Properties

Label 1682.2.b.a.1681.1
Level $1682$
Weight $2$
Character 1682.1681
Analytic conductor $13.431$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4308376200\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1681.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1682.1681
Dual form 1682.2.b.a.1681.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{10} +3.00000i q^{11} -1.00000i q^{12} +1.00000 q^{13} +2.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +1.00000 q^{20} -2.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000i q^{26} +5.00000i q^{27} +2.00000 q^{28} -1.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} -3.00000 q^{33} -8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} +1.00000i q^{39} -1.00000i q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000i q^{43} -3.00000i q^{44} -2.00000 q^{45} -4.00000i q^{46} +13.0000i q^{47} +1.00000i q^{48} -3.00000 q^{49} +4.00000i q^{50} +8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} -3.00000i q^{55} -2.00000i q^{56} +1.00000i q^{60} +8.00000i q^{61} +3.00000 q^{62} -4.00000 q^{63} -1.00000 q^{64} -1.00000 q^{65} +3.00000i q^{66} +12.0000 q^{67} +8.00000i q^{68} +4.00000i q^{69} -2.00000i q^{70} -2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} +8.00000 q^{74} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{78} -15.0000i q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +2.00000i q^{84} +8.00000i q^{85} +11.0000 q^{86} -3.00000 q^{88} +10.0000i q^{89} +2.00000i q^{90} -2.00000 q^{91} -4.00000 q^{92} -3.00000 q^{93} +13.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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 \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 1.00000i 0.316228i
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 2.00000i 0.534522i
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) − 8.00000i − 1.94029i −0.242536 0.970143i \(-0.577979\pi\)
0.242536 0.970143i \(-0.422021\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.00000 0.223607
\(21\) − 2.00000i − 0.436436i
\(22\) 3.00000 0.639602
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) − 1.00000i − 0.196116i
\(27\) 5.00000i 0.962250i
\(28\) 2.00000 0.377964
\(29\) 0 0
\(30\) −1.00000 −0.182574
\(31\) 3.00000i 0.538816i 0.963026 + 0.269408i \(0.0868280\pi\)
−0.963026 + 0.269408i \(0.913172\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) −3.00000 −0.522233
\(34\) −8.00000 −1.37199
\(35\) 2.00000 0.338062
\(36\) −2.00000 −0.333333
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 1.00000i 0.160128i
\(40\) − 1.00000i − 0.158114i
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) −2.00000 −0.308607
\(43\) 11.0000i 1.67748i 0.544529 + 0.838742i \(0.316708\pi\)
−0.544529 + 0.838742i \(0.683292\pi\)
\(44\) − 3.00000i − 0.452267i
\(45\) −2.00000 −0.298142
\(46\) − 4.00000i − 0.589768i
\(47\) 13.0000i 1.89624i 0.317905 + 0.948122i \(0.397021\pi\)
−0.317905 + 0.948122i \(0.602979\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −3.00000 −0.428571
\(50\) 4.00000i 0.565685i
\(51\) 8.00000 1.12022
\(52\) −1.00000 −0.138675
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 5.00000 0.680414
\(55\) − 3.00000i − 0.404520i
\(56\) − 2.00000i − 0.267261i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 3.00000 0.381000
\(63\) −4.00000 −0.503953
\(64\) −1.00000 −0.125000
\(65\) −1.00000 −0.124035
\(66\) 3.00000i 0.369274i
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 8.00000i 0.970143i
\(69\) 4.00000i 0.481543i
\(70\) − 2.00000i − 0.239046i
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 2.00000i 0.235702i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 8.00000 0.929981
\(75\) − 4.00000i − 0.461880i
\(76\) 0 0
\(77\) − 6.00000i − 0.683763i
\(78\) 1.00000 0.113228
\(79\) − 15.0000i − 1.68763i −0.536633 0.843816i \(-0.680304\pi\)
0.536633 0.843816i \(-0.319696\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 8.00000i 0.867722i
