Properties

Label 2450.2.c.c.99.1
Level 2450
Weight 2
Character 2450.99
Analytic conductor 19.563
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\)
Character \(\chi\) = 2450.99
Dual form 2450.2.c.c.99.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +2.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +2.00000 q^{19} +2.00000 q^{24} -4.00000 q^{26} -4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -2.00000i q^{38} -8.00000 q^{39} -6.00000 q^{41} -8.00000i q^{43} +12.0000i q^{47} -2.00000i q^{48} -12.0000 q^{51} +4.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} -4.00000i q^{57} -6.00000i q^{58} -6.00000 q^{59} -8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} -2.00000 q^{76} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{82} -6.00000i q^{83} -8.00000 q^{86} -12.0000i q^{87} -6.00000 q^{89} -8.00000i q^{93} +12.0000 q^{94} -2.00000 q^{96} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + 2q^{16} + 4q^{19} + 4q^{24} - 8q^{26} + 12q^{29} + 8q^{31} - 12q^{34} + 2q^{36} - 16q^{39} - 12q^{41} - 24q^{51} - 8q^{54} - 12q^{59} - 16q^{61} - 2q^{64} + 4q^{74} - 4q^{76} - 16q^{79} - 22q^{81} - 16q^{86} - 12q^{89} + 24q^{94} - 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000i 0.577350i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 0 0
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 4.00000i 0.554700i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) − 6.00000i − 0.787839i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 8.00000i 0.905822i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000i 0.662589i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) − 12.0000i − 1.28654i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 12.0000i 1.18818i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 4.00000i 0.369800i
\(118\) 6.00000i 0.552345i
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 8.00000i 0.724286i
\(123\) 12.0000i 1.08200i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000i 0.162221i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000i 0.864242i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 8.00000i 0.609994i
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 6.00000i 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 16.0000i 1.18275i
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) − 12.0000i − 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) − 4.00000i − 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) − 4.00000i − 0.268462i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 10.0000i 0.641500i
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) − 8.00000i − 0.509028i
\(248\) 4.00000i 0.254000i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 16.0000i 0.996116i
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 18.0000i 1.11204i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 4.00000i 0.244339i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) − 14.0000i − 0.839664i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) − 24.0000i − 1.42918i
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) − 2.00000i − 0.117041i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) − 18.0000i − 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 8.00000i − 0.452911i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 12.0000i 0.672927i
\(319\) 0 0
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 4.00000i 0.221201i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 2.00000i − 0.109599i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000i 0.108148i
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 12.0000i 0.643268i
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000i 1.05118i
\(363\) 22.0000i 1.15470i
\(364\) 0 0
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000i 0.414781i
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) − 24.0000i − 1.22795i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 8.00000i 0.406663i
\(388\) − 10.0000i − 0.507673i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000i 1.81596i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.0000i − 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) − 20.0000i − 1.00251i
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 16.0000i − 0.797017i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) − 12.0000i − 0.594089i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 4.00000i 0.197066i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 28.0000i − 1.37117i
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) − 4.00000i − 0.191127i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.0000i 1.14156i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) − 36.0000i − 1.70274i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000i 0.282216i
\(453\) − 16.0000i − 0.751746i
