Properties

Label 1450.2.b.b.349.2
Level $1450$
Weight $2$
Character 1450.349
Analytic conductor $11.578$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5783082931\)
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 349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1450.349
Dual form 1450.2.b.b.349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} +2.00000i q^{18} +2.00000 q^{21} -3.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +5.00000i q^{27} +2.00000i q^{28} +1.00000 q^{29} -3.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -8.00000 q^{34} -2.00000 q^{36} +8.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} +2.00000i q^{42} +11.0000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +13.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -5.00000 q^{54} -2.00000 q^{56} +1.00000i q^{58} -8.00000 q^{61} -3.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -12.0000i q^{67} -8.00000i q^{68} +4.00000 q^{69} +2.00000 q^{71} -2.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} +6.00000i q^{77} -1.00000i q^{78} -15.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} -2.00000 q^{84} -11.0000 q^{86} +1.00000i q^{87} +3.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} +4.00000i q^{92} -3.00000i q^{93} -13.0000 q^{94} -1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 2.00000i 0.471405i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 3.00000i − 0.639602i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 5.00000i 0.962250i
\(28\) 2.00000i 0.377964i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) −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
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 11.0000i 1.67748i 0.544529 + 0.838742i \(0.316708\pi\)
−0.544529 + 0.838742i \(0.683292\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 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
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) − 1.00000i − 0.138675i
\(53\) 11.0000i 1.51097i 0.655168 + 0.755483i \(0.272598\pi\)
−0.655168 + 0.755483i \(0.727402\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 1.00000i 0.131306i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) − 8.00000i − 0.970143i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) − 2.00000i − 0.235702i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) − 1.00000i − 0.113228i
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 1.00000i 0.107211i
\(88\) 3.00000i 0.319801i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 4.00000i 0.417029i
\(93\) − 3.00000i − 0.311086i
\(94\) −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
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) − 8.00000i − 0.792118i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) − 2.00000i − 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) − 2.00000i − 0.188982i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 8.00000i − 0.724286i
\(123\) 2.00000i 0.180334i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −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 \(0.710612\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 16.0000i 1.29352i
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) − 15.0000i − 1.19334i
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000i 0.0785674i
\(163\) − 9.00000i − 0.704934i −0.935824 0.352467i \(-0.885343\pi\)
0.935824 0.352467i \(-0.114657\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 2.00000i − 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) − 11.0000i − 0.838742i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 2.00000i 0.148250i
\(183\) − 8.00000i − 0.591377i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) − 24.0000i − 1.75505i
\(188\) − 13.0000i − 0.948122i
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) − 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
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) − 8.00000i − 0.562878i
\(203\) − 2.00000i − 0.140372i
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) − 8.00000i − 0.556038i
\(208\) 1.00000i 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) − 11.0000i − 0.755483i
\(213\) 2.00000i 0.137038i
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 6.00000i 0.407307i
\(218\) − 5.00000i − 0.338643i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 8.00000i − 0.536925i
\(223\) 26.0000i 1.74109i 0.492090 + 0.870544i \(0.336233\pi\)
−0.492090 + 0.870544i \(0.663767\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) − 1.00000i − 0.0656532i
\(233\) 1.00000i 0.0655122i 0.999463 + 0.0327561i \(0.0104285\pi\)
−0.999463 + 0.0327561i \(0.989572\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) − 15.0000i − 0.974355i
\(238\) 16.0000i 1.03713i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 16.0000i 1.02640i
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 3.00000i 0.190500i
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 12.0000i 0.754434i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 13.0000i 0.810918i 0.914113 + 0.405459i \(0.132888\pi\)
−0.914113 + 0.405459i \(0.867112\pi\)
\(258\) − 11.0000i − 0.684830i
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 12.0000i 0.741362i
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 12.0000i 0.733017i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 8.00000i 0.485071i
\(273\) 2.00000i 0.121046i
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) − 13.0000i − 0.774139i
