Properties

Label 1440.2.b.a.431.2
Level $1440$
Weight $2$
Character 1440.431
Analytic conductor $11.498$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1440.431
Dual form 1440.2.b.a.431.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +4.24264i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +4.24264i q^{7} +1.41421i q^{11} -4.24264i q^{13} +2.82843i q^{17} +4.00000 q^{19} -6.00000 q^{23} +1.00000 q^{25} -6.00000 q^{29} +8.48528i q^{31} -4.24264i q^{35} +4.24264i q^{37} -9.89949i q^{41} -8.00000 q^{43} -11.0000 q^{49} -12.0000 q^{53} -1.41421i q^{55} +1.41421i q^{59} +8.48528i q^{61} +4.24264i q^{65} -8.00000 q^{67} +14.0000 q^{73} -6.00000 q^{77} +8.48528i q^{79} -2.82843i q^{83} -2.82843i q^{85} +7.07107i q^{89} +18.0000 q^{91} -4.00000 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} + 8 q^{19} - 12 q^{23} + 2 q^{25} - 12 q^{29} - 16 q^{43} - 22 q^{49} - 24 q^{53} - 16 q^{67} + 28 q^{73} - 12 q^{77} + 36 q^{91} - 8 q^{95} - 20 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) − 4.24264i − 1.17670i −0.808608 0.588348i \(-0.799778\pi\)
0.808608 0.588348i \(-0.200222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.24264i − 0.717137i
\(36\) 0 0
\(37\) 4.24264i 0.697486i 0.937218 + 0.348743i \(0.113391\pi\)
−0.937218 + 0.348743i \(0.886609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.89949i − 1.54604i −0.634381 0.773021i \(-0.718745\pi\)
0.634381 0.773021i \(-0.281255\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 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
\(55\) − 1.41421i − 0.190693i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421i 0.184115i 0.995754 + 0.0920575i \(0.0293443\pi\)
−0.995754 + 0.0920575i \(0.970656\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.24264i 0.526235i
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 8.48528i 0.954669i 0.878722 + 0.477334i \(0.158397\pi\)
−0.878722 + 0.477334i \(0.841603\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 2.82843i − 0.310460i −0.987878 0.155230i \(-0.950388\pi\)
0.987878 0.155230i \(-0.0496119\pi\)
\(84\) 0 0
\(85\) − 2.82843i − 0.306786i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) − 4.24264i − 0.418040i −0.977911 0.209020i \(-0.932973\pi\)
0.977911 0.209020i \(-0.0670273\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421i 1.36717i 0.729870 + 0.683586i \(0.239581\pi\)
−0.729870 + 0.683586i \(0.760419\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i 0.913708 + 0.406371i \(0.133206\pi\)
−0.913708 + 0.406371i \(0.866794\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.3137i 1.06430i 0.846649 + 0.532152i \(0.178617\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) − 4.24264i − 0.376473i −0.982124 0.188237i \(-0.939723\pi\)
0.982124 0.188237i \(-0.0602772\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.41421i 0.123560i 0.998090 + 0.0617802i \(0.0196778\pi\)
−0.998090 + 0.0617802i \(0.980322\pi\)
\(132\) 0 0
\(133\) 16.9706i 1.47153i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 5.65685i − 0.483298i −0.970364 0.241649i \(-0.922312\pi\)
0.970364 0.241649i \(-0.0776882\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) − 16.9706i − 1.38104i −0.723311 0.690522i \(-0.757381\pi\)
0.723311 0.690522i \(-0.242619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 8.48528i − 0.681554i
\(156\) 0 0
\(157\) − 4.24264i − 0.338600i −0.985565 0.169300i \(-0.945849\pi\)
0.985565 0.169300i \(-0.0541506\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 25.4558i − 2.00620i
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 4.24264i 0.320713i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 7.07107i − 0.528516i −0.964452 0.264258i \(-0.914873\pi\)
0.964452 0.264258i \(-0.0851271\pi\)
\(180\) 0 0
\(181\) − 8.48528i − 0.630706i −0.948974 0.315353i \(-0.897877\pi\)
0.948974 0.315353i \(-0.102123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 4.24264i − 0.311925i