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 4.00000i 0.186908i
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) − 6.00000i − 0.276172i
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 24.0000i 1.09773i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) − 10.0000i − 0.455488i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) − 12.0000i − 0.541002i
\(493\) − 36.0000i − 1.62136i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) − 18.0000i − 0.803379i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.00000i 0.266469i
\(508\) 16.0000i 0.709885i
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 16.0000 0.704361
\(517\) 0 0
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000i 0.262613i
\(523\) 2.00000i 0.0874539i 0.999044 + 0.0437269i \(0.0139232\pi\)
−0.999044 + 0.0437269i \(0.986077\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 0 0
\(527\) − 24.0000i − 1.04546i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) − 24.0000i − 1.03568i
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 40.0000i 1.71656i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 4.00000i 0.169334i
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) − 20.0000i − 0.829027i
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 42.0000i 1.73353i 0.498721 + 0.866763i \(0.333803\pi\)
−0.498721 + 0.866763i \(0.666197\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 2.00000i 0.0821995i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) − 40.0000i − 1.63709i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) − 2.00000i − 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) − 6.00000i − 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 4.00000i − 0.159617i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 8.00000i 0.317971i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) − 24.0000i − 0.947204i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 16.0000i − 0.626608i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) − 8.00000i − 0.310929i
\(663\) 48.0000i 1.86417i
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 12.0000i 0.460857i
\(679\) 0 0
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) − 8.00000i − 0.304997i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 36.0000i 1.36360i
\(698\) 28.0000i 1.05982i
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 16.0000i 0.603881i
\(703\) 4.00000i 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) − 12.0000i − 0.450988i
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) − 6.00000i − 0.224860i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 48.0000i 1.79259i
\(718\) − 24.0000i − 0.895672i
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000i 0.558242i
\(723\) − 20.0000i − 0.743808i
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 22.0000 0.816497
\(727\) − 44.0000i − 1.63187i −0.578144 0.815935i \(-0.696223\pi\)
0.578144 0.815935i \(-0.303777\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) − 16.0000i − 0.591377i
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) − 6.00000i − 0.220863i
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 36.0000i − 1.31191i
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 32.0000i 1.15924i
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 24.0000i 0.866590i
\(768\) − 2.00000i − 0.0721688i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) 14.0000i 0.503871i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 18.0000i 0.645331i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 24.0000i − 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 32.0000i 1.13635i
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 24.0000i 0.844840i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) − 16.0000i − 0.559769i
\(818\) − 14.0000i − 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) − 36.0000i − 1.25564i
\(823\) 40.0000i 1.39431i 0.716919 + 0.697156i \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) 56.0000 1.94496 0.972480 0.232986i \(-0.0748495\pi\)
0.972480 + 0.232986i \(0.0748495\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) 4.00000i 0.138675i
\(833\) 0 0
\(834\) −28.0000 −0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) − 6.00000i − 0.207267i
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000i 0.344623i
\(843\) 12.0000i 0.413302i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) −44.0000 −1.51008
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 44.0000i 1.50653i 0.657716 + 0.753266i \(0.271523\pi\)
−0.657716 + 0.753266i \(0.728477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 38.0000i 1.29055i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) − 2.00000i − 0.0677285i
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) 48.0000 1.61900