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 2.00000i 0.210819i
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) −3.00000 −0.311086
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 6.00000i 0.603023i
\(100\) 4.00000 0.400000
\(101\) 8.00000i 0.796030i 0.917379 + 0.398015i \(0.130301\pi\)
−0.917379 + 0.398015i \(0.869699\pi\)
\(102\) − 8.00000i − 0.792118i
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 1.00000i 0.0980581i
\(105\) 2.00000i 0.195180i
\(106\) 11.0000i 1.06841i
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −3.00000 −0.286039
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 16.0000i 1.46672i
\(120\) 1.00000 0.0912871
\(121\) 2.00000 0.181818
\(122\) 8.00000 0.724286
\(123\) −2.00000 −0.180334
\(124\) − 3.00000i − 0.269408i
\(125\) 9.00000 0.804984
\(126\) 4.00000i 0.356348i
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −11.0000 −0.968496
\(130\) 1.00000i 0.0877058i
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) − 12.0000i − 1.03664i
\(135\) − 5.00000i − 0.430331i
\(136\) 8.00000 0.685994
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 4.00000 0.340503
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) −13.0000 −1.09480
\(142\) 2.00000i 0.167836i
\(143\) 3.00000i 0.250873i
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) − 3.00000i − 0.247436i
\(148\) − 8.00000i − 0.657596i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) −4.00000 −0.326599
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) − 16.0000i − 1.29352i
\(154\) −6.00000 −0.483494
\(155\) − 3.00000i − 0.240966i
\(156\) − 1.00000i − 0.0800641i
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) −15.0000 −1.19334
\(159\) − 11.0000i − 0.872357i
\(160\) 1.00000i 0.0790569i
\(161\) −8.00000 −0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 9.00000i 0.704934i 0.935824 + 0.352467i \(0.114657\pi\)
−0.935824 + 0.352467i \(0.885343\pi\)
\(164\) − 2.00000i − 0.156174i
\(165\) 3.00000 0.233550
\(166\) − 4.00000i − 0.310460i
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 2.00000 0.154303
\(169\) −12.0000 −0.923077
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) − 11.0000i − 0.838742i
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) 3.00000i 0.226134i
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 2.00000 0.149071
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 2.00000i 0.148250i
\(183\) −8.00000 −0.591377
\(184\) 4.00000i 0.294884i
\(185\) − 8.00000i − 0.588172i
\(186\) 3.00000i 0.219971i
\(187\) 24.0000 1.75505
\(188\) − 13.0000i − 0.948122i
\(189\) − 10.0000i − 0.727393i
\(190\) 0 0
\(191\) 8.00000i 0.578860i 0.957199 + 0.289430i \(0.0934657\pi\)
−0.957199 + 0.289430i \(0.906534\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −2.00000 −0.143592
\(195\) − 1.00000i − 0.0716115i
\(196\) 3.00000 0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 6.00000 0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) − 4.00000i − 0.282843i
\(201\) 12.0000i 0.846415i
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) −8.00000 −0.560112
\(205\) − 2.00000i − 0.139686i
\(206\) − 14.0000i − 0.975426i
\(207\) 8.00000 0.556038
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) − 3.00000i − 0.206529i −0.994654 0.103264i \(-0.967071\pi\)
0.994654 0.103264i \(-0.0329287\pi\)
\(212\) 11.0000 0.755483
\(213\) − 2.00000i − 0.137038i
\(214\) 2.00000i 0.136717i
\(215\) − 11.0000i − 0.750194i
\(216\) −5.00000 −0.340207
\(217\) − 6.00000i − 0.407307i
\(218\) 5.00000i 0.338643i
\(219\) −4.00000 −0.270295
\(220\) 3.00000i 0.202260i
\(221\) − 8.00000i − 0.538138i
\(222\) 8.00000i 0.536925i
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 2.00000i 0.133631i
\(225\) −8.00000 −0.533333
\(226\) −6.00000 −0.399114
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 4.00000i 0.263752i
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) − 2.00000i − 0.130744i
\(235\) − 13.0000i − 0.848026i
\(236\) 0 0
\(237\) 15.0000 0.974355
\(238\) 16.0000 1.03713
