Properties

Label 2240.2.a.bb.1.1
Level $2240$
Weight $2$
Character 2240.1
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2240.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -5.00000 q^{11} +3.00000 q^{13} +3.00000 q^{15} -1.00000 q^{17} +6.00000 q^{19} +3.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} +9.00000 q^{29} +4.00000 q^{31} -15.0000 q^{33} +1.00000 q^{35} -2.00000 q^{37} +9.00000 q^{39} -4.00000 q^{41} +10.0000 q^{43} +6.00000 q^{45} +1.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} -4.00000 q^{53} -5.00000 q^{55} +18.0000 q^{57} -8.00000 q^{59} +8.00000 q^{61} +6.00000 q^{63} +3.00000 q^{65} +12.0000 q^{67} -18.0000 q^{69} -8.00000 q^{71} +2.00000 q^{73} +3.00000 q^{75} -5.00000 q^{77} -13.0000 q^{79} +9.00000 q^{81} -4.00000 q^{83} -1.00000 q^{85} +27.0000 q^{87} +4.00000 q^{89} +3.00000 q^{91} +12.0000 q^{93} +6.00000 q^{95} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −15.0000 −2.61116
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 9.00000 1.44115
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 27.0000 2.89470
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −19.0000 −1.87213 −0.936063 0.351833i \(-0.885559\pi\)
−0.936063 + 0.351833i \(0.885559\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 18.0000 1.66410
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 30.0000 2.64135
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) −15.0000 −1.25436
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) −15.0000 −1.16775
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 0 0
\(173\) 1.00000 0.0760286 0.0380143 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −24.0000 −1.80395
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 36.0000 2.53924
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −36.0000 −2.50217
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) 0 0
\(213\) −24.0000 −1.64445
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) −1.00000 −0.0663723 −0.0331862 0.999449i \(-0.510565\pi\)
−0.0331862 + 0.999449i \(0.510565\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 1.00000 0.0652328
\(236\) 0 0
\(237\) −39.0000 −2.53332
\(238\) 0 0
\(239\) −1.00000 −0.0646846 −0.0323423 0.999477i \(-0.510297\pi\)
−0.0323423 + 0.999477i \(0.510297\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 30.0000 1.88608
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 54.0000 3.34252
\(262\) 0 0
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 9.00000 0.544705
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 11.0000 0.656205 0.328102 0.944642i \(-0.393591\pi\)
0.328102 + 0.944642i \(0.393591\pi\)
\(282\) 0 0
\(283\) −31.0000 −1.84276 −0.921379 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 0 0
\(285\) 18.0000 1.06623
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −39.0000 −2.28622
\(292\) 0 0
\(293\) 5.00000 0.292103 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −45.0000 −2.61116
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) −57.0000 −3.24262
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −7.00000 −0.395663 −0.197832 0.980236i \(-0.563390\pi\)
−0.197832 + 0.980236i \(0.563390\pi\)
\(314\) 0 0
\(315\) 6.00000 0.338062
\(316\) 0 0
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) −45.0000 −2.51952
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 9.00000 0.497701
\(328\) 0 0
\(329\) 1.00000 0.0551318
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 42.0000 2.28113
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −18.0000 −0.969087
\(346\) 0 0
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 27.0000 1.44115
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 42.0000 2.20443
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 27.0000 1.39057
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 24.0000 1.22956
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) 60.0000 3.04997
\(388\) 0 0
\(389\) 31.0000 1.57176 0.785881 0.618378i \(-0.212210\pi\)
0.785881 + 0.618378i \(0.212210\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) −30.0000 −1.51330
\(394\) 0 0
\(395\) −13.0000 −0.654101
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −42.0000 −2.05675
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) −45.0000 −2.17262
\(430\) 0 0
\(431\) 9.00000 0.433515 0.216757 0.976226i \(-0.430452\pi\)
