Properties

Label 280.2.a.b.1.1
Level $280$
Weight $2$
Character 280.1
Self dual yes
Analytic conductor $2.236$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.23581125660\)
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 \(\chi\) \(=\) 280.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} -6.00000 q^{19} +1.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -9.00000 q^{29} +5.00000 q^{33} +1.00000 q^{35} +6.00000 q^{37} -1.00000 q^{39} +8.00000 q^{41} +6.00000 q^{43} +2.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} -12.0000 q^{53} +5.00000 q^{55} +6.00000 q^{57} +8.00000 q^{59} -4.00000 q^{61} +2.00000 q^{63} -1.00000 q^{65} -4.00000 q^{67} +6.00000 q^{69} +8.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} +5.00000 q^{77} -3.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -3.00000 q^{85} +9.00000 q^{87} -16.0000 q^{89} -1.00000 q^{91} +6.00000 q^{95} +7.00000 q^{97} +10.0000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 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
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 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
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 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
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) 0 0
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 0 0
\(173\) 19.0000 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 11.0000 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) −25.0000 −1.67412 −0.837062 0.547108i \(-0.815729\pi\)
−0.837062 + 0.547108i \(0.815729\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 3.00000 0.194871
\(238\) 0 0
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) 30.0000 1.88608
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 16.0000 0.979184
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) 29.0000 1.72387 0.861936 0.507018i \(-0.169252\pi\)
0.861936 + 0.507018i \(0.169252\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −7.00000 −0.410347
\(292\) 0 0
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −25.0000 −1.45065
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −27.0000 −1.54097 −0.770486 0.637457i \(-0.779986\pi\)
−0.770486 + 0.637457i \(0.779986\pi\)
\(308\) 0 0
\(309\) 9.00000 0.511992
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 45.0000 2.51952
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) −18.0000 −1.00155
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 11.0000 0.608301
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 0 0
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −33.0000 −1.75641 −0.878206 0.478282i \(-0.841260\pi\)
−0.878206 + 0.478282i \(0.841260\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −29.0000 −1.51379 −0.756894 0.653538i \(-0.773284\pi\)
−0.756894 + 0.653538i \(0.773284\pi\)
\(368\) 0 0
\(369\) −16.0000 −0.832927
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) 25.0000 1.26755 0.633775 0.773517i \(-0.281504\pi\)
0.633775 + 0.773517i \(0.281504\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 9.00000 0.449439 0.224719 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −18.0000 −0.881464
\(418\) 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 \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 0 0
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) 23.0000 1.10787 0.553936 0.832560i \(-0.313125\pi\)
0.553936 + 0.832560i \(0.313125\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 36.0000 1.72211
\(438\) 0 0
\(439\) −34.0000 −1.62273 −0.811366 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 0 0
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 0 0
\(453\) 19.0000 0.892698
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 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
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) −27.0000 −1.21602
\(494\) 0 0
\(495\) −10.0000 −0.449467
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) −31.0000 −1.38222 −0.691111 0.722749i \(-0.742878\pi\)
−0.691111 + 0.722749i \(0.742878\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) −30.0000 −1.32453
\(514\) 0 0
\(515\) 9.00000 0.396587
\(516\) 0 0
\(517\) −15.0000 −0.659699
\(518\) 0 0
\(519\) −19.0000 −0.834007
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −16.0000 −0.694341
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −9.00000 −0.386940 −0.193470 0.981106i \(-0.561974\pi\)
−0.193470 + 0.981106i \(0.561974\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) 3.00000 0.127573
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 0 0
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 60.0000 2.48495
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) −7.00000 −0.287456 −0.143728 0.989617i \(-0.545909\pi\)
−0.143728 + 0.989617i \(0.545909\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −33.0000 −1.34834 −0.674172 0.738575i \(-0.735499\pi\)
−0.674172 + 0.738575i \(0.735499\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 17.0000 0.690009 0.345004 0.938601i \(-0.387877\pi\)
0.345004 + 0.938601i \(0.387877\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) 0 0
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −30.0000 −1.19808
\(628\) 0 0
\(629\) 18.0000 0.717707
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) 47.0000 1.85350 0.926750 0.375680i \(-0.122591\pi\)
