Properties

Label 950.2.b.b.799.1
Level 950
Weight 2
Character 950.799
Analytic conductor 7.586
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\)
Character \(\chi\) = 950.799
Dual form 950.2.b.b.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -5.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} -1.00000 q^{19} -1.00000 q^{21} +6.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} -5.00000 q^{26} -5.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -5.00000 q^{39} +1.00000i q^{42} -8.00000i q^{43} +6.00000 q^{44} -3.00000 q^{46} -1.00000i q^{48} +6.00000 q^{49} +3.00000 q^{51} +5.00000i q^{52} +3.00000i q^{53} -5.00000 q^{54} +1.00000 q^{56} +1.00000i q^{57} +9.00000i q^{58} -9.00000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +5.00000i q^{67} -3.00000i q^{68} -3.00000 q^{69} -6.00000 q^{71} +2.00000i q^{72} +7.00000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{77} +5.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +1.00000 q^{84} -8.00000 q^{86} +9.00000i q^{87} -6.00000i q^{88} +12.0000 q^{89} -5.00000 q^{91} +3.00000i q^{92} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -6.00000i q^{98} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} - 12q^{11} - 2q^{14} + 2q^{16} - 2q^{19} - 2q^{21} + 2q^{24} - 10q^{26} - 18q^{29} - 8q^{31} + 6q^{34} - 4q^{36} - 10q^{39} + 12q^{44} - 6q^{46} + 12q^{49} + 6q^{51} - 10q^{54} + 2q^{56} - 18q^{59} - 20q^{61} - 2q^{64} + 12q^{66} - 6q^{69} - 12q^{71} + 4q^{74} + 2q^{76} + 20q^{79} + 2q^{81} + 2q^{84} - 16q^{86} + 24q^{89} - 10q^{91} - 2q^{96} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) −1.00000 −0.267261
\(15\) 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
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 6.00000i 1.27920i
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) − 5.00000i − 0.962250i
\(28\) 1.00000i 0.188982i
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 6.00000i 1.04447i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000i 0.154303i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 5.00000i 0.693375i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000i 0.132453i
\(58\) 9.00000i 1.18176i
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000i 0.508001i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 5.00000i 0.610847i 0.952217 + 0.305424i \(0.0987981\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 2.00000i 0.235702i
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 6.00000i 0.683763i
\(78\) 5.00000i 0.566139i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 9.00000i 0.964901i
\(88\) − 6.00000i − 0.639602i
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 3.00000i 0.312772i
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 5.00000i 0.481125i
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) − 10.0000i − 0.924500i
\(118\) 9.00000i 0.828517i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 10.0000i 0.905357i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) 1.00000i 0.0867110i
\(134\) 5.00000 0.431934
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) − 9.00000i − 0.768922i −0.923141 0.384461i \(-0.874387\pi\)
0.923141 0.384461i \(-0.125613\pi\)
\(138\) 3.00000i 0.255377i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000i 0.503509i
\(143\) 30.0000i 2.50873i
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) − 6.00000i − 0.494872i
\(148\) − 2.00000i − 0.164399i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 6.00000i 0.485071i
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) − 1.00000i − 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 8.00000i 0.609994i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 9.00000i 0.676481i
\(178\) − 12.0000i − 0.899438i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 5.00000i 0.370625i
\(183\) 10.0000i 0.739221i
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) − 18.0000i − 1.31629i
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 12.0000i 0.852803i