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 16.0000i 1.02640i
\(244\) − 8.00000i − 0.512148i
\(245\) 3.00000 0.191663
\(246\) 2.00000i 0.127515i
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) 4.00000i 0.253490i
\(250\) − 9.00000i − 0.569210i
\(251\) − 27.0000i − 1.70422i −0.523359 0.852112i \(-0.675321\pi\)
0.523359 0.852112i \(-0.324679\pi\)
\(252\) 4.00000 0.251976
\(253\) 12.0000i 0.754434i
\(254\) −8.00000 −0.501965
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) 13.0000 0.810918 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) 11.0000i 0.684830i
\(259\) − 16.0000i − 0.994192i
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) − 3.00000i − 0.184637i
\(265\) 11.0000 0.675725
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −5.00000 −0.304290
\(271\) − 13.0000i − 0.789694i −0.918747 0.394847i \(-0.870798\pi\)
0.918747 0.394847i \(-0.129202\pi\)
\(272\) − 8.00000i − 0.485071i
\(273\) − 2.00000i − 0.121046i
\(274\) 12.0000 0.724947
\(275\) − 12.0000i − 0.723627i
\(276\) − 4.00000i − 0.240772i
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 6.00000i 0.359211i
\(280\) 2.00000i 0.119523i
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 13.0000i 0.774139i
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) − 4.00000i − 0.236113i
\(288\) − 2.00000i − 0.117851i
\(289\) −47.0000 −2.76471
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) − 4.00000i − 0.234082i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −15.0000 −0.870388
\(298\) 15.0000i 0.868927i
\(299\) 4.00000 0.231326
\(300\) 4.00000i 0.230940i
\(301\) − 22.0000i − 1.26806i
\(302\) 2.00000i 0.115087i
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) − 8.00000i − 0.458079i
\(306\) −16.0000 −0.914659
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 14.0000i 0.796432i
\(310\) −3.00000 −0.170389
\(311\) 8.00000i 0.453638i 0.973937 + 0.226819i \(0.0728326\pi\)
−0.973937 + 0.226819i \(0.927167\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) 18.0000 1.01580
\(315\) 4.00000 0.225374
\(316\) 15.0000i 0.843816i
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) −11.0000 −0.616849
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) − 2.00000i − 0.111629i
\(322\) 8.00000i 0.445823i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) −4.00000 −0.221880
\(326\) 9.00000 0.498464
\(327\) − 5.00000i − 0.276501i
\(328\) −2.00000 −0.110432
\(329\) − 26.0000i − 1.43343i
\(330\) − 3.00000i − 0.165145i
\(331\) − 23.0000i − 1.26419i −0.774889 0.632097i \(-0.782194\pi\)
0.774889 0.632097i \(-0.217806\pi\)
\(332\) −4.00000 −0.219529
\(333\) 16.0000i 0.876795i
\(334\) − 2.00000i − 0.109435i
\(335\) −12.0000 −0.655630
\(336\) − 2.00000i − 0.109109i
\(337\) − 32.0000i − 1.74315i −0.490261 0.871576i \(-0.663099\pi\)
0.490261 0.871576i \(-0.336901\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 6.00000 0.325875
\(340\) − 8.00000i − 0.433861i
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −11.0000 −0.593080
\(345\) − 4.00000i − 0.215353i
\(346\) − 6.00000i − 0.322562i
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) − 8.00000i − 0.427618i
\(351\) 5.00000i 0.266880i
\(352\) 3.00000 0.159901
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) − 10.0000i − 0.529999i
\(357\) −16.0000 −0.846810
\(358\) − 10.0000i − 0.528516i
\(359\) 25.0000i 1.31945i 0.751507 + 0.659725i \(0.229327\pi\)
−0.751507 + 0.659725i \(0.770673\pi\)
\(360\) − 2.00000i − 0.105409i
\(361\) 19.0000 1.00000
\(362\) − 7.00000i − 0.367912i
\(363\) 2.00000i 0.104973i
\(364\) 2.00000 0.104828
\(365\) − 4.00000i − 0.209370i
\(366\) 8.00000i 0.418167i
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 4.00000 0.208514
\(369\) 4.00000i 0.208232i
\(370\) −8.00000 −0.415900
\(371\) 22.0000 1.14218
\(372\) 3.00000 0.155543
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) − 24.0000i − 1.24101i
\(375\) 9.00000i 0.464758i
\(376\) −13.0000 −0.670424
\(377\) 0 0
\(378\) −10.0000 −0.514344
