Properties

Label 1050.2.g.f.799.1
Level 1050
Weight 2
Character 1050.799
Analytic conductor 8.384
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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.f.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} -10.0000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} -1.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} -1.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +1.00000 q^{56} +10.0000i q^{58} -4.00000 q^{59} -6.00000 q^{61} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +6.00000i q^{68} -8.00000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +6.00000i q^{73} +2.00000 q^{74} +4.00000i q^{77} +2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} -1.00000 q^{84} -8.00000 q^{86} -10.0000i q^{87} -4.00000i q^{88} -14.0000 q^{89} +2.00000 q^{91} -8.00000i q^{92} -8.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +1.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} - 2q^{14} + 2q^{16} + 2q^{21} - 2q^{24} + 4q^{26} - 20q^{29} - 16q^{31} - 12q^{34} + 2q^{36} - 4q^{39} - 4q^{41} + 8q^{44} + 16q^{46} - 2q^{49} + 12q^{51} - 2q^{54} + 2q^{56} - 8q^{59} - 12q^{61} - 2q^{64} - 8q^{66} - 16q^{69} - 24q^{71} + 4q^{74} + 16q^{79} + 2q^{81} - 2q^{84} - 16q^{86} - 28q^{89} + 4q^{91} + 8q^{94} + 2q^{96} + 8q^{99} + O(q^{100}) \)

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
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 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
\(22\) 4.00000i 0.852803i
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 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
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 2.00000i − 0.277350i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 2.00000i 0.226455i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) − 10.0000i − 1.07211i
\(88\) − 4.00000i − 0.426401i
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 8.00000i − 0.834058i
\(93\) − 8.00000i − 0.829561i
\(94\) 4.00000 0.412568
\(95\) 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
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 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
\(116\) 10.0000 0.928477
\(117\) − 2.00000i − 0.184900i
\(118\) 4.00000i 0.368230i
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 6.00000i 0.543214i
\(123\) − 2.00000i − 0.180334i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 8.00000i 0.681005i
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 12.0000i 1.00702i
\(143\) − 8.00000i − 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) − 1.00000i − 0.0824786i
\(148\) − 2.00000i − 0.164399i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000i 0.609994i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) 14.0000i 1.04934i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) − 2.00000i − 0.148250i
\(183\) − 6.00000i − 0.443533i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 24.0000i 1.75505i
\(188\) − 4.00000i − 0.291730i
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000i 0.422159i
\(203\) 10.0000i 0.701862i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) − 8.00000i − 0.556038i
\(208\) 2.00000i 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 10.0000i 0.686803i
\(213\) − 12.0000i − 0.822226i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 8.00000i 0.543075i
\(218\) − 18.0000i − 1.21911i
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000i 0.134231i
\(223\) 24.0000i 1.60716i 0.595198 + 0.803579i \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) − 10.0000i − 0.656532i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000i 0.519656i
\(238\) 6.00000i 0.388922i
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) − 8.00000i − 0.508001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 32.0000i − 2.01182i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 14.0000i − 0.873296i −0.899632 0.436648i \(-0.856166\pi\)
0.899632 0.436648i \(-0.143834\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) − 4.00000i − 0.247121i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 14.0000i − 0.856786i
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 2.00000i 0.121046i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 28.0000i − 1.66443i −0.554455 0.832214i \(-0.687073\pi\)
0.554455 0.832214i \(-0.312927\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 2.00000i 0.118056i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) − 6.00000i − 0.351123i
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000i 0.232104i
\(298\) − 6.00000i − 0.347571i
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) 40.0000 2.23957
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) − 8.00000i − 0.445823i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 18.0000i 0.995402i
\(328\) − 2.00000i − 0.110432i
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 2.00000i − 0.109599i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) − 6.00000i − 0.326841i −0.986557 0.163420i \(-0.947747\pi\)
0.986557 0.163420i \(-0.0522527\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 10.0000i 0.536056i
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000i 0.213201i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 6.00000i − 0.317554i
\(358\) − 4.00000i − 0.211407i
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 18.0000i − 0.946059i
\(363\) 5.00000i 0.262432i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 8.00000i 0.414781i
\(373\) 30.0000i 1.55334i 0.629907 + 0.776671i \(0.283093\pi\)