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) − 16.9706i − 1.20301i −0.798869 0.601506i \(-0.794568\pi\)
0.798869 0.601506i \(-0.205432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 25.4558i − 1.78665i
\(204\) 0 0
\(205\) 9.89949i 0.691411i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.65685i 0.391293i
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −36.0000 −2.44384
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 21.2132i 1.42054i 0.703929 + 0.710271i \(0.251427\pi\)
−0.703929 + 0.710271i \(0.748573\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.6274i 1.50183i 0.660396 + 0.750917i \(0.270388\pi\)
−0.660396 + 0.750917i \(0.729612\pi\)
\(228\) 0 0
\(229\) − 16.9706i − 1.12145i −0.828003 0.560723i \(-0.810523\pi\)
0.828003 0.560723i \(-0.189477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.7990i 1.29707i 0.761183 + 0.648537i \(0.224619\pi\)
−0.761183 + 0.648537i \(0.775381\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.0000 0.702764
\(246\) 0 0
\(247\) − 16.9706i − 1.07981i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.41421i 0.0892644i 0.999003 + 0.0446322i \(0.0142116\pi\)
−0.999003 + 0.0446322i \(0.985788\pi\)
\(252\) 0 0
\(253\) − 8.48528i − 0.533465i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.3137i 0.705730i 0.935674 + 0.352865i \(0.114792\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) −18.0000 −1.11847
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.41421i 0.0852803i
\(276\) 0 0
\(277\) − 4.24264i − 0.254916i −0.991844 0.127458i \(-0.959318\pi\)
0.991844 0.127458i \(-0.0406817\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 9.89949i − 0.590554i −0.955412 0.295277i \(-0.904588\pi\)
0.955412 0.295277i \(-0.0954120\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 42.0000 2.47918
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) − 1.41421i − 0.0823387i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.4558i 1.47215i
\(300\) 0 0
\(301\) − 33.9411i − 1.95633i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 8.48528i − 0.485866i
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) − 8.48528i − 0.475085i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.3137i 0.629512i
\(324\) 0 0
\(325\) − 4.24264i − 0.235339i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.65685i 0.303676i 0.988405 + 0.151838i \(0.0485192\pi\)
−0.988405 + 0.151838i \(0.951481\pi\)
\(348\) 0 0
\(349\) 8.48528i 0.454207i 0.973871 + 0.227103i \(0.0729255\pi\)
−0.973871 + 0.227103i \(0.927074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.82843i 0.150542i 0.997163 + 0.0752710i \(0.0239822\pi\)
−0.997163 + 0.0752710i \(0.976018\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 29.6985i 1.55025i 0.631809 + 0.775124i \(0.282313\pi\)
−0.631809 + 0.775124i \(0.717687\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 50.9117i − 2.64320i
\(372\) 0 0
\(373\) − 12.7279i − 0.659027i −0.944151 0.329513i \(-0.893115\pi\)
0.944151 0.329513i \(-0.106885\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.4558i 1.31104i
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) − 16.9706i − 0.858238i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 8.48528i − 0.426941i
\(396\) 0 0
\(397\) − 12.7279i − 0.638796i −0.947621 0.319398i \(-0.896519\pi\)
0.947621 0.319398i \(-0.103481\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 18.3848i − 0.918092i −0.888413 0.459046i \(-0.848191\pi\)
0.888413 0.459046i \(-0.151809\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 2.82843i 0.138842i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.8701i 1.31269i 0.754462 + 0.656344i \(0.227898\pi\)
−0.754462 + 0.656344i \(0.772102\pi\)
\(420\) 0 0
\(421\) 33.9411i 1.65419i 0.562063 + 0.827095i \(0.310008\pi\)
−0.562063 + 0.827095i \(0.689992\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.82843i 0.137199i
\(426\) 0 0
