.2.a.j.1.1 1 3.2 odd 2
490.2.e.a.361.1 2 21.2 odd 6
490.2.e.a.471.1 2 21.11 odd 6
560.2.q.d.81.1 2 84.47 odd 6
560.2.q.d.401.1 2 84.59 odd 6
630.2.k.e.361.1 2 7.5 odd 6
630.2.k.e.541.1 2 7.3 odd 6
2450.2.a.f.1.1 1 15.14 odd 2
2450.2.a.p.1.1 1 105.104 even 2
2450.2.c.f.99.1 2 105.83 odd 4
2450.2.c.f.99.2 2 105.62 odd 4
2450.2.c.p.99.1 2 15.8 even 4
2450.2.c.p.99.2 2 15.2 even 4
3920.2.a.g.1.1 1 12.11 even 2
3920.2.a.be.1.1 1 84.83 odd 2
4410.2.a.c.1.1 1 1.1 even 1 trivial
4410.2.a.m.1.1 1 7.6 odd 2