0.216757 + 0.976226i \(0.430452\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 27.0000 1.29455
\(436\) 0 0
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 0 0
\(453\) −15.0000 −0.704761
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) 0 0
\(459\) −9.00000 −0.420084
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) 5.00000 0.231372 0.115686 0.993286i \(-0.463093\pi\)
0.115686 + 0.993286i \(0.463093\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) −50.0000 −2.29900
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −24.0000 −1.09888
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) −18.0000 −0.819028
\(484\) 0 0
\(485\) −13.0000 −0.590300
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) −30.0000 −1.35665
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −30.0000 −1.34840
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −9.00000 −0.402895 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) 3.00000 0.133763 0.0668817 0.997761i \(-0.478695\pi\)
0.0668817 + 0.997761i \(0.478695\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) −19.0000 −0.837240
\(516\) 0 0
\(517\) −5.00000 −0.219900
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 0 0
\(543\) 60.0000 2.57485
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) 0 0
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) −13.0000 −0.552816
\(554\) 0 0
\(555\) −6.00000 −0.254686
\(556\) 0 0
\(557\) −40.0000 −1.69485 −0.847427 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) 0 0
\(559\) 30.0000 1.26886
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −9.00000 −0.375980
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 13.0000 0.541197 0.270599 0.962692i \(-0.412778\pi\)
0.270599 + 0.962692i \(0.412778\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 0 0
\(585\) 18.0000 0.744208
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 0 0
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −15.0000 −0.612883 −0.306442 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 72.0000 2.93207
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) 27.0000 1.09410
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) 0 0
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −90.0000 −3.59425
\(628\) 0 0
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −47.0000 −1.87104 −0.935520 0.353273i \(-0.885069\pi\)
−0.935520 + 0.353273i \(0.885069\pi\)
\(632\) 0 0
\(633\) −33.0000 −1.31163
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) 30.0000 1.18125
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) −10.0000 −0.390732
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 31.0000 1.20759 0.603794 0.797140i \(-0.293655\pi\)
0.603794 + 0.797140i \(0.293655\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) −9.00000 −0.349531
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 0 0
\(669\) 15.0000 0.579934
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) 9.00000 0.346410
\(676\) 0 0
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) −13.0000 −0.498894
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −12.0000 −0.457829
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) −30.0000 −1.13961
\(694\) 0 0
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) −78.0000 −2.92523
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 0 0
\(717\) −3.00000 −0.112037
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −19.0000 −0.707597
\(722\) 0 0
\(723\) −78.0000 −2.90085
\(724\) 0 0
\(725\) 9.00000 0.334252
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −10.0000 −0.369863
\(732\) 0 0
\(733\) 47.0000 1.73598 0.867992 0.496578i \(-0.165410\pi\)
0.867992 + 0.496578i \(0.165410\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 54.0000 1.98374
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 19.0000 0.693320 0.346660 0.937991i \(-0.387316\pi\)
0.346660 + 0.937991i \(0.387316\pi\)
\(752\) 0 0
\(753\) 90.0000 3.27978
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) 90.0000 3.26679
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 3.00000 0.108607
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 42.0000 1.51259
\(772\) 0 0
\(773\) −45.0000 −1.61854 −0.809269 0.587439i \(-0.800136\pi\)
−0.809269 + 0.587439i \(0.800136\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 81.0000 2.89470
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 31.0000 1.10503 0.552515 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 0 0
