Properties

Label 378.2.a.c.1.1
Level $378$
Weight $2$
Character 378.1
Self dual yes
Analytic conductor $3.018$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.01834519640\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 378.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} +5.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{28} -4.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +5.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} +9.00000 q^{41} -10.0000 q^{43} -5.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} -12.0000 q^{53} +5.00000 q^{55} +1.00000 q^{56} +4.00000 q^{58} +14.0000 q^{59} +9.00000 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} -1.00000 q^{70} +13.0000 q^{71} -2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{76} +5.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} -9.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} +10.0000 q^{86} +5.00000 q^{88} +9.00000 q^{89} +1.00000 q^{92} +6.00000 q^{94} +1.00000 q^{95} +16.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 9.00000 1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 13.0000 1.54282 0.771408 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −9.00000 −0.993884
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) −5.00000 −0.476731
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −14.0000 −1.28880
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) −9.00000 −0.808224
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −13.0000 −1.09094
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 5.00000 0.410997
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −5.00000 −0.402911
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) −6.00000 −0.477334
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) −7.00000 −0.532200 −0.266100 0.963945i \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) −9.00000 −0.674579
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) 10.0000 0.731272
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −13.0000 −0.921546 −0.460773 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 1.00000 0.0696733
\(207\) 0 0
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) −7.00000 −0.474100
\(219\) 0 0
\(220\) 5.00000 0.337100
\(221\) 0 0
\(222\) 0 0
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 9.00000 0.571501
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 0 0
\(262\) 22.0000 1.35916
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −13.0000 −0.792624 −0.396312 0.918116i \(-0.629710\pi\)
−0.396312 + 0.918116i \(0.629710\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 13.0000 0.771408
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) −5.00000 −0.290619
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −5.00000 −0.285365 −0.142683 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) 5.00000 0.284901
\(309\) 0 0
\(310\) −9.00000 −0.511166
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −9.00000 −0.496942
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 10.0000 0.547176
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 27.0000 1.47078 0.735392 0.677642i \(-0.236998\pi\)
0.735392 + 0.677642i \(0.236998\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 45.0000 2.43689
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 7.00000 0.376322
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −13.0000 −0.689968
\(356\) 9.00000 0.476999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 35.0000 1.82699 0.913493 0.406855i \(-0.133375\pi\)
0.913493 + 0.406855i \(0.133375\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 5.00000 0.259938
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −17.0000 −0.880227 −0.440113 0.897942i \(-0.645062\pi\)
−0.440113 + 0.897942i \(0.645062\pi\)
\(374\) −10.0000 −0.517088
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) 10.0000 0.510976 0.255488 0.966812i \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 16.0000 0.812277
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 13.0000 0.651631
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) −25.0000 −1.23920
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 9.00000 0.444478
\(411\) 0 0
\(412\) −1.00000 −0.0492665
\(413\) −14.0000 −0.688895
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) −5.00000 −0.244558
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 22.0000 1.07094
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 7.00000 0.335239
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −5.00000 −0.238366
\(441\) 0 0
\(442\) 0 0
\(443\) −11.0000 −0.522626 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) −5.00000 −0.236757
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −45.0000 −2.11897
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000 0.608114 0.304057 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −31.0000 −1.44381 −0.721907 0.691990i \(-0.756734\pi\)
−0.721907 + 0.691990i \(0.756734\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 26.0000 1.20314 0.601568 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) −14.0000 −0.644402
\(473\) 50.0000 2.29900
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) −9.00000 −0.404112
\(497\) −13.0000 −0.583130
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 5.00000 0.222277
\(507\) 0 0
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 27.0000 1.19092
\(515\) 1.00000 0.0440653
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 5.00000 0.219687
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −22.0000 −0.961074
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 1.00000 0.0433555
\(533\) 0 0
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 13.0000 0.560470
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −16.0000 −0.683486
\(549\) 0 0
\(550\) −20.0000 −0.852803
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 19.0000 0.807233
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −13.0000 −0.545468
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 60.0000 2.48495
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 14.0000 0.576371