\(427\) −36.0000 −1.74216
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 16.9706i 0.809961i 0.914325 + 0.404980i \(0.132722\pi\)
−0.914325 + 0.404980i \(0.867278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 11.3137i − 0.537531i −0.963206 0.268765i \(-0.913384\pi\)
0.963206 0.268765i \(-0.0866156\pi\)
\(444\) 0 0
\(445\) − 7.07107i − 0.335201i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 26.8701i − 1.26808i −0.773302 0.634038i \(-0.781396\pi\)
0.773302 0.634038i \(-0.218604\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −18.0000 −0.843853
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) − 12.7279i − 0.591517i −0.955263 0.295758i \(-0.904428\pi\)
0.955263 0.295758i \(-0.0955723\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 19.7990i − 0.916188i −0.888904 0.458094i \(-0.848532\pi\)
0.888904 0.458094i \(-0.151468\pi\)
\(468\) 0 0
\(469\) − 33.9411i − 1.56726i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 11.3137i − 0.520205i
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) − 21.2132i − 0.961262i −0.876923 0.480631i \(-0.840408\pi\)
0.876923 0.480631i \(-0.159592\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.41421i 0.0638226i 0.999491 + 0.0319113i \(0.0101594\pi\)
−0.999491 + 0.0319113i \(0.989841\pi\)
\(492\) 0 0
\(493\) − 16.9706i − 0.764316i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 59.3970i 2.62757i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.24264i 0.186953i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.5563i 0.681536i 0.940147 + 0.340768i \(0.110687\pi\)
−0.940147 + 0.340768i \(0.889313\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −42.0000 −1.81922
\(534\) 0 0
\(535\) − 14.1421i − 0.611418i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 15.5563i − 0.670059i
\(540\) 0 0
\(541\) 8.48528i 0.364811i 0.983223 + 0.182405i \(0.0583883\pi\)
−0.983223 + 0.182405i \(0.941612\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 8.48528i − 0.363470i
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −36.0000 −1.53088
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 33.9411i 1.43556i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.5980i 1.66886i 0.551117 + 0.834428i \(0.314202\pi\)
−0.551117 + 0.834428i \(0.685798\pi\)
\(564\) 0 0
\(565\) − 11.3137i − 0.475971i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1.41421i − 0.0592869i −0.999561 0.0296435i \(-0.990563\pi\)
0.999561 0.0296435i \(-0.00943719\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) − 16.9706i − 0.702849i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.1421i 0.583708i 0.956463 + 0.291854i \(0.0942722\pi\)
−0.956463 + 0.291854i \(0.905728\pi\)
\(588\) 0 0
\(589\) 33.9411i 1.39852i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 5.65685i − 0.232299i −0.993232 0.116150i \(-0.962945\pi\)
0.993232 0.116150i \(-0.0370552\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.00000 −0.365902
\(606\) 0 0
\(607\) 4.24264i 0.172203i 0.996286 + 0.0861017i \(0.0274410\pi\)
−0.996286 + 0.0861017i \(0.972559\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12.7279i 0.514076i 0.966401 + 0.257038i \(0.0827465\pi\)
−0.966401 + 0.257038i \(0.917253\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.7696i 1.48029i 0.672449 + 0.740143i \(0.265242\pi\)
−0.672449 + 0.740143i \(0.734758\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) − 8.48528i − 0.337794i −0.985634 0.168897i \(-0.945980\pi\)
0.985634 0.168897i \(-0.0540205\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.24264i 0.168364i
\(636\) 0 0
\(637\) 46.6690i 1.84909i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.0122i 1.61988i 0.586510 + 0.809942i \(0.300502\pi\)
−0.586510 + 0.809942i \(0.699498\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) − 1.41421i − 0.0552579i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.89949i 0.385630i 0.981235 + 0.192815i \(0.0617617\pi\)
