Properties

Label 1050.2.g.i.799.1
Level 1050
Weight 2
Character 1050.799
Analytic conductor 8.384
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\)
Character \(\chi\) = 1050.799
Dual form 1050.2.g.i.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} +1.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +5.00000 q^{29} +7.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} -1.00000i q^{42} -11.0000i q^{43} -2.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} +1.00000i q^{52} -1.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} -5.00000i q^{58} +5.00000 q^{59} -3.00000 q^{61} -7.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} +3.00000i q^{68} +1.00000 q^{69} +12.0000 q^{71} -1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} -2.00000i q^{77} -1.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} -7.00000i q^{82} -11.0000i q^{83} -1.00000 q^{84} -11.0000 q^{86} +5.00000i q^{87} +2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} +1.00000i q^{92} +7.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} - 2q^{14} + 2q^{16} + 2q^{21} - 2q^{24} - 2q^{26} + 10q^{29} + 14q^{31} - 6q^{34} + 2q^{36} + 2q^{39} + 14q^{41} - 4q^{44} - 2q^{46} - 2q^{49} + 6q^{51} - 2q^{54} + 2q^{56} + 10q^{59} - 6q^{61} - 2q^{64} + 4q^{66} + 2q^{69} + 24q^{71} + 4q^{74} - 20q^{79} + 2q^{81} - 2q^{84} - 22q^{86} + 20q^{89} - 2q^{91} - 16q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) − 2.00000i − 0.426401i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 1.00000i − 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) − 11.0000i − 1.67748i −0.544529 0.838742i \(-0.683292\pi\)
0.544529 0.838742i \(-0.316708\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 1.00000i 0.138675i
\(53\) − 1.00000i − 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) − 5.00000i − 0.656532i
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) − 7.00000i − 0.889001i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) − 1.00000i − 0.113228i
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 7.00000i − 0.773021i
\(83\) − 11.0000i − 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 5.00000i 0.536056i
\(88\) 2.00000i 0.213201i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 1.00000i 0.104257i
\(93\) 7.00000i 0.725866i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) − 11.0000i − 1.08386i −0.840423 0.541931i \(-0.817693\pi\)
0.840423 0.541931i \(-0.182307\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 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 \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) 1.00000i 0.0924500i
\(118\) − 5.00000i − 0.460287i
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 3.00000i 0.271607i
\(123\) 7.00000i 0.631169i
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 22.0000i 1.95218i 0.217357 + 0.976092i \(0.430256\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 1.00000i − 0.0851257i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) − 12.0000i − 1.00702i
\(143\) − 2.00000i − 0.167248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) − 1.00000i − 0.0824786i
\(148\) − 2.00000i − 0.164399i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 3.00000i 0.242536i
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) − 1.00000i − 0.0785674i
\(163\) − 1.00000i − 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) −7.00000 −0.546608
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 11.0000i 0.838742i
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 5.00000i 0.375823i
\(178\) − 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
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) − 3.00000i − 0.221766i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 7.00000 0.513265
\(187\) − 6.00000i − 0.438763i
\(188\) 8.00000i 0.583460i
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 17.0000i 1.21120i 0.795769 + 0.605600i \(0.207067\pi\)
−0.795769 + 0.605600i \(0.792933\pi\)
\(198\) 2.00000i 0.142134i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) − 12.0000i − 0.844317i
\(203\) − 5.00000i − 0.350931i
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) 1.00000i 0.0695048i
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 1.00000i 0.0686803i
\(213\) 12.0000i 0.822226i
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 7.00000i − 0.475191i
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 2.00000i 0.134231i