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 8.00000 0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.00000i 0.305788i
\(386\) −14.0000 −0.712581
\(387\) 22.0000i 1.11832i
\(388\) 2.00000i 0.101535i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −1.00000 −0.0506370
\(391\) − 32.0000i − 1.61831i
\(392\) − 3.00000i − 0.151523i
\(393\) −12.0000 −0.605320
\(394\) − 18.0000i − 0.906827i
\(395\) 15.0000i 0.754732i
\(396\) − 6.00000i − 0.301511i
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 12.0000 0.598506
\(403\) 3.00000i 0.149441i
\(404\) − 8.00000i − 0.398015i
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 8.00000i 0.396059i
\(409\) − 30.0000i − 1.48340i −0.670729 0.741702i \(-0.734019\pi\)
0.670729 0.741702i \(-0.265981\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) − 8.00000i − 0.393179i
\(415\) −4.00000 −0.196352
\(416\) − 1.00000i − 0.0490290i
\(417\) − 20.0000i − 0.979404i
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) − 2.00000i − 0.0975900i
\(421\) 32.0000i 1.55958i 0.626038 + 0.779792i \(0.284675\pi\)
−0.626038 + 0.779792i \(0.715325\pi\)
\(422\) −3.00000 −0.146038
\(423\) 26.0000i 1.26416i
\(424\) − 11.0000i − 0.534207i
\(425\) 32.0000i 1.55223i
\(426\) −2.00000 −0.0969003
\(427\) − 16.0000i − 0.774294i
\(428\) 2.00000 0.0966736
\(429\) −3.00000 −0.144841
\(430\) −11.0000 −0.530467
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 5.00000i 0.240563i
\(433\) − 16.0000i − 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) 0 0
\(438\) 4.00000i 0.191127i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 3.00000 0.143019
\(441\) −6.00000 −0.285714
\(442\) −8.00000 −0.380521
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 8.00000 0.379663
\(445\) − 10.0000i − 0.474045i
\(446\) 26.0000i 1.23114i
\(447\) − 15.0000i − 0.709476i
\(448\) 2.00000 0.0944911
\(449\) 10.0000i 0.471929i 0.971762 + 0.235965i \(0.0758249\pi\)
−0.971762 + 0.235965i \(0.924175\pi\)
\(450\) 8.00000i 0.377124i
\(451\) −6.00000 −0.282529
\(452\) 6.00000i 0.282216i
\(453\) − 2.00000i − 0.0939682i
\(454\) − 18.0000i − 0.844782i
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 10.0000 0.467269
\(459\) 40.0000 1.86704
\(460\) 4.00000 0.186501
\(461\) 2.00000i 0.0931493i 0.998915 + 0.0465746i \(0.0148305\pi\)
−0.998915 + 0.0465746i \(0.985169\pi\)
\(462\) − 6.00000i − 0.279145i
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 1.00000i 0.0463241i
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −24.0000 −1.10822
\(470\) −13.0000 −0.599645
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −33.0000 −1.51734
\(474\) − 15.0000i − 0.688973i
\(475\) 0 0
\(476\) − 16.0000i − 0.733359i
\(477\) −22.0000 −1.00731
\(478\) 0 0
\(479\) − 5.00000i − 0.228456i −0.993455 0.114228i \(-0.963561\pi\)
0.993455 0.114228i \(-0.0364394\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 8.00000i 0.364769i
\(482\) 17.0000i 0.774329i
\(483\) − 8.00000i − 0.364013i
\(484\) −2.00000 −0.0909091
\(485\) 2.00000i 0.0908153i
\(486\) 16.0000 0.725775
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −8.00000 −0.362143
\(489\) −9.00000 −0.406994
\(490\) − 3.00000i − 0.135526i
\(491\) − 33.0000i − 1.48927i −0.667472 0.744635i \(-0.732624\pi\)
0.667472 0.744635i \(-0.267376\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) 0 0
\(495\) − 6.00000i − 0.269680i
\(496\) 3.00000i 0.134704i
\(497\) 4.00000 0.179425
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −9.00000 −0.402492
\(501\) 2.00000i 0.0893534i
\(502\) −27.0000 −1.20507
\(503\) 19.0000i 0.847168i 0.905857 + 0.423584i \(0.139228\pi\)
−0.905857 + 0.423584i \(0.860772\pi\)
\(504\) − 4.00000i − 0.178174i
\(505\) − 8.00000i − 0.355995i
\(506\) 12.0000 0.533465
\(507\) − 12.0000i − 0.532939i
\(508\) 8.00000i 0.354943i
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 8.00000i 0.354246i
\(511\) − 8.00000i − 0.353899i
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) − 13.0000i − 0.573405i
\(515\) −14.0000 −0.616914
\(516\) 11.0000 0.484248