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −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
\(297\) − 15.0000i − 0.870388i
\(298\) − 15.0000i − 0.868927i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 22.0000 1.26806
\(302\) 2.00000i 0.115087i
\(303\) − 8.00000i − 0.459588i
\(304\) 0 0
\(305\) 0 0
\(306\) −16.0000 −0.914659
\(307\) − 7.00000i − 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) − 9.00000i − 0.508710i −0.967111 0.254355i \(-0.918137\pi\)
0.967111 0.254355i \(-0.0818632\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 15.0000 0.843816
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) − 11.0000i − 0.616849i
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) − 8.00000i − 0.445823i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 9.00000 0.498464
\(327\) − 5.00000i − 0.276501i
\(328\) − 2.00000i − 0.110432i
\(329\) 26.0000 1.43343
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 16.0000i 0.876795i
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 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
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) − 1.00000i − 0.0536056i
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) − 3.00000i − 0.159901i
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 16.0000i 0.846810i
\(358\) 10.0000i 0.528516i
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 7.00000i 0.367912i
\(363\) − 2.00000i − 0.104973i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) − 32.0000i − 1.67039i −0.549957 0.835193i \(-0.685356\pi\)
0.549957 0.835193i \(-0.314644\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 22.0000 1.14218
\(372\) 3.00000i 0.155543i
\(373\) 21.0000i 1.08734i 0.839299 + 0.543669i \(0.182965\pi\)
−0.839299 + 0.543669i \(0.817035\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 13.0000 0.670424
\(377\) 1.00000i 0.0515026i
\(378\) 10.0000i 0.514344i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 8.00000i − 0.409316i
\(383\) − 14.0000i − 0.715367i −0.933843 0.357683i \(-0.883567\pi\)
0.933843 0.357683i \(-0.116433\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 22.0000i 1.11832i
\(388\) 2.00000i 0.101535i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) − 3.00000i − 0.151523i
\(393\) 12.0000i 0.605320i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) − 17.0000i − 0.853206i −0.904439 0.426603i \(-0.859710\pi\)
0.904439 0.426603i \(-0.140290\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0 0
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 3.00000i − 0.149441i
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) − 24.0000i − 1.18964i
\(408\) 8.00000i 0.396059i
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 14.0000i 0.689730i
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) − 3.00000i − 0.146038i
\(423\) 26.0000i 1.26416i
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 16.0000i 0.774294i
\(428\) 2.00000i 0.0966736i
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 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 \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) −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 \(0.658375\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) − 8.00000i − 0.380521i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) − 15.0000i − 0.709476i
\(448\) 2.00000i 0.0944911i
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) − 6.00000i − 0.282216i
\(453\) 2.00000i 0.0939682i
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −40.0000 −1.86704
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) − 6.00000i − 0.279145i
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) − 27.0000i − 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) − 33.0000i − 1.51734i
\(474\) 15.0000 0.688973
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) 22.0000i 1.00731i
\(478\) 0 0
\(479\) 5.00000 0.228456 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 17.0000i 0.774329i
\(483\) − 8.00000i − 0.364013i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) − 22.0000i − 0.996915i −0.866914 0.498458i \(-0.833900\pi\)
0.866914 0.498458i \(-0.166100\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 9.00000 0.406994
\(490\) 0 0
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) − 4.00000i − 0.179425i
\(498\) 4.00000i 0.179244i
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 2.00000 0.0893534
\(502\) 27.0000i 1.20507i
\(503\) − 19.0000i − 0.847168i −0.905857 0.423584i \(-0.860772\pi\)
0.905857 0.423584i \(-0.139228\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 12.0000i 0.532939i
\(508\) − 8.00000i − 0.354943i
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −13.0000 −0.573405
\(515\) 0 0
\(516\) 11.0000 0.484248
\(517\) − 39.0000i − 1.71522i
\(518\) 16.0000i 0.703000i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 24.0000i − 1.04945i −0.851273 0.524723i \(-0.824169\pi\)
0.851273 0.524723i \(-0.175831\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) − 24.0000i − 1.04546i
\(528\) − 3.00000i − 0.130558i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000i 0.0866296i
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 10.0000i 0.431532i
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) − 13.0000i − 0.558398i
\(543\) 7.00000i 0.300399i
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 38.0000i 1.62476i 0.583127 + 0.812381i \(0.301829\pi\)
−0.583127 + 0.812381i \(0.698171\pi\)
\(548\) 12.0000i 0.512615i