\(223\) 9.00000i 0.602685i 0.953516 + 0.301342i \(0.0974347\pi\)
−0.953516 + 0.301342i \(0.902565\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 17.0000i 1.12833i 0.825662 + 0.564165i \(0.190802\pi\)
−0.825662 + 0.564165i \(0.809198\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 5.00000i 0.328266i
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −5.00000 −0.325472
\(237\) − 10.0000i − 0.649570i
\(238\) 3.00000i 0.194461i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) 3.00000 0.192055
\(245\) 0 0
\(246\) 7.00000 0.446304
\(247\) 0 0
\(248\) 7.00000i 0.444500i
\(249\) 11.0000 0.697097
\(250\) 0 0
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 2.00000i − 0.125739i
\(254\) 22.0000 1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 23.0000i − 1.43470i −0.696713 0.717350i \(-0.745355\pi\)
0.696713 0.717350i \(-0.254645\pi\)
\(258\) − 11.0000i − 0.684830i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 8.00000i 0.494242i
\(263\) − 21.0000i − 1.29492i −0.762101 0.647458i \(-0.775832\pi\)
0.762101 0.647458i \(-0.224168\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) − 12.0000i − 0.733017i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) − 1.00000i − 0.0605228i
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −7.00000 −0.419079
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) − 7.00000i − 0.413197i
\(288\) 1.00000i 0.0589256i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 6.00000i 0.351123i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 2.00000i − 0.116052i
\(298\) 15.0000i 0.868927i
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) 18.0000i 1.03578i
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) − 3.00000i − 0.168497i −0.996445 0.0842484i \(-0.973151\pi\)
0.996445 0.0842484i \(-0.0268489\pi\)
\(318\) − 1.00000i − 0.0560772i
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 1.00000i 0.0557278i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 0 0
\(328\) 7.00000i 0.386510i
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 11.0000i 0.603703i
\(333\) − 2.00000i − 0.109599i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 27.0000i 1.47078i 0.677642 + 0.735392i \(0.263002\pi\)
−0.677642 + 0.735392i \(0.736998\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 14.0000 0.758143
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 2.00000i 0.107366i 0.998558 + 0.0536828i \(0.0170960\pi\)
−0.998558 + 0.0536828i \(0.982904\pi\)
\(348\) − 5.00000i − 0.268028i
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 2.00000i − 0.106600i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 5.00000 0.265747
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) − 3.00000i − 0.158777i
\(358\) − 10.0000i − 0.528516i
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000i 0.946059i
\(363\) − 7.00000i − 0.367405i
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) −3.00000 −0.156813
\(367\) − 23.0000i − 1.20059i −0.799779 0.600295i \(-0.795050\pi\)
0.799779 0.600295i \(-0.204950\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −7.00000 −0.364405
\(370\) 0 0
\(371\) −1.00000 −0.0519174
\(372\) − 7.00000i − 0.362933i
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) − 5.00000i − 0.257513i
\(378\) 1.00000i 0.0514344i
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −22.0000 −1.12709
\(382\) 3.00000i 0.153493i
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 11.0000i 0.559161i
\(388\) − 2.00000i − 0.101535i
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) − 1.00000i − 0.0505076i
\(393\) − 8.00000i − 0.403547i
\(394\) 17.0000 0.856448
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 7.00000i − 0.348695i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) 4.00000i 0.198273i
\(408\) 3.00000i 0.148522i
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 11.0000i 0.541931i
\(413\) − 5.00000i − 0.246034i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) − 20.0000i − 0.979404i
\(418\) 0 0
\(419\) −25.0000 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 13.0000i 0.632830i
\(423\) 8.00000i 0.388973i
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 3.00000i 0.145180i
\(428\) − 2.00000i − 0.0966736i
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) −7.00000 −0.336011
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) − 6.00000i − 0.286691i