\(799\) −1.00000 −0.0353775
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) −54.0000 −1.90089
\(808\) 0 0
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) 60.0000 2.09913
\(818\) 0 0
\(819\) 18.0000 0.628971
\(820\) 0 0
\(821\) 39.0000 1.36111 0.680555 0.732697i \(-0.261739\pi\)
0.680555 + 0.732697i \(0.261739\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 0 0
\(825\) −15.0000 −0.522233
\(826\) 0 0
\(827\) 50.0000 1.73867 0.869335 0.494223i \(-0.164547\pi\)
0.869335 + 0.494223i \(0.164547\pi\)
\(828\) 0 0
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) 0 0
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 33.0000 1.13658
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) −93.0000 −3.19175
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 36.0000 1.23117
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 1.00000 0.0340010
\(866\) 0 0
\(867\) −48.0000 −1.63017
\(868\) 0 0
\(869\) 65.0000 2.20497
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 0 0
\(873\) −78.0000 −2.63990
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 0 0
\(879\) 15.0000 0.505937
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −45.0000 −1.50756
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −54.0000 −1.80301
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 30.0000 0.998337
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 0 0
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) −10.0000 −0.330229
\(918\) 0 0
\(919\) 31.0000 1.02260 0.511298 0.859404i \(-0.329165\pi\)
0.511298 + 0.859404i \(0.329165\pi\)
\(920\) 0 0
\(921\) −69.0000 −2.27363
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −114.000 −3.74425
\(928\) 0 0
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 54.0000 1.76788
\(934\) 0 0
\(935\) 5.00000 0.163517
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) −21.0000 −0.685309
\(940\) 0 0
\(941\) −32.0000 −1.04317 −0.521585 0.853199i \(-0.674659\pi\)
−0.521585 + 0.853199i \(0.674659\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 9.00000 0.292770
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 78.0000 2.52932
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 0 0
\(957\) −135.000 −4.36393
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 0 0
\(969\) −18.0000 −0.578243
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 9.00000 0.288231
\(976\) 0 0
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) 0 0
\(983\) 43.0000 1.37149 0.685744 0.727843i \(-0.259477\pi\)
0.685744 + 0.727843i \(0.259477\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 84.0000 2.66566
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −11.0000 −0.348373 −0.174187 0.984713i \(-0.555730\pi\)
−0.174187 + 0.984713i \(0.555730\pi\)
\(998\) 0 0
\(999\) −18.0000 −0.569495
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 2240.2.a.bb.1.1 1
4.3 odd 2 2240.2.a.c.1.1 1
8.3 odd 2 140.2.a.b.1.1 1
8.5 even 2 560.2.a.a.1.1 1
24.5 odd 2 5040.2.a.bd.1.1 1
24.11 even 2 1260.2.a.h.1.1 1
40.3 even 4 700.2.e.a.449.2 2
40.13 odd 4 2800.2.g.c.449.1 2
40.19 odd 2 700.2.a.b.1.1 1
40.27 even 4 700.2.e.a.449.1 2
40.29 even 2 2800.2.a.be.1.1 1
40.37 odd 4 2800.2.g.c.449.2 2
56.3 even 6 980.2.i.j.961.1 2
56.11 odd 6 980.2.i.b.961.1 2
56.13 odd 2 3920.2.a.bl.1.1 1
56.19 even 6 980.2.i.j.361.1 2
56.27 even 2 980.2.a.b.1.1 1
56.51 odd 6 980.2.i.b.361.1 2
120.59 even 2 6300.2.a.bf.1.1 1
120.83 odd 4 6300.2.k.p.6049.2 2
120.107 odd 4 6300.2.k.p.6049.1 2
168.83 odd 2 8820.2.a.n.1.1 1
280.27 odd 4 4900.2.e.a.2549.2 2
280.83 odd 4 4900.2.e.a.2549.1 2
280.139 even 2 4900.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.a.b.1.1 1 8.3 odd 2
560.2.a.a.1.1 1 8.5 even 2
700.2.a.b.1.1 1 40.19 odd 2
700.2.e.a.449.1 2 40.27 even 4
700.2.e.a.449.2 2 40.3 even 4
980.2.a.b.1.1 1 56.27 even 2
980.2.i.b.361.1 2 56.51 odd 6
980.2.i.b.961.1 2 56.11 odd 6
980.2.i.j.361.1 2 56.19 even 6
980.2.i.j.961.1 2 56.3 even 6
1260.2.a.h.1.1 1 24.11 even 2
2240.2.a.c.1.1 1 4.3 odd 2
2240.2.a.bb.1.1 1 1.1 even 1 trivial
2800.2.a.be.1.1 1 40.29 even 2
2800.2.g.c.449.1 2 40.13 odd 4
2800.2.g.c.449.2 2 40.37 odd 4
3920.2.a.bl.1.1 1 56.13 odd 2
4900.2.a.u.1.1 1 280.139 even 2
4900.2.e.a.2549.1 2 280.83 odd 4
4900.2.e.a.2549.2 2 280.27 odd 4
5040.2.a.bd.1.1 1 24.5 odd 2
6300.2.a.bf.1.1 1 120.59 even 2
6300.2.k.p.6049.1 2 120.107 odd 4
6300.2.k.p.6049.2 2 120.83 odd 4
8820.2.a.n.1.1 1 168.83 odd 2