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) − 18.0000i − 1.26648i
\(203\) 9.00000i 0.631676i
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) − 6.00000i − 0.417029i
\(208\) − 5.00000i − 0.346688i
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) − 3.00000i − 0.206041i
\(213\) 6.00000i 0.411113i
\(214\) −9.00000 −0.615227
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 4.00000i 0.271538i
\(218\) 11.0000i 0.745014i
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 15.0000 1.00901
\(222\) − 2.00000i − 0.134231i
\(223\) − 26.0000i − 1.74109i −0.492090 0.870544i \(-0.663767\pi\)
0.492090 0.870544i \(-0.336233\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 15.0000i − 0.995585i −0.867296 0.497792i \(-0.834144\pi\)
0.867296 0.497792i \(-0.165856\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) − 9.00000i − 0.590879i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −10.0000 −0.653720
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) − 10.0000i − 0.649570i
\(238\) − 3.00000i − 0.194461i
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) − 16.0000i − 1.02640i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000i 0.318142i
\(248\) − 4.00000i − 0.254000i
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 18.0000i 1.13165i
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 8.00000i 0.498058i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 1.00000 0.0613139
\(267\) − 12.0000i − 0.734388i
\(268\) − 5.00000i − 0.305424i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 5.00000i 0.302614i
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 8.00000i 0.480673i 0.970690 + 0.240337i \(0.0772579\pi\)
−0.970690 + 0.240337i \(0.922742\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 30.0000 1.77394
\(287\) 0 0
\(288\) − 2.00000i − 0.117851i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 7.00000i − 0.409644i
\(293\) 21.0000i 1.22683i 0.789760 + 0.613417i \(0.210205\pi\)
−0.789760 + 0.613417i \(0.789795\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 30.0000i 1.74078i
\(298\) 0 0
\(299\) −15.0000 −0.867472
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 10.0000i 0.575435i
\(303\) − 18.0000i − 1.03407i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) − 5.00000i − 0.283069i
\(313\) 19.0000i 1.07394i 0.843600 + 0.536972i \(0.180432\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) − 9.00000i − 0.505490i −0.967533 0.252745i \(-0.918667\pi\)
0.967533 0.252745i \(-0.0813334\pi\)
\(318\) − 3.00000i − 0.168232i
\(319\) 54.0000 3.02342
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 3.00000i 0.167183i
\(323\) − 3.00000i − 0.166924i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 11.0000i 0.608301i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) 4.00000i 0.219199i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 4.00000i − 0.217894i −0.994048 0.108947i \(-0.965252\pi\)
0.994048 0.108947i \(-0.0347479\pi\)
\(338\) 12.0000i 0.652714i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 2.00000i 0.108148i
\(343\) − 13.0000i − 0.701934i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) − 9.00000i − 0.482451i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −25.0000 −1.33440
\(352\) 6.00000i 0.319801i
\(353\) 15.0000i 0.798369i 0.916871 + 0.399185i \(0.130707\pi\)
−0.916871 + 0.399185i \(0.869293\pi\)
\(354\) 9.00000 0.478345
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) − 3.00000i − 0.158777i
\(358\) 0 0
\(359\) −21.0000 −1.10834 −0.554169 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 2.00000i − 0.105118i
\(363\) − 25.0000i − 1.31216i
\(364\) 5.00000 0.262071
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) − 3.00000i − 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) − 4.00000i − 0.207390i
\(373\) − 23.0000i − 1.19089i −0.803394 0.595447i \(-0.796975\pi\)
0.803394 0.595447i \(-0.203025\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) 45.0000i 2.31762i
\(378\) 5.00000i 0.257172i
\(379\) 7.00000 0.359566 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) − 3.00000i − 0.153493i
