Properties

Label 2450.2.c.h.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: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} -7.00000 q^{19} -3.00000i q^{22} +1.00000 q^{24} -2.00000 q^{26} -5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} -8.00000i q^{37} +7.00000i q^{38} -2.00000 q^{39} +9.00000 q^{41} +8.00000i q^{43} -3.00000 q^{44} -6.00000i q^{47} -1.00000i q^{48} +3.00000 q^{51} +2.00000i q^{52} -12.0000i q^{53} -5.00000 q^{54} +7.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} +10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -3.00000 q^{66} +7.00000i q^{67} -3.00000i q^{68} +6.00000 q^{71} +2.00000i q^{72} -5.00000i q^{73} -8.00000 q^{74} +7.00000 q^{76} +2.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} +9.00000i q^{83} +8.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} -15.0000 q^{89} -4.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + 6q^{11} + 2q^{16} - 14q^{19} + 2q^{24} - 4q^{26} + 12q^{29} + 8q^{31} + 6q^{34} - 4q^{36} - 4q^{39} + 18q^{41} - 6q^{44} + 6q^{51} - 10q^{54} + 24q^{59} + 20q^{61} - 2q^{64} - 6q^{66} + 12q^{71} - 16q^{74} + 14q^{76} - 28q^{79} + 2q^{81} + 16q^{86} - 30q^{89} - 12q^{94} - 2q^{96} + 12q^{99} + O(q^{100}) \)

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
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 3.00000i − 0.639602i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 5.00000i − 0.962250i
\(28\) 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
\(33\) − 3.00000i − 0.522233i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 7.00000i 1.13555i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 2.00000i 0.277350i
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000i 0.927173i
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 2.00000i 0.235702i
\(73\) − 5.00000i − 0.585206i −0.956234 0.292603i \(-0.905479\pi\)
0.956234 0.292603i \(-0.0945214\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) 2.00000i 0.226455i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 9.00000i − 0.993884i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) − 6.00000i − 0.643268i
\(88\) 3.00000i 0.319801i
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) − 20.0000i − 1.97066i −0.170664 0.985329i \(-0.554591\pi\)
0.170664 0.985329i \(-0.445409\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 5.00000i 0.481125i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) − 9.00000i − 0.846649i −0.905978 0.423324i \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) 7.00000 0.655610
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) − 4.00000i − 0.369800i
\(118\) − 12.0000i − 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 10.0000i − 0.905357i
\(123\) − 9.00000i − 0.811503i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 0 0
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) − 21.0000i − 1.79415i −0.441877 0.897076i \(-0.645687\pi\)
0.441877 0.897076i \(-0.354313\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 6.00000i − 0.503509i
\(143\) − 6.00000i − 0.501745i
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) −5.00000 −0.413803
\(147\) 0 0
\(148\) 8.00000i 0.657596i
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 7.00000i − 0.567775i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 20.0000i 1.59617i 0.602542 + 0.798087i \(0.294154\pi\)
−0.602542 + 0.798087i \(0.705846\pi\)
\(158\) 14.0000i 1.11378i
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −14.0000 −1.07061
\(172\) − 8.00000i − 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) − 12.0000i − 0.901975i
\(178\) 15.0000i 1.12430i
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 9.00000i 0.658145i
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 5.00000i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −20.0000 −1.39347
\(207\) 0 0
\(208\) − 2.00000i − 0.138675i
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 12.0000i 0.824163i
\(213\) − 6.00000i − 0.411113i
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 0 0
\(218\) 14.0000i 0.948200i
\(219\) −5.00000 −0.337869
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 8.00000i 0.536925i
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 7.00000i − 0.463586i
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 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
\(236\) −12.0000 −0.781133
\(237\) 14.0000i 0.909398i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 2.00000i 0.128565i
\(243\) − 16.0000i − 1.02640i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 14.0000i 0.890799i
\(248\) 4.00000i 0.254000i
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) 15.0000i 0.917985i
\(268\) − 7.00000i − 0.427593i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) −21.0000 −1.26866