\(517\) −39.0000 −1.71522
\(518\) −16.0000 −0.703000
\(519\) 6.00000i 0.263371i
\(520\) − 1.00000i − 0.0438529i
\(521\) 13.0000 0.569540 0.284770 0.958596i \(-0.408083\pi\)
0.284770 + 0.958596i \(0.408083\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) − 12.0000i − 0.524222i
\(525\) 8.00000i 0.349149i
\(526\) −9.00000 −0.392419
\(527\) 24.0000 1.04546
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) − 11.0000i − 0.477809i
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000i 0.0866296i
\(534\) 10.0000i 0.432742i
\(535\) 2.00000 0.0864675
\(536\) 12.0000i 0.518321i
\(537\) 10.0000i 0.431532i
\(538\) 0 0
\(539\) − 9.00000i − 0.387657i
\(540\) 5.00000i 0.215166i
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) −13.0000 −0.558398
\(543\) 7.00000i 0.300399i
\(544\) −8.00000 −0.342997
\(545\) 5.00000 0.214176
\(546\) −2.00000 −0.0855921
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 16.0000i 0.682863i
\(550\) −12.0000 −0.511682
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) 30.0000i 1.27573i
\(554\) 2.00000i 0.0849719i
\(555\) 8.00000 0.339581
\(556\) 20.0000 0.848189
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 6.00000 0.254000
\(559\) 11.0000i 0.465250i
\(560\) 2.00000 0.0845154
\(561\) 24.0000i 1.01328i
\(562\) − 27.0000i − 1.13893i
\(563\) − 11.0000i − 0.463595i −0.972764 0.231797i \(-0.925539\pi\)
0.972764 0.231797i \(-0.0744606\pi\)
\(564\) 13.0000 0.547399
\(565\) 6.00000i 0.252422i
\(566\) 4.00000i 0.168133i
\(567\) −2.00000 −0.0839921
\(568\) − 2.00000i − 0.0839181i
\(569\) − 30.0000i − 1.25767i −0.777541 0.628833i \(-0.783533\pi\)
0.777541 0.628833i \(-0.216467\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) − 3.00000i − 0.125436i
\(573\) −8.00000 −0.334205
\(574\) −4.00000 −0.166957
\(575\) −16.0000 −0.667246
\(576\) −2.00000 −0.0833333
\(577\) 8.00000i 0.333044i 0.986038 + 0.166522i \(0.0532537\pi\)
−0.986038 + 0.166522i \(0.946746\pi\)
\(578\) 47.0000i 1.95494i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) − 2.00000i − 0.0829027i
\(583\) − 33.0000i − 1.36672i
\(584\) −4.00000 −0.165521
\(585\) −2.00000 −0.0826898
\(586\) −14.0000 −0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000i 0.740421i
\(592\) 8.00000i 0.328798i
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 15.0000i 0.615457i
\(595\) − 16.0000i − 0.655936i
\(596\) 15.0000 0.614424
\(597\) − 10.0000i − 0.409273i
\(598\) − 4.00000i − 0.163572i
\(599\) 5.00000i 0.204294i 0.994769 + 0.102147i \(0.0325713\pi\)
−0.994769 + 0.102147i \(0.967429\pi\)
\(600\) 4.00000 0.163299
\(601\) − 2.00000i − 0.0815817i −0.999168 0.0407909i \(-0.987012\pi\)
0.999168 0.0407909i \(-0.0129877\pi\)
\(602\) −22.0000 −0.896653
\(603\) 24.0000 0.977356
\(604\) 2.00000 0.0813788
\(605\) −2.00000 −0.0813116
\(606\) 8.00000i 0.324978i
\(607\) 3.00000i 0.121766i 0.998145 + 0.0608831i \(0.0193917\pi\)
−0.998145 + 0.0608831i \(0.980608\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) 13.0000i 0.525924i
\(612\) 16.0000i 0.646762i
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) 7.00000 0.282497
\(615\) 2.00000 0.0806478
\(616\) 6.00000 0.241747
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 14.0000 0.563163
\(619\) − 35.0000i − 1.40677i −0.710810 0.703384i \(-0.751671\pi\)
0.710810 0.703384i \(-0.248329\pi\)
\(620\) 3.00000i 0.120483i
\(621\) 20.0000i 0.802572i
\(622\) 8.00000 0.320771
\(623\) − 20.0000i − 0.801283i
\(624\) 1.00000i 0.0400320i
\(625\) 11.0000 0.440000
\(626\) − 9.00000i − 0.359712i
\(627\) 0 0
\(628\) − 18.0000i − 0.718278i
\(629\) 64.0000 2.55185
\(630\) − 4.00000i − 0.159364i
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 15.0000 0.596668
\(633\) 3.00000 0.119239
\(634\) −12.0000 −0.476581
\(635\) 8.00000i 0.317470i
\(636\) 11.0000i 0.436178i
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) − 1.00000i − 0.0395285i
\(641\) 8.00000i 0.315981i 0.987441 + 0.157991i \(0.0505015\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 8.00000 0.315244