\(383\) − 18.0000i − 0.919757i −0.887982 0.459879i \(-0.847893\pi\)
0.887982 0.459879i \(-0.152107\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) − 16.0000i − 0.813326i
\(388\) 10.0000i 0.507673i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 6.00000i 0.303046i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 12.0000 0.603023
\(397\) 20.0000i 1.00377i 0.864934 + 0.501886i \(0.167360\pi\)
−0.864934 + 0.501886i \(0.832640\pi\)
\(398\) 11.0000i 0.551380i
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) − 5.00000i − 0.249377i
\(403\) 20.0000i 0.996271i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) − 12.0000i − 0.594818i
\(408\) 3.00000i 0.148522i
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 14.0000i 0.689730i
\(413\) 9.00000i 0.442861i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) − 4.00000i − 0.195881i
\(418\) − 6.00000i − 0.293470i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) − 5.00000i − 0.243396i
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 10.0000i 0.483934i
\(428\) 9.00000i 0.435031i
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) − 5.00000i − 0.240563i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 3.00000i 0.143509i
\(438\) − 7.00000i − 0.334473i
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) − 15.0000i − 0.713477i
\(443\) 18.0000i 0.855206i 0.903967 + 0.427603i \(0.140642\pi\)
−0.903967 + 0.427603i \(0.859358\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000i 0.282216i
\(453\) 10.0000i 0.469841i
\(454\) −15.0000 −0.703985
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) − 6.00000i − 0.279145i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 10.0000i 0.462250i
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) − 9.00000i − 0.414259i
\(473\) 48.0000i 2.20704i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 6.00000i 0.274721i
\(478\) − 21.0000i − 0.960518i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) − 8.00000i − 0.364390i
\(483\) 3.00000i 0.136505i
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) − 27.0000i − 1.21602i
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000i 0.269137i
\(498\) − 6.00000i − 0.268866i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) − 6.00000i − 0.267793i
\(503\) 21.0000i 0.936344i 0.883637 + 0.468172i \(0.155087\pi\)
−0.883637 + 0.468172i \(0.844913\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) 12.0000i 0.532939i
\(508\) − 2.00000i − 0.0887357i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) − 1.00000i − 0.0441942i
\(513\) 5.00000i 0.220755i
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) − 2.00000i − 0.0878750i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 18.0000i 0.787839i
\(523\) − 11.0000i − 0.480996i −0.970650 0.240498i \(-0.922689\pi\)
0.970650 0.240498i \(-0.0773108\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) − 12.0000i − 0.522728i
\(528\) 6.00000i 0.261116i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) − 1.00000i − 0.0433555i
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) 0 0
\(538\) − 6.00000i − 0.258678i
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) − 11.0000i − 0.472490i
\(543\) − 2.00000i − 0.0858282i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 5.00000 0.213980
\(547\) 44.0000i 1.88130i 0.339372 + 0.940652i \(0.389785\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) 9.00000i 0.384461i
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) − 3.00000i − 0.127688i
\(553\) − 10.0000i − 0.425243i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 24.0000i 1.01691i 0.861088 + 0.508456i \(0.169784\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(558\) 8.00000i 0.338667i
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) − 1.00000i − 0.0419961i
\(568\) − 6.00000i − 0.251754i
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 30.0000i − 1.25436i
\(573\) − 3.00000i − 0.125327i
\(574\) 0 0