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 7.00000i 0.419832i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 1.00000i 0.0594438i 0.999558 + 0.0297219i \(0.00946217\pi\)
−0.999558 + 0.0297219i \(0.990538\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) − 2.00000i − 0.117851i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 5.00000i 0.292603i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) − 15.0000i − 0.870388i
\(298\) 12.0000i 0.695141i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 7.00000i − 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 12.0000i 0.672927i
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) − 21.0000i − 1.16847i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 5.00000 0.276924
\(327\) 14.0000i 0.774202i
\(328\) 9.00000i 0.496942i
\(329\) 0 0
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) − 16.0000i − 0.876795i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 14.0000i 0.757033i
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 21.0000i 1.12734i 0.826000 + 0.563670i \(0.190611\pi\)
−0.826000 + 0.563670i \(0.809389\pi\)
\(348\) 6.00000i 0.321634i
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) − 3.00000i − 0.159901i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) 3.00000i 0.158555i
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 2.00000i 0.105118i
\(363\) 2.00000i 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000i 0.207390i
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 6.00000i 0.306987i
\(383\) 30.0000i 1.53293i 0.642287 + 0.766464i \(0.277986\pi\)
−0.642287 + 0.766464i \(0.722014\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) 16.0000i 0.813326i
\(388\) 10.0000i 0.507673i
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 14.0000i − 0.701757i
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) − 7.00000i − 0.349128i
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) 3.00000i 0.148522i
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 20.0000i 0.985329i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 7.00000i 0.342791i
\(418\) 21.0000i 1.02714i
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) − 17.0000i − 0.827547i
\(423\) − 12.0000i − 0.583460i
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 3.00000i 0.145010i
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) − 5.00000i − 0.240563i
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 5.00000i 0.238909i
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6.00000i − 0.285391i
\(443\) 21.0000i 0.997740i 0.866677 + 0.498870i \(0.166252\pi\)
−0.866677 + 0.498870i \(0.833748\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) 9.00000i 0.423324i
\(453\) − 8.00000i − 0.375873i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) − 17.0000i − 0.795226i −0.917553 0.397613i \(-0.869839\pi\)
0.917553 0.397613i \(-0.130161\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 12.0000i 0.552345i
\(473\) 24.0000i 1.10352i
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) 0 0
\(477\) − 24.0000i − 1.09888i
\(478\) − 12.0000i − 0.548867i
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) − 25.0000i − 1.13872i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 9.00000i 0.405751i
\(493\) 18.0000i 0.810679i
\(494\) 14.0000 0.629890
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) − 9.00000i − 0.403300i
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 15.0000i 0.669483i
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 2.00000i 0.0887357i
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 35.0000i 1.54529i
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) − 18.0000i − 0.791639i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) − 12.0000i − 0.525226i
\(523\) 7.00000i 0.306089i 0.988219 + 0.153044i \(0.0489077\pi\)
−0.988219 + 0.153044i \(0.951092\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 12.0000i 0.522728i
\(528\) − 3.00000i − 0.130558i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) − 18.0000i − 0.779667i
\(534\) 15.0000 0.649113
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) 3.00000i 0.129460i
\(538\) 6.00000i 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 2.00000i 0.0859074i
\(543\) 2.00000i 0.0858282i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) − 35.0000i − 1.49649i −0.663421 0.748246i \(-0.730896\pi\)
0.663421 0.748246i \(-0.269104\pi\)
\(548\) 21.0000i 0.897076i
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 7.00000 0.296866
\(557\) − 36.0000i − 1.52537i −0.646771 0.762684i \(-0.723881\pi\)
0.646771 0.762684i \(-0.276119\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 18.0000i 0.759284i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 1.00000 0.0420331
\(567\) 0 0