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) − 8.00000i − 0.267860i
\(893\) 24.0000i 0.803129i
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 18.0000i 0.600668i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 18.0000i 0.597351i
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 24.0000i 0.792118i
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 12.0000i 0.395199i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 4.00000i 0.131377i
\(928\) − 6.00000i − 0.196960i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 6.00000i − 0.196537i
\(933\) − 48.0000i − 1.57145i
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) − 8.00000i − 0.260654i
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000i 0.779895i 0.920837 + 0.389948i \(0.127507\pi\)
−0.920837 + 0.389948i \(0.872493\pi\)
\(948\) − 16.0000i − 0.519656i
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 54.0000i 1.74923i 0.484817 + 0.874616i \(0.338886\pi\)
−0.484817 + 0.874616i \(0.661114\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 36.0000i 1.16311i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 8.00000i − 0.257930i
\(963\) − 12.0000i − 0.386695i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) − 32.0000i − 1.02325i
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 12.0000i 0.382935i
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 16.0000i − 0.507745i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 8.00000 0.253109
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 2450.2.c.c.99.1 2
5.2 odd 4 2450.2.a.t.1.1 1
5.3 odd 4 98.2.a.a.1.1 1
5.4 even 2 inner 2450.2.c.c.99.2 2
7.6 odd 2 350.2.c.d.99.1 2
15.8 even 4 882.2.a.i.1.1 1
20.3 even 4 784.2.a.b.1.1 1
21.20 even 2 3150.2.g.j.2899.2 2
28.27 even 2 2800.2.g.h.449.1 2
35.3 even 12 98.2.c.b.79.1 2
35.13 even 4 14.2.a.a.1.1 1
35.18 odd 12 98.2.c.a.79.1 2
35.23 odd 12 98.2.c.a.67.1 2
35.27 even 4 350.2.a.f.1.1 1
35.33 even 12 98.2.c.b.67.1 2
35.34 odd 2 350.2.c.d.99.2 2
40.3 even 4 3136.2.a.z.1.1 1
40.13 odd 4 3136.2.a.e.1.1 1
60.23 odd 4 7056.2.a.bd.1.1 1
105.23 even 12 882.2.g.d.361.1 2
105.38 odd 12 882.2.g.c.667.1 2
105.53 even 12 882.2.g.d.667.1 2
105.62 odd 4 3150.2.a.i.1.1 1
105.68 odd 12 882.2.g.c.361.1 2
105.83 odd 4 126.2.a.b.1.1 1
105.104 even 2 3150.2.g.j.2899.1 2
140.3 odd 12 784.2.i.c.177.1 2
140.23 even 12 784.2.i.i.753.1 2
140.27 odd 4 2800.2.a.g.1.1 1
140.83 odd 4 112.2.a.c.1.1 1
140.103 odd 12 784.2.i.c.753.1 2
140.123 even 12 784.2.i.i.177.1 2
140.139 even 2 2800.2.g.h.449.2 2
280.13 even 4 448.2.a.g.1.1 1
280.83 odd 4 448.2.a.a.1.1 1
315.13 even 12 1134.2.f.l.379.1 2
315.83 odd 12 1134.2.f.f.757.1 2
315.223 even 12 1134.2.f.l.757.1 2
315.293 odd 12 1134.2.f.f.379.1 2
385.153 odd 4 1694.2.a.e.1.1 1
420.83 even 4 1008.2.a.h.1.1 1
455.83 odd 4 2366.2.d.b.337.2 2
455.363 even 4 2366.2.a.j.1.1 1
455.398 odd 4 2366.2.d.b.337.1 2
560.13 even 4 1792.2.b.c.897.1 2
560.83 odd 4 1792.2.b.g.897.2 2
560.293 even 4 1792.2.b.c.897.2 2
560.363 odd 4 1792.2.b.g.897.1 2
595.118 even 4 4046.2.a.f.1.1 1
665.398 odd 4 5054.2.a.c.1.1 1
805.643 odd 4 7406.2.a.a.1.1 1
840.83 even 4 4032.2.a.r.1.1 1
840.293 odd 4 4032.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 35.13 even 4
98.2.a.a.1.1 1 5.3 odd 4
98.2.c.a.67.1 2 35.23 odd 12
98.2.c.a.79.1 2 35.18 odd 12
98.2.c.b.67.1 2 35.33 even 12
98.2.c.b.79.1 2 35.3 even 12
112.2.a.c.1.1 1 140.83 odd 4
126.2.a.b.1.1 1 105.83 odd 4
350.2.a.f.1.1 1 35.27 even 4
350.2.c.d.99.1 2 7.6 odd 2
350.2.c.d.99.2 2 35.34 odd 2
448.2.a.a.1.1 1 280.83 odd 4
448.2.a.g.1.1 1 280.13 even 4
784.2.a.b.1.1 1 20.3 even 4
784.2.i.c.177.1 2 140.3 odd 12
784.2.i.c.753.1 2 140.103 odd 12
784.2.i.i.177.1 2 140.123 even 12
784.2.i.i.753.1 2 140.23 even 12
882.2.a.i.1.1 1 15.8 even 4
882.2.g.c.361.1 2 105.68 odd 12
882.2.g.c.667.1 2 105.38 odd 12
882.2.g.d.361.1 2 105.23 even 12
882.2.g.d.667.1 2 105.53 even 12
1008.2.a.h.1.1 1 420.83 even 4
1134.2.f.f.379.1 2 315.293 odd 12
1134.2.f.f.757.1 2 315.83 odd 12
1134.2.f.l.379.1 2 315.13 even 12
1134.2.f.l.757.1 2 315.223 even 12
1694.2.a.e.1.1 1 385.153 odd 4
1792.2.b.c.897.1 2 560.13 even 4
1792.2.b.c.897.2 2 560.293 even 4
1792.2.b.g.897.1 2 560.363 odd 4
1792.2.b.g.897.2 2 560.83 odd 4
2366.2.a.j.1.1 1 455.363 even 4
2366.2.d.b.337.1 2 455.398 odd 4
2366.2.d.b.337.2 2 455.83 odd 4
2450.2.a.t.1.1 1 5.2 odd 4
2450.2.c.c.99.1 2 1.1 even 1 trivial
2450.2.c.c.99.2 2 5.4 even 2 inner
2800.2.a.g.1.1 1 140.27 odd 4
2800.2.g.h.449.1 2 28.27 even 2
2800.2.g.h.449.2 2 140.139 even 2
3136.2.a.e.1.1 1 40.13 odd 4
3136.2.a.z.1.1 1 40.3 even 4
3150.2.a.i.1.1 1 105.62 odd 4
3150.2.g.j.2899.1 2 105.104 even 2
3150.2.g.j.2899.2 2 21.20 even 2
4032.2.a.r.1.1 1 840.83 even 4
4032.2.a.w.1.1 1 840.293 odd 4
4046.2.a.f.1.1 1 595.118 even 4
5054.2.a.c.1.1 1 665.398 odd 4
7056.2.a.bd.1.1 1 60.23 odd 4
7406.2.a.a.1.1 1 805.643 odd 4