\(645\) 11.0000 0.433125
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 4.00000i 0.156893i
\(651\) 6.00000 0.235159
\(652\) − 9.00000i − 0.352467i
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) −5.00000 −0.195515
\(655\) − 12.0000i − 0.468879i
\(656\) 2.00000i 0.0780869i
\(657\) 8.00000i 0.312110i
\(658\) −26.0000 −1.01359
\(659\) − 15.0000i − 0.584317i −0.956370 0.292159i \(-0.905627\pi\)
0.956370 0.292159i \(-0.0943735\pi\)
\(660\) −3.00000 −0.116775
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −23.0000 −0.893920
\(663\) 8.00000 0.310694
\(664\) 4.00000i 0.155230i
\(665\) 0 0
\(666\) 16.0000 0.619987
\(667\) 0 0
\(668\) −2.00000 −0.0773823
\(669\) − 26.0000i − 1.00522i
\(670\) 12.0000i 0.463600i
\(671\) −24.0000 −0.926510
\(672\) −2.00000 −0.0771517
\(673\) −9.00000 −0.346925 −0.173462 0.984841i \(-0.555495\pi\)
−0.173462 + 0.984841i \(0.555495\pi\)
\(674\) −32.0000 −1.23259
\(675\) − 20.0000i − 0.769800i
\(676\) 12.0000 0.461538
\(677\) 38.0000i 1.46046i 0.683202 + 0.730229i \(0.260587\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 4.00000i 0.153506i
\(680\) −8.00000 −0.306786
\(681\) 18.0000i 0.689761i
\(682\) 9.00000i 0.344628i
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) − 12.0000i − 0.458496i
\(686\) − 20.0000i − 0.763604i
\(687\) −10.0000 −0.381524
\(688\) 11.0000i 0.419371i
\(689\) −11.0000 −0.419067
\(690\) −4.00000 −0.152277
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) − 12.0000i − 0.455842i
\(694\) − 2.00000i − 0.0759190i
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 15.0000i 0.567758i
\(699\) − 1.00000i − 0.0378235i
\(700\) −8.00000 −0.302372
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 5.00000 0.188713
\(703\) 0 0
\(704\) − 3.00000i − 0.113067i
\(705\) 13.0000 0.489608
\(706\) − 26.0000i − 0.978523i
\(707\) − 16.0000i − 0.601742i
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) − 2.00000i − 0.0750587i
\(711\) − 30.0000i − 1.12509i
\(712\) −10.0000 −0.374766
\(713\) 12.0000i 0.449404i
\(714\) 16.0000i 0.598785i
\(715\) − 3.00000i − 0.112194i
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 25.0000 0.932992
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −28.0000 −1.04277
\(722\) − 19.0000i − 0.707107i
\(723\) − 17.0000i − 0.632237i
\(724\) −7.00000 −0.260153
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −13.0000 −0.481481
\(730\) −4.00000 −0.148047
\(731\) 88.0000 3.25480
\(732\) 8.00000 0.295689
\(733\) 24.0000i 0.886460i 0.896408 + 0.443230i \(0.146168\pi\)
−0.896408 + 0.443230i \(0.853832\pi\)
\(734\) 32.0000 1.18114
\(735\) 3.00000i 0.110657i
\(736\) − 4.00000i − 0.147442i
\(737\) 36.0000i 1.32608i
\(738\) 4.00000 0.147242
\(739\) 5.00000i 0.183928i 0.995762 + 0.0919640i \(0.0293145\pi\)
−0.995762 + 0.0919640i \(0.970686\pi\)
\(740\) 8.00000i 0.294086i
\(741\) 0 0
\(742\) − 22.0000i − 0.807645i
\(743\) 44.0000i 1.61420i 0.590412 + 0.807102i \(0.298965\pi\)
−0.590412 + 0.807102i \(0.701035\pi\)
\(744\) − 3.00000i − 0.109985i
\(745\) 15.0000 0.549557
\(746\) 21.0000i 0.768865i
\(747\) 8.00000 0.292705
\(748\) −24.0000 −0.877527
\(749\) 4.00000 0.146157
\(750\) 9.00000 0.328634
\(751\) 32.0000i 1.16770i 0.811863 + 0.583848i \(0.198454\pi\)
−0.811863 + 0.583848i \(0.801546\pi\)
\(752\) 13.0000i 0.474061i
\(753\) 27.0000 0.983935
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 10.0000i 0.363696i
\(757\) − 8.00000i − 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) −20.0000 −0.726433
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) 10.0000 0.362024
\(764\) − 8.00000i − 0.289430i
\(765\) 16.0000i 0.578481i
\(766\) 14.0000i 0.505841i
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) − 20.0000i − 0.721218i −0.932717 0.360609i \(-0.882569\pi\)
0.932717 0.360609i \(-0.117431\pi\)
\(770\) 6.00000 0.216225
\(771\) 13.0000i 0.468184i
\(772\) 14.0000i 0.503871i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 22.0000 0.790774
\(775\) − 12.0000i − 0.431053i
\(776\) 2.00000 0.0717958