\(568\) 6.00000i 0.251754i
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 6.00000i 0.250873i
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 5.00000 0.207793
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000i 0.414513i
\(583\) − 36.0000i − 1.49097i
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 39.0000i 1.60970i 0.593477 + 0.804851i \(0.297755\pi\)
−0.593477 + 0.804851i \(0.702245\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 8.00000i − 0.328798i
\(593\) 27.0000i 1.10876i 0.832265 + 0.554379i \(0.187044\pi\)
−0.832265 + 0.554379i \(0.812956\pi\)
\(594\) −15.0000 −0.615457
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) − 14.0000i − 0.572982i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 7.00000 0.285536 0.142768 0.989756i \(-0.454400\pi\)
0.142768 + 0.989756i \(0.454400\pi\)
\(602\) 0 0
\(603\) 14.0000i 0.570124i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 44.0000i 1.78590i 0.450151 + 0.892952i \(0.351370\pi\)
−0.450151 + 0.892952i \(0.648630\pi\)
\(608\) 7.00000i 0.283887i
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) − 6.00000i − 0.242536i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 20.0000i 0.804518i
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 18.0000i − 0.721734i
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 21.0000i 0.838659i
\(628\) − 20.0000i − 0.798087i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) − 14.0000i − 0.556890i
\(633\) − 17.0000i − 0.675689i
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) − 18.0000i − 0.712627i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 3.00000i 0.118401i
\(643\) 40.0000i 1.57745i 0.614749 + 0.788723i \(0.289257\pi\)
−0.614749 + 0.788723i \(0.710743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 −0.826234
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.00000i − 0.195815i
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 25.0000i 0.971653i
\(663\) − 6.00000i − 0.233021i
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 9.00000i 0.345643i
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) − 12.0000i − 0.459504i
\(683\) 3.00000i 0.114792i 0.998351 + 0.0573959i \(0.0182797\pi\)
−0.998351 + 0.0573959i \(0.981720\pi\)
\(684\) 14.0000 0.535303
\(685\) 0 0
\(686\) 0 0
\(687\) − 26.0000i − 0.991962i
\(688\) 8.00000i 0.304997i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 19.0000 0.722794 0.361397 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 21.0000 0.797149
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 27.0000i 1.02270i
\(698\) − 8.00000i − 0.302804i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 10.0000i 0.377426i
\(703\) 56.0000i 2.11208i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) −28.0000 −1.05008
\(712\) − 15.0000i − 0.562149i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 3.00000 0.112115
\(717\) − 12.0000i − 0.448148i
\(718\) − 6.00000i − 0.223918i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 30.0000i − 1.11648i
\(723\) − 25.0000i − 0.929760i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 34.0000i − 1.26099i −0.776193 0.630495i \(-0.782852\pi\)
0.776193 0.630495i \(-0.217148\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 10.0000i 0.369611i
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 21.0000i 0.773545i
\(738\) − 18.0000i − 0.662589i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 14.0000 0.514303
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 18.0000i 0.658586i
\(748\) − 9.00000i − 0.329073i
\(749\) 0 0
\(750\) 0 0
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 15.0000i 0.546630i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000i 1.23575i 0.786276 + 0.617876i \(0.212006\pi\)
−0.786276 + 0.617876i \(0.787994\pi\)
\(758\) 17.0000i 0.617468i
\(759\) 0 0
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 2.00000i 0.0724524i
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) − 24.0000i − 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) − 5.00000i − 0.179954i
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 16.0000 0.575108
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) − 24.0000i − 0.860442i
\(779\) −63.0000 −2.25721
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) − 30.0000i − 1.07211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000i 0.213201i
\(793\) − 20.0000i − 0.710221i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 27.0000i 0.953403i
\(803\) − 15.0000i − 0.529339i
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 2.00000i 0.0701431i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) − 56.0000i − 1.95919i
\(818\) 25.0000i 0.874105i
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 21.0000i 0.732459i
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 20.0000 0.696733