−0.629907 + 0.776671i \(0.716907\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) − 20.0000i − 1.03005i
\(378\) 1.00000i 0.0514344i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) − 12.0000i − 0.613973i
\(383\) 28.0000i 1.43073i 0.698749 + 0.715367i \(0.253740\pi\)
−0.698749 + 0.715367i \(0.746260\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 8.00000i 0.406663i
\(388\) − 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
\(391\) 48.0000 2.42746
\(392\) − 1.00000i − 0.0505076i
\(393\) 4.00000i 0.201773i
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) − 8.00000i − 0.396545i
\(408\) 6.00000i 0.297044i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 16.0000i − 0.788263i
\(413\) 4.00000i 0.196827i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 4.00000i − 0.194487i
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 6.00000i 0.290360i
\(428\) 4.00000i 0.193347i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 6.00000i 0.286691i
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) − 12.0000i − 0.570782i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 6.00000i 0.283790i
\(448\) 1.00000i 0.0472456i
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.00000i − 0.280668i −0.990104 0.140334i \(-0.955182\pi\)
0.990104 0.140334i \(-0.0448177\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) − 4.00000i − 0.184115i
\(473\) 32.0000i 1.47136i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 10.0000i 0.457869i
\(478\) − 12.0000i − 0.548867i
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 14.0000i 0.637683i
\(483\) 8.00000i 0.364013i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 60.0000i 2.70226i
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 12.0000i 0.538274i
\(498\) 4.00000i 0.179244i
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 20.0000i 0.892644i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) 9.00000i 0.399704i
\(508\) − 16.0000i − 0.709885i
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) − 16.0000i − 0.703679i
\(518\) − 2.00000i − 0.0878750i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) − 10.0000i − 0.437688i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) − 4.00000i − 0.173259i
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 2.00000i 0.0862261i
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 18.0000i 0.772454i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) − 8.00000i − 0.340503i
\(553\) − 8.00000i − 0.340195i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 22.0000i 0.928014i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) − 1.00000i − 0.0419961i
\(568\) − 12.0000i − 0.503509i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 12.0000i 0.501307i
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 2.00000i 0.0829027i
\(583\) 40.0000i 1.65663i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 1.00000i 0.0412393i
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 2.00000i 0.0821995i
\(593\) 38.0000i 1.56047i 0.625485 + 0.780236i \(0.284901\pi\)
−0.625485 + 0.780236i \(0.715099\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 24.0000i − 0.982255i
\(598\) 16.0000i 0.654289i
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) − 6.00000i − 0.242536i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 16.0000i 0.643614i
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 14.0000i 0.560898i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 10.0000i 0.399043i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 4.00000i − 0.158986i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) − 2.00000i − 0.0792429i
\(638\) − 40.0000i − 1.58362i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 16.0000i 0.626608i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) − 6.00000i − 0.234082i
\(658\) − 4.00000i − 0.155936i
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) − 20.0000i − 0.777322i
\(663\) 12.0000i 0.466041i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) − 80.0000i − 3.09761i
\(668\) 12.0000i 0.464294i
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) − 1.00000i − 0.0385758i
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) − 32.0000i − 1.22534i
\(683\) − 28.0000i − 1.07139i −0.844411 0.535695i \(-0.820050\pi\)
0.844411 0.535695i \(-0.179950\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 22.0000i 0.839352i
\(688\) − 8.00000i − 0.304997i
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 18.0000i 0.684257i
\(693\) − 4.00000i − 0.151947i
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 10.0000 0.379049
\(697\) 12.0000i 0.454532i
\(698\) 18.0000i 0.681310i
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 6.00000i 0.225653i
\(708\) 4.00000i 0.150329i
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) − 14.0000i − 0.524672i
\(713\) − 64.0000i − 2.39682i
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 12.0000i 0.448148i
\(718\) − 20.0000i − 0.746393i
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 19.0000i 0.707107i
\(723\) − 14.0000i − 0.520666i
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 6.00000i 0.221766i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) − 2.00000i − 0.0736210i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 10.0000i 0.367112i
\(743\) 8.00000i 0.293492i 0.989174 + 0.146746i \(0.0468799\pi\)
−0.989174 + 0.146746i \(0.953120\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 30.0000 1.09838
\(747\) − 4.00000i − 0.146352i
\(748\) − 24.0000i − 0.877527i
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 4.00000i 0.145865i