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 0 0
\(552\) − 4.00000i − 0.170251i
\(553\) 30.0000i 1.27573i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) − 6.00000i − 0.254000i
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 27.0000i 1.13893i
\(563\) 11.0000i 0.463595i 0.972764 + 0.231797i \(0.0744606\pi\)
−0.972764 + 0.231797i \(0.925539\pi\)
\(564\) 13.0000 0.547399
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) − 2.00000i − 0.0839921i
\(568\) − 2.00000i − 0.0839181i
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 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
\(573\) − 8.00000i − 0.334205i
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) −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
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 8.00000i 0.328798i
\(593\) − 39.0000i − 1.60154i −0.598973 0.800769i \(-0.704424\pi\)
0.598973 0.800769i \(-0.295576\pi\)
\(594\) 15.0000 0.615457
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 10.0000i 0.409273i
\(598\) 4.00000i 0.163572i
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 22.0000i 0.896653i
\(603\) − 24.0000i − 0.977356i
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) 3.00000i 0.121766i 0.998145 + 0.0608831i \(0.0193917\pi\)
−0.998145 + 0.0608831i \(0.980608\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) −13.0000 −0.525924
\(612\) − 16.0000i − 0.646762i
\(613\) 31.0000i 1.25208i 0.779792 + 0.626039i \(0.215325\pi\)
−0.779792 + 0.626039i \(0.784675\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 14.0000i 0.563163i
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) − 8.00000i − 0.320771i
\(623\) − 20.0000i − 0.801283i
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 9.00000 0.359712
\(627\) 0 0
\(628\) − 18.0000i − 0.718278i
\(629\) −64.0000 −2.55185
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 15.0000i 0.596668i
\(633\) − 3.00000i − 0.119239i
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) 3.00000i 0.118864i
\(638\) − 3.00000i − 0.118771i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 2.00000i 0.0789337i
\(643\) − 34.0000i − 1.34083i −0.741987 0.670415i \(-0.766116\pi\)
0.741987 0.670415i \(-0.233884\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.0000i − 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 9.00000i 0.352467i
\(653\) 26.0000i 1.01746i 0.860927 + 0.508729i \(0.169885\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 5.00000 0.195515
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) − 8.00000i − 0.312110i
\(658\) 26.0000i 1.01359i
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) − 23.0000i − 0.893920i
\(663\) − 8.00000i − 0.310694i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) − 4.00000i − 0.154881i
\(668\) 2.00000i 0.0773823i
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 2.00000i 0.0771517i
\(673\) − 9.00000i − 0.346925i −0.984841 0.173462i \(-0.944505\pi\)
0.984841 0.173462i \(-0.0554955\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −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
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 9.00000i 0.344628i
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) − 10.0000i − 0.381524i
\(688\) 11.0000i 0.419371i
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 12.0000i 0.455842i
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 16.0000i 0.606043i
\(698\) 15.0000i 0.567758i
\(699\) −1.00000 −0.0378235
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) − 5.00000i − 0.188713i
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 16.0000i 0.601742i
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −30.0000 −1.12509
\(712\) − 10.0000i − 0.374766i
\(713\) 12.0000i 0.449404i
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 25.0000i 0.932992i
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −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 \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −88.0000 −3.25480
\(732\) 8.00000i 0.295689i
\(733\) − 24.0000i − 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 36.0000i 1.32608i
\(738\) 4.00000i 0.147242i
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 22.0000i 0.807645i
\(743\) − 44.0000i − 1.61420i −0.590412 0.807102i \(-0.701035\pi\)
0.590412 0.807102i \(-0.298965\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −21.0000 −0.768865
\(747\) − 8.00000i − 0.292705i
\(748\) 24.0000i 0.877527i
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 13.0000i 0.474061i
\(753\) 27.0000i 0.983935i
\(754\) −1.00000 −0.0364179
\(755\) 0 0
\(756\) −10.0000 −0.363696
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −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
\(763\) 10.0000i 0.362024i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) −13.0000 −0.468184
\(772\) 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
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 16.0000i 0.573997i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 32.0000i 1.14432i
\(783\) 5.00000i 0.178685i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 6.00000i 0.213201i
\(793\) − 8.00000i − 0.284088i
\(794\) 17.0000 0.603307
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) − 32.0000i − 1.13350i −0.823890 0.566749i \(-0.808201\pi\)
0.823890 0.566749i \(-0.191799\pi\)
\(798\) 0 0
\(799\) −104.000 −3.67926
\(800\) 0 0
\(801\) 20.0000 0.706665
\(802\) 27.0000i 0.953403i
\(803\) 12.0000i 0.423471i