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000i 0.0358057i
\(781\) − 6.00000i − 0.214697i
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) − 18.0000i − 0.642448i
\(786\) 12.0000i 0.428026i
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −18.0000 −0.641223
\(789\) 9.00000 0.320408
\(790\) 15.0000 0.533676
\(791\) 12.0000i 0.426671i
\(792\) −6.00000 −0.213201
\(793\) 8.00000i 0.284088i
\(794\) 17.0000i 0.603307i
\(795\) 11.0000i 0.390130i
\(796\) 10.0000 0.354441
\(797\) 32.0000i 1.13350i 0.823890 + 0.566749i \(0.191799\pi\)
−0.823890 + 0.566749i \(0.808201\pi\)
\(798\) 0 0
\(799\) 104.000 3.67926
\(800\) 4.00000i 0.141421i
\(801\) 20.0000i 0.706665i
\(802\) − 27.0000i − 0.953403i
\(803\) −12.0000 −0.423471
\(804\) − 12.0000i − 0.423207i
\(805\) 8.00000 0.281963
\(806\) 3.00000 0.105670
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 1.00000i 0.0351364i
\(811\) 18.0000 0.632065 0.316033 0.948748i \(-0.397649\pi\)
0.316033 + 0.948748i \(0.397649\pi\)
\(812\) 0 0
\(813\) 13.0000 0.455930
\(814\) 24.0000i 0.841200i
\(815\) − 9.00000i − 0.315256i
\(816\) 8.00000 0.280056
\(817\) 0 0
\(818\) −30.0000 −1.04893
\(819\) −4.00000 −0.139771
\(820\) 2.00000i 0.0698430i
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 12.0000i 0.418548i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) 14.0000i 0.487713i
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 13.0000i 0.452054i 0.974121 + 0.226027i \(0.0725738\pi\)
−0.974121 + 0.226027i \(0.927426\pi\)
\(828\) −8.00000 −0.278019
\(829\) − 40.0000i − 1.38926i −0.719368 0.694629i \(-0.755569\pi\)
0.719368 0.694629i \(-0.244431\pi\)
\(830\) 4.00000i 0.138842i
\(831\) − 2.00000i − 0.0693792i
\(832\) −1.00000 −0.0346688
\(833\) 24.0000i 0.831551i
\(834\) −20.0000 −0.692543
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) −15.0000 −0.518476
\(838\) − 10.0000i − 0.345444i
\(839\) 45.0000i 1.55357i 0.629764 + 0.776786i \(0.283151\pi\)
−0.629764 + 0.776786i \(0.716849\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 0 0
\(842\) 32.0000 1.10279
\(843\) 27.0000i 0.929929i
\(844\) 3.00000i 0.103264i
\(845\) 12.0000 0.412813
\(846\) 26.0000 0.893898
\(847\) −4.00000 −0.137442
\(848\) −11.0000 −0.377742
\(849\) − 4.00000i − 0.137280i
\(850\) 32.0000 1.09759
\(851\) 32.0000i 1.09695i
\(852\) 2.00000i 0.0685189i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) − 2.00000i − 0.0683586i
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 3.00000i 0.102418i
\(859\) − 25.0000i − 0.852989i −0.904490 0.426494i \(-0.859748\pi\)
0.904490 0.426494i \(-0.140252\pi\)
\(860\) 11.0000i 0.375097i
\(861\) 4.00000 0.136320
\(862\) − 32.0000i − 1.08992i
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) 5.00000 0.170103
\(865\) −6.00000 −0.204006
\(866\) −16.0000 −0.543702
\(867\) − 47.0000i − 1.59620i
\(868\) 6.00000i 0.203653i
\(869\) 45.0000 1.52652
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) − 5.00000i − 0.169321i
\(873\) − 4.00000i − 0.135379i
\(874\) 0 0
\(875\) −18.0000 −0.608511
\(876\) 4.00000 0.135147
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 14.0000 0.472208
\(880\) − 3.00000i − 0.101130i
\(881\) − 42.0000i − 1.41502i −0.706705 0.707508i \(-0.749819\pi\)
0.706705 0.707508i \(-0.250181\pi\)
\(882\) 6.00000i 0.202031i
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 8.00000i 0.269069i
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) − 33.0000i − 1.10803i −0.832506 0.554016i \(-0.813095\pi\)
0.832506 0.554016i \(-0.186905\pi\)
\(888\) − 8.00000i − 0.268462i
\(889\) 16.0000i 0.536623i
\(890\) −10.0000 −0.335201
\(891\) 3.00000i 0.100504i
\(892\) 26.0000 0.870544
\(893\) 0 0
\(894\) −15.0000 −0.501675
\(895\) −10.0000 −0.334263
\(896\) − 2.00000i − 0.0668153i
\(897\) 4.00000i 0.133556i
\(898\) 10.0000 0.333704
\(899\) 0 0
\(900\) 8.00000 0.266667
\(901\) 88.0000i 2.93171i
\(902\) 6.00000i 0.199778i
\(903\) 22.0000 0.732114
\(904\) 6.00000 0.199557
\(905\) −7.00000 −0.232688
\(906\) −2.00000 −0.0664455