\(753\) − 20.0000i − 0.728841i
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 28.0000i − 1.01701i
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 16.0000i 0.579619i
\(763\) − 18.0000i − 0.651644i
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) − 8.00000i − 0.288863i
\(768\) 1.00000i 0.0360844i
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 10.0000i 0.359908i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 2.00000i 0.0717496i
\(778\) 10.0000i 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) − 48.0000i − 1.71648i
\(783\) 10.0000i 0.357371i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 22.0000i 0.783718i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 4.00000i 0.142134i
\(793\) − 12.0000i − 0.426132i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) − 34.0000i − 1.20058i
\(803\) − 24.0000i − 0.846942i
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) − 2.00000i − 0.0704033i
\(808\) − 6.00000i − 0.211079i
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) − 10.0000i − 0.350931i
\(813\) 8.00000i 0.280572i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 26.0000i 0.909069i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 8.00000i 0.278019i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) − 2.00000i − 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 28.0000i 0.967244i
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 26.0000i 0.896019i
\(843\) − 22.0000i − 0.757720i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) − 5.00000i − 0.171802i
\(848\) − 10.0000i − 0.343401i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 12.0000i 0.411113i
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) − 38.0000i − 1.29806i −0.760765 0.649028i \(-0.775176\pi\)
0.760765 0.649028i \(-0.224824\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −24.0000 −0.818869 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 36.0000i 1.22616i
\(863\) 8.00000i 0.272323i 0.990687 + 0.136162i \(0.0434766\pi\)
−0.990687 + 0.136162i \(0.956523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) − 19.0000i − 0.645274i
\(868\) − 8.00000i − 0.271538i
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) 18.0000i 0.609557i
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 32.0000i 1.07995i
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 20.0000i 0.671534i 0.941945 + 0.335767i \(0.108996\pi\)
−0.941945 + 0.335767i \(0.891004\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 24.0000i − 0.803579i
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 16.0000i − 0.534224i
\(898\) 26.0000i 0.867631i
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) − 8.00000i − 0.266371i
\(903\) − 8.00000i − 0.266223i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) − 16.0000i − 0.529523i
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) − 4.00000i − 0.132092i
\(918\) 6.00000i 0.198030i
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) − 18.0000i − 0.592798i
\(923\) − 24.0000i − 0.789970i
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) − 16.0000i − 0.525509i
\(928\) 10.0000i 0.328266i
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 18.0000i − 0.589610i
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) − 10.0000i − 0.325818i
\(943\) − 16.0000i − 0.521032i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) − 6.00000i − 0.194461i
\(953\) 50.0000i 1.61966i 0.586665 + 0.809829i \(0.300440\pi\)
−0.586665 + 0.809829i \(0.699560\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 40.0000i 1.29302i
\(958\) − 8.00000i − 0.258468i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 4.00000i 0.128965i
\(963\) 4.00000i 0.128898i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 6.00000i 0.191957i 0.995383 + 0.0959785i \(0.0305980\pi\)
−0.995383 + 0.0959785i \(0.969402\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) − 20.0000i − 0.638226i
\(983\) − 28.0000i − 0.893061i −0.894768 0.446531i \(-0.852659\pi\)
0.894768 0.446531i \(-0.147341\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 60.0000 1.91079
\(987\) 4.00000i 0.127321i
\(988\) 0 0
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 20.0000i 0.634681i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 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.f.799.1 2
3.2 odd 2 3150.2.g.t.2899.2 2
5.2 odd 4 1050.2.a.q.1.1 1
5.3 odd 4 210.2.a.a.1.1 1
5.4 even 2 inner 1050.2.g.f.799.2 2
15.2 even 4 3150.2.a.t.1.1 1
15.8 even 4 630.2.a.i.1.1 1
15.14 odd 2 3150.2.g.t.2899.1 2
20.3 even 4 1680.2.a.o.1.1 1
20.7 even 4 8400.2.a.m.1.1 1
35.3 even 12 1470.2.i.n.961.1 2
35.13 even 4 1470.2.a.g.1.1 1
35.18 odd 12 1470.2.i.t.961.1 2
35.23 odd 12 1470.2.i.t.361.1 2
35.27 even 4 7350.2.a.bo.1.1 1
35.33 even 12 1470.2.i.n.361.1 2
40.3 even 4 6720.2.a.z.1.1 1
40.13 odd 4 6720.2.a.cg.1.1 1
60.23 odd 4 5040.2.a.bg.1.1 1
105.83 odd 4 4410.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.a.1.1 1 5.3 odd 4
630.2.a.i.1.1 1 15.8 even 4
1050.2.a.q.1.1 1 5.2 odd 4
1050.2.g.f.799.1 2 1.1 even 1 trivial
1050.2.g.f.799.2 2 5.4 even 2 inner
1470.2.a.g.1.1 1 35.13 even 4
1470.2.i.n.361.1 2 35.33 even 12
1470.2.i.n.961.1 2 35.3 even 12
1470.2.i.t.361.1 2 35.23 odd 12
1470.2.i.t.961.1 2 35.18 odd 12
1680.2.a.o.1.1 1 20.3 even 4
3150.2.a.t.1.1 1 15.2 even 4
3150.2.g.t.2899.1 2 15.14 odd 2
3150.2.g.t.2899.2 2 3.2 odd 2
4410.2.a.bc.1.1 1 105.83 odd 4
5040.2.a.bg.1.1 1 60.23 odd 4
6720.2.a.z.1.1 1 40.3 even 4
6720.2.a.cg.1.1 1 40.13 odd 4
7350.2.a.bo.1.1 1 35.27 even 4
8400.2.a.m.1.1 1 20.7 even 4