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) 0 0
\(808\) 8.00000i 0.281439i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) − 13.0000i − 0.455930i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) − 30.0000i − 1.04893i
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 12.0000i 0.418548i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 0 0
\(827\) 13.0000i 0.452054i 0.974121 + 0.226027i \(0.0725738\pi\)
−0.974121 + 0.226027i \(0.927426\pi\)
\(828\) 8.00000i 0.278019i
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) − 1.00000i − 0.0346688i
\(833\) 24.0000i 0.831551i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) − 15.0000i − 0.518476i
\(838\) 10.0000i 0.345444i
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 32.0000i 1.10279i
\(843\) 27.0000i 0.929929i
\(844\) 3.00000 0.103264
\(845\) 0 0
\(846\) −26.0000 −0.893898
\(847\) 4.00000i 0.137442i
\(848\) 11.0000i 0.377742i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) − 2.00000i − 0.0685189i
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) − 27.0000i − 0.922302i −0.887322 0.461151i \(-0.847437\pi\)
0.887322 0.461151i \(-0.152563\pi\)
\(858\) 3.00000i 0.102418i
\(859\) 25.0000 0.852989 0.426494 0.904490i \(-0.359748\pi\)
0.426494 + 0.904490i \(0.359748\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 32.0000i 1.08992i
\(863\) 46.0000i 1.56586i 0.622111 + 0.782929i \(0.286275\pi\)
−0.622111 + 0.782929i \(0.713725\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −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
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 6.00000i 0.202031i
\(883\) 26.0000i 0.874970i 0.899226 + 0.437485i \(0.144131\pi\)
−0.899226 + 0.437485i \(0.855869\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 33.0000i 1.10803i 0.832506 + 0.554016i \(0.186905\pi\)
−0.832506 + 0.554016i \(0.813095\pi\)
\(888\) 8.00000i 0.268462i
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) − 26.0000i − 0.870544i
\(893\) 0 0
\(894\) 15.0000 0.501675
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 4.00000i 0.133556i
\(898\) 10.0000i 0.333704i
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −88.0000 −2.93171
\(902\) − 6.00000i − 0.199778i
\(903\) 22.0000i 0.732114i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) − 18.0000i − 0.597351i
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) − 24.0000i − 0.792550i
\(918\) − 40.0000i − 1.32020i
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 2.00000i 0.0658665i
\(923\) 2.00000i 0.0658308i
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 28.0000i − 0.919641i
\(928\) 1.00000i 0.0328266i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 1.00000i − 0.0327561i
\(933\) − 8.00000i − 0.261908i
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) − 24.0000i − 0.783628i
\(939\) 9.00000 0.293704
\(940\) 0 0
\(941\) 37.0000 1.20617 0.603083 0.797679i \(-0.293939\pi\)
0.603083 + 0.797679i \(0.293939\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) − 8.00000i − 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 33.0000 1.07292
\(947\) 33.0000i 1.07236i 0.844105 + 0.536178i \(0.180132\pi\)
−0.844105 + 0.536178i \(0.819868\pi\)
\(948\) 15.0000i 0.487177i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) − 16.0000i − 0.518563i
\(953\) 1.00000i 0.0323932i 0.999869 + 0.0161966i \(0.00515576\pi\)
−0.999869 + 0.0161966i \(0.994844\pi\)
\(954\) −22.0000 −0.712276
\(955\) 0 0
\(956\) 0 0
\(957\) − 3.00000i − 0.0969762i
\(958\) 5.00000i 0.161543i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 8.00000i − 0.257930i
\(963\) − 4.00000i − 0.128898i
\(964\) −17.0000 −0.547533
\(965\) 0 0
\(966\) 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
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) − 16.0000i − 0.513200i
\(973\) − 40.0000i − 1.28234i
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 13.0000i 0.415907i 0.978139 + 0.207953i \(0.0666802\pi\)
−0.978139 + 0.207953i \(0.933320\pi\)
\(978\) 9.00000i 0.287788i
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 33.0000i − 1.05307i
\(983\) − 49.0000i − 1.56286i −0.623995 0.781429i \(-0.714491\pi\)
0.623995 0.781429i \(-0.285509\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 26.0000i 0.827589i
\(988\) 0 0
\(989\) 44.0000 1.39912
\(990\) 0 0
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) − 3.00000i − 0.0952501i
\(993\) − 23.0000i − 0.729883i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) 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 1450.2.b.b.349.2 2
5.2 odd 4 1450.2.a.c.1.1 1
5.3 odd 4 58.2.a.b.1.1 1
5.4 even 2 inner 1450.2.b.b.349.1 2
15.8 even 4 522.2.a.b.1.1 1
20.3 even 4 464.2.a.e.1.1 1
35.13 even 4 2842.2.a.e.1.1 1
40.3 even 4 1856.2.a.f.1.1 1
40.13 odd 4 1856.2.a.k.1.1 1
55.43 even 4 7018.2.a.a.1.1 1
60.23 odd 4 4176.2.a.n.1.1 1
65.38 odd 4 9802.2.a.a.1.1 1
145.28 odd 4 1682.2.a.d.1.1 1
145.128 even 4 1682.2.b.a.1681.2 2
145.133 even 4 1682.2.b.a.1681.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 5.3 odd 4
464.2.a.e.1.1 1 20.3 even 4
522.2.a.b.1.1 1 15.8 even 4
1450.2.a.c.1.1 1 5.2 odd 4
1450.2.b.b.349.1 2 5.4 even 2 inner
1450.2.b.b.349.2 2 1.1 even 1 trivial
1682.2.a.d.1.1 1 145.28 odd 4
1682.2.b.a.1681.1 2 145.133 even 4
1682.2.b.a.1681.2 2 145.128 even 4
1856.2.a.f.1.1 1 40.3 even 4
1856.2.a.k.1.1 1 40.13 odd 4
2842.2.a.e.1.1 1 35.13 even 4
4176.2.a.n.1.1 1 60.23 odd 4
7018.2.a.a.1.1 1 55.43 even 4
9802.2.a.a.1.1 1 65.38 odd 4