0.926750 + 0.375680i \(0.122591\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) −20.0000 −0.780274
\(658\) 0 0
\(659\) −25.0000 −0.973862 −0.486931 0.873441i \(-0.661884\pi\)
−0.486931 + 0.873441i \(0.661884\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 54.0000 2.09089
\(668\) 0 0
\(669\) 25.0000 0.966556
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 33.0000 1.26829 0.634147 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 13.0000 0.498161
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) −10.0000 −0.379869
\(694\) 0 0
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) −36.0000 −1.35777
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −41.0000 −1.53979 −0.769894 0.638172i \(-0.779691\pi\)
−0.769894 + 0.638172i \(0.779691\pi\)
\(710\) 0 0
\(711\) 6.00000 0.225018
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) 0 0
\(717\) 7.00000 0.261420
\(718\) 0 0
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) 9.00000 0.335178
\(722\) 0 0
\(723\) −18.0000 −0.669427
\(724\) 0 0
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 18.0000 0.665754
\(732\) 0 0
\(733\) 5.00000 0.184679 0.0923396 0.995728i \(-0.470565\pi\)
0.0923396 + 0.995728i \(0.470565\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) 24.0000 0.878114
\(748\) 0 0
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) −35.0000 −1.27717 −0.638584 0.769552i \(-0.720480\pi\)
−0.638584 + 0.769552i \(0.720480\pi\)
\(752\) 0 0
\(753\) −14.0000 −0.510188
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 0 0
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) −46.0000 −1.66750 −0.833749 0.552143i \(-0.813810\pi\)
−0.833749 + 0.552143i \(0.813810\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 1.00000 0.0359675 0.0179838 0.999838i \(-0.494275\pi\)
0.0179838 + 0.999838i \(0.494275\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) 0 0
\(783\) −45.0000 −1.60817
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 11.0000 0.392108 0.196054 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) 0 0
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) 35.0000 1.23976 0.619882 0.784695i \(-0.287181\pi\)
0.619882 + 0.784695i \(0.287181\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 32.0000 1.13066
\(802\) 0 0
\(803\) −50.0000 −1.76446
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −23.0000 −0.808637 −0.404318 0.914618i \(-0.632491\pi\)
−0.404318 + 0.914618i \(0.632491\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −7.00000 −0.244302 −0.122151 0.992512i \(-0.538979\pi\)
−0.122151 + 0.992512i \(0.538979\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 2.00000 0.0695468 0.0347734 0.999395i \(-0.488929\pi\)
0.0347734 + 0.999395i \(0.488929\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 9.00000 0.311458
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 13.0000 0.447744
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) −29.0000 −0.995277
\(850\) 0 0
\(851\) −36.0000 −1.23406
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) −19.0000 −0.646019
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 0 0
\(879\) 1.00000 0.0337292
\(880\) 0 0
\(881\) −16.0000 −0.539054 −0.269527 0.962993i \(-0.586867\pi\)
−0.269527 + 0.962993i \(0.586867\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 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
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) 0 0
\(909\) 28.0000 0.928701
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 60.0000 1.98571
\(914\) 0 0
\(915\) −4.00000 −0.132236
\(916\) 0 0
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 0 0
\(921\) 27.0000 0.889680
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) 18.0000 0.591198
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 14.0000 0.458339
\(934\) 0 0
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) −11.0000 −0.355952
\(956\) 0 0
\(957\) −45.0000 −1.45464
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 18.0000 0.578243
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) −18.0000 −0.577054
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 80.0000 2.55681
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.00000 −0.0316703 −0.0158352 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) 0 0
\(999\) 30.0000 0.949158
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 280.2.a.b.1.1 1
3.2 odd 2 2520.2.a.p.1.1 1
4.3 odd 2 560.2.a.e.1.1 1
5.2 odd 4 1400.2.g.e.449.2 2
5.3 odd 4 1400.2.g.e.449.1 2
5.4 even 2 1400.2.a.k.1.1 1
7.2 even 3 1960.2.q.m.361.1 2
7.3 odd 6 1960.2.q.e.961.1 2
7.4 even 3 1960.2.q.m.961.1 2
7.5 odd 6 1960.2.q.e.361.1 2
7.6 odd 2 1960.2.a.k.1.1 1
8.3 odd 2 2240.2.a.j.1.1 1
8.5 even 2 2240.2.a.v.1.1 1
12.11 even 2 5040.2.a.be.1.1 1
20.3 even 4 2800.2.g.m.449.2 2
20.7 even 4 2800.2.g.m.449.1 2
20.19 odd 2 2800.2.a.i.1.1 1
28.27 even 2 3920.2.a.r.1.1 1
35.34 odd 2 9800.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.b.1.1 1 1.1 even 1 trivial
560.2.a.e.1.1 1 4.3 odd 2
1400.2.a.k.1.1 1 5.4 even 2
1400.2.g.e.449.1 2 5.3 odd 4
1400.2.g.e.449.2 2 5.2 odd 4
1960.2.a.k.1.1 1 7.6 odd 2
1960.2.q.e.361.1 2 7.5 odd 6
1960.2.q.e.961.1 2 7.3 odd 6
1960.2.q.m.361.1 2 7.2 even 3
1960.2.q.m.961.1 2 7.4 even 3
2240.2.a.j.1.1 1 8.3 odd 2
2240.2.a.v.1.1 1 8.5 even 2
2520.2.a.p.1.1 1 3.2 odd 2
2800.2.a.i.1.1 1 20.19 odd 2
2800.2.g.m.449.1 2 20.7 even 4
2800.2.g.m.449.2 2 20.3 even 4
3920.2.a.r.1.1 1 28.27 even 2
5040.2.a.be.1.1 1 12.11 even 2
9800.2.a.n.1.1 1 35.34 odd 2