−0.981235 + 0.192815i \(0.938238\pi\)
\(660\) 0 0
\(661\) 25.4558i 0.990118i 0.868859 + 0.495059i \(0.164853\pi\)
−0.868859 + 0.495059i \(0.835147\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 16.9706i − 0.658090i
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) − 42.4264i − 1.62818i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14.1421i 0.541134i 0.962701 + 0.270567i \(0.0872111\pi\)
−0.962701 + 0.270567i \(0.912789\pi\)
\(684\) 0 0
\(685\) 5.65685i 0.216137i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 50.9117i 1.93958i
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 16.9706i 0.640057i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 25.4558i − 0.957366i
\(708\) 0 0
\(709\) 50.9117i 1.91203i 0.293320 + 0.956014i \(0.405240\pi\)
−0.293320 + 0.956014i \(0.594760\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 50.9117i − 1.90666i
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 4.24264i 0.157351i 0.996900 + 0.0786754i \(0.0250691\pi\)
−0.996900 + 0.0786754i \(0.974931\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 22.6274i − 0.836905i
\(732\) 0 0
\(733\) 12.7279i 0.470117i 0.971981 + 0.235058i \(0.0755281\pi\)
−0.971981 + 0.235058i \(0.924472\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 11.3137i − 0.416746i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −60.0000 −2.19235
\(750\) 0 0
\(751\) − 42.4264i − 1.54816i −0.633087 0.774081i \(-0.718212\pi\)
0.633087 0.774081i \(-0.281788\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.9706i 0.617622i
\(756\) 0 0
\(757\) − 4.24264i − 0.154201i −0.997023 0.0771007i \(-0.975434\pi\)
0.997023 0.0771007i \(-0.0245663\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 26.8701i − 0.974039i −0.873391 0.487019i \(-0.838084\pi\)
0.873391 0.487019i \(-0.161916\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 8.48528i 0.304800i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 39.5980i − 1.41874i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.24264i 0.151426i
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.7990i 0.698691i
\(804\) 0 0
\(805\) 25.4558i 0.897201i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 18.3848i − 0.646374i −0.946335 0.323187i \(-0.895246\pi\)
0.946335 0.323187i \(-0.104754\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 4.24264i 0.147889i 0.997262 + 0.0739446i \(0.0235588\pi\)
−0.997262 + 0.0739446i \(0.976441\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.65685i 0.196708i 0.995151 + 0.0983540i \(0.0313578\pi\)
−0.995151 + 0.0983540i \(0.968642\pi\)
\(828\) 0 0
\(829\) − 8.48528i − 0.294706i −0.989084 0.147353i \(-0.952925\pi\)
0.989084 0.147353i \(-0.0470753\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 31.1127i − 1.07799i
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.00000 0.172005
\(846\) 0 0
\(847\) 38.1838i 1.31201i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 25.4558i − 0.872615i
\(852\) 0 0
\(853\) 29.6985i 1.01686i 0.861104 + 0.508428i \(0.169773\pi\)
−0.861104 + 0.508428i \(0.830227\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 48.0833i − 1.64249i −0.570574 0.821246i \(-0.693279\pi\)
0.570574 0.821246i \(-0.306721\pi\)
\(858\) 0 0
\(859\) 46.0000 1.56950 0.784750 0.619813i \(-0.212791\pi\)
0.784750 + 0.619813i \(0.212791\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 33.9411i 1.15005i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 4.24264i − 0.143427i
\(876\) 0 0
\(877\) − 12.7279i − 0.429791i −0.976637 0.214896i \(-0.931059\pi\)
0.976637 0.214896i \(-0.0689412\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 32.5269i 1.09586i 0.836524 + 0.547930i \(0.184584\pi\)
−0.836524 + 0.547930i \(0.815416\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 7.07107i 0.236360i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 50.9117i − 1.69800i
\(900\) 0 0