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) −18.0000 −0.597351
\(909\) 16.0000i 0.530687i
\(910\) − 2.00000i − 0.0662994i
\(911\) − 13.0000i − 0.430709i −0.976536 0.215355i \(-0.930909\pi\)
0.976536 0.215355i \(-0.0690907\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) − 2.00000i − 0.0661541i
\(915\) 8.00000 0.264472
\(916\) − 10.0000i − 0.330409i
\(917\) − 24.0000i − 0.792550i
\(918\) − 40.0000i − 1.32020i
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) − 4.00000i − 0.131876i
\(921\) −7.00000 −0.230658
\(922\) 2.00000 0.0658665
\(923\) −2.00000 −0.0658308
\(924\) −6.00000 −0.197386
\(925\) − 32.0000i − 1.05215i
\(926\) 4.00000i 0.131448i
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) − 3.00000i − 0.0983739i
\(931\) 0 0
\(932\) 1.00000 0.0327561
\(933\) −8.00000 −0.261908
\(934\) 27.0000 0.883467
\(935\) −24.0000 −0.784884
\(936\) 2.00000i 0.0653720i
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 9.00000i 0.293704i
\(940\) 13.0000i 0.424013i
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 18.0000i 0.586472i
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) 10.0000i 0.325300i
\(946\) 33.0000i 1.07292i
\(947\) − 33.0000i − 1.07236i −0.844105 0.536178i \(-0.819868\pi\)
0.844105 0.536178i \(-0.180132\pi\)
\(948\) −15.0000 −0.487177
\(949\) 4.00000i 0.129845i
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) −16.0000 −0.518563
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) 22.0000i 0.712276i
\(955\) − 8.00000i − 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) −5.00000 −0.161543
\(959\) − 24.0000i − 0.775000i
\(960\) 1.00000i 0.0322749i
\(961\) 22.0000 0.709677
\(962\) 8.00000 0.257930
\(963\) −4.00000 −0.128898
\(964\) 17.0000 0.547533
\(965\) 14.0000i 0.450676i
\(966\) −8.00000 −0.257396
\(967\) 13.0000i 0.418052i 0.977910 + 0.209026i \(0.0670293\pi\)
−0.977910 + 0.209026i \(0.932971\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 28.0000i 0.898563i 0.893390 + 0.449281i \(0.148320\pi\)
−0.893390 + 0.449281i \(0.851680\pi\)
\(972\) − 16.0000i − 0.513200i
\(973\) 40.0000 1.28234
\(974\) 22.0000i 0.704925i
\(975\) − 4.00000i − 0.128103i
\(976\) 8.00000i 0.256074i
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) 9.00000i 0.287788i
\(979\) −30.0000 −0.958804
\(980\) −3.00000 −0.0958315
\(981\) −10.0000 −0.319275
\(982\) −33.0000 −1.05307
\(983\) 49.0000i 1.56286i 0.623995 + 0.781429i \(0.285509\pi\)
−0.623995 + 0.781429i \(0.714491\pi\)
\(984\) − 2.00000i − 0.0637577i
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 26.0000 0.827589
\(988\) 0 0
\(989\) 44.0000i 1.39912i
\(990\) −6.00000 −0.190693
\(991\) −22.0000 −0.698853 −0.349427 0.936964i \(-0.613624\pi\)
−0.349427 + 0.936964i \(0.613624\pi\)
\(992\) 3.00000 0.0952501
\(993\) 23.0000 0.729883
\(994\) − 4.00000i − 0.126872i
\(995\) 10.0000 0.317021
\(996\) − 4.00000i − 0.126745i
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) − 20.0000i − 0.633089i
\(999\) −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 1682.2.b.a.1681.1 2
29.12 odd 4 58.2.a.b.1.1 1
29.17 odd 4 1682.2.a.d.1.1 1
29.28 even 2 inner 1682.2.b.a.1681.2 2
87.41 even 4 522.2.a.b.1.1 1
116.99 even 4 464.2.a.e.1.1 1
145.12 even 4 1450.2.b.b.349.2 2
145.99 odd 4 1450.2.a.c.1.1 1
145.128 even 4 1450.2.b.b.349.1 2
203.41 even 4 2842.2.a.e.1.1 1
232.99 even 4 1856.2.a.f.1.1 1
232.157 odd 4 1856.2.a.k.1.1 1
319.186 even 4 7018.2.a.a.1.1 1
348.215 odd 4 4176.2.a.n.1.1 1
377.12 odd 4 9802.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 29.12 odd 4
464.2.a.e.1.1 1 116.99 even 4
522.2.a.b.1.1 1 87.41 even 4
1450.2.a.c.1.1 1 145.99 odd 4
1450.2.b.b.349.1 2 145.128 even 4
1450.2.b.b.349.2 2 145.12 even 4
1682.2.a.d.1.1 1 29.17 odd 4
1682.2.b.a.1681.1 2 1.1 even 1 trivial
1682.2.b.a.1681.2 2 29.28 even 2 inner
1856.2.a.f.1.1 1 232.99 even 4
1856.2.a.k.1.1 1 232.157 odd 4
2842.2.a.e.1.1 1 203.41 even 4
4176.2.a.n.1.1 1 348.215 odd 4
7018.2.a.a.1.1 1 319.186 even 4
9802.2.a.a.1.1 1 377.12 odd 4