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) 11.0000i 0.457936i 0.973434 + 0.228968i \(0.0735351\pi\)
−0.973434 + 0.228968i \(0.926465\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 10.0000i 0.414513i
\(583\) − 18.0000i − 0.745484i
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 30.0000 1.23091
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0000i 0.450200i
\(598\) 15.0000i 0.613396i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 10.0000i 0.407231i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) − 22.0000i − 0.892952i −0.894795 0.446476i \(-0.852679\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.00000i − 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 14.0000i 0.563163i
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 21.0000i 0.842023i
\(623\) − 12.0000i − 0.480770i
\(624\) −5.00000 −0.200160
\(625\) 0 0
\(626\) 19.0000 0.759393
\(627\) − 6.00000i − 0.239617i
\(628\) 22.0000i 0.877896i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 10.0000i 0.397779i
\(633\) − 5.00000i − 0.198732i
\(634\) −9.00000 −0.357436
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) − 30.0000i − 1.18864i
\(638\) − 54.0000i − 2.13788i
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 9.00000i 0.355202i
\(643\) 22.0000i 0.867595i 0.901010 + 0.433798i \(0.142827\pi\)
−0.901010 + 0.433798i \(0.857173\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) 27.0000i 1.06148i 0.847535 + 0.530740i \(0.178086\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 54.0000 2.11969
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 20.0000i 0.783260i
\(653\) − 24.0000i − 0.939193i −0.882881 0.469596i \(-0.844399\pi\)
0.882881 0.469596i \(-0.155601\pi\)
\(654\) 11.0000 0.430134
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 45.0000 1.75295 0.876476 0.481446i \(-0.159888\pi\)
0.876476 + 0.481446i \(0.159888\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 1.00000i 0.0388661i
\(663\) − 15.0000i − 0.582552i
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 27.0000i 1.04544i
\(668\) − 12.0000i − 0.464294i
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 60.0000 2.31627
\(672\) 1.00000i 0.0385758i
\(673\) − 44.0000i − 1.69608i −0.529936 0.848038i \(-0.677784\pi\)
0.529936 0.848038i \(-0.322216\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) − 33.0000i − 1.26829i −0.773213 0.634147i \(-0.781352\pi\)
0.773213 0.634147i \(-0.218648\pi\)
\(678\) 6.00000i 0.230429i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) − 24.0000i − 0.919007i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) − 22.0000i − 0.839352i
\(688\) − 8.00000i − 0.304997i
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 12.0000i 0.455842i
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 0 0
\(698\) − 10.0000i − 0.378506i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 25.0000i 0.943564i
\(703\) − 2.00000i − 0.0754314i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) − 18.0000i − 0.676960i
\(708\) − 9.00000i − 0.338241i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 12.0000i 0.449719i
\(713\) 12.0000i 0.449404i
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 0 0
\(717\) − 21.0000i − 0.784259i
\(718\) 21.0000i 0.783713i
\(719\) −39.0000 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) − 1.00000i − 0.0372161i
\(723\) − 8.00000i − 0.297523i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) − 37.0000i − 1.37225i −0.727482 0.686127i \(-0.759309\pi\)
0.727482 0.686127i \(-0.240691\pi\)
\(728\) − 5.00000i − 0.185312i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) − 10.0000i − 0.369611i
\(733\) − 32.0000i − 1.18195i −0.806691 0.590973i \(-0.798744\pi\)
0.806691 0.590973i \(-0.201256\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) − 30.0000i − 1.10506i
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 5.00000 0.183680
\(742\) − 3.00000i − 0.110133i
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) 12.0000i 0.439057i
\(748\) 18.0000i 0.658145i
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) − 6.00000i − 0.218652i