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.00000i − 0.312961i −0.987681 0.156480i \(-0.949985\pi\)
0.987681 0.156480i \(-0.0500148\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 2.00000i 0.0693375i
\(833\) 0 0
\(834\) 7.00000 0.242390
\(835\) 0 0
\(836\) 21.0000 0.726300
\(837\) − 20.0000i − 0.691301i
\(838\) 3.00000i 0.103633i
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 20.0000i − 0.689246i
\(843\) 18.0000i 0.619953i
\(844\) −17.0000 −0.585164
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) − 12.0000i − 0.412082i
\(849\) 1.00000 0.0343199
\(850\) 0 0
\(851\) 0 0
\(852\) 6.00000i 0.205557i
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 15.0000i 0.512390i 0.966625 + 0.256195i \(0.0824690\pi\)
−0.966625 + 0.256195i \(0.917531\pi\)
\(858\) 6.00000i 0.204837i
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −11.0000 −0.373795
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) − 14.0000i − 0.474100i
\(873\) − 20.0000i − 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 5.00000 0.168934
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 4.00000i 0.134993i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) 47.0000i 1.58168i 0.612026 + 0.790838i \(0.290355\pi\)
−0.612026 + 0.790838i \(0.709645\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 21.0000 0.705509
\(887\) 6.00000i 0.201460i 0.994914 + 0.100730i \(0.0321179\pi\)
−0.994914 + 0.100730i \(0.967882\pi\)
\(888\) − 8.00000i − 0.268462i
\(889\) 0 0
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 14.0000i 0.468755i
\(893\) 42.0000i 1.40548i
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 15.0000i − 0.500556i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) − 27.0000i − 0.899002i
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 7.00000i 0.231793i
\(913\) 27.0000i 0.893570i
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) − 15.0000i − 0.495074i
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 18.0000i 0.592798i
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) − 40.0000i − 1.31377i
\(928\) − 6.00000i − 0.196960i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 6.00000i − 0.196537i
\(933\) − 18.0000i − 0.589294i
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 29.0000i 0.947389i 0.880689 + 0.473694i \(0.157080\pi\)
−0.880689 + 0.473694i \(0.842920\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) − 20.0000i − 0.651635i
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) − 14.0000i − 0.454699i
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 57.0000i 1.84641i 0.384307 + 0.923206i \(0.374441\pi\)
−0.384307 + 0.923206i \(0.625559\pi\)
\(954\) −24.0000 −0.777029
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) − 18.0000i − 0.581857i
\(958\) − 18.0000i − 0.581554i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 16.0000i 0.515861i
\(963\) − 6.00000i − 0.193347i
\(964\) −25.0000 −0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) 34.0000i 1.09337i 0.837340 + 0.546683i \(0.184110\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 16.0000i 0.513200i
\(973\) 0 0
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 3.00000i 0.0959785i 0.998848 + 0.0479893i \(0.0152813\pi\)
−0.998848 + 0.0479893i \(0.984719\pi\)
\(978\) − 5.00000i − 0.159882i
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) 12.0000i 0.382935i
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) − 14.0000i − 0.445399i
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 25.0000i 0.793351i
\(994\) 0 0
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 62.0000i 1.96356i 0.190022 + 0.981780i \(0.439144\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) −40.0000 −1.26554
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.h.99.1 2
5.2 odd 4 2450.2.a.x.1.1 1
5.3 odd 4 2450.2.a.m.1.1 1
5.4 even 2 inner 2450.2.c.h.99.2 2
7.6 odd 2 350.2.c.c.99.1 2
21.20 even 2 3150.2.g.f.2899.2 2
28.27 even 2 2800.2.g.i.449.1 2
35.13 even 4 350.2.a.a.1.1 1
35.27 even 4 350.2.a.e.1.1 yes 1
35.34 odd 2 350.2.c.c.99.2 2
105.62 odd 4 3150.2.a.m.1.1 1
105.83 odd 4 3150.2.a.x.1.1 1
105.104 even 2 3150.2.g.f.2899.1 2
140.27 odd 4 2800.2.a.h.1.1 1
140.83 odd 4 2800.2.a.x.1.1 1
140.139 even 2 2800.2.g.i.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 35.13 even 4
350.2.a.e.1.1 yes 1 35.27 even 4
350.2.c.c.99.1 2 7.6 odd 2
350.2.c.c.99.2 2 35.34 odd 2
2450.2.a.m.1.1 1 5.3 odd 4
2450.2.a.x.1.1 1 5.2 odd 4
2450.2.c.h.99.1 2 1.1 even 1 trivial
2450.2.c.h.99.2 2 5.4 even 2 inner
2800.2.a.h.1.1 1 140.27 odd 4
2800.2.a.x.1.1 1 140.83 odd 4
2800.2.g.i.449.1 2 28.27 even 2
2800.2.g.i.449.2 2 140.139 even 2
3150.2.a.m.1.1 1 105.62 odd 4
3150.2.a.x.1.1 1 105.83 odd 4
3150.2.g.f.2899.1 2 105.104 even 2
3150.2.g.f.2899.2 2 21.20 even 2