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 3.00000i 0.142695i
\(443\) − 16.0000i − 0.760183i −0.924949 0.380091i \(-0.875893\pi\)
0.924949 0.380091i \(-0.124107\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) − 15.0000i − 0.709476i
\(448\) 1.00000i 0.0472456i
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) 6.00000i 0.282216i
\(453\) − 18.0000i − 0.845714i
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 0 0
\(457\) − 33.0000i − 1.54367i −0.635820 0.771837i \(-0.719338\pi\)
0.635820 0.771837i \(-0.280662\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) − 2.00000i − 0.0930484i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) − 33.0000i − 1.52706i −0.645774 0.763529i \(-0.723465\pi\)
0.645774 0.763529i \(-0.276535\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 5.00000i 0.230144i
\(473\) − 22.0000i − 1.01156i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 1.00000i 0.0457869i
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 22.0000i − 1.00207i
\(483\) − 1.00000i − 0.0455016i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) − 3.00000i − 0.135804i
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) − 7.00000i − 0.315584i
\(493\) − 15.0000i − 0.675566i
\(494\) 0 0
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) − 12.0000i − 0.538274i
\(498\) − 11.0000i − 0.492922i
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) − 7.00000i − 0.312425i
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) 12.0000i 0.532939i
\(508\) − 22.0000i − 0.976092i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −23.0000 −1.01449
\(515\) 0 0
\(516\) −11.0000 −0.484248
\(517\) − 16.0000i − 0.703679i
\(518\) − 2.00000i − 0.0878750i
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 5.00000i 0.218844i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) − 21.0000i − 0.914774i
\(528\) 2.00000i 0.0870388i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) −5.00000 −0.216982
\(532\) 0 0
\(533\) − 7.00000i − 0.303204i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 10.0000i 0.431532i
\(538\) − 10.0000i − 0.431131i
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 18.0000i − 0.772454i
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) 37.0000i 1.58201i 0.611812 + 0.791003i \(0.290441\pi\)
−0.611812 + 0.791003i \(0.709559\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 3.00000 0.128037
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000i 0.0425628i
\(553\) 10.0000i 0.425243i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 7.00000i 0.296334i
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) − 32.0000i − 1.34984i
\(563\) − 31.0000i − 1.30649i −0.757145 0.653247i \(-0.773406\pi\)
0.757145 0.653247i \(-0.226594\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) − 1.00000i − 0.0419961i
\(568\) 12.0000i 0.503509i
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) − 3.00000i − 0.125327i
\(574\) −7.00000 −0.292174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −11.0000 −0.456357
\(582\) 2.00000i 0.0829027i
\(583\) − 2.00000i − 0.0828315i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 27.0000i 1.11441i 0.830375 + 0.557205i \(0.188126\pi\)
−0.830375 + 0.557205i \(0.811874\pi\)
\(588\) 1.00000i 0.0412393i
\(589\) 0 0
\(590\) 0 0
\(591\) −17.0000 −0.699287
\(592\) 2.00000i 0.0821995i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 0 0
\(598\) 1.00000i 0.0408930i
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 11.0000i 0.448327i
\(603\) − 12.0000i − 0.488678i
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 0 0
\(609\) 5.00000 0.202610
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) − 3.00000i − 0.121268i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) − 38.0000i − 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) − 11.0000i − 0.442485i
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 18.0000i 0.721734i
\(623\) − 10.0000i − 0.400642i
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) − 13.0000i − 0.516704i
\(634\) −3.00000 −0.119145
\(635\) 0 0
\(636\) −1.00000 −0.0396526
\(637\) 1.00000i 0.0396214i
\(638\) − 10.0000i − 0.395904i
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 2.00000i 0.0789337i
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0000i 0.864909i 0.901656 + 0.432455i \(0.142352\pi\)