\(591\) 0 0
\(592\) 5.00000 0.205499
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −10.0000 −0.407570
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −15.0000 −0.605844 −0.302922 0.953015i \(-0.597962\pi\)
−0.302922 + 0.953015i \(0.597962\pi\)
\(614\) 5.00000 0.201784
\(615\) 0 0
\(616\) −5.00000 −0.201456
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −11.0000 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(620\) 9.00000 0.361449
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −20.0000 −0.791808
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) −70.0000 −2.74774
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 22.0000 0.859611
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) −17.0000 −0.662226 −0.331113 0.943591i \(-0.607424\pi\)
−0.331113 + 0.943591i \(0.607424\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) −10.0000 −0.386912
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −45.0000 −1.72314
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −7.00000 −0.266100
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −3.00000 −0.112906
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 13.0000 0.487881
\(711\) 0 0
\(712\) −9.00000 −0.337289
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) −18.0000 −0.664845 −0.332423 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) −35.0000 −1.29187
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 40.0000 1.47342
\(738\) 0 0
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) −5.00000 −0.183804
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 17.0000 0.622414
\(747\) 0 0
\(748\) 10.0000 0.365636
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 18.0000 0.656829 0.328415 0.944534i \(-0.393486\pi\)
0.328415 + 0.944534i \(0.393486\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 14.0000 0.508503
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) −7.00000 −0.253417
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −10.0000 −0.361315
\(767\) 0 0
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 5.00000 0.180187
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −19.0000 −0.683383 −0.341691 0.939812i \(-0.611000\pi\)
−0.341691 + 0.939812i \(0.611000\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) −65.0000 −2.32588
\(782\) 2.00000 0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) 6.00000 0.213470
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 0 0
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) −13.0000 −0.460773
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) 20.0000 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) 0 0
\(811\) 1.00000 0.0351147 0.0175574 0.999846i \(-0.494411\pi\)
0.0175574 + 0.999846i \(0.494411\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) 25.0000 0.876250
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 10.0000 0.349856
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) 14.0000 0.487122
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 10.0000 0.346064
\(836\) 5.00000 0.172929
\(837\) 0 0
\(838\) 6.00000 0.207267
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 27.0000 0.930481
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) 5.00000 0.171398
\(852\) 0 0
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 10.0000 0.340997
\(861\) 0 0
\(862\) −15.0000 −0.510902
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 7.00000 0.238007
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) 9.00000 0.305480
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 0 0
\(872\) −7.00000 −0.237050
\(873\) 0 0
\(874\) 1.00000 0.0338255
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −24.0000 −0.809961
\(879\) 0 0
\(880\) 5.00000 0.168550
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 11.0000 0.369552
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 9.00000 0.301681
\(891\) 0 0
\(892\) 5.00000 0.167412
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 45.0000 1.49834
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 30.0000 0.996134 0.498067 0.867139i \(-0.334043\pi\)
0.498067 + 0.867139i \(0.334043\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) −13.0000 −0.430002
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 22.0000 0.726504
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 1.00000 0.0329690
\(921\) 0 0
\(922\) 31.0000 1.02093
\(923\) 0 0
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 14.0000 0.460069
\(927\) 0 0
\(928\) 4.00000 0.131306
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) −26.0000 −0.850746
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) −13.0000 −0.423788 −0.211894 0.977293i \(-0.567963\pi\)
−0.211894 + 0.977293i \(0.567963\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) −50.0000 −1.62564
\(947\) 17.0000 0.552426 0.276213 0.961096i \(-0.410921\pi\)
0.276213 + 0.961096i \(0.410921\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 0 0
\(964\) 14.0000 0.450910
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 0 0
\(977\) −60.0000 −1.91957 −0.959785 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −10.0000 −0.317982
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 9.00000 0.285750
\(993\) 0 0
\(994\) 13.0000 0.412335
\(995\) 13.0000 0.412128
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −10.0000 −0.316544
\(999\) 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 378.2.a.c.1.1 1
3.2 odd 2 378.2.a.f.1.1 yes 1
4.3 odd 2 3024.2.a.m.1.1 1
5.4 even 2 9450.2.a.dc.1.1 1
7.6 odd 2 2646.2.a.i.1.1 1
9.2 odd 6 1134.2.f.c.757.1 2
9.4 even 3 1134.2.f.n.379.1 2
9.5 odd 6 1134.2.f.c.379.1 2
9.7 even 3 1134.2.f.n.757.1 2
12.11 even 2 3024.2.a.t.1.1 1
15.14 odd 2 9450.2.a.bx.1.1 1
21.20 even 2 2646.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.c.1.1 1 1.1 even 1 trivial
378.2.a.f.1.1 yes 1 3.2 odd 2
1134.2.f.c.379.1 2 9.5 odd 6
1134.2.f.c.757.1 2 9.2 odd 6
1134.2.f.n.379.1 2 9.4 even 3
1134.2.f.n.757.1 2 9.7 even 3
2646.2.a.i.1.1 1 7.6 odd 2
2646.2.a.v.1.1 1 21.20 even 2
3024.2.a.m.1.1 1 4.3 odd 2
3024.2.a.t.1.1 1 12.11 even 2
9450.2.a.bx.1.1 1 15.14 odd 2
9450.2.a.dc.1.1 1 5.4 even 2