\(901\) − 33.9411i − 1.13074i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.48528i 0.282060i
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) − 25.4558i − 0.839711i −0.907591 0.419855i \(-0.862081\pi\)
0.907591 0.419855i \(-0.137919\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.24264i 0.139497i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 41.0122i 1.34557i 0.739840 + 0.672783i \(0.234901\pi\)
−0.739840 + 0.672783i \(0.765099\pi\)
\(930\) 0 0
\(931\) −44.0000 −1.44204
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 59.3970i 1.93423i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.6274i 0.735292i 0.929966 + 0.367646i \(0.119836\pi\)
−0.929966 + 0.367646i \(0.880164\pi\)
\(948\) 0 0
\(949\) − 59.3970i − 1.92811i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.2843i 0.916217i 0.888896 + 0.458109i \(0.151473\pi\)
−0.888896 + 0.458109i \(0.848527\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −41.0000 −1.32258
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) − 21.2132i − 0.682171i −0.940032 0.341085i \(-0.889205\pi\)
0.940032 0.341085i \(-0.110795\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 43.8406i 1.40691i 0.710739 + 0.703456i \(0.248361\pi\)
−0.710739 + 0.703456i \(0.751639\pi\)
\(972\) 0 0
\(973\) − 8.48528i − 0.272026i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.82843i 0.0904894i 0.998976 + 0.0452447i \(0.0144068\pi\)
−0.998976 + 0.0452447i \(0.985593\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) − 16.9706i − 0.539088i −0.962988 0.269544i \(-0.913127\pi\)
0.962988 0.269544i \(-0.0868729\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.9706i 0.538003i
\(996\) 0 0
\(997\) 21.2132i 0.671829i 0.941893 + 0.335914i \(0.109045\pi\)
−0.941893 + 0.335914i \(0.890955\pi\)
\(998\) 0 0
\(999\) 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 1440.2.b.a.431.2 2
3.2 odd 2 1440.2.b.b.431.2 2
4.3 odd 2 360.2.b.a.251.2 yes 2
5.2 odd 4 7200.2.m.b.3599.2 4
5.3 odd 4 7200.2.m.b.3599.4 4
5.4 even 2 7200.2.b.b.4751.1 2
8.3 odd 2 1440.2.b.b.431.1 2
8.5 even 2 360.2.b.b.251.2 yes 2
12.11 even 2 360.2.b.b.251.1 yes 2
15.2 even 4 7200.2.m.a.3599.1 4
15.8 even 4 7200.2.m.a.3599.3 4
15.14 odd 2 7200.2.b.a.4751.1 2
20.3 even 4 1800.2.m.a.899.3 4
20.7 even 4 1800.2.m.a.899.1 4
20.19 odd 2 1800.2.b.b.251.1 2
24.5 odd 2 360.2.b.a.251.1 2
24.11 even 2 inner 1440.2.b.a.431.1 2
40.3 even 4 7200.2.m.a.3599.2 4
40.13 odd 4 1800.2.m.b.899.3 4
40.19 odd 2 7200.2.b.a.4751.2 2
40.27 even 4 7200.2.m.a.3599.4 4
40.29 even 2 1800.2.b.a.251.1 2
40.37 odd 4 1800.2.m.b.899.1 4
60.23 odd 4 1800.2.m.b.899.2 4
60.47 odd 4 1800.2.m.b.899.4 4
60.59 even 2 1800.2.b.a.251.2 2
120.29 odd 2 1800.2.b.b.251.2 2
120.53 even 4 1800.2.m.a.899.2 4
120.59 even 2 7200.2.b.b.4751.2 2
120.77 even 4 1800.2.m.a.899.4 4
120.83 odd 4 7200.2.m.b.3599.1 4
120.107 odd 4 7200.2.m.b.3599.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.b.a.251.1 2 24.5 odd 2
360.2.b.a.251.2 yes 2 4.3 odd 2
360.2.b.b.251.1 yes 2 12.11 even 2
360.2.b.b.251.2 yes 2 8.5 even 2
1440.2.b.a.431.1 2 24.11 even 2 inner
1440.2.b.a.431.2 2 1.1 even 1 trivial
1440.2.b.b.431.1 2 8.3 odd 2
1440.2.b.b.431.2 2 3.2 odd 2
1800.2.b.a.251.1 2 40.29 even 2
1800.2.b.a.251.2 2 60.59 even 2
1800.2.b.b.251.1 2 20.19 odd 2
1800.2.b.b.251.2 2 120.29 odd 2
1800.2.m.a.899.1 4 20.7 even 4
1800.2.m.a.899.2 4 120.53 even 4
1800.2.m.a.899.3 4 20.3 even 4
1800.2.m.a.899.4 4 120.77 even 4
1800.2.m.b.899.1 4 40.37 odd 4
1800.2.m.b.899.2 4 60.23 odd 4
1800.2.m.b.899.3 4 40.13 odd 4
1800.2.m.b.899.4 4 60.47 odd 4
7200.2.b.a.4751.1 2 15.14 odd 2
7200.2.b.a.4751.2 2 40.19 odd 2
7200.2.b.b.4751.1 2 5.4 even 2
7200.2.b.b.4751.2 2 120.59 even 2
7200.2.m.a.3599.1 4 15.2 even 4
7200.2.m.a.3599.2 4 40.3 even 4
7200.2.m.a.3599.3 4 15.8 even 4
7200.2.m.a.3599.4 4 40.27 even 4
7200.2.m.b.3599.1 4 120.83 odd 4
7200.2.m.b.3599.2 4 5.2 odd 4
7200.2.m.b.3599.3 4 120.107 odd 4
7200.2.m.b.3599.4 4 5.3 odd 4