\(754\) 45.0000 1.63880
\(755\) 0 0
\(756\) 5.00000 0.181848
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 7.00000i − 0.254251i
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) − 2.00000i − 0.0724524i
\(763\) 11.0000i 0.398227i
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 45.0000i 1.62486i
\(768\) − 1.00000i − 0.0360844i
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 14.0000i 0.503871i
\(773\) − 51.0000i − 1.83434i −0.398493 0.917171i \(-0.630467\pi\)
0.398493 0.917171i \(-0.369533\pi\)
\(774\) −16.0000 −0.575108
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) − 2.00000i − 0.0717496i
\(778\) 18.0000i 0.645331i
\(779\) 0 0
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) − 9.00000i − 0.321839i
\(783\) 45.0000i 1.60817i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) − 31.0000i − 1.10503i −0.833503 0.552515i \(-0.813668\pi\)
0.833503 0.552515i \(-0.186332\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 12.0000i − 0.426401i
\(793\) 50.0000i 1.77555i
\(794\) 20.0000 0.709773
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) − 39.0000i − 1.38145i −0.723117 0.690725i \(-0.757291\pi\)
0.723117 0.690725i \(-0.242709\pi\)
\(798\) − 1.00000i − 0.0353996i
\(799\) 0 0
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) − 42.0000i − 1.48215i
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) − 6.00000i − 0.211210i
\(808\) 18.0000i 0.633238i
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) − 9.00000i − 0.315838i
\(813\) − 11.0000i − 0.385787i
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 8.00000i 0.279885i
\(818\) 32.0000i 1.11885i
\(819\) −10.0000 −0.349428
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 9.00000i 0.313911i
\(823\) − 41.0000i − 1.42917i −0.699549 0.714585i \(-0.746616\pi\)
0.699549 0.714585i \(-0.253384\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 9.00000 0.313150
\(827\) 33.0000i 1.14752i 0.819023 + 0.573761i \(0.194516\pi\)
−0.819023 + 0.573761i \(0.805484\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 5.00000i 0.173344i
\(833\) 18.0000i 0.623663i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 20.0000i 0.691301i
\(838\) − 12.0000i − 0.414533i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) − 17.0000i − 0.585859i
\(843\) 0 0
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.0000i − 0.859010i
\(848\) 3.00000i 0.103020i
\(849\) 22.0000 0.755038
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) − 6.00000i − 0.205557i
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) − 30.0000i − 1.02418i
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 6.00000i − 0.204361i
\(863\) 18.0000i 0.612727i 0.951915 + 0.306364i \(0.0991123\pi\)
−0.951915 + 0.306364i \(0.900888\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) − 8.00000i − 0.271694i
\(868\) − 4.00000i − 0.135769i
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 25.0000 0.847093
\(872\) − 11.0000i − 0.372507i
\(873\) − 20.0000i − 0.676897i
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) −7.00000 −0.236508
\(877\) 23.0000i 0.776655i 0.921521 + 0.388327i \(0.126947\pi\)
−0.921521 + 0.388327i \(0.873053\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) − 12.0000i − 0.404061i
\(883\) 34.0000i 1.14419i 0.820187 + 0.572096i \(0.193869\pi\)
−0.820187 + 0.572096i \(0.806131\pi\)
\(884\) −15.0000 −0.504505
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) − 42.0000i − 1.41022i −0.709097 0.705111i \(-0.750897\pi\)
0.709097 0.705111i \(-0.249103\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 26.0000i 0.870544i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 15.0000i 0.500835i
\(898\) − 18.0000i − 0.600668i
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) − 37.0000i − 1.22856i −0.789086 0.614282i \(-0.789446\pi\)
0.789086 0.614282i \(-0.210554\pi\)
\(908\) 15.0000i 0.497792i
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) − 36.0000i − 1.19143i
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) − 15.0000i − 0.495074i