−0.901656 + 0.432455i \(0.857648\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) 1.00000i 0.0391630i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.00000 0.273304
\(657\) 6.00000i 0.234082i
\(658\) 8.00000i 0.311872i
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) − 17.0000i − 0.660724i
\(663\) − 3.00000i − 0.116510i
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) − 5.00000i − 0.193601i
\(668\) 18.0000i 0.696441i
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) − 1.00000i − 0.0385758i
\(673\) 29.0000i 1.11787i 0.829212 + 0.558934i \(0.188789\pi\)
−0.829212 + 0.558934i \(0.811211\pi\)
\(674\) 27.0000 1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 32.0000i 1.22986i 0.788582 + 0.614930i \(0.210816\pi\)
−0.788582 + 0.614930i \(0.789184\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −17.0000 −0.651441
\(682\) − 14.0000i − 0.536088i
\(683\) − 46.0000i − 1.76014i −0.474843 0.880071i \(-0.657495\pi\)
0.474843 0.880071i \(-0.342505\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 10.0000i 0.381524i
\(688\) − 11.0000i − 0.419371i
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) − 24.0000i − 0.912343i
\(693\) 2.00000i 0.0759737i
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) − 21.0000i − 0.795432i
\(698\) 15.0000i 0.567758i
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) − 12.0000i − 0.451306i
\(708\) − 5.00000i − 0.187912i
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 10.0000i 0.374766i
\(713\) − 7.00000i − 0.262152i
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 25.0000i 0.932992i
\(719\) 10.0000 0.372937 0.186469 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 0 0
\(721\) −11.0000 −0.409661
\(722\) 19.0000i 0.707107i
\(723\) 22.0000i 0.818189i
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 13.0000i − 0.482143i −0.970507 0.241072i \(-0.922501\pi\)
0.970507 0.241072i \(-0.0774989\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 3.00000i 0.110883i
\(733\) − 11.0000i − 0.406294i −0.979148 0.203147i \(-0.934883\pi\)
0.979148 0.203147i \(-0.0651170\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 24.0000i 0.884051i
\(738\) 7.00000i 0.257674i
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00000i 0.0367112i
\(743\) 29.0000i 1.06391i 0.846774 + 0.531953i \(0.178542\pi\)
−0.846774 + 0.531953i \(0.821458\pi\)
\(744\) −7.00000 −0.256632
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) 11.0000i 0.402469i
\(748\) 6.00000i 0.219382i
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 7.00000i 0.255094i
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 25.0000i − 0.908041i
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 22.0000i 0.796976i
\(763\) 0 0
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) − 5.00000i − 0.180540i
\(768\) 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
\(771\) 23.0000 0.828325
\(772\) − 14.0000i − 0.503871i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 2.00000i 0.0717496i
\(778\) 10.0000i 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 3.00000i 0.107280i
\(783\) − 5.00000i − 0.178685i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) − 17.0000i − 0.605600i
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 2.00000i − 0.0710669i
\(793\) 3.00000i 0.106533i
\(794\) −13.0000 −0.461353
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 8.00000i 0.282490i
\(803\) − 12.0000i − 0.423471i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −7.00000 −0.246564
\(807\) 10.0000i 0.352017i
\(808\) 12.0000i 0.422159i
\(809\) 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 \(0.602351\pi\)
\(812\) 5.00000i 0.175466i
\(813\) − 28.0000i − 0.982003i
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) − 10.0000i − 0.349642i
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 12.0000i 0.418548i
\(823\) 54.0000i 1.88232i 0.337959 + 0.941161i \(0.390263\pi\)
−0.337959 + 0.941161i \(0.609737\pi\)
\(824\) 11.0000 0.383203
\(825\) 0 0
\(826\) −5.00000 −0.173972
\(827\) 2.00000i 0.0695468i 0.999395 + 0.0347734i \(0.0110710\pi\)
−0.999395 + 0.0347734i \(0.988929\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 5.00000 0.173657 0.0868286 0.996223i \(-0.472327\pi\)
0.0868286 + 0.996223i \(0.472327\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 1.00000i 0.0346688i
\(833\) 3.00000i 0.103944i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) − 7.00000i − 0.241955i
\(838\) 25.0000i 0.863611i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) − 22.0000i − 0.758170i