\(919\) 7.00000 0.230909 0.115454 0.993313i \(-0.463168\pi\)
0.115454 + 0.993313i \(0.463168\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 12.0000i 0.395199i
\(923\) 30.0000i 0.987462i
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 28.0000i − 0.919641i
\(928\) 9.00000i 0.295439i
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) − 6.00000i − 0.196537i
\(933\) 21.0000i 0.687509i
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) − 7.00000i − 0.228680i −0.993442 0.114340i \(-0.963525\pi\)
0.993442 0.114340i \(-0.0364753\pi\)
\(938\) − 5.00000i − 0.163256i
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 22.0000i 0.716799i
\(943\) 0 0
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) − 48.0000i − 1.55979i −0.625910 0.779895i \(-0.715272\pi\)
0.625910 0.779895i \(-0.284728\pi\)
\(948\) 10.0000i 0.324785i
\(949\) 35.0000 1.13615
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 3.00000i 0.0972306i
\(953\) − 30.0000i − 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −21.0000 −0.679189
\(957\) − 54.0000i − 1.74557i
\(958\) 36.0000i 1.16311i
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 10.0000i − 0.322413i
\(963\) − 18.0000i − 0.580042i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 25.0000i 0.803530i
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 16.0000i 0.513200i
\(973\) − 4.00000i − 0.128234i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 20.0000i 0.639529i
\(979\) −72.0000 −2.30113
\(980\) 0 0
\(981\) −22.0000 −0.702406
\(982\) 36.0000i 1.14881i
\(983\) 30.0000i 0.956851i 0.878128 + 0.478426i \(0.158792\pi\)
−0.878128 + 0.478426i \(0.841208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −27.0000 −0.859855
\(987\) 0 0
\(988\) − 5.00000i − 0.159071i
\(989\) −24.0000 −0.763156
\(990\) 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
\(993\) 1.00000i 0.0317340i
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 10.0000 0.316386
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 950.2.b.b.799.1 2
5.2 odd 4 950.2.a.d.1.1 1
5.3 odd 4 38.2.a.a.1.1 1
5.4 even 2 inner 950.2.b.b.799.2 2
15.2 even 4 8550.2.a.m.1.1 1
15.8 even 4 342.2.a.e.1.1 1
20.3 even 4 304.2.a.c.1.1 1
20.7 even 4 7600.2.a.n.1.1 1
35.13 even 4 1862.2.a.b.1.1 1
40.3 even 4 1216.2.a.m.1.1 1
40.13 odd 4 1216.2.a.e.1.1 1
55.43 even 4 4598.2.a.p.1.1 1
60.23 odd 4 2736.2.a.n.1.1 1
65.38 odd 4 6422.2.a.h.1.1 1
95.3 even 36 722.2.e.e.389.1 6
95.8 even 12 722.2.c.c.653.1 2
95.13 even 36 722.2.e.e.245.1 6
95.18 even 4 722.2.a.e.1.1 1
95.23 odd 36 722.2.e.f.415.1 6
95.28 odd 36 722.2.e.f.423.1 6
95.33 even 36 722.2.e.e.595.1 6
95.43 odd 36 722.2.e.f.595.1 6
95.48 even 36 722.2.e.e.423.1 6
95.53 even 36 722.2.e.e.415.1 6
95.63 odd 36 722.2.e.f.245.1 6
95.68 odd 12 722.2.c.e.653.1 2
95.73 odd 36 722.2.e.f.389.1 6
95.78 even 36 722.2.e.e.99.1 6
95.83 odd 12 722.2.c.e.429.1 2
95.88 even 12 722.2.c.c.429.1 2
95.93 odd 36 722.2.e.f.99.1 6
285.113 odd 4 6498.2.a.f.1.1 1
380.303 odd 4 5776.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 1 5.3 odd 4
304.2.a.c.1.1 1 20.3 even 4
342.2.a.e.1.1 1 15.8 even 4
722.2.a.e.1.1 1 95.18 even 4
722.2.c.c.429.1 2 95.88 even 12
722.2.c.c.653.1 2 95.8 even 12
722.2.c.e.429.1 2 95.83 odd 12
722.2.c.e.653.1 2 95.68 odd 12
722.2.e.e.99.1 6 95.78 even 36
722.2.e.e.245.1 6 95.13 even 36
722.2.e.e.389.1 6 95.3 even 36
722.2.e.e.415.1 6 95.53 even 36
722.2.e.e.423.1 6 95.48 even 36
722.2.e.e.595.1 6 95.33 even 36
722.2.e.f.99.1 6 95.93 odd 36
722.2.e.f.245.1 6 95.63 odd 36
722.2.e.f.389.1 6 95.73 odd 36
722.2.e.f.415.1 6 95.23 odd 36
722.2.e.f.423.1 6 95.28 odd 36
722.2.e.f.595.1 6 95.43 odd 36
950.2.a.d.1.1 1 5.2 odd 4
950.2.b.b.799.1 2 1.1 even 1 trivial
950.2.b.b.799.2 2 5.4 even 2 inner
1216.2.a.e.1.1 1 40.13 odd 4
1216.2.a.m.1.1 1 40.3 even 4
1862.2.a.b.1.1 1 35.13 even 4
2736.2.a.n.1.1 1 60.23 odd 4
4598.2.a.p.1.1 1 55.43 even 4
5776.2.a.m.1.1 1 380.303 odd 4
6422.2.a.h.1.1 1 65.38 odd 4
6498.2.a.f.1.1 1 285.113 odd 4
7600.2.a.n.1.1 1 20.7 even 4
8550.2.a.m.1.1 1 15.2 even 4