\(843\) 32.0000i 1.10214i
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 7.00000i 0.240523i
\(848\) − 1.00000i − 0.0343401i
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) − 12.0000i − 0.411113i
\(853\) 29.0000i 0.992941i 0.868054 + 0.496471i \(0.165371\pi\)
−0.868054 + 0.496471i \(0.834629\pi\)
\(854\) 3.00000 0.102658
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) − 2.00000i − 0.0682789i
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 7.00000 0.238559
\(862\) 3.00000i 0.102180i
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) 8.00000i 0.271694i
\(868\) 7.00000i 0.237595i
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 35.0000i 1.18119i
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) 39.0000i 1.31245i 0.754563 + 0.656227i \(0.227849\pi\)
−0.754563 + 0.656227i \(0.772151\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) − 58.0000i − 1.94745i −0.227728 0.973725i \(-0.573130\pi\)
0.227728 0.973725i \(-0.426870\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) − 9.00000i − 0.301342i
\(893\) 0 0
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 1.00000i − 0.0333890i
\(898\) 20.0000i 0.667409i
\(899\) 35.0000 1.16732
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) − 14.0000i − 0.466149i
\(903\) − 11.0000i − 0.366057i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) 17.0000i 0.564476i 0.959344 + 0.282238i \(0.0910767\pi\)
−0.959344 + 0.282238i \(0.908923\pi\)
\(908\) − 17.0000i − 0.564165i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −53.0000 −1.75597 −0.877984 0.478690i \(-0.841112\pi\)
−0.877984 + 0.478690i \(0.841112\pi\)
\(912\) 0 0
\(913\) − 22.0000i − 0.728094i
\(914\) −33.0000 −1.09154
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 8.00000i 0.264183i
\(918\) 3.00000i 0.0990148i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) − 12.0000i − 0.395199i
\(923\) − 12.0000i − 0.394985i
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 11.0000i 0.361287i
\(928\) − 5.00000i − 0.164133i
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 24.0000i − 0.786146i
\(933\) − 18.0000i − 0.589294i
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) − 7.00000i − 0.227951i
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) −22.0000 −0.715282
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 10.0000i 0.324785i
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) − 3.00000i − 0.0972306i
\(953\) − 16.0000i − 0.518291i −0.965838 0.259145i \(-0.916559\pi\)
0.965838 0.259145i \(-0.0834409\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) 0 0
\(957\) 10.0000i 0.323254i
\(958\) − 20.0000i − 0.646171i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) − 2.00000i − 0.0644826i
\(963\) − 2.00000i − 0.0644491i
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 20.0000i 0.641171i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 1.00000i − 0.0319765i
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) − 2.00000i − 0.0638226i
\(983\) − 16.0000i − 0.510321i −0.966899 0.255160i \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) −7.00000 −0.223152
\(985\) 0 0
\(986\) −15.0000 −0.477697
\(987\) − 8.00000i − 0.254643i
\(988\) 0 0
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) − 7.00000i − 0.222250i
\(993\) 17.0000i 0.539479i
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) −11.0000 −0.348548
\(997\) − 58.0000i − 1.83688i −0.395562 0.918439i \(-0.629450\pi\)
0.395562 0.918439i \(-0.370550\pi\)
\(998\) 25.0000i 0.791361i
\(999\) 2.00000 0.0632772
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 1050.2.g.i.799.1 2
3.2 odd 2 3150.2.g.h.2899.2 2
5.2 odd 4 1050.2.a.r.1.1 yes 1
5.3 odd 4 1050.2.a.b.1.1 1
5.4 even 2 inner 1050.2.g.i.799.2 2
15.2 even 4 3150.2.a.n.1.1 1
15.8 even 4 3150.2.a.y.1.1 1
15.14 odd 2 3150.2.g.h.2899.1 2
20.3 even 4 8400.2.a.ck.1.1 1
20.7 even 4 8400.2.a.d.1.1 1
35.13 even 4 7350.2.a.bi.1.1 1
35.27 even 4 7350.2.a.cb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.b.1.1 1 5.3 odd 4
1050.2.a.r.1.1 yes 1 5.2 odd 4
1050.2.g.i.799.1 2 1.1 even 1 trivial
1050.2.g.i.799.2 2 5.4 even 2 inner
3150.2.a.n.1.1 1 15.2 even 4
3150.2.a.y.1.1 1 15.8 even 4
3150.2.g.h.2899.1 2 15.14 odd 2
3150.2.g.h.2899.2 2 3.2 odd 2
7350.2.a.bi.1.1 1 35.13 even 4
7350.2.a.cb.1.1 1 35.27 even 4
8400.2.a.d.1.1 1 20.7 even 4
8400.2.a.ck.1.1 1 20.3 even 4